The Science of Hysteresis : 3-volume set

Example Consider the system

L (~)x y(t) rl(t)

-

fit, x,y(t)),

=

(I)(r/(t)),

=

(F[rl(to)]x)(t),

(2.83)

with

L(p) = p~ + alp ~-1 + . . . q- a~, where x(t) and y(t) are inputs and outputs of the Preisach nonlinearity and the functionf(t, x, y) is supposed to be continuous and satisfiesflt, x, y) _-fit + T, x, y) for some T > 0. We split the set of all T-periodic solutions (x(t), r/(t)) = (x(t + T), tl(t + T)) (t>~to) of system (2.83) into disjoint equivalence classes, assigning two periodic solutions to the same class if they have the same first component and their second components ~lj(t) satisfy (I)(t/1 (t)) = q)0/2(t)), i.e. the variable states tlj(t) define the same periodic output of the Preisach nonlinearity. We denote the equivalence class of periodic solutions (x(t), rl(t)) by Ix(t), y(t)] where y(t) = (I)(r/(t)) for all its representatives. Classes [x(t), y(t)] are natural for several reasons. Firstly, the differential equation in system (2.83) contains inputs and outputs rather than

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184

variable states r/(t) of the hysteresis nonlinearity. This situation is typical since states of the hysteresis nonlinearity are basically considered as values that one can neither observe nor control. Secondly, the second components of all periodic solutions (x(t), q(t)) that belong to the same class Ix(t), y(t)] are related by very simple explicit formulas, which we discuss below. Let for all t, x e R

[fit, x, y)l<<.qlxl + b,

y e [0, 1],

(2.84)

(here [0, 1] is the range of output values of the Preisach nonlinearity). Suppose that all the zeros of the polynomial L(p) are different from the numbers nwoi (n 9 7/)where w0 = 21t/T. Set

(

k - max IL(nwoi)1-1,

kl - T -1/2 IL(O)1-2 + 2

neZ

~ )1/2 IL(nwoi)1-2 . n--1 (2.85)

In fact, these formulas define the norms k = IIH IIC2----~C2 and kl = IIH IIL2-,C of the solution operator H = H(T) sending a function u = u(t) to the solution

x

Hu of the T-periodic problem for the linear Eqn. L ( d ) x = u(t) in the

spaces L2 = L2[0, T] and C - C[0, T]. To keep things simple, we shall always assume in this subsection that the measure It of the Preisach nonlinearity has a bounded density h = h(~, t) with respect to the Lebesgue measure. Define the scalar function ~(r) = It({(~,//)'~ < - r , r
r>~0,

(2.86)

and denote by Ru the least nonnegative solution of the Eqn. ~(r) = 0 if any exists. If ~(r) > 0 for all r>~0 then we set Ru = c~. The value Ru plays an important role in all the statements below. PROPOSITION2.3.16. Let estimate (2.84) hold and

k l b V/-T 1 - qk < Ra.

qk < 1,

(2.87)

Then there exists a one-parameter continuum of classes [x,~(t), y2(t)] (0~<2~1) of T-periodic solutions to system (2.83), whose classes are different for different 2. Assume additionally that the density of the measure It satisfies h(~, fl)<.v(fl- ~),

~ < fl,

(2.88)

v(s) ds < cxz.

(2.89)

where v is integrable on the positive semiaxis, i.e. v~-

f

OO

0

185

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

Moreover, assume that the function fit, x, y) satisfies the Lipschitz condition Lf(t, Xl, yl) - f i t , x2, y2)l~qlxl - x21 if- qllyl - y2l

(2.90)

for all t, xj ~ JR, O<~yj<~l and 1 - qk ql < 2v~kl ~/T"

(2.91)

Then the set of all T-periodic solutions of system (2.83) is the join of the classes [x,~(t), y,~(t)] (0~<2~<1) above and there are numbers cj such that - 221,

IIx2~ (.) - x~ 2 ( . ) I I c ~ c l l , ~ l

IIY21(') - Y22(') IIc~c2121 - 221,

2j e [0, 1]. (2.92)

For example, consider Eqn. (2.9) of Section 2.1, i.e. x" + x = sin(~/2t) + p(2y(t) - 1),

(2.93)

where x and y are the input and output of the Preisach nonlinearity, p > 0. Here T - ~/2~z, estimate (2.84) holds with q = 0, b - 1+ p and the quantities (2.85) equal

k-

1 ~ v/2~z+ sin(V~n) ~. 0.829.

1,

kl - ~

1 - cos(~/2~z)

Therefore the first of the estimates (2.87) holds. Assume that the density h = h(c~, fl) of the measure/~ is defined by h(~, fl) = 1/(2r 2)

if

- r~<~ < fl<<.r;

h(~, fl) = 0

otherwise.

Then R~ = r and the second of the estimates (2.87) has the form r

> ~2~z 2 + ~/2~z sin(~/27r)

1+ p

4(1 - cos(~/~z))

1.747.

(2.94)

By Proposition 2.3.16 this estimate implies that Eqn. (2.93) has a oneparameter continuum of classes [x,~(t), y2 (t)] of periodic solutions with the period ~/2~z. Furthermore, the Lipschitz estimate (2.90) holds with ql = 2p and from

h(c~, fi)<~l/(2r 2)

if

0 < f l - c~<~2r,

h(c~, fi) = 0

otherwise

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186

it follows that Voo = 1/r, and therefore estimate (2.91) takes the form 4/t/2~ 2 q- ~/-2~sin (w/2~) P > V 4(1 -- c--0-os(---~~)) ~ 6.9876.

r

(2.95)

The second part of Proposition 2.3.16 implies that if estimates (2.94) and (2.95) are valid, then the set of all v/2~-periodic solutions of Eqn. (2.93) is the join of the classes [x2(t), y,l(t)] (0~<2~<1). In fact, these classes are related here by the simple formulas

x,~(t) = xo(t) + 2p6(2),

y~(t) = yo(t) + 6(2),

2 e [0, 1],

where the continuous function 6 = 6(2) strictly increases and 6(0) = 0. Note that estimate (2.94) implies r>1.747. One can use operator norms different from (2.85) to conclude that all Tperiodic solutions of system (2.83) constitute a one-parameter set of classes [x~(t), y,l(t)]. For example, that conclusion and estimates (2.92) are valid if the relation k2(q + 2vooql) < 1 holds with k2 - IIHIIc--.c. This relation and relation (2.91) of Proposition 2.3.16 do not imply each other and give different estimates for the coefficients q and ql of the Lipschitz condition (2.90). We remark that the norm k IIHIIL2__+L2 equals the spectral radius of the operator H = H(T) and that if qk < 1 (i.e., the first relation of (2.87) holds) then estimate (2.84) alone implies that system (2.83) has at least one T-periodic solution, which may be eventually unique. =

Structure of the Set of Periodic Variable States and Outputs The following proposition, describing the set of all possible periodic variable states and outputs of the Preisach nonlinearity for a given periodic input, explains why the one-parameter continua of equivalence classes of periodic solutions appear and play the main role in the proofs of Proposition 2.3.16 above and further statements of this subsection on such continua. Consider a continuous input, which is periodic on the semiaxis t>~to, i.e. x(t) - x(t + T) (t>~to). For some admissible initial states r/0 = r/0(~, fl), the variable state r/(t) = (F[r/0]x)(t) and the output y(t) = O(r/(t)) of the Preisach nonlinearity are also periodic on the semiaxis t>,to. To describe the class of all such variable states and outputs for a given input x(t), let us define the numbers Xm = min x(t), XM = max x(t) and the sets

G(xm, XM) = {(0r fl) : 0r < Xm~XM < fl}, Gc(xm, XM) = 1-I\G(xm, XM) on the half-plane FI = {(a, fl) : a < fl}. From the relay definition it follows that if (a, ]/) e Gc(xm, XM) then the relay output (R~,/~[t0, .]x)(t) is periodic

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

187

on the semiaxis t~to for exactly one of the two initial relay states 0 a n d 1, for the other initial state the o u t p u t is either not periodic or not defined. A u n i q u e (for a given x(t)) periodic relay o u t p u t is d e n o t e d by r/per(t, (x, fl) a n d we set

s s (J,~x)(t) - I ] r/per(t, (X, fl)d#(a:,//) + ,~#(G(Xm, XM)), d JGc(Xm,XM)

t~to,

(2.96) w h e r e 0 ~ 2 ~ 1. For each ,l, this operator is defined on the class of all continuous periodic functions x : [to, co) ~ R, acts in this class and sends a function x = x(t) with a period T to the function J , tx with the same period. Denote by (9 the class of all functions g : ~+ --> ~ such that Ig(yl) g(y2)l~lYl - y2[ (yl, y2~0) and define the function @(e) =

sup # {(~, fl) ~ 1-I : I~ + fl - g(fl - ~)lKe}, g=g(y)~q)

e~0,

(2.97)

w h i c h is nondecreasing, nonnegative a n d satisfies ~(e) ---> 0 as e ---> 0, since we a s s u m e that the m e a s u r e # has a b o u n d e d density. PROPOSITION2.3.17. The following statements hold:

(i) For any periodic input x(t) =_ x(t + T) (t~to) the formula y(t) = (J2x)(t) with 2 ranging over the segment 0 ~ 2 ~ 1 defines the class of all outputs of the Preisach nonlinearity that are periodic on t ~ to. (ii) For any periodic input x(t) =_ x(t + T) (t~to) the formula r/(t, a, fl) = {

r/,r/per(t fl)(X''(fl) (x, ififeithero~< Xm~XM fl > O~Xm<

[~,~ XM~fl

> O~or both,

(2.98) where r/~(~, fl) is an arbitrary measurable (with respect to the measure #) function with the values O, 1, defines the class of all periodic variable states r/(t) = r/(t, ~, fl) of the Preisach nonlinearity. Their periods equal the period of the input , i.e. r/( t) = r/( t + T) ( t ~ to ) , and the periodic output y(t) = ~(r/(t)) for the variable state (2.98) is defined by y(t) = (J2x)(t) with

2#(G(xm, XM)) -- #({(~, fl) 6 G(xm, XM): r/~(~, fl) -- 1}).

(2.99)

(iii) Each variable state r/(t) = (F[.]x)(t) (t~to) equals one of the variable states (2.98) on the semiaxis t~to + T. (iv) For every pair of periodic inputs xj = xj(t) (t~to) with a period T and every ,l ~ [0, 1] II,J2Xl - J2x211C[to,to+T]~26(llXl - x211C[to,to+T]).

(2.100)

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CHAPTER 2 Differential Equations with Hysteresis

From conclusion (iv) it follows that operator (2.96) is uniformly continuous in the space C[to, to + T] for each 2 (we identify periodic inputs and outputs with their restrictions to the period). If relations (2.88) and (2.89) hold, then I~,(E)I~
IIJ x X 1

- - ,.,']'2X2

]]C[to,to+T]<~2v~ ]Ix1 -

x2

[[C[to,to+T].

(2.101)

By Proposition 2.3.17 for every class [x(t), y(t)] of T-periodic solutions of system (2.83) the relation y(t) = (J,~x)(t) with some 2 E [0, 1] is valid. Under the conditions of Proposition 2.3.16 such classes [x(t), (J,~x)(t)] exist for all 0K2~<1 and are pairwise disjoint. In fact, the dependence of the periodic problem on the parameter 2 rather than any special behavior of the function fit, x, y) is the reason for the existence of a one-parameter continuum of classes of periodic solutions. Simple assumptions like estimates of the measure # density and Lipschitz continuity of the nonlinearity fit, x, y) used in Proposition 2.3.16 may guarantee that the set of all classes [x(t), y(t)] of T-periodic solutions is the image of the segment 0~2K 1 under a one-to-one continuous mapping. The easiest way to construct examples of periodic solution sets with any number of connected components is to consider open-loop systems consisting of the differential Eqn. L(d/dt)x --- fit, x) without y, and the relations r/(t) = (F[rl(to)]x)(t), y(t) = @(x(t)) that define variable states and outputs of the hysteresis nonlinearity. Here every periodic solution xj(t) of the differential equation (if any exists) defines a one-parameter set of classes [xj(t), (J,~xj)(t)] (0K2~1) of periodic solutions for the system.

Classes of Periodic S o l u t i o n s Now we consider individual T-periodic solutions (x(t), rl(t)) of the system (2.83) from any class [x(t), y(t)]. Simple relations between their second components r/(t) follow from Proposition 2.3.17. Recall that, by definition, (x(t), rl(t)) belongs to the class [x(t), y(t)] if and only if r/(t) = (F[tl(to)]x)(t), y(t) = (I)0/(t)) and r/(t) = r/(t + T) for all t>~to. Statements (i) and (ii) of Proposition 2.3.17 imply that for every periodic solution (x(t), rl(t)) E Ix(t), y(t)] the component r/(t) = r/(t, a, t) coincides on the domain (~, t) E Gc(xm, XM) for all t~to with the function ~/per(t, ~x, t) uniquely defined by the input x(t). Thus, the components ~/(t) = r/(t, a, t) of the representatives (x(t), rl(t)) of the class Ix(t), y(t)] differ from each other on the domain G(xm, XM) only, where these components do not change with time. That means that r/(t, :r fi) - ~/. (a, t) on G(xm, XM) for each individual solution (x(t), r/(t)). If #(G(xm, XM)) > 0, then one obtains all representatives of the class [x(t), y(t)] inserting here characteristic functions r/. (~, t) of all measurable subsets of G(xm, XM) that

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

189

satisfy a unique restriction, namely (2.99) should hold with the 2 ~ [0, 1] defined by the equality

y(t) - f f r/per(t, a, fl)d#(a, fl)+ 2#(G(xm, XM)), d JGc(Xm,XM) which is nothing other than as y - J, tx. Since there is a continuum of such functions r/. = r/~(a, ]/), each class Ix(t), y(t)] consists of a continuum of periodic solutions (x(t), r/(t)) if l~(G(xm, XM)) > 0. For example, under the conditions of Proposition 2.3.16, system (2.83) has a one-parameter continuum of classes [x,t(t), Y,t(t)] of T-periodic solutions and each of those classes is a continuum parameterized by the functional parameter r/~ Formula (2.96) implies that J2x - J0x for all 2 and the Preisach nonlinearity assigns a unique periodic output to a given periodic input x - x(t) if and only if #(G(xm, XM)) -- 0. In contrast to the situation considered above, it means that the a class [x(t), y(t)] of periodic solutions is not included in the one-parameter set of such classes [x,~(t), y2(t)] if #(G(xm, XM)) = O. Moreover, formula (2.99) implies that the class [x(t), y(t)] contains a unique element (x(t), r/per(t)) in this case, since we consider functions r / = r/(~, fl) that are, with respect to the measure #, equal almost everywhere as the same state of the Preisach nonlinearity. Thus, periodic solutions (x(t), r/(t)) such that #(G(xm, XM)) -- 0 are basically isolated (if the nonlinearity f itself is not so complicated that it 'produces' nonisolated solutions). For example, the relation #(G(xm, X M ) ) - - 0 holds if Xm < - R ~ and XM > Ra.

Guiding Functions It is instructive to modify the statements of Subsection 2.3.3 that guarantee the existence of at least one T-periodic solution for systems with the Preisach nonlinearity in order to formulate sufficient conditions for the existence of the continuum of such solutions. Under the hypotheses of each of the Propositions 2.3.5, 2.3.6, 2.3.7 the periodic solutions admit an a priori estimate llz(t)licKp of the z-component. Such an estimate implies the estimate llx(t)llc~P0 for the corresponding periodic inputs x = (c, z) of the Preisach nonlinearity, for example with P0 - plcl. If we assume that P0 < R~, in addition to the conditions of any of above propositions, then the set of periodic solutions will include the one-parameter continu u m of disjoint classes [z2(t), y,t(t)], 0K2K1. Here and henceforth we denote by [z(t), y(t)] the class of all periodic solutions (z(t), r/(t)) satisfying y(t) = (I)(r/(t)) for systems (2.104) below, similar to the notation Ix(t), y(t)] used for system (2.83). We consider in more detail how this approach works in another application based on the method of guiding functions.

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190

A scalar-valued, continuously differentiable function V(z) is called a guiding function for the system of ODEs z' = F(t, z) with z ~ Ra if the gradient VV(z) of this function satisfies for some p > 0 (VV(z), F(t, z)) < 0

if

Izl > p, t ~ E.

(2.102)

Let the continuous function F(t, z) be T-periodic with respect to t. The simplest fact illustrating the role of guiding functions in the study periodic problems is that estimate (2.102) and the relation IV(z)I --> oe as IzI --* c~ imply the existence of at least one T-periodic solution of system z' = F(t, z). An important idea that leads to less restrictive existence conditions is to use several guiding functions. From (2.102) it follows that outside some ball in Ra the vector fields VV(z) and -F(to, z) are linearly homotopic and do not have zeros for every to. Therefore if Vj(z) (j = 1 . . . . . k) are guiding functions for the system z' = F(t, z), then the index at infinity is defined for each gradient vector field VVj(z) and all these indices coincide. The common value of these indices is called the topological index of the finite set of guiding functions 111(z) . . . . . Vk (z). Such a set is called complete if IWl(Z)i-}-..--}-IWk(z)l--+

oo

as

[zl--*

oo.

(2.103)

There is a classical theorem which states that if there exists a complete finite set of guiding functions with a nonzero topological index, then the system has at least one T-periodic solution. If one uses a straightforward modification of the definition of a guiding function, then the same statement holds for systems with hysteresis nonlinearities. So, we consider the system

dz/dt y(t) ~l(t) x(t)

= = = =

F(t, z(t), y(t)), (I)0/(t)), (F[rl(to)]x)(t), (c, z(t))

Z E [~d,

(2.104)

with the Preisach nonlinearity, where F(t, z, y) =- F(t + T, z, y) is a continuous function. Assume that for j = 1 . . . . . k

(VVj(z), F(t, z, y)) < 0

if

Izl > pj, 0KyK1, t ~ ~,

(2.105)

the relation (2.103) holds and the common value of the index at infinity of the gradient vector fields VVj(z) is nonzero. Then the system (2.104) has at least one T-periodic solution (see, e.g. [61]). Combining this statement with Proposition 2.3.17 in accordance with the basic approach of this subsection, we arrive at the following conditions

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

191

for the existence of a continuum of periodic solutions. We use the notations mj and Mj for the smallest and the largest values of the guiding function Vj(z) on the ball Izl~Pj and the notation c j - {z ~ ~d " m j < . V j ( z ) < . M j } .

PROPOSITION2.3.18. Let scalar-valued continuously differentiable functions V] (z), . . . . Vk(z) satisfy (2.103) and (2.105) and the index at infinity of the gradient vector fields VVj(z) in ~cl be nonzero. Let a ball {z E ~ 9 Izl~p.} contain the set G1 D ... D Gk and let R~ > p.Ici. Then the set of T-periodic solutions of system (2.104) includes a one-parameter continuum of disjoint classes [z2 (t), y2 (t)] (0~2~1). We remark that relation (2.103) implies that the set G1D... NGk is bounded. The conditions of Proposition 2.3.18 imply the inclusion z(t) E G1Vi... DGk, t>~to for the z-component of each periodic solution of system (2.104) and therefore the a priori estimate Ilz(t)Ilc~p. for all such solutions. If IV(z)] --~ cx~ as ]z] --> cx~ (i.e., k = 1) then the index at infinity of the vector field VV(z) is either I or -1. There are many other simple conditions which guarantee that the topological index of the complete set of guiding functions Vl(Z) . . . . . Vk(z) is different from zero. For example, if one of these guiding functions is even, then the topological index is odd. The proofs of Propositions 2.3.16 and 2.3.18 are based on the same idea. Using the operator (2.96), one constructs an operator equation depending on 2, whose solutions define the classes Ix(t), y(t)] of T-periodic solutions of the system (2.83) (the classes [z(t), y(t)] for system (2.104), respectively), then proves the existence of a solution for each 2 E [0, 1] and shows that the solutions are different for different 2. For example, one can use the operator equation

~(t) =fit, (H~)(t), (,.72H~)(t))

(2.106)

in the space L2[0, T] to prove Proposition 2.3.16. From the definition of the linear operator H = H(T) and Proposition 2.3.17 it follows that each solution ~ = ~(t) of (2.106) defines the class Ix(t), y(t)] of T-periodic solutions of the system (2.83) with x(t) -- (H~)(t), y(t) = (J,~x)(t). The existence of a solution ~,~ of (2.106) for each 2 E [0, 1] follows from (2.84) and the first of the estimates (2.87) by the Schauder principle. The second of estimates (2.87) and the a priori estimate ]lxllcKklb~/T/(1 - qk) of all T-periodic solutions of (2.83) imply that all the classes [x,l(t), y,~(t)] of such solutions defined by the solutions ~,~ of (2.106) are disjoint. If the additional assumptions of the second part of Proposition 2.3.16 hold, then Eqn. (2.106) satisfies the conditions of the contraction mapping principle and therefore has a unique

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CHAPTER 2 Differential Equations with Hysteresis

solution r for each ;~ ~ [0, 1]; moreover, the map 2 --, r is Lipschitz continuous. For the proof of Proposition 2.3.18, one can use the operator equation

z(t) = z(O) +

~0t F(s, z(s), (J2x)(s))

ds

with

x(t) = (c, z(t))

in the space C T ( [ 0 , T]; •a) of continuous functions z - z(t) satisfying z(0) = z(T) with the uniform norm. The proof of the existence of a solution z2 of this operator equation for each ;~ ~ [0, 1] follows standard lines of the guiding functions method. For details of the proofs of Propositions 2.3.16 and 2.3.18 and for the proof of Proposition 2.3.17 we refer to [162].

A u t o n o m o u s Systems Our approach also works in periodic problems for autonomous systems, and here it leads to statements on the existence of continua of cycles. For the same reasons as in problems on forced periodic oscillations, the continua of cycles have a similar type of one-parameter structure as described above if the a priori estimate lixllc < R~ holds for the cycles. We give one particular statement. Consider the system

L (-~)x y(t) ~l(t)

= = =

f(x, y(t)), (I)(r/(t)), (F[~l(to)]x)(t)

(2.107)

with the Preisach nonlinearity. Its solutions of the form (x(t), ~l(t)) =(x0, r/0) are called stationary, since for such solutions y(t) = Yo with Y0 = 9 (r/0). We are interested in conditions that guarantee the existence of nonstationary periodic solutions (x(t), rl(t)) -- (x(t + T), ~l(t + T)) of any period T > 0. Since system (2.107) is autonomous, those periods are u n k n o w n a priori. Furthermore, one should take into account that the set of nonstationary periodic solutions contains, together with any solution (x(t), r/(t)), all its time shifts (x~(t), rl~(t)) = (x(t + ~), r/(t + ~)) with 0 < ~ < T. Passing to the equivalence classes [x(t), y(t)] = {(x(t), ~/(t)) : y(t) = (I)(r/(t))} of periodic solutions of the system (2.107), we see that the time shifts of solutions from a class Ix(t), y(t) ] by the time r form the class [x(t + z), y(t + ~)]. In order to simplify the description of shift invariant structures of periodic solution sets, we introduce closed curves 7 = {(x(t), y(t)) ~ [~2 . 0Kt K T} on the i n p u t output plane of the Preisach nonlinearity and say that the curve 7 represents the class of all periodic solutions from the join Uo<.~
2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

193

The curve 7 is called degenerate if (x(t), y(t)) =_ (xo, Yo) and nondegenerate otherwise. We consider stationary solutions in more detail. For 0<~A~
-

#({(a,/~)

9 ~ < fl < ,o}) + ,~ ~ ( { ( ~ , / ~ )

9 a < ,o < fl}).

(2.108)

This function is continuous with respect to the set of arguments A, p. As we know, the Preisach nonlinearity assigns stationary outputs to stationary inputs for any admissible initial states. Moreover, from conclusion (i) of Proposition 2.3.17 it follows that the values of all the outputs y(t) -- Yo corresponding to the input x(t) - xo form the segment {Y0 = ~b,t(x0) : 0<~A~
(2.109)

If L(0) ~- 0 and the continuous functionf(x, y) is bounded, then Eqn. (2.109) has at least one solution p - p*(2) for each 2 e [0, 1] with Ip*(2)l~pl = sup ~(x, y)llL(0)1-1, and therefore the system (2.107) has stationary solutions. In particular, if there is a unique solution p = p* (A) of (2.109) for each 2 and #({(a,//) 9 ~ < p < fl}) > 0 whenever p e [ - P l , Pl], then the points representing all the classes of stationary solutions of system (2.107) in the plane (x, y) form the continuous curve F -- {(x0, Y0) - (p*(2), q~,t(p*(A))) 9 0~<~.~<1} without self-intersections. N o w we are ready to formulate a statement on the existence of nonstationary solutions. Let fix, y) be a bounded continuously differentiable function satisfying the global Lipschitz condition

~(Xl, Yl) --f(x2, y2)l<~qlxl -- x21-+- qllyl -- Y21,

xj, yj 6 ~.

(2.110)

We assume that the relations (2.88), (2.89) are satisfied for the Preisach nonlinearity. It is easy to see that if there are no zeros of the polynomial L(p) on the imaginary axis and the coefficients q, ql in (2.110) are small enough, then the system (2.107) does not have nonstationary periodic solutions. Let us assume that the polynomial L(p) has a pair of the conjugate imaginary zeros 4-woi, and the so-called nonresonance condition L(nwoi) g= 0

for

n - 0, 4-2, 4-3 . . . .

(2.111)

holds. Define on the positive semiaxis r > 0, the function d(r) -

/0

sin t fir sin t, 1) dt -

/0

sin t f ( - r sin t, O) dt.

(2.112)

194

CHAPTER 2 Differential Equations with Hysteresis

PROPOSITION2.3.19. Let the polynomials L(wi) and ~mL(wi) of the variable w have a pair of common zeros • (wo > O) of the same odd multiplicity. Let the nonresonance condition (2.111) hold. Suppose that estimates (2.88) and (2.89) are valid, that the bounded continuously differentiable function fix, y) satisfies the Lipschitz condition (2.110) and function (2.112) satisfies Id(r)l~c > Ofor r~r,. Then there are numbers q0 > 0, Wl e (w0/2, w0) and w2 > w0, such that if q + vooql < q0, the following statements are valid:

(i) The set of all stationary solutions of the system (2.107) is represented on the plane (x, y) by a continuous curve F, which may degenerate to a unique point. (ii) If there is a point (x*, y*) o f f such that

ef

d(r.) ~xx(X*, y*) < 0,

(2.113)

then the system (2.107) has at least one class of nonstationary periodic solutions represented by a nondegenerate closed curve ~; = {(x, y) e R 2 9 x = x(t), y = y(t), t e [0, T)} with a period satisfying 2~/w2~T~2rc/w]. (iii) If estimate (2.113) is valid for all points of the curve F, then there exist numbers r], ro > 0 such that the x-components of all periodic solutions of the system (2.107) with periods 2~/w2<~TK2rc/wl satisfy IIx(t) - YClIL2[O,T]~rl,

]Ix(t)IIc[o,T]~ro,

(2.114)

where Ycdenotes the average value of the function x(t) over its period. If in addition Ra > ro, then the set of all nonstationary periodic solutions of the system (2.107) includes a one-parameter continuum of disjoint classes represented at the plane (x, y) by nondegenerate closed curves ?,~ where 2 ranges over the segment 0K2~l. These curves are different for different 2 and the periods of all periodic solutions represented by them satisfy 2rc/w2 ~ T K2rC/Wl. By conclusion (iii), the x-components of all periodic solutions represented by the curves 7,l satisfy the estimates (2.114) and consequently are separated from the one-dimensional subspace of constant functions and from infinity in L2, C and other functional spaces, which implies, in particular, that such periodic solutions are separated from stationary ones. Additional assumptions about the functionf and the measure # of the Preisach hysteresis nonlinearity guarantee that the one-parameter set of curves ?,t represents all periodic solutions of the system (2.107) and these curves depend continuously on 2. Formulas for the numbers q0, w], w2 (they are defined by the polynomial L(p)) and estimates (2.114) can be given explicitly. We do not present

195

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

these cumbersome explicit expressions here, 3 but consider, as an example, the equation x'" + x" + x' -t- x =fix, y(t)). (2.115) The polynomial L(p) = p3 + p2 + p + I has the pair of simple roots +i,

L(wi) = (wi + 1)(1

3mL(wi) = w(1

- w2),

- w2),

w0-1

and 1/2 < Wl < 1 < w2. For example, if the Lipschitz estimate (2.110) holds with q = 0.305 and v~q1~<0.295, the Eqn. p = tip, qbx(p)) has a unique solution p = p~ for some 2 ~ [0, 1] and for all sufficiently large r

d(r) -~x

<

o,

[d(r) l~c > O,

then Eqn. (2.115) with the Preisach nonlinearity has nonstationary T-periodic solutions with 5.575~
L ( d ) x - g(x) + qly(t). Then

d(r) = 2ql + 4

f rcl2 J0

sin t godd (r sin t) dt,

where godd (x) -- (g(x) -- g(--x))/2 is the odd part of g. If godd (x) ~ go as x --, +oe, then d(r) ~ do = 2ql + 4g0 as r --~ +cx~. If do ~ 0, then the condition Id(r)l>~c > 0 of Proposition 2.3.19 is satisfied for all sufficiently large r.

2.3.6 AVERAGING PRINCIPLE General Description The averaging principle, see [146] and bibliography therein, is a powerful tool in nonlinear dynamics. Here we apply averaging theory to differential equations with hysteresis operators on the right-hand side. We remind the reader, that the standard averaging principle is applicable to the ordinary differential equation of the form z ' = ef(t, z) 3Further details, statements and proofs can be found in [163].

(2.116)

196

CHAPTER 2 Differential Equations with Hysteresis

where the function f is periodic or almost periodic in t. The idea is to introduce the function

F(z)-

lim Z1 IJoT f(t, z) dt

T~oo

(2.117)

and then to consider the equation

z' = eF(z)

(2.118)

as an approximation to (2.116). It has been proved by N. Bogoljubov, that the averaged Eqn. (2.118) is a good approximation to (2.116) over intervals of the size O(1/e), which is the characteristic time for Eqn. (2.116). Now we consider the system with hysteresis given by

~(t) = el(t, z, y(t)), y(t) = (W[rlo]x)(t),

z ~ ~N,

(2.119)

x(t) = g(t, z(t)).

(2.120)

Here y(t) is the output of the hysteresis operator W with the initial state t/0, and e > 0 is a small parameter. The functions f and g are almost periodic in t. We note that the system (2.119), (2.120) is not a particular case of (2.13). We explain how these Eqns. (2.119), (2.120) 'come to life' in the analysis of Eqn. (2.13) below.

Averaged Hysteresis Operators First we need some analysis of hysteresis nonlinearities for inputs that are almost periodic functions depending on a slowly changing parameter s ~ ~+. Formally, the function u(t, s) is given in such a way that for each fixed s ~ R+ it is almost periodic as a function of t ~ ~ and the map s --+ u(.; s) is uniformly continuous on any bounded interval [0, T]. For each s ~ ~+ we define the sequence of inputs

VSk(t) = u(t, min{t/k, s}),

t~>0,

to the operator W[~/0] and denote z~ (t) = W[r/0]v~(t). Since v~(.) converges to an almost periodic function, the same is true for the output z~ (t); by y~ (t) we denote the almost periodic limit of z~ (t). Note that y~ (t) is defined for all t ~ R. If, for each s ~ ~+, the sequence y~ (t) converges to a function ys(t) as k --+ oo uniformly in t ~ 0~, then ys(t) is almost periodic and we say that the map y(s) = ys(') from ~+ to the space of almost periodic functions is the output of the averaged hysteresis operator W'[r/0]. We return to the system (2.119), (2.120) The solutions are traced on a finite interval of the 'slow' time ~ = et. The functions fit, x, z) and g(t, x)

197

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

are assumed to be almost periodic in t and uniformly continuous in all the variables. We denote 1 T + ~ -T /0 fit, z(s), W*[~lo]X s (t)) dt, (I)z(s) = T ~lim

x s(t) = g(t, z(s)).

The averaging m e t h o d replaces (2.119) by the equation

~(t) = ~(bz(t),

(2.121)

whose solutions converge to the corresponding solutions of (2.119) as e --+ +0, if the averaged operator W* exists. Existence of the averaged operator W* has been proven for m a n y classical models, including the Preisach nonlinearity and hysterons. We note especially Vladimirov's results concerning the multidimensional hysteresis operator [111] and their applications to IT problems.

Averaging of Preisach Nonlinearity Let us give a simple example of an explicit description of the limit nonlinearity. Let P be a Preisach nonlinearity and let function u(t, s) be T(s)periodic. We suppose also that the set f~ coincides with the half plane It is sufficient to consider the case w h e n the initial state ~/0 is the characteristic function of a set S, which looks like the shaded set in Fig. 2.35(a). That is, S is located to the left of the line a - ] / a n d below a continuous linear curve L = Ls, with a non-positive slope. We denote the set of states ~/describing the sets S as above by (~.. Additionally, we assume that the coordinates of the intersection of L with the diagonal coincide with the current value of the input x. We will discuss this type of initial condition in more detail in Section 2.4.1. N o w we will give a correspondence between a given function u(t, s) and a varying set S(s). For each s>~0 introduce u_ (s) = min u(t; s) t

and

u+(s) = max u(t; s).

(2.122)

t

For simplicity we consider only the case w h e n both these functions are piecewise monotone. We define S(0) as the part of S which is located to the left from the vertical line :~ = u_ (0), that is,

s(0)see Fig. 2.35(b).

< u_(0)}.

CHAPTER 2 Differential Equations with Hysteresis

198

FIGURE 2.35 The sets S, S(0). Suppose that S(cr) is defined for 0~cr~s0 and that the functions (2.122) are monotone at the interval so Ks. Let us consider four cases: Case Case Case Case

1. 2. 3. 4.

u_(s)~u_(so), u_(s)~u_(so), u_(s)Ku_(so), u_(s)Ku_(so),

and and and and

u+(s)Ku_(so). u+(s)~u_(so). (u_(s), u+(s)) r S(so). (u_(s), u+(s)) ~ S(so).

Figure 2.36 provides the 'picture definitions' for the corresponding set S(s), which is the union of the hatched and shaded regions. The bold line is given by the parametric Eqn. ~ = u_ (rT),//-- u+ (rr). For instance in Case 1, S(s) is the union of the set S(so) and the set {(a, ]/) "u_(so)~o~ < u_(s), and a < / / < u+(a)}. The limit Preisach nonlinearity P*[~/0] can be described by

(P*[~lo]xS)(t) = (P[rl(s)]xS)(t), where x s (t) = u(s, t) and r/(s) is the characteristic function of the set S(s).

Numerical Example Here we demonstrate how the techniques described above work in a particular example. Consider our standard scalar differential-operator equation of the form x"+x = ey(t), y(t) -- (I)0/(t)) 2, (2.123) r/(t) = (F[r/0])x(t).

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

FIGURE 2.36

199

Set S(s).

w h e r e O(r/(t)) is a Preisach n o n l i n e a r i t y w i t h a m e a s u r e #. We s u p p o s e that p is a L e b e s g u e m e a s u r e on the triangle A ( - 1 , 1) = {-1K0cKflK + 1} a n d zero a n y w h e r e else. U s i n g the s u b s t i t u t i o n x = r cos(0 + t),

x' = - r sin(0 + t)

the Eqn. (2.123) can be r e w r i t t e n in the f o r m of the s y s t e m

dr/dt dO/dt y(t)

= =

r/(t)

=

-

- e s i n ( t 4- O)y(t), ecos(t 4- O)y(t)/r, O(r/(t)) 2, (F[~/0])(r(t) cos(0(t) + t)).

C o n s i d e r the initial c o n d i t i o n s r(O) = ro e (0, 1],

0(0) = O,

rt(O) = ~o.

(2.124)

CHAPTER 2 Differential Equations with Hysteresis

200

FIGURE 2.37 Set S(s) for (2.124) is hatched; all possible sets described by characteristic function r/belong to the union of the filled areas.

For these initial conditions, the system (2.124) has a unique solution. The initial state of the Preisach nonlinearity ~/0 is chosen to be u n i o n of the s h a d e d and hatched areas on Fig. 2.37. Using the averaging m e t h o d from the previous subsection and the properties of the Lebesgue measure, we can write a differential equation for the average ~ of r

f2~+o

1 d~/dt= _ m2~ Jo

e Sin(S -}- 0)(I)((1P[r/0])(~ cos(0 if- S))) 2 ds

1 fo ~ e sin(s)(c q- -~~.2 (cos(s) - 1)2) 2 ds = - 2---~ ~2 1 f2~ 2re e sin(s)(c + 2~z - ~ ( 1 - cos(s))2) 2 ds

4~2(C if_ ~2) 3~ Here c > 0 and ~ is decreasing with time, that is the case of the evolution of a Preisach nonlinearity s h o w n in Fig. 2.35(a). The solution x - r sin(t + 0) has symmetric m i n i m a and m a x i m a which means that c is the area in the hatched p o l y g o n in Fig. 2.37. Thus, c - 1 - ~2 and the equation above can

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

201

be written as d~ dt

4~ 2 =

3rt

~(0) - r0

and has the solution 3~zr0 ?(t) = 3rr + 4erot"

Similarly, one can write d o w n and solve the equation for the average of 0. From the averaging principle d0 1 f020 ~ cos(s + 0)(I)((F[r/0])(r cos(0 + s))) 2 ds d--~ = 2~z~ = 2 1~ fo ~ e cos(s)(c q- -~~2 (cos(s) -- 1)2) 2 ds

1 ~2~r 2~z~

~2 e cos(s)(c + 2r 2 - ~-(1

- cos(s))2) 2 ds

= e?(c + 72). N o w the equation for dO dt

= ~

3~ro 3~z + 4roet '

o(o) - o

has the solution 0(t) - -3/4(zt log(3r0 - zr log(3rc + 4er0t)). In Fig. 2.38 we c o m p a r e the results of the averaging m e t h o d with straightforward numerical integration. We consider two values of p a r a m e t e r e: E = 0.01 a n d e = 0.001. Initial conditions are the same in both cases, r0 = 1 a n d we c o m p a r e the solutions on the time interval [0, 1000].

A p p l i c a t i o n to Q u a l i t a t i v e A n a l y s i s of S y s t e m s with Hysteresis We will d e m o n s t r a t e h o w it could w o r k by a particular example. We consider the scalar differential-operator equation of the form x" + x -

qb(t) + ey(t),

y(t) - (I)0/(t)),

(2.125)

r/(t) - (F[r/0])x(t),

(2.126)

CHAPTER 2

202

Differential E q u a t i o n s w i t h H y s t e r e s i s

in ! ! ....

0.2

0.5 0.1

o

!

C 0.1

-0.5

0.2

-10

|

--

4'0

i

6'0

i

i

900

(a) c = 0.01, t r [0, 100] 1

i

0.5

f

nir,f

|

1

10(

920 940 960 980 (C) c=O.O01, t E [900, 1000]

1000

-

0.5 :

i

0

2

-0.5 -1

920 940 960 980 (b) ~ -- 0.01, t E [900, 1000]

i 0

i

20

I

40

60

-0.5

I i

i~

.

,

80

-1 90O

100

(C) c=O.O01, t E [0, 100]

FIGURE 2.38 Dashed oscillating-numerical solution; solid oscillating-solution of averaging equations. w i t h the Preisach nonlinearity. We will s u p p o s e that the c o r r e s p o n d i n g m e a s u r e # is finite, positive, absolutely continuous w i t h b o u n d e d density, a n d defined in the half plane a < ft. Suppose that the external force ~b(t) is a s u m of periodic functions qbi(t), i = 1 . . . . . n, w i t h periods Ti, for w h i c h n

2~

Z ki-~ii + kn+l ~ 0

(2.127)

i=1

for a n y integers kin, m = 1 . . . . . n + 1. As w e know, this equation for a n y initial conditions x(0) = x0, x'(0) = x 0' has a u n i q u e solution x(t , x(O) , x'(O) , e). We denote b y ui(t), i = 1 . . . . . n, the periodic solution (it is unique) of the equation u" + u = qbi(t ), and we set n

u(t) = Z i=1

ui(t).

(2.128)

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

203

By m e a n s of the usual substitution x = u(t) + r cos(0 + t),

x' = u' (t) - r sin(0 + t)

the Eqn. (2.125) can be rewritten in the form of the system dr d--t = - ~ sin(t + O)y(t), dO d---t-= e cos(t + O)y(t)/r, y(t) = ~(r/(t)),

~/(t) = (F[~/0])x(t), x(t) = u(t) + r(t)cos(0(t) + t).

For any initial conditions r(0)=r0>0,

0(0)=00,

~/(0)=~/0

this system has a u n i q u e solution (r(t; ro, 0o, tl o, e), O(t; ro, 0o, ~1o, e), r/(t; r0, 00, ~/0, e)). We denote by f~(r), r > 0 the following d o m a i n in the half plane ct < fl: f~(r) =

~ <

m a x ui(t) + r or e > i=1

m i n ui(t) - r

.

i=1

For r > O, (e, fl) ~ f~(r) and 0 < Oi < 2re, i = 1 . . . . . n + 1, we set F(r; ~, fl, 01 . . . . On+l) =

lim

to---+ - - ~

(R[z(e, fl); ~, fl)x) (0)

n ( x(t) = ~

ui

t -- to -t- - ~ T i

-t- r cos(t - to -t- On+l).

i=1

The limit of the right-hand side always exists. For fixed values of r > 0 a n d (e, fl) e f~(r) it is convenient to consider this function to be defined on the (n + 1)-dimensional torus H n+l. For r > 0 we set P(r) = ~

(r)

d/.t(0c,fl)fH

n+l

F(r; ct, [3, 6)) sin On+l d |

Since in any d o m a i n 0 < r < R, P(r) satisfies a Lipschitz condition, each Cauchy problem d? m

= -ePff),

dt has a unique solution ?(t; ?0, e).

r(-0) = r0

204

CHAPTER 2 Differential Equations with Hysteresis

THEOREM2.3.4. The function P(r) is strictly positive for r > O. COROLLARY2.3.10. Let R > O. Then to each ~ > 0 there correspond numbers l eo > 0 and L > 0 such that for 0 < e < e0, and Ix0[ + Ix0] < R Ix(t; x0, x~) - u(t)] + Ix'(t; x0, x~) - u'(t)] < 6,

t > L/e.

2.3.7 BIFURCATIONANALYSIS In Section 2.2 we observed a rich bifurcation behavior of Eqn. (2.9) depending on the parameter b. Here we survey some statements about bifurcation points. We restrict ourselves to one type of bifurcations, namely Hopf bifurcations in autonomous systems, discussing mainly sufficient conditions for some parameter value to be a bifurcation point and the structures of solution sets, to some extent, and omitting the proofs.

H o p f Bifurcation Let some autonomous system depending on a scalar parameter b have an equilibrium z.(b) for each parameter value from some interval (b_, b+) and let those equilibria form a smooth curve in the phase space Z of the system. We say that Hopf bifurcation occurs and a parameter value b0 is a Hopf bifurcation point if for any sufficiently small r > 0 one can find a parameter value br E (bo - r, bo + r) such that the system with b -- br has a cycle which lies in the ball of radius r centered at the equilibrium z.(br) (this type of weak definition of Hopf bifurcation was introduced in [164], where nonsmooth problems were studied for the first time). Freely speaking, we consider bifurcation of a small cycle from an equilibrium point of the autonomous system. The simplest example is the equation x" + bx' + x = fix, x', b)

(2.129)

with f satisfying flZl, Z2, b) = o(]Zll q-IZ21) as Zl, Z2 --+ 0 uniformly with respect to b. Here the origin is an equilibrium of the equivalent planar firstorder system for every b; the value b = 0 is a unique Hopf bifurcation point. Figure 2.39 presents a typical structure of the set of cycles in the product of the phase space ~2 of the equivalent system and the axis of parameter values b in the case of a sufficiently smooth f. The cycles form a smooth two-dimensional surface ~ (a 'cup') passing through the origin (note that the picture is local both in b and z = (Zl, z2), i.e. we consider cycles lying in a small vicinity of the zero equilibrium and b small in absolute value only). Appropriate nondegeneracy conditions guarantee that ~ is tangent

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

205

I Z2 s ss

E1 %

FIGURE 2.39 Hopf bifurcation.

to the plane b = 0 and lies on one side of it. In Fig. 2.39, s lies to the right of this plane, which means that the system has no small cycles for b < 0 and has such cycles for small b > 0; for b = b' the cycle is a cross section of with the plane b = b'. Under the conditions of the classical Hopf Theorem (see e.g. [165]) a similar picture is observed for a u t o n o m o u s systems

d z / d t = F(z, b),

z c ~a,

(2.130)

of ODE of any dimension in some vicinity of the equilibrium z,(b). Let us recall more precisely the algorithm to determine bifurcation points by linearization of (2.130). Since the change of variables z ~ z + z, (b) places the equilibrium at the origin, let us assume without loss of generality that z, (b) ~ 0, i.e. system (2.130) has zero equilibrium for all b. In this situation, we call b0 a Hopf bifurcation point for system (2.130) with a limit period To > 0 if for any sufficiently small r > 0 the system has a periodic solution zr(t) of a minimal period Tr > 0 for some b = br and max Izr(t)l ~ O,

br ~ bo,

Tr ~ TO

as

r ~ 0.

t~

We remark that periods Tr are not k n o w n a priori and that we use an auxiliary parameter r. The reason w h y the parameter b is not good for the parameterization of the set of small cycles becomes clear if one considers linear equations, for example x" + bx' + x = 0, i.e. Eqn. (2.129) w i t h f ~ 0.

206

CHAPTER 2 Differential Equations with Hysteresis

Here all the cycles exist for one parameter value b0 = 0, which is a unique Hopf bifurcation point (the surface ~ in Fig. 2.39 becomes a piece of the plane b = 0). Suppose F is differentiable with respect to z at the point z = 0 uniformly with respect to the parameter b, i.e. F(z, b) = A(b)z + g(z, b) where suPb ]g(z, b)l/JzJ ~ 0 as izi --* 0, and let the matrix-valued function A(b) and the vector-valued function g(z, b) be continuous. Assume that the matrix A(bo) has a pair of simple imaginary eigenvalues 4 +iwo (w0 > 0) for some b0, which implies that A(b) has a unique pair of simple eigenvalues rr(b) 4- iw(b) close to 4-iwo for b close to b0. Let the derivative rr'(b0) of the real part of these eigenvalues at the point b = b0 exist and the so-called transversality condition cr'(b0) ~= 0 (2.131) and nonresonance condition

iwok ~ Sp(A(b0))

for all

k ~ 77, k ~= 4-1

(2.132)

hold, where Sp(A) is the spectrum of A. Then b0 is a Hopf bifurcation point for system (2.130) with the limit period 2rc/wo (for example, this follows from the results of [164]). If in addition F is sufficiently smooth, then according to the Hopf Theorem the small cycles form a two-dimensional smooth surface in the product of the phase space ~c/of (2.130) and the axis b. Moreover, information about the Taylor expansion of F at the origin allows us to study further important problems including stability of the cycles, but we do not consider those problems here.

Bifurcation in Systems with Preisach Hysteresis Consider the autonomous system

dz/dt y(t) rl(t) x(t)

= = = =

F(z(t),y(t),b), (I)(r/(t)), (F[rl(to)]x)(t), (c, z(t))

z ~ ~a (2.133)

with the Preisach nonlinearity, where the first equation depends on the scalar parameter b; the function F is supposed to be continuous. We use notation similar to that of Subsection 2.3.5. Solutions of the form (z(t), r/(t)) = (z0, r/0) are called stationary, the point (z0, ~/0) in the product ~a x (~. is called an equilibrium of the system. By a cycle we mean the trajectory of any nonstationary periodic solution (z(t), rl(t)) =_ (z(t + T), rl(t + T)) 4The existence of imaginary eigenvalues of A(bO) is a necessary condition for b0 to be a Hopf bifurcation point.

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

207

in ~a x (~., with any T > 0; actually, the cycle is a c o m m o n trajectory for all the time shifted periodic solutions (z~(t), ~/~(t)) = (z(t + ~), rl(t + ~)), 0~ (c, z0). Since the Eqn. y0 = ~0/0) with a given Y0 has a solution r/0 in this class of admissible states if and only if #({(~, f l ) : ~ < fl < (c, z o ) } ) ~ y o ~ # ( { ( ~ , fl): ~
(2.134)

we conclude that system (2.133) has an equilibrium if and only if the Eqn. F(zo, Y0, b) = 0 has a solution (z0, Y0) satisfying (2.134). Similarly to the definition for ODEs above, we call b0 a H o p f bifurcation point for system (2.133) with a limit period To if for any sufficiently small r > 0 the system has a nonstationary periodic solution (zr(t), rlr(t)) with a period Tr > 0 for some b = br and max I z r ( t ) - z o l ~ 0, max/9 l(r/r(t ), r/0) ~ 0, br -+ bo, Tr ~ TO t t

as

r --, 0,

where (z0, ~/0) is an equilibrium for b = b0 and/91 (., .) is the metric (2.20) in the state space @. of the Preisach nonlinearity. To formulate the simplest statement on H o p f bifurcation for the system (2.133), suppose that there are z0, Y0 satisfying (2.134) such that F(z0, y0, b) - 0

for all b.

Therefore system (2.133) has an equilibrium at the same point (z0, r/0) ~a x @. for all b. A s s u m e that the function F can be linearized at the point (z0, Y0), more precisely F(z, y, b) - A ( b ) ( z - zo) + a(b)(y - Yo) 4- g(z, y, b),

(2.135)

with the matrix A = A ( b ) and the vector a - a(b) d e p e n d i n g continuously on b and lim maxlb-b01~R Ig(z, y, b)l - 0 (2.136) z--,zo, y-*yo Iz - z01 + lY - Y01 for some R > 0. In this situation, H o p f bifurcation points for system (2.133) are d e t e r m i n e d by the matrix A (b) in the same w a y as for ODE. PROPOSITION 2.3.20. Let relations (2.135), (2.136) be valid and strict estimates #({(a, fl) 9 a < fl < (c, z0)}) < Y0
CHAPTER 2 Differential Equations with Hysteresis

208

hold in place of (2.134). Let the measure # have a bounded density with respect to the Lebesgue measure. Suppose that the matrix A(bo) has a pair of simple eigenvalues • that the transversality condition (2.131)and the nonresonance condition (2.132) hold. Then bo is a Hopf bifurcation point for the system (2.133) with a limit period 2~/wo. As an example, consider the equation x" + bx' + x = 2y(t) - 1 where x(t) and y(t) are the input and output of the Preisach nonlinearity. If #({(a, f l ) : a < f l < 0 } ) < l / 2 < # ( { ( a , fl):a 0 for some b = br and max [zr(t)- Zr(Z)[--+ cx~, br ~ bo, Tr ~ TO as t,'~

r ~ cx~.

(2.138)

In other words, the system has cycles with arbitrarily large amplitudes of the z-component and periods close to To for some parameter values arbitrarily close to b0. Suppose that the function F can be represented as

F(z, y, b) = A(b)z + g(z, y, b),

(2.139)

lim max Ig(z, y, b)] = 0 Izl-~c~ Ib-bol<~R,lYI~R~ ]Z[

(2.140)

where

for some R > 0 and every R1 > 0. Then the derivative A(b) of F with respect to z at infinity plays the same role in problems on Hopf bifurcation of large cycles as the derivative at the equilibrium point in local problems above.

209

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

PROPOSITION2.3.21. Suppose that relations (2.139), (2.140) hold, the matrix A (bo) has a pair of simple eigenvalues +iwo, the transversality condition (2.131) and the nonresonance condition (2.132) are valid. Then bo is an asymptotic Hop]: bifurcation point for system (2.133) with a limit period 2n/wo. This proposition is a consequence of a more general statement of [166]. Proposition 2.3.20 and some further statements on Hopf bifurcation from an equilibrium can be found in [167].

Bifurcation with Respect to Internal Parameter As in Subsection 2.3.5, let [z(t), y(t)] denote the class of periodic solutions (z(t), rl(t)) = (z(t + T), rl(t + T)) (t~to) such that y(t) = (I)(r/(t)) and let us say that the closed curve 7 - {(z(t), y(t)) 9 0~t < T} in the space Ea+l of pairs (z, y) represents the periodic solutions of the join Uo<~
dz/dt y(t) rl(t) x(t)

= = = =

F(z(t), y(t)), (I)(r/(t)), (F[ri(to)]x)(t), (c, z(t))

ZE[~ d, (2.141)

that does not depend explicitly on parameters. Nevertheless, as we already know from considerations in Subsection 2.3.5, problems on cycles of this system contain the parameter 2, which we call 'internal'. This results in the fact that each closed curve 7 c ~a+l representing a class of cycles (if any exists) such that the amplitude of their z-component is smaller than the characteristic size of the Preisach nonlinearity is generically included in the one-parameter continual set of such curves 7,~; a typical example is presented by the conclusion (iii) of Proposition 2.3.19. It turns out that the curves 7 representing small cycles of the system (2.141) can accumulate at its equilibrium and form the structure similar to that observed for Hopf bifurcation problems for ODEs with a scalar parameter. Moreover, sufficient conditions for the existence of those structures for system (2.141) with hysteresis are close to the classical assumptions of the Hopf Theorem. Let us consider one example. Let the Eqn. F(z, 2) = 0 define a continuous branch z = ~p(2) of the implicit function on some open interval A*. Denote by F the curve (z, y) = (~p(2), 2), 2 E A* in [~d+l and assume that for each point (z0, Y0) ~ F relations (2.137) hold, which implies that this point represents the class [z0, Y0] of equilibria of system (2.141). We shall suggest sufficient conditions that

CHAPTER 2 Differential Equations with Hysteresis

210

guarantee the existence of a one-parameter set of closed curves ~ with vanishing diameters that accumulate at one of the points Ado = (z0, Y0) 6 F. The density of the m e a s u r e / t of the Preisach nonlinearity will be everywhere assumed to be bounded. Let us suppose that the function F(z, y) is continuously differentiable with respect to z and Lipschitz continuous with respect to y in some neighborhood of the curve F. As usual, set A(2) = 8F/8z(q)(2), 2),

2 e A*.

Suppose that at some point .Ado = {q~(20), 2o} e F the spectrum Sp(A(20)) of the d x d matrix A(20) = 0F/8z(q~(20), 2o) contains a pair of simple eigenvalues +iwo (w0 > 0) and the nonresonance condition iwok r Sp(A(20)) holds for all integer k 4= +1. Consider the pair of the simple eigenvalues cr(2) + iw(2) of A(2) satisfying cr(20) 4- iw(20) = +iwo, where the real continuous functions a(2), w(2) are defined in some neighborhood of 2o. PROPOSITION2.3.22. Let the function r~(2) take values of both sign in each neighborhood of 2 = 20. Then for every small r > 0 system (2.141) has a class [Zr(t), yr(t) ] of nonstationary periodic solutions of a period Tr, all the closed curves ~ r = { ( Z , y) 6 [ ~ d + l . z - - Zr(t), y -- yr(t), t ~ [0, Tr)} are different for different r and max ]zr(t)- q~(,~0)] ~ 0, t

max lyr(t)- ,~0] ~ 0, t

Tr --* TO = 2~/wo

(2.142) as r --+ 0, i.e. the curves ~r shrink to the point .Ado and periods approach To. Particularly, the conditions of this proposition are satisfied if the function r~(2) is differentiable at the point 2 = 20 and the transversality condition r~'(20) ~= 0 holds, which one can consider as the main case. The simplest example of system (2.141) satisfying the conditions of Proposition 2.3.22 is x" + (y(t) - K)x' + x = g(x, y), where x and y are the input and output of the Preisach nonlinearity and g(x, y) = o(]xl + lYl) as x,y --+ 0. If u ( { ( ~ , / ~ ) : a < / ~ < 0}) < ~: < u ( { ( a , / ~ ) : a < / ~ , a < 0})

then this second-order equation has the continuum of disjoint classes [Xr(t), yr(t)] (0 < r < r0) of nonstationary periodic solutions with periods Tr such that maxt Ixr(t)l --+ 0, maxt lYr(t) - tel ~ 0 and Tr --+ 2~z as r --+ 0. If the function F(z, y) is sufficiently smooth in a neighborhood of the point M0, then the functions r~(2), w(2) are continuously differentiable and F(q~(2) + ~, 2 + v) = A(2)~ + a(2)v + g(2, ~, v)

(2.143)

2.3 EXAMPLESOF RIGOROUS CONCLUSIONS

211

for 2 close to 20 and ( ~ ~a, v ~ R small in absolute value. Let the matrixvalued function A(2) and the vector-valued function a(2) be Lipschitz continuous and let the residual term g satisfy uniformly in 2 ]g(2, ~1, Vl)]~ o(p),

(2.144)

]g(21, ~1,121) - g(22, ~2,122)]~ o(P)]~,l-22]nus(P) (]~1-~2] if- ]121-122])

(2.145)

whenever ]~jl, ]vj]~fi with e(p) = o(1), p ~ 0. In this situation, Proposition 2.3.22 is supplemented by the following statement. PROPOSITION2.3.23. Let relations (2.143)-(2.145) hold and r~'(20) ~= 0. Let the density of the measure t2 of the Preisach nonlinearity be Lipschitz continuous with respect to both the arguments ~, ft. Then there is an rl > 0 such that the curves ~/r -- {(Z, y) E ~ d + l zr(t), y -- yr(t), t ~ [0, TF)} representing cycles of (2.141) in Ra+l and satisfying (2.142) as well as their periods Tr depend Lipschitz continuously on r for 0 < r < rl. Moreover, there is a neighborhood U c Ra+l of the point ;~do ~ F such that each closed curve ~ lying in U and representing a class of cycles of (2.141) with a period sufficiently close to To belongs to the set of curves ?r, 0 < r < rl, i.e. there are no other such curves in U. .

Z

--

We stress the difference between the conclusion of this proposition and that of the statements in the begining of this section on Hopf bifurcations in systems without hysteresis. System (2.141) with the Preisach nonlinearity has a continuum of small cycles in the phase space (this continuum is the join of the classes of cycles represented by the curves 7r with 0 < r < rl), while system (2.130) has typically at most one small cycle in the phase space for each value of the parameter, the surface like that shown in Fig. 2.39 in the product ~a+l of the phase space and the parameter axis consists of cycles existing for different parameter values. Another remarkable difference is seen if we consider the asymptotics of the cycles. Small cycles of system (2.130) form a smooth surface 5 in the space [~d+l of pairs (z, b). The curves ~/r representing cycles of system (2.141) in the space R~a+l of pairs (z, y) form a structure close to a cone, as is shown in Fig. 2.40. More precisely, the curves ?r are asymptotically close to the ellipses gr = Ado + r(z,, y,) + r(C, 0) for small r > 0, where Ado + r(z,, y,) is a ray tangent to the curve F at the point ,Ad0 and C is an appropriate cycle of the linear system z' = A(20)z (the cycles rC form a plane in the phase space ~d of z' = A(20)z). The Hausdorf distance between 7r and gr is o(r) as r ~ 0; Fig. 2.40 shows the curves ~r and the 5We consider sufficiently smooth nonlinearities F(., .).

212

CHAPTER 2 Differential Equations with Hysteresis

~Z2

/ z1

FIGURE2.40 Curves 7r. two-dimensional cone surface formed by the ellipses gr with the vertex at the point .Ado. Some further details can be found in [117].

D i m e n s i o n of the Set of Cycles of System with a Parameter Let us consider again small cycles of the system (2.133) in the case of Hopf bifurcation from an equilibrium (z0, ~/0) at the Hopf bifurcation point b = b0. Here the Problem includes the 'external' parameter b and the 'internal' parameter 2. So, one might expect that the set of closed curves 7 = {(z(t), y(t), b) ~ [ ~ d + 2 . t E •} representing small cycles of (2.133) in the space R a+2 of triples (z, y, b) in a vicinity of the point.M0 - (z0, Y0, b0) with Y0 = (I)0/0) is generically two-parametric and that these curves accumulate at the points of some curve $ passing through the point.M0 and representing equilibria of (2.133). Namely, if F(zo, Y0, b0) = 0 with Y0 = cI)(r/0) satisfying (2.137), the matrix A(.,., .) = cgF/cgz(., .~ .) has a pair of simple eigenvalues • iwo (w0 > 0) at the point AF0 and proper smoothness and nondegeneracy conditions are satisfied, then the equality F(z, y, b) = 0 defines a two-dimensional surface | in some vicinity of the point d~0 in ~a+2, each point (z~, y~ b~) ~ | defines the stationary solution (z(t), y(t)) = (z~, y~) of (2.133) for b = b~ and the curve 8 c | is defined by the relations Sp(A(z, y, b)) ~ iw, Iw - w0J < ~, w > 0 with a sufficiently small ~. Yet, we do not know exact statements of this type. In problems on Hopf bifurcation from infinity for system (2.133) one should distinguish between the cases R~ = cx~ and Ra < cx~. For example, suppose that the conditions of Proposition 2.3.21 are satisfied. If R~ -- cx~

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

213

then the set of closed curves 7 - {(z(t), y(t), b) e ~d+2. t e R~}representing large cycles of (2.133) for b close to the asymptotic Hopf bifurcation point b0 is generically two-parametric; if R~ < cx~ then this set is generically (for 'nonexotic' nonlinearities F) one-parametric. One can see it more precisely from operator equations equivalent to the problem. We outline the approach roughly, not presenting any particular operator equation, which can be constructed in different ways in the form W r (z, y, b, T) = 0. Here r is an auxiliary parameter; the u n k n o w n s are the functions z = z(t), y = y(t), the parameter value b for which the class Uo<.~ r0 and every 0~<2~<1, the relations m a x I Z r , 2 ( t ) - Zr,2('c)l---> cx~, br, 2 ---> bo, Tr,2 ~ t,'c

TO = 27r/wo

as

r ~ cx~

similar to (2.138) hold uniformly with respect to 2 e [0, 1], and Zr, 2 -Zr,,2', Yr,2 -- Yr',2' if and only if r = r', 2 = 2 I. This implies that there is the two-parameter set of c u r v e s 7r,2 - {(Zr,2(t), Yr,2(t), br,2) 6 [~d+2 . t e [0, Tr,,~)} (r > r0, 0K2~<1) which are different for different (r, ;~) and that the diameter of 7r,,~ goes to infinity as r increases. Moreover, it turns out that the m a x i m u m max{d(b, 2, b,,V) " 0~2, 2' <~1} of the Hausdorf distance d(., .) between the c u r v e s 7r,2 and 7r,2' with the same r vanishes as r increases, i.e. the set U0~<,~<~l7r,;~ shrinks to the individual curve 7r,0 as r ~ cx~. If the vector c does not belong to the invariant plane of the matrix A(bo) corresponding to the pair of its simple eigenvalues +iwo, then the relations mint (c, z(t) ) ~ - c x t maxt (c, z(t) ) ~ ooasb ~ bo hold a priori for the cycles bifurcating from infinity. In case R~ < cx~ this implies (J,~x)(t) = (Jox)(t) with x(t) = (c, z(t)) for all 0~2~<1 and all the large cycles. Therefore in place of (2.146) one arrives at the system Wr(z, y, b, T) = 0, y = J0x, x = (c, z) with only the parameter r. One proves that for all r > r0 this system has solutions (Zr, yr, br, TF) satisfying (2.138). In this case, large cycles of system (2.133) form the one-parameter set of classes represented by the curves 7r (r > r0) with the diameters going to infinity as r increases. The auxiliary parameter r can be chosen in different ways. For example, a ' g o o d ' choice (which allows one to construct topologically nondegenerate systems (2.146)) is r -- Ilzllc. With this choice, the above statements

CHAPTER 2 Differential Equations with Hysteresis

214

guarantee the existence of cycles for all sufficiently large amplitudes of the z-component. Statements of this type on Hopf bifurcations from infinity for systems with the Prandtl-Ishlinskii hysteresis nonlinearities (a class of nonlinearities close to a particular class of the Preisach models) in case Ra - ~ are presented in [116]. Case Ra < cx~ for systems with the stop hysteresis nonlinearity was studied in [168], including stability analysis of large cycles; a similar approach can be applied here.

2.3.8 CHAOTIC BEHAVIOR Definition of Chaos Letf : S ~ S be a continuous mapping in a subset S of an Euclidean space Re. Important attributes of chaotic behavior include sensitive dependence on initial conditions, an abundance of periodic trajectories and an irregular mixing effect describable informally by the existence of a finite number of separated subsets U0. . . . . Urn-1 of Re which can be visited by trajectories of some fixed iteratef k off in any prescribed order. Let/d = {U0 . . . . . Urn-l}, m > I be a family of disjoint subsets of S and let us denote the set of onesided sequences co = o90, o91. . . . " ogi 9 {0, 1 . . . . . m - 1} by f~R m. Sequences in f~R m will be used to prescribe the order in which sets Ui are to be visited. m-1 For x 9 Ui=0 Ui we denote by I(x) the number i satisfying x 9 Ui. A mapping f is called (/d, k)-chaotic (k is a positive integer) if there exists a compact f-invariant set Z C ~Ji Ui with the following properties: (p l) for any o9 9 f2R there exists x 9 Z with fik (x) = (fk')i 9 Uo~i for i~1; (p2) for any p-periodic sequence o9 9 f~R m, there exists a pk-periodic point x 9 Z with fik(x) 9 U~i; (p3) for each ~/> 0 there exists an uncountable subset Z(r/) of S, such that the simultaneous relationships lim supi__,~

I(fik(xl)) -- l(fik(x2))] ~1,

lim infi~ c~

~ik(xl) -- fik(x2)l < rI

h o l d for all Xl, x2 6 I ( r ] ) , Xl # x2.

The above-defined properties of chaotic behavior are similar to those in the Smale transverse homoclinic trajectory theorem with an important difference being that we do not require the existence of an invariant Cantor

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS TABLE 2.2 i

V~1) (i)

V~2) (i)

0 1 2 3 4 5 6 7 8

2.595 2.597 2.606 2.641 2.808 3.045 2.677 2.622 2.602

1.286 1.278 1.258 1.189 0.9016 1.150 1.599 1.366 1.304

Va(1) (i) 0.001328 0.001194 0.001099 0.001086 0.001535 -0.002766 0.001789 0.001615 0.001465

215

Parallelograms Va(2) (i) -0.002690 -0.002419 -0.002190 -0.002059 -0.002426 0.004905 -0.002620 -0.003255 -0.002989

v~1) (i) 0.003545 0.004168 0.004869 0.005637 0.007217 0.0003166 0.001140 0.002219 0.002922

v~2) (i) 0.009351 0.01097 0.01284 0.01472 0.01352 -0.009191 0.006788 0.007069 0.008053

set. Instead, the definition includes property (p2), which is usually a corollary of uniqueness, and (p3), which is a form of sensitivity and irregular mixing as in the Li-Yorke definition of chaos [169], with the subset/701) corresponding to the Li-Yorke scrambled subset So. A similar definition was used previously in [170]. Note also that the one-sided left shift cr+ is a factor of the restrictionfkls (See [171], p. 68).

Chaos in an ODE with Preisach Nonlinearity Consider Eqn. (2.9) with b = I and an appropriate Poincare map F1 : FI 1-I generated by it. Let v = (v (1), v (2)) denote the element of FI. A part of the co-limit set of F that was constructed with help of the computer is given in Fig. 2.21, (p). Introduce the parallelogram sets R(i) ~ l-I, i = 0 . . . . . 8 given by the center points vc(i), and the halves of the side vectors va(i) and vb(i):

R(i) -- {vc(i) 4- ~va(i) 4- flvb(i) : --1 < ~, fl < 1}.

(2.147)

Values of vc(i), va(i) and vb(i) are given in Table 2.2. Let/d denote the family of connected components of the set ~8=0 R(i). 2.3.6. The family bl has nine components, i.e. R(i) N R(j) = O, i ~ j, i , j = O . . . . . 8.

LEMMA

PROOF. This is clear from the definition of set R(i). See also Fig. 2.41.

THEOREM2.3.5. Mapping F1 is (ld, 16)-chaotic. Proof of this theorem is given in Section 2.4.4.

[]

CHAPTER 2

216

Differential Equations with Hysteresis

I

I

'

I

'

I

'

1.35 1.8 1.3 1.6 1.25 1.4

-

,

I

,

2.59

I

,

2.6

I

,

2.61

I

,

2.62

I

2.63

1.2 -

1-

0.8 2.5

2.6

F I G U R E 2.41

2.7

2.8

2.9

3

Sets R(i), i = 0 . . . . . 8 in c o o r d i n a t e s t m o d V ~ r , z (2).

2.5 -

~,,~

-

1.4

r :Y~'G

1.5

v-,,,, #?.. , : , . ~ , - 4 d ~ ".

:V,

..

~" \'~

2.6

2.7

2.8

2.9

0.5

1 F I G U R E 2.42

2

3

4

Sets R(i), i = 0 . . . . . 8 a n d a c h a o t i c a t t r a c t o r .

5

,

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

217

Note also the following. PROPOSITION2.3.24. The map F1 has a fixed point v, ~ R(0) and for any positive

integer p greater than eight there is a periodic point v ~ R(O) which has the minimal period p, and which is not asymptotically stable. Moreover, Eqn. (2.9) has periodic solutions of minimal periods v/2rc 9 (3kl + 29k2), where kl, k2~>0, kl + k2 > 0. In particular there exist solutions of all periods x/-2rcn, with n > 55. Proof of the proposition closely follows ideas in [172,173].

Interpretation In this subsection we discuss another interpretation of our main result from the previous section. It is convenient to begin with some definitions in terms of Eqn. (2.13). We will always suppose that the functionf(t, z, u) is periodic in t with minimal period T > 0. Thus any periodic solution of that equation must have a period which is a multiple of T. Let no be a positive integer and zo(t), - ~ < t < ~ , (2.148) be a function with the minimal period noT. This function will play the role of an approximate n0T-periodic solution of Eqn. (2.13). We will always deal with the situation when the function zo(t) is piecewise continuous, and the limits z(t - 0) = lim z(t)

rSt

are well defined at the points of discontinuity. In our setting the wording 'the function zo(t) can be considered as an approximate n0T-periodic solution of Eqn. (2.13)' means that, firstly, the jump Iz0(t) - zo(t - 0)l at any discontinuity point t is rather small, and, secondly, the function almost satisfies the equation between successive points of discontinuity (the quantitative bounds for the corresponding discrepancies will be specified below). Suppose that for a positive integer nl < no, the number Iz0(0) - z0(nlT - 0)1 is small. Then we can treat the niT-periodic function Zl(t), - o e < t < oe, which is defined by zl(t) = zo(t), 0 <~t < niT, as an approximate niTperiodic solution of Eqn. (2.13). Since no is the least period of zo(t) and

CHAPTER 2 Differential Equations with Hysteresis

218

z

Zl.

$_;

,..G/ Zo

x

t

!

FIGURE 2.43 Functions z0 (solid) and Zl (dotted). Here z0 has period 2T, z I has period T; thus z0, Zl coincide on 2nT<.t < (2n + 1)T, n = 0, 4-1, 4-2. . . . in this case. nl < no, w e define the distance r b e t w e e n the c o r r e s p o n d i n g functions

zo(t), zl (t) b y r = sup Izo(t) - zl(t)l > 0.

(2.149)

t

Figure 2.43 illustrates the situation for the case no = 2, nl - 1: the functions z0 a n d Zl are s h o w n by a solid a n d a dotted line, respectively. We can use as a p p r o x i m a t e patterns for possible b o u n d e d o u t p u t profiles, not only the functions zo(t) a n d zl(t) themselves, b u t also all possible 'shuffiings of these functions'. To explain w h a t this m e a n s w e need additional notation a n d definitions. Let f~ be the totality of all bi-infinite oo b i n a r y sequences o9 = {o9i}i=-oo w i t h ogi E {0, 1} for i = 0, 4-1, 4-2 . . . . . To each sequence co e f2 there corresponds the sequence of n u m b e r s ti to be defined b y to = 0, i-1

ti = T E no~] j=0 for i = 1, 2, 3 . . . . . a n d b y i

ti--T

E

no~j

j=-I

for i = - 1 , - 2 , - 3 . . . . . . That is, the real line R 1 is partitioned into the intervals Ii = [ti, ti+l) w h e r e the length of the interval Ii equals no~iT. Figure 2.44 illustrates this idea: here 09= . . . . 0 , 1 , 0 , 1 , 1 , 0 . . . . . a n d the element coo is in bold.

(2.150)

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

-T

0

T

219

2T

3T

4T

FIGURE 2.44 Partition of R 1 corresponding to o~ . . . . . 0, 1, 0, 1, 1, 0 . . . . ; w e show the intervals corresponding to coi = 0 by the solid lines and intervals I i, corresponding to coi = 1, by the dotted line, with co0 = 0 in bold.

Z l ZI.

9

M

-.~.. .~"

Z1"

Z1.

ir

".~

If ~

l

,

'0

i

r

CT

l

~3T

i

/

/

if

,

,

I

~4T

t

I

!

z0 FIGURE 2.45

Function zo~ for o~ . . . . .

0, 1, 0, 1, 1, 0 . . . .

C o r r e s p o n d i n g to a n y o9 ~ f~ t h e f u n c t i o n zoo(t) is d e f i n e d b y

zoo(t) - zo~i(t - ti),

for

t i ~ t < ti+l.

F i g u r e 2.45 s h o w s a f r a g m e n t of t h e f u n c t i o n zo~ for t h e case w h e n co satisfies (2.150). We will u s e t h e n o t a t i o n Z ~ = {zm(t) 9 o9 ~ f~}. PROPOSITION 2.3.25. A n y two functions Zm~l), zo)(2) 09(2) . Moreover, sup

IZco(1)--Z(o(2) I -- r(09 (1), to(2))~rmin > 0,

~_

Z ~ are different if 09(1)

09(1) 09(2) E ~'-~, ~(1) ~: 09(2).

-c~
inf r(09 (1), 09(2))/~rmin > O, co(l) ~_co(2)

09(1), 09(2) E ~

is strictly positive. We call r ( Z n ) t h e s e p a r a t i o n t h r e s h o l d for t h e set Z n .

(2.152)

CHAPTER 2 Differential Equations with Hysteresis

220

Z

zo)(t) + e / I ~" ~ " ~ ,

II l ~

,/

"

~ '",

,'

'" .... ',,

"

,' ,'

"',

'" ....

"~, 9 ,

'

"'"

r

, i

,

2~T.... "'/

',,

!

I

.

I

~

9 ~ aL

~" ~

'

,' I 9

~

IL

""

9 ',.: v

:'T--'" " "

t ,"

",

'

zto(t)- e

FIGURE 2.46 A function z(t) (solid dark curve) satisfying (2.155), with o~ . . . . . O, 1, O, 1, 1, 0 .... If the sequence o9 is p-periodic in the sense that ogi = 0, 4-1, +2 . . . . . then the function z~(t) is T~-periodic where

ogi+p

for i =

p-1

To~ - T Z

n~i"

(2.153)

i=0

The number (2.153) is the minimal period of the function zoo, provided that p is the minimal period of the sequence co. Thus, among the elements of the set Zfl there are periodic functions with arbitrary large minimal periods. However, the main bulk of the set Zfl consists of functions which do not follow any particular pattern. Let e be a positive number satisfying e < r(Zn)/2.

(2.154)

We will say that Eqn. (2.13) is (zo, Zl, e)-compatible if the following two conditions hold: (C1) For any function zoo(t), o9 ~ f2 there exists a solution z(t) of Eqn. (2.13) satisfying the uniform estimate max

-cx~
Iz(t) - z~(t)l < e.

(2.155)

(C2) Additionally, if the sequence o9 is p-periodic, then there exists a T~periodic solution z(t) of Eqn. (2.13) satisfying the uniform estimate (2.155), where To~ is given by (2.153). Figure 2.46 graphs a function z(t) that satisfies (2.155), where o9 satisfies (2.150).

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

'

i

i

!

'

i '

2

"

221

i

'

i

'

'

1'0

'

i

'

_

0-1 -2

,...

,

0

|

20

,

i

40

,

!

60

,

i~ , 80 100 120 140

-20

;

1;

'

20

FIGURE 2.47 A function zo(t) on the left and a function z l(t) on the right. In the scalar function z for small e, functions z(t) satisfying (2.155) follow closely the 'guidance' of a c o r r e s p o n d i n g function zoo(t). In the term i n o l o g y of hyperbolicity theory, a function z(.) is an e-shadow of zoo(-). Since e is strictly less than half of the separation threshold r(Zr any solution z(t) can be considered as an e-shadow of not more than for one guide zoo. Thus, the m e a n i n g of (z0, Zl, e)-compatibility is that for any pattern given b y a function zo~ ~ Zr we can find a b o u n d e d solution profile which is an e-shadow of zoo. Consider Eqn. (2.9) with b = 1. As zo(t) = (xo, x'o) we choose the numerical solution of (2.9) c o r r e s p o n d i n g to the pseudo-trajectory of F1 of the length nine, which is given as the u n i o n of nine numerical trajectories with initial conditions x(ti) = 0.5, x'(ti) v~2) (i) on the intervals [ti, ti+l), i - 0 . . . . . 8; V~1) , V~2) are given in Table 2.2 and ti is given b y -

ti =

-

v~1) (i) + 3 i - v ~ n , v~1) (i) + (3i + i - 4 ) . ~/2n, v~1) (i) + (3i + 2). ~/2n,

i - 0 . . . . . 4, i - 5 . . . . . 6, i = 7 . . . . . 8,

vf 1) (0) + 29. v ~ n ,

i - 9.

Thus no = 29. Let us choose z] (t) as zo(t) on the interval [to, to + 3- ~/2n), in which case n l -- 3. The pictures of the w a v e g u i d e s are given in Fig. 2.47. LEMMA2.3.7. The following relation holds: r(Zn) > 0.1. By this l e m m a the (z0, Zl, e)-compatibility is well defined w h e n e v e r e<0.05. The principal result of the previous section, T h e o r e m 2.3.5, m a y be rewritten n o w as follows.

222

CHAPTER 2 Differential Equations with Hysteresis

THEOREM2.3.6. The equation is (zo, Z l , O.O1)-compatible.

Why It is Interesting and Important The (z0, zl, s)-compatibility implies, in particular, that the totality of bounded solutions is spatially chaotic in a topological sense. We recall, as an aside, that the three main aspects of chaotic behavior are equally important: the measure theory aspect such as the existence of the Sinai-Ruelle-Bowen invariant measures [174]; the 'sensitivity' aspect, such as positivity of Lyapunov exponents [171]; and the topological aspect describing irregular mixing properties in terms of symbolic dynamics [171]. At an informal level the topological aspect of the spatially chaotic behavior can be described as follows. Suppose that we register the forms of the solutions within a certain precision ~, where ~ is greater than s, but significantly less than the separation threshold (see the relevant numerical information in the next section). Then a 'shadow solution' z(t), satisfying (2.155), is indistinguishable from its guide zo~(t), although different guides are quite distinguishable. Thus any chaotic shuffling zo~ of two basic patterns z0, Zl provides a possible steady profile (within the accuracy ~). The next level of formalization requires ideas of symbolic dynamics. Denote by n the least common multiple of the numbers no, nl. Also let e [0, noT) be a fixed number such that Iz0(z) - Zl(Z)l > 2s (such a ~ exists by (2.154)). Firstly, we characterize a function z(t), -cx~ < t < cx~,by the bi-infinite sequence of its values Zi - -

z(inT + z),

i = O, 4-1, 4-2 . . . . .

(2.156)

That is, we measure the values of z(t) 'stroboscopically' with the uniform interval nT between the measurements. Further, encode the function z(t) by a binary sequence o9 e f~, which is constructed as follows. Let

Io = {z:lz- z00:)l~e},

I1 -- {z:lz- Zl('01- ~s}.

The sets I0 and 11 do not intersect since Iz0(z) - Zl('01 > 2S. If for any i the value zi belongs either to I0 or to I1, then we define the code sequence o9 - co(z(.)) by setting o g i = 0 in the first instance, and ogi = 1 in the second. In other words, we indicate the measurement positions inT, i = 0, 4-1, 4-2 . . . . where the function z(t) goes through a window I0 or I1 (see Fig. 2.48 for one-dimensional interpretation). If the value zi does not belong to one of these two windows for some i, then our coding procedure fails, and we are not interested in such 'unencodable' functions z(t). Since I0 N I1 -- 0, not more than one code can correspond to a particular function z(t). N o w the compatibility condition (C1) implies that any random

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

223

Z

I I f

t

FIGURE 2.48 A function z(t) with the code .... 0, 1, 0, 1, 1, 0 . . . . . sequence o9 a p p e a r s as a code for some b o u n d e d profile. (Indeed, let us introduce a sequence o9* b y substituting n/no ones (n/nl zeros) instead of a n y one (any zero) entry of the sequence o9. Then, the function z~. (t) ~ Z~ has the required properties.) For a given function z(t) w e denote b y ;:(-) = anTZ(.) its (left) nT-shift to be defined b y ;:(t) = z(t + nT). Let us denote also b y a the s t a n d a r d (left) shift on f2 given b y a(og) = o91 -- ( . . . . . o9I 1' 090' r 091t . . . . ) !

w h e r e ogi = o9i+1. For instance, if o9 - . . . , 0, 1, 0, 1, 1, 0 . . . . , then ogl

...,0,1,0,1,1,0

.....

PROPOSITION 2.3.26. Suppose that Eqn. (2.13) is (zo, Zl, e)-compatible. Then to

each o9 ~ ~ there is a solution z~(x) of Eqn. (2.13), such that z~(.) has the code co, and this correspondence og~-+zo~is shift invariant: O'nTZo~(') = Zao~(') for og~f~. Proposition 2.3.26 can be p r o v e d in a s t a n d a r d w a y using the Z o r n L e m m a . In technical terms, the above proposition m e a n s that the restriction O'nT of the shift an0T to the set {z~ : o9 ~ f2} is semi-conjugate to the shift a in f2. Such semi-conjugacy is w i d e l y recognized as the m o s t i m p o r t a n t attribute of topologically chaotic behavior, see, for example, [171]. It is also i m p o r t a n t to note that (z0, Zl, e)-compatibility implies the existence of solutions w i t h a w i d e range of nontrivial regular patterns. For instance, one more basic characteristic of chaotic b e h a v i o r is an a b u n d a n c e of periodic solutions and, in particular, an exponential rate of increase of the m a x i m a l n u m b e r of pairwise e-separated 6 kT-periodic solutions w h e n k increases, see [ 171]. 6Two functions z(x), y(x) are E-separated if sup Iz(x) - y(x)l > e.

224

CHAPTER 2 Differential Equations with Hysteresis

PROPOSITION2.3.27. Suppose that Eqn. (2.13) is (z0, Zl, e)-compatible and denote by n the least common multiple of the numbers no, nl. Then it has at least 2 k periodic solutions of period kn which are pairwise e-separated. This assertion follows immediately from the definitions. Another example of functions with a regular pattern is given by the socalled recurrent functions [97], which exhibit fractal-type features. The recurrence property of a function z(t) with - c ~ < t < o0 means that for each pair of positive numbers (e, a) there exists A > 0 such that for each real r, any interval which is longer than A contains r such that the function t --, z(t + r) is e-close to the function t ~ z(t + a), t ~ [-T, T]. Recurrent nonperiodic functions are important today in the analysis of desynchronized systems, various new models in biology, etc. The notion of a recurrent sequence co is defined analogously. Recall, in particular, the classical Morse example of a recurrent sequence. We begin from the one-element sequence Sl = 0 and then substitute at each step the two-element sequence 0, 1 instead of 0 and two-element sequence 1, 0 instead of 1. Thus, Sl = 0 , 1 ; s2 = 0 , 1 , 1 , 0 ; $3 = 0 , 1 , 1 , 0 , 1 , 0 , 0 , 1 ; S4 = 0, 1, 1,0, 1,0,0, 1, 1,0,0, 1,0, 1, 1,0; etc.

After infinitely many iterations we obtain the famous forward Morse sequence M. Considering backward shifts of this sequence and their weak limits we arrive at the set A//of bi-infinite Morse sequences. PROPOSITION2.3.28. Suppose that Eqn. (2.13) is (zo, zl, e)-compatible and the sequence co is recurrent in the Morse sense (for instance, one of the set .M). Then there exists a recurrent, nonperiodic, solution satisfying (2.155). The proof follows the standard constructions [175]. 2.3.9 RECURRENT OSCILLATIONS

We will be interested in recurrent solutions of the general equations

x' =fit, x, y(t)), y(t) = (Fit0, y(to)]x) (t).

(2.157) (2.158)

Here x ~ ~a, and y belongs to a complete metric space 3; with the metric py. Also, for a fixed Y0, the operator F[t0, Y0] transforms functions

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

225

x 9 [to, oo) ~ ~a into functions y 9 [to, oo) ~ 3;. For a particular function x(.), the notation (Fit0, y0]x) (t) refers as usual to the value of the corresponding function y(.) at the m o m e n t t. The operators describing the hysteresis nonlinearities can be treated with a typically infinite dimensional set 3; of internal states y. We will use the definition of recurrent property introduced on the previous page. Periodic, quasiperiodic and almost-periodic functions are recurrent, however the class of recurrent functions is m u c h wider. The typical examples of recurrent, but not almost-periodic behavior are given by various modifications of the classical Morse construction [176]. Such functions are getting rather important today due to analysis of desynchronized systems [177], various new models in biology, etc. We will establish, u n d e r some natural technical restriction, that the system (2.157)-(2.158) has a recurrent solution, providing that the function f in the right-hand side of Eqn. (2.157) is recurrent in t, and that there exists at least one solution (x(t), y(t)) which is uniformly b o u n d e d at t --+ oo. Surprisingly, some of our results seem to be new even for Eqns. (2.157)(2.158) without hysteresis.

(W, C)- and s

Functions

Let us introduce some notation. 9 Let a function a of scalar a r g u m e n t be measurable at the interval [r, r~] and take values in a finite-dimensional Euclidean space. We denote II~ll&.~ =

I~(t)l dt.

(2.159)

9 If a function ~ is additionally absolutely continuous at [~, rr], denote

Ilallw~,~ = la(~)l + Ila'lls

(2.160)

9 For any continuous functions yl, Y2 : [z, rT] --+ 3;, we define

P:,cr(Yl, y2)

-- max P3;(yl(t), y2(t)). te[~,cr]

(2.161)

We denote by Wr,r aa the Banach space of absolutely continuous functions x 9 [r, cr] --+ R a with the n o r m (2.160). Analogously, denote by C~,~;3; the complete metric space of continuous functions y : [~, or] ~ 3; with the uniform metric (2.161).

CHAPTER 2 Differential Equations with Hysteresis

226

Denote the z-shift of a function y by 0(r)y:

(O(r)y) (t) = y(t + r). We say that a set ,5' c ~ is A-dense if any interval [t, t + A] contains at least one element from $; that is ,_q~[t, t + A] ~a 0, for all t ~ ( - ~ , ~ ) . Let a function x 9 R ~ R~a be absolutely continuous on any finite interval, whereas a function y : ~ ~ 3; is continuous. The pair (x, y) is said to be (W, C)-recurrent if for any positive T, ~ there exists A = A(T, e) such that the set of rT ~ ( - ~ , o0) satisfying

II0(~)x- O(~)xlIw_T,T + P-T,T(O(r)Y, O(cr)y) < e is A-dense for each z ~ (-cx~, cx~). Let a functionf be measurable in t and continuous in x, y (that is, it satisfies the Caratheodory condition). Let also f satisfy the following growing condition: TT Ixl,
T< ~

(2.162)

for any T, R > 0. The function f is s in t if for any positive numbers T, R, e there existsA = A(T, R, e) such that the set of cr ~ (-oo, o0) satisfying T

max Ll:(t + ,, x, y) - f i t + or, x, y)l dt < e T ]xI<<.R,Y~3;

(2.163)

is A-dense for each 9 ~ (-oo, oo).

(W, C)-Correct Hysteresis Nonlinearities We will use a standard approach to describe hysteresis nonlinearities, s e e [15] and the references therein. We suppose that the operator F[t0, Y0] can be understood as a description of an autonomous control dynamical system F. That is, the family of operators F[t0, Y0] must satisfy the following properties: 9 the Volterra property: X(S) =-- Xl(S), S E

[to, t],

implies

(F[t0, y0]x) (t) = (F[t0, y0]xl) (t);

9 the semigroup property: (F[tl, (F[t0, y0]x)(tl)]X)(t) - (F[t0, y0]x)(t), for to < tl < t;

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

227

9 the autonomy property: (Fit0, y0]x) (t) = (F[t0, y0]Xl) (t -- t o q- tl), where Xl(t) = x(t - to + tl) and to < tl < t. N o w we introduce some additional properties of the hysteresis operators F. We will suppose that the operators F[t0, Y0] are defined for all absolutely continuous functions x 9 [0, oe) --~ ~ct and that their outputs are continuous. Using the Volterra property, the operators Fit0, Y0] are also well defined at any finite interval [to, tl]. Recall, that a set $1 of measurable functions x 9 [0, t] ~ ~a is uniformly integrable if the relationship f lim / [x(t)[ dt = 0 2--,oo dlx(t)l>2 holds uniformly in x 6 51. A set $2 of Y-valued continuous functions is said to be equicontinuous if for any e > 0 there exists 6 > 0 such that p y ( y ( t l ) , y(t2)) < e,

It1 - t 2 ] < 6, y 6 $2,

holds. We say that a nonlinearity F is (W, C)-correct if for any uniformly bounded set $1 of absolutely continuous functions, such that the set of derivatives {x', x 6 $1} is uniformly integrable, the set $2 -- {l-'[0, yo]x : Yo ~ 3;, x

~

$1}

is equicontinuous, and if the relationships [IXn - X]lWto,t 4- P 3; (,YO(n) ' Yo) ---> 0

imply p y ((F[t0, y~n)]xn)(t), (F[t0, yo]x)(t)) ~ O.

for any - ~ < to < t < c~. As examples of (W, C)-correctness we mention the Prandtl, Besseling, Ishlinskii, Mizes, Tresca, etc. models of plasticity, Madelung, Preisach, Giltay, Mayergoys, etc. models in ferromagnetism and many, many others, see [7,14,15,157] and the bibliographies therein. Actually all hysteresis nonlinearities with continuous output signals known to us belong to the class of (W, C)-continuous nonlinearities.

228

CHAPTER 2 Differential Equations with Hysteresis

Main Result Let us come back to the system (2.157)-(2.158). The function f in the right-hand side of (2.157) will always satisfy the Caratheodory condition. We will understand the solutions of Eqns. (2.157)-(2.158) in the classical sense: solution is a pair (x, y) where x 9 [to, cx~) "---> RJ is a locally absolutely continuous function and y : [to, cx~) :--+ 32 is continuous; the relationships (2.157) should hold almost everywhere and the relationship (2.158) everywhere. (Recall [178] that the function x 9 [~, a] ~ ~d is absolutely continuous if x' exists almost everywhere, is integrable on [r, a], and x'(t)dt = x(tl) - x(t2) for r < tl < t2 < a.) The pair (x, y) where the functions x and y are defined over the whole interval (-cxz, oo) is said to be a solution of the system (2.157)-(2.158) if the restrictions of the functions x, y to the interval [to, ~ ) provide a solution of this system for each to 6 (-cx~, oo). The last definition is correct by the Volterra property.

ft~

THEOREM2.3.7. Let f be s

in t and let F be (W, C)-correct. Let (x,(t), y,(t)),

t>~O,

be a solution of the system (2.157)-(2.158) with the function x,(t) uniformly bounded. Then there exists a (W, C)-recurrent solution (x(t), y(t)) of the system (2.157)-(2.158). N o w we will give some general comments concerning the use of the preceding theorem. Under simple additional assumptions, that hold for all practically important hysteresis nonlinearities, any initial problem

x(to) = xo ~ Rd,

y(to) -- Yo ~ Y,

(2.164)

has at least one solution (x, y) defined at the maximal interval [to, T) where either T = ~ or Ix(t)l --* ~ as t -+ T. It is sufficient, for instance, to suppose that the hysteresis nonlinearity F is continuous from C to C (as in the case of Prandtl, Besseling, Tresca, Mises, Giltay, etc. nonlinearities) or is continuous from W to W (as is the case for the Madelung nonlinearity). The proof proceeds in the usual way using the Showder fixed-point theorem. We will say in this situation that the initial problems for Eqns. (2.157)-(2.158) have local solution. If the initial problems have local solutions, we say that the system (2.157)-(2.158) is dissipative if for any R > 0 there exists R1 > 0 such that lim sup Ix(t)l~R, t--, oo ix01~
2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

229

where x(t) is a solution of Eqn. (2.164); the sufficient conditions of dissipativity are easy to suggest in terms of Lyapunov functions, etc. COROLLARY2.3.12. Let f be s in t and let F be (W, C)-continuous. Let the system (2.157)-(2.158) be dissipative. Then there exists a (W, C)-recurrent solution (x(t), y(t)) of the system (2.157)-(2.158). Note that for analysis of stability of recurrent solutions, the method suggested in [83] is applicable without trouble.

2.3.10 HYSTERESIS VERSUS TIME RELAXATION Generalities Time relaxation and hysteresis are common strongly nonlinear phenomena which occur in many industrial, physical and economic systems. Often these two types of phenomena manifest different stages of the same or similar processes. A number of fundamental hysteresis models can be considered as limit cases of time relaxation processes, or admit an approximation by a differential equation which is singular with respect to a particular parameter. However, the level of interaction between theories of systems with time relaxation and systems with hysteresis (and between the 'relaxation' and 'hysteresis' research communities) is quite low. The aim of this section is to discuss, at the very basic level, relationships between those two topics. An instructive singularly perturbed system is the following

x' =fit, x, y), ey' = g(x, y),

(2.165) (2.166)

where e > 0 is a small parameter. We concentrate first on the auxiliary equation

~y' = g(x(t), y), where x(t) is an arbitrary function. We denote the solution of this equation, with a given initial condition y(0) = Y0, by

y(t) = VE[t0, yo]x(t). Under suitable technical conditions it can be proved that for a generic function x(t) there exists a limit

y~(t) = V[to, yo]x(t) = lim V~[t0, yo]x(t). 6--+0

CHAPTER 2 Differential Equations with Hysteresis

230

g(x,y)
/s x,~ X

g(x,y)>O

FIGURE 2.49 Relay as a singularly perturbed ODE. The solutions of the original system are then approximated as e ~ 0 by the solutions of the differential-operator equation

x' =fit, x, y(t)),

y(t) = V[to, yo]x(t).

(2.167)

The following observation is straightforward: LEMMA2.3.8. The limit operator V is rate-independent. Thus, the global analysis of singularly perturbed equations becomes, in the first approximation, an investigation of equations with hysteresis nonlinearities. This is the first point of contact between equations with hysteresis and singularly perturbed equations. We will demonstrate how this observation works below. Let the function g be described by the picture in Fig. 2.49. Suppose x(t) does not have local minima equal to a or local maxima equal to ft. Then the operator V is well defined on x(t) and

V[to, yo]x(t) = R~,fl[to, yo]x(t) for all t > to such that

x(t) # a

and

x(t) # fl.

(See Section 2.1.2 for the definition of Ra,fl.) This example shows that singularly perturbed equations can approximate equations with hysteresis. This is the second point of contact between equations with hysteresis and singularly perturbed equations. This approach is especially important if there are some physical reasons to believe that the relay (or any other discontinuous hysteresis nonlinearity) is an idealization of a limit operator similar to V. Sometimes, however,

2.3 EXAMPLESOF RIGOROUS CONCLUSIONS

231

discontinuous hysteresis operators are idealizations of quite different continuous operators. The application of methods used in systems with singular perturbations to these new types of singular approximations is the third point of contact between equations with hysteresis and singularly perturbed equations. We note in conclusion that numerous mathematical models of physical, mechanical and other systems can be described by the equations of the form

x' =fit, x, y, z)~ y = Fx(t),

ez' = g(x, y, z) where F is a continuous hysteresis nonlinearity acting on function x(t) (for, instance, Preisach nonlinearity P a with an absolutely continuous measure #). The analysis of such systems will require the combination of ideas both in the area of systems with hysteresis and in the area of singularly perturbed systems. The main technical problem lies in the fact that hysteresis operators are not smooth in a standard sense: they only satisfy a Lipschitz condition as operators in function spaces. Thus, even the notions of stable and unstable manifolds must be adjusted accordingly.

Generalized Play Ferromagnetic monocrystals can be considered as an idealization of an interesting hysteresis nonlinearity: the generalized play [15]. This will be denoted by L~, and is illustrated in Fig. 2.50. Formally, this nonlinearity can be described as follows. We denote by f2 the set of pairs (x, y) satisfying

?_(x)~y~7+(x). For a continuous input x(t) and for initial state r/0 satisfying (x(to), rio) ~ f2, the output y(t) = L~[to, r/0]x(t) is the unique continuous function such that, firstly, y(to) = r/0, secondly,

(x(t), y(t)) ~ f2,

t~to,

and, thirdly, the inclusion

(x(t), y(~)) ~ f2,

~t~r~

implies y(t) =_ y(r), z~tKr~. Thus, the output behaves 'lazily': it prefers to be unchanged when the phase pair (x(t), y(t)) belongs to f2. Fig. 2.51 shows a typical phase trajectory.

232

CHAPTER 2 Differential Equations with Hysteresis

V_(x)

,

1

='? / 13+~

x

(x) FIGURE 2.50 Generalized play Le.

Generalized play L~ can be also represented by the Preisach m o d e l with an appropriate measure. LEMMA2.3.9. Suppose x(t) has no local minima equal to ~ or local maxima equal

to fl, and let Yo equal either 0 or 1. Then the limit y,(t) = lim L~[to, yo]x(t) e-+ O

is well defined, and y,(t) - 2R_Kl~[t0, yo]x(t) - 1 for all t > to such that x(t) # ~

and

x(t) # fl.

In other words, for generic input functions x(t) the discontinuous nonlinearity 2R_K~x(t) - I is a limit of continuous nonlinearities LE[to, Y0] as E ----~ 0.

233

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS y

ar

ill

v

X

x~

FIGURE 2.51 Phase portrait of a typical trajectory for the generalized play Le.

Hysteretic Canards We consider the equation x" + x - b sin(t) + y(t),

y(t) - 2 R _ K ~ x ( t ) - 1.

(2.168)

By L e m m a 2.3.9 this is to be considered as an idealization of the equation x" 4- x = b sin(t) 4- y(t)~

y(t) = L~x(t)

(2.169)

where Le is a generalized play from the previous subsection, and e > 0 is a small parameter. We will say that a pair (x,(t)~ y~(t)) is a limit periodic solution of Eqn. (2.169), if it can be approximated by some 2~-periodic solutions (x~ (t), ye (t))

234

CHAPTER 2 Differential Equations with Hysteresis

of the system (2.169) with arbitrary small e. That is, if for any a > 0 there exists e0 > 0 such that for e < e0 Eqn. (2.169) has a periodic solution xe(t), ye (t) satisfying max [xe(t)- x,(t)[ < a. THEOREM 2.3.8. Let 0 < b < 4/n, then Eqn. (2.169) has exactly three limit periodic solutions:

(xo(t), yo(t)),

(xl(t), yl(t)),

(x2(t), y2(t)).

(2.170)

We n o w give explicit formulas for the solutions (2.170). Introduce the functions b x0(t, b) = - f l cos t + ~ (sin t - t cos t)

nb +-v-(cos t - 1) sin t, 4

(

Xl t + n-~:.b 2 '

)

=-(l+fl)

+I+

cost+~

- n K t < n,

sin~sint-tcos

t+

2

nb (cos t - 1)(sin t + 1), ~ - 2nKt < ~, 4 sin(z/2)

x2(t) = - x l ( t - n). Here ~ = v(b), 0 < b < 4 / n is defined by the relationships 1-fl arccosl_Ffl
b=2

f l - 1 + (fl+ 1)cosz sin(r/2) (sin ~ + "c- 2n)

Introduce also 2n-periodic functions y0 (t), yl(t), y2(t) by the equalities nb ~ sint,

y0(t;b)=

Yl t +

n-r.b)= 2 '

1

-nKt
nb r --~-sin~(l+sint),

y2(t) = - y l ( t - n).

r-2n~
235

2.4 EXAMPLES OF TECHNICAL TOOLBOXES PROPOSITION2.3.29. The functions given above satisfy Theorem 2.3.8.

Theorem 2.3.8 and Proposition 2.3.29 are similar to those proved in [179]. Figures 2.52, 2.53 show x0(t), yo(t) w h e n / / = 1. Similarly, Figs. 2.54, 2.55 show the functions xl(t), Yl (t). We claim that these limit solutions are analogs of canard solutions, see [180-182]. In particular, (xo(t), yo(t)) is an analog of the symmetric canard and (xl (t), Yl (t)) is an analog of the asymmetric canard. 9 The first reason is that classical canards and these solutions are mathematical explanations of the same physical phenomena: the limit behavior of periodic solutions of a family of differential equations approximating a given equation with a discontinuous, relaytype nonlinearity. 9 Secondly, in both cases the limit solutions demonstrate strange behavior: in the classical canard situation for a time they follow unstable manifolds; in the 'hysteretic canard' situation, the solutions can stop in the course of a jump, and then hover for a while at an intermediate level. 9 Thirdly, similar methods could probably be applied to analyze both situations. For example, the topological degree approach used to prove the last theorem can be useful for the rough location of classical canards. On the other hand, the typical 'canard community' methods of investigation will be useful in a more subtle asymptotic analysis of hysteresis canards.

2.4 E X A M P L E S

OF TECHNICAL

TOOLBOXES

2.4.1 DESCRIPTIONS OF PREISACH NONLINEARITY Standard Representation States and Outputs Consider the set f2 on which the internal states ~/ : f2 ~ {0, 1} of the Preisach nonlinearity P are defined. Under natural technical restrictions this set can be chosen to be the two-dimensional half-plane FI = {(~, fl) : fl > ~} (see [15]). Moreover, the formula (2.3) is replaced by:

y(t) - y[to, r/0](t) - f~<~ Ra,~[to, r/0(co)]x(t) d#,

t~to.

(2.171)

Here # is a probability Borel measure in FI. Below, unless the opposite is stated explicitly, we suppose that this measure is absolutely continuous

236

CHAPTER 2

Differential Equations with Hysteresis

1

0.5

.

.

-3

FIGURE 2.52

.

.

.

-2

.

.

2

-1

3

Functions xo(t) (sine-like function) and yo(t) (step-like function).

l-

v

-1-

FIGURE 2.53

Parametric plot of (x0(t), yo(t)).

X

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

237

2

1.5 1

0.5

9

/-5

FIGURE 2.54

,

-4

-3

,

.

|

.

.

.

.

.

.

-2

.

.

.

1

Functions x l(t) (sine-like function) and yo(t) (step-like function).

1 I I I I I

-1

J

1

-1-

FIGURE 2.55

Parametric plot of (x l(t), yl(t)).

X

238

CHAPTER 2 Differential Equations with Hysteresis

with respect to the standard Lebesgue measure. That is, for S c II #(S) = L h(~x, fl)d~dfl, where h is nonnegative, and satisfies

/ii

h(~, fl) dadfl = 1.

It is sufficient to consider the case when the initial and varying states r/0, 1/(t) are characteristic functions of sets S, which look like the shaded set in Fig. 2.56. That is, S is located to the left of the line a - fl and below a continuous curve L = Ls, with a nonpositive slope. Additionally, in this standard case the coordinates of the intersection of L with the diagonal coincide with the current value of the input x. In other words such sets S admit the representation S = {(a,/~) :/~ + a ~ < ~ s ( / ~ - a)}

where 6s(U), 0~
- vl.

The curve Ls is connected with the function ~'s by

LS = {(a, t) : fl + ~ = ~/s(fl- a)}.

(2.172)

We denote by (~, the set of states r/of the Preisach nonlinearity the characteristic functions of which are described above (see Fig. 2.42).

Dynamics

The set ~, is convenient because the evolution of the every state r/e ~, admits a simple geometrical interpretation in terms of the area described by i/, see Figs. 2.57, 2.58. Here the input x(t), moves along the horizontal scroll-bar, and controls the point on the diagonal a = fl above itself. When moving toward the upper right corner, this point on the diagonal drags the horizontal line, and hatches the domain below this line and above the diagonal. (For instance, if, in Fig. 2.57, x increases from the value u to the value v, the filled area is increased by the hatched triangle.) When moving towards the bottom-left comer, the diagonal point drags the vertical line, and hatches everything to the right of this line and above the diagonal as in Fig. 2.58. The output y(t) is the area of both shaded and hatched domains with respect to the measure #.

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

239

FIGURE 2.56 The set S given by typical characteristic function r/E (~,. In particular the set @, is invariant with respect to piecewise continuous inputs. Clearly, after some time the 'tail' of the curve Ls(t) becomes a continuous piecewise linear curve L, whose links are parallel to one of the coordinate axes.

The Krej~i Theorem The 'picture definition' of the dynamics given above works formally only for piecewise monotone functions. There exists also a simple and very convenient general analytic definition, first discovered, probably, by Pavel Krej~i, which we will also use below. We recall that the play nonlinearity describes a mechanical device illustrated by Fig. 2.59. Here a bean B can be moved along the string S. We control a frame F which can be moved along a string S forward and backwards. The position x(t) of the handle is a varying input. The bean moves only if the left end of the frame pushes it forward, or if the right end of the frame pushes it backward. The position y(t) of the center of the bean is the output. This 'device' is the play operator with the width w = s - r/2 where s is the width of the frame, and r is the radius of the bean. We will illustrate the relationship between the play operator and the Preisach model first by Fig. 2.60 below.

240

CHAPTER 2 Differential Equations with Hysteresis

FIGURE 2.57 Evolution of the set S corresponding to the state r/of the Preisach nonlinearity; increasing input.

Here we have many (ideally infinitely many) strings all parallel to the diagonal fl = a. There are also (infinitely) small beans, one on each string. There is no gravity, so beans do not move without external influence. We control the position of the angle-shape frame F, whose origin O is always at the diagonal a - ft. The position of the origin with respect to the scale x is the input. Initially, let the beans be located along a certain line L = L0, which bounds the set described by the initial state 1/= r/(0) of the Preisach system. We suppose that the initial position of O coincides with the intersection of L with the diagonal. Since r/~ ~, the whole line L is within the initial position of the frame. Then, when time passes, the beans will be moved (to the north-east when the input increases, and to the south-west, when the input decreases). It is easy to see that at any timemoment the beans will be positioned along the line L~l(t). This sketches the picture proof of the following Krej~i theorem. THEOREM2.4.1. Let P[to, x(t0); u] denote a one parameter family of the play nonlinearities with the initial state x(to) E ~ at to and the parameter u~O. Let rl(t) ~ if,,, t~to be the varying state of a Preisach nonlinearity for an input x(t), t~to and let the set S(t), corresponding to r/(t), be defined by a continuous

241

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

FIGURE 2.58 Evolution of the set S corresponding to the state r/of the Preisach nonlinearity; decreasing input.

FIGURE 2.59 The play nonlinearity.

function

~S(t), see

~l(t)(U)

(2.172). Then

-- ~S(t)(U)

-- P[t0, ~S(to)~ u]y(t),

where x(t) is the input of the Preisach nonlinearity.

y(t) - 2x(t),

t~to.

242

CHAPTER 2 Differential Equations with Hysteresis

FIGURE 2.60 The Preisach nonlinearity as a family of play nonlinearities. Interestingly, with some minor restrictions a play nonlinearity m a y be considered also as a Preisach nonlinearity.

Algorithmic Description Restricted Set of States In this text we are mainly interested in an even more restricted set of states of the Preisach model. These are the states ~/ corresponding to the sets S ~ FI b o u n d e d by the lines L with an infinite horizontal link, and with a finite n u m b e r of vertical and horizontal links, see Fig. 2.61. We denote the height of the infinite horizontal link by w (1). Let w (2k), 1 < 2kKK, be the ascending sequence of the horizontal coordinates of the vertical links. That is, the vertical links belong to the straight lines L2k = (W(2k), fl), 1 < 2 k ~ K . Let also w(2k+l), 1 < 2k + I ~ K , be the descending

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

243

0

f

I

I

161

14

I21

i

w2

w4

w6

w5

i

w3

w1

FIGURE 2.61 State r/ ~ ~, that defines a set S bounded by the piecewise-linear curve L. sequence of the vertical coordinates of the finite horizontal links. By the definition numbers w (k), l <.k <<.K, satisfy the inequalities W (1) > . . .

> w(2k0 - 1 ) > w(2k0) > . . .

> W (2),

W (1) >

> w(2k0 +1) > w(2k0) > . . .

> W (2),

...

if K - 2k0; if K = 2k0 + 1.

(2.173)

The totality W of all states ~/, described by the finite n u m b e r of links as above, m a y be naturally identified with the family of all finite sequences w = {w k }lK

satisfying the inequalities (2.173). We emphasize that the lengths K - #w of different elements from W m a y be different. We introduce also the semiclosed intervals I0 = (cx~, w(1)], Ii=(-cx~, W(2)] and !k--[W (k-l), w(k+l)), 2Kk
CHAPTER 2 Differential Equations with Hysteresis

244

(TR) Let s be defined by x(a) ~ Ir Then assign K = K(a) = #w(a) := s + 1; afterwards assign w (K(~)) (a) := x(a). Note that the function K(t) = #w(t), zKtKa, is piecewise constant with the jump-points ti given by ti -- x - 1 ( w (ki) ('~)),

where

w (ki) ('~) E [x('~), x(o)].

Moreover, all components of w(t) except the last one are unchanged between the successive jumps. This rule (TR), together with the semigroup property, can be conveniently used as a base for an algorithmic description of the evolution of the state of the Preisach model, and in numerical algorithms for solving ODEs with the Preisach model.

Transformation Rule for Outputs

Let us also discuss a similar transformation rule to compute the varying outputs of the Preisach model together with its states. Let A(a, b) be the triangle with the vertices (a, a),(a, b),(b, b): A(a, b ) = {(a, fl) : aKa < fl~b}. Denote for a~b #(a, b) = #(A(a, b)) = f h(~, fl) d~ dfl, dA(a,b)

(2.174)

and define/z(b, a) = -#(a, b). In some cases there can be found an explicit formula for #(a, b). For a sequence w ~ W, with #w = K we correspond the array #(0) . . . . . #(K) by the equalities #(0) = 0, fl(1) : fl(--OO, W (1)) and the recurrent relationships fl(k) = ]z(k-1) if_ fl(w(k-1), w(k)). Clearly, (I)(w) = #(K). Since we are interested in outputs of the Preisach model for piecewise monotone inputs, then the Transformation Rule (TR), should be changed. Let the input x(t) be monotone at the interval r ~ t ~ a . Then w(a), K(a), and the output u(a) are obtained from the state w(r) via the following rule (TR*) below, see aslo Fig. 2.62: (TR*) Let ~ be defined by x(a) ~ I~. Then assign K = K(a) = #w(a) := ~ + 1; assign w (K)(a) := x(a); assign #(K)(a) "- fl(K-1)('0 -}- fi(w(K-1) , w(K)), with w (~ = -cx~; assign u(a) "- #(K) (a).

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

245

Let the function x(t) have all different local extrema whose values are different from the components of w(~). Then K(t) jumps up only at extremal points and points equal to w (i) (~). Note that if x(t) is increasing in the interval t ~ (r, a), i.e. x1(t) > 0, then K(t) is odd. Similarly, if x'(t) < 0 for t ~ (r, a), then K(t) is even. It is convenient to introduce the following notations K t = K~(t) - K t ( K ( t ) ) - K 4- 1 - ( K m o d 2 ) ; K S - K S (t) = K S (K(t)) = K 4- (K m o d 2).

Sometimes it is also convenient to describe the future dynamics of the output in terms of the current state w. LEMMA2.4.1. Let x(t) be increasing and x(t) ~ I~ for r K t < a. Then u(t) = y(K~-I)(r) + #(w(K~-I)(~), x(t)) u(t) -- # ( - ~ , x(t))

if if

K(r) > 1; K(r) - 1.

Let x(t) be decreasing and x(t) ~ I~ for r ~ t < or. Then u(t) - fl(K4-1)(1;) 4- fl(w (K4-1) (T,), x(t)).

Transformation Rules as Restrictions of Smooth Functional Dependencies Smooth Density It is important to treat the i n p u t - o u t p u t relationships as localizations of some smooth functions. Below we describe the convenient modifications of the transformation rule (TR*). We note initially that the function it(a, b), as defined for all pairs a, b in formula (2.174), is often n o n s m o o t h enough at the line a = b. We suppose that in the domain of its definition the function g(a, b), F ~ a ~ b ~ b coincides with some smooth function ~(a, b), -cx~ < aKb < cx~. Usually, this function/t can be defined as: /, ft(a, b) - / h(~, fl) d~ dfl; JA (a,b) where h(a, fl) - h(max{~, a}, min{b, fl}). Introduce the families of functions #+ (x), #w (x), w ~ W, x ~ ~1 as follows: g+(x)-

/ /t(-oc, x),

/

~(w(Kt -1),x),

if

K-1

if

K>I

246

CHAPTER 2

Differential Equations with Hysteresis

FIGURE 2.62 Evolution scenarios (TR*) for the state of the Preisach nonlinearity depending on parity of K=K(to) and direction of the change of the input x.

and

#w (X) = fl(X, w(K4-1)). These functions are s m o o t h in the x variable. LEMMA2.4.2. Let x(t) be increasing and x(t) E I~ for T~t < a. Then a smooth

function u(t) = fl (Kt-1) (-c) q- flw(T) + (x) coincides with the output u(t) of the Preisach nonlinearity for r ~ t < a. Let x(t) be decreasing and x(t) E I~ for TKt < a. Then a smooth function

~(t) = fi(K4-1)(.~) _ #w(T)(x). coincides with the output u(t) of the Preisach nonlinearity for r ~ t < r

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

247

N o n - S m o o t h Densities Sometimes the density h(a, fl) defining #(a, fl), see (2.174), is not smooth itself. In this case the description of the functions should be modified. Here we describe how it could be done. We restrict ourselves to the case when the density h(~, fl) coincides with some smooth function h 9 ~2 ~ [~2 inside a given triangle A(~, b), and is zero outside of this triangle. Consider the function

~(a, b) =/A(a,b)

h(~, fl) d~ dfl

which is smooth on {(~, fl) : -cx~ < aKb < cx~}. The function #(a, b) coincides with ~(a, b) on the triangle A(~, b). Introduce the families of functions #+(x), #w (x), w ~ W, x ~ [~1 as follows.

#+ (x) --

~(~, x)

if

[t(max{w (K1"-1), ~}, x)

if

0

otherwise

K-

1 and a ~ w (1) < b, ~Kw (K~ < b,

and

/ It(X, W(K4-1)) /~w(x)

/

0

if

~ w (K) < b,

otherwise

These functions are smooth in the x variable. LEMMA2.4.3. Let x(t) be increasing and x(t) E I~ for r ~ t < a. Then a smooth function

~(t) = fl (Kt - 1)(.~) if_ flw(r) +

(x)

coincides with the output u(t) of the Preisach nonlinearity for rKt < r Let x(t) be decreasing and x(t) E I~ for r ~ t < r Then a smooth function f~(t) = I~(K,-I) (T) -- l~w(r)(x). coincides with the output u(t) of the Preisach nonlinearity for r ~ t < a. In this sense i n p u t - o u t p u t relationships of the Preisach model are restrictions of some smooth functional dependencies.

248

CHAPTER 2

Differential Equations with Hysteresis

2.4.2 LIPSCHITZ CONSTANT OF THE SHIFT OPERATOR

Let us consider Eqn. (2.13) with the initial conditions (2.14); we repeat them below for the reader's convenience: dz/dt u(t) ~/(t) x(t)

= = = =

fit, z, u(t)), (I)(r/(t)), (F[r/0]x)(t), (c, z(t)) ;

(2.175)

and to, z(to) = z0, r/(t0) = ~/0 ~ (~..

(2.176)

Here t ~ R+, z e ~n, the scalar function u(t) is the output of the Preisach nonlinearity and the operator F describes the evolution of the state ~/of the nonlinearity depending on the scalar input x(t). We assume t h a t f is differentiable in all variables and satisfies [fit, z, u) - f i t , z, ~t)l<~2fu iu - ~tl.

(2.177)

Recall that every r/ e (~. is uniquely defined by a graph L given by the function 0r : N+ ~ N, see (2.172). Thus a metric between functions 0r(u), Oo(u) can be used as a metric between the states ~/, ~ ~ (~. of the Preisach nonlinearity. To estimate the Lipschitz constant of the shift operator we use the metric given by

P'(~' ~)

=

max

0
J6.(u)

-

O~(u)l.

(2.178)

Let (I)0/) satisfy the Lipschitz condition with the constant 20:

I~(~) - ~(~)l~2~p~(~, ~).

(2.179)

In this case the system (2.175), (2.176) has a unique solution. The state of the system (2.175) is uniquely defined by t, z, r/. We define a metric, r the space of all possible pairs (z, r/) as Q( (z, r/), (~, ~) ) = max{21cl. Iz - zl, Pn(~/, ~)}.

(2.180)

Lipschitz Constant of the Shift Operator The logarithmic norm A/'(Q) of a square matrix Q (with respect to a norm i l 9 l i) is defined by .M(Q) =

III + h Q i I - 1 h---~0, h>0 h lim

249

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

where I is the identity matrix and [[-[[ is the operator norm of the matrix. In the Euclidean metric the logarithmic norm A/'(Q) of the matrix Q coincides with the largest eigenvalue of the symmetric matrix 1 (Q, + Q) where Q' is a transposed Q (see [183], p. 59). Let E be a subset of Ea. PROPOSITION 2.4.1. Let (z(t), rl(t)), (~(t), O(t)) be two trajectories of(2.175) with

the initial conditions at time to being (zo, ~1o) and (zo, rio) respectively. Let the initial conditions be coherent, i.e., (F[r/0]z)(t0) = r/0

(F[O0]~)(t0) = ~0,

and let z(t), ;:(t) ~ E for all t ~ [to; tl]. Let A>~ sup~2~QAf((~) where (2 is a family of matrices given by

sup

Q - { f ~ - ( F / / j ) ' ~ / / j ~ [ [.zea,t,u inf -~z] 0j~ (t' z, u), z~E,t,u -&jzj(t' z, u)

]/

.

Then the following inequality holds O((z(t), rl(t)), (~(t), ~l(t)) )~e (A+22fu2~)[cl)(t-t~ O( (z0, t/0), (z0, ~0) )- (2.181) PROOF. Introduce the difference r(t) = z(t) - ~(t). To make the formulas below more compact we use z, w, u, x, r to denote z(t), r/(t), u(t), x(t), r(t), respectively, and x0, r0 to denote x(to), r(to). Similar notations are used for ~(t), ~(t), ~(t), ~:(t), ~(t). Let us estimate [r[':

1 (r, flt, z, u) - f i t [ r [ ' - (v/(r, r))' = {r,[r[r') = [r--[

~, ~))

_- 1 (r, f(t, z, u) - f i t , ~, u)) + ~[1 (r,f(t , ~, u) - f i t , ~, ~)) Irl

<.~1 (r,f(t , z, u) - f i t ,

~, u)) + [fit, ~, u) - f i t , ~ ~)l.

The distance between ri and ~ changes depending on the distance between inputs x, 2. From the definition of p~ it follows that

P~(r/' O)K max { p~(r/~ ~~ 2 maxs~[to,I(c,z(s)-~(s))l}-p(t). t]

(2.182)

CHAPTER 2 Differential Equations with Hysteresis

250

Using the Lipschitz properties off on u we can rewrite if(t, ~, u) -fit, ~, ~)[ as

If(t, ;:, u) -fit, ;:, ~)1 ~< 2Su lu - ~1 ~<XS,, I@(~) - @(4)1

'~fuX~Prl(rl, ["l)~Xfu 2@p(t). Let us rewritef(t, z, u) -fit, ;:, u) as

f(t, z, u) - f(t, ~, u) -

= Q(t, z, 5, u)r,

where Q(t, z, ;:, u) is a square matrix given by the integral. Using properties of the inner product we can write

Irl'

<~ ~1 (r, Q(t, z, ;:, u)r) + ,1.f~2@p(t) Af'(Q(t, z, Y:, u))lrl + 2fu 2@p(t).

(2.183)

Taking the m a x i m u m over the time interval [to, t] in (2.183) we get

d(maxse[to,t] Ir(s)l) ~ A max Ir(s)l + 2fu2<s)p(t). dt s~[to,t] Letus introduce the function Q(t) = max{21cl "maxse[to,t] Ir(s)l, p(t)}. Using the inequality p'(t) = (2 max I(c, r(s))I)'K21cl( max Ir(s)l)'

s~[to,t]

s~[to,t]

that follows from the definition of p(t), we can estimate Q(t)'r

~ max{21cl(maxse[to,t] Ir(s)l)', p(t)'}~21cl(maxse[to,t] Ir(s)l)' ~< 2AIcl maxs~[to,t] Ir(s)l + 22f~2@lclp(t) (A + 22f~2@1cl)- max{21cl maxse[to,t] Ir(s)l, p(t)} (A + 22f~2@1cl). r

By the Gronwall inequality we can write

r

(A+22f~2@lcl)(t-t~ Q(to).

To obtain the statement of the proposition it remains to note that

~( (z(t), ~l(t)), (;:(t), ~(t)) )Kr and that since the initial conditions are coherent,

p ( t o ) - max {p~(r/0, ~0), 21 (c,z(to)- ~(to))I} = P~(r/0, ~0) and r

- O((z(to), r/(t0)), (z(t0), ~(t0)) ). The proposition is proved.

[]

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

251

Lipschitz Constant Estimates for the Pendulum Example As an illustrative example of Proposition 2.4.1 we estimate the Lipschitz constant of the shift operator along the trajectories of (2.9). We estimate the constant L for the trajectories in z(t) ~ ~2 and for the trajectories {z 9 Iz(1)1~0.5} separately. The results will be used in the computer-assisted proof in Subsection 2.4.4. We begin with the estimate of A, which turns out to be the same for all z(t) ~ ~2. Consider the vector form of the Eqn. (2.9)

d z / d t =fit, z, (P (c, z))(t)), where z = (z (1), z (2)) and

fit, z, (P (c, z))(t))

-

(Z (2), --Z (1) q- sin(V~t) + 2b(Pz(]))(t) - b) .

From here

Q(t,z,~,u)-

-~z =

-1

0

"

Thus A = 0 as 0.5(Q + Q') is a zero matrix and has two eigenvalues equal to 0. For Eqn. (2.9) c = (1, 0), Icl = I and constant 2du is equal to 2b. Let (z(t), r/(t)) and (~(t), ~(t)) be the solutions of Eqn. (2.9) with the coherent initial conditions z(to) = z0, r/(t0) = r/0 and z(t0) = z0, ~(t0) = ~0Firstly, we consider the situation w h e n z(t), ~(t) ~ {z : Iz(1)1~0.5}. The output of the nonlinearity u = (I)(r/) is constant as long as Ix(t)[ = Iz(1) I~0.5. Thus 2~ = 0, L = 0 and using (2.181) we may write r

r/(t)), (~(t), ~(t)) )KQ( (z0, r/0), (~0, ~0) ).

(2.184)

Secondly, let z(t), ~(t) ~ ~2. In this case 2~ is estimated as the largest possible distance between two curves in triangle (2.8), a = {(a, fl) : -0.5~<~
Thus 2~ - 0.5. Substituting all the constants to (2.181) we get r

r/(t)), (~(t), ~(t)) ) ~ e 2b(t-t~

(z0, r/0), (z0, ~0) ).

(2.185)

2.4.3 NUMERICAL INTEGRATION

Eqn. (2.175) can be numerically integrated by a method of the first-order as the right-hand side of the differential equation is Lipschitz-continuous. The standard example of the explicit first-order method is the Euler method.

CHAPTER 2 Differential Equations with Hysteresis

252

In general, numerical integration by first-order methods is neither efficient nor precise and this section is devoted to the scheme that allows the application of high-order methods for the integration of Eqn. (2.175). For the purposes of the numerical integration we are going to use algorithmic description of the Preisach nonlinearity given in Subsection 2.4.1 in terms of w. To simplify the notation we assume that r/is equivalent to w and we loosely use the representation r] = W -- {W ( 1 ) , . . . , W (K)}

if there is no ambiguity. We assume that the initial states of the Preisach nonlinearity r/0 for the initial-value problems like (2.175), (2.176) always admit algorithmic representation.

Euler Method The Euler numerical integration method is given by the formula Zn+l tn+l

---

Znf(tn, Zn, Un), tn Jr-'Cn, (F[r/n ] (C, Z n + l ) ) ( t n + l ) ,

r/n+1 Un+l

------ (I)(r/n+ 1).

Here Zn, Un, tin are the values of the numerical trajectory at tn and Zn is the size of the nth time step. Here the state r/n+1 = w and the output Un+l of the Preisach nonlinearity are computed according to the transformation rule (TR*). We denote the result of the z-size numerical integration step of (2.175) with the Euler method as Veul(tO, zo, r/o; to + z) -- (Zeul;1,//eul;1)" The local error of the Euler method with the time step z~h and z0 ~ E can be estimated as h2 ~f ~_~_f/( sup + t,, z, u,) [z(t0 if- Z) -- Zeul;ll ~ -~- z**E,t,,u,

sup zo~E,to,u, z~h

--~

q-h2

'

sup 2fuA~[c[.Lf(t,,z,,u,) [ z, ~E, t,, u,

< eeul;z(h, E). Since the Zeul;1 is linear as a function of z, it is possible to write sup

~( Veul(t0, z0, r/0; to + z)), (z(to + z), r/(t0 + z)) ) < 2[C[eeul;z(h, E).

z ~h, to, rlo,Zo~E

(2.186)

253

2.4 EXAMPLESOF TECHNICAL TOOLBOXES

Thus the number eeul(h, E) = 2]Cleeul:z(h, E) estimates the local error of the method. The algorithm for the numerical integration of (2.175) offered below is going to rely on the Euler method steps in some 'complicated' situations.

Local Error of the Higher-Order Methods Locally Equivalent Systems

Let K(t) denote the length of the sequence {w (i) (t) } that describes ~/(t) and let K denote K(to). Let the initial conditions (2.176) be coherent, i.e. (F[r/0] (c, z))(t0) = r/0. If x'(to) > 0 then by Lemma

2.4.2 the system (2.175) is locally equivalent to dz/dt u(t) w(Kt)(t)

= --

fit, z, u(t)), #K(tO) + #-~o(x), x -- (c, z) .

(2.187)

If x'(t0) < 0, the system (2.175) is locally equivalent to dz/dt u(t) W (K~)(t)

= --

fit, z, u(t)), #K(tO) -- t~qo(x), x -- (c, z).

(2.188)

Both (2.187) and (2.188) are smooth and can be numerically integrated by standard methods. Denote the shift operator along the trajectories of the Eqns. (2.187), (2.176) for the time z by q~t(to, zo, r/0; to + z) = (zt(to + z), ~l~(to + z))

where t/1,(t) = {w(1) . . . . . W (Kt-1), (c, zt(t))}. Denote a step of size z of the numerical integration of the Eqns. (2.187), (2.176) by vt(to, zo, q0; to + z) - (Zvt(to + z), ~lv~(to + z))

where rlv~ (t) = { w (1) . . . . , w ( K i ' - 1 ) , (c, Zv~ (t))}. Notations v4 (to, z0, r/0; to + z) and q~ (to, z0, r/0; to + z) are introduced analogously for Eqns. (2.188), (2.176).

Numerical Step Error for the Equivalent Systems

Let etz(h, E) estimate

the local error of the numerical integration of (2.187) sup zonE, rlo, to~tl ~to+h

[Zvt(tl) - z t ( t l ) l ~ e t z ( h ,

E)

CHAPTER 2 Differential Equations with Hysteresis

254

and let e+z(h, ~) estimate the local error of the numerical integration of (2.188). Let ez(h, E) = max{etz(h, ~), e4z(h, E)}. (2.189) Note that by the definition of r/vt (t) and r/t (t) Q(vt (to, zo, r/o; tl), (Pt (to, zo, r/o; tl)) = max{21c[. Izvt(tl) - zt(ta)[, p,(r/v,(tl), r/t(tl))} ~< max{21cl Izvt(tl) - zt(tl)[, ]2(C, Zvt(tl) - zt(tl))l}" <~ 2[c[]zvt(tl) - z t ( h ) I. Taking s u p r e m u m over ~ 6 E, ~, t0~>0, to<~tl<~to + h on both sides of the inequality above, we may write sup r

t (to, z0, r/0; tl), ~Pt (to, z0, r/0; tl)) < 2]c]etz(h, E).

(2.190)

Similarly for (2.188), sup ~o(v+(to, zo, r/o; h), q~+(to, zo, r/o; h)) < 2[cle+z(h, E).

(2.191)

Thus the number e(h, E) = 2]c[ max{etz(h, E), e4z(h, E)} = 2Iclez(h, ~.) estimates the step error of the numerical integration of both (2.187) and (2.188) in terms of metric Q. With some technical restrictions it is possible to substitute the numerical integration of the system (2.175), (2.176) with that of the appropriate local equivalent. Introduce the notations x(t) = 0 on [to, tl], tl - to~h. If Kt(to) < 3 or xvt(tl) + [clez(h, E) < w(Kt(to)-2)(to) then z t (t) = z(t),

r/t (t) = r/(t)

and

Q((z(h), r/(tl)), vt(to, zo, %; tl)) < e(h, E). Let x'(t) < 0 on [to, tl], tl - to~h. If K4(to) < 3 or Xvj,(tl) - [cIez(h, E) > w(K4(t0)-2) (to) then z~ (t) = z(t), rl~ (t) = ~l(t) and

•((z(tl), r/(tl)), v4 (to, z0, r/0; tl)) < e(h, E).

255

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

Integration Algorithm General Description

The idea of the numerical algorithm is simple. During each cycle of the algorithm we are trying to make a time step h ~ [hmin, hmax] along either (2.187) or (2.188) in such way that the conditions of Lemma 2.4.4 are satisfied. If, as a result of the step, we are too close to the zone w h e r e f of (2.175) is not smooth enough as a function of t, we try to 'jump' over the area of non-smoothness, see Figs. 2.63, 2.64. In this way the local error associated with 'jump' steps is slightly bigger then usual. If some technical restrictions for the 'jump' are not satisfied, the algorithm makes hmin time-step using the Euler method. The advantages of the offered algorithm over the straightforward Euler integration are clear: 'most of the time' we use a high-order integration method thus significantly decreasing global error estimates and the total CPU time for the integration. Since the offered algorithm only 'describes' the conditions in which it is possible to use high-order integration methods, it can be easily coupled with an integration method best suited for the given ODE. The main disadvantage of the algorithm is that the effective global error estimate needs to be computed incrementally along the trajectory, due to the mixture of different integration steps. One particular application of the algorithm, a computer-aided proof of chaotic behavior, is discussed in Section 2.4.4.

Required Constants

To formulate the algorithm we require the follow-

ing constants: 9 the size of the 'jump' zone: Y = Nmax{1, A + Ic[2fu2~} 9e(h, E),

N>I;

9 minimal hmin and maximal hmax possible time steps; 9 set (Wv) of horizontal(vertical) lines on which the Preisach measure under consideration is discontinuous; 9 the maximum length Kmax of the vector w in the algorithmic representation of the Preisach nonlinearity.

Wh

Dit,

dix/dt i

Let i = 1, 2, 3, estimate the derivatives for all solutions of interest z t (t) of the Eqn. (2.187); derivatives are taken along the trajectories of the system. Dlt > to. t. z~E. t/o D2t >

I df(t,z,

sup I C, ~-~ to, t, z~, r/0

+(x))/I ~K(t0 ) + #,o

256

CHAPTER 2

sup

Differential Equations with Hysteresis

I c,=+

+

to, t, z~E, r/o

D31' >

sup

c3u

d2d

9

c3x

cgt

I

'

+ / I.

I c, -d-~(t, z, laK(tO) + #rio (X))

to, t, z~.~., r/o

I n t r o d u c e Di - m a x { D / t , Di4 } w h e r e n u m b e r s DiS are i n t r o d u c e d a n a l o g o u s l y for Eqn. (2.188). Let vl(to, zo, ~/o; to + r) b e d e f i n e d as v 1,(to, zo, r/o; to + ~), Vl(.) v4 (to, zo, ~/o; to + ~),

if if if if

x 1(to) x 1(to) x 1(to) x 1(to)

> < =

0 0 0 0

or a n d x~ (to), x~ (to) > 0 or a n d x "(t0), t x4" (to) < 0.

Finally, as t h e r e are explicit f o r m u l a s for xi, x i,' x i" in t e r m s of locally e q u i v a l e n t s y s t e m s (2.187), (2.188) w e a s s u m e t h a t for e a c h step t h o s e valu e s are c o m p u t e d for t h e e q u i v a l e n t s y s t e m u s i n g c u r r e n t v a l u e s of ti, rli, zi. N o t e , t h a t if a locally e q u i v a l e n t s y s t e m is n o t d e f i n e d , v a l u e x n c a n n o t b e c o r r e c t l y c o m p u t e d , b u t it is n o t u s e d either.

Integration algorithm

INTEGRATE10 Initial values to, r/0, z0; time step range [hmin, hmax]" 'jump' tolerance Y; arrays }'Vh, }iVy. O u t p u t : Result of the integration is an array of {(ti, Zi, rl i, gi) }in_~0 (1) n = 0, t o, z o, r/0 (2) GETNEWZPLANEO (3) while n < = n o (4) hi = ([Xn - Xhyp[- T ) / D 1, h 2 - ([Xn[- Y)/D 2 (5) if hl < h2 then ~n -- min{hmax, h 2 } (6) else ~n = min{hmax, h 1, [x"nl/D3} (7) if TnKhmi n or vI (tn, Zn, tin; .) is not defined (8) EULERSTEP0 (9) else (10) Zn+l = vl (tn, Zn, tin; Zn) (11) ~n=0 (12) if sign(xn - Xhyp) 9 (Xn+ 1 - Xhyp) < T (13) JUMPTOZ0 (14) if sign(x~) 9 Xn+ f 1 ~
Input:

2.4 E X A M P L E S OF T E C H N I C A L T O O L B O X E S

257

GETNEWZPLANE0 I n p u t : Set of planes ]/Vh, ~/Vv on whichf from (2.175) is not differentiable O u t p u t : Xhyp defining the position of the hyperplane (1) Xdisc h -- min{ff; ~ Wh U {~} : ~ > x} (2) Xdiscv "- m a x { ~ E W v U { - cx~} : Co < x} (3) if (c, f(tn, Zn, (I)l 0/; Xn))) < 0 (4) if KS < 3 t h e n Xhyp = Xdisch

(5) (6) (7)

else Xhyp = min{Xdisc h, W(nK*-2) else if K 1, < 3 t h e n Xhyp -- Xdisc v

(8)

else Xhyp -- max{Xdisc v, W(nKt-2)

EULERSTEP0

I n p u t : Current step n O u t p u t : Updated values of Zn, Zn+l, r/n+ 1, qn (1) ~n = 2 Zn -- hmin (2) Zn+l -- Veul(tn, Zn, tin; Zn) JuMPTOZ0 I n p u t : Current step r/ Output: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Updated values of "On, Zn+l, tin+ 1 , ~n if n = = 0 or qn-1 - - = 1 EULERSTEP0 return w h i l e (Xn - Xhyp) 9 (Xn+l - Xhyp)~<0 or ]Xn+ 1 -- Xhyp] > ~I" or ]Xn+ 1 -- Xhyp] < ]c[ez(h, E) Modify Zn Zn+l = vl (tn, Zn; Zn) Project Zn+ 1 orthogonally on x -- Xhyp ;n = 1

JUMPTOZP0 I n p u t : Current step n Output: (1)

Updated values of 77n, Zn+l, Yln+l, gn if n ==

(2) (3) (4) (5) (6)

0 or ~n-1

==

1

EULERSTEP0 return

!

!

0} (7)

I

Choose the largest Zn such that x n 9Xn+ 1 > 0 and IXn+ 1 [~>(A + 21cl2f 2,)lclez(h, E) Compute r/n+1 Letz. betheclosesttOZn+ 1 pointof{(z, r/) : (c, f(tn + Zn, z, rl.)) = if

]Zn+l

-- Z , [ / > F t h e n

EULERSTEP0

258

CHAPTER 2

Differential Equations with Hysteresis

(8) else Zn+ 1 = z,, qn -- 1 UPDATEPREISACH(r/old,X) Input: Old state r/old, new input X Output: Updated value r/ -- {W (i) } and appropriate K (1) r / = r/old, K -- Kol d (2) if Ix[ > = ? (3) K= 1 w (1) = s i g n ( x ) . ? (4) return ~/ (5) if x > w (K~ (6) if K mod 2 = = 0 then K + + (7) while K>~3 and x > w (K-2) (8) K= K- 2 (9) else if x < w (K~ (10) if K mod 2 = = I then K + + (11) while K~>3 and x < w (K-2) (12) K -- K - 2 (13) if K~>Kmax then error (14) w (K) = x (15) return r/ Let us c o m m e n t on the m e t h o d JumpToZPO. In principle, if (c,f(t, z, u)) d e p e n d s o n u it m a y be v e r y h a r d to localize p o i n t s (z,, r/,) s u c h t h a t at a g i v e n t = t,, (c,f(t,, z,, u,)) = 0. In this case JumpToZPO c a n be r e p l a c e d b y the call of the E u l e r S t e p m e t h o d , t h a t w o u l d p e r f o r m the i n t e g r a t i o n at the step n b y the E ul e r m e t h o d .

Properties of Numerical Solution PROPOSITION 2.4.2. Consider a successful step n of the algorithm Integratel O. The

error of the integration step satisfies Q(q)(tn, Zn, r/n; tn+l), (Zn+l,r/n+l)) < e(hmax, E),

/f~n = 0 ;

Q(q~(tn, Zn, r/n; tn+l), (Zn+l, r/n+l)) < e(hmax, E) + 2 1 c l T ,

(2.192)

if qn = 1; (2.193)

Q(q2(tn, Zn, r/n; tn+l), (Zn+l, r/n+1)) < eeul(hrnin, '~"),

if; n = 2.

(2.194)

Error Propagation Let the shift o p e r a t o r ~o(t0, z0, r/0; t) satisfy Q (~a(to, zo, r/o; t), q~(to, zo, rio: t)) ~ e L(t-t~ •((zo, r/o), (z0, ~o)) as l o n g as for all toKtKt,, ~Oz(to,zo, r/o; t), ~Oz(t0, z0, ~0; t) ~ E.

(2.195)

2.4 EXAMPLES OF T E C H N I C A L TOOLBOXES

259

/\x - --~ numerical traj. real traj. Ioc. err.

I

Jumping

Ioc. err.

~

Ioc. err. time

Z> F I G U R E 2.63

Behavior of the a l g o r i t h m close to the h y p e r p l a n e x - Xhyp.

X

--'> numerical traj. -real traj. j

x'=O

-

~

_

~

/// ....-~"

~

~ _

"

"

--

...~._..,.,.;.::.~ ..... L . . . . . . . . . . . . . . ~-.--: " Ioc. err.

"~'~'~"~' \ Ioc. err. ".(~ ::> ti m e

~x'

o,.

11! Jumping

-y

F I G U R E 2.64

"'-

Ioc. err.

oc;er:

f

time

Behavior of the a l g o r i t h m close to h y p e r s u r f a c e x t = 0.

260

CHAPTER 2 Differential Equations with Hysteresis Let 8appr estimate

sup [~z(t,, z,, r/,; t , + ~ z ) - ( 1 - a ) z , - a v z ( t , , O<<.z<<.h,t,,z,~G,rl,

z,, r/,; t,+z)l < 8appr. (2.196)

PROPOSITION2.4.39 Let the algorithm Integratel O terminate successfully and let 110 the output of the algorithm be a set {(tn, Zn, qn, ~n)}n=0" Let ~ be a convex open set. Let e > 0 be a given number. Introduce 811 by the following recursive formula:

80 = en+l 8n+1 8n+l

8, -- 8ne L~n + e(hmax, ~) = 8ne Lzn 4- e(hmax, E) 4- 21cIY = 8ne Lzn 4- eeul (hmin, E)

if q n - - 0 if gn -- 1; if qn - 2.

Let the points Zn satisfy Zn ~ E(Sn 4- 8appr), where E(a) = {z ~ E ' l z -

z,I > a for all z, ~ ~9E}.

Then for each (zo, rlo) such that Q( (zo, flo), (zo, rlO) ) ~ e ~Oz(to, zo, q0; t) ~ E,

for all to<<.t<~tn

(2.197)

and Q( ~p(t0, z0, q0; tn), (Zn, rln) ) < en,

for all O<<.n<<.no.

(2.198)

PROOF. Introduce the notation r ;50, q0; t) = (~(t), ~(t)). Let us s h o w the statement of the l e m m a b y induction on n. The statement of the l e m m a holds for n = 0. A s s u m e that the statement of the l e m m a holds for some n. Let us s h o w that it holds for n 4- 1. A s s u m e that (2.197) is not true. Then there is a m o m e n t t, such that t,-

min {t'f:(t) ~ 0E}. tE[tn,tn+l]

Let v(t) = (1 - a)Zn + ~Zn+l where ~ = (t - tn)/Zn. Then by the triangle inequality [~(t,) -- v(t,)l ~ ICpz(tn, Z(tn), ~(tn); t,) -- r

4.1r

Zn, tln; t,)[

Zn, tin; t,) -- v(t,)l.

By (2.195) and (2.196)

ICPz(tn, Z, tl; t,) -- v(t,)l<~ene L(t*-tn) 4. eappr.

2.4

EXAMPLES OF TECHNICAL

261

TOOLBOXES

Let d(., 0E) denote the distance to the b o u n d a r y of E. By the convexity of we can write

d(v(t.), cgE)~>(1-c0d(zn, OE)q-c~d(Zn+l, OE) > e.apprq-en((1-cOq-o~e L(tn+l-tn)) where c~ - (t-tn) /Zn. Since (1--CO-}-o~eL(tn+l-tn) >/e L(t*-tn), point qOz(tn , ~:, fl; t. ) belongs to E. We have arrived at a contradiction and (2.197) holds. The inequality (2.198) for n + I holds by (2.197) and Proposition 2.4.2. []

Plane Intersection Algorithm To construct a numerical analogue of the first return map on the hyperplane, one needs a procedure to estimate the area in which the intersection of the solution with the hyperplane happens. Setting aside the question of whether it is the first intersection or not, we offer a simple algorithm that allows us to localize the point of intersection on a given hyperplane using the information on the numerical solution and integration errors. Let (p, z) - a be a hyperplane in R a. Let {(tn, Zn, r/n, ~n)}n~ 0 be a numerical solution produced by IntegratelO. Let (z(t), r/(t)) denote the solution of (2.175) with the initial conditions at to being (z0, r/0). Let there exist 0~
9 Iz(tn) -

Znl~81

9 a - ( p , znl)

;> 81

for all nl<~n<<.n2, and (p, Zn2)- a > el.

Let np be defined as

np -

min {n " (p, Zn) ~ a<<.(p, Zn+l)}. nl <<.n<<.n2

Let us find the intersection point of the line segment O~Znpif- ( 1 - - 0 0 Z n p + l , 0 ~ C t ~ 1 with the hyperplane (p, z) - a: Zbase - -

(XZnp -}- (1

-

~)Znp+l

where,- /P Znp+ll (p Znp+l Znpl For all Zn, n ~ [nl, n2] we compute the projection ~:n on the hyperplane (p, z / = a along the vector Zproj - - Znp+l -- Znp"

Zn = Zn -Jr-flZproj,

fl -- (a -- (p, Zn})/ (p, Znp+l -- Znp).

Let 82 m maXnl<<.n<<.n2 IZn -- Zbasel.

CHAPTER 2 Differential Equations with Hysteresis

262

LEMMA2.4.5. In the conditions of this subsection, z(t) intersects the hyperplane (p, z) = a at t, ~ (tnl, tn2) in the direction 'from (p, z) < a to (p, z) > a'. The

intersection point z(t,) belongs to the open ball Bo of radius

81

(C,C)(Zproj,Zproj)

(C,Zproj)

-}-

82 q- 8appr centered at Zbase, where 81, 82 are given above.

If, additionally, 0<

inf

rl,zEBo,tn 1 ~t~tn 2

(p,flt, z, (I)(F[r/0] (c, z) (t)))),

then t, ~ (tnl, tn2) s u c h that (p, z(t,)) - a is unique. PROOF. The proof follows from the construction method.

[]

Computation of the Topological Degree of a Plane Vector Field Consider a continuous map F 9 ~2 ~ ~2 defined on Jordain set f2 c R 2 with the piecewise smooth boundary. Since we are mainly interested in fixed points of F we restrict ourselves to the problem of computing deg(f2, F - id, 0) = 7(f2, F - id), where id stands for identity function. There are various algorithms for computing two-dimensional rotation of the vector field. Here, just for consistency, we offer one of them. Let B(v; r) ~ ~2 be an open ball satisfying 3r2 > 0 such that F(B(v; r)) c B(F(v); r2); F(B(v; r)) n B(F(v); r2) -- f~. (2.199) Then for every v, ~ B(v; r) the angle between the vectors F(v,) - v, and F(v) - v is strictly less than n/2. Let F(s), OKsKs,, F(0) = F(s,) be a 'natural' parameterization of 8fl; (i.e. s is length of the curve 0f2 between the points F(0) and F(s) measured counterclockwise).

(1) (2) (3) (4) (5) (6) (7)

Choose so = O, v0 = F(s0) and rmin; n = O. while (true) if B(vn; rmin) not satisfies (2.199) then error Choose rn~rmin such that Bn = B(vn; rn) satisfies (2.199) if Of2 ~ U~=0Bk then terminate successfully Choose Sn+l > = Sn + rn such that F([Sn; Sn+l]) C Bn Xn+l --

F(Sn+l); n + +

LEMMA2.4.6. If the algorithm above terminates successfully, then

/(F(vn)_Vn, 7(f2, F) = [ 1~nn ~i=1

f(Vn+l)_Vn+l)q_~l,

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

263

where /(v, v.) stands for the angle between the vectors v, v. measured counterclockwise. PROOF. The proof easily follows from the algorithm.

Proofs System Equivalence Lemmas LEMMA2.4.7. Let (z(t), rl(t)) be the solution of (2.175), (2.176) and let x(t) = (c,z(t)). Let x'(t) > Ofor all t ~ (to, tl]. If Kt(to) < 3 or if Kt(to)~3 and x(t) < w(Kt (t~ then the solution (z(t), ~/(t)) coincides with the solution of (2.187), (2.176) on the time interval [to, tl]. Let x'(t) < Ofor all t ~ (to, t]]. IrKs(to) < 3 or if K4(to)~3 and x(t) > w(K4 (t~ then the solution (z(t), ri(t)) coincides with the solution of (2.188), (2.176) on the time interval [to, t]]. LEMMA2.4.8. Let (z(t), ~l(t)) be the solution of (2.187), (2.176) and let x(t) =

(c, z(t)). Let x'(t) > 0 for all t ~ (to, tl]. If K t (to) < 3 or if K t (to) >>,3and x(t) < W (Kt(t~ then the solution (z(t), rl(t)) coincides with the solution of (2.175), (2.176) on the time interval [t0, tl]. Let (z(t), rl(t)) be the solution of(2.188), (2.176) and let x(t) = (c, z(t)). Let x'(t) < Ofor all t ~ (to, tl]. IrKs(to) < 3 or if K(to)~3 and x(t) < w (K~(t~ then the solution (z(t), ~l(t)) coincides with the solution of(2.175), (2.176) on the time interval [to, t]]. Proofs are omitted. PROOF OF LEMMA 2.4.4. As cases with x'(t) > 0 and xt(t) < 0 are similar we consider only the first one. If Kt(t0) < 3, the statement of the lemma immediately follows from the first part of Lemma 2.4.7. By (2.189) Ixvt (tl) - xt(tl)l -- I Ic, zvt(tl) - z$(tl))I~iclizvt(tl) - z$(tl)l~lclez(h, E). The last inequality means that

x t ( t l ) ~ X v t ( t l ) + Iclez(h, E)~w(t) (Kt(t~ and solutions (zt(t), r/t(t)) and (z(t), rl(t)) coincide on the time interval [to, tl] by the first part of Lemma 2.4.7. The error estimate ~((z(tl), r/(tl)), vt(t0, z0, ~/0; tl)) < e(h, E) is n o w trivial. The lemma is proved.

[]

264

CHAPTER 2 Differential Equations with Hysteresis

PROOFOF PROPOSITION2.4.2. The case w h e n qn the description of the Euler method. Let us consider the situation w h e n qn E nient to formulate the following lemma.

--

{0,

2 is trivial and follows from 1}. Technically it is conve-

LEMMA2.4.9. Let the algorithm Integratel O terminate successfully. Then each step n of the algorithm such that ~n E {0, 1} satisfies Lemma 2.4.4. PROOF. Consider a step n. This m a y be either a step along (2.187) or (2.188). We are going to consider only the case of a step along (2.187) as the situation with (2.188) is exactly the same. Firstly, we show that the sign of x~ (t) is constant on the interval (tn, tn+l]. If x~ (tn) > D2"cn then the condition holds by the fact that D2 estimates sup Ix~ (t)I. If x~ (tn) <<,D2h there are two situations possible: there is a call to JumpToZPO on lines 14, 15 and qn = 1, or there is no such call. In the first c a s e tn+l is chosen so that

X~v(tn+l ) > (A q- 21clAfuA~)lClez(h, E), see lines 4, 5 of the

(2.200)

JumpToZPO; while in the second case

Xl, v' (tn+l) > Y > (A if-21Cl2du2r

E)

(2.201)

see line 14 of IntegratelO. Note that by the choice of ~ n , lines 4-6 of IntegratelO, the call of JumpToZO is impossible. Let us estimate [X~v(tn+l) - x 't (tn+l)[. Since "Cn~
[X~v(tn+l ) - x~ (tn+l)[

<~(c,flt, Ztv(tn+l), ~/1.v(tn+l)) -fit, Ztv(tn+l), r/tv(tn+l))) E).

~
~< x~(tn+l) q-[X~v(tn+l) - x~(tn+l)[ <~ x~(tn+l) q-IcI(A q- 21cl2fu2r E).

(2.202)

Inequalities (2.200), (2.201) and (2.202) imply that x~ (tn+l) > 0. From line 6 of IntegratelO IXnl > D3"cn, which means that x ft (t) is monotone on the > 0 since x 't (tn+l) > 0 and x~>0. interval [tn , tn+l] and x '(t) t

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

265

Secondly, if K t ~>3 we need to show Xv~(tn+l) -Jr-JcJez(h, E) < Xhyp

(2.203)

where Xhyp stores the value of W (Kt-2) . If Xhyp -- X~ (tn) > D l h m a x q- Y then the condition holds by the fact that D1 estimates sup Ix~ (t)l. If x '1,(tn)<<,Dlhmax there are two situations possible: there is a call to JumpToZO on lines 12, 13 and qn -- 1 or there is no such call. In the first case "Cn+l is chosen so that Xhyp-

xtv(tn+l) > Iciez(h, •),

see lines 4-6 of the JumpToP(); while in the second case Xhyp- xtv(tn+l) > Y > Iclez(h, E),

(2.204)

see line 12 of IntegratelO. Note that by the choice of rn, lines 4-6 of IntegratelO, the call of JumpToZP() here is impossible. Inequalities (2.200), (2.201) imply that (2.203) holds.

[]

We can n o w complete the proof of the proposition. Again since cases w h e n x' (t) > 0 and x'(t) < 0 are similar, we consider only the former case. Consider the step n. If ~n -- 0 then v1,(tn, Zn, YIn; tn+l) -- (Zt (tn+l), r/1,(tn+l)) -(Zn+l, r/n+ 1) and the proposition is proved. If qn -- 1, which only h a p p e n s due to a call from either JumpToZO or JumpToZP(), that w o u l d result in

IZn+l - zvt i~Y. In the case of JumpToZPO the condition above is explicitly checked, see lines 6-8, while in the case of JumpToZ() it follows from the orthogonal projection to the hyperplane (line 9) and choice of/:n (lines 5, 7). By the properties of the metric

O(vt(tn, Zn, tin;

t n + l ) , (Zn+l, r/n+l))~<21ci Y

and the proposition is proved.

[]

Example: Integration of Eqn. (2.9) Pendulum Example

To avoid confusion we rewrite Eqn.

notation of this section

dz(1)/dt dz(2)/dt u(t) rl(t)

--

Z(2)

= = =

--Z (1) + sin(v/2t) + b(2u(t) - 1), ~0/(t)), (F[~/.]z(1)) (t).

(2.9) in the

CHAPTER 2 Differential Equations with Hysteresis

266

The measure of the Preisach nonlinearity #(a, fl) is defined as #(a, fl) -- (a, -- fl,)2;

a, -- max{a, --0.5}, fl, -- min{fl, 0.5}.

Thus we have a one-parameter (b) family of equations. Similarly to (2.187), (2.188) we need to introduce locally equivalent systems. If x'(t0) > 0 then by L e m m a 2.4.2 the system (2.9) is locally equivalent to

I

dz/dt u(t) w(Kt)(t)

(Z (2), --Z (1) q- sin(~/2t) + b(2u(t) - 1)), x

--

--

,i

+ (x), ~K (t0) + #,0 X -- (C, Z) -- Z(1).

(2.205)

If x I (to) < 0, the system (2.9) is locally equivalent to

I

dz/dt u(t) w(K,)(t)

(z (2) , --Z (1) -}- sin(~/2t) + b(2u(t) - 1)),

--

#K(tO) -- I~o (X),

"-

X-- (C,Z) = Z (1).

(2.206)

From the formula for the Preisach measure # it follows that u(t) has the form

u(t) = kl q- k2(x - k3) 2

in (2.205),

(2.207)

u(t) - k l - k2(x - k3) 2

in (2.206);

(2.208)

where kl E [0, 1], k2 -- {0, 1}, k3 E [-0.5, 0.5]. Note that ki are determined based on #K(t0), #~0 (x), and u(t) is always smooth. Note that (2.205) and (2.206) are equivalent for Iz(1)l~>0.5. R u n g e - K u t t a M e t h o d of Fourth Order Let q~(t0, z0; t) denote a shift operator for a time t - to along the trajectories of

z' = g(t, z),

(2.209)

with the initial conditions z(to) = zo. We assume that all the trajectories of interest z(t) belong to an open convex set E. Let hi~hmax be the time-step size for numerical integration at the step i and let ti = to + Y~=0 hk. We approximate (p(t0, z0; ti) by vi that are calculated recursively by the Runge-Kutta m e t h o d of fourth order: v0 = z0,

h 4 Vi+l =- Vi -t- -~ E bkkk(vi' ti; hi) -Jr coi. k=l

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

267

Here bl - b4 -- 1, b2 = b3 = 2, coi is the total error of the c o m p u t a tions a c c u m u l a t e d on the step i due to the usage of IEEE double-precision floating-point n u m b e r s . M a p s ki are c o m p u t e d according to the following formulas k] (z, k2(z, k3(z, k4(z,

t; t; t; t;

2:) 2:) 2:) 2:)

= = = =

g(t,z), g(t + 0.52:, z + 0.52:k1(z, t; 2:)), g(t + 0.52:, z + 0.52:k2(z, t; 2:)), g(t + Z, z + )r t; Z)).

We will refer to the numerical integration m e t h o d described above as RK4. General estimates of the local error for R u n g e - K u t t a m e t h o d s can be found, for example, in T h e o r e m 3.1 from [183], p.157, and w e reformulate t h e m for our particular case. LEMMA 2.4.10. The local error e(h, E) of the RK4 with the step size h and initial condition z(t) ~ E at time t can be estimated as h5 ~o(z5) h 5 ~ Ibil (4) e(h, E)<~-~. sup l (t, z; t + z)l + ~ i=1 --6-- sup Iki (z, t; 2:)1 + co. (2.210) Here suprema are taken over all t, z ~ E, 2: E [0, h]; bi are from the definition of RK4: bl = b4 = 1, b2 -- b3 = 2; and co estimates m a x iogii for the given range of parameters. Note that all derivatives in the l e m m a above are taken w i t h respect to the system (2.209). For example, d egg (t, z) + -~z c~g (t, z)g(t, z). k~ 1) (z, t; X) = -~g(t, z) - --~

Poincare Section M e a s u r e # is equal to 0 outside the triangle {(~, fl) : -0.5~fl~0.5}. Thus (I)0/(t)) = 0 (or 1) for every z(1)(t)~ - 0.5 (z(])(t)~ 0.5), m o r e than that, the history of z (1) (t) w h e n Iz(1) (t)l ~0.5 does not affect the behavior of the system w h e n Iz(1) (t) i < 0.5. We a s s u m e that (F[~/]x) (t) = r/0.5 = {0.5} for all t : x(t) > 0.5 and that (F[~]x)(t) = ~/-0.5 = {-0.5} for all t: x(t) > 0.5. We can introduce a Poincare section for Eqn. (2.9) on any plane z (]) = a, lal~0.5. The full state of the s y s t e m on such a plane is described by z (2), t m o d ~/2~ since the r i g h t - h a n d side of (2.9) is ~/2~-periodic in time. We restrict ourselves to the situation w h e n z (2) ~0.

CHAPTER 2 Differential Equations with Hysteresis

268

We introduce a discrete mapping Fb from the semi-infinite cylinder (z (2) ~>0, t mod ~/2=) to itself as a first return map of the trajectories of (2.9); of course, such a mapping may be partially defined. Global Error Propagation for Eqn. (2.9) Note that the surface dz (1)/dt=0 is given by the plane z (2) = 0. We assume that 8n Jr eappr < 0.01 (2.211) Later we will be interested not in the behavior of a particular trajectory but rather the behavior of the set of trajectories in the e-neighborhood of a given one. Numerical construction of the Fb consists of the three phases: 9 numerical integration of (2.9) until Zn < 0.5 - 0.01 and Zn+m > 0.5 + 0.01; 9 error estimates for the set of solutions from e-neighborhood of the initial conditions of (2.9) 9 computation of Fb based on the data from the previous two steps Introduce s

E_, E0.5 as s

= E n {z 9z(1)~>0.5 + 0.01},

9"~,_ -- E n {z" z (1) N< -- 0.5 -- 0.01}, E0.5 -- E n {z. [z(1)[~0.5 -+- 0.01}.

LEMMA2.4.11. Let the array of (tn, Zn, ~]n, gn), n = 0 . . . . . no be the result of successful termination of lntegratelO and let the assumption (2.211) hold. If zi ~ s for all il <.i<.i2, then

z(t)

e E n {z" Z(1)/>0.5}(, "~. n {Z" Z(1)
for all til 0 be a given number. Introduce en by the following recursive formula: 80 - - g ;

Ln =

0, 2b,

iff Zn, Zn+l E ~+ U ~ _ , iff Zn or Zn+l ~ ,'~,+ O E_;

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

8n+1 "-

{ Ene Lnrn q- e(hmax, E), ~n eLnzn q- e(hmax, E) q- 21clY, ~neLnzn if- eeul (hmin, E)

269

if-f ~n = O, iff ;n -- 1, ill;n--2.

Let the points Zn satisfy Zn E E(~n + Eappr), where E(~) - {z ~ E" l z - z l > a for all ~ ~ OF,}. Then for each (z0, ~0) such that r

~0), (z0, r/0))~e

q0z(t0, z0, ~0; t) ~ E for all to~t~tn

(2.212)

and p(q~(to, zo, ~0; tn), (Zn, tin)) < en for all OKnKno.

(2.213)

PROOF. The proof follows from the previous lemma, Lipschitz constant estimates (2.184), (2.185), and Proposition 2.4.3. [] Estimates for the Integration of (2.9) w i t h b - 1 In this subsection we fix the parameter value b = 1. In particular, for this value of the parameter we observe chaotic-like behavior of the map F1 = F, see Subsection 2.2.4. The estimates of this subsection will be used in the computer-assisted proof of Theorem 2.3.5. We define E to be an open set (-3, 3) x (-2.5, 2.5).

LEMMA2.4.13. Let h~0.01 and b - 1. Consider the RK4 method applied to (2.205) and (2.206). A local error of the numerical integration of both (2.205) and (2.206) is estimated by ez(h, E) - 59h 5 + 10 -13 < 6-10 -9.

(2.214)

PROOF. To show the statement of the lemma we need to estimate the terms of the inequality (2.210) of Lemma 2.4.10. To do so we follow the computeraided technique described in [2.153]. The basic idea is simple: using a powerful symbolic computations package (like 'Mathematica') one can obtain the formulas for the time derivatives of ki, that contains hundreds of thousands of terms, see [161] for details. Next, using the computer implementation of the interval arithmetics one could 'fold up' those gigantic formulas and obtain surprisingly reasonable estimates. The Mathematica program to perform such computations can be downloaded from http://phys.ucc.ie / - oll/hysteresis/. To reduce the estimates for k~4) , ~p(5)we consider three separate situations, and in the case of each estimate we present only the maximum over

CHAPTER 2 Differential Equations with Hysteresis

270

three values. The situations are 9 numerical step along ODE (2.205) for the initial condition z ~ • 9 Iz(~) 1~<0.5; 9 numerical step along ODE (2.206) for the initial condition z ~ E 9 IZ(1) I~0.5;

9 numerical step along ODE (2.205)/(2.206) for the initial condition Z E E " ]Z(1)i~0.5.

The estimate (2.214) follows immediately from (2.210) and the inequalities (2.215)-(2.220) that can be verified by a program in 'Mathematica': Ik~4)(z, t; Z)I < 1179,

z ~ E, 0 ~ ; ~ h ,

(2.215)

Jk(24)(z, t; ;~)l < 1 070,

z ~ E, 0KzKh,

(2.216)

Ik(34)(z, t; X)I < 1070,

z ~ E, 0 ~ z ~ h ,

(2.217)

Ik4(4)(z, t; Z)I < 1101,

z ~ E, 0~x~h,

(2.218)

Icp}s)(t, z; Z)I < 1039,

z ~ E, 0 ~ ; ~ h ,

(2.219)

o ~ 10 -13.

(2.220)

In particular, the inequality (2.220) follows from the description of IEEE arithmetic, and the manner in which trigonometric and exponential functions are realized in the computer (See [184] and bibliography therein). The lemma is proved. [] In the lemma we give the estimates for the other constants required for the numerical integration of (2.9) for b = 1. We choose hmin - 5 . 1 0 -5, hmax - 0.01, Kmax = 10 and Y = 10e(hmax, E). LEMMA2.4.14. Thefollowing estimates hold: D1 < 3, D2 < 5, D3 < 14, eeul(5" 10 -5, E) < 8 . 1 0 -8.

2.4.4 PROOFSOF SUBSECTION 'CHAOTIC BEHAVIOR' Topological Hyperbolicity In this section we remind the reader of the definition of some topological tools that play a major role in the proof of chaotic behavior.

2.4 EXAMPLESOF TECHNICAL TOOLBOXES

271

Let 19 denote the closure of the set V. Fix two positive integers du, ds with du + ds = d. Let V and W be bounded, open and convex product-sets V = V (u) x V (s) C R clu x ~Cts,

W = W (u) x W (s) C ~'~" x R ';is,

satisfying the inclusions 0 ~ V, W and letg 9 17 ~ ~au x Ras be a continuous mapping. It is convenient to treat g as the pair (g(U), g(S)) where g(U). V R~u and g(S). V ~ R as . The mapping g is topologically hyperbolic on the pair of sets (V, W), if the equations g(U) (SV(U) x V(S)) N lTV(U) - O, (2.221) hold, and deg (g(U)(., 0), V (u), O) 4= O.

(2.222)

Here S denotes the closure of a set S. The first relationship in (2.221) means geometrically that the image of the 'u-boundary' c~V(u) x 12(s) of V does not intersect the infinite cylinder C - 17V(u) x R~as;similarly, the second part of (2.221) means that the image g(l/) of the whole set V can intersect the cylinder C only by its central fragment I,V(u) x W (s). Thus the first equation (2.221) means that the mapping expands in a rather weak sense along the first coordinate in the Cartesian product Ran x •as, whereas the second one confers a type of contraction along the second coordinate (the indices' (s)' and'(u)' refer to the adjectives 'stable' and'unstable', respectively). Figure 2.65 illustrates the geometrical meaning of the relationships (2.221) in the two-dimensional case. The application of the concept of topological hyperbolicity is simplified significantly if du = 1. In this case the mapping g(U)(x(U), 0) is onedimensional, V (u) is an interval (a, fl) with 0r fl < 0, and verification of the inequality (2.222) is trivial. LEMMA2.4.15. The inequality (2.222) holds if and only if g (u) (~, 0)g (u) (fl, 0) < 0.

The concept of topological hyperbolicity allows one to show the existence of chaotic behavior. Here we rephrase Corollary 3.1 from [155] for our particular notation. Recall that the matrix is called k-transitive, if all entries of its kth power are positive.

272

CHAPTER 2

Differential Equations with Hysteresis

FIGURE 2.65 The horizontal axis represents R du and the vertical one represents R ds for du = ds = 1. The rectangle represents the set W = W (u) x W (s). The shaded areas are parts of the infinite cylinder C = 17 x Rds . This cylinder cannot be intersected by the image g(17); the images of OV(u) x 17(s) also cannot intersect W. The deformed quadrilateral represents an admissible location of g(V). The solid parts of its border are images of the sets OV(u) x V (s), whereas images of V (u) x OV(s) are dashed.

COROLLARY 2.4.1. Let a matrix A be k-transitive and let F : ~a __, Ra be a continuous mapping. Let there exist homeomorphisms "}'[i and the product set V such that c/-[71FC]-[i is (V, V)-hyperbolic if ai,j -- 1, and let the family ld of connected components of the union set ~ c/-li(V) have more than one element. Then any mapping F~, which is sufficiently close to F in the uniform metric, is (b/, k)-chaotic.

Proof of Theorem 2.3.5 (V, W)-Hyperbolicity and Poincare Map F1 To p r o v e t h e t h e o r e m w e n e e d to satisfy the c o n d i t i o n s of C o r o l l a r y 3.1 f r o m [155]. D e n o t e V = {(~(u) ~(s)). i~(U)l~0,003, i~(S)l~0.01} w h e r e i - 0 . . . . . 8. I n t r o d u c e ~ i " (~(u)~(s)) ~ va(i) + ~(U)va(i) + ~(s)vb(i) w h e r e i - 0 . . . . . 8 a n d vc(i), vb(i), Va(i) are g i v e n in Table 2.2. N o t e t h a t "t-li(V) - R(i) for all

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

273

i = 0 . . . . . 8. It is easy to see that each "]-/i is a h o m e o m o r p h i s m so we can introduce m a p s gi,j - 9r{71F19r{i. Finally let us define (9 x 9)-matrix A = (ai,j) i, j = 0 . . . . . 8: I1

a

~.

1

...

0

0

0

0

...

0

0

9

9

9

o

0

0

...

0

1

1

0

...

0

0

.9

.

(2.223)

LEMMA2.4.16. The matrix A is 16-transitive and not k-transitive for k < 16. PROOF9 The proof consists of straightforward calculations. Note that according to L e m m a 2.3.6 there are more than two connected components of b/, matrix A is 16-transitive according to L e m m a 2.4.16 and to finalize the proof we need the following statement. LEMMA2.4.17. The mapping F1 is defined and continuous on Ri and the mappings gi,j - "]-//1Fl'}-[i are (V, V)-hyperbolic if ai,j = 1. The proof is computer assisted and given in the next subsection.

Computer-Assisted Proof of Lemma 2.4.17 Let us fix i and j. To prove that F1 is defined on R(i) and that gi,j is topologically hyperbolic for (V, V) we cover R(i) with a n u m b e r of balls of the given radius and show the required properties of F1 and gi,j for each ball separately. Let Vc, Va, Vb be some vectors in S and the parallelogram R = R(vc, Va, Vb) C S be given by

R -- {Vc + O~Va if- fly b

:

I~1~1, Ifll~l},

see Fig. 2.66. Denote an open (closed) ball with the radius r centered at by B(~; r) (B[~; r]). We define the parallelogram grid with the step hg, G(R) = {~(k, l) = Vc + khgva + lhgVb E S}, k E [-N1, N1], l E [-N2, N2]

CHAPTER 2 Differential Equations with Hysteresis

274

z//'"

. . . . . ~ ... -..

-../

"-- . ~ / .

-.. -~ / /

/

;" Y

-.-

Va

/

FIGURE 2.66 Parallelogram R = R(c, y, z) and the grid G(R).

where N1 = [[Va[/hg] + 1, N2 = [IVb]/hg] + 1 and the operation [.J stands for the maximum integer that is not greater than the number in the 'floor brackets'. Introduce A I ( j ) = {V ~--- Vc(j) if- O~Va(j) nt- flVb(j) : [fl[~l} and A2(j) = {v = vc(j) + ~va(j) + flvb(j) : [~[~>1}. Let dist(v, A) = supo,A Iv - ~[ be the Euclidean distance between the point v and the set A. LEMMA2.4.18. Letr > Oand 13(v; r) ~ H. Ifthealgorithm Getlmage(v, r, v,, r,) returns 'True" then the map F1 is defined on 13(v; r) and FI (B(v; r)) c/3(v,; r,) where the values v,, r, are returned by the algorithm. PROOF. Here we just sketch the proof as there is no formal definition of some steps of the algorithm GetlmageO. By line 1 the point v is transformed into the initial conditions for the problem (2.9). There exists a numerical trajectory described by the set no { (tn, Zn, ~ln, qn)}n=0 which is an output of the algorithm IntegratelO. By lines 5, 6 and Proposition 2.4.3, the error in numerical trajectory at the moment tn satisfies en~eno ~0.01 - eappr. This, together with lines 3, 4 guarantees that the numerical trajectory in the extended phase, space including cyclic time, satisfies conditions of Lemma 2.4.5 and FI(B(v; r)) is well defined. More than that, there exist v,, r, such that FI(B(v; r)) c 13(v,; r,). The lemma is proved. []

275

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

Let R be a parallelogram satisfying ~ (R) c V. If the algorithm VWHypPartl (R, hg, j) returns 'True', then F1 is defined and continuous on R and

LEMMA 2.4.19.

7-/71F I(R) r3

(V -(u) x (~ \ V(S))) = 0.

(2.224)

GETIMAGE(V,r,v,,r,) Input: Ball on the plane FI centered at v with radius r Output: True if F 1 is well defined on/3(v; r)" in this case F 1 (B(v; r)) C/3(v,; r,) (1) Lett 0 = v(1),z0 = (0.5, v(2)), r/0 = 0 . 5 . no (2) Use IntegratelO to compute (tn, Zn, rln, qn)n= 0 (3) (4) (5) (6) (7) (8) (9) (10)

Value n o should satisfy Zno _(1) > 0.5 + 0.01 , and 3n, < no " z n, (1) < 0.5 -- eappr. if there is no such n o then return false Compute sequence en according to Lemma 2.4.12 if value eno + eappr > 0.01 then return false Use plane intersection algorithm in the extended phase space (including cyclic time) for the plane (z, (1, 0)) = 0.5 to compute B(v,; r,) if Plane intersection algorithm fails then return false Make sure that this is the first intersection and intersection before tnl is not possible return true

VWHYPPARTI(R, hg, j) Input: Parallelograms R, R(j) mesh hg Output: True if part 2 of (2.221) 'partially' holds Let G(R) be the grid (2N 1 + 1) x (2N 2 + 1) on R with the mesh size hg (1) (2) for k = - N 1 to N 1 for l = - N 2 to N 2 (3) if GETIMAGE(~(k,l),hg,v,,rad)==false then return false (4) if dist(v,, E2 \ AI(j)) < rad t h e n return false (5) return true (6)

PROOF. First w e m e n t i o n t h a t R c a n b e c o v e r e d b y balls of r a d i u s at t h e n o d e s of t h e grid: R C

U /3(~;hg). ~e~(R)

hg c e n t e r e d (2.225)

I n d e e d , m e s h e s of t h e g r i d G (R) are r h o m b u s e s w i t h s i d e s of l e n g t h hg a n d a r h o m b u s e s w i t h s i d e s of l e n g t h a c a n b e c o v e r e d b y t h e f o u r c l o s e d balls w i t h c e n t e r s in its c o r n e r s a n d r a d i i of 4 a . Let t h e a l g o r i t h m VWHypPartl(R, hg, j) r e t u r n 'True'. By line 4 of t h e a l g o r i t h m a n d L e m m a 2.4.18 t h e m a p F1 is d e f i n e d o n all ~ ~ ~(R) (lines 1, 2, 3). T h u s F1 is d e f i n e d a n d c o n t i n u o u s o n R b y (2.225) a n d

gi,j is d e f i n e d a n d c o n t i n u o u s o n 7/I -1) (R).

276

CHAPTER 2 Differential Equations with Hysteresis

By line 5 of the algorithm B(~; hg) C .AI(j) for all ~ e G(R). Thus F1 (a) C ,,41 (j). Since, by the definition 71~ 1 (A1 (j)) C [~ x W(s) and ~ 1F1 (a)

C R x V (s), formula (2.224) follows. The lemma is proved. LEMMA2.4.20. Let R

[]

- "}-[i(V). If the algorithm VWHypPart2(hg, i, j) returns

"True', then gi,j ((9

x

F]

: 0.

(2.226)

VWHYPPART2(hg,i, j) Input: ParallelogramsR(i), R(j) mesh hg Output: True if part 1 of (2.221) holds for gi,j (1) Let G(R(i)) be the grid (2N1 + 1) x (2N2 + 1) on R(i) with the mesh size hg (2) for l = - N 2 to N 2 (3) if GETIMAGE(~(--N1,l),hg,v,,rad)==false t h e n return f a l s e (4) if dist(v,, R2 \ A2(j)) < rad t h e n return f a l s e (5) (6) if GETIMAGE(~(N1,l),hg,v,,rad)==false t h e n return f a l s e (7) if dist(v,, •2 \ A2 (j)) < rad t h e n return false (8) return true

PROOF. Let ~, = {~(k, l) 9 k e {-N1, N1}, l e [-N2, N2]}. We notice that by the definition of the grid ~(R),

c}-~i(~V(U) x V (s)) C U B(~,hg).

(2.227)

~e~, Let the algorithm VWHypPart2(R, hg, j) return 'True'. By lines 3, 6 of the algorithm, Lemma 2.4.18 and (2.227) the map F1 is defined on r (cOR(i)(u) x R(i)(s)). By lines 4, 7 of the algorithm B(~; hg) c A2(j) for all ~ e G,. Thus

"]'{i (0V(u) x

W(S)) C ,A2(j).

Since, by the definition of A2 (j)

"]-{)-I)(A2(j)) C ( ~

\ V (u)) x [~

formula (2.226) follows. The lemma is proved.

2.4 EXAMPLES OF TECHNICAL TOOLBOXES TABLE 2.3

Computational details:

277

hg(i), #({Rk(i)}), average time

to process

Rk(i)

(i, j)

(0,0)

(0,1)

(1,2)

(2,3)

(3,4)

(4,5)

(5,6)

(6,7)

(7,8)

(8,0)

hg (i)

10 -4 1 59

10 -4 1 59

10 -4 1 62

10 -4 1 67

10 -4 1 74

5.10-7 4 694 996

10 -6 1381 951

10 -4 1 56

10 -4 1 54

10 -4 1 57

Pieces Average time (s)

LEMMA2.4.21. Let {Rk} be a finite set of parallelograms such that R(i) C UkRk.

Let hg be fixed. The successful terminations of VWHypPartl (Rk, hg, j)for all k and VWHypPart2(hg(i), i, j) together with g!U) z,] ( - 0 . 0 0 3 , "" u)gi,(u) j (0.003, 0) < 0

imply that gi,j is defined on V and is (V, V)-hyperbolic. PROOF. By Lemma 2.4.19, the map gi,j is defined on all arLiRk, which means that gi,j is defined on the whole of V. By (2.224)

gi,j(~7) n (fz ~u) x (~, \ V(S))) - 0, that is, exactly the second part of (2.221). The first part of (2.221) holds due to successful termination of VWHypPart2(hg, i, j) and Lemma 2.4.20. Finally (2.222) follows from the supposition of the lemma and Lemma 2.4.15. [] Our implementations of VWHypPartl(Rk(i),hg(i),j) and VWHypPart2 (h~(i), i, j) terminated successfully for all i,j such that ai,j -- 1 and appropriate partitions Rk(i) of R(i). The additional verification shows that (2.221) holds for all i, j such that ai,j -- 1. This, together with Lemma 2.4.21, proves Lemma 2.4.17. Theorem 2.3.5 is proved. The value hg(i) and total number of Rk(i) for all i can be found in Table 2.3. On the intuitive level the proof may be seen in Fig. 2.67. However, to justify that picture proof we used parallel computations on a cluster containing 100 CPU P-III 1.2GHz. The total processor time used for each R(i) is given in Table 2.3 as well. The total computational time is just below 6 000 000 seconds, that is, 70 days of computing on one PC. []

278

CHAPTER 2 Differential Equations with Hysteresis

0.01

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FIGURE 2.67 The first picture: g0,0(V) against V; other eight pictures are gi, i+l (V) against V for i = 0 . . . . . 7 and the final one is g8,0(V) against V.

ACKNOWLEDGEMENT This work was partially supported by the Enterprise Ireland, Grant S C / 2000/138, grant S C / 2 0 0 3 / 3 7 6 and Post-doctoral fellowship P D / 2 0 0 4 / 2 6 . D. Rachinskii was partially supported by the Russian Science Support Foundation, grant MD-87.2003.01 of the President of Russia and grants 03-01-00258, 04-01-00330 of the Russian Foundation for Basic Researches.

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134. M. Brokate and J. Sprekels, 'Existence and optimal control of mechanical processes with hysteresis in viscous solids', IMA J. Appl. Math. 43(3), 219-229 (1989). 135. C. Cavaterra and F. Colombo, 'Automatic control problems for reaction-diffusion systems', J. Evol. Equ. 2(2), 241-273 (2002). 136. S. Gatti, 'Automatic control of the temperature in phase change problems with memory', Z. Anal. Anwendungen 20(4), 883-914 (2001). 137. M. Grasselli, 'Automatic control of the temperature in thermoviscoelasticity', Math. Methods App1. Sci. 22(16), 1447-1464 (1999). 138. K.-H. Hoffmann, J. Sprekels, and A. Visintin, 'Identification of hysteresis loops', J. Comput. Phys. 78(1), 215-230 (1988). 139. M.A. Krasnosel'skii and A. V. Pokrovskii, 'Simulation of transformers with hysteresis by means of continual relay systems', Dokl. Akad. Nauk SSSR 227(3), 547-550 (1976). 140. R. C. Smith, 'Hysteresis modeling in magnetostrictive materials via Preisach operators', J. Math. Systems Estim. Control 8(2), 23 pp. (1998). 141. P. Colli, M. Grasselli, and J. Sprekels, 'Automatic control via thermostats of a hyperbolic Stefan problem with memory', Appl. Math. Optim. 39(2), 229-255 (1999). 142. W. Desch and J. Turi, 'The stop operator related to a convex polyhedron', J. Differential Equations 157(2), 329-347 (1999). 143. L. R. Dupr6, R. Van Keer, and J. A. A. Melkebeek, 'Complementary 2-D finite element procedures for the magnetic field analysis using a vector hysteresis model', Int. J. Numer. Methods Engrg. 42(6), 10051023 (1998). 144. M.A. Krasnosel'skii and A. V. Pokrovskii, 'The absolute stability of systems with several nonlinear components', Dokl. Akad. Nauk SSSR 271(6), 1314-1316 (1983). 145. R.E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill Book Co., New York (1969). 146. M.A. Krasnosel'skii, V. S. Burd, and Y. S. Kolesov, Nonlinear Almost Periodic Oscillations, Halsted Press (A division of John Wiley & Sons), New York, Toronto (1973). (Translated from the Russian by A. Libin, translation edited by D. Louvish). 147. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985). 148. M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin (1984) (Translated from the Russian by Christian C. Fenske).

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149. N. A. Bobylev, Y. M. Burman, and S. K. Korovin, Approximation Procedures in Nonlinear Oscillation Theory, de Gruyter Series in Nonlinear Analysis and Applications, vol. 2, Walter de Gruyter & Co., Berlin (1994). 150. A. V. Pokrovskii, 'The limit norm of a linear operator and its applications', Dokl. Akad. Nauk SSSR 249(3), 548-551 (1979). 151. A. V. Pokrovskii, 'Limit norm of a linear element', Avtomat. i Telemekh. (11), 62-69 (1985). 152. A. V. Pokrovskii and M. E. Semenov, 'Stable periodic regimes in control systems with monotone nonlinearities', Avtomat. i Telemekh. (2), 31-37 (1990). 153. M.A. Krasnosel'ski/, The Operator of Translation Along the Trajectoriesof Differential Equations, American Mathematical Society, Providence, RI (1968) (Translations of Mathematical Monographs, Vol. 19. Translated from the Russian by Scripta Technica). 154. A. Pokrovskii, 'Shuttle algorithm in investigation of systems with hysteresis nonlinearities', in Models of Hysteresis, Trento, 1991, Longman Sci. Tech., Harlow (1993), pp. 124-142. 155. A. Pokrovskii, S. J. Szybka, and J. G. McInerney, 'Topological degree in locating homoclinic structures for discrete dynamical systems', Preprints of INS, UCC, Ireland (January 2001). 156. M. Krasnosel'ski/, J. Mawhin, and A. Pokrovskii, 'New theorems on forced periodic oscillations and bounded solutions', Soviet Physics Doklady 36(11), 743-745 (1991). 157. P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho, Tokyo (1996). 158. J. Hale, Theory of Functional Differential Equations, second edition, Applied Mathematical Sciences, vol. 3, Springer-Verlag, New York (1977). 159. L. Padulo and M. A. Arbib, System Theory. A Unified State-space Approach to Continuous and Discrete Systems, W. B. Saunders Co., Philadelphia, PA (1974). 160. A. PokrovskiL 'Topological shadowing and split-hyperbolicity', Funct. Differ. Equ. 4(3-4), 335-360 (1998), (1997). 161. O. Rasskazov, 'Methods of geometrical analysis of complicated dynamics, with applications to models of semi-conductor lasers', PhD thesis, University College Cork, Cork, Ireland (2003). (Webpage: http: //phys.ucc.ie/~oll/phdthesis.pdf).

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162. D. RachinskiL 'On natural continua of periodic solutions of systems with hysteresis', Automat. Remote. Control. 64(3, part 1), 420-438 (2003). 163. D. Rachinskii and A. M. Krasnosel'skii, 'On a nonlocal condition for existence of cycles of systems with hysteresis', Automat. Remote. Control. 64(2, part 1), 231-251 (2003). 164. V. S. Kozjakin and M. A. Krasnosel'skii, 'The method of parameter functionalization in the Hopf bifurcation problem', Nonlinear Anal. 11(2), 149-161 (1987). 165. J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Applied Mathematical Sciences, vol. 19, Springer-Verlag, New York (1976). (With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale). 166. A.M. Krasnosel' skii and M. A. Krasnosel'ski/, 'Large amplitude cycles in autonomous systems with hysteresis', Dokl. Akad. Nauk SSSR 283(1), 23-26 (1985). 167. M. A. Krasnosel'skiL D. I. Rachinskii and M. G. Yumagulov, 'Hopf bifurcation for autonomous systems with hysteresis', Dokl. Akad. Nauk 355(4), 446-449 (1997). 168. P. Diamond, N. Kuznetsov, and D. RachinskiL 'On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity', J. Differential Equations 175(1), 1-26 (2001). 169. T. Y. Li and J. A. Yorke, 'Period three implies chaos', Amer. Math. Monthly 82(10), 985-992 (1975). 170. P. Diamond, P. E. Kloeden, M. A. Krasnosel'skiL and A. PokrovskiL 'Chaotic dynamics in nonsmooth perturbations of bishadowing systems', Arab J. Math. Sci. 6(1), 41-74 (2000). 171. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge (1995) (With a supplementary chapter by A. Katok and L. Mendoza). 172. E. Cox, M. Mortell, A. PokrovskiL and O. Rasskazov, 'On chaotic and recurrent travelling wave patterns in a periodically forced and extended kdev', Preprints of School of Math., Appl. Math. and Stat., UCC, Ireland (February 2002). 173. O. Rasskazov, G. Huyet, J. G. McInerney, andA. PokrovskiL 'Rigorous analysis of complicated behaviour in a truncated Lang-Kobayashi

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CHAPTER 3

Scalar Preisach Models of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 3.1 3.2

The Classical Preisach Model of Hysteresis Generalized Scalar Preisach Models of Hysteresis References

293 360 443

3.1 T H E C L A S S I C A L P R E I S A C H M O D E L OF HYSTERESIS 3.1.1 DEFINITION OF THE CLASSICAL PREISACH MODEL The phenomenon of hysteresis has been with us for ages and has been attracting the attention of many investigators for a long time. The reason is that hysteresis is ubiquitous. It is encountered in many different areas of science. Examples include magnetic hysteresis, ferroelectric hysteresis, mechanical hysteresis, superconducting hysteresis, adsorption hysteresis, optical hysteresis, electron beam hysteresis, economic hysteresis, etc. However, the very meaning of hysteresis varies from one area to another, from paper to paper and from author to author. As a result, a stringent mathematical definition of hysteresis is needed in order to avoid confusion and ambiguity. Such a definition will serve a twofold purpose: first, it will be a substitute for vague notions, and, second, it will pave the road for more or less rigorous treatment of hysteresis. We begin with the definition of scalar hysteresis and, for the sake of generality, we adopt the language of control theory. Consider a transducer that can be characterized by an input u(t) and an output flt). This transducer is called a hysteresis transducer (HT) if its input-output relationship is a multibranch nonlinearity for which branch-to-branch transitions occur after input extrema. For the most part, the case of rate-independent hysteresis nonlinearity will be discussed. The term 'rate-independent' The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

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means that branches of such hysteresis nonlinearities are determined only by the past extremum values of input, while the speed (or particular manner) of input variations between extremum points has no influence on branching. It is important to stress that the notion of rate-independent hysteresis implies three distinct time scales. The first is the time scale of fast internal dynamics of the transducer. The second is the time scale on which observations (measurements) are performed. This time scale is much larger than the time scale of internal transducer dynamics so that every observation can be identified with a specific output value of the transducer. The third is the time scale of input variations. This time scale is much larger than the observation time scale so that every measurement can be associated with a specific value of input. In the existing literature, the hysteresis phenomenon is, by and large, linked with the formation of hysteresis loops (looping). This may be misleading and create the impression that looping is the essence of hysteresis. In this respect, the given definition of hysteresis emphasizes the fact that history-dependent branching constitutes the essence of hysteresis, while looping is a particular case of branching. Indeed, looping occurs when the input varies back and forth between two consecutive extremum values, while branching takes place for arbitrary input variations. From the given definition, it can also be concluded that scalar hysteresis can be interpreted as a nonlinearity with a memory which reveals itself through branching. In the given definition of hysteresis, the physical meanings of the input u(t) and the outputf(t) were left unspecified. It was done deliberately, for the sake of mathematical generality. However, it is not difficult to specify the meanings of u(t) and fit) in particular applications. For instance, in magnetism u(t) is the magnetic field and fit) is the magnetization, in mechanics u(t) is the force andf(t) is the displacement (length), in adsorption u(t) is the gas pressure and fit) is the amount of material adsorbed. The notion of hysteresis transducer may have different interpretations as well. For instance, in magnetism the HT can be construed as an infinitesimally small volume of magnetic material, and the corresponding input-output hysteresis nonlinearity can be interpreted as a constitutive equation for this material. All rate-independent hysteresis nonlinearities fall into two general classifications: (a) hysteresis nonlinearities with local memories, and (b) hysteresis nonlinearities with nonlocal memories. The hysteresis nonlinearities with local memories are characterized by the following property. The value of output f(t0) at some instant of time to and the values of input u(t) at all subsequent instants of time t>~to uniquely predetermine the

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295

value of outputf(t) for all t > to. In other words, for hysteresis transducers with local memories the past exerts its influence upon the future through the current value of output. This is not the case for hysteresis transducers with nonlocal memories. For such transducers, future values of output fit) (t~to) depend not only on the current value of outputflt0) but on past extremum values of input as well. Our discussion is solely concerned with mathematical models of hysteresis with nonlocal memory. The question arises, why are these models needed? The answer is that the hysteresis transducer is usually a part of a system. As a result, its input is not known beforehand, but is determined by the interaction of the transducer with the rest of the system. Since the input of HT is not predictable a priori, it is impossible to specify ahead of time the branches of hysteresis nonlinearity that will be followed in a particular regime of the system. This is the main impediment as far as selfconsistent mathematical descriptions of systems with hysteresis are concerned. To overcome the difficulty mentioned above, mathematical models of hysteresis are needed. These models represent new mathematical tools that themselves (due to their structure) will detect and accumulate input extrema and will choose appropriate branches of the hysteresis nonlinearity according to the accumulated histories. Coupled together with the mathematical description of the rest of the system, these models will constitute complete and self-consistent mathematical descriptions of systems with hysteresis. Without such models, the self-consistent mathematical descriptions of systems with hysteresis are virtually impossible. Next, we proceed to the discussion of the classical Preisach model of hysteresis. This model has a long and instructive history that can be best characterized by the following eloquent statement of J. Larmor made in his preface to the book [1]: ...scientific progress, considered historically, is not a strictly logical process, and does not proceed by syllogisms. New ideas emerge dimly into intuition, come into consciousness from nobody knows where, and become the material on which the mind operates, forging them gradually into consistent doctrine, which can be welded on to existing domains of knowledge. This is exactly what has happened with the Preisach model. The origin of this model can be traced back to the landmark paper of F. Preisach [2] published in 1935. Preisach's approach was purely intuitive. It was based on some plausible hypotheses concerning the physical mechanisms of magnetization. For this reason, the Preisach model was first regarded as a physical model of hysteresis. It has remained primarily known in the area of magnetics where this model has been the focus of considerable research for many years. This has resulted in the further development of

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the Preisach model, and many improvements and valuable facts have been accumulated in this area. These developments and results are recorded, for instance, in the publications [3-12,51]; it is important to note that the given list of references is not complete by any standards. Somewhat in parallel with the mentioned developments in magnetics, the Preisach model was independently invented and then extensively studied and tested for adsorption hysteresis by D. H. Everett and his collaborators [13-16]. This clearly indicated that the applications of Preisach's model were not limited only to the area of magnetics. The next decisive step in the direction of better understanding of the model was made in the 1970s and 1980s when the Russian mathematician M. Krasnoselskii and his colleagues undertook a comprehensive mathematical study of systems with hysteresis. It was then gradually realized that the Preisach model contained a new general mathematical idea. As a result, this model was separated from its physical connotation and was represented in a purely mathematical form that is similar to a spectral decomposition of operators [17]. In this way a new mathematical tool has evolved that can now be used for the mathematical description of hysteresis of various physical nature. At the same time, this approach has strongly revealed the phenomenological nature of the Preisach model. This has put this model into a new perspective and has led to the clarification of many controversies that have long surrounded the Preisach model. We next proceed to the purely mathematical description of the Preisach model. Consider an infinite set of simplest hysteresis operators ~aB. Each of these operators can be represented by a rectangular loop on the input-output diagram (see Fig. 3.1). Numbers c~and fl correspond to 'up' and 'down' switching values of input, respectively. It will be assumed in the sequel that a~>fl. Outputs of the above elementary hysteresis operators may assume only two values, +1 and -1. In other words, these operators can be interpreted as two-position relays with 'up' and 'down' positions corresponding to ~ u ( t ) = +1 and }~u(t) - -1, respectively. As the input, u(t), is monotonically increased, the ascending branch abcde is followed. When the input is monotonically decreased, the descending branch edfba is traced. It is clear that the operators ~1~ represent hysteresis nonlinearities with local memories. Along with the set of operators ~a]/consider an arbitrary weight function #(a, fl) that is often referred to as the Preisach function. Then, the Preisach model can be written as follows: A

fit) - Fu(t) -

/~(c~, fl)~a~u(t) de dfl.

(3.1)

A

Here F is used for the concise notation of the Preisach hysteresis operator that is defined by the integral in (3.1).

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

297

A

f +1

d

e

-

~U

-1

FIGURE 3.1

It is apparent that the model (3.1) can be interpreted as a continuous analog of a system of parallely connected two-position relays. This interpretation is illustrated by the block diagram shown in Fig. 3.2. According to this diagram, the same input u(t) is applied to each of two-position relays. Their individual outputs are multiplied by #(~, fl) and then integrated over all appropriate values of a and ft. As a result, the output, fit), is obtained. Discrete approximation to the above block diagram can be used as device realizations of the Preisach model (3.1). It is clear from the above discussion that the Preisach model is constructed as a superposition of simplest hysteresis operators ~ . These operators can be construed as the main building blocks for the model (3.1). The idea that a complicated operator can be represented as a superposition of simplest operators is not entirely new and was exploited before in mathematics, particularly in the functional analysis [17]. For instance, according to the spectral decomposition theory for self-adjoint operators, any self-adjoint operator can be represented as a superposition of projection operators that are, in a way, the simplest self-adjoint operators. The above analogy shows that the Preisach model (3.1) can be interpreted from the mathematical point of view as a spectral decomposition of a complicated hysteresis operator F into the simplest hysteresis operators ~ . There is also an interesting parallel between the Preisach model and wavelet transforms that are currently very popular in the area of signal processing.

CHAPTER 3 ScalarPreisach Models of Hysteresis

298

A

A

u(t)

/ FIGURE 3.2

Indeed, all rectangular loop operators ~//can be obtained by translating and dilating the rectangular loop operator ~1,-1, that can be regarded as the 'mother loop operator'. Thus, the Preisach model can be viewed as a 'wavelet operator transform'. The Preisach hysteresis nonlinearity (3.1) is constructed as a superposition of elementary hysteresis nonlinearities ~a/~with local memories, nevertheless, it usually has a nonlocal memory. (This fact will be proved in the next section.) It is remarkable that a new qualitative property of nonlocal memory emerges as a collective property of a system having a huge (infinite) number of simple and qualitatively similar components. Having defined the Preisach model, it is appropriate to describe the directions along which the further discussion will proceed. The subsequent discussion will be centered around the following five topics. 1. How does the Preisach model work? In other words, how does this model detect local input extrema, accumulate them and choose the appropriate branches of hysteresis nonlinearity according to the accumulated histories? The answer to this question will reveal the mechanism of memory formation in the Preisach model. It will require the development of a special diagram technique that will constitute the mathematical foundation for the analysis of the Preisach model. 2. What experimental data are needed for the determination of the Preisach function, #(a, fl), for a given hysteresis transducer?

3.1 THE CLASSICALPREISACH MODEL OF HYSTERESIS

299

This is the so-called identification problem. The solution to this problem is very important as far as the practical applications of the Preisach model are concerned. 3. What are the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model? The significance of this problem is that its solution will clearly establish the limits of applicability of the Preisach model. 4. How can the Preisach model be implemented numerically? This is an important question because it seems at first that the numerical evaluation of two-dimensional integrals is required for the numerical implementation of the Preisach model. However, it will be shown in the sequel that the evaluation of the above integrals can be completely avoided. 5. How can the Preisach model be useful for the computation of hysteretic energy dissipation? It is well known that the hysteretic energy dissipation can be easily evaluated for the case of periodic (cyclic) input variations. However, the problem of computing hysteretic energy losses for arbitrary (not necessary periodic) input variations has remained unsolved. It will be shown (see Section 3.1.5) that the Preisach model can bring about the solution to this problem. This solution can be useful for the computation of entropy production and, in this way, may facilitate the development of irreversible thermodynamics of hysteretic media. There is another problem which has not been mentioned above but has been extensively studied by Krasnoselskii and his colleagues. This problem is related to the fact that hysteresis nonlinearities are naturally defined on the set of continuous and piecewise monotonic inputs. However, the above set of functions does not form a complete function space. This constitutes the main difficulty as far as the rigorous mathematical treatment of differential (or integral) equations with hysteresis nonlinearities is concerned. Thus, the problem of continuous extension of hysteresis operators from the above set of piecewise monotonic inputs to some complete function spaces presents itself. The essence of the problem is in finding (or description) of such complete function spaces. The solution to this problem is important because these function spaces form the natural 'environment' for the rigorous mathematical study of equations with hysteresis nonlinearities. Nevertheless, this problem is more or less of a purely mathematical nature. It is not directly related to practical applications, and for this reason will not be discussed here. The reader interested in the

CHAPTER 3 ScalarPreisach Models of Hysteresis

300

discussion of this problem is referred to Krasnoselskii and Pokrovskii [18] as well as more recent publications [19-24]. It is apparent from the above discussion that the Preisach model has been defined without any reference to a particular physical origin of hysteresis. This clearly reveals the phenomenological nature of the model and its mathematical generality. To further emphasize this new approach to the Preisach model, it is instructive to compare the definition of the model given above with the definition regularly used in magnetics. In magnetics, separate magnetic 'particles' (or 'domains') are introduced. These particles have rectangular hysteresis loops and they play the same role as elementary hysteresis operators ~/~. A loop typical for such particles is shown in Fig. 3.3. Here Hu and Ha are 'up' and 'down' switching magnetic fields, respectively, and ms is the magnetic moment (magnetization) that is identical for all particles. The notation ~(Hu, Ha) is used for the particle having the hysteresis loop shown in Fig. 3.3. The magnetic material is considered to be composed of many such particles. It is also assumed that the different particles have some distribution of reversal field Hu and Ha that can be characterized by the distribution function dp(Hu, Hcl). This function plays the same role as ~(~, t) in (3.1). It is typical to speak about statistical nature of the distribution function dp(Hu,Hi), although the author is not aware of any rigorous justification or experimental evidence for this statistical interpretation. By using the magnetic particles and their distribution function, the Preisach model is usually defined in M

+m

s

-

Hd

-m

iHu

s

FIGURE 3.3

=H

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

301

magnetics as follows

M(t) - ffUu>Ha

dp(Hu, Hd)~(Hu, Hd)H(t) dHu dHd,

(3.2)

where M is the magnetization, while

~(Hu, Ha)H(t) = +ms

(3.3)

if the particle is switched up, and

~(Hu, Ha)H(t) = -ms

(3.4)

if the particle is switched down. The given definition is framed in terms of magnetics and can naturally be called the 'magnetic' definition. It is obvious that the 'magnetic' definition of the Preisach model is mathematically equivalent to the previous definition (3.1). The'magnetic' definition suggests some relation of the Preisach model to certain physical realities such as magnetic particles, their distribution, etc. For this reason, the Preisach model has long been regarded as the physical model. However, there are some intrinsic difficulties with respect to this interpretation. Indeed, if the magnetic particles are to have some features of reality, then their geometric shapes and mutual spatial locations should be important and affect the value of magnetization. But, in the 'magnetic' definition of the Preisach model there is no reference whatsoever to the spatial locations of the particles or to their shapes. In the mathematical definition (3.1), this difficulty does not appear because there is no need to relate the operators, ~ , to some real physical objects. Another difficulty comes from the fact that many (almost all) particles in the Preisach model have asymmetrical rectangular loops (Hu ~: -Ha). Such loops are not physical for single particles. The use of asymmetrical loops is commonly justified on the grounds that asymmetry is caused by the interaction between the particles, and the amount of asymmetry is a measure of the interaction. More precisely, this means that each particle 'feels' not only the applied (external) magnetic field H(t), but also the interaction magnetic field that is due to the magnetization of adjacent particles. The interaction magnetic field may, of course, vary from one particle to another. These interaction magnetic fields result in shifted (asymmetrical) hysteresis loops when these loops represent the dependence of magnetization on the applied field alone. Although the given explanation sounds plausible, it is not completely satisfactory. This explanation does not account for the fact that the

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CHAPTER 3 Scalar Preisach Models of Hysteresis

~U

FIGURE 3.4

interaction fields depend on particular states of magnetization, shapes, and lbcation of the particles. This information is not represented in the Preisach model at all. In the case of purely mathematical definition (3.1) of the Preisach model, the above difficulty does not appear. This is because there is a straightforward phenomenological explanation for the need of operators ~a/~ with asymmetrical loops. These operators are needed in order to describe asymmetrical minor loops exhibited by actual hysteresis nonlinearities (see Fig. 3.4). Indeed, if only operators ~ _ ~ with symmetrical loops were used, then the model (3.1) being constructed as a superposition of such ~-operators would also describe only symmetrical minor loops. In other words, ~-operators with symmetrical loops do not form a complete set of operators in the sense that hysteresis nonlinearities with asymmetrical minor loops cannot be represented as superpositions of ~-operators from this set. From the discussion presented above the following conclusion can be reached. The 'magnetic' definition of the Preisach model is possible and historically was developed first. However, this definition obscures the model, conceals its mathematical and phenomenological nature, and narrows the area of applicability of this model to the field of magnetics. The definition (3.1), on the other hand, interprets the Preisach model as a new mathematical tool whose importance may well extend beyond the area of magnetics. For this reason, the purely mathematical definition (3.1) seems to be more attractive and will be used throughout our discussion.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS (O~o,~o)

(X

303

A

/

FIGURE 3.5 3.1.2 GEOMETRIC INTERPRETATION A N D MAIN PROPERTIES OF THE PREISACH MODEL

The mathematical investigation of the Preisach model is considerably facilitated by its geometric interpretation. This interpretation is based on the following simple fact. There is a one-to-one correspondence between operators ~/~ and points (~, fl) of the half-plane ~>~fl (see Fig. 3.5). In other words, each point of the half-plane a~ fl can be identified with only one particular ~-operator w h o s e ' u p ' and 'down' switching values are respectively equal to a and fl coordinates of the point. It is clear that this identification is possible because both ~-operators and the points of the half-plane ~ f i are uniquely defined by pairs of numbers, a and ft. Consider a right triangle T (see Fig. 3.5). Its hypotenuse is a part of the line ~ = fl, while the vertex of its right angle has the coordinates ~0 and fl0 with fl0 = -~0. In the sequel, this triangle willbe called the limiting triangle and the case when #(a, fl) is a finite function with a support within T will be discussed. In other words, it will be assumed that the function #(a, fl) is equal to zero outside the triangle T. This case covers the important class of hysteresis nonlinearities with closed major loops. At the same time, the above case will not essentially limit the generality of our future discussions. To start the discussion, we first assume that the input u(t) at some instant of time to has the value that is less than rio- Then, the outputs of all ~-operators which correspond to the points of the triangle T are equal to -1. In other words, all ~-operators are in the 'down' position. This corresponds to the state of 'negative saturation' of the hysteresis nonlinearity represented by the model (3.1). Now, we assume that the input is monotonically increased until it reaches at time tl some maximum value Ul. As the input is being increased, all ~-operators with 'up' switching values a less than the current

CHAPTER 3 Scalar Preisach Models of Hysteresis

304

input value u(t) are being turned into the 'up' position. This means that their outputs become equal to +1. Geometrically, this leads to the subdivision of the triangle T into two sets: S + (t) consisting of points (a, fl) for which the corresponding ~-operators are in the 'up' position, and S-(t) consisting of points (a, ]/) such that the corresponding ~-operators are still in the 'down' position. This subdivision is made by the line a = u(t) (see Fig. 3.6) that moves upwards as the input is being increased. This upward motion is terminated when the input reaches the maximum value Ul. The subdivision of the triangle T into S + (t) and S- (t) for this particular instant of time is shown in Fig. 3.7. Next, we assume that the input is monotonically decreased until it reaches at time t2 some minimum value u2. As the input is being decreased, all )-operators with 'down' switching values ]/above the current input

m

FIGURE 3.6

/

S-(t)

/(X-- U1 / ~ ~

s§

/

FIGURE 3.7

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

305

(x

s§

[~ = u(t)

/

FIGURE 3.8 (x

FIGURE

3.9

value, u(t), are being turned back into the 'down' position. This changes the previous subdivision of T into positive and negative sets. Indeed, the interface L(t) between S + (t) and S-(t) now has two links, horizontal and vertical. The vertical link moves from right to left and its motion is specified by the e q u a t i o n / / = u(t). This is illustrated by Fig. 3.8. The above motion of the vertical link is terminated when the input reaches its m i n i m u m value u2. The subdivision of the triangle T for this particular instant of time is shown in Fig. 3.9. The vertex of the interface L(t) at the above instant of time has the coordinates ~ = Ul and fl = u2. Now, we assume that the input is increased again until it reaches at time t3 some m a x i m u m value u3 that is less than Ul. Geometrically, this increase results in the formation of a new horizontal link of L(t) which

CHAPTER 3 Scalar Preisach Models of Hysteresis

306

/

s*

FIGURE

3.10

/ ( u~, u 2) s§

i (u~u,)

FIGURE

3.11

moves upwards. This upward motion is terminated when the maximum u3 is reached. This is shown in Fig. 3.10. Next, we assume that the input is decreased again until it reaches at time t4 some minimum value u4 that is above u2. Geometrically, this input variation results in the formation of a new vertical link that moves from right to left. This motion is terminated as the input reaches its minimum value u4. As a result, a new vertex of L(t) is formed that has the coordinates = u3 and fl = u4. This is illustrated by Fig. 3.11. By generalizing the previous analysis, the following conclusion can be reached. At any instant of time, the triangle T is subdivided into two sets: S+ (t) consisting of points (a, fl) for which the corresponding ~-operators are in the 'up' position, and S-(t) consisting of points (a,//) for which the corresponding ~-operators are in the 'down' position. The interface

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

I

307

/

/ F I G U R E 3.12

s-(t)

F I G U R E 3.13

L(t) between

S + (t) and S-(t) is a staircase line whose vertices have a and fl coordinates coinciding respectively with local maxima and minima of input at previous instants of time. The final link of L(t) is attached to the line a = fl and it moves w h e n the input is changed. This link is a horizontal one and it moves u p w a r d s as the input is increased (see Fig. 3.12). The final link is a vertical one and it moves from right to left as the input is decreased (see Fig. 3.13). Thus, at any instant of time the integral in (3.1) can be subdivided into two integrals, over S + (t) and S-(t), respectively: A

fit)

= F#(t)=

+(t)

#(o~, fl)~u(t) d~ dfl

q - / ~ s - (t)

#(a,

fl)~u(t) da dfl.

(3.5)

CHAPTER 3 Scalar Preisach Models of Hysteresis

308

Since

~&~u(t) = +1,

if (~, fl) ~ S+(t)

(3.6)

~u(t)

if (~, fl) ~ S-(t),

(3.7)

and - -1,

from (3.5) we find

fit) = fS +(t) #(a, fl) da dfl - f L

-(t)

#(a, fl) d0r dfl.

(3.8)

From the above expression, it follows that an instantaneous value of output depends on a particular subdivision of the limiting triangle, T, into positive and negative sets S+ (t) and S-(t). This subdivision is determined by a particular shape of the interface L(t). This shape, in turn, depends on the past extremum values of input because these extremum values are the coordinates of the vertices of L(t). Consequently, the past extremum values of input shape the staircase interface, L(t), and in this way they leave their mark upon the future. To make the above point perfectly clear, consider two inputs ul(t) and u2(t) with two different past histories for t < t'. This means that they had different local extrema for t < t t. It is next assumed that these inputs coincide for t~t'. Then according to (3.8), the outputs fl (t) andf2(t) corresponding to the above inputs are given by the formulas fl(t) -/fs~-(t) #(0r fl) d~ dfl - / L 1 (t) #(a, fl) da dfl,

(3.9)

f/" f2(t) = U /z(a, fl)da d f l - ] ] /~(0r fl)d0r dfl, dds -~(t) d l S2 (t)

(3.10)

where S + (t) and S l(t), S+ (t) and S2 (t) are positive and negative sets of two subdivisions of T associated with ul(t) and u2(t), respectively. The above two subdivisions are different because they correspond to two different input histories. Thus, from (3.9) and (3.10) we conclude

fl(t) ~-f2(t)

fort > t'.

(3.11)

It is clear that inequality (3.11) holds even if the outputsfl(t') andf2(t') are somehow the same at time t I. This means that the Preisach model (3.1) describes, in general, hysteresis nonlinearities with nonlocal memories. The above discussion reveals the mechanism of memory formation in the Preisach model. The memory is formed as a result of two different rules for the modification of the interface L(t). Indeed, for a monotonically increasing input, we have a horizontal final link of L(t) moving upwards,

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

309

while, for a monotonically decreasing input we have a vertical final link of L(t) moving from right to left. These two different rules result in the formation of the staircase interface, L(t), whose vertices have coordinates equal to past input extrema. It is apparent from the previous analysis that the Preisach model can be defined in purely geometric terms, without any reference to the analytical definition (3.1). Indeed, the formula (3.8), along with the above two rules for the modification of L(t), can be interpreted as an independent definition of the Preisach model. This definition is fully equivalent to the previous one. However, the geometric definition may be convenient for further generalization of the Preisach model. For instance, new and more general rules for the subdivision of T into positive and negative sets, S+ (t) and S-(t), may be introduced. In these rules, the links of L(t) may not necessarily be the segments of straight lines parallel to coordinate axes. Furthermore, different functions/~+ (~, fl) and #-(~, fl) may be defined on the positive and negative sets, respectively. All these modifications may result in some meaningful generalizations of the Preisach model. However, the above possibilities have not been examined enough in the existing literature. Having described the geometric interpretation of the Preisach model, we are now well equipped for the discussion of the main properties of this model. We begin with the simplest property, which expresses the fact that the output value, f+, in the state of positive saturation is equal to the minus output value, - f - , in the state of negative saturation. In the state of positive saturation, the input u(t) is more than a0 and all ~-operators are in the 'up' position. Hence, according to (3.8), we find: f + - JJT/~(a' fl) da dfl.

(3.12)

Similarly, in the state of negative saturation the input u(t) is less than fl0 and all ~-operators are in the 'down' position. As a result, we obtain

f - - - ffT l.t(~x, fl) da dfl.

(3.13)

From (3.12) and (3.13), we have f+= -f-.

(3.14)

It is important to keep in mind that the saturation valuesf + and f - remain constant for any value of input u(t) above ~0 and below fl0, respectively. In other words, after ascending and descending branches merge together, they become flat. Partly for this reason, it is often said that the Preisach

CHAPTER 3 Scalar Preisach Models of Hysteresis

310

(~ / .

~ .

t

.

.

u 1 __

u3

I

U8

/

FIGURE 3.14

model does not describe reversible components of hysteresis nonlinearities. These components are regarded as being responsible for finite slopes of ascending and descending branches after they merge together. The inability of the Preisach model to describe the reversible components of hysteresis nonlinearities has long been viewed as a deficiency of the model. It will be shown later in the book that this deficiency along with some others can be removed by the appropriate generalization of the Preisach model. We next proceed to the more interesting property which further elucidates the mechanism of memory formation in the Preisach model. It turns out that this model does not accumulate all past extremum values of input. Some of them can be wiped out (erased) by subsequent input variations. To make this property clear, consider a particular past history that is characterized by a finite decreasing sequence {Ul, U3, U5, U7} of local input maxima and an increasing sequence {u2, u4, u6, us } of local input minima. A typical a - ] / d i a g r a m for this kind of history is shown in Fig. 3.14. Now, we assume that the input u(t) is monotonically increased until it reaches some maximum value u9 that is above u3. This monotonic increase of input u(t) results in the formation of a horizontal final link of L(t) that moves upwards until the maximum value u9 is reached. This results in a modified a-//diagram shown in Fig. 3.15. It is evident that all vertices whose a-coordinates were below u9 have been wiped out. It is also clear that the wiping out of vertices is equivalent to the erasing of the history associated with these vertices. Namely, the past input maxima and minima that were respectively equal to ~- and fl-coordinates of the erased vertices have been wiped out. We have illustrated how the wiping out of vertices occurs for monotonically increasing inputs. However, it is obvious that the wiping out of vertices occurs in a similar manner for monotonically

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

311

U 1

/

FIGURE 3.15

decreasing inputs as well. Thus, we can formulate the following property of the Preisach model. WIPING-OUTPROPERTY. Each local input maximum wipes out the vertices of L(t) whose a-coordinates are below this maximum, and each local minimum wipes out the vertices whose fl-coordinates are above this minimum. The wiping-out property is asserted above in purely geometric terms. This makes this property quite transparent. However, the same property can also be described in analytical terms. The analytical formulation complements the geometric one because it is directly phrased in terms of time input variations. Consider a particular input variation shown in Fig. 3.16 for the time interval toKt~t I. We assume that at the initial instant of time to the input value u(to) was below fl0. This means that the initial state is the state of negative saturation. Consequently, the whole history has been written by the input variation after time to. We would like to specify explicitly local input extrema that will be stored by the Preisach model at time Y. Consider the global maximum of the input at the time interval [to, t']. We will use the notation M1 for this maximum and t~- for the instant of time the maximum was reached: M1 - max u(t), u(t-~) - M1. (3.15) [t0,Y] It is clear that all previous input extrema were wiped out by this maximum. Now, consider the global minimum of the input at the interval [t +, Y]. We will use the notation ml for this minimum and t I for the time itwas reached: ml - min u(t), [t~-,t']

u ( t l ) - ml.

(3.16)

312

CHAPTER 3

Scalar Preisach Models of Hysteresis

u(t)

mk m2 ml

FIGURE 3.16

It is apparent that all intermediate input extrema that occurred between t + and t I were erased by the minimum ml. Next, consider the global m a x i m u m of the input at the interval [t 1, tl]. The notations M2 and t~- are appropriate for this maximum and the time it occurred: M2 - max u(t), [t~-,t']

u(t~)

-- M2.

(3.17)

It is obvious that this maximum wiped out all intermediate input extrema that occurred between t I and t~. As before, consider the global minimum of input at the time interval [t +, t'] and the notation m2 and t 2 will be used for this minimum and the time it was achieved: m2 = min u(t),

u(t2) = m2.

(3.18)

[t~-,t']

It is clear that this minimum wiped out all intermediate input extrema. Continuing the above line of reasoning, we can inductively introduce the global maximum, Mk, and global minimum, mk"

Mk -- max u(t), [tk_l ,t']

u(t-~) = Mk

(3.19)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

313

and

mk = min u(t),

u(t-~) -- mk.

(3.20)

[ t k+ ,t']

Only these input extrema were accumulated by the Preisach model, while all intermediate input extrema were erased. It is natural to say that Mk and mk (k = 1, 2 . . . . ) form an alternating series of dominant maxima and minima. It is evident from the above analysis that a- and//-coordinates of vertices of the interface L(t') are equal to Mk and mk, respectively. It is also clear that the alternating series of dominant extrema is modified with time. This means that new dominant extrema can be introduced by the time-varying input, while the previous ones can be erased. In other words, Mk and mk are functions of t f as it is clearly suggested by their definitions (3.19) and (3.20). Now, the wiping-out property can be stated in the following form. WIPING-OUTPROPERTY. Only the alternating series of dominant input extrema are stored by the Preisach model. All other input extrema are wiped out. It is worth noting that the wiping-out property is somewhat natural and consistent with experimental facts. Indeed, experiments in the area of magnetics show the existence of major hysteresis loops whose shapes do not depend on how these loops are approached. In other words, the major hysteresis loops are well defined. It means that any past history is wiped out by input oscillations of sufficiently large magnitude. This is in complete agreement with the wiping-out property. Consider another characteristic property of the Preisach model that is valid for periodic input variations. Let ul(t) and u2(t) be two inputs that may have different past histories (different alternating series of dominant extrema). However, starting from some instant of time to, these inputs vary back and forth between the same two consecutive extremum values, u+ and u_. It can be shown that these periodic input variations result in minor hysteresis loops. Let Figs 3.17 and 3.18 represent ~-fl diagrams for the inputs ul(t) and u2(t), respectively. As the inputs vary back and forth between u+ and u_, the final links of staircase interfaces Ll(t) and L2(t) move within the identical triangles T1 and T2. This results in periodic shape variations for L1 (t) and L2(t), which in turn produce periodic variations of the outputs, fl (t) and f2(t). This means that some minor hysteresis loops are traced in t h e f - u diagram for both inputs (see Fig. 3.19). The positions of these two loops with respect to the f-axis are different. This is because the above two inputs have different past histories that lead to different shapes for staircase interfaces, Ll(t) and L2(t). As a result, the values of

314

CHAPTER 3

Scalar Preisach Models of Hysteresis

/

FIGURE 3.17

___

U+_

FIGURE 3.18

2

~U U

U+

FIGURE 3.19

315

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

outputs for the same values of inputs are different. This is easily seen from the formula (3.8). However, it can be proved that the above two hysteresis loops are congruent. This means that the coincidence of these loops can be achieved by the appropriate translation of these loops along the f-axis. The proof of the congruency of the above loops is equivalent to showing that any equal increments of inputs ul(t) and u2(t) result in equal increments of outputs fl(t) and f2(t). To this end, let us assume that both inputs after achieving the same minimum value u_ are increased by the same amount: Aul = Au2 = Au. As a result of these increases, the identical triangles AT1 and AT2 are added to the positive sets S~-(t) and S~-(t) and subtracted from the negative sets S l(t) and S2 (t) (see Figs 3.17 and 3.18). Now, using the formula (3.8), we find that the corresponding output increments are given by

afl a f2

-

II #(~, fl) d~ dfl, J,I AT1

(3.21)

2 ]/ g(c~,/~)dor dfl. JJA T2

(3.22)

2

Since AT1 -- AT2, we conclude All = Af2.

(3.23)

The equality (3.23) has been proved for the case when inputs Ul(t) and u2(t) are monotonically increased by the same amount after achieving the same m i n i m u m value u_. Thus, this equality means the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (3.23) holds when the inputs Ul (t) and u2(t) are monotonically decreased by the same amount Au after achieving the maximum value u+. This means that the descending branches of the above minor loops are congruent as well. Thus, we have established the following property of the Preisach model. CONGRUENCYPROPERTY. All minor hysteresis loops corresponding to back-andforth variations of inputs between the same two consecutive extremum values are congruent.

Possible Applications of the Preisach Model as a Neural Network We conclude this section with a remark on possible applications of the Preisach model beyond the area of hysteresis modeling. It is clear from the previous discussion that the Preisach model has the ability to detect

316

CHAPTER 3 Scalar Preisach Models of Hysteresis

and store the alternating series of dominant extrema of input. In other words, the Preisach model is endowed with memory. For this reason, the Preisach model might have appeal as the mathematical model of memory with some interesting properties, and its device realization (see Fig. 3.2) might be utilized as an unusual storage device. We will discuss below only a few peculiarities of this memory. First, the mechanism of memory formation in the Preisach model is quite simple and results from the parallel connection of qualitatively similar elements (two-position cells) ~a/~. Since this model employs very simple elements and has little structure, its memory formation can be interpreted as a 'spontaneously emerged collective effect'. (This kind of terminology is in vogue in the area of neural networks; see, for instance, [25].) Second, storage of information is not localized. Indeed, the model (3.1) stores the information (extremum values of input) not in particular separate cells (as in the case of computer storage devices), but some ensembles of the cells ~/~ participate in storage of each bit of information. As a result, if some of the cells are destroyed, the stored information might still be preserved. This opens the way for cell replenishment without affecting the currently stored information. Third, some subsequent events may erase the information previously stored in the memory (wiping-out property). This erasing occurs only if new events make stronger impacts (characterized by larger input extrema) than previous ones. Fourth, it can be shown that the Preisach model exhibits 'fading' memory. This means that the stored information is slowly and gradually erased in the presence of small random perturbations (intrinsic noise). The properties discussed above are somewhat similar to those being observed (or suspected) for memories in biological systems. However, it would be imprudent to speculate how far this similarity goes. Nevertheless, it may be expected that the mathematical tool (3.1) might find some interesting applications beyond the conventional area of hysteresis modeling. 3.1.3 IDENTIFICATION PROBLEM FOR THE PREISACH MODEL REPRESENTATION THEOREM

Next, we proceed to the discussion of the identification problem for the Preisach model. The essence of this problem is in the determination of weight function #(~, fi). The set of first-order transition curves will be used for this purpose. These curves can be defined as follows. First, the input u(t) should be decreased to the value that is less than rio- This brings a hysteresis nonlinearly to the state of negative saturation. Next, the input is

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

317

/

FIGURE 3.20

I I

"I

I

,

,.u

FIGURE 3.21

monotonically increased until it reaches some value c~'. The corresponding a-fl diagram is shown in Fig. 3.20. As the input is increased, an ascending branch of a major loop is followed (see Fig. 3.21). This branch will also be called the limiting ascending branch because usually there is no branch below it. The notationfa, will be used for the output value on this branch that corresponds to the input value u = a t. Afirst-order transition (reversal) curve is formed as the above monotonic increase of the input is followed by a subsequent monotonic decrease. The term 'first-order' is used to emphasize the fact that each of these curves is formed after the first reversal of input. The notationf~,/~, will be used for the output value on the transition curve attached to the limiting ascending branch at the pointfa,. This output value corresponds to the input value u = fl' (see Fig. 3.21). The above

318

CHAPTER 3 Scalar Preisach Models of Hysteresis (x / o~'

---

T (0~,, 13, )

i

FIGURE 3.22

monotonic decrease of input modifies the previous a-fl diagram shown in Fig. 3.20. A new a-fl diagram for the instant of time when the input reaches the value fl' is illustrated by Fig. 3.22. Now, we define the function 1

(3.24)

=

This function is equal to one-half of the output increments along the firstorder transition curves. The next step is to express this function in terms of the Preisach function #(a, fl). To this end, we compare the a-fl diagrams shown in Figs 3.20 and 3.22. It is clear from these diagrams that the triangle T(a', fl') is added to the negative set S- and subtracted from the positive set S + as a result of the monotonic input decrease from the value u = a' to the value u = fl'. Using the above fact and the formula (3.8), we find that the Preisach model will match the output increments along the first-order transition curves if the function #(~, fl) satisfies the equation

fa, fl, - fa, = -2 f/T(a',fl') #(a, fl) da dfl.

(3.25)

By comparing the formulas (3.24) and (3.25), we obtain

F(a', fl') = [I /~(a, fl)da dfl. .liT (~,,~')

(3.26)

The integral over the triangle T(a ~, fl~) can be written as the following double integral: F(a', fl') =

,

#(a, fl)da

dfl.

(3.27)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

319

By differentiating the last expression twice (first with respect to fiit and then with respect to air), we find

#(ait fiit) = _

a2F(ait, fiit) 0ait0fl it .

(3.28)

Invoking (3.24), the expression (3.28) can be written in another equivalent form: 1r ' (3.29) #(a', fl') - 2 0a'0fl'" The formula (3.29) allows for a simple geometric interpretation of the function #(a', fl'). Indeed, for the first derivative of fa,~, with respect to fl' we have ~f~,/~' (3.30) @' = tan 0(a', fl'), where 0(a it, fiit) is the angle b e t w e e n the axis u and the tangent to the firstorder transition curvefa, fl, at the point u = fiit. From (3.29) and (3.30), we find 1 0 tan 0(a it, fiit)

(3.31)

-

From (3.31), we conclude that the Preisach function #(a', fl') is positive if tan 0(a', fl') is a monotonically increasing function of a' for any fixed fiit. The last condition is satisfied if all first-order transition curves are monotonically increasing functions of fiit, and they do not intersect inside the major loop but merge together at the point where the descending and ascending branches of the major loop meet one another (see Fig. 3.23). 1 To secure the above merge of first-order transition curves, tan 0(a it, fiit) should increase as a function of a it for any fixed fit in order to compensate for larger values off~,/~,. It is worthwhile noting that in the 'magnetic' definition of the Preisach model (see Section 3.1.1) the function #(a, fl) is treated as a distribution function with some statistical connotation. For this reason, this function is tacitly a s s u m e d to be positive. In our treatment, the last property is directly related to experimental facts. For positive #(a, fl), F(a it, fiit) is a monotonically increasing function of a it for any fixed fit and a monotonically decreasing function of fl' for any fixed a it. This follows directly from (3.26).

lit is useful to keep in mind that the descending branch of the major loop can be interpreted as a particular case of the first-order transition curve for which a I = a 0.

CHAPTER 3 Scalar Preisach Models of Hysteresis

320

f

U

i F I G U R E 3.23

If the Preisach function is determined from (3.29) (or (3.28)), then the expression (3.25) is satisfied. This means that the Preisach model matches the output increments along the first-order transition curves. We next show that this also implies that the first-order transition curves themselves are matched. To this end, consider the particular case when fl' - fl0. For this case, we have f=,& = f - . (3.32) From (3.25) and (3.32), we obtain: f/" f - -f~, = - 2 I I

,liT (~',3o)

#(e, fl) da dfl.

(3.33)

Thus, the Preisach model matches the output increment, f - - f ~ , , along the limiting ascending branch. For a t - ~0, we have f~0 - f + . (3.34) From (3.34) and (3.33), we find

f - - f + = - 2 f f T #(o~, fl) dcz dfl,

(3.35)

where T = T(a0, rio) is the limiting triangle. Since major hysteresis loops are usually symmetric, the following equality holds: f+= -f-. (3.36)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

321

From (3.35) and (3.36), we derive:

f - - - ffT #(~' fl) da dfl.

(3.37)

This means that the Preisach model matches the output value in the state of negative saturation. From the last fact and (3.33), we conclude that the model matches the limiting ascending branch. Since the limiting ascending branch and the output increments along first-order transition curves are matched, we deduce that the first-order transition curves themselves are matched. The Preisach function, #(~, fl), has been found by using the first-order transition curves. These curves are attached to the limiting ascending branch, and each of them is formed when a monotonic increase along this branch is followed by a subsequent input decrease. For this reason, these curves can be named first-order decreasing transition curves. However, by almost literally repeating the previous reasoning, a similar expression for #(~, fi) can be found by using the first-order increasing transition curves. These curves are attached to the descending branch of the major loop. Each of these first-order increasing transition curves is formed as a monotonic decrease along the limiting descending branch is followed by a subsequent input increase. The notationf]~,, will be used for the output value on the limiting descending branch. This value is achieved when the input is monotonically decreased from some value above a0 to the value u = fit,. The corresponding a-/J diagram is shown in Fig. 3.24. The notation f~,,e, will be used for the output value on the first-order increasing transition curve that is attached to the limiting descending branch at the point f/~,,. This output value corresponds to u - :r (see Fig. 3.21). The corresponding a-fl diagram is shown in Fig. 3.25. By using the function #(~, fl) d0{ dfl,

=

(3.38)

from Figs 3.24 and 3.25 and the formula (3.8), we obtain F(a", fl") - 1 (/,,,=,, -f,,,)

(3.39)

From (3.38), as before we find cq2F(o~", ]~') _

_

&,,@,,

(3.40)

CHAPTER 3

322

Scalar Preisach Models of Hysteresis

FIGURE 3.24 / s+

T (o~",ly' )

/

FIGURE 3.25

It is clear on the physical grounds (symmetry considerations) that firstorder decreasing and increasing transition curves are congruent. In mathematical terms, this means that if fl"=-~'

and

~"=-fl',

(3.41)

then f//,,- -f~,

and f/~,,~,,- -f~,/~,.

(3.42)

From (3.24), (3.39), and (3.42) we find that if (3.41) holds, then F(a", fl") = F(a', fl').

(3.43)

By substituting (3.41) into (3.43), we obtain

F(-fl', -o() - F(o(, #').

(3.44)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

323

/

,, ,,~,,(.u;~,) X (- u',- ~') " , , , ~

FIGURE 3.26

Next, substituting (3.43) and (3.41) into (3.40), we derive #(-fl" -~') - -

c92F(~', fl') a]/'a~, "

(3.45)

By comparing (3.45) and (3.28), we find # ( - f l ' , - a ' ) = #(a', fl').

(3.46)

The formulas (3.44) and (3.46) express the mirror symmetry of functions F(~, fl) and #(a, fl) with respect to the line ~ = -fl (see Fig. 3.26). This symmetry is a consequence of the congruency of the first-order decreasing and increasing transition curves. If the line a = -fl is the interface between the positive and negative sets, S + (t) and S-(t), then, according to the above symmetry and (3.8), we find that the output is equal to zero:

fit) = 0.

(3.47)

For this reason, the state corresponding to the above interface is called in magnetics 'the demagnetized state'. However, this state cannot be exactly achieved. This is because an interface L(t) is always a staircase line whose links are parallel to the a and//axes. As a result, an actual staircase interface can only approximate the line a = -ft. Such an approximation can be achieved, for instance, if the input u(t) is an oscillating function whose amplitude is slowly decreased to zero starting from some value above a0 (see Fig. 3.27). The corresponding staircase interface is shown in Fig. 3.28. We next proceed to the formulation and the proof of the fundamental theorem which gives the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model.

CHAPTER 3 Scalar Preisach Models of Hysteresis

324

U

I

FIGURE 3.27 fx /

/

FIGURE 3.28

The wiping-out property and the congruency property constitute the necessary and sufficient conditions for a hysteresis nonlinearity to be represented by the Preisach model on the set of piecewise monotonic inputs.

REPRESENTATION THEOREM.

PROOF. Necessity: Let a hysteresis transducer be representable by the Preisach model. Then, this transducer should have the same properties as the model. In particular, it should have the wiping-out and congruency properties. Sufficiency: Consider a hysteresis transducer that has both the wipingout property and the congruency property. It can be recalled that the wiping-out property means that the future values of output do not depend on all past extremum values of input but only on those that form an alternating series of dominant input extrema. The congruency property, on the other hand, suggests that all minor hysteresis loops formed as a result of back-and-forth input variations between the same two consecutive input extrema are congruent. We intend to prove that the above properties

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

325

f

I

FIGURE

7

~u

3.29

imply that the hysteresis transducer can be represented by the Preisach model. The proof of the last statement is constructive. First, it is assumed that the weight function, #(~, fl), is found for the given transducer by matching its first-order transition curves. This can be accomplished by using the formula (3.28). This formula is equivalent to (3.25), which means that the integrals of/~(~, fl) over triangles T(~', fl') are equal to one-half of the output increments, (1/2)Af = (1/2)(f~, -f~,F), along the first-order transition curves. Next, it will be proved that if the above weight function is substituted in (3.1), then the Preisach model and the given transducer will have the same input-output relationships. This statement is true for the first-order transition curves due to the very way the weight function, /~(~, fl), is determined. The induction argument will be used to prove that the same statement holds for higher-order transition curves as well. Let us assume that the above statement is true for transition curves with numbers 1, 2 . . . . . k. Then, for the induction inference, we need to prove that this statement holds for a transition curve number k 4- 1. Let a be a point at which the transition curve number k 4- 1 starts (see Fig. 3.29). The point, a, corresponds to some input value u = a'. According to the induction assumption, the output values of the transducer and the Preisach model coincide at this point. Thus, it remains to be proved that the output increments along the transition curve number k 4-1 are the same for the actual transducer and for the Preisach model. Consider an arbitrary input value u = fl' < ~t. The output increment for the transducer will be equal to the increment off along some curve ab (see Fig. 3.29). Let Figs 3.30 and 3.31 represent a-fl diagrams for the Preisach model at the time instants

CHAPTER 3 Scalar Preisach Models of Hysteresis

326

/ s-

FIGURE 3.30

/ s-

/ FIGURE 3.31

when u = ~ and u - ]/~, respectively. From these diagrams we find that the input decrease from ~' to ]/' results in adding the triangle T(~', ]/') to the negative set, S-, and subtracting the same triangle from the positive set, S +. Using the above fact and the formula (3.8), we find that for the Preisach model the output increment along the transition curve number k + 1 is given by

af

2 [[ #(~, fl) d~ dfl. d.l Z(~',3')

(3.48)

However, according to the way the function, #(a, ]/), is defined, the righthand side of (3.48) is equal to the increment of the transducer output along the first-order transition curve cd (see Fig. 3.29). Thus, it remains to be shown that the output increments along the curves ab and cd are the same. It is here that the wiping-out and congruency properties will be used. The

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

327

proof proceeds as follows. If starting from point b we monotonically increased the input value from fl' back to a', then, according to the wiping-out property, we would arrive at the same point a by moving along some curve ba (see Fig. 3.29). Indeed, the wiping-out property implies that as soon as the input exceeded the value a', the history associated with back-and-forth input variations between ~' and fl' should be erased and the subsequent output variation should follow the transition curve number k. But, this would be possible only if for u = ~' we arrived back at the point a. Similarly, if starting from the point d we monotonically increased the input value from fl' to a', then, according to the same wiping-out property, we would arrive at the point c moving along some curve dc (see Fig. 3.29). Now, by invoking the congruency property, we conclude that the hysteresis loops bab and dcd are congruent. This is true because both loops result from back-and-forth input variations between the same two consecutive input extrema, a' and fl'. From the congruency of the above loops, we find that the output increments along the curves ab and cd are the same. Consequently, the output values of the transducer and the Preisach model coincide along the transition curve number k + 1. The last fact has been proved for any fl'~?' (see Fig. 3.29). However, according to the wipingout property, the transition curve number k 4- 1 should coincide with the transition curve number k - 1 for fl' < 7'. Thus, the case//' < 7' falls in the domain of the induction assumption. In the above discussion, we have considered the transition curve number k + 1 corresponding to the monotonically decreasing input. However, the case when the transition curve number k 4- 1 is formed as a result of monotonically increasing input can be treated similarly. The only difference in the proof will be that the first-order increasing transition curves must be employed instead of the first-order decreasing transition curves used in the above reasoning. This completes the proof of the theorem. [] Discussion

It is easy to see that the essence of the given proof is in the reduction of higher-order transition curves to the first-order transition curves. This reduction rests on both the wiping-out property and congruency property. The proven theorem is important because it clearly establishes the limits of applicability of the Preisach model. These limits are formulated in purely phenomenological terms, without any reference to the actual physical nature of hysteresis. This reveals the physical universality of the Preisach model. The theorem also explicitly indicates the factors which affect the accuracy of the Preisach model. Indeed, deviations from the

328

CHAPTER 3 ScalarPreisach Models of Hysteresis

conditions of the representation theorem may serve as the measure of model accuracy. The above theorem allows for a simple explanation of so-called statistical instability of the Preisach model. This instability has been a very popular topic in the 'magnetic' literature. It usually means that weight functions, #(a,//), determined from different experimental data are not identical. Since these functions have been construed in magnetics as 'particle distributions' (with some statistical connotation), the dependence of /x(a,//) on a particular experimental way of its determination has been termed as 'statistical instability'. The origin of this 'statistical instability' can be easily understood from the following discussion. The formula (3.28) for #(a,//) has been derived from the expression (3.26). This means that the weight function, #(a, fl), can always be determined if the integrals of #(a, fl) over the triangles T(a', fl') are somehow experimentally found. It has been proposed before (see (3.25)) to find these integrals by matching the first-order transition curves. However, this is not the only way it can be done. Indeed, according to the formula (3.48), the same integrals can be found by matching the output increments along any particular transition curves. But, we will end up with the same values for these integrals if and only if higher-order transition curves and the first-order transition curves are congruent. According to the proof of the representation theorem, this congruency takes place if and only if the wiping-out and congruency properties are valid for hysteresis transducers. If these properties are not valid, then, by matching different transition curves, we will end up with different values for the integrals over the triangles T(~',//') and, consequently, with different values of #(~, fl). This is exactly what happened when 'statistical instability' was discovered (see [3.8]). Thus, from the phenomenological point of view, the origin of 'statistical instability' comes from the fact that the wiping-out and congruency properties may not be exactly satisfied for actual hysteresis nonlinearities. Under these circumstances, the Preisach model cannot serve as an absolutely accurate representation for actual hysteresis nonlinearities, but it can still be used as an approximation. The quality of this approximation will depend on the extent to which the wiping-out and congruency properties are satisfied. It has been mentioned before that the weight function, #(~,//), can be found by matching output increments along any transition curves. Thus, the question arises: 'Which transition curves should be used for the above purpose?' If the wiping-out and congruency properties are valid, then it really does not matter which transition curves are used for the determination of/~(~, fl). All of them will theoretically lead to the same result. However, from the practical point of view, the first-order transition curves have some

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

329

clear advantages. First, it is easier to find these curves experimentally than higher-order transition curves. Second, measurements of these curves start from a well-defined state, namely, the state of negative (or positive) saturation. This is not the case for some experimental techniques described in the literature. For instance, it has been suggested in the literature to use a demagnetized state as a starting state in the experimental determination of #(e, fl). But, as was pointed out before, the demagnetized state is not well defined for the Preisach model. This state depends on the particular way it has been prepared. As a result, some errors and discrepancies may be introduced if the demagnetized state is used as the initial state for the experimental determination of/2(e, fl). It is often asserted in the literature that the Preisach model describes reversal curves with zero initial slopes. This statement is based on the following reasoning. Let f~,/~, be a reversal curve which is traced for a monotonically decreasing input. This curve starts at the point f~,, which corresponds to the local input maximum u - e'. The difference f~,fl, -f~, is then given by (3.25) and the current slope of this reversal curve at the point fl = fl' can be found from (3.30). By using (3.25), (3.26), (3.27), and (3.30), it is easy to derive the following expression for the initial slope lim tan O(e', fl') = 2 lim

#(e, if)de.

In the literature, the function #(e, fl) is usually construed as a distribution function for magnetic particles and, for this reason, it is tacitly assumed to be bounded. This assumption and the last formula result in zero initial slopes for the reversal curves f~,fl,. This property is often regarded as an intrinsic property of the Preisach model. However, this is not true. Indeed, if the phenomenological approach is adopted, then there is no need to interpret/2 as a distribution for magnetic particles and the assumption that /2 is bounded can be removed. The freedom of choice of function #(e, fl) should be used in order to match as many experimental facts as possible. It is easy to show that experimentally observed nonzero initial slopes of reversal curves f~,~, can be matched by the Preisach model if we allow for delta-type (Dirac function) singularities of #(e, fl) along the line e = ft. It is easy to see that these singularities can be interpreted as an additional (and fully reversible) term in the Preisach model consisting of degenerate (zero width) rectangular loop (or step) operators ~,~:

fit) =

fl(e, fl)~flu(t) de dfl +

k(e)~au(t) de,

where function fl(e, fl) is now free of singularities.

CHAPTER 3 Scalar Preisach Models of Hysteresis

330

f

ms I

I fo~13

/

'/Li I I I

I I I

w

v

1~

t~ 2

~u

-m/ r

w F I G U R E 3.32

This form of the Preisach model leads to the following modification of formulas (3.25)-(3.27)-

f~'B' -f~'

= -2

,

#(a, fl) da

dfl - 2

, k(a) da.

From the last formula we derive: ~t

tan O(e', fl') =

c~fl'

= 2

ffl

, #(a, fl')da + 2k(fl'),

and lim tan 0(a', fl') = 2k(a'). Thus, the actual nonzero initial slopes of the first-order reversal curves can be matched by the appropriate choice of function k(a). This section will be concluded by the discussion of a simple example that illustrates the application of the formula (3.29). Consider a hysteresis nonlinearity with a local memory shown in Fig. 3.32. This hysteresis nonlinearity has a set of inner curves that are parallel to the u-axis. The above hysteresis nonlinearity is completely characterized by the following parameters: ms, ~Xl, 0~2, W. O u r goal is to find the Preisach representation

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

331

(3.1) for this nonlinearity. To this end, we have to find the weight function #(~, fl). This requires the specification of first-order transition curves f~/~ and the subsequent application of the formula (3.29). It is clear from Fig. 3.32 that for any a such that 0r < 0r < OO the corresponding first-order transition curve f~/~ consists of two parts: (a) a particular inner curve, and (b) a part of the limiting descending branch if al - w ~ ] ~ 0 ~ 2 - w , tiff- w ~ 0 r K oo. It is apparent thatf~/~ varies only along the second part. From the above remarks, we find 2ms

f~ - -ms + ~ ( f l -

o~1 +

W),

(3.49)

0r -- 0~1

0r1 -- W ~ f l ~ O ~ 2 -- W,

~

+ w~cz~oo,

(3.50)

and f~/~ -- const

(3.51)

for all other possible values of ft. It is worthwhile to keep in mind that, depending on a particular value of fl, the constant in (3.51) may assume one of the following two values: - m s , - m s + ((2ms)/(~z2 - 0r162 - 0r From (3.49), (3.50), and (3.51), we derive: c~f~

2ms

=

~

,

(3.52)

0~2 -- 0~1

if fl + w ~ a ~ o o ,

0~1 -- W ~ f l ~ O ~ 2 -- W,

(3.53)

and 8f~/~ = 0

(3.54)

for all other values of a and ft. The last three formulas can be written in the following equivalent form: c~f~

2ms

=

~fl

0r

,

if (a, fl) ~ f~,

(3.55)

- - 0r

and ~f~/~ = 0,

M

if (~, fl) ~ f~,

(3.56)

where f] is the region defined by the inequalities (3.53) (see also Fig. 3.33).

CHAPTER 3 Scalar Preisach Models of Hysteresis

332

0~ /

/

//

/

/ /

I !

I i

.p

3.33

FIGURE

From (3.55) and (3.56), we obtain:

c92f ~ 630~6~~

2ms

~&(a-

O~2 -- O~1

f l - w),

(3.57)

(3.58)

~1 ~ 0 r

and (32f~/~)/ (cgacgfl)is equal to zero otherwise. From (3.29), (3.57), and (3.58), we find #(~, fl) = m------L--s&(~- f l - w) (3.59) 0r -- 0r

for (3.60)

~1 ~ ~ 2

and #(a, fl) is equal to zero otherwise. Thus, the weight function #(a, fl) has the support on the line a - fl- w = 0. This support is the bold segment (see Fig. 3.33) of the above line. By substituting (3.59) and (3.60) into (3.1), after simple transformations we obtain fit)

-

ms 0~2 -- ~1

~~2 1

~,~_wU(t) d~.

(3.61)

This is the Preisach representation of the hysteresis nonlinearity shown in Fig. 3.32. It is left to the reader as a useful exercise to verify that the formula (3.61) indeed describes all branches of the above hysteresis nonlinearity.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

333

,' , [ -

U m

"',2 x4 I I I I I I

Urn

U

I I I I I t

FIGURE 3.34

It is clear from Fig. 3.33 that by varying 0r 0r and w we can freely move around the support of the function # in the half-plane a~fl. This fact and the expression (3.61) suggest that the equivalent representation for the Preisach model can be obtained as a superposition (parallel connection) of an infinite number of hysteresis nonlinearities shown in Fig. 3.32. However, the representation (3.1) is preferable due to the most elementary nature of hysteresis operators ~a/~. It is not accidental that in the above example the function #(a, fl) has a 'line support'. This is related to the fact that the above hysteresis nonlinearity has a local memory. It can even be proved in general that if we have a hysteresis nonlinearity with a local memory that can be represented by the Preisach model, then the weight function #(~, fl) has a support along some particular curve. This curve may intersect all possible staircase interfaces L(t) only once. It was emphasized before that the Preisach model can describe (and usually does describe) hysteresis with nonlocal memories. These nonlinearities are much more complex than those with local memories. However, it would be a mistake to think that any hysteresis nonlinearity with local memory can be represented by the Preisach model. In fact, there are very simple hysteresis nonlinearities with local memories that cannot be described by the model. For instance, this is true for the hysteresis nonlinearity shown in Fig. 3.34. Indeed, let the input vary back and forth between two consecutive extremum values a' and fl',--Um < fi' < ~' < Um. Then, depending on the past history, the bold reversible parts of the descending and ascending branches will be traced back and forth. These parts can

334

CHAPTER 3 ScalarPreisach Models of Hysteresis

be construed as degenerate minor loops. Since these parts are not congruent; the congruency property of the representation theorem is not satisfied. Consequently, the hysteresis nonlinearity shown in Fig. 3.34 cannot be represented by the Preisach model. However, this hysteresis nonlinearity is typical for single Stoner-Wohlfarth magnetic particles. It is known from the theory of these particles that the reversible parts of their hysteresis loops result from uniform rotations of magnetization within these particles, that is, from reversible processes. Incongruence of the above parts is the reason why the Preisach model cannot describe the hysteresis loops of single Stoner-Wohlfarth particles. For this reason, it is often asserted that the Preisach model does not fully account for reversible processes. It will be shown in Section 3.2 that this deficiency of the Preisach model can be removed by its appropriate generalization.

3.1.4 NUMERICAL IMPLEMENTATION OF THE PREISACH MODEL

The Preisach model can be numerically implemented by using the formula (3.8) for the computation of the output, fit), and the formula (3.29) for the determination of the weight function, #(~, fl). Although the above approach is straightforward, it encounters two main difficulties. First, it requires the numerical evaluation of double integrals in (3.8). This is a time-consuming procedure that may impede the use of the Preisach model in practical applications. Second, the determination of the weight function #(a, fl) by employing the formula (3.29) requires differentiations of experimentally obtained data. These differentiations may strongly amplify errors (noise) inherently present in any experimental data. It turns out that another approach can be developed for the numerical implementation of the Preisach model. This approach completely circumvents the above two difficulties. It is based on the explicit formula for the integrals in (3.8). This formula directly involves (without any differentiation) the experimental data used for the identification of #(~, fl). Moreover, the above formula will be a valuable tool for the theoretical investigation of the Preisach model, and it is also useful in the discussion of vector models of hysteresis. The starting point for the derivation of the explicit formula for fit) is the expression (3.8). It is worthwhile to remind here that the positive (S + (t)) and negative (S-(t)) sets in (3.8) are separated by the staircase interface L(t). This interface has vertices whose a and fl coordinates are equal to Mk and mk, respectively (see Fig. 3.35). As discussed in Section 3.1.2, numbers Mk and mk form the alternating series of dominant input extrema.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

335

/

~-~1 (Mk, mk_~)

(t)

FIGURE 3.35

By adding and subtracting the integral of/~(a, fl) over S + (t), the expression (3.8) can be represented in the form:

f(t) - - ffz#(O~, fl) do~dfl + 2 fL+(t) #(~, fl) da dfl,

(3.62)

where, as before, T is the limiting triangle. According to (3.26), we find:

ffTlz(~z, fl) dcz dfl - F(ao, rio).

(3.63)

The positive set, S + (t), can be subdivided into n trapezoids Qk (see Fig. 3.35). As a result, we have n(t)

fL

+(t)

p(a, fl)dcz dfl - ~

k=l

fro k(t)

fl)da dfl.

(3.64)

It is clear that the number, n, of these trapezoids and their shapes may change with time. For this reason, n and Qk are shown in (3.64) as functions of time. Each trapezoid Qk can be represented as a difference of two triangles, T(Mk, mk-1) and T(Mk, mk). Thus, we obtain

ffQ k(t)

fi) dfl = ff r(Mk,mk-1) p(a, fl) d~ dfl ff - ]/

JJT (Mk,mk)

/z(~, fi)da dfl,

where, for the case of k - 1, m0 in (3.65) is naturally equal to rio.

(3.65)

CHAPTER 3

336

Scalar Preisach Models of Hysteresis

According to (3.26), we find:

ffT

(Mk,mk-1)

fl) da dfl = F(Mk, mk-1),

(3.66)

and

fIT

[1(~, fl) dc~ dfl - F(Mk, ink). (Mk,mk) From (3.65), (3.66), and (3.67), we derive:

/Q

fl(a, [1) da dfl = F(Mk, mk-1) -- F(Mk, mk).

(3.67)

(3.68)

k(t)

From (3.62), (3.63), (3.64), and (3.68), we obtain n(t)

f(t) = -F(~o, rio) 4- 2 ~ [ F ( M k , mk-1) -- F(Mk, mk)].

(3.69)

k=l

It is clear from Fig. 3.35 that mn is equal to the current value of input

mn = u(t).

(3.70)

Consequently, the expression (3.69) can be written as n(t)-I

fit) = -F(~o, rio) 4- 2 ~

[F(Mk, ink-l) -- F(Mk, mk)]

k=l

4- 2[F(Mn, ran-l) - F(Mn, u(t))]. (3.71) The last expression has been derived for monotonically decreasing input, that is, when the final link of interface L(t) is a vertical one. If the input u(t) is being monotonically increased, then the final link of L(t) is a horizontal one and the a-fl diagram shown in Fig. 3.35 should be slightly modified. The appropriate diagram is shown in Fig. 3.36. This diagram can be considered as a particular case of the previous one. This case is realized when mn(t) = Mn(t) -- u(t). (3.72)

According to the definition (3.26) of F(a, fl), we find

F(Mn, mn) - F(u(t), u(t)) - O.

(3.73)

From (3.69), (3.72), and (3.73), we derive the following expression for fit) in the case of monotonically increasing input: n(t)-I

fit) - -F(~o, rio) 4- 2 ~_, [F(Mk, mk-1) -- F(Mk, mk)] 4- 2F(u(t), mn-1). k=l

(3.74)

3.1 THE CLASSICAL P R E I S A C H M O D E L OF HYSTERESIS

337

/

]il F I G U R E 3.36

The function F(a, fl) is related to experimentally measured first-order transition curves by the formula (3.24). Using this formula, expressions (3.71) and (3.74) can be written in terms of the above experimental data as follows: n-1

fit) - -f +

+ ~(fMkm k --&kmk_l ) "J-&nu(t) --&nmn-1,

(3.75)

k=l n-1

fit)-

-f +

+ ~--~(fMkmk --&kmk_X ) -4-f--mn_l -- f--mn_x,--u(t)"

(3.76)

k=l

Here, f+ is, as before, the positive saturation value of output, and the last term in (3.74) has been transformed using the formulas (3.24) and (3.44). Thus, we have derived the explicit expressions (3.75) and (3.76) for f i t ) in terms of experimentally measured data. These expressions constitute the basis for the numerical implementation of the Preisach model. Using the above expressions, the digital code that implements the Preisach model has been developed. This code determines the output of a hysteresis transducer from an input, a set of first-order transition curves and an input history that are all specified by the user. The code also allows the user the flexibility of specifying an analytical expression for the weight function,/~(~, fl), instead of first-order transition curves. In this case, the function F(~, fl), defined by (3.26), is precomputed and then used in the formulas (3.71) and (3.74) for the evaluation of output values. As usual, the code performs three main functions: preprocessing, processing, and postprocessing. During the preprocessing stage, a square mesh covering the limiting triangle T (see Fig. 3.37) is created and a discretized set of first-order transition curves is entered. This set consists of mesh values of the functionf~fl. In the case when an analytical expression

338

CHAPTER 3 ScalarPreisach Models of Hysteresis

-§247247 _#-,-§

FIGURE 3.37

for #(a, fl) is specified, the mesh values of F(a, fl) are computed during the preprocessing. At the processing stage, an input history and current values of input are entered. Using these data and the formulas (3.19) and (3.20), the alternating series of dominant input extrema {Mk, ink} is first determined and then continuously updated for each new instant of time. Using Mk, mk, and mesh values off~]~ (or F(~fl)), all terms in the formulas (3.75) and (3.76) (or (3.71) and (3.74)) are computed. This is done by first determining particular square (or triangular) cells to which points (Mk, mk), (Mk, mk-1), and (Mn, u(t)) belong, and then computing the values off~/~ (or F(a, fl)) at these points by means of interpolation of the mesh values off~]~ (or F(a, fl)) at the vertices of the above cells. In the code, the interpolation is based on the following local approximation to fa~ in each square mesh cell: f~fl "~" CO if- C10~ q-- C2fl -}- C30~fl. (3.77) The C-coefficients in (3.77) are found by matching the values of f~/~ in the cell vertices. For triangular cells that are adjacent to the line a = fl, the linear local approximations to f~/~ are used: fail "" CO q- Clo~ q- C2~.

(3.78)

After computing all necessary values of f~/~, the current values of output are evaluated by employing the formulas (3.75) and (3.76). At the postprocessing stage, the computed output values are stored in a special file along with the corresponding input values and the values of Mk and ink. The user has the option to plot the output and input as functions of time, the path traced on t h e f - u plane and the set of first-order transition curves used in computations.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS .....i ' ' ' ' I

....

1 ' ' ' ' i ' ' ' ' 1

....

I ' ' ' ' 1

....

339

I ' ' ' ' I ' ' ' ' [ ' ' ' ' l

-

.04

.02

- -

-

J 4.J

:

0

o=

-.02

-

-.04

-

-2500

-2000

- 1500

- 1000

-500

0

500

1000

1500

2000

2500

Input

FIGURE 3.38

Below, we present some examples computed by using the above code. Figures 3.38, 3.39, and 3.40 show the set of first-order transition curves, the initial staircase interface L(t), and the input u(t), respectively. These data have been used to compute the outputflt). The computed outputflt) and t h e f - u path of the transducer are given in Figs 3.41 and 3.42, respectively. As was pointed out before, the derived formulas (3.75) and (3.76) (or (3.71) and (3.74)) are useful not only for the numerical implementation of the Preisach model but for its theoretical investigation as well. For instance, by using (3.75) and (3.76), the following proposition can easily be proven. PROPOSITION. A n y f -u path of hysteresis transducer representable by the Preisach model is piecewise congruent to first-order transition curves. In other words, any f - u path of such a transducer consists of pieces, which are congruent to some particular first-order transition curves. PROOF. Suppose that the input u(t) is being monotonically decreased from its previous maximum value u = Mn until it reaches the value u = mn-1. Then, the formula (3.75) is valid. In this formula, all terms exceptfMmu(t) do not vary. Consequently, the f - u path traced during the above input variation is congruent to the first-order transition curvefMnu that is attached to the limiting ascending branch at the point fMn" As the input reaches

CHAPTER 3

340

,

w

I

~

~'

~ ...... '

"i' ''~

~

'

'

Scalar Preisach Models of Hysteresis

I

~

'~

~

'~

J

i.

J

l

,

"r

,

~ ............ I '

i

I

2000

1000

I

/

- 1000 -2000 l.

I

i ..... i

,

,

-2000

1

,

~

J

,

I

- 1000

I

0

1

~

I

I...,

1000

I

,

,

2000

F I G U R E 3.39

i-,

2000

.... ,

.., .... i ..... i . ,

~

,

,

i

,

,

.v

,

I.

,.

,

,

-T

i..~.,.~

...... ,

~

9i

!

1000

-1000

-2000

0

100

200

300 Time

FIGURE 3.40

400

500

,

341

3.1 T H E C L A S S I C A L P R E I S A C H M O D E L OF HYSTERESIS

.04

.02

o

-.02

-.04

0

lO0

200

300

400

500

Time

F I G U R E 3.41

'

'

I

. . . .

I

'

'

'

'

I

'

'

'

i

I

'

'

'

'

1

.04

.02

.a

0

o

-.02

-.04

-2000

- 1000

0 Input

F I G U R E 3.42

1000

2000

'

'

CHAPTER 3 ScalarPreisach Models of Hysteresis

342

the value mn-1, the cancellation of the last two terms in (3.75) occurs. If the input is being further decreased, remaining between u = ran-1 and u -~ ran-2, then the formula (3.75) is modified as follows n-2

fit) - - f + + ~_, (fMkmk -- fMkmk_l) -}- fMn-lU(t) -- fMn-lmn-2 9 k-1

(3.79)

From this formula, as before, we conclude that the f - u path traced during the above input variation is congruent to the first-order transition curve fM,_lU that is attached to the limiting ascending branch at the point f = fMn-l" By continuing the same line of reasoning, we find that if the input is being monotonically decreased between u = mk and u = mk-1, then the corresponding f - u path is congruent to the first-order transition curve fMkU. For the case when the input is monotonically increased, the formula (3.76) is valid. From this formula, we find that if the input is monotonically increased between the values u = Mk and u -- Mk-1, then the time-varying part of the output is described by the t e r m f_mk_l_u(t). Thus, the corresponding f - u path is congruent to the first-order transition curvef-mk_l.-u that is attached to the limiting ascending branch at the point f = f-mk-l" This completes the proof. [] It is evident from the proven theorem that initial slopes of reversal curves described by the Preisach model coincide with initial slopes of experimentally measured first-order transition curves and, consequently, in general they are not equal to zero. This is consistent with our discussion of initial slopes of reversal curves presented in the previous section. Another important application of the formulas (3.75) and (3.76) is the derivation of the expressions for linear extrapolations of output corresponding to small input increments. Such extrapolative expressions can be useful for numerical analysis of systems with hysteresis by time-marching techniques. The derivation of these expressions proceeds as follows. Consider the case when the input has been monotonically increased for t t~t and this monotonic increase continues just after the time t. Then, the formula (3.76) is valid for the output before and after the time t. From this formula, by using the local Taylor expansion at time t for t t > t, we find f ( t ' ) - f i t ) "" -

ff-mn-l ~

[u(t') - u(t)].

(3.80)

fl=-u(t)

It is convenient to represent the last formula in terms of input and output increments: A f -- --

(~f--mn-lfl ] AU. Off fl=-u(t)

(3.81)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

343

It is important to keep in m i n d that mn-1 in (3.81) m a y change with time. This occurs because of the wiping out property. It is easy to see that, at any instant of time, mn-1 in (3.81) is the largest mk in the series of Mk and mk. If a monotonic increase of input for t ~Kt is followed by a monotonic input decrease for t t > t, then the input value u(t) can be treated as the last m a x i m u m , Mn, and from (3.75), we find: n-1

fit) -- --f+ -t- ~-~(fMkmk -- &kmk+l) q- fu(t)u(t) -- fu(t)mn-1

(3.82)

k=l

and

n-1 f (t') = - f + + ~_, (fMkmk -- &kink-1 ) q- fu(t)u(t') -- fu(t)mn-l"

(3.83)

k=l

From (3.82) and (3.83), we obtain f(t')

-

fit) --fu(t)u(t,)

--

fu(t)u(t).

(3.84)

Using the last expression, we derive

Af "~" Cgfu(t)fl Au. ~fl fl=u(t)

(3.85)

Now, we consider the case w h e n the input has been undergoing a monotonic decrease for t'~ t. If this decrease continues for t' > t, then the formula (3.75) is valid. From this formula as before we find

'

Af ~"

(~fMnfl I Au. ~fl fl=u(t)

(3.86)

In the last formula, Mn m a y change with time. It is clear that at any instant of time Mn in (3.81) is the smallest Mk in the alternating series of Mk and

mk. If the monotonic decrease of input for t IK t is followed by a subsequent monotonic input increase for t I > t, then the input value u(t) can be considered as mn-1 and the formula (3.76) can be used for t' > t. From this formula, as before we derive

Af "~ -

~f-u(t)fl I AU. ~fl fl=u(t)

(3.87)

Thus, for the previously increasing input, the output can be linearly extrapolated by using the formulas (3.81) and (3.85) for Au > 0 and Au < 0,

CHAPTER 3 Scalar Preisach Models of Hysteresis

344

respectively. For the previously decreasing input, the output can be linearly extrapolated by using the formulas (3.86) and (3.87) for Au < 0 and Au > 0, respectively. It is important to note that the above formulas are given in terms of slopes of experimentally measured first-order transition curves. The formulas (3.81), (3.85), (3.86), and (3.87) give local linearizations of output increments at time t. In other words, these linearizations are fairly accurate only for small time difference t - t'. However, by employing the formulas (3.75) and (3.76), it is possible to find global linearizations of output increments which are valid for finite (not necessary small) time intervals. The simplest way to do this is to use a special set of continuous inputs u(t). Consider an arbitrary time interval [to, tl]. We define the set K[to,t'] of inputs that are continuous on this interval and have the properties: (a) any two inputs from the set assume their maximum and minimum values at the same time instances, (b) maxima and minima that constitute alternating series of dominant extrema are also assumed at the same times. For two inputs u(t) and ~(t), the last condition can be expressed as

u(t-~) = Mk,

~(t~-) =/VIk,

(3.88)

u(t~) -- mk,

~(t~) = Tnk,

(3.89)

where the same notation as in Section 3.1.2 is used above. It is obvious that K[to,t'] is a convex set of functions. Indeed, if the above two properties are satisfied for inputs u(t) and ~(t), they are also satisfied for the input 2u(t) + (1 - 2)~(t) where 0~2K1. It is also clear that if u(t) belongs to K[to,t,], then 2u(t) also belongs to K[to,t'] for any 2~0 and it does not belong to K[to,t,] for 2 < 0. This means that the set K[to,t,] is a cone in the Banach space of continuous functions. It is clear that the set of inputs K[to,t'] depends on the alternating sequence of t k+ and tk- By specifying different times t k+ and t~-, we end up with different sets of inputs. For this reason, it would be more precise to adopt the n o t a t i o n K[to,t-~,t-~ ..... t+tn,t,]. But for the sake of conciseness, we preserve the previous notation. From the above definition of K[to,t'], it follows that any two inputs from K[to,t'] are monotonically decreased (or increased) on the same time intervals. This means that at any instant of time the same formula (3.75) (or (3.76)) can be applied to the computations of outputs. Thus, assuming that at time t both inputs, u(t) and ~(t), are monotonically decreased, we can use (3.75) for u(t) and the similar expression for ~(t): n-1

(3.90) k=l

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

345

By subtracting (3.75) from (3.90) and retaining only linear terms of Taylor expansions, we derive

n-1 ( ~fafl

bMk q- (~fafl

- - k=l ~(~]M

k'mk-1(~Mkq-~lMk,m k 1~mk-1)

8f(t)=~\ k-1 8~ Mk,mk

8f~

(~mkt

- ~ Mk,mk

(~Mn + 8f~

-'~-~ Mnu(t)

bu(t),

(3.91)

- ~ Mnu(t)

where the following notations are used:

6f(t) =d?(t) - fit),

6u(t) - ~(t) - u(t),

6 M k - Mk - M k ,

6mk -- ~nk -- mk.

(3.92) (3.93)

If both inputs are monotonically increased at time t, then the formula (3.76) is applied and, as before, we derive n-1

( Sf~fl

~Mk + 8f~

af(t)= k=l ~_.,--~ Mkmk

~mk

--~ Mkmk

8f~fl ~fafl 80~ IMkmk_1bMk ----~-IMkmk_l (~mk-l)

(Sfa

~ --mn-1

Ofa~

Of~fl

~(~mn-1

~0~ --mn-1,--u(t)Y 8u(t).

if- - ~ -mn-1,-u(t)

(3.94)

The expressions (3.91) and (3.94) represent a global linearization of output increment on the interval [to, t']. These expressions may be useful in the analysis of periodic regimes in systems with hysteresis by perturbation techniques. We conclude this section by discussing another example that demonstrates the usefulness of the above explicit formulas for the output f(t). Consider an input u(t) that has an infinite number of oscillations as time progresses to infinity. We assume that the amplitude of these oscillations (around some value u(cx~)) monotonically diminishes to zero. In other terms, this means that l i m M k = lim m k = u ( r k--,~ k~c~

(3.95)

346

CHAPTER 3 Scalar Preisach Models of Hysteresis

The described input variations are typical for anhysteretic magnetization processes. We are going to prove that there exists the limit for the output flt) as time goes to infinity. By using (3.75), we find oo

lim fit) = -F(ao, rio) + 2 ~ [ F ( M k , mk-1) -- F(Mk, mk)].

(3.96)

t--~ o o

k=l

Thus, it remains to be proven that the infinite series in (3.96) converges. To this end, we introduce the following notations: F(Mk, mk-1) = a 2 k ,

F(Mk, mk) -- a2k+l

(3.97)

and rewrite (3.96) as follows: oo

lim fit) = -F(ao, rio) + 2 y~(--1)kak .

(3.98)

t---~ o o

k=2

Since usually p(a, fl)>~0, from (3.26) and (3.97) we find that ak > 0. Moreover, since mk-1 < mk and Mk > Mk+l, (3.99) we conclude that a2k -~ F(Mk, ink-l) > F(Mk, ink) -- a2k+l,

(3.100)

a2k+l -- F(Mk, mk) > F(Mk+I, ink) = a2k+2.

(3.101)

In addition, from (3.95) and (3.26) we have lim a2k -- lim a2k+l -- F(~, ~) = 0.

k--+oo

k--,oo

Thus, the expression (3.98) is the alternating series with monotonically decreasing to zero coefficients. According to the well-known theorem, this series converges. This proves that there exists the limit off(t) for t ~ ~ . 3.1.5 THE PREISACH MODEL AND HYSTERETIC ENERGY LOSSES A hysteresis phenomenon is associated with some energy dissipation, which is often referred to as hysteretic energy losses. The problem of determining hysteretic energy losses is a classical one. It has been attracting considerable attention because the hysteresis energy loss is an important component of 'core losses' occurring in almost all electromagnetic power devices as well as in many high frequency microwave devices. For this reason, the

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

347

~U

FIGURE 3.43

means for accurate predictions of hysteresis losses and their reduction are important for optimal design of various equipment. The solution to the above problem has long been known for the particular case of periodic (cyclic) input variations. In magnetics, this solution is most often associated with the name of C.P. Steinmetz. This solution shows that a hysteretic energy loss per cycle is equal to an area enclosed by a loop resulting from periodic input variations (see Fig. 3.43). However, energy dissipation occurs for arbitrary (not necessary periodic) variations of input. The problem of computing hysteretic energy losses for arbitrary input variations has remained unsolved. A solution to this problem would be of both theoretical and practical importance. From the theoretical point of view, the solution to the above problem will allow for the calculation of internal entropy production that is a key point in the development of irreversible thermodynamics of hysteretic media. In magnetics, it may also allow one to separate a dissipated energy from an energy stored in a magnetic field. This eventually may lead to expressions for electromagnetic forces in hysteretic media. From the practical viewpoint, the solution to the above problem may bring new experimental techniques for the measurement of hysteretic energy losses occurring for arbitrary input variations. It should not be surprising that the expression for hysteretic energy losses has been found only for the case of periodic input variations. The reason behind this fact is that the hysteretic energy losses occurring for periodic input variations can be easily evaluated by using only the energy conservation principle; no knowledge of actual mechanisms of hysteresis

348

CHAPTER 3 ScalarPreisach Models of Hysteresis

or its model is required. The situation is much more complicated when arbitrary input variations are considered. Here, the energy conservation principle alone is not sufficient, and an adequate model of hysteresis should be employed in order to arrive at the solution to the problem. It turns out that the Preisach hysteresis model is very well suited for this purpose. In this section, the Preisach model will be used for the derivation of general expressions for hysteretic energy losses. These expressions will be given in terms of the weight function, #(c~, fl), as well as in terms of experimentally measured first-order transition curves. Furthermore, a formula that relates the hysteretic energy losses occurring for arbitrary input variations to the losses occurring for certain periodic input variations will be found. This formula may result in simple techniques for the measurement of hysteretic losses occurring for arbitrary input variations. The application of the mentioned results to the irreversible thermodynamics of hysteretic media will be discussed as well. We begin by defining the input, u(t), and the output, fit), as work variables. This means that the infinitesimal energy supplied to the transducer (media) in the form of work is given by 6W = u dr.

(3.102)

In magnetics, u is the magnetic field H, f is the magnetization M, and the formula (3.102) becomes the classical expression for the work done in magnetizing a unit volume of magnetic media: 6W = H dM.

(3.103)

Similarly, in mechanics, u is the force F, f is the specific length L, and from (3.102) we find the standard formula 6W = F dL.

(3.104)

Now, we proceed to the derivation of expressions for hysteretic energy losses. We first consider the case when a hysteresis nonlinearity is represented by a rectangular loop shown in Fig. 3.44. If a periodic variation of input is such that the whole loop is traced, then the hysteretic energy loss for one cycle, Qcycle, is equal to the area enclosed by the loop Qcycle = 2 ( a - fl).

(3.105)

It is clear that the horizontal links of the loop are fully reversible and, for this reason, no energy losses occur as these links are traced. Thus, it can be concluded that only' switching-up' a n d ' switching-down' result in energy losses. It can be assumed (on physical grounds) that there is symmetry

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

-}-11

349

--

~U

-I

T. 1

FIGURE 3.44

between the above two switchings. In other words, these switchings are identical as far as energy losses are concerned. Consequently, the same energy loss occurs for each of these switchings. As a result, we conclude that the energy loss per switching, q, is given by q = ~ - ft.

(3.106)

The product # ( a , / / ) ~ can be construed as a rectangular hysteresis loop with output values equal to 4-#(a, fl). For this reason, switchings of such loops will result in energy losses equal to #(a, fl)(a - fl). In the Preisach model, any input variation is associated with switchings of some rectangular loops #(~, fl)~fl. These switchings represent irreversible processes occurring during input variations. Consequently, it is natural to equate the hysteretic energy loss occurring for some input variation to the sum of energy losses resulting from the switching of rectangular loops during this input variation. Since in the Preisach model we are dealing with continuous ensembles of rectangular loops, the above summation should be replaced by integration. Thus, if f2 denotes the region of points on ~fl diagram for which rectangular loops were switched during some input variation, then the hysteretic energy loss, Q, for this input variation is given by Q - ~.to #(~' fl)(~ - fl) d~ dfl.

(3.107)

This is the fundamental formula for hysteretic energy losses, and all subsequent results will follow from this expression. It is clear from the above

CHAPTER 3

350

Scalar Preisach Models of Hysteresis

/

tS FIGURE 3.45

reasoning that the derivation of the formula (3.107) rests on the following two facts: (a) for rectangular hysteresis loops hysteretic losses can be evaluated for arbitrary input variations, (b) the Preisach model represents complicated hysteresis nonlinearities as superpositions of rectangular loops. The above two facts make the Preisach model a very convenient tool for the solution of the problem at hand. However, it has to be kept in mind that the formula (3.107) cannot be applied to any hysteresis nonlinearity. It has certain limits of applicability that are the same as for the Preisach model itself. A typical shape of the region f~ is shown in Fig. 3.45. It is clear from this figure that f2 can be always subdivided into a triangle and some trapezoids. The trapezoids, in turn, can be represented as differences of triangles. Thus, if the integral in (3.107) can be evaluated for any triangular region, then it will be easy to determine this integral for any possible shape of f2. For this reason, it makes sense to compute the values of the above integrals over various triangles. By using these values, hysteresis losses can be easily found for any input variations. In the case when f2 is a triangle, the integral (3.107) can be evaluated in terms of first-order transition curves. The derivation proceeds as follows. Consider the function (a - fl)F(a, fl). By differentiating this function, we find (92 02F(a, 3a3]/[(a - ]/)F(a, fl)] = 0F(a, fl) _ 3F(a, ]/) + ( a - ]/)]/-------~)

(3.108)

By using (3.28) and (3.108), we derive:

~F(a, ~) #(a, ]/)(~- ]1) --

~fl

~F(a,fl) -

~

~2 8e~fl[(a- ]/)F(a,fl)].

(3.109)

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

351

/

i

/~/ T(u§ ] FIGURE 3.46

Let T(u+, u_) be a triangle (see Fig. 3.46) swept during the monotonic increase of input from u_ to u+. According to (3.107), the above input variation results in the hysteretic loss Q ( u _ , u+) that is given by ?f Q (u _ , u + ) - ] ] la ( oe, fl ) ( oe - fl ) d oed fl. .]J T (u+,u_)

(3.110)

Substituting the expression (3.109) into (3.110), we find: c')F ( oe, fl ) doe d fl Q ( u _ , u+)

-

ff:r(u+,u_) -

lfT

(u+,u_ )

0F(oe, fl) doe dfl cgoe

02 [(oe - fl)F(oe, fl)] doe dfl. - ffT (u+,u_) cgoeOfl

(3.111)

The first integral in (3.111) can be evaluated as follows:

aF(~, fl) d~ dfl

dfl d~

-

M

_

-

f(e, ~) d~ -

1 u+

- since F(oe, a) = O.

_

F(oe, u_) de

/uu+

fuu+F(oe, u_) doe,

(3.112)

CHAPTER 3 Scalar Preisach Models of Hysteresis

352

Similarly, for the second integral in (3.111) we obtain U+

OF(a, fl) da dfl 0~

/T(u+,u_ )

F(u+, fl) dfl.

(3.113)

Finally, for the third integral in (3.111) we derive

//T(u+,u_) 3~fl ~2 [(a - fl)F(a, fl)] d~dfl u+

(~2

-- ~u_ ( f u+ O~-Ofl[(~x- fl)F(~' fl)] d0r -

luU+

[(u+ - fl)F(u+, fl)] dfl - - ( u + - u_)F(u+, u_).

(3.114)

Substituting (3.112), (3.113), and (3.114) into (3.111), we find

Q(u_, u+) = (u+ - u_)F(u+, u_) -

luu+

f(u+,

fl) dfl -

/uu+f(~,

u_) d~.

(3.115)

The last formula has two main advantages over (3.110). First, its application requires the evaluation of one-dimensional integrals. Second, this formula expresses the losses directly in terms of experimentally measured firstorder transition curves. By using the expression (3.115), the hysteretic energy losses can be evaluated for arbitrary input variations. Consider some input variation for which the region f2 has the shape shown in Fig. 3.45. Then, the corresponding energy loss is given by (3.107) that can be written as follows: k

Q - Q(Mn, mn) q- ~,[Q(Mn-i, mn-i) 1 Q(Mn-i, mn-i-1)],

(3.116)

i=1

where it is assumed that

mn_k_ 1 -- u(t)

(3.117)

and each term in (3.116) can be evaluated by employing the formula (3.115). Next, we discuss some interesting qualitative properties of energy losses occurring in hysteresis transducers described by the Preisach model. Consider a cyclic variation of input between two consecutive extremum values u_ and u+. During the monotonic increase of input from u_ to u+, the final horizontal link of L(t) sweeps the triangle T(u+, u_) (see Fig. 3.46). Consequently, the losses occurring during this monotonic increase

3.1 THE CLASSICALPREISACH MODEL OF HYSTERESIS

353

I

,

lu2; /

FIGURE 3.47

I

u2

FIGURE 3.48

are given by (3.110). On the other hand, during the monotonic decrease of input from u+ to u_, the final vertical link of L(t) sweeps the same triangle, and the corresponding losses Q(u+, u_) will be given by the same integral in (3.110). Thus, Q(u_, u+) -- Q(u+, u_) (3.118) and we obtain the following result.

For any loop, the hysteretic losses occurring along ascending and descending branches are the same.

CHAPTER 3 Scalar Preisach Models of Hysteresis

354

The above result can be used to find the formula which relates a hysteretic loss occurring for arbitrary input variations to certain cyclic hysteretic losses. Suppose that the input u(t) is monotonically increased from some minimum value u_ and it reaches successively the values Ul and u2 with u2 > Ul (see Fig. 3.47). We are concerned with the hysteretic loss Q(ul, u2) during the monotonic input increase between Ul and u2. For this input increase we have (see Fig. 3.48)

f2 = T(u2, u_) - T(Ul, u_).

(3.119)

Q(Ul, u2) -- Q(u_, u2) - Q(u_, Ul).

(3.120)

Consequently, Using the above result (see formula (3.118)), losses Q(u_, U2) and Q(u_, Ul) can be expressed in terms of cyclic losses

Q(u_, Ul)

1 m

-

~Q(u_, Ul),

Q(u_,

z

1--

U2) "-

~Q(u_, A

U2),

(3.121)

where Q(u_, ul) is the hysteretic loss per cycle when the input is periodically varied between u_ and Ul; the notation Q(u_, u2) has a similar meaning. By substituting (3.121) into (3.120), we obtain

Q(Ul, u2)

-- x [ Q ( u _ , z ~

u2) -

Q(u_, Ul)].

(3.122)

The last formula expresses the loss occurring during the monotonic increase of input. By literally repeating the same line of reasoning a similar formula can be derived for the case of monotonic input decrease between Ul and u2 (Ul > u2)" m

Q(Ul, U2)

= X[ A"

Q(u+,

m

U2) --

Q(u+,

Ul)],

(3.123)

where u+ is the last input maximum (see Fig. 3.48). The formulas (3.122) and (3.123) may be useful from the practical point of view, because it is much easier to measure cyclic losses than those occurring for nonperiodic input variations. It is instructive to show that the derived expressions for hysteretic energy losses are consistent with the classical result:

A hysteretic energy loss occurring for a cyclic input variation is equal to the area enclosed by the loop resulting from this cyclic input variation.

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

355

Consider a cyclic input variation between u_ and u+. According to (3.118) and (3.115), we find that the hysteretic loss per cycle for the above input variation is given by

Q(u_, u+) - 2Q(u_, u+)

[

= 2 (u+ - u_)f(u+, u_) -

L

f(u+, fl) dfl

f(e, u_) de .

(3.124)

On the other hand, the area enclosed by the corresponding loop is given by /.8

W = $

u df.

(3.125)

d u _It+U_

Since

~u_u+u udf + ~u_u+u f dU- ~u_u+u_ d(uf)

- 0,

(3.126)

f du.

(3.127)

we find

w=-

fu

_U+u_

f du--

fu

_U+

f d u - fu

+lt_

To evaluate the last two integrals, we shall use the formulas (3.71) and (3.74). In the case of monotonic input increase, the formula (3.74) is appropriate. This formula can be written in the form

fit) = C + 2F(u(t), u_), where u_ is used instead of

mn-1

(3.128)

and the constant C is given by

n-1 C - -F(~o, rio) + 2 ~ [ F ( M k ,

ink-l) -- F(Mk, mk)].

(3.129)

k=l

From (3.128), we find

/u

f du = C(u+ -

-U+

u_) + 2

L

F(u, u_) du.

(3.130)

_

In the case of monotonic input decrease from u+ to u_, the formula (3.71) is appropriate. This formula can be rearranged as follows

fit) - C + 2[F(u+, u_) - F(u+, u(t))],

(3.131)

CHAPTER 3 Scalar Preisach Models of Hysteresis

356

where the notation u+ is used instead of Mn, and C is the same historydependent constant as in (3.128). From (3.131), we obtain: u

f du - C(u_ - /,/+) + 2(u_ - u+)F(u+, u_)

+U_

- 2

F(u+, u) du.

f +u_

(3.132)

Using (3.127), (3.130), and (3.132), we derive W = 2[(u+ - u_)F(u+ , u _ ) -

fuu, F(u+, u) d u - /uu, F(u, u_) du

. (3.133)

It is apparent that the expressions (3.124) and (3.133) are identical. This proves that the expressions for hysteretic energy losses derived above are consistent with the classical result (3.125). We next discuss the applications of the above results to the irreversible thermodynamics of hysteretic media. It is clear that any hysteresis phenomenon is accompanied with energy dissipation. This means that hysteretic processes are irreversible, and consequently they fall in the domain of irreversible thermodynamics. Irreversible thermodynamics is the farreaching extension of classical thermodynamics that describes reversible processes. For this reason, it is appropriate to begin with the brief review of the formal structure of classical thermodynamics. Classical thermodynamics is based upon three main principles. The first principle of classical thermodynamics is the law of energy conservation. According to this principle, there exists a function of state, called the internal energy, U, of the closed system. This state function is such that its infinitesimally small change d U may occur as a result of energy exchange with the surroundings in the form of heat cl Q, as well as a result of energy cl W added to the system (or spent by the system) in the form of work. Mathematically, the first principle is expressed by dU = fl Q 4- fl W,

(3.134)

where a stroke is put across the symbol d in (3.134) to emphasize that the quantities d Q and cl W are path-dependent infinitesimals, which are sometimes called imperfect differentials. In other words, the above quantities depend on a particular path traced by the system during its transition from one equilibrium state to another. The second principle of classical thermodynamics postulates the existence of another state function, called the entropy S. This principle also

3.1 THE CLASSICALPREISACH MODEL OF HYSTERESIS

357

relates cl Q to the differential of entropy dS for reversible processes by the formula dQ dS- T ' (3.135) where T is the absolute temperature. For irreversible processes, the above equality is replaced by the inequality dQ dS > ~ . (3.136) T The formulas (3.134) and (3.135) are often combined into one formula: 1

dS = ~ ( d U - d W) 1

(3.137)

that constitutes the mathematical foundation of classical thermodynamics. The factor 1/T in (3.137) and (3.135) can be mathematically interpreted as an integrating factor for the imperfect differential cl Q. The first and second principles of classical thermodynamics are complemented by the Nernst-Plank postulate which is sometimes called the third principle of thermodynamics. According to this principle, the entropy of any system vanishes at zero temperature. This principle provides some useful information concerning the asymptotic behavior of entropy. However, the bulk of phenomenological thermodynamics does not require this principle. For this reason, the above postulate cannot be compared in importance with the first and second principles of classical thermodynamics. Classical thermodynamics has by and large been extended in two main directions. The first extension is based on the introduction of new variables describing the composition of the system. This approach has been very successful in applications of thermodynamics to chemical reactions. It has led to the development of chemical thermodynamics. The second extension is based on the generalization of the second principle of classical thermodynamics; it has led to the development of thermodynamic theory of irreversible processes that is often called irreversible thermodynamics. This theory has been mainly developed by I. Prigogine and his collaborators. As mentioned above, the basic difference between classical and irreversible thermodynamics lies in the way in which the second principle is stated. In irreversible thermodynamics, the second principle is formulated as follows. 1. The change in entropy dS can be split into two parts: dS = ~e S if- ~i S,

(3.138)

358

CHAPTER 3

Scalar Preisach Models of Hysteresis

where de S is due to the flow of entropy into the system from its surroundings, while di S is the generation of entropy by irreversible processes within the system. The term di S is often called internal entropy production. 2. The internal entropy production di S is never negative. It is zero if only the system undergoes a reversible process and positive if it undergoes an irreversible process. Thus, ~i S/~0.

(3.139)

3. For closed systems, the term de S is related to the energy de Q received in the form of heat by the formula that is similar to (3.135):

de Q

d e S -- ~ ,

T

(3.140)

where T is the absolute temperature that is assumed to be definable (at least locally) for nonequilibrium situations. It is also assumed in irreversible thermodynamics that the formula (3.137) holds with dS meaning the total change in entropy. From (3.138) and (3.140), we find T dS = d e Q + T ~i S.

(3.141)

The last expression allows one to relate the entropy production to the dissipated energy within the system. Indeed, from (3.137) and (3.141) we find dU = d W 4- ~e Q 4- T di S. (3.142) The first two terms in (3.142) have the same meaning as in (3.134) while the last term can be interpreted as the energy supplied to the system as a result of dissipating (irreversible) processes. This interpretation is especially clear for adiabatically isolated (de Q = o) systems that undergo cyclic changes ( / d W = 0). The increase in internal energy during one cycle ( / d U ) of such systems is only due to dissipating processes within the system ( / T di S). From (3.142) and the foregoing discussion, we find that the entropy production can be related to the dissipated energy di Q by the formula: ~i Q d i S -- ~ . (3.143) T It is apparent that the novel part of the above formulation of the second principle is the introduction of internal entropy production di S. However,

3.1 THE CLASSICAL PREISACH MODEL OF HYSTERESIS

359

this new quantity is useful only if it can be evaluated (in a mathematical form) for different irreversible processes. This is the central problem of irreversible thermodynamics and it has been emphasized in many books on the subject. For instance, in the book [26; p. 90] by I. Prigogine we find: 'The main feature of the thermodynamics of irreversible processes consists of the evaluation of the entropy productions...' Similarly, in [27; p. 21] we read: 'In thermodynamics of irreversible processes, however, one of the important objectives is to relate the quantity di S, the entropy production, to the various irreversible phenomena which may occur inside the system'. Finally, in [28; p. 69] it is noted: 'A central problem of irreversible thermodynamics, in fact, is the development of formulas for the entropy production ~i S in specific cases'. The above problem has been resolved for various irreversible processes that are caused by macroscopic non-uniformities of the system. Examples of such processes include heat flows due to temperature gradients, diffusion flows due to density gradients, electric current flows due to potential gradients, and so forth. For the above processes, the so-called entropy balance Eqn. (3.138) is written in local (differential) form in which the entropy production, eli S, is replaced by the entropy source. The entropy source is then found as a sum of several terms, each being a product of a flux characterizing a particular irreversible process and a quantity, called thermodynamic force, which is related to a particular macroscopic nonuniformity of the system (temperature gradient, for instance). In this way, many useful results have been established. The celebrated Onsanger reciprocity principle for phenomenological coefficients is the most known example of these results. However, the above developments cannot cover hysteresis phenomena. This is because hysteresis is not caused by macroscopic nonuniformities and therefore cannot be linked to gradients of some physical quantities. As a result, different approaches should be developed for the calculation of entropy production in the case of irreversible hysteretic processes. It is logical to expect that mathematical models of hysteresis may help to solve the above problem. It is shown below that by using the Preisach model and the expressions for hysteretic energy losses derived on the basis of this model, the entropy production for hysteresis processes can indeed be found. Consider the input u(t), which is monotonically increased from its previous local minimum value u_. If the current input value, u(t), is lower than M n , then by using (3.115), for the dissipated energy we find

Q ( u _ , u) = (u - u _ ) F ( u , u _ ) -

fun F(u, fl) dfl - fuu F(a, u) da.

(3.144)

360

CHAPTER 3 Scalar Preisach Models of Hysteresis

Employing (3.144), it is easy to conclude, that the energy dissipation, ~i Q, that occurs as a result of the input increase from u to u 4- du, is given by ~i Q 1

3Q(u_, u) du. Ou

(3.145)

From (3.145) and (3.143), we derive: 1 OQ(u_, u) di S - ~ cgu du.

(3.146)

By differentiating (3.144) with respect to u and substituting the result into (3.146), after simple transformations we obtain du fli g - ---Z- fu _n ~0 [F(u , u_) - F(u, fl)] dfl.

(3.147)

If the input is monotonically decreased from its previous local maximum value, then the internal entropy production occurring as a result of the input decrease from u to u - du is given by {1i S -

du {u+ ~ ~UU[F(u+, u) - F(a, u)] da.

(3.148)

The derivation of the last formula is similar to that for (3.147). It has been tacitly assumed in the previous derivation that the monotonic input variations are such that no previous history is wiped out. However, by using (3.116), it is easy to extend (3.147) and (3.148) to the most general case. The details of this extension are left to the reader. Very interesting discussion of thermodynamic aspects of hysteresis can be found in [29,30]. The presentation of the material in this section is largely based on references [31-36].

3.2 G E N E R A L I Z E D S C A L A R P R E I S A C H M O D E L S OF HYSTERESIS 3.2.1 'MOVING' PREISACH MODEL OF HYSTERESIS The classical Preisach model of hysteresis has been discussed in detail in the Section 3.1. It has been repeatedly emphasized that this model has some intrinsic limitations. The most important of them are the following: 1. The classical Preisach model describes hysteresis nonlinearities that exhibit congruency of minor loops formed for the same reversal

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

361

values of input. However, many experiments show that actual hysteresis nonlinearities may substantially deviate from this property. 2. The classical Preisach model is rate-independent in nature and does not account for dynamic properties of hysteresis nonlinearities. However, for fast input variations these properties may be essential. 3. The classical Preisach model describes hysteresis nonlinearities with the wiping-out property. This property is tantamount to the immediate formation of a hysteresis loop after only one cycle of back-andforth variation of input between any two reversal values. However, experiments show that hysteresis-loop formation may be preceded by some 'stabilization process' that may require a large number of cycles to achieve a stable minor loop. This process is also called in the literature the'accommodation' or'reptation' process. 4. In the classical Preisach model, a scalar output exhibits hysteretic variations with respect to only one scalar input. However, in the case of magnetostrictive hysteresis, the strain is a hysteretic function of two variables: magnetic field and stress. Thus, the problem of developing the Preisach-type models with two inputs presents itself. These models may find applications beyond the area of magnetostrictive hysteresis, for instance in the modeling of piezoelectric hysteresis. To remove (or relax) the above-mentioned limitations, essential generalizations of the classical Preisach model are needed. These generalizations are discussed in this section. In this section, experimental testing of various scalar Preisach-type models of hysteresis is presented as well. The presentation of the material is largely based on our publications [31,32,37--46]. We begin with an interesting modification of the classical Preisach model. This modification will reveal that the Preisach model does describe, to a certain extent, reversible properties of hysteresis nonlinearities. This fact has been overlooked in the existing literature. Apart from the mentioned fact, this modification will also be instrumental in the further generalizations of the classical Preisach model that are discussed in subsequent sections. The classical Preisach model has been defined as

f(t)

A

= Fu(t) -

#(e,

fl)~u(t) de d[3,

(3.149)

where T is the limiting triangle specified by inequalities flo~fl~e~eo. This triangle is the support of the function #(e,/~) and does not change with input variations.

CHAPTER 3 Scalar Preisach Models of Hysteresis

362

(~o

FIGURE 3.49

We next subdivide the triangle T into three sets Fig. 3.49), that are defined as follows:

Su+(t) (ce,fl) ~ Ru(t) (ce,fl) ~ Su(t) (ce, fl) E

Su+(t),Ru(t) and Su(t) (see

if fl0~flKceKu(t),

(3.150)

u(t)~ce~ceo, if u(t)KflKceKceo.

(3.151)

iffl0~fl~u(t),

(3.152) By using the above subdivision, we can represent (3.149) in the form

fit) =//R

#(ce'fl)~u(t)dcedfl +/fs+ /a(ce,fl)~u(t)dcedfl u(t)

u(t)

+//s

#(ce,fl)~u(t)

dce dfl.

(3.153)

u(t)

Since

+ u(t)~ce for any (ce,fl) ~ Su(t), then ~u(t)

/fs

#(ce'fl)~u(t)

ace 4]3 = / f s

#(ce']3) ace aft.

(3.154)

u(t) +

u+(t)

Similarly,

= +1 and

u(t)Kfl for any (ce,fl) E Su(t) and

/fs

/a(ce,fl)~&~u(t)dceaft -

-/fs

u(t)

#(ce']3)dce dfl.

(3.155)

u(t)

By substituting (3.154) and (3.155) into (3.153), we obtain

f(t)=ff nu(t) U(~,fl)G,u(t)d~dfl+ffs+u(t) #(ce,fl)dcedfl - f/s

#(ce'fl) dcedfl. u(t)

(3.156)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

363

We next find a simple expression for the last two terms in (3.156). Consider a monotonic increase of input from some value below fl0 (state of negative saturation) to some value u(t). Then the output will change along the ascending branchf,+L,, and according to (3.156) we find u~,t) .pp

:-

PP

11 #(~, fl)d~x dfl + 1 1 #(oc,fl)da dfl JJR u(t) dd Su+(t) - ff

diS

/~(~,fl)da dfi.

(3.157)

Similarly, if we consider a monotonic decrease of input from some value above a0 (state of positive saturation) to the same value u(t), then by using (3.156) we obtain

-- H #(a, fl)d0~ dfl + H /~(a, fl)d~ dfl J JR u (t) dds+u(t) - ff #(~, fl)d~ dfl. JJs ~lt) By summing up (3.157) and (3.158), we derive

f~Su+(t)

fl)

aft - ffSu ,) #(~

fl) da aft = ~'1-(fu~t) + f~t)).

(3.158)

(3.159)

By substituting (3.159) into (3.156), we finally obtain

f(t) =

la(o~, fl)~flu(t) dadfl

1

+ -~(fu+((t)+ f~t))"

(3.160)

u(t)

The last expression is formally equivalent to the classical one (3.149). However, in this expression the integration is performed not over the fixed limiting triangle T but over the rectangle Ru(t) which changes along with input variations. For this reason, the expression (3.160) is termed here as a ,moving , Preisach model. It is also clear from (3.160) that 2(fu(t) 1 + -}-f~t)) represents a fully reversible component of hysteresis nonlinearity described by the classical Preisach model. In this respect, the first term in the right-hand side of (3.160) can be construed as an irreversible component of the classical Preisach model. To make the last point transparent, consider some output increment Afu corresponding some input value u (see Fig. 3.50). This increment depends on a particular history of input variations and can be regarded as a measure of irreversibility of hysteresis nonlinearity for this particular history. It is clear from the diagram shown in Fig. 3.51 and the expression (3.160) that the above increment is given by

Afu - 2 f f #(~, fl) d~ dfl. JJ f~

(3.161)

364

CHAPTER 3 ScalarPreisach Models of Hysteresis f

L

i ~U

FIGURE 3.50

FIGURE 3.51

The region of integration f~ belongs to Ru(t). This is true for any past history. This fact clearly suggests that the first term in the right-hand side of (3.160) describes an irreversible processes. Thus the expression (3.160) gives the decomposition of hysteresis nonlinearity described by the classical Preisach model into irreversible and reversible components. It is stressed in the above discussion that the expression (3.160) is mathematically equivalent to the classical definition (3.149). However, it is apparent that this equivalence holds only for input and output variations confined to the region enclosed by a major hysteresis loop. Outside this region, the classical Preisach model prescribes flat saturation values for output, while the moving model (3.160) prescribes the actual experimentally observed values.fu+(t) and f~t) for the states of negative and positive saturation, respectively. This is the case because for these states ascending

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

(~,P) s-

365

/,

,

(a,13o)

FIGURE 3.52

and descending branches merge together and consequently fult) "-- f~t)" As far as the first term in (3.160) is concerned, it is clear from (3.151) that this term vanishes. Thus, the moving model (3.160) can be regarded as a generalization of the classical model as far as the description of hysteresis nonlinearities beyond the limits of major loops is concerned. It is instructive to consider directly the identification problem for the model (3.160) without invoking our previous discussion of this problem for the classical model (3.149). The essence of this problem is in determining the function ~ by fitting the model (3.160) to some experimental data. Suppose that, starting from the state of negative saturation, the input u(t) is monotonically increased until it reaches some value e. Then the input is monotonically decreased until it reaches some value ft. The corresponding diagram is shown in Fig. 3.52. By using (3.160) and this diagram, we find that the output valuef~fl corresponding to the final value of output is given by

#(~', fl') da' dfl' - ffR(~o,~o,~,~) #(cr fi') do~'dfl' + f; +fff.

(3.162)

2

Here R(a0, rio, a, fi) is the rectangle whose opposite vertices are the points (c~0, fl0) and (c~,fi); the same meaning as the notation R(c~, fl0, fl, fl). Now suppose that, starting from the state of positive saturation, the input u(t) is monotonically decreased until it reaches the value ft. The corresponding c~-fl diagram is shown in Fig. 3.53. From this diagram and

CHAPTER 3 ScalarPreisach Models of Hysteresis

366

FIGURE 3.53

(3.160) we find the following expression for the resulting value of output:

f~- =

f•R

lt(e', fl') de' dfl' + f ; + f~

(=o,~o,~,~)

---T~"

(3.163)

Next we introduce the function T(e, fl) =f~- -f~/~,

(3.164)

which is equal to output increments between the limiting descending branch and first-order transition curves. From (3.162), (3.163) and (3.164) we find

T(e, fl) = 2 f f

,IJR (~o,&,~,/~)

#(e', fl') de' dfl'

- 2 f~~ (f~i #(0r fl') d]/t de.

(3.165)

By differentiating the last expression twice, we obtain #(e, fl) =

1 c32T(e, fl) ~ . 2 cgec3fl

(3.166)

By recalling the definition (3.164) of T(e, fl), from (3.166) we derive

#(~' fl) = 2 8e 8fl'

(3.167)

which coincides with the expression (3.29) obtained for the classical Preisach model.

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

367

0{ s-(t)

I I I

I

Rk

FIGURE 3.54

Next, we show that the integration in (3.160) can be avoided and that the explicit expression forf(t) in terms of experimentally measured function T(~, fl) can be derived. To start the derivation, consider the cr diagram shown in Fig. 3.54. Then, according to (3.160), we have

fit) -

IL

+(t) la(a, fl) da dfl -

=2iL

SL

-(t)

u(t)

l~(cr

#({z, fl)d{zdfl-i/R +(t)

#(~, fl) da dfl +

u(t)

+

q2 f~t) +f~t) 2 "

(3.168) To evaluate the integral over Ru(t) in (3.168), we consider a monotonic increase of input from the state of negative saturation to some value u(t) Then from (3.160) we find

fu+(t) = - ifI~ u(t)

P(~' ~) d(x dfl + fu(+) + f~t) 2 "

(3.169)

By substituting (3.169) into (3.168), we conclude

fit) = 2 i L

+(t)

/z(oc, fl) dcz df1-4- fu+(t).

(3.170)

It is clear that the moving model (3.160) is equivalent to the classical Preisach model as far as description of purely hysteretic behavior is concerned. For this reason, the wiping-out property and congruency property of minor loops are valid for the moving model. According to the wipingout propertj6 the staircase interface L(t) in Fig. 3.54 has vertices whose

CHAPTER 3 Scalar Preisach Models of Hysteresis

368

a and fl coordinates are equal to Mk and mk, respectively, and, as before, numbers Mk and mk form the alternating series of past dominant input extrema. Now we subdivide S+(t) into rectangles Rk and represent the integral in (3.170) as n(t)

/fs +(t)

#(a, fl)de dfl = ~ / / R k=l

fl(~'/J) de dfl.

(3.171)

k

It is easy to see from the expression (3.165) for the function T(a, fl) that the integrals over Rk can be represented in the form

f/R #(a, fl) da dfl = T(Mk+I, ink) -- T(Mk, ink).

(3.172)

k

From (3.170), (3.171) and (3.172) we obtain n(t)

f(t) = 2 ~[T(Mk+I, mk) - T(Mk, mk)] + fu+(t).

(3.173)

k-1

This is the final result which expresses explicitly the outputf(t) in terms of experimentally measured function T. Up to this point, we have discussed the modeling of 'counterclockwise' hysteresis. For this type of hysteresis, the positive saturation state is achieved for a larger input value than the negative saturation state, and for any two consecutive descending and ascending branches the descending branch is above the corresponding ascending branch (see, for instance, Fig. 3.50). This type of hysteresis is typical for magnetic materials and it can be modeled by using the classical Preisach model with positive measure #(a, fl). It is apparent that the classical Preisach model with negative measure can be used for the modeling of 'clockwise' hysteresis when the positive saturation state is achieved for a smaller input value than the negative saturation state. This type of hysteresis occurs in superconductors due to their diamagnetic nature. There exists, however, the 'clockwise' hysteresis with the property that the positive saturation state is achieved for a larger input value than the negative saturation state. This is typical for hysteresis that occurs for certain front propagation problems such as nonlinear diffusion of electromagnetic fields in magnetically nonlinear conductors. The modeling of this type of hysteresis is not readily available within the framework of the classical Preisach model, however it can be easily accomplished by using the 'moving' Preisach model (3.160). Indeed,

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

369

the proper saturation states can be modeled by the appropriate choice of the reversible terms 1 (fu+(t)ff-f~t))" while the 'clockwise' nature of hysteresis is imposed by choosing a negative measure #(a, fl) in the irreversible component. In other words, the decomposition (3.160) of hysteresis nonlinearity on reversible and irreversible components makes it possible to control separately the (clockwise or counterclockwise) orientation of hysteresis and the location of saturation states. 3.2.2 PREISACH MODEL OF HYSTERESIS WITH INPUT-DEPENDENT MEASURE

The Preisach model with input-dependent measure is discussed in this section. It has the following advantages over the classical model. First, the congruency property of minor loops is relaxed for this model. This results in a broader area of applicability of this model as compared with the classical model. Second, the model with input-dependent measure allows one to fit experimentally measured first- and second-order reversal curves. Since higher-order reversal curves are' sandwiched' between firstand second-order reversal curves, it is natural to expect that this model will be more accurate than the classical one. The Preisach model with input-dependent measure can be mathematically defined as

fit) - ffR

u(t)

#(a' fl'

u(t))~flu(t) d~ dfl +

du+(t)if-f ~t)

(3.174)

It is clear that a new feature of this model in comparison with the 'moving' model (3.160) is the dependence of the distribution function # on the current value of input, u(t). For this reason, the model (3.174) is also termed as the 'nonlinear' Preisach model. Owing to the new feature mentioned above, the first term in the right-hand side of (3.174) can be construed as a partially reversible component of hysteresis nonlinearity. Indeed, for each pair (~, fl), the integrand #(~, fl, u(t))~u(t) is reversible for input variations between ~ and ft. The 'nonlinear' Preisach model admits a geometric interpretation, which is similar to that for the classical model. In other words, it could be easily demonstrated that at any instant of time t the rectangle Ru(t) is subdivided into two sets (see Fig. 3.55): S+ (t) and S- (t) consisting of points (~, fi) for which ~aflu(t) = 1 and ~ u ( t ) - - 1 , respectively. The interface L(t) between S+ (t) and S-(t) is a staircase line whose vertices have a and fl coordinates equal (respectively) to past dominant extrema Mk and mk. Using the above geometric interpretation, the model (3.174) can be represented

CHAPTER 3 Scalar Preisach Models of Hysteresis

370

,

/

u(t)

/ FIGURE 3.55

in the form

fit) = f~S+(t ) #(a, fl, u(t)) da dfl - / ~ s - ( t ) 11(0r fl' u(t)) da dfl +

f u+((t) -Jr-f ~ t)

. (3.175) 2 It can be shown that the following two properties are valid for the 'nonlinear' model. PROPERTYA (WIPING-OUTPROPERTY). Only the alternating series of subsequent global extrema Mk and mk are stored by the "nonlinear" Preisach model. The proof of this property is identical to that for the classical model.

PROPERTYB (PROPERTYOF EQUAL VERTICALCHORDS). All minor loops resulting from back-and-forth input variations between the same two consecutive extrema have equal vertical chords (output increments)for the same input values (Fig. 3.56). PROOF. Consider a minor loop formed as a result of back-and-forth input variations between u+ and u_ (Fig. 3.57). Letfu f andfu be output values on descending and ascending branches of this loop, respectively. These values correspond to the same value of input u from the interval u_ < u < u+. By using the expression (3.175),fu I andfu can be computed as f ~ = ffs~

~(~, fl, u) d~ dfl - f/s" l~(~,fl, u)dadfl

+ ~fu+ ,+ f u 2

(3.176)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

U_

U

U+

371

--U

F I G U R E 3.56

fu'

U_

U

U+

F I G U R E 3.57

fu' - / ~ s p(~, fl, u) &z dfl - f f s #(a, fl, u) d~ dfl ~_ + fu+ + f u , (3.177) 2 where the sets S~_, St__, S~_ and S'_~ are shown on the diagrams presented in Fig. 3.58. It is clear from these diagrams that S+ - S+ ' - R(u+, u , u _ ) ,

S ~_ - S"_ - R(u+, u, u_).

(3.178)

From (3.176), (3.177) and (3.178), we derive fu' - f u - 2 I I p(~, fl, u) da dfl. J JR (u+,u,u_)

(3.179)

According to (3.179), we find that for any u ~ (u+, u_) the corresponding vertical chord does not d e p e n d on a particular past history preceding the formation of the minor loop. This proves that all comparable minor loops, that is, the loops with the same reversal input values u+ and u_, have equal vertical chords. It is left to the reader as a useful exercise to prove that comparable minor loops described by the nonlinear model (3.174) are not necessarily congruent.

CHAPTER 3 Scalar Preisach Models of Hysteresis

372 (X

/

/

I

8"

I S'+

' I

R,u_

~

u

R(u_,u,u+)

I

u

u

u+

(a)

u+

(b) F I G U R E 3.58

Now we turn to the identification problem of determining the distribution function #(a,/}, u) by fitting the model (3.174) to some experimental data. It turns out that for the solution of this identification problem the sets of first- and second-order reversal curves are required. These curves can be measured experimentally as follows. We first decrease the input, u(t), to such a negative value that the outputs of all operators ~a~ are equal to - 1 (state of negative saturation). Then we monotonically increase the input until it reaches some value a. As we do this, we will follow the ascending branch of the major loop (see Fig. 3.59). As we already know, the first-order reversal curves are attached to this ascending branch and they are formed when the above monotonic increase of u(t) is followed by a subsequent monotonic decrease. The notation f + will be used for the output values on the first-order reversal curve attached to the ascending branch of the major loop at the pointf +. The second-order reversal curves are attached to the first-order reversal curves, and they are formed when the above monotonic decrease is followed by a monotonic increase. The notation f~u will be used for the output values on the second-order reversal curve attached to the first-order reversal curve f~u at the point f ~ (Fig. 3.59). Consider the function

P(~, ~,

u) - f~u - f ~ , u ,

(3.180)

which has the physical meaning of output increments between the firstand second-order reversal curves.

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

373

I

u

FIGURE 3.59

It is clear from the definition of this function that

P(o~, u, u) = P(u, fi, u) = P(u, u, u) = O.

(3.181)

This property will be employed later in our discussion. Now we will try to relate the function, P(a, fl, u), to the distribution function, #(a, fl, u). To this end, we will use the diagrams shown in Fig. 3.60. From these diagrams and (3.175) we conclude

f~u = ff~+ l~(~x', fi', u) da' dfl' - ff~ la(a', fl', u) d~x' dfl' + f : , , u - ff~ ~(~', fl', u)d~' a f t ' - f J i #(~'' fl'' u)d~' dfl +

fu+ +f; (3.182)

-

f++fu 2 S+ - S+ - R(~, fl, u),

S_ - S_ - R(~, fl, u).

(3.183)

By using (3.182) and (3.183), we derive

P(~, fl, u)

-

2 ffR

= 2

~,fl,u)

#(~" fl'' u) d~' aft'

fu (f, u#(e', fl', u)dfl' ) d~'.

(3.184)

374

i

CHAPTER 3 Scalar Preisach Models of Hysteresis

i

d

d

/

I

H~ u L

U

(a)

~,

(b)

FIGURE 3.60

By differentiating the last expression twice, we obtain 1r p(a,//, u) -

2

]~, u) r

.

(3.185)

By recalling the definition (3.180) of P(a, fl, u), from (3.185) we find 1 02fa~u # ( ~, ~6, u) = 2 &z cgfl

(3.186)

It is clear from the above derivation, that the expressions (3.185) and (3.186) are valid if ]/< u < a. If uK~fl or u~>a, then we define p(a, fl, u) = 0,

(3.187)

which is consistent with (3.181). Thus, if the distribution function, p(a, fl, u), is determined from (3.184) or (3.186), then the nonlinear model (3.174) will match the increments between first- and second-order reversal curves. Next we shall show that the limiting ascending branch will be matched by this model as well. Indeed, it is easy to see that the output values, fu, on the limiting ascending branch predicted by the nonlinear model (3.174) are equal to

fu = -

IfR

#(a" fl'' u) da' dfl' + f+ + f u (~0,&,u) -----2---"

(3.188)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

375

I

1

foq3u

FIGURE 3.61

According to (3.184) and (3.180), we have

ffR

(~o,&,u) #(a'' fl'' u) d~dfl' - ~P(ao, rio, u) = -~(f~ou -f~o&u) 1 1 1

= i 0 u-fu+). By substituting (3.189) into (3.188), we find

(3.189)

fu -- fu+. Thus, if the function,/~(a, fl, u), is determined according to (3.184) or (3.186), then the nonlinear model (3.174) fits: (a) the output increments between the first- and second-order reversal curves, and (b) the ascending branch of the major loop. Since the ascending branch of the major loop can be construed as a second-order reversal curve, we conclude that the nonlinear model (3.174) fits the sets of first- and second-order reversal curves. In the above discussion we have used second-order increasing reversal curves (see Fig. 3.61) in order to determine the distribution function, #(~, fl, u). However, by almost literally repeating the previous line of reasoning, a similar expression can be found for #(a, fl, u) by using secondorder decreasing reversal curves f ~ . One of these curves is shown in Fig. 3.62. By using this figure, we can introduce the function (3.190)

CHAPTER 3 Scalar Preisach Models of Hysteresis

376

I

I I

i

i

I

I I

,/

',

o,,

FIGURE 3.62

In the same way as before, we can show that

P ( -fl, ~ , fi ) = 2 f f # ( o{ , fl' , fi ) d o{ d fl' 3JR ~,~,~

(3.191)

and

1 (92P(~, ~, U) #(~' fl' u) =

2

O~Ofl

"

(3.192)

If --a,

&=-fl,

fi--u,

(3.193)

then due to the symmetry between increasing and decreasing second-order reversal curves (see Fig. 3.62) we find (3.194) By using (3.193) and (3.194) in (3.192), we derive 1 02P(a, fl, u) /~(-fl, -c~, - u ) =

2

0c~cgfl

.

(3.195)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

377

From (3.195) and (3.185), we conclude #(~, fl, u) = # ( - / / , - ~ , - u ) .

(3.196)

The last formula can be regarded as a generalization of the mirror symmetry (3.46) previously established for the classical Preisach model. [] We next proceed to the proof of the following important result. REPRESENTATION THEOREM. The wiping-out property and the property of equal

vertical chords for minor loops constitute the necessary and sufficient conditions for the representation of a hysteresis nonlinearity by the nonlinear Preisach model on the set of piecewise monotonic inputs. PROOF. Necessity: If a hysteresis nonlinearity is representable by the nonlinear Preisach model, then this nonlinearity should have the same properties as the model. This means that this nonlinearity should exhibit the wipingout property and the property of equal vertical chords for comparable minor loops. Sufficiency: Consider a hysteresis nonlinearity that has both the wipingout property and the property of equal vertical chords. For this nonlinearity we find the distribution function #(a, fl, u) by using formula (3.185) (or (3.38)). Then the nonlinear Preisach model (3.174) will match exactly the sets of all first- and second-order reversal curves. We intend to prove that this model will match all possible higher-order reversal curves as well. The proof is based on the induction argument. Let us assume that the above statement is true for all possible reversal curves up to the order k - 1. Then, for the induction inference to take place, we need to prove that the same statement holds for any reversal curve of order k. Let a be a point at which a reversal curve of order k is attached to a reversal curve of order k - 1 (see Fig. 3.63). According to the induction assumption, the output values for the actual hysteresis nonlinearity and for the nonlinear Preisach model coincide at each point of the reversal curve of order k - 1. Thus, it remains to be proven that the output increments between the reversal curves of orders k and k - 1 are the same for the actual hysteresis nonlinearity and for the nonlinear model (3.174). It is here that the wiping-out property and the property of equal vertical chords will be used. According to the wiping-out property, the kth reversal curve should meet the (k - 1)th reversal curve at the point b, which is the point of inception of the latter curve. As a result, a minor loop is formed. Consider a comparable minor loop (loop with the same reversal input values), which is attached to the limiting ascending branch (Fig. 3.63). This loop is formed by some first- and second-order transition curves. According to the property of equal vertical chords, this

378

CHAPTER 3 Scalar Preisach Models of Hysteresis

U

FIGURE 3.63 o~

u

FIGURE 3.64

loop has the same vertical chords as the loop formed by (k - 1)th and kth reversal curves. Consider an arbitrary value u of input such that fl < u < a. Using the diagram shown in Fig. 3.64, it is easy to derive the following expression for the output increment

f(k) _ F(k- 1) _ 2 ffR

#(a', fl', u ) d e ' dfl',

(3.197)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

379

wherefu(k) andfu(k-l) are the output values on the kth and (k - 1)th reversal curves, respectively, corresponding to the input value u and predicted by the model (3.174). From (3.184) we find that the right-hand side of (3.197) is equal to the output increment el between the first- and second-order transition curves. Consequently, f(u k) -- f(u k-l) -- el.

(3.198)

According to the property of equal vertical chords we have

el = cd.

(3.199)

From (3.198) and (3.199), we conclude f(u k) -- ju'c(k-1) _ c d .

(3.200)

Thus the nonlinear model predicts the correct output increments between the kth and (k - 1)th reversal curves. From this fact and the above induction assumption we conclude that the nonlinear model predicts the correct output values on the reversal curve of order k. This concludes the proof of the theorem. [] The proved theorem establishes the exact bounds of applicability of the nonlinear Preisach model. It is apparent that the property of equal vertical chords is more general than the congruency property. Indeed, if comparable minor loops are congruent, then they have equal vertical chords. However, if comparable minor loops have equal vertical chords, they are not necessarily congruent. This clearly shows that the nonlinear Preisach model (3.174) has a broader area of applicability than the classical Preisach model. Next we shall show that the nonlinear model (3.174) contains the classical Preisach model as a particular case. The exact statement of this fact is given by the following theorem. REDUCTIONTHEOREM. If all comparable minor loops of hysteresis nonlinearity are congruent, then the nonlinear Preisach model for this nonlinearity coincides with the classical Preisach model as far as purely hysteretic behavior of this nonlinearity is concerned. PROOF. Since all comparable minor loops are congruent, these loops have equal vertical chords. Assuming also that the wiping-out property holds, we can represent the mentioned hysteresis nonlinearity by the nonlinear Preisach model (3.174). We next show that, because of the congruency property, the distribution function # in (3.174) does not depend on u. It is

CHAPTER 3 Scalar Preisach Models of Hysteresis

380

f

II I I I

I I I

0~

0~

P

U

FIGURE 3.65 clear from Fig. 3.65 that the congruency property results in the congruency of second-order reversal curves fa'Bu, fa~u and fa"~u. This means that the derivative -N--u G~u does not depend on cr Consequently,

c~f~u - O.

(3.201)

&~~//Ou From (3.201) and (3.186) we find

~(~,/~, u) 0u

--- 0.

(3.202)

Thus/~ does not depend on u and is only a function of ~ and fl: /~(~, ]3, u) - v(c~, fl).

(3.203)

By substituting (3.203) into (3.174), we obtain

fit) = ffR

v(a, fl)~flu(t) dec dfl +

du~t) q- du~t)

(3.204)

u(t)

It remains to be proved that v(ce, fl) in (3.204) coincides with #(a, ]3) in (3.160). The proof is straightforward. By using (3.160) and (3.204) we find

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

381

that for any a, fl and u such that flKu~a we have

P(~, fl, u) = f~u -f~flu - 2

ff. (~,~,u),(~,, fl,) da' dfl',

(3.205)

F/"

P(:r fl, u) =f~u - faflu - 2

__JJR(a,fl, u) v(a', fl') da' dfl'.

(3.206)

From (3.205) and (3.206), as before, we derive #(~, fl) = v(~, fl) =

1 02P(a, fl, u) 2

#~afl

.

(3.207)

Thus, it is proved that under the congruency condition, the nonlinear Preisach model (3.174) coincides with the moving Preisach model (3.160). On the other hand, the moving model coincides with the classical Preisach model as far as the description of purely hysteretic behavior is concerned. From here we conclude that under the congruency condition the nonlinear model (3.174) coincides with the classical Preisach model, and this concludes the proof of the theorem. We next turn to the discussion of numerical implementation of the nonlinear model (3.174). We intend to show that double integration in (3.174) can be completely avoided and that explicit expressions for the outputflt) in terms of the experimentally measured function P(a, fl, u) can be derived. The starting point of our derivation is the expression (3.175) that can be modified as follows:

f(t) -

ffRu(t)

/~(~, fl, u(t)) da dfl - 2

-(t) Ia(a, fl, u(t)) doc dfl

+ fu+(t) -}-f ~t) 2

(3.208)

According to (3.189), we have

~fRu(t ) [.1(0~,~, u(t)) da dfl -

f ~t) -- f u+(t)

(3.209)

By substituting (3.209) into (3.208), we obtain

fit) -- f ~ t ) - 2 fL-(t) #(~, fl, u(t)) da dfl.

(3.210)

We next subdivide S-(t) into rectangles Rk (see Fig. 3.66) and represent the integral in (3.210) as

n(t) Ia(:r fl, u(t)) d~dfl - E ffR IJ(~, fl, u(t)) d~dfl.

fL -(t)

k=l

k

(3.211)

CHAPTER 3 Scalar Preisach Models of Hysteresis

382

s-(t)

o~

W___]_', ~-R~

/

FIGURE 3.66

It is easy to see that

Rk = R(Mk, mk, u(t)) -- R(Mk+I, mk, u(t)).

(3.212)

From (3.212) and (3.184), we conclude

fRk ~(~' fl' u(t)) de aft = ff

ddR (Mk,mk,u(t))

lz(or fl, u(t))dadfl - f f

#(~,

dd R(Mk+l,mk,u(t))

-- l [p(Mk, ink, u(t)) -- P(Mk+I, ink, u(t))]. 2

u(t)) dadfl

(3.213)

By substituting (3.213) into (3.211) and then (3.211) into (3.210), we obtain n(t)

fit) = f~t) + ~-'~.[P(Mk+I, mk, u(t)) -- P(Mk, mk, u(t))].

(3.214)

k=l

This is the final formula which expresses explicitly the output f(t) in terms of the experimentally measured function P. This formula has been used to develop a digital code which numerically implements the nonlinear Preisach model. This code computes output values by using input values, a set of second-order reversal curves and an input history which are all specified by the user. The structure of the code is very similar to the structure of the code which implements the classical Preisach model. For this reason, the detailed discussion of this code is omitted. We next proceed to the discussion of hysteretic energy losses for hysteresis nonlinearities described by the nonlinear Preisach model (3.174). We

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

383

A

~ ( a, [~ u ) T~, u

.(a,~,u) b

v

~

h

~

~

U

\ a

FIGURE 3.67

begin with the case of elementary hysteresis nonlinearity represented by the expression #(a, fl, u(t))~u(t). It follows from (3.187) that the input-output relationship for this nonlinearity is described by a loop shown in Fig. 3.67. It is clear that branch I of this loop is fully reversible except for its part a along which the irreversible switching-up occurs. Similarly, branch 2 of the loop is fully reversible except its part b along which the irreversible switching-down occurs. If a periodic variation of input is such that the whole loop is traced, then the hysteretic energy loss for one cycle, Qcycle, is equal to the area enclosed by the loop Qcycle - 2

p(a, ]/, u) du.

(3.215)

It is clear that this energy loss occurs as a result of irreversible switchings along the vertical parts of the above loop. Consequently Qcycle -- q~ + q~,

(3.216)

where q~ and q~ are losses occurred as a result of irreversible 'up' and 'down' switchings, respectively. Owing to the symmetrybetweenbranches I and 2, it can be concluded that q~ - q~.

(3.217)

CHAPTER 3 Scalar Preisach Models of Hysteresis

384

J

FIGURE 3.68

From (3.215), (3.216) and (3.217) we find q= - q/~ =

/~(a, fl, u) du.

(3.218)

In the nonlinear Preisach model, any input variation is associated with irreversible switchings of some elementary loops #(a, ]/, u(t))~=~u(t). It is natural to equate the hysteretic energy loss occurring for some input variation to the sum of energy losses resulting from the irreversible switchings of the above elementary loops. Since in the nonlinear Preisach model we deal with continuous ensembles of the elementary hysteresis loops, the above summation should be replaced by integration. Thus, if f~ denotes the region of points on the ~-//diagram for which the elementary hysteresis loops are irreversibly switched during some monotonic input variation, then the hysteretic loss, Q, for this input variation is given by Q=

/~(~, ]/, u) du d= d]~.

(3.219)

In the particular case when the distribution function # does not depend on u, from (3.219) we obtain the expression

Q -//n #(~'fl)(a-//)da dfl,

(3.220)

which has been previously derived for the classical Preisach model (see Section 3.1). A typical shape of the region f2 is shown in Fig. 3.68. The region f2 can be subdivided into a triangle and some trapezoids which in turn can

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

385

{2

(u,,u_)

+,

_

FIGURE 3.69

"8 u

(X

FIGURE 3.70

be represented as differences of triangles. Thus, if the integral (3.219) can be evaluated for any triangle region, then it will be easy to determine this integral for any possible shape of f2. In the case when f2 is a triangle, a very simple expression for the integral (3.219) in terms of experimentally measured function P can be derived. The derivation proceeds as follows. Let be a triangle (see Fig. 3.69) swept during a monotonic increase of input from some local m i n i m u m u_ to some value u+. According to (3.219), this input variation results in the hysteretic loss:

T(u+,u_)

Q(u_, u+) -

(u+,u_)

/a(~,fl, u) du d~ dfl

=/ui+(/ui(/flaP(cx'fl'u)du)dfl) d~"

(3.221)

By using Fig. 3.70, we can change the order of integration in the two interior integrals as follows:

/ui (/fl~P(cx,fl, u)du)dfl- /ui (/ulP(~' fl'u)dfl) du"

(3.222)

CHAPTER 3 ScalarPreisach Models of Hysteresis

386

FIGURE 3.71

Now, by invoking the formula (3.185), from (3.222) we find

fu~(ffl ~

#(o~,fl, u) du

)

l fu~(fuU~ap(~x'[3'u)

d~ = --~

O~c]fl

d]/) du

lfu~[aP(~,u,u) 2

_

a~

]

c~P(a, u_, u) du. l

(3.223)

By substituting (3.223) into (3.221) and by changing this time the order of integration according to Fig. 3.71, we derive

lfuU+(fu~[C]P(cx'u'u) Q(u_, u+) = --~ _ _ O~

OP(cx,u_,u)]3~ du)doe

l fuU+(fuU+[cqP(cx'u'u) _ c]P(cx'u-'u)]&x)du 2

-~

_

~

l f u+[P(u+, u, u) -

~

P(u, u, u) - P(u+, u_, u)

+ P(u, u_, u)] du.

(3.224)

By using (3.181) in (3.224), we finally obtain

l fun+P(u+, u_, u) du. Q(u_, u+) - -~

(3.225)

The formula (3.225) is remarkably simple if compared with a similar formula (3.115) derived for the classical Preisach model. The formula (3.225) allows one to evaluate hysteretic energy losses for arbitrary input variations. Indeed, consider some input variation for which the region f] has

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

387

the shape shown in Fig. 3.68. Then, the corresponding hysteretic energy loss is given by (3.219), which can be written as k Q - Q(Mn, mn) + ~-~[Q(Mn-i, mn-i) - Q(Mn-i, mn-i-1)].

(3.226)

i=1

Each term in (3.226) can be evaluated by employing the formula (3.225). The formula (3.225) admits the following geometric interpretation. Consider a minor loop formed as a result of back-and-forth input variations between two consecutive extrema u_ and u+. If a hysteresis nonlinearity is representable by the nonlinear Preisach model, then all comparable minor loops of this nonlinearity have equal vertical chords. Consequently, P(u+, u_, u) can be construed as a vertical chord of the above minor loop. This implies that the integral in (3.225) is equal to the area A of this loop. Thus, the expression (3.225) can be written as A Q(u_, u+) - ~.

(3.227)

From (3.227) we immediately infer the following result.

For any loop of hysteresis nonlinearity representable by the nonlinear Preisach model, hysteretic losses occurring along ascending and descending branches are the same. By using this result we can relate hysteretic losses occurring for arbitrary input variations to certain cycle losses. This can be accomplished in exactly the same way as was done for the classical Preisach model. In particular, the expressions (3.122) and (3.123) can be derived. We conclude this section by the discussion of two facts that may facilitate further appreciation of the advantages of the nonlinear Preisach model over the classical one. Consider the hysteresis nonlinearities exhibited by the Stoner-Wohlfarth magnetic particles (see Fig. 3.72). It is pointed out in Section 3.1 that these nonlinearities cannot be described by the classical Preisach model. The reason is that degenerate minor loops traced as the input varying back and forth between ar and fl' are not congruent. It is easy to see that the hysteresis nonlinearities exhibited by Stoner-Wohlfarth magnetic particles can be described by the nonlinear Preisach model. First, this is clear from the fact that the property of equal vertical chords is satisfied because all minor loops are degenerate and, consequently, they have equal (zero) vertical chords. Second, the explicit representation (3.174) for the Stoner-Wohlfarth hysteresis nonlinearities can be found. Indeed, it can be

CHAPTER 3 ScalarPreisach Models of Hysteresis

388

fu

,,, ,,, ~N~,~

-hrn

I

h m U

I

II 1I I

I

I I

FIGURE 3.72

shown that

[a(a, fl, u(t))

= f~t)

- fu+(t) c~(o~ -

2

hm, fl q- hm)

(3.228)

where ~ is the Dirac function. By substituting (3.228) into (3.174), we find

f(t) = f~t) - fu+(t) ~hm,_hmU(t ) q_ f~t) + fu+(t) 2 2 "

(3.229)

It can be easily checked that the expression (3.229) indeed represents the hysteresis nonlinearities depicted in Fig. 3.72. Finally, the following observation may help the reader to appreciate the extent to which the nonlinear Preisach model is more general than the classical one. The classical Preisach model represents hysteresis nonlinearities that, for any reversal point, have (regardless of past history) only one branch starting from this point (see Fig. 3.73(a)). This branch is congruent to one of the first-order reversal curves. This is an apparent consequence of the congruency property of the classical model. On the other hand, the nonlinear Preisach model (3.174) describes hysteresis nonlinearities with the property that for any reversal point there are infinite possible branches starting at this point (see Fig. 3.73(b)). A particular realization of these branches is determined by a particular past history. This property follows, for instance, from the expression (3.214) where all terms are input-dependent. For the classical model, the similar expressions (3.71)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

389

f

Ji ///

/1 I'

I

- -

/

U

/ U

max

max

(b)

(a)

FIGURE 3.73

and (3.74) have only the last terms which are input-dependent. The above observation shows that the nonlinear model is endowed with a much more general mechanism of branching than the classical model. 3.2.3 'RESTRICTED' PREISACH MODELS OF HYSTERESIS

In the previous section we have discussed the nonlinear Preisach model of hysteresis which is a far-reaching generalization of the classical Preisach model. This generalization has been achieved by assuming that the distribution function y is dependent on the current value of input u(t). Another approach to the generalization of the classical Preisach model is to assume that the function y depends on stored past extremum values of input, {Mk, mk}. This approach was briefly explored in [47]. Here, we shall follow this approach as well, however, our treatment of the model itself and the identification problem for this model will deviate appreciably from the discussion in [47]. We begin with the simplest case when the function y depends only on the first global maximum, M1. Thus we define a new Preisach-type model as

fit) - l I T

#(~' fl' M1)~aflu(t) da dfl + C M 1

,

(3.230)

,~ ,J l M 1

where M1 is the largest input maximum since the departure from the state of negative saturation, and the support of #(a, fl, M1) is the triangle TM1 defined by inequalities flo~flKa~M1 (see Fig. 3.74). We shall use the following interpretation of the above model. For any fixed extremum M1, the model (3.230) will be used to describe hysteresis behavior in the region confined between an ascending branchfu~t ) and a first-order reversal curve fMlfl (see Fig. 3.75). For this reason, the model (3.230) can be termed a

CHAPTER 3 Scalar Preisach Models of Hysteresis

390

TM1

FIGURE 3.74

'restricted' Preisach model of hysteresis. However, the above'restriction' does not diminish at all the region of applicability of the model. This is because M1 may assume any value between fl0 and a0. The constant CMI in (3.230) can be easily determined from the condition that the model matches the output values f+l for u(t) - M1 and f_ for u(t) = flo. Indeed, from (3.230) we easily derive

ft~l = f f ~ , It(a, ]~, M1) da dfl +

CM1 ,

(3.231)

9,,,1 M 1

f

fl, M1) da d ] / + CM1.

= - ff.

(3.232)

.J,J I M1

From (3.231) and (3.233) we find

CM1 =

fM1 -'~"f -

2

'

(3.233)

which leads to the following representation for the model (3.230):

fit) - ffTM It(a, fl, M1)~&~u(t) da d~ +

f~l +f-

(3.234)

1

Next, we shall be concerned with the solution of the identification problem. The essence of this problem is in determining the function #(a, fl, M1) by fitting the model (3.234) to some experimental data. It turns out that It(a, fl, M1) can be found by matching second-order reversal curves f M l ~ shown in Fig. 3.75. To do this, we introduce the function

F(a, fl, M1) "-

1

-~(fMlfl ~ --&l/J).

(3.235)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

391

f

we

~/

/

L//.'///

~

--.

M1

fMI~OC fMll3 /

~__~........~---~:~

,L_______second-.order reversal curve

FIGURE 3.75

d

d

-

(a)

(b) FIGURE 3.76

This function is equal to one-half of the output increments along the secondorder reversal curves. To relate this function to #(~, fl, M1), consider the diagrams shown in Fig. 3.76.

CHAPTER 3 Scalar Preisach Models of Hysteresis

392

By using exactly the same line of reasoning as in the case of the classical Preisach model (see Section 3.1.3), from these diagrams and (3.234) we derive F(~, fl, M1) --~

,liT (~,fl)

=

#(0r fl', M1) da' aft'

, #(a', fl', M 1 ) d s

aft'.

(3.236)

By differentiating (3.236) twice, we obtain #(a, fl, M1) = -

~2F(a, fl, M1)

.

(3.237)

By recalling the definition (3.235) of F(a, fl, M1), from (3.237) we find #(a, fl, M1) =

1 ~.02fM~fl~ 2 Oa c~fl

(3.238)

Thus, if the distribution function # is determined from (3.237) or (3.238), then the restricted Preisach model will match the output increments along second-order reversal curvesfM1/~a attached to the first-order reversal curve fM~fl. This implies that the limiting ascending branch f + will be matched because this branch coincides with the second-order reversal curvesfMlfloa and because the value f_ is matched by the model. From (3.235) we also find that the output increments along the first-order reversal curve fMlfl will be matched because these increments are twice F(M1, fl, M1). Since the limiting ascending branch, output increments along the first-order transition curvesfMlfl and the output increment along the second-order reversal curves fM~fla will all be matched, we conclude that the second-order transition curves fM~fl~ themselves will be matched. Up to this point it has been tacitly assumed that input variations are started from the state of negative saturation (u(t)<,flo). This is a natural assumption because the state of negative saturation is a well-defined state. However, another well-defined state is the state of positive saturation (u(t)>~ao). If input variations start from this state, then it is natural to assume that the distribution function depends on the first global m i n i m u m ml. This leads us to the following restricted Preisach model:

f(t)

-

ff_ fa(a, fl, ml)~/~flu(t) da dfl + Cml,

(3.239)

,I ,l l ml

where the support of ~(~, fl, ml) is the triangle ml~
Zml

defined by inequalities

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

393

/

Tm1

/

FIGURE 3.77

The model (3.239) can be regarded as a counterpart of the model (3.230) and our immediate goal is to establish some connections between distribution functions # and ~. To this end, we first find the constant Cm~ in (3.239) by matching the output values fml and f+:

fml -- - []~ ddl f+ - dff~ dl

~(:<,fl, ml) d~ dfl + Cm~,

(3.240)

ml

~(~, fl, ml) da dfl + Cml.

(3.241)

ml

From (3.240) and (3.241) we find Cml

fml 2+f+ '

=

(3.242)

and the restricted model (3.239) can be represented as

fit) -- fiT

~(~' fl' ml)7~3u(t) d:r dfl +

fml+f+

(3.243)

mI

To determine the distribution function ~(~, fl, ml), we shall use the decreasing second-order reversal curvesfm 1~ which are attached to the increasing first-order reversal c u r v e s f m 1~ (see Fig. 3.78). As before we can introduce the function 1 F(~, fl, ml) (3.244) -

-

-2(fml~--fmlSc-fl)

and show that F(~, fl, ml) - [ [

JJT

~(a', fl')d~' dfl'.

(3.245)

CHAPTER 3 Scalar Preisach Models of Hysteresis

394

f

mj

T2

T//q/

/

"~

FIGURE 3.78 From (3.245), we derive 632F(&, fl, ml) ~(~, fl, ml) = -

(3.246)

By invoking (3.244) in (3.246), we obtain 2

1 c~

~(~,fl, m l ) -

~ r

7ml~fl "

(3.247)

Next, we will utilize the symmetry between the ascending and descending second-order reversal curves. In mathematical terms, this symmetry means that if ml---M1, ~---fl, fl - -0r (3.248) then F(~, fl, rex) = F(~, fl, MI).

(3.249)

The last formula can also be expressed as F ( - f l , - ~ , - M 1 ) - F(~x, fl, M1).

(3.250)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

395

By substituting (3.249) and (3.248) into (3.246) and taking into account (3.237), we derive ~t(~, fl, ml) = --02F(~'/~, M1) __ #(0r 8~8//

M1).

(3.251)

From (3.248) and (3.251) we finally obtain U(a, fl, M1) = ~(-fl, - a , -M1).

(3.252)

The last expression can be regarded as a generalization of the mirror symmetry property (3.46) previously derived for the classical Preisach model. Indeed, if M1 = a0 and ml = fl0, (/30 = -a0), then the models (3.234) and (3.243), respectively, coincide with the classical Preisach model. Consequently, we have U(a, fl, ~0) = U(a, fl),

ft(a, fl,-a0) = U(a, fl),

(3.253)

and the equality (3.252) can be rewritten as ~(a, fl) = ~ ( - f l , - a ) ,

(3.254)

which is the property of mirror symmetry for the classical Preisach model. The formula (3.252) also shows that, having determined the function /,t(a, fl, M1), we have solved the identification problem for the model (3.234) and its counterpart (3.243). This suggests that the model (3.243) is not independent of the model (3.234), but rather complements the latter model. In a way, we deal here with the same model written in two different forms. These two different forms correspond to two different initial states (states of positive or negative saturation). For this reason, in our subsequent discussion we will not draw any distinction between the models (3.234) and (3.243). In the sequel, all results will be discussed for the model (3.234), and it will be tacitly implied that they are also valid for the model (3.243). It is clear that the model (3.234) has an almost identical structure to the classical Preisach model. For this reason, it is apparent that the wipingout property holds for the restricted model (3.234). However, the congruency property of minor hysteresis loops undergoes some modification. This modification can be described as follows. We call minor hysteresis loops comparable if they are formed as a result of back-and-forth input variations between the same consecutive reversal values and these input variations take place some time after the same largest input maximum was achieved. The given definition of comparable minor loops implies the possibility of different past input histories between the time when the largest maximum M1 was achieved and the time when the above-mentioned backand-forth input variations commence. By using the same reasoning as in

CHAPTER 3 Scalar Preisach Models of Hysteresis

396

Section 3.1.2, we can prove that all comparable minor loops described by the restricted model (3.234) are congruent. It is clear from the above statement that the congruency of minor hysteresis loops prescribed by the model (3.234) for the same input reversal values are to a certain extent history dependent. This is not the case for the classical Preisach model. It turns out that the wiping-out property and the modified congruency property are characteristic of the restricted model (3.234) in the sense that the following result is valid. REPRESENTATIONTHEOREM. The wiping-out property and the congruency property

of comparable minor loops constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the restricted Preisach model (3.234). The proof of this theorem is very similar to the proof of the representation theorem for the classical Preisach model. For this reason, this proof is omitted. It is clear that the modified congruency property is more general than the congruency property of the classical Preisach model. Thus the restricted model (3.234) is more general than the classical one. In addition, the restricted model (3.234) allows one to fit experimentally measured firstand second-order reversal curves, while the classical Preisach model is able to fit only first-order reversal curves. Since higher-order reversal curves are 'sandwiched' between first- and second-order reversal curves, it is natural to expect that the restricted model will be more accurate than the classical Preisach model. We next turn to the discussion of numerical implementation of the restricted model (3.234). As before, we will show that double integration in (3.234) can be completely avoided and that an explicit expression for the output, fit), in terms of the experimentally measured function, F(~, fl, M1), can be derived. The derivation proceeds as follows. From (3.234) by means of simple transformations we find

fit)- ffs ~l ~t~~(a' fl' M1) d~ dfl - ffs ;~ ~t~la(a, fl, M1) do~dfl + f~l + f 2

=2f sM1(t) q_ f/~l

+f-.

2

fl, M1) d~xdfl -

fWM 1

fl, M1) d~x dfl (3.255)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

397

F I G U R E 3.79

By invoking (3.231) and (3.232), we derive

f/rM l~(a, fl, M1) dc~ dfl

-- f~ll

1

-f-

(3.256)

2

By substituting (3.256) into (3.255), we obtain

fit) - 2 H /a(c~,fl, M1) dc~ dfl + f _ . dds ~VI1(t) The s e t S~dI (t) can be subdivided into n trapezoids As a result, we have

(3.257)

Qk shown in Fig. 3.79.

n(t) /fs+- (t) #(~X,fl, Ml)d~xdfl - y~ f/Q la(~x,tiM1) dc~ dfl. M1

k=l

(3.258)

k

Each integral in the right-hand side of (3.258) can be represented as

ffQ k

fi, M1)

d, - ff r (Mk,mk-1) l~(a, fl, M1) da dfl - [[ #(~, fl, M1) dc~ dfl. Jdr (Mk,mk)

(3.259)

By recalling (3.236), from (3.259) we find

ffQ #(~, fi, M1) dc~dfl - F(Mk, mk-1, M1) - F(Mk, mk, M1). k

(3.260)

CHAPTER 3 ScalarPreisach Models of Hysteresis

398

By substituting (3.260) into (3.258) and then (3.258) into (3.257), we finally obtain n(t)

fit) - 2 E[F(Mk, mk-1, M1) - F(Mk, mk, M1)] + f - .

(3.261)

k=l

Thus, the formula (3.261) expresses the output, fit), explicitly in terms of experimentally measured second-order reversal curves which are related to the function F by Eqn. (3.235). This formula has a two-fold advantage over the expression (3.234). First, double integration is avoided. Second, the determination of ~(~, fl, M1) through differentiation of experimentally obtained data (see (3.237) or (3.238)) is completely circumvented. This is a welcome feature because the above differentiation may amplify noise inherently present in any experimental data. So far we have discussed the restricted Preisach model for which the distribution function/~ depends only on the first global extremum M1 or ml. It is natural to call it a first-order restricted Preisach model. However, further generalizations in this direction are possible. In these generalizations, the function # is assumed to be dependent on some finite sequence of past dominant extrema: la(~, fl, M1, ml, M2, m2 . . . . . Mk, mk). It is natural to call these models high-order restricted Preisach models. To make the general idea of these models clear, consider a second-order restricted Preisach model. In this model, the function/~ is assumed to be dependent of M1 and ml, and the model itself is defined as

fit) - ffTM

#(~' fl' M1, ml)~;aflu(t) da dfl + CMlml,

(3.262)

lml

where the support of/~ is the triangle ZMlml specified by inequalities ml<<,fl<<,o~M1 (see Fig. 3.80). For any fixed extrema M1 and ml, the model (3.262) will be used to describe hysteresis behavior in the region confined between the firstorder reversal curve fMl2 and the second-order reversal c u r v e fMlmlOr (see Fig. 3.81). This explains the origin of the term 'restricted', used for the name of this model. However, the described restriction does not narrow at all the region of applicability of this model. This is because M1 and ml are free parameters that may assume any values between fl0 and a0. These values depend on a particular past input history and may even change with time when the previous extrema M1 and ml are wiped out by subsequent input variations.

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

399

1,

FIGURE 3.80 f

fM1m10~p

/ m,

/

,m,o /

/ I "i-;f~f/J/',

~U

FIGURE 3.81 The c o n s t a n t CMlml in (3.262) can be determined from the condition that the model matches the output values fM1 and fMlml"

fM1 -- iirM

la(Sr fl, M1, ml) de dfl + CMlml,

(3.263)

lml

fMlml -- -- ffTM

/./(0~,fl, M1, lml

ml) de dfl + CMlml.

(3.264)

CHAPTER 3 Scalar Preisach Models of Hysteresis

400

From (3.263) and (3.264), we find

CMlml --

f~I1 if- fM1 ml

(3.265)

which leads to the following expression for the model (3.262)"

fit) -

ffr

lml

~(~' ~' M1, ml)~aflu(t) d~ dfl +

f~I 1 -[-fM1 ml 2

(3.266)

To determine the distribution function #(~, fl, M1, ml), the experimentally measured third-order reversal curves fMl,ml~x~ (see Fig. 3.81) should be employed. By using these curves, we can introduce the function F(or ]~, M1, ml)

--

1 ~(fMlmlO~ --&lml~/J).

(3.267)

Exactly in the same way as before it can be shown that the second-order restricted Preisach model will fit the third-order reversal curves if the function F(a, ]/, M1, ml) is related to the function p(a, fl, M1, ml) by the equation

F(0r

M1, ml) - ] / ddT

#(0r fl', M1, ml) da' dfl'.

(3.268)

From the last equation we easily derive p(~,//, M1, ml) - -

02F(0r

M1, ml)

(3.269)

and

1 ~2fM 1ml or /2(cz, fl, M1, ml) - ~ c]~z~gfl

(3.270)

By its design, the model (3.262) is valid under the condition that the initial state of the hysteresis transducer is the state of negative saturation. If the initial state is the state of positive saturation, then the following counterpart of the model (3.262) can be used:

f(t) = ffTK 4

[_l(O~,fl, 17111,~/I 1)~a~u(t) d~ dfl +

ft~1 "q-fffl1~I1

(3.271)

lrhl

where rhl is a global m i n i m u m and M1 is a subsequent global m a x i m u m of the input.

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

401

As before, we can establish the following relationship between the function # and ~: (3.272)

/~(~, fl, M1, ml) - ~(-fl, -~x, -M1, - m l ) ,

which shows that (3.266) and (3.271) are just different forms of the same model. These two different forms correspond to two different well-defined initial states of the hysteresis transducer. It is apparent that the congruency property of minor hysteresis loops undergoes further modification (further relaxation) in the case of the secondorder restricted Preisach model (3.266). The essence of this modification is that for minor loops to be congruent the corresponding inputs must vary back and forth between the same consecutive reversal values and also assume in the past the same values of M1 and ml. It can be proven that the wiping-out property and the above-mentioned modified congruency property of minor loops constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the model (3.266). This is the so-called representation theorem for the second-order restricted Preisach model. As before, it can be shown that the following expression can be used for the numerical implementation of the model (3.266): n(t)

fit)

-

2 E[F(Mk, mk_ 1, M1, ml)

-

-

f(Mk, mk, M1, ml)]

-~&lm

I9

(3.273)

k

The main advantage of this expression over the formula (3.266) is that it represents the output,fit), directly in terms of the experimentally measured third-order transition curves. It is clear from the previous discussion that higher-order restricted Preisach models of hysteresis can be defined as

fit) -- f (_

[a(~, fl, M1, ml ..... Mk)~u(t) d~ dfl

J J l M lml""Mk

q'- CMlml...Mk.

(3.274)

It is also apparent now how higher-order transition curves can be used for the determination of the function #(~, fl, M1, ml . . . . . Mk). It goes without saying that by increasing the order of the restricted Preisach model we can increase the accuracy of this model. However, this increase in accuracy is amply paid for by the increase in the amount of experimental data required for the identification of higher-order models. For this reason, the use of restricted Preisach models of order higher than two does not seem to be practically feasible or attractive. It is instructive to note that another

402

CHAPTER 3 Scalar Preisach Models of Hysteresis

o~

X

R(Z,~.u) FIGURE

3.82

approach to the construction of Preisach models that can match any desired number of higher-order reversal curves has been recently proposed in [48-50]. We conclude this section with a brief discussion of the 'restricted nonlinear' Preisach model of hysteresis. It is clear from the name of this model that it combines the main features of the 'restricted' and 'nonlinear' models. The first-order restricted nonlinear (input-dependent) Preisach model can be defined as

fit) = /R M1U

#(:e, fl, u(t), M1)4&flu(t) d~ dfl + fMlu(t) q-fu+((t) 2

(3.275)

where the moving support of #(a, fl, u(t), M1) is the rectangle RMIu specified by inequalities flo~flKu(t), u(t)KaKM1 (see Fig. 3.82), and the meaning offMlu(t) is clear from Fig. 3.83. By introducing the function (3.276)

P(CX, ]~, U, M1) = fMlflOe u -- f M l f l u

and by using the same reasoning as in the previous section, we can establish that

P(0~, ]~, u, M1) -- 2 [ ] #(a', fl', u, M1) doe' dfl'. ddR (o4fl,u) From (3.277) we can derive the following expressions: /~(0e, ]3, u, M1) =

1 c~2P(0e, ]3, u, M1) 2

0a3fl

,

(3.277)

(3.278)

and

/~(0e, j~, U, M1) --

1 ~.c92fMlfl~u 2 c~e aft

(3.279)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

403

I

61

FIGURE 3.83

We can also derive the following explicit expression for the output, fit), of the model (3.275): n(t) f ( t ) -- fMlu(t) -Jr- ~[P(Mk+I, k

mk, u(t),M1) - P(Mk, mk, u(t),M1)]. (3.280)

It is worthwhile to remember here that the function P(~, fl, u, M1) is directly related to experimental data and has the meaning of output increments between the second- and third-order reversal curves. It is easy to formulate and to prove the representation theorem for the model (3.275). This is left to the reader as a useful exercise. 3.2.4 DYNAMIC PREISACH MODELS OF HYSTERESIS

In all the previous sections our discussion has been centered around the classical and generalized Preisach models of hysteresis that are rateindependent in nature. The term 'rate-independent' implies that in these models only past input extrema leave their mark upon the future values of output, while the speed of input and output variations has no influence on branching. The intent of this section is to relax the 'rate-independence

404

CHAPTER 3 Scalar Preisach Models of Hysteresis

property' of Preisach-type models. For this reason, new Preisach-type models that are applicable to the description of dynamic hysteresis are introduced. The identification problem of fitting these models to some experimental data is then studied. Finally, some discussion is presented concerning numerical implementation of the dynamic Preisach models of hysteresis. The main idea behind the dynamic Preisach-type models of hysteresis is to introduce the dependence of/x-functions on the speed of output variations, df This leads to the following dynamic Preisach models of hysteresis:

f(t) - ff~>>.3;(a, fl, d--fdt)9=3u(t)da dfl, fit) -

l/R( /x u(t)

df)

a, fl, u(t), -~ ~;~flu(t) da dfl +

(3.281)

f~t)-I-fu+(t). (3.282) 2

The above models are 'dynamic' generalizations of the classical Preisach model and the nonlinear Preisach model, respectively. Similar generalizations for the restricted Preisach models are apparent. They will not be discussed in this section because their treatment mostly parallels that of the model (3.281). The direct utilization of the models (3.281) and (3.282) is associated with some untractable difficulties. First,/x-functions depend on the unknown quantity df and this complicates numerical implementations of the models (3.281) and (3.282). Second, it is not clear how to pose the identification problems for these models. The above difficulties can be completely circumvented by using the power series expansions for/x-functions with respect to ~df.. (

d f ) -- //0 (~, fl) + ~-~//1(0r df ]~) q - . . . ,

(3.283)

/x ~, fl, ~

(

dr)

/X a, fl, u(t), ~

df (~, fl, u(t))+

-/X0 (a, fl, u(t))+ ~--~/Xl

"''.

(3.284)

By retaining only the first two terms of the above expansion, we arrive at the following dynamic models:

fl t ) - / f ~>>,~ #o(~, fl)9~3u(t) da dfl

-}--~dfffa)fl/xl(~,fl)~a3u(t) do~dfl,

(3.285)

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

fit) = ffI~

t~~176fl' utt))~a~u(t) da dfl +

405

f~t) q-fu+(t)

u(t)

q- --~df//Ru(t) #1 (or flu(t))~a~u(t)

da

dfl.

(3.286)

It is clear that in the case of very slow output variations, the above models are reduced to the corresponding rate-independent hysteresis models. This means that #0-functions in (3.285) and (3.286) should coincide with the/~functions of rate-independent models (3.149) and (3.174), respectively. In other words, the #0-functions in (3.285) and (3.286) can be determined by matching first-order and second-order transition curves, respectively. The above reasoning suggests that the models (3.285) and (3.286) can be represented in the following equivalent forms:

df

fit) =)~(t) + ~-~ JJ~>~/~fll (or f l ) ~ u ( t ) d~ dfl,

(3.287)

fit) =)~(t)+ ~df //R u(t) fl l (:r fl , u ( t ) ) ~a~ u ( t ) d :r d fl ,

(3.288)

where the f-terms stand for the 'static' components of hysteresis nonlinearities:

f (t) = ff~ >>-fl#o(~, fl)}~flu(t) d:c dfl, J:(t) - [[_ JJl<

u(t)

#o(O~,fl, u(t))}a~u(t) d~ dfl + fu+(t)q-Juft) f- . 2

(3.289) (3.290)

The expressions (3.287) and (3.288) are transparent from the physical point of view. They show that the instant speeds of output variations are directly proportional to the differences between instant and 'static' output values. We next turn to the identification problems of determining the #lfunctions by fitting the models (3.287) and (3.288) to some experimental data. We first consider the identification problem for the model (3.287). The following experiments can be used to solve this problem. Starting from the state of negative saturation, the input u(t) is monotonically increased until it reaches some value a at t = to and is kept constant for t ~>to. As the input is being kept constant, the output relaxes from its value fa at t = to to its static value )?a. According to the model (3.287), this relaxation process is described by the differential equation df + f =)~

(3.291)

406

CHAPTER 3 ScalarPreisach Models of Hysteresis

FIGURE 3.84

where

"ca = f f s ; #l (~X',fl') do~' dfl' - f f s + #l(Or

(3.292)

and S~- and S+ are negative and positive sets on the Preisach diagram (see Fig. 3.84) corresponding to the above input variation. The solution to Eqn. (3.291) is given by

fit) = (f~ - L ) e - G

t

'~

+ f~.

(3.293)

Thus % has the meaning of relaxation time and can be experimentally measured. Next, the hysteresis nonlinearity is brought back to the state of negative saturation. Starting from this state, the input is again monotonically increased until it reaches the value ~. Then the input is monotonically decreased until it reaches some value fl at time t - t or and is kept constant for t > t~. As the input is being kept constant, the output relaxes from its value f ~ at t = t~ to its static value)?~/~. The model (3.287) yields the following differential equation for the above relaxation process: -ca/~df ~ + f - f~/~,

(3.294)

where according to the model (3.287) and the Preisach diagram shown in Fig. 3.85 the coefficient za/~ is given by the expression za/~ --/fS~-~ #l(a" fl') d~162 dfl' -/fs+8 #1 (a', fl') d~' dfl'.

(3.295)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

407

c~

s~ {~.~1

FIGURE 3.85

By solving (3.294), we find t

fit) = (f~ - f~a)e ~ + f~a.

(3.296)

Thus, ra~ also has the meaning of relaxation time and can be experimentally measured. It is apparent that the relaxation time ra can be construed as the relaxation time ~aa, and, consequently, it belongs to the set {raft}. We next show that by knowing these relaxation times, we can find the function /A1(0~, ]~). To this end, we introduce the function q(~, ]/) = ~a - za/~.

(3.297)

From (3.292), (3.295) and (3.297), we find q(~, fl) = -2 ff_

/~1(~', fl') d~' dfl'.

(3.298)

J Jl

where Taft is a triangle shown in Fig. 3.85. From (3.298), we can easily derive the expression 1 02qa]~ #1(a,//) - 2 ~ ~fl"

(3.299)

By recalling (3.297), from (3.299) we find

/A1 ( a , fl) - -

1 02raft 2 ~ ~fl"

(3.300)

The solution to the identification problem for the input-dependent (nonlinear) model (3.288) can be found in a similar way. For this reason, our comments will be concise. We consider two types of relaxation

CHAPTER 3 Scalar Preisach Models of Hysteresis

408

processes. A first type process occurs after a monotonic input increases to some value a and a subsequent monotonic decrease of input to some value u. According to the model (3.288), the first type relaxation processes are described by the differential equation df + f =f~u

(3.301)

where

l~u = f f

JJs a

#1 (0r fl', U)da' d f l ' - / /

JJs+~

~1 (0r ]~', U)da' dfl'.

(3.302)

From the previous differential equation, we find fit) = (f~u-f~u)e - t

+f~u.

(3.303)

A second type relaxation process occurs when the input is first monotonically increased to a, then decreased to fl, again monotonically increased to u, and kept constant afterwards. These relaxation processes are described by the equation q

-

r~flu - ~ + f = f ~ u ,

(3.304)

where ~ c
(3.305)

As before, we have t

fit) = (f=flu - J:=flu)e "~'" + J:aflu.

(3.306)

It is clear from (3.302) and (3.305) that Z=u and z~/~uhave the physical meaning of relaxation times and can be measured experimentally. Knowing these relaxation times, we can define the function Q(~, fl, u) = ~u - r=/~u.

(3.307)

From (3.301), (3.304) and (3.306), we derive Q(~, fl, u) - - 2 / /

J JR (o~,fl,u)

fll (~', fl', u) d~' dfl'.

(3.308)

From (3.307), we obtain

I ~2Q(0r fl, u) ~0~cg/Y "

fll (0~, fl, U) -- ~

(3.309)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

409

By invoking (3.306), from (3.308) we have the alternative expression for/'1: 1 (?2ra/~_______2u" 2 {?~c~fl

/21 ((X,/J, U) --

(3.310)

Consider the relaxation processes that are symmetric to those discussed above. Suppose that we start from the state of positive saturation and reduce the input to some value fl, then increase the input to some value and keep it constant thereafter. Suppose also that the subsequent output variations are characterized by the relaxation times r ~ . If fl--~,

~--fl,

(3.311)

then due to the symmetry we have r ~ - r~/j.

(3.312)

1 02r~ /11 (~' ~) -- 2 {?&{?~"

(3.313)

As before, we can derive that

By substituting (3.311) and (3.310) into the right-hand side of (3.312) and by recalling (3.300), we obtain /11(~, fl) =

1 ~2"cafl __ -/11 (0{, fl). 2 0fl {?~

(3.314)

From (3.313) and (3.310), we conclude /11 (--/J, --0~) -- --/11(0r fl).

(3.315)

Similarly, we can introduce the relaxation times r~aa and prove that ~2 1 r~ /11(~, fl, U) -- 2 {?~0fi"

(3.316)

~=-fl,

(3.317)

If ~=-u,

then due to symmetry we have r.~

- "C~lJu.

(3.318)

CHAPTER 3 ScalarPreisach Models of Hysteresis

410

By using the same line of reasoning as before, from (3.309), (3.315), (3.316) and (3.317) we derive #i (-/~, -~, -u) - -#i (a,/~, u).

(3.319)

We next proceed to the numerical implementation of the dynamic models. The models (3.287) and (3.288) can be represented as the following differential equation:

df +fit)=;~(t), ?l(u(t))-~

(3.320)

where the hysteretic coefficient a(t) is defined by the formulas

?~(u(t)) = - [ [

#1(a, f l ) ~ u ( t ) d~ dfl,

(3.321)

?t(u(t)) = - ]]..

#1 (~, fl, u ( t ) ) ~ u ( t ) d a dfl

(3.322)

dJ~

J J l~u(t)

for the models (3.287) and (3.288), respectively. The explicit solution to Eqn. (3.319) is well known and can be expressed

as f(~) d~] ' fit) -- b(t) If0 + f0t a(~)b(~)

(3.323)

where f0 is the initial output value, and b(t) is given by the formula

b ( t ) - e x p ( - f0 t ad~ -~ ) "

(3.324)

The expressions (3.322) and (3.323) can be efficiently used for computing fit), iff(t) and a(t) are known. The calculation of static f-components in the case of the classical Preisach model as well as the nonlinear Preisach model has been discussed in detail in the previous sections. Since the mathematical structure of a(t) is similar to the mathematical structure of the rate-independent Preisach models, similar explicit formulas can be derived for the numerical evaluation of a(t). In particular, it can be shown that the following formulas are valid for a(t) in the case of models (3.287) and (3.288), respectively: n(t)

?z(u(t)) -- 2q(~0, fl0) - 4 y~[q(Mk, mk-X) -- q(Mk, mk)],

(3.325)

k=l 1

?l(u(t)) = --~Q(ao, flo, u(t)) n(t)

- E [ Q ( M k + I , mk, u(t)) -- Q(Mk, mk, u(t))].

k=l

(3.326)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

411

The above formulas are convenient not only because they give explicit expressions for integrals in (3.320) and (3.321), but also because these expressions are presented in terms of experimentally measured data. It has been tacitly assumed in the foregoing discussion that the relaxation processes used in the identification procedures are well characterized by single relaxation times. If this is not the case and several relaxation times have to be employed to describe the above relaxation processes, then the discussed dynamic models must be generalized. The natural way to generalize these models is to use higher-order differential equations with hysteretic coefficients in order to account for several relaxation times. We demonstrate such a generalization for the second-order dynamic model: d2f fl(1) (u(t)) df ~(2)(u(t))h- ~ + +fit)-f(t)

(3.327)

where

37(t) = ff~

po(a, fl)~u(t)dadfl,

~(1) (u(t)) -- l~

111 (:r

fl)~=flu(t) d:r dfl,

~(2)(u(t))= ff~~>~ #2(~,fl)~=~u(t)da dfl.

(3.328) (3.329) (3.330)

To find #1 (0r fl) and/A 2(0r fl), we shall use the same two relaxation processes as for the identification of the model (3.287). In the first process, we start from the state of negative saturation and monotonically increase input to some value a and keep it constant thereafter. As the input is being kept constant, the output relaxes. According to (3.326), this relaxation is described by the differential equation a(2) d2f

(1)df

~

--~t~ + a~ --~ + f - f~.

(3.331)

A solution to this equation has the form t

t

fit) = c(~l)e ~(1~+ C(2)e ~(2~+ ~ ,

(3.332)

where r(~1) and r(~2) are the roots of the characteristic equation a (2)"C2 Jr- a(~l)r + 1 - O.

(3.333)

CHAPTER 3 Scalar Preisach Models of Hysteresis

412

Consequently, a(2) =

1 (1) (2)'

(3.334)

a(~1 ) - _,c~1) _}_1:(2)

(3.335)

The constants r(1) and z~2) have the physical meaning of relaxation times and can be measured experimentally. This leads to the experimental determination of a(~2) and a(~1) according to (3.333) and (3.334). N o w we consider the second relaxation process. In this case we start from the state of negative saturation, increase input to some value a, then decrease input to some value fl and keep it constant thereafter. As the input is being kept constant, the output relaxes and this relaxation is governed by the equation a(2) d2f _(1) df ~fl ~ + u ~ ~ + d - f ~ ,

(3.336)

whose solution is given by t

c~(1)e fit)-..~

t

,~(2) ~ + t. ~e

3(1)

r (2)

~

~ +f~.

(3.337)

The constants 1:(1)a/~and f~/~-(2)have the meaning of relaxation times and can be measured experimentally. As soon as this is done, we can find a (2) and a(1). a(2) __ 1 a// 1:(1)1:(2)'

a(1)-~]/

.L.(1) .C(2) ~]~ + ~// 1:(1)1:(2) "

(3.338)

(3.339)

Next we introduce the functions (1) _ a(1) _ a(1)

(3.340)

(2) _. a(2) _ a(1) qafl aft'

(3.341)

which are directly related to the above-mentioned experimental data. Using the same reasoning that we have used many times before, we can

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

413

establish the formulas (1)-2fjf

(2) _ 2

ffT

111 (~', fl') d~' dfl',

(3.342)

112(0r fl') da' dfl',

(3.343)

from which we derive

111(0{, fl) --

,,2 (1) 1 v q~/~ 2 0~ Off'

#2(~, fl) -

lvq~-----~fl.

(3.344)

,,2 (2) (3.345) 2 0~ 0fl Thus, the identification problem for the second-order model (3.326)-(3.329) is solved. Extensions to higher-order dynamic models are straightforward. 3.2.5 PREISACH MODEL OF HYSTERESIS WITH ACCOMMODATION

The Preisach-type models described in previous sections exhibit the wipingout property. This property results in an immediate formation of minor hysteresis loops after only one cycle of back-and-forth input variations between any two consecutive extremum values. However, experiments show that in some cases hysteresis loop formations may be preceded by some stabilization process that may require an appreciable number of cycles before a stable minor loop is achieved. This stabilization process is often called in the literature the 'accommodation' or 'reptation' process. Sometimes this accommodation process can be appreciable and then it is important to model it. In this section we discuss a certain modification of the moving Preisach model which allows one to account for the accommodation process. Another approach to the Preisach modeling of accommodation is described in [51]. We define the Preisach model with accommodation as

fit) -

fro

fl,d(m))5;aflu(t)do~ dfl + r(u(t)).

(3.346)

J J l~ u(t)

In the previous formula r(u(t)) stands for a fully reversible component which is represented by some single-valued function of u(t), while

d (m) - Mfk

if

(a, fl), Rk(t),

where Mkf are local extremum values of o u t p u t f and

(3.347)

Rk(t)

are rectangular

regions formed after the extremum M f has been achieved (see Fig. 3.86).

CHAPTER 3 Scalar Preisach Models of Hysteresis

414

a 0

R2k+l

/

R2k

_ _ _

FIGURE 3.86 ao

I

i

v'

/

FIGURE 3.87

It is easy to see that the regions Rk(t) are reduced with time and can even be completely wiped out. This fact is reflected in (3.346) by indicating that Rk are dependent of t. It is also clear from Fig. 3.86 that MkI are local output minima for even k and local maxima for odd k. We next prove that the model (3.345) does account for the accommodation process. For the sake of clarity, we will show this for the simplest case when the initial state of a hysteresis transducer is the state of negative saturation, and the input u(t) is monotonically increased to some value u+ and is varied back and forth between u+ and u_ thereafter. More general cases can be treated in a similar way. As the input is increased to the value u+, the output reaches a maximum value M( which, according to (3.345) and Fig. 3.87, is equal to

M( = -//~R #(~' fl' Mr~ da dfl + r(u+), 0

(3.348)

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

415

(X Ro i

FI

FIGURE 3.88

where M0f =f_,

(3.349)

and f_ is the output value in the state of negative saturation. As the input is monotonically reduced to the value u_, the output reaches minimum value M2f. According to (3.345) and Fig. 3.88, this minimum value is given by

ffR1

Mr2 =

fl' M() dadfl -

ffRo

fl'MfO) dadfl

+ r(u_).

(3.350)

It is clear that M2f ~=M0f - f_.

(3.351)

As the input is monotonically increased again to the value u+, the output reaches some maximum value M3f. According to (3.345) and Fig. 3.89, for this value we find

Mf = -

ffRo

fl' MfO) d e d f l - / / a 2 #(a, fl, M f) dadfl

+ r(u+).

(3.352)

By comparing Figs 3.87 and 3.89 as well as formulas (3.347) and (3.351) and by taking the inequality (3.350) into account, we conclude M3f # M(.

(3.353)

As the input is monotonically decreased again to the value u_, the output reaches some minimum value M f, which, according to Fig. 3.90, is

416

CHAPTER 3 ScalarPreisach Models of Hysteresis (~

Ro

u

i

1

Ro

FIGURE 3.89 (x

/

R

FIGURE 3.90 given by

Mf = ffe3 /~(a' fl'Mf3) d a d f l -

ffRo

fl'MfO) d~dfl

+ r(u_).

(3.354) By comparing formulas (3.349) and (3.353) and by taking into account that R1 = R3 and the inequality (3.352), we find M f ~: M f.

(3.355)

By continuing the same line of reasoning, we can establish that

M /2k+l # Mf2k-1 '

(3.356)

Mf2k 7~ Mf2k_2.

(3.357)

and

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

417

f I

i

i

U+

U

F I G U R E 3.91

The last two inequalities show that some stabilization (accommodation) process is going on before a stable minor loop is asymptotically reached. This accommodation process is illustrated by Fig. 3.91. We next proceed to the discussion of the identification problem. It turns out that the model (3.345)-(3.346) is very general in nature and it is not clear at this point how to solve the identification problem for this model. However, the identification problem becomes tractable if the dependence of the #-function on f(m) is factored out. In this way, we arrive at the following model:

v(f(m))#(O~,fl)~&~u(t) dadfl

fit) - ffR

+

r(u(t)),

(3.358)

u(t)

where f(m) are defined in the same way as in (3.346), and v-function is assumed to be known. Thus the identification problem consists of determining the functions r(u) and #(0r fl) by fitting the model (3.357) to some experimental data. We begin with the determination of the function r(u). Suppose that starting from the state of negative saturation the input is monotonically increased to some value u. As a result of this monotonic increase, the output will reach some value f + on the limiting ascending branch. According to (3.357), for this output value we have

fu+ - -v(f_) ffR(~o,#o,U)

#(~, fl) d~ dfl + r(u).

(3.359)

Now consider the state of positive saturation and suppose that the input is monotonically decreased to the same value u. In this case, the limiting descending branch will be followed and some output value f u on this branch will be reached. According to (3.357), for the above-mentioned

CHAPTER 3 Scalar Preisach Models of Hysteresis

418

output value we find

fu

v(f+) ffR(~o,flo,U) #(a, fl)da dfl 4- r(u),

-

(3.360)

where as before f+ is the output value in the state of positive saturation. From (3.358) and (3.359), we derive

r(u) =

v (y+ ) f u+ + v (f _ ) f ;

vq+) + vif_)

.

(3.361)

Consider a particular case when v-function is even. In this case, since f_ = -f+, we find v(f+) = v(f_), (3.362) and from (3.260) we conclude 1

r(u) = -~ (f+ 4- fu).

(3.363)

This coincides with the previous result for the moving Preisach model. We next turn to the determination of the #-function in (3.357). Consider first-order reversal curves f~fl which are attached to the limiting ascending branch. By fitting these curves to the model (3.357), we arrive at the following equation:

f~fl = v(f~)

-

(~,flo,fl,fl)

d s dfl'

+ v(f_) ffR (~o,flo,~,fl)~(~" fl') d~' dfl'

F ( fl) .

(3.364)

vq_) ffR(~o,flo,~,fl) "(~" fl') da' dfl. v (f~)

(3.365)

From (3.363), we obtain

f~fl - r(fl)

v(f~)

[[

= JJR (~,flo,fl,fl) #(a',fl')da'dfl'

On the other hand, by fitting the limiting descending branch ffl- to the model (3.357), we find

f ~ = v(f+) ffR(~o,flo,fl, fl)

#(s fl') da' dfl' + r(fl).

(3.366)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

419

The expression (3.365) can be transformed as follows:

J;

f[ ~(~,, fl')da' aa R(~o,flo,~,fl)

r(fl)

v(f+)

_-

+ ffR(o~,flo,fl,fl)

dfl

#(a', fl') da' dfl'.

(3.367)

By introducing the function T(o~, fl)

- ffR(~o,flo,~,fl)#(a', fl')d~' dfl'

(3.368)

and by using (3.364) and (3.367), we derive the following expression for T(c~, fl) in terms of experimentally measured data: T(~, fl) =

v(f~)[f~-

r(fl)]-

v(f+)[fafl -

r(fl)]

v(f+)[vq;) + v(f-)l

.

(3.369)

By knowing the function T(a, fl) and by employing the same reasoning as usual, we can derive from (3.367) the following expression for/2: #(~, fl) = -~92T(a' fl). & Sfl

(3.370)

For the purpose of numerical implementation of the model (3.357) we do not need to know the function/2. This is because an explicit expression for the output in terms of the function T(c~, fl) can be derived. The derivation proceeds as follows. From (3.357) and Fig. 3.86, we find n(t)

f(t)- Ev(Mfk_l)ffR k=l

fl) dadfl 2k-1

n(t)

P l,

v(M~k) [[.. k=0

/~(~, fl) de aft.

(3.371)

,J,J l,(2k

Owing to the definition of T(a, fl), we have

fiR

#(0~,fl) da dfl

- T(Mk+I, ink) -- T(Mk, ink).

(3.372)

2k-1

Similarly, we find

ffR #(~' fl) d~ dfl 2k

- T(Mk+I, m k + l ) -- T(Mk+I, mk),

(3.373)

CHAPTER 3 Scalar Preisach Models of Hysteresis

420

where it is assumed that

T(M1, m0) = 0.

(3.374)

By substituting (3.371) and (3.372) into (3.370), we derive n(t)

f(t) = y~ v(Mfk_l)[T(Mk+l, mk) -- T(Mk, mk)] k=l

n(t)

- ~ v(Md2k)[T(Mk+l, mk+l) -- T(Mk+I, mk)].

(3.375)

k=0

This is the final expression for the output in terms of the function T which is directly related to the experimental data by the formula (3.368). We will conclude this section by the discussion of one sufficient condition which guarantees the convergence of the stabilization (accommodation) process. This condition is given by the following inequality:

maxlv'(Md)l .mffx ffR MY

#(a, fl) da dill - q < 1,

(3.376)

where vI stands for the derivative of v-function, and R is any rectangle within the limiting triangle. The proof that the condition (3.375) guarantees the convergence of the stabilization process is given below for the simplest case. This is the same case that was discussed in the beginning of this section when we proved that the model (3.345) did account for accommodation. More general cases can be treated similarly. From (3.357), we find 2k+1 -- --v(f_)

#(~, fl) de dfl 0

- v(M;)//R

l~(e, fl)dedfl + r(u+),

(3.377)

+

where the notation R+ is used for R2k because they do not change from one cycle to another. Similarly we have M2fk = v ( M f - 1 ) f f R _

#(~' fl) dadfl

- v(f_) [[_ #(a, fl)dadfl + r(u_), d,I R 0

where R_ stands for R2k+l.

(3.378)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

421

From (3.376) and (3.377), we derive Mf2k+l--Mf2k_l----[v(Mf2k)--v(Mf2k_2)]

x [[_

/~(0r fl) dc~dfl,

(3.379)

JJN +

Mf2k --Mf2k_2--[v(Mf2k_l) - v(M2fk_3)]

x fro

fl) de dfl.

(3.380)

J,JN

It is apparent that

]v(Mf2k) -- v(Mf2k_2) l<~max v' (Mf ) ]lMf2k - Mf2k_21, Mf

(3.381)

]v(Mf2k_l) -- v(Mf2k_3) l~ maxlv'(MJ) llM~2k_l -

(3 382)

Mf'

M 2k_31

From (3.375) and (3.378)-(3.381), we derive

[Mf2k+1 -- M ; _ 11~ q[Mf2k -- M;_21

(3.383)

]M; - Mf2k_21 <~qlMf2k_l - Mf2k_31.

(3.384)

By combining (3.382) and (3.383), we obtain ]M2fk+l _

Mf2k_l 1<~q2 IMf2k_l _ Mf2k_3 ],

IM;+2 - M ; I ~

q2[M; - Mfk-2

["

(3.385) (3.386)

Inequalities (3.384) and (3.385)mean that extrema {Mfk+l } and {Mfk } form two contracting sequences. For this reason, there exist the limits: lim M f = Mf+ k--~cx~ 2k+1

(3.387)

lim Mf2k = Mfu_. (3.388) k--~ Consequently, the condition (3.375) guarantees the convergence of the accommodation process. 3.2.6 MAGNETOSTRICTIVE HYSTERESIS AND PREISACH MODELS WITH TWO INPUTS

The essence of the magnetostrictive phenomenon is the dependence of strain on magnetization. Since the magnetization can be varied by applied magnetic fields, this opens the opportunity to control the strain of

CHAPTER 3 ScalarPreisach Models of Hysteresis

422

magnetostrictive rods by magnetic fields. For this reason, magnetostrictive materials (especially Terfenol-type materials with giant magnetostriction) are very attractive as actuator materials for many applications, such as robotics, active vibration damping, micromotors, etc. However, magnetostrictive materials exhibit hysteresis, which represents a problem for fine-positioning applications. If hysteretic effects of magnetostrictive materials could be predicted, then actuator controllers could be designed to correct for these effects. This would result in high precision actuators powered by magnetostrictive materials. It is clear that mathematical models for magnetostrictive hysteresis could facilitate the design of the above controllers. The Preisach modeling of magnetic hysteresis of Terfenol-type materials was first attempted in [52]. It is known that magnetostrictive materials exhibit hysteretic behavior with respect to variations of two variables: magnetic field and stress. Thus, the problem of developing the Preisach-type models with two inputs presents itself. For the sake of generality, we consider a hysteresis nonlinearity that can be characterized by two inputs u(t) and v(t) and an output fit). In magnetostriction applications u(t) is the magnetic field, v(t) is the stress, whilef(t) is the strain. We shall discuss the following Preisach-type model with two inputs:

fit) = ['[" la(~, fl, v(t))~aflu(t)d~ dfl ,Ida

+ f~

v(a, fl, u(t))~aflv(t)dadfl.

(3.389)

In the above model, the dependence of the functions # and v on v(t) and u(t), respectively, reflects the cross-coupling between two inputs. The investigation of the model (3.388) is greatly facilitated by its equivalent representation as a model with moving supports for [a(a, fl, v(t)) and v(cr fl, u(t)). To arrive at this representation, we introduce the sets + + . . Su( t) , Ru(t), Su(t), Sv(t), Rv(t) and Sv(t), which can be defined by the same formulas as (3.150)-(3.152). By using these sets, the expression (3.388) can be transformed as follows:

fit)- ffR

u(t)

+ JJs II

#(~' fl' v(t))ya~u(t) dec dfl + ffI~

u+(t)

JJs v+(t)

II

J J S-u(t)

J J Sv(t)

v(t)

v(~, fl, u(t))~&~v(t) da dfl

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

423

We next find a simple expression for the last four terms in (3.389). Consider first a state of negative saturation and a subsequent monotonic increase of inputs until they reach some values u(t) and v(t) 9 Letfu(t)v(t + ) be the resulting output value. Then, according to (3.389), we find:

--

f

ff~

,Idle, u(t)

+ dds II

#(~, fl, v(t))d~

aft- [1~

e I Kv(t)

,, v(t))dadfl

-

u+(t)

II

v(~, fl, u(t))da

dfl

vt , ,, u(t))dadfl

J,]Su(t)

+ ff v(a, fl, u(t)) dadfl dds v~t)

-

ff

JJSv(t)

v(a, fl, u(t)) dad//. (3.391)

Similarly, starting from the state of positive saturation, we can definef~t)v(t) and derive the following expression: --

f~t)v(t)

If l~(a, fl, v(t)) dadfl ddRu(t)

+

fl v(a, fl, u(t)) dadfl d,I Rv(t)

+ ff l~(a,fl, v(t))dad//- ff l~(a,fl, v(t))dadfl dis u+(t) JJS~(t) + ff v(a, fl, u(t)) dadfl - ff v(a, fl, u(t)) dad//. JJS v(t) + ,],Y gu(t) (3.392) From (3.390) and (3.391) we derive

-21 +(fu(t)v(t) + f~t)v(t)) -- ~

Su+(t)

#(~, fl, v(t)) dadfl -

/s u(t)

~(~, fl, v(t)) dadfl

+ f v(~, fl, u(t))dadfl- f v(a, fl, u(t))d~dfl. d.qSv+(t) J Sv(t) (3.393) By substituting (3.292) into (3.389), we arrive at the following representation of the model (3.388):

fit)

-

ff

fl. v(t))~aflu(t)dadfl

JJ l
+

fro v(=.fl. u(t))~,flv(t)dadfl

d d tKv(t)

+

if- -2 (fu(t)v(t) "[- f~t)v(t))"

The last expression can be regarded as a generalization of (3.160).

(3.394)

424

CHAPTER 3 Scalar Preisach Models of Hysteresis

(U2,V2)

(u~ ,v~) ,-v FIGURE 3.92

By using the same line of reasoning as in previous sections, it can be established that the model (3.393) has the following two characteristic properties. WIPING-OUT PROPERTY. Only the alternating series of past dominant extrema of

u(t) and v(t) are stored by the model (3.393), while all other past extrema of u(t) and v(t) are wiped out. PROPERTYOF EQUALVERTICALCHORDS. All

minor hysteretic loops corresponding to the same consecutive extremum values of u(t) for the same fixed value of v have equal vertical chords regardless of the past history of variations of u(t) and v(t). The same is true for minor hysteretic loops formed as a result of back-and-forth variations of v(t) for any fixed value of u. It turns out that the model (3.393) has also the following distinct property. PATHINDEPENDENCEPROPERTY. Consider two points (Ul, Vl) and (u2, v2) on the u-

-v plane and a set of paths connecting these points and corresponding to monotonic variations of both u(t) and v(t) (see Fig. 3.92). Then, the output increment predicted by the model (3.393) does not depend on a particular monotonic path between the points (Ul, Vl) and (u2, v2). PROOF. For any instant of time the model (3.393) can be represented as

fit) =/~S+(t ) p(a, fl, v(t)) da d f l - /~S-(t) #(a, fl, v(t)) dadfl

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

q- fff2 + (t)

v(oe, fl, u(t)) doe dfl -

~ff2-(t)

425

v(oe, fi, u(t)) doedfl

1 + q- -2 G(t)v(t) if- fff((t)v(t))"

(3.395)

Here S+ (t), S- (t), f2+ (t) and f2- (t) are positive and negative sets on which Ru(t) and Rv(t) are subdivided, respectively. It is clear that the above subdivision on positive and negative sets will be the same at the point (u2, v2) for all monotonic paths connecting this point with (Ul, Vl). This is because the monotonic variations of u(t) and v(t) will result in the same modification of negative and positive sets upon achieving the values u2 and v2, respectively. From here it is easy to conclude the validity of the above property. We next proceed to the discussion of the identification problem for the model (3.393). In order to determine the functions ~(oe, fl, v) and v(oe, fl, u) we shall use the first-order transition curves fa~v and fua/~. As before, these curves are measured for piecewise monotonic input variations started from the state of negative saturation. Namely, f~v is the output resulting from monotonic increases of two inputs to the values oeand v, respectively, and subsequent monotonic decrease of u(t) to the value ft. The output value fu~ can be determined in a similar way. We shall define the functions 1

F(oe, fl, v ) - ~ (f~/~v -ffl+), 1

p, u)=

-&).

(3.396) (3.397)

By using (3.393) and the diagram technique, it is easy to show that ,

v) -

(~,fl)

fl', v)doe' dfl'

= ffla(fflila(oe',fl',v) dfl')doe',

(3.398)

G(oe, fl, u) - ffR (~,~) v(oe', fl', u)doe' dfl' - ffla (ffl/~ v(oe', fl', u)dfl')doe',

(3.399)

0

where R(oe, fl) is the rectangle shown in Fig. 3.93. From (3.397) and (3.398) we derive 02F(oe, fl, V) ~(~,/~, v)=

a a,

(3.400)

CHAPTER 3 Scalar Preisach Models of Hysteresis

426

(a,~)

/,

R(O~,I~) I

FIGURE 3.93

v(a, t, u) =

c~2G(0r t, u)

.

(3.401)

As far as the numerical implementation of the model (3.393) is concerned, it can be accomplished without the formulas (3.398) and (3.400) but by employing explicit expressions for the integrals in (3.393) in terms of the functions F(~, t, v) and G(a, t, u). The derivation of these expressions proceeds as follows. For the first integral in (3.393) we find

([ #(a, t, v(t))~u(t) dadfl ddRu(t)

= 2 [[

JJS+(t)

#(a, t, v(t)) dadfl

-//R u(t)

/a ( a , fl, v ( t ) ) d a d fl.

(3.402) Similarly,

//R

u(t)

v(~, fl, u(t))4&~v(t) d~ dfl = 2 //~

+(t)

v(a, fl, u(t)) dadfl

-- /fav(t, V(~, j~, u(t)) da dfl. (3.403) From (3.390) and (3.391), we derive

1+ -~(l:u(t)v(t)- f ~t)v(t)) = -

u(t)

#(o~, [3, v(t)) do~aft

- ffRv(t~ v(o~,fl, u(t))de aft.

(3.404)

3.2 GENERALIZEDSCALAR PREISACH MODELS OF HYSTERESIS

427

(X

s-It)

I

1

,

FIGURE

3.94

By substituting (3.401) and (3.402) into (3.393) and taking (3.403) into account, we obtain

f(t) -- 2[ f~s+(t) #(e, fl, v(t))

dedfl +

ffn

+(t) v(e,

fl, u(t)) de dfl]

+

(3.405)

ff- fu(t)v(t)

A typical geometry for the set S+ (t) is shown in Fig. 3.94. From this figure follows

n(t)

ffs

l~(e, fl, v(t)) dedfl +(t)

= ~ k=l

ffR fl'v(t)) dedfl.

(3.406)

k

By using (3.397), we find

ffa #(e, fl, v(t)) de dfl = F(M~u), m~u), v(t)) - F[M(u)v,,k+l,m~u)v(t))'

(3.407)

k

where M~u) and m~u) form the alternating series of past dominant extrema of u(t). From (3.405) and (3.406) we conclude

f +(t)f #(e,sfl, v(t)) de dfl n(t)

= z..,LF,M, u) , m k(u), v(t))- F(M~U)_F1 , m~u) , v(t))]. k=l

(3.408)

CHAPTER 3

428

Scalar Preisach Models of Hysteresis

By using the same line of reasoning as before, we can derive the expression

fff~

+(t)

v(~, fl, u(t)) d~dfl

l(t)

= ~ [ G ( M k (v), m~v) , u(t)) - Gt^/t(V)v,.k+X,m~v) , u(t))] .

(3.409)

k=l

By substituting (3.408) and (3.407) into (3.404), we obtain (t)

+ 2 ~ [ F ( M ~ u), mp ) , v(t)) - F(M~U)+I,m~u) , v(t))] fit) --fu(t)v(t) + k=l

t(t)

+ 2 y~[G(M~ v) , m k(v) ,u ( t ) ) - G ( M 2 1 , m k(v) ,u (t))].

(3.410)

k=l

This is the final expression for the output in terms of experimentally measured functions F and G defined by (3.395) and (3.396), respectively. We next formulate the following result. REPRESENTATIONTHEOREM. The wiping-out property, the property of equal vertical

chords and the path independence property constitute the necessary and sufficient conditions for the representation of actual two-input hysteresis nonlinearities by the model (3.393) on the set of piecewise monotonic inputs. The proof of this theorem closely parallels the proof of the representation theorem from Section 3.2.2. The only new element is the utilization of the path-independence property. By using this property, it can be easily shown that an arbitrary simultaneous piecewise monotonic variation of u(t) and v(t) can be reduced to two consecutive piecewise monotonic single-input variations during which the other input remains constant. Further details of the proof of the theorem are left to the reader. In conclusion of this section, we consider one peculiar property of strain hysteresis in comparison with magnetic hysteresis. This peculiarity stems from even symmetry of strain hysteresis which exhibits butterflyshaped major hysteretic loops with respect to variations of magnetic field (see Fig. 3.95). It is clear that the first-order transition curvesf~v will also exhibit even symmetry. This can be expressed mathematically as

f~'~'v - fa~v

and f~v - f~+

(3.411)

if fl'--a

and

a'--fl.

(3.412)

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

429

f

'--.

.

.

.

.

FIGURE 3.95

By using (3.410) and (3.411), it can be easily proven that

~(a,/~, v) = - ~ ( - f l , - ~ ,

v).

(3.413)

Thus, the function #(a, fl, v) is of odd symmetry with respect to the line = -ft. In the case of magnetic hysteresis, the/2-function has an even symmetry with respect to the same line. It follows from (3.412) that the function/2(a, fl, v) cannot be positive everywhere on the a-fl plane. Actually, this function changes its sign across the line ~ = -ft. As a result, first-order and higher-order transition curves are not enclosed by major hysteretic loops. This property has been experimentally observed and it also distinguishes strain hysteresis from magnetic hysteresis. 3.2.7 EXPERIMENTAL TESTING OF PREISACH-TYPE MODELS OF HYSTERESIS

In our previous discussions of Preisach-type models of hysteresis we consistently tried to establish the necessary and sufficient conditions under which actual hysteresis nonlinearities can be represented by these models. These conditions establish the limits of exact applicability of Preisach-type models. As a result, these conditions allow one to judge to what extent one or another Preisach-type model is applicable. These conditions also clearly reveal the phenomenological nature of Preisach-type models and

430

CHAPTER 3 Scalar Preisach Models of Hysteresis

their physical universality. However, it is unrealistic to expect that actual hysteresis nonlinearities will satisfy these conditions exactly. For instance, the experimental results presented in [53] show that neither the congruency property nor the property of equal vertical chords of comparable minor loops are satisfied for some typical materials used in magnetic recording. For this reason, the classical and 'nonlinear' Preisach models, as well as some other Preisach-type models, cannot be expected to be absolutely accurate. However, we may try to use these models as approximate ones. Then the question arises how to judge the accuracy of these models. It has been emphasized throughout our discussion that branching is the essence of hysteresis. Consequently, any mathematical model of hysteresis is expected to predict this branching. In other words, models of hysteresis are expected to predict higher-order reversal curves (branches) which occur after several (or many) reversals of input. This prediction should be based on some limited experimental data used for the identification of hysteresis models. Thus, the accuracy of the classical, 'nonlinear' and 'restricted' Preisach models can be measured by their abilities to predict higher-order reversal curves on the basis of information provided by the first- and second-order reversal curves. This is exactly the criterion that has been chosen for the experimental testing of the accuracy of the Preisach-type models. This testing was first performed for two typical particulate materials: Co-coated ~;-Fe203 and iron magnetic-tape materials. The experimental work consisted of two main stages. During the first stage, the first- and second-order transition curves were measured by using a vibrating sample magnetometer. Experimentally measured first-order transition curves for the above two materials are shown in Figs 3.96 and 3.97, respectively. For each first-order transition curve there is a set of second-order transition curves which are attached to the above first-order curve. Experimentally measured sets of second-order transition curves are shown in Figs 3.98 and 3.99, respectively. These first- and second-order transition curves were used for the identification of the classical, 'nonlinear', and 'restricted' Preisach models. During the second stage of the performed experiments, several sets of higher-order (with the order up to six) transition curves were measured. The sequences of field reversal for these higher-order transition curves were chosen in such a way that these higher-order transition curves were more or less uniformly (and densely) distributed over the area enclosed by major hysteresis loops. This was important in order to get the information about uniform accuracy of Preisach-type models. Then, by using the developed computer software, which implements the Preisach-type models, various sets of higher-order transition curves predicted by the

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

431

.04

.02

-.02

-.04

-2000

-1500

-1000

-500

0

500

1000

1500

2000

FIGURE 3.96

Preisach-type models were computed for the same sequences of field reversals as in the experiment. The comparison of the higher-order transition curves predicted by the classical Preisach model (3.149) with the experimentally measured higher-order transition curves is demonstrated in Figs 3.100 and 3.101. Similar comparisons for the nonlinear Preisach model (3.174) and the restricted Preisach model (3.234) are shown in Figs 3.102 and 3.103. It is evident from these figures that the actual (experimentally measured) transition curves are 'sandwiched' between the predictions of the nonlinear and restricted Preisach models. This observation has prompted the idea to use an 'average' (superposition) model

1//

fit) -- -~

4-

[#(~, fl, u(t)) 4- #(a, fl, M1)]~a]~(t) d~ d]~

4

"

(3.414)

CHAPTER 3 Scalar Preisach Models of Hysteresis

432

1L

0.-

-.05-

-Z

: -,!

-3000

-2000

-tO00

0

1000

2000

3000

FIGURE 3.97

The numerical implementation of this model can be carried out by using the formula 1 n(t) fit) - f - + f~t) + ~.[P(Mk+I mk u(t)) -- P(Mk mk, u(t))] 2 -2 ' ' ' k=l

n(t)

+ ~ [ F ( M k , mk-1, M1) - F(Mk, ink, M1)],

(3.415)

k=l

which follows from (3.214) and (3.261). It is clear that the 'average' model inherits the common wiping-out property of the 'nonlinear' and 'restricted' models. Functions/~(~, fl, u) and ~(~, fl, M1) can be individually determined by matching first- and second-order reversal curves exactly as in the case of those models. On the other hand, the 'average' model further relaxes the congruency property of the 'restricted' Preisach model and the property of equal vertical chords of the nonlinear model. Namely, the 'average' model describes hysteresis nonlinearities whose minor loops have

,;,,,~

,~' o.

3.2

.

I

N o

.

~,,,~

GENERALIZED

~,

o

.

(nuIa) UOl",,~z.t:laul~l~

oN

,

~

SCALAR

i

~

9

,

PREISACH

~

o.

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MODELS

.o I"

o.

I

O 0

O O

I

-o

e2

o

OF HYSTERESIS

I"

o,,. I

~o

I

?,

9

(nuaa) uo!~z!lau~eR

"

o~o

q

-'-T-

o

(nu,te) uo~.~,~zr~au~l~

~.

433

434

,,-:

~ q (nu,.m)

(n~e)

o

uo1"*,mzr

uol~ez;'~uhR

i

,~ o

i

9

,-, ,"

i

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CHAPTER 3

I

o

,-:

-

.

i|i|

o

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~

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w|l!

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Scalar Preisach Models of Hysteresis

ill|

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(nt.u~) uol1,~zr~u3ux

~n

i i

9

(nuze) uo!'~zp,~uleR

q

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9

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I' I

I

I

l

I

NoLDrZLT~ND'C'R

I 'C

t"

I

i

l

1

I

o

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!

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c~

iI

o

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i

'

f 0

''

I

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(n=a)

"

~ (nuaa)

9

g

I

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NOU.Y'ZLL,3NOYR

o

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g

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"o NOLI,V'ZLT~N DV'R

'

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oi

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

I

' i ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 '

~

. 1

.

, (nru=~) NOU,Ir"ZU,3NDYR

435

436

....

I ....

~i

(ntua) NOIJ.V21J~N~VM

q

' ' l ' ' ~ ' ' l " ' ' " l

_

9

''1

CHAPTER 3

'

tli

I ' ' t ~

t"

I ....

'I ....

I ....

o

I ~'

f

i

"

i

Scalar Preisach Models of Hysteresis

....

.

(nma) NOIJ,VZI.~NgYI~

'_

.

I

.

.

NOIJ.Y"LLL3NDYI'I

| f

I

. ~

!t

.....

!-' 9

1 ~ 1 7 ,

'

,

'

,

I

;

,

, i"

1

NOIJ.Y'ZI.L.qNOYI~

I

( n m ~ ) NOIJ.V'ZLI.3.qDYN

(nmo)

"

1'1',

,

,

,

3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS

t-'

x

(nm~)

I

( n m a ) NOIJ.Y'ZIJ,,.qNDYR

I i

I

,

o

- o

437

438

1

, ~,

i

"7

I

~

,

, ~l

i

,,,,gl,Ill

,o o.

, t , ,

(ntn~)

9

(nm~)

o

! ....

, I ....

,i

CHAPTER

o

5'

~'

~

t

I,

!

.-.

I ,

I ....

|

I ....

9 I"

o NOI.L'rZLI~i~I~

3

L

Scalar

-

z

~ I

~.

Models

NOl~ll~tl~'CM

I'

~

NOI.LWZII,~NDWI

Preisach

ul

"

(nma)

~

o

..................

(nma)

~

~~ \\~

i o

~

of

Hysteresis

7

o

o o

o

oi I

/t~

~

~\\\ ~o~

.

,

~ 5

.

~,,,

(nmo)

(nine)

1,,,

.

t

1,,,

0 I

N OI.l.'krz LL;3 INIDVIW

.

NOI.I,YZLI.~NDYI~

.

i

.

~ ,

I

r~-

2

o

i

:I

i

IF'' i ....

gl

Ir

I ~ I ' P , , I ,

l~.

i Li~ . (nuae)

,

I,

~

,

i

i

~

439

0 .-.

o~

'~I ~

O. I

I , , , 1 , | I INIOLI,Y'ZI,I,..~NDY]I~

~

3.2 GENERALIZED S C A L A R PREISACH MODELS OF HYSTERESIS

i , ~ , , ,

R ~

E ~ FR L~

L;'

.

C~

[.,T.1

I,-,nq

440

i

''1''''1''''1''"'!''''1''

i

I" (nu~a) HOLLYZIJ.3NDV~

'

z

''l''''!

-

-

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o

8

.r

I''''I''~'IT~-I~

..,

~

!

P.

~

~=I

~,

Scalar Preisach Models of Hysteresis

....

i

i" NOM.~LI~.~NDYI~

(.u~a) ~otLvzu~xov~

"

i

(nuar

.., 9

i

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3.2 GENERALIZED SCALAR PREISACH MODELS OF HYSTERESIS 0.06

'

1

'

'

'

'

I

'

'

'

'

I

'

441

-

0.04 0.02 o

-0.02 -0.04 -0.06

-0.6

0 H [KOe]

.6

FIGURE 3.106

equal vertical chords only if they are formed after the same largest input maximum M1 was achieved. In other words, the property of equal vertical chords becomes history-dependent. It is clear that the average model relaxes much further the congruency property of the classical Preisach model than the 'nonlinear' or 'restricted' models. At the same time, the average model requires the same experimental data for its identification as the two aforementioned models. The 'average' model turned out to be remarkably accurate for all used (and densely distributed) sequences of reversal values of magnetic fields. The comparison of the higher-order transition curves predicted by this model with the experimentally measured higher-order transition curves is demonstrated in Figs 3.104 and 3.105. The above comparison suggests a higher accuracy of the 'average' Preisach model. This prompted the attempt to test this model for other magnetic materials. This testing was performed for six different magnetic-tape materials: Ampex641, Ampex-797, Ampex-D1, TDK-YHS, Maxell-Beta, and Memorex-3480. Sample results of the comparison of the higher-order transition curves predicted by the 'average' model with the experimentally measured higherorder transition curves are demonstrated for the above materials in Figs 3.106-3.111, respectively. Thus, it can be concluded that the extensive experimental testing presented above has revealed a remarkable accuracy of the 'average' Preisach model of hysteresis. It is believed that this accuracy is related to the fact that the average model provides the far-reaching generalization of the congruency property of the classical Preisach model. Actually, this generalization goes much further than in the cases of 'nonlinear' and 'restricted' Preisach

442

CHAPTER 3

.a.

,

,

0.02

I

'

'

'

S c a l a r P r e i s a c h M o d e l s of H y s t e r e s i s

'

I

'

'

'

'

I

Exp. -

Model

0.01 ~E

0

-0.01 -0.02 -0.5

0 H [KOe]

FIGURE 3.107 '

0.01 _- -

;

'

'

I

'

;

;

;

I

;

;

'

'

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o

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-2

-1

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0.02

Exp. Model

0.01 o

0

-0.01 -0.02 -1

0 H [KOe] F I G U R E 3.109

1

'

'

;-

REFERENCES

443 __

'

'

I

'

'

'

'

I

'

'

'

'

I

'

'

4

0.02 0.01 o

-0.01 -0.02 -1

0 H [KOe]

1

F I G U R E 3.110 '

0.02

'

'

I

'

'

'

'

I

'

'

'

'

I

I

I

I

'

'

'

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I

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0.01 ,~, o

-0.01

-0.02 -1

0 H [Koe]

1

F I G U R E 3.111

models, although the average model needs the same experimental data for its identification as the two aforementioned models.

References 1. H. Poincar6, Science and Hypothesis, Dover, New York (1952). 2. F. Z. Preisach, Ober die magnetische nachwirkung: Zeitschrift f~ir Physik, 94, 277-302 (1935). 3. L. N6el, Sur les effects d'un couplage entre grains ferromagn6tiques dou6s d'hyst6r6sis, Comptes rendus hebdomadaires des seances de 1 academie des sciences, 246(16), 2313-2319 (1958). 4. G. Biorci and D. Pescetti, Analytical Theory of the behaviour of ferromagnetic materials, Nuovo Cimento 7(6), 829-842 (1958).

444

CHAPTER 3 ScalarPreisach Models of Hysteresis

5. G. Biorci and D. Pescetti, Some consequences of the analytical theory of the ferromagnetic hysteresis, Journal de Physique et le Radium 20(2-3), 233-236 (1959). 6. G. Biorci and D. Pescetti, Some remarks on hysteresis, J. Appl. Phys. 37(1), 425-& (1966). 7. W.F. Brown Jr, Failure of local-field concept for hysteresis calculations, J. Appl. Phys. 33(3), 1308 (1962). 8. G. Bate, Statistical Stability of preisach diagram for particles of gammaFE203, J. Appl. Phys. 33(7), 2263 (1962). 9. J. G. Woodward and E. Della Torre, Particle interaction in magnetic recording tapes, J. Appl. Phys. 31, 56-62 (1960). 10. E. Della Torre, Measurements of Interaction in an Assembly of ?-Iron Oxide Particles, J. App1. Phys. 36(2), 518-522 (1965). 11. J.A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and minor loops -models and experiments, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 386 (1791), 251-261 (1983). 12. A. Damlanian and A. Visintin, A Multidimensional generalization of the preisach model for hysteresis, Comptes rendus de 1 academie des Sciences serie I-Mathematique 297(7), 437-440 (1983). 13. D. H. Everett and W. I. Whitton, A General Approach to hysteresis, Trans. Faraday Soc. 48(8), 749 (1952). 14. D.H. Everett, A General approach to hysteresis. 3. A Formal treatment of the independent domain model of Hysteresis, Trans. Faraday Soc. 50(10), 1077-1096 (1954). 15. D. H. Everett, A General approach to hysteresis. 4. An Alternative formulation of the domain model, Trans. Faraday Soc. 51(11), 15511557 (1955). 16. J.A. Enderby, The domain model of hysteresis. 2. Interacting domains, Trans. Faraday Soc. 52(1), 106-120 (1956). 17. A. Friedman, Foundation of Modern Analysis, Dover, New York (1982). 18. M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis, Nauka, Moscow (1983). 19. MI Brokate, Some Mathematical properties of the preisach model for hysteresis, IEEE Trans. Magn. 25(4), 2922-2924 (1989). 20. M. Brokate and A. Visintin, Properties of the preisach model for hysteresis, J. Reine Angew. Math. 402, 1-40 (1989).

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445

21. M. Brokate and A. Friedman, Optimal-design for heat-conduction problems with hysteresis, SIAM J. Control Optim. 27(4), 697-717 (1989). 22. A. Visintin, On the preisach model for hysteresis, Nonlinear Analysistheory methods & Applications 8(9), 977-996 (1984). 23. A.A. Visintin, A model for hysteresis of distributed systems, Ann. Mat. Pura Appl. 131, 203-231 (1982). 24. A. Friedman and K.-H. Hoffmann, Control of free-boundary probles with hysteresis, SIAM J. Control Optim. 26(1), 42-55 (1988). 25. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proceedings of the National Academy of Sciences of the United States of America-Biological Sciences 79(8), 2554-2558 (1982). 26. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, Wiley, New York (1961). 27. R. deGroot and P. Mazur, Non-Equilibrium Thermodynamics, NorthHolland, Amsterdam (1963). 28. W.G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York (1965). 29. G. Bertotti, Energetic and thermodynamic aspects of hysteresis, Phys. Rev. Lett. 76, 1739-1742 (1996). 30. G. Bertotti, Hysteresis in Magnetism, Academic Press, Boston (1998). 31. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, New York (1991). 32. I.D. Mayergoyz, Mathematical-Models of Hysteresis, Phys. Rev. Lett. 56 (15), 1518-1521 (1986). 33. I. D. Mayergoyz, Hysteresis models from the mathematical and control-theory points of view, J. App1. Phys. 57(8), 3803-3805 (1985). 34. I. D. Mayergoyz, Mathematical-Models of hysteresis, IEEE Trans. Magn. 22(5), 603-608 (1986). 35. T. Doong and I. D. Mayergoyz, On numerical implementation of hysteresis models, IEEE Trans. Magn. 21(5), 1853-1855 (1985). 36. I.D. Mayergoyz and G. Friedman, The preisach model and hysteretic energy-losses, J. App1. Phys. 61(8), 3910-3912 Part 2B (1987). 37. I. D. Mayergoyz and G. Friedman, Generalized preisach model of hysteresis, IEEE Trans. Magn. 24(1), 212-217 (1988). 38. I. D. Mayergoyz, Dynamic preisach models of hysteresis, IEEE Trans. Magn. 24(6), 2925-2927 (1988).

446

CHAPTER 3 ScalarPreisach Models of Hysteresis

39. I. D. Mayergoyz, G. Friedman, and C. Salling, Comparison of the classical and generalized preisach hysteresis models with experiments, IEEE Trans. Magn. 25(5), 3925-3927 (1989). 40. I. D. Mayergoyz, G. Friedman, and A. A. Adly, Superconducting hysteresis and the preisach model, J. Appl. Phys. 67(9), 5466-5468 Part 2B (1990). 41. I. D. Mayergoyz, The Classical Preisach model of hysteresis and reversibility, J. Appl. Phys. 69(8), pp. 4602-4604 (1991). 42. A. A. Adly, I. D. Mayergoyz, and A. Bergqvist, Preisach modeling of magnetostrictive hysteresis, J. Appl. Phys. 69(8), 5777-5779 Part 2B (1991). 43. G. Friedman and I. D. Mayergoyz, Input-dependent preisach model and hysteretic energy-losses, J. Appl. Phys. 69(8), 4611-4613 Part 2A (1991). 44. A.A. Adly and I. D. Mayergoyz, Experimental testing of the Average preisach model of hysteresis, IEEE Trans. Magn. 28(5), 2268-2270 Part 2 (1992). 45. I. D. Mayergoyz and A. A. Adly, Numerical implementation of the feedback preisach model, IEEE Trans. Magn. 28(5), 2605-2607 Part 2 (1992). 46. I. D. Mayergoyz and P. Andrei, Preisach modeling of clockwise hysteresis and its application to front propagation problems, J. Appl. Phys. 91(10), 7645-7647 Part 2 (2002). 47. K. Wiesen and S. H. Charap, A better scalar preisach algorithm, IEEE Trans. Magn. 24(6), 2491-2493 (1988). 48. G. Friedman and K. Cha, Power series expansion of hysteretic processes with wipe-out memory, Journal of Material Processing and Manufacturing Science 9 (1), 70-78 (2000). 49. G. Friedman and K. Cha, Representation of hysteresis with wipe-out memory, J. Appl. Phys. 89(11), 7236-7238 Part 2 (2001). 50. G. Friedman, Second-order Preisach model of scalar hysteresis, Physica B 275 (1-3), 173-178 (2000). 51. E. Della Torre, Magnetic Hysteresis, IEEE Press, Piscataway (1999). 52. J. B. Restorff, T. H. Savage, A. E. Clark, and M. Wum-Fogle, Preisach modeling of hysteresis in terfenol, J. Appl. Phys. 67(9), 5016-5018 Part 2A (1990). 53. C. Salling and S. Schultz, Using minor loop behaviour to investigate the preisach distribution for particulate media, IEEE Trans. Magn. 24 (6), 2877-2879 (1988).

CHAPTER 4

Vector Preisach Models of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Classical Stoner-Wohlfarth Model Vector Hysteresis Definition of Vector Preisach Models of Hysteresis and their Numerical Implementation Some Basic Properties of Vector Preisach Hysteresis Models Identification Problem for Isotropic Vector Preisach Models Identification Problem for Anisotropic Vector Preisach Models Dynamic Vector Preisach Models of Hysteresis Generalized Vector Preisach Models of Hysteresis. Experimental Testing References

4.1 C L A S S I C A L S T O N E R - W O H L F A R T H V E C T O R HYSTERESIS

447 458 467 485 494 504 511 526

MODEL

In magnetics, research on the modeling of scalar and vector hysteresis has been pursued along two quite distinct lines. Modeling of scalar hysteresis has been dominated by the Preisach approach. This approach can be traced back to a landmark paper [1]; it has been proved to be very successful and has won many followers. On the other hand, phenomenological modeling of vector hysteresis has long been centered around the classical Stoner-Wohlfarth model (S-W) model [2]. As a result, this model has further been developed and used in the area of magnetic recording [3-5]. The The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

448

CHAPTER 4 Vector Preisach Models of Hysteresis

attractiveness of the S-W can be attributed to its strong appeal to physical intuition. This appeal is, in turn, based on the fact that the S-W model is designed as an ensemble of single-domain, uniaxial magnetic particles. Since these particles have some features of physical realities, the S-W model is usually regarded as a physical (not mathematical) model. Owing to its popularity in magnetics, the S-W model is a natural benchmark for comparison with other vector hysteresis models. This is the main reason w h y we precede our discussion of vector Preisach models by the discussion of the S-W model. Since single-domain, uniaxial magnetic particles are the main building blocks of the S-W model, we begin with the discussion of hysteresis of these particles. We consider a single-domain, uniaxial magnetic particle with magnetization (magnetic momentum) M which may change its orientation under the influence of an applied field but has a constant magnitude. Such a particle is now commonly called a Stoner'Wohlfarth (S-W) particle. It is clear from the symmetry consideration that the vector M of this particle lies in the plane formed by the easy axis x and the applied magnetic field H (see Fig. 4.1). The orientation-dependent part of free energy g of the S-W particle is given by g = K sin 2 0 -/VI./7/,

(4.1)

where K is the anisotropy constant and 0 is the angle between the easy axis and M. The first term in the right-hand side of (4.1) represents the anisotropy energy, while the second term in (4.1) is the energy of interaction of magnetic m o m e n t u m M with the applied magnetic field. By using the Cartesian coordinates shown in Fig. 4.1, the expression (4.1) can be represented as g = K sin 2 0 - MHx cos 0 - MHy sin 0.

(4.2)

Equilibrium orientations of/VI correspond to minima of s and they can be found from the equations ~g --

~0 (~2g

=0,

(4.3)

~0 2 ~ 0.

(4.4)

2K sin 0 cos 0 + MHx sin 0 - MHy cos 0 - 0.

(4.5)

From (4.2) and (4.3) we derive

4.1 CLASSICAL STONER-WOHLFARTH MODEL VECTOR HYSTERESIS

449

M

H

,,

•

F I G U R E 4.1

By introducing the so-called switching field a: 2K - M'

(4.6)

the expression (4.5) can be rewritten as sin 0 cos 0 + Hx sin 0 - Hy cos 0 = O,

(4.7)

which is equivalent to sin 0

cos 0

= a.

(4.8)

Equation (4.7) (as well as (4.8)) is a quartic equation with respect to cos 0. For this reason, this equation m a y have two or four real solutions. Which of these two cases is realized depends on the applied magnetic field, H. In the first case, we have only one m i n i m u m and, consequently, only one equilibrium orientation of M. In the second case, there are two minima and this results in two equilibrium orientations of M. Thus, on the H-plane there are two different regions where M has one and two equilibrium orientations, respectively. On the b o u n d a r y between the above two regions one m i n i m u m and one m a x i m u m merge together. In the case of a minim u m , the inequality (4.4) holds, while for a m a x i m u m we Consequently, on the above b o u n d a r y we have 0g 0---0 = 0

have (~2~/~02~0.

02g and

~02

= 0

9

(4.9)

The first condition in (4.9) leads to Eqn. (4.8). Now, let us derive the second equation by using both conditions in (4.9). From (4.2), by using the notation (4.6), we derive

1

egg

sin 0 cos 0 00

= a -t

Hx

Hy

cos 0

sin 0"

(4.10)

CHAPTER 4 VectorPreisach Models of Hysteresis

450

Hy (Hxl ,

H

FIGURE 4.2

By differentiating (4.10), we obtain c9( 1 )egg 1 cgagHxsinOHycosO 0---0 sin 0 cos 0 ~ + sin 0 cos 0 002 = cos 2 0 -ff ~ ' s i n 2 0

(4.11)

By using both conditions (4.9), from (4.11) we find

HI

cos3---~ +

Hy sin 3 0

= 0.

(4.12)

Thus, on the boundary which separates regions with one and two minima, Eqns (4.8) and (4.12) are satisfied. These are two linear equations with respect to Hx and Hy. By solving these equations, we find

Hx =

- a c o s 3 (9,

Hy

-- 0r s i n 3 0.

(4.13)

From (4.13) we find the equation for the boundary separating the above two regions: Hx2/3+--y /_/2/3 _. 0~2/3 (4.14) This equation represents the astroid curve shown in Fig. 4.2. This astroid curve helps to visualize the solution of the quartic equation (4.7). This solution can be found by using the following geometric construction [6,7]. For a given external magnetic field H with components Hx and Hy the directions of M that satisfy Eqn. (4.7) are parallel to those tangent lines to the astroid that pass through the point H (see Fig. 4.2). The proof of the above statement proceeds as follows. Let Hx and Hy be Cartesian components of f/, and let Hxl and Hy1 be Cartesian coordinates of the point on the astroid at which the abovementioned tangent line touches the astroid. For the slope of this tangent

4.1 CLASSICAL STONER-WOHLFARTH MODEL VECTOR HYSTERESIS 451 line we have m=

I-ly1 - I-Iy Hx~ - H ~ '

(4.15)

which is tantamount to Hy 1 - Hy = m(Hxl - Hx).

(4.16)

Since the point (Sx1 , S y 1) belongs to the astroid, we have 2/3

/_/2/3 __ ~2/3

(4.17)

By using implicit differentiation of (4.17) with respect to Hxl, we derive 2 H~11/3 + 2-H,,--11/3 dHyl = 0 -3 3-dHxl "

(4.18)

From (4.18) we obtain m ~-~

ally 1 Hy I 1/3 " dHxl = - - ( ~ )

(4.19)

L e t / / b e the angle formed by the above tangent line with the easy axis of the particle. Then, m = tan//, (4.20) and from (4.19) and (4.20) we find HYl -- - tan 3 ft.

(4.21)

Hx I

Expressions (4.17) and (4.21) can be construed as two simultaneous equations with respect to two unknowns: Hxl and Hy 1. By solving these equations, we obtain Hxl -- - ~ c o s 3 fl,

Hy 1 -- cxsin 3 ft.

(4.22)

By substituting (4.20) and (4.22) into (4.16), we find cxsin 3 f l - Hy = tan/J(-cx cos 3 f l - Hx).

(4.23)

A trivial transformation leads to 0r

3 fl cos/J + cos 3 fl sin fl) q- Hx sin fl - Hy cos fl - 0,

(4.24)

CHAPTER 4 VectorPreisach Models of Hysteresis

452

Hy 6

a

Hx

FIGURE 4.3

which can be simplified as sin fl cos fl 4- Hx sin fl - Hy cos fl = 0.

(4.25)

It is clear that Eqns (4.7) and (4.25) are identical. Consequently, fl = 0.

(4.26)

This proves the validity of the above-described geometric construction. It can be shown that equilibrium orientations of M correspond to the tangent lines with smallest slopes. It is clear that, when the point H is outside the astroid, only two tangent lines_,are possible, and therefore there is only one equilibrium orientation of M. When H is inside the astroid, there are four tangent lines. However, only two of these tangent lines represent equilibrium orientations of the magnetization M. Which one is realized depends on the previous history of the magnetization. The described geometric rules allow one to compute hysteresis loops of a S-W particle for the case when the applied magnetic field is restricted to vary along one arbitrary chosen direction. Suppose that this direction is specified by the line a-a' (see Fig. 4.3) and that the magnetic field is first monotonically increased from its value H_ corresponding to point I to the value H+ corresponding to point 6 and then is monotonically decreased back to H_. The dependence of the magnetization projection along the line a-a t on the value of the magnetic field H exhibits hysteresis that is shown in Fig. 4.4. This is clear from Fig. 4.3. Indeed, as we move up along the line a-a t, the equilibrium (stable) orientations of M coincide with directions of tangent lines to the right-hand side of the astroid until we reach point 5.

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 453 Ma_a,

H 8-8'

/

FIGURE 4.4

At this point a 'switch' from the right-hand side to the left-hand side of the astroid occurs and, after that point, stable orientations of M coincide with directions of tangent lines to the latter part of the astroid. As we move down along the line a - a I from point 6, we use the tangent lines to the lefthand side of the astroid to determine the stable direction of/~'i. However, at point 2 a 'switch' from the left-hand side to the right-hand side of the astroid occurs and, after that point, stable orientations of M coincide with directions of tangent lines to the latter part of the astroid. Thus for the points of the line a - d , which are inside the astroid, there are two stable orientations of M that result in two different branches of the hysteresis loop shown in Fig. 4.4. It is clear from the preceding discussion that if the applied field varies along the easy axis x then a S-W particle exhibits the rectangular hysteresis loop shown in Fig. 4.5. It is also clear that if the applied magnetic field is varied along the direction perpendicular to the easy axis, then due to symmetry there is no hysteresis effect and a S-W particle exhibits the single-valued magnetization curve shown in Fig. 4.6. Thus, the shapes of hysteresis loops depend on the direction along which the applied field is being varied. It has been shown in Chapter 3 (see formula (3.221)) that the loop shown in Fig. 4.4 can be represented in terms of the rectangular ~-loop. G. Friedman found a very interesting generalization of the formula (3.221) (see reference [8]). This generalization represents the magnetization of a S-W particle for arbitrary (not only collinear) variations of magnetic fields in terms of rectangular ~-loops. The basis for this representation is the notion that there are two distinct states (vector branches) for any S-W

454

CHAPTER 4 VectorPreisach Models of Hysteresis

MX

HX

F I G U R E 4.5

,'H

F I G U R E 4.6

particle. In the first state, stable orientations of the magnetization coincide with directions of tangent lines to the left-hand side of the astroid. The notation/vl + (r H(t)) will be used for the magnetization in the first state, where ~(t) is the angle formed by the applied magnetic field with the easy axis and H(t) is the magnitude of the magnetic field. In the second state, the stable orientations of M coincide with directions of tangent lines to the right-hand side of the astroid. The notation M-(q~(t), H(t)) will be used for the magnetization in the second state. It is clear that ~r + (q~(t), H(t)) and/~I- (~(t), H(t)) can be found geometrically by using the previously described 'astroid' rule or by solving the quartic equation (4.7).

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 455 The rectangular loop operator ~a,_~ will be employed to describe switching from the first state to the second state and vice versa. The input v(t) for this operator is given by cos q0(t) [Hx(t)2/3 + Hy(t)2/313/2 I cosq~(t)l

v(t)-

(4.27)

It is clear from (4.27) and (4.14) that v(t) reaches the value 0r as the tip of /7/(t) crosses the right-hand side of the astroid, and v(t) reaches the value -c~ as the tip of H(t) crosses the left-hand half of the astroid. This shows that the switching of the rectangular loop ~a,_~ occurs at the same time as the switching of the S-W particle from one state to another. By using this fact we can represent the magnetization M(t) of the S-W particle as

~r

= -~(t)~a,_~v(t) + ~t(t),

(4.28)

where 1

3(t) - ~[/~+ (~0(t), H(t)) - ~l-(qo(t), H(t))],

(4.29)

-. - ~[/~I 1 d(t) + (q0(t) , H(t)) + ~ i - (q0(t) , H ( t ) ) ] .

(4.30)

It is clear from (4.28), (4.29) and (4.30) that/~(t) =/VI + (q0(t), H(t)) when ~a _~v(t) - 1, and M(t) - M - (qo(t), H(t)) when ~a,_~v(t) - - 1 . Switchings of r from I to - 1 and vice versa occur at the times when the tip of H(t) crosses the astroid. This proves that formulas (4.27)-(4.30) give the right representation for the magnetization of the S-W particle. Having described the basic properties of a S-W particle, we can now proceed to the discussion of the S-W hysteresis model. This model is designed as an ensemble of S-W particles. Consider an infinite set of S-W particles with different orientations of their easy axes and different values of switching field c~. The notation So, a will be used for a S-W particle whose switching field is equal to c~and whose easy axis forms the angle 0 with the x-axis. By using this notation, the S-W model can be represented mathematically as

A/i(t) - f f ~(0, a)'So,~H(t) dO da,

(4.31)

where ~(0, ~) is a distribution function that should be determined by fitting the model to some experimental data.

456

CHAPTER 4 VectorPreisach Models of Hysteresis

The expression (4.31) defines the S-W model in terms of magnetic quantities such as magnetization M and magnetic field H(t). However, it is possible to interpret the S-W model as a general mathematical model of vector hysteresis by writing this model in the form

A f" (t) - fl ~(0, a)So,~(t) dO da,

(4.32)

wheref(t) is the vector output, while fi(t) is the vector input. The output of the operator So,~ can be formally determined by using the astroid rule or the quartic equation (4.7) in which Hx and Hy are replaced by ux(t) and uy(t), respectively. In other words, the S-W model can be defined in purely mathematical terms without using any connections of this model to some physical objects such as uniaxial, singledomain magnetic particles. Such a purely mathematical point of view of the S-W model may have two-fold advantages. First, it suggests some possibilities of using this model not only in the area of magnetics. Secondly, it may open some opportunities for further generalization of this model. By using representation (4.28)-(4.30) for the S-W particles, the S-W model can be written in terms of ~-operators. This suggests some connections between the S-W model and the Preisach-type models. In particular, Preisach-type diagrams can be used to keep track of switching of different S-W particles. The S-W model has been known and used in magnetics for a long time. Gradually, it has been realized that this model has certain limitations. The most important of them can be summarized as follows. Since the S-W model is designed as an ensemble (superposition) of particles (hysteresis nonlinearities) with symmetric loops, this model does not describe nonsymmetric minor loops. This limitation is often attributed to the fact that the S-W model does not account for 'particle interactions'. This limitation can be somewhat corrected by expanding the set of elementary hysteresis operators So, a and by introducing the operators So,~,~ with shifted astroids (Fig. 4.7). Then, the generalized S-W model can be represented as A

f(t) - f f / f r

~, ~)'So,~,~u(t) dO da d~.

(4.33)

However, this generalization will require much more computational work for the numerical implementation of the S-W model. Even without this

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 457 Y -(X \\

,

FIGURE 4.7

generalization, the S-W model is computationally slow. This is in partbecause the calculations of outputs of individual elementary operators So,~ require the solution of quartic equations associated with the astroid construction. In addition, individual outputs should be integrated over some distributions of S-W nonlinearities (particles) with respect to their easy axis directions and switching fields. This requires the evaluation of double integrals in the case of the classical two-dimensional S-W model (4.32). The last difficulty is magnified in the case of the generalized S-W model (4.33). Furthermore, the identification problem of finding the distribution function ~(0, a) by fitting the S-W model to some experimental data has not been adequately addressed yet. Solutions to this problem are usually achieved by some artwork rather than by using a well-established procedure. Our research has been motivated by the desire to circumvent the limitations of the S-W model described above. To achieve this goal, we have turned to the Preisach approach and tried to extend it to the vector case. The guiding idea in our efforts has been the notion that the scalar hysteresis is a particular case of vector hysteresis. As a result, many important and characteristic properties of vector hysteresis can be exhibited in the scalar case. By exploring this notion, new vector Preisach models of hysteresis have been developed. These models have many of the desirable features of the scalar Preisach hysteresis models, and they represent a viable alternative to the S-W model.

458

CHAPTER 4 VectorPreisach Models of Hysteresis

4.2 D E F I N I T I O N O F V E C T O R P R E I S A C H M O D E L S OF HYSTERESIS AND THEIR NUMERICAL IMPLEMENTATION For the sake of generality, a vector hysteresis nonlinearity will be characterized below by a vector input ~(t) and a vector outputf(t). In magnetic applications, ~(t) is the magnetic field, whilef(t) is the magnetization. The most immediate problem we face is how to define vector hysteresis in a mathematically rigorous as well as physically meaningful way. To do this, it is important to understand what constitutes in the case of vector hysteresis the essential part of the input history that affects the future variations of output. In the case of scalar rate-independent hysteresis, experiments show that only past input extrema (not the entire input variations) leave their mark upon future states of hysteresis nonlinearities. In order words, the memories of scalar hysteresis nonlinearities are discrete and quite selective. There is no experimental evidence that this is the case for vector hysteresis. As a result, we must resign ourselves to the fact that all past vector input variations may affect future output values. The past input variations can be characterized by an oriented curve L traced by the tip of the vector input ~(t). Such a curve can be called an input 'hodograph'. Vector rate-independent hysteresis can be defined as a vector nonlinearity with the property that the shape of curve L and the direction of its tracing (orientation) may affect future output variations, while the speed of the input hodograph tracing has no influence on future output variations. Next, we shall give another equivalent definition of rate-independent vector hysteresis in terms of input projections. This definition is very convenient in the design of mathematical models of vector hysteresis. Consider input projection along some arbitrary chosen direction. As the vector ~(t) traces the input hodograph, the input projection along the chosen direction may achieve extremum values at some points of this hodograph. In this sense, the extrema of input projections along the chosen direction sample certain points of the input hodograph. If the projection direction is continuously changed, then the extrema of input projections along the continuously changing direction will continuously sample all points on the input hodograph. In this way, the past extrema of input projections along all possible directions reflect the shape of the input hodograph and, consequently, the past history of input variations. Thus, we arrive at the definition of vector rate-independent hysteresis as a vector nonlinearity with the property that past extrema of input projections along all possible directions may affect future output values. It is clear that mathematical models of vector hysteresis are imperative for self-consistent descriptions of systems with

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

459

i1 iI i

s

s

s S

,'

I III

Lr

" J"

o ~. *" ~, ~

.~ ~ ' '

s

i I

. . o

i S

i

II

i i

s

II

s

i

t

%

I

% %

i

s 9

...."

i

s s s

s

s S

]-~r-~(-~.U-~(t)).. ,~, *

r

..

A

%

i

s

t

i

I

II

9

I

i i s i

I

e

I

FIGURE 4.8 vector hysteresis. These models should be able to detect and store past extrema of input projections along all possible directions and choose the appropriate value of vector outputs according to the accumulated history. To detect and accumulate the past extremum values of input projections along all possible directions, the scalar Preisach models (Preisach's particles) can be employed. These scalar models are continuously distributed along all possible directions (see Fig. 4.8). Thus scalar Preisach models are the main building blocks of the vector model, which is constructed as a superposition of scalar models. This can be expressed mathematically in two dimensions as )7(t) - ~~1=1 ~F~(~. ~(t)) ds

(4.34)

and the integration in (4.34) is performed over a unit circle. Similarly, a three-dimensional vector Preisach model can be written in the form f(t) - ~~1 =1 ~ FF(r. ~(t))dsr, where the integration is performed over a unit sphere.

(4.35)

460

CHAPTER 4 VectorPreisach Models of Hysteresis A

The scalar Preisach models F~ are defined by

F~(~ 9 fi(t)) = fl)]~ v(e, fl)~/~(~, fi(t))de dfl

(4.36)

for isotropic vector models, and

F~(~ 9 fi(t)) - ff~>/~ v(e, fl, ~)~//(? 9 fi(t)) de dfl

(4.37)

for anisotropic vector models. Ideas of the construction of vector Preisach models that are somewhat similar to those described above have been briefly mentioned (without any analytical details) in [9,10]. Some similarities can also be found between our definition of the vector Preisach models and a purely mathematical vector generalization of the scalar Preisach model discussed in [11]. The following proposition further elucidates the above definition of vector Preisach models. PROPOSITION. The integration in (4.35) over a unit sphere can be reduced to the

integration over a unit hemisphere. PROOF. Consider the partition of the unit sphere into two hemisphere C+ and C-. For every point ~ + ~ C + there is a corresponding opposite point ~- ~ C- (see Fig. 4.9) ~+ = - ~ - . (4.38) From (4.35) we find

f(t) = ffc+ ?+F~+ (? +" fi(t)) ds+ + fJc- ?-F~- (?-" fi(t)) ds_,

(4.39)

where F~+ (~ +. fi(t)) - ff~

v(e, fl, ~ +)~a/~(~+. fi(t)) de dfl,

(4.40)

F ~ - ( ~ - . fi(t)) = ff~

v(e, fl, ~ - ) ~ ( ~ - .

(4.41)

fi(t))dedfl.

The following identity can be verified for any continuous piecewise monotonic function v(t)" ~ v ( t ) = -~_~,_~ (-v(t) ). (4.42)

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

461

+

FIGURE 4.9

Indeed, if v(t) < fi, t h e n - v ( t ) > -fl and

~aflv(t)- -1,

while

~_~,_a(-v(t)) - +1.

(4.43)

If v(t) increases and exceeds ~, then -v(t) decreases and becomes smaller than-c~. Consequently,

~v(t)-

1,

while

~_~,_~(-v(t)) - -1.

(4.44)

Similarly, it can be shown that the identity (4.42) holds for any monotonic decrease of v(t). In the last integral in (4.39) we change the variable ~- to -~ +. As a result, we find

l / c - -f -F~- (~- " ~(t)) as_ - - f/c+ ~+F_~+ ( - ~ + . ~(t))ds+,

(4.45)

where F_~+ (-~ + 9~(t)) -

)/~ v(e, fl, -~ +)~c~(-~ +. ~(t)) de aft.

(4.46)

By using the identity (4.42), the last integral in (4.46) can be transformed as follows" ff~ ~>~v(~,fl,-~ - - ~ ddc~

+. ~(t)) d~ dfl

v(c~, fl, -~+)~_1~_~(~+. ~(t)) da dfi.

(4.47)

CHAPTER 4 VectorPreisach Models of Hysteresis

462

In the last integral we will change a to -]7 and ]7 to -a, then according to (4.46) and (4.47) we obtain F_~+ ( - 7 + . fi(t)) = - ~

dda >~

v(-fl, -~, -~+)~3(7+. fi(t)) d~ aft. (4.48)

By substituting (4.48) into (4.45), we find as_

fc

=

fZ >.3

+

ds+"

(4.49)

By substituting (4.40) into the first integral in (4.39), we have

f f + ~+?~+ (F+. fi(t)) ds+ ,tIC =ffc~+(fZ +

>.3

v(e, fl,-f+)~aB(-f+.fi(t))&zdfl)ds +.

(4.50)

From (4.39), (4.49) and (4.50) we obtain

f"(t) =

+ ff + 7[fZ tvt=, >>.

fl, ~ +) + v(-~, -~, --f +))

x9~3(~ +. fi(t)) dad//] ds+.

(4.51)

By introducing a new function

v' (o~,fl,-f +) = v(o~, fl,-f +) + v(-fl, -a, -~ +),

(4.52)

from (4.51) we derive

f(t) -

ffc 7 +"F~+ (~+. fi(t)) ds+, +

(4.53)

where

~'~+(~+ 9~(t)) - f L ~ v'(~, fl, ~+)~3(~+. fi(t)) da dfl. This completes the proof.

(4.54) []

The proven proposition suggests that (4.53) and (4.54) can be regarded as an equivalent definition of the vector Preisach model. This definition will

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

463

be used in our subsequent discussions and the superscript ' will be omitted. It is also clear that a similar proposition is valid for the two-dimensional vector Preisach model (4.34) as well. Consequently, this model can be represented as )7(t) = f

~ +F~+ (~ +. fi(t)) dl+,

(4.55)

,/L +

where L+ is a semicircle and F'~+ is defined by (4.54). It is apparent from the above proof that particular choices of semispheres and semicircles in (4.53) and (4.55), respectively, are unimportant; all these choices will lead to equivalent vector Preisach models. From the above proof we can also find some interesting symmetry properties of the function v'. Indeed, if in (4.52) we change a to - f l , / / t o - a , and ~ + to - ~ +, then we find =

(4.56)

+

From (4.56) and (4.52), we obtain (4.57) This expresses the property of mirror symmetry of the function vf with respect to the line a - -//. This also extends the definition of the function vI from a unit semisphere or unit semicircle to the entire unit sphere or unit circle, respectively. In the case of isotropic vector models, the expression (4.57) can be simplified as follows: v' (a, fl) = v'(-fl, - a ) , (4.58) which is similar to the symmetry property of the #-function for the classical scalar Preisach model. Up to this point, the vector Preisach models have been defined in coordinate invariant forms (4.34), (4.35), (4.53) and (4.55). However, in many applications it is more convenient to use the expressions for the vector Preisach models in spherical and polar coordinates for three and two dimensions, respectively. In the case of spherical coordinates we have ds+

-

sin 0 dO d~o,

~+

- e 0 ,-~ r

-~ r +

.fi(t)

-

uo,~o

(t) ,

(4.59)

where e0,~0 is a unit vector along the direction specified by angles q~ and 0, and uo,~o(t) is the projection of fi(t) along the direction of e0,~0. By using (4.59), the three-dimensional vector Preisach model (4.53)(4.54) can be written as

f (t) =

f0270

A euo, e(t) sin 0 dO dq~, -~o,eFo,

(4.60)

CHAPTER 4 VectorPreisach Models of Hysteresis

464

where A

Fo, q)UO,q)(t) =

v(e, fl, O, q))4&flUO,q)(t) dedfl. >>.~

(4.61)

The last two formulas can be combined into one expression:

( ff~>~ v(e,fl, O,q));&~uo,~o(t)dedfl)

f(t) =

-do,~o

x sin 0 dO dq~.

(4.62)

Similarly, the two-dimensional vector Preisach model (4.55) can be represented in polar coordinates as A

f(t) =

~FCu~(t) dq~,

(4.63)

where A

Fq~uq~(t) -

v(e, fl, q))~flUq~(t) de dfl.

(4.64)

In the previous formulas, ~r is a unit vector along the direction specified by a polar angle q~, Fr is the scalar Preisach model (operator) for this direction, and uq~(t) is the projection of fi(t) along the direction of ~q~. By combining (4.63) and (4.64), we have

f (t) =

-do

v(e, fl, qo)~flu~o(t) de dfl dq).

(4.65)

The expressions (4.62) and (4.65) are written for anisotropic models. In the isotropic case, the function v should be independent of 0 and q~ (or of q~in the two-dimensional case). This leads to the following three-dimensional and two-dimensional isotropic vector Preisach models:

-"eo,~o( ff~ >~ v(e, fl)4&~uo,~o(t)dedfl) sinOdOdqo, (4.66)

f(t) = f (t) -

-@

v(a, fl)~fluq~(t) doe dfl dq~.

(4.67)

In the models described above, the functions v have not yet been specified. These functions should be determined by fitting the vector models to some experimental data. This is an identification problem. It is apparent that

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

465

d

(~, ~)

T| (0, ,~)

FIGURE 4.10

the identification problem is the central one as far as practical applications of the above vector hysteresis models are concerned. This problem will be discussed in sufficient detail in the subsequent sections. However, it is appropriate to comment already here that the solution of the identification problem is significantly simplified by the introduction of the auxiliary function P(~, fl, O, q~). For any fixed 0 and q~, consider a triangle T(~, fl) shown in Fig. 4.10. Then by definition, we have

P ( o~, fl, O, r ) -//T{~,fl) v(a', fl', O, q~)da' dfl'.

(4.68)

By using (4.68), it can easily be shown that P is related to v by the formula

v(c~, fl, O, qo) = -

a2P(~, fl, O, q~)

a~afl

.

(4.69)

Thus, if the function P is somehow determined, then the function v can be easily retrieved. However, from the computational point of view, it is more convenient to use the function P than v. This is because the double integrals with respect to a and fl in expressions (4.62), (4.65), (4.66) and (4.67) can be explicitly expressed in terms of P, and in this way the above double integration can be completely avoided. Indeed, for any fixed direction e0, we can consider the corresponding a-fl diagram. A typical example of such a diagram is shown in Fig. 4.11, where Mo, q~,kand mo, q~,kform an alternating series of dominant maxima and minima of input projections along the direction specified by e0, ~. By using these maxima and minima, the output

CHAPTER 4 VectorPreisach Models of Hysteresis

466

0'~'k)

FIGURE 4.11

of the scalar Preisach model Fo, q~uo,q~(t) associated with the direction e0,q~ can be evaluated as follows A

Fo, q~uo,r

= - P ( a o , flo) no,r +2 ~ [P(Mo, q~,k, mo, q,,k-1, O, r -- P(Mo, q~,k, mo, q~,k, O, r k=l

(4.70) The proof of (4.70) literally repeats the proof of the similar expression for the classical Preisach model and, for this reason, it is omitted. In the case of the two-dimensional vector Preisach model (4.63)-(4.65), a similar expression is valid for Fq~ur A

Feud(t) = - P ( a 0 , rio) %(t) +2 y~ [P(M~,k, mq~,k-1, q~) -- P(Mq~,k, mq~,k, q~)].

(4.71)

k=l

By using formulas (4.70) and (4.71) the numerical implementation of the three-dimensional vector model (4.60)-(4.62) and the two-dimensional vector model (4.63)-(4.65) can be reduced to the evaluation of double and single integrals, respectively. We note here that the numerical implementation of the classical two-dimensional Stoner-Wohlfarth model (4.32) requires the evaluation of double integrals. In this respect, numerical implementation of the two-dimensional vector Preisach model can be accomplished more efficiently than the numerical implementation of the

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

467

two-dimensional Stoner-Wohlfarth model. Another advantage of using the formulas (4.70) and (4.71) is that the function P can be directly related to experimental data. This will be demonstrated when we study the identification problems for the vector Preisach models. Using the expressions (4.70) and (4.71), digital codes that implement the vector Preisach models (4.60)-(4.62) and (4.63)-(4.65) have been developed. In these codes finite meshes of directions e0,~ and ~ are used to evaluate double and single integrals in (4.60) and (4.63), respectively. For each mesh direction the integrands in (4.60) and (4.63) are computed by using (4.70) and (4.71). Some numerical examples computed by using the developed digital codes will be given in the next section.

4.3 S O M E B A S I C PROPERTIES OF V E C T O R P R E I S A C H HYSTERESIS MODELS In the previous section we have defined the vector Preisach models of hysteresis and discussed their numerical implementation. The purpose of this section is to study some basic properties of these models and to show that these properties are qualitatively similar to those observed in experiments. We begin with the property of reduction of vector hysteresis to scalar hysteresis. This property is experimentally observed when an input is restricted to vary along an arbitrary chosen direction. We shall show below that a similar reduction property holds for the vector Preisach models; this property is stated more precisely below. For the sake of notational simplicity it is formulated and proven only for the two-dimensional Preisach model (4.63)-(4.65), although it holds for the three-dimensional Preisach model (4.60)-(4.62) as well. REDUCTION PROPERTYOF THE VECTOR PREISACH MODEL TO THE SCALAR PREISACH

MODEL.Consider an input fi(t) restricted to vary along some direction -~ofor times t>~to. Suppose that during t>~to, Ur (t) = u(t) consecutively reaches values u+ and u_ (with u+ > u_) and remains thereafter within these bounds. Then,for the Preisach vector model (4.63)-(4.65), the relationship between the output projection f~o(t) along the direction -e~o and the input u(t) exhibits the wiping-out and congruency properties. Since these properties constitute necessary and sufficient conditions for the representation of hysteresis nonlinearities by the classical scalar Preisach model, we conclude that the vector Preisach model is reduced to the scalar Preisach model.

CHAPTER 4 Vector Preisach Models of Hysteresis

468

(z+cos~ ~mj 'ucos~o FIGURE 4.12

PROOF. Without impairing the generality of our discussion, we can assume that q~0 = 0. Then, for any r the input projection u~(t) varies between u+ cos ~ and u_ cos ~. This means that for any ~ input variations m a y affect a - ] / d i a g r a m s only within the triangle T(u+ cos ~, u_ cos r (see Fig. 4.12). It is also clear that all input projections ur reach m a x i m u m or minim u m values at the same time and that these extremum values are 'cos qamultiples' of the corresponding extremum values of uo(t) = u(t). Consequently, if {Mk} and {ink} constitute an alternating series of dominant extrema of u(t), then {M~,k} and {m~,k} defined as

Mr

= Mk cos q~,

m~,k = mk cos ~

(4.72)

constitute the corresponding alternating series of dominant extrema of

u~(t). From (4.63) we find A

fx(t) -

cos ~FCu~(t) de.

(4.73)

2

From (4.71), (4.72) and (4.73) we conclude that only the alternating series of dominant extrema Mk and mk of u(t) affect the value offx(t). All other input extrema are wiped out. This is tantamount to the wiping-out property of the hysteretic relation betweenfx(t) and u(t). We shall next prove that the above hysteretic relation also exhibits congruency of minor loops. Let u (1) (t) and u (2) (t) be two inputs which vary between u+ and u_ for t~to and which m a y have different past histories I these inputs vary for to~t~t~o . However, starting from instant of time t 0,

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

469

(x

u~cos~o .

.

.

.

u cosq~

FIGURE 4.13

u+cosq~

/ ulcos "13 u_cosq~

FIGURE 4.14

back and forth between the same two consecutive extrema, u+I and u t_. As a result of these back-and-forth input variations, some minor loops are formed. We intend to show that these minor loops are congruent. The proof of the congruency of the above loop is equivalent to showing that any equal increments of inputs u (1) (t) and u (2) (t) result in equal increments of outputs fx(1) (t) andfx(2) (t). To this end, let us assume that both inputs after achieving the same value u'_ are increased by the same amount: Au (1) = Au (2) - Au. As a result of these increases, the identical triangles T~1) and T~2) are added to positive sets S + qo,1 (t) and S + qo,2 (t) and subtracted from the negative sets S~, 1(t) and S~, 2 (t) (see Figs 4.13 and 4.14). Since

F~ou~o(t) -

v(o~, fl, qo)de d f l +(t)

;(t)

v(~, fl, ~o)de dfl,

(4.74)

CHAPTER 4

470

Vector Preisach Models of Hysteresis

from (4.73) we derive

Af(x1) _ 2 f _ ~ (cos / / T q~ ( l V ( ~ f l ),

,

qo)dadfl)dq ~,

(4.75)

2

Afx(2) = 2

f

_~COS q)

(2)

)

v(~, fl, q~) da dfl dq~.

2

(4.76)

Since T (1) = T (2) for any q~, we conclude that Af(1) -- Af(2).

(4.77)

The equality (4.77) has been proven for the case when inputs U(1) (t) and u (2) (t) are monotonically increased by the same amount after achieving the same m i n i m u m value u'_. Thus, this equality shows the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (4.77) holds when the inputs u O) (t) and u (2) (t) are monotonically decreased by the same amount Au after achieving the same maximum value u~. This implies the congruency of descending branches of the above minor loops. Thus, the congruency property for minor loops is established. This completes the proof of the validity of the reduction property. [] In our discussion of the reduction property, we have proved the congruency of 'scalar' minor loops described by the vector Preisach models. The last result admits the following generalization. CONGRUENCYPROPERTYOFVECTORMINORLOOPS. Let the tips of two inputs Ft(1)(t) and ~(2)(t) trace the same closed curve for t>~to (see Fig. 4.15). Then the tips of the corresponding outputs ](1)(t) and/(2)(t) of the vector Preisach models trace congruent closed curves for t>~to (see Fig. 4.16). These curves may be noncollocated in space because of possibly different past input histories prior to to. PROOF. Consider the three-dimensional Preisach model (4.60)-(4.62). Since the tips of both inputs ~(1)(t) and ~(2)(t) trace the same closed curve, we conclude that for any direction eo,~0 the corresponding a-fl diagrams are modified with time in an identical way within the same triangles T(u+,o,~o, u-,o,~o). For instance, as the tips of the inputs ~ (1) (t) a n d v~(2) (t ) move from the point a to the point b (see Fig. 4.15), the same regions ftl,0,~0 and f22,0,~0m a y b e added to the positive sets S + 0,(p,1 and S + 0,q~,2 and subtracted

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

471

Uz

1

, Uy

Ux

FIGURE 4.15

from the negative

sets So,(pI and So, q~,2. Since

-Fo, q~UO,q~(t) - I/s ~,+~o~(t) v(c~,fl, O, qo)de d[3 -

[[ JJs ~,r

v(c~, fl, 0, ~o)d~ dfl,

(4.78)

we conclude that AFo, q~A u(ll0,q~(t) -- 2

ffolo

AFo, q ~ u(2)O,q~(t) = 2 [[_ J d.s

v(or fl, 0, q~) de dfl,

(4.79)

v(c~, fl, 0, qo) de dfl.

(4.80)

2,0,tO

By using (4.60), (4.79) and (4.80), we find that the corresponding output increments which connect the points A (1), B (1) and A (2), B (2) (see Fig. 4.16) are given by

af(l) -2 f2~f~o,~( ff n JO

Af (2) = 2

JO

fOarCfo rce0,~o(ff

v(a, fl, O,q))do~dfl)sinOdOdq),

(4.81)

1,0,rp

)

v(c~, fl, 0, qo) d~ dfl sin 0 dO dqo. (4.82) 2,0,(p

Since f21,o,~o - f22,o,~o for any 0 and q~, from (4.81) and (4.82) we conclude that A f ( l ) -- A f (2) . (4.83) The equality (4.83) holds for arbitrary chosen points a and b, and this proves that the vector minor loops shown in Fig. 4.16 are congruent. []

CHAPTER 4

472

Vector Preisach Models of Hysteresis

B2

2~

A2 " fy

fx FIGURE 4.16

We next proceed to the discussion of one remarkable property which is valid for the two-dimensional isotropic vector Preisach model (4.67). ROTATIONALSYMMETRYPROPERTY. Consider a uniformly rotating input (that is, one of constant magnitude and angular velocity)

Ft(t) - {ux(t) - UmCOscot, Uy - - Um Sin cot}.

(4.84)

Then the output of the two-dimensional isotropic Preisach model (4.67) can be represented as N

f (t) =f0 + f (t),

(4.85)

where fo does not change with time, while f (t) is a uniformly rotating vector. PROOF. It is clear from the very definition of the uniformly rotating input ~(t) and Fig. 4.17 that

u~o(t) -

U m

cos(cot - (p).

(4.86)

According to the proposition proved in Section 4.2, the integration over any semicircle can be used in the definition of the two-dimensional vector Preisach model. This fact allows one to modify the definition (4.67) of the two-dimensional isotropic Preisach model as follows: -

1

2re

f (t) - -~ f o

-G( ff~>~v(c~,fl)P~u~o(t)d~dfl) d~~

(4.87)

It is clear from (4.86) that for all directions e~0 the corresponding a-fl diagrams are modified with time within the same triangles T - T(um, -urn). Outside these triangles, the a-fl diagrams remain unchanged. These unchanged parts of c~-fl diagrams contribute to the term f0 in (4.85), while the

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

~

473

~(t)

FIGURE 4.17

time-varying parts of ~-fl diagrams result in the time-varying term By using the above comment as well as (4.86) and (4.87), we find: 1

f(t).

2~

f (t) - -i f ~ -dq,( f f v(a, fl)~t3Um COS(O~t- q)) da dfl) dq).

(4.88)

The expression (4.88) can be represented in terms of Cartesian components as

1 fO2re cos q)(ff v(~, fl)~Um COS(Ogt- qo)dadfl) dq~, fx(t)-~

(4.89)

1 fO2re sin q)(/f v(~, fl)~UmCOS(O~t- q~)d~dfl) dq~. fy(t) -- -i

(4.90)

Consider some instant of time t. For this instant of time, all directions ~ can be divided into two sets such that O~
(4.91)

~ < o 0 t - q~<2~.

(4.92)

and For the first set, all input projections u~o(t) are monotonically decreasing, and this results in the 0r diagram shown in Fig. 4.18. For the second set, all input projections uq,(t) are monotonically increasing, and this results in the ~-fl diagram shown in Fig. 4.19. Next we introduce new variable 0 - ogt- q~.

(4.93)

It is apparent that the double integral over T in (4.89) and (4.90) is the function of 0. This justifies the following notation

G(O) = ff~ v(e, fl)F~flUm A cos(o~t -

q~) d~ dfl.

(4.94)

CHAPTER 4 Vector Preisach Models of Hysteresis

474

(X

FIGURE 4.18

If 0~0~,

(4.95)

then the diagram shown in Fig. 4.18 is valid, and from this diagram and formulas (4.68) and (4.94) we find G(O) = P(um,

- - U m ) --

2P(um,

Um

COS 0).

(4.96)

If ~KOK2z~,

(4.97)

then the diagram shown in Fig. 4.19 is valid, and as before we find G(O) = 2P(um cos O, -urn) - P(um, -urn).

(4.98)

By using the change of variables (4.93), the notation (4.94), and by taking into account that dO - -d~o, (4.99) and c o t - 2z~O~cot,

(4.100)

we transform (4.89) as follows:

lj02

fx(t) - -~

where

cos(cot - O)G(O) dO - A cos cot + B sin cot,

lf02

(4.101)

cos OG(O) dO,

(4.102)

B -- ~1 f02~ sin OG(O) dO.

(4.103)

A- ~

475

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

/

FIGURE 4.19

Similarly, the expression (4.90) can be reduced to the form (4.104)

fy(t) - A sin cot - B cos cot. The formulas (4.101) and (4.104) can be modified as follows:

VIA2 q- B 2 cos(cot- ~),

(4.105)

y~y( t ) - v/A 2 + B2 s i n ( c o t - ~),

(4.106)

B tan ~ = A"

(4.107)

fx(t)

-

where

From (4.105) and (4.106), we conclude t h a t f is a uniformly rotating vector with the magnitude equal to v/A 2 + B2. We next express A and B in terms of the function P and prove that the angle ~ in (4.105) and (4.106) is acute. From (4.95)-(4.98) and (4.102) we find

1{/0 ~2~

cos O[P(um, -urn) - 2P(um, Um cos 0)] dO

A- ~

q-

}

cos O[2P(um cos 0, -urn) - P(um, -urn)] dO . (4.108)

In the second integral in (4.108), we will use the change of variables 0'= 0- ~

(4.109)

CHAPTER 4 Vector Preisach Models of Hysteresis

476

and take into account that

P(-um cos 0, -urn) ~- P(um, Um cos 0).

(4.110)

The last formula easily follows from the symmetry property (4.58) and the definition (4.68) of the function P. By using (4.109) and (4.110), we can transform (4.108) as follows: A = -2

~0 ~

cos OP(um, Um COS 0) d0.

(4.111)

From (4.95)-(4.98) and (4.103), we obtain

11/0

B= ~

+

sin O[P(um, -urn) - 2P(um, Um cos 0)] dO sin O[2P(um cos 0, -urn) - P(um, -urn)] d0 . (4.112)

To transform the second integral in (4.112), we will use the change of variables (4.109) and the identity (4.110). This eventually leads to the following expression for B:

/z

B - 2P(um, --Urn) -- 2

f0

sin OP(um, Um COS 0) d0.

(4.113)

Thus, we have found explicit expressions (4.111) and (4.113) for A and B in terms of the function P. In the next section, it will be shown that P can be related to some experimental data. In this way we can relate A and B to the experimental data, and, consequently, find the magnitude and phase of the uniformly rotating vectorf(t) in terms of these data. Now, we will use formulas (4.111) and (4.113) to prove that the phase angle ~ is acute under some general conditions. The first condition can be expressed mathematically as

P(a, fl)~P(a, fl')

if fl>~fl'.

(4.114)

The condition (4.114) means that P(a, fl) is a monotonically decreasing function of fl for any fixed ~. It is clear from the definition (4.68) of P that the condition (4.114) is satisfied if v(a, fl) is positive. From (4.111), we find

A 2[/0

cos OP(um, Um cos 0) dO q-

cos OP(um, Um cos 0) dO . (4.115)

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

477

By using the change of variables 0t = ~ - 0 in the second integral in (4.115), we derive /T

A

-

-2

f0

cos O[P(um, Um COS 0)

-- P ( u m , - u r n c o s 0)] dO.

(4.116)

From (4.114) and (4.116), we conclude A > 0.

(4.117)

O[P(um, -Um) -- 2P(um, Umcos 0)] d0 > 0.

(4.118)

The second condition is f0 ~rsin

From (4.114) and (4.118), we find B > 0.

(4.119)

From (4.107), (4.117) and (4.119), we conclude that the phase angle ~ is acute. This completes the proof of the rotational symmetry property. [] It is worthwhile noting that in spite of the nonlinear structure of the vector Preisach model (4.67) the time harmonic input (4.84) produces (up to a history-dependent constant term f0) a time harmonic output of the same frequency. In other words, no generation of higher-order harmonics is caused by the nonlinear structure of the vector Preisach model. This remarkable fact admits the following physical explanation. The isotropic vector Preisach model (4.67) has a mathematical form which is invariant with respect to any rotation of Cartesian coordinates. The mathematical form of uniformly rotating input (4.84) is also invariant with respect to any rotation of Cartesian coordinates. Thus, on symmetry grounds we expect that the mathematical form of the resulting output should also be invariant with respect to any rotation of Cartesian coordinates. This is possible only if the output is a uniformly rotating vector. The above discussion clearly reveals the meaning of the term 'rotational symmetry property'. The rotational symmetry property has been confirmed by numerical computations. By using a digital code that implements the vector Preisach model (4.67) and that has been briefly described in the previous section, the output of the Preisach model has been computed for the input shown in Fig. 4.20. This input gradually approaches the regime of uniform rotation (4.84). The results of computations shown in Fig. 4.21 demonstrate that the output also gradually attains the regime of uniform rotation. The property of rotational symmetry has also been observed in the experiments. The --4

CHAPTER 4 VectorPreisach Models of Hysteresis

478

5.0

2.5

_

o.o_

t,,._j

,

,

,)/, i

-5.0 -5.0

-2.5

0.0

2.5

5.0

FIGURE 4.20

results of these experiments are shown in Fig. 4.22. In these experiments, the applied magnetic fields were stationary, while the magnetizable sample of Isomax material was uniformly rotated. Thus, in these experiments, the magnetization was measured in the coordinate frame uniformly rotating with respect to the sample. The component of magnetization measured along the applied field is called 'parallel magnetization' while the component perpendicular to the field is named 'transverse magnetization'. The results shown in Fig. 4.22 clearly suggest that for large applied fields the whole magnetization of the sample moves in synchronism with the uniformly rotating coordinate frame. This means that we are dealing with uniformly rotating magnetization. As the applied field is reduced, the time constant term of magnetization appears, which reveals itself (in the uniformly rotating frame) as sinusoidally changing components of magnetization. PROPERTY OF CORRELATION BETWEEN MUTUALLY ORTHOGONAL COMPONENTS OF

OUTPUTANDINPUT. Suppose that the input ~(t) is first restricted to vary along the y-axis. It is increased from an infinitely negative value to some positive value u+, and then it is decreased to zero. Some remnant value of output fr = -eyfr results from the above input variations. After reaching zero, the input is restricted to vary

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH 20

I

I

0

10

479

10

-10

-20

-10

20

F I G U R E 4.21

along the orthogonal x-direction (see Fig. 4.23). It is asserted that, by increasing the input in x-direction, it is possible to reduce the orthogonal remnant component of the output to any, however small, value. PROOF. All directions ~r can be subdivided into sets: 0Kq~K~

and

~q0K0.

(4.120)

The input variations along the y-axis result in the a-fl diagrams shown in Figs 4.24 and 4.25 for the first and second sets of directions, respectively. As the input is increased along the x-axis, the above diagrams are modified. These modifications are shown by dashed lines. They result in continuous expansions of positive sets S~ (t) at the expense of negative sets S~ (t). From (4.67), we find

fy(t) = f_~ sinq~( / L 2

v(~, fl)~[~Ux(t)cosq~d~dfl) dq~.

(4.121)

~fl

By introducing the notation

G(qo, t) - ~ v(c~,fl)~ux(t) cos (p d~ dfl, dJa>~

(4.122)

480

'

I'

'

'

'

I

'

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Vector Preisach Models of Hysteresis

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uop, uzp, a u g g t u

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CHAPTER 4

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4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

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Vector Preisach Models of Hysteresis

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CHAPTER 4

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e,i r

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

f~

483

/~(t) ~X

FIGURE

4.23

u cos~

(~o)

u.~in~o

FIGURE 4.24

•u

Usn'q l

cos~

(~o)

FIGURE 4.25

formula (4.121) can be represented as follows:

fy(t) =/o ~ sin ~G(~,

t)do +/0

sin qaG(~, t)do. 2

(4.123)

CHAPTER 4 Vector Preisach Models of Hysteresis

484

Changing q~ to -q~ in the second integral in (4.123), we obtain 2

fy(t) -

Jo

sin q~[G((p, t) - G(-q~, t)] dq~.

(4.124)

If Ux cos q~a0, then according to the diagrams shown in Figs 4.24 and 4.25 we have G(q~, t) - G(-q~, t). (4.125) From (4.124) and (4.125), we find

fy (t) =

rccos

=0

sin q~[G(q~, t)

-

G(-q~, t)] dq~.

(4.126)

ux(t)

Since arccos Ux(t) ~0 ~ ~ as ux(t) ~ cx~,we conclude that

fy(t) ~ O.

(4.127)

This completes the proof of the above property. This property has been confirmed by numerical computations performed by using a digital code that implements the model (4.67). The results of the computations are shown in Fig. 4.26 for different remnant values of the

5

0

--5

-lO

1

~,~1~1,~,~1,,,,i,,,,1,,,,

0

1000

2000

3000

F I G U R E 4.26

4000

-

5000

6000

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

485

o

-5

-lo

j L i i 1 , l , l I i l , , 1 l o

.05

.1

.15

FIGURE 4.27

output. The property of correlation between orthogonal components of input and o u t p u t has also been observed in the experiments. The results of these experiments are presented in Fig. 4.27. There is an apparent qualitative similarity between the computational and experimental results. It is worthwhile noting here that the above property of orthogonal correlation has long been regarded as an important 'testing' property for vector hysteresis models in magnetics.

4.4

IDENTIFICATION PROBLEM FOR VECTOR PREISACH MODELS

ISOTROPIC

The essence of the identification problem is in determining the function v(a, fl) or P(a, fl) from some experimental data. It turns out that this problem can be reduced to the solution of a special integral equation that relates the function P(~, fl) to some 'unidirectional' (scalar) hysteresis data. We first present the derivation of this equation for the two-dimensional model (4.67). The derivation proceeds as follows. Consider the projectionfx(t) off(t) along the direction q~ = 0. According to Eqn. (4.67), we have

fx(t)

=

I' (o(iL'>~' v(a', fl')~,~,u~o(t) d~' aft' )de. cos

(4.128)

CHAPTER 4 Vector Preisach Models of Hysteresis

486

d

s:,

FIGURE 4.28

Now we restrict ~(t) to vary along the direction r = 0, which means that Ft(t) = -~xu(t). First, we assume that u(t) is made 'infinitely negative'. Then for any a,I fl! and (p we have ~,/~,u,(t) = -1. Next, we assume that the input is monotonically increased until it reaches some value a. Let fa denote the corresponding value offx(t). As a result of the above input increase, we find that for any q~ we have ~,/~,u,(t) = +1 if a' < a cos q~ and ~a,/~,u,(t) = - 1 if a t > a cos q~. This is shown geometrically in Fig. 4.28. Consequently, for any r we have

ff~,>>.~,v(a', fl')~,ffUq~(t) da' dfl' = ffs+ * v(a', i f ) d a ' d f f - ffs;.,

v(a', fl') da' dff.

(4.129)

Finally, we assume that the above input increase is followed by a subsequent monotonic decrease of u(t) until it reaches some value ft. Let f~/~ denote the corresponding value offx(t). The above input decrease modifies the previous geometric diagram that now for any q~ assumes the form shown in Fig. 4.29. According to this figure, we have ff~ ,>>.~'v(a',

= ffsL

fl')~,~,ur

da' dfl'

v(a', fl')da' d f l ' - f f

dis ~.~

v(a', fl')da' dfl'.

(4.130)

Consider the function 1

F(a, t) = ~(f~ -f~/~).

(4.131)

By using formulas (4.68), (4.128)-(4.131), it is straightforward to show that

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

487

Td (~) / ,

13cos~ s~*~,,I-

FIGURE 4.29

P(a, fl) is related to F(a, fl) by the expression

f

~ cos q~P(a cos q~, fl cos q~) dq~ - F(a, fl).

(4.132)

2

It is apparent that f~ and f~/~ can be found by measuring the first-order transition curves. Of course, the input u(t) cannot be made infinitely negative as was assumed in the derivation. However, by making the input 'sufficiently negative', a reasonable accuracy can be secured. The formula (4.132) can be construed as the integral equation which relates P(~, fl) to the first-order transition curves. This integral equation has a peculiar structure that is revealed by the following result. THEOREM. Consider the operator

AP = / ~ cos qoP(a cos q~, fl cos q~) d~0.

(4.133)

2

Monomials ~k fls are eigenfunctions of the operator A. PROOF. From (4.133) we find -

~ cos Oa k cos k Oti s cos s r d o 2

~xkfls f ~ cos k+s+l q~dq~ - 2k+lak fls, d-

(4.134)

CHAPTER 4 VectorPreisach Models of Hysteresis

488

where

(2n-l)!!

ifk + s + 1 = 2n, (4.135)

2k+s --

2 (2n)!!

ifk + s + 1 - 2n + 1.

By using the above theorem, the solution of Eqn. (4.132) can be easily found when the right-hand side F(a, fl) is a polynomial M F(a, f l ) = ~ ~-(m)akfls aks m=0 k+s=m

(4.136)

Indeed, looking for the solution of the integral equation in the form M P(a, fl) -- y ~ ~-~__"(m)akfls,laks m=0 k+s=m

(4.137)

from the above theorem we find ~(m) p~m) = Uks 2m"

(4.138)

By using this fact, the following algorithm for the solution of Eqn. (4.132) can be suggested. We first extend the function F(a, fl) from the triangle -a0~
(4.139)

where the Chebyshev polynomials are defined as To(a) = 1,

1

Tt(a) - ~--;--ccos(s

(4.140)

By assuming for the sake of notational simplicity that a0 = 1, for the expansion coefficients agq in (4.139) we have the formula (see [12])

agq=

2e+qf; f/ F(a,fl)Tt(a)Tq(fl)dadfl. /z2

1

1

v/l_a2v/l_fl2

(4.141)

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

489

Let teq(O~,fl) be the solution of the integral equation

/

7[

~ cos (pts

cos (p, fl cos q~) dq~ - Te(~) Tq(fl),

(4.142)

then the solution of the integral equation (4.132) can be represented in the form P(a' fl) - E as163 fl). (4.143)

s Thus, there are two major steps in finding the solution (4.143) of the integral equation (4.132). The first step is to solve Eqn. (4.142). This can be accomplished by using the technique (4.136)-(4.138). The second step is to evaluate the expansion coefficients as This can be handled by using the formula (4.141). However, for the sake of computation it is desirable to transform this formula as follows. By introducing new variables 0 and 0I = cos 0,

fl = cos 0',

(4.144)

cos qO' 2q_1 ,

(4.145)

F(cos 0, cosO') cosf.OcosqO' d0d0'.

(4.146)

we find cos t~0 , 2s

Ts

4/070

as - -~

Tq(fl)

-

Thus the expansion coefficients as can be evaluated by using fast cosinetransform schemes. There is another approach to the solution of the integral equation (4.132) that leads to a closed-form expression for P(a, fl). This approach is based on the following change of variables: x = a cos q~,

2 = ilia,

(4.147)

which after simple transformation yields the following integral equation of the Abel type:

f0 40r x-- X2 P(x, 2x) dx = ~F(~, 0~ 2a).

(4.148)

Some simplification of the above equation is achieved by introducing new auxiliary functions:

N(x) - xP(x, 2x),

R(a) - 2F(a, 2a).

(4.149)

CHAPTER 4 Vector Preisach Models of Hysteresis

490

oc

x

FIGURE 4.30

Then, from (4.148) we obtain

fo ~ N(x) dx

= R(~).

(4.150)

/0r _ x 2

The solution of the above equation can be found by using the following trick. We multiply both sides of (4.150) by ~/s2_~2 and integrate with respect to e from 0 to s:

f0s

o~ x/S 2 __ CX2

(f0

N(x) dx

) d ~ = f0

da.

(4.151)

~zdcz ~ dx. V/(S2_~2)(-~2_x 2) ]

(4.152)

~/0r __ X 2

&2

__ 0r

By using Fig. 4.30, we find

fos

cx (fo~ N(x)dX)dcz V/S2 -- 0~2

X/C(2 -- X 2

[~ ~N(x)dx d~ JJ. v/(S2 -- 0r162 2 -- X2) s =f0

s N(x)(L

Next, by using simple transformations, we obtain fx s

0rdcz

v/(S2_ CX2)(CX2_X2)

_ 1 fx s 2 ~

d(cz2)

v/(s22x2_)2 _(cx 2 -

~

(4.153)

s2+x22)2

By introducing a new variable of integration S2 + X2 09 -- O~2

(4.154)

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

491

from (4.153) we derive s2_x 2

/x s

o~da v/(S2 -- X2)(0r2 -- X 2)

_ 1 ~x22s2 2

do~

~ ~/ (S2-X2) 2 __ 092

1 209 = - arcsin 2 s 2 -- 09 2

s2 _x 2 09----

--. x2 -s 2 09----

2

(4.155)

From (4.151), (4.152) and (4.155) we obtain f0 s N(x) dx - -2 f0 s

c~R(c0 da.

(4.156)

v/S2 _ ~2

By differentiating (4.156) with respect to s and then by replacing s by and a by s, we find N(~) -

2 d [~ d~ ~0

sR(s)

ds.

(4.157)

V/or2 -- S2

This is the closed-form solution to the integral Eqn. (4.150). Now, by substituting (4.149) into (4.157), we find the closed-form expression for P:

P(c~, ;ccO- 1 d fo ~ s2F(s, ,~s) ds. ~:o~d~x

(4.158)

V/0r _ s 2

The last expression is valid for any value of 2. By varying 2 from 0 to 1 and using (4.158), we can compute all required values of P. However, for actual computations, it is convenient to transform (4.158) by integration by parts. The final expression is then given by

P(a, 2a)

1~

F(s, 2s) + s dF(s, ,ls)

7C Y0

V/0r 2 -- S 2

ds.

(4.159)

Now, we turn to the three-dimensional isotropic model (4.66). Consider the projection fz(t) of f(t) along the direction 0 - 0. According to (4.66), we have

fz(t) -

cos 0 x sin 0 dO dcp.

'~>/~'v(a', fl )?a,~,Uo, q~(t) da' aft' (4.160)

We restrict ~(t) to vary along the direction 0 - 0, which means that ~(t) -dzu(t). As before, we first assume that u(t) is made infinitely negative. Then

CHAPTER 4 Vector Preisach Models of Hysteresis

492

d ..

+

l~coso

/ FIGURE 4.31

I d (o , , ) / , +

FIGURE 4.32

for any a, fl, 0 and q~ we have r q~(t) - - 1 . Next, we assume that the input is monotonically increased until it reaches some value ~. Letf~ denote the corresponding value of fz(t). As a result of the above input increase, we find that for any 0 and r we have ~a,~,UO,q~(t) = +1 if a' < c~cos 0 and r q~(t) - - 1 if a' > a cos 0. This is illustrated geometrically in Fig. 4.31. Consequently, for any 0 and (p we have

fL ,>>.~'v(a', ff

dis

do~'dfl'

v(a', fl')da' a f t ' - f f

JJs;,o

v(a', fl')da' aft'.

(4.161)

Finally, we assume that the above input increase is followed by a subsequent monotonic decrease of u(t) until it reaches some value ft. Let f~/~ denote the corresponding value of fz(t). The above-mentioned input decrease results in geometric diagrams shown in Fig. 4.32. According to this

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

493

figure, we have /f~ '>~]~'

- dffi s

fl')~,/~, , u 0,q~(t) da' dfl'

v(~x',fl')&x' aft'- [[ v(~x',fl')d~x' dfl'. JJs;, ,o

(4.162)

By substituting (4.161) and (4.162) into (4.160), then by subtracting one expression from another and by using (4.68) and (4.131), we derive

f0

2~0 ~ COS OP(a cos 0, fl cos 0) sin 0 dO d o = F(a, fl).

(4.163)

Since the integrand does not depend on q~, we obtain 7r

/0

cos 0 sin OP(a cos O, fl cos O) dO -

1

F(a, fl).

(4.164)

This is the integral equation that relates P(a, fl) to the experimentally measured first-order transition curves. It turns out that this equation is easily solvable. By using the change of variables x = c~cos O,

2 = fl/~,

(4.165)

Eqn. (4.164) can be represented in the form

fo

C~XP(x, 2x) dx - ~nnF(a, 200.

(4.166)

By differentiating the last equation with respect to c~, we find 1 d [~2F(~ ' ,~)] c~P(~, 2~) - 2~z dc~

(4.167)

which leads to the following final expression: P(c~, 2cr -

1 d [0r162 ' ~0r 2rcc~ dc~

(4.168)

It is remarkable that the solution of the identification problem for threedimensional isotropic Preisach models of vector hysteresis turns out to be m u c h simpler than the solution of the same problem for two-dimensional isotropic models.

CHAPTER 4 VectorPreisach Models of Hysteresis

494

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR PREISACH MODELS We shall first discuss the identification problem for the two-dimensional anisotropic model (4.65). As before, we shall use the function P(~, fl, q0) defined by (4.68). We shall relate this function to the experimental data represented by the sets of first-order transition curves measured along all directions e~0. These experimental data can be characterized by the function 1

F(or fl, r = ~(f~r -f~/~r

(4.169)

which will be assumed to be known in the subsequent discussion. Of course, it is not feasible to measure first-order transition curves for all directions e~0. However, it is possible to measure these curves for some finite meshes of directions e~0and to use subsequent interpolation for computing F(~, ]~, (p) for all 9). Consider local polar coordinates (p, ~) with the polar axis directed along the vector e~o'. The model (4.65) in local polar coordinates can be written as

f (t) =

-de/

v(~, fl, q~' + d/)~ue/(t) d~ dfl dO,

(4.170)

where the relationship between the angles q0, opt and ~ is illustrated by Fig. 4.33. By using the expression (4.170) and by repeating almost literally the same reasoning as in the derivation of the integral Eqn. (4.132), we find that the function P(a, fl, q0) is related to the function F(a, fl, q0) by the expression

f

~ _~cos OP(a cos ~, fl cos ~, q0' + O)dO - F(a, fl, (p'). 2

FIGURE 4.33

(4.171)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

495

Next we shall use the following Fourier series expansions: (x)

F ( ~ , fl, (p' ) -

E

Fn ( o~, fl) e in~~,

(4.172)

Pn((X, fl)e inq~

(4.173)

Vn ( Ot, fl) e inq).

(4.174)

tl-----r

(x)

P(r fl, q ) ) -

E tl'----(X)

oo

V( Ot, fl, q)) -

E tl-'--O0

It is clear from (4.68) and (4.69) that Pn and

Pn(ot, fl) - / f T Vn(O~, f l ) - -

(~,~)

Vn are related by the expressions:

Vn(O~',fl') da' dfl',

~2pn(~, fl)

.

(4.175)

(4.176)

By substituting (4.172) and (4.173) into (4.171), we find

E

einqr / ~~ einr

n----oo

I[IPn(c~cos ~, fl cos I]/) dO

2 oo

-- E

Fn(c~, fl) e inq~

(4.177)

tl----(X)

From (4.177) we obtain

f

2 einr/ COSI~Pn(or cos ~, fl cos ~) dO = Fn(or fl) (n -- O, +1, 4-2 . . . . ).

(4.178)

Since sin n~ cos 6Pn(a cos 6, fl cos ~) is an odd function of ~, from (4.178) we derive

/2

_n

cos $ cos n$Pn((Z cos $, fl cos $) d $ - Fn(O~, fl)

(n = 0, 4-1, +2 . . . . ).

(4.179)

Thus we have obtained the infinite set of decoupled (separate) integral equations for Pn. The right-hand sides Fn(~X, fl) of these equations can be computed by using fast Fourier transform (FFT) algorithms.

CHAPTER 4 Vector Preisach Models of Hysteresis

496

For the case n = 0, from (4.179) we find _n 2 COS ~/P0( 0~c o s ~/, ]~ c o s ~/)

d6

= F0(a,

fl),

(4.180)

which naturally coincides with the integral Eqn. (4.132) for the isotropic vector Preisach model. If n = 4-1, from (4.179) we obtain

f_

~ c o s 2 ~P+l(a cos ~, fl cos ~) d~ = F+l ((x, fl). _n 2

(4.181)

Closed-form solutions to integral Eqns (4.180) and (4.181) can be found by using the change of variables x - a cos ~,

2=

fl/~

(4.182)

and by reducing the above equations to the Abel-type integral equations. In the case of Eqn. (4.180), the result is readily available and can be expressed by the formula P0(~, 2~)

1 d fa~ s2Fo(s, 2s) ~a d~ vu ~--~2~i2s as.

(4.183)

In the case of Eqn. (4.181), the change of variables (4.182) leads to the integral equation f0 a

X 2 dx 0r2 P+I (x, 2x) ~/0r2 _ x 2 -- -~-F+I (0r 200.

(4.184)

These equations can be solved in exactly the same way as Eqn. (4.148). The final result is given by the following formula P+I (a, 2a)

1

d [ ~ s3F+l(S, 2s) ds.

~0~2 d~ J0

(4.185)

V/o~2 -- S 2

Thus, if w e are interested only in the firstthree terms of the Fourier expansion for P (this is a first-order approximation for anisotropic media),

then the solution of the identification problem can be found by using the closed-form expressions (4.183) and (4.185). In the case of higher-order terms, the change of variables (4.182) leads to the following Abel-type integral equations

fo ~ Pn(x, 2x) ~/O~ xTn(X) ~ 2 _ X2 d x - -i-dFn((X, 2a),

(4.186)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

497

where Tn are Chebyshev polynomials defined by (4.190). Unfortunately, we have not been able to find closed-form solutions to these equations. However, there are many efficient numerical techniques developed for the solution of integral equations of this type (see, for instance, [12]). The analytical solutions to integral Eqn. (4.179) can also be found by exploiting the following property of this equation. THEOREM. Consider operators lr

~4nP - / 2~ cos ~ cos n~P(~ cos Ifi, fl cos ~) d~.

(4.187)

2

Monomials ~kfls are eigenfunctions of these operators. PROOF. It is straightforward to check the validity of the above statement and to get the following expressions for the eigenvalues "~k+s" j(n) . 2k+s (n) - 2

fo

cos nO cos k+s+l 0 dO

(4.188)

The explicit formula for the integral in (4.188) can be found in the literature (see [13]). [] By using the above theorem, polynomial solutions to the integral Eqn. (4.179) can be easily found. Indeed, if the right-hand sides of these equations are polynomials in the form (4.136), then the solutions to these equations will be polynomials in the form (4.137). Polynomial coefficients of the right-hand sides and the solutions will be related by the expression (4.138) where instead of eigenvalues ~m the eigenvalues ~(n) should "~k+s be employed. By taking advantage of this fact, Chebyshev polynomial expansions of type (4.139) for Fn(~, fl) can be used and the solutions can be found in the form (4.143), where the expansion coefficients can be computed by using expressions similar to (4.146). Now, we proceed to the identification problem for the three-dimensional anisotropic vector Preisach model (4.62). This problem is technically more complicated than the corresponding problem for two-dimensional models and, for this reason, it requires special treatment. It turns out that some facts from the theory of irreducible representations of the group of rotations of three-dimensional Euclidean space are instrumental in the treatment of this identification problem. As before we shall use the function P(~, fl, 0, q~) defined by (4.68) and we shall relate this function to the first-order transition curves experimentally measured along all possible directions e0,~0. These curves can be used

CHAPTER 4

498

Vector Preisach Models of Hysteresis

to define the function 1

F(o~, fl, O, (p) = -~(f~or -f~oq,).

(4.189)

It is convenient to use spherical harmonic expansions for the functions v, P andF: oo

V(O~,fl, O, q))= E k=0

k

E Vkm(O~'fl)Ykm(O' q))' m=-k

(4.190)

k

P(~, fl, O, r

E

E Pkm(~ fl)Ykm(O, r k=O m=-k o0 k

F(~, ,8, O,q~)= E

E Fkm(a,fl)Ykm(O,q~),

k=O

(4.191)

(4.192)

m=-k

where Ykm(O, q~) are spherical harmonics, while Vkm, Pkm a n d Fkm are corresponding expansion coefficients. It is clear from (4.68) and (4.69) that Pkm and Vkm are related by the following expressions:

Pkm(O~,fl) = f f Vkm(Off, if) ,liT (~,~) Vkm(e, f l ) = -

a2pkm(O~,fl)

d e ' dfl',

9

(4.193)

(4.194)

It turns out that the identification problem can be reduced to the solution of special integral equations which relate Pkm(~, fl) to Fkm(Ct, fl). The derivation of these equations proceeds as follows. Consider an arbitrary direction specified by the angles ff and q~'. We shall use a local coordinate system xyz, which is obtained from the system XYZ by Euler rotations Rz(0), Rx(0') and Rz(g7~ + q~') (see Fig. 4.34). The Euler angles for the inverse rotation from xyz to XYZ are equal to grc - ~0', 0 f and 7r, respectively. It is clear that the direction of the z-axis coincides with the direction of-do,e,. By using local spherical coordinates ~ and ~, we can represent the model (4.62) as

(/I., x sin r d~ d~.

o,,

d.d,) (4.195)

By using the last expression and by repeating almost literally the reasoning that was used in the derivation of the integral Eqn. (4.163), we can show

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

499

z z

y

o! i

x

,u

X

FIGURE 4.34

that the model (4.195) matches first-order transition curves measured along the direction e0'~0' if the function P satisfies the integral equation

J0270: cos ~P(a cos ~, fl cos ~, ~, ~, 0t, q~') sin ~ de d6 = F(~, fl, 0', ~0').

(4.196)

Next, we shall relate the function P in local coordinates to the same function in spherical coordinates 0 and q0. To achieve this, we shall use the spherical harmonic expansion (4.191) and the following facts from the theory of irreducible representations of the group of rotations (see [14-16]). Linear combinations of spherical harmonics Wkmof the fixed order k form a linear space Hk of the irreducible representation of the group of rotations. This means that any spherical harmonic Ykm(O, q)) in coordinates 0 and ~0 can be represented as a linear combination of the spherical harmonics Ykm(~, ~) of the same order k: k

Ykm(O,,q),)

-

-

~ k atom, ( l'r. -- q)') Ykm'(~, 0), ~, 0', -~ ml'--k

(4.197)

where atom, k are the matrix elements of the irreducible representation of the rotation group in Hk. These matrix elements are functions of Euler angles that determine the rotation from the local xyz coordinate system to the original XYZ system. These functions are sometimes called generalized spherical harmonics because they are reduced to spherical harmonics when m or m t is equal to zero.

CHAPTER 4 Vector Preisach Models of Hysteresis

500

By using (4.197) and (4.191), we find the expression for P in local coordinates

P(~, ~, ~, r 0', ~') k

k

--

amm, ~z, , -~ - q;' Pkm(O~,fl)Ykm'(~, 6). (4.198) m=-k m'=-k By substituting (4.198) into (4.196), we find k=0

F(~, ~, 0', ~') c~ -

k

k amm,

-

k=0

~,

, -~ -- qJ

m=-k m'=-k

x ~O2~z~O~ cos~Pkm(O~cos~, ~cos~)Ykm,(~, 0 ) s i n ~ d ~ d 0 .

(4.199)

In (4.199) the integral with respect to 6 is equal to zero if m' r 0, and it is equal to 2Ir if m' = 0. Thus we obtain

F(~, ~, 0', ~') oo

k

k=0

m=-k

= 27r

x

)

amo re, 0', ~ - q~'

~0 arc

COS~Pkm(~Xcos ~, fl cos ~)Yko(~) sin ~ d~.

(4.200)

It is known from group theory (see [16]) that

k ( 7r ) m / 4re amo ~z, 0',~--q0' - - ( - - 1 ) ~ 2 k + 1

Ykm(O', ~o').

(4.201)

It is also known (see [12]) that ~/2k + 1 Lk(cos ~), 4~r

Yko(~) =

(4.202)

where Lk are Legendre polynomials. From (4.200), (4.201) and (4.202), we derive oo

F(~, fl, 0', ~0') - 2Ir ~ k=0

k

~ (-1)mykm(O ', q)') m=-k

/[

x

Pkm(O~cos ~, ~ cos ~)Lk(cOS ~) cos ~ sin ~ d~.

(4.203)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

501

By comparing (4.203) with (4.192), we find f0 ~ Pkm(~Xcos ~, fl COS ~)Lk(cos ~) cos ~ sin ~ d~ = (-1)m 2----~Fkm(O~,fl). (4.204) These are the final integral equations that relate Pkm to Fkm. These equations can be reduced to Volterra integral equations by using the following change of variables x - ~ cos ~,

2 = -fl.

(4.205)

Indeed, after simple transformations, we derive

f0 ~ Pkm(X, 2x)XLk (X) dx --

~2 (--1)m-~-~Fkm(C~, ~,~x).

(4.206)

Explicit solutions to Eqn. (4.206) can be found for the cases k = 0 and k - 1. If k = 0, from (4.206) we obtain

~0~xPoo(x, 2x) dx = ~-~F00(c~, ~ 2~).

(4.207)

By differentiating (4.207) with respect to ~, we find P00(a 2a) -

1

d [a2F00(~ ,2~)].

2 ~ d~

'

(4.208)

As expected, this result coincides with the one obtained for the isotropic model (see (4.168)). If k = 1, according to (4.206) we have

f0 c~X2plm(X,

0r3 Flm(~X, ,~00 )~x) dx - ( - 1 ) m -~--~

(m = 4-1, 0).

(4.209)

By differentiating (4.209), we derive Plm( 0~, ,~,~) _

(-1) m d 27r~ 2

d~

[~Z3Flm(0~' ~)]

(m - +1 0).

(4.210)

Thus, if we are interested only in the first four terms of spherical harmonic expansion for P (this is a first-order approximation for anisotropic media), then the explicit analytical solution for the identification problem is given by formulas (4.208) and (4.210). For k > 1, Eqn. (4.206) can be solved numerically. Discretization procedures can be applied directly to Eqn. (4.206), or the equation can first be

502

CHAPTER 4 VectorPreisach Models of Hysteresis

reduced by differentiation to the Volterra equation of the second kind

Pkm(~Z,,~0~)-- fo ~Pkm(X, ,~x)-~x Lk,(X)d x = (-1) m d [o~2Fkm(~,~)]. (4.211) 2z~a da In both cases we shall end up with simultaneous algebraic equations with triangular matrices which are easy to solve. However, the reduction to the second kind of integral equation may be desirable as far as computational stability is concerned. Finally, the integral Eqn. (4.204) can be solved analytically if polynomial approximations for the right-hand side are employed. These analytical solutions can be found in exactly the same manner as for twodimensional identification problems discussed before. Up to this point, we have used scalar hysteresis data (4.169) and (4.189) measured for unidirectional variations of input ~(t) in order to solve the identification problems. However, for anisotropic media these input variations result in vectorial data that for two- and three-dimensional problems can be represented in the following forms, respectively 1

F(~, fl, ~#)= ~ ( f ~ -)?~/~),

(4.212)

F(a, ]/, 0, q~)= ~ 07~0~ -)?~//0r

(4.213)

The above vectorial data account for the output components that are orthogonal to the directions of input variations. There is a natural desire to utilize these vectorial data in the identification of vector Preisach models. This can be achieved by generalizing the models themselves. The essence of generalization is in employing vectorial functions ~. This leads to the following vector Preisach model:

f(t) - ~lTl:l ( Sf~>>.~7(~'fl")~a~(-f "u(t))d~

dsr"

(4.214)

By repeating the same line of reasoning as in Section 4.2, we can show that by means of redefining ~ the integration over a unit sphere in (4.214) can be reduced to the integration over a unit hemisphere

f(t) - //c+ ( ff~>>.~v(~z'fl'-f )~;a~(-f"u(t))d~

dsr"

(4.215)

It can also be shown that the redefined function ~ has the following symmetry property ~(~, fl, ~) - -~(-fl, -~, - ~ ) . (4.216)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

503

By using spherical coordinates, the generalized model (4.215) can be represented as

-* - ~02rc~0~( f ~ f(t)

~(a, fl, O, q~)~a~uoq~(t) da dfl ) sin 0 dO dq~.

(4.217)

Similarly, the two-dimensional model can be expressed in the form /r

f (t) =

~(a, fl, q~)~13uo(t) dadfl

dq~.

(4.218)

To solve the identification problems for the models (4.217) and (4.218), we as before introduce the following auxiliary functions P(~, fl, O, q~) = P(a, fl, q~) =

(~,/~) (~,/~)

?(a', fl', O, q~) dor aft,

(4.219)

~(a', fl', q~) de' dfl'.

(4.220)

These functions can be related to the experimental data (4.213) and (4.212) by the following equations, respectively

fo2O

P(0r cos ~, fl cos ~, ~, ~, 0', q~') sin ~ d~ dO = F'(a, fl, 0', q/), (4.221) /r

f ~ P(acos$, flcos$, q~' + ~ ) d $ - F ( ~ , fl, q~').

2

(4.222)

By employing spherical harmonic expansions in the three-dimensional case and Fourier expansions in the two-dimensional case, we derive as before the following integral equations cos O, fl cos O)Lk(cos ~) sin ~ d~ = (-1) mF-km(~, [3) (4.223) f0~_Pkm(O~ . 2~r /r

2

COSnl~Pn(o~ cos ~, fl cos ~) d o = Fn(o~, fl).

(4.224)

By using the change of variables (4.182) and (4.205), the above integral equations can be reduced to the following forms, respectively

f0

()

(4.225)

~0 ~ -" Zn(X) d x - 1 F-n(O~,,~,o0 Pn(x, ~,x) V/0~2 _ x 2 n2 .

(4.226)

Pkm(X, 2x)Lk -dx d x - (--1) m a ~_km(a ' 2~x),

504

CHAPTER 4 Vector Preisach Models of Hysteresis

When k and n are equal to one, the following explicit solutions of Eqns (4.225) and (4.226) can be derived: Plm(~, ,~or = (--1)m d [o~2~71m(Or ~,or 2rc~ dc~

(m = O, +1),

P1 (a, 2a) = 1 d f0 ~ SaFl(S, 2s) ~ dc~ v/---2-2 82 ds.

(4.227)

(4.228)

For other values of k and n, polynomial expansion techniques or numerical techniques can be employed for the solution of the above integral equations. For isotropic media, the data (4.212) and (4.213) are reduced to (4.131). It can be shown that in this case ~(~, fl) = ~q~V(~,fl) and thus the generalized model (4.218) is reduced to the model (4.67). The proof of the above statement is left to the reader as a useful exercise.

4.6 D Y N A M I C

VECTOR PREISACH MODELS

OF

HYSTERESIS The vector Preisach models of hysteresis that have so far been discussed are rate-independent in nature; they do not account for dynamic properties of vector hysteresis nonlinearities. The purpose of this section is to develop dynamic vector Preisach models of hysteresis. We shall discuss only isotropic dynamic vector models of hysteresis. We begin with two-dimensional dynamic models; a straightforward extension to three dimensions will then follow. The main idea of the design of the dynamic vector hysteresis models is to introduce the dependence of the function v for scalar Preisach models for all directions q0 on the speed of output variations, dd~, along these directions. This can mathematically be expressed as

f (t)

-

f ~ -de ff~ >>-~v a, fl, c3t ,I ~ u ~ o ( t )

d~ dfl) dqg.

(4.229)

The direct utilization of the above model is associated with some untractable difficulties. These difficulties can be circumvented by using the power series expansion of the v-function with respect to df~. dt

v(~, B, --~ df~) = vo(~,~) + --~-Vl(~, df~ fl) q - . - . .

(4.230)

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

505

By retaining only the first two terms of the above expansion, we arrive at the following dynamic model: 7[

f(t) =fo(t)+ L~-dodf~ ( f L 2

"-~

Vl(a, fl)~;afluo(t)dlzdfl)dq~,

(4.231)

~fl

where

fo(t) -

-do

vo(cc,fl)~3uo(t) d~ dfl dq).

(4.232)

It is clear that in the case of very slow input variations the second term in the right-hand side of (4.231) becomes negligible. Thus fo(t) can be construed as a rate-independent component. This means that the function v0(~, fl) should coincide with the v-function of the rate-independent model (4.67). In other words, the function v0(a, fl) can be determined by matching rate-independent first-order transition curves measured for unidirectional variations of the input ~(t). We next represent the model (4.231) in Cartesian coordinates. To this end, we shall use the following expressions: eo - ex cos q~ + ~y sin q~, (4.233)

df 0 =G df dt

dfx

9 ~-~ = cos q~~

dfy

(4.234)

+ sin q~ dt "

By substituting (4.233) and (4.234) into (4.231), after simple transformations we arrive at the following form of the model (4.231): .

-

--~ -- f (t) - fo(t),

(4.235)

where the matrix,4 is given by

fii_(?Zxx(FZ(t)) ?Zxy(~(t))) ?Tyx(~(t)) ~yy(~(t)) '

(4.236)

and the matrix entries are specified by the expressions: --

f

COS 2 q9

(If

Vl(a,

)

fl)9~3uo(t) da dfl dq~,

(4.237)

/z

--f

sin2 r ( / L

Vl(~Z,fl)7afluo(t)d~zdfl)dcP,

(4.238)

?Zxy(F~(t))- ?Zyx(~(t)) 7[

cos O sin q~

/z

2

Vl

>/3

(0~,fl)~flUo(t) de dfl)

do.

(4.239)

CHAPTER 4 Vector Preisach Models of Hysteresis

506

Thus the dynamic model (4.231) can be interpreted as a set of two coupled ordinary differential equations (4.235) with hysteretic coefficients (4.237)-(4.239). The expression (4.235) also suggests that the instant speed of output variations is directly proportional to the difference between instant and rate-independent output values. The last fact is transparent from the physical point of view. We next turn to the identification problem of determining the function Vl(a, t) by fitting the model (4.235) to some experimental data. The following experiments are used to solve this problem. We restrict ~(t) to vary along the direction r = 0, which means that ~(t) = -~xu(t). First, we assume that u(t) is made 'infinitely negative' and then it is monotonically increased until it reaches some value a at t = to. Afterwards, the input is kept constant. As the input is being kept constant, the output relaxes from its valuef~ at t = to to its rate-independent value f0~. Owing to symmetry, we have fo(t) - -~xfo(t),

f (t) - -~xf(t).

(4.240)

Thus, according to the model (4.234)-(4.239), the above relaxation process is described by the differential equation ~T~ ~d f

= f i t ) - foa

(4.241)

where "~a =

COS2 (p

(ff

V1(a~, fl') da t dfl'

- ffs+ ~ vl (a', fl')de ~dfl') do,

(4.242)

and geometrical configurations of S+~,r and S;, r are the same as in Fig. 4.28. The solution to Eqn. (4.241) is given by t

fit) -- (f~ - f o ~ ) e - ~

+ fo~.

(4.243)

Thus z~ has the meaning of relaxation time and can be experimentally measured. Next the input u(t) is made again 'infinitely negative'. Then it is monotonically increased until it reaches the value a. Afterwards, the input is ! monotonically decreased until it reaches some value fl at time t = t o and it is kept constant for t > t~. As the input is being kept constant, the output relaxes from its value f~/~ at t - t~ to its 'static' value f0~3. Owing to

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

507

symmetry, expressions (4.240) hold and the model (4.235)-(4.239) yields the following differential equation for the above relaxation process:

df =fit) - fo~fl,

(4.244)

-r~-d-i where 7C

~ COS 2 q)

~/~ --

2

(ffs

Vl(~Z', fl') &z' dfl'

_

~,fl,~o

~,/~,~

and the geometry of regions S + 0~,(p and S~,~ is the same as in Fig. 4.29. By solving (4.244), we find: t

f(t) = (f~/~-f0~/~)e ~ + f 0 ~ .

(4.246)

Thus, %/~ has the meaning of relaxation time and can be experimentally measured. We next show that by knowing relaxation times % and %/~ for all possible a and fl we can determine the function Vl (~, fl). To this end, we introduce the functions 1

q(~, fl)= ~(% - %/~),

(4.247)

P1 (~, fl) - I f

Vl (~, if) d~ ~dff, JJT (~,fl) where T(~, fl) is the triangle shown in Fig. 4.10. It is clear as before that P1 and Vl are related by the formula Vl(~, fl) - - ~2P1 (~' fl).

(4.248)

(4.249)

8~ 8fl Thus, if the function P1 (~, fl) is found, then the function Vl (~, fl) can be retrieved. However, from the computational point of view, it is more convenient to deal with the function PI(~, fl) rather than with Vl(~, fl). This is because the double integral with respect to ~ and fl in expressions (4.237)(4.239) can be explicitly expressed in terms of PI(~, fl) by using formulas similar to (4.71). Another advantage of using PI(~, fl) is that this function can be directly related to the experimental data (4.247). Indeed, by using the expressions (4.242), (4.245), (4.247), (4.248) and Figs 4.28 and 4.29, we derive

f2

n COS2 (PP1 (a c o s r

fl c o s ( p ) d o

= q(a, fl).

(4.250)

CHAPTER 4 VectorPreisach Models of Hysteresis

508

The expression (4.250) is the integral equation that relates the function P1 (a, fl) to the experimental data q(a, ]/). This equation is similar to the integral Eqn. (4.132) and, consequently, the same techniques can be used for the solution of Eqn. (4.250) as for Eqn. (4.132). Namely, by using the change of variables x = ~ cos q), 2 = ]//~, (4.251) Eqn. (4.250) can be reduced to the following Abel-type integral equation: fo ~

x 2 dx 0r2 P1 (x, ,,Ix)V/(x2_ x2 -- --~-q(~x,,~).

(4.252)

By using the technique discussed in Section 4.4, the following closed-form solution of the above equation can be found:

PI (O~,,~o0 -

1 d fo ~ s3q(s, ,~s) ds. ~0r2 d ~

(4.253)

~/0r _ s2

We next turn to the discussion of three-dimensional dynamic vector Preisach models of hysteresis. Similar to (4.231), these models can be represented in the following mathematical form

df~176 ( fl >~ vl(a,fl)%~uo~o(t)dadfl)

f_.(t) = fo(t) _. + foaZ~fO~ -dOq)-~-[x sin 0 dO do,

(4.254)

where f0 (t) represents a'static' component of hysteresis nonlinearities defined by the expression ?~

- f02~f0~eoq~-"( /f~)~ Vl(~, fl)~Uoq~(t) d~ d]/ ) sin 0 dO dq~. (4.255)

By using Cartesian coordinates, we have e0~o- ex cos ~osin 0 + ~y sin q~sin 0 + ez cos 0,

dfo, dt

= cos ~0sin 0~-~f; + sin q0sin 0~-~f~4- cos 0 dfz dt "

(4.256) (4.257)

By substituting (4.256) and (4.257) into (4.254), after simple transformations we find df - f ( t ) -f0(t),

(4.258)

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

509

where the matrix,4 and its entries are given by

~l-

(

?lxx(~(t)) ?Zxy(~(t)) ?Zxz(~(t))) ?Zyx(~(t)) ~yy(~(t)) ?lyz(~(t)), ?Zzx(~(t)) ?~z~(F~(tl) ?~zz(~(t))

(4.259)

?~xx(F~(t))= foar~fO~ cos 2 qosin 3 0 X (/~

~yy(~(t)) =

Vl(~ ,

~>~

f020 x ( /L

(4.260)

sin 2 qosin 3 0

x (/f~

~lzz(U(t))--

fl)~&~uo~o(t)dc~d~)dOdq0,

Vl(~x,fl)~fluo~o(t)doraft)dO dqo,

(4.261)

COS2 0 sin 0

vl (a, fl)~a~uom(t)do~dfl) dOdq~,

(4.262)

?~xy(~(t))= ?Zyx(~(t))- ~O2'~fO-~cos q~sin q~sin 3 0

axz (u(t)) -- azx (u(t)) --

x (ff~

Vl(~X,fl)~&BUOq~(t)&xdfl)dOdq~, (4.263) ~>B

f02f0

cos q~cos 0 sin 2 0 Vl(C~,fl)~BUOq~(t)&xd~)dOdq~, (4.264)

x (/f~ ~yz (~(t)) - ~zy (~(t)) =

f0210

sin q~cos 0 sin 2 0

x (/f~

Vl(C~,~)}~,Uoq)(t)&xd[3)dOd~0. (4.265) ~>~

Thus the three-dimensional dynamic vector Preisach model of hysteresis (4.254) can be interpreted as a set of three coupled ordinary differential equations (4.258) with hysteretic coefficients (4.260)-(4.265). We next proceed to the discussion of the identification problem for the model (4.258)-(4.265). The solution to this problem is very similar to

510

CHAPTER 4 Vector Preisach Models of Hysteresis

that for the model (4.235)-(4.239) except that the final form of the solution is much simpler in three dimensions than in two dimensions. The experimental data used for the identification of the model (4.258)-(4.265) are measured when the input is restricted to vary along the z-axis, that is, when ~(t) - -~zU(t). As before, two types of relaxation processes are considered. The relaxation processes of the first type occur when the input is made 'infinitely negative' and then monotonically increased to some value a and kept constant thereafter. The relaxation processes of the second type occur when, starting from the state of negative saturation along the z-axis, the input is first monotonically increased to some value a, then monotonically decreased to some value fl and kept constant afterwards. The relaxation times r~ and r~/~ of the above processes can be measured and used for computing the function q(a, fl) defined by (4.247). On the grounds of symmetry, for both types of relaxation processes we have f0 - ~zf0(t),

f(t) - -~zflt).

(4.266)

This results in the reduction of the model (4.258)-(4.265) to the following equation

df

~zz (~(t)) ~ = fit) - fo (t).

(4.267)

Now, by using the same line of reasoning as in the derivation of Eqn. (4.250), we can show that the function P1 (cx, fl), defined by (4.248), is related to the experimental data q(a, fl) by the integral equation f0 ~ Pl(a cos 0, fl cos 0) cos 2 0 sin 0 dO =

q(a, 2~

"

(4.268)

By using the change of variables x = a cos 0,

;~ = -fl,

(4.269)

the integral Eqn. (4.268) is reduced to the form ~0 a x2p1 (x, ,~x) dx - -~-~ ~3 q(a, ,~),

(4.270)

from which we derive the final expression PI(X, ,~x) --

1 d [cx3q(or ,~or 27ca2 da

(4.271)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

511

4.7 GENERALIZED VECTOR PREISACH M O D E L S OF HYSTERESIS. EXPERIMENTAL TESTING There are two ways in which the above vector Preisach models can be further generalized. The first way is to use generalized scalar Preisach models as the main building blocks for the construction of vector models. The second way is to generalize the notion of input projection Uq~(t). We begin with the first approach. To be specific, we shall use the nonlinear (input-dependent) scalar Preisach models. Analysis of generalized vector Preisach models of this type is very similar to that for the 'classical' vector Preisach models. For this reason, our discussion will be concise and will be centered around the description of final results, while filling in the details will be left to the reader. We begin with two-dimensional isotropic models that can be represented in the following mathematical form

f (t) =

-

f (ff-~q~

v(~, fl, u~o(t))~Uq~(t) d~ dfl dq)

)

u,p(t)

4- ~(u(t))fi(t).

(4.272)

The numerical implementation of the above model is substantially facilitated by the use of the following function P(~, fl, u) - f f

v(~', fl', u) d~' dfl',

(4.273)

, I l K aflu

where R~/~u is the rectangle shown in Fig. 4.12. By employing the above function we can find explicit expressions for the double integral with respect to a and fl in (4.272). In addition, the function P(a, fl, u) can be directly used for the identification of the model (4.272). First- and second-order transition curves measured along any fixed direction (for instance, along the direction (0 -- 0) will be utilized for the solution of the identification problem. By using these curves, the following function can be constructed

1

F(a, fl, u) - -~(f~u - f~u),

(4.274)

where f~u and f~u have the meaning of output values along the first- and second-order transition curves, respectively. It can be shown that the function P(a, fl, u) is related to the experimental data (4.274) by the following integral equation 7~

f

~ cos q~P(a cos r 2

cos q~, u cos q~) dq~

F(a, fl, u).

(4.275)

CHAPTER 4 Vector Preisach Models of Hysteresis

512

This equation is very similar to the integral Eqn. (4.132). Thus, the same techniques can be employed for the solution of Eqn. (4.275) as for the solution of Eqn. (4.132). For instance, an approximate polynomial solution of the above equation can be found. The finding of this solution is based on the fact that monomials akfl sum are eigenfunctions of the operator ti: 7~

AiP = f 2 cos q~P(~ cos ~0, fl cos ~o, u cos q~) d~o.

(4.276)

7~

2

In other words, we have A o~kfl s U m -- ,~k+s+mO~k fl s u m,

(4.277)

,~k+s+m -- /_~ (COScp)k+s+m+l dcp.

(4.278)

where: 2

Now, by expanding the right-hand side of the integral Eqn. (4.275) into the series of Chebyshev polynomials

F(a, fl, u) = y ~ a~qmT~(~)Tq(fl)Tm(u),

(4.279)

f.,q,m

we can represent the solution of the above equation in the form P(a, fl, u) = y ~ aeqmt~qm(a, fl, u),

(4.280)

~,q,m

where the polynomials t~qm(a, fl, u) are m a p p e d by operator A into polynomials T~(~)Tq(fl)Tm(u). Polynomials t~qm(a, fl, u) can be determined by using expressions (4.277) and (4.278). As far as expansion coefficients a~qm are concerned, they can be computed by using a formula similar to (4.146):

8 f0/0/0 F (cos 0, cos 0', cos

ae.qm = ~

x cos s cos qO~cos toO" dO dO~dO".

(4.281)

It is also possible to find a closed-form solution to Eqn. (4.275). To this end, the following change of variables is used: x = ~ cos q~,

2 = fl/~,

;~ = u/~

(4.282)

and the above equation is reduced to the Abel-type integral equation fo x v/cz2x_ x 2 P(x, 2x, ;~x) dx = ~F(~, ~ 2a, ;~a).

(4.283)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

513

Using the same reasoning as in Section 4.4, we obtain the following solution of Eqn. (4.283):

1 d fo ~ s2F(s, 2s, Zs) ds. P(~, 2~, Z~) - ~ d~ V~2 _ S2

(4.284)

As is seen from the above discussion, function P(~, fl, u) (and consequently function v(a, fl, u)) can be determined by matching the increments (4.274) between the first-order and second-order transition curves. The function ~(u) in (4.272) can be found by matching an ascending branch of a major loop. This leads to the expression ~(u) - f + + f u . 2u

(4.285)

We next proceed to the discussion of three-dimensional generalized vector Preisach models of hysteresis. These models can be represented as follows

f-"(t) -

/0270rt-dOq)(/[

v(~, fl, UOq)(t))~UOq~(t) d~ aft ) sin 0 dO dq0 uoq~(t)

+ ~(u(t))F~(t).

(4.286)

By introducing function P defined by (4.273) and by using experimental data (4.274) the identification problem for the model (4.286) can be reduced to the following integral equation 2re f0 ~ cos 0 sin OP(oecos O, fl cos O, u cos O) dO - ~(~,/~, u) "

(4.287)

The explicit solution of the above equation can be obtained by using the change of variables

u/~

(4.288)

f0~xP(x, 2x, Zx) dx - ~-~F(a, 0~2 2a, Za).

(4.289)

x = ~ cos O,

2 =///~,

Z=

and by reducing (4.287) to the integral equation

Differentiation of (4.288) yields 1 d [a2F(~ ' 2a, Za)]. P(~, 2~, Z~) - 2 ~ dc~

(4.290)

We next turn to generalized dynamic vector Preisach models of hysteresis. For the sake of notational simplicity, we consider only two-dimensional

CHAPTER 4 VectorPreisach Models of Hysteresis

514

models; extensions to three-dimensional models are straightforward. The two-dimensional generalized dynamic Preisach models can be defined as follows \

f (t) - fo(t) +

Vl (e, fl, Uq)(t))~BUq~(t) de dfl) do,

-b~o --~

u~o(t)

(4.291) where f0 is a 'static' component of hysteresis nonlinearity which coincides with (4.272). By using Cartesian coordinates, the model (4.291) can be reduced to the following ordinary differential equations df -. -. A~-~ =f(t) -fo(t),

(4.292)

where the matrix,4 has the following entries:

xxlO t t/ =

Vl (e, fl, U~o(t))~aBUq)(t)de dfl) dq~, (4.293)

COS 2 (p

u~o(t)

V1(e, fl, Uq~(t))~;~BUq~(t)de dfl) do, (4.294) 2

U~o(t)

?Zxy(~(t)) = ~yx(u(t)) = / _ ~ cos ~osin qo 2

X (ffR

uq~(t)

vl(e, fl, Uq~(t))~;aflUq)(t)dedfl)dq~.

(4.295)

To solve the identification problem for the model (4.292)-(4.295), we restrict input to vary along the x-axis. Then, on the grounds of symmetry, it is easy to conclude that f (t) = -Grit), fo(t) - -exfo(t). (4.296) This results in the reduction of the model (4.292)-(4.295) to the following equation

^ df[ =fit) -fo(t) axx(U(t))--d-

(4.297)

We next introduce the function Pl(e, fl, u): P1 (e, fl, u) - / f R

Vl(e',/J', u) de' dfl', aflu

(4.298)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

515

and experimental data: 1

Q(:r fl, u) = ~(%u - %flu),

(4.299)

where %u and r~flu are relaxation times for the processes, which are similar to those described in detail in the previous section. By using the expressions (4.298)-(4.299) and the same line of reasoning as in the previous section, it can be shown that the function P1 (cx, fl, u) is related to the experimental data (4.299) by the following integral equation

f

2 COS2 (PP1(~zcos (.p, fl cos q~, u cos qo) dq~ - Q(a, fl, u).

(4.300)

/c

By using the change of variables (4.282), the above equation can be reduced to the Abel-type integral equation f0 a

V/~x2X2_ x2P1 (x, ~x, )~x) d x

-

~X2 -~--Q(a, 2a, za),

(4.301)

whose solution is given by 1 d fo a s3Q(s, ,~s, zs) ds. P1 (0r 20r Z~x) - ~0r d~ ?~-2 _ s2

(4.302)

Finally, we shall discuss generalized anisotropic vector Preisach models. Again, for the sake of notational simplicity, we consider only twodimensional models. These models can be defined as /-r ..+

f(t)-L2(ffR

u,p(t)

~(oe, fl, cp, ur

d~ dfl d~p + ~(~(t)). (4.303)

The experimental data defined by the function -*

1 -*

fl, u,

-*

- i(f u ,

(4.304)

will be used for the identification of the model (4.303). As before, we introduce the auxiliary function

P(o~, fl, cp, u) = ffR

:r

~(~'' fl'' cp, u) da' dfl'.

(4.305)

CHAPTER 4 Vector Preisach Models of Hysteresis

516

We employ the following Fourier expansions: OO

P(a, fl, u, q)) -- ~

rJn(a, fl, u)e imp,

(4.306)

F-(a, fl, u, q))= y~ F-n(a, fl, u)e inc.

(4.307)

n~--OO OO

By using (4.304)-(4.307), it can be shown that Pn is related to Fn by the following integral equation cos

n~Pn(a cos ~, fl cos ~, u cos ~) d~ - Fn(a, fl, u).

(4.308)

This equation can be reduced to the Abel-type integral equation /0 ~ Pn(x, -" ,~x, Zx) ~rTn(X) 2 _ x 2 dx = ~1 Fn(a, ,~a, Za)

(4.309)

Various numerical techniques can be used for the solution of this equation. In particular cases when n --- 4-1, the following closed-form solutions can be found: -~ 1 d f ~ s2F+l(S, ,~s, Zs) P+I (a, 2a, Za) = ~a da ]0 2---7~/a - s2 ds. (4.310) As far as the function ~(~) in (4.303) is concerned, the following expression (similar to (4.285)) can be derived: ~(~pU) = f+~p +f~cp 2 '

(4.311)

where the notations in (4.311) are self-explanatory. Next we proceed to the discussion of the second (and probably most fruitful) approach to the generalization of vector Preisach models. This approach is based on the notion of generalized input projection. The corresponding vector model can be written as follows:

f (t)

= ,i-~12

-J4~

~fl

v(a, fl)~l~(t)lg[O(t) - ~b] dad]/ d~b,

(4.312)

where O(t) is the angle between ~(t) and the polar axis. In the case when g(O-q5) = cos(0-q)), the above model is reduced to vector Preisach models extensively studied in this chapter. This justifies the following' cosine-type'

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

517

constraints on the function g(~b): g'(~b)~<0 for 0~r g(0) - 1, g(~) = - 1 ,

g(2)

(4.313) = 0,

(4.314)

g ( ~ - ~b) = - g ( r

(4.315)

and the product Ifi(t)Ig(O- q5) can be construed as a generalized projection of vector input fi(t) on the direction specified by the angle r Function

1

g(r = I cos r (cos ~) is an example of a 'cosine-type' function that satisfies the constraints (4.313)-(4.315). In the above model, functions v(a, fl) and g(r are not specified in advance but rather should be determined by fitting this model to some experimental data. This is an identification problem. To perform the identification of the model, the following experimental data will be used. (a) First-order transition curves that are measured when the input fi(t) is restricted to vary along one, arbitrary fixed direction. By using these curves, we can introduce the function:

1

(4.316)

7(~,/~) = ~(f~ - f ~ ) .

(b) 'Rotational' experimental data measured for the case when the input is a uniformly rotating vector: fi(t) = {urn cos cot, um sin cot}. It can be shown that for isotropic hysteretic media, the output has the form f(t) =f0 +fl(t), where f0 does not change with time, whilef~ (t) is a uniformly rotating vector that lags behind the input by some angle. By using the rotational experimental data, the following function can be introduced:

R(um) =

fi(t) .fl(t)

Um

,

(4.317)

that has the meaning of the projection of fl (t) on the direction of input. Functions v(a, fl) and g(r will be recovered from experimental data (4.316) and (4.317). For the identification as well as computational purposes, it is convenient to introduce the function P(~, fl):

P(~' fl) --/fT (~,~) v(a', fl')d~' dfl',

v(a, fl) = -

O2p(~, fl)

.

(4.318)

518

CHAPTER 4 Vector Preisach Models of Hysteresis

By using the same line of reasoning as before, it can be shown that model (4.312) will match the experimental data (4.316) and (4.317) if functions P(a,//) and g(~b) satisfy the equations: 2

cos ~bP(eg(4~), ~g(~b)) d4~ = ~'(c~, ~),

(4.319)

-2

cos d~P(um, umg(d~)) dd~ = R(um).

(4.320)

Since v(e,/~) is nonzero only within the triangle r = {-e0~
.Jz'(O~,fl) "- y ~ n--0

y~ Uks-(2n+l)o~kfls. k+s=2n+l

(4.321)

Approximation (4.321) contains only odd terms because of the mentioned symmetry properties of ~'(a, ]/). Owing to these symmetry properties, it is also clear that Uks _(2n+1) -- --Usk ..(2n+1)9 An actual polynomial approximation (4.321) can be found by using a Chebyshev polynomial series expansion of ~(~, ]/) as described before. We shall also use the odd extension of R(um) from [0, a0] to [-a0, a0] and the following polynomial approximation: N

a(um)

(4.322)

= y ~ R2n+lU 2n+1 m 9 n--0

We shall next look for P(cr ]/) in the form N

P(a, fl) - y~ n=0

y~.

C:ks"(an+l)o~k~s.

(4.323)

k+s--2n+l

By substituting (4.321), (4.322) and (4.323) into (4.319) and (4.320) and by equating similar terms, we derive ..(2n+1) f ~ ..(2n+1) 2~'ks JO COS ~bg2n+l (~b) d~b = Uks

-2 ~ k+s=2n+l

_(2n+1)~07rCOS~gS (~b) d~b = Cks

(4.324) a2n+l.

(4.325)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

519

From (4.315) we find that the integrals in (4.325) are equal to zero if s is even and

f0

cos qbg s ( q5) d q5 = 2

/0

cos ~gS (~b) d~b

if s is odd.

(4.326)

Next, from (4.324) we obtain (2n+1) ao ' 2n+1 ~0 ~ COS ~g2n+l (~b) d~b. (2n+1) -- ~(2n+1) = 2 Cks ~0,2n+1

(2n+1) aks

(4.327)

From (4.325), (4.326) and (4.327) we derive . (2n+1) ~0 ~ cos ~g2n+l (~) d~b R2n+l -- -~c0,2n+l

-

~0 ~ cos c~gs (~b) d~b, 4 y ~ Ic(2n+1) ks

(4.328)

where Y-~' stands for the s u m m a t i o n over all s that are odd and less than 2n + 1 and such that k + s = 2n + 1. By using (4.327) in (4.328), we find .. ( 2 n + 1 ) _ 2 ~ , . , ( 2 n + l ) ~ a ~ 2 s n + l ) ) ( a ~ ) s ' ~ R2n+l -- -za0,2n+l ~'0,2n+1/ (2n+1) ~(s) ] ' \a0,2n+l COs / which leads to ~(2n+1) c0,2n+l - - _

., (2n+1) R2n+l q- za0,2n+l _(2n+1) (s) " ) ( a6s~ 2 ~ ' ( Uks _(2n+1) (s) ,] UO,2n+l COs-

(4.329)

(4.330)

By assuming that C0,1(1) 1 = 2,

(4.331)

where 2 is some u n k n o w n constant, from (4.330) we derive c(2n+1) _ ~h(2n+l) (4.332) 0,2n+1 . . . . 0,2n+1" ~(2n+1) According to (4.330), constants u0,2n+l can be recursively determined by using the formula ~,(2n+1) u0,2n+l - - _

_(1) and the fact that ,o0,1

_

1.

,. (2n+1) R2n+l -}- za0,2n+l

2E'(a(2n+l))(aoS)s ' k s "~ ..(2n+1) (s) u0,2n+l bos ,]

(4.333)

CHAPTER 4 Vector Preisach Models of Hysteresis

520

After h(2n+l) ~'0,2n+1 are computed, from (4.326), (4.327) and (4.332) we find f0 z~COS ~bg2n+1 (~) d~b - $2n+1 2 '

(4.334)

(2n+1) a0'2n+l $2n+1 -- ~(2n+1) " u0,2n+l

(4.335)

where

According to (4.313)-(4.315), there exists a function z(g) which is inverse to g(~b). By using this function, we shall use the following change of variables in integrals (4.334): ~b = ;~(g),

dq~ = Z ( g ) d g ,

z(g(~b)) = ~b.

(4.336)

By substituting (4.336) into (4.334) and integrating by parts, we transform (4.334) as follows: f_ +1 g2n sin )~(g) dg = 2(2n $2n+1 + 1)" 1

(4.337)

We shall next make another change of variables: g=cos~,

dg=-sin~d~,

(4.338)

which leads to f0 z~COSan ~ sin

6T(6) d~

=

$2n+1 ;~(2n + 1)

(4.339)

where T(6) - sin Z(cos ~).

(4.340)

We shall extend T(~) from [0, ~] as a periodic odd function with half-wave symmetry. As a result, this function can be expressed as N T(~t) - E t2n+l sin(2n + 1)~, n=0 where

2f0 T(6)

t2n+l -- -7~

sin(2n + 1)~ d~.

(4.341)

(4.342)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

521

It is known (see [13]), that

sin(an+1)~_sin~[a2ncos2n~_(n-1)a2n-2 1 COS2n-2 if/ (n--3)22n-4cos2n-4d/

+ 2

--(n--4)aan-6cosan-6~4-...]

3

where (Pk) --

(4.343)

P! . From (4.339), (4.342) and (4.343), we derive k,(p-k)

t2n+l

1[

tn-)aan_2.S2n-1 2

~ 22n

2n 4- 1

1

2n- 1

4-(n-3) $ 2 n22n-4" - 3 2 an-3 --

(n-4) $ 2 n22n-6 - 5 3"2-dn~-511 {- . . . . ~s

(4.344)

From (4.314), (4.336) and (4.340) we find T(2)-sinz(0)-sin(2)=1.

(4.345)

From (4.341), (4.344) and (4.345) we obtain N , ~ - ~--~(--1)ns n--0

(4.346)

Now, the identification procedure can be summarized as follows. First, we find polynomial approximations (4.321) and (4.322). Then, by using coefficients , a2n+l and (4.333), we recursively compute b(2n+1)0,2n+1and $2n+1. Next, by using (4.344) and (4.346), we determine s ,~ and t2n+l. By using (4.327) and (4.332), we compute C ~(2n+1) ks 9 Finally, by using (4.341), (4.340) and (4.336), we compute T(q), ;~(g) and then retrieve the inverse function g(~b). In the particular case when

Uks"(2n+l)

g(~b) = [cos ~p[1/nsign(cos ~)

CHAPTER 4 Vector Preisach Models of Hysteresis

522

the solution of the identification problem can be simplified. Indeed, in this case, the integral Eqn. (4.319) can be written as follows: F(a, fl) = 2

f0~cos (~)P(a COS1/n t~, fl COS1/n r

d~b.

(4.347)

By introducing the following change of variables, ,~ =

X = a COS1/n t~,

fl/a,

(4.348)

and by using the same transformations as before, the following integral equation is obtained from (4.347):

a(a) -

fo ~

(4.349)

N(x) dx, v/a2n _ x 2n

where the following notations are adopted:

N(x) = x2n-lp(x, J,x),

(a2/2n)F(a, ,a.a).

R(a) =

(4.350)

Multiplying both sides of Eqn. (4.349) by a 2n-1/v/S 2n -- a 2n and integrating with respect to a from 0 to s, and then interchanging the order of the double integration for the right-hand side, the following result can be derived: ~0 s

a2n-1

/S2n _ a2n

R(a) da

/ o sN(x) (/x lf0S

an

2n-ld

x/(san _ a2n)( a2n _ xan)

N(x)(~x s ~/( san-xana2 )

)dx

d(a 2n) -

-

(a 2n

(s2n+x2n)2 ) dx. (4.351) 2

Then, by introducing the following variable of integration,

W - a 2n -- (S2n if- x2n ) / 2 ,

(4.352)

we obtain fO s

a 2n-1 R(a) d a /S2n _ a2n

1loS N(x) t f ~n

(s2n-x2n)/2

dw

"J(xRn-s2n)/2 V/( s2n-xan ) _ W2 ) dx

~f0 s N(x) dx. 2n

(4.353)

4.7 GENERALIZEDVECTOR PREISACH MODELS OF HYSTERESIS.

523

By using the notation of (4.350), we obtain

P(o~, 200

1 ['~ nsn-lF(s, 2s) + s n (d/ds)F(s, ,~s) I"C

Jo

ds.

(4.354)

V/~2n _ s2n

From Eqn. (4.354) we conclude that, no matter what value of n is chosen, the model can always match the data obtained from the 'scalar' experiments. In the specific case when n = 1, the final result (4.354) coincides with the final result for the case of the identification of vector models discussed in Section 4.4. In order to complete the identification procedure, the optimum value of the unknown n should be determined by making use of the experimental data obtained from the 'rotational' experiments. One way to do this is by adopting a 'least-squares' approximation. Next, we shall illustrate the above discussion by the following experimental identification and testing of the model (4.312). The identification of the model has been performed for a typical Ampex-641 (7Fe203) magnetic-tape material by using a vibrating sample magnetometer equipped with an orthogonal pair of pickup coils. As mentioned above, the identification problem has been solved by using two sets of experimental data. First, the set of first-order transition curves was measured (see Fig. 4.35). Then, the unknown function P(a, fl) was found as a function of n. Next, the set of 'rotational' experimental data were measured. This was performed by first applying a negative input field, sufficiently large to drive the sample into the state of negative saturation, then increasing it gradually until it reached a certain positive value where it was kept constant while sample (in-plane) rotation was introduced. This was repeated for different positive field values and the o u t p u t / i n p u t phase-lag relationship as a function of the rotating input amplitude was obtained. It was found that the model gives very good matching to the 'rotational' phase-lag relationship when the value of n is chosen to be 3. A comparison between the computed and measured o u t p u t / i n p u t phase-lag values is shown in Fig. 4.36 for two different cases of n in order to demonstrate the impact of the choice of n on the model capability to match 'rotational' experimental results. After the identification process was performed, the experimental testing of the model was carried out in order to estimate its quantitative ability to mimic the property of correlation between mutually orthogonal components of output and input. This property has long been regarded as an important 'testing' property for any vector hysteresis model in the area of magnetics. This experiment was carried out as follows. First, the input field was restricted to the y-axis and was increased from some negative

CHAPTER 4

524

Vector Preisach M o d e l s of Hysteresis

0.06

0.04

3' E .~.

0.02-

r

.9

0.m

(1) E o)

-0.02

--

r

- i

-0.04

-0.06

I , , , I , , , I , , , I , , , I , , , I , -400 -200 0 200 400 600

-600

Magnetic

field

[Oe]

F I G U R E 4.35

60

I

I

t

I

I

I

I

I

I

I

I ,I

I

I

I

I

I

I

I

I

Experiment

(/)

o o

_

,

....

Model

when

n=l

Model

when

n=3

-

I I \X

o

40

__

/

~ \

E~r) _.1

--

6ffl

--

e.12.

--

II

\\\

II

\ \

II / ~

\ x x

"5 20 e~ E Q

//

d--,

0

--

d/

0 I

0

I

I

I

t

t

0.2

t

I

i

i

t

0.4

Rotating

I

0.6

field amplitude

F I G U R E 4.36

i

i

t

I

0.8 [KOe]

t

I

I

1

4.7 G E N E R A L I Z E D VECTOR PREISACH MODELS OF HYSTERESIS.

0.06 ~, E .o

I:' '~'r"'i'r'''

I ''

''

I~

r l ' '-i'~: !

0.04 0.02

0n -0.02

.

.

0

.

.

.

0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

1 1.5 2 Orthogonal Field H x [KOe]

2.5

0.04

O E 5;

"11

0.02 0~

~~'e=~--'~-

'~'~"~'"-"~

- ~

_oo~;,,,, i i~-i i ~i i,,,,,, ii ii, ,l 0

0.5

1

1.5

2

2.5

Orthogonal Field H x [KOe] 0.01 0 E -0.01 :~ -0.02 -0.03

,...,. ,_.I. ,_, _ L , I , _ _ , , -0.04 ----__,., J~_!. ,., j_j_l, 0 0.5 1 1.5 2 Orthogonal Field H x [KOe] 0.02

'' E 0

J'l'

~'~I

,-'2.5

" " '-"'-~[') i--i-'l ~-~ I ~ .-4

~, E 9~. -0.02

~; -0.04 -0.06 0

0.5

1 1.5 2 Orthogonal Field H x [KOe]

[] Experiment

Z~ n = 1

n=3 F I G U R E 4.37

2.5

525

526

CHAPTER 4 VectorPreisach Models of Hysteresis

value (sufficient to drive the sample into the negative saturation state) to some positive value Hy+, and then decreased to zero. This resulted in some remnant magnetization Myr. Then, the magnetic field was restricted to vary along the x-axis. By increasing the field value (now Hx) gradually from zero to very high values, the orthogonal remnant component Myrw a s reduced and the Myr versus Hx relationship was recorded. This experiment was repeated for different values of Hy+ (and consequently different values of Myr). Then by using the model (4.312) the computational results were obtained for the same sequences of field variations as in the experiments. Some sample comparisons between the computed and measured results of this experiment are shown in Fig. 4.37. It is clear from Fig. 4.37 that the model (4.312) gives appreciably better matching with the experimental data than the vector Preisach models discussed in previous sections. This may be due to the presence of an additional unknown function g(~0), which can be determined from the identification process. As a result, the opportunity appears to incorporate more experimental data in the identification process, leading to a more accurate model. The presentation of the material in this chapter is largely based on publications [17-27].

References 1. F. Preisach, Uber die magnetische nachwirkung, Zeitschrift fiir Physik 94, 227-302 (1935). 2. E.C. Stoner and E. P. Wolhfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Philosophical Transactions of the Royal Society of London Series A-Mathematical and Physical Sciences 240 (826), 599-642 (1948). 3. I.A. Beardsley, Modeling the record process, IEEE Trans. Magn. 22 (5), 454-459 (1986). 4. T. R. Koehler, A computationally fast, two-dimensional vector hysteresis model, J. App1. Phys. 61 (4), 1568-1578 (1987). 5. G. Bertotti, Hysteresis in Magnetism, Academic Press, New York (1998). 6. J. C. Slonczewski, IBM Research Memorandum, No. RM 003.111.224 (1956). 7. M. Prutton, Thin Ferromagnetic Films, Butterworth, Washington, DC (1964). 8. G. Friedman, New formulation of the Stoner-Wohlfarth hysteresis model and the identification problem, J. Appl. Phys. 67 (9), 5361-5363 (1990).

REFERENCES

527

9. M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis, Nauka, Moscow (1983). 10. J. A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and Minor loops-Models and Experiments, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 386 (1791), 251-261 (1983). 11. A. Damlamian and A. Visintin, A multidimensional generalization of the Preisach model for hysteresis, Comptes Rendus De L Academie Des Sciences Serie I-Mathematique 297 (7), 437-440 (1983). 12. C. Lubich, Fractional linear multistep methods for Abel-Volterra integral-equations of the 1st kind, IMA J. Numer. Anal. 7 (1), 97-106 (1987). 13. I. S. Gradstein and I. M. Ryzkik, Tables of Integrals, Series and Products, Academic Press, New York (1980). 14. A.W. Joshi, Elements of Group Theoryfor Physicists, Wiley Eastern, New Delhi (1982). 15. N.J. Vilenkin, Special Functions and the Theory of Group Representation, American Mathematical Society, Providence, RI (1968). 16. G.Y. Lyubarski, Applications of Group Theory in Physics, Pergamon Press, Oxford (1960). 17. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, Berlin (1991). 18. I.D. Mayergoyz, Mathematical-Models of hysteresis, IEEE Trans. Mag. 22 (5), 603-608 (1986). 19. I. D. Mayergoyz and G. Friedman, Isotropic vector Preisach model of hysteresis, J. Appl. Phys. 61 (8), 4022-4024 (1987). 20. I. D. Mayergoyz and G. Friedman, On the Integral-Equation of the vector Preisach hysteresis model, IEEE Trans. Mag. 23 (5), 2638-2640 (1987). 21. I. D. Mayergoyz, Vector Preisach hysteresis models, J. Appl. Phys. 63 (8), 2995-3000 (1988). 22. I. D. Mayergoyz, Identification problem for 3-D anisotropic Preisach model of vector hysteresis, IEEE Trans. Magn. 24 (6), 2928-2930 (1988). 23. A.A. Adly and I. D. Mayergoyz, A new vector Preisach-type model of hysteresis, J. App1. Phys. 73 (10), 5824-5826 (1993). 24. I. D. Mayergoyz and A. A. Adly, A new isotropic vector Preisach-type model of hysteresis and its identification, IEEE Trans. Magn. 29 (6), 2377-2379 (1993).

528

CHAPTER 4 VectorPreisach Models of Hysteresis

25. I.D. Mayergoyz, 2D vector Preisach models and rotational hysteretic losses, J. Appl. Phys. 75 (10), 5686-5688 (1993). 26. A.A. Adly and I. D. Mayergoyz, Accurate modeling of vector hysteresis using a superposition of Preisach-type models, IEEE Trans. Magn. 33 (5), 4155-4157 (1997). 27. A. A. Adly, I. D. Mayergoyz, and A. Bergqvist, Utilizing anisotropic Preisach-type models in the accurate simulation of magnetostriction, IEEE Trans. Magn. 33 (5), 3931-3933 (1997).

CHAPTER 5

Stochastic Aspects of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 5.1 5.2 5.3 5.4 5.5 5.6

5.7

Preisach Model with Stochastic Input as a Model of Thermal Relaxations Experimental Testing. Scaling and Data Collapse in Magnetic Viscosity Preisach Model Driven by Continuous-Time Noise. Origin of the Universality of Long-Time Thermal Relaxations Noise in Hysteretic Systems and Stochastic Processes on Graphs Analysis of Spectral Noise Density of Hysteretic Systems Driven by Stochastic Processes Modeling of Temperature-Dependent Hysteresis within the Framework of Randomly Perturbed Fast Dynamical Systems Functional (Path) Integration Models of Hysteresis References

5.1 P R E I S A C H M O D E L W I T H S T O C H A S T I C

529 541 550 562 573

578 587 601

INPUT AS

A M O D E L OF T H E R M A L RELAXATIONS It is well known that the physical origin of hysteresis is due to the multiplicity of metastable states exhibited by hysteretic materials or systems. At temporally constant external conditions, large deviations of random (thermal) perturbations may cause a hysteretic system to move from one metastable state to another. This may result in gradual (slow) changes of the output variable. This temporal loss of memory of a hysteretic system is generally referred to in the literature as 'after effect', 'viscosity' The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

530

CHAPTER 5 Stochastic Aspects of Hysteresis

or 'creep'. This phenomenon can be of practical significance in various engineering applications where hysteresis is utilized. One important example is the magnetic storage technology where the viscosity effect is detrimental as far as the time reliability of recorded information is concerned. Traditionally, the viscosity phenomenon has been studied by using thermal activation-type models (see, for example, [1,2]). It has been realized that these models have some intrinsic limitations. For instance, they are usually regarded as 'noninteracting particle' models and they are valid for hysteretic systems with energy barriers much larger than the thermal activation energy. In this section, we shall explore a new approach [3-6] to the modeling of thermal relaxations based upon the Preisach model of hysteresis. In this approach, random thermal agitations will be modeled by a stochastic input. We shall compare this approach with thermal activation-type models and with some known experimental facts. Consider a deterministic input u(t), which at time t = 0 assumes some value u and remains constant thereafter. In a purely deterministic situation, the output flt) would remain constant for t~0 as well. However, in order to model thermal relaxations, we assume that some noise described by the stochastic process Xt is superimposed on the constant input u. Consequently, the Preisach model is driven by the following random process: Xt = U -Jr-Xt,

Xt = 0,

(5.1)

where Xt stands for the expected value of Xt. The output will also be a random process given by

f t - /f~ >>-~#(~

dad]/.

(5.2)

It is instructive to note that adding noise Xt to the deterministic input u(t) is mathematically equivalent to subtracting the same noise from switching thresholds ~ and//. This is true because (~_Xt,fl_xt)U(t)

--

~ ( u ( t ) + Xt).

(5.3)

Imposing noise on the switching thresholds (barriers) may be more transparent (or justifiable) from the physical point of view, while adding noise to the deterministic input makes the problem more tractable from the mathematical point of view. To simplify the problem, we shall first model the noise by a discrete-time independent identically distributed (i.i.d.) random process Xn. In this case, it can be assumed that process samples remain constant at time intervals At = tn+l - tn and undergo monotonic step changes

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

531

at time tn+l. Accordingly, the Preisach model is driven by the process Xn -- U -}- X n ,

(5.4)

X n - O.

The output process is given by /f~

(5.5)

[.t(~, fl)~flXn da dfl.

We will be interested in the time evolution of the expected value of the output process. Since integration is a linear operation, from (5.5) we derive f n = E{fn} -

#(o~, fl)E{~;~flXn} dadfl. >>,~

(5.6)

Thus, the whole problem is reduced to the evaluation of the expected value, E{~a~Xn}. Since ~ X n may assume only two values, +1 and - 1 , we find -- +1} - P{~aflXn -

E{~aflXn} - P { ~ f l X n

-1}.

(5.7)

It is also clear that P{~aflXn - +1} + P{~aflXn -

- 1 } - 1.

(5.8)

By introducing the notation P{~Xn

- +1} = qa~(n),

(5.9)

- 2qa]~(n) - 1.

(5.10)

from (5.7) and (5.8) we obtain E{~Xn}

We next derive the finite difference equation for q ~ ( n ) . According to the total probability theorem, we have P{~flXn+l - + 1 } - - P { ~ X n + l = q-ll~flXn = q-1}P{~;~Xn = +1} +P{~aflXn+l = +ll~aflXn - -1}P{~flXn - - 1 } . (5.11) It is convenient now to introduce the switching probabilities P++(n)-P{~flXn+l

- +ll~flXn - +1}

(5.12)

P-~; (n) - P{~aflXn+l - q-ll~aflXn - - 1 } .

(5.13)

CHAPTER 5 Stochastic Aspects of Hysteresis

532

Similar meanings hold for the switching probabilities P~fl- (n) and P+fl (n). It is clear that ++ p+P~fl (n)4- ~/~ ( n ) = 1, (5.14) P~-; (n) + P~-~ (n) = 1,

(5.15)

P{~flXn = -1} = 1 - q~fl(n).

(5.16)

By using (5.9) and (5.12)-(5.16), we can transform (5.11) to +-(n))] + P-d;(n) qafl(n + 1) = qa/~(n)[1 - (P2; (n) + P~/~

(5.17)

We next proceed to the evaluation of switching probabilities P 2 ; (n) and P+-(n) e/l 9 It is clear that these probabilities can also be defined as

P-~; (n) = P{Xn+l > ~l~;~flXn = -1}, +-

P~fl (n) = P{Xn+l < fli?aflXn = +1}.

(5.18) (5.19)

In general, these probabilities are difficult to evaluate because multidimensional conditional probability density functions are required. However, the problem is significantly simplified if the noise is modeled by the i.i.d. process. In this case we find

P-~; (n) = P-~+ = P{Xn+l > or -= ~fl (n) = P-~- = P{Xn+l P+-

<

/J} - -

p(x) dx,

(5.20)

/~oo fi(X) dx,

(5.21)

where p(x) is a probability density function. By using (5.20) and (5.21), the finite difference equation (5.17) can be represented as q~fl(n + 1) = r~flq~fl(n) + P;+, (5.22) where

r,fl = 1 - (P;+ + P-~-) =

p(x) dx.

(5.23)

If the probability density function p(x) is strictly positive, then 0 < r~fl < 1.

(5.24)

Equation (5.22) is a constant coefficient first-order finite difference equation whose general solution has the form

q~fl(n) = Arnfl 4- B.

(5.25)

533

5.1 PREISACH MODEL WITH STOCHASTIC INPUT By substituting (5.25) into (5.22), we find p~-+ Pa-+ B - 1 - r~/~ = P~-+ + P J - "

(5.26)

From the initial condition we obtain q~/~(0) - A 4- B,

(5.27)

where qa/~(0) -

1 if (a, fl) ~ S+(0), 0 if (a, fl) e S-(0),

(5.28)

and S + (0) and S-(0) are positive and negative sets on the ~-fl diagram, respectively, at the instant of time t = 0. From (5.25)-(5.27) we derive qa~(n) - [q~/~(O) - q~/~(oo)]rn/~ +

p~-+ P~-+ + P~--'

(5.29)

where q~/~(oo) = lim q ~ ( n ) = n - - + OO

P-~+

(5.30)

P~+ + P~-"

By substituting (5.29) into (5.10), we obtain p~-+ _ p ~ E{~;aflXn} - 2[q~fl(0)- q~/~(oo)]rn/~ +

P~-+ + P~-"

We next introduce the functions p~-+ _ p ~ ~(~, fl)= p~_+ + p + - '

(5.31)

(5.32)

y.

~9(~, fl)= [ 1 ! -1

if (~, fl) E S + (0), if (~, fl) E S-(0).

(5.33)

By using (5.28), (5.32) and (5.33), we can transform (5.31) to E{~a~Xn} - [t9(0r fl) - ~(0r fl)]rn/~ + r

fl).

(5.34)

By substituting (5.34) into (5.6), we finally find

J~n -)?o~ + f L

#(~' fl)[tg(~, fl) - ~ ( ~ , fl) ] rnfl d o~d fl,

(5.35)

CHAPTER 5 Stochastic Aspects of Hysteresis

534

where )~oo = n_,oofJ lim n

#(~, fl)~(a, fl)d~ dfl.

=

(5.36)

It is apparent from (5.36) and (5.32) that the limiting expected value of output foo does not depend on the history of input variations prior to the time t = 0. In this respect, the value, foo, bears some resemblance to an anhysterefic output value. This resemblance is enhanced by the fact that foo = 0 if the expected value u of Xn is equal to zero, and p(x) = p(X) is an even function. This fact can be proved as follows. According to (5.20) and (5.21) we find P~-+ = P+~-. (5.37) By using (5.37), it is easy to check that r

fl) = - r

(5.38)

On the other hand, we recall that (5.39)

~(a, fl) = ~ ( - f l , - ~ ) .

From (5.38) and (5.39), we conclude that #(~, fl)~(~, fl) is an odd function with respect to the line a = -ft. From the last fact and (5.36), we find that )~oo = O.

(5.40)

Next, we shall compare the result (5.35) with thermal activation-type models for viscosity. For this purpose, we replace discrete time n by continuous time t and rewrite (5.35) as

ft

= foo

+ fL

%(or fl)e -r

do~dfl,

(5.41)

where

Z(~, ,8)= #(o~, ,8)[~](o~, ,8) - ~'(~, fl)], 1

~(~,/~)- - ~ In r~.

(5.42) (5.43)

According to thermal activation-type models, metastable equilibrium states of hysteretic systems are separated by energy barriers EB. It is assumed that there is a continuum of these energy barriers and it is postulated (with some physical justification) that the viscosity phenomenon is described by the model (see [1]): fit) = flee) + A

/o

g(EB)e -'~(EB)t dEB,

(5.44)

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

535

where g(EB) is some density of states, 2(EB) --

d.oeEB/kT,

(5.45)

k is Boltzmann's constant, T is the absolute temperature, while A and 20 are some constants. It is clear by inspection that there is some similarity between our result (5.41) and the thermal activation model (5.44). Actually, our model (5.41) can be reduced to (5.44) in the particular case when only symmetrical loops (operators) ~ _ ~ are used in the Preisach model. In this case p(~, fl) = 0(~,/3)6(~ + fl),

(5.46)

where 6(a 4-/3) is the Dirac delta function. By substituting (5.46) into (5.42) and (5.41), after simple transformations we can represent (5.41) as

ft = foo +

2(~z)e-~(~)t da.

(5.47)

Now, by using the change of variables ~ ( ~ ) - 2(EB),

~-

~-1 (,~(EB)),

(5.48)

the expression (5.47) can be reduced to (5.40). This shows that the Preisach model of viscosity (5.41) is reduced to the thermal activation model in a very particular case. This case occurs when only symmetrical rectangular loops are used in the Preisach model. Since it is generally believed that nonsymmetrical loops in the Preisach model account for 'particle interactions', the last reduction is consistent with the generally held opinion that the thermal activation model (5.44) is a 'noninteracting particle' model. We next show that under some assumptions about X(~, fl) the model (5.41) describes in t-type variations for ft. Another (purely stochastic) justification for In t-asymptotics will be given in Section 5.3. The In t-type variations have been observed in many experiments and they are considered to be characteristic of viscosity phenomena. The expression (5.41) can be written as follows E

ft - f~ + f (/Z(~, fl)e-~(~"e)td(z) dfl.

(5.49)

For any fixed fl, we make the change of variables from ~ to ~ in the integral with respect to a:

ft - f~ + f (f ~,(~,fl)e-r

dfl.

(5.50)

CHAPTER 5 Stochastic Aspects of Hysteresis

536

We assume that the function 6(~, fl) can be approximated as

6(r ]/)~

D(fl) for r < r < r r

(5.51)

and zero otherwise. Then, we derive

f

~/(~, fl)e -~t d~ ,~, D(fl)

~ ~2 e-~t - ~ d~ = D(fl)

1

~~2te-V 1t

dr.

(5.52)

Consider such instants of time t that 1

1

(5.53)

~2 << t << ~119 For these instants of time, from (5.52) we derive

f

~/(~, fl)e -r d~ ,~ D(fl)

-

D(fl)

~oo e-V dv

1t

If eV r ltev] -a

V

dv -

e -a

V

dv .

(5.54)

Here a is a fixed small positive n u m b e r and the last two integrals exist in the sense of the Cauchy principal value. Since ~lt << 1, we can use only the first two terms of the Taylor expansion of e -v in the second integral. This yields

f

~(~, fl)e -~t d~ ~ S(]/)lnt 4- C(fl).

(5.55)

By substituting (5.55) into (5.50), we find

ft ~" S ln t 4-C,

(5.56)

which is a famous In t-type variation for ft. It has been observed in experiments (see, for instance, [7]) that the temperature dependence of the decay rate of;~t is satisfactorily approximated by T 1/2 at low temperatures. In the framework of the model (5.41) this fact can be explained as follows. From (5.43) and (5.23), we find

dx]

(5.57)

We subdivide all exponents in (5.41) into two groups: (a) exponents for such ~'s and fl's that u = xt ~ [fl, ~], and (b) exponents for such ~'s and fl's

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

537

that u = xt E [fl, 0r Consider a thermal noise with small variance 0-2. If U = ~:t ~ [fl, ~], then

~

~ p ( x ) d x --> 0

as 0- ---> 0.

(5.58)

This means that according to (5.57) the exponents of the first group decay very fast and they do not contribute appreciably to the 'long-time' decay of the integral in (5.41). If u = ~:t E []~, ~], then

ffl

~ p ( x ) d x --, 1

as 0- ---> 0.

(5.59)

This means that the exponents of the second group decay slowly and they are by and large responsible for the 'long-time' decay of ft. For this reason, only these exponents are discussed below. It is apparent that p(x) d x - 1 -

p(x) d x -

p(x) dx.

(5.60)

For small 0- the integrals in (5.60) are small quantities, which we assume can be approximated as f cx: p(x) d x ~ a(:r

//

p(x) d ~ b(]/)0-.

(5.61)

By substituting (5.61) into (5.60) and then into (5.57), we find ~(or fi) ~ - l n { 1 - [a(~) + b(fl)]0-} ~ [a(a) + b(fl)]0-.

(5.62)

It is reasonable to assume on physical grounds that the variance is proportional to the temperature 0-2 ,~, T. (5.63) From (5.62) and (5.63) we conclude that, for the exponents of the second group, the rate of decay is approximated by ~(~, fl) ~-, T 1/2.

(5.64)

This suggests that the overall rate of 'long time' decay offt may be approximated by T 1/2 at low temperatures. It is also interesting to examine the dependence of the limiting expected value of the output, f ~ , on the variance 0-2 of the noise. Expression (5.36)

CHAPTER 5 Stochastic Aspects of Hysteresis

538

'

1

'

I

'

'

'

"

'

i

"

'

'

m

f~

0.5 m

.

n

O o

,

n

O n

E <

~

-

--0.5

-1

,

,

I

,

i

i

-200

l,

i

I

0

I

I

,,

200

Mean input

FIGURE 5.1

is to be used for this purpose. The noise can be chosen to have normal distribution and ~(a, ]~) in (5.36) can be computed by using formulas (5.32) and (5.20) and (5.21). For illustrative purposes, Preisach function #(a, ]/) can be chosen as

Aexp{

40"2

2+

40"2

2Uc21},

(5.65)

where A is some normalization constant, Uc is the coercive value of a major hysteretic loop, while 0.1 and 0"2 are parameters that are used to control the spread of the function/~(a, fl) along and around the line a = -ft. This Gaussian form of/~(a, ]/) is often used in magnetics [8]. The computations have been performed for different values of 0"1,0"2 and the ratio Z - G. The computations show that for arbitrary fixed values of 0"1 and 0"2 the dependence o f f ~ on u is not sensitive to wide variations of 0.. This is demonstrated in Figs 5.1 and 5.2 where the computed curvesf~ versus u are plotted for the following values of X: 0.5, 0.35, 0.2, 0.15. These curves were computed for the following values of 0"1 and 0"2,respectively: 0"1 "-- 0"2 - - 40, 0"1 - - 80 and 0"2 = 40. The fact that fc~ is insensitive to the variance 0"2 suggests that f ~ can probably be used as another definition of an anhysteretic value. We next consider vector Preisach models driven by vector stochastic inputs as models for viscosity [4,9]. Below we shall only discuss the

5.1 PREISACH MODEL WITH STOCHASTIC INPUT

539 [

1

I

1

L 0.5

o

0

(~

< -0.5

I _J -1

I

5

i

0 Mean input

-200

uc i

I

; l

I-

200

FIGURE 5.2

case of two-dimensional vector Preisach model. Extensions to the threedimensional case will be straightforward and, for this reason, they will be omitted. Consider a deterministic vector input ~(t), which at time t - 0 assumes some value ~ and remains constant thereafter. We assume that some zero mean vector noise described by a discrete-time r a n d o m process is superimposed on ~(t). As a result, the Preisach model is driven by the process

Vn

(5.66)

~dn --- U -}- Wn.

For the output process we have

fn-'- /_~2-~q~(ff~), v(~x,fl, cp)~Vnq~d~xdfl)dq~.

(5.67)

From (5.68), we derive the following expression for the expected value of

fn"

lr

flfn}

--

/__'2-~q~(/~~fl V(~X,fl, q~)E{~,flVnq~}d~xd,)dq~.

(5.68)

By modeling the noise by the i.i.d, vector process with independent Cartesian components and by repeating the same line of reasoning as before, we derive

E{~aflVnq~} -

[O(a, fi, r

- ~(a, fl, q~)]rn/~ + r

fl, q~),

(5.69)

CHAPTER 5 Stochastic Aspects of Hysteresis

540

where ~9(a, fl, q~)=

+1 -1

if (a, fl) e S~(O), if (a, fl) e S~ (0),

(5.70)

P;~ -P~f ~(~, fl, q~)= p~-~- + P ~ '

(5.71)

P~-+ =

pq~(x) dx,

(5.72)

P~-~- =

~ pq~(X)dx,

(5.73)

r~q~ =

pq~(x) dx.

(5.74)

In expressions (5.73)-(5.75), pq~ is a probability density function for Vnq~. This function can be expressed in terms of probability density functions for Cartesian components of vn. Indeed, for Vnq)we have

Vnrp-~-Xncos r -}-yn sin r

(5.75)

where Xn and yn are independent Cartesian components of Vn with probability density functions px(X) and py(y), respectively. According to the well-known result from probability theory, from (5.76) we derive

pq~(V) = cos q~1sin q~ f_~-~ Px( cosx q~) Py( vsin- xq~t dx. o0

(5.76)

By substituting (5.70) into (5.69) and by replacing discrete time n by continuous time t, after simple transformations we obtain

(/L

E{fy} - E l f , } +

~m

)

dq~,

(5.77)

v(a,fl, q~)~(~z,fl, q~)dadfl) dq~,

(5.78)

Z(a, fl, q~)e-~(a'/~'m)t dadfl

where E { f o o } = L ~ "6q~(f L

(5.79) ~(~, #, q~) - - ~

1

In r~#~.

(5.8o)

5.2 EXPERIMENTALTESTING. SCALING AND DATA COLLAPSE

541

By comparing (5.78) with (5.44), it can be concluded that the model (5.78) can be regarded as a vector generalization of the classical thermal activationtype models. For the isotropic model the symmetry property similar to (5.39) holds for v. It can also be checked that for ~ - 0, p~ is an even function if Px and py are even. By using these facts and the same line of reasoning as before, it can be proved that E{f~} = 0 . (5.81) Therefore, for ~ = 0 and regardless of the past history, the limiting expected value of output is equal to zero. In conclusion of this section, it can be remarked that the model (5.35) (or (5.41)) has certain attractive features in comparison with the thermal activation-type model (5.44). First, the model (5.35) explicitly accounts for the specific hysteretic nature of the system as well as for specific input histories. Second, stochastic characteristics of thermal noise explicitly appear in the model, whereas the thermal activation-type model is formulated in purely deterministic terms. Third, the thermal activation-type model (5.44) is intrinsically scalar, whereas the model (5.35) can be generalized to the vector case.

5.2 EXPERIMENTAL TESTING. S C A L I N G A N D DATA COLLAPSE IN M A G N E T I C V ISC OSITY As discussed before, at constant external magnetic fields, large-deviation internal random (thermal) perturbations may cause a hysteretic system to move from one metastable state to another. This may result in slow time variations of magnetization, which are the essence of viscosity. The described temporal variations of magnetization are usually characterized by the following intermediate 'ln t' asymptotics: M ~ Mo - S(H, T ) I n t,

(5.82)

where S(H, T) is called the viscosity coefficient. This coefficient depends on the values of magnetic-field H, temperature T, as well as the past history of magnetic-field variations. The latter means that the viscosity coefficient is also a function of the current state of hysteretic material. The term 'intermediate asymptotics' means that formula (5.82) describes quite well the long-time behavior of magnetization. However, it does not describe properly the ultimate (at t ~ cx~)value of magnetization. In other words, the asymptotic behavior (5.82) breaks down for very long times, at which In t diverges.

542

CHAPTER 5 Stochastic Aspects of Hysteresis

The ability of the model (5.41) to predict the viscosity coefficient (relaxation rate) was first experimentally tested by C. E. Korman and P. Rugkwamsook [10] for ~;-Fe302 magnetic recording materials. By using a vibrating sample magnetometer (VSM), first-order reversal curves were measured. These experimental data were used for the identification of the Preisach model (determination of #(a, ]/)). Then, viscosity measurements were performed under the conditions that the sample was first brought to positive saturation and afterwards the applied field was reversed and held constant at some fixed value, and gradual decay of magnetization was observed. These magnetization decay curves (experimentally measured at constant temperature T and various fixed values of magnetic field H) were approximated by Eqn. (5.82) and the viscosity coefficient as a function S(H) of magnetic field was found. The described viscosity experiments were numerically simulated by using model (5.41) and experimentally measured first-order transition curves. The noise was assumed to be Gaussian and calculations were performed for different values of variances 0-2. The calculations revealed the 'In t' intermediate asymptotics as given by formula (5.82). The slopes of these asymptotics were computed, then they were normalized and it was observed that the normalized c u r v e s S(H)/Smax computed for different variances practically collapsed into one curve. This computed universal curve is almost identical to the normalized curve S(H)/Smax found from the experimental data previously described (see Fig. 5.3). This suggests that if the variance 0-2 of the noise is found by matching Smax, then the model (5.41) will predict the same curve S(H) as observed in the experiments. The 'computational collapse' of c u r v e s S(H)/Smaxprompted the idea to experimentally test the scaling and data collapse of viscosity coefficient S(H, T) for various magnetic materials, and especially for materials used in magnetic recording [11]. Our experiments have been conducted for high-coercivity thin-film CoCrPt recording media by using a vibrating sample magnetometer (VSM) with a temperature-controlled furnace (model Micro Mag 3900 from Princeton Measurements Corporation). First, experiments were performed to demonstrate clearly that the viscosity coefficient strongly depends on the past history of magnetic-field variations. To this end, the viscosity measurements were conducted for two distinct states of magnetic material, which had almost the same values of magnetization achieved at the same value of the external magnetic field but corresponding to two different past histories of magnetic-field variations. These two distinct states and histories of their formations are shown in Fig. 5.4. Figure 5.5 presents the observed time variations of magnetization for the above two states plotted on an In t scale. It is apparent from Fig. 5.5 that the above two

5.2 EXPERIMENTALTESTING. SCALING AND DATACOLLAPSE Simulation with sigma = 10 Oe

543

5e-005

-;

.,I ~ Simulation with sigma = 15 Oe . . . . . . S i m u l a t i o n with sigma = 20 Oe Experiment - - e . - -

0.9

4e-005

,., 0.8 cO

r

m

O

"O

.o_ 0.7

E

"O

O

o 0.6

3e-005

I II

o

"O "O

N ....

0.5

c-

2e-005 ._o

0.4

O o

E o.3

0 Z

9

0.2

.'

o

,,~;

1 e-005 a

0.1 0 - 1 0 ( )0

-ooo

-600

-400

-aoo

o

260

0 400

Applied field, Oe

F I G U R E 5.3 M

P.H

S F I G U R E 5.4

magnetization variations have different (in magnitude and sign) viscosity coefficients. This clearly reveals the strong dependence of the viscosity coefficients on past history.

CHAPTER 5 StochasticAspects of Hysteresis

544

x 10 . 4

-7

Magnetization vs Ln(Time) Measurement of a Hard Disk Sample

!

-7.2 ........................................................ :....................................... -7.4 ............................... ~ E -7.6

................

~"" 1 .......

..................................................................................................

eo ..~

-7.8

N -,~

...............................................................................................

r-

-8 -8.2 -8.4 -8.6

0

9

.

.

i

i

2

4

i 8

6

10

Ln (Time(sec))

F I G U R E 5.5

In subsequent experiments, the viscosity coefficients were measured as a function of H for various temperatures T for the same (and quite simple) histories. Namely, specimens were first driven into the state of positive saturation by applying sufficiently strong external magnetic fields and then the magnetic field was gradually decreased to a desired value at which viscosity measurements were performed. Thus, the viscosity measurements were performed for the states corresponding to the points on the descending branch of the major hysteresis loops. For various fixed values of T, viscosity coefficients S(H, T) are 'bell-shaped' functions of H with the maximum near (but not necessarily equal to) the coercivity. These bell-shaped curves measured for two different samples are shown in Figs 5.6 and 5.7. The samples had coercivities of about 1600 and 2 600 Oe at a temperature of 25 ~ respectively. The material of the second sample (with higher coercivity) had a small value of remanence (Mrt ~ 0.85 x 103 emu/cm2). For this reason, and in order to increase the signal level, the second sample was formed by stacking several layers of the same material. By using the experimental data shown in Figs 5.6 and 5.7, the hypothesis that S(H, T) admits the following scaling: S(H, T) - Smax(T) f ( H , ( T ) )

(5.83)

5.2 EXPERIMENTALTESTING. SCALING AND DATA COLLAPSE x 10 -5

Sample#01

o

M a g n e t i c V i s c o s i t y C o e f f i c i e n t vs M a g n e t i c F i e l d

o

T=25oc

.1~

9

m=5o~

/"

"

T=~O0~

4 /~ Io

~

T=,~ooc

o

T--~oooc

4.5

3.5

~ .......

i..........

i........

!

i

..... !.~...i

I T~

~ ....

!1

.~:..~.i ........ -I

i I li I i I

'I!

I /

"",~ .........!.......... !~~!1~1! ........1 11, i ~! ..... I i 1 ..... ~ _1 ". . . . . . . . . . :. . . . . . . . . . : : " ........

-. . . . . !

545

!

i

A

i

3~ 3 ~ 2.5

----

} ,.s

if

i

:

0.5 0 -1900

i -1800

i,

i

-1700

-1600

i -1500

i

i

-1400

-1300

i -1200

i -1100

I -1000

-900

Magnetic Field H (Oe)

FIGURE

5.6

was tested. In the above formula, Smax(Z)is the maximum viscosity coefficient as a function of T, while H* (T) is the value of the magnetic field at which Smaxis achieved. The last formula suggests that S versus H curves experimentally measured for different temperatures must collapse onto one universal curve when plotted in coordinates: s --

S Smax(t) '

H h -- ~ . H*(T)

(5.84)

This phenomenon of data collapse is the principal significance of scaling and its occurrence was observed for the collected experimental data. Namely, by performing the scaling described above, it was found that the curves shown in Figs 5.6 and 5.7 practically collapsed onto the single curves shown in Figs 5.8 and 5.9, respectively. In addition, the following scaling hypotheses: Smax(Z) = a T ~,

(5.85)

H* (T) - bT ~,

(5.86)

were also experimentally tested and verified with some accuracy. The results of this testing for samples I and 2 are shown in Figs 5.10 and 5.11,

CHAPTER 5 Stochastic Aspects of Hysteresis

546

x 10 -5 4

Sample#02 i

Magnetic i

Viscosity !

o

o

T = 25~

9

3.5

9

9

T = 50~

.

3

~

a

T = 100~

.'

2.5,

v

v

T=150~

0

0

T = 175~

8

Coefficient

vs Magnetic !

Field

x z~

.......................

....................

: .... ilk .... / ~ - I . I . . . . . . . . . . . . . . . . . . . . .

~ 1.5

0.5 ~

i

a

i

=' =,

/

0/ -4000

-3500

-3000

~

~',/ -2500 Magnetic

r

,

-2000

- 1500

- 1000

-500

Field H (Oe)

FIGURE 5.7

respectively. It was found that, for thin-film recording media with coercivity below 2 000 Oe, the exponent ]/was fairly close to -1. For recording media with coercivity above 2 000 Oe, substantial deviations from the above value of/Y were observed. Hypotheses (5.85) and (5.86) along with the formula (5.83) lead to the following self-similar expression for the viscosity coefficient

S(H, T) - aT~f ~

.

(5.87)

The last formula (as well as formula (5.83)) reveals an interesting and peculiar mathematical structure of the viscosity coefficient as the function of two variables: H and T. The essence of this structure is that the normalized viscosity coefficient S/Smax is a function of one variable h - H/H* (T). This structure is quite different from the traditional representation of S(H, T) in terms of irreversible differentiable susceptibility; the representation that is widely analyzed and discussed in the existing literature. This structure is a direct and concise reflection of experimental observations and is not based on any ad hoc assumptions.

5.2 EXPERIMENTALTESTING. SCALING AND DATACOLLAPSE

547

Normalized S vs H Curve of Hard Disk Sample#01 i

i

i

i

i

i

i

i

v_.

T = 25~

~= .... T = 50~ ~-

T = 100~ T = 150~

0.8

--e--

T = 200~

x

E

0.6

0.4

0.2

.5

I

I

I

I

I

-1.4

-1.3

-1.2

-1.1

-1

I

-0.9 H/H*

I,

I

I

I

-0.8

-0.7

-0.6

-0.5

FIGURE 5.8

The experimental evidence of scaling of S(H, T) and data collapse has been presented here only for two samples. However, the described scaling and data collapse were observed for several other samples as well. Those samples had coercivities between 1600 and 2 600 Oe. Scaling and data collapse are typical for critical phenomena where their physical origin is due to the divergence of correlation length [12]. The physical origin of scaling in magnetic viscosity is not clear at this time. However, one may speculate that this scaling is somehow related to the 'granular' structures of hysteretic materials, which ultimately lead to multiplicity of metastable states. In the experimental studies of magnetic viscosity reported above, magnetic samples were subject to external magnetic fields that were constant in time and spatially uniform. In these experiments, the applied fields and sample magnetization were usually aligned, while demagnetizing fields were negligible. This type of experimental study is usually performed by using magnetometers and reveals the intrinsic scalar thermal relaxation properties of the materials. In magnetic recording, thermal relaxations of magnetization patterns recorded on hard disks are of interest. These are data-dependent thermal

CHAPTER 5 StochasticAspects of Hysteresis

548

N o r m a l i z e d S vs H C u r v e of Hard Disk S a m p l e # 0 2 I

I

I

I

I

I

I

I

I

I I I

_

0.8 _

X

E

0.6

----e---

T = 25~

-

T = 50~

-"

T = 75~

=

T = 125~

v--

T=150~

~,

T = 175~

~

_

T=100~

-

~_

_

00 or)

0.4 i

I

0.2

0

i

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

H/H*

FIGURE 5.9

relaxations and they occur under quite different conditions than in the case of intrinsic thermal relaxations. First, the recording media are subject to strong local demagnetizing fields. These fields depend on recorded patterns and they vary with time due to the temporal deterioration of these patterns. Second, the demagnetizing fields are spatially nonuniform. As a result, data-dependent thermal relaxations occur with different local rates. Third, the local demagnetizing fields and magnetization are not aligned. Therefore, data-dependent thermal relaxations are inherently vectorial. Finally, data-dependent thermal relaxations are measured by using a giant magneto-resistive (GMR) head. Signals in these heads are generated by magnetic fields produced by virtual magnetic charges that are proportional to the divergence of magnetization (divM). It is quite possible that, due to the spatially inhomogeneous and vectorial nature of thermal relaxations, very small temporal changes in local magnetization may result in appreciable magnetic charges and GMR head signals. Data-dependent thermal relaxations are usually studied by using spinstands. However, traditional spin-stand studies have only examined the harmonic content (mostly the fundamental harmonic) of recorded patterns.

5.2 EXPERIMENTAL TESTING. SCALING A N D DATA COLLAPSE

549

Ln (Smax) vs Ln (T) of Hard Disk Sample -9.9 Io -10

o'Sample#'01 .

.

.

.

.

.

.

.

.

.

.

.

I .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-10.1

-10.2 o3 v r ._1 -10.3

-10.4

-10.5

5.6

5.7

5.8

5.9

6

6.1

6.3

6.2

Ln (T)

F I G U R E 5.10

Ln (H'ax) vs Ln (T) of Hard Disk Sample 7.9 ......

7.8

o

~ o Samp-'le #01

7.7 7.6 7.5 I,I 7.4 t.._1 7.3

iiiiiiiiiii

i iiiiii

7.2 7.1

6.9 5.6

i

5.7

5.8

5.9

6 Ln (m)

F I G U R E 5.11

6.1

6.2

6.3

550

CHAPTER 5 Stochastic Aspects of Hysteresis

For this reason, the spatially inhomogeneous and vectorial nature of datadependent thermal relaxations has not been captured. It turns out that the spatial and vectorial characterization of data-dependent thermal relaxations can be performed by using the spin-stand imaging technique [13-15]. The detailed exposition of this technique is beyond the scope of this discussion.

5.3 PREISACH M O D E L D R I V E N BY C O N T I N U O U S TIME NOISE. O R I G I N OF THE U N I V E R S A L I T Y OF L O N G - T I M E T H E R M A L RELAXATIONS In Section 5.1, the Preisach model driven by the discrete-time i.i.d, random process is used as a model for thermal relaxations in hysteretic systems. However, actual random thermal agitations are better described as continuous-time noise. From the mathematical point of view, this makes the problem quite complicated. It is shown next that these difficulties can be largely overcome by using the mathematical machinery of the 'exit problem' [16]. The noise Xt in Eqns (5.1) and (5.2) will be modeled by a (continuoustime and continuous-sample) diffusion process, which is a solution to the Ito stochastic differential equation [17]

dXt = b(Xt) dt q- cr(Xt) dWt.

(5.88)

In this equation, Wt is the Wiener process, and its formal derivative is white noise. Formula (5.88) can be construed as a generic equation for dynamical systems driven by white noise, and trajectories of such dynamical systems can be viewed as samples of the stochastic diffusion process. From the purely mathematical point of view, the Ito stochastic differential equation generates complicated diffusion processes by using the Wiener process, which is one of the simplest and most studied diffusion processes. Now we shall return to Eqn. (5.2). Since integration is a linear operation, from (5.2) we derive

ft-- f ~ ~fl #(~, fl)E{4/~xt} do~dfl.

(5.89)

Thus, the problem is reduced to the evaluation of the expected value,

E l ~flxt }. Let

q~,B(t) = Prob{~/~xt- +1}.

(5.90)

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

551

A ~oq~Xt

/k +1 , , JL IF

~

~

Xt

XO

\/

-1 1t`. 1`. i`. i`. i`. i`. i ' / ` . 1 / / ~ / / 1 /

13

11

~-

Xo

~

i

i

I

11

i

i

i

i

i

i

r-

F I G U R E 5.12

Since

~o~,flXtmay assume only two values +1 and -1, we find E { ~ x t } - 2%,fl(t) - 1.

(5.91)

In this way, the problem is reduced to the calculation of qa,~(t). The last quantity can be expressed in terms of switching probabilities P-~(t) and P~-(t), which are defined as follows: P ~ ( t ) - Prob{

k switchings of ~/~ during } time interval (0, t)l~/~x0 = +1 '

P~- (t) - Prob{

k switchings of ~/~ during time interval (0, t)l~/~x0 = - 1 / "

/

(5.92) (5.93)

By using these switching probabilities, we derive

%,~(t) - { Y~.~=0P~ (t) ~ = 0 P2k+l (t)

if ~a/~x0 = +1, if ~a/~x0 - - 1.

(5.94)

The last expression is valid because occurrences of different numbers of switchings are nonintersecting (disjoint) events. Next, we shall discuss the mechanism of switching. It is clear from Fig. 5.12 that the first switching occurs at the moment when the stochastic process Xt, starting from the point x0, exits the semi-infinite interval (fl, ~ ) . Then, the second switching occurs at the moment when the process xt, starting from the point x = fl, exits the semi-infinite interval (-oo, a). The third switching takes place at the moment when the process xt, starting from the point x = a, exits the semi-infinite interval (//, cx~). It is apparent

552

CHAPTER 5 Stochastic Aspects of Hysteresis

that the mechanism of all subsequent even switchings is identical to the mechanism of the second switching, while all subsequent odd switchings occur in the same manner as the third switching. Thus, switchings of rectangular loops ~a]~ are closely related to the exit problem for stochastic processes. This problem is one of the most studied problems in the theory of diffusion processes and the mathematical machinery developed for the solution of this problem will be utilized in the calculation of probabilities 4Pk (t). The exit problems just described can be characterized by exit times rxi , which are random variables. In the above notation for the exit times, subscript 'x' means that process xt starts from point x, while superscripts '4-' mean that upward and downward switchings, respectively, occur at these exit times. Next, we introduce the functions v+(t, x) = Prob{~xi~t},

(5.95)

V + (t, x) - e(t) - v + (t, x),

(5.96)

where e(t) is a unit step-function. It is clear that V+(t, x) = Prob{z x4-Kt},

(5.97)

which means that V+(t, x) has the meaning of a cumulative distribution function for the random variable ~x~. This, in turn, implies that /9+ (t, x) = 3V+ (t, x) 0t

(5.98)

is the probability density function for the random variable Zx ~. It is apparent from (5.96)-(5.98) that p+(t, x) can be easily computed if v + (t, x) are somehow found. It turns out (and this is a well-known result from the theory of stochastic processes) that v4-(t, x) is the solution to the following initial boundary value problem for the backward Kolmogorov equation: ~V40"2(X) ~2V+ ~V4= }- b ( x ) - (5.99) c~t 2 cgx2 3x ' v(O, x) - 1,

v(t, c4-) - O,

(5.100)

where c+ are the exit points for the process Xt, which are equal to a - u0 and f l - u0, respectively. Next, we shall show that switching probabilities P~ (t) can be expressed in terms of v + (t) and p+ (t). Note that, according to (5.96)-(5.98), p4- (t) are related to v+(t) as follows: p+(t) = ~[e(t) - v+(t)].

(5.101)

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

553

A ~or ~Xt /\

+1

I X

I X+dX

~

t

-1 - -

F I G U R E 5.13

It is clear from the very definition of v • (t, x) that P ~ (t) - v m (t, 0).

(5.102)

It is apparent from Fig. 5.13 that the occurrence of exactly one downward switching is the union of the following disjoint elementary events: downward switching occurs in the time interval (2, 2 + d2) and then no upward switching occurs up to the time t. Owing to the strong Markov property of Xt, the probability of this elementary event is given by p-(2, 0)v + ( t - 2 , / / - u0)d2.

(5.103)

Now the probability P+ (t) of exactly one downward switching can be found by integrating (5.103) from 0 to t: P+ (t) = fo t p-(2, O)v+ ( t - 2, f l - u0)d2.

(5.104)

In other words, P~-(t) is the convolution of p-(t, O) and v + (t, fl - u0) P~- (t) - p- (t, 0) ,v + ( t , / / - u0).

(5.105)

By using similar reasoning, we can derive P1 (t) - p+ (t, 0),v-(t, a - u0).

(5.106)

Next, consider the probability P~-(t) of the occurrence of exactly two switchings starting from the initial state ~flx0 - 1. According to Fig. 5.14 this occurrence can be considered as the union of the following disjoint elementary events: downward switching occurs in the time interval (2, 2+d2) and then exactly one upward switching occurs up to the time t. The probability of these elementary events is given by p- (2, 0)P l ( t - 2) d2.

(5.107)

CHAPTER 5 Stochastic Aspects of Hysteresis

554 /k

~{o~13Xt J\ +1 = t

i X+dX -1

t

--

FIGURE

5.14

Now, by integrating (5.107), we find (5.108)

P~- (0 = fO t p - (,~, 0)P i- (t - ,b d,~. From (5.106) and (5.108) we obtain

(5.109)

P~ (t) = p - (t, 0) ,p+ (t, ] / - u0) , v - (t, a - u0). By using the same line of reasoning, we derive

(5.110)

P2 (t) = p+ (t, O ) , p - ( t , ~ - u o ) , v + (t, f l - uo).

For the sake of conciseness, we introduce the notations: p+ (t, 0) = p~ (t),

p+ ( t , / / - uo) = p+ (t),

p- (t, a - u0) = p - (t), (5.111)

v+ (t, O) = v:~ (t),

v + (t, fl - uo) = v + (t),

v-(t, ~ - u0) = v-(t). (5.112)

Now, by using the same line of reasoning as before and the induction argument, we can easily derive the following expressions for the switching probabilities: 2k-2 terms ,,

P ~ ( t , uo) - P o ( t ) , p + ( t ) ,

p-(t),p+(t),.

.. , p - ( t ) , p + ( t )

, v - (t), (5.113)

2k terms P2k+l(t, uo) = p-~(t), p - ( t ) ] , p - ( t ) ,

.. . , p - ( t ) , p + ( t )

,v-(t).

(5.114)

By substituting (5.113) and (5.114) into (5.94), we obtain the expression for q ~ ( t ) in terms of infinite series of iterated convolutions. These series can

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

555

be reduced to geometric ones by employing Laplace transforms: ~(s)~(s)-

It is clear that

p(t)e -st dt

Jo

(Re s > 0),

(5.115) (5.116)

v(t)e -st dt.

]~(s) I < 1.

(5.117)

By using these Laplace transforms, from (5.102), (5.113), and (5.114) we obtain / ~ (s) - ~ ( s ) ,

(5.118)

~P~k(S) - ~9o(S)~9+(s)~;-(s)[~-(s)~+(s)] k-l,

(5.119)

P2k+l (s) -- ~ (s)~-(s)[~-(s)~ + (s)] k.

(5.120)

From (5.120) and (5.94), we derive

~/a/~(s)

-

~-~(s)~- (s) 1 - ~- (s)~ + (s)

if ?~flx0 = - 1 .

(5.121)

A similar expression can be derived for the case 9~/3x0 - +1. According to (5.101) ~+ (s) - 1 - s~ + (s).

(5.122)

Thus, the problem of computing F/a/~is reduced to the problem of determining ~+(s). This can be accomplished by using the initial boundary value problem (5.99)-(5.100). The complexity of this task will depend on the nature of the stochastic process Xt, which models the noise in hysteretic systems. It is natural that the stochastic process that models the noise must be a stationary Gaussian Markov process. According to the Doob theorem [17], the only process that satisfies these requirements is the Ornstein-Uhlenbeck process. This process is the solution to the following Ito stochastic differential equation: dXt = - b X t dt + a d Wt,

(5.123)

where 1/b has the meaning of the correlation time. (This means that Xt and Xt, are only significantly correlated if it - t'l<,l/b.) The backward Kolmogorov equation for the Ornstein-Uhlenbeck process has the form (~V+

0-2 0-2V+

c~t = - 2

c~x2 - b x

(~V+

Ox

(5.124)

CHAPTER 5 StochasticAspects of Hysteresis

556

This equation should be considered jointly with initial and boundary conditions (5.100). By applying the Laplace transform to (5.124) and (5.100), we arrive at the following boundary value problem for ~+(s):

2 d2v+(s, x) _ bxd~;+(s, x_______~) 0" _ s~;+(s ' x) - - 1 2

dx 2

~;+(s, c +) - O,

dx

~;+(s, (x~) - 1/s.

'

(5.125) (5.126)

The solution to the boundary value problem (5.125)-(5.126) can be written in the form

~)&(S,X)_ !(1_e[X2-(c+)2]/4,~2 E)-s/b(x/~) ) ,3 ~D_s/b(C&/,~,) '

(5.127)

where 1)_s/b(X/2) are parabolic cylinder functions, while ,,1.= o-/J~.

Expressions (5.122), (5.121), and (5.127) jointly with (5.89) and (5.91) outline the main steps of computing)~t. It is apparent that the case of continuous-time noise is computationally expensive. Some sample examples of computations for this case can be found in [18,19]. In spite of its complexity, the case of continuous-time noise is convenient for the discussion of the origin of the universality of intermediate asymptotics M ",, M o - S(H, T ) I n t. (5.128) The very fact that intermediate asymptotics (5.128) has been observed for various magnetic materials as well as for superconductors and other hysteretic systems reveals its universality. This raises the question of the physical origins of this universality. The purpose of the subsequent discussion is to demonstrate that this universality can be traced back to the asymptotic temporal behavior of internal thermal noise [20]. This demonstration also suggests that ~F~ t-type relaxations may actually occur in the case of stationary Gaussian thermal noise. Our analysis also yields a simple expression for the viscosity coefficient in terms of differential susceptibility and temperature T. Experimental data supporting the theoretical analysis are also presented. Consider a hysteretic material that is first driven into the state of positive saturation. Then, the external magnetic field is reduced to some value -H0 close to coercivity - H c and maintained constant (in time) thereafter. Owing to the inherently present thermal noise, gradual relaxations

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE

557

FIGURE 5.15

of magnetization take place. To mathematically describe these relaxations, we shall use the Preisach model: f/"

M t - JJ #(~z, fl)~a~Ht d~z dfl,

(5.129)

Ht -- -Ho if- ht,

(5.130)

and ht is a stochastic process that models the thermal noise. It is instructive to note again that adding noise ht to -H0 is mathematically equivalent to adding noise -ht to switching thresholds a and ]~. It seems natural to assume that ht is a stationary Gaussian process with decreasing in time correlations. In this case, the following temporal asymptotics is valid for ht with probability one [21] h t - max Ihrl ~ v/2R(0)lnt, 0Kr<~t

(5.131)

where R(0) is the value of the autocovariance function of ht at zero. It is this asymptotic behavior of ht that is ultimately responsible for long-time relaxations of magnetization. Our reasoning is based on the diagram technique for the Preisach model. According to this technique, the state of the Preisach model at the moment when the external magnetic field is reduced to - H 0 is represented by the diagram shown in Fig. 5.15(a). Let M0 be the magnetization in this state. At subsequent instants of time, this diagram will be modified due to the time variations of noise process ht. This will lead to fluctuations

558 AMt

CHAPTER 5 Stochastic Aspects of Hysteresis of magnetization

Mt - Mo - ~ t .

(5.132)

Consider the nature of these fluctuations during the time interval 0~
~

(5.133)

2 f/f~ #(o~, fl) do~dfl.

The trapezoid f] can be represented as a difference of two right triangles: F ( - H 0 - h~n) and F(-H0). This means that

AMt "~2(//F(_Ho_h,m ) #(a,fl)dad~-ffr(-H0)

#(a, fl) da dfl).

(5.134)

It is known from the theory of the Preisach model that p/.

2 ]/ #(c~,fl) da dfl - M~- - M + (H), JJF (S)

(5.135)

where M~ is the positive saturation value of the magnetization, while M + (H) is the magnetization value on the descending branch of the major hysteresis loop. By using Eqns (5.134) and (5.135), we derive AMt ~ M + (-S0)

- M + (-S0

-

hm)

dM+ (-Ho)h' m. dH

(5.136)

By invoking formula (5.131) and the fact that ~aM + (-H0) is the differential susceptibility )~(-H0, T) evaluated at -H0 and temperature T, we transform formula (5.136) as follows: ~/It

"~" v/2R(0)

Xa(-Ho, T)v/-~ t.

(5.137)

Furthermore, R(0) can be interpreted as being proportional to the total energy of the thermal noise. Indeed, R(0) -

F

oo

S(co) d o - 7T,

(5.138)

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE x 10 - 4

559

CoCrPt Material Hard Disk Sample at T=25~

3

2 .............................:

................................................. ~.....................................,..........................~

.............

"~" 0 E

(I) "~ cO ~N C

-I

-2

--3

-4 ............................................................................. ~

..........................................................................................

-5 .......................................... ~

.......................................~ i

-7 2

2.2

2.4

2.6

i 2.8

............. :

i 3

3.2

Sqrt (Ln(Time(sec)))

FIGURE 5.16

where S(~) is the spectral density of noise process ht. By using Eqn. (5.138) in Eqn. (5.137) and taking into account Eqn. (5.132), we arrive at Me ~ MO - ,~~ / ~ Za ( - Ho , T) ~ n

t,

(5.139)

where 2 is some constant. This is the final expression for the intermediate asymptotics of thermal relaxations in hysteretic systems. The immediate question is how this expression stands the test of experiment. To test the ~ dependence of long-time thermal relaxations, the measurements of these relaxations for thin-film CoCrPt magnetic recording materials and YBaCuO superconducting materials have been performed by using a vibrating sample magnetometer (VSM) (Micro Mag 3900 from Princeton Measurements Corporation). The measured values of M t were plotted on ~ t scale and the linear dependence of M t versus ~ t has been consistently observed. Some sample results of these measurements are presented in Figs 5.16 and 5.17. To test the temperature dependence of intermediate asymptotics (5.139), we point out that for very low temperatures Za(-H0, T) does not depend on T for magnetic materials. This means that asymptotics (5.139) predicts dependence of the ~ t decay of magnetization. This prediction was made earlier (see [1]) on the basis of the thermal activation model and is

CHAPTER 5 Stochastic Aspects of Hysteresis

560

High Tc Superconductor 0.15

E

9

I

Sample

at T=30

0.14

................................... ! .................................... i ..........

0.13

.......................................................................-~....................................~.....................................i..................................

0.12

~

0.11

................................... . ....................................--....................................,.....................................i..................................

................................................................................................................~..................................

O

,= ~

"

N

" O'}

K

I

~

0.1

................................................

~

0.09

................................... ! .................................... .~.........................

.

0.08

................................... .................................... ~.................................... ~

}

2

9 99

.

.

9

.

.

i

i

2.1

2.2 Sqrt

..............................................................

.................... :.................

"-'--'--

J

2.3

2.4

2.5

(Ln(Time(sec)))

F I G U R E 5.17

consistent with the reported experimental observations (see [7]). The temperature dependence given by Eqn. (5.139) has also been tested for high temperatures in the following manner. Equation (5.139) can be written in the form Mt ~" Mo - S(-Ho, T)~/--~, (5.140) where the slope S(H, T) can be retrieved from time observations of thermal relaxations for various values of H and T. Equations (5.139) and (5.140) suggest that S(H, T) - 2 ~ z a ( H , T). By integrating the last equality with respect to H over a sufficiently large range H_ KH~
fHH+ S(H, T) dH 2-

~/T [M+(H+, T) - M + ( H _ , T)]

.

(5.141)

If Eqn. (5.139) predicts the correct temperature dependence, then 2 in Eqn. (5.141) must remain constant. This has been verified over the temperature range of about 150 ~ for samples of high- and low-density CoCrPt recording materials. The results of this verification are presented in Fig. 5.18. It is apparent from Fig. 5.18 that 2 remains fairly constant and exhibits slight decay with increasing T. This decay can be attributed to the fact that S(H, T) becomes very sharply peaked around coercivity with the increase

5.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE 10

!

!

i

!

561

! I

High Density Hard Disk Samp,el

9

[]

I Low Density Hard Disk Sample I i

8 ............................................................................................................................................................................ i.........

7 ............................................................................................................................................................................ i ........ . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . .

. . . . . . . .

"~

5

. . . . . . . . . . . . . . . . . . . i ..................... i ..................... i ..................... ~..................... i ..................... i ..................... i ..................... i . . . . . . . .

~

4

.....................................................................................................................................................................................

1

o

280

300

320

340

360

380

400

420

440

Temperature (K)

F I G U R E 5.18

in T. This makes the measurement of max S(H, T) less accurate and eventually leads to the underestimated values of the integral in Eqn. (5.141). The derivation of Eqn. (5.139) is based on asymptotics (5.131). The proof of Eqn. (5.131) rests on the Gaussian nature of 'low probability tail' of the probability density function for ht. The Gaussian nature of stationary noise is customarily justified on the grounds of the central limit theorem. This theorem holds for normal and moderately large deviations. However, very low probability tails may appreciably deviate from Gaussian tails. If the very low probability tails have asymptotics p(x) = e -anxlv and the process ht has sufficiently strong mixing properties, then it can be shown that Eqn. (5.131) can be modified as follows: max Ih~l ~ b(ln t) 1Iv.

(5.142)

0 ~ t

By using Eqn. (5.142) and literally repeating the previous line of reasoning, the following intermediate asymptotics can be derived Mt ~ Mo - S(ln t) 1/v.

(5.143)

Particularly, in the case of exponential (v - 1) tails, Eqn. (5.143) leads to In t intermediate asymptotics. Experimentally, it may be quite difficult to distinguish between (in t) 1Iv and v / ~ t (or In t) thermal relaxations. This

CHAPTER 5 Stochastic Aspects of Hysteresis

562

is because In t is a slowly varying function within the time range of the validity of intermediate asymptotics.

5.4 N O I S E IN HYSTERETIC SYSTEMS A N D S T O C H A S T I C PROCESSES O N G R A P H S In the previous section, the Preisach model driven by a (continuous-time and continuous-sample) diffusion process is discussed. The mathematical treatment of this model is reduced to the analysis of random switching of ~flxt which, in turn, is reduced to the 'exit problem' for diffusion processes. Although the concept of random switching as an exit problem is quite transparent from the physical point view, its mathematical implementation leads to an infinite series of iterated convolutions. It turns out that the mathematical treatment can be appreciably simplified by using an entirely different approach based on the theory of stochastic processes on graphs. This theory has only recently been developed [22] and applied to the study of random perturbations of Hamiltonian dynamical systems [23]. The main purpose of this section is to demonstrate that the mathematical machinery of this theory is naturally suitable for the analysis of random output processes of hysteretic systems. As a by-product of this demonstration, we derive analytical expressions for stationary characteristics of these processes. These expressions are of interest in their own right. The discussion presented below closely follows the paper [24]. This discussion is based on the following simple fact. The output it ~Xt is a random binary process. This process is not Markovian. However, the two-component process yt (itt) is Markovian. This is because the rectangular-loop operators describe hysteresis with local memory. This means that joint specifications of current values of input and output uniquely define the states of this hysteresis. The two-component process yt is defined in the four-edge graph shown in Fig. 5.19. The binary process it assumes constant values on each edge Ik of the above graph. This justifies the following concise notation for the transition probability density: -

-

-

-

p(t, YlY0)I~*xk -- p(k)(t, x[~0). It is obvious that p(1) _ p

for

x~fl,

p(4) _ p

(5.144)

for x~>a,

p(2) q_ p(3) = p for x ~ [fl, a], where p is the transition probability density of the process assumed to be known.

xt,

(5.145) which is

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES 13

i=+1

11

/ i=-1

563

12

i=-1

14

i=+1

FIGURE 5.19

According to the theory of Markovian processes, the following equality is valid for

p(k).

4

4

E fkfCgP(k)dx--E fk(Lf)p(k)dx.

k=l

0t

(5.146)

k=l

Here,

1

~2

L = -~cr2 (x) ~

if- b(x)

0---~

is the generator of the semigroup of the process Xt, while f is a function that is continuous on the entire graph and sufficiently smooth inside the edges and satisfies certain 'gluing' (interface) conditions at a and ft. These interface conditions follow from the Markovian nature of the process yt on the entire graph [22]. In our case, the process yt 'spends zero time' at the graph vertices. In this situation, the interface conditions can be written as follows [22]:

E '~kJ-~x k

-- O,

,~,kj~O.

(5.147)

oj

Here, fIk = flIk, summation is performed over all edges connected to a graph vertex while the derivatives are taken along the edges in outward directions with respect to It is known [22] that constants are (roughly speaking) proportional to the probability that the process will 'move' from vertex along the edges It is clear that in our case there is zero probability that the process yt will move from the vertex fl along the edge I3, while the random motions along the edges I1 and 12 are equally probable. A similar assertion is valid for the vertex a. As a result, we arrive at the following interface conditions:

Oj,

Oj.

Oj

|

dx ~

2kj

Ik.

_

|

~x

- df141 dx

(5.148)

CHAPTER 5 StochasticAspects of Hysteresis

564

while the values of the derivatives dflB/dXlB and dfi2/dxl~ are entirely arbitrary. It is understood that differentiation in (5.148) is performed in the direction of increasing values of x. By integrating by parts in the equality (5.146) and by taking into account the interface conditions (5.148) and the choice inherent in f, dfI3/dxlB and dfI2/dxl~, one finds that the transition probability density p(3) satisfies the forward Kolmogorov equation (~ (b(x)p ~xx

~p(3) 1 c32 r = 2 ~?x2 ( ~

(3))

(5.149)

and the following boundary conditions: P(3)ie = PI~-

p(3)l] / = 0,

(5.150)

It is also tacitly understood that the standard r initial condition is imposed at Y0. A similar initial boundary value problem can be stated for p(2). However, p(2) can also be found by using formula (5.145). The solution to the initial boundary value problem (5.149)-(5.150) can be found in terms of parabolic cylinder functions and their Laplace transforms in the case when xt is the Ornstein-Uhlenbeck process (dxt -- b ( x t - xo) dt + e dWt) with expected value x0. Simpler analytical results can be obtained for stationary densities Pst _(3) and Pst _(2)" In this case, we have to deal with the following boundary value problem for the ordinary differential equation 1 d2 d (5.151) 2 dx 2 (Cr2(X)Pst)(3) (3) "- 0, ~xx(b (X)Pst) p(3)

_(3)

(5.152) Although the analytical solution to the above boundary value problem can be written out for any stationary diffusion process xt, below we present this solution only for the case of the Ornstein-Uhlenbeck process: st (fl)--" 0,

p(3) st (X) -- Pst(X)q0(X, 0r fl)'

Pst ( ~ ) - - P s t (00"

_(2) (X) -- Pst (x)[1 -- q0(X, 0r ]~)], Pst

(5.153)

where /)st(X)-qP(X,

/ ~ff2b e_b(x_xo)2/~2

0~,fl) -- /; eb(y-x~162 dy f; e b(y-xO)2/rr2 dy

(5.154) (5.155)

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES

565

If we consider the probability current

0-2 dp (k)(x) _ b(x - xo)p (k) (x) 2 dx

Jk(X) --

then it is easy to conclude from formulas (5.153), (5.154), and (5.155) that Jl(X) -- J4(x) -- 0, while J2(x) = -J3(x) ~- 0. Thus, there exists a nonzero probability current circulating in the loop formed by edges 13 and 12. The existence of the circulating-loop current can be considered as the manifestation of the lack of detailed balance in the two component process yt. The existence of this circulating current can also be traced to energy losses associated with the random switchings of rectangular loop 9~. This dissipated energy is extracted from noise, which is the only source of energy present in the discussed problem. The situation here is somewhat analogous to the one observed for stochastic resonance, where a feeble deterministic signal alone cannot affect switchings. These switchings are assisted by the internal noise and they are accompanied by the extraction of energy from the noise [25]. It is instructive to compute the expected value it of the binary output process and its variance 0-2. It is clear that -- P~tfl(it - 1)

it -- Est(9~flxt)

-

P~flt (it -

-1),

(5.156)

which is equivalent to it --

2P~flt (it - 1) - 1.

(5.157)

It is apparent that

P~tfl( i t

-- 1)

-

_(3) Pst (x) dx +

fist (X) dx,

(5.158)

which leads to

it -- 2

Pst-(3)(X) dx -}-

Pst(X) dx - 1

(5.159)

When it is computed, 0-2 can be calculated as follows: 0-2 _ 1 - 72.

(5.160)

Calculations are substantially facilitated by the observation that

foX ey2 dy - --~erfi(x),

(5.161)

f e-Xaerfi(x) dx - ~x2 2F2 ( 1, 1; ~, 3 2; --X 2) ,

(5.162)

CHAPTER 5 StochasticAspects of Hysteresis

566

/ i/,;':. .i 0.8 r ........... ::............ I ........... i ............ i~;[i.-"- .... i

o.o ~ ........... ~............ I ........... ~............ ~i':....:...d.... !

!

~:..

i ....... ~

=~ ..... -I

.~........', ........... 4

!

!

o.,

........... i .......................

i ............ !J: ........ ! ....................

i...........

0.2

...........

::. . . . . . . . . . . . . . . . . . . . . . .

: ............

:. . . . . . . .

9 ....................

-o-

o.s

!

:

:"

:

-e-

1.0

:

:

i:

: ::............

_o..~ :

:

: ::-

: -it i r.--.'-..:i;r.:

_o,

.......

-0.8

. . . . . . . . . . . . . . . . . . . . ........ .. . .-. .

-0.6

............

!:." .:"i] .

-1

,.. v -2

v

v _ _

' iF

~-- ~----

! ......................

.,.~j:!................................................. . . . . . . . :,

...."

-

-1.5

-1

-().5

0

0.5

1

1.5

2

V

FIGURE

5.20

where erfi(x) and 2F2 are 'imaginary error function' and 'generalized hypergeometric function', respectively. By using the formulas presented above, the expected value it and variance cr2 have been computed as functions of the expected ('bias') value x0 of the input process xt for the case of hysteretic nonlinearities represented by symmetric rectangular loops ~,_~. The results of the calculations are shown in Figs 5.20 and 5.21, respectively. These results are plotted for normalized values of input bias v = x0/a and normalized values of switching thresholds ~ = a/2, where ~2 _ ry2/b is the variance of the stationary distribution of the input Ornstein-Uhlenbeck process xt. The dependence of it on x0 can be interpreted in magnetics as an 'anhysteretic' magnetization curve. This anhysteretic curve depends on the noise variance. This dependence is especially appreciable when the noise standard deviation 2 is comparable with the switching threshold value a. Next, we shall apply the obtained results to the case of a hysteresis loop with 'curved' ascending and descending branches (Fig. 5.22). This type of hysteresis loop is exhibited, for instance, by Stoner-Wohlfarth particles when the applied magnetic field is restricted to vary along one direction. Suppose that the loop shown in Fig. 5.22 is driven by the Ornstein-

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES 1

567

m ~ m

0.9 0.8 0.7 0.6

%

0.5

0.4 0.3 0.2 0.1 0 -2

~~'~" -1.5

-1

-0.5

0 v

0.5

1

1.5

- v

" ~ ,~ 2

FIGURE 5.21

-Uhlenbeck process Xt and we are interested in the stationary distribution of the output process tt. T h e process Yt is not a binary one. However, it admits the following representation in terms of the binary process

it -- ~o~flXt: yt =

d + (Xt) --d- (Xt) 2

it +

d + (Xt) if-f- (Xt) 2

'

(5.163)

where the meaning o f f + (xt) and f - ( x t ) is clear from Fig. 5.22. By using formula (5.163) and the appropriate change of variables, we obtain the following expression for the stationary distribution density ~s+ (y) of the process yt along the descending branch: ~s+t(y)

_(3)

--Pst(g+(Y))

d g + (Y)

dy

(5.164)

Here, g+ (y) is the inverse of the function y - f+(x). The last formula is valid for fl < x < ~. For x > ~, density/9 st (3) should be replaced by Pst. In a similar way, the density ~t (Y) along the ascending branch can be computed.

CHAPTER 5 Stochastic Aspects of Hysteresis

568

f+(zt)

S

~

J

A

< FIGURE

J

f -(xt)

5.22

The results obtained for ~a/; operators can also be used to compute the stationary characteristics of the output processft of the Preisach model driven by the stochastic process xt:

~t _ [ [ ~(0~,~)Est(7~xt) dd~

d~ dfl,

(5.165)

where Est(7a~xt) can be evaluated by using expressions (5.153)-(5.162). If it is desired to evaluate the stationary value of the second moment Est (ft2) of the output processft and its variance, the following integral must be evaluated:

Est(f2) -- /~x>.~l /~a>/~aEst(7~176

xt)

xfl(~162]~1)fl(~162f12)d0r dill d0r dfl2,

(5.166)

where Est(.) stands for the stationary expected value. To compute Est (70r1/~1x t ~ 0~2fl2xt )" we consider the three component Markovian process

(:i) xt where i~ - ~ l & X t and i 2 - ~2&xt. Depending on the relation between c~1, //1, e2 and//2, this process is defined on graphs shown in Fig. 5.23. By using the same line of reasoning as before, one can easily arrive at the following _(k) expressions for the stationary densities Pst (zt)IIk = Pst (x).

5.4 NOISE IN HYSTERETIC SYSTEMS A N D STOCHASTIC PROCESSES 13

16 ~~ ( X

(X1/

llxx ~ 1 ~

569

2 /17

14 12

(a)

15

15 ~2N~/~lX2

13 O~1

14

,/ ~//NN

(b)

16

~2 11

\

12

14

O~1

Ill

O~2

~2

12

/

17

(X1

(c) FIGURE 5.23

In the first case (Fig. 5.23(a)) w h e n two rectangular loops do not overlap (ill < ~1 < f12 < ~2), w e have p(3) p(stk) -- Pst f o r k - 1, 4, 7, st (X) -- Pst(X)q~(X, 0r i l l ) , (5.167) p(6) st ( X ) -

Pst(X)qO(X, 0r

f12),

/9(2) ^(3) st - - P s t - - P s t ,

p(5) (6) st -- /gst -- /)st 9

(5.168) In the second case (Fig. 5.23(b)) w h e n two rectangular loops completely overlap (ill < f12 < 0r < C~1),we have p(k) _(3) (x) -/9st (x)q)(x, ~1, ill) (5.169) st --/)st for k - 1, 8, Pst _(2) _(3) Pst -- [ s t -- Pst '

p(st4) - - P s_(3) t

^(5) --Pst '

-(5) (X) _ Pst ~(3)(x)q)(x, 0r Pst

f12)

(2) (X)q)(X, C~2, f12) ' /)st(7) (X) -- l) st

(5.170)

CHAPTER 5

570

S t o c h a s t i c A s p e c t s of H y s t e r e s i s

(7) p(st6) m P^(2) st --Pst "

(5.171)

Finally, in the third case (Fig. 5.23(c)) when two rectangular loops partially overlap (ill < / J 2 < ~ < ~162 w e have

p(stk) -- Pst

for k = 7,

p(3) ^(2) ' st -- Pst -- Pst

^(2) (X) -- Pst (X) [1 -Pst

q~(x,~l , ill)], (5.172)

(4) (X) -- P s t ( X ) q ) ( x , Pst

0r

p(5) ^(2) (4) st - - P s t - - P s t --Pst '

^(6) (4) Pst m P s t _ P s t 9

]~2),

(5.173) (5.174)

It is worthwhile noting that in the last case there is no graph edge corresponding to i~ -- - 1 and it2 = 4-1 because these simultaneous values of i~ and it2 are not consistent with the definition of rectangular-loop operators ?~1#1 and ~2/J2. By using the above expressions for Pst^(k),Est(~alfllXl~o~2flaxt) can be computed. As an example, consider the particular case of the Preisach model when all rectangular ~-loops are symmetric: ~_~xt = ~=xt. Mathematically, this case is obtained when the 'weight' function #(a, t) has the form

/~(a, t) = ~(a)6(a 4- fl),

(5.175)

and the Preisach model is reduced to

ft =

f0 ~(e)~xt

de.

(5.176)

For many magnetic materials, the weight function #(e, fl) is usually narrowly peaked around the line e = -ft. For these materials, formulas (5.175) and (5.176) can be regarded as fairly good approximations. At the same time, the calculations are considerably simplified because, in the case of symmetric loops, any two loops completely overlap. As a result, one has to deal with the three-component Markovian process ~.t defined only on the graphs shown in Fig. 5.23(b). For the case of model (5.176), formulas (5.165) and (5.166) can be written as follows: --

f0

~(~x)Est(~xt)

d~,

(5.177)

Est(f2)-a~o~~ Est(~;axt), formulas (5.153)-(5.162) Est(~;axt~;a,xt) in (5.178), we first remark that

To compute

Est(~axt~a, xt) - 2P(~;axt~a,x t -

(5.178) can be used. To evaluate 1) - 1.

(5.179)

5.4 NOISE IN HYSTERETIC SYSTEMS AND STOCHASTIC PROCESSES ! 9

! .

.

.

571

I . .

. . . . . . . . _

.: .

.

.

.

.

.

.

.... 9

.,_~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~,'.

, *9

/ i

9

/

1

...........

:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

..~-

9

; ......

.....,

,i, . .

.

9

.,-,

9

~,,

.

.

.

.

.

,,

_,...,

:

.

':~:7~ .

:

,

.

_

.

--

,--

--

w

.

J

l

.

'

?~" . . . . . ,; . . . . ; . . . . . . . :l

~. . : . : . . :

.

.........

!. . . . . . . . . . . .

.

.

"

.

w 0 ~ ~ ' / l : . ~

I

'

"'i

j

9.

..

.j.

-1

9

9

..

I

9

!

9

.

9

-2

~-

""

,d.

_

/L - - - - : -

.,.,

.,-

-.-

9

"

#

.:

,,.

--tl-0.5 ~ 1.0

I

9

"-".-:

'...J'9"

-:3

t

-2

9 9

9 ,....

_..

_-

,,...,,.,

-

-'~"

"9

--

9

i

i

i

-1.5

-1

-0.5

--

1.5 2 . 0 2 . 5

3.0

I

0

0.5

1.5

1

V

FIGURE 5.24 To find P(~axt~;ovxt), one has to integrate the appropriate Pst-(k)(X) over those edges of the graph shown in Fig. 5.23(b) on which ~xt and ~&,xt have the same signs. This leads to the formula

Est(~xt~a, xt) = 2

[f_~' Pst [a Pst(3)(x) dx 4- [ c o ~(5) (x) dx +

Pst (x) dx

!

0C!

A(6)(x)d x

4-

Pst 0~ I

A(2)

Pst(X)d x

Pst (x) dx +

+ 0~

- 1.

OO

(5.180)

The integrals in the last expression can be evaluated by using formulas (5.169)-(5.171) and taking advantage of relations (5.161) and (5.162). Some sample results of calculations are shown in Figs 5.24 and 5.25 for ?tst and Est(f2), respectively. In these calculations, it is assumed that ~(~) - I and ?tst and Est(f 2) are computed as functions of normalized values of input bias v - xo/~o for various normalized values of ~0 - ao/2, where ~2 is the variance of the stationary distribution of xt. The values ofy~tst and Est(f2)

CHAPTER 5

572

9

.....

Stochastic Aspects of Hysteresis

~.

8

.........................................

~. . . . . . .

....... i .... i ................. i

I

I

I

I

....... ~ ! i ! ,

~ ...........

,.. : - \....,.,.,.,.,.../. ,:::,

i

!

.... :,,,,,,^,~, 2 ' --.--.-.'---.--.~-;

-2

I

! ~ : :

i

!

!

-1.5

-1

.

.

.

.

.

.

.

.

.

.

.

.

.

\

I

....... ~ - :

~!i!,

~!i!~ ...... \..~.l.i.R

~

........................................

i,'

I

i

.

I

~.... i.. . . i . . . t ',!

.......................

.

!

x

3-

.

'7..~!,,' ..... ~......... !. ...... ,~!-...7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 ......................................... 4

--2.0 - -2.5 .-. 3.0

i

.

I

~1.5

"

I

'

I ~ 1". 0 0.5

7 ..................................... , ...... ,I........... t: ...... .,

III

.

/ -.

/ ~,

6 ................................

'

..4. . . . . .

x ,~

\

-0.5

........................

/ /

\ qii~/

~ ...........

::

i

i

i

i ....

i .....

!

i

1

1.5

~

0

.

0.5

2

V

FIGURE

5.25

12k+1

13

t1"

/

I I I I I I I

I1

FIGURE 5.26

h a v e also b e e n n o r m a l i z e d b y 2 a n d ,~2, respectively. It is w o r t h w h i l e to n o t e that, as e v i d e n t f r o m Fig. 5.25, the a s y m p t o t i c v a l u e s of E s t ( f 2) are e q u a l to ~ a n d coincide w i t h a s y m p t o t i c v a l u e s of (ftst) 2. This g u a r a n t e e s zero a s y m p t o t i c v a l u e s for v a r i a n c e 0-2

ft"

5.5 ANALYSISOF SPECTRALNOISE DENSITYOF HYSTERETIC SYSTEMS 573 The described formalism of stochastic processes on graphs can be further extended to compute higher-order moments of the output process ft. This extension is more or less straightforward in the case of the model (5.176). In this case, the relevant multicomponent Markovian processes are defined on the graph shown in Fig. 5.26 and, by using the same line of reasoning as before, we can derive the following explicit expression for p st (2k+1)(x) for edges I2k+1 k P ((2k+1) x)st

-- Pst (X) I-I ~(X, ~j, --~j) . j=l

(5.181)

Similar expressions can be derived for other graph edges.

5.5 ANALYSIS OF SPECTRAL N O I S E D E N S I T Y OF HYSTERETIC SYSTEMS D R I V E N BY S T O C H A S T I C PROCESSES The behavior of hysteretic systems (materials) is usually affected by the internal noise that causes thermal relaxations in these systems. This implies that outputs of hysteretic systems are stochastic processes. In this section, a method for the calculation of the spectral density of such output processes is presented. The discussion is centered on the case of a rectangular hysteretic loop driven by a diffusion stochastic process. This case is important because complex hysteretic systems can be modeled through the Preisach formalism as weighted superpositions of rectangular loops. The problem of calculation of the spectral density of the output process is of considerable mathematical complexity because this process is not Markovian in nature. This difficulty is overcome by using the mathematical machinery of stochastic processes on graphs. In addition, the so-called 'effective' distribution function is introduced that appreciably simplifies the calculation of spectral density. As a result, remarkably simple explicit expressions for the spectral density in terms of parabolic cylinder functions are derived for the case when the input stochastic process is the Ornstein-Uhlenbeck process. The discussion in this section follows the paper [26]. Consider a rectangular loop hysteretic nonlinearity driven by a diffusion stochastic process xt: it = ~flxt. (5.182) The input stochastic process (noise) Xt leads to random switching of rectangular loop ~ . Thus, the output it is a random binary process. The main goal of this section is to compute the spectral density of it. The process it is not Markovian. The efficient way to study the random process it is to

574

CHAPTER 5 Stochastic Aspects of Hysteresis

consider as before the two-component Markovian process yt:

yt -- xtit) '

(5.183)

which is defined on the four-edge graph (see Fig. 5.19). Process Yt can be analyzed by using the mathematical theory of Markovian processes on graphs. This approach will be pursued in our calculations of the spectral density of it. To start these calculations, we shall first define the autocovariance matrix for yt. Consider random vectors ( ix)

Y~-Y=

and

y 0 - y ' = ( i ' ) x' '

(5.184)

generated by the process yt at times t -- r and t = 0, respectively. Then, the autocovariance matrix C(z) can be defined as follows: C(r) = ((y - (y}) (y' - (y'))T),

(5.185)

where the symbol '{ }' is used for the notation of average value, while superscript 'T' denotes a transposed vector. By using the joint probability density function p(i, x, r; i~, x ~, 0) and stationary distribution function ps(i, x) of the process yt, the autocovariance matrix can be computed as follows: C(~)

- f f ~ ~ y(y')Wp(i, X, ~; i', X', O) dx dx' J J

i

i~

-(/~yps(i,x)dx)(f i

E(y')Tps(i',x')dx'). i'

(5.186)

Since the process yt is Markovian, its joint probability function can be represented as the product of the transition probability function p(i, x, r.]i~, x ~, O) and the stationary probability function. This leads to the following simplification of formula (5.186):

"C(r) = ff ~ ~y(y')T[p(i,x, r]i',x', O)- Ps(i,x)]Ps(i',x') dx dx'. i i'

(5.187) To further simplify the calculations, the following two-component 'effective' distribution function is introduced:

g(i'x"c)-(gl(i'x'~) )ga(i,x,"c)

- f Ey'[p(i,

x, r[i', x', 0) -

ps(i, x)]Ps(i', x') dx'.

(5.188)

5.5 ANALYSISOF SPECTRAL NOISE DENSITY OF HYSTERETIC SYSTEMS 575 If this 'effective' distribution function is somehow found, then the autocovariance matrix can be computed as follows: C(r) - 1 ~

ygT (i, x, r) dx.

(5.189)

i

Next, we shall derive the initial boundary value problem for g(i, x, ~) on the graph shown in Fig. 5.19. On each edge of this graph, the transition probability density function satisfies the forward Kolmogorov equation

8p(i, x, z[i', x', O) 8~

+'Lxp(i, x, rli', x', O) - 0,

(5.190)

A

where Lx is the second-order differential operator whose structure is determined by the nature of the process xt. The transition probability density function satisfies the obvious initial condition

p(i, x, 0[i', x', 0) = ~ii,~(x- X')

(5.191)

and certain boundary conditions at graph vertices x = ~ and x = ft. These 'vertex' boundary conditions are discussed in the previous section and they express the continuity of the transition probability function when the transition from one graph edge to another through a vertex occurs without switching of the rectangular loop ~/~, and a zero boundary condition is imposed on the third graph edge connected to this vertex. In addition, the probability current is conserved at each vertex. The stationary probability function satisfies the equation A

LxPs(i,x) - 0

(5.192)

on each edge of the graph and the 'vertex' boundary conditions. By using the formulas (5.190)-(5.192) and the definition (5.188) of g(i, x, ~), the following initial boundary value problem on the graph can be derived for the 'effective' distribution function: g(i, x, ~) satisfies the equation ?g(i, x, "c)

A

+ Lxg(i, x, "c) = 0

(5.193)

on each graph edge, the following initial condition g(i, x, ~:)l~=o = (Y - (Y))Ps(i, x) and the 'vertex' boundary conditions.

(5.194)

CHAPTER 5 StochasticAspects of Hysteresis

576

It is well known that the matrix of spectral density is related to the autocovariance matrix through the Fourier transform:

A S(co) = F

C(z)e -j~~ dr

(5.195)

(30

A

The element $11 (09) of the matrix S(co) can be construed as the spectral density of the output process it. By introducing the Fourier transform of the' effective' distribution function g(i, x, ~:)e-jc~ dr, G(i'x'c~176 t G 2 ( ix,, o9) -- fo ~176

(5.196)

and using formulas (5.189) and (5.195), we find: $ 1 1 ( c o ) - 2 R e ( f y ~ i G l ( i , x , co)dx). i

(5.197)

Thus, if Gl(i, x, co) is found, then the spectral density of the process it can be computed. To find G1 (i, x, co), we perform Fourier transformation of the initial boundary value problem (5.193)-(5.194). As a result, we arrive at the following boundary value problem for G1 (i, x, co): it satisfies the following equation

jCOGl(i, x, 09) + L"xGl (i, x, 09) - (i - (i))ps(i, x)

(5.198)

on each graph edge and the 'vertex' boundary conditions. Thus, by solving the boundary value problem (5.198) and by using formula (5.197), the spectral density of the output process it can be computed for any input process xt. Analytical results can be obtained when xt is the OrnsteinUhlenbeck process. This process is very appealing as a noise model because of its stationary and Gaussian nature. In the case of the OrnsteinUhlenbeck process the operator Lx has the form: A

LxGl(i, x, co) -

0-2

2

d2G1 (i, x, o9) b d dx 2 q- dx [(x - x0)GI(/, x, o9)],

(5.199)

where x0 is the expected value of Xt. Formulas (5.198)-(5.199) lead to a linear second-order, inhomogeneous differential equation whose solution has two distinct components: particular solution of the inhomogeneous

5.5 ANALYSISOF SPECTRALNOISE DENSITYOF HYSTERETICSYSTEMS 577 equation and general solution of the homogeneous equation: j09G 0

0"2 d 2 G ~

if-

2 dx 2

b d [(X -- x0)G ~

-

(5.200)

0

dx

It is apparent from formula (5.192) that a particular solution of the inhomogeneous equation has the form - J ( i - (i))ps(i, x). Because the particular solution is purely imaginary, it does not contribute to the spectral density computed in accordance with formula (5.197). Thus, G1 in formula (5.197) can be replaced by GO that satisfies Eqn. (5.200) and can be expressed as follows:

GO-Jl

-~

a2 daG~ 2 dx 2

b d~

u~ ((x

- x~176

"

(5.201)

By substituting expression (5.201) into formula (5.197), performing integration and using the 'vertex' boundary condition, the following formula can be derived for the spectral density: $11 (09) - -

2a 2 dG 0 dG~ ] - - -09 Im[F~dG0 (1, a+, 09) - - ~ x (1, a-, 09) + d x - (1, fl+, 09) " (5.202)

The solution of Eqn. (5.200) can be expressed in terms of parabolic cylinder functions. This fact, along with formula (5.202), leads to the following analytical expression for the spectral density

$11 (09) - -

20"2 09

Im

I~ (

dzk (~X,09) + Ck(09)-d-xx dzk (fl, 09)t l , bk(09)-d-xx

(5.203)

k=l

where functions Zk(X, 09) are related to the parabolic cylinder functions yk(x, 09) by the formula:

Zx(X, 09) - Yk

(x -- Xo), 09 e -~2

(5.204)

and coefficients bk and Xk are found from the 'vertex' boundary conditions. The functions yl and Y2 are the parabolic cylinder functions that vanish at +cx~ and -cx~, respectively. The sample results of the computation of the spectral density are shown in Fig. 5.27.

CHAPTER 5 StochasticAspects of Hysteresis

578

3.5 \

.

.

.

.

.

3 "'",.~ 2s

o~=b=c~=l

',.\

"- , :",.,~-"',, \:\ "',,, ",..\ ", '..X 1.5 -",, '",XX,

Xo=0.2 .... Xo=O.4 ---e- Xo=0.6 ---- Xo=0.8 4 - Xo=l.0

2

~,,. "".~,.

.....

0.5 0.5

1

1.5

2

2.5

03

FIGURE 5.27

5.6 M O D E L I N G O F T E M P E R A T U R E - D E P E N D E N T HYSTERESIS WITHIN THE FRAMEWORK OF RANDOMLY PERTURBED FAST DYNAMICAL SYSTEMS In the previous sections, the theory of stochastic processes is used to study noise-induced thermal relaxations in hysteretic systems within the framework of the Preisach model of hysteresis. In this and the next sections, the theory of stochastic processes will be used to construct novel models of hysteresis. We begin in this section with the discussion of models for temperature (noise)-sensitive rate-independent hysteresis. It has been experimentally observed that hysteretic properties of materials are sensitive to temperature, i.e. to the level of internal thermal noise. At a given temperature, various materials exhibit rate-independent deterministic (that is, fully reproducible) hysteresis. However, many hysteretic properties, such as coercivity, saturation magnetization, loop shape and others, change appreciably with temperature. Usually, temperature dependence of hysteresis is taken into account within the framework of the Preisach model in ad hoc ways without any effort to understand how sensitivity to noise and deterministic properties of rate-independent hysteresis can be compatible and coexist. In the following discussion, the modeling of temperature (noise)-dependent deterministic hysteresis is

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

579

developed within the framework of randomly perturbed fast dynamical systems. By using the mathematical machinery of the large deviation theory, it is demonstrated that, for specific interplay of the fast system dynamics and the noise strength, randomly perturbed fast dynamical systems may exhibit rate-independent deterministic hysteresis sensitive to noise. The theory of large deviations of randomly perturbed dynamical systems is extensively developed in [26]. The subsequent discussion closely follows the papers [27,28]. To emphasize the sensitivity of deterministic hysteresis to noise, the discussion is deliberately centered around the case of dynamical systems that under purely deterministic conditions do not exhibit hysteresis. It is demonstrated that noise-induced deterministic hysteresis may occur in such systems. This phenomenon is of interest in its own right, and it clearly reveals sensitivity of deterministic hysteresis to noise. First, the case of dynamical systems with two attractors is studied. Then, the case of dynamical systems with many attractors is discussed and the mathematical machinery for identifying the attractors through which such systems evolve in time is presented. In the case of temperature (noise)-sensitive rate-independent deterministic hysteresis one deals with four distinct time scales. The first is the time scale of fast internal dynamics of the hysteretic system. The second is the time scale on which observations (measurements) are performed. This time scale is much larger than the time scale of systems dynamics so that every observation can be identified with a specific stable state of the system. The third is the time scale of input variations. This time scale is much larger than the observation time scale so that every measurement can be associated with a specific value of input. In this way, every measurement establishes the connection between a specific input value and the respective state of hysteretic system. Finally, the largest time scale is the time scale on which thermal relaxations occur. It is demonstrated that for a given noise level all attractors can be subdivided into two distinct groups. The first group consists of attractors that participate in fast switchings that occur on the first time scale. These switchings lead to deterministic hysteresis. The second group consists of attractors that participate in slow switchings that occur on the fourth time scale. These switchings are known as thermal relaxations. As temperature (noise level) is changed, the 'boundary' between these two groups is shifted and this results in temperature-dependent deterministic hysteresis. To start the technical discussion, let us consider one of the possible mechanisms that leads to the occurrence of rate-independent hysteresis. Suppose that the state of a system is defined by a set of interior parameters X = ( X 1 . . . . . X n) E [~n, and the evolution of the system is governed by the

CHAPTER 5 Stochastic Aspects of Hysteresis

580

g

T A

(a)

gl

c

-..~_ ,. )X

l

(b)

FIGURE 5.28

equations:

~ X t - b(Xt, gt),

Xo E ~n

(5.205)

where gt is an exterior parameter-input, while b(x, g) is a sufficiently smooth vector field in R n+l. It is further assumed that 6 << 1, which implies the fast dynamics of interior parameters in time. The case when this dynamic is appreciably faster than the temporal variations of input gt is discussed below. Let us also assume that the system (5.205) with a 'frozen' input gt -- g has a finite number s of asymptotically stable equilibriums Kl(g) . . . . . K~(g)(g) and that any trajectory of this system is attracted to one of Ki(g) (with the possible exception of trajectories belonging to separatrix surfaces). An observation (measurement) Yt performed on the system can be mathematically interpreted as a certain function of interior parameters Xt. Since there are no instantaneous observations, the above function will include some integration (averaging) over some time h. If h is small on the time scale of input variations and large on the time scale of Xt-variations, then the relation between Yt and gt will be essentially rate-independent. When input gt is changed in time, the vector field b(x, g) may have bifurcations that lead to hysteresis. A typical example of this situation for the one-dimensional case is shown in Fig. 5.28(a), where curve g is defined by equation b(x, g) = 0, and it is assumed that b(x, g) < 0 and b(x, g) > 0 above and below this curve, respectively. It is apparent that the vector field b(x, g) has one stable equilibrium for gt ~ (gl, g2) and two stable equilibria for gt ~ (gl, g2). If the initial value of input go > g2, then the evolution of the system for decreasing gt essentially follows the left branch MBA of the curve E. The state of the system is switched to point D on the right branch

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

581

of g when gt reaches (from above) the value of gl, and the system follows the right branch (DN) towards N as the input is further decreased. When the variation of input is reversed, the state of the system moves upward along the right branch of g until the switch to the left branch occurs at point C. It is apparent that the evolution of this system results in a rateindependent hysteretic relation between observation Yt and input gt. Next, consider the case when the field b(x, g) has no bifurcations. The one-dimensional example of this situation is shown in Fig. 5.28(b), where the set of g of solutions of equation b(x, g) = 0 consists of three disjoint curves x = ?0(g), x = ~2l ( g ) , x = ?2(g) with ]2l(g) ( ? o ( g ) ( 7 2 ( g ) . It is assumed that points of x -- 71 (g) and x = 72(g) correspond to stable equilibria, while points of ?o(g) are unstable equilibria. It is apparent that, for sufficiently slow input variations, the system will essentially move only along one curve (?l(g) or 72(g)) and no hysteretic relation between Yt and gt can be observed. It will be demonstrated below that small random perturbations may result in observable deterministic hysteresis between Yt and gt, although the original deterministic system has no bifurcations. Hysteresis appears due to bifurcations in the randomly perturbed system. First, the case of hysteresis induced by noise in systems with two asymptotically stable equilibria will be discussed. Then, the case of many equilibria will be treated. Consider small white-noise perturbations of dynamical system (5.205)

(~J(y - b(X~, gt) + v/-d o'(X~)IfVt.

(5.206)

Here, Wt is the vector Wiener process in R n, while r~(x) is a nondegenerate square matrix of order n with bounded Lipshitz-continuous entires, while e is a small parameter. We next assume that the following relations for small parameters and e hold e, ~ $ 0, l i m ~ - 1 ] In ~] -- c > 0. (5.207) By introducing the time r - ~, eI - ~ and by fixing gt - g, the stochastic differential equation (5.206) can be written as follows

'g - b(X "g, g) +

(5.208)

Next, we shall introduce the (normalized) action functional defined as follows [26]: Sg0,T((p ) _ 21 ~0 T [o._l(q)s) (~bs _ b((fis,

g))]2 as,

(5.209)

if ~ ~ Co, T is absolutely continuous, and Sg0,T is infinite for all other functions from CO,T. The action functionals is the main mathematical tool of

CHAPTER 5 Stochastic Aspects of Hysteresis

582

the theory of large deviations. It is, roughly speaking, a measure of the 'difficulty' for a sample of stochastic process X~'g to pass through a small neighborhood of function qh. This function determines the'most probable' transition route between equilibrium points as well as the asymptotics of switching (transition) times between these equilibria. By using this functional, functionals V12 (g) and V21 (g) can be introduced: V12(X) -- inf{g0,T(~0)" q~0 -- },l(g), (PT -- },2(g), T > 0},

(5.210)

V21(g) = inf{g0,T(q~)" q~0 -- },2(g), OPT = },l(g), T > 0}.

(5.211)

When the vector field b(x, g) is potential: b(x, g) = -VxU(x, g) (which is always true for the one-dimensional case) and a is the unit matrix, then the above functionals can be expressed as the following potential differences: V12(X) -- 2[U(},0(g), g) - U(}, l(g), g)] and V21(g) - 2[U(},0(g), g) -

U(},2 ( g), g) ]. Finally, let us introduce the sets: G12 - {g: V12(g) < c, V12(g) < V21(g)},

(5.212)

G21 -- {g: V21 (g) < c, V21(X) < V12(g)},

(5.213)

where c is defined in (5.207). It is known from the theory of large deviations [26] that for the process X g'e' the transition times between equilibrium points },i(g) and ?7(g) (i, 7 e {1, 2}, and i # 7) have (in the sense of logarithmic equivalence) the exponential asymptotics exp{ F1 FiT(g)} for ~' $ 0. This means that the transition times TiT for the process Xt 'g have the asymptotics b exp{ 7'~Vd( g) }. By taking into account relation (5.207), we find In ~ } _ (~1 vii(g) c .

T d ,-, ~exp ViT(g)---d-

(5.214)

It follows from the last expression that TiT tends to infinity if ViT(g) > c, and it tends to zero if ViT(g ) < c. Thus, it can be concluded that if the input gt is monotonically increased and the state of the system moves upwards along the curve },2(g), it is switched to the curve },1(g) as soon as the boundary of the set G21 is crossed. Similarly, if the variation of the input is reversed and the system tracks the curve }'1 (g), it is switched to the curve },2(g) as soon as the boundary of the set G12 is crossed. It is apparent that the above switchings result in a rate-independent hysteretic relation between observation Yt and input gt. It is also clear that the input switching ('coercive') values as well as the shape (width) of the corresponding hysteresis loop are determined by constant c in (5.207) and, consequently, by

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

583

the interplay of fast system dynamics and the noise strength. It is worthwhile to remark that trajectory X~ of randomly perturbed system (5.206) is close to 71 (gt) or 72(gt) in some integral norms that filter out 'short-lived' excursions of stochastic process X[. Such integral norms naturally appear in functionals that relate observation Yt to the state X[ of the system due to some finite time of any observation. Now, consider the case when for any fixed gt = g the dynamical system (5.205) has an arbitrary finite number s of asymptotically stable equilibria Kl(g), K2(g) . . . . . K~(g)(g). Let us suppose that as the input is changed with time, these equilibria trace disjoint curves 71 (g), 72(g) . . . . . 7e(g) (g). It is apparent that, for sufficiently slow input variations, the deterministic system essentially moves along one of those curves and no hysteresis is observed. The situation may change when our dynamical system is randomly perturbed and its motion is governed by Eqn. (5.206). In this case, the presence of noise may result in switching from one 7-curve to another and, in this way, produce hysteretic behavior of the system. The immediate task is to describe the mathematical machinery that allows one to determine the switching values of input and to identify the 7-curves involved in switchings. This mathematical machinery is based on the notion of the action functional (5.209) that is used to define the set of functions

Vij ( g )

-

inf { S g0,T(~)" ~0 -- 7i(g), ~(T) -- 7j(g)}.

(5.215)

It turns out that the time sequence (order) of noise-induced switchings in the randomly perturbed dynamical system (5.206) is in a way not random and completely governed by deterministic functions Vij(g). To fully describe the time sequence of noise-induced switchings, it is necessary to introduce the hierarchy of cycles [26] defined by functions Vij(g) on the set of L of equilibrium points and the notion of/-graphs defined on L. Each point i from L is considered as rank 0 cycle and the exit rate ei from this cycle is given by the formula

ei - min Vij. j: j#i

(5.216)

The 'next' state to i is defined as the state k = Jl(i), where the above m i n i m u m is reached. The transition i --~ k is the most probable (with probability close to one). It is assumed below that our dynamical system (5.205) is generic in the sense that the above m i n i m u m as well as all similar minima considered in the subsequent discussion are achieved only at one equilibrium state. A cycle rank I is defined as an ordered subset of v equilibrium points from L with the property that for any point i of this subset we have J~(i) - i. It is clear that the ordering on rank-1 cycles is introduced by

CHAPTER 5 Stochastic Aspects of Hysteresis

584

the transition operation Jl(i). To introduce such characteristics as the main state of a cycle, the stationary distribution rate on a cycle and the exit rate from a cycle, the concept of/-graphs [26] is very instrumental. Consider an arbitrary subset A of L and some equilibrium point i ~ A. A set of directed edges (arrows) connecting some points j ~ A is called an /-graph qi if one edge starts from any point of this graph j ~ A (j # i), and from any point of this graph there is a directed path to i along the edges of the graph. The set of all/-graphs is denoted as Qi(A). Now, the main state of rank-1 cycle C is defined as the state j" = M(C) where the following minimum is reached

2 - min min

j~C g~Qj(C)

y~

Vmn.

(5.217)

(m--+n)~q

The stationary distribution rate mc(i) of the rank-1 cycle C is defined by the formula mc(i) = min ~ Vmn -- ,~. (5.218) q~Qi(C) (m--,n)~q

Finally, the exit rate ec from the cycle C is given by the expression

ec(i) =

min [mc(i) + Vij].

i~C, j~C

(5.219)

For any rank-1 cycle C (p) there is such i* ~ C (p) and such j* belonging to the rank-1 cycle C (s) = J2(C (p)) that the above minimum is reached. In this sense, the cycle C (s) is the 'next' to the cycle C (p), and the transition C (p) --+ C (s) is the most probable (with probability close to one) among all possible transitions between C (p) and other rank-1 cycles. A cycle rank 2 is defined as an ordered subset of ~ rank-1 cycles with the property that for any rank1 cycle C (p) of this subset J~2(C(p)) = C (p). It is clear that the ordering on rank-2 cycles is introduced by the transition operator J2(C(P)). For rank-2 cycles, we can introduce main states, stationary distribution rates and exit rates by using formulas (5.217), (5.218), and (5.219), respectively. The only qualification is that C in the above-mentioned formulas is a rank-2 cycle. By literally repeating the same line of reasoning, we can introduce cycles of higher ranks until we reach some rank at which there exists only one cycle consisting of all equilibrium points. For this cycle the exit rate is defined as infinite. The described hierarchy of cycles is illustrated by Fig. 5.29 where four rank-1 cycles, two rank-2 cycles and one rank-3 cycle are depicted. Now, the important notion of observable equilibrium states of dynamical systems (5.206) can be introduced. Consider an arbitrary value g of input and some equilibrium state i that exists for that input value. For this

5.6 MODELING OF TEMPERATURE-DEPENDENT HYSTERESIS

585

FIGURE 5.29

state, we can introduce the ordered sequence of cycles Ck of all possible ranks that describes the sequence of the most probable (with probability close to one) transitions initiated from i. The above sequence of cycles generates the sequence of exit rates ek given by formula (5.219). Let ej be the smallest exit rate from the above sequence such that

c < ej,

(5.220)

where c is the constant from (5.207). Let j* be the main state of the cycle Cj. Then, by using asymptotics similar to (5.214), it can be shown that a very fast (theoretically instantaneous) transition from the equilibrium state i to the equilibrium state j* is induced by the noise. This transition occurs via very fast intermediate transitions between equilibrium states that form cycles Ck with k < j. In this sense, the equilibrium state i, as well as the intermediate states participating in the transition, are not observable, while the state j* is observable. It is clear that the observability of the state is determined by the value of constant c in (5.207) and, consequently, by the interplay of fast system dynamics and the noise strength. It is also clear that for any instant of time there is a finite number of observable equilibrium states Jn (n = 1, 2 . . . . . e * ( g t ) ) . A particular observable state, occupied by the system at a given instant of time, is determined by the past history of system dynamics. Suppose that the system (5.206) is in some observable state Jn and suppose (for the sake of being specific) that the input is monotonically decreased. If this input decrease does not violate inequality (5.220) and the condition that Jn is the main state of the cycle Cj, then the state of the system will move along the ?-curve to which the observable state Jn belongs. However, as soon as the input gt reaches the

586

CHAPTER 5 StochasticAspects of Hysteresis

value at which the state Jn is no longer observable, the system is switched to another observable state Jm belonging to another ~-curve. The new observable state Jm is determined by using the same algorithm as above with only the difference that the ordered sequence of cycles Ck describes now the sequence of the most probable transitions initiated from Jn" It is this change in the initial state of transitions that leads to branching and hysteresis phenomena. Indeed, if after switching to the state Jm input is somewhat further decreased and then reversed, then for the monotonically increasing input the system (5.207) may not switch back to the observable state Jn at the same input value as the switching from Jn to Jm occurred. This is because the equilibrium state Jm may loose its 'observable' status at a different input value than happened for the state Jn" It is apparent from the above discussion that inequality (5.220) subdivides all equilibrium states (attractors) into two distinct groups: equilibrium states that participate in fast switchings that lead to deterministic hysteresis and equilibrium states that form cycles Ck with k > j. These equilibrium states participate in slow switchings that are usually known as thermal relaxations. In the previous discussion, emphasis has been given to the noiseinduced hysteresis in dynamical systems that under purely deterministic conditions do not exhibit hysteresis. However, it is clear from the presented reasoning that in dynamical systems, which under purely deterministic conditions do exhibit hysteresis, the presence of noise may substantially alter the structure of hysteresis. For instance, in one-dimensional dynamical systems (5.205), which exhibit hysteresis due to bifurcations between two existing asymptotically stable equilibrium states, the presence of noise may appreciably alter switching input values and the shape of the hysteresis loop. In dynamical systems (5.205), which exhibit hysteresis due to bifurcations between many existing asymptotically stable equilibrium states, the presence of noise may make only some of those states actually observable and, in this way, appreciably change the structure of hysteresis. It is apparent from the presented discussion that the effect of small noise on hysteresis in dynamical systems is very pronounced for fast systems. Since fast systems are typical for rate-independent hysteresis and since some small noise is always inherently present in physical systems, the study of rate-independent hysteresis in fast dynamical systems without noise may be inadequate. The presented discussion has been based on relation (5.207) between small parameters ~ and e that characterize the fast dynamics and the noise strength, respectively. The question is to what extent this relation is natural. By using results from [29], it can be demonstrated that for other relations between ~ and e deterministic hysteresis either does not exist (strong noise) or it is insensitive to noise (very weak noise). Under the condition (5.207),

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

587

hysteresis still exists in a deterministic sense and it is quite sensitive to noise. This is consistent with many experimental facts which demonstrate that deterministic hysteresis is temperature sensitive. In this sense, the condition (5.207) can be regarded as natural for fast dynamical systems employed to describe noise-sensitive deterministic hysteresis. Finally, it is important to note that usually observation Yt is a scalar or a vector whose dimension is much smaller than the dimension of the state vector Xt. For this reason, the observation of the (Yt) versus input (gt) relation gives a 'reduced' description of the dynamical system. This reduced description may often lead to the situation where many different state vectors Xt result in the same observation Yt. This, in turn, may be the origin of the nonlocal memory of the hysteretic relation between Yt and gt.

5.7 F U N C T I O N A L

(PATH) INTEGRATION

MODELS

OF

HYSTERESIS The Preisach model is designed as a continuous superposition of the simplest rectangular loop operators ~ . These operators can be construed as elementary building blocks of the Preisach model. A natural way to generalize the Preisach model is to consider more sophisticated elementary hysteresis operators and to design hysteresis models as continuous superpositions of such elementary operators. In this section we pursue this approach and consider functional (path) integration models of hysteresis that are designed as superpositions of elementary hysteresis operators generated by continuous functions. A physical interpretation of the path integration models as well as their various connections with the classical Preisach model are presented. The discussion in this section follows (to a certain extent) the paper [30]. Consider a continuous function g(x) on some closed interval [x_, x+] that satisfies the condition

u_ ~g(x_) ~g(x) Kg(x+) = u+.

(5.221)

Such a function will be called a generating function, while x_ and x+ can be termed as lower and upper saturation values, respectively. An elementary hysteresis operator ~gu(t) can be associated with each generating function by traversing its upper or lower envelopes (see Fig. 5.30). This can be done as follows. Suppose that at time to the input u(t) assumes some extremum value u0 and

uo = g(xo).

(5.222)

CHAPTER 5 StochasticAspects of Hysteresis

588

Uo

| | |

X_

X0

X+

FIGURE 5.30

If u0 is some minimum value, then for the subsequent monotonic increase of input the upper envelope g+0 (x) = max g(x) [x0,x]

(5.223)

is traversed. On the other hand, if u0 is some maximum value, then for the subsequent monotonic decrease of the input the lower envelope guo (x) = m i n g(x) [x,x0]

(5.224)

is traversed. This means that for monotonic input variations the elementary hysteresis operator ~gU(t) is defined as follows:

~gU(t) =

x (t) x-(t)

if u(t) is monotonically increased, if u(t) is monotonically decreased.

(5.225)

Here x + (t) and x-(t) are the solutions of the following equations, respectively: g+ (x + (t)) - u(t),

(5.226)

guo (x- (t)) - u(t).

(5.227)

Since upper g+ (x) and lower gu0 (x) envelopes usually have 'horizontal' parts parallel to the x-axis, solutions of Eqns (5.226) and (5.227) may not be unique for some values of u(t). This difficulty can be removed by using minimal and maximal solutions of Eqns (5.226) and (5.227), respectively: x+(t) = min{x: g+(x) - u(t)},

(5.228)

x - ( t ) - max{x: guo(X) - u(t)}.

(5.229)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

589

Ut

U2 x

U_ X_

X2

Xt

X1

FIGURE 5.31

The elementary operator ~gu(t) has been so far defined for monotonic input variations. This definition can be extended to the case of piecewise monotonic inputs by consecutively applying the definition (5.225) for each time interval of monotonic variation of u(t). The ambiguity of choosing x0 in Eqn. (5.222) can be removed if it is agreed that the evolution is started from the state x_ (or x+) of negative (or positive) saturation. The given definition of elementary hysteresis operator ~gu(t) is illustrated by Fig. 5.31. It is clear that elementary operator ~gu(t) is rate-independent. This is because the output value x(t) depends only on the current value of input u(t) and the past history of input variations but does not depend on the rate of input variations. It is also clear that the operator ~;~u(t)has local memory. This is because the simultaneous specification o t output and input uniquely defines the state of elementary hysteretic nonlinearity ~gu(t). Finally, it is clear that the operator ~gu(t) exhibits the 'wiping-out' property (see [3]). In a way, the wiping-out property can be regarded as a consequence of local memory. It is important to note that not all parts of g(x) are accessible. For instance, point A shown in Fig. 5.31 is not accessible. This part of g(x) will not be traversed for any input variations. In this sense, the same elementary hysteresis operator ~gu(t) is defined on the equivalence class of functions g(x) with the same accessible parts. Now, consider some set G of generating functions g(x) and some measure/~(g) defined on this set. Then, the functional (path) integration model of hysteresis can be formally defined as follows:

fit) -/c ~gu(t) d#(g).

(5.230)

CHAPTER 5 Stochastic Aspects of Hysteresis

590

The above model is quite general. Its structure depends on the measure #(g) introduced on the set G. Next, we demonstrate that the classical Preisach model is a particular case of the path integration model (5.230). To this end, consider the subset Gp of G that consists of functions with two vertical parts (x_ = - 1 , u ~ ) and (x+ = 1, u~fl) separated by inaccessible parts 2(x) with ] / < 2(x) < a (see Fig. 5.32). It is clear that for such functions

~;gu(t) - ~;~u(t)

for

g(x) ~ A~,

(5.231)

where A~/~is the equivalence class of functions from Gp that have the same values of a and ft. Now, consider the measure #(g) that is concentrated on the subset Gp. Since Gp is the union of nonintersecting equivalence classes A~/~, the model (5.230) can be written as follows: (5.232)

f(t)= f~>~(fA~ ~gu(t)d#~(g))d~

where #~/~(g) are the measures on the equivalence classes A~/~ induced by the measure #(g) on Gp. By using (5.231) in (5.232), we obtain

fA

~gu(t)d#~(g)=(fAd#~(g))~u(t)=#(a,fl)~u(t),

(5.233)

where the following notation is introduced (5.234)

#(~'//) = fA d/~/~(g).

By substituting (5.233) into (5.234), we end up with the classical Preisach model

- [ [ #(~, ~)~&~u(t)d~ dfl. ,Ida

(5.235)

In the case when the vertical parts in Fig. 5.32 are replaced by curved paths (see Fig. 5.33), the corresponding elementary hysteresis operator ~/gu(t)can be represented as follows:

x(t) -

1;+ (Ut) -- 1;- (Ut) ,, ~~

2

+

V+ (Ut) -}- V - (Ut)

2

'

(5.236)

where v+ (u) and v- (u) are the inverses of g+ (x) and g - (x), respectively.

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

591

u (x

-1

m,,,..

X

FIGURE 5.32

u

g+(x)

~X

/~ g-(x) FIGURE 5.33

By choosing measure ~(g) concentrated on the set of functions shown in Fig. 5.33 and literally repeating the same line of reasoning as before, it can be demonstrated that the path integration model (5.230) is reduced to the Preisach model with the input-dependent measure (see [3]). It has been demonstrated above that the path integration model (5.230) is reduced to the classical Preisach model if the measure #(g) is concentrated on functions g(x) such that ~gu(t) - ~a~u(t). Below, it will be shown that this reduction is also possible in cases when the measure #(g) is concentrated on functions g(x) such that ~gu(t) ~ ~a~u(t). This will further emphasize the generality of the classical Preisach model. In general, it is not immediately obvious how to generate the measure on the functional set G and how to carry out functional integration in (5.230), in other words, how to compute output fit). It turns out that the above difficulties can be appreciably circumvented if the set G is interpreted as a set of samples of a stochastic diffusion process, that is, the Markovian process with continuous samples generated by the Ito

CHAPTER 5 StochasticAspects of Hysteresis

592

Ul

X

U_ X_

U_

Xt

i

T X_

x

I

xt

Xl

FIGURE 5.34

stochastic differential equation

dgx = b(gx, x) dx + a(gx, x) d Wx.

(5.237)

Here x must be construed as fictitious 'stochastic' time that must not be confused with real physical time t. In the above case, the measure #(g) is the stochastic measure that, in principle, can be generated by using the transition probability density function for the process defined by Eqn. (5.237). This is the consequence of the Markovian nature of the process gx. However, there is no need of doing this because, as will be demonstrated below, the output f(t) can be interpreted as an average level-crossing (stochastic) time. As a result, the mathematical machinery developed for the solution of level-crossing (exit) problems can be extensively used for the output calculations of functional integration-type models (5.230). First, we shall discuss the meaning of ~;gu(t) when the generating function g(x) is a sample of the diffusion stochastic process. This meaning is different for monotonically increasing and monotonically decreasing inputs. To illustrate this, let us consider Fig. 5.34 where a particular sample g(x) of the diffusion stochastic process that starts from the negative saturation value u_ is shown. It is clear from this figure and the definition of ~gu(t) that for monotonically increasing input u(t) the elementary operator ~gu(t) has the meaning of the first level-crossing time, where the level is equal to the current value of input u(t). This time is a random variable

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

593

and, consequently, the output of the functional integration model (5.230) is equal to the average value of the first level-crossing time. This is true for any value of input until u(t) reaches some maximum value Ul. It is clear from Fig. 5.34 and the definition of ~gu(t) that for the subsequent monotonic decrease of input u(t) the elementary operator ~gu(t) has the meaning of the last level-crossing time, where the level is equal to the current input value u(t). More precisely, this is the last time of crossing the level u(t) before the first time of crossing the level Ul. This last level-crossing time is also a random variable. Thus, for monotonically decreasing input, the output of the functional integration model (5.230) is equal to the average value of the last level-crossing time. It is this difference in the meaning of output values of the path integration model (5.230) for monotonically increasing and decreasing inputs u(t) that results in hysteresis. Next, we shall present a mathematical formalism that supports the statements outlined above. Let T(~, x~]fl, xfl) be the notation for the probability density of the first crossing time x~ of the level a under the condition that the sample of the stochastic process crossed the level fl < a at the time xfl. It is clear that

Jxx~ T(~x, x~]fl, xfl) dx~ 1. (5.238) fl It is apparent that the probability densities p~(x~) of xa and pfl(xfl) of xfl are related by the expression

p~(x~) -

f0x~T(~x, xalfl, xfl)pfl(xfl) dxfl.

(5.239)

Consider the set G of all samples of the diffusion process gx that satisfy the condition (5.221). Let us analyze mathematically how the output of the path integration model (5.230) changes when the input ut is increased from u_ to some maximum value Ul and then is decreased to some minimum value u2. During the monotonic input increase, the elementary operator xt - ~gut has the meaning of the first time of crossing the level ut. By using the notations pfl(xfl) = pu_ (x_) and p~(x~) = Put (Xt), according to (5.239) we find

Put (Xt)

--

~

Xt

T(ut,

XtlU-,

x-)pu_ (x_) dx_.

(5.240)

The probability density function Pu_ (x_) must be chosen as a part of the characterization of the initial state of lower saturation. For instance, it can be chosen as ~(x - x_). After that, Eqn. (5.240) permits one to compute the unknown probability density Put (Xt) provided that the first-time

CHAPTER 5 StochasticAspects of Hysteresis

594

level-crossing problem has been preliminarily solved and the function T(ut, xtlu-, x_) has been found. By using the probability density Put (xt), the output value of the path integration model (5.230) can be computed as follows:

f(ut) =

f0

xtput (xt) dxt.

(5.24:1)

Formulas (5.240) and (5.241) can be used to compute Put (xt) and f(ut) for all values of ut between u_ and Ul. In this way, the ascending branch of the major loop can be computed. For ut = ul, we have

pu~(Xl) =

r(ul, x l l u - , x - ) p u _ ( x - ) d x - .

f0

(5.242)

Next, consider the monotonic decrease of input from Ul to u2. For this input variation, the elementary operator xt = ~xut has the meaning of the last time of crossing the level ut before the level Ul is reached for the first time. This means that probability density Put (xt) of the last level-crossing time xt satisfies the integral equation

Pul (Xl) --

0 xl

T(Ul, Xl Jut, xt)Put (xt) dxt.

(5.243)

This integral equation can be (in principle) solved for any value of Ut between ul and u2. In this way, Put(Xt) c a n be found and used in formula (5.241) for the calculation of the output value f(ut) along the descending branch attached to the previous ascending branch at the point ut = Ul. After Pu2(X2) is found by solving integral equation

Pul (Xl) -- ~0xl T(Ul, X1 lU2, x2)Pu2 (X2) dx2,

(5.244)

it can be used in the formulas

Put (xt) -

f0xtr(ut, xtlu2, x2)Pu2(x2) dx2,

(5.245)

PuB(x3) -

fox3r(u3, x31u2, x2)Pu2(x2) dx2

(5.246)

for the computations of Put (xt) and PuB(x3) for the third hysteresis branch when input u(t) is monotonically increased from u2 to u3. Similarly, PuB(x3) can be used in the integral equations

PuB(x3) -

f0x3r(u3, x31ut, xt)Put (xt) dxt,

(5.247)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

Pu3 (X3) --" ~0X3Z(u3, x31u4, x4)Pu4 (x4) dx4

595

(5.248)

for the computations of Put (Xt) and Pu4(X4) for the fourth branch of the hysteresis loop when input u(t) is monotonically decreased from u3 to u4. It is clear that the computations described above can be recursively used to find any branch of hysteresis described by the path integration model (5.230) with stochastic measure. It is important to note that in the case when u3 = Ul the probability densities Pu3(x3) and Pul (Xl) coincide. This directly follows from the coincidence of Eqns (5.244) and (5.246) for the above case. The coincidence of Pu3(X3) with Pu~ (Xl) for u3 - Ul implies the validity of the 'wiping-out' property for the path integration model. This fact can also be deduced from the validity of the 'wiping-out' property for each elementary hysteresis operator ~gu(t). Now, consider the particular case when the stochastic process gx is homogeneous (translationally invariant) with respect to 'stochastic time' x. Such a process is described by the Ito stochastic differential equation dgx = b(gx) dx 4- cr(gx) dWx.

(5.249)

Owing to the translational invariance, the conditional first-time levelcrossing probability density T(~, x~]fl, xfl) has the property T(~, x~lfl, xfl) = T(~, x~ - xfllfl, 0).

(5.250)

Next, we shall use this property to compute the expressions for ascending f + (ut) and descending f - ( u t ) branches of hysteresis loops formed when the input u(t) is monotonically increased from some minimum value U2k to some maximum value U2k+l and is then monotonically decreased back to U2k. For the calculation off + (ut), formulas similar to (5.241) and (5.245) are appropriate. This leads to the expression

f+ (lit) --

XtPu+t(Xt) dxt

- fo ~176 xt (fo xt T(ut, xt - X2k]U2k, O)Pu2k(X2k) dX2k ) dxt. (5.251) By using Fig. 5.35, the above double integral can be transformed as follows: f + (ut) --

xtT(ut, xt - X2klU2k, O) dxt Pu~ (X2k) dx2k

2k --

2k

(Xt -- X2k)T(ut, Xt -- X2klU2k, O) dxt Pu2k(X2k) dx2k

CHAPTER 5 StochasticAspects of Hysteresis

596

Xt

Xt = X2k

" X2k

FIGURE 5.35

-t-

T(ut, xt - X2klU2k, O) dxt X2kPu2k(X2k) dx2k. 2k

(5.252) By using the change of variables (5.253)

Z -~ Xt -- X2k,

from (5.252) we find T(ut, ZIU2k, O) dz X2kPu2k(X2k) dX2k

f + (Ut) --

q-

zT(ut, Z[U2k, O) dz Puak(X2k)dx2k.

(5.254)

By using formulas (5.238) and (5.241) in the first integral and the normalization condition

/o ~ Puak(X2k) dX2k

--

1

(5.255)

in the second integral, from (5.254) we find J:+ (Ut) -- f(U2k) q- F(ut, U2k),

(5.256)

where ~(Ut, U2k) --

zZ(ut, ZlU2k, O) dz.

~0~176

(5.257)

It is clear from (5.256) that the current value of output on the ascending branch is determined only by the current value of input ut and the last minimum value U2k. We shall next establish a similar result for the descending branchf-(ut) of the hysteresis loop. According to (5.241) and (5.247)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

597

Xt X2k+l ~ / t / /

// /

/

/

= X2k+1 FIGURE 5.36

we have f(U2k+l)

/0

X2k+lPu2k+l(X2k+l) dX2k+l \

oo (foX2k+l Z(U2k+l, X2k+l [Ut, xt)Put (Xt) dxt) dX2k+l O X2k+l fO

Z(U2k+l, X2k+l -- Xt[Ut, O)put (Xt) dxt) dX2k+l.

X2k+l

(5.258) By using Fig. 5.36, the last double integral can be transformed as follows:

f(U2k+l) --

X2k+lZ(U2k+l, X2k+l -t

Xt[Ut, O) dX2k+l Put (xt) dxt. (5.259)

By using the change of variables Z ~- X2k+l --

Xt,

(5.260)

from formula (5.259) we derive

(/0

f(U2k+l) --

)

zr(u2k+l, zlut, O) dz p~ (xt) dxt

Jr-

T(u2k+l, Z]Ut, O) dz XtPut(Xt) dxt.

(5.261)

Now, by using the same reasoning that was used to simplify formula (5.254), we obtain d(U2k+l) --

.~'(U2k+l, ut) + f-(ut),

(5.262)

CHAPTER 5 StochasticAspects of Hysteresis

598

which leads to

f - (Ut) - -

f(U2k+l)

-- .T(Uak+l,

Ut).

(5.263)

Formulas (5.256) and (5.263) show that the shapes of generic ascending and descending branches of a minor hysteresis loop are the same regardless of the past input history. The past history is reflected in the values of f(u2k) and f(U2k+l ). In other words, the minor hysteresis loops corresponding to different past histories are congruent. Since the wiping-out property and the congruency of minor loops (formed for the same back-and-forth input variations) represent the necessary and sufficient conditions for the description of hysteresis by the Preisach model, we conclude that the path integration model (5.230) with the stochastic measure corresponding to the homogeneous diffusion process homogeneous diffusion process is equivalent to the Preisach model. It is clear from the previous discussion that the output calculations for the path integration model can be performed if the function T(~, x~l//, x/~) is known. This function can be computed by solving the exit problem for the stochastic process defined by Eqn. (5.237). This, in turn, requires the solution of initial boundary value problems for the backward Kolmogorov equation similar to those discussed in Section 5.3. In particular, by using the mathematical machinery of the exit problem, the closed-form expressions can be derived for the weight function #(~, fl) of the Preisach model, which is equivalent to the path integration model (5.230) with the stochastic measure generated by the process (5.249). Below, we present the final results; the mathematical details of the derivation can be found in the paper [30]. Consider the function

(/0=

0(g) = exp - 2

o.2(g,)

d=)

(5.264)

then the function Y'(a, ]/) from (5.257) and #(~, ]/) can be computed as follows: Y'(~, fl) - ~

0(g') dg'

)(/=" 0(g') dg' ) r~2(g)0(g d= ) ,

(5.265) #(~x, fl)= ~(~x)~,(fl) Y'(a,/~),

(5.266)

where K~/~ =

~(g) dg.

(5.267)

5.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS

599

8

.0

/

-4

-8 -4

I 2

0

-2

FIGURE

L

5.37

As examples, consider the following cases: (a) g x is the Wiener process (b - 0, r~ - 1). Then 6(g) - 1, Karl - o ~ - fl and 1 1 #(~, fl) = 3r(~, fl) = ~ ( ~ _ fl)2. (5.268) o

Thus, the Preisach weight function is simply a constant and all hysteresis branches are parabolic (see Fig. 5.37). (b) g x is the diffusion process with constant drift [b(g) - 2~ (Z > 0), r~ = 1]. In this case we have 6(g) - e-g/x,

(5.269)

Karl - e - f l / z - e - a / z ,

(5.270)

a-flcoth(~xfl ) - 1 fl(0r

]~) - -

2Z

(5.271)

,

sinh2 (~zfl) 3r(~,fl)-4[(

~fl)coth(

a -2Z fl)-11

.

(5.272)

Typical hysteresis branches computed for this case are shown in Fig. 5.38. (c) g x is the Ornstein-Uhlenbeck process (b(g) - - ~, cr - 1). In this case, ~(g) - e g 2 / Z , (5.273)

CHAPTER 5 Stochastic Aspects of Hysteresis

600

_

0

-4-

-8

I

-4

i

-2

'

0

I

2

U

FIGURE

5.38

and #(a, fl) and Z(a, fl) are obtained by inserting the last expression in the formulas (5.265)-(5.267). The described functional (path) integration model (5.230) admits the following physical interpretation. It is known that hysteresis is due to the existence of multiple metastable states in the system free energy F(X) (the temperature dependence is tacitly understood), which means that the system may be trapped in individual metastable states for long times. Consider a simple case where the state variable X is a scalar quantity and the relevant free energy in the presence of the external magnetic field H is ~(X; H) = F(X) - HX. The metastable states available to the system are represented by ~-minima with respect to X for which O~/OX = 0, 0~/r 2 > 0. When H is changed with time, the number and the properties of these minima are modified by the variation of the term - H X . The consequence is that previously stable states are made unstable by the field action and the system moves to other metastable states through a sequence of (Barkhausen) jumps. Because the condition 3~/OX = 0 is equivalent to H = OF/OX, one can analyze the problem by using the field representation shown in Fig. 5.39. The response of the system, expressed in terms of H(X), is obtained by traversing the upper and lower envelopes of OF/OX for increasing and decreasing H, respectively. From the physical viewpoint, this construction amounts to assuming that the system, once made unstable by the action of the external field, jumps to the nearest available energy minimum, which means that one excludes dynamic effects that could aid the system to reach more distant minima. It is clear from the above description

REFERENCES

601

FIGURE 5.39

that OF/OX and H are similar to the generating functional g(x) and input u(t), respectively, within the framework of the function (path) integration model (5.230). The functional integration model itself can be interpreted as the average hysteresis response of a statistical ensemble of independent (elementary) systems evolving in a random free-energy landscape. This interpretation can be of importance in applications where randomness due to structural disorders plays a key role in the appearance of hysteretic effects. A particularly important example is the motion of magnetic domain walls in ferromagnets, where various forms of structural disorder (point defects, dislocations, gain boundaries, etc.) are responsible for the random character of OF/OX. There are classical papers in the literature [31,32] where the domain wall picture has been applied to the prediction of coercivity and magnetization curve shapes, starting from some assumption about the properties of F(X). Equations (5.264)-(5.267) provide a general solution for the case where the process OF/OXis Markovian, continuous, and homogeneous. In particular, the proven equivalence of Markovian disorder to the Preisach model gives a sound statistical interpretation of the latter.

References 1. S. H. Charap, Magnetic viscosity in recording media, J. App1. Phys. 63(6), 2054-2057 (1988). 2. R. Street and J. C. Wooley, A study of magnetic viscosity, Proc. Phys. Soc. London A 62(357), 562-572 (1949). 3. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, Berlin (1991).

602

CHAPTER 5 StochasticAspects of Hysteresis

4. I. D. Mayergoyz and C. E. Korman, Preisach model with stochastic input as a model for viscosity, J. Appl. Phys. 69(4), 2128-2134 (1991). 5. I.D. Mayergoyz and C. E. Korman, On a new approach to the modeling of viscosity in hysteretic systems, IEEE Trans. Mag. 27(6), 4766-4768 Part 2 (1991). 6. I. D. Mayergoyz and C. E. Korman, The input dependent preisach model with stochastic input as a model for aftereffect, IEEE Trans. Mag. 30, 4368-4370 Part I (1994). 7. V. M. Tobin, S. Shultz, C. H. Chan, and S. B. Oseroff, Magnetization time decay in CrO2 tape, IEEE Trans. Mag. 24(6), 2880-2882 (1984). 8. J.A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and minor loops-models and experiments, Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences 386(1791), 251-261 (1983). 9. G. Friedman and I. D. Mayergoyz, Stoner-wholfarth hysteresis model with stochastic input as a model of viscosity in magnetic-materials, IEEE Trans. Mag. 28(5), 2262-2264 Part 2 (1992). 10. C. E. Korman and P. Rugkwamsook, Identification of magnetic aftereffect model parameters: Comparison of experiment and simulations, IEEE Trans. Mag. 33(5), 4176-4178 Part 2 (1997). 11. I.D. Mayergoyz, A. A. Adly, C. Korman, M. W. Huang, and C. Krafft, Scaling and data collapse in magnetic viscosity, J. App1. Phys. 85(8), 4358-4360 Part 2A (1999). 12. N. Goldenfeld, Lectures on Phase Transmitions and the Renormalization Group, Addison-Wesley, Reading, MA (1992). 13. I. D. Mayergoyz, C. Serpico, C. Krafft, and C. Tse, Magnetic imaging on a spin-stand, J. Appl. Phys. 87(9), 6824-6826 Part 3 (2000). 14. I. D. Mayergoyz, C. Tse, C. Krafft, and R. D. Gomez, Numerical technique for integration of the Landau-Lifshitz equation, J. Appl. Phys. 89(11), 6991-6993 Part 2 (2001). 15. C. Tse, D. I. Mircea, I. D. Mayergoyz, P. Andrei, and C. Krafft, Spatial and vectorial characterization of thermal relaxation using the spinstand imaging technique, J. Appl. Phys. 91(10), 8846-8848 Part 3 (2002). 16. I.D. Mayergoyz and C. E. Korman, The Preisach model with stochastic input as a model for aftereffect, J. Appl. Phys. 75(10), 5478-5480 Part 2A (1994). 17. C.W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, Berlin (1983).

REFERENCES

603

18. C.E. Korman and I. D. Mayergoyz, Preisach model driven by stochastic inputs as a model for aftereffect, IEEE Trans. Mag. 32(5), 4204-4209 Part I (1996). 19. C. E. Korman and I. D. Mayergoyz, Review of Preisach type models driven by stochastic inputs as a model for after-effect, Physica B 233(4), 381-389 (1997). 20. M. I. Freidlin and I. D. Mayergoyz, Rotationally symmetric solutions of the Landau-Lifshitz and diffusion equations, J. App1. Phys. 87(9), 5511-5513 Part 2 (2000). 21. H. Cramer and M. Leadbetter, Stationary and Related Stochastic Processes, Wiley, New York (1967). 22. M. I. Freidlin and A. D. Wentzell, Diffusion-processes on graphs and the averaging principle, Ann. Prob. 21(4), 2215-2245 (1993). 23. M. I. Freidlin, Markov Processes and Differential Equations: Asymptotic Problems, Birkh~iuser-Berlin, Berlin (1996). 24. M. I. Freidlin, I. D. Mayergoyz, and R. Pfeiffer, Noise in hysteretic systems and stochastic processes on graphs, Physical Review E 62(2), 1850-1855 Part A (2000). 25. L. Gammaitoni, P. H~inggi, P. Jung, and F. Marchesone, Stochastic resonance, Review of Modern Physics 70(1), 223-287 (1998). 26. I.D. Mayergoyz and M. Dimian, Analysis of spectral noise density of hysteretic systems driven by stochastic processes, J. Appl. Phys. 93(10), 6826-6828 (2003). 27. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin (1998). 28. M.I. Freidlin and I. D. Mayergoyz, Hysteresis in randomly perturbed fast dynamical systems, Physica B 306(1-4), 15-20 (2001). 29. M. I. Freidlin and I. D. Mayergoyz, Modeling of temperaturedependent hysteresis within the framework of dynamical systems, J. App1. Phys. 91(10), 7640-7642 Part 2 (2002). 30. M.I. Freidlin, On stochastic perturbations of dynamical systems with fast and slow components, Stochastic and Dynamics 1(2), 261-281 (2001). 31. G. Bertotti, I. D. Mayergoyz, V. Basso, and A. Magni, Functional integration approach to hysteresis, Physical Review E 60(2), 1428-1440 Part A (1999). 32. L. Ne61, Theories des lois d'aimantation de Lord Rayleigh, Cah. Phys. 12, 1-20 (1942). 33. H. Kronmuller, in Magnetic Hysteresis in Novel Magnetic Materials, Kluwer, Dordrecht (1997).

CHAPTER 6

Control of Systems with Hysteresis Stavros A. Belbas, Mathematics Department, University of Alabama, Tuscaloosa, AL 35487-0350, USA

Contents 6.1 6.2

Introduction Optimal Control of Systems with Simple Hysteretic Elements 6.3 Optimal Control of Systems with Hysteron-Type Operators 6.4 Optimal Control of Systems with Preisach Hysteresis 6.5 Hysteresis from Optimal Control, and Hysteresis for Optimal Control References

605 607 614 622 632 634

6.1 I N T R O D U C T I O N Classical control theory deals with systems described by ordinary differential equations that involve a control function. The problems of classical control theory are concerned with obtaining specified performance of the solution by choosing an appropriate control function, or by choosing a control strategy in the form of a feedback operator, or some sort of decision rule based upon total or partial observations of the current state of the system, as well as on other types of information. It has been understood that realistic systems cannot always be described by smooth and memoryless models, like systems of ordinary differential equations, but they can, and in many cases must, include operators with discontinuities, discrete control actions coupled with continuously evolving state and continuous controls, and memory terms like Volterra integral terms or general causal functional terms. It is an ongoing challenge to extend the results and methods of smooth and memoryless control systems to systems with discontinuities and memory. The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

606

CHAPTER 6 Control of Systems with Hysteresis

Among such systems, hysteresis plays a prominent role, because of its ubiquitous appearance in many applications, including modern actuator and sensor technology, hysteretic modeling of certain mechanical systems like systems with friction or systems with rheological properties that exhibit hysteresis, in environmental problems like groundwater motion and contaminant transport with hysteresis in the relationships between quantities of hydraulic significance during wetting and de-wetting of a porous medium, hysteresis in biological systems, in economic systems, etc. For applications to controlled ODEs, with hysteresis modeling of mechanical systems, see [1]. For rheological models with hysteresis, see, e.g., [2]. For applications to groundwater models, see, e.g., [3-5]. For applications to the modeling of actuators in elastic systems, see [6]. For applications to 'smart materials', see [7,8]. For identification problems (which are closely related to control problems), see [9,10]. Hysteresis exhibits these characteristics in the context of optimal control theory: it involves discontinuities and non-smooth nonlinearities; it has memory effects (in the case of Preisach hysteresis and its extensions and generalizations); and it involves set-valued controlled systems (for Preisach and related operators), since the evolution of a hysteretic nonlinearity depends on the evolution of the active set (relays that are in the 'on' position) on the Preisach plane, and consequently the study of optimal control problems for such systems involves set-valued analysis. Just like most branches of mathematics and the exact sciences, the study of control systems with hysteresis nonlinearities has a prehistory and a history. The prehistory is not easy to trace, but we can mention, for example, the classical older works on nonlinear controlled systems, that typically include some techniques and results for the analysis of systems with hysteretic relays. A typical representative of such works is the classical book of Popov [11] on nonlinear control systems. The history of control of systems with hysteresis is, in our opinion, easy to trace. We take as a starting point the work of Krasnosel'skii and Pokrovskii, which is described in their monograph [12], but which was also available, in journal articles, before the publication of [12]. We consider as 'history' of the control of hysteresis the works that appeared after the work of Krasnosel'skii and Pokrovskii, which make use of results, concepts and methods that are either available in [12] and related works or were developed in works following the impetus of [12], and also make use of the essential ideas of optimal control theory, namely, the extremal principle (usually called 'maximum principle') of Pontryagin, Boltyanskii, Mishchenko, and Gamkrelidze and Bellman's method of dynamic programming for systems of ordinary differential equations with hysteresis nonlinearities. With these stipulations, it is easy to pin-

607

6.2 OPTIMALCONTROL OF SYSTEMS

point the beginning of the history of the control of hysteresis, it is the work of Brokate [13] on control of ODE systems with hysteresis nonlinearities. Naturally, there is a 'gray area' in the transition from prehistory to history, but any further historical inquiry would be beyond the scope of this introduction. Subsequent contributions to the optimal control of systems of ordinary differential equations with hysteresis, and their discrete-time analogs, include the works [3,14-22]. In particular, [14] addresses the question of existence of optimal controls for systems with hysteresis. A number of other authors have worked on particular cases of systems with hysteresis. The mathematics of hysteresis is covered mostly in [12], [23] and its precursor [24], and [25-27]. Additional, state-of-theart information can be found in [28,29]. Early works include [30-34]. The references given in this work are not presented as an exhaustive or nearly exhaustive bibliography on control of hysteresis; rather, they represent the works (among those works that are known and available to us) that seem to us relevant to the specific issues that we present in this report. The type of optimal control problems that arise in systems with hysteresis nonlinearities belong to the category of systems with both continuous state and discrete changes in system behavior, i.e. to the general category of hybrid and discontinuous systems [35-37]. However, the specific properties of hysteresis nonlinearities are such that previously existing methods are not applicable, and a direct study of such systems is necessary.

6.2 O P T I M A L C O N T R O L O F S Y S T E M S HYSTERETIC ELEMENTS

WITH

SIMPLE

As is well known, the standard Preisach model of hysteresis involves a weighted superposition of a family of rectangular relays. Every one of these rectangular relays is a 'primitive' element in that it cannot be further decomposed into simpler elements. We use the term 'simple hysteretic elements' to refer to operators that cannot be reduced to simpler terms and, at the same, have been traditionally labeled as 'hysteresis' in the optimal control literature. 6.2.1 CONTROL OF A SINGLE RELAY

First, we consider a simple rectangular element R ~/~, an operator that depends on two real-valued parameters ~ and fl that satisfy a > ft. The effect of R a~ on a continuous input signal z(.) is defined by the conditions:

9 (R~flz)(t) is continuous at points t where ~ ~- z(t) ~ fl;

CHAPTER 6 Control of Systems with Hysteresis

608

9 (Ra~z)(t) takes the values 0 or 1, and is a right-continuous function 9

9

that has a limit from the left at every point t; if, at some time-instant t, (R~z)(t -) - 0andz(t) = ~, then (R~z)(t) (R~z)(t +) - 1; if, at some time-instant t, (Ra~z)(t -) - I andz(t) - t, then (Ra~z)(t) -

(Ra~z)(t +) - O. Naturally, the definition of (R~z)(t) requires an initial value, 0 or 1, at some initial time to in the case z(to) ~ (fl, ~). Therefore, (R~z)(t) is really (R~z)(t; to, v0), defined for t>~to, with (R~z)(to; to, vo) = vo. Unless we specifically w a n t to emphasize this dependence of (Ra~z)(t) on some initial value, we shall suppress the variables to and v0, and we shall write simply

(Ra~z)(t). This operator R ~/~ is usually called a non-ideal relay. A different but equivalent definition is given in [12], where also several properties of nonideal relays are established. We consider a controlled ordinary differential equation that includes a simple hysteresis operator acting on the control: dx(t)

dt

=fit, x(t), u(t), (R~u)(t)); x(to) = xo.

(6.1)

The control u is one-dimensional, and it is postulated to be a continuous function. We consider the problem of minimizing a cost functional of the form: J := fo(x(T)) 4-

f;

F(t, x(t), u(t), (R~u)(t)) dt.

(6.2)

We seek the appropriate set-up for obtaining dynamic p r o g r a m m i n g equations for this model. Clearly, the quantity (R~u)(t) takes values in the set {0, 1}, and the admissible values of u(t) that are consistent with the values of (R~u)(t) are: if (R~u)(t) = 0, then u(t) ~ ( - ~ , a); if (Ra~u)(t) = 1, then u(t) ~ (fl, +oo).

(6.3)

We set U0 := ( - ~ , ~), U1 " - (fl, 4-(X)). We parameterize the system (6.3) by initial time, initial state, and initial value of R~u; thus, we consider the parameterized system:

3x(t, s, ~, v) =fit, x(t, s, ~, v), u(t), (Ra~u)(t)), for s < tKT; Ot x(s +, s, ~, v) - ~; (R~u)(s) - v ~ {0, 1}; u(s) ~ Uv.

(6.4)

6.2 OPTIMAL CONTROL OF SYSTEMS

609

We define the parameterized cost functional

Is T F(t, x(t, s, ~, v), u(t), ( R ~ u ) ( t ) ) dt.

J(s, ~, v" u(.)) " - Fo(x(T, s, ~, v)) +

(6.5) subject to the same initial conditions at time s as in (6.4). The HamiltonJacobi functions (value functions) are denoted by Wv, v E {0, 1} and they are defined by Wv(s, ~) = inf l(s, ~, v; u(.)). (6.6) u(.):u(s)~uv Then a formal application of the principles of dynamic programming yields the following equations for Wv: min

inf ~Uv

C~Wv(S,~) + ~s

f(s, ~, a, v) + f(s, ~, a, v) ,

~

W v ( S - ~ ~) -- W(vq_l)mod2(Sq-~

r

-- 0;

W v ( Z , ~) -- Fo(r

v E

{0~ 1}. (6.7)

The symbol (v + 1)mod 2 refers to arithmetic modulo 2, i.e. (2k)mod 2 = 0, (2k + 1)mod 2 = 1. Another approach is possible. If the controls that are used for the derivation of the dynamic programming equations are taken as absolutely continuous functions, then we can use the current value of the control as part of the state, and the derivative of the control as a new control variable; thus the state dynamics equation (6.4) is augmented with u(t) -- u(t; s, a, b); cgu(t; s, a, b)

0t

= p(t); u(s; s, a, b) - a; p(s) = b.

(6.8)

The corresponding Hamilton-Jacobi functions will now depend on three variables (in addition to the index v): Wv = Wv(s, ~, a). Again, the principle of dynamic programming gives the following equations: inf / c~Wv(s, ~, a) cgWv(s, ~, a) ~gWv(s, ~, a) b b . -~s + c9~ f(s, ~, a, v) + cga +F(s, ~, a, v) } - 0; a ~ Uv, v - O, 1; W0(s-, ~, a-) = Wl(S +, ~, a); Wl(S-, ~,/1+) = W0(s +, ~, fl).

(6.9)

610

CHAPTER 6 Control of Systems with Hysteresis

6.2.2 A VARIANT OF A SINGLE RELAY

Thus far, we have used a relay that operates on the principle that a switching may occur when the input reaches the boundary of a certain set, namely, the set (-cx~, ~), or the set (fl, +cx~). This can be formalized by defining the access time, subsequent to time t, of the signal's u (.) reaching the boundary of an open set I, say r(t, u,/), by r(t, u,/) "= sup{s 9s > t, and u(s1) E IVs1 E (t, s)}.

(6.10)

Clearly, the definition (6.10) is non-trivial only when u(t) ~ I. According to the definition of the operator R ~/~, if, at some time t, u(t) < a and (R~u)(t) = 0, then a switching in the value of Ra~u will occur, for the first time after time t, at the time r(t, u, ( - ~ , a)), and, similarly, if u(t) > fl and (R~u)(t) = 1, then the first time, after t, that Ra~u switches values is the instant ~(t, u, (fl, + ~ ) ) . There is, however, another related but different operator, which we shall denote by Qa/~. This operator is based on the exit times r~(t, u, I), defined by rift, u, I) := inf{s : s > t and u(s) r I}. (6.11) In a way analogous to the case of access times, the definition of the exit times is also non-trivial only when u(t) ~ I. The operator Q~/~ is defined in a way analogous to R aB, by substituting exit times in lieu of access times; thus, if, at some time t, u(t) < a and (Q~u)(t) = 0, then a switching in the value of Qa~u will occur, for the first time after time t, at the time rift, u, (-cx~, a]), and also, if u(t) > ]/and (Q~u)(t) = 1, then the first time, after t, that Qa~u switches values is at the exit time r~(t, u, [fl, +cx~)). We consider the same system dynamics as (6.1) and the same cost functional as in (6.2), but with Qa]~ instead of R a]~. Then the dynamic programming equations become quite different from those that correspond to the relay R ~/~. In this case, it seems appropriate to increase the regularity of the control functions that are used in the derivation of the dynamic programming equations, and to require those control functions to be twice continuously differentiable. Thus, the controlled system becomes dx(t)

dt

= fit, x(t), u(t), (Q~u)(t))~

du(t) = p(t); dt dp(t) = q(t). dt

(6.12)

6.2 OPTIMAL CONTROL OF SYSTEMS

611

The Hamilton-Jacobi functions Wv are now parameterized by s, ~, a, b, which have the same meaning as before, except that now b, representing the value of the state component p(t) at t = s, is part of the variables inside Wv. The possible values of q(s) will be denoted by c. For definiteness we shall assume that p(.) takes values in an open interval Ip, and q(.) takes values in a closed interval Iq. According to the definition of Q~fl, a switching in the values of Q~flu takes place if (i) u(t) - ~ and p(t) > 0, or (ii) u(t) = fl and p(t) < 0. The dynamic programming principle now gives inf

+

c~Wv(s, ~, a, b) 4- cgWv(s, ~, a, b).f(s, ~, a, v) cgWv(s, ~, a, b) I c~b q + F(s, ~, a, v) J - 0, for a ~ Uv, b ~ Ip"

Wv(T, ~, a, b) = F0(~); W0(s-, ~, ~-, b) = Wl(S +, ~, a, b) for b > 0; Wl(S-, ~, fl+, b) = W0(s +, ~, fl, b) for b < 0.

(6.13)

Two remarks are in order about the system (6.13). First, at points where ~Wv(S,~,a,b) # 0, the dynamic programming equation gives optimal controls cgb of 'bang-bang' type, i.e. a minimizer of the Hamiltonian (the expression in curly brackets in the first equation in (6.13)) takes only the extreme possible values of q, the maximum value (say, qmax) if ~Wv(S,~,a,b) < 0, and cgb the minimum value (say, qmin) if OWv(s,~,a,b) ~b > 0. Second, a slight change in the definition of the relay, from switching at access times to switching at exit times, increases the dimensionality of the dynamic programming equations by inserting the additional variable b in the Hamilton-Jacobi function. 6.2.3 MULTIDIMENSIONAL SYSTEMS

The above treatment can be extended to the case of systems with multidimensional state and multidimensional control, with simple hysteretic elements acting on 'observation states' (to be defined below), which are functions of both the original state and the control. The resulting dynamic programming equations are based on an extended state that includes the original state, the original control, and certain derivatives of the 'observation state'. We consider the following system of controlled ordinary differential equations: dx(t) dt =fit, x(t), u(t), (Q~y)(t), (R~z)(t));

CHAPTER 6 Control of Systems with Hysteresis

612

y(t) = g(t, x(t), u(t)); z(t) = h(t, x(t), u(t)); x(to) = x0; x(t) = (x i(t) 9 l ~ i ~ n ) , f ( t ,

x0 = (x~ 9 l ~ i ~ n ) ; u 1

x, u, Q, R)

(u i .

= ~ i (t, x , u ,

Q, R) 9 l K i K n ) ;

l~i~m);

y(t) = (yJ(t) " I K j K M ) ; g(t, x, u) = (gJ(t, x, u) " I<~jKM): = (aJ'IKj~M):

1KjKM); aJ > ]~u

fl-(/~"

Q~fl y _ (Qd ~ yj . 1 Kj ~ M); z(t) - (zJ(t) 9 1KjKN): h(t, x, u) = (hJ(t, x, u) 9 I ~ j ~ N ) ;

= (?J" 1KjKN); ~ - ( ~ J " I~jKN); ~ > 6JYj; (6.14)

R ~ 6 z - (R~JYz 9 I K j K N ) . We set

p(t) "=

cgu(t) 0t

We call the functions y(.) and z(.) the observation states, by analogy with a similar situation in the theory of partially observable systems, even though, in the present context, they do not have the meaning of actual observations. We shall need the time-derivatives of y and z; these are easily calculated

as c~yJ( t ) cgt

cgt

cggJ + ~cggJ - c f k (t x, u Q , R ) + ~T pk(t) UU ~ dX,~-

cgzJ(t) c~hJ cghJ k (~hJ ~pk cgt = cgt + -~xkf ( t ' x ' u ' Q ' R ) + -~uk

~ ( t , x, u, p Q, R) ;

(t) =_ pJ (t, x, u, p, Q, R).

(6.15) In (6.15) above, we use strictly the tensor convention of summation over repeated upper and lower indices, and we have suppressed some of the variables to avoid messy-looking formulas. The functional that is to be minimized is given by l " - Fo(x(T)) + ~0 T F(t, x(t), u(t), (Q~fly)(t), (R?6z)(t)) dt.

(6.16)

We shall use two multi-indices v = ( v J ' l ~ j K M ) ~ {0, 1}M, w - ( w J : l ~ j ~ N ) ~ {0, 1}N; the symbols v and w represent the possible values of (QC~fly)(t),

6.2 OPTIMAL CONTROL OF SYSTEMS

613

and (R~&z)(t). The symbol (1)j will denote the M- or N-dimensional array (M or N, according to the context) with the jth entry equal to I and all other entries equal to 0. The Hamilton-Jacobi function contains the variables s, ~, r/, f, a, b, which represent the values of t, x(t), y(t), z(t), u, p, respectively, and is also indexed by two subscripts v and w, which represent the values of (Q~fly)(s), (R~flz) (s), respectively. We set

M Nx (Uvi

Uvw "= x

j--lj'=l

x Uj).

(6.17)

/,/int,~int

The symbols signify limits of a quantity, depending on r/', f', as 01', f') --* (r/, f) while (r/', f') ~ Uvw. For 01, f) ~ c~Uvw, we shall d e n o t e b y BvwO1, f) the set of those values of b = p(s) that guarantee exit of y; from Uvj, provided yJ (s) ~ CgUvj; those values of b are characterized by the condition pJ > 0 if vJ = 0 and vJ < 0 if vJ = 1; it is clear that Bvw(rl, f) also depends on all the other variables that appear inside Wvw, but we do not explicitly show that dependence. Finally, we shall denote addition modulo 2 by the symbol @2. The dynamic p r o g r a m m i n g equations are:

~,~f J cgWvw(S, ~, r/, f, a, b) 4c' / c~s

+

4-

~Wvw(S, ~

""

.)

(~Wvw(S, .~.... ).fi (s, ~, a, v, w)

q)J(s,~,a,b,v,w)+

~Wvw(S, ~

""

.)

pJ(s,~,a,b,v,w)

OWvw(S, ~.... ) bk -Jr-OWvw(S, ~.... ) ck 4- F(s, ~ a v, w) } - 0 Oak

cgbk

' ,

for (r/, f) ~ Uvw;

Wvw(S-, ~, //int ~int, a, b) - Wv, w,(S +, ~, 11, f, a, b) if (r/, f) ~ cgUvw and b ~ Bvw(r !, f),

where v " - v •a (j ~ 2 (1)j) , w' - w Oa (j (~E)2 (1)j) :rlJE((gUvw)J :zJE(~Uvw)J Wvw(T, ~, ri, f, a, b) - Fo(~); ri - g(s, ~, a), f - h(s, ~, a).

(6.18)

614

CHAPTER 6 Control of Systems with Hysteresis

We note that certain simplifications are possible if the operator Q ~ is omitted from the state dynamics, or if the hysteresis operators depend only on the state x but not on the controls u. The problems we have examined up to this point are related to the' exit time problems' of optimal control; however, the problems above are different from exit time problems in a substantial way. In exit time problems, the exit event depends only on the behavior of the state and its time-derivative; in the problems we have examined up to this point in this section, the exit events depend on the choice of the controls, i.e. it is a part of the control policy to decide whether a switching (exit from a set Uvw) will take place. The papers [15-17] deal with optimal control problems that are closer to the standard exit time problems of classical optimal control. In those problems, the relays depend only on the state, but not explicitly on the control.

6.3 O P T I M A L C O N T R O L O F S Y S T E M S W I T H HYSTERON-TYPE OPERATORS For the purposes of this exposition, we use the non-standard terminology 'hysteron-type operators' to denote a class of hysteresis operators that includes the so-called 'stop' and 'play' operators, and are akin to the 'hysteron' nonlinearity of [12]. This class of simple hysteresis operators can be defined in a general way as described below, with possible additional conditions for specific hysteresis models. 6.3.1 A CASE OF MARKOVIAN HYSTERESIS

We consider one-dimensional input signals, say z(t), OKtKT, and onedimensional output, say w(t), O~t~T. On the zw-plane, we consider an open domain f2 whose boundary, 0f2, consists of two connected nonintersecting arcs, which we conventionally denote by (c~), (Co). The open set f2 may be bounded or unbounded. There are two families of curves, parameterized by a one-dimensional parameter p, say (c~), (c~), 0 < p < 1, with the property that, for every point P in f2, there exists a unique + p+(P) ~ (0, 1) and a unique p-(P) ~ (0, 1), such that both c u r v e s (Cp+(p)) and (cp_(p)) contain the point P. Each of the curves (c~), (Co), (c~), (Cp), can be described as w = g(z) for some continuous function g; g depends on p and on the superscript + or - , thus we have the functions g~, go, g+, gp" The effect of the hysteresis operator is that, as long as the input signal z(t) takes values in the projection of h onto the z-axis, the response w(t)

6.3 OPTIMAL CONTROL OF SYSTEMS

615

is such that the point (z(t), w(t)) moves on a curve with a superscript ' + ' dz(t) < 0. However, if _~(t d t_) > 0, and on a curve with a superscript ' - ' if --a/this description is not limited to differentiable inputs; if z(.) is piecewise monotone, then (z(t), w(t)) moves on a curve with superscript ' + ' when z is non-decreasing, and on a curve with superscript ' - ' when z(.) is nonincreasing. The definition of the associated hysteresis operator clearly also depends on some initial conditions, as was the case for the relays of Section 6.2. Sometimes, a hysteresis operator of the 'hysteron- type' class is extended, by a substitution operator, to the exterior of the domain f~, i.e. w(t) = h(t, z(t)), for some function h, when z(t) falls outside the projection of h onto the z-axis; such extensions are described in [38], and also have been used in [19 ]. A hysteresis operator of 'hysteron type', as described above, is Markovian in the sense that knowledge of the current value of input and output, and knowledge of the time-derivative of the input, implies knowledge of the forward time-derivative of the output. Therefore, the associated optimal control problems fall within the scope of dynamic programming methods. Here, we shall outline the dynamic programming equations for a controlled system with a hysteron-type operator acting on the control. We shall follow the ideas of [19] but we shall present the optimality conditions for a continuous-time problem, whereas the problem in [19] was a discrete-time problem. We consider a controlled ordinary differential system dx(t) = f i t , x(t), u(t), (Hu)(t)); x(O) = xo dt

(6.19)

where the state x(.) is n-dimensional and the control u(.) is one-dimensional (i.e. a real-valued function). The objective is to minimize a cost functional J of the form

J "- Fo(x(T)) 4- ~0 T F(t, x(t), u(t), (Hu)(t)) dt.

(6.20)

The Hamilton-Jacobi function depends on four variables: s, the current time; ~ = x(s), the current state; a = u(s), the current value of the control; and b = (Hu)(s), the current value of the hysteretic function of the control. The formulation of the dynamic programming equations is facilitated by introducing some assumptions and notation. We assume that each arc (c~) (i.e. the closure of (c~))intersects both curves (c~-) and (Co) transversely. We denote by (Z#io, W#io) the points of intersection of (c~) with

CHAPTER 6 Control of Systems with Hysteresis

616

(c~). We denote by (cpi0) the following sets: (c~0) "= {(z, w) ~ h ' ( z , w) ~ (c~)or [z>~Z-~oand (z, w) ~ (c~)]}; (c~-0) "= {(z, w) ~ h ' ( z , w) ~ (c;) or [z<.Z-~o and (z, w) ~ (Co)]}.(6.21) For consistency of notation, we set

Co0)-= Co)

(6.22)

We extend the functions g~,_ 0 < p < I to functions gp~0_defined on (c~0)ras follows: +

for z < Zoo +r for z)Zpo '

g,+o(z)= { _

gp0 (z) =

{

g

go (z)

I,

for z > z~0 for zKZ-~o "

(6.23)

For consistent notation, we also set g~0 - g ~

(6.24)

With this notation, the dynamic programming equations are:

~W(s, 3s

a,

b)

inf [V~W(s , ~, a, b)f(s , ~, a', g++(a,b),O(a')) + min /|a':a'>~a

-t-F(s, ~, a', gp+(a,b),O(a'))], inf [V~W(s, ~, a, b)f(s, ~, a", gp_(a,b),O(a"))

a" :a" ~
+F(s, ~, a', gp_(a,b),O(a'))l

W(O, ~, a, b) = Fo(s, ~).

}= 0 (6.25)

The first equation in (6.25) holds for (a, b) ~ ~. The two infima in (6.25) are taken over a', a" in the projection of f~ on the z-axis and also satisfying the indicated inequalities a'>~a, a"~
6.3 OPTIMAL CONTROL OF SYSTEMS

617

6.3.2 BROKATE'S OPTIMAL CONTROL PROBLEM AND ITS CONNECTION TO THE METHOD OF DYNAMIC PROGRAMMING

Brokate [13,39] initiated the study of optimal control problems of ODE systems with hysteresis nonlinearities by considering a system with the hysteron-type operator acting only on the state. Brokate's model fits into the framework of the problems introduced above in Section 6.3.1. The hysteresis operator is defined within a set h on a zw-plane, if z denotes the input signal and w the output of the hysteresis operator. Instead of parameterizing the interior curves (c~) by a real parameter p, as we have done in Section 6.3.1, Brokate describes those curves as families of solutions of two ordinary differential equations. Now, this is just a matter of the manner of description of those curves. In principle, given a family of curves (Cp) described by w - ~p(z, p) (6.26) we can calculate the derivative of w with respect to z along the curves (Cp) as

dw d z (Cp)

=

0~p(z, p) Oz

(6.27)

and then eliminate p between (6.26) and (6.27) to come up with the differential equation dw - ~(z, w). (6.28) dz Conversely, the general solution of a differential equation like (6.28), under standard conditions on the function 6, is a one-parameter family of curves. We shall describe a model that is, in some respects, more general than Brokate's model, but in some other respects less general, and we shall do so in a way consistent with our notation. An open set f2 on the zw-plane is bounded by two curves (c~) given by equations w - ~p~(z) (6.29) and such that r > q0o (z)Vz. Through every point (~, o9) in f2, two curves (c+(~, o9)), (c-(~, o9)) are defined as solutions of the initial value problems: dw = ~+ (z, w); WIz=~ - o9. (6.30) dz It is assumed that each problem in (6.30) has a unique solution, for every initial point (~, o9) in f2, and that a solution exists forwards and backwards in z, and it gives an arc in h extended from some point on (Co) to some point on (c~-).

CHAPTER 6 Control of Systems with Hysteresis

618

For a piecewise monotone continuous input signal, the hysteretic output w(t) = (Hz)(t) is defined by the requirement that the point (z(t), w(t)) fl moves on one of the curves with superscript '+' when z(.) is increasing, and one of the curves with superscript ' - ' when z(.) is decreasing. Now we consider an ODE system dx(t) dt =fit, x(t), y(t), u(t)); x(0) = x0;

y(t) := (H(r~))(t), (~(t) : - g(t, x(t)). (6.31) Here, x(.) takes values in IR n, u(.) takes values in a compact subset U of IR m, and cr and y are real-valued functions, and H is the hysteron operator defined above. The objective of the optimal control problem is to minimize a functional J given by J "= f0(x(T), y(T)) +

J0' f(t, x(t), y(t), u(0) dt.

(6.32)

This problem can be treated by the method of dynamic programming. We assume that the function g is continuously differentiable with respect to all its variables. Then, for x in f~, the evolution of y(.) can be described by dy(t) = ~,+(g(t x(t)) y(t))

dt

'

'

[ 3g(t,3tx(t)) + Vxg(t, x(t))f(t, x(t), y(t), u(t))] + -~,- (g(t, x(t)), y(t))

l

Og(t, x(t)) + Vxg(t x(t))f(t, x(t), y(t), u(t)) 3t

l-

(6.33)

where, for any real number r, r + := max(r, 0), r- := m a x ( - r , 0). We parameterize the problem by initial time s and initial values ~ : x(s) and r/:= y(s). The corresponding Hamilton-Jacobi function is W(s, ~, r/). At each point (s, ~, r/), we partition the set U of admissible control values into three mutually disjoint subsets Uo(s, ~, ~1., U1 (s, ~, r]), U-1 (s, ~, r]), defined by: ! Og(s, + V~g(s, ~)f(s, ~, ~l, a) = 0 } ; UO(S, ~, ~I) := {a U" / Os ! + V~g(s, ~)f(s, ~, r/, a) > 0 } ; UI(S, ~ t]) := {a U" '

l

OS

U-l (S, ~, r]) "-- {a ~ U" Og(S,os~) + V~g(s, ~)f(s, ~, r/, a) < 0 }. (6.34)

6.3 OPTIMAL CONTROL OF SYSTEMS

619

For brevity, we shall denote by # the expression ag(s,~) + V~g(s ~)f(s, ~, ~I, a) r and by 9~ - ~(s, ~, r/, (#)+, (~)-, a) the right-hand side of (6.33). Then the dynamic programming equations take the form: min

inf

i~{0,1,-1} a~Ui(s,~,rl)

{ c~W(s, ~, rl) + V~W(s, ~, rl)f(s, ~, rl, a) dS

+VnW(s, ~, ~)V/(s,~, ~, (#)+, (#)-, a) + F(s, ~, ~, a)}

- O;

(6.35)

W(T, ~, ~) = Fo (~, ~).

In addition, we need boundary conditions for W(s, ~, ri) on the boundary 0 f 2 - (c~)U(Co). For (g(s, ~), r/) ~ (c~-), we have r/ = q~(g(s, ~)),andfora U1 (s, ~, r/), where ~/- q~- (g(s, ~)), the total derivative (D/Ds) W(s, x(s), y(s)), evaluated at x(s) - ~, y(s) - q;-~(g(s, x(s)), is:

,~W(s,

~,~(g(s,~)))

8s + v,~ W(s, ~,~ (g(s,

(ag(s,as ~) +

+ V~W(s, ~, q~-~(g(s, ~)))f(s, ~, q~ (g(s, ~)), a)

,;)))(~-)' (g(s, ~))

V~g(s, ~)f(s, ~, qY~(g(s, ~)), a) ) .

(6.36)

On the other hand, for a ~ Uo(s, ~, rl) U U-1 (s, ~, r/), with r / - q~-(g(s, ~)), the total derivative (D/Ds)W(s, x(s), y(s)) is:

aW(s, ~, ~) + V~W(s, ~, rl)f(s, ~, ri, a) + V~W(s, ~, rl)~P(s, ~, rI, (#)+, ~s (6.37) (6)-, a) [ rl=r ~(s, ;)) 9 Consequently, the boundary conditions for W on (c~-) are: min{

inf

{ 0W(s, ~, q~- (g(s, ~)))

aEU1 (s, ~, q~-~(g(s, ~))

~S

+V~W(s, ~, q~-~(g(s, ~)))f(s, ~, q~-~(g(s, ~)), a)

+v~w(s, ~, G-(g(s, ,;)))(G-)'(~(s, ,;)) (ag(s, ~) 8s

+ V~g(s, ~)f(s, ~, q~ (g(s, ~)), a)

+F(s, ~, q~-~(g(s, ~)), a},

)

CHAPTER 6 Control of Systems with Hysteresis

620

a~Uo(s,r

inf

(S,r

{ OW(s, ~, ~1) + YeW(s, ~, ~l)f(s, ~, ~1,a)

r

+V,W(s, r rl)W(s, r rl, (~)+, (~)-, a) + F(s, r rl, a) } ],=r162

O.

}

(6.38) The boundary conditions on (co ) are obtained from (6.38) above by interchanging the roles of Ul(S, ~, r/) and U-l(S, ~, ~/) and replacing c#~ by cp0 9 The dynamic programming equations obtained above allow a formal derivation of Hamiltonian equations and an extremum principle akin to Pontryagin's m a x i m u m principle. We define three Hamiltonians:

hl (s, r r/, a, p, q) := pf(s, r rl, a) +q6+ (g(s, r rl) [ 3g(s'

+ Vcg(s' r

r rl' a)] + F(s' r rl' a)'

for a ~ UI(S, r r/);

h-l (S, r rl, a, p, q) "= pf(s, r rl, a) +q6-(g(s, r r/)[ 3g(s' r

+ Vcg(s, r

r rl, a)] + F(s, r rl, a),

for a ~ U-1 (s, r r/);

ho(s, r rl, a) "= pf(s, ~, 11, a) + F(s, ~, 11, a), for a ~ Uo(s, r ~1).

(6.39)

The variable p is an n-dimensional row vector; p plays the role of Vr W(s, ~, 11) and q plays the role of 3W(s, ~, ~1)/&l. Further, we define the minimized Hamiltonians,

hm,i(s, ~, rl, p, q) := hm(s, ~, rl, p, q) :--

inf aEUi(s,r

hi(s, ~, ~/, p, q, a), i = 0, 1, -1;

min hm,i(s, ~, rl, p, q). iE{0,1,-1}

(6.40)

Now, we assume that there is an optimal control function u*(t), OKtKT, with corresponding trajectory (x*(t), y* (t)), O~tKT. We set p*(t) := VxW(t, x*(t), y*(t));

q*(t) := 3W(t,x*(t), y*(t)) ~y

(6.41)

assuming existence and continuity of the indicated derivatives. Then we have, from (6.38), dp*(t) = -Vxhm(t, x* (t), y* (t), p* (t), q* (t)); dt

6.3 OPTIMAL CONTROL OF SYSTEMS

621

cghm(t, x*(t), y*(t), p*(t), q*(t)) cgy

dq* (t)

dt

hm(t, x* (t), y* (t), p* (t), q* (t)) =

min

a~Ui(t,x* ( t),y* ( t) )

hi(t, x* (t), y* (t), p* (t), q* (t))

= hi(t, x* (t), y* (t), u* (t), p* (t), q* (t)), for u* (t) ~ Ui(t, x* (t), y* (t)) (6.42) for OKt < T, (x*(t), y*(t)) ~ f2. On the boundary curve (c~), the Hamiltonian hi is replaced by h~where h + (s, ~, r/, a, p, q) := pf(s, ~, ~1,a)

1

+q(~)' C4(s,O) I ~g(s,_ds~) + V~g(s, ~)f(s, ~, ~1,a)~ + F(s, ~, rI, a) (6.43) and we define

h +, (s, ~, rI p, q) := 1

'

inf

a~Ul(S,~,rl)

h~- (s, ~, r/, a, p, q);

h + : - min{h +, h-i, h0}. (6.44) Then the Hamiltonian equations and the extremum principle (6.42) hold with hi, hm replaced by h~-, h +, for O~t < T, (x*(t), y*(t)) ~ (c-~). Analogous statements hold for OKt < T, (x*(t), y*(t)) ~ (Co). At the final time T, we have

p*(T) - VxFo(x*(T), y*(T)), q*(T) - cgFo(x*(T), y*(T)). cgy

(6.45)

In the particular case in which the function g(t, x) is independent of t, we can obtain a somewhat different form of the Hamiltonian equations. We define the Hamiltonians:

H(s, ~, ri, a, r) := rf(s, ~, rl, a) + F(s, ~, rl, a) for a ~ U; Hm,i(s, ~, rl, r) :=

inf

H(s, ~, ri, a, r);

aEUi(s,~,rl)

Hm(s, ~, ~1, r) := inf H(s, ~, rI, a, r). aEU

(6.46)

Here, the variable r plays the role of the 'total derivative' of W with respect to ~,i.e. theroleofD~W(s, ~, r/) - - V~W(s, ~, ~/)+ 0W(s,~,,) 0, 6+ (g(~), r/) V~g(~). If we set r*(t) := DxW(t, x*(t), y*(t)) (6.47)

622

CHAPTER 6 Control of Systems with Hysteresis

then we find, from (6.38), dr*(t)

dt

= -DxH(t, x* (t), y* (t), r* (t))

(6.48)

at points of differentiability of r* (t), and the extremal condition

H(t, x* (t), y* (t), u* (t), r* (t)) = Hm(t, x* (t), y* (t), r* (t))

(6.49)

dy at points of continuity of ~ with (x, y) in f2. The definition of r* implies the followingjump conditions at times tij at which u* (t~ ) ~ Ui(tij, x* (tij), y* (tij) ) and u*(t~) ~ Uj(tij, x*(tij), y*(tij)), i ~=j, with i and j taking the values 0, 1, and -1:

r* (t~) - r* (tT~) = 2ijVxg(x* (tij)); ,~ij .__ cgW(tij, x* (tij), y* (tij)) cgy [qJ(g(x* (tij)), y* (tij)) - 6~ (g(x* (tij)), y* (tij))]; 61 ~ ~y+~6 -1 ~ ~-~ 60 ~- 0.

(6.50)

The problem formulated and studied in [39] has g(t, x(t)) = bx(t) for a constant row vector b, so that Vxg(t, x) = b and jump conditions of the form (6.50) hold with the constant vector b in place of Vxg(x* (tij)). The problem in [39] involves additional ingredients such as a final time constraint (which modifies the final conditions on the co-state) and a change of the time scale. (The relationship between dynamic programming and the effect of final time constraints on the extremum principle has been analyzed in [37].) The problems in [13] contain more general hysterons, such as intersecting or coalescing boundary curves (c~).

6.4 O P T I M A L C O N T R O L OF S Y S T E M S W I T H P R E I S A C H HYSTERESIS Systems with Preisach hysteresis operators are the most crucial category of hysteretic control systems. The Preisach model of hysteresis, and certain variants of the standard Preisach model, have been used to model many kinds of physical systems with hysteresis. From the point of view of control theory, an important category is the control of elastic systems via actuators that exhibit hysteresis.

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

623

In this section, we shall make reference to, without repeating all the details of, the definition and basic properties of the Preisach hysteresis operator. We shall take for granted the information contained in the essential reference [23], and we shall also make reference to other works that contain results about the Preisach operator and systems that include Preisach hysteresis nonlinearities. The Preisach hysteresis operator has global memory, it stores information about the entire history, from some initial time to the current time, of the dominant local extrema of the input signal. For this reason, it is clear that the use of dynamic programming arguments necessitates the extension of the original state space to a substantially larger space, so as to include, in the extended state, relevant information about the past history of the input signal. It turns out that a suitable way to produce a Markovian model (in a substantially broader space than the original state space) is to formulate the Preisach operator in terms of the active set on the Preisach plane, and to formulae the controlled dynamical system as a system with a set-valued state. This creates some conceptual questions, because (i) we need to define the concept of a derivative of a set-valued function of time, and (ii) the Hamilton-Jacobi function, as a function of the extended state, is a real-valued set function, not necessarily additive, and we need a suitable concept of derivatives for non-additive set-functions. The concept of derivatives for set-valued functions that we introduce here, and the almost identical concept introduced in [21], are new concepts, different from other definitions of derivatives of a set-valued functions. The one paper that is conceptually somewhat close to our treatment is [40]. It is expected that the concept of a set-valued state may be useful for more general systems with a set-valued state, and not only for Preisach hysteresis. The papers [15-18], and particularly [17], deal with controlled hysteretic systems with hysteresis acting on the state, and the analysis is carried out from the point of view of 'viscosity solutions', a form of generalized solutions for dynamic programming equations. We remark that, some cases of Preisach operators with a singular measure, instead of a continuous function, supported on a monotoneconnected curve on the Preisach plane, such as the version of StonerWohlfarth models introduced by G. Friedman [41], result in a dynamical system description of the corresponding optimal control problems, with one-dimensional (rather than set-valued) state, which can be treated by easier methods than the set-valued state evolution that we consider here. The double and multiple extensions of the Preisach model, introduced in [42,43], can be treated by the method described in [20], which we shall not repeat here.

CHAPTER 6 Control of Systems with Hysteresis

624

6.4.1 SET-VALUED DERIVATIVES

In this section, we shall give an account of an approach that is conceptually based on [21]. Our exposition below is technically different from [21], however. Let H denote a Preisach hysteresis operator with weight function w(a, fl), flo~fl < ~ 0 . This means that, for a continuous input z(.), the hysteresis output is

(Hz)(t)=/fp(~o,~o) w(a, fl)(R~z)(t) da dfl

(6.51)

where P(a0, fl0) := {(a,

fl)

~ Ia2 " ~0 ~0r > fl~/J0}

(6.52)

and each R ~/~is an operator of the type defined in Section 6.2. It is assumed that initial conditions for R~z are chosen so as to make (R~z)(s), at some initial time s, a measurable function. Now, if the input signal z(.) is piecewise monotonic, we assume that the specific construction given in [23] is used to define the hysteresis output Hz. This means that, if at some initial time to we have z(to) < rio, then all relays (R~z)(to), (~, fl) e P(~o, fl0), are in the output state 0; analogously, if z(to) > s0, then all relays (R~z)(to) are in the output state 1; for other values of z(t0), all relays (R~z)(to) with aKz(to) are in the output state 1, and all other relays, at time to, are in the output state 0. As z(t), t > to, increases or decreases, new relays are being set to the 1 or the 0 output state, producing, at each time instant t, the staircase curve that separates the values of (~, fl) ~ P(a0, rio) for which (Ra~z)(t) = I from those values for which (R~z)(t) = 0. The rule is this: if z(t) increases from z(tl) to z(t2), t2 > tl, i.e. if z(.) remains increasing for t ~ (tl, t2), then all relays with z(h) < a < z(t2) will have been switched from output state 0 (at time tl) to output state I (at time t2); similarly, if z(t) is decreasing for t2 < t < t3, then the relays that correspond to (a, fl) with z(tg)~fl < z(t2) will have been switched from output state I (at time t2) to output state 0 (at time t3). If we denote by S(t) the active set at time t, i.e.

S(t) "- {(~, fl) ~ P(~o, rio) " (R~z)(t) - 1}

(6.53)

then we plainly have

(Hz)(t) = / fs

(t)

w(~, fl)d~dfl.

(6.54)

Thus the hysteretic output Hz is completely characterized by the active set S(t). Consequently, a dynamical system that includes Preisach hysteresis

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

625

nonlinearities can be interpreted as a system in which part of the state is set valued, i.e. the active set S(t) becomes part of the state of a dynamical system. We are led, therefore, to consider dynamical systems with a set-valued state. Consequently, we need a suitable concept of derivatives of set-valued functions of one real variable. The research literature contains m a n y concepts of derivatives of set-valued functions. However, we need a concept of derivatives that is suitable for the situation that arises in Preisach hysteresis operators and we develop such a concept from first principles. Also, in anticipation of having to apply dynamic p r o g r a m m i n g arguments, we need a concept of derivatives of real-valued set functions, which m u s t have the property of compatibility with the derivatives of set-valued functions, in the sense that a version of a chain rule should hold. It will be seen that the appropriate concepts of derivatives can be extended to more general cases than just the Preisach hysteresis operator. Given a set-valued function, say Q(t), whose values are open sets in IR n, we represent the two set-theoretic differences Q(t+~t) \Q(t), Q(t) \ Q ( t + &t) as unions of simpler sets. The m a g n i t u d e of a set is generally measured via w h a t we shall call a gauge function g, i.e. a function defined on b o u n d e d open sets in Ia n and taking values in [0, +cx~), that has the following properties:

A c B =~ g(A)~g(B); g(Q) = g({x}) - O, Vx ~

I a n"

An ,[, A =~ g(An) --> g(A).

(6.55)

We note that, if we omit the condition g({x}) - 0, the remaining properties of g are the defining properties of Choquet capacities. The gauge function is used to measure the 'size' of a set. We shall say that a set Z(t, ~t) is og(l~tl ~) if g(Z(t,~t)) i~tl~ --+ 0 as ~t ~ 0. We have: DEFINITION6.4.1. We say that the set-valued function Z(t) is differentiable at t - to if, for all ~St sufficiently small, we have

Z(to + ~t)\Z(to) -

U (a + + A+(~St)U +) Uog(l~tI); i

Z(to)\Z(to + ~St) -

u (a~- + A-j-(~St)U~-) Uog(l~Stl)

/

(6.56)

CHAPTER 6 Control of Systems with Hysteresis

626

where the points a +, a7 and the sets U +, U~- are independent of ~t, and the matrices A + (St), A 7 (~t) satisfy det(A + (~t)) = O(l~tln), det(A 7 (~t)) -- O(l~tJn). The definition above parallels the definition of derivatives of an ordinary function. The usual requirement that an ordinary derivative should be a linear function of ~t is replaced by an affine 'function' of ~t, of the form a~ + A~(~t)U~. The sets U~ specify the 'shape' of the affine function, the matrices A~ (~t) specify the size of this affine function (where by 'size' we mean different sizing along different directions), and the points a~ specify the position of the affine set-valued function. In the case of derivatives of ordinary functions, the 'position' attribute of the derivatives is irrelevant, but for set-valued functions it is necessary to include this ingredient, and it is necessary to change the concept of derivative from a linear operator to an affine operator. Simple examples, which we shall present below, show that this is the minimal necessary adjustment we need to make to the concept of derivatives to make it applicable to set-valued functions. The requirement det(A~(6t)) = O(t~tl n) may seem counterintuitive, as it appears to make the set-valued derivative a higher degree, rather than an affine, function in the increment bt. The reason for this will be explained through the utilization of this concept of derivatives in the derivation of dynamic programming equations for systems with Preisach hysteresis nonlinearities. Next, we present a concept of derivatives of a set function, i.e. a realvalued function with a set-valued independent variable. DEFINITION6.4.2. A set-function V has a Frdchet-like derivative if there exist bounded measurable functions q~+, q~- such that fA

W(Z CJc~Z) - V(Z) - / 9

V(Z \ 6Z) - V(Z) - f 9

q)+ (~),~,n(d~) + 0(,~n ((~Z)),

,15Z

q)- (~),~n (d~) + o(~ n (c~Z))

,/SZ

where ,~n is an n-dimensional Lebesgue measure, and the symbol \ signifies the inclusion of ~Z into a Z modulo set of n-dimensional Lebesgue measure zero, i.e. 2 n (~Z\Z) - O. We say that V has a G~teaux-like derivative if, for variations 5Z of the form ~Z = u (ai q- Ai(~)Ui) with det(Ai(~)) - o(~n), we have i~I V(Z U c~Z) - V(Z) - ~(D+V)(Z, {ai, Ai, Ui" i E I}) q-o(~); V ( Z \ c~Z) -

V(Z) - e,(D-V)(Z, {ai, Ai, Ui " i ~ I}) + o(~)

for some functions D +, D - that depend on the indicated variables.

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

627

The two concepts of derivatives above are very different from ordinary Fr6chet and G~teaux derivatives; we use the names 'Fr6chet-like' and 'Gateaux-like' only for mnemonic purposes. There is, in general, no comparison between the two concepts of derivatives introduced in Definition 6.4.2; the existence of either of those derivatives does not imply the existence of the other. 6.4.2 CALCULATION OF SET-VALUED DERIVATIVES In the case when the set-valued function is S(t), the active set of a Preisach hysteresis operator, we can give an explicit calculation of the derivatives in the sense of the definitions of Section 6.4.1. We consider a continuous piecewise monotone signal u(.), and the corresponding evolution of the active sets on the Preisach plane with coordinates a and fl, in the part P(a0, rio) := {(a, fl):fl0 < fl < a < a0}. The symbol P(a0, rio) signifies the closure of the open set P(a0, fl0). We consider active sets S(t) such that the relative boundary of S(t) in P(ao, rio) is a piecewise smooth non-increasing curve ?(t), where the closure ?(t) extends from some point on the part {(a0, fl):flo~fl~ao}U{(a, fl0) : fl0~a~a0} ofthe boundary 0P(a0, fl0) to some point on the diagonal {(a, fl) : fl0~a = flKa0}. The point on the diagonal will be denoted by D(7(t)). We shall obtain explicit formulae for the derivatives of S(t) under smooth monotone variations of the input u(.). The curve ?(t), of the type described above, will fall into one of three categories: (i) A curve which we shall call of a-type, which contains a nontrivial (i.e. of positive length) straight line segment parallel to the a-axis with one endpoint at D(7(t)), and the maximal line segment has its other endpoint at a point Da(?(t)); we shall denote that oriented line segment by seg(D(7(t)), D~(?(t))). (ii) A curve which we shall call of fl-type, which contains a nontrivial straight line segment parallel to the fl-axis with one endpoint at D(?(t)), and the maximal line segment has its other endpoint at D~(7(t)); this oriented line segment will be denoted by seg(Dfl(7(t)), D(7(t))). (N.B. The orientation of the segments in cases (i) and (ii) is such that they are oriented in the direction of increasing a or increasing ft.) (iii) A curve that is neither of a-type nor of fl-type; in that case we shall say that 7(t) is of oblique type. For a curve of a-type, there exists a neighborhood V of D~(7(t)) such that the part (V A~/(t))\seg(D(7(t), Da(7(t))) of the curve ?(t) can be represented by an equation a = (fi(fl). Similarly, for a curve of fl-type, there exists a

CHAPTER 6 Control of Systems with Hysteresis

628

neighborhood V of D[~(y(t)) such that the part (VNy(t)) \seg (D/~(~ (t), D(7(t))) of the curve y(t) can be represented by an equation fl = @(a). The curves -- r and fl --- @(a) are both non-increasing and, for the purposes of this derivation, they are postulated to be differentiable at the points D~(y(t)) and D/~(y(t)), respectively. A curve of oblique type can be locally represented, at and near D(y(t)), by both equations a = q~(fl) and fl = ~(~). We consider an active set S(t) at time t; S(t) is a set of the type described above. The point D(~(t)) has coordinates, in the aft-coordinate system, (a, a), i.e. the present value u(t) of the input signal u(.) is u(t) = a. We shall use the following notation: R(al, ill; ~2, f12) will be the rectangular domain, on the ~fl-plane, with vertices (al, ill), (al, f12), (a2, ill), (a2, f12); T(a3, f13; a4, f14) will be the triangular domain, on the aft-plane, with vertices (0, 0), (a3, f13), (a4, f14). In the case where two of the vertices of either a triangle or a rectangle coincide, we shall identify that domain with the empty set. The gauge function g will be the diameter of a set,

g(A) = sup{dist(x, y) : x e A, y ~ A}. For an active set with a b o u n d a r y curve 7(t) of a-type, we shall denote by a~ the a-coordinate of the point Da(y(t)), and, similarly, for the case of a domain with a b o u n d a r y curve of fl-type, we shall write b/~ for the fl-coordinate of the point D~(y(t)). For a domain with a b o u n d a r y curve that is not of a-type, we identify D~(y(t)) with D(y(t)), and, by the same token, if y(t) is not of fl-type, we identify D~(y(t)) with D(y(t)). These conventions allow us to present unified formulas for the derivatives of S(t). We consider a variation vbt of the input signal u over the time interval It, t + bt) (this notation incorporates the condition that 6t > 0), with v # 0. We have: THEOREM6.4.1. Under the conditions and notational conventions stated above,

the derivatives of S(t) are given by the following formulas:

a~--

b/~

'Af(bt)=

0

0] 1 ' U+ - R(0, b/~; 1, a); 0 ] v +dt , U f - T(q~'(a), 1; 1, 0);

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

[a]a

[v'to

o]

vdt

, U~- -- T(1, 0; 1, 1);

_ = [o] ,Vjl A C,~t~-- [1 - v - ~t I ' [v t - v -0d t ] ' U2 -

a1

a 2 --

a 3 --

0

a

'

[a][v

' A3=

a

]

Ul=R(a~,0;a,1);

0

' A2 -

a

629

0

0

tO]

-v-dt

T(1, 1; 1, 6'(a));

' U3 = T ( 1 ' 1;0,1)

where v + "- max(v, 0), v - - m a x ( - v , 0). The i m p o r t a n c e of this definition of derivatives of set-valued functions is also s h o w n in the following theorem: THEOREM 6.4.2. For piecewise monotone inputs u(.), which are also piecewise

continuously differentiable over each interval of monotonicity, if the derivatives as calculated in Theorem 6.4.1 above are specified at each time instant t, and if the active set S(O) is specified at an initial time t = 0, then the active set S(t) is uniquely determined for all t. The above t h e o r e m is an existence a n d u n i q u e n e s s t h e o r e m for a type of differential equation w i t h set-valued functions as u n k n o w n s . We consider a controlled s y s t e m of the form dx(t) dt

=fit, x(t), S(t), v(t)); x(0) - x0, S(0) - So

(6.57)

w h e r e v(t) is the control, a n d the set-valued function S(t) evolves according to the derivatives described in T h e o r e m 6.4.1 above. This type of s y s t e m includes the case of Preisach hysteresis operators w i t h inputs u(t) that satisfy du(t) dt

= v(t)

(6.58)

w h e n the Preisach operator is

(Hu)(t) - / fs(t) w(~' fl) d~ dfl

(6.59)

or, in a more general sense,

(Hu)(t) - / f s

(t)

w(t, a, fl) dadfl

(6.60)

630

CHAPTER 6 Control of Systems with Hysteresis

with a time-varying weight function w. This model does not include the case of Preisach operators with Borel measures (with possibly a singular part) instead of the weight function w. In the latter case, some modifications would have to be made to the concept of derivative of a set-valued function to include special treatment to the support of the singular part of that Borel measure. The control function u(.) itself does not need to appear explicitly in the controlled dynamical system (6.5.7), because the value u(t) is uniquely determined by the coordinates of D(~;(t)) on the Preisach plane P(ao, rio). The optimal control problem is the minimization of a functional J :=

J0

f(t, x(t), u(t), (Hu)(t)) dt + f0(x(r), (Su)(r)).

(6.61)

The dynamic programming equations, under conditions of differentiability of the Hamilton-Jacobi function W(t, ~, S), take the form

OW(t, ~, S) Qt

+ min{inf{VcW(t, ~, S)f(t, ~, S, v) v~>0

+D+'VW(t, ~, S) + F(t, ~, S, v)}, inf{VcW(t, ~, S)f(t, ~, S, v) + D-'VW(t, ~, S) + F(t, ~, S v)} = 0; v~<0

W(T, ~, S) = Fo(~, S). (6.62) The superscript "v" in the G~teaux-like derivatives D • denotes that the value of the variation v is used in the definition of the derivatives of the Hamilton-Jacobi function W with respect to the set-valued variable S. Further, the essential concept of weak solutions in terms of extremal approximations (this process is most commonly called 'viscosity solutions') can be applied once the appropriate concept of derivatives is available. This has been summarized in [21]. 6.4.3 OTHER RELATED TECHNIQUES

A second approach to the problem of defining, calculating and utilizing a concept of derivatives, is based on a concept of variations of curves. The relevant curve, for the Preisach model, will be the relative boundary of the active set. Our definition of the appropriate concept of derivatives is inspired by Hadamard's work in the calculus of variations: reference [44] contains, among lots of other things, the calculation of the variation of Green's functions for the Dirichlet and the Neumann boundary value problems of potential theory, relative to perturbations of the domain on which the boundary value problems are to be solved. (There exist modern offshoots of Hadamard's ideas, in the context of shape optimization

6.4 OPTIMAL CONTROL OF SYSTEMS WITH PREISACH HYSTERESIS

631

and sensitivity analysis, relative to domain perturbations, for partial differential equations and variational inequalities. For the purposes of the present exposition, it is more efficient to refer directly to H a d a m a r d ' s work.) H a d a m a r d defined the perturbation of a surface as the normal distance bn between two nearby surfaces; cf. articles 249-252 of [44]. Following the ideas, but not the specific technicalities, of [44], we may define the derivative of a curve with respect to a parameter in the following way: DEFINITION6.4.3. Let (7o) be a continuously differentiable curve in !R n with equation P = P(t), t o ~ t K t l , where P stands for a generic point on that curve. (We identify a point P with the vector 0--~, where 0 stands for the origin of a coordinate system.) Let a family of continuous curves (7~), 0 < ~ < ~0, be described by an equation P~ = P(t) + rl(t, ~)--~ (t), where P~ is a generic point on (7~), -it (t) is the vector field on unit normals to the curve (70) at each point P(t) of (70), and the function rl is continuous in both variables and differentiable at ~ = 0 +, with r/(t, 0 +) ~_ lim r/(t, ~) = 0. Then we define the derivative ~--,0 +

~/~(t, 0 +) = lim ~rl(t, ~) as the normal derivative of (7~) with respect to the pa~--,0 +

rameter ~ at ~ -- 0 +. More generally, if (70) is a continuous curve with equation P = P(t), t o K t ~ t l , if--d (t) is a vectorfield of vectors of constant length (for technical convenience, we may want to use a length other than unity) defined on (?0), and if a family of curves (7~), 0 < ~ < ~o is defined by P~ - P(t) + rl(t, ~)-d (t), 1 where rl has the same properties as above, then we define r/~(t, 0 +) = lira ~/(t, ~) ~--,0 +

as the derivative relative to the vector field --d (t) of (7~) with respect to the parameter ~ at ~ = 0 +.

On the Preisach plane P(a0, rio), let (~0) be a piecewise smooth nonincreasing curve, with diagonal endpoint (i.e. endpoint on the diagonal a-flofP(a0

flo)) denoted b y D(7o) W e u s e thevector field -d~ (t) _ [ l 1 I..

_!

which suffices to describe variations due to a small increase or decrease of the input u(.). The calculations of the variations for domains of different types (atype, fl-type, or oblique type can be carried out by techniques analogous to Section 6.4.2 above, and the related dynamic programming equations can be written down. This alternative definition of the derivatives of the active set may be useful for the description and analysis of more general controlled systems with a set-valued state. For the specific issue of control of Preisach hysteresis, this second concept of derivatives of set-valued functions will be presented in a future work, and is not reported here.

632

CHAPTER 6 Control of Systems with Hysteresis

6.5 H Y S T E R E S I S F R O M O P T I M A L C O N T R O L , A N D HYSTERESIS FOR OPTIMAL CONTROL In this section, we outline certain cases in which a controlled system, in its original formulation, does not contain any hysteresis elements, but hysteresis is later introduced into the system through the design of optimal control policies. Sometimes, an optimal control policy, expressed as a feedback law, leads to hysteretic behavior of the corresponding optimal control function or the corresponding optimal state trajectory. Sometimes, hysteresis is intentionally part of the design of a control policy. The general phenomenon of control systems that acquire new, nonsmooth, behavior, or some other kind of behavior that is qualitatively different from the original description of the system, is well known and well understood in many parts of optimal control theory. For instance, we may start with a system that we originally intend to describe as an ODE system with controls that are continuous in time; however, an optimal control problem may lead to a feedback law that creates discontinuities, as in the case of 'bang-bang' controls in (certain classes of) minimum time problems. An analogous situation occurs in connection with the emergence of hysteresis operators as a consequence of optimality.

6.5.1 HYSTERESISARISING FROM DISCRETE-VALUED CONTROLS IN DETERMINISTIC AND STOCHASTIC SYSTEMS A controlled ODE system with a binary control decision, say 0 or 1, can be described by dx(t) =fit, x(t), u(t)); u(t) ~ {0, 1}; x(0) = x0. dt

(6.63)

There are two possible cost functionals that are of interest for control applications: ]1 "-

f(t, x(t), u(t)) dt + fo(x(T))

(6.64)

and

J2 "- f0 T F(t, x(t), u(t)) dt + ~

k(:i, x('ci),

U('C~), U(':+)) q- FO(x(T)).

(6.65)

i In the case of h , the Hamilton-Jacobi function Wl(t, x) satisfies, in the appropriate sense of generalized solutions, the dynamic programming

6.5 HYSTERESISFROM OPTIMAL CONTROL

633

equation: C~Vl (t, x) ~t

+ min {VxV1 (t, x)f(t, x, a) + F(t, x, a) } = 0; a~{0,1}

V1 (T, x) = Fo(x).

(6.66)

In the case of J2, the Hamilton-Jacobi function V2 (t, x, a) is parameterized also by the current value of the control, a = u(t), and satisfies, in the appropriate generalized sense, the dynamic programming equation: min

OV2(t, x, a) -Ot 4- VxV2(t, x, a)f(t, x, a) + F(t, x, a),

V2(t, x, b) - V2(t, x, a) 4- k(t, x, a, b) } = O; V2(T, x, a) = Fo(x)

(6.67)

where, in (6.67), a and b take values in 0, 1 and b ~- a. In the case of minimizing J1, if an optimal control policy exists, it can be described in feedback form in terms of continuation sets C a, a ~ {0, 1}. Each set C a consists of all values of (t, x) such that, when the value x(t) of the state trajectory (solution of (6.63)) satisfies (t, x(t)) ~ Ca, then an optimal decision is to take u(t) = a. According to this decision rule, switching from one value of the control to another is consistent with optimality if (t, x(t)) E C a N C b. The situation is a little more involved for the case of minimizing J2, but an analogous principle of optimal feedback applies to that case. Now, restricting our attention, for simplicity, to the problem of minimizing J1, we remark that, in general, the intersection C a A C b contains an open set, which creates the analog of a nonideal relay (Preisach relay) for the optimal control policy. Switchings become mandatory when the point (t, x(t)) escapes from a continuation set. The switchings that are mandatory for optimality are those that are delayed for as long as possible, i.e. those that satisfy the analog of Visintin's 'laziness principle' or minimum bounded variation for the Preisach relays (cf. [27]). In the particular case of one-dimensional state x(t), with continuation sets that are independent of t and are of the form C O ~-- ( - o o , 0~], C 1 = [fl, ff-oo) in the state-space (-oo, +oo), with ~ > fl, we get precisely a relay of the Preisach type as the optimal feedback law that describes many possible optimal switching policies (in general, an optimal policy, in this context, is not unique), with the 'laziness principle' dictating when switchings become inevitable. Thus, we have, in this case, arrived at the following situation: a simple control system, with binary control decisions, through a requirement of optimality, becomes a system with a nonideal relay: a system that was originally dx(t) dt = fit, x(t), u(t)) has dx(t) been transformed into a system -aN = f i t , x(t), (R~flx)(t)). We remark that

634

CHAPTER 6 Control of Systems with Hysteresis

this principle is not limited to systems governed by ordinary differential equations, but applies to all kinds of systems for which a generator (in the sense of evolution semigroups) can be defined, for example, systems governed by Ito stochastic differential equations (cf., for instance, [45]). 6.5.2 HYSTERESIS FOR THE CONTROL OF DYNAMICAL SYSTEMS

The situation described in Section 6.5.1 above is sometimes put in reverse. Starting with a control system and some optimality criterion, a binary decision structure is introduced, and a feedback control policy is postulated in the form of a general choice about the nature of the continuation sets. For example, one may postulate that CO = (-cx~, a], C 1 -- [/~, q - o o ) , and seek an optimality condition. Under these scenarios, an optimality condition then leads to a conclusion (among other things) that ~ > fl, and thus hysteresis has been intentionally designed for the purpose of controlling a dynamical system. We must point out that the description of the process of arriving at a hysteretic control policy is our interpretation, and the researchers who apply equivalent methods may proceed in ways that are technically different, or conceptually explained in different ways, from what we have described. For example, see the paper [46] for hysteresis arising in 'rational' (economic) institutions, or the paper [47] for hysteresis introduced for the design of optimal control of a queueing system. Further, we can consider a continuous collection of controlled systems with binary controls, each system having its own (locally) optimal policy defined by a pair of continuation sets, with the collective outcome being a weighted superposition of the 'locally' optimal outcomes; this leads naturally to a Preisach hysteresis operator. This last step in the description of an instance of 'hysteresis from control' was inspired by the paper [48] on hysteresis in economic systems.

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CHAPTER 6 Control of Systems with Hysteresis

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35. L. T. Ashchepkov, Optimal Control of Discontinuous Systems, Nauka, Novosibirsk (1987) (in Rusian). 36. B.M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-continuous Systems, Kluwer, New York (2003). 37. S. A. Belbas and V. M. Matrosov, 'Optimal control of systems in continuous-discrete time scales', Int. Congress on Nonlinear Analysis and its Applications, Moscow, International Academy of Nonlinear Sciences (1998); preprint at arxiv.com. 38. J. W. Macki, P. Nistri, and P. Zecca, 'Mathematical models for hysteresis', SIAM Review 35(1), 94-123 (1993). 39. M. Brokate, 'Optimal control of ODE systems with hysteresis nonlinearities', in Trends in Mathematical Optimization, K.-H. Hoffmann et al. Eds, Birkh~iuser-Verlag, Basel (1998), pp. 25-41. 40. V. F. Demyanov, C. Lemarechal, and J. Zowe, 'Trying to approximate a set-valued mapping', Technical Report MRC-2786, Mathematics Research Center, Madison, WI (1985). 41. G. Friedman, 'New formulation of the Stoner-Wohlfarth hysteresis model and the identification problem', J. Appl. Phys. 67(9), 5361-5363 (1990). 42. G. Friedman, 'Second-order Preisach model of scalar hystresis', Physica B 275, 173-178 (2000). 43. G. Friedman and K. Cha, 'Representation of hysteresis with wipe-out memory', J. App1. Phys. 89, 7236-7238 (2001). 44. J. Hadamard, Ler sur le Calcul des Variations, A. Hermann et ills, Paris (1910). 45. S. A. Belbas, 'The dynamic programming equations for stochastic games with discrete actions', Systems and Control Letters 11, 235-242 (1988). 46. R. Di Tella and R. Mac Culloch, 'Rational institutions yield hysteresis', Report No. B9, Zentrum f. Europ. Integrationsforschung, Bonn (2000). 47. A. N. Dudin, 'Optimal hysteresis control for an unreliable BMAP/SM/1 system with two operation modes', Automation and Remote Control 68, 1585-1596 (2002). 48. R. Cross, J. Darby, J. Ireland, and L. Piscitelli, 'Hysteresis and unemployment: a preliminary study', preprint, Strathclyde University (1998). 49. F. Bagagiolo and A. Visintin, 'Hysteresis in filtration through porous media', Zeitschrift f. Analysis u. Anwendungen 19, 977-998 (2000).

CHAPTER 7

Binary Detection and Rectangular Hysteresis Operators Can E. Korman, Department of Electrical and Computer Engineering,

The George Washington University, Washington, DC 20052, USA

Contents 7.1 7.2 7.3 7.4

Introduction General System Description and Outline Input Process and Mid-Span Repeater Receiver Integrator Output: Probability Density Function Calculation 7.5 Summary 7.6 Concluding Remarks References

639 641 644 646 662 663 664

7.1 I N T R O D U C T I O N The subject of hysteresis has been studied and is of considerable interest in a broad range of disciplines in engineering, material science, finance, economics, etc. Hysteresis arises in such fields either due to the nature of the problem or it may be deliberately incorporated for the purpose of improving system performance. Fields of science where hysteresis is due to the physical nature of the material include magnetics, superconductivity, magnetostriction, etc. Applications where hysteresis is specifically introduced to improve system performance, or minimize some cost function, include the setting of trade prices for the buying and selling of stocks, control systems, etc. For instance, in a thermostat-based temperature control system, hysteresis is generally introduced for the purpose of reducing multiple switching of equipment due to randomly fluctuating inputs near The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

640

CHAPTER7 Binary Detection and Rectangular Hysteresis Operators

a threshold. In such systems, hysteresis helps minimize the system's cost function by preventing multiple rapid switching due to the fluctuation of inputs (stock price, temperature, etc.) around a set threshold. The original impetus for this work was to investigate the effect of hysteresis on the performance of an optical communication link. In order to set the problem in the proper context, it is beneficial to briefly review the relevant requirements of an optical network. For the purpose of reliable optical communication links over long distances, optical repeaters are placed at appropriate intervals to detect the incoming optical signal, amplify it and then retransmit it. Naturally, the optimal detector in a repeater should include a filter matched to the bit rate of the optical signal. However, owing to network design considerations, the optical links may be required to carry signals with different bit rates. Considering the fact that the same repeater must handle signals with multiple bit rates, optical repeaters are required to be bit-rate transparent. The issue of bit-rate transparent repeaters has been investigated by several groups in the recent past [1-4]. The repeaters in such networks are required to detect and retransmit rectangular pulses that have different bit rates. In an optical link these signals may be detected and retransmitted by such mid-span repeaters several times. If these repeaters were designed for fixed pulse widths, the optimum design would naturally be a matched filter. Unfortunately, the design of such repeaters is complicated by the fact that matched filters are only optimal for a given fixed pulse width. This being the case, it is necessary to employ suboptimum filters in the design of such mid-span repeaters in optical network links [5]. Arguably, the simplest possible binary bit-rate-independent detector is a step threshold device that can distinguish the presence or absence of light at its input. In such a device, hysteresis can either be a natural part of the optical physical layer or, if desirable, may be introduced deliberately to improve system performance. In light of this possibility, one begs the question of whether hysteresis in a binary threshold-type detector offers any advantages over a simple threshold scheme, whether in an all-optical network link or another application. This work explores the possibility of employing a binary threshold device in a binary detection problem and outlines the mathematical framework that is necessary to calculate various system performance measures. In this sense, it is meant to be a general mathematical approach even if the original motivation came from the analysis of optical network links. Therefore, it is applicable for the analysis of diverse applications that can be modeled by the system and noise constraints stated below. Problems of this nature have been studied before but have largely focused on nonlinearities with no hysteresis. In particular, the problem

7.2 GENERAL SYSTEM DESCRIPTION AND OUTLINE

641

of the detector where a low-pass filter is cascaded to a hard-limiter has been investigated by several groups [6-12]. In this context a hard-limiter is nothing more than a special case of a hysteresis switch with zero width. Therefore, the study of a detector with a hysteresis switch can be viewed as a generalization of prior work done in this field. To our knowledge no analytical investigations have been performed where the detector has a hysteresis-type nonlinearity. The work presented here is largely based on our prior research in this field [9]. While the work is partly motivated by all-optical networks, where network links are required to be bit-rate transparent, the analysis presented here is quite general.

7.2 GENERAL SYSTEM D E S C R I P T I O N A N D OUTLINE The general block diagram of the system to be considered in this chapter is shown in Fig. 7.1. The system consists of a binary transmitter, a number of mid-span repeaters and additive noise sources, all terminated at a receiver matched to the bit rate of the transmitter. Each mid-span stage can be considered to be identical to the others where the respective signals are perturbed by mutually independent noise sources. For clarity, it is beneficial to consider a communication link with only one midspan repeater. This one-stage link can be modeled by a system whose block diagram is shown in Fig. 7.2. Here the system is modeled by a one-stage bit-rate-transparent repeater that consists of a rectangular hysteresis loop and a receiver that is tuned to the specific bit rate of the binary transmitter. The input source of the system is modeled by a series of rectangular pulses perturbed by additive Gaussian noise. For mathematical

oJ Bin~ Midspan Repeater

noise

Midspan Repeater

"'" ~

Receiver

noise

y

Cascade of midspan repeaters

FIGURE 7.1 General system block diagram mith multiple mid-span repeaters terminated by a receiver tuned to a specific bit rate.

642

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators Bit-Rate Independent Mid-span Repeater White Noise Hysteresis Loop

nt

~t

Non- ] xt [ hysteretic[....~ filter

I-I

~(t)

JL2

O-Binary Input Signal T >T/2 ~
1 0

I

(*)dt

0

Decision Rule

Integrator

Receiver Module

FIGURE 7.2 Blockdiagram of the system to be modeled in greater detail with a one-stage mid-span repeater consisting of a hysteresis-type nonlinearity and binary detector. convenience, the binary signal perturbed by white noise is considered to be an input first to a non-hysteretic filter whose output then drives a second filter with hysteresis. The hysteresis in the mid-span repeater is modeled by a rectangular hysteresis loop, an elementary hysteresis operator that can be employed as a building block to model more complex hysteretic systems (see, for instance [13-16]). The output of the hysteresis operator is terminated by a receiver that is tuned to the bit rate of the transmitter. The receiver is composed of an integrator, which integrates the output signal from the rectangular loop over one bit duration, and this is followed by a threshold detector that makes the binary decision. At this last stage, the integrator averages the sum of the output of the hysteresis loop and a second independent noise source, which models noise in the receiver. The analysis of the system with a rectangular hysteresis loop driven by stochastic processes poses certain mathematical difficulties. This is due to the fact that the state of the hysteresis loop depends on the history of past input extrema of the input stochastic process. Fortunately, the problem of stochastic inputs to hysteretic systems has been considered previously in the somewhat unrelated problem of aftereffect modeling and viscosity (for instance, see [17-19]). In such problems, the stochastic input models thermal perturbations in metastable systems that lead the system to relax from higher energy states to lower energy states [20]. Examples of

7.2 GENERALSYSTEM DESCRIPTION AND OUTLINE

643

such problems include magnetic viscosity, aftereffects in superconducting materials, shape memory alloys, etc. In the analysis of such aftereffect problems it was shown by the author that many of the mathematical difficulties can be overcome when the stochastic inputs to the rectangular hysteresis loop are considered to be a stationary Markov process. Based on such an assumption it was shown that key calculations can be reduced to the well-known mathematical tool of 'exit problem' of stochastic differential equation (SDE) theory [21]. This mathematical tool is also known in the mathematics and physics literature as the 'first passage problem'. The mathematical tool of 'exit problem' is employed in the analysis of binary detectors with hysteresis. Employing this tool, a method is developed to compute the probability density function (pdf) of the output of the receiver integrator in terms of exit time probability density functions of the input stochastic process [9]. Owing to the nature of the calculations, it turns out that the pdf of the integrator output can be represented in terms of generating functions (Laplace transform domain). In terms of the Laplace transforms of the exit time pdfs (expressions (7.44) and (7.45)) compact closed-form expressions of the integrator output pdf are derived in the transform domain. Subsequently, it is shown that the derived expressions can be employed to compute system performance measures, such as bit error rates. The chapter is organized as follows. The characteristics of the stochastic noise process that is driving the hysteretic system will be described first. For practical application considerations and, to some extent, mathematical convenience, it will be stressed that the analysis presented here requires the input noise to the rectangular hysteresis loop to be a stationary Markov process. In the following section, calculations will be presented to compute the probability density function of the integrator output in terms of the exit time probability density functions of the input process. It will be shown that compact expressions of the receiver integrator output can be derived in terms of the Laplace transforms of the aforementioned exit time probability density functions. The preliminary analysis will be presented for the general case of any stationary Markov input process. Subsequently, a particular case will be presented for which the additive noise process is stationary, Markov and Gaussian. This process can be modeled when the non-hysteretic filter is considered to be a first-order low-pass filter driven by white Gaussian noise (AWGN). This case will be presented as a particular example where the Laplace transforms of the relevant exit time probability density functions can be explicitly calculated in terms of parabolic cylinder functions [21]. Finally, conclusions will be presented, including a brief discussion of how this work can be employed beyond the scope outlined in this chapter.

644

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

7.3 INPUT PROCESS A N D M I D - S P A N REPEATER A binary signal ~(t) in the presence of additive noise nt is considered to be the input to the mid-span repeater. Over a one-bit duration T the binary signal takes the values of zero or one. The mid-span repeater is modeled by a cascade of two filters as shown in Fig. 7.2: the first one is a nonhysteretic filter while the second one is a filter that exhibits hysteresis. The latter is modeled by a rectangular hysteresis loop with output values of zero and one. Therefore, the output of the mid-span repeater is a binary process whose value depends on the past switching history of the rectangular loop. This point will be elaborated further in the discussion below. The output of the first (non-hysteretic) filter is denoted by xt. The second stage of the mid-span is driven by this stochastic process. The analysis presented in this work requires: 9 the exit-time (or first passage time) pdfs of this process, either analytically or numerically, from semi-infinite intervals, and 9 that this stochastic is Markov and stationary around a known mean. The analysis presented in the following section is for any stationary Markov process with arbitrary exit-time pdfs. Although this approach is fairly general, there is a class of problems where a well-understood methodology exists in order to calculate the underlying exit-time pdfs of the stochastic process: the case of the so-called diffusion process of stochastic differential equation theory. In the problem considered here xt is modeled as a diffusion process, that is, the output of the non-hysteretic first-order filter driven by the superposition of white Gaussian noise and the input signal ~(t) is given as follows: dxt c Jr b(xt) = c ( x t ) | ~ ( t ) q- r 1. dt

dWt 1

|. cU _l

(7.1)

In the above stochastic differential equation, b(x) and c(x) characterize the first-order filter, r~n(X) is the measure of noise and Wt denotes the standard Wiener process. Solutions of this SDE are diffusion processes and are continuous in time. Most notably, there is a well-known particular case of this problem when the system is modeled by a first-order low-pass filter of bandwidth b. The stochastic output process of this low-pass filter xt satisfies the following first-order constant coefficient stochastic differential equation: dWt dxt r -F bxt -- bk~(t) Jr ~n - ~ - ], (7.2) d---T where cr2 - N0/2 and No denotes the spectral density of the driving white noise process. From the definition of the input data signal ~(t), the mean

7.3 INPUT PROCESS AND MID-SPAN REPEATER

645

A

7x Switch from 1 to 0

--t-0

( ~

>X

Switch from 0 to 1 FIGURE 7.3 Hysteresis nonlinearity employed in the mid-span repeater: a rectangular hysteresis loop with switching-up value a and switching-down value/3.

value of Xt is a series of pulses with rise and fall time constants of 1/b and the zero mean stochastic part is the well-known Ornstein-Uhlenbeck process. It is well known that such a stochastic process is Gaussian, Markov and stationary, according to the Doob theorem. For the Ornstein-Uhlenbeck process, well-known analytical (closed-form) solutions exist for the exittime pdfs in the transform domain. This aspect will be elaborated further in the subsequent discussions. The hysteretic filter for which the above stochastic process is an input is modeled by the rectangular hysteresis loop shown in Fig. 7.3. This rectangular hysteresis loop is characterized by the output values of one and zero, switching-up value a and switching-down value ft. Therefore, the output of this hysteresis loop, ~a~xt, is a binary stochastic process whose value depends on the random exit times of the input stochastic process from semi-infinite intervals (-cx~, a) and (fl, +cx~). Since a conventional threshold-type detector is a special case of a rectangular loop with equal switching up and down values, the analysis presented in this paper enables one to ascertain any performance gain due to hysteresis. The analysis of the hysteresis detector presented here is for arbitrary switching up and down values. However, for the particular binary signal ~(t) considered here, rectangular hysteresis loops centered on one half would be of particular interest. The output to the binary receiver is the sum of the output of the rectangular hysteresis loop and a second source of noise, ~t, that is modeled to be white Gaussian with the spectral amplitude Mo/2. This additional noise source is assumed to be uncorrelated to nt. The receiver integrates the sum of the output of the rectangular hysteresis loop and the second

646

CHAPTER7 BinaryDetection and Rectangular Hysteresis Operators

uncorrelated noise source over one bit duration T. The result of this integration is as follows: RT = [YT q- ZT], (7.3) where, YT--

f0T~aflxtdt

and

ET--

f0T~tdt.

(7.4)

As shown in Fig. 7.2, once this integration is performed over one bit interval a threshold detector makes the decision: zero if the integrator output RT is less than T/2 and one if the integrator output is greater than T/2. Therefore, in order to determine the performance of the system, such as probability of error, the probability density function of the integrator output RT needs to be computed. Since nt and ~t are uncorrelated stochastic processes, YT and ET are independent random variables, as well. Therefore, the pdf of RT can be determined from the convolution of the pdfs of the random variables TT and ET: ~'DRT "- ~'DyT * ~'D~T. (7.5) The probability density function of the latter is straightforward since, from (7.4), this random variable is normal with zero mean and variance TMo/2. Therefore, the remaining problem in the evaluation of the system performance is the computation of the pdf of YT.

7.4 RECEIVER I N T E G R A T O R OUTPUT: PROBABILITY DENSITY FUNCTION CALCULATION Let us consider the general problem of an arbitrary stochastic process as input to a rectangular hysteresis loop shown in Fig. 7.3. Since the output of this rectangular loop is only one or zero at any given time, it is clear from expression (7.4) that O
Y T < U +du},

if O<,u<~T, elsewhere.

(7.6)

In order to calculate this probability density function, it is useful to know the probability distribution function of the state of the rectangular hysteresis loop. Let us define the probability distribution of the state of the rectangular hysteresis loop employing the following notation:

q;(x, 7; t)dx - Pr{x

< Xt

< x + dx, ~flxt = 7 }.

(7.7)

7.4 RECEIVERINTEGRATOR OUTPUT

647

Let (x0, ~ x 0 ) denote the state of the rectangular hysteresis loop at the start of the current bit interval 0~
+~ 8vyT(u) - f

oo

1 dx E

q0(x, 7; 0) ~VrTl(Xo,~xo ) (UlX, 7)"

(7.8)

7=0

Here, ~DyTI(Xo,~a~Xo) denotes the conditional pdf of YT when the initial state of the rectangular hysteresis loop is (x0, ~/~x0). In order to compute the conditional pdf of YT for a given initial state (x0, ~a~x0), one must find the pdf of all input sample trajectories starting from a given initial state under the constraint that the output of the rectangular hysteresis loop is equal to one for a total duration of u and zero for a duration of T - u during one bit interval of duration T. This is the most general definition of the problem. In order to solve the problem, several basic observations can be made. First, the aforementioned constraint on the input sample trajectories may be satisfied for an arbitrary number switchings of the rectangular hysteresis loop. Second, the spaces of sample trajectories of different numbers of switchings are disjoint events. Employing these two observations, it is possible to calculate the conditional pdf of YT for a given initial state in terms of the following sum of probabilities:

oo ~DyTI(Xo,~;aflXo)(U) -- ~

n=O

" (n) ~,9yTl(xo,~aflxo ) (U),

(7.9)

where

(n) rrl(xo,?~xo) (u)du -- Pr { u < YT < u + du, n switchings } of ~/~/~in [0, t] I (xo, ~/~/~xo) "

(7.10)

Therefore, from expressions (7.6)-(7.10), it can be observed that the computation of 69yT is reduced to the computation of ~'J(n) Yzl(xo,7a~xo)" (u)du, the probability of u < YT < u + du for a given initial state ~al~x0with n switchings of the rectangular hysteresis loop. In the subsequent discussions it will be shown that this probability, together with the initial distribution, can be computed by employing the machinery of the 'exit problem' and the Kolmogorov-Feller equation of SDE theory.

648

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

7.4.1 DEFINITION OF THE EXIT PROBLEM The sample trajectories of the rectangular loop output for the two possible initial values of ~/~x0 - +1 and ~/~x0 = 0 are shown in Fig. 7.4. For example, w h e n ~a~x0 = 0, the output remains at zero until the input starting at x0 exits the semi-infinite interval ( - ~ , a]. At the time instant of switching, x = a and the output switches to one. The output of the rectangular hysteresis loop remains in this state until the input exits from the semi-infinite interval [fl, + ~ ) . At this time instant of switching, x = fl and the output switches d o w n to zero. The output of the loop remains at this zero state until the input stochastic process exits the semi-infinite interval ( - c ~ , a]. From this description, it is clear that all subsequent switchings of the rectangular hysteresis loop correspond to the last two exit problems and the switching process repeats this sequence of events. The switching events for the rectangular hysteresis loop occur at a sequence of r a n d o m time instants and corresponding exit times. Consider an input stochastic process x t that results in n~>1 switchings of the rectangular loop at times 0 < tl < t2 < ... < tn < T . For such a trajectory, the following exit times can be defined: Zl ----tl, "Ci =--ti -- t i - 1 , Zn+l - T -

2<.i<~n,

(7.11)

tn.

If an input trajectory does not switch in the current bit interval then the first exit time Zl > T and n = 0 by definition. The set of exit times defined in (7.11) are summarized in Table 7.1.

A

Initial State ofl

~')r 13Xt ~

101' '171

>1'

~2

~4 I >t T

A

Initial State of O t>

___ <

"r

>]<

'~3

~]<

~4

>i

I' >t T

FIGURE 7.4 Possible trajectories of the rectangular hysteresis loop output for the initial conditions ~ x 0 = I and ~ x 0 = 0, respectively.

7.4 RECEIVER INTEGRATOR OUTPUT TABLE 7.1

"ci

649

List of Exit Times

~;aflx 0 = 0

~flx 0 = 1

D e p e n d e n t on (x 0, ~c~flx0)?

x 0 --, ~x fl +-- ~ fl --+ ~

fl ,-- x 0 ]~ --+ ~ fl +-- 0r

YES NO NO

i= 1 i/>2, even i/>3, o d d

Based on the above definition of exit times, the r a n d o m variable YT can be expressed as follows:

07

if ~aflx0 - 0 and n - 0,

y]n+l Z.i i=2, even '

if ~/~x0 - 0 and n>~1,

T~

if ~/~x0 - I and n - 0,

y]n+l i=1, odd "Ci'

if ~/~x0 - I and n~>l.

(7.12)

YT --

This expression shows that I'T can be expressed as a sum of even and odd sequences of exit times. Since the underlying input process that results in these exit times is assumed to be Markov, this set of exit times consists of mutually independent r a n d o m variables. Therefore, the probability density function of their sums can be calculated by convolution of the pdfs of all the r a n d o m variables in a given sum. Consequently, in order to compute the pdf of Y T , the pdfs of the above exit times need to be determined. In order to compute these exit-time pdfs, it will be convenient to define the following exit-time probabilities: V + (r, x) - Pr

~a/~ does not switch in time interval(t', t'

+~)l}

I Xt, -- X, ~o~fl Xt' -- 1

(7.13)

V - ( r , x) - Pr

~B does not switch in time interval(t', t' I Xt, - - X ,

+~)1

~

~ f l X t, - - 0

(7.14) Furthermore, the corresponding pdfs of these exit times, v +, can be defined as follows: V +(r, x) - fO~ v +0:,l x)d~:,I ~:>~0. (7.15)

650

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

Based on the above definitions, the probabilities for the exit times summarized in Table 7.1 can be stated as follows: P~.--xo ('0 = V + ('c, xo), Pxo~a('c) = V - ('c, xo),

P~,__~('c) = V + ('c, ~), P ~ a ( ' c ) = V - ('c, fl).

(7.16)

From expressions (7.15) and (7.16), the corresponding exit-time pdfs can be calculated from the following expressions: P~.--xo (~') = v+ (~'' xo) , P x o ~ ( ~ ) = v-(~, xo),

p~._~(~) = v + (~, ~), p~__.~(~) - v-(~, fl).

(7.17)

7.4.2 EXIT-TIME PROBABILITY IN THE LAPLACE TRANSFORM DOMAIN

As eluded to in the previous discussion, the computation of the integrator output probability density function requires the convolutions of exit-time probability density functions. Consequently, it is mathematically convenient to express such convolutions in the Laplace transform domain, which enables one to simplify some of the resulting expressions. Therefore, at this stage it is also useful to state the definitions of the corresponding probability functions in the Laplace transform domain: V•

x) -

V•

f0 ~176

x)e-S~d~,

(7.18)

and (7.19)

0 + (s, x) = 1 - sV • (s, x). 7.4.3 CASE OF THE ORNSTEIN-UHLENBECK PROCESS

The so-called Ornstein-Uhlenbeck process is employed in this section as a source of noise to the hysteresis detector filter. As is well known, this is an asymptotically stationary Gaussian stochastic process. In general, for the stochastic process given in (7.1), it is well known that the exit time probabilities satisfy the backward Kolmogorov equation (for instance, see [21]). In particular, for the Omstein-Uhlenbeck stochastic process given in (7.2) there is a closed-form solution to the corresponding backward Kolmogorov equation in the Laplace transform domain. During a time interval when the binary input signal is constant, ~(t) = z, the exit-time probabilities are the solutions of the following partial differential equation: OV+ - b ( x - z) ~ v • (b~ cgz = -~x q 2

c92V+ cgx2

(7.20) '

7.4 RECEIVER INTEGRATOR OUTPUT

651

subject to the following boundary and initial conditions: V + (~, x)Ix=/~= O, V + (~, x)IT=o- 1, fl < x < ~ , V-(r, X ) ] x = - 0,

V-(r,

x)]~=0= 1, - o e < x < c~.

(7.21) (7.22)

In the time domain this differential equation does not lead to a closed-form solution directly. In order to obtain closed-form solutions, it is possible to rewrite the above differential equation by employing Laplace transforms. Applying the Laplace transform to expressions (7.20)-(7.22) results in the following boundary value problem: (bo'n)2 d2V + 2 dx 2

~+

d~ +

b(x -

z) dx - s

19+ (s, c+) = 0,

(s, x) = -1, 1

V + (s, -t-ec) = -,

(7.23/ (7.24)

S

Here, c+ denote the exit points for the process Xt that are equal t o / / a n d c~, respectively. The solution of such a boundary value problem is well known (for instance, see [21]) and can be expressed by the following expression: l)+(s, x) - -sl [1 - e

(x-z)2-(c+-z)2~)-sIb(~-'~)1 422

(7.25)

lg_s/b ( ~ f -~) where lg_s/b is the parabolic cylinder function and 2 - br~n/~/~. One can calculate the probabilities that characterize the exit times summarized in Table 7.1 based on the solutions of the backward Kolmogorov equation. In the following, it will be shown that PYT can be expressed in terms of the convolutions of exit-time probabilities. Therefore, by enabling such convolutions to be expressed as products, it will be shown that the closed-form solutions of the exit-time probabilities in terms of their Laplace transforms are very useful. 7.4.4 PROBABILITY DENSITY FUNCTION FOR A KNOWN

INITIAL STATE The next step in the calculation is to compute the conditional pdf of TT for a given initial state in terms of the aforementioned exit-time probabilities. According to expression (7.9), this probability density function can be computed from the probability of all trajectories that result in n~>0 number of switchings of the rectangular hysteresis loop. Therefore, one needs to compute the probabilities of all sample paths that start from (x0, 9~/~x0) and result in the output of the hysteresis loop being equal to one for a total

652

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

duration of (u, u + du) during one bit interval T. For clarity of the presentation, the cases for the two different initial conditions of the rectangular loop output are analyzed separately.

Initial State of ~ x 0 - 0 The required probabilities can be determined separately for all possible input trajectories that result in n number of switchings. For n = 0 (no switching), the output of the rectangular loop remains zero during the bit duration. Therefore, it is clear that:

t;j(o) YTI(Xo,O)(u)

(7.26)

- Pxo__,.(T) 6(u)

For n = 1, the output of the rectangular loop is equal to zero during T1 and then remains one at least until the end of the bit interval. Therefore, one obtains: 8(1) YTI(xo,O)(u)du =

Pxo--*o~( T - u) P~,_~(u)du.

(7.27)

For n = 2, the output of the rectangular loop is zero during 171, one during z2 and remains zero at least until the end of the bit interval. This results in the following convolution in terms of exit-time probability density functions:

$(2) (u)du YTI(X0,0) -- ~0T-u d'ClPx0_,a(l:l) = [Px0--*~ * P~_.,~](T-

p&_~(u)du P~__,~(T- u -

1:1)

u)p~,_~(u)du.

(7.28)

For n = 3, the output of the rectangular loop is one during z2 and T4, and zero elsewhere. This results in the following two convolutions:

. (3) (u)du~JYT[(X0,0)

/0 9p f l ~ ( T -

d'ClPxo__,a(/:l)

dz2p//,__o~(T2) /0 u u - zl)du Pp,_.~(u- 1:2)

= [Px0~ * p~_,~](T-

u)[p/~,__~* P~](u)du.

(7.29)

The computations above can easily be generalized for an arbitrary number of switchings n. This can be accomplished by constraining all exit times for which the rectangular hysteresis loop is equal to one to be of duration u and constraining all exit times for which the rectangular hysteresis loop is equal to zero to be of duration T - u. Consequently, it can be shown that the following set of iterated convolutions holds for all odd and even

7.4 RECEIVER INTEGRATOR OUTPUT

653

numbers of switchings, respectively: . (n) ~OYT [(x0,0) (U) --

[

Pxo--*~ *

E'.

fl--~a * ' "

1]

* p~__,

(T-

w~ n-1 2

u)

(7.30) n--1 2

and,

rTl(X0,0)

xo~

B-*~ ~r n--2 2

9[p/~,__a,.--,..

pfl,_~](u),n=2,4,6

.....

(7.31)

n

Initial State of ~ x o

- 1

Similar to the 9~]~x0 - 0 case, the required probabilities can be determined separately for all possible input trajectories that result in n n u m b e r of switchings. For n - 0, the output of the rectangular loop remains one during the bit duration. Therefore, it is clear that:

~j(0) (u) - P/~<--xo (T) $(u - T) " YTI(Xo,1)

(7.32)

The derivations for higher n u m b e r of switchings can be done in a very similar manner to the ~a/~x0 = 0 case and result in the following iterated convolutions:

Yvl(x0,1)(u) -

P~,-xo *

~_~ ,...

,

(u)

n-1 2

(7.33) n--1 2

654

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

and, rTl(X0,1 ) ( U ) - -

Pfl~---X0 *

~+--0~* ' ' " * tOt+--

*

(U)

n-2 ---2-

9

/~Z~,..., ,

,,

p//_,~ ( T -

u), n = 2 , 4 , 6 . . . . .

(7.34)

y

tl

Reformulation in the Transform Domain From expressions (7.26) and (7.30)-(7.34) it can be seen that the conditional probabilities for an arbitrary number of switchings can be expressed in terms of iterated convolutions of exit-time probabilities p/~__,~and p/~,__a. Therefore, according to (7.9) and the above formulas, the conditional pdf for a given initial state is an infinite sum of iterated convolutions and has the following form: oo

~rTI(Xo,~a~sXo) (U) -~ E g (n) (T - u)h (n) (u)

(7.35)

n--O

where the functions g and h denote the iterated convolutions of exit-time probability density functions. Clearly, computations based on the above expression are not practical, since this is an infinite sum of iterated convolutions. However, it turns out that this difficulty can be overcome by employing the two-dimensional Laplace transform of the above conditional probability density function. Consequently, the above sum of iterated convolutions takes the following form in the transform domain: oo

~(x0,~/~x0)($1, $2) -- E ~(n)($2) ~/(n)(S1 if- S2) n=0

(7.36)

where

~'D(Xo,~)aflXo) ($1,

$2)

= f / ~'DyTl(XO,~)o~flXO ) (u)e-(SlU+S2T)du dT

(7.37)

is the two-dimensional Laplace transform of the conditional probability density function and ~ and h denote the Laplace transforms of the respective functions. Applying the two-dimensional Laplace transformation to formulas (7.26) and (7.30)-(7.34) results in the following

7.4 RECEIVER INTEGRATOR OUTPUT

655

expressions: ~(0) (xo,O)(S1, $2) --/~x0__~(S2)

(7.38)

(xo,o)(sl,s2)-

(s2)

_ 1 (S1 9 [ ~ f l , _ _ = ]n--1 T pfl,__=

[

~(n)

(xo,O)(S1, S2) =

9

(7.39)

if- S2), n = 1, 3, 5 .....

n-2

~ ]

/9x0___~c~[/9//__~0rT Pfl--,c~ (s2)

H fl,--o~

(81+ s2),

(7.40)

n = 2, 4, 6 ....

and ~(o) (xo,1) (S1, S2) -- Pfl+--xo (s1 "4- $2) ,

(xo,1)(S1,S2)--

9

(7.41)

/9//,__xo[/9//,__ct]~-! (Slff-S2)

[ ill,ft._.O~ _]>n~l~P#__,~ ] (s2),

(7.42)

n - 1, 3, 5 .....

~(n) [ n~2 ] (x0,1)(SI' S2) -- /gfl~_xo[/gfl,__0r P//,--0r (s1 if- s2) 9[/gfl___~a]~~ ~ (S2), n -- 2, 4, 6 .....

(7.43)

By e m p l o y i n g expressions (7.38)-(7.43), it is n o w possible to express the Laplace transform of the conditional probability density function given in (7.36) in a more compact form. For the case of the initial state ~flxo - 0, it can be s h o w n that:

~(x0,0) (S1, S2) -- Px0__~c~(s2) /gxo__,c~(s2) q- [i~fl,__~(Sl -}-82)if- Pfl__,o~(sa)Dfl,__o~(s1 -ff 82)] +

1

-- Dfl__,o~(S2)~)fl,__o~(S1 -Jr-S2)

" (7.44)

656

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

Similarly, for the case of the initial state ~ x 0 - 1, one obtains: ~(x0,1) ($1, $2) --

Pfl*--xo($1 -}- $2)

(Sl q- 82)if- -[Pfl '*~162 if- Pfl*--~ +

if-s2)[gfl-~a(s2)la

1 -- p//__,~(S2)p]/~__~(S1 q- $2)

(7.45) As a result, it can be seen that in the transform domain these expressions provide a closed-form solution of the conditional probability density function of YT for a given initial state (x0, ~px0). In this form, the Laplace transform of the corresponding conditional probability density functions can easily be computed for any process, xt, for which the corresponding exit-time probability density functions are known, either analytically or numerically. The closed-form solutions of exit-time probabilities given in (7.25) can be used directly in the particular case of the Ornstein-Uhlenbeck process. 7.4.5 INITIAL STATE DISTRIBUTION

As can be seen from expression (7.8), the probability density function of YT is given in terms of its conditional pdf for a given initial state and the probability distribution of this initial state ~. Therefore, the next step in the computation of the probability density function of YT is to determine the initial distribution by employing the machinery of the 'exit problem' developed above. In this analysis it will be assumed that the input stochastic process to the rectangular hysteresis loop is continuous in time. Indeed, this assumption is naturally satisfied for the diffusion process defined by the SDE given in (7.1). Based on this property, it can easily be shown that the joint probability density function satisfies the following initial condition: ~(x, 7; 0+) = ~o(x, 7; 0-). (7.46) For clarity, it will be assumed that the system reaches a steady state by the end of each bit interval. Therefore, the joint probability density function of the state of the rectangular hysteresis loop at the end of the previous bit interval, q0(x, 7; 0-), may be approximated to be the steady-state probability density function for the prior bit. Since the steady-state joint distribution function is of interest here, one can begin the derivation from an arbitrary initial state (xt0, ~flXto) at some time to << 0. Without loss of generality, Xto < a and ~o~Xto - 0 can be chosen as an initial state. Next, consider some time interval [to, t] that is within the duration of the previous bit interval and let n denote the number of times the

7.4 RECEIVERINTEGRATOR OUTPUT

657

rectangular hysteresis loop switches in this time interval. Since the space of sample paths of the stochastic process can be divided into sets of disjoint events where sample paths have different numbers of switchings it is easy to show that oo

q~(X, 7~ t) -- E

q~(n)(X, 7~ t),

(7.47)

n--0

where

q~(n)(x, 7; t ) d x =_ P r

x < x t < x nt- d x , ~;o~fl x t - 7, } n switchings of ~ in [to, t] "

(7.48)

Furthermore, it is also clear that all sample paths with zero or even numbers of switchings result in the hysteresis loop being in the '0' state, and all sample paths with odd numbers of switchings result in the hysteresis loop being in the '1' state. Consequently, the joint probabilities can be expressed as follows: (x)

(p(x, 7 - O; t) -

E q~(n) (x, ? - 0; t), n=0, even oo

~(x, ? - 1" t ) -

E

q~(n)(X, ~; -- 1; t).

(7.49)

n=l, odd

Green's Function Formulation It is clear that in order to compute the above expressions it is necessary to compute the probability density function of the input xt conditioned on the premise that it does exit the semi-infinite intervals [fl, +cx~) and (-cx~, ~]. Towards this end, let ~b+ and ~b- denote these conditional probability density functions, respectively. Naturally, these functions depend on the characteristics of the input stochastic process xt. In the case of a diffusion process (7.1), there is a well-established mathematical procedure to calculate these conditional probability density functions. For the particular case when the system is modeled by a first-order low-pass filter given in (7.2), the aforementioned conditional probability density functions satisfy the following differential equation: C9C~+

c9"c -

-b(x

- z) c9c~+

- ~ x -}

(bo_n)2 ~2(~+ 2 c~x2 ,

(7.50)

658

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

subject to the boundary and initial conditions: O+(z, x; X')lx=/~= 0,

(7.51)

~b+(~, x;

(7.52)

X !

)[3=0 = c~(x - x ! ), / / < x, x ! < ~ ,

~b- (z, x; x')Ix=~ - 0, ~b-(~, x; x !) I3=0 =

(7.53)

6(x -

x !), - ~

< x, x ! < :r

(7.54)

where x' denotes the value of the input at the beginning of the interval. Based on this definition, it is clear that 4~+ is Green's function of the backward Kolmogorov equation. Similar to the 'exit problem' formulation, its solution is greatly facilitated by employing the Laplace transform. Applying the Laplace transform to equations (7.50)-(7.54), one obtains:

(bc~n)2 d2~ + 2 dx 2

b(x-z)dgP+

s dP+(s, x) = - c ~ ( x - x 1,

~+(s, c+; x') = 0, where ~+(s, x; x') :

Loo ~+ (~, x; x')e-s'~dr.

(7.55) (7.56)

(7.57)

Here, c+ are the exit points for the process xt, which are equal to fl and a, respectively. The solution of the above differential equation is the summation of the homogeneous and particular terms. The homogeneous solution is given in terms of parabolic cylinder functions (for instance, see [18,19,21]). The particular solution (in the time domain) is simply a Gaussian pdf with mean z 4- (x' - z)exp [-b~], and variance a2/2b[1 - exp (-2b~)]. Consequently, the overall expression for the pdf can easily be found by combining the homogeneous and particular solutions and imposing the boundary condition (7.56). In the Laplace transform domain, this results in the following expression:

~•

~ 4x; x') = ~p (s, x;

X'

~'4- ,X'[ (x-z)24"-2(~+-z)a'l~)-s/b(X~--~z)] --(~p (S, C+" ) e .D_slb(~ )

(7.58)

In the above expression, ~p denotes the Laplace transform of the particular solution, whereas the expression in the square bracket is the Laplace transform of the homogeneous solution.

659

7.4 RECEIVER INTEGRATOR OUTPUT

Probability for a Given N u m b e r of Switchings The next step is to compute the probability for n number of switchings of the rectangular hysteresis loop in the time interval [to, t]. In this section, only the case of ?~Xto - 0 will be considered. It is clear that the derivation for ~/~Xto - I can be done in a similar manner. For n = 0, the output of the rectangular loop does not switch and remains at zero. Consequently, it is clear that: (p(0) (x, 0;

t ) d x - Pr

{ x<xt<xq-dx,~flxt-O]

}

no switching in [to, t]

9Pr{no switchings in [to, t] }.

(7.59)

By employing the definitions of Green's function and exit-time probabilities, one obtains the following expression: (p(~

- to, x; Xto)Pxto_,a(t - to)

0" t) = r

- [q~-(-,

x" Xto)Pxto__,~](t- to).

(7.60)

For the case of n = 1, the rectangular loop switches after some random exit time rl and remains in the '+1' state, while the input exits the interval (-cx~, a] and remains in the interval [fl, cx~) at least until time t. Therefore, the joint probability for this case can be expressed as follows: (p(1) (x, 1; t)

-

fo t-to d'clV-('cl, 9r

Xto)V+(t - to

-

"Cl, c~)

( t - to - "cl, x; a)

= Pxto--*a * [P/~,--a~b+( " x; c0](t - to).

(7.61)

For the case of n = 2, the rectangular loop switches to one and back to zero after some random exit times 'cl and "c2, respectively. The input remains in the ( - o c , ~] interval and the output remains in the zero state thereafter. The probability of this event can be computed as follows: (p(2) (x, 0; t)=

~0 t-tO

dr1

~0 t-tO-r1

9[ V - ( - , f l ) r

dr2v- ('cl,

Xto)v+ ('c2, 0r

x; f l ) ] ( t - to - 'cl

= Pxto--,a * Pfl.--~ * [ P f l ~ r

-- "Ca)

x; f l ) ] ( t - to).

(7.62)

Similar to the above formulations, for n = 3 the rectangular loop switches at three random exit times. After the third switching the input remains in the interval [fl, ec) and the output remains in the '+1' state. The

660

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

probability of this event is as follows: r

(X, 1; t)=

~ot-toft-to-zlft-to-~l dzl

dz2

dO

-z2 dO

d1:3

9v- (1;1, Xto)V + (1;2, ~)v- (1;3, ]~) 9Iv + (., ~)~+ (., x; or

to - 1;1 - 1;2 - 1;3)

= Pxto__,~ * Pfl,_~ * P f l ~

9 [P/~,__a~b+ (., x; a ) ] ( t - to).

(7.63)

From the above formulation, it is clear that all even and odd terms can be obtained recursively by the following convolutions: r

(X, ~;; t) = [p/~,__~9 p/~a] 9 r

~/; t), n~2.

(7.64)

In the Laplace transform domain, this results in the following recursive relationship:

~(n) (X, ~; S) -- [pfl~_ccpfl__,~]~p(n-2) (X, ~/; S), n~2.

(7.65)

Calculation of the Joint Probability Density Function in the Transform Domain In summary the joint probability density function of the state variables (the input and output of the rectangular hysteresis loop) can be expressed by the summations given in expression (7.49). Furthermore, each term in these summations is given in terms of the iterated convolutions (7.60)(7.64). However, it is quite cumbersome to evaluate the joint probability density function in the time domain. This difficulty can be rectified by using Laplace transformations where these summations can be expressed in a more compact form. Indeed, by employing the recursive relationship (7.65) and applying the geometric summation to all common even and odd terms, a closed-form expression can be obtained in the Laplace domain. The summation of the even terms results in the following: ~(X, 0; S) --

[pxtO__,~(s) * ~

(S, X; Xto)

]

PXto_+ ~ (S)/9/J~___~ (S) + [ P I ~ ( s ) 9 ~-(s, x; fl)]l 7 ~]~,_~(s-----~~5,----~(s)'

(7.66)

whereas the odd summation results in: q~(x, 1 ; s ) = [P/~,__~(s). ~+(s, x; a ) ] ' l -

79~.__~(s)~~(s) 9

(7.67)

7.4 RECEIVERINTEGRATOR OUTPUT

661

Consequently, the Laplace transform of the joint pdf can be expressed in a compact form and can be computed in a straightforward manner in terms of the closed-form solutions for the exit-time pdfs. Since it is assumed that the system reaches steady state by the end of a bit interval, one is actually interested in the asymptotic value of the joint probability density function. Therefore, by employing the final-value theorem, one can compute the joint probability density function of the initial state from expressions (7.66) and (7.67), as follows: (p(x, 3;; 0+) - (p(x, 3;; 0-) ~ lim sO(x, 3;;s), s---,0

(7.68)

7.4.6 THE PROBABILITY DENSITY FUNCTION OF INTEGRATOR

OUTPUT It is now possible to compute the probability density function of TT and RT in the transform domain. By applying the two-dimensional Laplace transform defined in (7.37) to expression (7.8), it can be found that:

~VYT(Sl, S2) --

oo

dx0 (p(X0, 0; 0)~(x0,0)(S1, $2) +oe

+

ffl

dx0 q0(x0, 1; 0)~(x0,1)(S1, S2).

(7.69)

Here, the two-dimensional probability density function of YT in the transform domain is defined as follows"

~VyT(Sl,S2) ~ f f ~yz(U)e-(SlU+S2T)du dT.

(7.70)

Hence, we have shown that the probability density function of YT can be computed in the transform domain by combining expressions (7.44), (7.45) and (7.66)-(7.69). As described earlier, the input to the binary receiver is the addition of two independent random variables: YT and a Gaussian random variable ET. Therefore, according to Eqns. (7.3) and (7.5), the relevant probability density function for the decision rule is simply the convolution of the probability density functions of these two random variables. In other words, the two-dimensional Laplace transform of the probability density function of RT is given by the product of the two-dimensional Laplace transform ~Syr (Sl, s2) with the one-dimensional Laplace transform of the probability density function of ET"

~RT(S1,$2) -- ~yT(S1,

S2)~,T(S1).

(7.71)

662

CHAPTER 7 Binary Detection and Rectangular Hysteresis Operators

Consequently, the probability density function of the integrator output RT can be found by the following two-dimensional inverse Laplace transform of this product:

g~RT(U)-- //~TT(S1, S2)~ET(S1)e(SlU+S2T)ds1 ds2.

(7.72)

It is worthwhile to note here that by only taking the one-dimensional inverse Laplace transform in terms of the variable s2 and substituting Sl = jco results in the so-called generating function of the random variable RT.

7.5 S U M M A R Y In summary, it has been shown that the probability density function of the integrator output ~VRr can be calculated from Eqns. (7.44), (7.45), (7.66)-(7.69), (7.71) and (7.72). Once this function is determined, it is possible to calculate various system performance measures for this system. For instance, the bit error probability of the system can be calculated as follows:

Pr {Error} = Pr {~ -- 0}

/T+OO /2 ~RTI~=o(U) du+Pr {~ -

f T/2

1} e-c~ ~RTI~=I(U) du. (7.73)

where ~VRTI~=0and g~azI~=1are the conditional probability density functions of the random variable RT for ~ = 0 and ~ = 1, respectively. In order to determine the possible advantages of hysteresis, a series of simpler Monte Carlo-based simulations for two signal-to-noise ratios and four hysteresis-loop widths were performed. These simulations showed that a gradual increase in the probability of error results with increasing width in the hysteresis-loop. These results indicate that there exist system parameters where hysteresis limiters may not be beneficial compared with a hard-limiter when the cost function depends only on the probability of bit error. On the other hand, alternative cost functions that also include the switching rate may easily render a hysteretic system advantageous compared with a hard-limiter system. Indeed, for system performance cost functions where switching has a cost, or in problems with different sets of noise parameters, detectors with hysteresis may prove to be advantageous. The mathematical formalism presented in this discussion may be very useful to investigate such problems. However, a detailed numerical study of such problems is beyond the scope of the work presented here. The accurate numerical calculation of the expressions presented here may pose certain challenges depending on the system noise parameter. In

7.6 CONCLUDING REMARKS

663

principle, numerical calculations of the probability density functions presented above can be accomplished by most mathematical software packages and mathematical function libraries. As an illustrative example, sample calculation details of a similar calculation in hysteresis and aftereffect can be found in [18] and the references contained therein. Surprisingly, the most challenging aspect of such calculations turns out to be the evaluation of the numerical inverse Laplace transformations. This is due to the fact that the Laplace transforms of the probability density functions do not in general have closed-form inverse transformations. Moreover, most calculations of interest for such probability density functions involve cases where the bit error rate is very small. This naturally implies that the probability density functions, ~VyT(U), have very small amplitudes, except for small intervals of support near u = 0 and u = T, depending on the initial state of the rectangular loop and the binary input signal. Since an accurate calculation poses a substantial numerical computation problem, the scope of this discussion is limited to the derivation of key expressions and leaves the practical numerical computations to other researchers.

7.6 C O N C L U D I N G

REMARKS

In conclusion, a binary detector with hysteresis-type nonlinearity has been investigated and closed-form expressions in the Laplace transform domain have been derived for probability density functions that are necessary to compute bit error rates and probability of error. Naturally, in a typical communication system there are several other important issues related to system performance, such as jitter and clock recovery. In this discussion the system is considered to have perfect synchronization between the detector and the noisy input data. Therefore, it is assumed that there is no performance degradation in the receiver due to errors in the timing of bit intervals. Although this important topic is not elaborated in this discussion, it turns out that the machinery of the' exit problem' and the associated techniques presented here may be very useful in determining jitter error, hence, overall system performance. This is due to the fact that timing errors in the detector are directly related to the level-crossing problem of the noisy input signal in a clock recovery circuit. Since such a phenomenon can be posed by employing the formalism of an exit problem, a coupled approach may prove to be very beneficial where the integrator limits [0, T] are themselves random variables that depend on the level crossing of the input process. As was mentioned earlier, hysteresis is present (or is introduced) in many engineering and scientific problems. Therefore, it is expected that

664

CHAPTER7 Binary Detection and Rectangular Hysteresis Operators

the mathematical framework presented in this work may form the basis for several future directions of research and not simply be limited to the application of bit-rate transparent detection in optical-network links.

References 1. P. B. Hansen, C. Jorgensen, S. L. Danielsen, B. Mikkelsen, and K. E. Stubkjoer 'Cascadibility of broadcast and select switch blocks with interferometric wavelength converters at 10 Gbit/s', Proc. of Optical Fibre Communications '97, Dallas, USA, February 16-21 1997, p. 6. 2. K. Inoue and M. Jinno, 'Signal improvement effect in O / E - E / O conversion using an electro-optic absorption (EA) modulator', IEEE Photonics Technology Letters 9(1), 109-111 (1997). 3. P. Ohlen and E. Berglin, 'Noise accumulation and BER estimates in concentrated nonlinear optoelectronic repeaters', IEEE Photonics Technology Letters 9(7), 1011-1013 (1997). 4. P. E. Green, F. J. Janniello, and R. Ramaswami, 'WDM protocoltransparent distance extension using R2 remodulation', IEEE Journal on Selected Areas of Communications 14(5), 962-967 (1996). 5. R. A. Barry and M. Medard, Lasers and Electro-Optics, 1998, CLEO '98. Technical Digest, 3-8 May, 455-456 (1998). 6. L. D. Davisson and P. Papantoni-Kazakos, 'On the distribution and moments of RC-filtered hard-limited RC-filtered white noise', IEEE Trans. Inform. Theory 19(4), 411-414 (1973). 7. J. E. Mazo, R. F. Pawula, and S. O. Rice, 'On a nonlinear problem involving RC Noise', IEEE Trans. Inform. Theory 19(404-411), 63-72 (1973). 8. R.F. Pawula, 'The probability density and level-crossings of first-order non-linear systems driven by the random telegraph signal', Int. J. Control 25(2), 283-292 (1977). 9. C. E. Korman, R. Barry, and U. C. Kozat, 'On binary detection with hysteresis', SIAM Journal on Applied Mathematics (SIAP) 62(5), 1794-1809 (2002). 10. R.F. Pawula and S. 0. Rice, 'On filtered binary processes', IEEE Trans. Inform. Theory 32(1), 63-72 (1986). 11. R.F. Pawula, 'Level crossings of filtered dichotomous noise', Physical Review A 37(5), 1731-1735 (1988). 12. R.F. Pawula, J. M. Porra, and J. Masoliver, 'Mean first-passage times for systems driven by gamma and McFadden dichotomous noise', Phys. Rev. E 47(1), 189-210 (1993).

REFERENCES

665

13. M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer-Verlag, Berlin (1980). 14. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, New York (1991). 15. E. Della Torre, Magnetic Hysteresis, Wiley-IEEE Computer Society Press, (2000). 16. G. Bertotti, Hysteresis in Magnetism for Physicists, Materials Scientists, and Engineers, Academic Press, (1998). 17. I.D. Mayergoyz and C. E. Korman, 'The Preisach model with stochastic input as a model for aftereffect', J. App1. Phys. 75(10), 5478-5480 (1994). 18. C. E. Korman and I. D. Mayergoyz, 'Switching as an exit problem', IEEE Trans. on Magnetics 31(6), 3545-3547 (1995). 19. C. E. Korman and I. D. Mayergoyz, 'Presiach model with stochastic inputs as a model for aftereffect', IEEE Trans. on Magnetics 32(5), 4204-4209 (1996). 20. C. E. Korman and E. Della Torre, 'Definition of the magnetic ground state using Preisach-based aftereffect models', Physica B 275, 28-33 (20O0). 21. A. V. Balakrishnan and C. W. Helstrom, Communication Theory, InterUniversity Electronics Series, Vol. 6, Ch.2, McGraw-Hill, New York, (1968), pp. 26-87.

CHAPTER 8

Hysteresis and Unemployment: A Preliminary Investigation, Julia Darby, Rod Cross, and Laura Piscitelli, Department of Economics, University of Strathc/yde, UK

Contents 8.1 8.2 8.3 8.4 8.5 8.6

Introduction Hysteresis An Illustrative Model Computing the Hysteresis Variable Hysteresis and UK Unemployment Concluding Remarks Appendix References

667 669 670 677 685 687 687 696

8.1 I N T R O D U C T I O N The natural rate hypothesis applies the axiom of monetary neutrality to the equilibrium rate of unemployment. This means that monetary shocks can affect the actual rate of unemployment but not the equilibrium rate, which is determined by a set of real variables, z, which represent the endemic 'structural characteristics of labor and commodity markets' ([1], p. 8). The models that embody the natural rate hypothesis (see [2] for a brief survey) tend to have the stronger implication that only sustained real shocks can generate sustained changes in the equilibrium rate of unemployment.

*This chapter benefited from numerous discussions with Jonathan Ireland.

The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

668

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

In a 'battle of the mark-ups' model, for example, an increase in the z index of mark-up pressures arising from unemployment benefits, minimum wages and so on, followed by a fall in the z index back to its previous value, would see the natural rate return to the status quo ante. Similarly, in a 'structural' model of Phelps-type shocks [3] to real variables such as relative oil prices, real interest rates or exchange rates that are subsequently reversed would generate only a temporary perturbation in the time path for equilibrium unemployment. The question is whether the natural rate decomposition of variables that can and cannot change the equilibrium rate of unemployment is evidentially coherent. For European countries it is by no means clear that natural rate variables can explain the ratchets, predominantly but not exclusively, upwards in unemployment rates since the 1970s (see [4,5]). The present paper focuses on the UK experience. Here some investigators have failed to find cointegration between unemployment and natural rate variables ([6], for example), with other studies ([7], for example) finding cointegration only when 'unnatural' variables such as unemployment duration are introduced. The present paper outlines an alternative approach to the explanation of equilibrium unemployment. Instead of ruling out nominal variables, or temporary shocks to real or nominal variables, we allow aspects of the past profile of such variables to help determine the equilibrium rate of unemployment. The analytical innovation is to introduce hysteresis into the processes underlying the determination of equilibrium unemployment. This is done by specifying that economic agents respond in a nonlinear way to shocks; and by respecting the heterogeneity of economic agents by allowing them to respond differently to aggregate shocks. The key implications are that the economic system displays remanence, in that the application and removal of a shock will not be accompanied by a return to the status quo ante; and that the equilibrium rate of unemployment contains a selective memory of past shocks, retaining only the nondominated extremum values of the shocks experienced. This analysis of hysteresis produces some sharp contrasts with treatments of hysteresis effects elsewhere in the economics literature. In the 'hysteresis as persistence' usage, shocks can produce only persistence in the deviations of actual unemployment from unperturbed or homeostatic natural rate equilibria; whereas the presence of hysteresis actually means that shocks can change unemployment equilibria. In the' unit root hysteresis' usage, the equilibrium rate of unemployment is a palimpsest bearing the marks of all past shocks, with a positive shock followed by a negative shock of equal size leaving no net effect; whereas hysteresis actually implies a selective, erasable memory of shocks, and remanence,

8.2 HYSTERESIS

669

in that positive and negative shocks of equal size do not cancel each other out. The chapter is organized as follows. Section 8.2 discusses hysteresis and its relevance to economic shocks and unemployment. Section 8.3 outlines a simple model of hysteresis based on firms having two separate triggers for market entry and exit (see [8]). Only active firms employ labor, so the entry-exit decisions determine employment. The possible implications for the equilibrium rate of unemployment are sketched. Section 8.4 outlines the steps involved in calculating a hysteresis memory index, and presents a program for calculating such hysteresis indices for economic time series. Section 8.5 presents some tentative, preliminary econometric results, which are designed to investigate the explanatory power of hysteresis variables with regard to UK unemployment.

8.2 H Y S T E R E S I S The term hysteresis comes from the Greek 'to be late, or come behind'. The term was first coined for application to scientific explanation by the physicist Ewing (1881) to refer to effects (in terms of magnetization) that remain after the initial cause (the application of a magnetizing force) is removed. Such effects have subsequently been discovered or invoked in relation to a wide array of physical, biological and social phenomena. A general account of hysteresis as a system property has been provided in [9]. The key elements required to produce hysteresis are some form of nonlinearity in the way the elements in a system respond to shocks; and heterogeneity in the elements and therefore in their responses to shocks. The key implications of hysteresis are remanence, in that the application and reversal of a shock will not be followed by a return to the status quo ante; and a selective memory, in which only the nondominated extremum values of shocks remain in the memory bank, dominated extremum values being wiped out (see [10] for a general account of hysteresis in economic systems). Standard economic analysis assumes that economic equilibria are homeostatic, in that the reversal or removal of a temporary shock will be accompanied by a return to the initial equilibrium. The issue of hysteresis raises the question of whether this assumption holds in economic systems. Marshall [11] thought that this assumption was likely to be violated in actual market processes, citing the effects of the shock to the supply of cotton during the American Civil War as an example. At a more aggregate level Keynes [12] answered the question 'are economic systems self-adjusting?' in the negative. If temporary shocks can have permanent effects, economic equilibria become characterized by heterostasis, there now being a range

670

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

of possible equilibrium values, with the actual equilibrium realized being determined by the temporary shocks experienced. Hysteresis thus involves stronger properties than those conveyed by the use of the term to describe persistence or zero/unit roots. In the persistence case the natural rate equilibrium is unchanged by shocks affecting actual unemployment, whereas hysteresis implies that each new extremum value of the shocks experienced will lead to a new unemployment equilibrium. In the zero/unit root case all the shocks experienced shape the equilibrium, whereas hysteresis involves only the non-dominated extremum values of the shocks counting in the equilibrium selection process. In the policy literature the key distinction is usually perceived as being between structural and actual unemployment: '...economic analysis generally distinguishes between the actual unemployment rate prevailing, at any time, and the "natural" (or "structural") unemployment rate' ([13] Part 1, p. 66). The presence of hysteresis implies that temporary shocks can change the structural dynamics which help determine equilibrium unemployment (see [14]). Thus, in contrast to the natural rate hypothesis, the shocks associated with the peaks and troughs of actual unemployment are themselves part of the process determining equilibrium unemployment. 8.3 A N I L L U S T R A T I V E

MODEL

The simplest form of nonlinearity in the Krasnosel'skii and Pokrovskii [9] hysteresis analysis is the piecewise linear case analyzed in [15]. This framework is well suited to micro foundations based on discontinuous adjustment [16]. The presence of fixed costs of adjustment in the form of, for example, the sunk costs associated with investment or market entry [8], including entry into export markets [17], implies the existence of separate triggers for upward and downward adjustment. The following exposition is based on [18], which analyzes market entry and exit under sunk costs in a Dixit-Pindyck [8] framework. The market has M potential suppliers. The number of active firms is N. When active, that is, in the market, each firm produces one unit of output and employs one unit of labor. While out of the market firms produce zero output and employ zero units of labor. Each firm faces sunk costs of market entry, the ith firm requiring a market price of p>~ato induce entry and a price of p~b to induce exit. Figure 8.1 illustrates the two switching points. In the range b < p < a the firm will either be active or inactive depending on its previously acquired propensity, which turns on whether this range has been approached from above or below.

8.3 AN ILLUSTRATIVE MODEL

671

P

b

a MARKET PRICE, P

FIGURE 8.1 Switching points.

Heterogeneity is introduced by allowing the a and b switching points to differ between firms: '...different firms have different technologies or managerial abilities ...historical accidents may leave different firms with stocks of capital that are differently situated relative to their action thresholds ...then they will have different action thresholds...' ([8], p. 421). Each firm is thus identified by a pair of a and b switching points, which define its hysteron or hysteresis operator Fab which maps from shocks to prices into output and employment. The market price is specified as: pt - x t f ( q t - 1 ) , t = 1, 2, 3 ....

(8.1)

where x is an aggregate shock, to interest rates or exchange rates for example, faced by all firms, and f(q) is the deterministic component of the inverse demand function, with qt-1 ~- N t - 1 / M . The dynamics of 8.1 turn on how pt+l determines qt+l, which can be written: 1

M

qt+l -- -~ ~_~ Faibi(Yt+lf(qt))

(8.2)

i=1

In a shockless economy, i.e. where x is constant, it can be shown that every initial condition converges to a fixed point or a two-period solution in which q swings between two points ([18], p.571). The interesting question is what happens in an economy with shocks.

672

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

; / I I

I I

I I

I I I I I

1

'

2

I I I I

I I I I

II I I

II I I

I

I I

I I

3

4

5

I

Time, t

FIGURE 8.2 A sequence of shocks. Consider the effects of a sequence of aggregate shocks that generate the price fluctuations illustrated in Fig. 8.2. As the price rises to Pl firms with aKpl enter the market and employ labor, as the price falls to P2 firms with b~p2 leave the market and cease to employ labor, and so on. The division between active and inactive firms is illustrated in Fig. 8.3, which uses the Mayergoyz [15] half-plane diagram in which each firm is represented by its (a, b) switching characteristics. The (ao, b0) vertex of the triangle is determined by boundary conditions. The distribution of firms within the triangle can be seen as depending, inter alia, on the 'structural characteristics of labor and commodity markets', such as the 'z' variables stressed in the natural rate literature. Thus a more favorable set of 'z' characteristics would tend to shift the (a, b) values south-westwards in the triangle, so yielding higher activity and employment levels for any given set of aggregate shocks. Referring to Fig. 8.3, the rise in price to Pl serves to create a horizontal partition between the N1 active firms below the Pl line, and the (M - N1) inactive firms above the line. The subsequent p2, P3 and P4 shocks then trace out a staircase partition between N4 and (M - N4), the coordinates being (Pl, P2), (P3, P2) and (P3, P4). This illustrates the memory property of systems with hysteresis: only the extremum values of the shocks experienced count. The wiping-out property can be seen by considering the effect of the rise in price to P5, illustrated in Fig. 8.2. This dominates the previous local maximum price at P3 and so wipes the effect of this dominated extremum value from the memory bank. Thus, the coordinates of the staircase partition between N and (M - N), (P3, P2) and (P3, P4) are removed from the memory. This leaves the new staircase partition between

8.3 AN ILLUSTRATIVE MODEL

673

FIGURE 8.3 Staircase partition between active and inactive firms.

the unshaded area in Fig. 8.3, N5, and the shaded area, (M - N5). Thus the memory of systems with hysteresis is selective: only the nondominated extremum values of the shocks experienced retain an effect. The contrast with the unit root characterization of the memory properties of time series is interesting. Unit root tests characterize time series as short or long in memory depending on whether the root is closer to zero or unity, the extreme unit root case implying an infinitely long memory. With hysteresis the memory is a derivative of the pattern of shocks. If the major expansionary and contractionary shocks have occurred recently the memory will be short, the previous shocks having been dominated and therefore eliminated from the memory. Otherwise undominated major shocks from the more distant past can impart a long memory. Figure 8.4 reproduces the results of simulating the model in (8.1) and (8.2), with the aggregate demand shock x and the distributions of a and b switching values being specified by random number generators [18]. The nonstochastic component of the inverse demand function was specified as f(qt) - a(flqt + 1) -1, with the a intercept being set at 0.8 and the fl slope parameter at 1.4, and 1,200 iterations were used. In response to the price fluctuations indicated by the dark line the proportion of active firms N / M follows the pattern indicated by the lighter line.

674

CHAPTER8 Hysteresis and Unemployment: A Preliminary Investigation 0.8 | k / , Pt

Nt/M: Proportion of Active Finns Pt: Price Fluctuations

0.7

0.6

,,._,

Z

~'~

os

2

0.4

0.3

O. 0

1O0

200

300

400

500

600

700

800

900

1000

11O0

1200

TIME, t

FIGURE 8.4 Price fluctuations and the proportion of active firms. 8.3.1 RANGES FOR EQUILIBRIUM UNEMPLOYMENT

It is, of course, a major step to move from the simple model of hysteresis above to the equilibrium rate of unemployment. The model, however, does offer some guidelines. The 'structural characteristics of labor and commodity markets' or 'z' variables stressed in the natural rate literature can be seen as affecting the (a, b) values of firms. Thus, a more favorable set of z characteristics would tend to reduce the a and b values of firms, making firms more likely to be active and employing labor for any given sequence of aggregate shocks, thus reducing the rate of inactivity or' unemployment' (M - N)/M. T h e innovation is that the sequence of shocks experienced, in the form of the nondominated extremum values, also shapes the rate of inactivity or unemployment. The implication is that, for any given set of 'z' characteristics, there will be a range of feasible equilibrium unemployment rates. Each new extremum value of the shocks will, by changing the partition between active and inactive firms, change the equilibrium unemployment rate. The inflation-unemployment interaction suggested by this hysteretic equilibrium can be written as: pt - pet = A a p t - F [ u t - uh~],

(8.3)

g[zt, X t ] ,

(8.4)

Ut - -

8.3 AN ILLUSTRATIVE MODEL

675

SHOCKS Xt>0

Z+

4

5

v

Time, t Z-

SHOCKS xt<0

FIGURE 8.5 Contractionary and expansionary shocks.

uh~ ~- f[zt, ht ( x ) ] .

(8.5)

Equation (8.3) follows the natural rate hypothesis, except in specifying the equilibrium as hysteresis-haunted; Eqn. (8.4) is standard in saying that actual unemployment depends on the contemporary value of the aggregate shock x as well as on the z variables; and Eqn. (8.5) specifies the hysteresis equilibrium for unemployment, uh*, as depending on a hysteresis index of the past nondominated extremum values of the shocks h(x), as well as o n z.

To illustrate how this system works, consider the sequence of aggregate shocks, to interest rates or exchange rates say, illustrated in Fig. 8.5. Whether or not a particular interest rate or exchange rate is expansionary or contractionary depends on z. The horizontal line on the diagram is thus conditioned on a particular value of z, ~. A less favorable set of z characteristics, z +, would shift the horizontal line upwards, making any given interest rate or exchange rate more contractionary or equivalently less expansionary; and vice versa for a more favorable shift in z characteristics to Z--o

Figure 8.6 illustrates the implications for unemployment and inflation of the shocks given in Fig. 8.5. The contractionary shock reaching a trough in period 2 sees unemployment rise from 1 to 2. The actual is above the equilibrium rate of unemployment, so the rate of inflation falls. As the contractionary shock fades in the face of real balance effects, the system does not retrace its steps back to the original equilibrium, as the natural rate hypothesis would imply, but instead reverts to a higher equilibrium

676

CHAPTER8 Hysteresis and Unemployment: A Preliminary Investigation u= uh* UNEMPLOYMENT, u

i

2

3

u
A2p >0 7

6

h*

,9[

EXPANSIONARY SHOCK,

CONTRACTIONARY SHOCK, xt<0

xt>O

Z-

Z

Z+

FIGURE 8.6 Hysteresis and ranges for equilibrium unemployment. unemployment rate in period 3; the extremum value of the shock experienced in period 2 remains in the memory bank. The expansionary shock in period 4 sees unemployment fall below the equilibrium rate, the rate of inflation rises, which stimulates adverse real balance effects. Unemployment rises to a new equilibrium rate in period 5, which is lower than the preceding equilibrium rate because the period 4 shock is retained in the memory bank. And so on. Take this economy through a further sequence of shocks and a range of equilibrium unemployment rates is traced, corresponding to the intersections with the vertical axis. The position of the vertical axis itself in Fig. 8.6 is conditioned on a particular value of z, ~. If the z characteristics move favorably to z- the equilibrium rates of unemployment would be lower for any given x shocks, and vice versa if there are unfavorable developments to z +. Thus, in an economy exposed to shocks there is a range of possible values for equilibrium unemployment. Rather than leaving the equilibrium indeterminate, the hysteresis hypothesis specifies the specific subset of the values of the shocks that determine the realized equilibrium. The 'z' variables invoked by the natural rate hypothesis are also included, and would be the sole determinants of equilibrium unemployment in a

8.4 C O M P U T I N G T H E HYSTERESIS VARIABLE

677

shockless economy. The implications for policy in a world of shocks, however, are substantially different, macro policy measures being reinstated as having potentially lasting influences on unemployment as well as on inflation.

8.4 C O M P U T I N G THE HYSTERESIS VARIABLE In this section the procedure for computing a hysteresis index based on the model of the previous section is outlined. The output of the system considered in the previous section can be expressed as

y(t)-ffg(a,b)dadb-~ffg(a,b)dadb N(t)

(8.6)

k--1 Qk(t)

where the set N is considered as the union of n(t) rectangular trapezoids Qk. From visual inspection of Fig. 8.3, it is clear that calculating the area N requires at least four steps. Step I involves specifying the (a0, b0) vertex of the right-angled triangle above a -- b. Step 2 requires selecting the nondominated extremum values from the time series for the input variable. Step 3 involves calculating the N area under the staircase partition between N and (M - N) as the union of the rectangular trapezoids specified as Qk in (8.6) above. Step 4 requires the specification of a weight function g(a, b) that specifies how much each (a, b) switching combination contributes to aggregate output. This last step is perhaps the most troublesome. Ideally, we would have cross-sectional information on the distribution of (a, b) switching points amongst agents. Then, by applying the means of the order statistics, the distribution for the ordered set N of (a, b) points would be determined. Without such information, a sensible procedure would start with a simple specification for the weight function g(a, b), which could then be varied to test for sensitivity to the assumption used. Accordingly, in what follows it is assumed initially that the weight function is uniform, i.e. takes a constant value 1. In this case, the output is given by the unweighted sum of the areas of the trapezoids n(t)

y(t) - ~ Qk(t).

(8.7)

k=l

Step 1. This step is made only once, at the beginning of the procedure, whereas Steps 2 and 3 are performed recursively, so that, at any point in time, the sequences of extrema and the value of the hysteresis output are updated. Mayergoyz [15] defines a0 = max{a, (a, b) ~ T} and b0 =

678

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

to (a)

t~

t2

t3 t

t4

t5

t ot~

t2

t3

(b)

t4

t5 t6

t

FIGURE 8.7 Possible inputs for the Preisach nonlinearity. min{b, (a, b) e T}, where T is the right-angle triangle. Given that we cannot compute ao and bo on the basis of experimental evidence, we scale the triangle T by assuming b0 -- min {x(t), t = 1, 2 . . . . .

Zmax }

(8.8)

a0 = max{x(t), t = 1, 2 . . . . .

Zmax}.

(8.9)

and These assumptions would be problematic if they caused the hysteresis output measure to be highly sensitive to the initial value of the input variable. To see, however, that this is not a major problem, consider the alternative initial values for the input variable illustrated in Fig. 8.7. The time evolution of the set N for the input sequence in Fig. 8.7(a) is given in Fig. 8.8, while the equivalent evolution for the sequence in Fig. 8.7(b) is provided in Fig. 8.9. By comparing Fig. 8.8 and Fig. 8.9 it can be seen that the different initial conditions only affect N as long as the global m a x i m u m and minimum have not been hit. Step 2. Let M be the matrix of elements M ( i , t), i = 1, 2 ..... t = 1, 2 ..... Tmax, where M ( i , t) is the ith nondominated maximum at time t, where we assume to = I for computational reasons. And let m be the matrix of elements m(i, t), i = 1, 2 ..... t = 1, 2 ..... tmax, where m(i, t) is the ith nondominated minimum at time t as illustrated in Fig. 8.10. For a given time t, defined by x() = M(1, t). Obviously, this is the first nondominated maximum since it dominates all previous maxima and is not dominated by any following maxima. By definition, it also removes the previous minima, so that the first nondominated minimum is: m(1, t) = min x(j),

j~[t-~ t]

(8.10)

8.4 COMPUTING THE HYSTERESIS VARIABLE

679 a

a

!,

x,

)b

(a)

(b)

a

a

X3

-b

(c)

X2

-

X5

,/

b

i/

(e)

(d)

X~

(f)

FIGURE 8.8 State of the Preisach nonlinearity for the input from Fig. 8.7(a). w h e r e t I is defined by m(1, t') = x(t 1) and m(1, t) removes all n o n d o m i nated extrema that are b e t w e e n M(1, t) and itself. The next n o n d o m i n a t e d m a x i m u m is then M(2, t) = m a x x(j), (8.11) j~[tl, t] w h e r e t~- is defined by M(2, t') - x(t-~). The second n o n d o m i n a t e d minim u m is: m(2, t) = m i n x(j), (8.12) j~[tf t] with t 2 defined by m(2, t') = x(t2). The complete sequence of n o n d o m i nated m a x i m a a n d m i n i m a is then:

M(k, t) -

m i n x(j), j~[tk_l t]

m(k, t) = m i n x(j), j~[t-~ t]

such that M(k, t') = x(t~-); such that m(k, t') = x(tk).

(8.13) (8.14)

Step 3. Once the sequence of n o n d o m i n a t e d extrema has been selected, the c o m p u t a t i o n of the areas of the trapezoids in Fig. 8.10 is relatively straightforward. Referring to Fig. 8.11, s u p p o s e at time t we have comp u t e d the sequence of n o n d o m i n a t e d extrema up to i = k - 1, and let T(t)

680

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation a

a

a

b

(a)

b

(b) a

a

X~

(d)

(c) a

b

~b

b x5

(e)

(f)

(g) FIGURE 8.9 State of the Preisach nonlinearity for the input from Fig. 8.7(b).

be the shaded area of the set N. Use Mk and mk as shorthand for M(k, t) and m(k, t), respectively. The p r o g r a m for calculating the hysteresis variable updates the sequence of extrema and the area T(t) at the same time: first, the m a x i m u m Mk is c o m p u t e d and then the area of the triangle having a vertex (Mk, mk-1) is a d d e d to T(t), so that

T(t) -- Tp(t) +

(Mk

-

-

ink-l) 2 2 '

(8.15)

8.4 COMPUTING THE HYSTERESIS VARIABLE

681

FIGURE 8.10 Detailed structure of a state of the Preisach nonlinearity.

FIGURE 8.11 Computing the output of the Preisach nonlinearity, part 1.

where Tp(t) denotes the previous value of T(t). Note that the p r o g r a m as constructed is more general since it allows for the cases where bo and / or the n o n d o m i n a t e d extrema are negative. Next, the m i n i m u m mk is c o m p u t e d and the area of the triangle with vertex in (Mk, mk) is subtracted from the

682

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

FIGURE 8.12 Computing the output of the Preisach nonlinearity, part 2. value computed in (8.15):

T(t) = T p ( t ) -

(Mk -- ink) 2 2 '

(8.16)

Thus the shaded area in Fig. 8.11 is augmented by the area of a trapezoid, as illustrated in Fig. 8.12. Applying the same procedure recursively, the evolution of the set N over the rest of the time series can be computed. The Program The following program, see Fig. 8.13, is written in RLAB and is available from the authors on request (the RLAB programming language is copyright 1993-94 Ian R. Searle). For programming reasons v denotes the input vector, tp(j) denotes t~ and tm(j) denotes t~. Step 4. The sensitivity of the hysteresis index variable as calculated in the program to the assumption of the form of the weight function g(a, b) is best investigated using an actual time series example. Fig. 8.14 graphs a time series for the UK real effective exchange rate as calculated by the IMF 1950-1996 (IMF, International Financial Statistics with the pre-1965 data kindly provided by the IMF Research Department). Figure 8.15(a) graphs the hysteresis index derived from this variable ht(x), under the assumption that g(a, b) is uniform. The h(x) variable

8.4 COMPUTING THE HYSTERESIS VARIABLE

683

Clear format(10) v=readm ( "v.dat" ); n-length(v);

u--v; ~0=~bs(m~x(u)); b0--abs(min(u)); for (t in l:length(u)) { k-l; M [k;t]-max (u [1:t]); tp [k]-maxi(u [1:t]); if (min(u)LO) T[t]-(M[k;t]-b0)2/2; else

Tit]= (M[k;t]4-b0)2/2; tt-tp[k]; while (ttit) { z--u[tp[k]:t]; m[k;t]-min(z); tm [k]-t p [k]-1+ mini(z); T[t] =W[t]- (M[k;t]-m[k;t]) f~/2; if (tm[k] it)

{

w--u[tm[k]:t]; M[k+l;t]=max(w); tp[k + 1]=tm[k]- 14-maxi(w); T[t]=W[t]4-(M[k4-1;t]-m[k;t])2/2; tt=tp[k4-1]; k-k+l; else

tt=tm[k]; k=k4-1; }}} FIGURE 8.13

Text of the program.

generated under the assumption of the uniform distribution can then be compared with those generated under alternative assumptions on g(a, b). In Fig. 8.15(b) a normal distribution is used, in Fig. 8.16(a) an exponential, and in Fig. 8.16(b) a Poisson distribution. Visual inspection suggests that h(x) is not hypersensitive to the specification of g(a, b), though there are, of course, an infinity of specifications that could be tried. Note that the independence of the output of the Preisach model with respect to the distribution of micro-units is well known to experimental physicists and is often referred to as the 'statistical stability' of the Preisach model [6,15,19,20]. The hysteresis variable computed by the program is the output of a nonlinear transformation of the input variable that detects the nondominated extrema and combines them in a nonlinear way. The hysteresis

684

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation 120

110

100

90

80 1

5

9

13

FIGURE 8.14

17

21 25 time

41

45

0"0015" I

~176176 1

(b)

6 11 16 21 26 31 36 41 46 time

Sensitivity of hysteresis index, part 1.

0.008T

0.008 T

0.0064

0.006 + ~"

~" 0.004

l

0 ~:::::::::::::::::::~: ................... 1 6 11 16 21 26 31 36 41 46 time

FIGURE 8.16

\,,,

0 ~:::::::::::::::::::~,,i ,,,,,,,,:,,,,,,: ,,,,:,,,,:::

,,,,,,,,,,,,,,,,,,,,,,,,,,,',,,,,,,,,,,,,,,,,,,,

1 6 11 16 21 26 31 36 41 46 time

FIGURE 8.15

(a)

37

0.002 T

~" 0.03 0.02

(a)

33

UK real effective exchange rate.

0.06 0.05 ~- 0.04

0.01 0

29

0.004

o

T-~:::::::::::::::::::l .................... 1

(b)

6 11 16 21 26 31 36 41 46 time

Sensitivity of hysteresis index, part 2.

variable appears smoother than the input variable, reflecting the fact that it records only the nondominated extrema. This is also confirmed by comparing a plot of the spectral density function of the input variable with that

8.5 HYSTERESISAND UK UNEMPLOYMENT

685

of its hysteretic transformation. Plots of the autocorrelation function and conventional unit root tests also indicate that the hysteresis variable tends to reflect the stationarity properties of the input variable.

8.5 HYSTERESIS A N D UK U N E M P L O Y M E N T Despite the conventional wisdom on the status of the natural rate hypothesis, evidence that the 'z' variables invoked by the hypothesis, such as unemployment benefits, minimum wages and wage bargaining power, cointegrate with unemployment is rather thin on the ground. With relation to UK unemployment, Darby and Wren-Lewis [6] failed to find cointegration, for example, though Westaway [7] was somewhat more successful in an equation involving the proportion of the population of working age not in work. In view of this it is interesting to see if hysteresis variables of the type suggested by the analysis of Section 8.3, and calculated according to the program in Section 8.4, figure in the cointegrating or equilibrium relationship for UK unemployment. In the econometric results that follow, Johansen's procedure was used to see if a unique cointegrating vector could be identified involving the log of the UK unemployment rate (LUPC) and to see if a valid error correction model (ECM) process could be estimated for LUPC. A vector error correction model underlies the estimation and testing framework employed. The data are annual and the estimation period is 1959-1996. Details are provided in the Appendix. We were able to identify a single cointegrating vector. Data admissable restrictions on the loading's 'alpha' matrix were imposed such that the cointegrating vector only feeds into the dynamic equation for LUPC, supporting the idea that there is a single valid ECM that determines movements in the log of the unemployment rate. The restricted cointegrating vector is reported below. 8.5.1 RESTRICTED COINTEGRATING VECTOR

Likelihood Ratio Test: The result of a likelihood ratio test of the null hypothesis that a single cointegrating vector exists in the data and feeds only into the LUPC equation is provided below.

LUPC

=

2.2977REPR(0.42) +constant

11.599HLNER+ 0.004704HRR+ 3.8666HLRPOIL (2.00) (0.00063) (0.40)

686

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

1.5

1.0

0.5

0.0 1959 1961 1963 1965 1967 1969 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989

Years

FIGURE 8.17 The restricted cointegrating relation.

Restricted alpha:

LUPC -0.86554(0.13)

REPR 0.0

HLNER 0.0

HRR 0.0

HLRPOIL 0.0

The 'natural' variable in the cointegrating vector is the replacement ratio (REPR). The 'unnatural' variables entering the cointegrating vector reflect the nondominated extremum values of two types of shock. The first type of shock is that registered in the nominal exchange rate. The program outlined in Section 8.4 was used to construct a hysteresis variable for the log of the nominal effective exchange rate (HLNER). The presence of this variable in the cointegrating vector suggests that money neutrality does not hold for equilibrium unemployment in the UK. The other type of shock registered in the cointegrating vector involves perturbations in real variables. Hysteresis variables were constructed for the log of the real price of oil (HLRPOIL), measured as the average of the Brent, Dubai and Alaskan prices in US$ per barrel, and converted to s using the spot exchange rate, then deflated using the GDP deflator; and for the real ex post interest rate, constructed from the three-month Treasury Bill rate and the CPI inflation rate (HRR). The presence of these hysteresis variables in the cointegrating vector suggests that perturbations in, and not necessarily sustained shifts in, such real variables change the equilibrium rate of unemployment in the UK. These results are preliminary, but certainly of some interest.

8.6 CONCLUDING REMARKS

8.6 C O N C L U D I N G

687

REMARKS

This paper has investigated the implications of hysteresis for the equilibrium rate of unemployment in the UK. The starting point was a simple model in which firms respond discontinuously and heterogeneously to some aggregate shock. The implication is hysteresis in aggregate output, employment, and by extension, aggregate unemployment, in that the equilibrium rates of output, employment and unemployment are shaped, inter alia, by the nondominated extremum values of the shocks experienced. A program for calculating hysteresis time series variables that capture the relevant profiles of the shocks experienced was then presented. The 'shocking' finding was that hysteresis variables reflecting both nominal shocks and perturbations in real variables figure in a cointegrating or equilibrium vector for UK unemployment 1959-1996. The empirical results are of interest for two reasons. The first is that they cast doubt on whether the monetary neutrality axiom, imposed by the natural rate hypothesis, applies to the equilibrium rate of unemployment. The second is that they suggest that the equilibrium rate of unemployment may display remanence in that when real shocks disappear their effects on equilibrium unemployment do not. It will be interesting to see if these results also apply to other countries and different time periods.

APPENDIX The results that follow relate to the cointegrating properties of the log of the UK unemployment rate (LUPC), the replacement ratio (REPR) and hysteresis transforms of the log of the real price of oil, the log of the nominal effective exchange rate, and the ex post real interest rate (these hysteresis variables are named HLRPOIL, HLNER and HRR, respectively). The data are annual, and the estimation period runs from 1959-1996. The oil price, GDP deflator, exchange rate and interest data are from the IMF International Financial Statistics CD-ROM, December 1997 release, while the unemployment rate and CPI inflation are from ONS Economic Trends and the replacement rate is based on DSS data and constructed by HM Treasury. The data are graphed in Fig. 8.18. Johansen cointegration results are conditional on the specification of the initial unrestricted VAR, known as the UVAR. As a result we initially performed a number of diagnostic checks on a five-variable VAR. The first set of tests were directed at determining the appropriate lag length. Akaike's information criterion (AIC) and the adjusted likelihood ratio test favor three lags, while Schwarz's Bayesian information criterion (SBC)

688

CHAPTER 8

Hysteresis and Unemployment: A Preliminary Investigation

LUPS

2.5

REPR

0.8

2.0

0.7

1.5 0.6 1.0 0.5

0.5 0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995

0.4

..................................

1959 1963 1967 1971 1975 1979 1983 1987 1991 1995

Years

Years

HLNER

HRR

0.08

250 200

0.06

150 0.04 100 0.02

5O

0.00 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995

0 ..................................... 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995

Years

0.30 -

Years

HLRPOIL

0.25 1

/

~

o'o T 1959 1963 1967 1971 975 1979 1983 1987 1991 1995 Years

FIGURE 8.18

Simulation results.

TABLE 8.1 Specification of the UVAR: Test Statistics and Choice Criteria for Selecting the Order of the VAR Model Based on 37 observations from 1960 to 1996. Order of VAR = 3. List of variables in the unrestricted VAR: LUPC, HLNER, HLRPOIL, HRR, REPR. List of deterministic a n d / o r exogenous variables" C. Order 3 2 1 0

LL 255.3574 221.8598 182.4560 -0.2713

AIC 175.357 166.860 152.456 -5.2713

SBC 110.921 122.560 128.292 -9.2986

Adjusted LR test n.a. Chi-square (25) = 38.02[0.046] Chi-square (50) = 82.75[0.002] Chi-square (75) = 290.2[0.000]

APPENDIX

689 TABLE 8.2

Significance of Dummies in UVAR Equations

DD73 Equation LUPC REPR HLNER HRR HLRPOIL

Coefficient -0.13474 -0.00042 -0.00227 2.96780 -0.09712

SE 0.10 0.02 0.00 12.04 0.02

t-value -1.40 -0.03 -1.12 0.25 -4.66**

t-probability 0.17 0.98 0.27 0.81 0.00

DX86 Equation LUPC REPR HLNER HRR HLRPOIL

Coefficient 0.06171 0.00016 -0.00447 38.81200 -0.05418

SE 0.11 O.O2 0.00 13.83 0.02

t-value 0.56 0.01 -1.93 2.81"* -2.26*

t-probability 0.58 0.99 0.07 0.01 0.03

DD93 Equation LUPC REPR HLNER HRR HLRPOIL

Coefficient 0.07719 0.00543 -0.00852 -6.41700 0.00144

SE 0.09 0.02 0.00 11.78 0.02

t-value 0.82 0.33 -4.31 -0.55 0.07

t-probability 0.42 0.74 0.00 0.59 0.94

Joint significance tests: All dummies in each equation Equation F-statistic F(3,24) LUPC 1.088 [0.373] REPR 0.038 [0.990] HLNER 7.266 [0.001]** HRR 2.794 [0.062] HLRPOIL 8.701 [0.000]**

Dummy DD73 DX86 DD93

Dummy in all equations F-statistic F(5,20) 5.67 [0.002]** 6.33 [0.001]** 4.56 [0.006]**

Joint significance of all three dummies: Chi-square (15) 80.08[0.000]**

"Indicates significance at the 5% level. **Indicates significance at the 1% level. f a v o r s a o n e - l a g VAR. SBC a l w a y s p e n a l i z e s a d d i t i o n a l lags m o r e t h a n AIC, b u t closer e x a m i n a t i o n of the d i a g n o s t i c s for the i n d i v i d u a l VAR e q u a t i o n s s u g g e s t s that a single lag is insufficient, in t h a t significant serial c o r r e l a t i o n r e m a i n s (as r e v e a l e d b y L a g r a n g e m u l t i p l i e r statistics). A t w o - l a g U V A R is m o r e satisfactory in this r e g a r d , b u t the d i a g n o s t i c s r e v e a l s o m e p r o b l e m s w i t h n o n n o r m a l i t y of the residuals. T h e s e n o n n o r m a , lity p r o b l e m s c a n be t r a c e d to outliers in the r e s i d u a l s of the U V A R e q u a t i o n s w h i c h m o d e l the b e h a v i o r of the h y s t e r e s i s t r a n s f o r m e d variables. Specifically, the U V A R r e s i d u a l s c a n be i m p r o v e d t h r o u g h the i n c l u s i o n of t h r e e 0 , 1 , - 1 , 0 d u m m y variables.

690

CHAPTER 8

Hysteresis and Unemployment: A Preliminary Investigation

TABLE 8.3 UVAR Diagnostics

Correlation of actual and fitted Portmanteau five lags AR1-2 F(2,22) Normality Chi-square (2)

LUPC

REPR

HLNER

HRR

HLRPOIL

0.990

0.959

0.996

0.973

0.978

7.534

5.765

7.516

5.793

10.65

0.722[0.50]

0.082[0.92]

3.340[0.06]

1.53010.24]

0.442[0.65]

3.67710.16]

1.57710.45]

2.71910.26]

1.00010.61]

4.20610.12]

System diagnostics: Vector portmanteau five lags Vector AR1-2 Vector normality F-tests on retained regressors

LUPC_I LUPC_2 REPR_I REPR_2 HLNER_I HLNER_2 HRR_I HRR_2 HLRPOIL_I HLRPOIL_2

118.48 F(50,48) = 1.221 [0.24] Chi-square (10) = 13.49 [0.20] F(5, 20)

6.69 [0.001] ** 3.74 [0.015] * 3.22 [0.027] * 0.94 [0.475] 21.6 [0.000] ** 2.92 [0.039] * 3.61 [0.017] * 6.36 [0.001] ** 20.7 [0.000] ** 6.53 [0.001] **

"Indicates significance at the 5% level. "Indicates significance at the 1% level.

The first of these dummy variables relates to the timing of the discrete jump in the real oil price following the first OPEC shock (required in the HLRPOIL equation). The second dummy relates to the rapid interest rate hike which occurred at the peak of the Lawson boom in the late 1980s (required in the HRR equation) and the final dummy is needed in the period following the UK's exit from the ERM, when sterling depreciated rapidly (in the HNER equation). Importantly, none of these dummies are statistically significant in the LUPC equation of the UVAR, nor are they statistically significant in any other equations of the UVAR, though each is clearly statistically significant in the UVAR as a whole (see Table 8.2). Normality and serial correlation tests on the residuals from the dummied UVAR are presented in Table 8.3.

FIGURE 8.19

Performance of the UVAR.

692

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation

2.0

1.5

1.0

. I 1 1959 1961

1 I 1 1 I I I I 1 1 1963 1965 1967 1969 1971

1 I 1 1 1 1 1 1 1 1 1973 1975 1977 1979 1981

1 I 1 1 1 1 1 1 1 I 1983 1985 1987 1989 1991

1 : 1 ', 1 1993 19951996

Years

Correlogram

Spectral Density (Parzen)

2.5-

1.00.8-

2.0.

0.6. 1.5

0.40.2-

1.0.

0.0 0.5

"0"40

I

12

3

14

/

/

7I

Order of lags

/

1110

1ll

11

. ~ . ~

0.0

, 0

,

~

,

I 1

,

,

~

~

I 2

,

,

,

,

/

3

,

,

Frequency

FIGURE 8.20 The Johansen-based cointegrating relation. Having estimated a UVAR with Gaussian residuals, it is feasible to test for the number of linearly independent cointegrating vectors among the five variables. The UVAR is still conditioned on the three d u m m y variables, none of which have a long-run effect, but all of which will affect the relevant critical values for the Johansen maximal eigenvalue and trace tests. Unfortunately, adjusted critical values are not available, though we know from work by Perron and Phillips that a positive adjustment to the standard critical values is probably appropriate. The cointegration tests on the d u m m i e d UVAR are reported in Table 8.4 and these are followed by cointegration tests based on a UVAR with no conditioning dummies. Whilst estimates based on the u n d u m m i e d UVAR are likely to be adversely affected by the nonnormality of the system residuals, the cointegration tests are less subject to bias, so may have some value in clarifying inference. Tests based on the unconditioned UVAR are certainly easiest to interpret, with the maximal eigenvalue and trace tests both supporting the conclusion of a single cointegrating vector using 99 per cent critical values, the small-sample corrected tests reject cointegration at the 95 per cent but not the 90 per cent level. In the case of the dummy-adjusted UVAR, we clearly need to adjust the critical values up by some u n k n o w n amount. The large sample tests suggest up to three cointegrating vectors at standard 99

APPENDIX

693 TABLE 8.4

Cointegration Analysis 1959-1996

Cointegration tests based on UVAR conditioned on three dummy variables Eigenvalue

Loglikelihood for rank

0.656235 0.580331 0.492231 0.243391 0.002610

482.257 0 502.546 1 519.043 2 531.920 3 537.219 4 537.269 5

Maximum eigenvalue test

Trace test

H 0" rank=p

-Tlog(l\mu)

Using T-nm

95%

- T \ s u m log(.)

Using T-nm

95%

p == p <= p <= p <=

40.58 ** 33.00 ** 25.75 ** 10.60

29.90 24.31 18.98 7.81

33.5 27.1 21.0 4.1

110.02** 69.45** 36.45** 10.70

81.07 ** 51.17 * 26.86 7.88

68.5 47.2 29.7 15.4

0 1 2 3

Cointegration tests based on unconditioned UVAR Maximum eigenvalue test

Trace test

H0: rank=p

- T log(l\mu)

Using T-nm

95%

- T \ s u m log(.)

Using T-nm

95%

p == 0 p <= 1 p <-- 2

41.65 ** 26.58 16.51

30.69 19.58 12.17

33.5 27.1 21.0

92.55** 50.90* 24.32

68.19 37.5 17.92

68.5 47.2 29.7

*Indicates significance at the 5% level. **Indicates significance at the 1% level.

p e r c e n t critical v a l u e s , t h o u g h p l o t s of t h e r e s i d u a l s w o u l d n o t s u p p o r t s u c h a c o n c l u s i o n . T h e a d j u s t e d tests are m o r e s u p p o r t i v e of a s i n g l e c o i n t e g r a t i n g vector, as is t h e i n f o r m a l g r a p h i c a l e v i d e n c e . T h e J o h a n s e n c o i n t e g r a t i n g v e c t o r is s h o w n in Fig. 8.20 a l o n g w i t h its c o r r e l o g r a m a n d s p e c t r a l d e n s i t y , all of w h i c h are s u p p o r t i v e of c o i n t e g r a t i o n . T h e l i k e l i h o o d ratio test p r e s e n t e d in Table 8.5 s u p p o r t s t h e h y p o t h e s i s t h a t t h e r e is a s i n g l e c o i n t e g r a t i n g v e c t o r t h a t o n l y f e e d s i n t o t h e D L U P C e q u a t i o n of t h e V E C - E C M . F o r m a l l y , i m p o s i n g this r e s t r i c t i o n c a n n o t b e rejected, a n d i m p o s i t i o n i m p r o v e s t h e efficiency of t h e e s t i m a t e d c o i n t e g r a t i n g v e c t o r a n d a l l o w s u s to i d e n t i f y v a l i d LR s t a n d a r d errors. T h e s e

694

CHAPTER 8 Hysteresis and Unemployment: A Preliminary Investigation TABLE 8.5

Restricted Cointegrating Vector

Likelihood Ratio Test: H 0" there is a single cointegrating vector which only feeds into the LUPC equation. Rank=l" Chi-square (4) = 5.6224 [0.23] - the restrictions cannot be rejected.

LUPC

=

2.2977REPR(0.42) +const

Restricted alpha: LUPC -0.86554 (0.13)

TABLE 8.6

REPR 0.0

11.599HLNER+ 0.0047HRR+ (2.00) (0.00063)

HLNER 0.0

HRR 0.0

3.8666HLRPOIL (0.40)

HLRPOIL 0.0

Error Correction Coefficient and the t ECM from the VEC-ECM ,

Equation

Coefficient

DLUPC DREPR DHLNER DHRR DHLRPOIL

-0.8901 -0.0338 -0.0025 3.6765 0.0324

SE

t-value

t-probability

0.138 0.029 0.003 24.48 0.039

-6.45** - 1.17 -0.84 0.15 0.84

0.00 0.25 0.40 0.88 0.41

standard errors reveal that each of the five variables plays a significant role in the vector. The conclusion that the cointegrating vector feeds into solely the DLUPC equation is also supported through examination of each equation of the unrestricted VEC-ECM (still conditioned on the dummies). The tECM statistics from the individual equations of the VEC-ECM are reported in table 8.6, and importantly, the only statistically significant ECM term appears in the LUPC equation. These results suggest that nonsystem-based estimation and testing procedures are applicable to the data. In Table 8.7 we present Granger-Engle estimates of the same cointegrating relation. Once again the results are supportive of the existence of cointegration and the parameter estimates are close to those obtained using Johansen's procedure. So, to summarize, using Johansen's cointegration testing and estimation procedure we were able to identify a single cointegrating vector involving all five variables LUPC, REPR, HLRPOIL, HLNER and HRR. This vector feeds significantly into a dynamic equation for unemployment, but is statistically insignificant in the other equations of the VEC-ECM.

APPENDIX

695

TABLE 8.7 Granger-Engle Estimates of the Cointegrating Vector. Ordinary least-squares estimation - dependent variable is LUPC 38 observations used for estimation from 1959 to 1996

Regressor

Coefficient

C REPR HLNER HRR HLRPOIL

-0.1583 2.0977 -14.8072 0.0039 3.1198

R-squared

0.9001

SE

T-ratio [Probability]

0.6405 0.8005 2.6794 0.0010 0.5875

-0.24720[ 0.806] 2.6205 [0.013] -5.5263 [0.000] 3.7592 [0.001] 5.3108 [0.000]

DW-statistic

1.3435

Unit root tests for residuals: 5% critical value for the Dickey-Fuller statistic - -4.8380 Test statistic DF

-3.9779

Log likelihood 3.4004

AIC

SBC

HQC

2.4004

1.6373

2.1402

LR test F(1, 36) =

10.8810.00] ADF(1)

-5.5319

8.1523

6.1523 4.6259

5.6318

ADF(2)

-3.9446

8.1613

5.1613

4.3805

F I G U R E 8.21

2.8717

F(1, 35) = 0.00110.97]

A c t u a l a n d ' e q u i l i b r i u m ' u n e m p l o y m e n t rates.

These results are additionally supported by Granger-Engle estimates of the cointegrating vector. Our empirical results are therefore supportive of an 'equilibrium' log unemployment rate, which is determined by the

696

CHAPTER8 Hysteresis and Unemployment: A Preliminary Investigation

replacement ratio, and by the nondominated extrema in the real price of oil, the nominal effective exchange rate and the real interest rate. The implied equilibrium rate of unemployment is graphed in Fig. 8.21, along with the actual unemployment rate over the period 1959-1996.

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CHAPTER8 Hysteresis and Unemployment: A Preliminary Investigation

30. M. G6cke, An approximation of the hysteresis loop by linear partial functions - econometric modelling and estimation, mimeo, University of M~inster (1994). 31. S.B. Kamin and N. Ericsson, 'Dollarisation in Argentina', Board of Governors of the Federal Reserve System, International Finance Discussion Paper no. 460 (November 1993). 32. R.G. King, J.H. Stock and M.W. Watson, 'Temporal instability of the unemployment-inflation relationship', Economic Perspectives of the Federal Bank of Chicago, May/June, 2-12 (1995). 33. R. Layard and S.J. Nickell, 'The labour market', in The Performance of the British Economy, R. Dornbusch and R. Layard Eds, Oxford University Press, Oxford (1987). 34. R. Layard, S. Nickell and R. Jackman, Unemployment: Macroeconomic Performance and the Labour Market, Oxford University Press, Oxford (1991). 35. N. Lawson, The Viewfrom No. 11, Bantam Press, London (1992). 36. A. Manning, 'Wage bargaining and the Phillips curve: the identification and specification of aggregate real wage equations', Economic Journal 103, 98-118 (1993). 37. S.J. Nickell, 'Why is wage inflation in Britain so high?', Oxford Bulletin of Economics and Statistics 49, 103-128 (1987). 38. OECD, 'The interactions between structural reform, macroeconomic policy and economic performance', Economic Outlook 59, 42-57 (1996). 39. E.S. Phelps, 'Phillips curves, expectations of inflation and optimal unemployment over time', Economica 34(3), 254-281 (1967). 40. E.S. Phelps, 'Money wage dynamics and labour market equilibrium', Journal of Political Economy 76(2), 678-711 (1968). 41. E.S. Phelps, Inflation Policy and Unemployment Theory, Macmillan, (1972). 42. E.S. Phelps, 'The origins and further development of the natural rate of unemployment', in The Natural Rate of Unemployment: Reflections on 25 Years of the Hypothesis, R. Cross Ed., Cambridge University Press, Cambridge (1995). 43. L. Piscitelli, 'Hysteresis in economics', PhD thesis, University of Strathclyde (1997). 44. I.R. Searle and P. Musumeci, RLAB Primer- version 1.0 (1993).

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CHAPTER 1

Thermodynamics, Hysteresis, and Micromagnetics Giorgio Bertotti, Vittorio Basso, Martino LoBue, and Alessandro MagnL Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada delle Cacce 91, 10135 Torino, Italy

Contents 1.1 1.2 1.3 1.4

Introduction Classical and Nonequilibrium Thermodynamics Thermodynamic Aspects of Hysteresis From Micromagnetics to Magnetic Hysteresis References

1 5 29 75 100

'...ask yourself whether our language is complete - whether it was so before the symbolism of chemistry and the notation of the infinitesimal calculus were incorporated in it; for these are, so to speak, suburbs of our language. (And how many houses or streets does it take before a town begins to be a town?) Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses.' Ludwig Wittgenstein. Philosophical Investigations.

1.1 I N T R O D U C T I O N Hysteresis and thermal relaxation are ubiquitous phenomena that characterize the behavior of many different classes of materials. Examples are ferromagnets, shape memory alloys, and superconducting materials. Various chapters of 'Science of Hysteresis' are devoted to the discussion The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All ri~ht~ re.gerved.

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics of this aspect. Whatever the physical system considered, hysteresis phenomena are always the sure indication that the system is living for long times in states that are far from thermodynamic equilibrium. Therefore, in principle the observed behavior should be interpreted in the frame of nonequilibrium thermodynamics [1-5]. This goal has not yet been fully achieved because of the many conceptual difficulties inherent in the extension of thermodynamic methods to far-from-equilibrium conditions. As a consequence, simplified approximate models based on reasonable phenomenological assumptions are still an unavoidable step in the interpretation of hysteresis phenonema. In this respect, a fundamental role is played by the notion of rate-independent hysteresis. By this we mean that the system response X to the driving force H is some history-dependent function X(H~ {Hhistory}) independent of the rate dH/dt at which H is varied. In particular, this implies that X is constant if the field is not varied, that is, the system remains for an infinite time in whatever state has been reached as a result of the H history, even if that state is not the thermodynamic equilibrium state under the current value of H. This makes clear that rate-independent hysteresis is an approximation in which the natural tendency of all systems to approach thermodynamic equilibrium is ignored. This approximation raises nontrivial thermodynamic issues. In particular, we do not know the general principles determining in what proportion the work performed on the system by external sources will be stored as free energy change or dissipated through entropy production mechanisms. Rate-independent hysteresis has generated considerable interest among mathematicians, because it is in the study of models containing rateindependent hysteretic elements that one encounters the most interesting mathematical challenges [6-8]. On the other hand, from the physical point of view, the fact that one ignores the natural tendency of thermodynamic systems to approach equilibrium is a serious shortcoming. When the temperature is not too large, it is possible to introduce thermal relaxation effects as a correction to the rate-independent behavior, where one assumes that the energy barriers separating the various states of the system can be overcome not only by the forcing action of H responsible for rate-independent hysteresis, but also by the random internal forces associated with thermal fluctuations. The main difficulty with this approach is that the energy surface in which the system evolves is usually of extreme complexity, with a huge number of local energy minima representing the states in which the system can be found at low temperature, connected by equally numerous saddle points identifying the possible paths through which thermal transitions from one state to another may occur. It has been often proposed in the literature to treat this problem by drastically simplified models,

1.1 INTRODUCTION where it is assumed that thermal relaxation involves an ensemble of statistically independent two-level subsystems, the two levels in each of them being separated by a single energy barrier. Under these conditions, thermal relaxation is fully controlled by the distribution of the energy barriers thus introduced. A well-known result in this respect is that thermal relaxation will be logarithmic in time if the energy barrier distribution is flat [9]. An important case where the description of the energy surface responsible for hysteresis can be expressed in precise mathematical form is in the so-called micromagnetic approach to ferromagnetic media [9-12]. In micromagnetics, the energy of the system is expressed as the volume integral over the body of terms involving the space-dependent magnetization M(r). From the mathematical point of view, this results in a functional of the vector field M(.), which has to be minimized in the infinite-dimensional functional space of all admissible realizations of M(.). In principle, the micromagnetic theory should be able to explain all the basic aspects of the nonequilibrium behavior of magnetic materials, including hysteresis phenomena. However, in practice this turns out to be a challenge of prohibitive difficulty, due to the complexity of the nonlinear equations (Brown's equations) associated with local energy minima and to the concurrent fact that the parameters themselves of the theory in general exhibit a very complex space dependence due to the presence of structural disorder [9,11]. Once more, drastically simplified models have been proposed in the literature. A key role has been played in this respect by models where magnetic domain walls are assumed to be the fundamental objects involved in magnetization processes, and where wall dynamics is reduced to the study of a certain number of statistically independent single degrees-offreedom evolving in random energy profiles associated with structural disorder. Starting from this scenario, in the present chapter of 'Science of Hysteresis' we will discuss the connection between thermodynamics, hysteresis, and micromagnetics, with the emphasis on some recent developments that show this connection from a new perspective. Section 1.2 contains a discussion of the basic concepts on which the various formulations of equilibrium and nonequilibrium thermodynamics are based. The subject is vast and no attempt will be made to give a comprehensive treatment. Conversely, attention will be paid only to those aspects where a closer connection to the problems and issues typical of hysteresis can be recognized. In particular, some emphasis will be given to nonequilibrium thermodynamic formulations in terms of internal variables. In fact, this approach provides a natural path, discussed in Section 1.3, to the introduction of rate-independent hysteresisis as an appropriate

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics limit of the general formulation and subsequently of thermal-activationtype relaxation as a correction to it. Section 1.3 is devoted to the analysis of idealized model systems for which a relatively general nonequilibrium thermodynamic description can be developed. These systems are ensembles of independent two-level subsystems (TLS), the two states of each TLS unit being separated by a suitably defined energy barrier hindering spontaneous transitions from one level to the other. By considering such systems we are not attempting any interpretation of specific experimental results, but rather the investigation of a relatively simple framework in which the connection between thermodynamics, hysteresis and thermal relaxation can be fully appreciated. In a sense our aim is to show how TLS ensembles might play a role in the comprehension of nonequilibrium phenomena and hysteresis similar to what the 'perfect gas' has represented in classical thermodynamics. The relaxation kinetics of the ensemble will be examined under the assumption that the relaxation of each individual TLS unit can be described by an Arrhenius-type thermal activation law. It will be shown that under appropriate approximations the nonequilibrium behavior of the ensemble can be described by a thermodynamic formulation with internal variables exhibiting many similarities with the thermodynamic description of chemical reactions. This formulation shows that, in the limit of low temperatures, the ensemble exhibits rate-independent hysteresis exactly describable by the Preisach model of hysteresis [9,13,14], a result creating a direct link between nonequilibrium thermodynamics and mathematical modeling of hysteresis. Thermal relaxation effects can be added to the rate-independent picture by investigating the kinetic behavior of the internal variables describing the nonequilibrium collective state of the TLS ensemble. A broad range of viscosity and rate-dependent effects will be predicted by this approach, as well as some results concerning the response of the system to small oscillating fields. Finally, Section 1.4 discusses the onset of hysteresis in micromagnetic systems. The general derivation of the average hysteretic response of the system from the properties of the micromagnetic free energy is a task of prohibitive difficulty for which no general solution is known. We will limit the discussion to a particular class of problems, where the response of the system to the external field is reduced to the motion of a certain number of independent degrees-of-freedom in random one-dimensional pinning-field profiles. Hystorically this approach has been extremely important in the development of theories of the coercive field and, more generally, of the effect of structural disorder on the magnetic properties of various classes of magnetic materials. Interestingly, the problem can be cast into a functional integration form amenable to full analytical treatment

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS in the case where the pinning-field profiles can be interpreted as sample functions of some Markovian stochastic process. In particular, one proves that when the pinning-field profiles are Markovian and spatially homogeneous, the ensuing average hysteretic behavior of the system is of Preisach type. This represents an interesting point of contact with the framework discussed in Section 1.3, where Preisach hysteresis emerges from TLS ensemble thermodynamics. It is not unreasonable that through this connection one might attempt the application of the nonequilibrium thermodynamic methods discussed in Section 1.3 to micromagnetic systems as well.

1.2 CLASSICAL A N D NONEQUILIBRIUM THERMODYNAMICS When attempting a thermodynamic approach to hysteresis, an accurate preliminary discussion is necessary in order to reduce, as much as possible, any ambiguity in the use of terminology. As a matter of fact, thermodynamics is one of the branches of physics where the direct connection between the contents of theories and the way in which they are exposed is more critically evident. The great generality of the laws of thermodynamics makes any attempt to reformulate them a challenge, involving some reinterpretation of the physical meaning of even the most basic and accepted concepts. A first look at modern theories of out-of-equilibrium phenomena shows an intriguing and, to some extent, disappointing aspect common to most of them; it seems that, in spite of their declared programs, none of these approaches eventually attains the same degree of universality of classical thermodynamics, or the same formal coherence of thermostatics. Most approaches, from the celebrated OnsagerPrigogine theory of irreversible processes to Coleman and Truesdell rational thermodynamics, seem to have some difficulty in freeing themselves from the context in which they were conceived. Although the smell of steam engines is still present in classical thermodynamics, there is no doubt that the first (conservation of energy) and the second (arrow of time, irreversibility) principles of thermodynamics are among the most universal laws formulated in natural sciences. Conversely, it would be difficult to find some result with a comparable level of generality within the various formulations of nonequilibrium thermodynamics. One is confronted with the frustrating situation where a new theory, aimed at including broader classes of phenomena (in fact equilibrium and reversibility are properties that characterize a very small and particular subset of the real world), produces the same level of universality of thermostatics,

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics that is, of a very particular case of the theory. As a matter of fact, the domain of nonequilibrium processes covers a range of phenomena, from deformable solids to biological systems, whose extension is such that one would hardly expect it to fit into a single coherent theory. The state-ofthe-art of nonequilibrium thermodynamics is far from the elegant formal structure made by a few general principles valid for all systems and processes and a set of constitutive laws describing each particular system. On the other hand, what probably contributed to this state of affairs is also the fact that the notions of thermodynamic processes and irreversibility have been, to some extent, removed from the formulation of thermodynamics - more precisely called thermostatics - inspired by the seminal work of Josiah Willard Gibbs about equilibrium in homogeneous substances [15]. The formal elegance of equilibrium thermodynamics is the result of a compromise between generality and formal rigor whose price has been the exclusion of irreversibility from the field of applicability of the theory. In this introductory section, no attempt will be made to give an exhaustive presentation of the conceptual foundations of modern thermodynamics. The scope will be limited to those key topics that will play a role in the approach to hysteresis discussed in Section 1.3. In particular, the following aspects will be considered: 9 Classical thermodynamics (Section 1.2.1). 9 From equilibrium properties to thermodynamic processes: principle of local equilibrium, entropy production, and a brief account of the main approachs to nonlinear and out-of-equilibrium phenomena (Section 1.2.2). 9 Internal-variable formulations, time dependence, and relaxation (Section 1.2.3).

1.2.1 CLASSICAL THERMODYNAMICS A common attitude when discussing the foundations of thermodynamics consists in paying attention mainly to the second law and notably to entropy and its microscopic interpretation. The second law marks the image of thermodynamics inside and outside the scientific community in such a deep way that the first law is sometimes relegated to a secondary role. This can be hardly justified, because the wide generality of the first law is the primary source of the great freedom with which one can define the object of every thermodynamic approach, this freedom being both a tool to resolve problems and a source of ambiguity.

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

The First Law and the Definition of Thermodynamic System The first law expresses the conservation of energy by asserting that for every thermodynamic system there exists a state function U, called the internal energy. In the original form of the first law, the internal energy change A U - UB - U A in a transformation from state A to state B is equal to the algebraic sum of the mechanical work L performed on the system and of the heat Q absorbed by the system from its surroundings. So, the change in internal energy AU consequent to the application of a frictionless work L on a system that cannot exchange heat with the external world is just AU = L. From this we can give a definition of heat as the difference between the change in internal energy and the mechanical work done in the case of a system not thermally isolated, Q = AU - L. The equivalence between heat and work, quantified by the classic Joule experiment, is at the origin of the word 'thermo-dynamics', where 'dynamics' refers to the interpretation of heat in terms of random molecular motions. The first law expressed in its differential form, dU=~L+~Q,

(1.1)

is remarkable in that it contains no specification of the kind of physical objects involved in the relation. While in dynamics we have material points and forces, and in electrodynamics charges and fields, here we have nothing but the balance of work L and of its thermal equivalent Q, defined over some region of space and some interval of time. Therefore, to define a thermodynamic system we need first of all to identify in some way a region of space and an interval of time. In classical thermodynamics textbooks, this is often accomplished by giving particular emphasis to the notion of 'wall'. Walls translate the description of the system into a rigorous definition of the separation of the system from its surroundings, this separation being characterized by detailed boundary conditions defining which kind of exchange the system can have with the outside world. Three main types of exchange can modify the state of a given system: matter exchange, energy exchange mediated by the generalized work terms previously mentioned (that is, not only mechanical work but any kind of work that can modify the state of the system), and heat exchange. Walls restricting matter exchange are called impermeable walls; rigid walls are walls restrictive with respect to one of the generalized work terms; and finally, walls restricting heat exchange are called adiabatic walls. A system for which no matter or energy exchange occurs with the

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics external world is an isolated system, whereas a system bounded by impermeable walls but in energy contact with the environment is a closed system. A system can be divided into subsystems by walls imposing any of the constraints previously described. Following Landsberg [16] we call 'partitions' the walls dividing a system into subsystems and 'enclosures' the walls separating the whole system from the outside world. The freedom in choosing the most appropriate scale for the description of a given thermodynamic problem together with the possibility of dividing the ensuing thermodynamic system into subsystems are powerful tools in the approach to complex, nonhomogeneous systems. The global scale defines which properties of the system will be averaged out and which ones will be described by their local values. On the other hand, partitions are important because they permit one to reduce a nonhomogeneous system to a set of many homogeneous subsystems. However, it should be kept in mind that spatial and temporal scales associated with subsystems and local processes are always bounded by the lower limit defined by the atomic scale. Thermodynamic quantities must be macroscopic, in the sense that the operations by which we measure them must be slow with respect to atomic relaxation times, and coarse with respect to atomic distances [1]. Once the scale of the problem has been defined, one needs to identify an appropriate set of state variables that characterize the state of the system. In the case of mechanical systems, the very laws of mechanics naturally identify the positions and velocities of individual mass points as the appropriate state variables. This is no longer the case for thermodynamic systems, where the identification of the state variables becomes a delicate issue involving a proper reinterpretation of known experimental facts. Indeed, at the root of classical thermodynamics there is the interest in the efficiency of thermal engines and the solid acceptance by the scientific community of the Boyle-Mariotte law as a fundamental aspect of the behavior of gaseous systems. The identification of appropriate state variables is far more difficult in complex inhomogeneous systems where different types of forces can be active, internal couplings can be a source of correlation on different scales, and the description of the system can be scale-dependent. The notion of state variable is intimately related to that of equilibrium state. The equilibrium state of a system is characterized by the following properties: 9 the state at time t is univocally identified by the values taken at the same time by an appropriate set of independent state variables; this

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS implies that history dependence, that is, dependence on past values of the state variables, cannot be a property of systems at equilibrium; 9 the state is time-independent, that is, at equilibrium the state variables remain constant in time. The set of independent state variables needed to describe equilibrium states defines the thermodynamic space of the system. State variables can be divided into two classes: extensive variables, proportional to the size of the system, describing properties like volume, energy, and magnetic moment; and intensive variables, independent of the size of the system, describing agents acting on the system like pressure, temperature, and electric and magnetic fields. A conjugate intensive variable Ii is associated with each extensive variable Xi. Conjugate variables contribute a work term to the first law of the form:

~L -- ~

Ii dXi,

(1.2)

i reflecting the fact that the action of Ii consists in modifying the value of Xi. The distinction between intensive and extensive variables is a key concept in Gibbs' formulation of thermostatics, where the fact that the sum of a given extensive variable over all partitions of an isolated system is an invariant plays a major role in the definition of stability. The notions of state variable and equilibrium state, together with the first law of thermodynamics, permit one to describe a variety of situations. However, the following key aspect remains unresolved. Imagine a closed, isolated system divided into subsystems by a partition restrictive with respect to some property (for example, a partition made of adiabatic walls that block heat exchange). Let us assume that the subsystems are individually in equilibrium, so well-defined values of the appropriate state variables are associated with each of them. In general, when we remove the partition the system as a whole will no longer be in equilibrium and the fundamental question is [1]: 'How can we determine the equilibrium state that will eventually result after removal of the internal constraints?' Despite the attempts made in the late nineteenth century ([17] p. 108), it is impossible to obtain some indication about the direction of the evolution of the system toward equilibrium from the first law of thermodynamics only. Some new concept is needed to break the symmetry, implicit in the first law, between two states connected by a transformation and to be able to discriminate between the initial and the final state of a process. This is the subject of the second law of thermodynamics.

10

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

The Second Law, Equilibrium and Irreversibility In the work of Clausius the introduction of a new state function, the entropy, derives from the generalization of the relation, holding for a Carnot cycle, between the heat Q1 exchanged during the isotherm at temperature T1 and the heat Q2 exchanged during the isotherm at temperature T2 (with T1 > T2), Q1 ~0, T1

Q2 T2

(1.3)

where the equality holds for reversible cycle. Based on Eqn. (1.3) Clausius introduced a new state function S called entropy, from the Greek word r,porcrl, transformation. If $1 and $2 are the two values of this function in the equilibrium states I and 2 then:

52 - 51~/

2 --~--, ~Q dS~--T--, ~Q

/1

(1.4)

where the equality refers to the particular case of reversible transformations. The inequality in Eqn. (1.4) reveals the deep nature of the second law of thermodynamics, that is, the breaking of the symmetry in the possible directions of evolution of a thermodynamic process. At the same time, however, the inequality gives a certain impression of incompleteness, because it does not provide any quantitative measure of the degree of irreversibility associated with a given process. This aspect was already recognized by Clausius, who, in his ninth memoir, went beyond Eqn. (1.4) by writing the equality:

N-

$ 2 - $1 -

2 ~Q T

~1

(1.5)

Clausius termed the quantity N - which in modern terms would be called entropy production - the uncompensated transformation and then restated the second law in the form: 'uncompensated transformations can only be positive'. There is no doubt that since Clausius' work irreversibility has played a dominant role in thermodynamics. In his 'Treatise on Thermodynamics' [17] written during the crucial transition between the nineteenth and the twentieth century, Max Planck devoted the entire third part of the volume to the discussion of the second law and its consequences. On pp.109-112 Planck enumerates three cases of processes that are intrinsically

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

11

irreversible: 9 processes where heat is produced by friction ([17] p. 109); 9 processes where a gas expands without performing work or absorbing heat ([17] p. 110); 9 processes involving conduction of heat from a hotter to a colder body ([17] p. 111). The last case is closely related to Clausius' formulation of the second law: heat can in no way and by no process be transported from a colder to a warmer body without compensation. Thereafter Planck writes ([17] p. 113): '... Whether reversible processes exist in nature or not, is not a priori evident or demonstrable .... but it can be demonstrated ... that if, in a single instance, one of the processes declared to be irreversible in p.109, p.110, and p.111 should be found to be reversible, then all of these processes must be reversible in all cases. Consequently, either all or none of these processes are irreversible. There is no third possibility. If those processes are not irreversible, the entire edifice of the second law will crumble.' The subsequent chapters of Planck's treatise are devoted to the proof of this statement, the main issue being the fact that if we were able to obtain a spontaneous decrease of entropy in one system, we would be able to decrease the entropy of any other system through appropriate interaction. So, we can summarize Planck's analysis by two fundamental statements about the nature of classical thermodynamics: 9 classical thermodynamics requires the existence of irreversible processes; irreversibility is a central, necessary element on which classical thermodynamics is founded; and 9 the second law of thermodynamics contains a definite prediction about the evolution of a thermodynamic system, which can be verified by experiments; its validity cannot be proven, however a single experimental violation would invalidate the whole theory; using the terminology of modern epistemology [18], we can say that classical thermodynamics is a falsifiable theory. Therefore, a sort of inherent conflict is present in classical thermodynamics, namely, the very same theory that attributes a central role to the notion of irreversibility of processes in nature has significant predictive power only for the very particular case of reversible processes. Mechanics, for a long time the most successful theory in natural sciences, developed the conceptual and the mathematical tools for the solution of static as well as dynamical problems during its early stages, whereas thermodynamics

12

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

exhibits this deceptive lack of symmetry between the problems posed by the theory (processes and irreversibility) and its ability to solve them. This is the reason for the conceptual shift, which occurred between the end of the nineteenth and the early twentieth century, of thermodynamics from a theory of processes to a theory of equilibrium, known as thermostatics after Gibbs' work, based on the analogy with the statics of mechanical systems [15]. Some of the relevant characteristics of Gibbs' formulation of thermostatics will be discussed in the next section. The axiomatic formulation developed in the early twentieth century (1909) by Carath6odory and afterwards by Born (1921) and Land6 (1926) will not be discussed here. A classical exposition of these approaches is the one by Chandrasekhar [19].

Gibbs' Thermostatics Gibbs exposed his approach to thermodynamics in a series of papers in the late nineteenth century, notably in 'A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces' and in the monumental 'On the Equilibrium of Heterogeneous Substances', both included in [15]. Gibbs' work on thermodynamics is mainly devoted to the formulation of a theory of stability of the equilibrium states of isolated systems. The distinction between extensive and intensive variables plays a major role in the theory. Indeed, in the case of isolated systems the intensive variables are the source of coupling between different parts of the system (subsystems), whereas extensive variables are additive-invariant, that is, their total value summed up over all subsystems is constant. Gibbs discusses in detail the generalization of the work expression in the first law (Eqn. (1.2)) in terms of products IibXi between a given extensive variable Xi and the conjugated intensive variable Ii. Moreover, he introduces specific state variables for the description of systems consisting of many phases, the mole number Nk of a certain (i.e., the kth) phase and the associated intensive variable, the chemical potential #k" For multiphase systems, characterized by multiple extensive variables, the first law, when applied to reversible transformations between equilibrium states, takes the form:

dU = T dS + ~ i

!i

dXi + E #k dNk.

(1.6)

k

To simplify the notation, we consider a system characterized by a single pair of conjugated work variables I and X. We can write Eqn. (1.6) in the

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS form (Gibbs' equation):

1(

dS = -~ d U - I d X - y ~ # k dNk

)

.

13

(1.7)

k

From Eqn. (1.7) one obtains the equations of state: (~9~U)

1

X,N1.....Nn ~S) -~

T _

-~k

(1.8)

r

U, N1 ..... Nn

(9S)

I

=

~k

U,X,N~#k

T

Knowledge of the equations of state permits one to integrate Eqn. (1.7). This yields to the so-called fundamental equation, S -- S(U, X, N~ . . . . . Nn).

(1.9)

Knowledge of the fundamental equation or, equivalently, of the equations of state provides the complete thermodynamic description of the system. The intensive variables defined by the equations of state are not all mutually independent. This derives from the fact that entropy is an extensive quantity, which implies that S is a homogeneous function of the variables U, X, N1 .... Nn: S -- S(2U, 2X, ,~N1 ..... ,~Nn) -- 2S(U, X, N1 ..... Nn).

(1.10)

By differentiating this expression with respect to 2, and setting 2 = 1, one obtains: 0S

0S

~S

S - - ~ U 4- - ~ X + y ~ - ~ k Nk.

(1.11)

k By using Eqn. (1.8) one finds:

s -

"1( ;

u-

)

Ix-

k

(1.12)

14

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

The differential of this expression is consistent with Eqn. (1.7) only if the Gibbs-Duhem equation holds: S d T + X dI 4- ~

Nk d#k -- 0,

(1.13)

k

which expresses the mutual dependence of intensive variables. Equation (1.9) is the fundamental equation for a given system in the so-called entropy representation. The entropy representation contains the same amount of information as the energy representation obtained by inverting Eqn. (1.9) with respect to U and S: U - U(S, X, N1 ..... Nn).

(1.14)

Gibbs' thermodynamic criteria of stability can be formulated by saying that the entropy tends to a maximum at constant internal energy or, equivalently, that the internal energy tends to a minimum at constant entropy. So all the information on the stability of equilibrium states can be inferred by studying the curvature of the thermodynamic space surface defined by Eqn. (1.9) or Eqn. (1.14). By applying appropriate Legendre transformations to Eqn. (1.14), it is possible to rewrite the fundamental equation in a form where the dependence on the intensive variables T and I appears instead of that on S and X. This is a fundamental step in the interpretation of the many experimental situations where the intensive variables, rather than the extensive ones, are under the control of the experimentalist. The state functions, which take the place of the internal energy in such reformulation, are called thermodynamic potentials. Different thermodynamic potentials are obtained depending on the particular Legendre transformation applied to the internal energy. These are (dependencies on mole numbers are understood): Helmholtz free energy F(T, X), defined as F = U - T S; enthalpy H(S, I), H -- U - I X; and Gibbs' free energy G(T, I), G = U - T S - I X. Each potential is characterized by the property that it never increases during a spontaneous transformation where its arguments are kept fixed. Thus the choice of a given potential is strictly related to the type of constraints that can be controlled in experiments. Analogous transforms can be applied to the entropy representation (Eqn. (1.9)): the resulting potentials are known as Massieu functions. Gibbs' approach led to an elegant geometrical interpretation of equilibrium thermodynamics based on the properties of certain surfaces representing the fundamental equation in the thermodynamic space of the system. However, a price must be paid for this geometrical reinterpretation, namely, the fact that the notions of process and of spontaneous evolution

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

15

of the system tend to disappear from thermodynamics. This difficulty can be overcome by introducing partitions of the system into subsystems [1], as a method to create a connection between equilibrium and nonequilibrium states. As an example, let us consider an isolated system consisting of two subsystems A and B with entropy, internal energy, volume, temperature, and pressure equal to SA, UA, VA, TA, PA and SB, UB, VB, TB, PB, respectively. The value of the extensive variables for the entire system is then: S = SA + SB, U = UA + UB, V = VA + VB. Each subsystem is at equilibrium and is separated from the other subsystem by a rigid and adiabatic partition. When these constraints are removed, some redistribution of entropy, energy, and volume will take place (we assume that no exchange of matter takes place). Since the system is isolated, dU = dUA + dUB = 0 and d V = d VA + d VB = 0 during the transformation. On the other hand, by applying Eqn. (1.7) to the two subsystems one finds:

1 1 PB dVB ' dS -- dSA + dSB -- ~AdUA + -~BdUB -- ~PA dVA - -~B

(1.15)

which, taking into account that dUB = -dUA, d VB = -dVA, yields:

dS-

(1TA

TB1)dUA - ( pATA TBPB) dVA"

(1.16)

The extremum principle for entropy implies that at equilibrium dS = 0 for any arbitrary variation d UA or d VA. This can be the case only if TA = TB, PA = PB.

(1.17)

These are the equilibrium conditions reached at the end of the transformation.

1.2.2 NONEQUILIBRIUM THERMODYNAMICS In principle, nonequilibrium thermodynamics is expected to provide general tools for the description of any arbitrary thermodynamic process. Classical thermodynamics should emerge from this framework as a very particular case. Reaching this goal for the totality of nonequilibrium processes is an extremely difficult and challenging task. The main approaches developed so far all include some limiting elements that impose some restriction on the type of phenomena described by the theory, in a way analogous to what makes thermostatics applicable to the particular case of reversible

16

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

transformations between equilibrium states. These limiting elements take the form of additional postulates or assumptions whereby definite predictions about specific systems can be extracted from the general principles of the theory and the general form of constitutive laws. This is the case for the principle of local equilibrium in Onsager-Prigogine classical irreversible thermodynamics [2,20,21], for the assumption of fading memory in Coleman-Truesdell rational thermodynamics [22,23], and for the notion of dissipative fluxes in Casas-Vasquez-Lebon extended thermodynamics [24]. In this connection, two aspects need to be addressed with particular care. 9 While thermostatics is limited to the comparison of the properties of different equilibrium states, thermodynamics must explicitly introduce time as a variable in the description of the transformation of a given system from the initial to the final state. This can be done in two ways: (i) in addition to the instantaneous values of state variables, one can explicitly introduce their rates of change for the description of the system; and (ii) one can implicitly take account of time effects by making the state of the system dependent not only on the instantaneous values of the state variables but also on their past history. The two possibilities are not mutually exclusive; in particular, history dependence may be the result of the superposition of many rate-dependent effects with a wide distribution of relaxation times. 9 In Section 1.2.1 we discussed the fact that in Gibbs' thermostatics the fundamental equation can be written either in the entropy representation, S = S(U, X, N1 ..... Nn), or in the energy representation, U = U(S, X, N1 ..... Nn), the two representations being equivalent. This symmetry is broken when one considers the spontaneous, irreversible evolution of an isolated system, because in that case energy is still a conserved quantity, whereas entropy becomes an increasing function of time. The entropy rate of change dS/dt characterizes the irreversible evolution of the system. One can write this rate in the form:

dS deS diS = + ~ dt dt dt '

(1.18)

where deS/dt is due to heat exchange between the system and the outside world whereas diS/dt describes the inherent entropy increase associated with the irreversible character of the process. In this frame, the second law takes the form: diS/dt~O. It clearly emerged from the earliest studies of irreversible processes that the

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

17

calculation of the rate of entropy production diS/dt is the central problem of irreversible thermodynamics [25-29].

Generalized Fluxes and Affinities The fundamental distinction between extensive and intensive variables holds also for irreversible processes. In order to characterize the evolution of the system during a certain process we need to know the rate of change dXi/dt of the system extensive variables. There are two mechanisms whereby these variables can change in time: Xi flow through the walls enclosing the system, that is, exchange of Xi with the outside world; or internal production of Xi inside the system. By definition, conserved quantities do not admit any internal production term. Thus, for a conserved variable, dXi/dt = q~i,where q~i is the inward flux of Xi through the walls of the system. For an isolated system in equilibrium, in the absence of internal partitions, intensive variables have the same value everywhere inside the system. If a given intensive variable Ii is not spatially uniform, some internal flux of the associated extensive variable Xi is expected to occur, for reasons similar to the ones discussed in Section 1.2.1 in relation to the entropy extremum principle. So the first step in order to give a quantitative account of irreversible processes must be the description of the fluxes of extensive variables bringing back the system to homogeneity. Some direct relation is expected between the degree of nonuniformity of the intensive variable Ik and the flux q9i of the associated extensive variable Xi. Let us express these considerations in quantitative form. We consider an isolated system in equilibrium, composed of two subsystems A and B separated by a partition restrictive with respect to the extensive variable X. The conjugate intensive variable is I. We denote by XA and XB the values of the extensive variable in the two subsystems, and by X = XA + XB the corresponding value for the entire system. A similar notation will be used for all other thermodynamic variables. When the partition is removed, a redistribution of XA and XB will take place under the constraint XA 4- XB -- const, until the system reaches the condition of equilibrium corresponding to the maximum of the total entropy S = SA + SB. Let us denote by d~/dt =_dXA/dt = -dXB/dt the rate at which the subsystem extensive variables change during the process. The corresponding rate of change of the total entropy S will be:

dS _ dS d~ _ d(SA + SB) d~ _ /'0SA dt d~ dt d~ dt 0XA

~gSB'~d~ _ I B - IA d~ 0XB dt T dt (1.19)

18

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

The system is isolated, so deS/dt = 0 which means (see Eqn. (1.18)) that dS/dt coincides with the rate of entropy production diS/dt. Therefore Eqn. (1.19) becomes:

diS dt

=~q~,

(1.20)

where (p = d~/dt is the flux of X between the two subsystems (positive when X flows from subsystem B into subsystem A) and

= IB T

IA. T

(1.21)

Equation (1.20) shows that entropy production is associated with the presence of a discontinuity in the value of the intensive variable from one subsystem to the other. The variable ~ , defined by Eqn. (1.21), is called the generalized force associated with X. Quantities like ~ were known under the name of 'affinity' from the fundamental works of Th6ophile De Donder in chemical thermodynamics (generalized forces associated with chemical reactions are usually expressed as ~ = d / T where d is the chemical affinity in the strict sense [3,4]; some authors, like [1,30], do not make any distinction between affinity and generalized force). In the particular case where the flowing extensive variable is the internal energy U, the associated force takes a form different from Eqn. (1.21), because in that case temperature itself is the conjugate intensive variable. One finds: ~-

=

1

1

TA

TB

.

(1.22)

On the other hand, when the relaxation process involves several variables Xi, Eqn. (1.20) becomes:

diS dt = ~ ~i(Di.

(1.23)

i The presence of nonzero generalized forces ~ i in a certain region of space (in the previous example this region corresponded to the system made up of A and B subsystems) is the sure indication that the system is not in equilibrium. The action of these forces is in the direction of bringing the system back to equilibrium through some irreversible relaxation process characterized by nonzero values of the fluxes q)i. Compatibility with the second law requires that the right-hand side of Eqn. (1.23) be positive under all circumstances. Any description of the connection between ~i and (fii must be consistent with this requirement.

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

19

With these last considerations we have reached the point where it is necessary to pass to a more refined description of irreversible processes in spatially distributed systems with space-time-dependent local inhomogeneities. This description will be based on some appropriate generalization of the assumptions that made it possible in this section to use equilibrium quantities, like temperature or entropy, to describe nonequilibrium processes. In the example that led to Eqn. (1.20), after removal of the wall separating the two subsystems, we implicitly assumed that the ensuing relaxation process involved only the interface between the two subsystems. In other words, we assumed that during the entire process the individual subsystems could be described as homogeneous systems in equilibrium with respect to their time-dependent state variables. This assumption is at the basis of the principle of local equilibrium for continuous systems, discussed in the next section.

Local E q u i l i b r i u m H y p o t h e s i s and Local Balance Equations The local equilibrium hypothesis states that the local thermodynamic properties of an inhomogeneous system are the same as for a macroscopic homogeneous system in equilibrium. In other words, this hypothesis guarantees that at each instant of time it is possible to subdivide the system into elementary volumes small enough that each of them can be assumed to be homogeneous and in equilibrium. Equilibrium thermodynamics holds for each elementary volume, and corresponding local state variables unambiguously define the thermodynamic state of each elementary volume. Accordingly, intensive variables become functions of position r and time t, e.g. T = T(r, t), p = p(r, t), or in general Ii = Ii(r, t). On the other hand, it is convenient to express extensive variables in terms of their local volume densities, e.g. entropy per unit volume s(r, t) or energy per unit volume u(r, t). Given the generic extensive variable Xi, its density will be denoted by xi(r, t). The description just outlined leads to a set of balance equations for the various extensive variables characterizing the system. For conserved variables, the continuity equation holds:

~xi ~t

+ V. Ji = 0,

(1.24)

where Ji is the current density associated with Xi flOW. The balance equation for the entropy takes a different form because it is not a conserved quantity.

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

20 One has:

~s

--

3t

+ V . Js = r~,

(1.25)

where Js represents the entropy current whereas cris the local rate of entropy production. According to the second law rr is always positive: ~0.

(1.26)

By integrating Eqn. (1.25) over a given volume V one obtains for the rate of change of the total entropy S associated with that volume: aS

=

r~dV-

(1.27)

V .JsdV.

By comparing Eqn. (1.27) with Eqn. (1.18) one concludes that:

ddtiS = Jv rrdV

,

dedtS = - /v

V.JsdV.

(1.28)

The term ~ rr dV corresponds to Clausius' uncompensated transformation N (see Eqn. (1.5)). It is possible to obtain an expression for the entropy current and the entropy production in terms of the forces and flows associated with the various thermodynamic variables characterizing the system. Let us consider the case of a system consisting of only one chemical species (the generalization to systems with components produced or destroyed by chemical reactions is straightforward) whose extensive variables are all conserved apart from entropy. For such a system Gibbs' equation (see Eqn. (1.7)) takes the local form:

cgs 1 c~u ~ = T ~t

Ii ~xi E

T 0t"

(1.29)

l

By using Eqn. (1.24) to express cgu/cgt and cgxi/cgt, and by expressing the divergence terms in a different form one obtains:

~S -- V ( 1ju E~Ji) ff-Ju v (Z) ~?t

i

EJi v (~t'

(1.30)

i

where Ju and Ji represent the current associated with the internal energy and the variables xi, respectively. Comparison of Eqn. (1.30) with

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

21

Eqn. (1.25) shows that the entropy current is given by: 1

Ii -~Ji,

-~Ju - ~

Js -

(1.31)

i

whereas the entropy production is equal to: r~-Ju.V(T

)-

z~. J i . V ( ~ ) .

(1.32)

Equation (1.32) represents the local form of Eqn. (1.23); the role of generalized force is now played by the gradient fu = V (l/T) for the energy flux (compare with Eqn. (1.22)) and by the gradient fi = - V (Ii/T) for the fluxes of the xi variables (compare with Eqn. (1.21)). With this definition, the entropy production is again reduced to the sum of products of generalized forces and associated fluxes: 0" -- y ~ fi" Ji, i

(1.33)

where it is meant that the sum now also includes the internal energy term.

Linear Processes and Onsager-Casimir Relations A first hint concerning the general relation between fluxes and affinities came from Henri-Louis Le Chatelier, the translator of Gibbs' works into French. The principle of Le Chatelier-Braun asserts that a system in chemical equilibrium undergoes, as a result of a variation in one of the factors governing the equilibrium, a compensating change involving all the factors governing equilibrium. In other words, a perturbation from equilibrium obtained by altering the homogeneity of a given intensive variable will activate processes in all the extensive variables whose flow can induce an effect contrary to the perturbation. In terms of generalized forces and fluxes, this means that the presence of the single generalized force fi0 may induce compensating fluxes for all the extensive variables, although only the individual current Ji0 will contribute to the entropy production (Eqn. (1.33)). The principle of Le Chatelier-Braun implies that a given current Ji depends on all the generalized forces fl ..... fn and not only on the one conjugate to the intensive variable Ii. For certain systems this dependence has no memory, in the sense that the fluxes at a given time instant depend only on the values of the forces at that instant. Such systems are referred to as 'purely resistive'. For other systems, fluxes may depend on the past history of the forces as well. It is a fact that several systems of physical interest are

22

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

purely resistive; their study represented a major step in the development of nonequilibrium thermodynamics. Let us discuss the functional dependence of fluxes on forces in purely resistive systems. The dependence must be such that when the generalized forces fi vanish (i.e. the system reaches equilibrium) the currents Ji vanish as well. In addition, if the system is not too far from equilibrium and the forces are consequently small, we can expand the currents in powers of the forces and truncate the expansion to first order. This gives linear relations of the form

Ji - ~ nijfj,

(1.34)

J where the coefficients Lij, termed 'kinetic coefficients' or 'phenomenological coefficients', are functions of the equilibrium values of the intensive variables. When Eqn. (1.34) holds, the entropy production (see Eqn. (1.33)) takes the quadratic form:

rT = ~ nkjf j 9 fk -- E njj [fjl 2 if- y ~ nkjf j 9 fk. j,k j k~:j

(1.35)

Since the entropy production must be positive under all circumstances, the kinetic coefficients must fulfill the inequalities:

Ljj~O; LjjLkk~ l (Ljk q- Lkj) 2.

(1.36)

In addition, they are characterized by the following fundamental symmetry relations:

Lkj = Ljk,

(1.37)

known as reciprocal, or Onsager, relations. These relations were demonstrated by Onsager [20,21] and then extended by Casimir [31] to include variables with odd parity. Onsager showed that Eqn. (1.37) is the direct consequence of time-reversal symmetry in the microscopic dynamics of the system. Casimir generalized the reciprocal relations to the case where the state variables exhibit odd parity with respect to time-reversal transformation. In that case the functional relation between the kinetic coefficients and intensive variables must be included explicitly in the reciprocal relations. For example, in the presence of a magnetic field Ha the reciprocal relations take the form:

Lkj(Ha) = Ljk(-Ha).

(1.38)

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

23

The Onsager-Casimir reciprocal relations led to the successful explanation of coupled phenomena like thermoelectric effects or cross-diffusion effects. These cross phenomena were well known from the late nineteenth century (Lord Kelvin tried, without success, to give a theoretical explanation of thermoelectric phenomena like Seebeck and Peltier effects) but they where studied case by case, outside of any unifying theoretical framework.

Alternative Approaches to Irreversible Thermodynamics Onsager-Casimir relations and their interpretation have been criticized, notably by scientists working in continuum mechanics. For example, a true philippic against these relations and all their derivations, called 'onsagerisms', can be found in the seventh chapter of [23] by Clifford Truesdell. Although all the proofs of Onsager relations are based on microscopic theories (statistical mechanics, fluctuations or kinetic theory), their consequences at the macroscopic level are generally accepted even when the generalized fluxes cannot be expressed as time derivatives of state variables, as is the case, for example, for the heat flux vector or the viscous pressure tensor. Moreover the very derivation of reciprocal relations from fluctuation theory is questioned because Onsager assumed, without demonstration, that the decay of fluctuations follows the same linear dynamical laws as the macroscopic equations. As a matter of fact Truesdell's criticisms are mainly methodological and concern the strict interplay between physical and stochastic postulates characteristic of the Onsager line of reasoning. In the end Truesdell's arguments challenge the very foundations of classical irreversible thermodynamics, that has its reference book in [4], and its leading representative in Ilya Prigogine. The approach discussed in the previous section is limited to processes so close to equilibrium that the dependence of fluxes on forces can be linearized. This is not always acceptable, given the ubiquity of nonlinearities in many of the contexts where the linearized approach was developed and extensively applied, as, for example, that of chemical reactions. Furthermore, linearized irreversible thermodynamics is based on the assumption that thermodynamic states depend on neither past history nor state-variable gradients. These two requirements are impossible to fulfill when studying systems with hysteresis or systems, like shock waves, where important gradients are present. Two alternative approaches have been proposed to go beyond these limitations. 9 Thermodynamics of materials with memory or rational thermodynamics. This approach is based on the formulation elaborated by Truesdell, Noll and Coleman between the late fifties and the early sixties in the field of mechanical properties of continuous media [32,33], and

24

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics subsequently developed into a more general form [22,23]. The study of dissipation in the mechanics of continuous media was at the origin of one of the most original ideas developed in this context, that is, the assumption that the present stress experienced by a solid is given by a functional of the past history of the deformation gradient and by a 'fading memory' function describing the attenuation of the influence of the history on the present state of the system. The ensuing thermodynamic formulation is inspired by nineteenth century rational mechanics and is developed in an axiomatic way. The basic postulates of the formulation are the following: 1. Absolute temperature and entropy are primitive concepts introduced a priori in the theory without any physical interpretation. 2. The state of the system at a given time is assumed to be completely determined by functionals of the state variables and of their history. 3. The balance equations for mass, momentum and energy are retained with the introduction of a specific rate of energy supply in the balance equation for internal energy. 4. The second law of thermodynamics, which serves as the main restriction in the formulation of constitutive relations, is introduced in the Clausius-Duhem form derived by Truestell and Toupin in [32]. Rational thermodynamics represents an elegant and powerful tool aimed at deriving constitutive relations for very complex continuous systems. However, the lack of a physical and operational analysis of the meaning of concepts like temperature and entropy for out-of-equilibrium systems raised strong criticism and, in a sense, seriously weakened Truesdell's ambitions of generality. Moreover, it was observed that, given the primitive nature of temperature and entropy, the Clausius-Duhem relation is deduced by generalizing Clausius' inequality (Eqn. (1.4)) to out-of-equilibrium systems without any proof. This problem was addressed by M~iller in a remarkable paper [34] where a nonequilibrium relation between the entropy flux and the heat flux is derived from the theory and applied to materials with memory. 9 Extended thermodynamics. This approach shows that it is possible to generalize several concepts based on the principle of local equilibrium to systems where this principle is no longer applicable. The key idea is that the gradients of the thermodynamic variables behave as independent state variables on short time scales. When

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

25

local equilibrium is attained, this independence is lost and the gradients become functions of the other state variables. In this frame, a nonequilibrium expression for the entropy and a generalized Gibbs equation are formulated and discussed using nonequilibrium statistical mechanics and kinetic theory. A detailed discussion of this approach, which is beyond the scope of the present chapter, can be found in the excellent book by Jou, Casas-V~squez and Lebon [24]. 1.2.3 THERMODYNAMICS WITH INTERNAL VARIABLES

One of the possible paths that can be explored to build a theory of irreversible processes in which the most useful aspects of the equilibrium formulation may be preserved consists in introducing new variables in the constitutive relations of the system and in using these additional variables to characterize nonequilibrium conditions. As discussed at the beginning of Section 1.2, thermodynamics is strongly dependent on the identification of the most suitable set of state variables for the description of a specific system. From this point of view it seems quite natural to attempt a description of nonequilibrium processes in terms of new variables that extend the thermodynamic space of the system. The new variables will appear in the constitutive relations but not in the work expression of the first law, since they will be related to some internal properties of the system not coupled to any external action. The origin of this approach goes back to Duhem's works on irreversible processes and to Bridgman's studies on thermodynamics [35,36]. Thermodynamics with internal variables has been used in many contexts [37-39]. We refer to the comprehensive book by Maugin [40] for a complete bibliography. In the internal variable approach, a system whose state at equilibrium is completely characterized by the absolute temperature T and by the set of state variables Xk, k = 1 . . . . n, under out-of-equilibrium conditions will be described by an additional set of variables ~j, j = 1 . . . . m, so that any nonequilibrium state function, like, for example, the internal energy u, will be expressed as u = u ( T , xl .... Xn; ~1 .... ~m). The variables ~j are extensive variables just as are the Xk ones, however they are not conjugate to any additional intensive variable. In fact, we assume that the system has been completely characterized as far as energy balance (first law) is concerned, and we do not want internal variables to alter this balance in any way. This means that internal variables will not appear explicitly in the expression of the first law and will not be explicitly coupled to any external actions (in fact this is the main reason for calling them 'internal'). It has been proven [35] that a variable appearing in constitutive relations but not in the first law must contribute

26

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

to dissipation, otherwise it would not be independent from equilibrium state variables. On the other hand, when equilibrium is reached the system must be completely characterized by its set of equilibrium state variables (T, X1 . . . . . Xn), which means that also the internal variables must be reduced to functions ~j(T, X1 . . . . . Xn) of those state variables. The use of internal variables to describe the out-of-equilibrium behavior of a system finds its justification in the complex interplay between internal processes taking place on widely different space and time scales. We saw that the first step in the description of a thermodynamic system is the choice of a suitable space and time scale and that this choice strongly affects the identification of the state variables that characterize the system. When the interplay between different scales becomes dominant it may happen that there is no obvious and unique choice for the best scale to describe the system. For example, there may exist internal spatial structures with their own relaxation times to equilibrium. If the intensive variables are changed at a rate comparable with the internal structure relaxation time the system will never attain equilibrium and its response will become history dependent. This type of behavior is characteristic of chemical reactions kinetics. As a matter of fact, the work of De Donder and his school about chemical thermodynamics [41] was so influential that we still use their notation and terminology when treating this subject.

C h e m i c a l R e a c t i o n s as an E x a m p l e of Internal Variable Formulation The kinetics of chemical reactions played a key role in the formulation of irreversible thermodynamics. One of the earlier attempts to develop a quantitative treatment of entropy production in chemical reactions is due to De Donder [41]. Starting from Gibbs' equation (Eqn. (1.7)), he formulated a theory in which he assumed that the entropy contribution due to the change in mole number dNk of a given substance can be divided into two terms: (1.39) dNk = deNk + diNk, where deNk describes the exchange of matter with the outside world whereas diNk represents the production (or annihilation) of the species caused by the chemical reaction. By using Eqn. (1.39) in Eqn. (1.7) for the case where pressure and volume are the appropriate work variables, one finds:

dS

dU + pdV T

1 T~ k

1 #k deNk - -~ ~ lak diNk. k

(1.40)

1.2 CLASSICALAND NONEQUILIBRIUM THERMODYNAMICS

27

The entropy variation is made up of the entropy exchange part: deS = dU + p d V T

1

T

~-~ IRk de Nk,

(1.41)

k

and the entropy production part:

1

diS = - - ~ ~_, #k diNk)O.

(1.42)

k

The entropy production was expressed by De Donder in terms of a single variable ~ measuring the degree of advancement of the reaction [3,41]. The role of the variable ~ is made clear by the following example. Let us consider a reaction involving three reacting species, which we term generically A, B and C: A 4- 3B ~ 2C.

(1.43)

No matter exchange with the environment takes place, so deNA = deNB -deNc = O, that is, dNA = diNA, dNB = diNs, d N c = diNc. The variations in mole numbers are not independent since they are controlled by the stoichiometry of the reaction. We can express them in terms of the single variable ~ as follows: dNA dNB dNc = = = d~. -1 -3 2

(1.44)

By using Eqn. (1.44) in Eqn. (1.42) the rate of entropy production takes the form: diS dt

#A q- 3#B -- 2#C d~ T dt

(1.45)

The quantity ar = - ~

Vk#k,

(1.46)

k

where Vk are the stoichiometric coefficients of the reaction (in this case, VA -- --1, VB = --3 and vc - 2) is a new state variable termed 'affinity' by De Donder. In terms of a~' the rate of entropy production becomes: diS ~r d~ = . dt T dt

(1.47)

Equation (1.47) is identical to Eqn. (1.20)" the rate d ~ / d t of advancement of the reaction acts as the generalized flux associated with the change in

28

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

mole number of the reacting species, whereas the quantity d / T plays the role of associated generalized force. The variable ~ has all the properties that were associated with the notion of internal variable in the previous section. The description of the reaction becomes complete once one specifies the equation of evolution obeyed by the internal variable: d~ dt = f i T , p, ~).

(1.48)

In particular, the solution of the equation fiT, p, ~) = 0 gives the value ~eq(T, p) of the internal variable in equilibrium. The description leading to Eqn. (1.47) can be generalized to the case where the internal structure of a generic system is described by a certain number of internal variables ~j, j = 1 . . . . . m, or by a continuous function of some parameter y, ~ = ~(y) [4]. Descriptions of this type are useful, for example, in relation to structural relaxation processes in binary alloys [35] or to nonlinear effects in viscoelastic materials [38,40]. A system characterized by the conjugated work variables (Xk, Ik), k = 1 . . . . . n and by the internal variable-affinity pairs (~j, g~ef'j),j = 1 . . . . . m, will be described by the following relations. I. First law: d U - TdS 4- ~

IkdXk.

(1.49)

k

If. Entropy relations: a S = ~1 d U diS dt

~ - ~ I k d x k 4 . ~ _ ~ J d- ~~ j k j

,

X--" s~'j d~j /--' T dt " J

(1.50)

(1.51)

Ill. Equations of motion for the internal variables: dt =fj(T, I1 ..... In; ~1 ..... ~m) , j -- 1 . . . . . m.

(1.52)

The extensive variables Xk are not necessarily conserved as were not conserved the mole numbers Nk in the chemical-reaction example. The variations dXk describe only the exchange of Xk with the external world, so they are analogous to the terms deNk introduced in Eqn. (1.39). The kinetics of the relaxation is described by Eqn. (1.52). One can attempt to derive

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

29

this equation from some microscopic model of the process, or, more often, from some critical analysis of the phenomenological laws describing the observed relaxation patterns.

Relaxation Times and Quenched Internal Variables Each internal variable ~j will be characterized by some relaxation time zj related to the properties of Eqn. (1.52). Depending on the nature of its internal variables, a system can exhibit relaxation times peaked around some dominant time constant or distributed over a wide range. Comparison of the relaxation times "cj with the characteristic time scale Zexp o v e r which the system is observed in a given experiment plays an important role in the nonequilibrium behavior of the system. In fact, one of the following situations will be encountered. Zexp. In this case equilibrium cannot be reached and the system internal structure remains quenched during the experiment. Internal variables with this type of behavior control hysteresis effects consequent to the presence of quenched, history-dependent, nonequilibrium states during the experiment. 9 "~j "~ Zexp. Internal variables of this type will relax during the observation time, generating measurable relaxation of the system properties toward equilibrium. 9 "Cj ~ "Cexp. In this case the internal structure is always fully relaxed and equilibrium thermodynamics can be applied to the description of the system.

9 Z,j ~

Complex types of time response can appear when the intensive variables present in Eqn. (1.52) are themselves time dependent as a consequence of driving actions imposed on the system. By interpreting the nonequilibrium properties of a system in terms of internal variables one is able to explain a broad range of phenomena, including history-dependent relaxation effects and rate-independent hysteresis. Aparticular internal variable model for the description of hysteresis and thermal relaxation will be presented and discussed in detail in the next section.

1.3 T H E R M O D Y N A M I C

A S P E C T S OF H Y S T E R E S I S

In this section, a particular class of hysteresis models is considered, which admits a natural thermodynamic interpretation. Our aim will not be to reproduce the detailed behavior observed in particular experiments, but rather to propose and develop an idealized framework for the description

30

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

of the basic aspects of hysteresis phenomenology. The idealized character of the approach will permit one to grasp the essential consequences of certain crucial assumptions. In addition, it will make it easier to identify the points where suitable additional assumptions could be introduced in order to develop more detailed models for particular systems. The systems we will analyze are collections of elementary two-level subsystems (TLS). In fact, such collections are rich enough in structure to reproduce well the essential features of hysteresis and thermal relaxation. At the same time, they can be described in relatively simple terms, which permit one to appreciate in intuitive terms the consequences of the assumptions on which the model is based [42]. In a sense, TLS collections play the role of model systems for hysteresis and relaxation in a way similar to what 'perfect gases' represent in the framework of classical thermodynamics. We will show that the extension to out-of-equilibrium conditions of the statistical mechanics of TLS ensembles leads to a thermodynamic formulation in terms of internal variables very similar to the approach discussed in Section 1.2 for the description of nonequilibrium thermodynamics of chemical reactions. The approach developed in the rest of this section will be presented by using the terminology typical of magnetic systems: the external driving force will be termed the magnetic field and denoted by H, the conjugate extensive variable (expressed in terms of its volume density) will be the magnetization M and so on. However, the treatment can be applied to other kinds of systems as well, after appropriate reinterpretation of the relevant variables and parameters. 1.3.1 COLLECTIONS OF ELEMENTARYTWO-LEVEL SUBSYSTEMS (TLS) Elementary TLS units characterized by two metastable states have been given special consideration in the literature, owing to the simplicity of their description. The properties of an individual TLS unit are fully controlled by the energy values associated with its two states. We will denote these states by (+) and ( - ) and we will assume that the TLS magnetic moment m in these two states is equal to +Am and -Am, respectively, where Am is a given constant parameter. The TLS energy in the field H is:

E+ - +#o (hu - H) Am,

(1.53)

where #0 represents the permeability of the vacuum. According to Eqn. (1.53), the TLS energy levels under zero field differ by the amount 2#ohuAm. The quantity -hu is equivalent to a local field acting on the TLS unit. On the other hand, the energy levels depend linearly on the field H, since we have assumed that Am is a constant parameter. One could envisage more

31

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

complex situations, where, for example, Am becomes field-dependent. Such possibilities will not be considered any further.

Equilibrium Properties TLS equilibrium properties are controlled by the partition function z(H, T). Given Eqn. (1.53), the TLS partition function takes the form:

2coshIHhuI

(1.54)

where

kBT H f - #0Am

(1.55)

is the so-called fluctuation field. Gibbs' free energy g(H, T) equal to:

= -kB T In z is

g(H,T)--kBTlnI2cosh(H-hu

(1.56)

and the equilibrium probabilities of occupation of the (+) states are:

1E

peq= 2

l+tanh

Hf

hUl

(1.57)

'

respectively. From the previous expressions one can derive the other thermodynamic state functions by standard methods. For the equilibrium TLS magnetic moment m(H, T) one finds:

m(H, T) =

1 egg

~0~H

= Am tanh

[H ~hu)

\ HI

.

(1.58)

Analogously, the TLS entropy s is given by the expression:

H-hu

HI

I ;ul I (1.59)

32

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

Finally, the TLS internal energy u can be calculated from the relation u -

g + Ts +/aoHm: u ( H , T ) - # o h u A m t a n h ( H -H/ hu) .

(1.60)

These results can be extended to a collection of No independent TLS units characterized by the different Am and hu values. To keep the discussion as simple as possible, we will assume that Am is the same for all units. In addition, we will assume that the system occupies a certain volume V0 and we will express the resulting extensive quantities in the form of volume densities, calculated as integrals of the individual TLS contributions over the function (No/Vo) Peq(hu), where Peq(hu) represents the probability density that a unit be characterized by the value hu. We will denote the resulting average quantities by the same symbols used for the individual TLS quantities, since individual TLS units will no longer be involved in the discussion. The overall Gibbs' free energy per unit volume is equal to:

=

in Iacosh

Hf

u)l Peq (hu) dhu,

(1.61)

where

Ms =

NoAm v0

(1.62)

represents the maximum (saturation) magnetization attainable by the system and HI is given by Eqn. (1.55). The magnetization M of the collection is equal to:

M = Ms

oo

tanh

H - hu Hf Peq (hu) dhu.

(1.63)

Finally, the entropy and the internal energy per unit volume become:

s - #~ Ms T

In 2 + In cosh

~

( ) H - hu Hf

HhUtanh(HhUlJ Peq'hu'dhu hu) u = l~oMs f_ x~ o0 hu tanh [\ H -~f

Peq

(hu) dhu 9

(1.64)

(1.65)

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

33

The previous expressions admit the following simple approximation that will be useful in the extension of the description to out-of-equilibrium conditions. The hyperbolic tangent multiplying Peq(hu) in the integrand of Eqn. (1.63) varies significantly only over a field interval of the order of H f around hu = H. If peq(hu) is approximately constant over intervals of this width, then the hyperbolic tangent can be safely approximated by a step function. The resulting expression is:

M ~ Ms

IF

oe Peq (hu) dhu -

F

]

Peq (hu) dhu .

(1.66)

Formally, this result is obtained by taking the limit Hd ---> 0 in Eqn. (1.63). The same limit can be used in Eqn. (1.61) to calculate the corresponding approximate free energy:

g ~" # ~

EF

oc (hu - H ) Peq (hu) dhu -

]

(hu - H ) Peq (hu) dhu .

(1.67) Under this approximation all temperature dependencies disappear, so no entropy contribution is present anymore. Relaxation Kinetics The relaxation toward equilibrium of the TLS ensemble may exhibit highly nontrivial features depending on initial conditions, temperature and field. In particular long-living memory and history-dependent properties can appear. A generic nonequilibrium state of the ensemble is defined by specifying the probabilities P+ of occupation of the (4-) TLS states for each TLS unit. The time evolution of these probabilities is governed by the master equation: dP+ dt

= w _ P _ - w+P+,

(1.68)

where P+ + P_ = 1 and (w+, w_) represent the transition rates from (+) to ( - ) and from ( - ) to (+), respectively. The transition rates contain the information on the mechanisms whereby the system approaches thermodynamic equilibrium. Whenever these rates become particularly small the TLS ensemble gets trapped in a nonequilibrium state for long times and develops long-living memory and history-dependent properties. To deal with such possibilities we assume for w+ the typical Arrhenius form

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

34

associated with thermal activation over energy barriers, namely: w+ = -

exp -

.

(1.69)

In the previous expression the factor 1/z0 represents the attempt frequency and hc characterizes the energy barrier involved in the process. Basically we are assuming that transitions between (+) and ( - ) states take place via an intermediate state of energy l~ohcAm > 0, independent of H. Under these assumptions, the TLS master equation for P+ (a similar expression holds for P_) can be written in the form: dP+ dt -

1 (p+ _ p+q), z

(1.70)

where P+q is given by Eqn. (1.57) and: 1

z

2

-w++w_=--cosh

"cO

\

Hf

exp

-

-~f

.

(1.71)

Equation (1.70) shows that our choice for the transition rates is consistent with the requirement that the system must eventually reach thermodynamic equilibrium under all circumstances. After assuming Eqn. (1.69) for the TLS transition rates, each TLS unit becomes a function of two parameters: hu and he > 0. Therefore, a collection of TLS units in a generic nonequilibrium state has to be described by the two-dimensional density p (he, hu) giving the probability density that a TLS be characterized by a certain pair of values (he, hu). The function p (he, hu) takes the place of the function Peq(hu) previously introduced. If one solves Eqn. (1.70) for the time-dependent TLS probability P+ (t; he, hu), then one can express the relaxing magnetization of the collection by the following expression, analogous to Eqn. (1.63):

M(t) = Ms

dhc

F

[2P+

(t; hc, hu) - 1] p (hc, hu) dhu.

(1.72)

Oo

The calculation of the magnetization of the TLS ensemble through Eqn. (1.72) is in general a demanding task if the initial condition for P+ is left completely arbitrary. However, a drastic simplification is obtained if the initial state of the ensemble at t = 0 has the following structure: there exists a line bo (he) such that P+ (t = 0; he, hu) = 0 for hu > bo (he) and P+ (t - O; he, hu) -- 1 for hu < bo (he) (Fig.l.1). The line bo (he) divides the (he, hu) plane into two simply connected regions representing the two TLS

35

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS hu

P+=0 bo(hc) beq(hc) H

FIGURE 1.1 Line b0 (hc) dividing the (hc, hu) plane into regions where TLS units are in (+) and (-) states with probability 1. The dashed line represents the equilibrium condition under the field H. sets that are initially in the (+) or ( - ) state with probability 1. Partitions of this type are important because, on the one hand, they are generated in a natural way by field histories that start from large initial values and, on the other hand, they are preserved, to a good approximation, by the TLS relaxation kinetics. Collective TLS states initially of the type shown in Fig. 1.1 evolve in the form of a moving probability front gradually approaching the stationary equilibrium front described by Eqn. (1.57). The front is represented by the region in the (he, hu) plane where P+ (t; he, hu) passes from I to 0. Therefore we can identify the location of the front by introducing the line b (he, t) defined by the condition: 1 P+ [t; he, b (he, t)] - ~.

(1.73)

With this definition, the line associated with the equilibrium state under the field H is

beq(hc) -- H, as shown by Eqn. (1.57).

(1.74)

36

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics The steepness of the front is measured by the quantity:

Q+ (he, t) - r0P+ 1 L--~u Jh=-b(hc,t)"

(1.75)

By taking the hu derivative of Eqn. (1.70) and by making use of Eqns (1.57), (1.71), (1.73), and (1.75) one finds:

(1)

0Q+

1 Q++

c~t

.

(1.76)

Equation (1.76) shows that the front approaches a width of the order of

2HI. The front velocity for a certain value of he can be obtained by writing Eqn. (1.70) for the particular TLS unit for which hu - b (he, t). By taking into account Eqns (1.73) and (1.75) one finds:

3b(hc, t) -Q+

,~t

l ( ~ _ p + t eq ' "c

_

-

(1.77)

where P+q (Eqn. (1.57)) must be calculated for hu - b (hc, t). If one divides Eqn. (1.76) by Eqn. (1.77) side by side, one obtains a separable differential equation for dQ+/db, which can be integrated by standard methods. The result is: = - 1 + sinh [(H - b)/Hl]

1

2HfQ+

(1.78)

sinh [(H - bo)/Hf]'

where bo (hc) is the initial step-front line shown in Fig. 1.1. Equation (1.78) shows that the relaxation of the front slope toward its final value Q+ = -1/2H I is exponentially faster than that of the front position toward the equilibrium line beq (he) - H. Therefore, one can take Q+ "~ -1/2Hf in Eqn. (1.77). By making use of Eqn. (1.57) and Eqn. (1.71) one finally arrives at the equation:

Ob(hc, t) ~t

~o

sinh

t,] HI

exp

-

,

(1.79)

with the initial condition b (hc, t - O ) - bo (hc). This is the fundamental equation for the description of the relaxation of the TLS ensemble. This equation can also be applied to situations where the field H is timedependent, as no assumption whatsoever was made about H in the derivation of the equation.

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

37

The fact that P+ (t; hc, hu) exhibits significant variations only in an hu interval of the order of Hf around b (he, t) permits one to treat Eqn. (1.72) similarly to Eqn. (1.63). In fact, if p (he, hu) is approximately constant over hu variations of the order of Hf one can safely approximate the term 2P+ - 1 in the integrand of Eqn. (1.72) by a step function, which gives:

M(t) ~ Ms

~oCX~[fb(hc't)p (he, hu) dhu - ~bCX~p (he, hu) dhu 1 dhc. -oc (hc,t) (1.80)

This equation, together with Eqn. (1.79), gives the complete description of the relaxation kinetics of the TLS ensemble. Before proceeding, a remark is appropriate about the validity of Eqn. (1.79). When we introduced the transition rates described by Eqn. (1.69) we made the implicit assumption that the intermediate state of energy #ohcAm represented indeed a barrier to the relaxation. Since the energy of the (4-) TLS levels is equal to +#0 (hu- H)Am, this will be the case only when [hu - H I ~ he. This condition involves the field H, so under varying field one can pass from a situation where the TLS states are separated by a barrier to a different one where no higher-energy intermediate state is present anymore. When Ihu - HI >~ he, it would seem natural to assume for the transition rates the values: 1/~0 for transitions from higher to lower energy; and exp ( - 2 IH - hu I/Hf) for transitions from lower to higher energy. This would give, for the relaxation time ~, instead of Eqn. (1.71), l = l [ l~0 +exp(r

-21H-hul)]Hf .

(1.81)

This equation predicts relaxation times of the order of r0, whereas Eqn. (1.71), under identical conditions, gives relaxation times smaller by the factor exp [(he - I H - hul)/Hf]. In this sense, Eqn. (1.79) is not accurate whenever IH - b (he, t) l >~hc. However, our main effort is to develop a suitable description of relaxation processes taking place on time scales many orders of magnitude larger than r0. The details of changes occurring over time intervals of the order of r0 play no role because those events are practically instantaneous on the time scale of interest. Therefore, no appreciable error is introduced if transitions on the r0 scale occur even faster, as is the case if one uses Eqn. (1.79) also in the region IH - b (hc, t) l>~hc. 1.3.2 THERMODYNAMIC FORMULATION The analysis of TLS ensembles carried out in the preceding section provides a suitable basis for the formulation of a thermodynamic description

38

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

of hysteresis and thermal relaxation. Basing the description on TLS units as elementary building blocks amounts to making the working hypothesis that collections of TLS units are rich enough in structure to reproduce the basic features of hysteresis and thermal relaxation observed in real systems. Historically, this working hypothesis appeared in the literature when F. Preisach proposed his model to interpret certain aspects of aftereffect phenomena [43]. The TLS model we are going to discuss is based on the same core assumptions, reformulated in the frame of nonequilibrium thermodynamics [44]. It provides a unified description where the same concepts and tools are at the basis of three main classes of phenomena: 9 rate-independent hysteresis under varying field a n d / o r temperature; 9 heat exchange and entropy production; 9 time-dependent response and relaxation effects. The interest and the usefulness of the model lies in the fact that it leads to interdependent predictions about all these mechanisms at the same time. There are cases, for example ferromagnetic and superconducting systems, where these predictions are in remarkable agreement with experiments. However, the model has a highly idealized character such that it should not be expected to provide a detailed description of real systems under all circumstances. The conditions under which a given system approximately behaves like a collection of TLS elementary contributions are by no means trivial and require a detailed study case by case. It is often obvious that the direct derivation or justification of the TLS ensemble approach from microscopic statistical-mechanics description is of prohibitive difficulty. However, a completely different strategy is possible, discussed in Section 1.3.2. In fact, it has been proven [13,45] that two certain properties, known as 'return-point memory' (or 'wiping-out') and 'congruency' properties, represent the necessary and sufficient conditions for the applicability of the TLS ensemble description, at least in its rate-independent limit. Therefore, if one can show that a certain system exhibits those properties then one knows for sure that system can be described by the model, even though one might have no idea how elementary TLS units might be defined for the system in question. In Section 1.4, we will discuss an important example where this strategy is used to show the emergence of Preisach-type hysteresis from micromagnetic stochastic processes.

Rate-Independent Hysteresis Rate-independent hysteresis is expected to emerge from the TLS ensemble approach in the limit of low temperatures, as a consequence of the fact

1.3 THERMODYNAMICASPECTS OF HYSTERESIS

39

that whenever the TLS ensemble gets trapped in some history-dependent collective state described by the state line b(hc), it will stay there for very long times (infinite in the limit of zero temperature), unless the external field is modified. Several models have been proposed in the literature to describe rate-independent hysteresis [6-8,13,14,46-51]. We will show that the rate-independent limit of the TLS ensemble behavior coincides with the Preisach hysteresis model (see other chapters of 'The Science of Hysteresis' on this topic), a conclusion that creates an interesting connection between the mathematical modeling of hysteresis and nonequilibrium thermodynamics. Let us consider the solutions of Eqn. (1.79) in the limit T ~ 0, assuming that H(t) is some given function of time. When T ~ 0, Hd --~ 0 as well (Eqn. (1.55)). Therefore all the exponential and hyperbolic terms appearing in Eqn. (1.79) become singular. The consequence is that c~b/Ot = 0 when I H - b (hc)l < he, and c~b/c~t --+ cx~ when IH - b (hc)l > he. In the latter case, the sign of the time derivative is always such that IH - b (hc) l ~ he. Therefore, c~b/c~t can attain finite nonzero values only when H - b (he) +he. In addition, the equality c~b/c~t = d H / d t necessarily holds, because this is the only way the condition H - b (he) - +he can be preserved over time. These results can be given the graphical representation shown in Fig. 1.2. Let us consider the half-plane (he ~ 0, hu) and let us introduce the region defined by the relation: IH - hul ~ hc.

(1.82)

In the T ~ 0 limit, the state line b (he) is confined to within this region, which will be termed the 'metastability region'. Any point of the state line that lies inside the metastability region stays fixed in time. However, the region itself moves in time as its vertex is located at hu -- H(t). When a point of the state line comes into contact with the moving boundary of the region, it gets stuck to the boundary and moves with it until the field is reversed. When the field oscillates back and forth the result of this action is that the state line acquires the typical oblique staircase shape shown in the figure. Thus, in the T ~ 0 limit the relaxation kinetics of the state line is reduced to the particular updating rule just described. This rule only involves geometrical considerations about the state line, with no reference to the rate of change of the external field, which means that the system response is rate-independent. Interestingly, this rate-independent limit coincides with the Preisach model of hysteresis, amply discussed in other chapters of 'Science of Hysteresis'. Two are the basic properties of the Preisach model: (i) differences

40

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

hu (-)

,/////////;i H -h u I<

H

hc (+)

FIGURE 1.2 State line b(hc) (bold line) lying inside metastability region

I H - hu I K hc.

from one system to another result in different choices for the probability density function p (he, hu), which in this context is usually referred to as the Preisach distribution or Preisach density function; and (ii) differences in past field histories result in different shapes of the Preisach state line b (he) in a way that is independent of the choice made for the Preisach distribution. Therefore, system-specific features and memory-storage features are completely separated and independent. As a consequence of this remarkable separation, the past-history-storage properties of the model will be universal, that is, they will hold for all systems described by the Preisach model, independently of the particular choice made for the Preisach distribution. Two properties in particular play a key role in this respect: they are known as the 'return-point memory' (or 'wiping-out') and the 'congruency' properties.

9 Wiping-out or return-point memory property.

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

41

Wiping-out property. According to the graphical representation shown in Fig. 1.2, the shape of the state line b (he) is determined by the sequence of inversion points at which the field is reversed, that is, by the sequence of local maxima and minima of the H(t) time function. As a matter of fact, this sequence of local extrema gives a redundant description [9]. It is possible to identify a sub-sequence containing the minimum complete set of information identifying the state line b(hc). This subset is obtained by extracting from the initial sequence of extrema the sub-sequence of monotonically decreasing maxima, HM1 > HM2 > "-, and of monotonically increasing minima, Hml < Hm2 < "" ", alternatively reached in time, that is, at times tM1 < tin1 < tM2 < tin2 < "'" 9 The other extrema have no influence on the final state of the system. The mechanism whereby this occurs is shown in Fig. 1.3. When the field is first increased to the value H1, then decreased to H2, and then increased again, the updating rule for the state line is such that, when the field reaches again the value H1, the state line acquires the same shape that it had when the value H = H1 was reached for the first time. The effect of the intermediate field oscillation is completely erased (wiped out). Return-point memory. The reason for using this term to describe the previous mechanism is that, when in Fig. 1.3 the field reaches the value H1 for the second time, the system returns to the exact same state it occupied at the time when the point H = H1 was reached for the first time. In other words, the system is able to store the information about its past states and is able to return to those exact states under appropriate conditions. This is the reason for the use of the term 'return-point memory'. The reason for the existence of return-point memory is quite general, and can be eventually traced back to the possibility of introducing an ordering relation among the states of the system. In fact, return-point memory is a property of any system with rate-independent evolution, for which the rule of evolution is such that states which are ordered with respect to some appropriate ordering relation, say A(0) ~ B(0), remain ordered when ordered field histories, HA(t)~ HB(t), are applied to them, that is, A(t)KB(t) (see [9] for a more detailed discussion). In the Preisach model one can naturally introduce the following partial ordering of states: two Preisach states A and B are ordered, A~B, if the corresponding state lines bA (he) and bB (he) are such that bA (he) ~ bB (he) for any he. It is easily verified by applying the updating rule shown in Fig. 1.2 that this ordering is preserved by the action of ordered field histories.

42

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

FIGURE 1.3 Connection between field history and shape of state line b(hc).

9 Congruency property. This property can be defined as follows. Consider two different field histories HA (t) and HB(t) ending up with the same final field value H1. The two histories will give rise to different state lines bA (hc) and bB (he) such that bA (he = O) = bB (he = 0) - H1. If we begin cycling the field between H1 and some other field H2, the same triangular region of the (he, hu) plane will be swept back and forth by the state line. The fact that bA(hc) ~ bB(hc) for he ~ 0 plays no role. As a consequence, the system will describe two geometrically congruent minor loops. The difference in the initial state lines will merely result in a vertical shift of the minor loops in the (H, M)plane. Any system for which minor loops between any two field extrema (H1, H2) are geometrically congruent is said to exhibit the congruency property. The importance of the wiping-out and congruency properties derives from the following fundamental theorem, due to Mayerogyz [45]. Applicability of the Preisach Model A scalar system with rate-independent

hysteresis can be described by the Preisach model if and only if it obeys the "returnpoint memory" and the 'congruency" properties. Return-point memory and congruency express the universal essence of the rate-independent limit of TLS ensemble thermodynamics. In Section 1.4, we will make use of these properties to prove that certain classes of microscopic processes give rise indeed to Preisach-type hysteresis when an appropriate average over structural disorder is carried out.

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

43

Internal Variable F o r m u l a t i o n In the analysis presented in Section 1.3.1, the parameters characterizing the individual TLS units, namely, Am, hc, and hu, were just constant quantities. A more general thermodynamic approach is obtained by interpreting each TLS unit as a unit itself resulting from some statistical mechanical average. In other words, the problem is characterized by well-separated relaxation time scales, such that the thermodynamic equilibrium condition associated with the individual TLS energy levels is reached much earlier than the time necessary for the TLS ensemble to reach thermodynamic equilibrium as a whole. In this framework, the TLS energy is reinterpreted as a free energy and the parameters describing the TLS energy levels, that is, Am, hc, and hu, become temperature-dependent thermodynamic parameters. To keep the discussion simple, we will assume that these parameters depend on temperature but not on the field H, although in principle such dependence cannot be ruled out. In addition, no volume effect will be considered. With these assumptions, the saturation magnetization Ms (Eqn. (1.62)), dependent on Am, and the probability density p (he, hu) introduced in Eqns (1.80) and (1.72) become temperature-dependent quantities. The relaxation kinetics of the TLS ensemble discussed in Section 1.3.1 is similar in many respects to the thermodynamics of reacting substances presented in Section 1.2.3. Let us first consider TLS units with identical hc. The generalization to a distribution of hc values will be straightforward. The TLS units in the (+) and ( - ) state represent two'species' whose 'concentrations' are:

N~ fbc~ N+ (hc) - N~ G fb__(hc) p (hc, hu) dhu , N_ (hc) - -~o (he) p (hc ' hu) dhu " (1.83) During the TLS relaxation process the state line b(hc) changes (Eqn. (1.79)) and a certain fraction of the N+ units is transformed into N_ ones or vice versa. Only the units in mutual 'contact', i.e. such that hu - b (he), are involved in the transformation. Accordingly, when the b(hc) line changes by the small amount 6b(hc), we have that dN+ = - d N _ = d~, where

No P [hc , b (hc)] ~b (hc). d~ - -~o

(1.84)

The energy change per TLS unit during the transformation represents the 'affinity' sO' for the 'reaction' (see Eqn. (1.46)): ~ = - ( E + - E_), where

44

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

E+ is given by Eqn. (1.53). One finds: d - 2#0 [H - b (hc)]Am.

(1.85)

Therefore, the entropy production associated with the transformation will be (see Eqn. (1.47)):

T dis -- d d~ -- 2#oM s [H - b (hc)] p [hc, b (hc)] (~b (hc).

(1.86)

The generalization to a distribution of hc values just amounts to an integration with respect to he, since there is no coupling between TLS units with different values of he. Thus we arrive at the following expression for the rate of entropy production:

dis = 2#~ T -d-[

Ob(hc, t) fo ~ [H - b(hc, t)]p[hc, b(hc, t ) ] ~ dhc, ~t

(1.87)

where the degree of advancement of the 'reaction' is measured by b(hc, t) through Eqn. (1.79). Equation (1.87) fulfills the fundamental requirement for the entropy production, namely, it is positive under all circumstances. In fact, Eqn. (1.79) shows that tVb/cgt always has the same sign as (H - b). Accepting the' reacting species' analogy is equivalent to assuming that the nonequilibrium Gibbs free energy g of the system is equal to the integral over the TLS ensemble of the individual TLS energies, E• = +#o(hu-H)Am for units in the (+) or ( - ) state, respectively:

g =fo + #oMs

-

(hc)

(hu - H ) p (hc, hu) dhu

(hu - H ) p (hc, hu) dhu

dhc.

(1.88)

This equation represents the natural out-of-equilibrium generalization of Eqn. (1.67). The function f0 takes account of additional effects that might be present. For example, it is possible to describe in this way mean-field effects due to magnetostatic or exchange interactions in magnetic media (see [9] for details). These complications will not be considered here. We will simply assume that f0 is a certain function dependent on temperature only. The precise form of this function will play no role in the rest of the discussion. The thermodynamic formulation becomes complete once one specifies the temperature dependence of the various quantities appearing in

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

45

Eqn. (1.88). This point must be treated with particular care, because temperature is hidden in b(hc) in quite a subtle form. As previously mentioned, temperature enters the model when one reinterprets the individual TLS units as free-energy units with temperature-dependent parameters. This interpretation necessarily requires the state line b (he) to be itself temperature dependent even when no TLS transition occurs. In fact b (he) describes the distribution of the TLS units between their two possible states. The fact that a certain TLS unit is in its (+) or ( - ) state will not be affected by a temperature change if no irreversible transitions occur. However, temperature will cause reversible distortions in the individual TLS energy levels, which will change the position of the TLS units in the (he, hu) plane and consequently also the location of the boundary b(hc) between units in the (+) and ( - ) state. The study of the thermodynamic properties of the TLS ensemble model for a generic form of the temperature dependence of the various quantities involved is by no means a straightforward task. Here we will limit our considerations to a particular case, which is, however, of quite general interest [52,53]. Let us assume that the statistical distribution of TLS (he, hu) values is the same at all temperatures, apart from temperature-dependent scale factors common to all TLS units. Physically this is expected to occur when, for example, in a magnetic system, the structural disorder responsible for magnetic hysteresis is not modified by temperature, in the sense that no irreversible microstructural relaxation occurs in the system. The effect of temperature is only to modify the value of the thermodynamic parameters coupling the magnetization to structural disorder, e.g. saturation magnetization, crystal anisotropy constants, and magnetostriction constants. Whenever this is the case, the distribution function p(hc, hu) and the state line b(hc) will become temperature-independent once renormalized to the mentioned scale factors. Let us term Qc(T) and Qu(T) the scale factors for hc and hu, respectively. Then appropriate temperature-independent variables are:

hc hu Zu = ~ . Zc = Qc (T) ' Qu (T)

(1.89)

Owing to the assumed scaling invariance, the distribution function and the state line can be expressed in the form:

p (hc, hu) dhc dhu - p (Zc, Zu) dzc dzu,

(1.90)

b (hc) - Qu(T) b (Zc),

(1.91)

46

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

where ]~ and b are temperature-independent functions of Zc and Zu. It is important to realize that the normalized state line/~ (Zc) changes if and only if some irreversible TLS transition occurs. Therefore this is the correct internal variable for the description of approach to equilibrium and entropy production, not the b line, which may change with temperature even if no entropy production is involved. The relaxation equation for/~ (Zc) is obtained by repeating the analysis presented in Section 1.3.1 in terms of the normalized TLS parameters Zc and Zu. The result is identical in structure to Eqn. (1.79):

2if-If

8t

~o

_ ' Hf

[-I-- Qu' H

Hf q=Qu Qc J=Qu'

sinh

I[-I- [~(Zc t)

]exp( )

cgb(Zc, t)

(1.92)

where (1.93)

Following this analysis the Gibbs free energyg given by Eqn. (1.88) is to be interpreted as a function of (H, T) and a functional of b(.): g[H, T, b(.)]. Once expressed in terms of the normalized variables (Zc, Zu) Eqn. (1.88) becomes: g =f0 + #oMsQu

IS::

(Zu -/2t)/? (Zc, Zu) dzu

~,Zu - HI ~ (Zc Zu) dzu dzc.

(1.94:)

The partial derivatives with respect to H, T and b(.) represent the magnetization, the entropy, and the entropy production, respectively. For the magnetization, one obtains" M-

1 ~g] /~0 8H r,~(.) -

= Ms

is::

p (Zc, Zu) dzu -

(zc)

]

p (Zc, Zu) dzu dzc (1.95)

if none of the TLS parameters depends on field. Equation (1.95) is identical to Eqn. (1.80) once expressed in terms of the (he, hu) variables. The Gibbs free energy is of the form:

g =f-

#oHM,

(1.96)

47

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS where ~

f =fo + #oMsQu

-

(Zc)

Zu ~ (Zc, Zu) dzu

Zu ~ (Zc, Zu) dzu

dzc.

(1.97)

Entropy is given by the partial derivative of Eqn. (1.94) with respect to T. According to Eqns (1.95) and (1.97),f - f o 4- MsQuAf and M = MsAM, where Af and AM are quantities independent of T. Therefore one finds from Eqn. (1.96): ~g S_=

~T H,~(.)

- so -

d ln(MsQu) d In Ms dT (f - f o ) 4- d----~#oHM,

(1.98)

where so = -Ofo/c~T. Finally, the rate of entropy production is given by:

dis _ dt -

~o ~

6g Ob(zc___~) dzc 6b(zc) cgt

(1.99) H,T

By calculating the variation 6g/6b(.) from Eqn. (1.94) one finds:

dis - 2#~ T --d-t

~o ~ [H - b(zc, t)] p[zc, b(zc, t)] Ob(zc, t) dzc, 0------~

(1.100)

where ~gb/c~t is given by Eqn. (1.92). This expression coincides with Eqn. (1.87) if one can identify Quc~b/c~twith Ob/c~t. This occurs indeed for isothermal transformations, because then Qu is a constant quantity and QuOb/cgt =_ O(Qub)/Ot =- Ob/c~t. This conclusion confirms that b(zc) is the correct internal variable for the description of dissipative processes. It is only for isothermal transformations that the distinction between b(hc) and b(zc) becomes irrelevant. Although we have used the TLS analogy to build up a consistent model, the result has a structure where there is no need anymore to refer to the TLS picture. The basic thermodynamic variables are the field H, the temperature T and the normalized state line b(zc). All thermodynamic functions are functionals of the state line, so they acquire well-defined values for any nonequilibrium state identified by a particular choice for the state line. The description is completed by the law of relaxation to equilibrium represented by Eqn. (1.79) or Eqn. (1.92). Once the direct connection to the

48

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

TLS picture is removed, one can envisage generalizations of the theory based on state-line functionals more complex than Eqn. (1.88) and relaxation equations different from Eqn. (1.79). This opens the door to a broad class of thermodynamic models farther and farther away from the seed provided by the initial TLS analogy. On the other hand, there is an important restriction in the applicability of the thermodynamic formulation leading to Eqns (1.94) - (1.100). In fact, one of the main problems arising when one tries to define nonequilibrium state functions like free energy and entropy is that some appropriate definition of nonequilibrium temperature should also be introduced [40,54,55] (see [56] for a comprehensive bibliography on the subject). In our formulation, this difficulty has been avoided by assuming that: 9 the system with hysteresis is constantly in thermal contact with a heat reservoir at temperature T; 9 all fast-relaxing degrees-of-freedom (e.g. the ones responsible for the saturation magnetization and for the individual TLS energy levels) are in thermodynamic equilibrium at temperature T; 9 all thermally activated relaxation processes taking place in the system are controlled by temperature T. Therefore, the formulation is not applicable to transformations where the temperature is not an externally controlled parameter, as it would be, for example, for adiabatic transformations in a thermally isolated system. This restriction is not particularly serious for ferromagnetic systems where the magnetic degrees-of-freedom are intermixed with the lattice degrees-offreedom on the atomic scale. The specific heat of the lattice is much larger than that of the magnetic degrees-of-freedom. Therefore, nonequilibrium magnetization processes occur with the magnetic degrees-of-freedom in thermal contact with the heat reservoir provided by the lattice. This makes it possible to apply the approach under quite general conditions (for example, even when the body is under adiabatic conditions) provided the temperature of the lattice is well defined at any time.

Entropy and Entropy Production According to the thermodynamic formulation discussed in the previous section the differential of the nonequilibrium Gibbs free energy is given by the expression: dg = -#oMdH

- s d T - T dis,

(1.101)

where M and s are given by Eqns (1.95) and (1.98), respectively, whereas T dis is obtained from Eqns (1.100) and (1.92). Equation (1.101) shows

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

49

that the Gibbs free energy decreases along any arbitrary transformation at constant H and T, because the third term on the right-hand side of the equation is negative-definite. The state of thermodynamic equilibrium is reached when g attains its minimum and the entropy production vanishes. This occurs when the state line becomes equal to: beq (Zc) = f t = H / Q u .

(1.102)

The equilibrium magnetization Meq associated with this state line can be calculated from Eqn. (1.95):

Meq = Ms

Peq (Zu) dzu

peq (Zu) dzu ,

-

(1.103)

where (X)

Peq (Zu) --

L

]9 (Zc, Zu) dzc.

(1.104)

From Eqn. (1.98), one obtains for the equilibrium entropy Seq: d ln(MsQu)

Seq - so -

dT

0Ceq-f0) +

d In Ms dT #0HMeq'

(1.105)

where (see Eqn. (1.97)):

feq -- fo -Jr-12oMsQu

Zu Peq (Zu) dzu -

Zu ]~eq(Zu) dzu .

(1.106)

The system response to a generic time-dependent field H(t) is seldom close to Eqn. (1.103), because the corresponding time-dependent solution of Eqn. (1.92) is usually quite different from Eqn. (1.102). It is only for infinitely slow variations of the field that the system responds according to Eqn. (1.102). An example of a situation where the system response gradually approaches the equilibrium one will be given in Section 1.3.3. Some entropy production is inevitably involved in any transformation for which [~(Zc, t) ~ H(t) and cgb/cgt r O. Let us discuss the consequences of this fact in a particular case amenable to a relatively simple analysis. We assume that the transformation is isothermal, so Eqn. (1.101) is reduced to: dg = - # o M d H - T dis.

(1.107)

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

50

The difference between b and b state lines becomes irrelevant when T is constant, so the equations for M,f, and the entropy production (Eqns (1.95), (1.97), (1.100)) can be written in the simpler form:

M=Ms

p(hc, hu) d h u -

p(hc, hu) dhu

dhc,

(1.108)

(hc)

,

hup(hc, hu) d h u -

7

rl~ hu p (hc, hu) dhu I dhc, Jb (hc) .3

(1.109)

T dis - 2#oMs

f0 ~[H -

b(hc)] p[hc, b(hc)] (~b(hc)dhc,

(1.110)

where cSb(hc) is the variation of the state line in the time increment dt. In addition, we assume that the rate-independent limit discussed in Section 1.3.2 holds. We know that in this limit the state line is confined to within the metastability region IH - hul<~hc (see Fig. 1.2). Furthermore, Ob/cgt = 0 unless the state line is in contact with the boundary of the region, i.e. H - b (he) = he and all~at > 0; or b (he) -- H + he and dH/dt < 0. In these two cases, c~b/St = dH/dt, i.e. ~b (he) = dH. Under these conditions Eqn. (1.110) can be written in the form:

Tdis = 21aoMs

fo

I

hcp (hc, H :F hc) bb (hc) dhc ,

(1.111)

where one must use p (hc, H - hc) when dH/dt > 0 and vice versa. The variation bb (hc) is different from zero only for those hc values where one of the two previous conditions is met, in which case it is simply equal to dH. Therefore, the entropy production depends on the field only, with no reference to the field rate. This is in agreement with the rate-independent character of the transformation. Let us apply Eqn. (1.111) to the particular case where the field oscillates between two fixed values H1 and H2, H1 < H2, thus driving the system along a closed hysteresis loop. By applying the updating rule for the state line shown in Fig. 1.2, one finds that bb (hc) ~= 0 for 0 <~hc <~ (H - H1)/2 when the field is increasing and for 0 <~hc <~ (H2 - H ) / 2 when the field is decreasing. Therefore, the total entropy production along the hysteresis loop can be written as the sum of two field integrals associated with the

1.3 THERMODYNAMICASPECTS OF HYSTERESIS

51

ascending and descending branches of the loop:

f T dis = 2#oMs f ~ 2 f dH (H-H1)/2 1

+2#oMs

dO

2

1

hop (he, H - he) dhc

f (H2-H)I2

dH

dO

hcp (hc, H + hc) dhc. (1.112)

The two field integrals are coincident because they involve the same triangular integration region: 0 ~ he ~ (H2 - H1)/2, H1 -J- he ~ hu ~ H2 - he. Accordingly, Eqn. (1.112) can be written as:

f

T dis - 4#oMs

So(H2-H1)/2

f

H2-hc

dHl+hc

hcp (hc, hu) dhc dhu.

(1.113)

It is a remarkable property of Preisach-type hysteresis that the entropy production along a generic hysteresis loop is the sum of two equal contributions produced along the ascending and descending branches of the loop. Equation (1.113) is an example of the general result that the entropy production along a cyclic isothermal transformation is equal to the area of the corresponding hysteresis loop. In fact, by integrating Eqn. (1.107) over the loop one finds:

f T dis - -#o / MdH,

(1.114)

since the integral of dg is equal to zero if the initial and final states coincide. However, the thermodynamic formulation developed so far is valid also for open transformations where the connection with loop areas is lost. This fact permits one to address a fundamental problem for which there is no obvious general solution, namely, the prediction of the heat exchanged by the system with the thermal bath in the course of an isothermal nonequilibrium transformation involving some entropy production. Heat flow is controlled by the entropy balance equation: ds = des + dis (see Eqn. (1.18)). The heat transferred from the system to the thermal bath is equal to the integral of - T des over the transformation. We will denote it by -TAes. This integral is equal to : -TAes = T(Ais - As), where, Ais and As represent the integrated entropy production and the change in entropy over the transformation, respectively. The heat exchanged with the thermal bath can be predicted by using Eqns (1.111) and (1.98) to calculate Ais and As.

52

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

We give an example of heat-exchange calculation for a system described by the Gaussian Preisach distribution:

p (hc, hu) = C exp [ (hc - hc~ ] -

2r~c2

I

exp -

h2@u]

,

(1.115)

where C is some appropriate normalization constant. We consider a cyclic isothermal transformation at temperature T in which the field H oscillates between the opposite values +Hm. Before the application of the field the system is in equilibrium under zero field, i.e. b (he) - O. We will assume that the rate-independent limit applies. Then, the shape of the state line is univocally determined by the value of H and by the sign of its rate of change. 9 Increasing H: b (hc) = b+ (hc), where

hc -Hm 4- he 0 H-

b+ (he) =

O<~hc<~(Hm 4- H)/2 (Hm 4- H)/2<~hc<~Hm . hc >/Hm

(1.116)

9 Decreasing H: b (hc) = b_ (hc), where

b_ (he) -

H + hc

O<~hc<~(Hm - H ) / 2

Hm - he 0

(Hm - H)/2~hc~Hm. hc>/Hm

(1.117)

By using these expressions and Eqn. (1.115) in Eqns (1.108), (1.109), (1.96), and (1.98) one can calculate any desired thermodynamic function at any given point of the hysteresis loop. For the entropy, it is necessary to make an additional assumption about the two logarithmic derivatives, d lnMs/dT and d In Qu/dT appearing in Eqn. (1.98). We will use for these derivatives values that are typical of iron-based ferromagnetic systems. In particular, hysteresis studies in these materials suggest that the scaling parameter Qu should be approximately proportional to the square root of the magnetocrystalline anisotropy constant. This yields to the following room temperature values: dlnMs/dT _~ - 5 x 10 -4 K -1 and d l n Qu/dT "" -0.05 K-1. The entropy variation As is equal to the difference between the entropy associated with the state line defined by Eqns (1.116), (1.117) and the initial entropy associated with b(hc) = 0. Conversely, the entropy production Ais can be calculated by the method previously used for the case of closed

1.3 THERMODYNAMICASPECTS OF HYSTERESIS

53

loops. Let us express TAis as

TAis = (TAis)o q- (TAis)B,

(1.118)

where (TAis)o represents the total heat produced in the past up to the latest field reversal point, and (TAis)B is the integral of T dis from that reversal point up to the current state of the system. One obtains the following expressions. 9 Increasing H: (TAis)B - (TAis)+, where

(TAis)+ - 2#oMs

f(Hm+H)/2 fH-hc hcp (hc, hu) dhc dhu. dO d-Hm+hc

(1.119)

9 Decreasing H: (TAis)B = (TAis)-, where

(TAis)_ - 2#oMs ~0 (Hm-H)/2

f Hm-hc , . hcp(hc hu) dhcdhu dH+hc

(1.120)

Notice that for a system described by Eqn. (1.115) (TAis)_ is transformed into (TAis)+when H is changed into - H , as the Preisach distribution is an even function of hu. Figure 1.4 shows the correspondence between the shape of the hysteresis loop traversed by the system and the behavior of the heat -TAes exchanged with the thermal bath. As the field oscillates back and forth, more and more heat (TAis)o is irreversibly produced and transferred to the thermal bath. The nontrivial field dependence of -TAes in the course of an individual half-loop comes from the interplay of TAs and TAis. Heat absorption by the system consequent to entropy increase can be as important as heat delivery consequent to entropy production. Indeed the behavior shown in Fig. 1.4 has been experimentally observed [57-60]. In these experiments, the temperature of the specimen was measured as a function of time in the course of a hysteresis loop. Temperature oscillations of the order of 10 -4 K were detected and used to estimate heat exchange. In fact, the crystal lattice acts as a thermal bath for the ferromagnetic subsystem, as its thermal capacity is exceedingly larger than that of the spin system. Thus, to a very good approximation, magnetic transformations are isothermal. On the other hand, if one is able to measure the extremely small temperature variations occurring during the transformation one can directly calculate the heat transferred to the thermal bath by multiplying the temperature variation by the thermal capacity of the lattice.

CHAPTER 1

54

,

,

\

,

Thermodynamics, Hysteresis, and Micromagnetics

,

"Uo ..................t ................................................................... ..........................

g_ i

,

i

0

,

i

,

i

,

i

,

i

i

,

I

,

i

i

,

i

,

i

,'

i

,

i

A,

i

-5

,

i

-2.5

,

i

0 H

X_

2.5

i

I

I

5

-5

|

I

-2.5

,

I

0 H

i

I

2.5

,

i

5

-5

I

-2.5

i

0

2.5

i

5

H

FIGURE 1.4 Isothermal hysteresis loops and corresponding heat exchanged with the thermal bath for three different parameter choices for the Gaussian Preisach distribution given by Eqn. (1.115). Left: hco = 1, rrc - 0.2, rru = 1; center: hco - 1, rrc - 0 . 2 , eu = 0.2; right: hco - 1, rrc - 1, rru = 0.2. For each case three diagrams are shown. Top: contour plot of the Preisach distribution. Middle: corresponding hysteresis curves originating from the initial state (b(hc) = 0 at H = 0) for the field sequence H = 0, 5 , - 5 , 5. Bottom: corresponding heat - T A e s transferred from the system to the thermal bath. The heat exchanged is -TAes = T (Ais - As), where the entropy change As is calculated from Eqn. (1.98) assuming d l n M s / d T = - 5 x 10 -4 K -1, d l n Q u / d T = - 0 . 0 5 K -1, whereas Ais is the entropy production (Eqn. (1.111)) integrated along the transformation.

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

55

Temperature-Dependent Generalization of Rate-Independent Hysteresis The scaling properties assumed for p (hc, hu) and b (hc) when the temperature is modified lead to definite predictions about the existence of hysteresis as a function of temperature, instead of field. It is reasonable to expect that there might exist a temperature interval where relaxation effects are not important, so that the zero-temperature limit applies with temperaturedependent parameters. This gives rise to two interesting possibilities: first, one can describe the temperature dependence of the rate-independent response to the field; and second, one can investigate temperature-driven hysteresis consequent to the application of temperature histories. The first point deserves no special consideration, as it simply amounts to using the appropriate temperature-dependent parameters in the rate-independent limit previously discussed. Conversely, the second point leads to new interesting features, appearing when one analyzes how the rate-independent limit of Eqn. (1.79) is affected by temperature. To carry out this analysis, we first need to get rid of any temperature dependence not directly related to hysteretic transitions. We know from Section 1.3.2 that this is accomplished by passing to the normalized state line b (Zc) (Eqn. (1.92)). The normalized line is temperature-independent, except for modifications due to irreversible transitions form (+) to ( - ) states or vice versa. The rate-independent limit of Eqn. (1.92) can be treated by the same method used for Eqn. (1.79). One introduces, instead of the plane (hc, hu), the normalized plane (Zc, Zu). In the rate-independent limit t7/I ~ 0 the line is confined to the interior of the renormalized metastability region:

IH - Zul ~ q Zc.

(1.121)

One has that c3b/c3t - 0 for any point of the state line lying inside the metastability region whereas c3b/ctt - d H / d t whenever a point of the state line touches the boundary of the region. The graphical representation of these results in the plane (Zc, Zu) (Fig. 1.5) is similar to the one in the plane (he, hu) (see Fig. 1.2). However, there are two important differences. Firstly, since the parameter q is temperature-dependent the inclination of the boundaries of the metastability region will be itself temperaturedependent; if, for example, Qc(T) decreases with temperature more rapidly than Qu (T), the inclination will decrease with temperature, which may result in irreversible state-line changes if the boundaries of the shrinking metastability region ever touch the state line at some point. Secondly, the normalized field/~/ - H/Qu will change with temperature even if H

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

56

zu

(-) ,I H - Zu I< q zo

H

zo

(+)

FIGURE 1.5 Normalized state line b(zc) (bold line) lying inside metastability region I/7/- Zul<~qzc.

is kept constant, thus acting as an additional driving force for irreversible transitions. The interplay of these two actions may lead to highly nontrivial types of hysteretic response under varying temperature. As an example, let us discuss the hysteresis effects that take place when the temperature oscillates between two given values under constant field. This is the situation typical of the so-called 'thermal remagnetization' experiments carried out for hard magnetic materials [61]. Let us consider a system described by the normalized Preisach distribution:

.., p (Zc, Zu) - C exp

[

1 (Zc-Zco 2 rr2

[ lz2]

exp -

E , 2 rr2u

(1.122)

where C is some appropriate normalization constant. To keep the discussion as simple as possible, we assume that the temperature dependence of the saturation magnetization Ms can be neglected, whereas that of the scale

1.3 THERMODYNAMICASPECTS OF HYSTERESIS

57

factors Qu(T) and Qc(T) can be approximated by the linear expressions:

Qu(T) = Qu(To) ( 1 + dlnQu dT ITo ( T - T o ) ) ,

(1.123)

(T-To)).

(1.124)

(

Qc(T) - Qc(To) 14-

dlnQc ]

dT

To

The system is initially at temperature T = To. The system is first brought to positive saturation, then the field is first decreased to the value H0 and subsequently increased to H1. Starting from this state, the system temperature is then increased to T1 and subsequently decreased to To under constant H = H1. The corresponding state-line changes in the normalized plane (Zc, Zu) are shown in Fig. 1.6. Three typical types of behavior are possible as a function of the value of the final field H1. When H1 is close to H0 the temperature variation mainly results in the change of the transformed driving field ~tl = H1/Qu. Conversely, when H1 is close to zero the transformed driving field/:/1 is also close to zero and plays no significant role in the process. In that case the temperature change mainly results in a change of the slope q(T) of the boundaries of the metastability region. When H1 attains intermediate values in the interval (H0, 0) these two different temperature actions have comparable effects and give rise to nontrivial types of hysteretic response. The interplay of the two effects is illustrated in Fig. 1.6. The contributions to the magnetization (Eqn. (1.95)) come from different parts of the normalized plane: from {A) to {B) regions (I) and (III) give negative contributions whereas region (II) gives a positive contribution to M. Conversely, from (B) to (C) region (I) gives an opposite positive contribution whereas regions (II) and (III) give no contribution at all. Consequently, when the temperature is cycled between (B) and (C) a small hysteresis loop is generated, corresponding to the periodic reversal of region (I). The resulting magnetization curves are shown in Fig. 1.7; the balance between the competing contributions just mentioned may result in a slightly positive or negative change in magnetization, depending on the field value H = H1. The two cases presented in Fig. 1.7 reproduce well the qualitative features of the experimental results reported by Mfiller et al. [61] for hard magnetic materials. Transformations like the ones considered in the previous example, where the temperature is changed but the field is kept constant, are associated with Gibbs free energy variations of the form (see Eqn. (1.101)):

dg = - s d T - T dis.

(1.125)

58

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

Zu

-"

Zu

s/ssSJSSSS ,,,'""

s/SS

q(T1) J

sj

I:11(To)

~.~

9

Zc

111(I) -

z

(B)

O)

(A)

"',,,," (~)

Zu

q(To) I:II(T O)

..... - .......

t'

(c)

....--'"'"

I(')

~.. Zc

\-(,if ..............

FIGURE 1.6 State line in normalized Preisach plane (Zc,Zu) consequent to the following field and temperature history. (a) after field sequence +co, H0, H1 at T = TO; (b) after temperature increase from TOto T1 at H = H1; (c) after subsequent temperature decrease from T1 back to TOat H = H1. Hatched regions (I), (II), and (III) represent regions where irreversible transitions are induced by the temperature variation.

Therefore the plane (T, s) plays a role similar to that played by the plane (H, M) for isothermal transformations (see Eqn. (1.107)). In particular, the area enclosed by the (T, s)-loop generated by a cyclic temperature change coincides with the entropy produced during the temperature cycle:

f T dis - - f s dT.

(1.126)

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

59

= To

cO .m

N . n

, , , ,

, , , ,

Ho ,

I

H1

~

I

,

Magnetic field H H=H 1 }(i)'

'

'

'

'

'

A, "

(ii)'

'

'

'

'

'

C

C

~

~

B

A

| 9

H 1 = -0.25 i

To

,

,

,

H 1 = -0.15 ,

I

T1

,

To

,

,

,

T1

Temperature T

FIGURE 1.7 Hysteresis curves u n d e r varying field and temperature calculated from Eqn. (1.95) by using the rate-independent limit in the normalized plane (Zc, Zu) to determine the state line b(zc) (Fig. 1.5). Equation (1.122) is used for the Preisach distribution p(zc, Zu), with Zco = 0.25, r~c = 0.5 and r;u = 0.5. Equations (1.123) and (1.124) are used for the scale factors, with Qu(To) = Qc(To) = 1, d l n Qu/dTITo = -0.05 K -1, d l n Qc/dTITo = - 0 . 1 K -1. Top: response to field sequence 0, +cx~, H 0, H1 at T = T O (H 0 = -0.7, H1 = -0.15). Final state is of the type s h o w n in Fig. 1.6(A). Bottom: subsequent temperature-driven hysteresis curves at H = H 1 obtained by cycling the temperature in the interval (T 0, T 1 = TO + 5). Two cases are shown: (i) H1 = -0.25; (ii) H1 = -0.15. The situation in the Preisach plane at T = TO and T = T 1 is s h o w n in Fig. 1.6. Magnetization is in arbitrary units, with the scale for the two bottom diagrams e x p a n d e d by a factor of 200 with respect to the top diagram.

60

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

1.3.3 THERMAL RELAXATION AND TIME-DEPENDENT EFFECTS

As discussed in Section 1.3.1, the relaxation kinetics of TLS ensembles can be predicted once one knows the probability P+ (t; he, hu) as a function of time for each TLS unit. This knowledge in principle permits one to calculate the system response, i.e. Eqn. (1.72), for any arbitrary initial state of the TLS ensemble. Interesting results in this direction have been obtained in [62] by studying the response of Preisach systems driven by stochastic inputs. However, the general study of the problem is mathematically challenging and computationally heavy. Two main approximations, inspired by the pioneering results obtained in this field by N6el [63], can be introduced to treat limit conditions of physical interest in relatively simple terms.

9 High-temperature approximation. This is the limit where thermal fluctuations dominate and a substantial fraction of the TLS units are in equilibrium at each instant of time, that is, they are characterized by P+ (t; he, hu) = P+q(hu) (see Eqn. (1.57)). This limit has been studied in considerable detail in connection with the relaxation kinetics of various types of magnetic systems [64,65]. 9 Low-temperature approximation. This is the limit considered in Section 1.3.1 and in the rest of this section, where the relaxing TLS units are concentrated in a relaxation front whose location is described by the state line b(hc, t). Although this approximation is expected to be valid only at low temperatures, in practice it is usually applicable in a broad temperature interval, because it holds whenever p(hc, hu) is approximately constant over hu intervals of the order of the fluctuation field Hd -- kBT/#oAm. In this section, we will analyze in some detail the low-temperature relaxation phenomena associated with the state-line kinetics described by Eqn. (1.79). We will show that the model predictions are extremely rich and complex, as well as strongly history-dependent. Indeed, they can explain many aspects of the connection between hysteresis and thermal relaxation observed in experiments. We will limit the analysis to the case where the temperature is constant and the system is driven away from equilibrium by time variations of the field H(t). An important property of the TLS ensemble kinetics is that the equation governing the evolution of the state line, Eqn. (1.79), does not contain the distribution function p(hc, hu). Therefore, the solution of the equation will hold for any TLS ensemble, no matter what its distribution function is. The function p(hc, hu) merely acts as a weight function for the integrals over the (he, hu) plane, which define macroscopic quantities like magnetization or entropy.

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

-I0

61

,

FIGURE 1.8 Field history considered for the analytical solution of Eqn. (1.127).

Exact Solutions for State-Line Kinetics Let us consider Eqn. (1.79) written in the form: c~b(hc, t)

~t

H/sinh rc

[

H(t) - b(hc, t) Sf

]

(1.127)

'

where

(1.128)

re = ~ exp

Equation (1.127) admits exact analytic solutions for any field history H(t) consisting of connected ramps (see Fig. 1.8) in each of which the field varies at the constant rate d H / d t (this also includes the case of constant field, d H / d t = 0). Let us first derive the solution for the single ramp starting from the field H0 at time t = to. The law of variation of the field along the ramp is: H(t) -- Ho + ( d H / d t ) ( t -

The corresponding solution of Eqn. b(hc, t = to) = bo(hc) is:

to).

(1.129)

(1.127) under the initial condition

b(hc, t) = Ho + ( d H / d t ) ( t - to)

62

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

-2

Arc

F• - to L 2" s

+ L

arcta ( s --

"cc

Zs

Zc

+ ~tanh

zH

,

2Sf

(1.130)

where the upper (lower) sign corresponds to positive (negative) dH/dt, and

Hf ~ H - IdH/dtl' Zs --

1

(1.131) 9

(1.132)

V/1/z 2 + 1/z 2 When the field history consists of a sequence of ramps covering the time intervals (to, tl), (tl, t2) . . . . (tk-1, tk) with field rates (dH/dt)l, (dH/dt)2 . . . . (dH/dt)k and initial field values Ho, H1 .... Hk-1, the state line bk(hc, t) in the interval (tk-1, tk) is still given by Eqn. (1.130) under the substitutions: to

tk-1, Ho ~ Hk-1, dH/dt ~ (dH/dt)k, bo(hc) ~ bk-1 (he, tk-1). Two limits of Eqn. (1.130) play a particularly important role: the stationary solution holding after all the transients due to the initial conditions have died out; and the full time-dependent solution in the particular case of constant field H0, i.e. when dH/dt = O.

9 Stationary state-line response under constantfield rate. This stationary solution is obtained by writing Eqn. (1.130) in the limit where to --* -cx~ and H0 --+ -cx~ for dH/dt > 0 (or H0 ~ +cx~ for dH/dt < O) in such a way that H(t) (Eqn. (1.129)) takes finite values at time t. One finds:

b(hc, t) = H ( t ) + 2HfArctanh ( zHzc ZH)zs"

(1.133)

Apart from the shift by the amount H(t), the shape of the state line is time-independent. In addition the dependence on the initial state bo(hc) has disappeared, whereas the dependence on the field rate dH/dt is in the ZH parameter. 9 State-line response under constantfield. The solution for the case where H(t) - Ho for t > to is obtained by taking the limit dH/dt --~ 0 in Eqns (1.130), (1.131), and (1.132). Equation (1.130) is reduced to:

b(hc t ) - H o - a H f A r c t a n h [ t a n h ( H ~ 1 7 6 ' 2Hf

exp

(

t-t~ - ~Zc

" (1.134)

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

63

In the limit t ~ ec, Eqn. (1.134) approaches the equilibrium state line under constant field, b(hc, oc) = beq(hc) = Ho (see Eqn. (1.74)).

Application of Analytical Solutions to Typical Experiments Let us now apply the solutions previously derived for b(hc, t) to typical experimental situations encountered in the study of hysteresis phenomena. The terminology used is that of magnetic hysteresis, but the analysis can be easily extended to other systems.

9 Response along the saturation loop traversed at the constant field rate dH/dt. Let us consider the case where the field H(t) is first brought to H ~ +cx~ (positive saturation) and then decreased at the constant rate dH/dt. At the time when the field takes the value H0 the state line is given by Eqn. (1.133) with H(t) = Ho. Owing to the exponential dependence of rc and rs on he, this solution is described to a very good approximation by straight lines (see Fig.l.9): b(hc) =

H0 hc < H~ Ho - H~ + hc hc > S?~'

(1.135)

where

S~ = S f In

( Hf lro ) IdH/dtl "

(1.136)

According to Eqn. (1.135), the state line is equal to the equilibrium line beq(hc) -- Ho in the interval hc < H~. Any contribution to the total magnetization coming from this hc interval will be reversible, with no energy dissipation involved. The parameter governing the balance between reversible and irreversible processes is the field H~ (Eqn. (1.136)), dependent on the field rate. The result of this balance becomes manifest when one studies the coercivity of the system as a function of the field rate. The coercive field Hc is defined as the field for which the magnetization attains zero value. Therefore, at the coercive field the state line b(hc) given by Eqn. (1.135) is placed in the (hc, hu)-plane in such a way that the opposite integrals defining M in Eqn. (1.108) are equal in magnitude. By definition Hc -- b(hc = 0). Figure 1.10 shows the position of the zero-magnetization state line for different values of the field rate. It is evident from the figure that the coercive field will increase under increasing field rate. The quantitative evaluation of this effect depends on the details of the distribution function p(hc, hu).

64

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics h~

........... .;--'2 ........... "0

H

.,'

i

,"3,# ....... 7" ......... -2""..... -"- ,,, '. i"

" I

Ho ss

.. . . . . . . . ,

,. ,

ss

~ .

..,:

',,.

k ~,

",.

\

) ; i

". . . . . . . .

.I

"....... .................. .-,

l

/

v

hc

.ooo -

"" ...,.~...................

s~o S

",.

"',

...,""

s

Ho- H*o

FIGURE 1.9 State line shape (Eqn. (1.134)) when the saturation loop is traversed at constant field rate.

h"T .

..........

/,,"

f,

\

i"., ', '-///../-'" );~ , ,. -- ,,~r. ~ .... ..-" ...... ,..'

I--1

/

II

, /

dH/dtt I ""

,' .......,~//...,

".-._-a".

..................

/'~i~iiii'. i~i~i~i~(hc)=Hco + "~" ......

.-"

,,

,.

,," -,c

,"

.........

FIGURE 1.10 State lines corresponding to zero magnetization for different field rates d H / d t . Lines are calculated from Eqn. (1.134)by setting b(hc = O) = HO equal to the value that makes M = 0 in Eqn. (1.108).

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

65

9 Relaxation under constant field. In the typical relaxation experiment carried out in magnetic viscosity studies the system is first brought to positive saturation (H --~ +cx~). The field is then decreased to the value H0 at the rate dH/dt. We set the time origin in such a way that t = 0 represents the time at which the field reaches the value H0. The field is kept constant for t~0. The relaxing state line is given by Eqn. (1.134), where the initial condition bo(hc) is equal to Eqn. (1.133) with H(t) = Ho. The resulting b(hc, t) line is shown in Fig. 1.11 for different times t. Similarly to Eqn. (1.134), this solution too is described to a very good approximation by the straight lines: b(hc t) ~ { Ho '

Ho - H * ( t ) 4-

where

he < H* (t) he he > H * ( t ) '

(1.137)

(t)

(1.138)

H* (t) - H~ + Hf In 1 + ~

,

and H~ is given by Eqn. (1.136). At the initial instant t = 0 the state line is already relaxed in the interval he < H*(0) = H~ as a consequence of the finite rate at which the system was prepared. For t > 0, this interval grows logarithmically in time, until the final equilibrium state b(hc) - Ho is reached for every value of hc. If Hf is small with respect to the typical scale over which the distribution p(hc, hu) exhibits significant variations, then by using Eqns (1.137) and (1.138) one can reduce Eqn. (1.108) to the form

M(t) = M(t = 0)

-

( t ) )~irrSf In 1 + ~HH '

(1.139)

where Zirr

~

21~s

E

p[hc, b(hc, t - 0)] dhc.

(1.140)

One obtains the logarithmic relaxation law typical of several viscosity phenomena. It is worth remarking that the time constant ~H appearing in Eqn. (1.139) depends on the field rate by which the initial state was approached (see Eqn. (1.131)). 9 Response along return branches originating from the saturation loop. This is the case where the field is first decreased from positive saturation to the reversal point HT at the rate (dH/dt)o < 0 and then increased at the rate (dH/dt)l > 0. We set the time origin in coincidence with the reversal point. Therefore, for t > 0 the field is equal to:

66

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

FIGURE 1.11 State line evolution for the field history considered in constant-field relaxation studies. The system is initially brought to positive saturation (H ~ +cx~). Then the field is decreased to the value H 0 at the rate dH/dt and subsequently kept constant in time. Let t = 0 be the time instant when the final field value H0 is reached. Continuous lines: sequence of state lines b(hc, t) for t < 0, calculated from Eqn. (1.133). Dashed lines: sequence of state lines b(hc, t) for t > 0, calculated from Eqn. (1.134) under constant field H 0. As time proceeds, b(hc, t) approaches the equilibrium state line b(hc, c~) = H 0 by a front that advances logarithmically in time.

H(t) = HT + ( d H / d t ) l t . The state line bT(hc) at t = 0 is given by Eqn. (1.133) with H(t) = HT and d H / d t = (dH/dt)o < 0. When the field is increased after the reversal point, b(hc, t) is given again by Eqn. (1.133), with d H / d t = ( d H / d t ) l > 0 and initial condition bo(hc) = bT(hc). We limit the discussion to the case of equal rates I(dH/dt)ol = I(dH/dt)ll = IdH/dtl. The resulting line b(hc, t) depends on time only through the field H(t). Its shape for different values of IH(t) HTI is s h o w n in Fig. 1.12. The line can be approximately divided into two parts A and B. Part A. In the interval 0 < hc~H~ + [H(t) - HT]/2 the line is approximately equal to the expression: bA(he, t) ~

H(t) H(t) + H~ - he

he < H~ he > H~ '

(1.141)

where H~ is given by Eqn. (1.136). This part moves u p w a r d at the same rate as the field H.

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS Part B. For

67

hc>H~ + [H(t) - HT]/2 the line is approximately equal

to:

bB(hc, t) - H*BO(t) + hc

(1.142)

where

H*Bo(t) - HT - H~ - Hf ln I2 - exp ( - IH(t) - HTI ) .

(1.143)

This part moves downward by the finite amount Hf In 2 when t increases from 0 to ~ . The magnetization after the reversal point results from the combination of contributions with opposite sign due to parts A and B. The relative value of these contributions depends on the shape of the distribution function p(hc, hu) and may result in a decrease of the magnetization after the reversal point, if part B dominates. This effect was indeed observed in relaxation experiments carried out for nanostructured magnetic materials [66].

9 Relaxation under constant field after a field reversal point. Let us reconsider the return branch just discussed and let us assume that at some time after the reversal point the field is stopped at the value Ho > HT and afterwards kept constant. Let us set the time origin in such a way that t = 0 represents the time instant when the field reaches the value H = H0. The state line b(hc, t) for t > 0 is equal to Eqn. (1.134) where the initial condition bo(hc) is given by the returnbranch solution previously discussed, with H(t = O) = Ho. The result is shown in Fig. 1.13 as a function of the time t elapsed after the field is stopped. One can identify two relaxing fronts (parts A and B of the relaxing state line) advancing logarithmically in time and progressively generating the equilibrium state line b(hc, 0o) = Ho. Part A. Relaxation of part A is described by the law (compare with Eqn. (1.137)):

b(hc t) "~ { Ho '

H o + H * (t) -

hc < H* (t) he he > H * (t)

'

(1.144)

where H* (t) is given by Eqn. (1.138). Part B. Relaxation of part B is approximately described by the expression:

bB(hc, t) - S~(t) + hc,

(1.145)

68

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

hu

HT /

/

/'

...,-" ...7 ,o ..,,-

/ ;"~

H"BO h,, v

hc

FIGURE 1.12 Shape of state line along the return branch starting from the reversal field H - HT. The two lines shown are the state lines corresponding to the reversal point HT and to the generic field H along the return branch.

where

H;(t) - H~o (t) - Hf In (1 +

1

t),

2 exp (IH0 - HTI/Hd) - 1 -~H

(1.146)

and H"B0(t) is given by Eqn. (1.143). The time constants controlling the relaxation are rH for part A (see Eqn. (1.138)) and "CH (2 exp (IH0 - HTI/Hf) - 1) for part B. They differ by a factor depending on the distance IH0 - HT[ between the field H0 at which the relaxation occurs and the field HT at which the return branch originated. Quantitative prediction of the magnetization behavior associated with the relaxing state line requires a detailed knowledge of the Preisach distribution for the specific system under study. In general, three distinct regimes are possible, depending on the observation

69

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

hL!

A

///~j~,-

//,//

,.-

Ho b(l%,~)=Ho

HT .,.. / : .," ' ./'

./

.,....

"'~'---- HB*(t) hc FIGURE 1.13 Relaxation of state line when field is reversed at H = HT and then stopped at H = H 0. Let t = 0 be the time at which the field H = H 0 is reached. Continuous lines: evolution of state line for t < 0. Dashed lines: evolution of state line for t > 0. Relaxation proceeds through the motion of the two fronts indicated by the gray arrows. time and the a m p l i t u d e of the difference IH0 - HTI: (i) m o n o t o n e decrease of M(t), if part B of the state line gives the d o m i n a n t contribution to the magnetization (this usually occurs for small IH0--HT I); (ii) m o n o t o n e increase of M(t), if part A dominates (large IH0 - HTI); and (iii) n o n m o n o t o n e relaxation where the magnetization initially increases and then decreases, if both fronts contribute to the magnetization on the same time scale. N o n m o n o t o n e relaxation has been observed in several magnetic systems [61,67-69]. The term 'anomalous' relaxation has been often used in this context, although it is evident from the previous discussion that the n o n m o n o t o n e behavior is the natural consequence of the internal structure of the system state described by the shape of the state line b(hc, t).

Law of Corresponding States Additional regularities are to be expected in the magnetization relaxation w h e n the Preisach distribution is equal to zero for certain he intervals. In

70

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

particular, we will consider the case of physical interest where p(hc, hu) is different from zero only beyond a certain he threshold, say he > hclim" These regularities arise from the fact that the region 0 < he < hclim gives no contribution to the magnetization, so the shape of the state line in this interval has no particular meaning and plays no role in the relaxation. This means that relaxation processes whose corresponding state lines are different only in the interval 0 < he < hclim will be indistinguishable. This fact gives rise to an interesting law of corresponding states for relaxation phenomena. To make this argument more quantitative, let us consider the fact that the state lines associated with the various relaxation mechanisms discussed in the previous section are all approximately of the form b(hc) = Hth +he for he > hco, where Hth and hco are quantities dependent on time and field rate in a way characteristic of the particular relaxation mechanism considered. Therefore, if the relaxation process is such that hco never exceeds the limit where the Preisach distribution is different from zero, i.e. hco < hclim, then the theoretical prediction is not modified if we change the true state line associated with the process into the state line equal to Hth 4- hc for all he > 0. This modified state line has the same functional form of rateindependent hysteresis. However, rate-dependent effects emerge from the fact that the field Hth is in general different from the field H acting on the system. This description naturally leads to a number of correspondence laws concerning: (i) the field-rate dependence of hysteresis loops; (ii) the scaling properties of time relaxation patterns and hysteresis curves; and (iii) the shape of return branches after a reversal point. All these laws involve only the fluctuation-field parameter Hf and are otherwise independent of the details of the p(hc, hu) functional form for he > hclim.

9 Rate-dependent coercivity. Let us consider Eqn. (1.135) for hc > H~. In this region the state line is of the form b(hc) - Hth + he, where: Hth = H - H~ - H - Hf ln ( Hf /z~ )

IdH/dtl "

(1.147)

With reference to Fig. 1.10, we have that if H~ < hclim, that is, IdH/dtI > (Hf/~o)exp(-hclim/Hf), the coercive field H = -Hc at which M - 0 will correspond to a unique, well-defined value of Hth, say Hth = -Hco. Therefore we immediately conclude that the coercive field will follow the law:

Hc(ldH/dtl) - Hco - H~ - Hco - Hf ln ( Hf /r~ ) IdH/dtl "

(1.148)

71

1.3 THERMODYNAMICASPECTS OF HYSTERESIS

Logarithmic dependence of the coercive field on the field rate has been observed in different magnetic materials: nanocrystalline alloys, hard magnets, metallic thin films, and magnetic recording media (see [66,67,70,71]). By using Eqn. (1.148) to fit experimental data one can obtain a direct estimate of the fluctuation field Hf. 9 Rescaling of relaxation and hysteresis curves. Corresponding states exist also for relaxation effects under constant field H0. In fact Eqn. (1.137) shows that in this case too the state line is of the form b(hc) = Hth + he for he > H* (t), with

Hth=So - Hfln( t

+

\ }

IdH/dtl /"

(1 149)

According to Eqn. (1.149) the same value of Hth may result from different combinations of field rate dH/dt, field H0, and time t. Whenever H*(t) < hclim all the relaxation curves M(t; Ho, dH/dt) resulting from different combinations of these parameters will collapse onto the single curve M(Hth) when plotted as a function of Hth. In addition, collapse onto the same curve will be obtained also when the descending branches M(H, dH/dt) of hysteresis loops traversed at different field rates are plotted as a function of Hth, Hth being given in this case by Eqn. (1.147). The universal curve M(Hth) coincides with the rate-independent descending branch of the hysteresis loop that would be obtained if thermal relaxation effects were not present. Magnetic viscosity has been the subject of detailed studies in hard magnetic materials [61]. Nonlogarithmic time relaxation patterns have been observed in various systems including ultrathin films and nanocrystalline materials [66,72,73]. Rescaling of measured relaxation curves similar to the law of correspondent states just discussed has been widely reported in the cited literature. 9 Dynamic effects after a reversal point. The b(hc, t) line after reversal at the field HT (see Fig. 1.12) is of the form b(hc) = Hth q- hc for he > H o* + (H - HT)/2, with Hth - H*B0(t) (Eqn. (1.143)) The resulting dynamic effect on the initial part of the return branch becomes evident by studying the differential susceptibility after the reversal point. The magnetization displays a sort of inertial effect and keeps on decreasing after the reversal point. The susceptibility d M / d H after the reversal point is obtained by inserting b(hc) - H*Bo 4-hc into Eqn. (1.108) and by taking the first derivative

72

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics with respect to H. The result is: dM dH

Zirr 2 exp(lH -

(1.150)

HTI/Hf) - 1'

where Zirr is the irreversible susceptibility before the reversal point. The susceptibility jumps from Zirr to -Xirr at the reversal field. Interestingly, its subsequent behavior as a function of the field is independent of the field rate. The effect described by Eqn. (1.150) provides an independent method for the determination of Hf. Its experimental verification has been recently reported in [66].

Response to Small Oscillating Fields The state line kinetic equation, Eqn. (1.79), can be used to investigate the system response to small oscillating fields. This is an important aspect that creates a direct connection between the TLS ensemble model and linear response theory. Let us assume that the system is subjected to a time-dependent field of the form:

H(t) -- Ho + H1 exp(icot),

(1.151)

where H0 is constant and H1 is a small quantity whose effect will be studied to first order. Let us look for a state-line solution of the form:

b (hc, t) = bo (hc, t) + bl (hc)exp(icot),

(1.152)

where bo (he, t) is the solution of the kinetic equation under the constant field H0 and bl (he) is of the same order as H1. By inserting Eqns (1.151) and (1.152) into Eqn. (1.79) and by keeping first-order terms only one finds:

ir

= 2 (Hl - bl) cosh [HO - bo(hc, t) ] exp ( HI

hc )

(1.153) "

Equation (1.153) shows that the assumption that bl (hc) be time-independent is not generally acceptable, because of the explicit time dependence in b0(hc, t). However, according to Eqns (1.137) and (1.138) Ho - bo(hc, t) ~" 0 for he < H*(t), where H*(t) is a logarithmic increasing function of time. After an initial time interval where the bo(hc, t) relaxation interferes with the bl response, H0 - b0(hc, t) becomes equal to zero in the entire he interval where bl is significantly different from zero. From this moment, the

1.3 THERMODYNAMIC ASPECTS OF HYSTERESIS

73

relaxation and the small oscillations get decoupled and we can solve Eqn. (1.153) by assuming Ho - bo(hc, t) = 0. This gives the solution: exp (-hc/Hf ) H1. bl (hc) - icoz0/2 + exp (-hc/Hf )

(1.154)

This result can be used to calculate the magnetization response to the oscillating field. According to Eqn. (1.108) one can write, to first order in bl:

M(t) = Mo(t) + 2Ms exp(icot)

bl (hc) p[hc, bo (hc, t)] dhc,

~0~176

(1.155)

where Mo(t) is the relaxing magnetization under the constant field H0. By taking bo (he, t) ~ H0, in agreement with our previous assumption, and by making use of Eqn. (1.154) one finds:

M(t) - Mo(t) - X(co)H1 exp(icot),

(1.156)

where:

)(co) -- 2Ms

f0 ~

exp (-hc/Hf ) (hc Ho) dhc icoz0/2 + exp (-hc/Hf) p ' "

(1.157)

Whenever co < < l/z0, the function multiplying p (hc, Ho) in the integrand of Eqn. (1.157) is fairly constant up to hc ~ -Hf ln(coz0/2) and then decays exponentially to zero. Therefore, to a very good approximation, the integral can be calculated by changing p (hc, Ho) into its average over the hc interval (0, -Hf ln(coz0/2)), which gives"

)(.(co) ~ -2MsHf po~ (Ho) In

icozo

(1.158)

where:

1 po~ ( H o )

-

-

fo-Hf ln(o)z0/2)

Hf In (coz0/2)

p (hc, Ho) dhc.

(1.159)

When co ~ 0, Eqn. (1.158) tends to the limit: X(co) ~ Xeq -- 2MsPeq(Ho),

(1.160)

74

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

where:

Peq ( N o )

--

p

(hc , Ho ) dhc .

~0~176

(1.161)

Equation (1.160) represents the so-called anhysteretic susceptibility, Zeq = r where Meq is the equilibrium magnetization (see Eqn. (1.103)). However, this limit is only attained for extremely low frequencies. In fact, assuming t0 "" 10-9s, the integration interval in Eqn. (1.159) is only a few times Hf when 09 "-" ls -1. If the distribution is approximately constant over hc intervals of the order of Hd then po~ (H0) "" p (0, H0), independent of o9. Under these conditions, the susceptibility is no longer real. By writing X = X' - ix" we find from Eqn. (1.158):

X'(09) '~ -2MsHf p (0, Ho) In (09zo/2),

(1.162)

X" (09) "" rcMsHf p (0, Ho).

(1.163)

Interestingly, the imaginary part of the susceptibility is frequencyindependent. This has important consequences on the noise properties of the system. In fact, once the long-time relaxation and the response to the oscillating field get decoupled (b0 - H0) the system is in thermal equilibrium on the time scale of the field-driven oscillations. Therefore, one can apply the fluctuation-dissipation theorem in order to predict the spectrum SM (09) of the spontaneous fluctuations of the magnetization:

SM (09) =

2kBT X"(09)

09

.

(1.164)

From Eqns (1.164) and (1.163) one concludes that the system will exhibit 1/f noise. The prediction expressed by Eqn. (1.164) has been confirmed by remarkably accurate experiments carried out on soft magnetic cores at low temperatures [74,75]. Both the linear-response susceptibility X(09) and the noise spectrum SM(09) were independently measured. It was verified that the imaginary part X" was indeed frequency-independent and that the noise spectrum was of 1/f type. By comparing them, it was possible to show the validity of the fluctuation-dissipation theorem with no adjustable parameters. It is worth remarking that some of these results were obtained by N6el [63] on the basis of the equivalence he derived between magnetic hysteresis at low fields and the Preisach model.

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

75

1.4 F R O M M I C R O M A G N E T I C S T O M A G N E T I C HYSTERESIS The aim of this section is to discuss a particular class of problems where hysteresis phenomena are dominant, that is, the behavior of ferromagnetic systems at temperatures not too close to the Curie temperature. Micromagnetics is the theoretical framework most often used for the description of these phenomena [9-11]. Micromagnetics leads to complex nonlinear equations for which only a few rigorous solutions are known for some idealized situations. In fact, many of the models proposed in the literature for the description of magnetization processes amount to simplifying the micromagnetic description from the beginning, in order to reduce drastically the number of degrees-of-freedom to be considered in the model. A particularly important role is played in this respect by models where magnetization dynamics is reduced to the study of an ensemble of single degrees-of-freedom moving in statistically independent random potentials describing the effect of structural disorder. Certain general properties of hysteresis emerge from this simple picture in terms that suggest a number of generalizations and extensions. In particular, one finds that under appropriate conditions the zero-temperature hysteresis behavior of the ensemble obtained by averaging over different realizations of the disorder is of Preisach type. This creates a direct conceptual connection between micromagnetics and Preisach-type hysteresis, a result which casts new light onto the methods and tools discussed in Section 1.3. 1.4.1 MICROMAGNETICS The principal magnetic terms that contribute to the energy of a ferromagnet are exchange, anisotropy, magnetostatic energy, and the energy of interaction with the external magnetic field [9]. It is the competition between these energies that is eventually responsible for the hysteresis phenomena observed in magnetic materials. The key step is to express these contributions in terms of the magnetization M(r) defined over the ferromagnetic body. The resulting free energy Gf (M(.); Ha, T) is a functional of the vector field M(.), as well as a function of the external field Ha applied to the body and of the temperature T. Its local minima represent the metastable magnetization configurations in which the system can be found. The energy minimization has to be carried out in the infinite-dimensional functional space of all possible magnetization vector fields, a point which leads to nontrivial conceptual and mathematical difficulties. This program is referred to by the name micromagnetics [9-11]. The equations that express the condition of minimum for a given magnetization configuration are

76

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

known as Brown's equations, after W.F. Brown, who formulated this approach in general terms and clarified its limits. Brown's equations give a proper mathematical and physical foundation to the description of magnetic materials. The description of ferromagnetic bodies crucially depends on the spatial resolution chosen in the development of specific models. A detailed quantum mechanical treatment at the atomic scale is of prohibitive difficulty. The micromagnetic approach is an attempt to describe the magnetization behavior with a spatial resolution of the order of nanometers in terms of partial differential equations for the magnetization M(r), viewed as a differentiable vector field defined over the ferromagnetic body. The vector M(r) represents the magnetic moment density averaged over elementary volumes AV, of dimensions in the nanometer range, centered around the position r. The volume A V must be large enough to contain many atomic moments so that thermodynamic methods can be used to describe the local properties of the volume. On the other hand, A V must be small enough with respect to the length scale over which appreciable variations of M are expected. In micromagnetics it is assumed that M(r) has variable orientation in space but fixed magnitude IMI -- Ms, where Ms is the spontaneous magnetization of the ferromagnet. This constraint amounts to assuming that the system is locally in equilibrium with respect to the atomic degrees-of-freedom responsible for the macroscopic appearance of Ms and that the temperature is low enough that the influence of any local field on Ms can be neglected. Under these conditions, a particular state of the system is identified by the local orientation of the unit vector m(r) = M(r)/Ms at each point in space. Accordingly, the free energy becomes a functional of re(r), i.e. GL(m(.); Ha, T). From the experimental point of view, measuring the magnetization vector with nanometer resolution and resolving its space-dependent orientation is far from a straightforward task. Whenever the experimental method used implies space averages over much larger linear dimensions, the resulting average magnetization will exhibit some dependence of both its orientation and its magnitude on the external conditions.

Micromagnetic Free Energy By analyzing the specific mechanisms responsible for exchange, anisotropy, and magnetostatic interactions, one finds that the micromagnetic free energy can be expressed in the following dimensionless form [9]:

gc= V

I'2x -'2

+

1'

~ M'm--ha'm

}

d3r,

(1.165)

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

77

where gL = GL/(#oM2V) and V is the volume of the body. The four terms appearing in the integrand of Eqn. (1.165) represent the exchange, anisotropy, magnetostatic, and applied field-interaction terms, respectively. In the exchange term, (Vm) 2 is a short-hand notation for IVmxl2 4-IVmy] 2 4_ IVmzl2 whereas the length parameter lEx is the so-called exchange length: IEX --

~2A

,

(1.166)

where A is the exchange stiffness constant. The anisotropy contribution is expressed in terms of the dimensionless parameter 2K K - #0M2,

(1.167)

where K is the anisotropy constant, and of the dimensionless function fAN(m), which is an appropriately normalized function describing the angular dependence of local anisotropy effects. For example, in the case of uniaxial anisotropy with the easy axis identified by the unit vector eAN, fAN(m) -- [1 -- (m. eAN)2]/2. Finally, the vector ha - Ha~Ms represents the renormalized applied field whereas hM -- I-IM/Ms is the renormalized magnetostatic field solution of magnetostatic Maxwell equations: V.hM = - - V . m ,

V xhM = 0 ,

(1.168)

with the usual interface conditions at the body surface. As a consequence of Eqn. (1.168), the dependence of the micromagnetic energy on magnetostatic effects is nonlocal, because the knowledge of m in the neighborhood of a certain point r is not sufficient to determine hM. Magnetic charges even far away from r contribute to the magnetostatic field at that point. It is only after defining the geometry of the problem and after selecting a particular magnetization distribution m(r) for the entire body that it is possible to calculate the magnetostatic energy contribution appearing in Eqn. (1.165). Equation (1.165) shows that the micromagnetic energy is controlled by one characteristic length, the exchange length IEX, and by the dimensionless parameter K. The existence of the exchange length gives the natural unit in terms of which one can decide whether a certain body is 'small' or 'large'. The main difference is that 'small' bodies prefer to be uniformly magnetized, whereas 'large' bodies tend to develop nonuniform, complicated magnetization patterns. On the other hand, the parameter K gives a natural measure of the relative strength of anisotropy and magnetostatic effects.

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CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

As previously mentioned, being able to write an expression like Eqn. (1.165) for the system free energy depends on the assumption that thermodynamic equilibrium exists in each elementary volume AV even if the system is not globally in equilibrium. In other words, one assumes that local relaxation times inside each volume AV are negligibly small with respect to the time scale over which the system globally approaches equilibrium. This separation of time scales permits one to calculate thermodynamic averages in two steps, by first averaging over all microscopic states corresponding to a certain local magnetization M(r) in each elementary volume, and then over all possible M(.) vector fields. The first step yields to the Landau free energy GL, of which Eqn. (1.165) is an approximate estimate, whereas the second step leads to the equilibrium Gibbs free energy G. Therefore one can write:

exp

( - G(na,4kZ)) B~ = ~y~.exp t - GL(M(');kBTHa' T)t .

(1.169)

M(.)

The sum appearing in Eqn. (1.169) is a formal notation for the complex path-integral operation whereby one can calculate the integral of the exponential over all M(.) vector fields compatible with the requirements of the problem. Introducing the intermediate statistical average expressed by GL is essential in order to describe the connection between hysteresis and metastability. In fact, local GL minima in M(.) functional space represent metastable states in which the system tends to remain for long times if thermal agitation is not important. In the limit of zero temperature, the set of all local minima represents the set of possible states that the system may occupy under a given external field. In general it is not true that the state of lowest energy will be the one actually realized in the body. Which of these states will be actually realized in the material depends on the applied field history, a fact that will result in the appearance of hysteresis effects. When thermal agitation is not negligible, the system can move from one metastable state to another, not only because of the action of the applied field but also because of thermal fluctuations. Then hysteresis and thermal relaxation are jointly present, a fact that substantially complicates the picture. These complications will be avoided in the discussion that follows by limiting the analysis to the zero-temperature limit.

Brown's Equations The problem of determining the magnetization configurations realized in a body whose Landau energy is given by Eqn. (1.165) amounts to solving the following variational problem: calculate the variation corresponding

~XL

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

79

to the arbitrary variation m ---> rn + 6m(r) under the constraint Im(r)12 = 1 and impose (~XL -- 0. The solution takes the form of a zero-torque condition to be fulfilled at each point inside the body: m x heft,

(1.170)

where the effective field heff is equal to minus the variational derivative of gL with respect to m(.): heft

-

~fAN

12xV2m- K c~m + hM + ha.

(1.171)

Equation (1.170) is accompanied by a boundary condition for m at the surface of the body. In the case where no surface anisotropy effects are present, the normal derivative of m must be zero: = 0. (1.172) 0n surface The micromagnetic program and the resulting Brown's equations are conceptually satisfactory. However, one encounters great difficulties when one tries to apply them to real magnetic materials. These difficulties have a double origin, one of mathematical nature and the other one of a more profound physical nature, related to the presence of structural disorder. From the mathematical viewpoint, Brown's equations are complicated nonlinear equations that are difficult to solve even in the simple case of a perfect single crystal of ellipsoidal shape. There is no chance to solve them analytically for a bulk material. One must resort to numerical methods, and even in this case only small portions of the ferromagnetic body can be simulated. The results obtained can at best represent some approximation to individual solutions of Brown's equations. They say very little about the general properties of the free-energy surface, the density and distribution of energy minima and so on, and offer little help in drawing general conclusions about the behavior of the system. To all of this, one should add the difficulties and approximations introduced in the numerical implementation of Brown's equations, which may be the source of spurious effects difficult to identify and to separate from the physical phenomena of interest. Even more important, however, is the physical fact that any realistic system always exhibits a substantial amount of structural disorder. In systems that are assemblies of fine particles, one can have complicated particle shapes and random distributions of particle size, orientation and position; in thin films and thin ribbons, surface roughness may play a role; and in bulk systems, one has complicated grain structures possibly endowed

80

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

with some crystallographic texture, random fluctuations of composition (in alloys), presence of defects, like dislocations, dislocation tangles, grain boundaries, inclusions, precipitates, and intrinsic atomic randomness (in amorphous structures) giving rise to random local anisotropy. All these random features are reflected in the space dependence of the parameters appearing in the micromagnetic free energy, that is, the spontaneous magnetization, the exchange stiffness constant, the anisotropy parameters, and the parameters describing the geometrical shape of the sample. The consequence is that the micromagnetic free energy acquires an extremely complicated structure in the magnetization functional space, with a huge number of local minima, maxima and saddle points. Even if one were able to solve Brown's equations in an exact and exhaustive way, that would probably be useless, because one would be overwhelmed by an enormous number of nearly equivalent solutions, differing only in a number of inessential details. Some simplified description is needed, where the unimportant details, eventually having no influence on the macroscopic magnetic properties of the system, are averaged out and neglected, and attention is focused on the few important, dominant features of the problem. In the next section, we will discuss attempts to overcome these difficulties by models where independent degrees-of-freedom evolve in random potentials.

D y n a m i c s and Relaxation Brown's equations express the conditions under which the system is in equilibrium, but they say nothing about two other important aspects: how the system will approach equilibrium if it is not initially in equilibrium; and how the magnetization will respond to time-varying driving fields. Out-of-equilibrium dynamics is most often described by means of the socalled Landau-Lifshitz-Gilbert equation [76-78]. This equation, originally proposed by Landau and Lifshitz in an attempt to describe domain wall dynamics, is based on the idea that in a ferromagnetic body the effective field hell(r, t) defined by Eqn. (1.171) will induce a precession of the local normalized magnetization re(r, t) in a way perfectly analogous to what an external field does for a free magnetic moment. In dimensionless form the precession equation is:

~m ~t

-- - m x

heft,

(1.173)

where time is measured in units of (TMs)-1, 7 being the absolute value of the electronic gyromagnetic ratio. Equation (1.173) gives rise to motions in which the value of the saturation magnetization Ms = IMI is preserved because m. ~gm/cqt= 0 by construction. In addition the dynamics is

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

81

nondissipative, as heft 9 Om/Ot = 0. This means that the torque exerted by heft gives no approach to equilibrium. Energy relaxation mechanisms can be taken into account by additional phenomenological terms, chosen through heuristic considerations. In their original paper, Landau and Lifshitz introduced a contribution to 0m/~]t proportional to the heft component perpendicular to the magnetization, of the form: ~m

= - m x heft + ~ (heft - m . heft).

~t

(1.174)

The rationale behind this choice is that heft identifies the direction in m space of steepest energy decrease, so this is the natural direction along which the magnetization is expected to relax. However, the magnetization magnitude must be preserved as well, which means that only the heft component perpendicular to m may contribute to ~]m/c3t. This component coincides with the vector - m x (m x heft) , so Eqn. (1.174) can be written in the equivalent form: cgm cgt

= -m

x

heft-

or

m

x

(m

x

heft).

(1.175)

This is the form in which the Landau-Lifshitz equation is ordinarily written in the recent literature. Energy dissipation and ensuing approach to equilibrium is controlled by the dimensionless parameter a. Its value in magnetic materials is always small, ~ << 1. Equation (1.175) is consistent with the two basic requirements of micromagnetics, that is, the magnetization magnitude is preserved under all circumstances since m . Om/cqt = 0, and the dynamics is reduced to Brown's equations when equilibrium is reached, i.e. when c3m/Ot = O. Another widely used equation for the description of magnetization dynamics is the so-called Landau-Lifshitz-Gilbert equation [78]: 0m

0m -- - m x heft q-cxm x ~ . 0t 0t

(1.176)

Interestingly, Eqn. (1.175) can be transformed into Eqn. (1.176) by applying the vector product operator, m x . . . . to both members of Eqn. (1.175) and by combining the result with Eqn. (1.175) itself. This calculation shows that Eqn. (1.176) is mathematically identical to Eqn. (1.175) provided the entire right-hand side of Eqn. (1.175) is divided by 1 + a2. Writing

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CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

Eqn. (1.176) in the form: 3t = - m x

heft - ~x--~

(1.177)

shows that the Gilbert form amounts to subtracting from the effective field in Eqn. (1.173) a viscous-like term proportional to the magnetization rate. The LLG equation is surprisingly rich in structure and quite difficult to solve in rigorous terms. Some aspects of this problem are discussed in two other chapters of 'Science of Hysteresis'. 1.4.2 RANDOM ENERGY LANDSCAPES

The micromagnetic approach discussed in the previous section has been particularly successful in the description of the magnetization configurations present in small magnetic elements of linear dimensions in the #m range or less. However, one encounters enormous difficulties when trying to describe larger elements up to bulk systems of macroscopic size. The common situation encountered in bulk systems is that the material is subdivided into magnetic domains [79]. The magnetization is practically uniform inside each domain, but varies from domain to domain. The interface layers separating one domain from the other are known as domain walls. In principle, micromagnetics should be the appropriate tool to use in order to explain the existence of domains. In practice, however, no one is able to overcome the mathematical difficulties encountered when micromagnetics is applied to realistic descriptions of actual materials. These difficulties come from the inherent complexity of the mathematical formulation, but also from the fact that real materials are usually systems where structural disorder dominates. The micromagnetic parameters describing the local state of the material (exchange stiffness constant, anisotropy parameters, etc.) are consequently random functions of space and it is hopeless to work out particular solutions of the ensuing stochastic differential equations. Furthermore, under these circumstances the knowledge of particular micromagnetic solutions is in the end of little interest. Macroscopic magnetic properties are determined by general aspects of the micromagnetic solutions, emerging from suitable averages over structural disorder, quite independently from the mass of unimportant details characterizing individual solutions. Domain walls acquire a privileged status in the description of magneticdomain structures. The internal wall structure can be studied by the micromagnetic approach, in order to estimate how much exchange and anisotropy energy is stored in the wall. Then the wall can be assimilated to a two-dimensional surface, where the wall thickness is neglected but

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

83

the wall is endowed with surface tension. This drastically reduces the complexity of the description. The magnetization changes induced by the external field are interpreted in terms of displacements of the domain walls, the estimate of the exchange and anisotropy energy stored in the system is reduced to the estimate of the total wall surface present under given conditions, and the magnetostatic energy can be expressed in terms of wall positions and orientations. In this sense, domain walls become the building blocks for more complex descriptions involving the superposition of many independent contributions associated with the motion of certain portions of wall surface that move rigidly and thus can be approximately interpreted as single degrees-of-freedom. Descriptions in terms of independent domain walls have many limits and should not be expected to yield accurate predictions for all aspects of the magnetization process. Nonetheless, they are conceptually stimulating in the search for general principles on which to found the description of magnetic hysteresis, and they are fruitful in suggesting more realistic generalizations. Their chief limits come from a basic difficulty inherent in the very concept of domain wall. Domain walls, if we neglect their internal structure and thickness, are interconnected geometrical surfaces extending across the whole magnetic body. The term 'single domain wall' is unclear and ambiguous, until we give some recipe on how to identify a certain portion of wall surface and how to refer to this portion as a single object. This difficulty reflects the fact that domain structures are governed by magnetostatic energy, which acts on large scales and makes domain walls strongly interacting parts of a unique global structure. However, there are cases, in particular the picture-frame single crystals investigated in [80], where the notion of a 'single domain wall' can be given physical plausibility. The study of these cases permits one to work out the mathematical structure of the equations describing domain-wall dynamics, at least in the limit of slow-field excitation. It is quite remarkable that, even in those cases where the domain structure is intricate and there is no chance to recognize in it individual wall contributions, this mathematical structure is conserved [81-84]. Certain aspects of the magnetization process turn out to be describable by a small number of indePendent degrees-of-freedom, each of which corresponds to the magnetic flux linked to some correlation region of the material. The detailed nature of these correlation regions may depend on specific structural aspects, but their dynamic behavior is largely independent of these details and closely resembles the dynamics of a 'single domain wall'. This approximation should be considered as a working hypothesis, of which, at present, it is still difficult to work out a direct derivation from fundamental micromagnetics. It is a fact, however, that real materials behave in a way not far from what is expected on the

84

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

basis of this assumption. The rest of this section will be devoted to the study of such domain-wall-like single degrees-of-freedom.

Equation of Motion The model system we will consider is a rigid domain wall moving in the direction perpendicular to its plane. In this case a single degree-of-freedom, namely, the position x of the domain wall along the direction of motion, is sufficient to characterize the state of the system. Abeautiful case where this situation is encountered is that of a Si-Fe single crystal cut in the form of a hollow picture frame [80]. The natural domain structure for this system is the one where four 90 ~ domain walls at the frame corners control the rotation of the magnetic flux from one frame limb to the next. Under particular preparation conditions, it is also possible to generate a metastable state where a 180 ~ wall is present all along the four frame limbs. The necessity to avoid the formation of magnetic charges keeps the wall closely parallel to the direction of magnetization. When one wraps an exciting coil around the frame limbs and applies a field, the wall loop shrinks or expands, depending on the sign of the field, and in each limb cross-section the wall moves inwards or outwards perpendicularly to the wall plane itself. During its motion, the wall will experience the various sources of structural disorder previously mentioned. The result will be the fact that the energy of the system gL (x) will exhibit a stochastic behavior as a function of x. On the one hand, the stochastic nature of gL (x) will give rise to rateindependent dissipation, measured by some finite coercive field. On the other hand, it will be the source of stochastic fluctuations in the domain wall velocity revealed by the Barkhausen effect [81,82]. The basic question to answer is what kind of dynamic equation can properly describe this phenomenology. The studies carried out in [80] and subsequent generalizations of those results suggested by Barkhausen effect investigations [81,82] show that the equation obeyed by the degree-of-freedom x, after proper renormalization of time and fields in order to get rid of inessential parameters, is of the form: dx = ha(t) I hp(x), dt

(1.178)

where ha(t) is the applied field and hp(x) represents the so-called pinning field. Equation (1.178) describes a viscous-like dynamics of the type 'velocity proportional to force', i.e. dx/dt = --cggL(x; ha)/Cgx, where the force -OgL(X~ha)/Cgx is due to spatial variations of the system energy.

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

85

Equation (1.178) is equivalent to assuming for the energy: gL(X; ha) = u(x) - ha x,

(1.179)

where u(x) includes exchange, anisotropy, and magnetostatic contributions, whereas -ha x represents the energy of interaction with the field. Thus the pinning field is equal to: du(x) hp(x) - d----~"

(1.180)

The energy gL (x; ha) and the field heff = ha - hp are the simplified forms in which the micromagnetic energy (see Eqn. (1.165)) and the associated effective field (see Eqn. (1.171)) appear in this description. Equation (1.178) itself can be viewed as a simplified form of Eqn. (1.177), although the passage from one equation to the other is far from trivial in metallic materials where Eqn. (1.178) results from the joint solution of micromagnetic equations and Maxwell equations. Stability conditions take a simple form in this one-dimensional representation where many of the complications present in higher-dimension systems [85,86] are ruled out. In fact, according to Eqn. (1.179), stable equilibrium states, i.e. states from which @L/Cgx -- 0 and 492gL/OX2 >/0, are simply characterized by the relations: dhp ha - hp, -~x >/0.

(1.181)

An x interval for each point of which d h p / d x > 0 will be called a stable pinning field interval, otherwise it will be termed unstable. Equation (1.178) describes a nonlinear and nonautonomous dynamical system. The general study of its properties can be quite complex when hp(x) is a generic function, possibly endowed with stochastic properties. However, some useful conclusions can be obtained by simple methods if the field ha(t) increases in time at the constant rate r > 0, i.e. d h a / d t = r. In that case Eqn. (1.178) can be rewritten in autonomous form by using (ha, x) as joint state variables: dx dha dt = ha - hp(x), dt = r.

(1.182)

By dividing the two equations in Eqn. (1.182) side by side one finds that the trajectory ha (x) described by the system, with no consideration of the time at which specific trajectory points are reached, follows the differential

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CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

FIGURE 1.14 Schematic representation of ha(x) trajectories associated with Eqn. (1.183) (dashed lines) for a given pinning-field profile hp(x) and for different values of the field rate r. equation: dha [ha dx

-

hp(x)]

=

r.

(1.183)

The solutions ha(x) of Eqn. (1.183) for different values of r are lines that tend to mimic the x dependence of hp(x) at a certain distance ha - hp(x), which is larger the larger r becomes, with the important restriction that dha/dx > 0 and ha - hp(x) > 0 for the entire solution, even in the unstable intervals where dhp/dx < 0 (see Fig. 1.14). In the limit r --, 0 the product appearing on the left-hand side of Eqn. (1.183) must beclose to zero. Wherever dhp/dx > 0, this requirement is fulfilled by imposing ha "~ hp (x). However, in the unstable x intervals where dhp/dx < 0 this type of solution is not acceptable because then dha/dx ~- dhp/dx < 0, so the left-hand side of Eqn. (1.183) would be negative, in contradiction with the right-hand side r > 0. Therefore, whenever dhp/dx < 0 one must have dha/dx "" 0 whatever the difference ha-hp(x) is. This is where an instability occurs and the system quickly moves to a different x interval where dhp/dx becomes positive again (see Fig. 1.14). Thus, when r --, 0 the ha (x) trajectory is made up of stable hp(x) intervals connected by horizontal segments associated with dynamic instability. The value of the field rate r is irrelevant in this limit, provided it is sufficiently small, so the system response becomes rateindependent. This limit is the starting point for the functional integration approach discussed in Section 1.4.3. The behavior of the system when the field rate r is appreciable will not be considered any further. A discussion of the ensuing rate-dependent effects can be found in [83].

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

87

Pinning-Field Properties The energy u(x) in Eqn. (1.179) describes the various sources of structural disorder that may be present in a given system. Its derivative, namely, the pinning field hp (Eqn. (1.180)), represents the force opposing the motion of the domain wall (Eqn. (1.178)). The function hp(x) controls all the main aspects of the behavior of the system, like equilibrium points, stability intervals, and hysteresis. In ordinary magnetic materials the energy profile u(x) can be extremely complex. This is due to the joint presence of two basic ingredients: disorder and frustration. Disorder refers to the presence of some random distribution of a large number of defects, e.g. nonmagnetic impurities, dislocations, etc., which may interact with the magnetization. Frustration, on the other hand, refers to the competition between the various micromagnetic energy terms that cannot be all simultaneously minimized by the same magnetization configuration. Depending on the material studied, different sources of disorder can be present [87], which can be investigated by electron microscopy and various types of electrical and magnetic measurements [82,88,89]. Typical examples are nonmagnetic inclusions, localized stress sources, surface irregularities, dislocations, and grain boundaries. An additional mechanism that can give rise to a position-dependent pinning field is the presence of a disordered domain structure. In this case, even if the medium is structurally homogeneous, during the wall motion the magnetostatic interactions between different domain walls can change with the applied field, which results in a contribution to the pinning field. This contribution is magnetization-dependent and will tend to be of higher intensity around zero magnetization, where the domain structure is more complex. The various forms of structural disorder present in different materials will correspond to different statistical properties of the pinning-field function hp(x). The central problem is to identify the core assumptions that should be made about this function in order to describe specific types of magnetic behavior. A first attempt in this direction was made by Kersten [90], who used a sinusoidal pinning field to interpret the origin of the coercive field in magnetic materials. Breakthrough progress was then achieved by N6el [91,92], who, in 1942, used the concept of random energy landscape to explain the origin of the Rayleigh law. This law, whose discovery dates back to 1887, states that at fields low with respect to the coercive field the relationship between magnetization and field is parabolic. In terms of the state variables (ha~ x) previously introduced one has: X -- aha 4- bh2,

(1.184)

88

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

where (ha, x) refer to the vertex of a symmetric hysteresis loop around the demagnetized state, whereas (a, b) are phenomenological coefficients. N6el recognized that the mechanism responsible for this law was domainwall motion in some random-energy profile. He treated the problem in the rate-independent limit of small field rates and he made a specific assumption for the energy profile. In terms of the associated pinning-field function hp (x) his model is equivalent to making the following assumptions: (i) the pinning-field values at the discrete points Xk = kL, k = 0, 1 . . . . . where L is some characteristic structural length, are independent random values extracted from a Gaussian distribution with zero average; and (ii) the complete pinning-field function hp (x) is obtained by joining these random points by straight-line segments. By classifying the stable pinning-field intervals and by carrying out a detailed calculation of the probability of finding the domain wall in one of these intervals after a given field history, N6el predicted the existence of Rayleigh law and obtained explicit expressions of the parameters (a, b) in terms of the parameters defining the pinning-field random function. In addition, he showed that the ensuing hysteretic behavior is equivalent to the Preisach model discussed in Section 1.3.2. We will show in Section 1.4.3 that this result is indeed a particular example of the general properties of rate-independent hysteresis associated with random pinning fields. Another approach to the characterization of pinning fields in magnetic materials was proposed in [88] based on the study of the stress field generated by dislocations. This analysis leads to an expression for the pinning field of the form: hp(x) - y ~ v ( x - xj), J

(1.185)

where v ( x - xj) is the individual contribution of an individual defect placed at the random position x = xj and the sum runs over all involved defects. By studying the statistical properties of the random function defined by Eqn. (1.185) definite expressions could be obtained for the susceptibility and the coercive field as functions of the parameters describing the defect arrangement [87,88,93-95]. These examples suggest a general approach to pinning-field studies: instead of trying to develop specific models for the prediction of hp(x), one can start from the idea that hp represents a generic stochastic process with respect to x and investigate the general connection between the statistical properties of the hp stochastic process and the resulting hysteresis properties of the system. This strategy leads to the functional integration approach discussed in Section 1.4.3.

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

%i~il~

89

, | li r

XO

X

FIGURE 1.15 Representation of random pinning-field profile and of associated system trajectory h(x; x O) out of initial state (h0, x0) when the field is first increased and then decreased.

Rate-Independent Limit The rate-independent limit introduced in Section 1.4.2 provides an interesting bridge between the dynamics of the single degree-of-freedom x and more abstract formulations of hysteresis [83]. Let us consider the stable state so = (h0, x0) shown in Fig. 1.15. When the field is increased the system follows the curve ha (x) - hp (x) until a pinning-field maximum is reached. At that point the system becomes unstable and executes a jump to the next stable x position. When the field is reversed and decreased, the onset of instabilities is no longer associated with hp maxima, but with hp minima. Therefore the system will follow a different ha(x) curve and hysteresis will appear. This graphical representation corresponds to the following rule for the evolution of the system from x0 to x:

h(x; x0) - { min[x,x0]g(x) maxix0,x]g(x)

if x<~xo if x>~xo'

(1.186)

where h(x; xo) denotes the system trajectory out of the initial position x - x0, min[a,bl g and max[a,b] g respectively represent the m i n i m u m and the maximum of the function g in the interval [a, b], and finally the symbols h and g have been used in the place of ha and hp, respectively, in order to simplify the notation. Thus the system evolution is given by the

90

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

nondecreasing upper envelope of the pinning-field function when the field is increased, and by the lower envelope when the field is decreased. From the physical point of view the function of main interest is the function x(h; xo) obtained by inverting h(x; x0), since h represents the control parameter and x describes the system response. With reference to Fig. 1.15, one can verify that, given the initial state (h0, x0), the value of x reached when the field is increased from h0 to h > h0 is given by the lowest possible solution of the equation g(x) = h. Analogously, for h < h0 one has to determine the largest possible solution of g(x) = h. The problem of finding these particular x solutions is known in the theory of stochastic processes as the level-crossing problem associated with g [96]. The behavior of the function x(h; xo) is intimately related to the subdivision of the x-axis into stable and unstable intervals. We previously defined a given x interval as stable with respect to the pinning-field function g(x) when dg/dx~O for every point inside the interval. It is useful to refine this definition by introducing the notions of strongly and weakly stable or unstable intervals [97]. 9 A certain x interval [a, b] is said to be strongly stable with respect to g if the function g(x) is monotonously increasing in that interval, whereas [a, b] is strongly unstable if g(x) is monotonously decreasing in it. 9 If the function g(x) is such that g(a) < g(x) < g(b) for any value of x inside the interval [a, b] without necessarily being monotone, then the interval is said to be weakly stable. Analogously, the interval will be weakly unstable if g(a) > g(x) > g(b) for any x inside the interval. Therefore, weakly stable (unstable) intervals may contain one or more strongly unstable (stable) subintervals. The evolution of the system is reversible inside a strongly stable interval, in the sense that the system revisits the same states in reverse order when the field is reversed. Therefore, the hysteresis properties of the system are governed by the sequence of jumps occurring from one strongly stable interval to another. The evolution rule shown in Fig. 1.15 selects well-defined portions of the pinning-field profile. If the system initially lies inside a weakly unstable interval it will never be able to come back to this interval if it ever leaves it under the field action. Therefore, only the strongly stable subintervals that do not belong to any weakly unstable interval control the permanent hysteresis properties of the system. Interestingly, a similar behaviour was found in [85,86] in the analysis of random walks in one-dimensional energy landscapes. In general, any hysteresis branch generated by a monotone field variation will correspond to a weakly stable x interval. When the system evolves under the action of a

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91

nonmonotone field history h(t), characterized by the sequence hi . . . . . hn of reversal points (that is, alternate h(t) maxima and minima), the intervals [X0, Xl] . . . . , [ X n - 1 , X n ] , where Xl . . . . . Xn are the values of x at the reversal points, will all be weakly stable. We can say that weakly unstable intervals are unreachable, in the sense that there is no field history that can force the system into them, whereas weakly stable intervals are reachable. The notion of reachable and unreachable states is quite general and can be useful in the description of other hysteretic systems as well. For example, it plays an important role in the description of the energetic aspects of the random-field Ising model [98] discussed in another chapter of 'Science of Hysteresis'. 1.4.3 FUNCTIONAL INTEGRATION APPROACH In this section we discuss how the rate-independent hysteresis mechanism shown in Fig. 1.15 can be translated into precise mathematical form. The approach will be based on the two following elements. 9 Given the field time-history h(t) and the pinning-field profile g(x), one defines an appropriate evolution operator T[g](h) translating into mathematical form the envelop construction shown in Fig. 1.15. In this context, the function g will be called the generating function of T[g](h). Given the initial state so = (h0, x0), the evolution operator transforms it into the time-dependent state s = (h, x): s -- Till(h) so.

(1.187)

9 One introduces the functional space G of all admissible generating functions g and one constructs a global evolution operator as the parallel connection S of the collection of evolved states obtained from Eqn. (1.187) by varying g over G with appropriate weights described by some measure # on G: S - fG T[g](h) so d#(g),

(1.188)

where the initial state so may itself depend on g. Equation (1.188) expresses in functional integration form [99,100] the average hysteresis response of a statistical ensemble of independent single degrees-of-freedom evolving in different pinning-field profiles. The space G acts as a probability space and the measure # describes the probability that one of these degrees-of-freedom will be characterized by a particular pinning-field profile.

92

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

The method whereby these elements can be translated into a precise hysteresis model is discussed in detail in [97]. In the rest of this section we will present the main results of that analysis. In particular, we will discuss the case where the generating functions g(x) are sample functions of a continuous Markovian process. This will create a direct connection between the pinning-field properties, described by the measure/~, and the ensuing hysteresis properties of the system. In particular we will show that homogeneous Markovian processes give rise to Preisach-type hysteresis.

Mathematical Formulation Let us consider the individual degree-of-freedom x characterized by the particular generating functiong(x) and acted on by the field history h(t), t~O. We shall consider piecewise monotone histories such that hLKh(t)Khu at any instant of time t. The parameters hL and hu are given fixed fields which delimit the field interval of interest. They will be termed the lower and upper saturation field, respectively. We impose some restrictions on the function g as well. Firstly, we want g(x) to be continuous in the interval [XL, xu], where XL and xu are solutions of the equations g(xL) -- hL and g(xu) = hu, respectively. The states SL = (hL, XL) and su = (hu, xu) will be called the lower and upper saturation states, respectively. Secondly, we assume that the function is such that hL < g(x) < hu for any x in the interval [XL, xU]. In other words, [XL, XU] is weakly stable in the sense previously discussed. The evolution of the degree-of-freedom starting from the initial state so = (h0, x0), where x0 is a solution of the equation g(xo) = h0, can be defined on the basis of the function: h[gl(X; x0) = / min[x,x0]g(x) / maX[xo,x] g(x)

if XL~x~xo if xoKx~xu'

(1.189)

which expresses in mathematical form the envelope construction shown in Fig. 1.15 (see Eqn. (1.186) for the meaning of the symbols). The inverse of hu(x; x0) will be denoted by x[gl(h; x0). It is defined as:

X[g](h; xo) =

max L[g](h, xo) min Uu(h, xo)

if hL KhKg(xo) if g(xo)Kh~hu'

(1.190)

where Lig](h, xo) (U[g](h, x0) ) is the set of all solutions of the equation g(x) = h in the interval [XL, x0] ([x0, XU]). The evolution operator introduced in Eqn. (1.187) is defined in terms of the function x[g](h; x0) as:

s = T j(h) so = (h, x j(h; xo)).

(1.191)

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

93

The evolution of the system for an arbitrary field history is obtained by applying the operator T[g](h) many times in sequence. More precisely, let us assume that at t = 0 the system is in the state so = (h0, x0) and let us apply the field history h(t). We denote by hi . . . . . hn the sequence of values at which the field is reversed and finally by ht the current field value at time t. The state of the system at the same time is: st -- (ht, xt) -- "-f[g](ht)T[g](hn) " " T[g](hl)s0.

(1.192)

The corresponding value of x is of the form: xt -~ X[g](ht; xn(hn . . . . . hi; x0)),

(1.193)

where the value Xn at the last reversal point depends in general on all past reversal points. An important aspect of the evolution law expressed by Eqns (1.192) and (1.193) is that it exhibits return-point memory (see Section 1.3.2). The proof, discussed in detail in [97], is based on the fact that return-point memory is a property of any system whose states can be ordered with respect to a partial ordering relation that is preserved by the rate-independent evolution law [9,101]. Let us now consider a statistical ensemble of independent degrees-offreedom, each described by Eqns (1.192) and (1.193). The global evolution of the ensemble is obtained by integrating these equations over the probability space G in the form shown by Eqn. (1.188). In particular, assuming that all the elements of the ensemble are initially in the lower saturation state SL = (hL, XL), the average ensemble response Xt obtained by integrating Eqn. (1.193) will be given by: Xt - f G X[g](ht; xn(hn . . . . . hi; XL))dp(g).

(1.194)

In Section 1.4.2 we mentioned the fact that the rate-independent evolution we are considering selects a sequence of weakly stable portions of the generating function g when the field goes through the reversal sequence hi . . . . . hn. Letus denote b y p ( x t , ht; hn . . . . . h l ) d x t the probability of having a function g in G with the following properties: (i) g(u) = ht for some u in the interval [xt, xt + dxt]; and (ii) there exists a sequence Xl . . . . . Xn such that g(xl) = hi . . . . . g(Xn) -- hn and [XL, Xl] . . . . . [Xn-1, Xn] are all weakly stable subintervals. Then one can formally rewrite Eqn. (1.194) as: Xt --

F oo

xt p(xt, ht ; hn . . . . . hi) dxt.

(1.195)

94

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

The interest of Eqn. (1.195) is that the probability density p can be explicitly calculated in the case where the measure/~ describes a continuous Markovian stochastic process.

Hysteresis for Markovian Pinning-Field Processes The main difficulty with the mathematical formulation outlined in the previous subsection lies in its abstract character. Some tool to define and manipulate the measure # in explicit form is necessary before one can attempt the application of the model to situations of physical interest. As previously mentioned, an interesting case where this becomes possible is when the functions g(x) are sample functions of some continuous and Markovian stochastic process [102-104], owing to the fact that the notion of a weakly stable interval is in direct correspondence with the solution of the so-called level-crossing problem [96,102]. In fact, we already know from Section 1.4.2 that the field reversal sequence hi . . . . . hn selects a sequence of weakly stable portions of the generating function g(x). Let us consider one of these portions and let us assume that it corresponds to the field interval [b, a], a > b and to the corresponding x interval [Xb, Xa], Xa > Xb. If we interpret g(x) as a sample function of some stochastic process, we can reinterpret the condition that [Xb, Xa] be weakly stable as the condition that the sample function must start from g(xb) = b and reach the level g(x) = a at x -- Xa for the first time without ever going back to g(x) = b. The quantity Xa is a random variable: the problem of determining its statistical properties is precisely what is meant by level-crossing problem for the stochastic process g. To clarify the consequences of this idea, let us assume that the levelcrossing problem just considered has been solved, and let us denote by T(a, xalb, Xb) the conditional probability density of having a level-crossing at x = Xa (i.e. g(Xa) = a) under the condition that g(xb) = b, b < a. If pa(Xa) and pb(Xb) represent the probability density functions characterizing the statistical distribution of Xa and Xb, respectively, then the equation holds:

pa(Xa) - - / x ~

T(a, xa]b, Xb) pb(Xb) dxb.

(1.196)

Let us define the probability space G as the space containing all those sample functions of the given Markovian process that satisfy the requirements previously introduced: (i) existence of two values, XL, Xu, XL < XU, such that g(xL) -- hL, g(xu) = hu; and (ii) weak stability of the [XL, Xu] interval. In general, XL and xu will be random variables, taking different values for each g in G.

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

95

The hysteretic evolution described by Eqn. (1.195) can be expressed in terms of T(a, xalb, Xb). Let us discuss how this can be done in the particular case where, starting from the lower saturation value h -- hL, the field is first increased to h = hi and then decreased (Fig. 1.16). The generalization to more complex field histories will be quite natural. Let us first consider the state St - - (ht, xt) along the ascending branch hL~ht~hl. One can apply Eqn. (1.196) to the weakly stable interval [XL, xt] (see Fig. 1.16) in order to obtain the probability density p(xt, ht) that appears in Eqn. (1.195):

p(xt, ht) -- ~xt T(ht, xt]hL, XL)pL(XL) dxL.

(1.197)

The probability distribution pL(XL) for the lower saturation state can be chosen at will, it is part of the initial characterization of the ensemble. By inserting Eqn. (1.197) into Eqn. (1.195) one obtains the average response of the system in terms of the known functions T(a, xalb, Xb) and pL(XL). Equation (1.197) can also be used to calculate the probability distribution Pl (Xl; hi) associated with the reversal point at hi: pl(Xl; hi) - f X l T(hl, xllhL,

XL)

pL(XL) dxL.

(1.198)

A similar analysis can be carried out for the descending branch hi ~ht ~h2. In this case, the weakly stable interval to consider is [xt, Xl] (see Fig. 1.16), where xl is distributed according to Eqn. (1.198). One obtains: pl(Xl; hi) -- f X l T(hl,xllht, xt)p(xt, ht; hl)dxt.

(1.199)

The main difference with respect to Eqn. (1.197) is that the u n k n o w n distribution p(xt, ht; hi) is now inside the integral, so Eqn. (1.199) is actually an integral equation for p(xt, ht; hi). This difference is due to the fact that the unknown xt corresponds to the lower end of the weakly stable interval in one case and to the upper end in the other. It is this difference that is eventually responsible for the onset of hysteresis in the ensemble response. The procedure that we have outlined can be iterated to calculate the distributions p2(x2; h2, hi) for the second reversal point, p3(x3; h3, h2, hi) for the third, and so on until the distribution pn(Xn; hn ..... hi) for the last reversal point is obtained. At that point, the probability density p(xt, ht; hn, ..., hi) for xt under the field ht is given by one of the two equations:

p(xt, ht; hn . . . . . hi) - fxt T(ht, xtlhn, xn)pn(Xn; hn . . . . . hi) dxn,

(1.200)

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

96

.IT"

jH hE

x XL

X

Jh

J

hE

x

XL

Xt

Xl

FIGURE 1.16 Level-crossing problems associated with ascending (top) and descending (bottom) branches of hysteretic response.

pn(Xn; hn . . . . .

hi) - i x ~

T ( h n , x n l h t , x t ) p ( x t , ht; hn, . . . ,

hi)

dxt,

(1.201)

depending on whether the current field is increasing or decreasing, respectively. The corresponding ensemble response can then be calculated through Eqn. (1.195). By this method the functional integral expressed by Eqn. (1.194) is reduced to a chain of integrals and integral equations dependent on the saturation distribution pL (XL), which can be arbitrarily chosen,

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

97

and on the transition density T(a, xaib, Xb). The key problem becomes the calculation of T(a, xalb, Xb) for a given pinning-field process.

C a l c u l a t i o n of P r e i s a c h D i s t r i b u t i o n s Particularly simple results are obtained when the statistical properties of the pinning-field process are translationally invariant with respect to x (homogeneous process). In fact, in that case the transition density T(a, xalb, Xb) is a function of the difference Xa -- Xb only: T(a, xa[b, Xb) ~ T(a, Xa -- Xblb, 0). By using this property in Eqns (1.200), (1.201), and (1.195) one finds that the average ensemble response is of the form:

Xt = Xn -4- XT(htihn)

(1.202)

under increasing field ht > hn, and

Xt -" Xn - XT(hnlht)

(1.203)

under decreasing field ht < hn, where:

Xr(alb) =

Xn --

u r(a, ulb, O) du,

Xn pn (Xn;

hn . . . . . hi) dxn.

(1.204) (1.205)

oo

Therefore, in order to predict the hysteresis properties of the ensemble it is sufficient to know the first T moment XT(aib), with no necessity to determine the complete function T(a, ulb, 0). In addition, as shown by Eqns (1.202) and (1.203), the geometrical shape of the hysteresis branch originating from a certain reversal point hn is given by the very function XT(aib). This shape is independent of the field history preceding hn, because different histories merely give rise to different shifts Xn along the X-axis. This means that the ensemble response satisfies the congruency property discussed in Section 1.3.2. It also exhibits return-point memory, because this is a property of the individual responses contributing to Xt (see comment to Eqn. (1.193)). Since return-point memory and congruency are the necessary and sufficient conditions for the representation of a given hysteretic system by the Preisach model (see Section 1.3.2) we come to the following remarkable conclusion.

Connection between Hysteresis and Disorder Hysteresis generated by homogeneous Markovian pinning-field processes is of Preisach type. The connection between the pinning-field properties, the function XT(alb) and the Preisach distribution #(a, b) describing the average hysteresis

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

98

properties of the system is discussed in detail in [97]. The main results of that analysis can be summarized as follows. 9 The pinning-field process is described through the associated Ito stochastic differential equation [102]:

dh -- A(h)dx + B(h)dWx,

(1.206)

where d Wx represents the increment of the Wiener process W(x), and A(h) and B(h) are independent of x because of the assumed homogeneity of the process. 9 By solving the level-crossing problem for this process one finds:

XT(alb) =

du 2 /b a lib u ~(u')du'][/ua~(u')dU']B2(u)~(u) ' (1.207) K[b,a]

where:

I /o

6(u) = exp - 2

K[b,a] =

B2(u,)

0(u) du.

,

1208 (1.209)

~b a 9 By proper differentiation of XT(alb) one obtains the associated Preisach distribution:

XT(aib). #(a, b) = ~(a)~/(b) K2

(1.210)

"[b,a]

Notice that the field variables (a, b) appearing in #(a, b) are different from the variables (he, hu) used for the Preisach distribution p(hc, hu) discussed in Section 1.3.2. The connection between the two sets is: a = hr + hu, b = hc hu. The two representations are equivalent and both are widely employed in the literature [13]. Equations (1.206)-(1.210) permit one to calculate in a straightforward manner the hysteresis curves produced by different types of structural disorder. We discuss three examples of particular physical interest.

9 Wiener-Ldvy process: A(h) = O, B(h) = 1. The experimental results obtained in [105,106] indicate that a good candidate to describe the disorder in magnetic materials, at least for small domain-wall displacements, is the Wiener-L6vy process W(x) [107,108], also called Brownian process. The same conclusion was reached in [81-83]

1.4 FROM MICROMAGNETICS TO MAGNETIC HYSTERESIS

99

when interpreting the essential aspects of the Barkhausen effect. By inserting A(h) = O, B(h) = 1 into Eqns (1.207)-(1.210) one immediately finds: #(a, b) -- 1,

(1.211)

( h - hn) 2 XT(hlhn) = - - . 3

(1.212)

All hysteresis branches are parabolic, in agreement with Rayleigh law. The fact that no term linear in the field is predicted is related to the nondifferentiable nature of the Wiener process. As a matter of fact, one expects that some correlation should exist in the pinning-field slopes over very short distances. When this refinement is introduced, a linear term is indeed obtained in Eqn. (1.212) in addition to the quadratic one [83]. 9 Wiener-L~vy process with drift: A(h) = 1/(2~), B(h) = 1, ~ > 0. This process describes pinning fields that on the average increase like x/(2~). This increase is expected when important demagnetizing effects are present in addition to structural disorder. From Eqns (1.207)-(1.210) one obtains: x cothx- 1 , #(a, b) sinh 2 x

x -

XT(hlhn) = 4(x c o t h x - 1),

a- b ~; 2~ x --

h - hn

2~

(1.213)

(1.214)

9 Ornstein-Uhlenbeck process: A(h) = -h/~,B(h) = 1, ~ > O. The Wiener-L6vy process is nonstationary, so it cannot provide a realistic description of the pinning field for large wall displacements. In the Ornstein-Uhlenbeck process [102], the additional correlation length ~ makes the pinning field stationary for large x variations. The appearance of ~ reflects the fact that the interaction range of the domain wall with local sources of structural disorder is finite. The equations for the hysteresis curves are definitely more complicated than for the previous cases. From Eqns (1.208) and (1.209) one finds: ~(u) = exp(u2/~),

K[b,a]--aO (1/2, 3/2; a2/~) - b O (1/2, 3/2" b2/~),

(1.215)

(1.216)

CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

100

where (I)(a, c; z) is the confluent hypergeometric function. The Preisach distribution and XT(hlhn) are obtained by inserting these expressions into Eqns (1.207) and (1.210). It is worth remarking that the applicability of the Preisach model has been proven in this section by taking advantage of return-point memory and congruency properties, rather than by identifying appropriate TLS units of the type introduced in Section 1.3. This leads to the interesting, still open, question as to whether the connection between random pinning fields and the Preisach approach can also be extended to include the thermal relaxation effects discussed in Section 1.3.3.

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66. V. Basso, C. Beatrice, M. LoBue, P. Tiberto, and G. Bertotti, 'Connection between hysteresis and thermal relaxation in magnetic materials', Phys. Rev. B 61, 1278-1285 (2000). 67. S. J. Collocot and J. B. Dunlop, 'Anomalous magnetic viscosity in the bulk-amorphous ferromagnet Nd60Fe20Col0All0', Phys. Rev. B 66, 224-420 (2002). 68. L. Wang, J. Ding, Y. Li, Y. P. Feng, and X. Z. Wang, 'Anomalous magnetic viscosity in the bulk-amorphous materials', J. Magn. Magn. Mater. 206, 127-134 (1999). 69. M. LoBue, V. Basso, G. Bertotti, and K. H. Mfiller, 'Magnetic aftereffect in spring magnets and the Preisach model of hysteresis', IEEE Trans. Magn. 33, 3862-3864 (1997). 70. P. Bruno, G. Bayereuther, and P. Beauvillain, et al., 'Hysteresis properties of ultrathin ferromagnetic films', J. Appl. Phys. 68, 5759-5766 (1990). 71. M. P. Sharrock, 'Measurement and interpretation of magnetic time effect in recording media', IEEE Trans. Magn. 35, 4414--4422 (1999). 72. S.D. Brown, R. Street, R. W. Chantrell, P. W. Haycock, and K. O'Grady, 'Time dependence and mechanisms of magnetization reversal in Tb-Fe-Co films', J. Appl. Phys. 79, 2594-2600 (1996). 73. R. A. Fry, A. Reimers, L. H. Bennett, and E. DellaTorre, 'Preisach modeling of aftereffect in a magneto-optical medium with perpendicular anisotropy', Physica B 257, 50-54 (2000). 74. S. Vitale, R. Tommasini, M. Cerdonio, M. Bonaldi, A. Cavalieri, and G. Durin, "Magnetic viscosity, thermal relaxation, and thermal equilibrium noise in Co-based amorphous alloys at milliKelvin temperatures', J. Appl. Phys. 72, 4820-4822 (1992). 75. G. Durin, P. Falferi, M. Cerdonio, G. A. Prodi, and S. Vitale, 'Low temperature properties of soft magnetic materials: magnetic viscosity and 1/f thermal noise', J. Appl. Phys. 73, 5363-5365 (1993). 76. L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon Press, Oxford (1980). 77. L. D. Landau and E. M. Lifshitz, 'On the theory of the dispersion of magnetic permeability in ferromagnetic bodies', in CollectedPapers of L. D. Landau, D. terHaar Ed., Pergamon Press, Oxford (1965), pp. 101-114. 78. T. L. Gilbert, 'A phenomenological theory of damping in ferromagnetic materials', IEEE Trans. Magn. 40, 3443-3449 (2004). 79. A. Hubert and R. Schaefer, Magnetic Domains, Springer-Verlag, Berlin (1998).

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105

80. H.J. Williams, W. Shockley, and C. Kittel, 'Studies of the propagation velocity of a ferromagnetic domain boundary', Phys. Rev. 80, 10901094 (1950). 81. B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, 'Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. I. Theory', J. App1. Phys. 68, 2901-2907 (1990). 82. B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, 'Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. II. Experiments', J. Appl. Phys. 68, 2908-2915 (1990). 83. A. Magni, C. Beatrice, G. Durin, and G. Bertotti, 'Stochastic model for magnetic hysteresis', J. Appl. Phys. 86, 3253-3261 (1999). 84. G. Durin, G. Bertotti, and A. Magni, 'Fractals, scaling and the question of self-organized criticality in magnetization processes', Fractals 3, 351-370 (1995). 85. D.L. Stein and C. M. Newman, 'Broken ergodicity and the geometry of rugged landscapes', Phys. Rev. E 51, 5228-5238 (1995). 86. C. M. Newman and D. L. Stein, 'Spin-glass model with dimensiondependent ground state multiplicity', Phys. Rev. Lett. 72, 2286-2289 (1994). 87. H. Kronmuller, 'Theory of the coercive field in amorphous ferromagnetic alloys', J. Magn. Magn. Mater. 24, 159-167 (1981). 88. H. Kronmuller, 'Magnetic techniques for the study of dislocations in ferromagnetic materials', Int. J. Nondest. Test. 3, 315-350 (1972). 89. G. Durin and S. Zapperi, 'Scaling exponents for Barkhausen avalanches in polycrystaline and amorphous ferromagnets', Phys. Rev. Lett. 84, 4705-4708 (2000). 90. M. Kersten, 'Versuche uber reversible und irreversible Wandverschiebungen zwischen antiparallel magnetisierten Weisschen Bezirken', Phys. Z. 9, 860-865 (1938). 91. L. N6el, 'Th6orie des lois d'aimantation de Lord Rayleigh. I: le d6placement d'une paroi isol6e', Cah. Phys. 12, 1-20 (1942). 92. L. N6el, 'Th6orie des lois d'aimantation de Lord Rayleigh. II: multiple domains et champ coercitif', Cah. Phys. 13, 18-30 (1943). 93. K. H. Pfeffer, 'Zur theorie der koerzitivfeldstarke und anfangssuszeptibilitat', Phys. Stat. Sol. 21, 857 (1967). 94. H. Kronmuller, J. Magn. Magn. Mater. 13, 53 (1979). 95. H. Kronmuller and T. Reininger, 'Micromagnetic background of magnetization processes in inhomogeneous ferromagnetic alloys', J. Magn. Magn. Mater. 112, 1 (1992).

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CHAPTER 1 Thermodynamics, Hysteresis, and Micromagnetics

96. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer-Verlag, New York (1994). 97. G. Bertotti, I. D. Mayergoyz, V. Basso and A. Magni, 'Functional integration approach to hysteresis', Phys. Rev. E 60, 1428 (1999). 98. V. Basso and A. Magni, 'Field history analysis of spin configurations in the random field Ising model', Physica B 343, 275-280 (2004). 99. A. Kolmogorov and S. Fomine, Elements de la Thdorie des Fonctions e de l'Analyse Fonctionelle, MIR, Moscow (1974). 100. I. Guikhman and A. Skorokhod, Introduction a la Thdorie des Processus Aldatoires, MIR, Moscow (1980). 101. J.P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, 'Hysteresis and hierarchies: dynamics of disorder driven first-order phase transformations', Phys. Rev. Lett. 70, 3347 (1993). 102. C.W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, Berlin (1985). 103. A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York (1991). 104. N.G. VanKampen, Stochastic Processes in Physics and Chemistry, NorthHolland, Amsterdam (1984). 105. M. Schroeder, Fractals, Chaos, Power Laws, Freeman, New York (1991). 106. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco (1983). 107. W. Grosse-Nobis, 'Frequency spectrum of the Barkhausen noise of a moving 180~ domain wall', J. Magn. Magn. Mater. 4, 247 (1977). 108. R. Vergne, J. C. Cotillard, and J. L. Porteseil, 'Quelques aspects statistiques des processus d'aimantation dans les corps ferromagn6tiques', Revue Phys. Appl. 16, 449-476 (1981).

CHAPTER 2

Random-Field Ising Models of Hysteresis James P. Sethna, Laboratory of Atomic and Solid State Physics (LASSP), Clark Hall, Comell University, Ithaca, NY 14853-2501, USA Karin A. Dahmen, Physics Department, University of Illinois at Urbana/Champaign, 1110 West Green Street, Urbana, IL 61801-3080, USA Olga Perkovi#., URS Corporation, 600 Montgomery Street, 26th floor, San Francisco, CA 94111-2727, USA

Contents 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Introduction Models The Renormalization Group and Scaling Critical Exponents, Scaling Functions Measuring Universal Properties Exponent Relations Finite Sweep Rate Subloops and History-Induced Critical Behavior Real Experiments Unsolved Problems Acknowledgement Appendix. Derivation of the Scaling Forms and Corrections References

107 112 117 122 128 141 144 145 149 158 159 160 167

2.1 I N T R O D U C T I O N Our group has devoted several years to the study of the dynamics of a simple model, the random-field Ising model at zero temperature. We do clever simulations involving the dynamics of billions of domains (Fig. 2.1), and we develop sophisticated analytical methods for extracting and explaining the properties of this model. We do this because we believe our model- despite its dramatically simplified nature - may well describe the properties of Barkhausen noise in many real physical systems. The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc.

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CHAPTER 2 Random-Field Ising Models of Hysteresis

FIGURE 2.1 Cross-section of all avalanches in a billion-spin simulation at the critical disorder [1]. The white background is the infinite, spanning avalanche, which would not exist above Rc.

We present here the results of our simulations and analysis, together with an explanation of why we believe they should (or could) describe real experiments. Our arguments for the applicability of our model are based on renormalization group and scaling theories, first developed to study continuous-phase transitions in equilibrium systems. To a large extent, these theories can be seen as the underlying reason why many, if not most, theories of nature apply to the real world, and (more specifically) why different magnets share common features in their dynamics despite having microscopically rather different morphologies and energetics. Why is the noise in magnets interesting? As one applies an external field to most magnetic materials, the magnetization changes through the nucleation and motion of domain walls. This motion is not smooth, and the resulting jumps in the net magnetization is termed Barkhausen noise. These jumps, corresponding to the reorganization (or avalanche) of a region of spins, usually span many decades in size; there will be many small avalanches of spins, and fewer and fewer avalanches of larger and larger sizes (what we call crackling noise). Indeed, the probability D of having

2.1 INTRODUCTION

109

9 Experiment ('~ = 1.46) Front propagation model (1: = 1.33) 9 Nucleation model ('~ = 1.60)

109

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108

S (Avalanche size) FIGURE 2.2 Avalanche size distributions [2]. The figure shows the number D(S) of avalanches of size S measured in an experiment [2] measuring near the center of the hysteresis loop, our (nucleation) model near Rc, and a related front propagation model. Notice the straight lines on a log-log plot indicate power-law behaviors D(S) ~ S -~.

a j u m p or avalanche of a given size S often decreases as a p o w e r law 1 in the size of the avalanche D ( S ) ~, S -~ for a significant range of sizes (Figs 2.2 a n d 2.1). Simple b e h a v i o r (like a p o w e r law) e m e r g i n g out of complicated microscopic d y n a m i c s is a good sign that s o m e t h i n g needs explanation! P o w e r laws in particular are the defining signature of continuous phase transitions, a n d a large fraction of the statistical physics c o m m u n i t y focuses on p r o b l e m s involving these p o w e r laws: a n e w class of s y s t e m s to s t u d y is exciting. Finally, similar p o w e r laws for crackling noise e m e r g e in m a n y other s y s t e m s [1], from flux-line avalanches in superconductors, to

1in our model, this law is obtained only at the critical field and at the critical disorder (Fig. 2.1: integrating over field yields a different power law Dint(S) ~ S -(r+r~fl~) at the critical disorder, see Eqn. (2.10).

CHAPTER 2 Random-Field Ising Models of Hysteresis

110

0.2

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FIGURE 2.3 Avalanche size distributions for our model [3], integrated over the hysteresis loop, for a range of disorders above Rc = 2.16: at Rc one gets a power law (the straight dashed line) Dint(S, Rc) (x s-(~+~1~). The inset shows a scaling collapse of the data (equation 2.10). In the main figure, the thin lines show the scaling function prediction for the associated curves. The true power-law behavior only emerges very close to Rc, but the universal scaling function governs a range of disorders spanning a factor of two above Rc. Notice also that we get six orders of magnitude of scaling at five per cent above Rc, without any self-organization. crumpling paper [4], to earthquakes (the Gutenberg-Richter law). Magnetic Barkhausen noise forms a manageable experimental prototype for a whole class of behavior. Why should crackling noise in magnets be comprehensible? Very small jumps in the magnetization, corresponding to avalanches on scales comparable to the microstructure, will certainly depend upon the details of the individual magnetic material. Large jumps, spanning the entire sample, will depend upon the sample geometry. Indeed, these extremes in sizes are not regions in which simple power laws should be observed experimentally. But these extremes are separated by many powers of ten in a typical experiment. Just as the complex microscopic properties of the molecules in a fluid only affect the viscosity and density on long length scales, so also one might expect the complicated microstructure in a magnet could be subsumed into a few constants when dealing with events of sizes much larger than the microstructure. The smooth power-law behavior suggests exactly this: something simple is happening on intermediate scales, independent of either the microscopic or macroscopic details.

2.1 INTRODUCTION

111

What should beexplicable? How ambitious can we be in our expectations of a successful theory? As we shall explain in this review, we cannot expect to be able to tell when and where to expect an avalanche. (Our tools, hence, will likely not be effective at predicting large earthquakes.) However, a successful theory should predict statistical averages of almost any quantity that is dominated by events on large length and time scales, up to certain overall parameter-dependent scales (analogous to viscosity and density for fluids). For example, the probability D(S, T, L, W, R, H) in our theory of having an avalanche of size S, duration T, long axis L, short axis W, at disorder R and external field H should be predicted by the theory, up to an overall size scale Ss, time scale Ts, field scale Hs and offset He, and disorder scale Rs and offset Re. Indeed, we shall see that functions like these take scaling forms Dmulti (S, T, L, W, R, H)

=DsS-(Z+avz)lgmulti( S r l / a T'Ts z vrL S, sW'rfl~h) ,

(2.1)

where r = R~sRC,h = - ~ sHe, Ds is a normalization factor, and z + avz is another universal critical exponent. We shall derive these scaling forms from self-similarity or scale invariance: the avalanche dynamics is inherently the same when observed (say) at microns and hundreds of microns. This scale invariance allows us to write functions of N variables as power laws times functions of N - 1 variables; hence, functions of one variable become pure power laws, as for D(S) above. What makes these theories predictive is that both the power law z and the entire multivariable function ~)multi are universal: they are independent of the microscopic model (within large classes of systems). If your theory is in the same universality class as the experiment, it will predict all critical exponents and scaling functions. There are three important qualifications about these predictions.

1. Analytic corrections. The various scales Ss, Ts, Rc, Hc . . . . will not be constants, but will depend smoothly and analytically on the control parameters in a given system. This is a serious issue, as the function we are describing (Dmulti above) will typically itself be a smooth and analytic function of its parameters except at a phase transition where something qualitatively changes in the behavior of the system. Indeed, it is precisely at these phase transitions when events of all scales arise, where we expect our theory to apply. These analytic corrections usually become less and less important as we confine our attention to large, slow events near the transition.

CHAPTER 2 Random-Field Ising Models of Hysteresis

112

c l a s s e s . T h e power of the theory rests in the prediction that quite different experimental systems, or a theoretical model and an experiment, may be in the same universality class. We shall see that universality classes are studied by considering a space of possible systems: if two systems flow towards the same fixed point as one coarse-grains to larger scales, then their long length scale behaviors must agree. In many cases, and for magnetic noise in particular, there can be more than one candidate theory. Our model, which incorporates short-range interactions and nucleation of new domains has competitors that allow only interface depinning [5,6] and which incorporate long-range fields in a mean-field fashion [712]. For many properties, the predictions of the various theories are not strikingly different: careful experiments may be needed to distinguish between the rival theories if it is not obvious whether nucleation or long-range interactions are relevant in a given experiment. 3. D y n a m i c cr i t i cal i t y. T h e static, equal-time properties of these models have historically been less fussy than the time-dependent behavior. More specifically, there will often be several dynamic universality classes for each static universality class, at least in the well-studied cases of equilibrium continuous phase transitions. We will see clear evidence that our theories are not predicting the dynamical behavior within the avalanches properly, but that the various theories are rather successful at describing the distributions of avalanches and other static quantities. 2. U n i v e r s a l i t y

2.2 M O D E L S

Several variations of the zero-temperature-random field Ising model have been proposed to explain the power laws in Barkhausen noise. They are differentiated on the basis of the presence of long-range forces, and the details of the dynamics. Our summary follows reference [13]. The model is composed of a large number of 'spins' si = 4-1 on a cubic lattice (square lattice in two dimensions, hypercubic for dimensions D > 3). These spins represent domains or small regions of the material. The model is subject to an external field H, which varies slowly in time, 2 usually ramping u p w a r d from H = -cx~ to H - ~ . The value of a spin si 2Except when otherwise mentioned, the field varies slowly enough that each avalanche finishes before the field changes.

2.2 MODELS

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FIGURE 2.4 Hysteresis loop and subloops. The magnetization in our model (for R > Rc), as the external field H is ramped up and down. Our focus will primarily be on the upper, outer loop as the external field is ramped from -cx~ to cx~.

represents whether that domain is aligned (+1) or anti-aligned (-1) with the final H - +cx~. The energy function, or Hamiltonian, for the models is ?-[ -- - ~ JnnsiSj - ~ Hsi - ~-~ his i nn i i

~_f q- ~ i

Si -- ~--~Jdipole {i,j}

3 cos2(0/j) - 1 sisj. r3"

(2.2)

V

Here Jnn is the strength of the ferromagnetic nearest-neighbor interactions,

hi is a random field representing the effects of compositional and morphological disorder, Jinf is the strength of an infinite range demagnetizing field [8], Oij is the angle between the positive spin direction and the difference vector between lattice positions i and j, and Jdipole is the strength of the dipole-dipole interactions. The critical exponents of the power laws are independent of the particular choice of random field distributions p(hi) for a large variety of distributions. We use a Gaussian distribution of random fields, with zero mean and standard deviation R. (When we refer to the

114

CHAPTER 2 Random-Field Ising Models of Hysteresis

strength of the disorder, we are referring to the width, R, of the random field distribution.) 3 Two different dynamics have been considered. The first is a frontpropagation dynamics in which a spin on the edge of an existing front flips as soon as it would decrease the energy to do so, introduced by Ji and Robbins [5]. Spins with no flipped neighbors cannot flip even if it would be energetically favorable. Second is the dynamics we use [14], which includes domain nucleation. Any spin can flip when it becomes energetically favorable to do so. In both cases, spins flip in shells - all spins which can flip at time t flip, then all of their newly flippable neighbors flip at time t + 1, causing an avalanche. The number of spins flipped in each shell gives the time series of the avalanche (Fig. 2.5), whose irregular fluctuations and near halts are typical also of experimental time series. Depending on which terms are included in the Hamiltonian, the behavior appears to fall into three different universality classes.

1. Front propagation model [5]. The front propagation model has a critical field Hc for a range of disorders R. As the field approaches

Hc, larger and larger avalanches arise. As one approaches Hc from below, these avalanches develop a power-law distribution in sizes with a cutoff which diverges at He. Above He the entire front is depinned, and the front moves forward with an inhomogeneous velocity that begins jerky on all length and time scales (echoing the avalanches below Hc) but which becomes smoother and more uniform at large H. 2. Our model (nucleation allowed, Fig. 2.4). For our model, where domain nucleation is allowed and only nearest-neighbor interactions are included, there is a continuous transition at a critical point with disorder Re and external field Hc. At very large disorder, the couplings between spins Jnn are a small perturbation and each spin si flips roughly when the external field H(t) is equal and opposite to the local random field hi. For all disorders above Re this qualitative behavior persists: all avalanches are finite, dying out after a number of spin flips that does not depend on the number of spins N, yielding a smooth hysteresis loop (Fig. 2.6). For a very small disorder, an early spin will trigger almost the entire system to flip over. This too extends qualitatively to all disorders below Re, where a finite fraction 3In this review we focus specifically on zero-temperature models; there are also finite temperature studies especially for pure systems [181,182],micromagnetic systems [50], interface depinning models in the presence of temperature [183,184], and hysteresis due to driven interfaces at finite temperature [100,101,185]. We are currently working on hysteresis at finite temperatures with randomness and nucleation [186].

115

2.2 MODELS

t I FIGURE 2.5 Typical avalanche time series [13]. Voltage (number of domains flipped) pulse during a single large avalanche (arbitrary units). Notice how the avalanche almost stops several times: if the forcing were slightly smaller, this large avalanche would have broken up into two or three smaller ones. The fact that the forcing is just large enough to, on average, keep the avalanche growing is the cause of the self-similarity: on average a partial avalanche of size S will trigger one other of size S. of the spins in the s y s t e m (even as the s y s t e m size b e c o m e s large) flip in a single event, the infinite or s p a n n i n g avalanche. The critical disorder Rc is defined as the disorder at w h i c h the infinite avalanche first arises, a n d w h e r e a j u m p in the m a g n e t i z a t i o n per spin m(H) first arises in the limit of an infinite system. 4 Figure 2.1 s h o w s avalanches in a cross-section of our s y s t e m at the critical disorder: the w h i t e b a c k g r o u n d is the spanning, or infinite avalanche. 5 O n e does not n e e d to be exactly at Rc to observe p o w e r laws in our model: one gets a large region of p o w e r - l a w scaling rather far from the critical point (Fig. 2.1 a n d Ref. [3]).

4Each avalanche produces a jump in the magnetization AM, but all finite avalanches produce jumps in the magnetization per spin AM/N which vanish as the number of spins N diverges. 5It may seem as if the spanning avalanche is occupying quite a large fraction of the volume. As the system size increases, the 'holes' formed by the finite avalanches (and other spanning avalanches) will gradually become larger and larger fractions of the total, and the spanning avalanche at Rc should grow with the number of spins N to a power below one.

CHAPTER 2 Random-Field Ising Models of Hysteresis

116 9

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FIGURE 2.6

Infinite avalanche, jump in M(H) for R < Rc.

3. I n f i n i t e range and dipolar fields. For disorders below Rc, or when domain nucleation is not allowed, the addition of an infiniterange demagnetizing field 6 [8] can self-organize the system to a different critical behavior. (Self-organization means that the system naturally sits at a critical point, without having to tune any parameters.) The infinite-ranged interaction is sometimes introduced to mimic the effects of the boundaries of materials with dipolar, or other long-ranged interactions [10]; these interactions also selforganize the model to the critical point [9]. Our use of the term 'infinite-range models' perhaps obscures the clear physical origin of this universality class of models. Zapperi et al. [10] argue that the addition of dipole-dipole interactions to the model lowers the upper critical dimension to three and produces mean-field exponents in three dimensions. Since large mean-field simulations are much easier than large simulations with dipole-dipole interactions, we will give results from mean-field simulations in this paper. Dipolar interactions without an infiniterange term were explored by Magni [15] in two dimensions, who 6This is not the same as a mean-field model, because the model contains nearest-neighbor interactions along with the infinite-range interactions.

2.3 THE RENORMALIZATION GROUP AND SCALING

117

found labyrinthine patterns and hysteresis loops similar to those seen in garnet films. The infinite-range model is apparently also equivalent to the rather successful single degree-of-freedom ABBM models [10,16,17]. Unless we specify otherwise, we will focus on our model, allowing nucleation with parameters Jnn, H, and hi but without long-range interactions (Jinf -- Jdipole -- 0).

2.3 THE R E N O R M A L I Z A T I O N G R O U P A N D S C A L I N G To study crackling noise, we use renormalization-group tools developed in the study of continuous phase transitions. 7 The word renormalization has roots in the study of quantum electrodynamics, where the effective charge changes in size (norm) as a function of length scale. The word group refers to the family of coarse-graining operations basic to the method: the group product is composition (coarsening repeatedly). The name is unfortunate, however, as the basic coarse-graining operation does not have an inverse, and thus the renormalization group does not have the mathematical structure of a group. The renormalization group studies the way the space of all physical systems maps into itself under coarse-graining (see Fig. 2.8). The coarsegraining operation shrinks the system, and removes degrees-of-freedom on short length scales. Under coarse-graining, we often find a fixed point S*: many different models flow into the fixed point and hence share longwavelength properties. To get a schematic view of coarse-graining, look at Figs 2.1 and 2.7: the 10003 cross-section looks (statistically) like the 1003 section if you blur your eyes by a factor of 10. Much of the mathematical complexity of this field involves finding analytical tools for computing the flow diagram in Fig. 2.8. Using methods developed to study thermodynamical phase transitions and the depinning of charge-density waves, we can calculate for our model the flows for systems in dimensions close to six (the so-called ~ expansion, where ~ = 6 - D, D being the dimension of the system). Interpolating between dimensions may seem a surprising thing to do. In our system it gives rather good predictions even in three dimensions (i.e. D = 3), but it is hard work, and we will not discuss it here. Nor will we discuss real-space renormalizationgroup methods or series-expansion methods. We focus on the relatively 7This section and the next follow closely the presentation in Ref. [1]. References to the broader literature may be found there.

118

CHAPTER 2 Random-Field Ising Models of Hysteresis

FIGURE 2.7 Cross-section of all avalanches in a million-spin simulation at the critical disorder [1]. Compare with the much larger simulation in Fig. 2.1.

simple task of using the renormalization group to justify and explain the universality, self-similarity, and scaling observed in nature. Consider the 'system space' for disordered magnets. There is a separate dimension in system space for each possible parameter in a theoretical model (disorder, coupling, next-neighbor coupling, dipolar fields, etc.) or in an experiment (temperature, annealing time, chemical composition...). Coarse-graining, however one implements it, gives a mapping from a system space into itself; shrinking the system and ignoring the shortest length scales yields a new physical system with identical long-distance physics, but with different (renormalized) values of the parameters. We have abstracted the problem of understanding crackling noise in magnets into understanding a dynamical system acting on the space of all dynamical systems. Figure 2.8 represents a two-dimensional cross-section of this infinitedimensional system space. We have chosen the cross-section to include our model; as we vary the disorder R, our model sweeps out a straight line in system space. The cross-section also includes a fixed point S', which maps into itself under coarse-graining. The system S" looks the same on all

2.3 THE RENORMALIZATION GROUP AND SCALING

119

Our model

x\

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R e n o r m a l i z a t i o n - g r o u p f l o w s [1]. The renormalization-group uses coarse-graining to longer length scales to produce a mapping from the space of physical systems into itself. Consider the space of all possible systems exhibiting magnetic hysteresis (including, in our imagination, both real models and experimental systems). Each model can be coarse-grained, removing some fraction of the microscopic domains and introducing more complex dynamical rules so that the remaining domains still flip over at the same external fields. This defines a mapping of our space of models into itself. A fixed point S* in this space will be self-similar; because it maps into itself upon coarse-graining, it must have the same behavior on different length scales. Points that flow into S* under coarse-graining share this self-similar behavior on sufficiently long length scales; they all share the same universality class. F I G U R E 2.8

length and time scales, because it coarse-grains into itself. We can picture the cross-section of Fig. 2.8 either as a plane in system space (in which case the arrows and flows depict projections, since in general the real flows will point s o m e w h a t out of the plane), or as the curved manifold swept out by our one-parameter m o d e l as we coarse-grain (in which case the flows above our m o d e l and below the horizontal curved line in Fig. 2.8 should be ignored). The flow near S* has one unstable direction, leading o u t w a r d along the horizontal curve (the unstable manifold). In system space, there is a

120

CHAPTER 2 Random-Field Ising Models of Hysteresis

surface of points C which flows into S" under coarse-graining. Because S" has only one unstable direction, C divides system space into two phases. To the left of C, the systems will have one large, system-spanning avalanche (a snapping noise). To the right of C, all avalanches are finite, and under coarse-graining they all become small (popping noise). Our model, as it crosses C at the value Rc, goes through a phase transition. Our model at Rc is not self-similar on the shortest length scales (where the square lattice of domains is still important), but because it flows into S" as we coarse-grain we deduce that it is self-similar on long length scales. Some phase transitions, like ice melting into water, are abrupt and do not exhibit self-similarity. Continuous phase transitions like ours almost always have self-similar fluctuations on long length scales. Also, we must note that our model at Rc will have the same self-similar structure as S" does. Indeed, any experimental or theoretical model lying on the critical surface C will share the same long-wavelength critical behavior. This is the fundamental explanation for universality. The flows in system space can vary from one class of problems to another: the system space for front-propagation models (Fig. 2.9(a)) will have a different flow, and its fixed point will have different scaling behavior (yielding a different universality class). In some cases, a fixed point will attract all the systems in its vicinity (Fig. 2.10). Usually at such attracting fixed points the fluctuations become unimportant at long length scales: the Navier-Stokes equation for fluids described earlier can be viewed as a stable fixed point. The coarse-graining process, averaging over many degrees-of-freedom, naturally smooths out fluctuations, if they are not amplified near a critical point by the unstable direction. Fluctuations can remain important when a system has random noise in a conserved property, so that fluctuations can only die away by diffusion; in these cases, the whole phase will have self-similar fluctuations, leading to generic scale invariance. Sometimes, even when the system space has an unstable direction like in Fig. 2.8, the observed behavior always has avalanches of all scales. This can occur simply because the physical system averages over a range of model parameters (i.e. averaging over a range of R including Re in Fig. 2.8). For example, this can occur by the sweeping of a parameter slowly in time, or varying it gradually in space either deliberately or through large-scale inhomogeneities. One can also have self-organized criticality, where the system is controlled so that it naturally sits on the critical surface. Self-organization to the critical p o i n t c a n occur via many mechanisms. In the front-propagation models with long-range fields (Fig. 2.9(b)), upon an increment of external field H the front will move forward until the long-range

2.3 THE RENORMALIZATION GROUP AND SCALING

121

FIGURE 2.9 Flows for a front-propagation m o d e l [1]. The front-propagation model has a critical field Hc at which the front changes from a pinned to sliding state. (a) Coarse-graining defines a flow on the space of depinning models. The fixed point S~fp will have a different local flow field from other renormalization-group fixed points, yielding its own universality class of critical exponents and scaling functions. The critical manifold C, consisting of models which flow into S*fp, separates the stuck fronts from those which slide forward with an average velocity v(H). (b) The velocity varies with the external force as a power law v ( H ) ,~ ( H - Hc) ~. Clever experiments, or long-range fields, can act to control not the external field, but the net magnetization; changing the magnetization slowly sets v ~ 0, thus self-tuning H ~ Hc. This is one example of self-organized criticality.

d e m a g n e t i z a t i o n force balances the increment; the net external field given by the s u m organizes itself to precisely the critical d e p i n n i n g field for the front.

122

CHAPTER 2 Random-Field Ising Models of Hysteresis

2.4 CRITICAL EXPONENTS, SCALING FUNCTIONS The renormalization group is the theoretical u n d e r p i n n i n g for understanding w h y universality and self-similarity occur. 8 Once we grant that different systems should sometimes share long-distance properties, though, we can quite easily derive some powerful predictions. To take a tangible example, let us consider the relation between the duration of an avalanche and its size. If we look at all avalanches of a certain duration T in an experiment, they will have a distribution of sizes S a r o u n d some average (S)exp(Z). If we look at a theoretical model, it will have a corresponding average size (S)th (T). If our model describes the experiment, these functions m u s t be essentially the same at large S and large T. We m u s t allow for the fact that the experimental units of time and size will be different from the ones in our model; the best we can hope for is that (S)exp(Z) -- A(S)th(T/B),

(2.3)

for some rescaling factors A and B. Now, instead of comparing to experiment, we can compare our model to itself on a slightly larger time scale. If the time scale is e x p a n d e d by a small factor B = 1/(1 - 6), then the rescaling of the size will also be small, say I + a. Hence, (S)(T) -- (1 + a 6 ) ( S ) ( ( 1 - ~)T).

(2.4)

Making 6 very small yields the simple relation a(S) = T d ( S ) / d T , which can be solved to give the power-law relation (S)(T) = SoT a. The exponent a is called a critical exponent, and is a universal prediction of a given theory. (That means that if the theory correctly describes an experiment, the critical exponents will agree.) In our work, we write the exponent a relating time to size in terms of three other critical exponents, a = 1/r There are several basic critical exponents, which arise in m a n y different combinations d e p e n d i n g on the physical property being studied. We have seen that at Rc and near Hc the probability of having an avalanche of size S goes as S-~. 9 The cutoff in the avalanche size distribution in Fig. 2.1 gets larger as one approaches the critical disorder as (R - Re) -~ (Fig. 2.1). The typical length of the largest avalanche goes as (R - Re) -v. The jump in the magnetization (Fig. 2.6) goes as ( R - Re) ~, and at Re the magnetization (M - Me) "~" (H - Hc)l/~. 10 Finally, we need to k n o w h o w time rescales; the duration of an avalanche of spatial extent L typically will go as L z. 8This section also follows closely the presentation in Ref. [1]. 9We will see that when the entire hysteresis loop is considered this changes to S-(Z+rTfl6). 10Do not confuse the small change in scale ~ earlier with the critical exponent ~ here.

123

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A n y physical property that shows singular behavior at the critical point will have a critical exponent that can be written in terms of these basic ones (Fig. 2.11). 11 To specialists in critical p h e n o m e n a , these exponents are central; whole conversations seem to rotate around various combinations of Greek letters. Critical exponents are one of the relatively easy things to calculate from the various analytic approaches, and so have attracted the most attention. They are derived from the eigenvalues of the linearized flows about the fixed point S* in Fig. 2.8. Figure 2.12 shows our numerical estimates for several critical exponents in our model in various spatial dimensions, together with our 6 - ~ expansions for them. Of course the key challenge is not to get analytical w o r k to agree with numerics; it is to get theory to agree with experiment. Figure 2.11 shows that our model does rather well in describing a wide variety of experiments, but that the two rival models (with different flows around their fixed points) also fit.

11Indeed, there are relations even between these exponents: see Section 2.6.

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CHAPTER 2 Random-Field Ising Models of Hysteresis

FIGURE 2.11 Universal critical exponents versus experiment [1]. Different experiments on crackling noise in magnets measure different combinations of the universal critical exponents. Here we compare experimental measurements (see Table I of Ref. [3]) to the theoretical predictions for three models: our model, the front-propagation model and the dipolar mean-field theory. Power laws giving the probability of getting an avalanche of a given size, duration, or energy at the critical point are shown; also shown is the critical exponent giving the power as a function of frequency [13] (due to the internal structure of the avalanches, Fig. 2.5). In each pair of columns, the first column includes only avalanches at external fields H near Hc where the largest avalanches occur, and the second column (when it exists) includes all avalanches. The various combinations of the basic critical exponents can be derived from exponent equality calculations similar to the one discussed in the text. Many of the experiments were done years before the theories were developed; many did not report error bars. All three theories do well (especially considering the possible systematic errors in fitting power laws to the experimental measurements; see in Fig. 2.1 how a bit away from Rc the effective slopes change). (A more systematic and critical review of the exponent measurements in the literature may be found in Table I of Ref. [11]. One may also find in section V.A of that paper a discussion of the relation between the experiments and the theoretical universality classes.)

Critical exponents are not the be all a n d end all; m a n y other scaling predictions, explaining w i d e varieties of behavior, are quite easy to extract from numerical simulations. Universality extends even to those long length scale properties for w h i c h one cannot write formulas. Perhaps the m o s t i m p o r t a n t of these other predictions are the universal scaling

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functions. For example, let us consider the time history of the avalanches, V(t), d e n o t i n g the n u m b e r of d o m a i n s flipping per unit time. (We call it V b e c a u s e it is u s u a l l y m e a s u r e d as a voltage in a p i c k u p coil.) Each avalanche has large fluctuations, b u t one can a v e r a g e over m a n y a v a l a n c h e s to get a typical shape. Figures 2.13 a n d 2.14 s h o w averages over all avalanches

126

CHAPTER 2 Random-Field Ising Models of Hysteresis

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(2.5)

C o m p a r i n g our m o d e l w i t h itself w i t h a shifted time scale becomes simple if w e change variables: let v(T, t/T) = (V)(T, t), so v(T, t/T) = Av(T/B, t/T).

127

2.4 CRITICAL EXPONENTS, SCALING FUNCTIONS 0.6

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FIGURE 2.14 Comparison of theoretical average pulse shape scaling functions for our nucleated model and the front-propagation model [2]. The front-propagation models have 1/~vz = 1.72 + 0.03 in this collapse; our nucleation model has 1/r~vz = 1.75 + 0.03 (in principle, there is no reason to believe these two should agree). The inset shows the two curves rescaled to the same height (the overall height is a nonuniversal feature). They are quantitatively different, but far more similar to one another than either is to the experimental curves in Fig. 2.13. The mean-field model apparently has a scaling function that is a perfect inverted parabola [32]. The ABBM model, which is also mean-field-like, interestingly has a different shape, that of one lobe of a sinusoid [33; equation 102]. H e r e t / T is a p a r t i c u l a r l y s i m p l e e x a m p l e of a scaling variable. N o w , if w e rescale t i m e b y a s m a l l factor B = 1/(1 - 6), w e h a v e v(T, t / T ) = (1 + b ) v ( t / T , (1 - 6)T). A g a i n , m a k i n g 6 s m a l l w e find bv = Tcgv/cgT, w i t h s o l u t i o n v - vo T b. H o w e v e r , the i n t e g r a t i o n c o n s t a n t v0 will n o w d e p e n d o n t / T , vo = V ( t / T ) , so w e arrive at the scaling f o r m

(V)(t, T) = T b V ( t / T ) ,

(2.6)

w h e r e the entire scaling f u n c t i o n V is a u n i v e r s a l p r e d i c t i o n of the theory.

128

CHAPTER 2 Random-Field Ising Models of Hysteresis

Figures 2.13 and 2.14 show the universal scaling functions V for two models and three experiments. For our model, we have drawn what are called scaling collapses, a simple but powerful way both to check that we are in the scaling regime, and to measure the universal scaling function. Using the form of the scaling equation, Eqn. (2.2), we simply plot Tb(V)(t, T) versus t/T, for a series of long times T. All the plots fall onto the same curve. This tells us that our avalanches are large enough to be self-similar. (If in the scaling collapse the corresponding plots do not all look alike, then any power laws one has measured are probably accidental.) It also provides a numerical evaluation of the scaling function V. Note that we use 1 / a v z - 1 for the critical exponent b. This is an example of an exponent equality, easily derived from the fact that

(S)(T)- f (v)(t,T)dt- f TbV(t/T)dt'~T b+l,

(2.7)

and the scaling relation (S)(T) --- T -1/r Notice that the two models and the three experiments have quite different shapes for V. How do we react to this? Our models are falsified if any of the predictions are shown to be wrong asymptotically on long length and time scales. Hence, our theory is either wrong (inapplicable to these systems) or somehow at least incomplete. Incorporating insights from careful experiments to refine the theoretical models has historically been crucial in the broad field of critical phenomena. The message we emphasize here is that scaling functions can provide a sharper tool for discriminating between different universality classes than critical exponents. Broadly speaking, most common properties that involve large length and time scales have scaling forms. Using self-similarity, one can write functions of N variables in terms of scaling functions of N - 1 variables: F(x, y, z) = z-~F(x/z ~, y/z~). In the inset to Fig. 2.1, we show the scaling collapse for the integrated avalanche size distribution (derived as Eqn. (2.10))" Dint(S, R) = S-(r+r - Rc)/S-a). (2.8) This example illustrates that scaling works not only at Rc but also near Rc; the unstable manifold in Fig. 2.8 governs the behavior for systems near the critical manifold C.

2.5 M E A S U R I N G

UNIVERSAL PROPERTIES

The simulations provide a rich variety of measured quantities, each with a characteristic universal scaling form and associated critical exponents. In

2.5 MEASURING UNIVERSAL PROPERTIES

129

this section we focus on a few key ones of particular experimental significance. 12 We will discuss the following properties obtained from the simulation: 9 the magnetization M(H~ R) as a function of the external field H; 9 the avalanche size distribution integrated over the field H, Dint (S, R)', 9 the avalanche correlation function integrated over the field

H, Gint (x~ R); 9 the distribution of avalanche durations/3(int) ~t (S, t) as a function of the avalanche size S, at R = Rc, integrated over the field H; and 9 the energy spectrum E(og) of the Barkhausen noise for various models at criticality. 2.5.1 MAGNETIZATION CURVES Unfortunately the most obvious measured quantity in our simulations, the magnetization curve M(H), is the one that collapses least well in our simulations. We start with it nonetheless. Figure 2.15 (left part) shows the magnetization curves obtained from our simulation in three dimensions for several values of the disorder R. As the disorder R is decreased, a discontinuity or jump in the magnetization curve appears where a single avalanche occupies a large fraction of the total system. In the thermodynamic limit this would be the infinite avalanche: the largest disorder at which it occurs is the critical disorder Rc. For finite size systems, like the ones we use in our simulation, we observe an avalanche which spans the system at a higher disorder, which gradually approaches Re as the system size grows. Figure 2.16 shows the slope d M / d H and its scaling collapse. By using this derivative, the critical region is emphasized as the peak in the curve, and the dependence on the parameter Me drops out. The right-hand graphs in Figs 2.15 and 2.16 show the scaling collapses of the magnetization and its slope. Clearly, in neither case are all the data collapsing onto a single curve. This would be distressing were it not for the fact that this also occurs in mean-field theory [34] at a similar distance from the critical point. Because the scaling of the magnetization is so bad, we use other quantities to estimate the critical exponents and the location of the critical point (Tables 2.1 and 2.3). Fixing these quantities, we use the collapse of the d M / d H curves to extract the rotation B, mixing the experimental variables r and h into the scaling variable h' =- h + Br (Eqn. (A13) and following discussion). Recent work suggests that much of the difficulty in collapsing 12This section follows closely the presentation in [19].

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2.5.2 AVALANCHE SIZE DISTRIBUTION In our m o d e l the spins flip in avalanches: each spin can kick over one or m o r e neighbors in a cascade. These avalanches come in different sizes.

131

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The integrated avalanche size distribution is the size distribution of all the avalanches that occur in one branch of the hysteresis loop (for H from -cx~ to oc). Figure 2.1 [3] s h o w s some of the r a w data (thick lines) in three dimensions. Note that the curves follow an approximate p o w e r law

132

CHAPTER 2 Random-Field Ising Models of Hysteresis

behavior over several decades. Even 50 per cent away from criticality (at R - 3.2), there are still two decades of scaling, which implies that the critical region is large. In experiments, a few decades of scaling could be interpreted in terms of self-organized criticality. However, our model and simulation suggest that several decades of power-law scaling can still be present rather far from the critical point (note that the size of the critical region is nonuniversal). The slope of the log-log avalanche size distribution at Rc gives the critical exponent ~ 4- r~fl6. Notice, however, that the apparent slopes in Fig. 2.1 continue to change even after several decades of apparent scaling are obtained. The cutoff in the power law diverges as the critical disorder Re is approached. This cutoff size scales as S ~ ]r] -1/~. These critical exponents can be obtained by using a scaling collapse for the curves of Fig. 2.1, shown in the inset. The scaling form for the avalanche sizes as a function of R and H is

D(S, R, H) ~ S-~lg(SrrlrI, h/r~6).

(2.9)

We can find the scaling form for the distribution of avalanche sizes integrated over H by integrating this formula, changing variables to y = h/r~, and rewriting the resulting formula in terms of S to a power times a function of SGr. For R > Re, Dint(S, R) - / D(S, R, H) dH CK

S-~D(SGr, h/r ~) dh

_ S-~r~,~f D(S~r, Y) dy = S-(~+~) (S~r)~ / D(SGr, y) dy -- S-(z+e]/~)T)~ nt)

(SGr),

(2.10)

w h e r e / 9 ~~nt) is the scaling function for the integrated avalanche size distribution (the + sign indicates that the collapsed curves are for R > Rc). We are sufficiently far from the critical point that corrections to scaling are important: as described in Ref. [34], we do collapses for small ranges of R and then linearly extrapolate the best-fit critical exponents to Rc. We estimate from this curve that the critical exponents ~ + crfl6 = 2.03 and e = 0.24. The scaling function "D(int) ~.+ (X) with X = S r~Irl is a universal prediction of our model. To facilitate comparisons with experiments, we fit a curve to the data collapse in the inset of Fig. 2.1. We have fit the scaling collapses in dimensions 3, 4, and 5 to a phenomenological form of an exponential

2.5 MEASURING UNIVERSAL PROPERTIES

133

times a polynomial. In three dimensions, our fit is 9(int) + ( x ) = e-0.789X1/e x (0.021 + 0.002X + 0.531X 2 - 0.266X 3 + 0.261X4),

(2.11)

where 1/a = 4.20. The distribution curves obtained using the above fit are plotted (thin lines in Fig. 2.1) alongside the raw data (thick lines). They agree remarkably well even far above Re. We should recall though, that the fitted curve to the collapsed data can differ from the 'real' scaling function even for large sizes and close to the critical disorder (in mean field [34] the error in the corresponding curve was about 10 per cent). The scaling function in the inset of Fig. 2.1 has a peculiar shape: it grows by a factor of 10 before cutting off. The consequence of this bump in the shape is that in the simulations it takes many decades in the size distribution for the slope to converge to the asymptotic power law. This can be seen from the comparison between a straight line fit through the R = 2.25 (billion spin) simulation in Fig. 2.1 and the asymptotic power law S -2"03 obtained from extrapolating the scaling collapses (thick dashed straight line in the same figure). A similar bump exists in other dimensions and mean field as well. Figure 2.17 shows the scaling functions in different dimensions and in mean field. In this graph, the scaling functions are normalized to one and the peaks are aligned (the scaling forms allow this). The curves plotted in Fig. 2.17 are not raw data but are fit to the scaling collapse in each dimension, as was done in the inset of Fig. 2.1. It is clear from the figure that the growing bump in the scaling curves as the dimension decreases is a foreshadowing of a zero in the scaling curve in two dimensions.

2.5.3 AVALANCHE CORRELATIONS The avalanche correlation function G(x, R, H) measures the probability that the initial spin of an avalanche will trigger, in that avalanche, another spin a distance x away. From the renormalization group description [18,36], close to the critical point and for large distances x, the correlation function is given by: 1 G(x, R, H) xa_2+~ G+(x/~(r, h)), (2.12) where r and h are respectively the reduced disorder and field, G+ (4- indicates the sign of r) is the scaling function, d is the dimension, ~ is the correlation length, and r/is called the 'anomalous dimension'. Corrections can be shown to be subdominant (see Appendix). The correlation length ~(r, h) is a macroscopic length scale in the system, which is of the order of the mean linear extent of the largest avalanches. At the critical field

134

CHAPTER 2

Random-Field Ising Models of Hysteresis

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(2.13)

where 3;+ is a universal scaling function. The avalanche correlation function should not be confused with the cluster or 'spin-spin' correlation

2.5 MEASURING UNIVERSAL PROPERTIES

135

which measures the probability that two spins a distance x away have the same value. (The algebraic decay for this other, spin-spin correlation function at the critical point (r = 0 and h = 0), is 1/x a-4+o [36].) We have mostly used, for historical reasons, a slightly different avalanche correlation function, which scales the same way as the 'triggered' correlation function G described above. Our function basically ignores the difference between the triggering spin and the other spins in the avalanche: alternatively, it calculates for avalanches of size S the correlation function for pairs of spins, and then averages over all avalanches (weighting each avalanche equally). We have checked that our correlation function agrees to within three per cent with the 'triggered' correlation function described above, for R > Re in three dimensions and above. (In two dimensions, the two definitions differ more substantially, but appear to scale in the same way.) We have measured the avalanche correlation function integrated over the field H, for R > Rc. For every avalanche that occurs between H - ~ and H -- + ~ , we keep a count of the number of times a distance x occurs in the avalanche. The spanning avalanches are not included in our correlation measurement. Figure 2.18 shows several avalanche correlation curves in three dimensions for L = 320. The scaling form for the avalanche correlation function integrated over the field H, close to the critical point and for large distances x, is obtained by integrating Eqn. (2.12):

Gint(X, R)

"

f

x d _ 21+ r !

(2.14)

(;+(x/~(r, h)) dh.

Using Eqn. (2.13) and defining u = h/[r[~,~, Eqn. (2.14) becomes: Gint(X, R) ~" ]r[fl6x -(d-2+rl)

f

G•

du.

(2.15)

The integral (Z) in Eqn. (2.15) is a function of x[r[ v and can be written as:

Z = (xlrlV) -fl(5/v ~+(xlrl v)

(2.16)

to obtain the scaling form: Gint(X, R)

xCl+B/v ~;•

(2.17)

where we have used the scaling relation (2 - q)v = f16 - fl (see [36] for the derivation). The right-hand part of Fig. 2.18 shows the integrated avalanche correlation curves' collapse in three dimensions for L = 320 and R > Re. The

136

CHAPTER 2 Random-Field Ising Models of Hysteresis

exponent v is obtained from such collapses by extrapolating to R = Rc as was done for other collapses [34]. The exponent fl/v can be obtained from these collapses too, but we found it better to use the jump in magnetization [19], which near the critical point involves several spanning avalanches. Recent work [37-39] suggests that the jump in magnetization may be dominated by only one of these spanning avalanches, and that work suggests that there may be two exponents related to our fl, one substantially larger (their fie ~ 0.15 + 0.08). The value of fl/v listed in Table 2.1 is derived exclusively from the magnetization discontinuity collapses. One of the most artificial features of our model is that the domains are arranged in a regular grid. It is thus important to check that this grid is not important for our scaling properties: if the avalanches looked noticeably cubical, we would have problems. We have looked for possible anisotropies in the integrated avalanche correlation function in two and three dimensions. The anisotropic integrated avalanche correlation functions are measured along' generalized diagonals': one along the three axes, the second along the six face diagonals, and the third along the four body diagonals. We compare the integrated avalanche correlation function and the anisotropic integrated avalanche correlation functions with each other, and find no anisotropies in the correlation, as can be seen from Fig. 2.19. Spherical symmetry emerges at the critical point on long length and time scales. 2.5.4 AVALANCHE DURATION MEASUREMENT

The scaling relation between the duration T of an avalanche (the time it takes to occur) and the linear size ~ of the avalanche defines the critical exponent z T ~ ~z. (2.18) The exponent z is known as the dynamical critical exponent. Equation (2.18) gives the scaling for the time it takes for a spin to 'feel' the effect of another a distance ~ away. Since the correlation length ~ scales like r -v close to the critical disorder, and the characteristic size S as r -1/Cr, the duration T then scales with avalanche size as:

T ,,, S r

(2.19)

In our simulation, we measure the distribution of durations for each avalanche size S. The probability Dt(S, R, H, T) that an avalanche of size S will be of duration T close to the critical field Hc and critical disorder Rc has the scaling form

Dt(S, R, H, T) ~, s-q D(t, ) (Sr

h/lrl ~ , T/Sr

(2.20)

137

2.5 MEASURING UNIVERSAL PROPERTIES .

.

.

.

.

.

.

.

!

.

.

.

.

.

.

.

.

!

10 4 cO .n O~ m

~ 102 o co c-

<~> ] 00

10 - 2 10 o

101

10 2

Distance (x)

Anisotropies in the avalanche correlation function. Four curves are shown on the graph: one is the avalanche correlation function integrated over the field H (as in Fig. 2.18), while the other three are measurements of the correlation along the three axes, the six face diagonals, and the four body diagonals. Avalanches involving more than four spins show no noticeable anisotropy: the critical point appears to have spherical symmetry. The same result is found in two dimensions. FIGURE 2.19

w h e r e q = r + rrvz, a n d is defined such that

~-oo~oo Dt(S, R, H, T) dH dt oo

__ S-(~+rrfl&) /)(,'lnt)(SrrITI)

(2.21)

w h e r e ~)('mt) w a s defined in the integrated avalanche size distribution section. The avalanche time distribution integrated over the field H, at the critical disorder (r - 0) is: DI int) (S, T) ~, t -(*+afla+avz)/Crvz D~int) (T/S avz)

(2.22)

as obtained from Eqn. (2.20) [34]. Figure 2.20 s h o w s the avalanche time distribution integrated over the field H for different avalanche sizes, a n d a collapse of these curves using the above scaling form.

138

CHAPTER 2 Random-Field Ising Models of Hysteresis 9

"

i

i

"

I

i

I

.

.

.

.

I

10 -2 cO ..Q . m i.,.

-o

o

10_4

E F--

10 -6 i

i

i

,

i

i

10 2

i

i

I

10 3

(a)

Time

10 8

v

GI 10 6

+

10 4 |

0.0

I

2.0

(b)

,

I

4.0

t S --~z

FIGURE 2.20 Distribution of avalanche durations for avalanches of fixed size S, for various sizes S, and their scaling collapse [19].

2.5.5 ENERGY SPECTRUM The energy spectrum of the voltage V(t) as a function of time for our model (given by adding up the voltage traces of each avalanche, as in Fig. 2.5),

E(co) = /

e -i~ (V(t)V(t + r)) = IW(~)l 2

(2.23)

2.5 MEASURING UNIVERSAL PROPERTIES

139

is a c o m m o n l y measured experimental quantity. 13 There are two distinct contributions to this power spectrum: (i) at high frequencies, the incoherent sum of the power spectra Einc of the individual avalanches, and (ii) at low frequencies, a term representing the correlations between avalanches. As the system is forced more and more slowly, the internal dynamics of the avalanches is unchanged but their separation in time increases, so these two contributions separate. Typical experiments seem to be dominated by the contribution (i) from the internal dynamics within an avalanche [13]. In this section, the results we discuss are valid for all three models of hysteresis. 14 See also [33] in this volume for a more complete discussion of our power spectrum theory and the experimental context. There have been a n u m b e r of naive arguments for the power law for E(Co), stretching back to 1972 [3,18,31,40], which yield a power law which is only valid if r > 2: Endive(Co) cK Co-(3-3)/r~vz (only for r > 2).

(2.24)

Let Smax be the typical largest avalanche in the system, say, because the system is finite in size. At root, these derivations fail for r < 2 because f0Smax SD(S) dS

=

fSma• dO

51-3

dS

2--3x = Sma

(2.25)

diverges. Since there are S spins in each avalanche, it w o u l d seem that Eqn. (2.25) should represent the n u m b e r of spins in the system. This is roughly correct for r > 2, but for r < 2 the amplitude of the power law D(S) = DoS -3 depends on this cutoff Do "" Sm2ax9 Continuing a detailed analysis [13], we get a different scaling form Endive(C0) cx

Co-1/crvz

(for r < 2).

(2.26)

It is not a surprise that this new prediction fixes a n u m b e r of discrepancies between theory and experiments [13]. Figure 2.21 shows the power spectrum P(co) (equal to E(co) divided by the duration of the measurement), together with four other curves, for both the front-propagation model and the infinite-range model [13]. The top two curves are the naive power law (Eqn. (2.24), obviously not correct) 13The power spectrum is E(CO)divided by the integration time. 14In the short-range model, we concentrate on the case where we measure only near Rc and Hc: we do not integrate over the loop. Integrating over the loop yields q: very close to two, where both forms of the scaling should compete.

140

CHAPTER 2 Random-Field Ising Models of Hysteresis .

.

.

.

.

.

.

.

.

.

.

.

j

10 5

.

.

.

.

.

.

.

.

.

.

.

~

_

% ,L

9 9 9 Oooooo

%" x~,...,,.~.~.~,,.,,.h,

10o

-~_

9 P(o, Us)~s 03-1/ovz 03(x-3)/ovz

10-5

.....

.......

,

i

10-3 . . . . . .

10 4

10 2

_

. . . . . . . .

i

10-2

--''.

i

~k

",~t,,!,

9

~-. 9 ...

. . . . . . .

|

'

'

",

~

'i~

i

'

]

~

~....

9....

i

10-1

\,

~',,,,

i

I

N,~,

10o

- s(]ff)

"",.."'--2"'--."~ ""~

P(03) 10 -2

9 P(031S)/S

-1fill

"'"

-

03 1/mcz 03(x-O)/o'vz

10--4

. . . . . . . .

10-3

,

10-2

,

,

,

,

,

,,,!

,

,

10-1

FIGURE 2.21 Power spectrum scaling and relatives. Top: short-range model; bottom: mean field (long-range dipolar forces [13]).

and the correct law (Eqn. (2.26)). We compare with the relation between the avalanche size S and the duration T (Eqn. (2.19)), which has the same combination of critical exponents S ~ (1/T) -1/rrvz. We also show that the energy spectrum of individual large avalanches is proportional to S and has the same power law in frequency o9 as that of the entire time series, so w e plot P(ogIS)/S for a variety of avalanche sizes S.

2.6

141

EXPONENT RELATIONS

TABLE 2.1 M e a s u r e d universal critical exponents. Values for the exponents extracted from scaling collapses in three, four, and five dimensions. The m e a n field values are calculated analytically [14,36]. v is the correlation length exponent and is f o u n d from collapses of avalanche correlations, n u m b e r of s p a n n i n g avalanches, and m o m e n t s of the avalanche size distribution data. The exponent 0 is a m e a s u r e of the n u m b e r of s p a n n i n g avalanches and is obtained from collapses of those data. ('~ + o']~c~ - 3)/o'v is obtained from the second m o m e n t s of the avalanche size distribution collapses. 1/(7 is associated with the cutoff in the p o w e r law distribution of avalanche sizes integrated over the field H, while "c + o'flc~ gives the slope of that distribution. "~ is obtained from the b i n n e d avalanche size distribution collapses [19]. d + ~ / v is obtained from avalanche correlation collapses and fl/v from magnetization discontinuity collapses, o'vz is the exponent combination for the time distribution of avalanche sizes and is extracted from those data. Error bars are based on variations in the results based on different approaches to the analysis, statistical fluctuations are typically smaller. Measured exponents 1/V 0 ('C + Cr~C~- 3)/CRY 1/0" "C+ CrflC~ "C

d 4- fl/v fl/v (Tvz

3D 0.71 4- 0.09 0.015 4- 0.015 -2.90 4- 0.16 4.2 4- 0.3 2.03 4- 0.03 1.60 + 0.06 3.07 4- 0.30 0.025 + 0.020 0.57 4- 0.03

4D

5D

Mean field

1.12 4- 0.11 0.32 4- 0.06 -3.20 4- 0.24 3.20 4- 0.25 2.07 4- 0.03 1.53 4- 0.08 4.15 4- 0.20 0.19 4- 0.05 0.56 4- 0.03

1.47 4- 0.15 1.03 4- 0.10 -2.95 4- 0.13 2.35 4- 0.25 2.15 4- 0.04 1.48 4- 0.10 5.1 4- 0.4 0.37 4- 0.08 0.545 4- 0.025

2 1 -3 2 9/4 3/2 7 (at dc = 6) 1 1/2

2.5.6 T A B L E S O F R E S U L T S

Here we summarize, from [19], our numerical estimates of the universal critical exponents and various nonuniversal quantities used in the scaling collapses above. 15

2.6 E X P O N E N T RELATIONS In the following sections we list various exponent relations for the nonequilibrium RFIM, for which we give detailed arguments in [18,41]. 15Some of the scaling collapses shown use exponents slightly different from those in the tables: see [19].

CHAPTER 2 Random-Field Ising Models of Hysteresis

142

TABLE 2.2 Calculated universal critical exponents. Values for exponents in three, four, and five dimensions that are not extracted directly from scaling collapses, but instead are derived from Table 2.1 and the exponent relations (see Section 2.6). The mean field values are obtained analytically [14,36]. Both o'/Jc~ and/Jc~ could have larger systematic errors than the errors listed here [19]. Recent work [37] on small systems, but with sophisticated scaling analysis, suggests a larger value for ~. Calculated exponents o'/Jc~ ~c~ fl cry rI = 2 + (fl - flc~)/v

3D 0.43 4- 0.07 1.81 4- 0.32 0.035 4- 0.028 0.34 4- 0.05 0.73 4- 0.28

4D

5D

Mean field

0.54 4- 0.08 1.73 4- 0.29 0.169 4- 0.048 0.28 4- 0.04 0.25 4- 0.38

0.67 4- 0.11 1.57 4- 0.31 0.252 4- 0.060 0.29 4- 0.04 0.06 4- 0.51

3/4 3/2 1/2 1/4 0

TABLE 2.3 Nonuniversal scaling variables. Numerical values for the critical disorders and fields, and the rotation parameter B (Eqn. (A13) and subsequent discussion), in three, four, and five dimensions extracted from scaling collapses. The critical disorder is obtained from collapses of the spanning avalanches and the second moments of the avalanche size distribution. The critical field is obtained from the binned avalanche size distribution [19] and the magnetization curves. Hc is affected by finite sizes, and systematic errors could be larger than the ones listed here. The mean field values are calculated analytically [14,36]. The rotation B is obtained from the d M / d H collapses. 3D Rc Hc B

2.16 4- 0.03 1.435 4- 0.004 0.39 4- 0.08

4D

5D

Mean field

4.10 4- 0.02 1.265 4- 0.007 0.46 4- 0.05

5.96 4- 0.02 1.175 4- 0.004 0.23 4- 0.08

0.79788456 0 0

2.6.1 EXPONENT EQUALITIES The exponents introduced above are related by the following exponent equalities: ]~- ]/~ = (~ - 2)/~ if ~ < 2, (2.27) ( 2 - ~/)v = / / 6 -

r,

(2.28)

/ / - ~(d - 4 + f/),

(2.29)

v

and = (d - 2r/4- ~/)/(d - 4 4- ~/).

(2.30)

(The latter three equations are not independent and are also valid in the equilibrium random-field Ising model).

143

2.6 EXPONENT RELATIONS 2.6.2 T w o VIOLATIONS OF HYPERSCALING

In the nonequilibrium RFIM there are two different violations of hyperscaling. (i) In [18,41], we show that the connectivity hyperscaling relation 1/rr = d v - fl from percolation is violated in our system. There is a new exponent 0 defined by 1/rr = (d - O)v - fi

(2.31)

with Ov = 1/2 - e/6 + 0032) and Ov = 0.021 + 0.021 in three dimensions. 0 is related to the n u m b e r of system-spanning avalanches observed during a sweep through the hysteresis loop: see also [37]. (ii) As we discuss in [18] there is a m a p p i n g of the perturbation theory for our problem to that of the equilibrium random-field Ising model to all orders in e. From that m a p p i n g we deduce the b r e a k d o w n of an infamous ('energy')-hyperscaling relation fl + f16 = ( d - 0)v,

(2.32)

with a new exponent 0, which has caused m u c h controversy in the case of the equilibrium random-field Ising model. In [18,41] we discuss the relation of the exponent 0 to the energy output of the avalanches. The e expansion yields 0 = 2 to all orders in e. Nonperturbative corrections are expected to lead to deviations of 0 from 2 as the dimension is lowered. The same is true in the case of the equilibrium RFIM. The numerical result in three dimensions is 0 = 1.69 + 0.28. (In the three-dimensional equilibrium RFIM [42] it is 0eq - 1.5 -}- 0.4). Another strictly perturbative exponent equality, which is also obtained from the perturbative m a p p i n g to the random-field Ising model is given by ~/= q. (2.33) It, too, is expected to be violated by nonperturbative corrections below six dimensions. 2.6.3 EXPONENT INEQUALITIES In [18,41] we give arguments for the following two exponent inequalities. 16 v/flf>~2/d,

(2.34)

16From the normalization of the avalanche size distribution D(s, r, h) it follows that "c > 1.

144

CHAPTER 2 Random-Field Ising Models of Hysteresis

which is formally equivalent to the 'Schwartz-Softer' inequality, ~<2v, first derived for the equilibrium random-field Ising model, and

v>~2/d,

(2.35)

which is a weaker bound than Eqn. (2.3) so long as f16>~1, as appears to be the case both theoretically and numerically, at least for d~>3.

2.7 F I N I T E S W E E P RATE Originally the nonequilibrium RFIM was studied in the adiabatic limit, where the external field is kept constant during each spin-flip avalanche and only increased between subsequent avalanches. In real experiments, however, the driving field H is typically increased at a finite rate f2 such that H = f2t where t is time. This finite rate allows for new avalanches to be triggered by the increasing external field while an earlier avalanche is still propagating. In [43,44] the effect of finite field sweep rates on the power spectra and avalanche size distributions is discussed in detail for a large class of systems with crackling noise. In particular, it is asked how the scaling behavior of the avalanche size and duration distribution and the power spectra of Barkhausen noise or crackling noise in general depend on the field sweep rate f2. One of the results is an exponent inequality as a criterion for the relevance of adding a small driving rate f2 > 0 to the adiabatic case f2 ---> 0. If in the adiabatic case the avalanche duration distribution scales as D(T) ~ T -a with a = 2 (such as in the ABBM mode] or the mean-field version of the RFIM), then at (small enough) finite field sweep rates the corresponding noise 'pulse' duration distribution is expected to scale as D(T, f2) ~ T 2-a(H)Cf~ where a(H) and C are nonuniversal constants [45-481. If, however, in the adiabatic limit the exponent ~ is either greater or smaller than two, then at (small) finite field sweep rates the pulse duration distribution exponent remains the same as in the adiabatic limit. Note that in the case a < 2 especially, f2 has to be particularly small in order to be able to still see distinct pulses even at finite field sweep rates, since at higher sweep rates the system will very quickly develop runaway events. In [43,44] quantitative criteria for the meaning of 'small sweep rate' are given. Also, the zero temperature nonequilibrium RFIM and recent variants are used to numerically test the analytic results, which are expected to be applicable to a much larger class of systems with crackling noise (the exact conditions are discussed in [43,44]). It is found that the results agree well both with simulations and recent experiments on Barkhausen noise in various soft magnetic materials with and without applied stress (with

2.8 SUBLOOPSAND HISTORY-INDUCED CRITICAL BEHAVIOR

145

< 2 and a = 2, respectively [11]). A brief review of other, related studies of the effects of finite field sweep rate is also given in [43].

2.8 S U B L O O P S A N D H I S T O R Y - I N D U C E D BEHAVIOR

CRITICAL

One of the characteristic features of magnetic hysteresis is the subloops, seen as the external field is changed up and down an amount insufficient to saturate the magnetization (Fig. 2.2).

2.8.1 RETURN-POINT MEMORY Much attention has been paid to modeling these subloops with the Preisach model [49]. Preisach models are quite different in spirit to ours: they represent the system as a large number of uncoupled hysteretic two-state domains, and fit the distribution to the observed behavior. They are able to model a hysteretic system if it possesses return-point memory [50], also known as wiping out [51]. Return-point memory states that the magnetization after a subloop rejoins a larger loop (perhaps itself a subloop) equals the magnetization at which the subloop left the outer loop. That is, if the subloop represents an excursion downward from a local maximum external field H, then when H returns to its previous maximum the magnetization M returns to its previous value - remembering its previous magnetization on returning, and wiping out all effects of the excursion. Return-point memory states that the subloops in Fig. 2.2 should close perfectly, without a gap and without crossing themselves (as observed). Indeed, our interacting, three-dimensional, disordered models exhibit the return-point memory in an even stronger form. Upon re-joining the larger loop, the entire state of the system is microscopically identical to the state it had when it left the larger loop (a microscopic return-point memory, as opposed to a macroscopic memory constraining only the net magnetization). In [14], we showed in great generality that return-point memory should be true of any system with: (i) A partial ordering of states. Here, one microstate A is 'ahead' of another B, A > B, if every spin in A is greater than or equal to the corresponding spin in B. (ii) No passing. That is, the dynamics preserves the partial ordering. (iii) Adiabatic. The external field is raised and lowered slowly enough that the system does not lag behind.

146

CHAPTER 2 Random-Field Ising Models of Hysteresis

Return-point memory can be quite remarkable: in our model (and in some experimental systems) repeating a subloop plays back precisely the same Barkhausen noise. Indeed, the return-point memory can be used as a way of retrieving analog magnetic memories, that is (at least theoretically) significantly superior to measuring the remanent magnetization [52]. On the other hand, the absence of exact (microscopic) return-point memory has also been observed [53-55], and there are various interesting experiments testing for the reproducibility of magnetic avalanches [56-58] and for microscopic return-point memory and complementary-point memory in magnets at various disorders [59].17 In [59] the authors experimentally study the influence of disorder on major loop return-point memory and complementary return-point memory in Co /Pt samples with varying interfacial roughness and find with increasing disorder the onset and saturation of both return-point memory and complementary-point memory. Disorder dependence of return-point memory versus reptation (i.e. gradual subloop closure [60,61]) has recently been reported for the zerotemperature random-coercivity model with antiferromagnetic-like interactions [62]. A related memory effect, recently reported for the hysteresis of spin glasses, is the reversal-field memory effect [63]. Katzgraber et al. show that this memory effect emerges in the nonequilibrium Edwards-Anderson spin glass (EASG) when the magnetic field is first decreased from its saturation value and then increased again from some reversal field HR. The authors find that EASG exhibits a singularity at the negative of the reversal field, -HR, in the form of a kink in the reversal of the magnetization of the reversal curve. They show that this memory effect is due to a local spin-reversal symmetry of the Hamiltonian. This symmetry and thus the reversal-field memory effect is present in spin glasses where the disorder is due to random couplings between spins. In general it is not expected in systems with random magnetic fields such as the RFIM, since there the disorder breaks local spin-reversal symmetry. As an extension to memory effects for a driving field that varies in two dimensions, a vector form of wipe-out memory, for situations when the magnetic field varies in both direction and magnitude, has been suggested based on the two-dimensional vector Preisach model in [64], and has been experimentally tested in [65].

17The complementary point memory relates the magnetic domains at one point on the major hysteresisloop to the domainsat the complementarypoint on the major loop the same and subsequent cycles.

2.8 SUBLOOPSAND HISTORY-INDUCED CRITICAL BEHAVIOR

147

0.5

-0.5

(a)

-1

-3

-2

'1

0

1

-'1

6

1

-

;2

3

~,-

3

0.5

0

-0.5 J

-% (b)

FIGURE 2.22

-

'~

H/J

Saturation hysteresis loops with concentric inner subloops. Top:

R = 2.225J > Rc; bottom: R = 2.1 < Rc [66]. 2.8.2 CRITICAL BEHAVIOR IN SUBLOOPS

The disorder-induced critical scaling for hysteresis loops in the RFIM that we have discussed so far all refers to the saturation hysteresis loop. However, often it is experimentally impractical to take magnets to their saturation point due to the large magnetic fields required, so the behavior of subloops (Fig. 2.22 [66]) is of great interest to experiments and applications. The RFIM can be used to model subloops, and in one dimension [67] and on a Bethe lattice [68,69] they have been computed exactly [70]. The magnetization curves of subloops have been collapsed near the demagnetized state using Rayleigh's law [12,71]. Interestingly, in the RFIM not only the saturation hysteresis loops but also subloops reflect the critical point

148

CHAPTER 2 Random-Field Ising Models of Hysteresis

described above, and at the critical disorder one finds history-induced critical scaling. Concentric inner subloops resemble rescaled saturation loops at effectively higher (possibly correlated) disorder. Avalanche size distributions for the inner subloops, too, look similar to rescaled avalanche size distributions of saturation loops at higher disorder. We have studied universal properties of this history-induced critical behavior using numerical simulations of systems with more than 100 000 000 spins and Widom scaling methods and extracted universal scaling exponents and scaling functions. These predictions are expected to apply to experiments on hysteresis and Barkhausen noise in hard magnetic materials. To model soft magnets on the other hand, we added demagnetizing fields to the model and obtained single-domain-wall propagation, as described in [13]. As a result the system was always critical, even for the subloops. The results of this study are discussed in [66,72]. A special, related kind of subloops so-called 'first-order reversal curves' (FORC) - have proven very useful in distinguishing between different magnetic materials, see [73]. 2.8.3 DEMAGNETIZATION CURVES

The RFIM also predicts disorder-induced critical behavior in the demagnetization curve, or initial magnetization curve, and the associated Barkhausen noise. Ferromagnetic materials with a remanent magnetization can be demagnetized by applying an oscillating magnetic field with amplitude slowly decreasing from a large initial value to zero. Sometimes the final state is also termed AC-demagnetized state. The oscillating external field with decreasing amplitude takes the system through concentric subloops. The line connecting the tips of the subloops is known as the normal or initial magnetization curve, or demagnetization curve. It is to be distinguished from the virgin curve, which is obtained by thermal demagnetization [50,74,75]. One can use the RFIM to simulate the demagnetization curve for different amounts of disorder, especially near the critical disorder mentioned above for the saturation loop. It turns out that this curve reflects much of the scaling behavior found for the saturation hysteresis loop. This may be surprising, since the metastable states encountered in the demagnetization curve are completely different from those of the saturation loop. In [76] the model is briefly described and the necessary exponent relations are derived, which show that the exponents for the demagnetization curve can actually be derived from those of the subloops. Predictions for the exponents from simulations of up to one million spins are presented. The demagnetization curve in the presence of long-range demagnetizing fields is also discussed, with resulting predictions for experiments on demagnetization curves and associated Barkhausen noise in soft magnetic materials.

2.9 REALEXPERIMENTS

149

For a discussion of the Rayleigh law and scaling of the demagnetization curve at small magnetic fields, see [69,71]. Zarand et al. recently proposed a new optimization method to search for ground states, which is based on the demagnetization procedure [77].

2.9 R E A L E X P E R I M E N T S 2.9.1 MAGNETIC HYSTERESIS LOOPS There are a few recent experiments on hysteresis and crackling noise where disorder is tuned and evidence of a phase transition related to that of the nonequilibrium RFIM is reported. In [78], Berger et al. describe a transition from smooth magnetic hysteresis loops to those with a jump in the magnetization in ultrathin epitaxial Gd-films with in-plane magnetization upon annealing the sample at successively higher annealing temperatures. The authors show that annealing at higher temperatures increasingly reduces the disorder in the crystal structure of the films. In [79-81] Berger et al. report a similar transition for thin Co/CoO films. In this system the antiferromagnetic CoO layer allows a reversible tuning of the magnetic disorder felt by the ferromagnetic Co layer, by simple temperature variation. The authors are able to extract scaling collapses for the magnetization hysteresis loops at different disorders, and find the values of the associated scaling exponents to be fl = 0.022 q- 0.006 and fl~ = 0.30 + 0.03. Note that this magnetic system is effectively two-dimensional. Incidentally, almost the same exponent values (fl = 0.03 4- 0.01 and fl~ = 0.4 4- 0.1) are reported by Marcos et al. in [82] for the magnetic hysteresis loops of Cu-A1-Mn alloys with different Mn content at low temperatures. The loops are smooth above a certain temperature, but exhibit a jump in the magnetization below that temperature. The authors perform a scaling analysis near this critical point with temperature as the effective 'disorder' tuning parameter. They point out that, contrary to the thin Co/CoO bilayers in Berger et al. [79], this system is effectively threedimensional. For these three systems it would be very interesting to also see a scaling analysis of the associated Barkhausen noise, since from simulations we know that the exponents extracted from avalanche size distributions and noise power spectra are often more reliable than those obtained just from magnetization curves alone [18]. For these three systems, so far, only the scaling of the magnetization curves has been reported.

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2.9.2 DISORDER EFFECTS ON BARKHAUSEN NOISE

Disorder effects on crackling noise (rather than hysteresis loops) has been seen in other systems. Barkhausen noise is commonly used for nondestructive testing - in engineering contexts we find careful studies of the effect of the amount of carbon content in steel on Barkhausen noise. One might speculate that varying the carbon content could be similar to varying the effective disorder in the system. It would be interesting to see a systematic scaling analysis for the available experimental data. In an early study [40] Lieneweg and Grosse-Nobis found that the distribution of Barkhausen pulse areas integrated over the hysteresis loop of an 81 per cent Ni-Fe wire was well described by a power law (with exponent ~ + rT]~ ranging from 1.73 to 2.1) up to a certain cutoff size. Annealing the sample at various annealing temperatures they found that the cutoff appeared to be smaller at higher annealing temperatures. It would be interesting to see whether the cutoff grows to take a (system-size dependent) maxim u m value at a critical annealing temperature Tann and decreases again at higher and lower annealing temperatures, as one would expect if the RFIM applies with temperature as the effective disorder-tuning parameter in this system. (In leading approximation, near Tann they should be linearly related to each other.) Near Tann the Barkhausen pulse area distribution should then be described by a scaling form that would allow a scaling collapse of all distributions onto one single curve for appropriate rescaling of the axes. Again, potentially universal critical exponents could be extracted from such a collapse. They would be predicted by the RFIM avalanche critical exponents if that model is in the same universality class. There are other experiments that reveal power-law decays for Barkhausen pulse-size distributions in various samples, as reviewed for example in [1] and in the article by Durin and Zapperi in this book [33]. Many of these are measurements in soft magnetic materials where long-range interactions are important and the main mechanism for Barkhausen noise is domain-wall propagation rather than new-domain nucleation. Disorder has usually not been varied in these experiments. Models predict that these systems naturally tend to operate close to the underlying critical domain wall depinning transition and thus produce power-law pulse-size distributions regardless of the underlying disorder, (at least as long as it is less than a critical value above which the domain wall may have overhangs [5,8-12]). The disorder-induced critical point of the nonequilibrium RFIM is expected to be relevant for hard magnetic materials instead, where either the disorder is very strong compared with the effects of long-range dipolar interactions, or the geometry of the sample and experiment is chosen such that the effects of long-range dipolar fields are small (as for thin

2.9 REALEXPERIMENTS

FIGURE 2.23

Fractal spatial structure of an avalanche.

151

This moderate-sized

avalanche contains 282 785 domains [1]. films with in-plane applied magnetic field, or for thin wires with parallel applied field). 2.9.3 IMAGING MAGNETIC AVALANCHES AND STATES The avalanche structures predicted by the theoretical models are visually interesting (Fig. 2.23); in our model they have fractal dimensions 1/r~v close to, but probably less than, three. Exciting experiments probing these spatial structures are coming on line. Recently, interesting experiments imaging avalanches in magnetic films have been reported. In [83] Kim, Choe, and Shin report full-field magnetooptical observations (via Kerr effect) of Barkhausen avalanches in Co polycrystalline thin films at criticality, i.e. their imaged avalanches are due to single domain-wall propagation rather than domain nucleation. In this experiment many avalanches extend far beyond the imaged region, so that their measured exponent appears to be an exponent combination of the exponents computed in the models. In addition to their results it would be interesting to also see time-resolved images of entire avalanches and

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direct measurements of the roughness exponent of the propagating domain wall, to compare with predictions from models for magnetic domain-wall propagation. The magnetization reversal dynamics of epitaxial Fe films grown on GaAs(001) and the field-dependent evolution of domain structure at various fields has also been studied using the magneto-optic Kerr effect, see for example [84]. In [85] Schwarz et al. report on visualization of the Barkhausen effect by magnetic force microscopy for a granular thin film with perpendicular anisotropy. The authors are able to localize individual Barkhausen volumes and distinguish them as either newly nucleated or grown by domain-wall propagation. They find a Gaussian size distribution of nucleated Barkhausen volumes indicating an uncorrelated random process, and a power-law distribution for grown Barkhausen volumes, reflecting critical behavior during domain-wall motion. Another very interesting experimental approach is X-ray speckle metrology [59]. In a recent preprint Pierce et al. use this method to study correlations between microstates with different histories and test for microscopic return-point memory of subloops in magnetic systems. 2.9.4 RANDOM BONDS AND RANDOM ANISOTROPY

Renormalization-group treatments, symmetry arguments [18], and simulations [86,87] show that the hysteretic, zero-temperature random-bond Ising model with ferromagnetic mean bond strength is expected to be in the same universality class as the nonequilibrium RFIM. Vives et al. also point out that their numerical scaling exponents in two dimensions of the two models also have overlapping error bars with the corresponding exponents of Blume-Emery-Griffiths models with disorder, in which the spins take three values 4-1, 0 [87]. An analysis of the short-time dynamics of first-order phase transitions in a disordered system, using the randombond Ising model, is given in [88]. The diluted ferromagnetic random Ising model [89] has both nonuniform bond strengths that reflect dilution due to nonmagnetic impurities, and random magnetic fields. Symmetry arguments [18] suggest that a model with uncorrelated random (ferromagnetic) bonds and random fields should again be in the same universality class as the RFIM, since it has no new symmetries compared with the RFIM. Similarly, daSilveira and Kardar developed an O(N) vector spin model with random fields and showed through a renormalization-group calculation that, in general, it can display the same continuous-phase transition in the hysteresis loop as the RFIM [90], i.e. the two critical points are in the same universality class. 18 The reason is that the applied external field breaks any added rotational symmetry [18] so that the behavior on long length 18Theyalso find a new universalityclass, see Section2.10.

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scales is determined by the same symmetries in both models. Similarly the random anisotropy model of [91] was shown numerically to be in the same universality class as the RFIM. This is again as expected, since the external applied field not only breaks rotational symmetry but also time-reversal symmetry. Perhaps somewhat surprisingly, a transition from hysteresis loops with a jump to those without, like the one of the RFIM, is also seen in a plastic depinning model for charge density waves [92]. If in the randomfield Ising model a certain fraction of the bonds between neighboring spins is enhanced to a very large value, one obtains a simple model for exchange bias [93]. Hysteresis loops observed for the Ising glass with zero ferromagnetic mean bond strength are smooth and continuous [94,95] - qualitatively this agrees with experimental hysteresis loops seen, for example, in AuFe systems [96]. For Ising glasses with strong ferromagnetic mean bond strength (compared with the width of the distribution of bond strengths, which is the disorder), a jump in the magnetization curve appears, just as in the low-disorder phase of the RFIM. The hysteresis loops resemble those seen for more concentrated AuFe systems [96] as well as CuMn and AgMn systems [97,98]. A brief review of these and other glassy spin systems, such as Ising spin glasses with long-range interactions and Heisenberg spin glasses with long-range interactions in both theory and experiment, is given, for example, in [95]. An interesting hysteresis model with rather different properties than ours, based on an infinite-range spin glass (Sherrington-Kirkpatrick model), is reported to show self-organized criticality; see [94] for details. 2.9.5 OTHER DYNAMICS AT HIGH SWEEP RATES

Much interesting work has been done in both theory and experiment on dynamic hysteresis loops at finite-field sweep rates in the presence of temperature. At field sweep rates higher than a critical value that depends on sweep amplitude and temperature, the system transitions to a symmetrybroken phase with a flatter hysteresis loop, centered around a nonzero average magnetization. Similar dynamic transitions also are seen for pulsed and stochastically varying fields. Chakrabarti and Acharyya published a review article on both models and experiments in this regime [99]. Some of the addressed questions deal with scaling of the hysteresis loop area with field sweep frequency, amplitude, and temperature. While, in the initial approaches to these issues, quenched randomness in the system was neglected, there are recent model approaches and experiments studying dynamic hysteresis of a moving domain wall in the presence of quenched disorder and temperature at finite-field sweep rates [100,101].

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2.9.6 NONMAGNETIC NOISY HYSTERETIC SYSTEMS In this section we briefly discuss a broad range of related work on avalanches and hysteresis in nonmagnetic systems. We start with a general overview, and then focus on a few particular systems.

Systems with Avalanches Magnets crackle (Barkhausen noise) when the external field is ramped upward slowly. Many systems exhibit similar crackling noise, when pushed slowly they respond with discrete events of a broad range of sizes. 19 The earth responds [102] with violent and intermittent earthquakes as two tectonic plates rub past one another. A piece of paper [4] (or a candy wrapper at the movies [103,104]) emits intermittent, sharp noises as it is slowly crumpled or rumpled. Just like the magnetization avalanches responsible for Barkhausen noise in magnets, these individual events span many orders of magnitude in size. Indeed, the distribution of sizes follows a power law with no characteristic size scale. In the past decades, scientists in a variety of fields have been making rapid progress in developing models and theories for understanding this sort of scale-invariant behavior in driven, nonlinear, dynamical systems. Interest in these sorts of phenomena goes back several decades. The work of Gutenberg and Richter [102] in the 1940s and 1950s established the well-known frequency-magnitude relationship for earthquakes that bears their names. A variety of many-degree-of-freedom dynamical models [105-116], with and without disorder, have been introduced in the years since to investigate the nature of slip complexity in earthquakes. More recent impetus for work in this field came from the study of the depinning transition in sliding charge-density wave (CDW) conductors in the 1980s and early 1990s [117-123]. Interpretation of the CDW depinning transition as a dynamic critical phenomenon sprung from Fisher's early work [117,118], and several theoretical and numerical studies followed. This activity culminated in the RG solution by Narayan and Fisher [120] and the numerical studies by Middleton [121] and Myers [122], which combined to provide a clear picture of depinning in CDWs and opened the doors to the study of other disordered, nonequilibrium systems. Bak, Tang, and Wiesenfeld inspired much of the succeeding work on crackling noise [124,125]. They introduced the connection between dynamical critical phenomena and crackling noise, and they emphasized how systems may naturally end up at the critical point through a process of self-organized criticality. (Their original model was that of avalanches in 19This subsectionfollow closelythe presentation in Ref. [1],with some added references.

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growing sand piles. Sand has long been used as an example of crackling noise [126,127]. However, it turns out that real sand piles do not crackle at the longest scales [128,129]). Researchers have studied many systems that crackle. Simple models have been developed to study bubbles rearranging in foams as they are sheared [130]; biological extinctions [131];20 fluids invading porous materials and other problems involving invading fronts [6,25,26,132-134]; 21 the dynamics of superconductors [135-137], and superfluids [138,139]; sound emitted during martensitic phase transitions [140]; fluctuations in the stock market [141-143]; solar flares [144]; cascading failures in power grids [145-147]; failures in systems designed for optimal performance [148-150]; group decision making [151]; crackling noise in mammalian lungs [152]; and fracture in disordered materials [153-158]. These models are driven systems with many degrees-of-freedom, which respond to the driving in a series of discrete avalanches, spanning a broad range of scales, that we are calling crackling noise. There has been healthy skepticism by some established professionals in these fields to the sometimes grandiose claims by newcomers proselytizing for an overarching paradigm. But, often confusion arises because of the unusual kind of predictions the new methods provide. If our models apply at all to a physical system, they should be able to predict all behavior on long length and time scales, independent of many microscopic details of the real world. This predictive capacity comes, however, at a price: our models typically do not make clear predictions of how the real-world microscopic parameters affect the long-length-scale behavior. In this paper and in [1] we have provided an overview of the renormalization group [159-163] that many researchers use to understand crackling noise and which is perhaps the most impressive use of abstraction in science. We now turn to some nonmagnetic systems that have been particularly well studied. Martensites

Hysteresis with crackling noise, and in certain cases even microscopic return-point memory, has long been seen in ferroelastic materials, such as shape memory alloys or martensites [164,165]. A martensitic transformation is a diffusionless first-order phase transformation where the lattice distortion is mainly described by a homogeneous shear. Many metals and 2~ models are controversial [187,188], they ignore catastrophic external events like asteroids. 21 The random-field model we describe was invented in this context [6,132]; as described in Section2.2.

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alloys with a bcc structure will, upon cooling (or under strain), undergo this transition to a low-temperature close-packed structure. In athermal martensites, such as Cu-Zn-A1, thermal fluctuations are not relevant- temperature acts as an external driving field. Similar to ferromagnets' ramping magnetic field, the martensitic transition takes place as a sequence of avalanches as the temperature is swept. The analog of Barkhausen noise is actual noise: acoustic emission due to elastic waves in the ultrasonic range, generated by propagating domain walls during the avalanches. Experimental studies of the acoustic emission generated during a thermally induced martensitic transformation of a Cu-Zn-A1 single crystal revealed an apparent absence of characteristic scales in the distribution of avalanches [166]. After an initial number of hysteresis cycles (ramping temperature), the system reaches a final attractor yielding two decades of power-law scaling in the time distribution of acoustic emission pulse durations. The authors argue that the cutoff at large times is probably due to experimental limitations (amplifier cutoff), and interpret the observed scaling behavior in terms of self-organized criticality. Of course, the system could have a tunable parameter that lies close to the critical point (two decades of power-law scaling arises in a range 50 per cent away from our critical point). In a more recent study, P6rez-Reche et al. [167] study the evolution of the kinetic features of the martensitic transition in a Cu-A1-Mn single crystal under thermal cycling. The authors use several experimental techniques, including optical microscopy, calorimetry, and acoustic emission, to perform an analysis at multiple scales. Focussing especially on avalanches, they find that there are significant differences between the kinetics at large and small scales during the initial cycles. Upon repeated temperature cycles, however, the system evolves from displaying a supercritical avalanche size distribution to a critical power-law distribution. In the language of the RFIM this could be described as the effective disorder being increased by the temperature cycles from R < Re to R = Re. (The critical exponents, and thus the universality class, are different from our RFIM.) Both of these systems demand repeated cycling to produce critical behavior. This is quite different from systems with return-point memory, where repeated cycles do not change the behavior. A recently introduced, simple two-dimensional phenomenological model for the martensitic transformation is shown to mimic near full reversal of morphology under thermal cycling in [168,169]. A crossover from the hysteresis loops with a jump or 'burst' to smooth hysteresis loops (ramping temperature) has been recorded in [170], as the grain size of a macroscopic, polycrystalline specimen of an Fe-Ni-C alloy is reduced. The crossover is explained in terms of finite size effects in

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defect-induced nucleation within each grain (due to extended defects like dislocation tangles, however, rather than random-point disorder). It seems clear that one should look for critical fluctuations near the crossover that would promote an interpretation of the grain size as the analog of the disorder parameter R in our model.

Liquid Helium in Nuclepore In [171] Lilly, Finley, and Hallock show a hysteresis loop for the capillary condensation of superfluid helium in the porous material Nuclepore. Nuclepore is a polycarbonate sheet perforated by a high density of nearly cylindrical holes with a nominal diameter of about 2 000 A. The pores intersect. The most probable number of intersections per pore is about five. Most pores participate in a macroscopic interconnected network, (i.e. they are apparently internally connected beyond the percolation threshold). In the experiment the authors measure the amount of helium absorbed and capillary condensed in the Nuclepore sample as a function of the number of helium atoms present (chemical potential) at fixed temperature T -- 1.52 K. Pore intersections result in weak interactions among the pores that lead to avalanches that have been observed upon slow withdrawal of helium from the cell. The upper branch of the hysteresis loop (upon withdrawal of He from the sample) actually has microscopic jumps. The largest avalanche observed involves 2.6 x 107 pores, which is about 2.6 per cent of the system. The distribution of avalanche sizes is described by power laws. This system has clear analogies to our model. The coupling between the pores is certainly 'ferromagnetic'. If a pore is emptied it becomes more likely for an intersecting pore to be emptied also. Different pore radii and different cross-sections of the pore intersections introduce a certain randomness, which might be used to tune the system to a critical point. In [172] the avalanches are studied for different characteristic pore sizes and the relevance of various theoretical models, which one might apply to the system, is discussed. In [173] a unifying theoretical approach to hysteresis phenomena associated with capillary condensation of gases in dis-ordered mesoporous solids is given, and the connection with the zero-temperature theoretical studies of magnetization-reversal hysteresis of driven interfaces in disordered media is discussed (see also [174]).

Superconductors Superconducting vortex avalanches. In an interesting experiment on the dynamics of superconducting vortices in the Bean state in Nb47%Ti53%,Field, Witt, and Nori recorded vortex avalanches as the system was driven to the threshold of instability by the slow ramping of the external magnetic field

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[135]. The individual avalanches were ranging in size from 50 vortices up to about 107 vortices, which is about 10 -6 of the total number of vortices in the system. The distribution of avalanche sizes was recorded for the events occurring in three 450 g wide windows of the external magnetic field range, centered at 2.25 kg, 5.33 kg, and 7.55 kg. At all three values for the field H there seems to be a power law up to some cutoff size, which varies with H. The authors suggest that the self-organized criticality is the underlying mechanism, despite the fact that their cutoff size changes as a function of H.

Ultra-thin granular superconducting films in a parallel field. Wu and Adams discovered avalanches in film resistance corresponding to the collapse of macroscopic superconducting regions involving up to 109 grains near the parallel critical field in ultra-thin granular superconducting Al-films [175]. The corresponding distribution of avalanche sizes displays four decades of power-law scaling (which in our simulations would correspond to a value of disorder about 10-15 per cent away from the critical point). Fewer decades are seen at lower external magnetic fields - the field may act as a tunable parameter for a critical point. The authors point out the relation to our model and the zero-temperature random-bond Ising model with the same dynamics. The scaling exponent they extract from the power-law decay is similar to the prediction from simulations of the two-dimensional random-bond Ising model.

Prewetting on a Disordered Substrate Blossey, Kinoshita, and Dupont-Roc have studied the effect of substrate randomness on the dynamics of a prewetting transition [176]. In their theoretical paper they first review several experiments and then suggest a mapping of the hysteretic prewetting system to a two-dimensional nonequilibrium RFIM in an external field H, by identifying the Ising spins 4-1 with a wet or nonwet patch of the surface. They find that the covering of the substrate appears to be a critical transition. The growth exponents that they extract from their simulations are reported to be compatible with those of percolation, see also [177].

2.10 U N S O L V E D PROBLEMS There are many open and challenging questions in this field.

1. Two dimensions. Except for the experiments in magnets and martensites, most of these experiments are in effectively two-dimensional systems. D = 2 appears to be the lower critical dimension of the transition in the RFIM [3,32]. Extracting reliable scaling behavior in

ACKNOWLEDGEMENT

159

two dimensions remains a challenge, and we have focused here on three dimensions where our understanding is reasonably complete. 2. Avalanche average pulse shapes. The average pulse shapes predicted by the various theories (Fig. 2.14) do not appear close to those measured in experiments, which themselves disagree with one another (Fig. 2.13). This is one of the few experiments where the scaling function has been extracted from the data. Are we mistaken about the universality of these functions? Does this reflect measurement artifacts, or perhaps unrecognized important control parameters? Are there analytical corrections to scaling (see the Appendix) that we are missing that skew the curves? Or are there many dynamical universality classes? Recent experiments [178] suggest that the skewness of the smaller avalanches can be affected by external stress. Signatures of time asymmetries have also been seen in high-order spectra studies of Barkhausen noise [179,180]. 3. Correlated disorder and long-range fields. Many of the experiments involve long-range fields, which may in principle alter the universality class - perhaps in ways that differ from that of the infiniterange model. Some experiments involve different kinds of disorder, such as correlated disorder rather than point disorder and random anisotropies or random bonds rather than random fields (which probably will not change the universality class). Also, there are new possible critical behaviors when one starts with other localorder parameters, for example, if we start not with Ising spins but with vectors [90]. 4. Extracting exponents and scaling functions. We need a more organized approach, both experimentally and computationally, for extracting critical exponents and reporting scaling functions. These hysteresis models represent a serious challenge, far more difficult than most of the critical points studied in equilibrium phase transitions. In part this is because the critical behavior is so much more important; because the avalanches become large so far from Re, one cannot get close to Rc without having severe finite size effects - hence the R dependence of the parameters becomes a problem.

ACKNOWLEDGEMENT The work (and many of the discussions) reviewed in this article represent collaborations with many of our students and colleagues. We acknowledge support from NSF DMR-0218475, NSF DMR-0314279 and NSF DMR9976550 (Materials Computation Center).

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A P P E N D I X . DERIVATION OF THE S C A L I N G F O R M S AND CORRECTIONS In our work we make extensive use of scaling collapses. 22 Many variations are important to us: Widom scaling, finite-size scaling, singular corrections to scaling, analytic corrections to scaling, rotating axes, and exponentially diverging correlation length scaling. The underlying theoretical framework for scaling is given by the renormalization group, in the context of equilibrium critical phenomena, and by now well explicated in a variety of texts. We have discovered that we can derive all the scaling forms and corrections that have been important to us from two simple hypotheses (found in critical regions): universality and invariance under reparameterizations. Universality is the statement that two completely different systems will behave the same near their critical point (for example, they can have exactly the same kinds of correlations). Reparameterization invariance is the statement that smooth changes in the units or methods of measurement should not affect the critical properties - or rather, that the predictions of our scaling theories are only valid up to a smooth change of coordinates in the control parameters (the analytic corrections to scaling we discussed in Section 2.1.) We use these properties to develop the scaling forms and corrections we use in this paper. Each example we cover will build on the previous ones while developing a new idea. Our first example repeats our discussion of size versus duration in Section 2.4. Consider some property F of a system at its critical point, as a function of a scale x. F might be the spin-spin correlation function as a function of distance x (or it might be the avalanche probability distribution as a function of size x, etc.) If two different experimental systems are at the same critical point, their Fs must agree. It would seem clear that they cannot be expected to be equal to one another, the overall scale of F and the scale of x will depend on the microscopic structure of the materials. The best one could imagine would be that Fl(Xl) = AF2(Bx2)~

(A1)

where A would give the ratio of, say, the squared magnetic moment per domain of the two materials, and B gives the ratio of the domain sizes. Now, consider comparing a system with itself, but with a different measuring apparatus. Universality in this self-referential sense would imply F(x) = AF(Bx), for suitable A and B. If instead of using finite constant A and B, we arrange for an infinitesimal change in the measurement of 22This Appendix follows closely the presentation in [34].

APPENDIX

161

length scales, we find: F(K) = ( 1 - ~ e ) F ( ( 1 - ~)x),

(A2)

where e is small and a is some constant. Taking the derivative of both sides with respect to e and evaluating it at e - 0, we find - a F = xF ~, so F(x) ,,~ x -~.

(A3)

The function F is a power-law! The underlying reason w h y power laws are seen at critical points is that power laws look the same at different scales. N o w consider a new measurement with a distorted measuring apparaF-

/

\ ' ~

tus. NowF(x)-~ . A i F ( B ( x ) ) i w h e r e . A a n d l 3 a r e s o m e n o n l i n e a r f u n c t i o n s . For example, one might measure the number of microscopic domains x flipped in an avalanche, or one might measure the total acoustic power B(x) emitted during the avalanche; these two 'sizes' should roughly scale with one another, but nonlinear amplifications will occur while the spatial extent of the avalanche is small compared with the wavelength of sound emitted: we expand/3(x) = Bx + bo + b l / x q - . . . Similarly, our microphone may be nonlinear at large sound amplitudes, or the absorption of sound in the m e d i u m may be nonlinear: .,4(F) -- AF + a2 F2 q- . . . So, L.

\

/.,.I

A ( F ( B x ) + F'(Bx)(bo + b l / x + . . . ) + F ' ( B x ) ( . . .) + . . . )

+ a2F2(Bx) + . . .

(A4)

We can certainly see that our assumption of universality cannot hold everywhere: for large F or small x the assumption of reparameterization invariance (A4) prevents any simple universal form. Where is universality possible? We can take the power-law form F(x) ~ x -~ = x l ~ 1 7 6 which is the only form allowed by linear reparameterizations, and plug it into (A4), and we see that all these nonlinear corrections are subdominant (i.e. small) for large x and small F (presuming ~ > 0). If ~ > 1, the leading correction is due to b0 and we expect x -~-1 corrections to the universal power law at small distances; if 0 < a < I the dominant correction is due to a2, and we expect corrections of order x -2~. Thus our assumptions of universality and reparameterization invariance both lead us to the power-law scaling forms and inform us as to some expected deviations from these forms. Notice that the simple rescaling led to the universal power-law predictions, and that the more complicated nonlinear rescalings taught us about the dominant corrections. This will keep happening with our other examples.

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For our second example, let us consider a property K of a system, as a function of some external parameter R, as we vary R through the critical point Re for the material (so r - R - Re is small). K might represent the second m o m e n t of the avalanche size distribution, where R w o u l d represent the value of the randomness; alternatively K might represent the fractional change in magnetization AM at the infinite avalanche ... If two different experimental systems are both near their critical points (rl and r2 both small), then universality d e m a n d s that the dependence of K1 and K2 on 'temperature' R must agree, up to overall changes in scale. Thus, using a simple linear rescaling K(r) = (1 - #~)K((1 - ~)r) leads as above to the prediction K(r) = r -~. (A5) ,/

N o w let us consider nonlinear rescalings, somewhat different from the one discussed above. In particular, the nonlinearity of our measurement of K can be dependent on r. So, A r ( K ( r ) ) - ao + air + a2 r2 + . . . 4 - a o l K ( r ) 4 - . . .

(A6)

If # > 0, these analytic corrections do not change the dominant power law near r = 0. However, if p < 0, all the terms an for n < - p will be more important than the singular term! Only after fitting them to the data and subtracting them will the residual singularity be measurable. For the fractional change in magnetization: AM ~ r~ has 0 < fl < 1 (at least above three dimensions), so we might think we need to subtract a constant term a0, but AM = 0 for R ~ R c , so a0 is zero. On the other hand, in a previous paper [18], we discussed the singularity in the area of the hysteresis loop: Area --- r 2-~, where 2 - ~ - fl+//~ is an analog to the specific heat in thermal systems. Since a is near zero (slightly positive from our estimates of ]/and in three, four, and five dimensions), measuring it w o u l d necessitate our fitting and subtracting three terms (constant, linear, and quadratic in r), we did not measure the area for that reason. For our third example, let us consider a function F(x, r), depending on both a scale x and an external parameter r. For example, F might be the probability Dint that an avalanche of size x will occur during a hysteresis loop at disorder r = R - Re. Universality implies that two different systems must have the same F up to changes in scale, and therefore that F measured at one r m u s t have the same form as if measured at a different r. To start with, we consider a simple linear rescaling: F(x, r ) = ( 1 - a ~ ) F ( ( 1 - ~)x, (1 + :~)r).

(A7)

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163

Taking the derivative of both sides with respect to e gives a partial differential equation that can be manipulated to show F has a scaling form. Instead, we change variables to a new variable y = xr (which satisfies y' - y to order e). If F(x, y) - F(x, r) is our function measured in the new variables, then

F(x, r ) = F(x, y ) = ( 1 - ~ e ) F ( ( 1 - e)x, y),

(A8)

and -o~F - x c~F/Ox shows that at fixed y, F ~- x -a, with a coefficient U(y), which can depend on y. Hence we get the scaling form

F(x, r) ~ x -a Y'(x~r).

(A9)

This is just W i d o m scaling. The critical exponents a and ~, and the scaling function.T(xr are universal (two different systems near their critical point will have the same critical exponents and scaling functions). We do not need to discuss corrections to scaling for this case, as they are similar to those discussed above and below (and because none were dominant in our cases). Notice that ifwe sit at the critical point r = 0, the above result reduces to equation (A3) so long as .T(0) is not zero or infinity. If, on the other hand, y.(y) ~ yn as y --+ 0, the two-variable scaling function gives a singular correction to the power law near the critical point: F(x, r) ~ x -a .T(x~r) ,~ x -a+nr for x << r-1/~; only w h e n x "~ r -1/~ will the p o w e r law x -a be observed. This is what h a p p e n e d in two dimensions to the integrated avalanche size distribution and the avalanche correlation functions (Figs 35, 36, and 37(b) in Ref. [34]). For the fourth example, we address finite-size scaling of a property K of the system, as we vary a parameter r. If we measure K(r, L) for a variety of sizes L (say, all with periodic b o u n d a r y conditions), we expect (in complete analogy to (A9))

K(r, L) ~, r -~ 1C(rL1/V).

(A10)

Now, suppose our 'thermometer' measuring r is weakly size-dependent, so the measured variable is C(r) = r + c/L + c2/L 2 + . . . The effects on the scaling function is

K(C(r), L) ~, r-'(1C(rL 1/v) q-(cL 1/v-1 -t- c2 L1/v-2) IC' (rL l/v) + . . . ) .

(All)

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CHAPTER 2 Random-Field Ising Models of Hysteresis

In two and three dimensions, v > 1 and these correction terms are subdominant. In four and five dimensions, we find 1/2 < v < 1, so we should include the term multiplied by c in Eqn. (All). However, we believe that this first term is zero for our problem. For a fixed boundary problem (all spins 'up' at the boundary) with a first-order transition, there is indeed a term like c/L in r(L). At a critical transition, the leading correction to r(L) can be c/L or a higher power of L (1/L 2 and so on). This seems to depend on the model studied, the geometry of the system, and the boundary conditions (free, periodic, ferromagnetic . . . . ). Furthermore, for the same kind of model, the coefficient c itself depends on the geometry and boundary conditions, and it can even vanish, which leaves only higherorder corrections. In a periodic boundary condition problem like ours, we expect that the correction is smaller than c/L. Our finite-size scaling collapses for spanning avalanches N, the second moments ($2), and the magnetization jump AM, were successfully done by letting c = 0. For the fifth example, consider a property K depending on two external parameters: r (the disorder, for example) and h (could be the external magnetic field H - Hc). Analogous to (A9), K should then scale as

K(r, h) ,,, r -~ ~(h/rB~).

(A12)

Consider now the likely dependence of the field h on the disorder r. A typical system will have a measured field that depends on the randomness: d_.(h) = h + b r + b2r2 + ... (Corresponding nonlinearities in the effective value of r are subdominant.) This system will have

K(r, d,(h)) = r -~ x ( ~ ( h / r ~ ) + (br + b2r2) r -B~ 1C'(h/r~)).

(A13)

Now, for our system 1 < fl~ < 2 for dimensions three and above. This means that the term multiplied by b is dominant over the critical scaling singularity: unless one shifts the measured h to the appropriate h' = h + b r, the curves will not collapse (e.g. the peaks will not line up horizontally). We measure this (nonuniversal) constant for our system using the derivative of the magnetization with field dM/dH(r, h). The magnetization M(r, h) and the correlation length ~(r, h) should also collapse according to Eqn. (A12) (but with h + b r instead of h); we do not directly measure the correlation length, and the collapse of M(r, h) in the right-hand side of Fig. 2.15 includes the effects of the tilt b. In two dimensions, fl~ is large (probably infinite), so in principle we should need an infinite number of correction terms. In practice, we tried lining up the peaks in the curves (with no correction terms). Because we did not know fl (which we usually obtain from

APPENDIX

165

AM, which gives fl/v = 0 in two dimensions), we failed to extract reliable exponents in two dimensions from d M / d H . For the sixth example, suppose F depends on r, h, and a size x. Then from the previous analysis, we expect

F(x, r, h) ~ x -~ Y(x~r, h/r~6).

(A14)

Notice that universality only removes one variable from the scaling form. One could, in practice, do two-variable scaling collapses (and we believe someone probably has done it), but for our purposes these more general scaling forms are used by fixing one of the variables. For example, we measure the avalanche size distribution at various values of h (binned in small ranges), at the critical disorder r = 0. We can make sense of Eqn. (A14) by changing variables from h/r~6 to x~6h:

F(x, r, h) .~ x-~.T(x~r, x~6h).

(A15)

Before we can set r - 0, we must see w h a t are the possible corrections to scaling in this case. If the disorder r depends on the field, then instead of the variable r, we m u s t use r + ah (the analysis is analogous to the one in example five; other corrections are subdominant). Setting r = 0, n o w leaves F d e p e n d e n t on its first variable, as well as the second:

F(x, r, h) ~ x -~ .T(x ~(ah), x ~ h )

.~. x -~

x (J'(0, x~6h) + ahx ~ ~-(1,0)(0, x ~ 6 h ) ) ,

(A16)

where .~(1,0) is the derivative of ~- with respect to the first variable (keeping the second fixed). For the binned avalanche size distribution, x ~ is S ~, where 0Ka < 1/2 as we move from two dimensions to five and above. Thus, the correction term will only be important for rather large avalanches, S > h -l/G, so long as we are close to the critical point. Expressed in terms of the scaling variable, important corrections to scaling occur if the scaling variable X -Saflbh > h l-fib. For us, fl~ > 3/2, and we only use fields near the critical field (h < 0.08), so the corrections will become of order one w h e n X = 4 for the largest h we use. In three and four dimensions, this correction does not affect our scaling collapses, while in five dimensions some of the data need this correction. We have tried to avoid this problem (since we do not measure our data such that they can be used in a two-variable scaling collapse) by concentrating on collapsing the regions in our data curves where this correction is negligible.

CHAPTER 2 Random-Field Ising Models of Hysteresis

166

A similar analysis can be done for the avalanche time distribution, which has two 'sizes' S and t, and one parameter r, which is set to zero; because we integrate over the field h the correction in (A16) does not occur, and other scaling corrections are small. Finally, we discuss the unusual exponential scaling forms we developed to collapse our data in two dimensions. If we assume that the critical disorder Rc is zero and that the linear term in the rescaling of r vanishes (~er in Eqn. (A7) vanishes), then from symmetry the correction has to be cubic, and Eqn. (A7) becomes:

F(x, r)

-- (1 - ors) F((1 - ~)x, (1 +

k~r2)r),

(A17)

with k (which is not universal) and ~ constants, and e small. Taking the derivative of both sides with respect to e and setting it equal to zero gives a partial differential equation for the function F. To solve for F, we do a change of variable: (x, r) ~ (x, y) with y - x e -a*/r2. The constant a* is determined by requiring that y rescales onto itself to order e: we find a* = 1/2 k. We then have:

0 = - a F(x, y)

~xx x,

(A18)

which gives

F(x,r) =x-~ ffC'(xe-1/2kr2).

(A19)

This is one of the forms we use in two dimensions for the scaling collapse of the second moments (S2)int, the avalanche size distribution Dint integrated over the field H, the avalanche correlation Gint, and the spanning avalanches N. We use another form too, which is obtained by assuming that the critical disorder Re is not zero but that the linear term in the rescaling of r still vanishes. Instead of Eqn. (A17), we have:

F(x, r) -

(1 - ~ ) F((1 - ~)x, (1 +

g~r)r).

(A20)

The function F becomes:

F(x, r ) - x -~ ffC'(xe-1/fr).

(A21)

The corrections to scaling for the last two forms (Eqns. (A19) and (A21)) are similar to the ones discussed above. They are all are subdominant.

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CHAPTER 3

The Barkhausen Effect Gianfranco Durin, Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada delle Cacce 91, 10135 Torino, Italy Stefano Zapped, INFM SMC and UdR Roma 1, Dipartimento di Fisica, Universita 'La Sapienza', Rle A. Moro 2, 00185 Roma, Italy

Contents 3.1 3.2 3.3 3.4 3.5 3.6

Introduction Experiments and Materials Experimental Results Models and Theories of Magnetization Dynamics Discussion of Theoretical Results Conclusions and Perspectives Acknowledgements Appendix. Translation of the Original Paper by Barkhausen References

181 184 190 211 239 251 253 253 257

3.1 I N T R O D U C T I O N 'Eisen gibt beim Unmagnetisieren ein Ger~iusch; bei ganz gleichm~it~igen Anderung der magnetomorischen Kraft klappen die Molekularmagnete sprungweise in ihre neue Lage und erzeungen dadurch in einer dar~iber geschobenen Spule unregelm~it~ige Induktionsst6f~e, die sich im Telephon als Ger~iusch erkenntlich machen.' [1] (Iron produces a noise when magnetized. As the magnetomotive force is smoothly varied, the molecular magnets flip in jumps to their new position. Because of this, they generate irregular induction pulses in a coil wound around the sample, that can then be heard as a noise in a telephone.) (See the Appendix). All the essential elements of the magnetization noise, as discovered by H. Barkhausen, are contained in his words introducing the famous paper of 1919. A piece of iron, the simplest magnetic material, produces a jerky noise when magnetized by a field smoothly changing in time as given, for instance, by the slow movement of a magnetic yoke driven by a hand. This noise is irregular, in contrast with the regularity of the applied field, and can be easily heard with a microphone. This was the first indirect The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

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CHAPTER 3 The Barkhausen Effect

evidence of the existence of magnetic domains, postulated a few years before by Weiss [2]: actually, Barkhausen believed that the noise was produced by the sudden reversal of the entire magnetic domains (the 'molecular magnets'). Only several years later, in 1938, did Elmore [3] make the first observation of the motion of domain boundaries in a cobalt crystal (described theoretically in 1932 by Bloch and thus referred to as Bloch walls) under the application of an external field. Curiously, Elmore did not recognize this motion as the source of the noise heard by Barkhausen. This instead was successively evidenced by Williams and Shockley measuring a SiFe single crystal [4]. It is worth noting that in this very year, Kittel, in his famous 'Physical Theory of Ferromagnetic Domains', warned 'it may be well to correct the widespread notion' that the Barkhausen effect is connected to the complete domain reversal, as Williams and Shockley showed 'quite clearly [...] that the Barkhausen discontinuities correspond to irregular fluctuations in the motion of a domain boundary (Bloch wall)' (from [5], p. 9). Once the origin of Barkhausen (BK) noise was understood, it was soon realized that it could be used as an effective probe to investigate and understand the magnetization dynamics in soft magnetic materials, and explain some of their hysteresis properties. At the same time, it was also clear that the task was far from easy. As significantly noted: 'the Barkhausen effect research is both simple and hard' ([6], p. 46), in the sense that it is relatively easy to perform a measurement, but it is much more difficult to interpret the results. This is due to the stochastic character of domain-wall (DW) motion, which proceeds in jumps, or avalanches, when slowly magnetized. This dynamics is strongly affected by material microstructure, and other effects such as the demagnetizing field, external stress, etc. For this reason, earlier studies were focused on materials having the simplest domain structure, like SiFe single crystals (see, for instance, [7]), where it is possible to obtain a single DW in frame-type samples. The intense activity of Barkhausen studies up to the 1970s, both from the theoretical and experimental side, is extensively described in the reviews of Rudyak [8], and of McClure and Sch6eder [6]. Both papers report a large series of experimental data on the statistical properties of the noise: distribution of duration and size of the avalanches, power spectra, pulse propagation along the sample, etc. The approach followed to interpret the data is fully phenomenological, without a precise connection with the DW dynamics. For instance, power spectra [9,10] and cross spectra, using two separated coils [11], are described by a proper superposition of elementary independent jumps, whose spectral characteristic are defined a priori. These elementary jumps are not strictly

3.1 INTRODUCTION

183

related to well-specified DW displacements. Another approach tried to mimic the active interactions on a DW considered the effect of a series of springs on a rigid wall, and calculated the resulting power spectrum [12,13]. Both reviews also present an exhaustive discussion on the possible effects of the measuring setup on the detected signal, considering the effect of the dimensions of both the solenoid and the pick-up coils, the different time constants related to the electrical circuits, and, more importantly, the effect of the eddy currents. This detailed analysis testifies to the accuracy and capability of doing BK experiments in that period. Most of the methodology is still valid and should be strictly followed, even if this has not always happened in the recent literature. The difficulties of correctly interpreting the experimental data were partially explained by the observation of Wiegman and Stege [14] first, and later of Bertotti and Fiorillo [15], who noticed that the statistical properties of the noise typically vary along the hysteresis loop. Only considering a region where the DW motion is the dominant or, better, the unique magnetization process, as around the coercive field, it is possible to obtain a stationary signal and to correctly perform a statistical analysis. This observation triggered a series of detailed experiments, which were quite well described by a new DW model, known as ABBM [16,17]. Even if still phenomenological, this model has the merit of describing the statistical properties of the noise on the basis of some simple assumptions verified experimentally, and to be analytically tractable. This new experimental determination of the noise properties put in evidence a series of interesting characteristics. The BK noise is self-similar, and shows scaling invariance and power laws. In other words, it has the typical features of a critical phenomenon, whose detailed nature has been the subject of an intense debate. Following this line of interpretation, in 1991 Cote and Meisel [18,19] claimed that the Barkhausen effect is a particular example of a 'self-organized critical system' [20], in which there is no need to fine tune a control parameter, as in usual secondorder transitions. Following this suggestion, new experiments and models were published quite regularly in the literature. After nearly 15 years, we think it is necessary to review this large production of data and ideas and, taking into account the important results previously appearing in the literature, to propose a single framework to interpret the phenomenon. This is particularly important in order to deal with recent observations of the Barkhausen noise lacking a coherent interpretation, such as for measurements in thin films, hard materials, and low-dimensional systems.

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CHAPTER 3 The Barkhausen Effect

3.2 E X P E R I M E N T S A N D M A T E R I A L S In the vast literature about the Barkhausen noise, there is such a large production and analysis of experimental data that there is a serious risk of confusion and misunderstanding. Data have been collected on nearly all possible soft magnetic materials, and considering many different properties, so that one can, in principle, assume them as sufficiently exhaustive and reliable. The absence of a coherent interpretation could then be ascribed to the lack of a proper theoretical model. Paradoxically, the situation is very different. Despite the large production of data, the number of reliable results is rather small, and is restricted, with only a few exceptions, to the last decades of study. There are many reasons for this, as we will discuss in detail in the following. One of the main points is that there is no general agreement about a 'standard' experimental setup, which would allow for a meaningful comparison of the results. In fact, different experimental procedures can all produce 'nice' Barkhausen data, but their interpretation could widely differ. The way the measurement is performed influences significantly the details of the magnetization process. Thus, there has been a strong effort in the recent years to set experiments in the simplest and most controlled way, avoiding unnecessary complications. This has opened the way to a convincing theoretical interpretation based on relatively simple models. It is only through well-defined experiments that parameters can be estimated and used in the theory. As discussed in the Introduction, a crucial observation is that the noise properties, such as the power spectrum, change along the hysteresis loop [14,15]. As a consequence, it is important to measure the noise only around the coercive field in the steepest part of the loop, where domain-wall motion is the dominant (and often the unique) magnetization mechanism, and the Barkhausen signal is stationary. This point questions the reliability of many experimental results reported in the literature. It is convenient to critically discuss the experimental procedures proposed in the literature, in order to identify a series of practical rules to establish a 'standard' setup. This will enable the direct comparison of different experimental data, and the development of a common theoretical framework to analyze and understand the magnetization process.

3.2.1 EXPERIMENTALSETUP As shown in the Introduction, earlier experiments, starting from the original one by Barkhausen, are all based on the induction of a flux change into

185

3.2 EXPERIMENTS AND MATERIALS

MAGN. l~p,1~ CURRENT STRESS

~~

J __.___

B. NOISE SIGNAL SOLENOID ] SPECIMEN STRESS

~="=~~,2

GAP"'~ ~:....... 1!

PICK-UPCOIL ,"

,,,

,,,,

'

,,

,

",,i,'

........

,

,

,,

11 ',,,'2119

i ........

L,

FLUX-CLOSINGYOKE

,,J,,,J

.....~GAP i

J

FIGURE 3.1 Schematic representation of a setup for inductive measurements. In this case, the sample is longer than the solenoid, so that the applied field is not uniform. The flux-closing yoke is used to change the apparent permeability (from [17], Fig. 1, p. 2909).

some concatenated coils in response to a slowly varying external field. The regularity of the applied field contrasts with the irregular character of the induced pulses, as a result of the random motion of the DWs. More recent experiments make use of magneto-optical methods based on the Kerr effect (MOKE) and are more suitable to investigate the noise properties of thin films, as we will describe in detail (see Section 3.3.3).

Inductive Measurements The essential elements of a inductive Barkhausen noise measurement are simply identified: an external solenoid or an Helmholtz coil, able to produce a sufficiently homogeneous field along the sample, and a certain number of pickup (secondary) coils w o u n d around the sample to detect the induced flux. In general, it is good practice to use a solenoid much longer than the sample to ensure a uniform applied field. Also, in the case of application of an external tensile stress, it is preferable to use a nonmagnetic ribbon glued to the sample and attached to a weight (as done in Ref. [21] using amorphous materials), so that the sample remains in the center of the solenoid. To make measurements at different permeability there are two alternatives both with advantages and disadvantages: as in [22], a magnetic yoke with a variable gap can be used, but, obviously, with a sample longer than the solenoid (see Fig. 3.1), otherwise the sample can be progressively cut, as in Ref. [21]. The induced signal (i) both detects the contribution of the applied field H across the pickup coil (having a cross-section Apickup) and of the material

186

CHAPTER 3 The Barkhausen Effect

itself, so that ~ = Apickup/~o/2/-ffAI, where A is the material cross-section, and I the magnetization. This expression is valid as long as the eddy current shielding is negligible, as happens for the usual measurements performed at very low frequency (usually up to no more than a few Hz). In high permeability magnetic materials and for A not too small in respect to Apickup, the contribution of the external field can be neglected for frequencies up to several kHz, thus the induced flux contains all the information about the material properties. On the other hand, with low permeability materials, or when A KKApickup , as, for instance, in thin films, it is necessary to compensate for the induced flux in air in order to have ~ ~ AI. This is usually performed using another pickup coil with the same number of turns and the same cross-section, but w o u n d in the opposite direction. A more refined solution requires a further coil put in air to get a finer compensation [14,23]. This solution is particularly important when it is necessary to measure the hysteresis properties of thin films using inductive methods [24]. In the simplest experimental conditions, as for a material having only two opposite domains and thus a single DW, the induced flux is simply proportional to the DW velocity v, as 9 -- N(2Isdv), where Is is the saturation magnetization, d the sample thickness, and N is the number of turns. In the more common case of a multiplicity of DWs, one can assume that v measures the velocity of the active DWs, even if this must be considered just as a rough approximation. In fact, the induced signal indirectly collects, in a very complicated and unpredictable way, the changes coming from all the rest of the sample because of long-range spatial interactions, a n d / o r propagation of local deformations of DWs. The latter effect has been quantitatively studied, showing that the flux variation decades roughly exponentially with a typical scale of the order of a few cm [11]. This suggests keeping the width of the pickup coil as small as possible (a few mm, in practice). A measurement obtained with N turns is thus a good approximation of a measurement in a single cross-section of the material, with the advantage of the amplification of the signal by the factor N. There is also an additional effect suggesting keeping the width of the coils small. The magnetostatic field in an open sample produces a counterfield known as demagnetizing field, which essentially depends on the sample shape and on the domain structure. This field is spatially constant only in a sample having the form of an ellipse and at saturation, otherwise it varies, as, for instance, when a domain structure is present. The demagnetizing effect has a strong effect on the DW dynamics, limiting for instance the maximum size of a jump (see Section 3.3.1): it is thus convenient to consider a constant demagnetizing field, and this is again

3.2 EXPERIMENTSAND MATERIALS

187

obtained by limiting the width of the secondary coils. A notable example not following this rule is the setup used by Spasojevi4 et al. [25] where the pickup coil has a length of 5.5 cm, while the sample is only 4 cm long. This seriously questions the interpretation of the data and the obtained scaling exponents given the strong spatial variation of the demagnetizing field. Even following all these experimental prescriptions, we must clarify that the inductive measurement is always a collective detection of many moving DWs, where propagation and long-range interactions still play a significant role. We emphasize it as the most successful models proposed in the literature (see Section 3.4) are based on the dynamics of a single domain wall. It is thus necessary to clarify what is the reason of their success and, at the same time, understand their limitations. In particular, we will describe in detail how the open problems, still under severe theoretical and experimental investigation, could be possibly understood, introducing more complicated magnetization models that have not been considered so far. A strict consequence of this collective detection of DW motion is that the measured signal cannot distinguish between single BK jumps and a superposition in space a n d / o r time of them. This limitation has strong consequences on the interpretation of the statistical distributions of BK jumps, and in particular on the effect of the external field rate (Section 3.3.1). Quite reasonably, it has been noted [26] that at zero driving rate (the socalled adiabatic limit) the signal should be considered as a sequence of Barkhausen avalanches, i.e. of elementary jumps without any superposition. On the other hand, at finite driving rates, the signal appears as a sequence of pulses containing a certain number of avalanches. We adopt this distinction, as it is necessary to understand theoretically the effect of the driving rate on the scaling exponents. Clearly, there is a practical difficulty in establishing the low-frequency regime that approximates the adiabatic limit. In order to further simplify the measurements, we finally note that the applied field has usually a triangular shape instead of the common sinusoidal waveform used in hysteresis-loop measurements. As the noise is taken around the coercive field, as said, this ensures having a constant magnetization rate, which is the fundamental parameter to describe the variation of the BK statistical distributions (Section 3.3.1), and of the power spectra (Section 3.3.2).

Magneto-Optical Kerr Effect Measurements As described above, earlier measurements on thin films were done using inductive setups, with fine compensation for air flux [14,23,27]. These

188

CHAPTER 3 The Barkhausen Effect

measurements focused both on the noise power spectra [27], and on the avalanche statistical distributions, detected along the hysteresis loop [14]. The authors were able to observe the noise in films down to a thickness of 40 nm. This is a very surprising result, as we must consider that the size of avalanches roughly diminishes with the sample thickness, and thus can be considerably small at such thicknesses. In [14] the largest avalanche size detected is of the same order of magnitude (~ 10 -8 Wb) found in ribbons (cf. Fig. 3.4), while the largest duration is two orders of magnitude smaller. This means that the 83-17 NiFe film used has DWs with an extremely high mobility. In other materials, the situation cannot be so favorable. For instance, we have recently used a similar setup to detect the noise in Co-based thin films. For thicknesses below 1 #m, the BK signal becomes indistinguishable from the instrumental background noise, so that measurement is practically impossible. Thus, the only valid alternative is to use optical methods, based on the magneto-optical Kerr effect (MOKE), which is the change of polarization of the light reflected by a magnetic material. This technique is extensively used to investigate the domain structure of most magnetic materials [28]. There are fundamentally two variations of the basic MOKE setup applied to the investigation of the Barkhausen noise and in particular to the statistical distributions. The first was introduced by Puppin et al. [29], who added an optical stage to tune the laser spot size onto the film, with values ranging from 20 #m up to 700 #m. The reduction of the background noise is obtained using a photoelastic birefringence modulator working at 50 kHz, while the signal is detected with a lock-in amplifier at 100 kHz to enhance the signal-to-noise ratio (see Fig. 3.2). This system allowed the estimation of the distribution of avalanche sizes in Fe thin films [30,31] (see Section 3.3.3). The second solution makes use of an advanced video-processing technique to directly observe the domains and their motion [32,33]. The main instruments used are an objective lens of numerical aperture 0.9 having a x 1000 magnification. The domain images are obtained at the rate of 30 frames/s in real time and a spatial resolution of 400 nm [33]. In this particular setup, the external field is applied with an electromagnet up to nearly 99 per cent of the coercive field starting from saturation. Barkhausen jumps are thus triggered only by thermal fluctuations, so that any sweep-rate effect is completely eliminated. The experiment is performed repeatedly, imaging the same film area, using up to 1000 repetitions. The BK jump size is defined as the area having two endpoints outside the image; this choice raises some problems in the interpretation of the experimental data, as we will discuss later (Section 3.3.3).

189

3.2 EXPERIMENTSAND MATERIALS ~C

'..

FL

j

~

M I i!

I

F

B

P

L : stabilized He-Ne laser P: polarizers B : beam expander F : best form focussing lens C 9coils for magnetic field generation S :sample FL : focussing lens M : photoelastic modulator A : second polarizer D silicon ohotodiode

FIGURE 3.2 The focused magneto-optical Kerr magnetometer with variable laser spot sizes used to investigate the avalanche size distributions in thin films (from [29], Fig. 1, p. 1753).

3.2.2 MAGNETIC MATERIALS

Nearly all soft magnetic materials have been studied using the Barkhausen noise technique. But despite the large amount of published papers, a systematic identification of material properties appears intrinsically hard to perform, because of the large differences between the same type of material, the possible annealing procedures, etc. Up to now, the only identification of different classes of behavior (the 'universality classes' deeply discussed in Section 3.4.4) has been reported for SiFe polycrystalline ribbons with high Si content and amorphous FeCoBSi alloys [21], which show two distinctive scaling exponents ruling the Barkhausen distributions. This difference is understood in terms of a different DW dynamics, as will be discussed extensively. Earlier fundamental studies have been performed in frame-type single crystals to eliminate the effect of the demagnetizing field, where a single DW is present (see [34-37] for 3% wt. SiFe, [4] for 3.8% SiFe, [7,38] for 4% SiFe, and [10] for 4.5% and 6.5% SiFe). The most studied polycrystalline materials are the traditional nonoriented 3% wt. SiFe strips [15,17,39-42], or the 1.8% ones [43], as well as the GO SiFe alloys [11,15,44--48]. Other interesting SiFe alloys are the ribbons (thickness about 50-60/~m) with high Si content (% 6.5-7) produced by planar flow casting, which reveal interesting noise characteristics [21,49,50]. Much attention has also been

190

CHAPTER 3 The Barkhausen Effect

given to the NiFe alloys, such as the Permalloy (Ni80Fe20) and similar [12,14,41,51,52] or the Perminvar (30% Fe, 45% Ni, 25% Co) [53,54], particularly when a strong disorder is induced by precipitation of some crystalline phases. In more recent years, many studies have been focused on amorphous materials, especially the iron-based family: some commercial ribbons, like the Metglass 2605 [18,19,55,56], the Metglass | 2826 [57], or the Vitrovac 6025 [25], and the vast productions of various laboratories with different compositions and additives [21,48-50,58-61]. On the other hand, Co-based amorphous alloys have received a very limited attention [41], due to their high permeability values connected with the usual presence of a single, large BK jump. Despite the large variety of compositions used, we should stress here that not all the soft magnetic materials are appropriate for BK measurements. Especially, materials having very nice'soft' properties (high permeability, small losses, etc.) show reduced or no BK noise, like the Co-based amorphous alloys or a well-prepared Permalloy, with limited structural defects. In fact, the essential ingredients for a 'nice' noise are the presence of a consistent structural disorder, hindering the motion of DWs, and a nonnegligible demagnetizing effect, limiting the size of avalanches. Also, the thickness appears to be an important factor to take into account, as it sets the limits of the eddy-current shielding effect. For instance, SiFe ribbons, having thicknesses of the order of hundreds of ~m, have to be magnetized at extremely low frequencies to detect well-defined BK jumps [17,40].

3.3 E X P E R I M E N T A L R E S U L T S 3.3.1 BARKHAUSEN SIGNAL AND AVALANCHE DISTRIBUTIONS

In Fig. 3.3, we show some typical experimental BK data obtained in a polycrystalline FeSi 7.8% wt. ribbon and in an amorphous Fe64Co21B15 ribbon, the latter measured under moderate tensile stress. The different sequences, acquired during the same time interval, correspond to increasing applied field rates. At low rates, the BK signal clearly comes out as a sequence of distinguishable and well-separated avalanches: at larger rates, this separation is progressively lost and the noise resembles a continuous sequence of peaks. This feature is common to both materials, even if this similarity is only seeming, as their statistical properties are very different and appear to be representative of two types of behavior, the 'universality classes', with different DW dynamics, as we will describe in detail.

191

3.3 EXPERIMENTAL RESULTS 4 FeSi7.8%wt.

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F I G U R E 3.3 Time sequences of B a r k h a u s e n noise m e a s u r e d in a polycrystalline FeSi 7.8% wt. ribbon (left) a n d an a m o r p h o u s Fe21Co64B15 u n d e r m o d e r a t e tensile stress (right). The labels indicate the f r e q u e n c y of the a p p l i e d field.

To investigate the statistical distribution of BK jumps it is necessary to univocally define the size and the duration of an individual jump. This poses a delicate problem, given the self-similar fractal nature of the BK signal, and the fact that, in addition, the background noise limits the detection of small avalanches. The common way to solve this problem is to define a threshold, a 'resolution coefficient' [19,23,51,62,63], in order to set the temporal limits of a single avalanche. This procedure has been justified in the context of the fractal properties of the signal, as it is possible to associate with it a random Cantor dust, with a fractal dimension ~<1, which can be used to calculate the critical exponents [63]. The determination of the threshold can vary, but in general we found the results are nearly insensitive to changes in its value [63], as long as it is not too small nor too large. Because the BK signal amplitude v follows a power law of the type P(v) - v - ( 1 - c ) f ( v / v o ) ,

(3.1)

192

CHAPTER 3 The Barkhausen Effect

where c is proportional to the driving rate, we usually assume the threshold between 5 and 15 per cent of v0. With this definition, the duration T and the size S (the integral over the duration) are unambiguously defined. Like the signal amplitude, the distributions of T and S also follow power laws P(T) = T-~g(T/To),

(3.2)

P(S) = S-~f(S/So),

(3.3)

and where To and So are two cutoffs, and ~ and 9 the critical exponents. It is interesting to note that, despite the large number of papers regarding BK noise, the reliable estimates of these critical exponents are very limited. We report in Table 3.1 the results presented in the literature we believe are sufficiently accurate to be considered and analyzed. We leave out, for instance, the results of the quite famous papers of Cote and Meisel [18,19], who first suggested the BK exhibits self-organized criticality [20], because they are based on a very poor statistics. At first sight, the measured exponents span a quite large range of values, and no general behavior can be easily guessed. But it is worth considering in detail each experiment analyzing the particular conditions of the material and of the measurement to find a possible explanation for this large variability. In particular, two key factors appear to affect the results, and should thus be carefully checked: the applied field-rate dependence of the critical exponents, and whether the data are taken in a small bin around the coercive field or along all the hysteresis loop. After this warning, we proceed to analyze the literature results in more detail: 1. The 81%NiFe used in [51] is the only wire that has been measured accurately. The authors use three different samples with different annealing conditions, but do not report any indication about the structural characteristics of the materials. They also noticed a dependence of the statistics on the applied field rate, but to simplify matters, they made the measurements at the same magnetization rate, which has since been established to be the correct parameter to take into account [16,17]. As it is not explicitly mentioned, the noise has been probably recorded along all the hysteresis loop and it is thus likely that it is not statistically stationary. 2. Similar critical exponents have been reported by Spasojevi4 et al. [25], with a completely different material. As mentioned in Section 3.2.1, this experiment has some serious drawbacks, due to the large extension of the pickup coil in respect the sample length, which

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3.3 EXPERIMENTAL RESULTS

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193

CHAPTER 3 The Barkhausen Effect

194

3.

4.

5.

6.

have been shown to affect the size distributions [6]. The data were taken in a small-field interval, presumably around the coercive field, but it is not clear from the paper if the applied field was large enough to saturate the material. The rate dependence of the critical exponents has not been discussed. The Perminvar used in [54] has been subjected to an annealing at 450~ for 24 h. This prolonged treatment has the effect of inducing the precipitation of microcrystalline phases, which create a large number of strong pinning centers for the DW, resulting in a low permeability of about 250. The estimation of the critical exponent is limited to the size exponent r, but is made with high accuracy. A comparable-size critical exponent has been measured for a sheet of interstitial-free, low-carbon, annealed steel [64]. The external field is applied both parallel and perpendicular to the rolling direction. The corresponding hysteresis loops show a different permeability, which affects the value of the cutoff So, and slightly the critical exponent ~. As matter of fact, the estimation at the lowest permeability (smaller So) appears much less accurate, due to the limited region of the power-law dependence. Reference [43] reports the first estimation of the critical exponents in polycrystalline Sire strips (1.8% wt.). Within the experimental errors, the critical exponents show a linear dependence on the applied field rate, expressed by a = 2 - c, and ~ = 1.5 - c/2, where c is the parameter appearing in Eqn. (3.1). In [21], we have reported measurements of the critical exponents in polycrystalline Sire ribbons (with high Si contents) and amorphous alloys with composition FexCo85-xB15. We also measured a partially crystallized Fe64Co21B15, obtained after annealing for 30 min at 350~ and then for 4h at 300~ under an applied tensile stress of 500 MPa. This induces the formation of a-re crystals of about 50 nm, with a crystal fraction of "~ 5% [67]. The Sire alloys and the partially crystallized alloy show r ~ 1.5 and a ~ 2, measured at the lowest possible frequency f = 3-5 mHz (see Fig. 3.4(left)). They also show a linear dependence on the applied field rate, as shown in Fig. 3.5(left) [43,63]. The highly magnetostrictive FexCoss-xB15 alloys (2s "~ 30 - 50 x 10 -6) have been measured under a tensile stress of a --- 100 MPa. The applied stress is found to enhance the signal-noise ratio, reducing biases in the distributions, but does not change the exponents [68], as long a small stress is applied (see below). These alloys yield r -~ 1.27 and a ~, 1.5 (Fig. 3.4(right)), independent of the applied field rate, as shown in Fig. 3.5(right).

3.3 EXPERIMENTALRESULTS

195

In contrast, the shape of the cutoff changes drastically as the rate increases. 7. The authors of [66] used the same type of amorphous alloy as [21], but without the application of a tensile stress. The resulting critical exponents are larger, and the authors have no simple explanation for the difference. As they claim, the residual stress may be the origin of the anomaly. We can confirm that unstressed amorphous materials have, in general, hysteresis loops with restricted regions of linear permeability, thus the determination of the exponents can be difficult and less accurate. In Table 3.1 we also report the values of another critical exponent 1/r~vz relating the average size (S) to the duration T as (S) ~ T 1/r This exponent has been shown to be in strict relation with the power spectrum exponent (see Sections 3.3.2 and 3.5.1). To understand the origin of the variability of the critical exponents it is important to first focus on the values that have been found to be close in different samples and experiments. In addition, it is better to consider the data recorded in a small bin around the coercive field that can thus be considered as stationary. Two sets of exponents emerge: one is measured in polycrystalline alloys or alloys with embedded grains, characterized by z ~ 1.5, and ~ ~ 2, with a linear dependence on the applied field rate. The other is measured in materials where the crystalline matrix is absent (amorphous samples under tensile stress) or progressively lost by induction of a strong disorder (like in the Perminvar of [54]). They show z ~, 1.3, and a ~ 1.5, and no rate dependence is found. These results support the idea that there are (at least) two kinds of behavior, the universality classes, to which correspond different sets of critical exponents. We will come back to this matter in more detail in Section 3.5.1, where we will also try to explain the occurrence of the other experimental exponents. Given the variability of the numbers shown above, it is quite clear that the sole comparison of the values of the exponents z and a can be misleading. In other words, it necessary to determine other experimental critical exponents that can confirm the existence of the two universality classes. In this respect, an important suggestion has been given by Perkovi4 et al., who pointed out that' an explanation for the experiment must involve collective motion of many domains; it must provide an explanation for the power-law scaling regions, and it must provide an explanation for the cutoff' (from [69], p. 4528). We have already stressed the importance of the demagnetizing effect in the DW dynamics, so we have investigated in detail the effect of the change of the demagnetizing factor k on the power-law distributions. We have used two samples belonging to the different universality

196

CHAPTER 3 The Barkhausen Effect

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FIGURE 3.4 Left: distributions of Barkhausen jump sizes measured in different materials. The solid line has a slope ~ - 1.5 while for the dashed one r -- 1.27. Right: similar plot for duration distributions. The solid line has a slope a = 2, while for the dashed one a = 1.5. (From [21], Fig. 1, p. 4706). classes, and have m e a s u r e d the avalanche distributions. Progressively cutting the samples to change the apparent permeability/lapp, w e have calculated the factor k using the well-known expression/lap p - - / 2 / ( 1 + k/l), w i t h / l the intrinsic permeability. We have verified that the cutoffs So and To strongly d e p e n d on k, as in Fig. 3.6, and follow the scaling relations So ~" k -1/Crk, and To ~ k - & . Best-fit estimation of the exponents gives 1 / a k ~ 0.57, Ak "" 0.30 for the polycrystalline class, and 1 / a k "~ 0.79, Ak "~ 0.46 for the a m o r p h o u s one.

197

3.3 EXPERIMENTAL RESULTS 10 8 i:~,,,,~:

106 104 ~" 102 10~

,

, FeSi 7.8 % wt.

~.,,._~~ ~ :200mHHz

10-2 10"12 10-11

10"10

10-9

I

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t

103 102 ~ 101 100 101 10"2 10-3 10-8 10"11 10"10

S (Wb)

O.o~5.

10-9

10-8

10-7

S (Wb)

FIGURE 3.5 Distributions of the avalanche sizes S for the polycrystalline SiFe (left) and the amorphous material (right) at different applied field frequencies. In the polycrystalline sample, the critical exponent z varies linearly with the frequency [43,63] between 1.5 and 1.15, while it is independent in the amorphous one with ~ 1.27. 20

lO

~q

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(a) 9

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9

!

9

.

9

i

.

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i

9

k "~176

10

.,.,,,,..

r

I0

(b)

I00

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FIGURE 3.6 The cutoff of the Barkhausen size (a) and duration (b) distributions as a function of the demagnetizing factor k in FeSi 6.5% wt. alloy (empty symbols) and an amorphous Fe21Co64B15 sample under constant tensile stress (filled symbols) (From [21], Fig. 3, p. 4707).

198

CHAPTER 3 The Barkhausen Effect

3.3.2 POWER SPECTRA AND AVALANCHE SHAPES

If we merely count the huge number of papers in the literature concerning the spectral properties of BK noise, the contrast with those considering scaling exponents results clearly. Since the earlier papers, but especially during the 1970s-1980s, almost all the authors publishing about the BK effect considered the problem of the detection and explanation of the power spectrum. The main reason for this abundant production is related to the practical need to reduce the spectral noise in applications where soft materials are used. In addition, it seemed easy to explain all the details of the power spectral shape in terms of a superposition of 'elementary' Barkhausen jumps (see, for instance, [9]). These elementary units were generally defined a priori as exponential or square events; a proper superposition of these independent jumps can give practically any measured power spectrum shape. Even if successful in giving a general description of the power spectrum shape, this approach was not able to account for the microscopic magnetization events and the details of the DW dynamics, so it really did not represent a significant step towards the fundamental understanding of BK noise. Remarkably, after the publication of the two seminal papers by Alessandro et al. [16,17], introducing the ABBM model (Section 3.4.2), the investigation of the power spectrum apparently lost interest, even though the problem remained basically a puzzle. Only recently did the subject, seen under a completely new point of view, receive a new impulse. As should be clear from the discussion in the preceding sections, the detection of stationary or nonstationary Barkhausen signals has a strong effect also on the estimation of spectral properties. More precisely, traditional power spectral calculations using FFT must be applied only to stationary data, otherwise they can give unpredictable results. A nonconstant value of the permeability, for instance, introduces low-frequency components unrelated to BK noise. Thus, we must consider only two types of spectral estimations: (i) those taken with a single DW at constant velocity, as in [10,36,57,70], where a feedback setup keeps the flux rate constant, and (ii) those taken in a small bin around the coercive field, as first performed in [14,15]. Measurements of power spectra of type (i) are particularly important. In fact, not only are they a measure of the time correlation of the BK signal (in the frequency domain), but more they are a measurement of the spatial fluctuation of local fields that are needed to unpin the DW. In other words, these measurements describe the spatial spectrum of the random pinning field (see Section 3.4.2). Measurements on a single crystal of SiFe give an f - 2 power spectrum, and thus a k -2 spectrum of the pinning field [10,36],

3.3 EXPERIMENTALRESULTS

199

because of the linear relation between f and k at constant v. It is curious to note that measurements on a toroidal Metglass sample gave a k -1"7 spectrum [57]. Even if a single measurement cannot be taken as conclusive, this result can be regarded as a manifestation of a different universality class, similar to that reported for the avalanche critical exponents. As anticipated, the most accurate report of power spectra in a polycrystalline material has been presented in [17]. A strip of nonoriented 3% SiFe, having a grain size of about 100 #m is measured under a small tensile stress (~ 5 MPa), and at variable apparent permeability, obtained by varying the air gap with a mumetal yoke (see Fig. 3.1). The effect of the grain size on similar materials is reported in [71]. Subsequently, a few other papers have reported the power spectra of single-crystal and polycrystalline SiFe materials [72], as well as of amorphous materials [59,73]. There some general common characteristics that can be summarized as follows: 1. The spectrum F(og) has a typical 1/f ~ shape at high frequency, with ~9 - 1.7 - 2, and scales linearly with the average magnetization rate I, so that the spectra normalized with the average flux F(~)/SI coincide at high frequency (Fig. 3.7). 2. At lower frequencies, the spectrum shows a marked peek at a frequency roughly proportional to ~1/2, with an amplitude that takes a constant value at low rates, as shown in Fig. 3.7. This amplitude roughly scales with the permeability, as #1.5 [17]. 3. At frequencies lower than the peek, the spectrum scales as f 6 , with 6 ~ 0.6, or ~ 1. These general features describe only qualitatively the observed results. More precisely, there are at least four different typical shapes, as shown in Figs 3.8 and 3.9. Single-crystal and polycrystalline SiFe strips show the simplest power spectrum, with ~9 -~ 2, and 6 ~ 0.6. The polycrystalline SiFe ribbons at high Si content keep the same exponents, but present a large region at intermediate frequencies where the spectrum strongly deviates from the simple 1/f 2 law. Amorphous materials under tensile stress show instead ~9 = 1.7 - 1.8, and 6 ~ 1, as shown in Fig. 3.9. Similar data are found in the unstressed case, ~9 = 1.73 4- 0.08 and 6 ~ 1, for the same amorphous alloy [66] (private communication). Amorphous materials, where a proper annealing induces a partial crystallization, show a sort of intermediate case: the high frequency part is more similar to the polycrystalline SiFe samples with ~9 ~ 2, while 6 ~ 1 keeps the values of a pure amorphous material. The former property is not merely a coincidence, as we observed a

CHAPTER 3

200

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1

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E

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The Barkhausen Effect

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FIGURE 3.7 Left: normalized power spectra F(~o)/S~Iat fixed permeability of a polycrystalline 3% SiFe strip. The solid lines are the experimental results while the dotted lines are the prediction of the ABBM model (Section 3.4.2). Right: normalized maximum of the power spectra FM/S~Ifor permeability ranging from 2 500 (o) up to 60000 (>). Continuous lines are a guide to the eyes only. The bold line is the boundary between the intermittent (5 < 1) and the continuous (5 > 1) DW motion. Here 5 is equivalent of the exponent c of Eqn. (3.1). (From [17], Figs 7 and 10, pp. 2912-2913.

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3.3 EXPERIMENTALRESULTS

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FIGURE 3.9 Normalized power spectra of the same amorphous material as in Fig. 3.5 (left), and of a partially crystallized (16%) sample induced by annealing !right) (reproduced from [59], Fig. 2, p. 300) as a function of the magnetization rate I. In the left-hand part, f* is roughly the frequency where the shape changes from f-1 tof-2.

corresponding change in the critical exponents of size and duration distributions towards the values measured in SiFe, as shown in Table 3.1. Curiously, all the spectra, despite these differences, show the same dependence of the peek frequency o n ~1/2. The interpretation of all these complicated details still represents a challenging puzzle, although some progress has recently been made [65,74]. Earlier estimates of the exponent t9 ([51], and similarly in [25,69,75]) led to the conclusion that 'the asymptotic decay of the spectrum is not given by the shape of the pulses, but is exclusively determined by the distribution of the pulse duration and the correlation between the size and the duration' (adapted from [51], p. 15). In our notation, t9 = 1 + 2avz - a, or analogously (3 - r)/avz (see also Section 3.5.2). Using the data of Table 3.1, this implies t9 ~ 3, much larger than the experiments. This result was successively found to be valid only for r > 2, while for r < 2, Kuntz and Sethna [74] calculated t9 = 1/avz, which is the same critical exponent relating the average avalanche size to the duration (Table 3.1). This prediction agrees pretty well with the experiments, as we can see in Fig. 3.10. In particular, it works for the amorphous material in an extended frequency range; the

CHAPTER 3 The Barkhausen Effect

202

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10 -13 10 0

. . . . . . . .

I

!

I

I

101

10 2

10 3

10 4

i

101

. . . . . . . .

l

10 5

. . . . . . . .

10 2

i

,

|

, , ,1,,I

10 3

10 4

, II,

= , |,,,

10 5

o)

FIGURE 3.10 Comparison of the power spectrum F(co) with the average size (S) as a function of the inverse of duration T, for the amorphous material and the polycrystalline SiFe ribbon (7.8%, inset) of Fig. 3.4. The theoretical prediction of [74] is also shown, with 1/r equal to 1.77 and 2 (from [65], Fig. 1, p. 1086). same is not valid for the polycrystalline SiFe ribbon, and the reason will be discussed after introducing the hypothesis at the base of the calculations by Kuntz and Sethna.

Avalanche Shapes The simple result z9 = 1/r~vz obtained by Kuntz and Sethna [74] is based on the existence of a certain number of scaling relations and on a few hypotheses, like the complete separation of the avalanches in time. We leave the discussion of these important aspects to a further section (see Section 3.5.2), focusing here on the newly proposed scaling relations that can be tested experimentally. These relations involve universal scaling functions whose shape could provide a critical test for the model, as 'a sharper tool for discriminating between different universality classes than critical exponents' [66,76]. The first relation states that the average avalanche shape scales in a universal way so that

v(t, T) - T 1/crvz-l fshape(t/z),

(3.4)

3.3 EXPERIMENTALRESULTS

203

where v is the BK signal, t is the time andfshape(t/T) is a universal scaling function. Similar to this relation, we can consider the voltage as a function of magnetization s - f0T v dt to get

V(S, S)

-- S 1-rTvz g s h a p e ( s / S )

(3.5)

A third relation deals with the fluctuations of avalanche sizes, and considers the probability P(vlS) that a voltage v occurs in an avalanche of size S. This probability scales as:

P(vlS) -

v -1 fvoltage(vS avz-1)

(3.6)

and, again, fvoltage is another universal scaling function. A partial experimental verification of these relations has been reported in a couple of papers [65,66]. In Fig. 3.11, we show the complete universal scaling laws of Eqns (3.4) and (3.5) for the same 7.8% SiFe ribbon and the amorphous sample of Fig. 3.3, scaled using the theoretical estimation of the critical exponent avz. The experimental data are compared with the theoretical average shape predicted using the ABBM model (see Eqns (3.101), (3.102)). In the case of the amorphous material, the scaling is pretty good, except for the magnetization signal at small sizes. In particular, both time signals show a marked time asymmetry, similar to that found in [25], where the average of the avalanches is made without any scaling. The asymmetry is also visible in the magnetization signal, even if less marked. Here, the shape calculated using the ABBM (see Eqn. (3.102) and Section 3.5.3) reproduces pretty well the experiments, except for the small sizes in the amorphous material. It is interesting to note that all the models presented in the literature predict symmetric time and magnetization signals. In Section 3.5.3, we will propose a possible explanation and discuss the relevance of the shape of the scaling function as an indicator of universality class. The scaling of time signal data of 7.8% SiFe ribbon using the theoretical exponent 1/crvz = 2 is clearly incorrect. The reason is illustrated in Fig. 3.12, where we plot the average avalanche size (S) as a function of duration T. The exponent 1/avz (the slope of the curve in the log-log plot) slightly varies from --- 2 down to about 1.75. If we use this 'variable' exponent 1/avz*, the rescaling of the time signal is highly satisfying. The third universal scaling function (Eqn. (3.6)) has been reported in [65] for an amorphous material, as shown in Fig. 3.13. This confirms the validity of scaling relations proposed in [74], at least for the amorphous material. In fact, the relation in Eqn. (3.6) does not hold so well for the 7.8% SiFe ribbon.

CHAPTER 3 The Barkhausen Effect

204

10

104 [-x 0 072 i- 0.025 . . . . \

6

F e S i 7 . 8 % wt.

Amorphous Fe64Co21B15

8

~"

~"

4

o

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

s/S 800 ~"

0.23,,,,0.16 ~

0.8

70r" 051

~00~ 0 ~ ~ ~ ~ ~ x ~.~

0.6

1.0

s/S

2.03

~0t

" , Iro81 ~

5oo

~

~.~ 40

~. 400

~

/

~.~

s~o

.2

~ 30

,~. 300 200

"~ 20

100

10

0 0.0

0.2

0.4

0.6

0.8

0 0.0

1.0

0.2

0.4

tiT

0.6

0.8

1.0

t/T

FIGURE 3.11 Average avalanche shapes for the 7.8% SiFe ribbon (left) and the amorphous sample (right) of Fig. 3.3. Time signal v(t, T) and magnetization signal v(s, S) are scaled according to Eqns (3.4), (3.5). Bold lines are the theoretical predictions of Eqns (3.101), (3.102), a semicircle and an arc of sinusoid, respectively (see Section 3.5.3). Numbers in the graphs denote avalanche sizes (in nWb, upper figures) and duration (in milliseconds, lower figures).

10 =

v

.N_ 0.1 oo (3o

0.01 < 1

1E-3 0.1

1 duration T(ms)

10

0 0.0

|

i

0.2

,

i

o

0.4

i

0.6

,

i

,

0.8

t/T

FIGURE 3.12 Average avalanche size as a function of the duration for the 7.8% SiFe ribbon (left). Time signals are rescaled as in Eqn. (3.4) using a variable exponent 1/r deduced by the plot on the left.

205

3.3 EXPERIMENTAL RESULTS 10 -7 r

~v

~%

10 -8

r s=2.19e-9

~"

o s=l.O9E-lO

I 0_ 9

"~ s = 2 . 9 4 e - l O

a.,

+ s=7.89e-lO a s=l.52e-11 v s=4.09e-ll ,to

a s=2.19e-12

10 - l o

.+,~ §

~, s = 5 . 6 8 e - 1 2

I~ #,

10 -11

.

10 -4

.

.

.

.

.

iJl

,

a

~

10 -3

. . . . .

|

10 -2

,

.8~'+ V ,

,

. . . .

10 -1

V s owz-I

FIGURE 3.13 Distribution of the signal at fixed size as predicted by Eqn. (3.6) for the amorphous material using the theoretical scaling exponent 1/r -- 1.77. (From [65], Fig. 3, p. 1087).

High-Order Power Spectra In [55], O'Brien and Weissman introduce a further statistical characterization of BK signal calculating power spectra of order higher than those shown in the previous section. This analysis is motivated by the possibility of proving the BK as a real example of SOC [20], lacking of any inertial effect unlike, for instance, grain falling in sand piles. In fact, the mere 'existence of pulses with a range of sizes and durations should not be taken as conclusive evidence for SOC' (from [55], p. 3446). Thus it is important to understand if the observed scaling properties reflect a real self-organization involving'cooperative avalanche processes invoked by SOC', or simply reflect the scaling properties of the pinning field, or, alternatively, the vague proximity of a disorder-driven critical transition [69]. The high-order spectra involve correlations between different temporal scales, so, for instance, could clarify if events on a certain scale systematically precede events on another scale, as in the case of systems showing precursors, such as earthquakes. Another important aspect, more specific to BK noise, is the possibility of distinguishing whether the high-frequency spectrum 1/fo primarily comes from short independent pulses, as often assumed, or conversely from the fine structure of longer pulses [73]. This is addressed considering the 1.5 power spectrum S1.5(f2,f1), involving a

206

CHAPTER 3 The Barkhausen Effect

third moment of time signal v(t), defined as: $1.5(f2,fl) =

F{v(t) }F{H(t, f~)}* , (H(t,fl))

(3.7)

where H(t,fl) is the time-dependent Haar power at the frequency fl, F{.} is the usual Fourier transform, and those components are calculated at frequency f2 [73]. Similarly, the second spectrum S2(f2,f1) is defined as:

F{H(t, fl)}F{H(t, fl) }* $2 (f2, fl) -~

(H(t,fl))2

9

(3.8)

A complete presentation of these spectra has been reported in [73] for an Fe21Co64B15 amorphous ribbon (in the unstressed state). In Fig. 3.14 we show the 1.5 power spectrum. The weak independence of Re{S1.5(f2,fl)} on fl indicates that the high-frequency spectrum l / f ~ contains a large contribution from pulses with duration much longer than 1/fl. This definitely demonstrates that, in this case, the power spectrum does not come from independent short pulses. The imaginary part, instead, is sensitive to the asymmetry of the time signal. The positive sign of Im{Sl.s(f2,fl)}/Re{Sl.5 0c2,fl)} suggests that, on average, high-frequency components precede the low-frequency ones. This is in accord with the time asymmetry of the average avalanche shape of Fig. 3.11. Interestingly, high-order power spectra of 1.8% SiFe single crystals have a completely different behavior [77], as shown in Fig. 3.15. Here $2 0c2,fl) is nearly flat as a function off2 and falls off as f11"2, which indicates that most of the power at fl comes from separated pulses of duration 1/fl. The 1.5 power spectrum confirms this hypothesis, as it strongly depends on fl, in contrast to the weak dependence in the amorphous ribbon. In this case, it is appropriate to assume the power spectrum is a superposition of individual independent pulses. 3.3.3 THIN FILMS

All the results presented so far are related to materials that are essentially three-dimensional, with the sole exception of the wire reported in [51] (see Table 3.1). The behavior of the BK noise in two-dimensional thin films is much less known, and the number of papers reporting a complete set of experimental results is very limited [14,23,27,30,32]. Many questions remain to be solved, both on the experimental and the theoretical sides, so that the topic represents an interesting and promising field of study for the near future. Contrary to bulk three-dimensional systems with a relatively simple magnetic structure of nearly parallel DWs, thin films show richer and often

3.3

EXPERIMENTAL

lif ~

......

,

207

RESULTS

........

,

1. .......

,

.....

-

%

I

- - ~.5~,

I

-

-

..... ~ "

",~,

......

,

........

,

9

.......

,

,

=0.085 [ [ - - ~t~=o.te8

0.5

..... '~'~"'1 ~

9

N

~i~

0.6

.......

9 ......

10 kl-lZ " [

5kHz

-

~'~,,.

7

. -

.~.~':-~.~

-

-

A

-

f,=~ou.~

I

f~v

0.4

.d ,"/,,'

10"9

>

~-

0.3

~-

0.2

~J j" r/J

,r o

i0 -l~

,-.c' ,9'

0.1 9' \ J ' - ~

-0.I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

lO-ll 10

100

(a)

1000

104

10

f2 (Hz)

lO0

(b)

1000

10'

f2 (Hz)

FIGURE 3.14 The 1.5 power spectrum defined in Eqn. (3.7) for an Fe21Co64B15 amorphous ribbon in the unstressed state, at five different frequencies fl. (From [73], Figs 6 and 9, pp. 6366-6367).

I"

'

' '"'"i"

'

.......

10 2 -

'"i'

I

-

101

I

_

_

.

L- 2_" \~ ~

!

r

~

[

~

-

[

:~"

~

,"tl,,,,

~\

-

I

'

' '""'1

-

2.5

--

1.25kHz

-

-

-

. . . . .

'~

104

kHz

10 .9

625 Hz 312

HZ

i0 -l~

.,~

-- 10-11

~"~

N

10 ~

q..~-i 0 "I

10 "2

m

!

10 .3

10 "13

10-14 1

I0

1 O0

1000

104

q (Hz) FIGURE 3.15 Second spectra S2 and 1.5 power spectra $1.5 for a 1.8% SiFe single crystal. (From [77], Fig. 3, p. 1172).

208

CHAPTER 3 The Barkhausen Effect

more complicated DW patterns. The film thickness plays a fundamental role because of the increasing importance of stray fields in the direction perpendicular to the sample, as first pointed out by N6el [78]. In addition to usual Bloch DWs, other different types exist, such as symmetric and nonsymmetric N6el walls, and cross-tie walls, a complicated pattern of 90 ~ N6el walls. A complete description of the existing configurations together with many experimental images can be found in the excellent book of Hubert and Sch~ifer [28]. To have an idea of the complexity, it is worth reporting their very words about the role of film thickness: 'The typical DW in very thin films, i.e. the symmetric N6el wall, needs basically a one-dimensional description of its magnetization structure. The stray field of this wall reaches out into the space above and below the film, however, and cannot be treated one-dimensionally. Cross-tie walls in thin films need at least a two-dimensional description for the magnetization and a three-dimensional treatment of the stray field. Bloch walls in thick films and their relatives are two-dimensional both in the magnetization and in the stray field. Most complex are cross-tie walls in thicker films (100 nm for Permalloy), which need a three-dimensional description of both magnetization and field and which have never been analyzed theoretically up to now' (from [28], p. 241). It is clear that the experimental BK data can be understood only through a precise knowledge of the active DWs, and their role in the magnetization process. Considering that the present research is still at a preliminary stage, the task will surely require some time to be fulfilled. Nevertheless, it is worth discussing in detail the main results presented in the literature. Quite impressive work has been done in the PhD thesis of N.J. Wiegman in 1979 [23], who explored the properties of Permalloy thin films using an inductive setup. A selected number of results had previously been reported in a couple of papers [14,27]. The simple visual comparison of typical noise data of Permalloy films with those of SiFe or amorphous samples (see Fig. 3.3) reveals striking differences. As shown in Fig. 3.16, the BK avalanches appear as sharp peaks, well separated in time, with nearly no inner structure. This behavior is confirmed by the data of the size and duration distributions, where the maximum duration is nearly two orders of magnitude less than the corresponding value in ribbons, as noted before, while the same does not occur for the maximum size. In other words, the avalanches in these films are rather sharp peaks of short duration, reversing a magnetization volume comparable to the value observed in ribbons. The behavior of the critical exponents ~ (~1 in Wiegman's notation) and a (~1), shown in Fig. 3.17, as a function of the film thickness is quite intriguing. The exponents do not show any strong thickness dependence, except for the increasing

209

3.3 EXPERIMENTAL RESULTS

Vincl(H )

I

IV1

= 104. V

T H [A/m] 1 A/m

FIGURE 3.16 A typical time signal for a Permalloy thin film (from [23], Fig. 3.2 p. 44). ,

,

,~

+

1.8

§ aI

l

1,6

.

.

.

.

,,

,

,

.

,

2.~

|

20

9

,O

§

|

Z3

9

O

+O

9 Jo;

*

*

2.1

+

I

+ §

**|

+

§

+

-i,

1 + +

§

1.t,

+%

1.;

12

"~

1.0 1

5o0

......

I~000

"'

i 1500

dm IJ~!

J 2000

=

: 2 S90 0. . . . . . . . .

iz_

:l

1

1

1

5OO

1000

1500

2000

dm |J~l

-=!=

2500

=

FIGURE 3.17 Values of the critical exponents ~ (c~1 in Wiegman's notation) (left) and c~ (61) (right) for Permalloy films as a function of the thickness (from [23], Figs. 3.10 and 3.14, p. 57,62).

fluctuations below 100 nm, which, by the way, is the region where complex cross-tie walls are observed. In the same region, the critical exponent 1/~rvz ()~) displays a sharp transition from 2 to 1.5, as clearly shown in Fig. 3.18. These interesting results should be considered as the basis for further investigations. In fact, the critical exponents have the same values observed in polycrystalline ribbons (r -~ 1.5, ~--~ 2, 1 / r ; v z ~, 2) for thicknesses down to 100 nm, suggesting a typical three-dimensional behavior with predominant long-range interactions. The situation at smaller thicknesses is more complicated and the role of DW patterns such as in cross-tie walls is not clear. While an accurate estimate of the critical exponents is desirable, it is also necessary to perform an extended scaling analysis, not restricted to a single exponent although this task is far from being easy. In [30], the critical exponent r is estimated in a Fe film of 90 nm using the MOKE setup (Section 3.2.1), and considering different laser spot sizes from 20 to 700 tim. A single distribution P(AM) is obtained properly rescaling the size of the avalanche AM and the distribution P, as shown in Fig. 3.19.

CHAPTER 3

210 !

!

. . . . . .

2.0

.

.

.

J_ .

.

.

. . . .

§. . . . . . .

.

x. 1.6

9- ~ 4 -

i

i

tl,

.

.

,

.

.

.

The Barkhausen Effect i

J r .

4=. . . . . .

.

i

!

.

.

.

.

.

.

.

.

.

-4.--.

4'

* - , - # - - -,

I 1./, -

t

* 4' 4.

-

m

l

....

t

" 500

,

,

1000

I

I

1500

2000

,,,

I

,

2500

FIGURE 3.18 The critical exponent 1/r (X in Wiegman's notation) for Permalloy films as a function of the thickness (from [23], Fig. 3.7, p. 53].

. . . . . . .

!

. . . . . . . .

I

. . . . . . . .

I

'

9

. . . . . .

'1

,'"

9

ii

......

P(aM)oc A M "1'1 r " t "

9

50 Nm I-/~!I t~

'",,'"

lOONm ~

Q.

!

\.NN,

250 Nm 70dpm t ,

,

= ,,l,,I

. . . . . . . .

I

. . . . . . . .

I

. . . . . . . . .

I

. . . .

o

| ....

l:

Amplitude ( a M ) (arb.units)

FIGURE 3.19 Distribution of magnetization steps &M in a 90 nm Fe film. Data taken at different laser spots are properly rescaled to show a single distribution (from [30], Fig. 5, p. 5418).

Unfortunately, data at small spot sizes are not averaged over different positions, so that the overall distribution appears quite unreliable under 100 #m. Given that, the estimated exponent is r - 1.1 4- 0.05. A successive estimation using a spot of 20 # m averaged over 10 different locations yields = 1.14 [31]. It worth noting that these values are m u c h smaller than the ones observed in Permalloy, a fact on w h i c h w e will c o m m e n t very soon.

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS

211

FIGURE 3.20 (a) Domain images of same 400 x 320 #m2 area for a 25-nm Co film between 0 and 4 sec. The sample was saturated downward first, and then a constant field was applied upward, denoted by the solid arrow, during observation. (b) Magnetization reversal curves obtained from the corresponding domain patterns of (a) (from [32], Fig. 2, p. 087203-2).

The paper of Kim et al. [32] reports an estimate of the critical exponent in Co films, using a different MOKE setup able to visualize directly the DWs. As shown in Fig. 3.20, both ends of the DWs are usually outside the frame, so that the authors deliberately neglect those jumps having one of the ends inside the frame. The resulting power-law distribution has an exponent close to 1.33, averaged over four different thicknesses, from 5 to 50 nm. As we explain more extensively in Section 3.5.4, this exponent is different from 7, which should be evaluated, in principle, considering only the jumps fully inside the frame. As a matter of fact, this appears quite difficult to perform, given the limited number of such jumps. It is worth noting that a similar comment could be appropriate also for the previous experiment of Puppin. In that case, it is not possible to distinguish between jumps fully inside the laser spot and the others. Moreover, the relative probability of the two types of events is not known, so that in principle we cannot be sure if the exponent r is correctly estimated. With many others, this could be one of the reasons for getting the lower value with respect to the Permalloy. But, as discussed above, all these matters have to be confirmed by further experiments.

3.4 MODELS A N D THEORIES OF M A G N E T I Z A T I O N DYNAMICS A theory of the Barkhausen effect should explain the statistical features of the noise, such as the power spectrum, the avalanche distributions, and the pulse shape. In particular, the theory should be able to predict the values

212

CHAPTER 3 The Barkhausen Effect

of the scaling exponents characterizing the distributions and possibly link the noise properties to microstructural details of the material under study. It is thus important to establish a controlled pathway from the microscopic reversal processes and the resulting macroscopic noise response. We can list here a set of crucial questions that need to be addressed by the theory: 1. an explanation for the occurrence of scaling and quantitative estimates for the exponents; 2. the origin of cutoff in the scaling regime; and 3. the effect of microstructural parameters (lattice structure, anisotropies, spin interactions) on the noise statistics. Clearly these questions are particularly broad and it is dubious that a single answer will be valid for all materials and experimental situations. The observation of scaling laws, however, can signify that relatively simple models could capture and reproduce a wide class of experiments. This is what is expected in critical phenomena, where universality implies that only the main symmetries and conservation laws influence the behavior of the system, while many other quantitative details are irrelevant. Thus, if the Barkhausen effect reflects the presence of an underlying critical phenomenon, one expects relatively broad universality classes spanning different materials, as are indeed observed. Discussing the main theoretical approaches proposed for understanding the Barkhausen effect, we will pay particular attention to the way each theory addresses the main questions reported above. In order to put the models into context, we first discuss the general properties of ferromagnetic materials in terms of the micromagnetic free energy. This represents a natural microscopic starting point to build up mesoscopic models and theories amenable to an analytic treatment and large-scale numerical simulations.

3.4.1 GENERAL PROPERTIES OF FERROMAGNETIC SYSTEMS

A ferromagnetic material can be represented as an ensemble of localized magnetic moments or spins, interacting between each other and with the external magnetic field H. The macroscopic properties of the material, such as the Barkhausen noise and the hysteresis loop, are due to microscopic rotations of the spins and could in principle be described by microscopic theory. As a first step in this direction, we will discuss the interactions governing the dynamics of the local magnetization and the associated energetic contributions. In particular, we can write the energy of a ferromagnetic

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS

213

material as a sum of different terms E = Eex + Em + Ean + Edis,

(3.9)

where Eex represents exchange interactions, Em the magnetostatic energy, Ean the anisotropy and Edis the disorder. The detailed form of these terms, as a function of the local magnetization M(~) will be discussed in the following sections.

Exchange Energy The most important energetic contribution comes from exchange interactions, which are typically short-ranged and tend to align spins. For a set of spins s(~i) on the lattice, the exchange energy can be written as Eex =

~l(]-fi-

rjl)s(ri), s(rj),

(3.10)

/j where J(x) decays rapidly for large x and the sum is over all atom pairs. Eqn. (3.10) can be approximated in the continuum limit, through an elastic approximation that involves the removal of all small-scale details. The set of spins s(ri) is replaced by a continuous field/~(~) and the exchange energy can be written as 3

Eex - A / d3r y~(VMa(~)) 2

(3.11)

0r

where A is the exchange coupling, which can be derived from J(x).

Magnetostatic Energy The magnetostatic energy is due to the interactions between the spins and the external field and to dipole-dipole interactions between different spins. For a uniformly magnetized sample the contribution of the external field to the magnetostatic energy is simply given by Em --

flo VA~./7/,

8~

(3.12)

where M is the magnetization and V is the volume. In addition, we should consider the energy due to the demagnetizing field Hdm, the magnetostatic field generated by the local magnetization. In order to compute Hdm, it is useful to define 'magnetic charges' associated with the discontinuities of the normal component of the magnetization.

CHAPTER 3 The Barkhausen Effect

214

For a surface separating two regions of magnetizations M1 and M2, the surface charge density is given by 6 --- ~. (M1 - M 2 )

(3.13)

where fi is the vector normal to the surface. For instance, at the boundary of the sample, where the magnetization varies abruptly from to zero, the charge density is given by cos 0, where 0 is the angle between the magnetization direction and ft. It is important to remark that magnetic charges are not physical charges, but are a convenient tool to compute the demagnetizing field and magnetostatic energy of magnetized bodies. Everything proceeds as in electrostatics provided we replace electric charges with magnetic charges and the electric field with the demagnetizing field, which is thus given by

Ms

Ms

-. f dS'rr Hdm(~) = - V . , I [~-~'['

(3.14)

where the integral is over the surface separating two regions of constant magnetization. For a uniformly magnetized ellipsoid Hdmis constant and proportional to magnetization vector Hdm =

-kM,

(3.15)

where k is the geometry-dependent demagnetizing factor. In this case the total_~magnetostatic energy is simply obtained by replacing H in Eqn. (3.12) b y H +Hdm. In general the demagnetizing field is not constant and Eqn. (3.12) has to be replaced by an integral Em -

#0 f dSr ~/I. (/7/-F f/dm(~)), 8n Y

(3.16)

where Hdmis given by the contribution of surface charges (i.e. Eqn. (3.14)) summed to the contribution due to the bulk variation of the magnetization. The bulk dipolar energy can be written explicitly as Em -

P0

8n

/

dSrdSr'

3( ~

E

~,fl=l

I~ - ~

,[3 --

3(r~-r~)(r~-r~)) IF --F 7i5

M~(F)M//(F ,)

(3.17) As in the case of the surface term, Eqn. (3.17) can also be rewritten in terms of the density of magnetic volume charges p(F) = V./VI(F) as Em =

Po f dBr dBr'p(~)p(F') 8n

I~ - F '1

(3.18) "

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS

215

Magnetocrystalline Anisotropy and Magnetoelastic Energies The magnetization in a ferromagnetic material typically has preferential directions corresponding to the crystallographic axis of the material. Therefore, it is easier to magnetize the sample along the easy directions. This observation can also be expressed in terms of the energy of magnetocrystalline anisotropy Ean

/ d3r ~

K~M~M~,

(3.19)

where M~ is the a component of the vector M and Ka]~is a symmetric tensor, describing the anisotropy of the material. In the simplest case of a uniaxial crystal, Eqn. (3.19) reduces to Ean -- / dSr K0(/~. ~ ) 2

f dSr

KoM2sin 2 4,

(3.20)

where q~is the angle between the easy axis ~ and the magnetization vector and K0 is the uniaxial anisotropy constant. The variations of the magnetization inside a ferromagnetic sample can cause deformation in the lattice structure, a phenomenon known as magnetostriction. Conversely, when an external mechanical stress is applied to the sample the magnetic structure can in principle be modified. To describe this effect it is useful to introduce the magnetoelastic energy, which in the most general form can be written as Ean =

f d3r E 2~&,~r~flM~M~,

(3.21)

where cr~ is the stress tensor and 2~&6 is the magnetoelastic tensor. For a crystal with isotropic magnetostriction, under a uniaxial stress r~ the anisotropy energy takes the simple form of Eqn. (3.20) with K0 replaced by K0 + 3/22r~, where 2 is the uniaxial magnetostriction constant.

Disorder In the previous discussion we have considered only a homogenous system, in which the interactions are globally defined and do not depend on position. In general, however, different sources of inhomogeneities are found in virtually all ferromagnetic materials. The presence of structural disorder is essential to understand the fluctuations in the Barkhausen noise, which would be strongly suppressed in a perfectly ordered system. The nature of the disorder can be inferred from the microscopic structure of the material under study.

216

CHAPTER 3 The Barkhausen Effect

We can thus distinguish several contributions to the magnetic free energy due to the disorder: in crystalline materials disorder is due to the presence of vacancies, dislocations or nonmagnetic impurities. In polycrystalline materials we should add to these defects the presence of grain boundaries and variations of the anisotropy axis in different grains. Finally, in amorphous alloys disorder is primarily due to internal stresses and the random arrangement of the atoms. It is important to notice that in the following we will consider the disorder as quenched (or 'frozen'): it does not evolve on the timescales of the magnetization processes under study. It is not always simple to quantify the energetic contributions of the different sources of disorder but we can highlight here the main effects. The presence of randomly distributed nonmagnetic inclusions gives rise to a magnetostatic contribution, due to the magnetic charges that form at the boundaries of the inclusions [79]. The energetic contribution due to a collection of those inclusions can be expressed as a local fluctuation of the dipolar coupling. A similar discussion can be repeated in the case of exchange interactions, leading to a fluctuating exchange coupling. The typical fluctuations of the coupling depend on the volume fraction Vnm of the nonmagnetic inclusions: the typical strength of the (exchange or dipolar) coupling g(~) will be of the order of ~ ~" (1 - Vnm)go and the fluctuations ((g_ (g))2) ~ Vnmg2, where go is the coupling without impurities. This type of disorder is conventionally called of random-bond type and should also be present in amorphous alloys, due to the random arrangements of the atoms. In polycrystalline samples, each grain has a different crystalline anisotropy. In particular, the direction of the anisotropy axes will fluctuate in space and, in the simple case of uniaxial anisotropies, the anisotropy energy will be given by Ean - ~ d3r K(/~I. ~(~))2 where ~ is a random function of position reflecting the grain structure of the material under study. In magnetostrictive samples internal stresses play a similar role to anisotropies, as we have discussed in the previous section. In particular, a random distribution of internal stresses produces an random energy of the type of Eqn. (3.20) with a random anisotropy constant K(~) cx K0 +3/22r~(~). While internal stresses can have multiple origins, it is possible to compute explicitly their distribution in some particular cases, such as in a crystal with a random distribution of parallel dislocations. The stress in due to a dislocation in the origin is given in cylindrical coordinates by cr~ - b#C~(O)/r, where b is the Burgers vector, # is the shear modulus and C~/~is an angular function depending on the dislocation type [80]. The

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS

217

distribution of the internal stresses is formally given by

P(~rafl) - f d2Nr D(rl . . . . -rN)~(~afl -- Y ~ oafl(r -- rk)),

(3.22)

k

where D is the distribution of the position of the dislocations. The internal stress distribution has been evaluated in the case of a random short-range correlated dislocation arrangement: P(a) is Gaussian for small stresses, with variance (Aa 2) proportional to the dislocation density and at larger stresses displays a power-law tail, scaling as a -3 [81]. It is interesting to remark that P(a) is an experimentally accessible quantity, since it is directly related to the X-ray spectrum [82]. We have discussed two common contributions to the disorder energy in ferromagnetic materials: random bonds and random anisotropies. To conclude the discussion, we remark that some diluted antiferromagnets in a field can be effectively described by a random-field type disorder, with energy given by

Edi s -- f dgr h(~)./~/i,

(3.23)

where h is randomly distributed in space. Despite the fact that random field disorder is not present in ferromagnetic materials, it provides a useful theoretical ingredient to understand the role of disorder in hysteresis, as we will discuss in the following.

Micromagnetic Equations In the previous sections we have discussed the different energetic contributions to ferromagnetic materials. Collecting all the terms, considering for simplicity a uniaxial material, we can write the energy

3

E = ~ f d3r[A(VMa) 2 + K(Maea) 2 _ (Ha q- H dem,~..a], (a) x^,~

(3.24)

a--1 A, K and ea in principle depend on position and H dem (a) is the component of the demagnetizing field, discussed in Section 3.4.1. The energy function reported in Eqn. (3.24) can in principle be used to compute the equilibrium properties of ferromagnetic materials. This problem is in general very complex and we do not attempt to treat it here. In fact, to describe the Barkhausen effect we are interested in the evolution of the magnetization in response to an increasing external field. The equation of motion for the magnetization can readily be obtained

CHAPTER 3 The Barkhausen Effect

218

from Eqn. (3.24) ~M = 7M x Heff ~t

(3.25)

where 7 is the charge to mass ratio and Heft ~- - ~ E / ~ M . Equation (3.25) does not include a dissipation mechanism and would predict an indefinite precession of the magnetization vector. This problem can be overcome introducing phenomenological laws for the dissipation. This results, for instance, in the Landau-Lifshitz-Gilbert equation whose numerical integration for different microstructures and boundary conditions is the subject of micromagnetism (see Chapters 4 and 5). It is tempting to use a micromagnetic approach to describe collective effects arising in the Barkhausen effect from first principles. An attempt in this direction was made in [83-85], where a Montecarlo version of the micromagnetic equation was used to simulate the Barkhausen effect in polycrystals. The model consists of a chain of N magnetic moments with orientations defined by two angles ~i and Oi. The total energy of the system is given by 1

E - ~ ~ ( s i n 2 o~i q- 2HcosOi + rosin 2 0isin 2 q~i - 2acos~i,i+l), i

(3.26)

where ~i is the angle between the magnetic moment and the local (random) easy axis, H is the magnetic field, a is the exchange-anisotropy energy ratio and m controls the magnetostatic energy. The last term takes into account the interaction between magnetic moments a n d fli,j is the angle between the moments in i and j. The evolution of the magnetization is dictated by a Montecarlo algorithm. As the external field is slowly increased the magnetic moments rotate in bursts, which are associated with the Barkhausen jumps. The results obtained for the jump distributions are in some range of parameters reminiscent of the experiments (i.e. a power law with a cutoff) but a quantitative comparison with experiments would be questionable. The model treats very accurately some aspects of a ferromagnetic system, namely, spin rotations, nucleation and random anisotropies, but completely neglects the fundamental aspects of the problem: dimensionality (the model is one-dimensional), the range of interactions (only nearest-neighbor interactions are present, while in reality dipolar interactions give rise to long-range terms decaying as 1/r3), and demagnetizing effects. For these reasons, this model is not able to reproduce quantitatively the scaling properties of the Barkhausen effect. To this end, it would be necessary to

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS

219

study three-dimensional models with long-range interactions, a task that promises to be computationally very demanding. 3.4.2 RANDOM-ENERGY MODELS The difficulties arising in the direct analysis of the micromagnetic equations naturally led to a search for simplified models that could capture the essential mechanism underlying the Barkhausen noise without excessive complications. The main ingredient of any model for the Barkhausen effect is the presence of disorder, thus the simplest approach is to neglect most of the details of the micromagnetic energy, keeping only a uniform magnetostatic term Em and replacing the rest with a random function of the magnetization m E -- F(m) - mHeff,

(3.27)

where H e f f = H + Hdem. The rationale behind this approximation can be understood considering a single rigid domain wall dividing the sample into two domains. In this case the magnetization is proportional to the wall position x: m = Ms ( 2 x / L - 1), where Ms is the saturation magnetization and L is the sample width. The sample is magnetized as the applied magnetic field pushes the domain wall across a disordered landscape represented by the random potential F(m). In order to define completely the problem, one needs to specify the statistical features of F(m), such as its distribution and correlations. N6el was the probably the first to use a random-energy model to study the hysteresis loop properties at low fields in his theory of Rayleigh loops [86,87]. In the N6el model the random function was schematized as a series of parabolas with randomly distributed curvatures. The Barkhausen noise can be constructed from the random-energy model by a visual inspection dF of the pinning field W(m) = dm as shown in Fig. 3.21. As the effective field is increased, the domain wall is pinned as long as Heff < W(m), when Heff reaches a local maximum of W(m) the domain wall jumps forward until the condition Heff < W(m) is met. The Barkhausen sizes S are just the changes in magnetization Am occurring after these jumps or, in more mathematical terms, the first returns of the stochastic process described by W(m). The random energy model in the form proposed by N6el, however, cannot describe correctly the Barkhausen noise because the pinning field is essentially uncorrelated. Thus, the first return distributions decay exponentially, while the experiments indicate a power-law. In order to obtain a power-law distribution of returns the pinning field has to be long-range correlated. This point was first realized by Bertotti [88,89], who included a Brownian pinning field in a random energy model.

220 0.02

9

|

Sarkhausen

-

jump

CHAPTER 3

The B a r k h a u s e n Effect

9

9

| /

~

9

~

0.015

.,~

0.01

0.005

.

.

0

. . . 100 200 m (magnetization)

300

FIGURE 3.21 The Barkhausen jumps (denoted by arrows) from the perspective of the random-energy model. The model, widely known as ABBM [16,17], yields a quite accurate description of the Barkhausen noise statistics. In the ABBM model the magnetization evolves according to an overdamped equation of motion dm dt

= ct - k m + W ( m ) ,

(3.28)

where the external field increases at constant rate H = ct, k is the demagnetizing factor, and the damping coefficient has been set to unity, rescaling the time units. The pinning field is a Brownian process where correlations grow as ( ( W ( m ) - W(m')) 2) = D i m m'l. (3.29) The main predictions of the ABBM model can be obtained by deriving Eqn. (3.28) with respect to time and defining v =_ d m / d t dv dt

= c - kv + vf(m),

(3.30)

whereflm) ~_ dW/dm is an uncorrelated random field with variance D. Expressing Eqn. (3.30) as a function of v and m only dv

dm

c =

v

k + tim),

(3.31)

we obtain a Langevin equation for a random walk in a confining potential, given by U ( v ) = k v - c log(v). Asymptotically, the statistics of v are ruled

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS

221

by the Boltzmann distribution P(v, m ~

cx~) ~ e x p ( - U ( v ) / D )

- v c/D e x p ( - k v / D ) .

(3.32)

The distribution in the time domain is obtained by a simple transformation and it is given by [16] P(v) - P(v, t ~

cx~) =

kC/Dv c/D-1 e x p ( - k v / D ) Dc/DF(c/D)

.

(3.33)

A consequence of Eqn. (3.33) is that the domain-wall average velocity is given by (v) = c/k. For c / D < 1 the velocity distribution in Eqn. (3.33) is a power law with an upper cutoff that diverges as k ~ 0. In this regime, the domain wall moves in avalanches whose size and durations are also distributed as power laws. For c / D > 1 the motion is smoother with fluctuations that decrease as c / D increases. The avalanche size distribution equals the distribution of first return times of a random walk in the confining potential U(v). In the limit k ~ 0, we can directly apply the exact result reported in [90] for the first return times of a random walk in a logarithmic potential, which implies that the avalanche exponents depend on c and are given by z=3/2-c/2D

a = 2 - c/D.

(3.34)

See also [43,63] for alternative but not rigorous derivations of these results. The scaling of the cutoff of the avalanche distributions in the ABBM model can be obtained similarly in the limit c ~ 0, solving for the first return probability of a biased random walk and given by So ~ k -2.

(3.35)

Using similar arguments one can also show the cutoff of avalanche duration scales as To "~ k -1 [50].

3.4.3 SPIN MODELS In this section we discuss the attempts to understand the statistical properties of the Barkhausen effect by models of interacting spins, that can either be simulated on a computer or in some cases analyzed theoretically. This approach is conceptually simple and theoretically appealing, since it tries to derive directly macroscopic properties from a microscopic model, replacing the vectorial micromagnetic equations by simplified rules for the evolution of integer-valued spins. The basis of this kind of approach is

222

CHAPTER 3 The Barkhausen Effect

the notion of universality, which is expected for critical phenomena: the values of the critical exponent and the shape of the scaling functions are independent of the microscopic details of the system, provided that the relevant symmetries of the problem are correctly taken into account. For this reason, we can expect to obtain the correct large-scale behavior, such as the scaling properties of the Barkhausen effect, from relatively simple models.

The Random-Field Ising Model Sethna et al. [91] have proposed the driven random-field Ising model (RFIM) as a prototype for hysteresis and avalanches at first-order phase transitions. The model was studied extensively analytically and numerically and the resulting scaling behavior was compared with the experimental results on the Barkhausen effect [69,75,76,92]. In this model, a spin si = 4-1 is assigned to each site i of a d-dimensional lattice. The spins are coupled to their nearest-neighbor's spins by a ferromagnetic interaction of strength J and to the external field H. In addition, with each site of the lattice is associated a random field hi taken from a Gaussian probability distribution with variance R, p(h) = exp(-h 2/2R 2)/~/~R. The Hamiltonian thus reads E - - ~ Jsisj - ~ (H q- hi)si, (i,j) i

(3.36)

where the first sum is restricted to nearest-neighbor's pairs. In the zerotemperature dynamics used by Sethna et al. [91], the external field is ramped from -o0 to cx~ and the spins align with the local field [94] (3.37) In this way a single spin-flip can lead the neighboring spins to flip, eventually triggering an avalanche. For small values of the disorder R the first spin-flips are likely to generate a big avalanche whose size is comparable to the system size, leading to a discontinuous magnetization reversal. On the other hand, a large disorder prevents the formation of large avalanches and the magnetization reversal is smooth. In fact, these two regimes are separated by a critical disorder Re, where the avalanches are distributed as a power law. The behavior of the model close to Re is very similar to that of equilibrium systems at the critical points and can thus be studied by standard methods, like mean-field theory and the renormalization group.

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS

223

A first qualitative picture of the behavior of the RFIM can be obtained by mean-field theory [75,91]. To this end, we consider Eqn. (3.36) extending the sums to all the pairs of sites and obtain E = - ~(JM

+ H + hi)si,

(3.38)

i

where M

=

~ i Si. The magnetization can then be obtained self-consistently

as

M = 1 - 2P(JM + H + hi < O) - 1 - 2

f -JM-Hp(h)dh.

(3.39)

,Y--(K)

This equation has a single-valued solution for R > Re = ~/-(2/n)J, while the solution becomes multivalued for R < Rc. In the second case, we have a hysteresis loop with a jump at He and a diverging slope d M / d H . For high disorder there is no hysteresis, but this is an artifact of mean-field theory that is not observed in low dimensions. At the critical point R = Rc the magnetization has a diverging slope, but no finite jump. In analogy with critical p h e n o m e n a it is possible to obtain scaling laws describing the singularities occurring at the transition. In particular, the magnetization obeys the scaling law M = [r[~g+(h/[r[~), (3.40) where r -- (R - Rc)/Rc and h - (H - Hc)/Hc are the reduced control parameters, and fl = 1/2, 6 -- 3. The scaling function g+ is obtained as the root of a cubic equation. The distribution of avalanche sizes near the critical point can be computed analytically in the framework of mean-field theory. The result can be s u m m a r i z e d in the scaling form P(S, r, h) - S - z P + ( S / I r [ -1/rr, h/[r[~),

(3.41)

where the exponents take the mean-field values ~ = 3/2, a = 1/2. The scaling function T'+ can be evaluated exactly in terms of g+ and is given by Th-(x, y) - (1/~2-~n)exp(-x(1 :F z~g+(y)2/4)2/2).

(3.42)

In addition to the size distribution one can also define the distribution of durations T, which obeys an analogous scaling form D(T, r, h) - T-~I)+(T/[r[ -1/(~vz), h / [ r [ ~ ) ,

(3.43)

where ~ - 2, v is the correlation length exponent (~ ~, r -v) and z is the dynamic exponent relating the correlation length to the characteristic time ( ~ T~).

224

CHAPTER 3 The Barkhausen Effect

Mean-field theory has the advantage of being easily tractable and provides a qualitative picture of the behavior of the model, but the numerical values of the exponents are typically inaccurate in dimensions lower than the upper critical dimension, that for the RFIM it is equal to dc -- 6. To overcome this problem, one can perform a renormalization-group analysis with an expansion in ~ = 6 - d, as in equilibrium critical phenomena. The renormalization group is quite involved and we do not discuss it here in detail, but just quote the main results (for a complete discussion the reader is referred to [75], and Tables 2.1 and 2.2). The exponents determining the scaling of the order parameter have been computed to first order in ~ and are estimated to be fl = 1/2 - ~/6 and 6 = 3 + ~. The avalanche exponent displays only corrections to order ~2 and the cutoff exponent is estimated as r~ = 1 / 2 - ~/12. Large-scale numerical simulations have been performed to obtain reliable estimates for the critical exponents in three dimensions. The avalanche size distribution is most naturally computed integrating the avalanches over the entire hysteresis loop. The exponent ~int measured in this way can be related to the other exponents by a scaling relation "~int = ~ -}- rYflC~. Typically the distribution displays an initial power-law behavior and a cutoff that depends on the disorder R (see Fig. 3.22). The distribution for different values of R can then be collapsed using the scaling form in Eqn. (3.41) [69,92]. The best numerical estimate for the exponents (or exponent combinations) in three dimensions is r = 1.604-0.06, ~int = 2.034-0.03, 1/r~ = 4.2 4- 0.3, r~vz = 0.57 4- 0.03, :t = 2.02, fl = 0.035 4- 0.03, 1/v = 0.71 [92]. Looking at the results above one may be tempted to perform a direct comparison with experiments. However, this should be done with a pinch of salt. The RFIM model in fact predicts scaling only close to a critical disorder Re while in general there is no reason to believe that this condition is fulfilled in experiments. To relate the Barkhausen scaling behavior observed in experiments to a disorder-induced phase transition, such as the one discussed here, it would be necessary to show that the cutoff to the scaling behavior scales with the disorder strength. Unfortunately such a measure has not yet been performed.

Random Bonds, Random Anisotropies and other Effects The RFIM represents probably the simplest model to study the effect of disorder in ferromagnetic hysteresis and Barkhausen noise. The model displays a disorder-induced transition with associated critical exponents, which has been recently observed in experiments [93]. While random fields can be a useful theoretical idealization, they are not present in

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS !

|

225

,

10 5

10o m~

~

0.2

10-5

10-10

10-15 10 ~

10 2

10 4

10 6

Avalanche size (S) FIGURE 3.22 The integrated avalanche distribution in the RFIM in d = 3 simulated in a lattice of 3203 spins. The inset shows the data collapse (from [69], Fig. 1, p. 4528).

ferromagnetic materials. Thus, an important question to address is whether the basic phenomenology of the RFIM carries over to more realistic types of disorder. While the answer to this question appears to be positive, it is still controversial whether the disorder-induced phase transition is universal with respect to the type of disorder. A second related question is whether the inclusion of more realistic ingredients in the RFIM results in a change of the scaling exponents or not. The first question is addressed by numerical simulations, considering random bonds, site dilution, and random (infinite) anisotropies. In the random-bond Ising model (RBIM), the disorder enters into the exchange coupling Jij, which is randomly distributed with mean J0 and standard deviation R. Hysteresis in the RBIM was first studied in the spin glass limit (J0 = 0) [94]. In analogy with the RFIM, the RBIM displays a disorderinduced phase transition in the hysteresis loop. Some of the scaling exponents, including those ruling the avalanche distributions, have been evaluated and their values differ slightly from those measured in the RFIM. For instance, some reported values are "tint ~" 2, z _~ 1.6 and v _~ 1 [95]. One should beware, however, that the lattice sizes used in this study are rather limited (up to 403 ) [95] compared with those used to estimate exponents in the RFIM (up to 3203) [92]. In the site-diluted Ising model (SDIM), the presence of nonmagnetic impurities is treated by removing a fraction of spins in the lattice. The

226

CHAPTER 3 The Barkhausen Effect

Hamiltonian is thus simply given by E ~- - ~_~lijcicj'3isj - )

/j

Hcirri,

(3.44)

i

where c i - - 0 , 1 indicates whether the spin is magnetic or not. Typically, the sum in the exchange interaction is not restricted to the nearest neighbors but extends to a larger range. Simulations of the SDIM in d = 3 yield exponents compatible with those of the RFIM, namely, "~int = 2.0 4- 0.2 [96]. The random (infinite) anisotropy model is obtained from a vectorial model (a lattice version of Eqn. (3.61): ~i)2 _}_/_~/.Si],

E = - ~ J-si"-sj - Y ~ [ K ( - s i " (ij) i

(3.45)

in the limit of K --~ oo. Thus, to minimize the anisotropy energy, in each site the spin should be parallel to the quenched random direction ni. The original vectorial model becomes then effectively of Ising-type as can be seen by introducing a fictitious spin O"i - - 4-1 defining the sign of Si along its random axis (i.e. -Si --- ~'lir7i). In terms of the new spin variables the Hamiltonian can be rewritten as E = - ~lijrrirrj (ij)

- E

Hir~i,

(3.46)

i

with effective random bonds Jij =- Jfti . ftj and random fields H i = nz. The model above has been simulated numerically with a zero-temperature dynamics analogous to the one employed in the RFIM and with two different distributions for the anisotropy axis. The measured exponents are in good agreement with those measured in the RFIM. In particular for the avalanche distribution, the reported values are T i n t = 2.06 4-0.05 and Z i n t = 2.10 4- 0.05 depending on the distribution [97] (see Fig. 3.23). A recently proposed analysis, based on the renormalization group, of the vectorial random anisotropy model shows that the model should be described generically by the RFIM fixed point [98], thus supporting the conclusions of the simulations. From the discussion above, we conclude that the disorder-induced critical point in the RFIM represents a broad universality class describing different sorts of spin models. This fact, if confirmed by larger-scale simulations and renormalization-group calculations, is particularly interesting because it allows us to concentrate on a relatively simple model (the RFIM) to understand quantitatively a more generic scaling behavior. There

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS

227

is, however, an important point still to be considered: the effect of longrange dipolar interactions. As we discussed in Section 3.4.1 the effect is twofold, interactions between spins in the bulk and surface charges at the boundary of the sample. The bulk interaction has not been studied deeply in the framework of spin models, apart from some numerical simulations in two dimensions [99,100]. From simple power-counting considerations, we would expect that dipolar forces are relevant in the renormalization group sense, thus modifying the numerical values of the critical exponents. Therefore, unless one can neglect dipolar forces (this could be the case in hard rare-earth alloys with strong anisotropies), the numerical results obtained in the RFIM would not apply to real materials. Surface effects have an even more drastic effect on the critical behavior discussed above. The presence of a demagnetizing field generates an effective correlation length, destroying scale invariance even at the critical point R -- Rc. Nevertheless, this effect is small, since the demagnetizing factor tends to vanish for very-large system sizes. Simulations of the RFIM in the presence of a demagnetizing field show remarkable features in the low-disorder phase, where otherwise one would expect the magnetization to change abruptly with a large jump. The demagnetizing field hinders the reversal process, which splits into a series of avalanches of different sizes [101,102]. The distribution of size is again a power-law distribution and the cutoff is controlled by the demagnetizing factor (see Fig. 3.24) [101]. It is important to observe that this power-law scaling is not related to the disorder-induced critical point, since this behavior is observed generically for R < Rc, but can be explained considering the dynamics of a domain wall. Indeed, for R < Re, at the coercive field a domain is nucleated and starts to grow. When the domain is large enough the demagnetizing field reduces the effective field and eventually the growth stops. At this moment the spins that will flip when the external field is increased are likely to be those at the domain boundary. To understand this process in detail one should thus analyze domain-wall dynamics, as will be discussed in the next section.

3.4.4 DOMAIN-WALL MODELS In order to minimize the exchange energy, a ferromagnetic material at sufficiently low temperature and zero field would display a uniform magnetization, oriented along an easy axis to lower the magnetocrystalline anisotropy energy. In most cases, however, this arrangement is hindered by a very high cost in magnetostatic energy, due to the discontinuities of the magnetization at the sample edges. Thus it is energetically favorable to create domains, which in the simplest situation span the sample from end

CHAPTER 3 The Barkhausen Effect

228

model A . . . . . . . .

_

i

. . . . . . . .

i

. . . . . . . .

I

. . . . . . . .

i

. . . . . . . .

0o-1.5 1.35

~

9

9 , 149149

>,

.m

J~ o

9 9

~.

"~-9

9 .~

10 ~

9

9

10"3

100

.

9 .

...."--.

"~

"'.o"""'~" .

! .

.

.

.. . . . . . .

101

! i

. . . . . . . .

10 2

i

. . . . . . . .

10 3

I.

i

10 4

. . . .

105

S

FIGURE 3.23 The integrated avalanche distribution in the random infinite anisotropy model in d - 3, simulated in a lattice of 303 spins. The different curves correspond to different values of 00, the average angle between the field axis and the anisotropy axis (from [97], Fig. 8, p. 134431-6). 100

i

|

o k=l

~

10 -2

[] k= 2 o k=4

a ~ ..

i x=1.28

10-4

10 -6 % z~ o

10-8

zx

o

10-1o

!

1

!

le+02

le+04

le+06

Ski/Ok FIGURE 3.24 Data collapse of avalanche size distribution P ( S ) ~ S - ~ f ( S / S o ) simulated in the RFIM for R < R c i n the presence of a demagnetizing field, using different values of the demagnetizing factor k. The critical exponent r is 1.28, while SO ~" k - 1 / ~ k with 1/r~ k "-, 0.7 (from [101], Fig. 1, p. 233).

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS

229

-M

..,.--

M

FIGURE 3.25 A domain wall separating two regions of opposite magnetization. The discontinuities of the normal component of the magnetization across the domain wall produce magnetic charges.

to end. In this condition, the magnetization process occurs prevalently by domain-wall motion, which can then be studied to recover the statistical properties of the Barkhausen noise [49,50,54,103-105].

Energetics of a D o m a i n Wall Let us consider a flexible 180 ~ domain wall separating two regions with opposite magnetization directed along the z-axis. In the absence of disorder the domain wall would be flat, but in general it will bend to accommodate pinning forces. If the surface has no overhangs, we can describe the position of the domain wall by a function h(~, t) of space and time (see Fig. 3.25). The energy associated with such a deformation field can be computed expanding the micromagnetic free energy discussed in Section 3.4.1, and we can thus split the energy into the sum of different contributions due to exchange and magnetocrystalline interactions, magnetostatic and dipolar fields, and disorder [50]. The contribution from the magnetostatic energy, if the external field H is applied along the z-axis is given by Em = -2#0(H +

Hdm)Ms/ d2r h(~, t),

(3.47)

where we have included the demagnetizing field Hdm, which is in general a complicated function of h(~, t), depending also on the sample shape. In the simplest approximation, however, the intensity of the demagnetizing

CHAPTER 3 The Barkhausen Effect

230

field will be proportional to the total magnetization. Considering the field constant through the sample, it can thus be written as Hdm --

kMs f

/ d2r h(~, t) V d

(3.48)

where the demagnetizing factor k takes into account the geometry of the domain structure and the shape of the sample, and V is a volume factor. The interplay between magnetocrystalline anisotropy and exchange interactions is responsible for the microscopic structure of the domain wall. While a very sharp change of the spin orientation has a high cost in exchange energy, a very smooth rotation of the spins between two domains is prevented by the magnetocrystalline anisotropy. The balance between these two contributions determines the width of the domain wall and its surface energy. The total energy due to these contributions is proportional to the area of the domain wall Edw = 7w / d2r~/1 + IVh(~, t)l 2,

(3.49)

where Yw "" 2 AV/-X-K0is the domain-wall surface energy. Expanding this term for small gradients we obtain

Edw = ?'wSdw+ ~ / d2rlVhff , t)l 2

(3.50)

where Sdw is the area of the undeformed wall. Thus the domain-wall energy represents an elastic interaction that tends to keep the domain-wall flat. An additional elastic interaction is due to dipolar forces, since the local distortions of the domain wall are associated with discontinuities in the normal component of the magnetization. As discussed in Section 3.4.1, this effect can be treated by introducing magnetic charges on the domainwall surface (see Eqn. (3.13)). The surface charge density is zero when the magnetization is parallel to the wall and for small distortions it can be expressed as ~?hff, t) a(~) = 2Ms cos 0 _~ 2Ms ~ , (3.51) 0z where 0 is the local angle between the vector normal to the surface and the magnetization. The energy associated with this distribution of charges is given by Ed

I d2r d2r '

Y

/~~ 2nl~-~'l

c3h(~,t) c3h(~', t) & &' '

(3.52)

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS

231

which integrating by parts can also be written as Ed

f d2r d2r'K(~ - ~ ')(h(~, t) - h(~', t)) 2,

(3.53)

where the nonlocal kernel is given by K(~ - ~ ') -

( 3(z - z')2 ) #~ 127z[r -- ~'[ 3 ~ -~;7 "

(3.54)

The interaction is long range and anisotropic, as can be seen by considering the Fourier transform /~oM2 K(p, q) -

p2

47z2 v/F2 if- 92,

(3.55)

where p and q are the two components of the Fourier vector along z and y. In the preceding derivation we have implicitly assumed an infinitely strong anisotropy, so that the magnetization never deviates from the easy axis. In practice, however, the magnetization will rotate slightly from the easy axis, producing additional volume changes as discussed in Section 3.4.1. This effect leads to a slight modification of the interaction kernel in Eqn. (3.53) 1 p2 [<(P' q) "" x / ~ & 2 q_ Qq2'

(3.56)

where Q - 1 + 2#oM2s/K is a material-dependent constant [79]. The disorder present in the material in the form of nonmagnetic impurities, lattice dislocations or residual stresses is responsible for the deformation and the pinning of the domain wall. In general, disorder can be modeled by introducing a random potential V(~, h), whose derivative gives the local pinning field r/(>, h) acting on the surface. In the particular case of point-like defects, such as nonmagnetic impurities, the random force is given by rl(-f , h) = - ~_.fpff - -fi, h - hi), (3.57) i where (ri, hi) are the coordinates of the pinning centers and fp(x) is the individual pinning force, which typically has a range comparable with the domain-wall width 6w ~ v/-K/A. After coarse-graining at a scale larger than the typical distance between the pinning centers, this disorder can be replaced by a Gaussian random noise with correlations (t](~, h)~](~ ', h')) - c~2(~- ~ ')R(h - h'),

(3.58)

232

CHAPTER 3 The Barkhausen Effect

where R(x) decays very rapidly for large values of the argument. It has been shown that the particular form of R(x) (i.e. due to random-bond or random-field types of disorder) has no relevant effect on the scaling laws associated with the domain-wall dynamics [106]. Another possible source of pinning is due to local variations of the domain-wall energy ?w [79], due, for instance, to random anisotropies. Thus the domain-wall energy becomes a function of position ?(~, h), which we can expand about its average 7w + r/(~, h). Introducing this expression in Eqn. (3.50), we obtain to lowest order an additional random term r/(~, h), whose distribution and correlations can be directly related to the random anisotropies.

Domain-Wall Dynamics and Depinning Transition In most cases, the motion of the domain wall is strongly overdamped, since eddy currents cancel inertial effects. The equation of motion for the wall is thus given by ~h(~, t) 6E({h(~, t)}) F c9------~=- 6h(-f,t) ' (3.59) where E({h(~, t) }) is the total energy functional, derived in the previous section, and F is an effective viscosity. We have neglected here thermal effects, since experiments suggest that these are not relevant for the Barkhausen effect in bulk three-dimensional samples [53]. This could be different in thin films [107], where thermal-activated motion can be described by adding an extra noise term to the equation of motion. Collecting all the energetic contributions, we obtain the equation of motion for the domain wall [50]. In order to avoid a cumbersome notation all the unnecessary factors can be absorbed into the definitions of the parameters. The equation then becomes 3h(~, t) = H - fch + ?wV2h(~, t) ~t

(3.60)

+ / d2r'K(~- ~ ')(h(~ ') - h(~)) + r/(~, h),

(3.61)

where the dipolar kernel K is reported in Eqn. (3.54), and the effective demagnetizing factor is given by k = 4[aokM2s/V and h ~ ~ d2r'h(~ ', t). Owing to the fact that the demagnetizing field term is just an approximation, the dependence of k on the sample shape and size can be quite complex. Variants of Eqn. (3.54) have been extensively studied in the past [49,50,54,79,103-105,108-113].

3.4 MODELSAND THEORIES OF MAGNETIZATION DYNAMICS

233

When the demagnetizing factor k is negligible, as, for instance, in a frame geometry, Eqn. (3.61) displays a depinning transition as a function of the applied field H and the domain wall moves only if the applied field overcomes a critical field Hc. For fields H > Hc the domain wall moves with an average velocity v that scales as v ~ ( H - Hc)flO(Hc - H ) ,

(3.62)

where O(x) is the step function. The critical behavior associated with the depinning transition has been studied using renormalization-group methods [106,114-116], which show that at large length scales the critical exponents take mean-field values [49,50]. This result is due to the linear dependence on the momentum of the interaction kernel (Eqn. (3.54)) in Fourier space [116,117]. In general, if we consider an interface whose interaction kernel in momentum space scales as K(q) = AKlq[ ~, the upper critical dimension is given by de = 2# and the values of the exponents depend on t2. Notice that here we define d as the internal dimension of the domain wall. Hence, in the present analysis, we have a two-dimensional interface (d = 2) moving in a three-dimensional medium. While in general we would expect meanfield behavior, there are situations in which the dipolar coupling could be neglected with respect to the domain wall tension 7w and one effectively could observe the critical behavior associated with # -- 2. In general, for interfaces close to the depinning transition, the response to small variations of the applied field occurs by avalanches whose sizes S are distributed as P(S) ~ S-~f(S/So), (3.63) where the cutoff scales as So ~ ( H - H c ) - 1 / ~ and is related to the correlation length ~ by So ~ ~+~,

(3.64)

where ( is the roughness exponent (Fig. 3.26). The correlation length diverges at the depinning transition as ~ ( H - Hc) - v ,

(3.65)

which implies 1

- = v(d 4- ~).

(3.66)

(7

The average avalanche size also diverges at the transition (S) ~ ( H - Hc) -~,

(3.67)

CHAPTER 3 The Barkhausen Effect

234

[

S

FIGURE 3.26 The domain wall moves between two pinned configurations in an avalanche of size S ~, ld+(. w h e r e 7 is related to z and r~ b y ( 2 - z) ?= ~ ,

(3.68)

? = (1 + v(),

(3.69)

a n d to v a n d z b y [106] thus i m p l y i n g l+v~ . (3.70) v(d + ~) The other e x p o n e n t relevant for the Barkhausen effect describes the distribution of avalanche durations z = 2-

P(T) ~, T-~ g(T/To),

(3.71)

w h e r e the cutoff diverges at the transition as To " (H - Hc) -1/A. From Eqn. (3.65) a n d the relation To ~" ~z w e obtain A - 1/zv and

vd- 1 = 1+ ~ .

(3.72)

ZV

Finally, an additional s y m m e t r y of the equation of m o t i o n can be used to reduce the n u m b e r of i n d e p e n d e n t exponents t h r o u g h the relation v(# - ~) - 1.

(3.73)

The r e n o r m a l i z a t i o n - g r o u p analysis allows for a d e t e r m i n a t i o n of the critical exponents in the f r a m e w o r k of the e = dc - d expansion. As discussed above, for # = I w e expect mean-field results z - 3/2 a n d a = 2, while for/~ - 2 to first order in e one obtains z = 5/4 and c~ - 11/7. Recently a t w o - l o o p expansion has been carried out, allowing for an estimate to order O(e 2) of the exponents (z = 1.24 and a - 1.51) [116]. Finally, n u m e r i c a l simulations yield z = 1.27 and a = 1.5.

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS 10 6

235

............................................

K..ff..

o

10 4

H* = 0.35

[] H * = 0 . 4 0 " ~

.

o

H* = 0.45

- "~,~ 1.73 10 2 r.~ o

o

10 0 o

o o

10 "2

10 -4

o

o

o

o

[] .

.

.

.

.

.

.

.

i

.

101

lO 0

.

.

.

.

.

.

.

|

.

.

.

.

.

10 2

.

.

.

i

10 3

.

.

.

.

.

.

.

.

,

o .

10 4

.

.

.

.

.

.

10 5

S

FIGURE 3.27 The distribution of avalanche sizes in the short-range domain-wall model at k = 0 when the field is swept from H = 0 to H = H* < Hc. For H* --~ Hc, we obtain Zint = z + ~. The discussion above applies strictly to the case k = 0 and scaling requires that H is close to H c . If one were instead to r a m p the field slowly, sampling the avalanche distribution over all the values of the field, the result w o u l d be different. We need to integrate Eqn. (3.63) over H [50] Pint(S) -with

f

Hc dH S -r f(S(H

- H c ) 1/a)

~

S-'~int ~

(3.74)

17-}- (9". Using the values of ~ and a reported above, we obtain 1.73 for # = 2 and "Cint --~ 2 for/~ = 1. A similar discussion can be repeated for avalanche duration, yielding ~ i n t - - ~ -}- A . The numerical result is ~ i n t = 2.3 and for/~ = 2 and ~ i n t = 3 for/~ = 1. In a typical Barkhausen experiment, however, the situation changes, since k > 0 and H is n o r m a l l y increased at constant rate. The effective field "tint

"~int - -

--~

acting on the d o m a i n wall is given by Heff - ct - kh. Thus, according to Eqn. (3.62), w h e n Heff < H c the d o m a i n wall is p i n n e d and the effective field increases. As soon as Heff > Hc the d o m a i n wall acquires a finite velocity and h increases. As a consequence, the effective field will be reduced until the d o m a i n wall stops. This process keeps the d o m a i n wall close to depinning transition, but criticality is only reached in the limit c ~ 0 and k ~ 0. A similar m e c h a n i s m occurs in models of self-organized criticality [118]. The d e m a g n e t i z i n g factor k represents a restoring force hindering the propagation of Barkhausen avalanches. Thus, w h e n k > 0, we expect to

CHAPTER 3 The Barkhausen Effect

236

7000 9

~o

9

L=60

9

L=IO0

0

.--- 1000 o

k-0.66

0

let= 1

~n~~..~

long range 100

,

,

,

,

,

,

. . . .

10-5

I

10-4 k/L 2

.....

I

. . . . . . . .

I

. . . . . . . .

I

. . . . . . .

a L=200 ~o 10 4 0 N

o

!

'I

3 t

1

~5 10a 0

10 2 10-6

.... . ,

short range

105

i

10.4

10-3

10"2

k/L 2

FIGURE 3.28 The cutoff SOof the avalanche size obtained from numerical simulations for different demagnetizing factors. The two universality classes correspond to a kernel in the equation of the [q["-type in Fourier space, with # = 1 (long-range dipolar interactions - top), and/~ - 2 (short-range elastic interactions - bottom).

find a finite avalanche size cutoff So. The d o m a i n wall cannot p r o p a g a t e over distances larger than ~, defined as the length at w h i c h the interaction term AK[q[ ~ is equal to the restoring force A K h ~ - ~ "-, k~ah,

(3.75)

w h i c h implies ",, (k/AK)-Vk

Vk -- 1/(# 4- d).

(3.76)

3.4 MODELS AND THEORIES OF MAGNETIZATION DYNAMICS

237

The avalanche size and duration distribution cutoffs So and To can be obtained using the scaling relations reported above So ~" D ( k / A k ) -1/~k,

1/ak -- (d + ()/(/2 + d),

(3.77)

and similarly To ~ (k/Ak) - & ,

A k - z/(/2 + d),

(3.78)

where D ~_ x / ~ 2) denotes the typical fluctuation of the disorder. Inserting in Eqs (3.77), (3.78) the renormalization-group results to first order in [106,114,115], ( = (2t2 - d)/3 and z =/2 - (4# - 2d)/9, we obtain 1/ak = 2/3 and Ak = (t2 - (4/2 - 2d)/9)/(/2 + d). Using the results obtained to order ~2 [116], one instead obtains for/2 = 2, 1/ak = 0.71 and Ak = 0.345. The numerical values obtained from simulations are in good agreement with this prediction (see Fig. 3.28). The mean-field theory, which provides a good qualitative description of the depinning transition and describes quantitatively the data for d - dc (apart from logarithmic corrections), is obtained by discretizing the equation of motion, coupling all the sites with the average domain-wall position [119]. The dynamics of such an infinite-range model is described by dhi dt

= ct - kh + l(h - hi) -Jr- lli(h),

(3.79)

where J is an effective coupling and i - 1 . . . . N. S u m m i n g over i for both sides of Eqn. (3.79), one obtains an equation for the total magnetization m dm

N = ~t - km + ~ tli(h ). dt i=1

(3.80)

This equation has the same form as the ABBM model (see Eqn. (3.28)) [16]. To make the similarity more stringent, we should interpret Y~i//i as an effective pinning W(m), with Brownian correlations. W h e n the d o m a i n wall moves between two pinned configurations, W changes as //

W ( m ' ) - W(m) - ~

Ar/i ,

(3.81)

i=1

where the sum is restricted to the n sites that have effectively m o v e d (i.e. their disorder is changed). The total n u m b e r of such sites scales as n --- Id and in mean-field theory is proportional to the avalanche size S - Im' - ml

CHAPTER 3 The Barkhausen Effect

238 '

'

......

I

........

I

........

i

. . . .

'"'1

'

''''"'1

'

''

....

,

~

_

D c=O

i

c=o.2

[

v

c = 0.32 .... ,c7~

10 ~

101

~q~v~/ ........

10 2

(a)

,

........

10 3

,

10 4

........

,

10 5

10 6

s

1

(b)

i

'i

10

1O0

1000

T

FIGURE 3.29 (a) The distribution of avalanche sizes in the infinite-range model for different driving rates when N = 32696. The fit of the power-law part yields = 3/2 - c/2, with c = ~,/D and 1/D ~" 1.2. (b) The corresponding distribution of avalanche durations. The power-law part is fit with an exponent a = 2 - c.

(since S ~, la+~ and ~ - 0). A s s u m i n g that the Ar/i a r e uncorrelated a n d have r a n d o m signs, w e obtain a Brownian effective p i n n i n g field ([W(m') - W(m)] 2) - D i m ' - mI,

(3.82)

w h e r e D quantifies the fluctuation in W. Thus the Brownian p i n n i n g field, i n t r o d u c e d p h e n o m e n o l o g i c a l l y in the ABBM model, is an effective description of the disorder resulting from the collective m o t i o n of a flexible d o m a i n wall. As a consequence of this m a p p i n g , w e expect that the infinite range m o d e l displays the same frequency d e p e n d e n c e of the exponents as in the ABBM m o d e l (Eqn. (3.34)). Simulations confirm this claim, as s h o w n in Fig. 3.29.

3.5 DISCUSSION OF THEORETICAL RESULTS 3.5 D I S C U S S I O N

239

OF THEORETICAL RESULTS

3.5.1 AVALANCHE DISTRIBUTIONS The models discussed in Section 3.4 provide different results and theoretical interpretations for pulse distributions. While we have already reported the results obtained for the various models, for the sake of clarity we summarize here again the main conclusions. We have distinguished between two general classes of models: spin models and domain-wall models. Rigorously speaking, the second class is contained in the first, since a domain structure is typically present in spin systems, but the analysis of domainwall dynamics would be needlessly complicated. It is useful instead to treat separately the case in which domain-wall propagation does not interfere with nucleation. The two classes of model yield different interpretations of the origin of scaling in the Barkhausen pulse distributions. In spin models, scaling is ruled by the proximity to a disorder-induced critical point. The cutoff to the power-law scaling is determined by the distance from this critical point. The broadness of the critical region would justify the wide occurrence of scaling in experiments, without any apparent parameter tuning [69]. In domain-wall models, scaling is dictated by a different critical point: the depinning transition. The interplay between slow driving (c ~ 0) and small demagnetization k ~ 0 keeps the domain wall close to the depinning critical point. The cutoff of the distributions is then controlled by the value of k, which in strip geometries is typically small. A collection of the numerical estimates of the Barkhausen noise distribution exponents for the RFIM and domain-wall models is reported in Table 3.2. Depending on the problem at hand, the appropriate model for the Barkhausen noise may vary. In soft magnetic ribbons the domain structure is typically controlled by dipolar interactions and the sample shape, imposing a parallel set of domain walls spanning the sample from end to end. In this condition, the depinning scenario should apply, as is testified to by the experimental findings. The avalanche exponents are in good agreement with the prediction of the depinning model and the cutoff is controlled by the demagnetizing factor. In particular, experiments on polycrystalline SiFe are well described by the ABBM model or, equivalently, by a long-range depinning model. The exponents measured in amorphous alloys are instead well fitted by the short-range interface model. This is due to the fact that in amorphous alloys the average anisotropy vanishes and thus dipolar interactions have no effect on the domain-wall dynamics (i.e. Q in Eqn. (3.56) is very large). On the other hand, in polycrystalline materials, every grain is the source

240

o~,,~

O

!

o~,,~

r

O @ o~,,~

a~

o,,~

o,,~

o3

CHAPTER

~

r

r

3

r

r162 r

r-~

Lr

r162 I ~ - Hr - H

tr3 o 3

r162

I ~-H-H

~tr3

The Barkhausen

tr3

~

r

-H ~ ~ - H r

O3 O

tr3

~

N

li

Effect

3.5 DISCUSSIONOF THEORETICAL RESULTS

241

of a relevant anisotropy and long-range effects occur. This is confirmed by the fact that long-range exponents are found when small crystallites are induced in an amorphous matrix by proper thermal annealing [21]. The scaling exponents measured in experiments on NiFe [51] wires and Vitrovac [25] (see Table 3.1) could also be explained in this framework if one assumes that the Barkhausen signal is not stationary. This could occur if the field is not swept up to saturation, so the effective field varies and is not kept around Hc as in most cases. Then one would obtain integrated exponents tint, 0 q n t instead of r and a. The striking similarity between the experimentally measured exponents and the theoretical ones (i.e. "~int = 1.72 and ~ i n t - - 2.3) supports this conclusion. Spin models could instead apply to cases in which the domain structure is more intricate and the effect of dipolar forces negligible. A typical example would be hard magnetic materials, with strong local anisotropies. Systematic investigation of the Barkhausen effect in hard magnetic materials has just been started and a quantitative comparison between experiments and theory is still not possible. An important problem to address, regardless of the theoretical framework, is the effect of the field rate on the Barkhausen-noise statistics. As we discussed in Section 3.4.2, the ABBM predicts a linear dependence from the exponents r and a from the field rate, encoded in the parameter c. This result also applies for the mean-field domain-wall model (see Fig. 3.29) and agrees well with experiments on FeSi alloys. Simulations of the shortrange domain-wall model show that the scaling exponents do not change continuously as in mean-field, but a peak appears for large values of the jump size and grows with the rate [105] (see Fig. 3.30). While one could fit the result for higher driving rate with different exponents, such a fit does not appear very reliable and a constant exponent coexisting with growing peak seems a more appropriate approximation. A similar result is found in amorphous soft magnetic alloys, as shown in Fig. 3.5. The effect of the driving rate on the avalanche distributions was analyzed in more general terms in [26]. A set of results can be derived under some basic assumptions: (i) The system is close to a critical point, so that in the adiabatic limit (c --, 0) the size and duration distributions are a power law. (ii) The avalanche signal is stationary under the time window considered. (iii) In the adiabatic limit, the average number of avalanche nucleation events per unit field increase is a smooth function of the driving field. (iv) For low c, the avalanche dynamics is independent of c.

CHAPTER 3 The Barkhausen Effect

242

FIGURE 3.30 The distribution of Barkhausen jump sizes from a short-range domain-wall model for different driving-field frequencies. While the scaling regime remains essentially unchanged, a peak occurs around the cutoff (see also [105]). (v) The field increases between avalanche nucleations are independent of the avalanche sizes. (vi) The avalanche sizes are uncorrelated in time. (vii) Avalanches are nucleated randomly in space. All these assumptions are reasonable both for the domain-wall models and for spin models. Notice, however, that assumption (ii) restricts the derivation to a stationary signal and thus does not apply to distributions measured integrating along the entire hysteresis loop. Under the assumptions above, one can show that the effect of the driving rate depends on the value taken by the exponent ~ in the duration distribution for c ~ 0. For ~ > 2 the distributions are unaffected by the driving rate, while for a < 2 a peak appears at large sizes and durations. The interesting case is a = 2, which corresponds to a linear dependence of the exponents from the driving rate c, recovering the ABBM result in Eqn. (3.34). Thisgeneral derivation provides an explanation of the results of experiments and numerical simulations, predicting the conditions under which drive-dependent exponents should be observed. 3.5.2 THE POWER SPECTRUM

Explaining the power spectrum of Barkhausen noise has been one of the main objectives of the past theoretical activity. In earlier approaches the

3.5 DISCUSSION OF THEORETICAL RESULTS

243

power spectrum was considered as the result of a superposition of elementary independent events (see, for instance, [121,122]) without a clear relation to the microscopic magnetization processes. A step forward in this direction was made in the framework of the ABBM model [16,17]. From Eqn. (3.28) one can derive a Fokker-Plank equation describing the evolution of the probability distribution P(v, tlv0) to have a velocity v at time t when the velocity at t = 0 is v0 [16] OP

~t

__( 0 Ov

(kv-

c)P + D

. Ov

J

(3.83)

Notice that, as in Section 3.4.2, we have rescaled time, setting a damping constant equal to one. The velocity correlation function is defined as G(t) =__{(v(t) - c/k)(v(O) - c/k)}

/ dv dv0 (v - c / k ) ( v o - c / k ) P ( v , t[vo)P(vo),

(3.84)

where P(vo) is givenby Eqn. (3.33). Deriving Eqn. (3.84) and using Eqn. (3.83), we obtain an evolution equation for G(t) dG = -kG, dt

(3.85)

which can be easily solved yielding G(t) I c / k exp(-kt).

(3.86)

The power spectrum has thus a Lorenzian shape F(og) =

2c o92 q- k2"

(3.87)

This simple result contains several interesting predictions: (i) At large frequencies the spectrum decays as o9-2. (ii) The scaling is cutoff below a frequency o90 = 1/k. (iii) The amplitude of the spectrum increases with the field rate c. These features describe qualitatively the shape of the experimentally measured power spectrum, but not quantitatively. For instance, the tail of the spectrum often decays with an exponent other than 2 and the cutoff frequency depends on k but not simply as 1/k.

CHAPTER 3 The Barkhausen Effect

244

To reproduce the low-frequency increase of the power spectrum, the ABBM model can be modified introducing a correlation length ~* in the pinning field [16]. This gives rise to a power spectrum of the type 0) 2

F(0)) oc

(0) 2 Jr-k2)(0) 2 q- Zc2) '

(3.88)

where Zc oc ~*. The ABBM predictions for the power spectrum can be compared with numerical simulations of the infinite-range domain-wall model (Eqn. (3.79)) for different values of c. The results in Fig. 3.31 are, as expected, in good agreement with the theory. Several attempts have been made in the past to link, through scaling relations, the power-spectrum exponent to the scaling exponents describing the avalanche distributions [25,50,51,75,103]. The comparison between these results and experiments is, in most cases, not satisfactory. In any case, a simple exponent would just describe the tail of the power spectrum, while a complete theory should explain the entire shape observed experimentally. The problem was recently revisited in [74] where all the assumptions underlying previous scaling approaches [25,50,51,75,103] were carefully scrutinized and tested. Through a combination of analytical methods and numerical simulations, Kuntz and Sethna [74] found a scaling relation describing the high-frequency tail of the spectrum, which seems to be in good agreement with experiments (see Fig. 3.10 and [65]). The derivation of [74] follows different steps involving some assumptions that can be tested numerically and experimentally. Close to criticality one expects that the average size for an avalanche of duration T scales as S(T) ,~ T 1/~vz and, consistently, the average shape scales as v(T, t) - T 1/crvz-1 fshape(t/Z),

(3.89)

where v is the signal voltage, t is the time and fshape(t/Z) is a (possibly universal) scaling function. The second assumption involves the probability P(vlS) that a voltage v occurs in an avalanche of size S. This probability scales as:

P(vlS) -

v -1 fvoltage(vS r~vz-1)

(3.90)

w h e r e fvoltage is another universal scaling function. Using this equation,

one can show that the energy E - (v2) of an avalanche of size S scales as E(S) ~ S 2-r The analysis proceeds with the calculation of the voltage correlation function G(t) - f dt'(v(t + t')v(t)),

(3.91)

3.5 DISCUSSION OF THEORETICAL RESULTS

245

FIGURE 3.31 The power spectrum measured in the infinite-range domain-wall model for different field rates c. The tail is well fitted by a Lorenzian function with amplitude proportional to c. The entire spectrum is in perfect agreement with the ABBM result.

FIGURE 3.32 The power spectrum measured in the short-range domain-waU model for different field rates c. The tail is well fitted by a power law o3-1"8, in good agreement with Eqn. (3.94), which predicts an exponent 1/r = 1.77.

CHAPTER 3 The Barkhausen Effect

246

which can be decomposed into various contributions for each avalanche G(t[S). This quantity obeys a scaling law

G(tlS)

= S 2-r~vz

f6(tS-r

(3.92)

The contribution to the power spectrum is then obtained by a cosine transformation F(ogIS) -

f0 ~

dt cos(ogt)G(tlS)

=

S2fenergy(og-1/r

(3.93)

To obtain the power spectrum of the signal one should average Eqn. (3.93) over different avalanche sizes. A naive estimate of the integral resulting from the average of Eqn. (3.93) using P(S) "~ S -z, leads to the incorrect prediction [25,51,75] F(og) o g - ( 3 - z ) / c r v z . The reason behind this incorrect result comes from the fact that fenergy(X) "~ 1/x for large x. Thus, if z < 2, the leading contribution to the averaged power spectrum comes from the upper cutoff in the size distribution, ultimately leading to the result [74]: ~

F(og) "~

og-1/r

(3.94)

Using mean-field exponents r~vz - 1/2, we recover the ABBM result F(og) o9-2. The result in Eqn. (3.94) agrees reasonably well with experiments (see Fig. 3.10) and is confirmed by numerical simulations of the domain-wall model (see Figs 3.31 and 3.32). 3.5.3 THE AVALANCHE SHAPE

The problem of the Barkhausen avalanche shape was first introduced in [74] and proposed as a strong test to characterize the universality classes of the Barkhausen noise. Subsequently, [66] analyzed more systematically the results of numerical simulations for the RFIM with and without a demagnetizing field. The conclusion reached is that the average avalanche shape is in both cases symmetric and close to a parabolic shape. The pulse shape changes slightly with and without a demagnetizing field, although a reasonable collapse of the two sets of curves can be obtained multiplying one set by a factor (see Fig. 3.33). Small differences emerge from a more detailed fitting procedure. A favorable comparison with experiments and a theoretical explanation of the numerical results is, however, still lacking. The general problem of the shape of a stochastic excursion has been recently addressed in more generality by Baldassarri et al. [123]. The authors show that for a broad class of stochastic processes the average shape of the excursion is a semicircle. This occurs irrespective of the value of the scaling exponent (the equivalent of r~vz). Thus, in this sense, the

247

3.5 DISCUSSION OF THEORETICAL RESULTS 0.6

0.4 A

V 1

- - - - Front Propagation Mode l Nucleation Model

1

w~ i

[,, 0.2

0

0.2

0.4

0.6

0.8

t/T

FIGURE 3.33 The avalanche shapes for the RFIM with and without a demagnetizing field (the second case is referred to as front propagation). The shapes for different times have been collapsed using the value 1/zav = 1.72 and 1/zav = 1.75 in the two cases. An additional normalization factor allows rescaling of the two sets of curves together (from [66], Fig. 1, p. 046139-3).

shape would appear to be more universal than the exponents. On the other hand, the shape and symmetry of the average shape can be changed at will, introducing correlations in the stochastic process [123], and it is even possible to construct two stochastic processes with the same scaling exponents, but different shapes. This calls into question the idea that the shape is a universal function [124]. The approach of [123] can be extended to compute the exact shape of the Barkhausen pulses in the ABBM model [125]. Neglecting the contribution of the demagnetizing factor k, Eqn. (3.31) describes the motion of a one-dimensional random walk in a logarithmic potential E(v) = - c log(v), where the magnetization s plays the role of time. The corresponding Fokker-Planck equation is:

c~P(v, s) ts

--

tv

+

v

(v s)

tv

)

(3.95)

CHAPTER 3 The Barkhausen Effect

248

where P(v, s) is the probability of finding the walk in v at s. We are interested in a solution of this equation with the initial condition P(v, O) = ~(v - v0) and an absorbing b o u n d a r y at the origin v = 0. Following [90], we can express this solution in terms of modified Bessel functions. For the case 0 < c < 1, which is the condition of the ABBM model to have p o w e r laws in the avalanche distribution (see Eqn. (3.1), and [16]), the probability P(v, s]vo, 0; c) for a walk starting at v0 to be at v after a 'time' s, in the limit v0 --* 0 is simply proportional to a power of v times a Gaussian with variance s, and the average excursion of the walk is then obtained as

(v) cx ~ / s ( S - s).

(3.96)

This result means that the universal scaling function gshape of Eqn. (3.5) is proportional to a semicircle. It is also possible to calculate the function fshape of Eqn. (3.89) in the time d o m a i n [125]. By definition, the avalanche size s at time t is given by the integral of v(t, T) from time zero to time t:

s=

f0tdtlv(t ' , T) cx T 1/avz fdo t/T f(x)dx,

(3.97)

which provides an expression for s -- s(t, T) as a function of t and T. If we n o w impose

v(t, T) = v(s - s(t, T), S(T)), we get an integral equation

(3.98)

for fshape involving gshape:

fshape(X) CKgshape(~oXg(x')dx').

(3.99)

Using the form of gshape c o m p u t e d previously, we can solve this equation with the b o u n d a r y conditions f(0) =f(1) = 1. The solution is

fshape(X) CKsin(nx),

(3.100)

3.5 DISCUSSION OF THEORETICAL RESULTS

249

i.e. the universal function fshape is proportional to an arch of sinusoid. Finally, for the normalized avalanche one obtains (see Fig. 3.11) v(s, S) - sl-evzrcv/(s/S)(1 - s/S),

(3.101)

v(t, T) - T1/evz-l~/2 sin(~t/T).

(3.102)

The average shapes obtained with the various models have all a notable difference with experimentally measured pulse shapes: models yield symmetric shapes while experiments give rise to asymmetries. In a typical experiment, the pulse rises fast and decays slowly (see Fig. 3.11). This behavior can be traced back to eddy-current propagation, as originally discussed in [6,126,127]. In the models analyzed so far, a direct relation between the instantaneous magnetization rate and the local effective field has been implicitly assumed. This is not strictly true in conductive materials where local magnetization changes are the source of eddy currents, which diffuse into the sample. This means that the damping term in Eq. (3.59) should be replaced by a convolution of the magnetization changes v(t) with an eddy current response function Fv(t) --,

i

t

dffl(t-

t')v(t').

(3.103)

In practice, the form of fl depends on the sample shape and on the domain structure and can be computed in some simple cases [6,126-128]. Once a non-local damping term is introduced in the model, the result is an asymmetric shape, whose features are, however, far from being universal. In conclusion, the average pulse shape obtained in conducting materials is not an appropriate indicator of universal scaling, being ruled by eddy-current propagation, which depends strongly on the sample geometry [128]. 3.5.4 T w o DIMENSIONS

The discussion presented so far has focused on three-dimensional systems, which is relevant for most Barkhausen-noise experiments. The experimental results obtained in thin magnetic films (see Section 3.3.3) require instead an analysis of two-dimensional models. This task is not straightforward, and several new features arise in two dimensions. The demagnetizing field that plays a major role in determining the statistical properties of the Barkhausen noise vanishes for very thin samples. For this reason, we often do not have a simple parallel domain structure, as in three-dimensional ribbons, and an analysis of single domain walls would not be appropriate. On

250

CHAPTER 3 The Barkhausen Effect

the other hand, an approach in terms of spin models suffers for different difficulties, since there is no consensus on the presence of a disorder-induced transition in two dimensions. The RFIM has been extensively simulated in d = 2, using extremely large system sizes (up to (7 000) 2) [69,92]. Despite this, it has not been possible to obtain a reliable estimate of the threshold Re for the disorder-induced transition. The simulations are consistent either with a vanishing threshold or with very large correlation-critical-exponents (i.e. v = 5.3 4- 1.4, 1/r~ = 10 4- 2 with Re = 0.54). An avalanche scaling exponent has also been estimated as " ~ i n t - - 2.04 -4- 0.04. Additional results and analysis have appeared in the literature, but are not particularly reliable, being based on much smaller system sizes. Typically, the presence of a disorder-induced transition is assumed and exponents are evaluated [60,129,130]. A similar discussion would apply to other models analyzed in the literature, like the RBIM [95,131], the RFIM with site dilution [132], and dipolar interactions [99]. In summary, the large-scale hysteretic behavior of disordered spin models in d = 2 is quite controversial, and quantitative predictions seem at the moment not to be available. A possibility is that d = 2 is the lower critical dimension, as is expected for these models in equilibrium [133]. If this is the case, power-law scaling may still be observed for low disorder, with the conjecture that r = 3/2 [69]. Nevertheless, we feel that, at present, the results of both experiments and models are still too unreliable to a make a comparison. Even in thin films there are several cases in which a domain-wall model may be appropriate. In particular, in the magneto-optical measurements of [32,33] one directly observes Barkhausen jumps due to domain-wall motion. The equation of motion for a single domain wall in two dimensions is similar to the one in three dimensions (e.g. Eqn. (3.61)) with some notable differences. For thin films with in-plane magnetization the demagnetizing factor k is virtually zero and in Fourier space the dipolar kernel (Eqn. (3.54)) does not scale as q but as q2 log(aq) where a is a small-scale cutoff [134,135]. Apart from the logarithmic correction, the dipolar term is similar to the domain-wall energy and thus, in principle, should not significantly affect the universality class of the depinning transition. Numerical simulations, however, yield slightly different results in the two cases (for instance, ~ = 1 with long-range forces and ~ = 1.25 without them [136,137]). While the short-range interface model has been studied extensively in two dimensions, less is known about the long-range problem. A result that could be relevant for the Barkhausen effect would be the avalanche distribution exponent, which has been numerically estimated for short-range models: = 1.13 4- 0.02 [138], r = 1.02 4- 0.29 [139]. The most reliable and precise estimate is probably r = 1.115 [140].

3.6 CONCLUSIONS AND PERSPECTIVES

251

Before comparing simulation data with the experiments, one has to be certain of the quantity effectively measured. When the demagnetizing factor is zero, a ramp up of the field does not lead to a Barkhausen stationary signal with Heff = Hc, but the avalanches grow up to He. One would then measure an effective exponent "Cin t = ~ -}- O'. Given the relation l/or = v/(d + ~) and ~ = 1.25 and v = 1.33 [115] we obtain " t i n t ~ " 1.6. If, instead of ramping up the field, one kept it fixed just below Hc as in [32,33], then we would expect that the avalanche distribution would scale with z. Notice, however, that the avalanche 'size' S defined in [32,33] is not what we have defined, but is instead proportional to the change in the displacement Ah. Avalanches are only counted when the transverse length I (see Fig. 3.26) is larger than the system size L. Hence S = LAh with a constant L. Using scaling relations one can show that the exponent measured Zh -- ((~ + 1)z -- 1)/~ ~- 1.2, which is still lower than the estimated value but could be consistent with the data. Finally, we mention that spin models have also been used to study the dynamics of a single domain-wall including overhangs. Simulations have shown a crossover from flat to fractal depending on the disorder strength [141]. The relation between these phenomena and the Barkhausen noise still remains to be explored. 3.6 C O N C L U S I O N S

AND PERSPECTIVES

The Barkhausen effect has been studied for almost a century. While the essence of the phenomenon, i.e. an irregular magnetization reversal process occurring at the microscopic scale, was already indicated in the original paper by Barkhausen, a quantitative and detailed understanding was gained only recently. The key to this achievement is twofold: the injection into the field of methods and theories drawn from statistical mechanics of nonequilibrium critical phenomena, and the identification of a standard experimental setup to collect reproducible and statistically significant data. These two aspects are tightly linked, since the necessity of rationalizing the experimental setup and performing a detailed statistical analysis of the data were dictated by the theory, which was stimulated in its turn by the indication of scaling behavior coming from experiments. The successful understanding outlined above is restricted mostly to the Barkhausen noise measured in bulk soft magnetic alloys, ribbons in particular. These are particularly instructive from the theoretical point of view because of the relatively simple parallel domain structure, induced by the interplay between dipolar forces and the sample geometry. The magnetization process is mainly due to domain-wall motion in a disordered landscape, which is responsible for the avalanche dynamics leading

252

CHAPTER 3 The Barkhausen Effect

to the observed noise. The scaling behavior describing the statistical properties of the noise (avalanche distributions, power spectrum) can be related to an underlying domain-wall depinning transition. Critical phenomena are typically associated with a certain degree of universality: the scaling exponents do not depend on the microscopic details of the system but only on general symmetries, dimensionality, and long-range properties of the interactions. One would thus expect to be able to classify materials in broad universality classes, sharing the same critical exponents, irrespective of the microscopic details, such as sample composition, grain size or disorder strength. To some extent experimental results confirm this expectation: polycrystalline SiFe alloys with different grain sizes and Si content, together with partially crystallized Co-based alloys, were all found to share the same critical exponents. Amorphous alloys of different composition are instead grouped in a different class, characterized by a different set of exponents. These two groups of exponents coincide with those associated with the depinning transition of a domain wall, with or without longrange dipolar interactions. The latter are enhanced by local anisotropies, induced by the polycrystalline grains and absent in the amorphous structure. The domain-wall depinning scenario successfully predicts additional features of the Barkhausen effect, such as the dependence on the field rate and the demagnetizing factor, and can thus be used as a basis for understanding the role of the microstructure on the macroscopic magnetization properties. The new frontier of the Barkhausen effect is represented mainly by thin films, as testified in the recent literature. Inductive methods, that have played a leading role in noise measurements in bulk materials, become less effective as the thickness of the film is reduced to smaller and smaller sizes. Magneto-optical techniques do not suffer from this problem and the improved resolution gained in time and space is allowing for precise noise measurements, joined to a direct visual inspection of domainwall dynamics. The effort in this direction, however, is still in progress and a systematic analysis of the noise statistics in different materials has not yet been completed. Other techniques, such as magnetic-force microscopy [142] or Hall sensors [143], are emerging as powerful tools for studying the Barkhausen noise at the nanoscale. For instance, a recent experiment revealed Barkhausen jumps due to a domain-wall depinning from the atomic Peierls potential [144]. As a note of warning, we note that these techniques reveal the magnetic properties of the surface and are thus not effective in studying the crossover from bulk three-dimensional to thin two-dimensional behavior. To this end, it would be desirable to improve inductive techniques, as recently proposed.

APPENDIX

253

Apart from the experimental problems outlined above, thin films pose new theoretical challenges. It is unlikely that a simple translation of threedimensional models to two-dimensions will provide a significant understanding of the Barkhausen effect in thin films. The domain structure is, in most cases, more complicated than in the bulk, and a single-domain-wall model will be inappropriate. The general framework of nonequilibrium critical phenomena and the understanding gained in domain-wall and spin models could nevertheless lead to a successful theoretical understanding. Finally, we would like to mention here hard ferromagnetic materials for which few Barkhausen noise measurements have been reported in the literature [145-147]. In these nucleation-type materials, the noise is not due to the motion of domain walls, but exclusively to grain reversal. Correlation between neighboring grains plays a significant role [145], as well as the temperature. These systems, having large local anisotropies and nucleation fields, could be well represented by disordered spin models such as the RFIM. Recent studies of these models have contributed to renewing the interest in the Barkhausen effect, but a clear experimental realization is still lacking.

ACKNOWLEDGEMENTS We wish to thank all our collaborators who have greatly contributed to our current understanding of the Barkhausen effect: M. J. Alava, V. Basso, G. Bertotti, P. Cizeau, F. Colaiori, C. Castellano, R. da Silveira, A. Magni, L. Santi, R. L. Sommer, H. E. Stanley, and M. B. Wiessmann. We thank Ingeborg Walter and Michael Zaiser for their help in the translation.

APPENDIX. T R A N S L A T I O N OF THE O R I G I N A L PAPER BY B A R K H A U S E N Two phenomena discovered with the help of the new amplifier by H. Barkhausen 1. Noise during magnetization of iron With the improvement of the new vacuum tube amplifier it is possible to generate easily a 10000-times larger current, which means a 100 million gain in power. With this apparatus, one can reveal electric or magnetic alternate fields that otherwise, due to their weakness, would be inaccessible to our knowledge. In some sense it is as if we discovered a microscope, and one with almost 10000-times magnification! Shortly after W. Shottky has indicated that, through a large amplification, one can, in some sense, hear the electrons flying in the amplifier tube, since the current they generate reveal the spontaneous fluctuations predicted by the kinetic gas theory.

CHAPTER 3 The Barkhausen Effect

254

G

si

FIGURE A1 Fig. I of the original paper.

I discovered a similar phenomenon two years ago on occasion of an experiment that I did in collaboration with Dr T. Tuczek. Iron produces a noise when magnetized. As the magnetomotive force is smoothly varied, the molecular magnets flip in jumps to their new position. Because of this, they generate irregular induction pulses in a coil w o u n d around the sample, that can then be heard as a noise in a telephone. I have now examined this phenomenon more closely. Fig. A1 shows the experimental apparatus. The iron sample E to be studied was put inside a small coil S with a diameter of 25mm and 300 turns. One end of the coil was connected to a telephone T through a 10000-times amplifier V and the other end to a mirror galvanometer G. The latter was so strongly d a m p e d that a deflection was staying almost unaltered for several seconds. Then each modification of the amplitude in practice is proportional to the corresponding modification of the induction flux of the coil. Displacing the U-shaped 10cm-long magnet M, one could observe at the same time the modification of the induction and the noise strength. I determined the contribution of the iron sample to the former, measuring and subtracting the smallest observable deflection of the galvanometer for a given displacement of the magnet, but in absence of the iron sample. The displacement was made by hand. For this reason, the precision was not so large. Next, it was shown that the noise became weaker for larger samples. 20mm large samples yielded effectively no more noise. As will be shown later, the reason for this is related to the too small variation of the induction (per unit area). The small magnet was not sufficient to magnetize strongly enough the large samples. So a pile of slabs gave rise to a weaker

256

CHAPTER 3 The Barkhausen Effect

FIGURE A2 Fig. 2 of the original paper.

FIGURE A3 Fig. 3 of the original paper.

of various sizes with different degrees of stability. Noise results only from the disruption or formation of greater assemblies. For a motion corresponding to Fig. A2, only small assemblies are modified in the region A-B and noise does not start. For a motion across B in the direction - R , little by little also medium and large assemblies get involved. In the direction +R, all small and middle size assemblies have been already dissolved in the first change coming from far away up to A. Only the largest and more stable assemblies in this direction are still remaining. Those will be first dissolved by a further change beyond A. In good agreement with this idea, moving across B one hears a uniform, low sough, while a loud crackling noise starts across A. For very slow variations, for instance when the magnet is moved far away from the sample, one can clearly distinguish isolated snapping hits. It appears that in steel the molecular magnets do not have the ability to join in large assemblies. In this material one only hears the low sough. Also the behavior described in Fig. A3 is in agreement with this idea. The weak assemblies, magnetized in the backward motion (from B to C in Fig. A2), are overcome again in the direction R, before A is reached.

APPENDIX

255

noise than a single slab coming from the same pile. The most suitable samples were found to be 1 to 1/2 mm large wires or 5 to 10mm thin ribbons. Employing stronger magnets one could also use stronger iron samples. Later it was established on a large number of different samples that the noise was stronger as the iron was softer. Hardened steel gave almost no noise, while soft annealed iron yielded a particularly strong noise. The cause of this, however, was not the well-known difficulty in magnetizing steel, that showed itself through a somewhat smaller deflection of the galvanometer given the same position of the magnet. The distinction in the noise existed also when soft iron was magnetized two to three times more weakly. Very soft iron still produced a noise when the magnet was moved at a 1 / 2 m distance. When it was placed in the close vicinity, the noise was so strong that it could be heard clearly in the telephone even without the amplifier. The distinction between different kinds of iron is so large that one could really build on this a method to investigate iron. The magnetic field variation was mostly done in such a way that the magnet was either displaced away by the side, turning it at 180 ~ and then moving it back, or slowly turned away for 180 ~ from a given position, in particularly close vicinity. The noise due to this motion was very strong in particular positions. The variations of the deflection of the galvanometer did not always correspond to the noise strength. When a variation in the direction R stopped at a point A (Fig. A2), all was suddenly quite and stayed perfectly quite as one made a variation back from A to B, even when the magnet was moved at will back and forth between A and B. As soon as the magnet overcame even slightly the position A, suddenly the full noise started again. In addition, by a further displacement in the direction R, the boundaries A and B were in some sense shifted away to the right. Then the boundary A remained again surprisingly sharp in correspondence to any motion between A and B. The boundary B was not sharp. The noise grew up slowly when one went back through B until C. At the same time, the boundary Abecame as well less sharp, in the sense sketched in Fig. A3. The galvanometer also showed quite strong variations in the region between A and B. No jump could be detected during its motion, when A was crossed and the noise suddenly started. Clearly the observation method was not very precise. For thick samples it would be perhaps the same for a stronger field, but the region A-B was not reachable with the changes produced by the present magnet. Thus, as remarked above, with this no noise could be generated. One could maybe explain this peculiar behavior of iron in the following way: the individual molecular magnets group themselves into assemblies

REFERENCES

257

A phenomenon occurring in a strip, thin as paper, of soft iron sheet is still not clearly understood. A turn of the magnet by 180 ~ gave rise to the stronger noise when the magnet was placed at a distance of 10cm. By turning it at shorter distance, the noise was surprisingly very weak, even when the magnet was turned very slowly. A similar analysis would be desirable with the magnetodetector of Marconi and with the telegraph, also in view of practical applications. Dresden, Institut f~ir Schwachstromtechnik, May 1919.

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CHAPTER 3 The Barkhausen Effect

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CHAPTER 4

Recent Analytical Developments in Micromagnetics Antonio DeSimone, SISSA-International School for Advanced Studies, via Beirut 4, 34014 Trieste, Italy Robert V. Kohn, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10023, USA Stefan MiJller, Max-Planck-lnstitut for Mathematik in den Naturwissenschaften, InselstraBe 24, D-04103 Leipzig, Germany Felix Otto, Institut for Angewandte Mathematik, Universit~t Bonn, WegelerstraBe 10, 53115 Bonn, Germany

Contents 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Introduction The Energy Functional and Pattern Formation Bulk Scalings A Hierarchy of Thin-Film Models A Two-Dimensional Model for Large Soft Films Higher-Order Terms and Wall Structures Pattern Formation in Micromagnetics: Mathematical Context Beyond Low-Energy States Appendix A. Magnetostatics Appendix B. F-convergence Appendix C. Frequently Used Symbols Acknowledgements References

269 273 278 282 290 300 337 346 366 367 372 374 374

4.1 I N T R O D U C T I O N Micromagnetics is a strikingly simple and remarkably powerful theory. At its core is the micromagnetic energy, whose local minima represent the stable magnetization patterns of a ferromagnetic body. This theory is now routinely used for the analysis and design of magnetic devices. It explains observations on many length scales, from nanometers to microns. It also explains magnetic hysteresis, through the multiplicity of local minima. The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rlcrhfc r,~,~r~7~fl

270

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

Micromagnetics has mainly been applied in two alternative ways. Before the advent of computing, analytical solutions were obtained by minimizing the energy over a suitable class of trial fields; a classic example is the Stoner-Wohlfarth theory of magnetic switching [1]. Since the advent of computing, micromagnetics has mainly been applied numerically. Simulation has been very successful, giving detailed information about pattern formation and hysteresis in samples up to about a micron in size. Both these approaches have limitations. When we minimize the energy over a restricted class of magnetizations, the answer is limited by the correctness of the ansatz. Many analytical solutions are now known to be wrong (though still informative), because the true solution - observed numerically a n d / o r experimentally - does not match the ansatz. For example, the Stoner-Wohlfarth theory matches experiments only for the smallest nanoscale particles. Numerical simulation is free from this difficulty; it is, in fact, the ideal tool for answering specific questions about submicron-scale phenomena. However, we seek understanding as well as answers. For example, why do we see boundary vortices but not interior vortices at some length scales? And what determines the familiar pattern of a cross-tie wall? Simulation alone cannot answer such questions. Moreover, simulation is limited to relatively small ferromagnets. Owing to the presence of small length scales (domain walls, Bloch lines) and the nonlocal character of the energy (dipolar interaction), fully resolved computations are limited - even using the most powerful computers - to dimensions of the order of a few microns or less. Thus the understanding and utilization of micromagnetics has remained a challenge. Recently two new approaches have begun to emerge: (a) identifying the scaling law of the minimum energy, and the character of magnetization patterns that achieve it; and (b) identifying simpler models, valid in appropriate regimes, whose behavior is easier to understand or simulate. This review article provides an introduction to these new methods, and a survey of the insight they provide. The new approaches are both based on asymptotic analysis. They take advantage of the presence of small nondimensional ratios, for example, exchange length or film thickness divided by diameter. The asymptotic analysis assumes that some or all of these ratios tend to zero, with specified relations between them. The existence of more than one nondimensional parameter makes the problem rich, since there are several distinct regimes. By identifying and exploring them we obtain a sort of 'phase diagram'. The new approaches are mathematically rigorous. To identify the scaling law of the minimum energy one must prove matching upper and lower

4.1 INTRODUCTION

271

bounds. The upper bound is usually familiar - obtained by the classical method of minimizing the energy over a suitable class of trial fields. The lower bound, however, is usually unfamiliar. By proving an ansatzindependent lower bound that matches an ansatz-dependent upper bound, we effectively prove the correctness (or at least the adequacy) of the ansatz. In some settings we get the energy scaling law but not the optimal prefactor. In other settings we identify the prefactor as well as the minimum value of a suitable 'asymptotic problem'. Some readers may wonder how one can ever hope to prove an ansatzindependent bound. For convex variational problems the answer is familiar: every test field for the dual problem gives a lower bound. For nonconvex problems there is no such universal tool. Most of our examples use the presence of a small parameter and draw from the theory of F-convergence. The goal of analysis is not to avoid simulation, but to maximize its utility. This is especially clear in Section 4.5, where the focus is on relatively large thin films. Fully-resolved simulations are simply not possible in this regime. Our asymptotic problem, on the other hand, is accessible numerically. It uniquely determines the magnetic charge. Moreover, there is a naturally associated domain structure, which appears to be the ground state. The achievements surveyed in this article include: (1) A systematic analysis of soft thin films (Section 4.4). We identify five distinct regimes corresponding to different relationships between the thickness, diameter, and Bloch line width. For each regime we identify an asymptotic variational problem. Its solution determines the ground-state magnetization pattern, and the leadingorder behavior of the energy. This topic encompasses work by Gioia and James [2], Moser [3,4], and Kohn and Slastikov [5]. (2) Nonhysteretic behavior in large thin films (Section 4.5). For relatively large thin films, which have many local minima and plenty of hysteresis, our asymptotic variational problem is degenerate but convex. As a result it determines certain quantities - in particular the magnetic charge and the region of field penetration- uniquely. We conclude that although a large thin film has many stable magnetization patterns and lots of hysteresis, some quantities (namely, those determined by the asymptotic problem) should be free of hysteresis. This topic encompasses our recent papers [6-8]. (3) The internal structure of a cross-tie wall (Section 4.6.5). The crosstie wall is a special type of domain wall seen in thin films. It is interesting due to its non-one-dimensional character, involving a periodic arrangement of vortices. The recent work of Alouges,

272

CHAPTER 4 Recent Analytical Developments in Micromagnetics Rivi6re, and Serfaty [9] uses matching upper and lower bounds to give the first analytical explanation of its structure. (4) The internal structure and stability of a Ndel wall (Sections 4.6.2, 4.6.3, 4.6.4, and 4.6.6). The one-dimensional N6el wall, seen in extremely thin films, is interesting due to its core-and-tail structure. Dipolar effects play a major role, so the variational problem for the wall profile is strongly nonlocal. The recent work of Melcher [10,11] uses matching upper and lower bounds to identify the structure of the tail. The energy of a one-dimensional N6el wall is greater than that of a cross-tie wall, so it must be a local not global minimum; the recent work of DeSimone, Kn~ipfer, and Otto [12] explains why. In thicker films the asymmetric Bloch wall has lower energy, and [13] proves that no other structure can do significantly better. (5) Nucleation of domain structure in a soft thin strip (Section 4.8.1). Under a sufficiently large applied field the magnetization of a strip is saturated and unique. As the applied field is reduced, the magnetization remains in a local m i n i m u m until one reaches the critical field, where an instability (bifurcation) occurs. The outcome of this instability is the formation of magnetic domains. Until recently, the spatial structure of the instability was thought to be limited to coherent rotation, curling, or smooth buckling. New work by Cantero-Alvarez and Otto [14-16] has shown the existence of a fourth alternative - 'oscillatory buckling' - which is probably the origin of the 'concertina' domain structure. This development demonstrates the value of seeking lower as well as upper bounds: the proof of the lower bound may point to the existence of a previously-unnoticed regime.

Micromagnetics was first formulated as a model over 60 years ago, but it remains in its infancy mathematically. The associated challenges are of course opportunities. In some areas, the recent analytical progress has taken advantage of apparently unrelated mathematical developments, for example, the theory of Landau-Ginzburg vortices and the framework of F-convergence. In other areas micromagnetics is driving the development of entirely new methods - for example, techniques for proving ansatzindependent lower bounds. As the reader has probably noticed by now, our story is coherent but not very orderly - and certainly not finished. The goal of this paper is not to review all of micromagnetics, but rather to give an overview of our recent work. We have included some closely related topics, including in Section 4.7 a fairly extensive survey of related mathematical tools and

4.2 THE ENERGY FUNCTIONAL AND PATTERN FORMATION

273

results. However, we have omitted many important topics - for example, Landau-Lifshitz-Gilbert dynamics - and we have barely scratched the surface concerning thermally-activated switching. For some recent progress in these areas see [17-20,110,111]. The paper begins, in Sections 4.2 and 4.3, with a brief introduction to micromagnetics and a discussion of the small-particle and large-body limits. Section 4.4 identifies five distinct thin-film regimes, discussing for each the form of the limiting energy and what it tells us about the associated magnetization. Section 4.5 discusses one of those regimes - corresponding to relatively large thin films - in greater depth. Section 4.6 assembles various results on domain-wall structures in thin films. Section 4.7 discusses some related mathematical problems, drawing connections with micromagnetics and setting our work in its natural mathematical context. All the preceding sections address local or global energy minimization. Section 4.8 is different: it addresses the nucleation of domain structure, and the relevance of saddle points for thermally-activated switching.

4.2 THE ENERGY F U N C T I O N A L A N D PATTERN FORMATION In this section we briefly describe the energy functional of micromagnetics, thereby also introducing the notation used in the rest of this article. We will mostly use the partially nondimensionalized form (4.17) in which the energy functional has units of volume and the important material parameters are the nondimensional quality factor Q and the Bloch line-width d, which has dimensions of length. In the last subsection we briefly discuss how the competition of the different terms in the energy leads to the formation of the complex magnetization patterns observed in experiments.

4.2.1 FORMULATION IN SI

UNITS

The energy functional of micromagnetics is given, in SI units, by the following expression: E(J) =

AIVml 2 +

s gap(m)

fl0 + T

fN3 I[H]indl2-- ~ Hext 9J.

(4.1)

Here f~ C [~3 is the region of space occupied by a ferromagnetic body and J is the magnetization density, i.e. a vector field of constant length inside f2, and extended by zero to N3. ij(x)l_|

/

Js inf2, 0 in R3 \ ~.

(4.2)

274

CHAPTER 4 Recent Analytical Developments in Micromagnetics

The positive scalar Js is called saturation magnetization. At fixed temperature, it is a material constant so that the (rescaled) magnetization

l Is

m :-- --

(4.3)

satisfies Im(x)l -

1 inf,, 0 in [~3 \ ~.

(4.4)

The four summands in (4.1) are called exchange energy, anisotropy energy, magnetostatic (or demagnetizing or stray field) energy, and Zeeman energy (or external field energy), respectively. The vector field Hext is an applied magnetic field, the constant P0 is the magnetic permeability of vacuum while Hind is the magnetic field generated by the magnetized body f~ (also called induced, or demagnetizing, or stray field). This is obtained from J by solving Maxwell's equations of magnetostatics curl Hin d -- 0, R3 div(#0Hind + J) = o, in . (4.5) From (4.5)1 we deduce the existence of a scalar potential U for Hind Hin d -- -V[LJ,

(4.6)

which, in view of (4.5)2 satisfies A U = div J . #0 The last two identities can be summarized by Hind -- - V ( A - l d i v J ) ,

(4.7)

(4.8)

which shows that divJ is the source of Hind, and hence the magnetostatic energy is minimized by divergence-free magnetization patterns. Since (4.5)2 holds in the sense of distributions

f

Hind 9Vq)-- - f j . C~(R3), (4.9) dfl dR3 the divergence of J may consist of two components which, by the electrostatic analogy, are called volume charges and surface charges. For a smooth J their densities are given, respectively, by div J (at points of f2) and - J . v (at a point of the boundary 0f2 of f2 where the outer unit normal is v). For a discontinuous J, a similar surface contribution arises also at all surfaces of discontinuity inside ~ across which the flux J. v of J is discontinuous.

4.2 THE ENERGY FUNCTIONAL AND PATTERN FORMATION

275

Thus, magnetization fields with vanishing magnetostatic energy describe domain patterns that achieve flux closure. The positive scalars A and Ka in (4.1) are material parameters which we will assume to be constant, for simplicity. They are called the exchange constant and the (magnetocrystalline) anisotropy constant. The energy density ~0 is a nonnegative function called the anisotropy energy density. It is typically a polynomial, with the symmetry properties inherited from those of the underlying crystalline lattice. The zeros of q~ form the set of preferred directions of magnetization (easy axes).

}~ := (p-l({0}).

(4.10)

For a uniaxial material (the typical example we will use in our discussion), there exists a distinguished axis identified through a unit vector e, and (a and/(7 are given respectively by CPuni(m) = 1 - (m. e) 2

(4.11)

](~uni-- { + e , - e } (4.12) Other expressions are of course possible, either for uniaxial materials (case of an easy plane, higher-order polynomials, etc.) or for materials with a different crystallographic symmetry, but we do not need to discuss them in detail. According to micromagnetics, stable magnetization patterns on a body f] are described by minimizers (global or local) of the energy functional (4.1). This is a nonconvex nonlocal variational problem in view of the nonconvex pointwise constraint lJR = Js in f] and of the nonlocal differential constraint linking Hind with J. 4.2.2 NONDIMENSIONAL FORMULATIONS AND MATERIAL PARAMETERS In our developments, we will focus on partially or fully nondimensional versions of (4.1). For example, let j2

(4.13)

;q) Ka

(4.14)

Ka- o

and define

E(m) := ~ . Setting N e x t - #0Hext Js '

U = #0 U Js

H i n d - #0Hind = - V U -

Is

- V ( A - l d i v m)

(4.15) (4.16)

276

CHAPTER 4 Recent Analytical Developments in Micromagnetics

we obtain

E(m) ~d2/~lVmi2-}-Q/~q)(m)-}-/R3 IVUI2

2 fn Next 9m. (4.17)

This is the form of the energy functional we will use most frequently. Here the 'energy' E(m) has the dimensions of volume (since Ka is an energy density). There are two important material parameters in (4.17). The socalled quality factor Q=

Ka

Ka

(4.18)

measures the strength of the anisotropy energy relative to that of the stray field. It is a nondimensional quantity, and it separates ferromagnetic materials into two broad classes: soft materials, for which Q < 1, and hard materials (Q> 1). The exchange length associated with the stray field energy d - dBc --

(K~) 1/2

(4.19)

measures the relative strength of the exchange and magnetostatic energies. It has dimensions of length, and it is also called Bloch line-width, and denoted with dBL, for reasons that will become clear later, see (4.195). In terms of the rescaled fields m, U, Hind, equations (4.7), (4.9) are written as AU -- div m

(4.20)

or, in weak form, as

/RVU'Vq)---/RSind'Vq)--/f~ 3 3 m'vq)" 'qq) E C0X~(R3)"

(4.21)

Another interesting partial nondimensionalization of (4.1) is obtained by dividing E by Ka

I(m) "-

E(J) E(m) Ka Q

(4.22)

hence obtaining

I(m) - d2w /~ IVm[2 + f~ ~(m) q- ~l f R 3 IVUI2 - ~2 f ~ Next" m.

(4.23)

The material parameter

dBW-

dBL

( A ) 1/2

Q1/2 - ~

(4.24)

is the exchange length associated with the anisotropy energy. It is also called Bloch wall-width, for reasons that will become clear later, see (4.193).

4.2 THE ENERGY FUNCTIONAL AND PATTERN FORMATION

277

TABLE 4.1 Material Parameter Values for Some Common Ferromagnetic Materials

Material Permalloy 0c-iron SmCo5 Ne2Fe14B Co

Q

dBL(nm)

dBW(nm)

2.5 x 10-4 2.5 x 10-2 38 4.9 0.3

5 3.7 5.2 2.7 3.8

316 23 0.8 1.2 7

It has dimensions of length, and it measures the strength of the exchange energy relative to that of the anisotropy energy. It is instructive to look at the values of the material parameters appearing in (4.17) and (4.22) for some concrete examples, see Table 4.1. The wide range of variability of the quality factor Q, which spans six orders of magnitude, is immediately apparent. By contrast, for the same set of materials, the exchange length dBL only varies by a factor two. 4 . 2 . 3 PATTERN SELECTION AS A RESULT OF

ENERGETIC C O M P E T I T I O N Pattern selection is controlled by the interaction of the energy terms in (4.17) as a compromise among conflicting trends. The most immediate example of such conflicts is the competition between exchange and magnetostatic energies. For m constant, Eqn. (4.20) is written more explicitly as AU

-- -(m

. v)(x)

d~r

(4.25)

where v is the outer unit normal to the boundary ~9f2of f2, and da is the surface measure. This shows that a constant magnetization always generates a nonzero magnetostatic energy if f2 is not an infinite cylinder with axis parallel to m. In other words, exchange energy and magnetostatic energy can never simultaneously achieve their minimal value zero. Actually, for soft materials, the competition between exchange and magnetostatic is the most fundamental driving force behind formation of domain patterns, at least in the absence of an external field. More generally, the features of the observed domain patterns are decided by the relative strength of all four s u m m a n d s in (4.1), and hence they depend on the values of the material parameters affecting them. The variability of these numbers over wide ranges explains the great variability and richness of observed configurations, see Figs 4.1 to 4.4. Essentially, understanding these patterns means understanding the interaction between the various s u m m a n d s in (4.1).

278

CHAPTER 4 Recent Analytical Developments in Micromagnetics

FIGURE 4.1 Iron whiskers. Reproduced with permission from Hubert and Sch/ifer, MagneticDomains,Springer (1998).

FIGURE 4.2 NdFeB domain patterns. Reproduced with permission from Hubert and Sch/ifer, MagneticDomains,Springer (1998).

4.3 BULK S C A L I N G S Consider a magnetic particle of fixed shape, e.g. a cubic particle. In this section we study how the minimizers of the micromagnetic energy (4.17) depend on the size of the particle. We will see that certain models that are often thought of as alternatives to the micromagnetic approach actually arise as rigorous limits of the micromagnetic formulation. Specifically,

4.3 BULKSCALINGS

279

FIGURE 4.3 Permalloy film with circular cross-section. Reproduced with permission from Hubert and Sch~ifer,Magnetic Domains, Springer (1998).

A cross-tie wall in a Permalloy film. Reproduced with permission from Hubert and Sch~ifer,Magnetic Domains, Springer (1998).

FIGURE 4.4

Stoner-Wohlfarth theory corresponds to the small-particle limit and N6el's theory of magnetic phases arises in the large-particle limit. The right notion of convergence is the so-called F-convergence of the corresponding energy functionals. This notion is explained in more detail in Appendix B. The main point is that F-convergence of the energy functionals is essentially equivalent to the convergence of the corresponding minimizing magnetizations, which are our main object of interest. The limits above correspond to the competition of two length scales, namely, a material-dependent scale (such as the Bloch line-width d) and the size of the magnetic particle. In the next section we will also discuss

280

CHAPTER 4 Recent Analytical Developments in Micromagnetics

thin magnetic films, which involve three length scales: the material scale d, the typical thickness t, and the typical lateral dimension l. In order to establish a consistent notation we consider from the beginning both t and l, but in this section the aspect ratio t = (4.26) l will be kept fixed, while in the later sections we will be interested in the limit ~ ~ 0. We consider for simplicity the case that the b o d y f2 is a p r i s m f2 = f2' x (0, t)

(4.27)

of height t a n d we denote by l a typical length of the base f2'. As an example one could take a square base, i.e. f2' = (0, i)2 x (0, t). For z = 1, f2 is a cubic particle of size l. The change of variables x ~ ~ = xT 6 co rescales f2 to a set co = (.ot x (0, z), w h e r e oY has size of order one, and it reduces the energy functional (4.17) to a fully n o n d i m e n s i o n a l form

e(m)=Kafo~lvmla+Qf,.,q)(m)+fRIvula-af~hext'm'3

(4.28)

where

e(m) =

1 ~

E(m),

(4.29)

d

(4.30)

te U(x),

(4.31)

u (/) =v hind

(x) ~-

----V~U

-- Hind(X) =

-VxU(x),

(4.32)

x

hext (-/) = Sext(X).

(4.33)

The static Maxwell equation (4.20) for u implies that Au - div m.

(4.34)

If we take a square base and aspect ratio z = 1 then r is simply the unit cube. Again, for simplicity, we will restrict our attention to the case of spatially uniform applied field, Hext ~ const. 4.3.1 THE SMALL-PARTICLE LIMIT Consider the case l < < d, i.e. K --+ +cx~, while ~, Q are fixed n u m b e r s of order one. Also, the strength of the applied field, if present, should be of order one. In this limit, the classical model of Stoner and Wohlfarth [1] is recovered, see [21].

281

4.3 BULK SCALINGS PROPOSITION4.3.1. As ~ --~ +cx~, the functional (4.28) F-converges in H 1 to

] Qq)(m) + m. Doom - 2hext" m for m - const in co, Ira] - 1, else, I +oc

ecx~ (m)

(4.35)

where Dco is the demagnetizing tensor of co. In this case the F-limit of the energy functionals agrees with the pointwise limit, but this is not true in general (see A p p e n d i x B). In the proposition above, the demagnetizing tensor Do is a traceless, positive-definite tensor d e p e n d i n g only on the shape of co. For co a unit cube or a unit sphere, Do = l i d , with Id the identity. For a more general co, Do is defined implicitly by

1 /co hm Vm = const in co, Icol

Doom =

(4.36)

where Icol is the volume of co and hm is the magnetic field induced by m, i.e., hm = - V A - l d i v m. REMARK4.3.2. The functional e~ is defined on a finite-dimensional space. It is

therefore easy to study its local minima, as was done by Stoner and Wohlfarth in their celebrated paper [1]. In fact, global and local minima of the function m e~ (m) characterize the asymptotic behavior of both global and local minima ore (see [21,22]). The Stoner-Wohlfarth model is thus relevant to gain some insight onto the hysteretic behavior of very fine particles. For a recent numerical investigation of the switching fields of small cuboidal particles, and of their deviations from Stoner-Wohlfarth predictions, see [23], where a nonmonotone dependence of the switching field on particle size has also been observed. 4.3.2 LARGE-BODY LIMIT

Consider n o w the case l > > d, i.e. K --~ 0, while ~, Q and hext are held fixed, at order one. In the limit, we obtain a mathematically rigorous version of a model which, for ellipsoidal specimens, coincides with the theory of magnetic phases of N6el ([24], see also [25-27]). PROPOSITION4.3.3. AS K ~ O, the functional (4.28) F-converges with respect to

the weak convergence in L 2 to e~

s

IVul2-2/ohext'm'

Iml~
(4.37)

282

CHAPTER 4 Recent Analytical Developments in Micromagnetics

where q~" is the convex envelope of q~ (extended by +cx~for Iml # 1): q~"(m) = min

2iq~(mi) " Imil = 1, m = i=1

2imi, 2i~0, i=1

,~i = 1 . i-1

(4.38) REMARK4.3.4. The notion of weak convergence appearing in the proposition above is essentially equivalent to convergence of "local" averages on arbitrary cubes C cut out of co:

mk ~ m weakly in L2(co) 4~ ~1 / c mk ~ -~l 1/c m

u cubes C c co.

(4.39) The loss of the unit length constraint, in the limit, is thus not surprising whenever mk is oscillating at aline scale. This is precisely the case in micromagnetics, because domain patterns of size smaller than l generate, when rescaled onto co,fine-phase mixtures with characteristic rescaled size tending to zero. Proposition 4.3.3 implies that, in the large-body limit, local averages of micromagnetic minimizers can be computed by minimizing (4.37). Since in computing magnetization curves the only relevant feature of a magnetic configuration is its average over the whole specimen, (4.37) can be used to compute magnetization curves in cases where the effects of hysteresis are negligible. For co an ellipsoid, minimizers of (4.37) are spatially uniform, and minimizing (4.37), (4.38) reduces to the problem of finding a mixture of at most four magnetic phases ml ..... m4 (with respective volume fractions 21 ..... 24), covering co and carrying minimal energy. This is the approach of N6el to the prediction of virgin magnetization curves, see [24,28,29]. REMARK4.3.5. The functional (4.37) is the "relaxation' (i.e. the lower semicontinuous envelope with respect to weak convergence in L 2) of the functional

Q/~oq~(m)+/R3 I V u l a - 2 / c o h e x t ' m '

(4.40)

(still endowed with the constraint Iml - 1 in fl). The study of (4.40), initiated with the paper [30] of James and Kinderlehrer, has received considerable attention in the mathematical literature, see [24-27,31].

4.4 A HIERARCHY OF THIN-FILM M O D E L S We now consider thin magnetic films. In this case there are three relevant length scales: the film thickness t, the lateral dimension l and the material

4.4 A HIERARCHY OF THIN-FILM MODELS

283

parameter d, the Bloch line-width (recall that for many materials d is of the order of 5 nm, see Table 4.1). Out of these one can form two independent dimensionless parameters. It turns out that for our discussion the most useful parameters are the aspect ratio r = t/l, already introduced above, and a dimensionless measure of the size of the film given by tl/d 2. We will always study the limit of small aspect ratio t - ~ --~ 0. (4.41) In this case we identify four different regimes depending on the relation of tl/d 2 and ln(1/r), which leads to four distinct two-dimensional limiting theories, see the first four rows in Table 4.2. For very small films (l/d fixed and still t/l ~ 0) one can also consider a different rescaling of the energy leading to a limiting theory which captures higher energy states (see also Table 4.2). The different limit theories are much simpler than the full threedimensional micromagnetic energy functional. They capture the leadingorder terms in the energy and lead to a drastic simplification of the very complex micromagnetic energy landscape. They thus provide a good description of the robust features of the magnetization patterns in the thinfilm limit. We will discuss this in more detail in the next section in the regime of large thin films (tl/d 2 >> ln(l/t)) and compare the predictions of the thin-film theory with experiments. In this section we first illustrate the separation of energy scales by investigating the energy contributions of certain prototypical magnetization patterns and then state the rigorous results which lead to Table 4.2. 4.4.1 HEURISTICS AND SEPARATION OF ENERGY SCALES

Here we briefly discuss the energy contributions of a prototypical magnetization configuration, following closely Section 4 of [6]. The results are summarized in Table 4.3. We first consider the energy due to variations of m in the x3 direction, i.e. normal to the film plane. If these variations are of order one they lead to an energy contribution of order 12d2/t. In all the relevant regimes this will be much larger than the energy scales we are interested in. Hence for the rest of this heuristic discussion we will suppose that m = re(x1, x2) in f2. (4.42) The rigorous convergence results actually show that this holds asymptotically. At finite thickness it is still possible that (4.42) is significantly violated on a small set (this occurs e.g. in asymmetric Bloch walls, cf. Section 4.6.2), but this turns out to be insignificant for the energy contributions we discuss below.

CHAPTER 4 Recent Analytical Developments in Micromagnetics

284

TABLE 4.2 The different scaling regimes in thin films, with the abbreviation 12 = ln(l/t) = ln(1/'O. The second column gives the factor by which the energy is divided before one passes to the limit. The limit energy in the b o u n d a r y vortex regime is formal, since it is actually infinite on all admissible magnetizations (see text). The limiting energies are invariant under the rescaling ~ ' ~ r ~ t, l ~ rl. Going down in the first four rows corresponds to a passage from small (compared with d) films to large films or, equivalently, to decreasing d. At least on the formal level, the different regimes are naturally connected (taking into account the prefactors in the ]energy' column). The second regime turns into the first as ,~ --~ 0, while the second and third regime merge as ~ --~ cx~ and /2 --~ 0. Finally, the third regime turns into the fourth as/2 --, c~ (if one looks at the limit in the sense of F-convergence, it is natural that the constraint Iml --- I is relaxed in the limit, since the Dirichlet integral drops out). Regime

Energy

Limit

Bulk constraint

tl/d 2 << 1 / s StonerWohlfarth

t2112

(1/2/1:l) f ( m ' . 1;')2

m 3 = O, ]m[ = 1 m = const

tl/d 2 = 2 / s Boundary penalty

t2112/,~

f ]Vim]2 + (,~/2~zl) f ( m ' . 1,")2

m 3 = 0, [ml = 1

tl/d 2 =/2 Boundary vortex

t2l//2

IV'me2 + ~ f IVV]2

tl/d 2 >> 12 Magnetostatic

t21

1I f IVWl2

K = d/l fixed

tl 2

/c2f iV'mi 2 + ~ f m2

TABLE 4.3

Boundary constraint

m3 = 0, ImI = 1

m ' . v' = 0

m3 = 0, I m l ~ l

m' 9 v' = 0

]ml-- 1

Estimates for the Key Energy Contributions in a Soft Ferromagnetic Film.

Out-of-plane component m3 Nontangential component m' 9v' In-plane divergence V' 9m'

l2 t l t 2 In l t2 IHext 112 t

External field Hext

Ql2t

Anisotropy Q N6el wall

it 2 ln-1 i t d2

Bloch line

d 2 t In I d2 t

Exchange energy for magnetization varying smoothly on scale l

In view of (4.42) the local energy contributions reduce to

d2t/ff ]V'ml2dx'+Qt/~,q~(m)dx ',

(4.43)

4.4 A HIERARCHY OF THIN-FILM MODELS

285

where we use the shorthand notation x' = (Xl, x2) and write V' for the in-plane gradient (01, 02). In the following we will always consider magnetizations that vary smoothly on a scale comparable to the lateral dimension l. Then, Vm is of order 1/l and the exchange and anisotropy energy contributions are of order d2t and Ql2t, respectively. The nonlocal magnetostatic energy can be computed by considering the Fourier transform in the tangential direction,

1 fR 2 e-i~"X'm(x') dx',

(4.44)

,/,]Z.(m) (~I) __ ~

where m is extended by zero outside f2', as usual. We note in passing that our normalization of Z differs from the choice made frequently in the physics literature; our choice makes Z an L2 isometry, i.e. [IZm[ [L2 = Ilm[ Ic 2 . A short calculation shows that the magnetostatic energy splits into a contribution due to the normal component m3 (which, in terms of the electrostatic analogy leads to a surface charge at the top and bottom surface of the film), and a contribution due to the in-plane divergence div'm I. We get

~f

t ~II) [-~

/~3 IvUladx-t feaf (2 I -q-t

2g

~l

2 .Z(N')]

I IZ(m3)

d~'

d .

(4.45)

Here the Fourier multipliers are given by

g(z) -

sinh(z)

Izl exp(z)

1 - exp(-2z) 2z

and

f(z) = 1- g(z).

Since we assume that m varies smoothly on a scale comparable to l, its Fourier transform is essentially concentrated on wave vectors ~' of order 1/l or less. In view of our assumption t/l << 1 the arguments of f and g are thus small in the relevant range of ~' and we expect that (4.45) is approximated as follows ~ 3 IVUI2 dx

t2/1~, I -2 2 [-~

9Z(m')

[2 d~' + t L [~(m3) [2 d~' 2

- t2 /~3 IV V [2dx + t f~, m2 dx',

(4.46)

286

CHAPTER 4 Recent Analytical Developments in Micromagnetics

where V is the single-layer potential of the surface charge density div ~m~. Classically, V is given as the solution of AV

=

[V] - 0 and

0

in [~3 \ (~, X {0})

[ov]~3x3 div'm'

}.

(4.47)

on f2' x {0}

Here [.] denotes the jump of a quantity '.' across the plane R 2 X {0}. A different heuristic argument giving the energy contribution of the in-plane divergence of m is as follows. The potential U t satisfies Maxwell's equation A U t - div(m(x')Z(o,t)(x3)), where Z(o,t) is 1 in the interval (0, t) and zero outside. Dividing by t and considering that Z(o,t)/t converges formally to the Dirac mass 60 we see that the limit V of u t / t is given by (4.47). For a rigorous argument using the relevant weak formulation of (4.47), see [6], p. 1439. The above characterization of V implicitly contains the assumption that the normal component of m' does not jump across a possible discontinuity line of m ~, in particular m'. v' = 0

on Of]',

(4.48)

where vt denotes the normal to the boundary 0f]' of the cross-section f2'. We will come back to this shortly. From (4.46) we first recover the classical argument of N6el that in thin films a nonzero normal component m3 leads to a large energy contribution of order tl 2. If the in-plane magnetization satisfies (4.48) then a scaling argument in connection with (4.47) shows that the magnetostatic energy is of order t21. If m ~does not satisfy (4.48) then an additional energy contribution due to the boundary 'charge' m ~9V arises. To compute it one has to go back to the exact representation (4.45) of the magnetostatic energy and a short calculation shows that for a constant magnetization and a straight boundar~y piece the extra contribution per unit length is of order t 2 l n ( l / t ) ( m ~. V) ~, (see e.g. [6], Section 4.4). For a general boundary one thus expects an extra contribution of the order t 2 ln(l/t) f~r~,(m'.

Vt) 2.

(4.49)

We will see in the next subsection that this is indeed the case, see (4.52) below. So far we have implicitly assumed that the magnetization is smooth and varies on a typical scale proportional to the lateral extension l of the film. Typical magnetization patterns, however, consist of regions of slowly

4.4 A HIERARCHY OF THIN-FILM MODELS

287

varying magnetization (domains) that are separated by domain walls, in which the magnetization changes rapidly. We will see below in Subsection 4.6.3 that the energy of a typical domain wall in thin films, i.e. the N6el wall, is of order 12t/ln(lt/d 2) and thus is asymptotically negligible compared with the magnetostatic energy contributions. One can also check that the energy of a Bloch line (where the magnetization develops a nontrivial m3 component on a region of diameter d) is of order d2t ln(l/d) and also asymptotically negligible, see (4.196) below and Section 4.5 in [6]. 4.4.2 RIGOROUS CONVERGENCE RESULTS-THE FIRST FOUR REGIMES The heuristic discussion on the separation of energy scales suggests that in the thin-film limit we get a strong simplification of the energy landscape. If we select a certain energy regime then all energy contributions below this level will become asymptotically irrelevant. Energy contributions that are asymptotically much larger are not allowed to be present, i.e. they lead to constraints in the limiting theory. Typical examples are the constraint that m should become independent of x3 or that the out-of-plane component m3 should vanish. We now describe the rigorous results that are motivated by this reasoning and that are summarized in Table 4.2. We first focus on the first four rows of this table. It is interesting to note that these four regimes capture very well the qualitative changes in the experimentally observed patterns for increasing size, even though they are mathematically only separated by logarithmic terms. For extremely small films one sees (nearly) uniform states (Stoner-Wohlfarth regime). With increasing size one finds flower states (boundary-penalty regime), states with vortices on the boundary, such as the C- and S-states in rectangular samples or the buckle-and-leaf states in square samples (boundary-vortex regime), and flux-closed patterns (magnetostatic regime). To state the results precisely it is convenient to work in a fixed twodimensional domain ~0' (e.g. the unit square) obtained from f~' by rescaling all lengths by l. We thus employ the scalings (4.28)-(4.33). For brevity we only consider the situation with zero applied field. Adding a suitably scaled applied field poses no additional difficulty since this represents a continuous perturbation and therefore passes through the F-limit, see Appendix B. As before we always consider the small-aspect-ratio limit, i.e. r ~ 0,

(4.50)

and we denote by e~ the minimal micromagnetic energy for a sample with aspect ratio z.

CHAPTER 4 Recent Analytical Developments in Micromagnetics

288

In the regime of very small films, i.e. if (lt/d 2) ln(l/t) ~ 0 or, equivalently, t

~c--gln(1/t) ~ 0.

(4.51)

Table 4.3 suggests that the most interesting term is the one arising from the b o u n d a r y energy (4.49). Indeed Kohn and Slastikov [5] showed (see also Carbou [109]) that 1 e~ F --, e, (4.52) t ln(1/t) where e

_ ~ ~ fon,(m 1. V') 2 if m - const, m3 = O, Iml - 1,

/

cx~

else.

(4.53)

To make the F-convergence result precise, we need to specify the convergence of three-dimensional magnetizations m r 9 co ~ [~3 to a twodimensional limit m. We say that convergence holds if the x3-averages rh~ -- ( l / t ) fo mrdx3 converge strongly to m in L2. If t --g ln(1/t) --, 2 (4.54) (small-body limit) then [5]

tin(l/t)

e~ ~F e,

(4.55)

where e --

2

ffl' IV'ml2 + ~ foff(m'" v') oo

if m3 = 0, Iml = 1, else.

(4.56)

On the other hand, if the body remains large in the sense that t

1 /r ln(1/t)

oe

(4.57)

then the magnetostatic energy dominates and we have, in the sense of F-convergence (with respect to weak L2-convergence of x3-averages) [6]" let r t

f•

IVVl 2

(4.58)

3

provided that m3 - 0 and that m satisfies the relaxed (convexified) constraint ImI ~<1 (else the limit energy is oo). Here v is the single-layer potential of div 1mI defined as in (4.47) (replacing fl~ by the nondimensionalized

4.4 A HIERARCHY OF THIN-FILM MODELS

289

domain cot). In particular this imposes the constraint m'. v' = 0 (for nonsmooth fields m t this has to be understood in the sense of traces, see [6]). A more detailed discussion of this result, its interpretation and comparison of the predictions of the limit theory with experiments appears in the next section. Finally for the critical scaling "C

K--~ = 1

(4.59)

the limit of e~/r is formally given by /

IVtml2 + s

IVvl2 3

(4.60)

subject to the constraints m3 -- 0 and [ml = 1. Here v is defined as in (4.47), replacing f2' by the nondimensionalized domain co'. The problem is that the limit functional is infinite on all admissible test functions, if cot is simply connected, e.g. if co' is a disk or a square. Indeed, the constraint (4.48) forces m' to be tangential at the boundary, but there is no unit vector field in H 1, which is tangential to the boundary. Moser [3,4] has nonetheless shown that the above formal limit captures the right behavior. More precisely he shows that 2nln ln(1/r) -C~< 1- mine~<~2nlnln(1/r) + C

(4.61)

and that the x3-averages of minimizers of e~ converge (strongly in the Sobolev space W l'p, for all p < 2) to a limit m that has two singularities (vortices) on &0t, is otherwise tangential to &d and satisfies the EulerLagrange equation corresponding to (4.60) in co'.

4.4.3 THE FIFTH REGIME The fifth regime corresponds to letting the aspect ratio r = t/l tend to zero, while keeping K = d/l fixed. It was historically the first regime for which a rigorous convergence result was obtained. In fact Gioia and James [2] showed that in this regime the energy functionals e~ F-converge to eGj given by eGj (m) --

{ K2 fco' IV'ml2 + fo~' m~ +oe

if m e H l(c0') ' else,

(4.62)

where V' is the in-plane gradient. Like the first regime this regime corresponds to very small films. The important difference is that one considers here just energy eT (without

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CHAPTER 4 Recent Analytical Developments in Micromagnetics

further rescaling) while in the first regimes one rescales the energy by 1/(z In(I/z)). The setting of Gioia and James emphasizes the fact that the nonlocal magnetostatic energy reduces to the local term m 2, which behaves like an effective anisotropy (this is thus an extreme case of shape anisotropy). The limiting functional has many (minimizing) zero energy states. In fact, all constant in-plane magnetizations have zero limiting energy. The first regime breaks the degeneracy between all these states and assigns to them a refined limiting energy according to foo~'(m'- v') 2. On the other hand, the limit functional for the first regime does not discriminate between limiting magnetizations that are not constant in-plane fields and assigns infinite energy to all of them.

4.5 A T W O - D I M E N S I O N A L MODEL FOR LARGE SOFT FILMS AND COMPARISON WITH EXPERIMENT In this section we discuss in detail the regime lt/d 2 ~ ln(l/t), which corresponds to large films. This regime is of special interest for at least two reasons. First, it covers a large range of concrete samples (including e.g. thicknesses from a few nm to hundreds of nm and lateral dimensions from 1/~m to several hundred #m and larger). This allows us to compare the predictions of the thin-film theory with a wealth of experimental data. Secondly this regime covers exactly a range of parameters that is not accessible by direct numerical simulation of the full micromagnetic functional E(m), since large lateral dimensions beyond a few/~m would require an excessive number of discretization points for an accurate resolution of all the relevant length scales.

4.5.1 STATEMENT OF THE LIMIT PROBLEM AND THE CONVERGENCE RESULT For the rest of this section we will neglect anisotropy, i.e. we will set Q = 0. Table 4.3 suggests that anisotropy is indeed negligible as long as Q KK z = t/l and this is established rigorously in [6]. For Permalloy we have Q = 2.5 x 10 -4 so that the condition Q KK z = t/l is indeed well satisfied, even for films with very small aspect ratio. To simplify the comparison with experiments we will explicitly include an in-plane applied field Hext (usually to be taken as constant). We thus study the three-dimensional energy functional (4.17)

E(m)-d2/fllVml2q-/R 3 IVUI2-a/f2Hext.m,

(4.63)

4.5 A TWO-DIMENSIONAL MODEL FOR LARGE SOFT FILMS

291

where (see (4.15)) Hext-- ~.#0Hext

(4.64)

Is

After the nondimensionalization (4.28)-(4.33) this becomes

e~(m)-Ka/o~lvmla+/R3

I v u ' a - a f ~ hext'm'

(4.65)

where 09 = 09f x (0, z) and z = t/l is the aspect ratio of the film. For both functionals we impose the constraint Iml = 1,

(4.66)

and the potentials U and u are determined from m through the static Maxwell equations (4.20) and (4.34), respectively. Table 4.3 suggests that a natural scaling for the applied field is Hext(X) - Zflext(x/l). We also assume that the applied field is in-plane. We thus suppose

Hext(lx) -- hext(X) -- z ( fl'ext(xl'x2) ) 0

.

(4.67)

THEOREM4.5.1. Suppose that K2 1 ~"c In -"c ---~ 0

-c~0,

(4.68)

-' t fixed. Then and that hext is given by (4.67) with hex

lez F T,

(4.69)

where e(m)

/

!

f~3 IVvl 2 -

2 fog' f/fext" m'

+cx~

if m3 -- 0, else

Im'lK1,

(4.70)

and where v is the single-layer potential of div 1m', i.e.

[v] -

0

Av and

0 in ~ 3 \ (09, x {0}) [ ~v ] -- div'm'

}.

(4.71)

0n 09' x {0}

We recall that the energy f IVvl 2 of the single-layer potential of div'm' is only finite if the normal component of m' does not jump across a possible discontinuity curve of m I, in particular if m'. v' -

0

on &o'.

(4.72)

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CHAPTER 4 Recent Analytical Developments in Micromagnetics

As before F-convergence is taken with respect to weak L2 convergence of thickness averages, i.e. we say that at a sequence of three-dimensional magnetizations m T converges to a two-dimensional magnetization m as 9 --, 0, if -l f 0 ~ m ~(', x3) dx3 ~ m

weakly in L2(co ', R3).

(4:.73)

In the following we will view e as a functional on tangential vector fields m' = (ml, m2), keeping in mind the constraint m3 = 0. One feature of Theorem 4.5.1, which may look unusual at first glance, is that the unit length constraint on m is replaced by the inequality constraint Im'l~
The reduced variational problem is convex. It is even strictly convex in the potential v. Consequently, v is uniquely predicted by energy minimization.

4.5 A TWO-DIMENSIONAL MODEL FOR LARGE SOFT FILMS

293

On the other hand, the reduced variational problem is highly degenerate ,-,! in m I" for constant external field hext - const the energy e depends on m ~ only through its in-plane divergence div ~m~. To extract further information from the reduced variational problem and to compare it with existing models in the physics literature, it is very instructive to look at the corresponding Euler-Lagrange equation (since the problem is convex, solutions of the Euler-Lagrange equations and minimizers of the functional agree). These can be written in the form ~

--V'V -}- hlext -- ,~m' on r

(4.74)

where 2 = 2(x~)~>0 is the Lagrange multiplier corresponding to the inequality constraint ]m'l~
(4.75)

It turns out that for sufficiently small applied fields the constraint is not active, i.e. one can satisfy (4.74) with 2 _-__0. Then the Euler-Lagrange equation reduces to ~!

--V'V q- hex t - 0,

(4.76)

i.e. in this case the stray field -V~v compensates the external field within the sample (in our thin-film framework). This is analogous to electrostatics, where charges on &o arrange themselves to shield the interior co of a conductor from an external field. The model of van den Berg [32] considers the case without external field and in view of (4.76) predicts that the stray field and thus div~m ' vanishes ('flux closure'). The model of Bryant and Suhl [33] allows for an external field and determines V'v and thus div'm t from (4.76). The constraint (4.75) on the magnetization, however, limits the 'charges', unlike the case in electrostatics. Hence, stronger external fields 'penetrate' into the sample and we have Y/pen -- --V'V if- ~/lext # 0. (4.77) Therefore, our model extends the model of Bryant and Suhl [33] and puts it in a natural variational context. In the region of field penetration !

~!

(_Open "-- { X' E (_0' [ hpe n (x') # 0 },

(4.78)

mt is uniquely determined by the Euler-Lagrange equation, in fact m t =

f/pen (X') /I]/pe n (X') ]. Moreover, the region 09pen is itself uniquely determined

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CHAPTER 4 Recent Analytical Developments in Micromagnetics

since it only depends on the uniquely determined potential v. We notice ! in passing that, in order to properly define (Open, we need some technical assumptions ensuring some regularity of v (in [6] it is shown that, for constant applied field, these are satisfied if od is simply connected; for the situation without anisotropy considered here, even this assumption can be dropped, see [34]). Using the additional information arising from the Euler-Lagrange equation of the reduced problem we can strengthen the results that naturally follow from Theorem 4.5.1. In particular, we can strengthen the notion of convergence of micromagnetic minimizers to minimizers of the reduced problem. COROLLARY4.5.2. Under the assumptions of Theorem 4.5.1 and assuming, in addition, that h'ext is constant, we have N

(i) Let m' and n' be two minimizers of the reduced variational problem with corresponding potentials v and w. Then w - v and

m'

--

n' -- hpen/Ihpen -' ~' I or/Ogpe ' n,

(4.79)

where hpe ~' n and (0pe n ' are defined in (4.76) and (4.78), respectively. (ii) Suppose that m ~ is a sequence of almost minimizers of e~/r, i.e.

lim -1(e~(m~) _ mine ~) = O.

(4.80)

"c--~0 "~

Then we have for any minimizer m' of the reduced variational problem and its potential v 2

lim-lf~

"C--+0 '1:

pent X (0,'0

lim

~--+0

3

Imp_ ( O ' )

dx-0,

(4.81)

1 Vu ~

dx=0,

(4.82)

"C

Vv

lim le~ ( m T)

-

e' (m').

(4.83)

7--,0 "~

REMARK4.5.3. We now come back to the point that the reduced problem only involves the inequality constraint [m'[~l while we impose the condition [m[ = 1 in the three-dimensional problem. This relaxation of the constraint might be surprising from a physics point of view - indeed both van den Berg, and Bryant and Suhl consider only magnetization fields of unit length. To understand this relaxation of the constraint better we first explain why it arises naturally in the F-convergence setting and then discuss how the unit length

4.5 A TWO-DIMENSIONAL MODEL FOR LARGE SOFT FILMS

295

constraint might be restored through a finer analysis. F-convergence analyzes the energy functional at a particular energy scale. Theorem 4.5.1 says in particular that given a two-dimensional magnetization field m' with e(m') < cx~, there exists a sequence of three-dimensional magnetizationfields m t, which converge m weakly to m' (in the sense of(4.73)) such that e~(m~) /~ converges to e(m'), even if lm'l < 1. In fact if m' is a minimizer ore (for a given applied field) then the fields m ~ can be chosen as approximate minimizers in the sense of (4.80). The reason that this is possible is, roughly speaking, the low energy of Ndel walls. To achieve, e.g. the limit m' = 0 one can consider magnetization patterns that consist of thin strips with magnetization • 0, 0), separated by Ndel walls. Since the energy of a Ndel wall is negligible compared with the other energy contributions (see Table 4.3) one can let the number of stripes and Ndel walls go to infinity as 9 --, 0 (at a sufficiently slow rate), thus obtaining a sequence of magnetizations whose volume average converges to zero at each fixed scale (this is exactly weak convergence) and for which the additional energy due to the walls is negligible (for a general limiting pattern the actual implementation of this idea is rather delicate and is discussed in detail in [6]). The low energy of Ndel walls thus allows one to construct highly oscillatory magnetization patterns that still have low energy. In contrast to the situation for the large-body limit, however, the limit functional does not favor magnetizations that satisfy Im'l < 1. In fact, we will see in the next section that given a minimizer m 'o with Im~l < I one can construct a new minimizer m' (for the same applied field) that satisfies Im'l = 1. Thus, one might speculate that rapid oscillations of m ~ are never necessary. In other words, if one considers minimizers ~ ore ~ (and not just approximate minimizers) then they may converge strongly to a limit m' that satisfies im'l -- 1. Mathematically this is so far an unsolved problem, even though some results are available for related, but different, problems [35-40]. To make progress on this question one could view the result in Theorem 4.5.1 as the zero-order term in an asymptotic expansion of the energy and one would then need to understand the higher-order terms that break the degeneracy of the zero-order limit functional (we have seen such a hierarchy ofF-limits in Section 4.4.3; a very simple one-dimensional example is discussed in Appendix B.3). Some results on such higher-order contributions are discussed in Section 4.6. 4.5.3 NUMERICAL IMPLEMENTATION The r e d u c e d m o d e l can be used for the c o m p u t a t i o n of d o m a i n patterns. Following the algorithm p r o p o s e d in [7], we proceed in two steps. First, we use to our a d v a n t a g e the fact that replacing the n o n c o n v e x constraint Im~l2 - 1 with the relaxed, convex constraint Im~12K1 does not change the m i n i m u m of eI. Thus, we minimize the r e d u c e d energy e' a m o n g all regular, in-plane vector fields of length less than or equal to one.

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CHAPTER 4 Recent Analytical Developments in Micromagnetics

Step 1. Find a regular m~0 minimizing e'. (We call an in-plane vector field m'(x') on co 'regular' if it satisfies [m'. v'] - 0 across all possible discontinuity lines and at &o). The in-plane divergence div'm' is known to have characteristic singularities near the edges and corners of o9. Since the energy eI depends on m' only through div'm f, this raises the question of how to choose a local refinement strategy for a triangulation of o9. In [41] we derive the necessary refinement rate to retain the optimal rate of convergence in the energy norm. In particular, we show that it is sufficient to refine the triangulation towards the edges (and not the corners). Computation of the magnetostatic part of the energy leads to matrixvector products with fully populated matrices due to the nonlocal character of the magnetostatic interaction. On translating invariant grids one can apply the Fast Fourier Transform (FFT) to reduce computational cost from O(N 2) to O(N log N). On locally refined triangulations we use the method of ?-/-matrices (hierarchical matrices) as introduced in [42]. This method relies on the fact that submatrices of the original matrix can be replaced by suitable low-rank approximations. The approximate matrix is then stored in a hierarchical data structure of size O(N log N), thus allowing an evaluation of matrix-vector products in O(N log N) time. We implement a new class of hierarchical matrices, the so-called ~2-matrices [43], with optimal complexity O(N) in the matrix-vector multiplication. An application of ~2-matrices to 3-d stray field computations in micromagnetics is given in [113], where the large body limit, cf. Section 4.3.2, is considered. In conjunction with a boundary integral formulation of the 3-d stray field, ?-/-matrices have been used in [114]. The m~ obtained at the end of the first step has the correct energy, but in general violates the unit length constraint. The second step postprocesses m~ to obtain a minimizer m ~of e~with unit length according to the following procedure. Step 2. For a given regular m'0 of at most unit length, find a scalar function ~ solving Iv'-L6 + m 0' 1 2 - 1 in o9

(4.84)

= 0 on 0o9. (4.85) The computed domain pattern m t, which we later compare with experiments, is obtained by setting m' - V'• + m~. The first step in the algorithm above involves a convex (but degenerate) variational problem. We solve it with an interior-point method [44]. In more detail, the convex constraint is enforced by adding to the physical energy e' a small multiple s of a logarithmic barrier B. The unique stationary point of the strictly convex functional et + sB is computed by Newton's method. This is then used as the initial guess for the minimizer of et + ~B,

4.5 A TWO-DIMENSIONAL MODEL FOR LARGE SOFT FILMS

297

where ~ < s. The parameter s is then slowly decreased by multiplicative increments. Within Newton's method, the Hessian of e' 4- sB is inverted by a preconditioned conjugate gradient method, where the magnetostatic part of the Hessian is evaluated with the help of ~2-matrices. For the second step, we recall that there are many Lipschitz-continuous solutions of (4.84),(4.85). However, there is a special solution, known as the viscosity solution, which has special mathematical properties. This can be computed efficiently using the level-set method [45]. This is the pointwise solution of (4.84),(4.85) selected by our algorithm. Our numerical scheme selects, automatically and robustly, one out of the many minimizers of the reduced energy e'. The selection principle implicit in this scheme is the same as the one often proposed in the physics literature. Indeed, it appears to pick minimizers with as few walls as possible. Clearly, a more physical selection mechanism should be based on minimizing wall energy. This requires, however, the characterization of energy terms that represent higher-order corrections of our model, a step that seems to be far from trivial. Understanding wall energies requires, in particular, the understanding of their structures. Preliminary steps in this direction are discussed in Section 4.6. Figure 4.5 shows the predictions of our numerical scheme for a square film of edge-length one, subject to a monotonically increasing field applied along the diagonal. This rather special geometry was chosen to guarantee that nontrivial domain patterns could persist beyond penetration of the external field, as illustrated by the 90 ~ domain wall emerging from the bottom right corner of the sample. 4.5.4 COMPARISON WITH EXPERIMENT

To check the predictions of the reduced theory, the responses of two ACdemagnetized Permalloy (Nis1Fe19, Js = 1.0 T) square samples of edge lengths L = 30 and 60 #m and thicknesses D = 40 and 230 nm, respectively, have been observed in a digitally enhanced Kerr microscope [7]. The observed domain patterns are given in Figs. 4.6, 4.7, where the field intensity Hext, measured in Tesla, is scaled according to l Hext H -- - ~ .

als

(4.86)

In comparing Figs. 4.5, 4.6, and 4.7 one may speculate that the small lag in the strength of the applied field exhibited by the thinner samples may be due the fact that, for these films, the walls are of N6el type and they are repelled by the lateral boundary, cf. Section 4.6.4 for a discussion of their repulsive effect. This effect, whose impact on the hysteretic response

298

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

FIGURE 4.5 Predictions of the two-dimensional theory: gray-scale plots of the vertical component of the computed magnetization.

of soft films is discussed in [8], could be captured by an enhanced twodimensional model in which higher-order terms in the film thickness are taken into account. Figure 4.8 examines more closely the predictions of our theory, and, in particular, the phenomenon of external-field penetration. In order to estimate the critical field strength at which penetration occurs, we have superimposed on each gray-scale plot the level curves of the potential Vpen

4.5 A TWO-DIMENSIONAL MODEL FOR LARGE SOFT FILMS

299

FIGURE 4.6 Permalloy films: L = 60 #m, D = 230 nm. of the penetrated field, defined by ~!

-- X71Vpen = hpen - - V ' v + hlext , see Eqn. (4.77). Regions where the field lines concentrate are regions where VVpen ~6 0, i.e. where the external field has penetrated the sample. Within them, the E u l e r - L a g r a n g e equation (4.74) implies that m I is parallel to Vv. The theory predicts that m I can have no walls in the penetrated region. The pictures confirm this, and show quite clearly that two apparently independent p h e n o m e n a - the expulsion of the d o m a i n walls from the interior of the sample and the penetration of the external field - are in fact two manifestations of the same event.

300

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

FIGURE 4.7 Permalloy films: L = 30/~m, D = 40 nm. The comparison between computed patterns and experimental observations shows that the material parameter regime in which the constructions of the theories of van den Berg, and of Bryant and Suhl, are relevant is correctly identified. Moreover, the successful comparison shows that the energy at leading order is captured in a satisfactory way by the reduced model described in this section.

4.6 H I G H E R - O R D E R T E R M S A N D WALL S T R U C T U R E S In this section, we discuss walls in ferromagnetic films of low anisotropy. Walls are transition layers in the magnetization that separate domains. What are the mechanisms which stabilize such a transition layer? Ordinary Bloch walls in bulk samples arise from the competition of exchange energy and crystalline anisotropy, cf. Section 4.7.2 and [29; Section 3.6.1 (A)]. As we shall see, walls in films rather arise from the competition of exchange and magnetostatic energy. Apart from the intrinsic interest in these structures, their energetic contributions are also expected to play a crucial role in breaking the degeneracy of the two-dimensional limit functional (4.70) discussed in the previous section. Following van den Berg, [32], we shall start by explaining why there are walls at all in low-anisotropy ferromagnetic films, cf. Subsection 4.6.1.

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

301

FIGURE 4.8 The transition between expulsion and penetration regimes. Lines show the level curves of the potential of the penetrated field, arrows indicate the magnetization direction.

Then we will discuss the wall structures themselves - as it turns out, the picture is quite complex. In Subsection 4.6.2, we introduce the two basic wall types, the N6el wall and the stray-field free asymmetric Bloch wall. The thickness-dependent crossover can be rigorously identified on the level of the specific line energy [13]. The N6el wall is a multiscale object in itself. In Subsection 4.6.3, we discuss the logarithmic tail of the N6el wall, and present its rigorous analysis by Melcher [10,11]. In Subsection 4.6.4, we discuss the core of the N6el wall in moderately thin films, and present some rigorous analysis on how the volume charge is distributed between core and tail [46]. In Subsection 4.6.5, we introduce the cross-tie wall, which is a compound wall made of a periodic microstructure of N6el wall segments. It is observed in moderately thin films. We present the rigorous analysis by Alouges, Rivi6re and Serfaty [9], which identifies an optimal pattern. In Subsection 4.6.6, we give an explanation of why the N6el wall in very thin films does not form a cross-tie pattern [12]. In Subsection 4.6.7, we address the mechanisms that determine the period of the cross-tie wall. In Subsection 4.6.8, we consider thick films with a perpendicular uniaxial anisotropy and analyze domain branching [47,48].

302

CHAPTER 4 Recent Analytical Developments in Micromagnetics

For each topic, we start with an informal discussion, then present the rigorous result and finally give some insight into its proof. The arguments of this section rely on the variational nature of the micromagnetic model: the ground-state structures are characterized as minimizers of the energy. Physically motivated constructions ('models') yield upper bounds on the minimal energy. A rigorous analysis can contribute by establishing that these upper bounds cannot be improved (in terms of scaling, asymptotically or exactly). Even if these ansatz-free lower bounds just capture the correct scaling, they allow one to unambiguously identify the various regimes. We shall try to give an idea of how such lower bounds are obtained. 4.6.1 WHY WALLS IN SOFT FERROMAGNETIC FILMS?

Based on the nonrelaxed 'magnetostatic regime', introduced in Section 4.4 we shall very informally explain how the principle of pole avoidance leads to the formation of walls in ferromagnetic films, even in the absence of crystalline anisotropy. This discussion follows the lines of van den Berg's work, see e.g. [32]. Let us think of a 'mesoscopic' magnetization m, where walls would be represented by line discontinuities. In the magnetostatic regime, the exchange energy suppresses the variation of m in the thickness direction, i.e. m = m(Z). The magnetostatic energy, on the other hand, suppresses the out-of-plane component, i.e. m = m ~ e 51, it suppresses a nontangential component, i.e. m ~. v' = 0 along Of]', and suppresses the in-plane divergence, i.e. V'. m' = 0 in f2'. This is the principle of pole avoidance. We collect these conditions on the two-dimensional unit length m' (x'): V'.m'=0

in f~

and

m ~.v ~ = 0 along ~?f2t.

(4.87)

(This is a special case of Eqns (4.74) and (4.85)) These conditions are too rigid for a smooth mesoscopic m ~. This can be seen by using the first property of (4.87) to write m' with the help of a stream function 6 as m t - V '• 6. Then is seen the fact that m ~ e ~1 and the second condition in (4.87) turn into the Dirichlet problem for the eikonal equation, i.e. IV'•

2 = 1 in f2'

and

q

-

const along Off.

(4.88)

It is well known that (4.88) does not admit a smooth solution for a simply connected, bounded f2t. On the other hand, (4.88) admits many continuous solutions that satisfy the first condition of (4.88) outside a set of lower dimensionality. The simplest one, 'the viscosity solution', is the distance function to c?f2t: ~(x') = dist(x', Of]'). (4.89)

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

303

v

FIGURE 4.9 The Landau state. The negative distance function ~/(x') - -dist(x', 0f~') is of course as good, or any distance function of the form ~/(x') = dist(x', 0f~I U F I) where F' is an arbitrary curve. Notice that unless f~ is a disk, all these functions feature lines across which the derivative jumps. Hence we learn that the solution concept for (4.88) has to be relaxed to allow for line singularities. This therefore applies also to (4.87): for (4.87) to be solvable, one has to admit for line singularities and interpret the first condition in (4.87) in the distributional sense. It is tempting to interpret these line singularities of the mesoscopic magnetization enforced by (4.87) as walls. Indeed, van den Berg has shown that a huge variety of domain and wall patterns in the 'magnetostatic regime' fit into the above framework. Let us just mention the 'Landau state', which can be associated with (4.89), see Fig. 4.9. Notice that the first condition in (4.87), when interpreted in the distributional sense, implies that the normal component of m' does not jump across the one-dimensional discontinuity lines - charge-free walls in the jargon of micromagnetics. As usual in the study of transition layers, we assume that the mesoscopic discontinuity lines are straight. In view of the above, straight discontinuity lines are characterized by a single (nonoriented) angle 2 0 ~ 0 , which is the angle between the constant magnetization to the left and to the right, see Fig. 4.10. In the sequel, we will always denote by Xl and x2 the in-plane variables normal and tangential, respectively, to the straight wall.

4.6.2 NI~EL VERSUS ASYMMETRIC BLOCH WALL In this subsection, we present the two basic wall types in soft ferromagnetic films. On a microscopic level, a straight discontinuity line is replaced by a smooth transition layer. We assume that the magnetization is independent of the tangential in-plane variable x2, but we do a priori allow for a

304

CHAPTER 4 Recent Analytical Developments in Micromagnetics

/~0oo

D X 1

FIGURE 4.10 The wall angle 0c~. dependence on the out-of-plane variable X3. Hence the considered class of admissible magnetizations is given by m

-

re(x1,

x3) E ~2

(4.90)

with lim m = (cos 0~, 4- sin 0~, 0). Xlt+OO

(4.91)

It is well known in the applied literature that there are two basic types of walls in films: 9 In thin films, the (symmetric) N6el wall is observed, see [29; Chapter 3.6 (C)]. If the film is sufficiently thin, the internal structure of the wall is subject to the same constraints, which are also imposed on the mesoscopic magnetization: variations in the x3-direction are effectively suppressed by the exchange energy, and the out-of-plane component m3 is effectively suppressed by the stray-field energy. Hence the magnetization is of the form m - (ml (Xl), m2(xl), 0),

(4.92)

see Fig. 4.11. In view of (4.91), these walls are not charge-free on a m i c r o s c o p i c level: din1 V. m = ~ 0. (4.93) dxl 9 In thicker films, the asymmetric Bloch wall (and its variant, the asymmetric N6el wall) are observed, see [29; Chapter 3.6 (D)]. Indeed, for sufficiently thick films the volume charges (4.93) lead to a

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

305

/ FIGURE 4.11 The basic form of a N6el wall.

S

y / /

t (/ ...o'. \ x ,

FIGURE 4.12 The asymmetric Bloch wall. prohibitively large magnetostatic energy. The ordinary Bloch wall, i.e. m = (cos0~, m2(xl), mg(xl)), which has no volume charges, would generate surface charges at x3 E {0, t}. For sufficiently thick films however, the exchange energy leaves enough room to avoid both volume and surface charges: V. m = 0 for X3 E (0, t)

and

m3 = 0 for X3 E {0, t}.

(4.94)

The nonobvious fact that there exists an m of the form (4.90), which exactly satisfies (4.94) has been discovered in [49], see Fig. 4.12. Simultaneously, numerical simulation in [50] showed that the relaxation of the symmetry assumption (4.92) allows for a considerable reduction of the energy. This distinction of two basic types of mesoscopically one-dimensional transition is reflected by the scaling of the specific line energy. We will first do

CHAPTER 4 Recent Analytical Developments in Micromagnetics

306

this heuristically and then present the corresponding rigorous result [13]. To fix ideas, in this and the next subsection we treat the case of adjacent domains of opposite magnetization, i.e. 0~ = ~z/2. For reasons which will be clarified in the sequel we require a small anisotropy Q << I whose easy axis stabilizes the magnetization (0, 4-1, 0) in the adjacent domains. Hence the relevant specific energy (that is, per length in x2) for configurations of the form (4.90) is given by

Exlxg(m)-d2 fN

x(O,t)

(Ic91m12q- lcgsm12)dxldx3

q-lR2 IVumladxldx3q-QfRx(0,t) (m2q-m2)dxldX3, (4.95) where the stray-field urn= urn(x1,x3) is defined in the usual way, cf. (4.16). Let us first heuristically discuss the case of the asymmetric Bloch wall. Because of the structure assumption (4.94), the magnetostatic contribution to (4.95) drops out. Let us assume that the anisotropy contribution is negligible. Then the remaining variational problem of minimizing the exchange energy among all magnetizations of the form (4.94) has a single length scale, namely, t. Hence, by dimensional analysis, we expect that the wall width w and its specific energy E scale as WasymBloch ""

t

and

EasymBloch ~" d 2.

(4.96)

In this case, the contribution from anisotropy would scale as Q t2, which is indeed small compared with d2 as long as t2d-2 << Q-1.

(4.97)

This purely dimensional argument, however, does not reveal how the exchange energy (in combination with the constraint (4.94)) manages to localize the transition layer. We will give the argument at the end of this subsection in connection with Theorem 4.6.1. Let us now consider the case of a N6el wall. For magnetizations of the form (4.92), the specific energy (4.95) reduces to

dm'ladxl+fR2 IVumladxldx3+QtfRm2dx1"(4"98) Eld(m) = d2tfR I-~x 1 The stray-field energy term becomes more transparent when considered in Fourier space (kl denotes the dual variable to Xl) IR 2

IVUml2dxldx3

- t

fR(sinh( 89 ) i~(m1).2 dkl. (4.99) 1 - 89 exp(ltlkll)

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

307

Note that this is the x2-independent version of (4.45). Here ~'(ml)(kl) -

~1

fR ml(Xl) e-iklXl dxl.

(4.100)

We note that the Fourier multiplier has a crossover at horizontal wavelengths 2rt of the order of the film thickness t: sinh(lt'kli)

1-

l tikl l exp(ltlklI)

{ 89 1

for for

tlkll<~ l "

(4.101)

Since we expect magnetizations of the form (4.92) to occur for sufficiently thin films it is natural to consider the tJkll << I approximation to (4.101). This yields Eldthin (m)

= d2t = dat

fo fo i-d-fix1

d m,12 dxl + ]-~Xl d m,12 dxl q-

t2f

Iklll~'(ml)l 2 dkl + Q t

II t2fRd

fo fo

11/2m11 2 dxl q- Q t

m 2 dXl

m 2 dxl. (4.102)

Let us make a deliberately naive attempt to guess the scaling of the width w of the N6el wall from (4.102) by a dimensional argument. The exchange energy would scale as d2t w -1, the anisotropy energy as Q t w, whereas the magnetostatic energy would not depend on w and scale like t 2. Hence, w would, like for the bulk Bloch wall, be determined by the balance of exchange and anisotropy contributions, i.e. w ~, d Q-1/2, see Subsection 4.6.8. But, as opposed to bulk Bloch walls, the main contribution would come from magnetostatics, since t 2 >> Q1/2 d t for sufficiently soft materials. The flaw of this argument lies in the assumption that the N6el wall has a single length scale w, when it actually has two: a core and a tail width Wcore ~ Wtail. Indeed, we think of the N6el wall centered at x] = 0 as being divided into two portions: a core Ix11~Wcore and two tails Wcore~iXl i~Wtail. This two-scale nature allows the N6el wall to decrease its magnetostatic energy by a logarithmic factor. In fact, the scalings of the energy contributions are contribution from exchange

d 2 t Wcore, -1

contribution from magnetostatics

t2 ln-1 Wtail Wcore Q t Wtail.

contribution from anisotropy

(4.103)

CHAPTER 4 Recent Analytical Developments in Micromagnetics

308

The logarithmic scaling of the magnetostatic contribution will be explained in Subsection 4.6.3. Now, optimizing in Wtail and Wcore yields Wcorethin ~ d 2 t-l, Wtai1 ~ Q-1 t and ENeelthin ~ t 2 In -1

Wtail Wcorethin (4.104)

For very soft materials like Permalloy (Q ~ 1), Q-1 t may be larger than the sample dimension s In this case Wtail has to be estimated by s and thus the specific N6el wall energy scales as ENeelthin ~ t2 in_ 1 d-s ~.

(4.105)

This is the model of what is called the ultrathin N6el wall, see [29; p.243], because it is the appropriate model for very thin films. The core width of the N6el wall in moderately thin films scales as d, as we shall explain in Subsection 4.6.4. In fact, the scaling (4.104) is only consistent with the thinfilm approximation (4.102) provided the smallest length scale, i.e. Wcorethin, is large compared with t, which yields t KK d. Notice also that the scaling (4.104) is consistent with the assumption that Wcorethin KKWtail as long as t2 d -2

~

Q.

(4.106)

The optimization of (Eqn. 4.103) in Wtail and Wcore also shows that, by a logarithmic factor, the main contribution ot the energy scaling comes from the magnetostatic energy. In particular (4.104) shows that the N6el wall not only requires a small-scale cutoff by the exchange energy but also a largescale cutoff, which in the present case is provided by anisotropy. Only this cutoff limits the tail width Wtail. The heuristics that lead to the energy scaling of asymmetric Bloch and N6el walls have been made rigorous in [13]. The analysis does not presuppose the special form (4.92) or (4.94). It thus confirms that there are only these two basic types of mesoscopically one-dimensional transition layers in soft materials - at least in terms of the leading-order scaling of the specific wall energy. THEOREM4.6.1 (I13]). In the regime (4.97) and (4.106) and for O~ = 7r/2 in (4.91) we have

d min(4.90) Exlx3 (m) ~ m satisfies

t 2 In -1

Qt2d2

for

t 2 d -2 ~ In ~ }

for

t 2 d -2 ~ In

(4.107) "

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

309

The notation 'A ~ B in the regime (4.97) and (4.106)' means that there exists a universal constant 0 < C < oo such that ~ A~B~CA provided t 2 d-2~ 1 Q-1 and t 2 d -2 ~C Q. Upper bounds are proved by construction based on the ansatz (4.92), respectively (4.94) and inspired by the physics literature, in particular [49] for (4.94). The main contribution of a mathematically minded analysis is to show that these models cannot be substantially improved, i.e. in terms of scaling. This is done by providing ansatz-free lower bounds. To give a flavor of a lower-bound argument, we show how to obtain the bound ~d 2 for m's which are of the form (4.94). This shows how the exchange energy in combination with (4.94), and the nonconvex constraint Iml2 = 1, manages to stabilize the transition layer. Indeed, the second condition in (4.94) implies

1z2/~

x(0,t)

m2 dxldx3 ~ t2 /~

x(0,t)

(~3m3)2dxldx3 .

(4.108)

Furthermore, (4.94) together with 0oo = ~ yields f0 ml dx3 - 0, which gives

~2 f~

x(0,t)

m2 dxldx3 ~ t2 fR

x(0,t)

(83ml)2dxldx3.

(4.109)

Similarly, we have for the vertical average/'h 2 -- } fO m2 dx3"

F2/~

( m 2 - rh2) 2 dxldx3 ~ t2J~ x(O,t)

(~3m2) 2 dxldX3.

(4.110)

x(O,t)

Now, because of m2 4- m2 4- m32 - 1, (4.108), (4.109) and (4.110) combine to ~2t/~

( 1 - rh2)dxl ~ t2/~

x(0,t)

x(0,t)

Ic~3ml2dxldx3.

(4.111)

On the other hand, we have t

1

1 -/'h22

1

((91ff/2)2 dxl

x (0,t) 1 J~

m2 (01m2) 2 dxldX3

101ml2 dxldx3,

(4.112)

x(0,t) where we again use m2 4- m2 4- m2 - 1. Now (4.111) and (4.112) combine by the Modica-Mortola argument (cf. Subsection 4.7.2) to give a

CHAPTER 4 Recent Analytical Developments in Micromagnetics

310 lower bound:

d2 IR

x(0,t)

Ic31ml2+ Ic33ml2dxldX3

(4"111)'(4112) d 2 ( t ZR

1

(Cqlff/2)2 dxl + /z2 IR (1 - rh2) dXl )

T

~/2~:d 2 JR 1~1ff/21dxl (4.91) /~ 41r d2 . 4.6.3 THE LOGARITHMIC TAILS OF THE NI~EL WALL

It is by now well established that the tails of a N6el wall decay logarithmically, see [29; Figure 3.76]. However, early treatments based on ansatz functions missed the very slow decay and led to an overestimation of the wall energy, in particular for low anisotropy Q (see e.g. the review article [51], pp. 397-401, where q = 1/Q, and [29], p. 242). A first formal asymptotic analysis predicting the logarithmic decay was carried out by Riedel and Seeger [52]; a detailed mathematical discussion and various refinements of their work can be found in the work of Garcia-Cervera [53,54]. Finally, Melcher [10,11] recently established the logarithmic behavior of the tails rigorously, see Theorem 4.6.2 below. We start by giving a very informal argument in favor of the logarithmic scaling (4.103) of the magnetostatic energy of the N6el wall, as well as in favor of a logarithmic tail in the range Wcore~lXl I~Wtail. We notice that due to (4.91) and (4.92), there must be an Xl such that Iml (Xl)l = 1, which we think of as the center of the wall. By symmetry we may assume ml(0) = 1. On the Fourier side, ml (0) = I is expressed as Ra'(m1) dkl -

~/~.

(4.113)

It helps to think of (4.113) as a constraint on the charge distribution

dml dxl r din1 dxl _ - f o ~din1 dxl - 1, rather than on the shape of m itself. Minimizing the thin-film magnetostaticenergy ~t2 fR Iklll~'(ml) 12dkl,

i.e. as f~

subject to the constraint (4.113), yields the optimality condition ~V(ml)(kl) -

Ikll

for kl r 0

(4.114)

for some Lagrange multiplier 2. Since the Fourier transform of - G2 (in the sense of Cauchy's principal value) is given by -V~-~ i sign(k1), we may

311

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

regard (4.114) as the Fourier transform of 2 In I-~l'Wwhere w is some length scale that can be identified with Wtail: ml(Xl) -- 22 In .Wtail.

(4.115)

Ixll

Note that changes in Wtail affect ml by an additive constant to which the dml is oblivious. charge distribution -~T1 dml reside in the tail If one assumes that most of the volume charges - -a-~ (this assumption will be questioned in Subsection 4.6.4), (4.113) turns into f{w

~'(ml)(kl) dkl ~ ~/2-~.

(4.116)

~a~l
This determines the constant 2 in (4.114) asymptotically; we obtain from (4.115): In Wtail ml(Xl) ~ IXll for Wcore ~ IXll KK Wtail (4.117) I n Wtail Wcore

and from (4.114): t2 [ ~ t2 1 Wtail -IkllI.T'(ml)l 2dkl ~ ln- ~ . 2 ,]{W~a~l~lkll~wclre} 2 Wcore

(4.118)

The asymptotic formula (4.118) justifies (4.103) and specifies (4.104) to ENeelthin ~

1r t2 ln_ 1

t2

(4.119)

Q d 2"

This very formal argument has been made rigorous in [10,11]. THEOREM4.6.2 ([11]) 9 In the regime (4.106) and for O~ - ~ in (4.91) one has t2 rc t2 1 min Eldthin (m) ~ -- l n m satisfies (4.92) 2 Q d 2" Furthermore, for any minimizer, ml is symmetric around 0, decreasing on the right half-axis and satisfies in

ml(Xl) "~

t QlXll t2

In Q d2

for

d2t -1 KK IXll ~

Q-it.

312

CHAPTER 4 Recent Analytical Developments in Micromagnetics

The precise meaning of 'A ~ B in the regime (4.106)' is the following: for any ~ > 0 there exists a ~ > 0 such that for Q~<~ t 2 d -2 it holds that 1 - ~<-~ ~<1 + e. Let us give an outline of the approach. One is interested in the tails, which are expected to carry most of the energy. Thus it seems promising to consider the linearization of Eldthin around ml = 0. The original nonlinear character is just preserved in the affine constraint dml dxl - - f o ~dml dxl = 1 on the charge distribution - ~ dml , which is expressed by (4.113). The linearized energy Eldthinli n can then be conveniently expressed in Fourier space:

fOo~

Eldthinlin (m) 2

t2

=dat/R ( d @ l m l ) dxl + ~ / N IkllI.T'(ml)ladkl+QtfRm2dx 1 /R( t2 ) = d2tlk112q- ~ Ikll + Q t I.T'(ml)l 2 dkl. (4.120) N o w (4.120) is minimized subject to the constraint (4.113). We introduce the rescaling Xl -- 1 Q - 1 t ~:1

and

Eldthinlin ---- ~1 t 2 Eldthinlin,

and denote by ~" a n d ]C1 the Fourier transform and the wave vector after rescaling. The solution is given by ~'(ml) =

~-~2

and

Eldthinlin = 2~2

with

~;Ikll 2 -}-Ikll -}- 1

e = 4

Qd2

t2 '

where the Lagrange multiplier 2 is determined through

2-1 _ ~O;l~Cll2 q_ Ikll q- 1) -1 dkl ~ 2 In 1. For e -- 0 the inverse Fourier transform can be carried out explicitly, defining rh - 2 -lm:

t~/0(~l):__~_1 (

1

)(Xl)

~2-~(IJCl I q- 1) cos ~:1 fl~oo cos ~______d~1~1 q- sin I~:lI ~x~ sin ~1 d~l,

for Xl 5~ 0.

313

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES The function r~~ has the expansions

thlO(~l) -- !/r { :~1 21n(1/1~11)+ 0(:~14)-7 -}- 0(~:1)

as IXl]K( 1 as IXll ~ 1

where 7 = 0.577... is Euler's constant. This shows that in the region d 2 t -1 KK Xl KK t Q-1 the optimal profile for the linearized energy Eldlinthi n indeed has a logarithmic shape (4.117). Theorem 4.6.2 shows that the minimizers of the linearized energy E ldthinlin already capture the correct leading-order asymptotics both for the tails and the minimal energy for Eldthin. In [46; Proposition 2], the correction has been shown to scale as min

m satisfies (4.92)

E I dthin (m)

2~ t2 ln-1Wtail Wcore ~

t2IlnlnWtail)ln-2Wtail Wcore

Wcore

in a slightly different context. 4.6.4 THE CORE OF THE NI~EL WALL IN MODERATELY THIN FILMS

In moderately thin films, the N6el wall has an even more complex structure than the analysis of the previous subsection reveals. We quote from [29; p. 243] 'In the centre of the wall, the flux into the wall normal direction (i.e. Xl) is ... 1 in reduced units. In the domains this flux is ... cos 0~. In each half wall the reduced total magnetic charge 4-(1 - cos 0~) has therefore to be distributed. Part of the charge is concentrated in the core, where it supports a low energy state by close interaction with its counterpart of opposite polarity. This part is limited by the exchange energy, which prevents an arbitrarily narrow core width. The other part of the charge gets widely spread in the tail.' We refer to Fig. 4.13. In this subsection we present a rigorous analysis [46], which brings this to the point. We start with the heuristics. The 'close interaction between charges in the core' cannot be captured within the thin-film approximation of the stray-field energy (4.102). As a heuristic, let us replace the Fourier multiplier in (4.99) by its second limiting expression in (4.101), since it is the asymptotic expression for horizontal scales << t. For subsequent use, we consider a general wall angle 0~ and neglect anisotropy, which is not relevant for the balance in the core. Notice that in this case ml - cos 0~ is the quantity that is square integrable for Ixll ~ 4-oo. Therefore, in (4.99) .T(ml) has to be replaced by

314

CHAPTER 4 Recent Analytical Developments in Micromagnetics

Tail

Core

Tail

FIGURE 4.13 The charge distribution in core and tail.

~v(ml - cos 0~). Hence, Eld is approximated by Eldthick (m) =

dm'1 la dxl + t fR l~(ml - cos O~)12 dkl d2t fR I-~x

dm'1 la dxl + t fR(mx - cos O~)a dXl. = d2 t fR I-~X It is convenient to express Eldthick in terms of the local angle, sin 0): Eldthick(m) -

0) d2t f R ( d~Xl

m'

= (cos O,

2 dxl + t fR (cos 0 - cos 0~) 2 dxl.

(4.121)

This standard variational problem has - up to translation - a unique minimizer for the b o u n d a r y conditions (4.91). A dimensional a r g u m e n t shows that the w i d t h of the transition layer scales as Wcorethick "~' d.

(4.122)

The energy of the optimal transition layer can be c o m p u t e d explicitly: ENeelthick -- 2 d t

F

Oo

(cos 0 - cos 0c~) dO -

4 d t (sin 0~ - 0~ cos 0~). (4.123)

Hence Eldthick(m ) indeed models the competition of magnetostatics and exchange in the N6el wall core as described in the quotation. This is consistent as long as the core w i d t h is small compared with t, i.e. d
(4.124)

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

315

so that the approximation of the Fourier multiplier (4.101) leading to (4.121) is justified. On the other hand, the magnetostatic part of Eldthin models the spreading of charges into logarithmic tails, as we have seen in Subsection 4.6.3. Let us now address the optimal mix suggested in the quotation and envision a N6el wall profile m I = (cos 0, sin 0) with 0~ = and center at Xl = 0. We think of the profile as being divided into a core governed by Eldthick(m) and tails governed by Eldthin(m ). The core extends o v e r 101~0core and the tails over the region described by 10]~10corel and IXll~Wtail. Observe that core and tails not only live in different parts of physical space but also in different parts of frequency space: in view of (4.122), the core has a peaked frequency spectrum around Ikll ~ d -1 whereas the tail has a spread-out frequency spectrum, cf. (4.114). This justifies thinking of the energy as the sum of the energies of core and tail. The energy of the optimal core is given by (4.123) with 0~ replaced by 0core. The magnetostatic energy of the optimal tail is given by (4.118), but with a factor of cos 2 0core. Indeed, the right tail has only to accomdin1 modate the charges --a-~l coming from the rotation from ml - cos 0core to ml -- 0 (instead of the rotation from ml - 1 to ml - 0 considered in Subsection 4.6.3). This means that in the informal discussion at the beginning of Subsection 4.6.3, the constraint (4.113) has to be replaced by ~ .T(ml) dkl - ~ cos 0core. The square in COS2 0core is a consequence of the fact that the magnetostatic energy is quadratic in ml, cf. (4.118). We summarize these findings: contribution from core

~

4 d t (sin 0core -- 0core COS 0core),

contribution from tail

~

Wtail ) COS2 0core, 2 t 2 (ln -1 Wcore

contribution from anisotropy

,--

Q t Wtail.

(4.125) We allow ourselves to be noncommittal about Wcore since it only appears in the logarithm; to fix ideas we take Wcore -- d ~ Wcorethick. N o w the quotation suggests that the optimal N6el wall profile is obtained by optimizing the sum of (4.125) in 0core and Wtail (as opposed to Wcore and Wtail for the ultrathin N6el wall, see (4.103)). This somewhat bold optimization principle has been made rigorous for a special case in [46]. It treats the case where the length Wtail of the N6el wall tails is not implicitly determined by the balance of magnetostatics and anisotropy as in (4.104), but is imposed by boundary conditions: m -

(0, 4-1, 0)

for

4- Xl ~Wtail.

(4.126)

CHAPTER 4 Recent Analytical Developments in Micromagnetics

316

The condition (4.126) models the confinement of the N6el wall tails by sample edges or neighboring N6el walls. The result of [46] is also restricted to the limiting regime where the core energy dominates the tail energy, i.e. d t -1 In -Wtail ~ << 1

with the understanding that

d KKWtail.

(4.127)

Notice that this contains (4.124). In the regime (4.127), the optimization principle in 0core based on (4.125) yields 0core ~

2 d t_l In Wtail d

2

and

ENeelmi x ~ 4 d t - 2~ d 2 In Wtail

d "

This is exactly what can be proved using the original magnetostatic energy in Eld.

([46]). Let Q - O. In the regime (4.127) and t << Wtail, we have

THEOREM4.6.3

min

m

satisfies (4.92) and (4.126)

Eld ,~ 4 d t - 2~ d 2 In Wtai-----~l. d

(4.128)

The precise meaning of the notation of Theorem 4.6.3 is the following: for any e > 0 there exists a ~ > 0 such that if d~wtail, dt -11n - ~ K~ and tK~ Wtail, one has 1

min Eld - 4 +

2rcdt-1 In Wtail ~ dt -1

In Wtail d

The statement in [46; Theorem 1] has an additional factor of two, since it considers the case of an array of winding N6el walls at distance Wtail and computes the energy of a pair of walls. The proof is based on a perturbation analysis around the well-characterized minimizer of (4.121). It uses the variational structure and, amongst others, the fact that Eld, when expressed in terms of ml alone:

Eld (m)=dat /R

1 - 1ml2

( ~Xl d ml )2

dx1-3L-f~ 2 IVuml2 dxldx3

+Qt /R m2 dxl,

is nonlinear but convex in ml. The comparison of (4.123) for 0o0 = ~ with (4.128) quantifies how the spreading of charges into the tails reduces the total wall energy. It can be seen from (4.128) that this effect decreases as the confinement length Wtail of the tails is decreased: (~ENeelmix < 0. It can also be read off that the rela~Wtail

tive importance of the repulsive term decreases with increasing thickness: OENeelmix/ENeelmi x ~ 7~d t -1 w -1 This quantifies the repulsion. (~Wtai1 2 tail"

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

317

It has to be noted, however, that for the full energy Exlx3 (i.e. when minimizing over the larger class (4.90)), there is no proper regime where the leading order energy scales as d t, as we learn from Theorem 4.6.1. The scaling in Theorem 4.6.3 is a consequence of the restriction to magnetizations of the form (4.92). Other restrictions of the magnetization have been considered, for instance, that of a bulk Bloch wall symmetry, i.e. (0, m2(xl), m3(xl)). This yields a scaling of the specific wall energy as d 4/3 t 2/3, see [54; Theorem 7.1]. 4.6.5 THE CROSS-TIE WALL

Formula (4.123) shows that the core energy of a N6el wall strongly degenerates for small wall angles 0oo: ENeelthick(0~)

--

4

3

4d t (sin 0~ - 0~ cos0~) ~ ~ d t 0~

for 0~ << 1. (4.129)

In particular, a ( 0 ~ = {)-wall 'costs' only a fraction of a ( 0 ~ = ~)-wall: ~ with ~-~ 4-1r ~ 0.152. In the physics literature, ENeelthick(~)-- ~4-Ir ENeelthick ( ~), this observation is evoked to explain the occurrence of the cross-tie wall, see Fig. 4.14. The cross-tie wall is a periodic pattern of small-angle N6el walls that macroscopically acts as a (0~ -- ~)-wall. This pattern is formed by perpendicular N6el-wall segments (the 'cross-ties'), which divide a central wall into segments. In this subsection, we report on a remarkable work by Alouges, Rivi6re and Serfaty [9], which, based on (4.129), identifies an optimal mesoscopic pattern. Motivated by van den Berg's view, cf. Subsection 4.6.1, all admissible mesoscopic patterns are described by (discontinuous) in-plane magnetizations m'(x') e ~1 that macroscopically act as a (0~ ~)-wall, which have some tangential period Wcross, and which are weakly divergence-free, i.e. limxl~+Oo m' = (0, +1), m'(xl, X2 -}- Wcross) -- m ' ( x l , x2), V'. m' = 0

(4.130)

distributionally.

We first describe the optimal mesoscopic magnetization m' and its line discontinuities, which are sketched in Fig. 4.15. If Xl - 0 denotes the position of the central discontinuity line, m' obeys the symmetry (ml, m2) (-Xl, x2) -- (ml,-m2)(Xl, x2). If x2 = 0 denotes the line half way between two perpendicular discontinuity lines, m t also obeys the symmetry (ml, m2)(Xl,--X2) -'- (--ml, m2)(Xl, x2). Hence, it suffices to specify m' in

CHAPTER 4 Recent Analytical Developments in Micromagnetics

318

FIGURE 4.14 Cross-tie wall in a 76-nm-thick Permalloy film. Reproduced with permission from Hubert and Sch~ifer, Magnetic Domains, Springer (1998).

J Wcross

\ X2

I

v

X1

FIGURE 4.15 The optimal mesoscopic pattern.

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

319

{Xl > 0, 0 < X2 < Wr }, the shaded region in Fig. 4.15. In this region, the magnetization is continuous (but not differentiable) and given by pasting a vortex-like magnetization to a constant magnetization: x2

m'(x') =

Xl

(-- .2---------2 2nt-x2 2) ~/xl -}-X2' ~/X1

v1f , ~ )1

(

for Xl>/X2 for

} .

Xl ~<X2

(4.131)

It is easy to check that m' satisfies (4.130). The angle 0oo of the line discontinuities covers the range from ~/t to zero. The line energy per unit length in the x2-direction of this pattern is easily calculated to be (we may set W c r o s s - 1) Ecrosstie - 2 ENeelthic k

~

if- 2

ENeelthick (arctan 2-~1) dxl

= 4d t ( v r i - 1).

(4.132)

Notice that this is indeed smaller than the line energy ENeelthic k (~) -- 4 d t of a simple (0oo= )-N6el wall. Surprisingly, it can be shown that the value (4.132) is optimal: THEOREM4.6.4 (I9]). It holds min

m satisfies (4.130) Wcros s

ENeelthick(0oo ) ds -

4d t (~ri - 1),

where F denotes the union of all line discontinuities of m' as characterized by the angle 0oo which varies with arc length s. Let us give the concept of the proof. It follows a strategy laid out in [55] for a mathematically related variational problem. This strategy makes use of continuous maps 9 I(mf) e [~2 with the following two properties: V'.O'(m') = 0 for every smooth m'(x') e ~1 with V'- m ' - 0

(4.133)

] ((I)' (mleft) ' V' I ~< ENeelthick(0oo) , _ (I)' (mright)).

(4.134)

and

for any mleft' i m Iright ~ ~1 that form the angle 0oo with a normal vI ~ ~1, see Fig. 4.16.

320

CHAPTER 4 Recent Analytical Developments in Micromagnetics

S

oo

m'left

m'right

FIGURE 4.16 Wall and normal.

By the divergence theorem, such a (I)' yields a lower bound independent of m~: 1

iF ENeelthick (0 ~) ds

Wcross (4.134)~

]

1 Wcross

_]

1 Wcross

fF (O' (mleft) -- O' (mtright)) 9Vt ds I

i~

V"O'(m')dxt x (0,Wcross)

-O'(mt(xl=+Cx~)) 9(1, O) - O'(m'(xl=-CX~)). (-1, 0)] (4.130),_(4.133) iO,((O ' 1))-(1, O) - 0 ' ( ( 0 , - 1 ) ) . (-1, 0)].

(4.135)

In order to yield an optimal lower bound, (I)~must be adapted to the form of ENeelthick (and the conditions imposed at Xl = -t-a). The choice of (IY in [9] is guided by 1 (~t

2dt

(m') = Ore'+

m~•

where

m' "- (cos 0, sin 0)

(4.136)

introduced in [39]. That paper treats the case where m t admits a global phase 0. It is not difficult to see that (~' satisfies (4.133) and (4.134). But it can not be used here since there is no global phase 0 for the cross-tie pattern (4.131), see also Subsection 4.6.6. For [9], symmetric variants of the (~' in

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

321

(4.136) are pieced together to form a map q)' (m') continuous in m' 6 ~1.

Om t + m '• 4- (0,-~/'2)

1

2dt

q)'(m') =

for (~ - 0)m' - m I• + (-v/2, O) for ( 0 - ~z)m' + m 1• + (0, ~/2) for ( ~ - O ) m ' - m '• 4- (~/2, O) for

Using the corresponding property of ~)', One checks that also q)t satisfies (4.133) and (4.134). Finally, one easily checks that for this choice of q)', the right-hand side of (4.135) indeed takes the value 4 d t (~/2 - 1). 4.6.6 T w o - D I M E N S I O N A L STABILITY OF THE NI~EL WALL IN VERY THIN FILMS

We return to the ultrathin N6el wall investigated in Subsections 4.6.2 and 4.6.3. In this case, the degeneracy in the angle is even more pronounced. Indeed,

rc t2 (ln-l Wcore Wtail ) (1_ cos 0oo)2

ENeelthin (0cx~) -- ~

~ _~zt2 ( ln-1 Wtail ) 0~ 4 8

Wcore

for0c~KK1,

(4.137)

cf. the argument which leads to (4.125). So the ultrathin N6el wall should be more prone to the formation of a cross-tie pattern Fig. 4.15 than the N6el wall in moderately thin films. But in fact, experiments do show stable ( 0 ~ = ~ )-N6el walls in sufficiently thin films. We first give the heuristic explanation from the physics literature why the ultrathin N6el wall does not suffer from the cross-tie instability. As we have noted in Subsection 4.6.5, the optimal mesoscopic cross-tie pattern m', see Fig. 4.15 and formula (4.131), does not allow for a global phase 0. More precisely, there are topological defects residing in the points indicated in Fig. 4.17. The black circles indicate a point singularity with winding number 1, the white circles a point singularity with winding number -1. Analogously to the line singularities, the point singularities are replaced by smooth structures on the microscopic level, the so-called Bloch lines ('line' refers to the x3-direction). Because of the topological reasons mentioned, these structures require an out-of-plane component m3. The inner structure of the Bloch line, which replaces a white circle, is easy, cf. Fig. 4.18: it is the regularization of a vortex, where the magnetization avoids the singularity by turning out of the plane when approaching the center.

CHAPTER 4 Recent Analytical Developments in Micromagnetics

322

\

,I A

/

/ / \ \

IIW

\ \

\ / () 'x /

/

t t t 7'

/

\

\

JILL,

X2

V

T

X1

I

FIGURE 4.17 Topological defects in the mesoscopic cross-tie pattern.

--f11\'-----.111\-..-X2 T I

3 X1

~- X l

FIGURE 4.18 The circular Bloch line. In [6] the following scaling of the energy for the Bloch line in ultrathin films is heuristically derived: EBloch ~"

d2t In

Wcross

see also Section 4.7.3. Since the N6el-wall energy tends quadratically to 0 for t $ 0, cf. (4.137), the relative cost of Bloch lines increases with decreasing film thickness. This is the common explanation why cross-tie walls are not observed in sufficiently thin films despite the pronounced angle degeneracy, cf. (4.137). This does not yet explain why the (0~-- ~)-N6el wall is stable in very thin films, even if an out-of-plane component is suppressed. This question has been addressed in [12]. There, a class of microscopic in-plane

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

323

7~ magnetizations m'(x') ~ ~1, which macroscopically act as a (0~ - ~)wall, and which have some period w in the tangential direction, is considered. The N6el-wall tails are constrained to have the width Wtail by a two-dimensional version of (4.126). This amounts to:

m'(x1, x2) -- (0, + l )

for 4- Xl ~ Wtail,

(4.138)

m'(xl, x2 if- w) -- mZ(x1, x2). For this type of m ~, the three-dimensional energy (per period w in x2) without anisotropy assumes the form

Exlx2(m')-- dat/R

x(0,w)

]V'm'12dx' + j ~

x(0,w)x•

IVumladx,

where the magnetostatic potential Um(XI, X2, X3) is generated by the volume charge V ' . m' in {0 < x3 < t}. For sufficiently thin films, this charge distribution is well approximated by the surface charge - t V'. m' on {x3 = 0}. This is the thin-film approximation of (4.46) and yields

Exlx2thin(m I) = d2t d2t

/~ /N

x(0,w)

IV'm'l 2 dx' +

taf~ t2fN

2

x-~;

Ik'l-11~(V '- m')l 2 dk'

IV'm'l 2 dx' + IIV'I-1/2v ' . m'l 2 dx'. (4.139) x(0,w) -2- x(0,w) This is the proper two-dimensional generalization of (4.102) with Q = 0. The relevant regime is when the core width of the ultrathin (0~ = ~)-N6el wall, cf. (4.104), is much smaller than the imposed tail width, i.e. -

-

d 2 t -1 ~ Wtail.

(4.140)

The energy of the ultrathin N6el wall approximately behaves as n t2 ln_ 1 Wtail t ENeelthin ~ ~ d2 , cf. the argument which leads to (4.104). In [12], it is shown that the onedimensional transition profile is asymptotically optimal under all twodimensional variations of the type (4.138): THEOREM4.6.5 ([12]). In the regime (4.140), 1

n t2 ln_ 1 Wtail t

min --Exlx2thin(m') ~ -~ m' satisfies(4.138) w

d2

(4.141)

CHAPTER 4 Recent Analytical Developments in Micromagnetics

324

lm

l

F

~=

1/2

X2

T I

~ X1

FIGURE 4.19 The mesoscopic wall position. Let us give the concept of the proof. For an arbitrary m' of the form (4.138) there is a curve F in (--Wtail, Wtail) X [~ with: F is w-periodic in x2

and

m' = v'

along F.

One may think of F as the wall center, see Fig. 4.19. The strategy is to show that for length (F) short for length ( r n (R x (0, w)))

Exlx2thin(m') >

= t 2 In -1

2

Wtailt) d2

length(F)

(4.142)

,

which yields (4.141) because of length(F)~>w. This is done by a duality argument based on the charge distribution - V ' . m'. The charge distribution is tested by the characteristic function ~b ~b -

- 89 left o f f

and

~b -

1 right o f F ,

see Fig. 4.19. By the construction of F, the definition of 4) and the divergence theorem, one has, on the one hand, R V'. m' ~bdx' = length(F). x(0,w) In order to conclude (4.142), one needs, on the other hand, that fR

x(0,w)

V'. m' ~bdx'

( 2 t- 2 In Wtail t ~ d2

1/2

Exlx2thin(m') "Jr-0(1))length(F)

. (4.143)

325

4.6 HIGHER-ORDERTERMS AND WALL STRUCTURES This inequality is based on Cauchy-Schwarz in Fourier space

/o

x

(0,w) ,m+dx

.

(1o

Ik' 1-1 I~-(v' 9m')12 dk'

x ~ 7/

Xl~x-~77

Ik ' l l"T"( qh)12 d k ' ) 1/2

(4.139)

2 t -2

Exlx2thin( m ' ) / R x

-~ 77

Ik'l 1~(~6)12dk') 1/2

on the fact that an interpolation estimate of the form f~

x -~7/

Ik'll.%(~b)l2 d k ' / ~ sup I~bl xt

IRx(0,w) IV'~61dx'

(4.144)

just fails logarithmically, and on the obvious identity

/R

sup I@I x'

x(0,w)

1

Iv'@l dx' - ~ length (A)

for our ~b. The failure of (4.144) is carefully controlled by a cutoff in k'-space and gives rise to the logarithmic scaling including the optimal prefactor in (4.143) via the interpolation estimate -1 12 dk' min{Wcore Ik'l2, Ik'l, Wtail}l~'(q~)

1[~ x u27r7]

_2 In Wtail sup I@1 f~ "~ ~

Wcore

Iv'@l dx',

x'

(4.145)

x (0,w)

which holds for all Wcore ~ Wtail and an arbitrary function ~b. The main ingredient for (4.145) is the inequality IE -~z 4 sin2 ( k' "2ax' ) lY(~b) I2 d k ' x

IR

(q~(X' -}- ~Y') -- q~(Y')) 2 dx'

x(0,w)

# [~x'. V' ~bldx' ~<2 sup I~1 ] x' JR x (0,w) (4.146)

326

CHAPTER 4 Recent Analytical Developments in Micromagnetics

for arbitrary 6x' e [~2. This inequality is multiplied with IC~Xf1-3 and integrated over the annulus {Wcore~ IC~X'I~Wtail }. 4.6.7 THE CROSS-TIE PERIOD

We return to the cross-tie wall of Subsection 4.6.5. The analysis of that subsection does not yield the cross-tie period Wcross. Indeed, the variational problem considered in Theorem 4.6.4 is oblivious to Wcross. In this subsection we heuristically address mechanisms that might determine Wcross, following [46]. In the physics literature, this is considered an open problem. 'An attempt to calculate the equilibrium period of cross-ties has to include the repulsive interaction between the main wall segments and the adjacent cross-ties... In addition, the extra energy connected ... [with] the Bloch lines has to be considered. No reliable estimates are available for either of these contributions, which means that a consistent theory of the cross-tie wall is still lacking.' [29; p. 245]. Let us address the proposed mechanisms. The repulsive interaction between the main wall segments and the adjacent cross-ties results from the following fact: the sense of winding of these N6el walls, cf. Fig. 4.20, is such that their tails compete for space. Since the average distance between the main wall segments and the adjacent cross-ties scales as Wcross,we have Wtail ~ Wcross in the language of Subsection 4.6.4. As Theorem 4.6.3 shows for (0 = )-walls, the specific wall energy decreases with increasing Wtail. The repulsive mechanism due to the Bloch lines is easy to understand: as stated in Subsection 4.6.6 for very thin films, every Bloch line carries a quantum of energy that only weakly depends on Wcross. But, the number of Bloch lines per unit length is given by 2 Wcros s.-1 There is no ambiguity about the relevant attractive mechanism. It is given by an in-plane uniaxial crystalline anisotropy, which favors the magnetization in the domains. Of course, we think of soft materials: Q << 1.

(4.147)

Since there is a band around the central wall segments of width "~ Wcross, where the magnetization deviates from the easy axis, see Fig. 4.21, anisotropy favors a small Wcross. In order to arrive at a more quantitative picture, we make the following assumption: the dominant energy contribution is the energy of the N6elwall core. According to Subsection 4.6.5, it completely determines the relative proportions of the optimal mesoscopic pattern, cf. (4.131) and Fig. 4.15, but not its length scale Wcross. We now identify the corrections to the energy from anisotropy, N6el-wall interaction, and Bloch-line contribution

328

CHAPTER 4 Recent Analytical Developments in Micromagnetics

now interested in the case of an arbitrary wall angle 0o0 and tails limited to a length Wtail, cf. (4.126). In this situation, the contribution from the tails in (4.125) has to be replaced by ~ t 2 (ln-1 w~i____)!l(cos {)core - cos 0o0) 2. Minimization of core and tail contributions in {)core gives the following expression for the specific N6el-wall energy:

ENeelmix ~ 4dt(sinOo0-0o0 cos 0 o 0 ) - 78d2 ( In wtail)~ {9~176 (4.149) in the regime d << Wtail and d t -1 in W~ai_J<< 1, cf. (4.127). The leading term in (4.149) is the expression on which the identification of the optimal mesoscopic pattern was based upon, see (4.129). We are interested in the correction term. It captures the reduction of the specific N6el-wall energy coming from spreading a small amount of the Ndel-wall charge in the form of logarithmic tails. More important, it quantifies how this reduction increases with decreasing confinement of the tails. We apply (4.149) to the N6el walls in the optimal mesoscopic pattern. As motivated above, we use Wtail ~ Wcross (SO that In ~ ~ In ~ ) and integrate the correction term along the discontinuity lines of the mesoscopic pattern (with varying 0o0 (s)). We so obtain the following correction to the total N6el-wall energy per length: Correction to total N6el-wall energy per length ~ _d2 in Wcross d ( n 4- 0.844 8 ) .

(4.150)

For the contribution of the Bloch lines, we focus on the simpler circular Bloch line. The circular Bloch line seamlessly extends into the vortex segment of the mesoscopic magnetization, cf. (4.131). By a logarithmic factor, the main contribution to the energy of a Bloch line comes from the exchange energy. We therefore calculate the exchange energy of the vortex segments outside the vortex core and assume that the vortex core size scales as d. Per unit length in x2 this yields:

Wcross -1 s In Total Bloch-line energy per length ~ rcd 2 t Wcros

(4.151)

We now minimize the sum of (4.148), (4.150) and (4.151) in Wcross. The minimizer is the solution of 0.643 Q t

5.29d 2 Wcros -1 s _ 3.14d 2 t Wcross -2 In

Wcross

0.

(4.152)

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

327

FIGURE 4.20 Winding N6el walls in cross-tie.

FIGURE 4.21 The effect of anisotropy.

based on this optimal mesoscopic pattern. The sum of these expressions will then be minimized in Wcross. Based on (4.131), the contribution from crystalline anisotropy per length in x2 is easy to quantify: anisotropy per length ~ ~ 1 + ~

Q t Wcross.

(4.148)

In order to assess the contribution of N6el-wall interaction, we need the result of Theorem 4.6.3 for an arbitrary wall angle 0oo. Recall the heuristics of Subsection 4.6.4 on the optimal mix of core and tail in a N6el wall. We are

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

329

Obviously, there are two regimes: Wcross ~"

ln-1 ~)1/2

8.23 Q-1 d 2 t-1

for

t d-1 << (~

2.21 (~ In ~)1/2 d

for

t d-1 ~ (~ in-1 ~)1/2

Regime I / Regime II /" (4.153)

In the first regime, the dominant repulsive mechanism is the N6el-wall interaction; in the second regime, it is the Bloch-line contribution. Let us check the consistency of the argument: for (4.150) and (4.151) we need to check Wcross >> d and dt -1 in - ~ << 1. This is consistent provided the film is sufficiently thick, i.e. td_ 1 >> In Q 1

(4.154)

so that Regime I is restricted to the band In ~ << td -1 K< (~ In -1 ~)1/2. Notice, however, that in regime (4.154), the asymmetric Bloch wall beats the N6el wall as we learn from Theorem 4.6.1. Hence, the above analysis only applies to a small range of film thicknesses. Let us now compare (4.153) with some experimental observations of cross-tie walls. The scaling of Wcross with respect to Q was experimentally investigated by Middelhoek in [56] for Permalloy films of thickness t = 60nm, for which the effective quality factor was varied through the application of mechanical stresses. The observations show that Wcross is proportional to Q-l, which is in agreement with Regime I. Concerning the dependence on thickness t, we turn to [29; p. 449]. We consider Figure 5.59(a)-(e) of [29], where we read off Wcross. To compare these experiments for Permalloy films of varying thickness with the theoretical prediction, we use d = 5nm and Q = 2.5 x 10 -4. Hence (4.147) is certainly satisfied while only the thicker films satisfy (4.154). In Table 4.4, the first column gives the film thickness, the second column the experimental cross-tie period, the third column the prediction based on Regime I, the fourth column the prediction based on Regime II and the fifth column the prediction based on the solution Wcross of (4.152). The order of magnitude is certainly well captured by the theoretical model (4.152). It tends to overestimate the repulsive mechanisms, especially for the larger thicknesses. The theoretical model also predicts the right trend in the t-dependence. The comparison of the last three rows shows that the experiments are in a parameter regime where the full theoretical model (4.152) is well approximated by Regime I. Hence the dominant mechanism for these experiments seems to be the N6el-wall interaction rather than the cost of Bloch lines.

330

CHAPTER 4 Recent Analytical Developments in Micromagnetics TABLE 4.4

t(nm) 10 30 40 60 76

Comparison of predictions and observations of Wcross.

exp Wcross (tim)

Reg I Wcross (tim)

Reg II Wcross (tim)

Wcross (tim)

62.4 29.2 16.4 8.0 2.8

82.3 27.4 20.6 13.7 10.8

2.01 2.01 2.01 2.01 2.01

82.4 27.7 20.9 14.2 11.5

Recent observations of cross-tie walls in films of CoFeSiB seem to provide an example of Regime II. The proportionality of Wcross to Q-1/2 for this set of experiments seems to be well established, while the dependence on the thickness is being currently investigated [57]. 4.6.8 DOMAIN BRANCHING In this section, we address thick films with a strong uniaxial perpendicular anisotropy, i.e. Q >> 1, (4.155) which favors the out-of-plane magnetizations m = (0, 0, 4-1). It is experimentally well documented and heuristically well understood that this leads to domains with m ~ (0, 0, 4-1), which branch and thus refine towards the sample edges {x3 = 0, t}, see Fig. 4.22. In this section, we present the heuristic arguments, which go back to Lifshitz [58], essentially following Privorotskii [112] Hubert [59] (see also [29; Section 3.7.1]). We also present a rigorous scaling analysis of the minimal energy [47,48], which confirms the heuristics. To fix ideas, we impose some period w in the in-plane variables, i.e. re(x1 q- w, x2, x3) -- re(x1, x2 q- w, x3) -- re(x1, x2, x3),

and consider the energy per area w2:

E(m)

Im'I2 dx. = d 2 f0,W)2X(0,t) IVmI2 dxq-/(0,w)2 xR IVui 2 dx+Q ~(0,w)2x(0,t)

In domains of infinite thickness, the uniform magnetizations m = (0, 0, 1) and m = (0, 0, -1) are the ground states since all energy contributions vanish. In films of a finite thickness, the magnetostatic and the anisotropy contributions to the energy compete: the magnetization favored by anisotropy generates surface charges m3. But the penalization of surface charges is 'soft': alternating domains with m = (0, 0, 1) and m = (0, 0 , - 1 ) (bubble

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

331

FIGURE 4.22 Domain branching in a cobalt crystal. Reproduced with permission from Hubert and Sch~ifer, Magnetic Domains, Springer (1998). domains) reduce the magnetostatic energy, see Fig. 4.23 for a sketch of the mesoscopic magnetization. This reduction is significant for d o m a i n widths ~ smaller than the thickness, i.e. ~

t,

(4.156)

since charge cancellations occur over the distance ~. A dimensional argument yields that the magnetostatic energy scales as w -2 x magnetostatic contribution ~ ~.

(4.157)

On the other hand, these domains require walls. We obviously have w -2 x wall area ~, s

(4.158)

In order to assess the total wall energy, we need the specific wall energy. For sufficiently thick films, the wall type is that of infinite films: the bulk Bloch wall. It is of the form: m -

m(xl),

m -

(0, m2, m3),

m ~

(0, 0, +1) as Xl ~ 4-cx3

and thus does not generate volume charges. Hence, the bulk Bloch wall is stabilized by a balance of exchange and anisotropy energy. The optimal wall profile and the specific wall energy is therefore found by

CHAPTER 4 Recent Analytical Developments in Micromagnetics

332

AU

-

I I I

-

+

X3

~

X1

FIGURE 4.23 Bubble domains. minimizing

Eldbulk(m) = da fR ~Xl dm 2 dxl + Q 1

fo

Im'I2 dXl

(dm3) 2

= dafR 1 - m 2 \ d x l

dxl+ Q

fo

(1-m~)dxl. (4.159)

Hence the wall profile is given by m3(xl) =- tanh WbulkBloch and wall energy per area defined by WbulkBloch

__

Q-1/2d,

EbulkBloc h

Xl

WbulkBloch

=

with wall width

4 Q1/2 d,

(4.160)

see also Section 4.7.2. From (4.158) and (4.160) we now deduce w -2 x wall contribution ~

Q1/2 d s

t.

(4.161)

Optimizing the sum of (4.157) and (4.161) in s yields the domain width s and total energy in the regime (4.156): s

~

Q1/4 dl/2 tl/2,

w-2 Ebubble ~,,

This is self-consistent as long as t >> s (4.160) and (4.162) translates into

t ~ Q1/2d.

Q1/4 dl/2 tl/2.

(4.162)

~ WbulkBloch,which in view of (4.163)

But, precisely in this regime one can do better! Indeed, the magnetostatic energy favors a small domain width s at the surfaces, whereas the

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

333

/ l / IV/ /V /I/V/

X1

FIGURE 4.24 Schematic domain branching. wall energy favors a large domain width s in the bulk. This suggests that the typical domain width ~ should be larger in the bulk than at the surfaces: t >> s

)~ s

))

(4.164)

WbulkBloch.

This refinement can be achieved by domain branching, see Fig. 4.24 for a schematic representation of the experimentally observed mesoscopic magnetization. For appropriate refinement, s appears in the wall energy contribution (4.161) but s in the surface magnetostatic contribution (4.157): w -2 x wall contribution

~-

Q1/2dg,-1 t, bulk

w -2 x surface magnetostatic contribution

(4.165)

~surface.

However, the mesoscopic structure in Fig. 4.24 is not charge-free in the interior, as indicated in Fig. 4.25. In the regime (4.164), each discontinuity of the mesoscopic pattern carries a surface-charge density scaling as s t-1 << 1. Hence the strength of the demagnetizing field scales as s t-1 so that w -2 x bulk magnetostatic contribution ~ s 2 The minimization of the sum of (4.165) and (4.166) in s s

~

Q1/6 d 1/3

t2/3.

t-1 9

(4.166)

yields (4.167)

334

CHAPTER 4 Recent Analytical Developments in Micromagnetics -VU

f

+1+l ++

FIGURE 4.25 Charges in the bulk. This is consistent with the first condition in (4.164) provided (4.163) is satisfied. We think of the surface width s to be (at least) so small that the surface magnetostatic contribution is dominated by the bulk magnetostatic contribution. In view of (4.165), (4.166) and (4.167) this means that s << Q 1/3d2/3t1/3. Notice that in regime (4.163) this necessarily implies s ~ Wbulk, the second assumption in (4.164), which thus is consistent. Finally, the third assumption in (4.164) is ensured by (4.155) and (4.163). Hence, we have w-2 Ebranch ~ Q1/3 d2/3 tl/3,

(4.168)

which beats (4.162) in regime (4.163). The energy scaling (4.168) has been made rigorous: THEOREM4.6.6 ([47,48]). In the regime (4.155) and (4.163) we have W-2 min E(m) ,~ Q1/3 d2/3 tl/3,

(4.169)

m

provided w >> Q1/6 d 1/3 t2/3. The last condition just ensures that the artificially imposed period w leaves enough room for the bulk domains of width ~bulk given by the heuristic discussion, cf. (4.167). Actually, [47,48] considers a slightly simplified model of E where the exchange energy has been replaced by a term that mimics the wall energy. The analysis in [47,48] is motivated by the work in [60,61] on a mathematically similar model for twin branching of martensite near a martensite-austenite interface. A finer analysis of this two-dimensional

4.6 HIGHER-ORDER TERMS AND WALL STRUCTURES

335

model, which captures the anisotropic self-similarity of the branched structure near the surface was carried out by Conti [62]. We sketch the argument for the lower bound, which is surprisingly elementary given the complexity of the pattern. At first, one uses the strong crystalline anisotropy to single out the m3-component:

Q1/2d ~0 f(

,w)2x(0,t)

[Vm31 dx ~ ~1 E(m),

kl I~'(m3)[ 2 dk <

~-7/) 2~ 2 x [~ Ik[ 2

(4.170)

E(m).

(4.171)

Inequality (4.170) follows from the standard Modica-Mortola argument (cf. (Eqn. B.8) below), i.e.

2Q1/2d f

IVm31 dx

J~o,w)2x(0,t)

l lvmsladx+QL ,W)2 X(0,t) ( 1 -

~daf( 0

,W)2 X(0,t) 1 - m~

m2) dx,

and the second identity of (4.159). The approximate inequality (4.171) follows from a triangle inequality in Fourier space (~(

)1/2

kl i.~-(m3)12 dk u2~ 7/)2x [k[2

2re u

1 Ik. )r(m)12 dk

2x~lkl 2

+

2re7/)2 X u

I.T'(m')]2 dk "

Notice that the right-hand side is (f IVul2 dx) 1/2 if- (f Im'l2 dx) 1/2, so that (4.171) follows from (4.155). The next step is to factorize inx3 the left-hand side of (4.170) and (4.171):

~ot(Q1/ad~o,w) 2 IV'm3[dx')dx3~Q1/2df(~,w)2x(0,t) IVm3ldx, (4.172)

fot(~ -w-27rZ)2 1 + t21k'l 1 213v'(m3)12dk' ) dx3 ~ 2 f 2~

-~. 7/)2 X ~

k2 13V(m3)12dk.

Ikl 2

(4.173) Inequality (4.172) is obvious. For inequality (4.173) we remark that since m3 vanishes outside {0 < x3 < t}, the left-hand side of (4.173) is

CHAPTER 4 Recent Analytical Developments in Micromagnetics

336

identical to

llotf(

2

1 2[ y (m3)(k', X3 -}- t) 2~7/)2 1 + t2lk'l = 2 f(

~, (m3)(k', X3) 12dk'dx3

sin2(k3t~ 2 ' [.T(m3)[2 dk, 2~7/)2X[~ 1 + t 2 [k' 12

so that (4.173) follows from sin2(~~) t2k~ 1+t2]k,[2 ~ t2k2_t_t2[k,]2 " Motivated by the heuristic discussion, we introduce a length scale s KKt and observe that for fixed x3 ']{[k'[~'bullk } - .~"(m3)[ 2 d k ' ~ 'bulk SUPx,[ms[ ,](0 [ , w ) 2 [V'm3[ dx',

(4.174)

flk, l<<.,~.},~l.~.,(ms)lS } dk, ~,~ < f_bulk 2 t2 ~ 2~

1 7 I2 ; 7/)2 1 -t-t 2 [k'I 2 I (m3) dk'.

bu,~.

(4.175) The approximate inequality (4.175) is obvious for s for (4.174) is identical to (4.146): f( arc

_ffj_ 7/)2

<< t.

The ingredient

4 sin2 ( k' "2c~/' ) [~' (m3)[2 dU

-- f(o ,w)2(m3(x'+c~x')-m3(x'))2dx'~2sup[m3[f(Ox, ,W) 2 [~x'.V'm3[dx'. This latter inequality has to be averaged over vectors 6x' of length [6xt[K s to obtain (4.174). We notice that (4.174) and (4.175) amount to a slight generalization of the interpolation inequality ~(0,W)2 m 2 dx t ~

(sup \ X'

[m3[

,W)2

[Vtm3[ dx'

y's(f0

,W)2

[[Vtl-IN3[ 2 dx'

)1,3

which holds for an arbitrary function m3. If we now choose s as in the heuristic argument, cf. (4.167), the estimates (4.170)-(4.175) combine to:

Q1/3d2/3t-2/3 f(o

,w)ax(0,t)

m2 dx- L tQ1/3d2/3 t-2/3[

< E(m) Qf(o,w)ax(o,t)[mr]2 dx~E(m)

Together with bound in the regime (4.155) and (4.163).

.J(_~7/)2

].T'(m3)[2 dk' dx3

this yields the desired lower

4.7 PATTERN FORMATION IN MICROMAGNETICS

337

4.7 P A T T E R N F O R M A T I O N I N M I C R O M A G N E T I C S : MATHEMATICAL CONTEXT

The analysis of magnetization patterns and their complex multiscale structure has only recently attracted the broader attention of the mathematics community. There has, however, been a lot of work on related problems (often motivated by questions in geometry), which capture certain crucial aspects of the micromagnetics functional. In this section we discuss three examples: the possible formation of singularities and harmonic maps, the relation between smooth wall profiles and sharp interface theories (such as domain theory), and the role of topological singularities in soft magnetic films. 4.7.1 STRUCTURE OF SINGULARITIES AT SMALL SCALES: MICROMAGNETICS AND HARMONIC MAPS

The different terms in the micromagnetic energy functional

E(m)-d2s

IVUI2-2/~Hext'm

(4.176)

scale differently under a rescaling of the domain. As we have already discussed in Section 4.3, at small scales the highest order term, namely, the exchange energy, is the dominant. Hence we expect that this term should control whether singularities (e.g. 'Bloch points') can arise in minimizing magnetization patterns and what their typical structure is. If we focus only on the exchange energy the minimization problem assumes an extremely simple form: minimize

D(m) - ff2 IVml2

subject to I m l - 1

(4.177)

and to suitable boundary conditions. In other words we seek to minimize the Dirichlet integral D(m) among maps m 9 f2 C ~n ~ S 2. Every minimizer m needs to satisfy the condition d dt t=0 D

m + qo ) = 0 Im + ~ol

for all qo ~ C~(f2; ~3),

(4.178)

where C~ denotes the space of smooth maps that vanish in a neighborhood of the boundary ~f2. A short calculation shows that this condition is equivalent to the (weak form of) the Euler-Lagrange equations -Am-

IVml2m,

(4.179)

338

CHAPTER 4 Recent Analytical Developments in Micromagnetics

where IVmi2 can be understood as a Lagrange multiplier arising from the constraint treK2 -- 1. Solutions of this equation are called harmonic maps. We say that a harmonic map is minimizing if D((m + q~)/im + q~l)~>D(m) for all r ~ C~ (f]; [~3). The study of harmonic maps and their (possible) singularities has been a central question in geometric analysis for many decades (see [63] for a recent introduction with many further references). In the following we first briefly summarize some of the key results and then discuss their extension to the micromagnetic functional. It turns out that the presence of singularities depends strongly on the dimension n of the domain f~. For n = 2 Morrey showed already in 1948 that minimizing harmonic maps must be smooth. Understanding the situation in higher dimensions was one of the major open question in geometric analysis in the 1960s and 1970s and the problem was finally solved by Schoen and Uhlenbeck. We state their result for the special case n = 3. THEOREM4.7.1 ([64]). Let n = 3 and let m be a minimizing harmonic map. Then there exists a discrete set S (which may be empty) such that m is smooth in f~ \ S. A set is discrete if it has no accumulation points in f~, in particular it can contain at most countably many points. If the boundary of the set f~ is smooth then one can show that m is smooth up to the boundary, with the exception of at most finitely many points. Singularities can indeed arise. A typical example is given by the map x

mo(x) = m

Ixl

from the three-dimensional unit ball B3 to its boundary. Indeed, it is easy to see that m0 solves (4.179). To prove that m0 is actually minimizing is much more delicate [65,66]. In the above example the singularity arises for topological reasons. There is no smooth map from B3 to S2 that agrees with m0 on the boundary. There are, however, also examples of singularities in cases were the boundary data are topologically trivial. The question whether nonminimizing harmonic maps may have singularities remained open even longer. Finally, H61ein [67,68] showed in 1990 that for n = 2 all solutions of (4.179) (with finite energy D(m)) are smooth. Shortly afterwards Rivi6re [69] destroyed all hope of a regularity theory for (4.179) in higher dimension. For n = 3 he constructed a harmonic map that is not continuous on any open subset. This led to a new notion of solution, which lies in between harmonic maps and minimizing harmonic maps. We say that a map is stationary if it satisfies (4.179) (or, equivalently, (4.178)) and if it is also well behaved with respect to variations

4.7 PATTERN FORMATION IN MICROMAGNETICS

339

of the independent variable, i.e. d

D(m dt It=0

o O t ) ~- O,

for all

Ot =

id + t(p

with q~ 9 C~(f2; ~3), (4.180)

where o denotes the composition of maps. For those maps Evans established a slightly weaker counterpart of the Schoen-Uhlenbeck result. THEOREM4.7.2 ([70]). Suppose that n - 3 and that m is a stationary map. Then there exists a closed subset S such that m is smooth on f2 \ S and such that S has z e r o one-dimensional Hausdorff measure, ,}_[1 ( S ) - - O. This in particular implies that the singular set cannot contain any segment of a line or a smooth curve. In contrast to the result of Schoen and Uhlenbeck the set S here could contain a set of small fractal dimension, e.g. a Cantor subset of a line segment. No such example is known, however. Now let us turn to micromagnetics. Carbou [72] showed that the above results can be extended to the micromagnetic functional E(m), given by (4.176) (in fact he took Hext = const, but a smooth variable field poses no additional difficulties). Independently, Hardt and Kinderlehrer [71] analyzed the regularity property of minimizing micromagnetic maps. The Euler-Lagrange equation for E(m) is 1

- d 2 A m + ~V(p(m)-(Hind +Hext) - (d2lVmI2-(Hind +Hext).m)m. (4.181) Here Hin d = --VU and V q~ denotes the gradient of q~ tangential to S 2. Denoting the left-hand side of (4.181) by He (the effective magnetic field acting on f2), and writing 2 for the scalar Lagrange multiplier of m on the right-hand side, we can rewrite the Euler-Lagrange equation as He(x) = ,~(x)m(x) or, equivalently, He A m = 0

in f2,

(4.182)

granting to (4.181) its physical meaning of equilibrium condition for the torques acting on m. We say that m is a stationary micromagnetic map if, in addition, d E(m dt It=0

o (~t)

for all

~t --

id + tq~ with (p 9 C~x~(~; [~3).

(4.183)

Finally, we say that m is minimizing if E((m + q~)/im + q~i)>~E(m) for all (fl e C~X~(fl; [~3).

340

CHAPTER 4 Recent Analytical Developments in Micromagnetics

THEOREM4.7.3 ([72]). Suppose that Next is smooth. (i) If n = 2 then every solution m of the Euler-Lagrange equation (4.181) is smooth. (ii) Suppose that n = 3 and that m is a stationary micromagnetic map. Then there exists a closed set S such that m is smooth in f~ \ S and the onedimensional Hausdorff measure of S is zero, ":]_[1(S) = O. (iii) Suppose that n = 3 and m is minimizing. Then there exists a discrete set S such that m is smooth in f~ \ S.

Statement (iii) can be refined. Hardt and Kinderlehrer [71] showed that minimizers are smooth up to the boundary (if the boundary is sufficiently smooth) with the exception of a finite set S. Near a singular point a the magnetization approaches the standard singularity R ~ , where R is a fixed rotation. Moreover, they derive a universal bound on the number of possible singularities in terms of the applied field, the domain and the anisotropy function. They also show that singularities can indeed arise for smooth applied fields, even for globally minimizing maps m. Examples of concrete physical situations, where singularities are expected, are reviewed in [29; Section 3.6.5 (D), pp. 268-271]. There, one can also find a discussion on how far the (continuum) micromagnetic energy can still be used near a singular point and what modifications of the energy functional might be needed in the immediate vicinity of a singular point. 4.7.2 WALLS VERSUS SHARP INTERFACES: MODICA-MORTOLA FUNCTIONALS AND DOMAIN THEORY

The exchange energy plays very different roles at different length scales. At very small scales (KK d) it determines the presence and structure of possible singularities, as we saw in the previous subsection. At scales comparable to the Bloch-line width d or the Bloch-wall width dBw -- d / ~ / ~ it determines the inner structure of domain walls. At larger scales the inner structure of the domain walls is not seen, but it is important to keep track of the fact that a domain wall carries a certain energy per unit area. This domain-wall energy is crucial in determining patterns at larger scales, as we saw in the discussion on domain branching in Section 4.6.8. The description of the magnetization pattern in terms of regions of constant magnetizations separated by sharp interfaces carrying a certain energy per unit area is exactly the content of domain theory, which is often used as an approximation of micromagnetics. In this subsection we discuss how domain theory arises as a rigorous limit of micromagnetics in a specific regime, corresponding in particular to

4.7 PATTERN FORMATION IN MICROMAGNETICS

341

hard materials Q >> 1. The idea that a simplified functional, which allows for sharp domain walls, should lead to the right description of the magnetization patterns on large scales should be applicable in a much broader regime. Mathematically, however, this so far remains an open question, and new tools to capture multiscale behavior might be needed (see e.g. [73]). The easiest setting to explain the relation between domain walls and sharp-interface theories is that of a hard uniaxial material. Macroscopic samples of such materials tend to exhibit complex domain patterns, including very fine magnetic domains. It is not surprising that, historically, the first attempts to understand pattern formation through the competition among the various terms of the energy functional (4.17) have involved cases in which the role of anisotropy is decisive. Consider, for simplicity, the case Hext = 0, and let l be the diameter of f2. By rescaling f2 to a set co of unit diameter through the change of x ~ co, and suitably normalizing the energy functional variables x ~ 2 - T (4.176) we obtain i~,Q(m) - ~ /co IVml2 4- -lfo) q0(m)4- 1 s ~

3

IVul 2

(4.184)

where u

(1)

d IQ1/2

=

dBw l

(4.185)

'

1

(4.186)

- -[U(x), 1 I(m)

i~,Q(m)-

1 E(m)

e l3 = e Q/3 9

(4.187)

For a sample that is large enough compared with the intrinsic length scale dBw, i.e. l >> dBw, (4.188) e is a small number. It is therefore natural to study ie,Q in the limit E ~ O. If, in addition, Q is large enough that it makes sense to assume

(~)2 Qe- 1 ,

"Q

-

, =

dBw

>> 1

(4.189)

we are naturally led to the study of minimizers of the functional i~(m) - ~

in the limit e ~ O.

fo

IVml2 + ~0(m)4lf~

3

IVul 2

(4.190)

342

CHAPTER 4 Recent Analytical Developments in Micromagnetics

Intuitively, as e tends to zero the second summand of (4.190) will force minimizers of ie to assume only values in/(2, the zero-level set of q0. In the uniaxial case KJuni = {-be}. Then, magnetostatic energy will favor nonuniform, divergence-free patterns (in order to reduce the effect of the charges m. v at the boundary of co). Since m(x) ~ {+e}, and since surface discontinuities of m are not allowed by the exchange energy, we expect to see transition layers joining the two values -be in a divergence-free fashion. In these layers, m(x) ~/Cuni: their thickness is then decided by the competition of anisotropy (which favors thin transition regions and thus high gradients of m) and exchange. This reasoning is precisely the one put forward by Landau and Lifschitz [74]. At least in the case when (4.188) and (4.189) hold, a rigorous mathematical justification of this argument is available. This has been provided in [75], by adapting to the micromagnetic setting an argument due to Modica and Mortola [76,77]. The essence of the argument is that in the limit e --, 0 minimizers of the functional s

L

IVml2 + -

1L

q~uni(m)

(4.191)

8

(possibly subject to suitable boundary conditions or augmented by other lower-order terms such as the magnetostatic energy) develop the structure outlined above. They take values very close to the set K;, except in an neighborhood of certain interfaces. Moreover, in this neighborhood the profile of the minimizer is essentially one-dimensional, varying mainly in the direction normal to the interface. Therefore, the effective energy due to the interface can be determined by solving a one-dimensional optimal profile problem. Performing a further rescaling of length by e this problem consists of minimizing

F

IVml2 +

(3O

F

q)uni(m)

(4.192)

(3O

among all one-dimensional profiles such that m(s) --, +e as s ~ 4-oo (and [m[ = 1). Using that qguni(m) = 1 - (m. e) 2 one can solve the corresponding Euler-Lagrange equation explicitly and one finds that minimizing profiles correspond to suitably parameterized great circles on S2. The optimal wall energy turns out to be four in this dimensionless formulation. In the physical domain f2, the typical scale of the wall profile, is the Bloch-wall width dBw = ~ L - ~ K~

(4.193)

4.7 PATTERN FORMATION IN MICROMAGNETICS

343

and its energy per unit area is given by Kal3~"

4 l---f -

-

4v/AKa

.

(4.194)

If we also consider the magnetostatic energy, then interfaces, whose normal v is perpendicular to e, play a special role. A sharp jump from e to - e at such an interface gives zero contribution, the distributional divergence div m, and hence does not lead to additional magnetostatic energy. If we replace the sharp jump by a transition profile, then there is a distinguished wall profile, which is optimal for (4.192), and which is in addition divergencefree - this is exactly the classical Bloch-wall profile, corresponding to the great circle with m. v = 0. Anzellotti, Baldo, and Visintin [75] extended the Modica-Mortola argument to the micromagnetic functional i~. The limit functional i0 allows only for magnetizations +e, penalizes the interfacial area between these two states by the Bloch-wall energy, and includes the magnetostatic energy of the magnetization pattern. This is the essence of domain theory, which is often considered as a useful, though only approximately valid, strategy to predict domain patterns. The following proposition shows that domain theory can be deduced from micromagnetics as a suitable limit theory, at least in the case that (4.188) and (4.189) hold. PROPOSITION4.7.4. AS e ~ 0, the functional i~ F-converges in L1 to i0 - { 4Per~{m(x)_}_cx~-- q--e} -q- f[~3 [VU[2

else,if m ~ BV(f~) and m(x) -~ +e,

where Perfl{m(x) = +e} measures the area of the interface between the region magnetized according to +e and that magnetized according to -e. REMARK4.7.5. In the result above, BV(f~) denotes the space of functions with bounded variation (see, e.g. [78-80]). For a function in this space, which in addition only takes the values +e and -e, the interface between the regions where each of the two values is taken has finite area (in a suitably generalized sense). As outlined above and argued in more detail in Appendix B, the essential part of the result is that, in the limit e ~ 0, exchange and anisotropy combine to define the energy per unit area of a domain wall. The additional term f[~3 IVul 2 is a continuous perturbation and it is simply reproduced in the limit (see Proposition B.4 and note that L 1 and L2 convergence are equivalent since [m] - 1). By the same reasoning, the argument can be extended to cover the presence of an applied field. There is also an extension to more complex anisotropy energies, such as those appropriate for materials with cubic symmetry.

344

CHAPTER 4 Recent Analytical Developments in Micromagnetics

The essence of domain theory is the elimination of a length scale (the thickness of domain walls), through the replacement of a transition layer with a sharp interface carrying the same energy. This scheme has a great appeal, and it should have a range of applicability much wider than the rather restrictive material parameter regime encoded in (4.188), (4.189). Clearly, it is essential that all dimensions of the sample should be large compared with dBw, and that the material be sufficiently hard. But it should be possible to relax the assumption Q >> 1, at the cost of having to overcome some technical difficulties. Nevertheless, the result above is important in that it demonstrates the possibility to derive domain theory as a rigorous consequence of micromagnetics. For soft materials, i.e. Q < 1, (4.189) certainly fails. In these cases, we cannot rely on q~to simplify the structure of the problem by allowing only finitely many values of m to compete for a low-energy state. In addition, for samples which are small in some dimension (thin films), the calculation proving that the optimal transition layer has the one-dimensional structure of a Bloch wall is no longer justified. In fact, the structure of walls in confined geometries is much richer, but also much more complex, see Section 4.6 above, [29] and K16man's review [81] on soft ferromagnetic materials. 4.7.3 THE ROLE OF TOPOLOGY: SOFT FILMS AND GINZBURG-LANDAU VORTICES

For a very soft ferromagnetic material, Q ~ 1, and in the absence of the aligning action of an applied magnetic field, pattern formation is controlled by the competition of magnetostatic and exchange energy. Minimization of the former promotes divergence-free magnetization patterns and, in particular, configurations that are everywhere tangential to the boundary of the sample. In a thin film f2, with cross-section f2t perpendicular to the coordinate axis x3, and neglecting variations of m along the thickness direction (which are strongly disfavored by the exchange energy) this requires m3 = 0, and that the in-plane component mt be tangential to the lateral boundary (see Section 4.4.1 and Table 4.3 for a more quantitative discussion). We are thus led to consider a two-dimensional unit vector field, defined on the cross-section of the film, and everywhere tangential to its boundary. There are, however, no continuous vector fields of this type, and there is no such field with finite exchange energy. In fact, the way the system beats this topological obstruction is by creating a localized region, called a Bloch line, where the magnetization becomes perpendicular to the plane of the cross-section (escape in the third dimension). The section of a Bloch line

4.7 PATTERN FORMATION IN MICROMAGNETICS

345

with the plane of the cross-section is a vortex. Its core size is decided by the competition between magnetostatics (penalizing the surface charges associated with m3 ~= 0, hence favoring a core as narrow as possible) and exchange (which clearly favors a core as spread out as possible). For a film of thickness t and circular cross-section of radius l, assume that the core of a Bloch line is a cylinder ~'~r - - ~'~fr• ( 0 , t) of radius r, coaxial with the sample f2l = f2't x (0, t). For thin films, it is safe to assume that the magnetization is constant along the thickness direction. The exchange contribution to the energy can be rougly estimated as

Eexch=d2 (ff~l\flr IVmla q- /Qr IVm[2) .~2ndRtln(1) - + ndZt r

I ~ 2nd2t In ( r ) 9

The magnetostatic contribution to the energy can be estimated through its leading-order term Ernst "~" t

l,, m2 = 7r,r2t. r

Optimizing Eexch q- Emst with respect to r shows that the competition between exchange and magnetostatic energies results in Bloch lines of characteristic width dBL -- d, where 1

d - ( K ~ ) ~,

(4.195)

each carrying a characteristic energy of order

d2tln

(') ~

,

(4.196)

where t is the film thickness, and l is the diameter of the cross-section. Since in typical applications d << l, one is confronted with the problem of studying vortices whose energy (after scaling by d2t) diverges while their size (relative to that of the sample) tends to zero. Vortex structures of the type just described are common in other physical systems that have been described using mathematical tools similar to the ones suitable for micromagnetics. Problems associated with topological obstructions are of great mathematical interest, and have received

346

CHAPTER 4 Recent Analytical Developments in Micromagnetics

considerable attention in the recent literature. A very well-studied example is that of the so-called 'Ginzburg-Landau vortices', see e.g. [82-84]. In two dimensions, this is the problem of minimizing an energy of the type /f2, IVml2 q- ~1 / ~ , (,Iml _ 1 )2

2

among vector fields in the space {m E H I ( ~ ') 9m = m0 on3f2'} in the limit ~ --+ 0. Here m0 is a prescribed vector field that has unit length on Off'. It is worth emphasizing a key difference between this problem, in which vector fields would be constrained to be tangential to c9f2pby a (hard) boundary condition, with the case of soft magnetic films, where only an energetic penalty against nontangential vector fields is in action. This results in an interesting qualitative difference for the solutions of the corresponding minimization problems. While Ginzburg-Landau vortices can only appear in the interior of f2~, in the case of soft magnetic films vortices at the boundary of co~will appear in certain material parameter regimes (see [3,4,85] and Section 4.4.2).

4.8 B E Y O N D L O W - E N E R G Y S T A T E S 4.8.1 MAGNETIZATION REVERSAL IN SOFT FILMS

Hysteresis means that an experimentally observed magnetization depends on the history of the external field. For an external field that varies at a sufficiently slow pace in time, we may think of the observed magnetization as a local minimizer of the micromagnetic energy at any moment. Hence, hysteresis can be related to a complex energy landscape with several local minima whose energy might be far from the global minimum. This complex energy landscape is probed by the external field. In fact, the Stoner-Wohlfarth astroid visualizes this point of view in a very special situation. It is based on the assumption that the sample is so small that the magnetization is spatially uniform at any moment, cf. Proposition 4.3.1. For somewhat larger samples, this is no longer true even if the remanent states (the local minimizers at zero external field) are almost uniform. This was recently brought to the point in [86] for an ellipsoid of revolution. For sufficiently strong external fields, the magnetization essentially saturates in (the fixed) direction of the external field. A further increase

348

CHAPTER 4 Recent Analytical Developments in Micromagnetics

!

FIGURE 4.27 Domains in the concertina pattern of Fig. 4.26. X3

Gxl X1

FIGURE 4.28 The geometry.

The Setting An Idealized Geometry

Motivated by the concertina pattern we consider the following idealized sample geometry f2 = ~ x ( 0 , ~ ) x ( 0 ,

t)

with s

see Fig. 4.28. The reasons for this choice are: 9 f2 mimics an elongated thin-film element of thickness t and w i d t h ~. Later, starting with Eqn. (4.221), a finite periodicity L in direction Xl will be imposed. 9 O w i n g to the translation invariance in Xl, f2 admits m* -- (1, 0, 0) as a stationary point for all external fields of the form Hext - (-hext, 0, 0), hext E [~ (note the change of sign), and for an easy axis in the direction of Xl or perpendicular to xl. In fact, we shall neglect crystalline anisotropy altogether without m u c h loss of generality.

The H e s s i a n

The local stability of the stationary point m" - (1, 0, 0) is described by the second variation of the energy E, its Hessian. O w i n g to the constraint Iml 2 - 1, infinitesimal perturbations of m* - (1, 0, 0) are of

4.8 BEYONDLOW-ENERGY STATES

347

FIGURE 4.26 Concertina pattern in a Permalloy film, thickness t = 300nm, width = 18#m. Reproduced with permission from Hubert and Sch~ifer, Magnetic Domains, Springer (1998).

in the external field strength will not affect the magnetization much. This defines the saturation branch, parameterized by the field strength. Imagine moving down the saturation branch by slowly decreasing the external field strength. Typically, one eventually encounters an instability. This event is termed nucleation and is at the onset of switching, but is not necessarily connected with an irreversible event. Nucleation has been theoretically and numerically investigated for some time, we shall mention some of the pertinent work below. In this section, we report on a series of mathematical papers [14-16] on nucleation in elongated thin-film elements. This work is motivated by the ubiquitous concertina pattern in elongated soft thin-film elements, see Fig. 4.26. The concertina pattern is the quasiperiodic pattern, which can be seen in the middle of the cross-section; it is framed by two Landau states. Figure 4.27 gives a sketch of the corresponding domain pattern with charge-free walls. The concertina pattern is experimentally generated in the following way: first, the element is saturated along the long axis, then the external field is slowly reduced, and eventually reversed. For some field strength, the uniform magnetization buckles into the concertina pattern. Let us give a short summary of this section. In Subsection 4.8.1.2, we present [14], which proves that there are exactly four regimes of instability in elongated thin-film elements. These regimes are identified in terms of the scaling of the critical field. To our knowledge, the oscillatory buckling regime (Regime III) has not been reported in the physics literature. In Subsection 4.8.1.3, we present [15], which focuses on Regime III and identifies the asymptotic form of the unstable mode by F-convergence of the Hessian. In Subsection 4.8.1.4, we present [16], which identifies the type of bifurcation in Regime III by F-convergence of a nonlinearly renormalized energy.

349

4.8 BEYOND LOW-ENERGY STATES the form

6m = (0, ~im2,6m3),

8m = 8m(x],

X2, X3).

The Hessian HessE(m*) in m* is given by

1_2HessE(m*)(cSm, (Sm) - 1 HessEO(m.)(bm, (Sm) - hext f,_, Ibml2 dx, (4.197) where HessE ~(m*) denotes the Hessian without Zeeman term, i.e.

12HessE~

8m) - d 2 If2 IV 8ml2 dx + f~3 IVUcSml2dx,

and the magnetostatic potential USm is determined by 8m in the usual way. The critical field hcrit is the value of hext for which HessE(m*) ceases to be positive definite. The unstable modes are the elements of the degenerate subspace of HessE(m*) at hext - hcrit. The following variational characterization of both can be inferred from (4.197): hcrit and the (normalized) unstable modes are the minimum and the minimizers, respectively, of the variational problem

Variational Characterization of Critical Field and Unstable Modes

!2 HessEO(m*)(8m, bin)

subject to

fn Ibml2 dx - 1.

(4.198)

It is natural to capitalize on the translation invariance of (4.198) in Xl by a partial Fourier transform in that variable. More precisely, the Hessian HessE ~(m*) (., .) factorizes into {HessE ~(m*) (kl,., .) }klan, where

1-HessE~ 2 -- d2 f(o,g)x (o,t)

(Sm, (Sm) k Ibml2 + Ic92cSml2 + 103 cSml2) dx2 dx3

if- /~2 (k 2 lU~ml2 if- Ic32U6m[2 q- Ic93U6m]2) dx2 dx3 ,

(4.199)

kl denotes the dual variable to Xl and cSm = c~m(x2, x3). Hence, we replace the variational problem (4.198) in (Sm(xl, x2, x3) by the variational problem in (Sm(x2, x3) and kl of minimizing 89HessE ~(m*) (kl, bm, bm)

subject to

/ 16ml2 dx2 dx3 - 1. J(o ,~)x(O,t) (4.200)

350

CHAPTER 4 Recent Analytical Developments in Micromagnetics

Notice that unstable modes can also be seen as the ground states for the operator s ~m

-

-d 2/XNeumann~m

-

C93 U~m.

(4.201)

A complete explicit diagonalization beyond the obvious factorization (4.199) of s seems not at hand. Indeed, the contribution from the exchange energy is diagonal with respect to cosine series in (x2, x3) whereas the contribution from the magnetostatic energy is diagonal with respect to the Fourier transform in (x2, x3). This lack of compatibility reflects the fact that the exchange energy is confined to the sample but the energy of the stray field extends into the ambient space. Different Types of Models and Modes Following [87] we distinguish between'models' and'modes'. A model is a special ansatz for an infinitesimal perturbation 6m. In view of (4.198), it gives an upper bound for hcrit. A mode is an eigenfunction of (4.201) - but only the eigenfunction corresponding to the smallest eigenvalue yields hcrit and the degenerate subspace. We now discuss the physics literature. There, besides infinite prisms like our fl, ellipsoids have been considered, since they also allow for constant stationary states m*. Brown [88] found two modes for ellipsoids of rotation: the first mode corresponds to a coherent rotation, the second mode corresponds to a curling of the magnetization. The characterizing feature of the curling mode is the complete absence of a stray field, i.e. no surface or volume charges are generated by this mode. Brown also found that for sufficiently small samples (with respect to d), the coherent mode has a lower eigenvalue than the curling mode. In [89], three models are compared for an infinite circular cylinder: coherent rotation, Brown's curling mode, and a model for buckling. It is found that only in a small size range the buckling model beats the two modes. In [90], the infinite cylinder is investigated more systematically: cylindrical coordinates (Xl, r, r reduce (4.201) to a one-dimensional problem in r parameterized by (kl, n) ~ 7/• ~. The case n - 0 is treated completely, the lowest eigenvalue occurs for kl - 0 and corresponds to the curling mode. By a lower bound estimate, the cases n~2 are discarded for nucleation. The case n = 1 is treated numerically, the buckling model of [89] is found to be close to an actual mode. In [91], an infinite prism with rectangular cross-section is considered. This is the geometry considered here, but in [91], no consideration was given to extreme aspect ratios of the rectangular cross-section, i.e. t << s

4.8 BEYOND LOW-ENERGY STATES

351

Guided by [90], modes and models that vary in the infinite direction Xl are ignored. Upper and lower bounds for hcrit are given. Based on these works, Aharoni [87; p. 202] emphatically rules out any other type of unstable mode besides coherent rotation, buckling and curling. Moreover, he claims that the buckling mode only plays a minor role [87; pp. 200-202]. We now report on [14], where it is shown that there are exactly four different regimes. As we shall discuss, two of these regimes (Regime II and Regime III) are buckling regimes, one of which (Regime III) covers a wide range in parameter space of thin films. Aharoni's analysis misses this regime since, implicitly guided by t -~ g, it discards modes that vary in the infinite direction Xl.

Identification of Four Scaling Regimes [14] is a rigorous analysis of the scaling of the critical field hcrit. By dimensional analysis, hcrit is a function of the nondimensional parameters

t/d,g/d: hcrit --

hcrit(t/d, e/d).

(4.202)

[14] identifies all regimes for the scaling of this function. THEOREM4.8.1. For t << g d2

~ln (~) for t ~ T ln-1 (~) (d) 2 hcrit "~

dt) 2/3

Regime I

for Td2 ln-l(~) ~< t ~< Td2 Regime II (4.203) d2

-g

for T <<,t <<,(de)

(d) 2

for

(de) 1/2

~ t

1/2

Regime III Regime IV

As in previous sections, the notation A ,~ B means that there exists a universal constant C < oo with ~A<<,B<,CA. Theorem 4.8.1 can best be visualized in terms of a phase diagram for (4.202) in parameter space, see Fig. 4.29. It is instructive to confront Theorem 4.8.1 with Theorem 4.5.1. The reduced convexified model of Theorem 4.5.1 implicitly treats the reduced critical field Thcrit e as being zero. Theorem 4.8.1 shows that 7hcrit e <( 1 is indeed true in the interior of Regimes III and IV which, up to a logarithm, coincide with the regime considered in Theorem 4.5.1. We notice that Regime III is broad in the sense that it spans a range of film thicknesses t, which includes both basic types of walls discussed in Subsection 4.6.2, cf. Theorem 4.6.1.

352

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

FIGURE 4.29 Phase diagram for hcrit(t/d, f/d). Theorem 4.8.1 is proved by establishing upper and lower bounds on (4.200), which match in terms of scaling. The upper bounds stem from physically motivated models. The main contribution of a mathematically minded analysis is to show that these models cannot be substantially improved, i.e. in terms of scaling. This is done by providing ansatz-free lower bounds, which rely on suitable interpolation inequalities. In this sense the analysis of [14] is half-way between proposing (new) models and identifying ground-state modes.

Heuristic Interpretation We now interpret Theorem 4.8.1. This is done by a discussion of the models that lead to the (optimal) upper bounds for (4.198). We go through the four regimes in the order of increasing thickness t (hence, from Regime I to Regime IV). The models are determined by a competition between the exchange energy and the magnetostatic energy. With increasing sample size the magnetostatic energy increases in relative importance: Regime I is completely dominated by exchange, Regime IV entirely by magnetostatics. The relative importance of surface charges and volume charges (which are at the origin of the magnetostatic energy) depends on size too: with increasing thickness the volume charges become more important. An interesting feature is dimensional reduction: the model in Regime I is constant, in Regime II it depends on x2, in Regime III it depends on Xl and x2, whereas in Regime IV it depends on x2 and x3. Regime h Coherent Rotation This regime is driven by the avoidance of an exchange contribution. The exchange energy favors a spatially constant

4.8 BEYOND LOW-ENERGY STATES X2

353 X2

I

hcrit"-~ ln (~f)

Surface charges

FIGURE 4.30 Coherent rotation. perturbation. But the coherent rotation necessarily creates a nontangential magnetization at the sample edges 0f2, which is penalized by magnetostatics. A coherent rotation in the film plane (the xl, x2-plane) only creates surface charges at the small lateral edges ~ x {0, e} x (0, t). Hence the model is of the form: c~m2 ~ (~t) -1/2, c~m3 ~ 0, where the constant (et) -1/2 is chosen such that the constraint in (4.200) is satisfied. The corresponding finite perturbation is sketched in Fig. 4.30. Let us comment on the scaling of hcrit- The only contribution to HessE~ *) comes from the surface charges. The surface charge at the two lateral edges ~ x {0} x (0, t) and ~ x {e} x (0, t) has density (~t) -1/2 and -(et) -1/2, respectively. On length scales larger than t, the surface charge behaves like a line charge on ~ x {0} x {0} and ~ x {e} x {0} with density g.-1/2t1/2 and - ~ - 1 / 2 t l / 2 , respectively. Hence the magnetostatic contribution per unit length in Xl scales as

2

IVubml 2 dx2 dx3 ~"

In ~

(g.-1/2t1/2)2 -- - In - . ~

t

g

The argument of the logarithm is 7 since t is the small-scale cutoff and s is the large-scale cutoff. This yields the scaling of hcrit.

Regime Ih Nonoscillatory Buckling This regime is driven by the avoidance of surface charges: the magnetostatic influence is already strong enough to suppress any normal component at 3f2. The exchange energy is still strong enough to suppress variations in the Xl- and x3-directions. The chosen model is (7~X2

6m2 -- ~/2(s -1/2 sin \ e /

and

~m3 ~ 0.

(4.204)

354

CHAPTER 4 RecentAnalytical Developments in Micromagnetics X2

X2 D

J J

~

J

D

J J

D

J J

J

+ Xl

( d ~2

hcrit -{j

+

+

+

Volume charges

x1

FIGURE 4.31 Nonoscillatory buckling. Figure 4.31 shows the corresponding finite perturbation, which displays 'edge pinning'. Let us again comment on the scaling of hcrit. Surface charges are completely suppressed, but there are volume charges: - V . 6m = -~2 c~m2 = -~/2rcs

(7~x2) cos - ~ .

We assess their contribution: the volume charge density in R x (0, s x (0, t) scales as s For t KKs these charges behave like surface charges on x (0, s x {0} with a density scaling as s The energy of a surface charge on ~ x (0, s x {0} with unit density would scale as s (always per unit length in Xl). Hence, we obtain for the magnetostatic contribution: 2

iVursm]2dx2dx3

.~ s163

= t. s

On the other hand, the exchange contribution scales as

d2 f(o,e) x (o,t)

]V~m[2dx2dx3

~" ( ~ ) 2

d2

Hence, in Regime II (i.e. t~
Regime III: Oscillatory Buckling This regime is driven by the competition of volume charges and exchange energy. As in Regime II, surface charges are suppressed at the expense of the exchange energy, which generates volume charges. As opposed to Regime II, volume charges do matter. Volume charges can be reduced by modulating the ansatz (4.204) in the

355

4.8 BEYOND LOW-ENERGY STATES

x2

X2 J

~

--

J

+

--

+

Long distance

/-.....I"-..

J

~

J N/ Short distance Volume charges

( dt ~2/3 hcrit ~ \--~]

FIGURE 4.32 Oscillatory buckling. Xl-direction by choosing a nonzero kl in (4.200), see Fig. 4.32. If the period 2~ of this modulation is much smaller than ~, volume charges cancel w = T~ll over a length scale of w instead of s cf. Fig. 4.32. We choose the following model: (2rCXl) (~zx2) c~m2 --- 2(s -1/2 cos sin and c~m3 - 0. (4.205) w -7We now address the scaling of w and hcrit. Provided w << s the dominant part of the exchange energy comes from

c31c~m2 = --4~z ( s

-1

sin

( )/I:X ( /rX2 ~. 2wl sin \--s

Thus, the exchange contribution scales as d2f0

,s

]V~ml2dx2dx3

~ (d) 2 w "

(4.206)

The volume charge density in ~ x (0, s x (0, t) is of the form - V 9 6m -

-632

c~m2 -- --2~ g.-3/2t-1/2 cos

(2~ZXl) (/rX2) cos w -T-"

Provided w >> t, this behaves like a surface charge density on ~ x (0, s x {0} of the form (2~ZXl) (/zx2) 2~ g.-3/2t1/2 cos cos (4.207) w -T-" To leading order in w << ~, the magnetostatic potential U6m generated by this surface charge density is of the form 1 (2~1)(/rX2 U6m ~ -~ g.-3/2t1/2w cos cos

~ (arc,x3]) s I exp . w

(4.208)

CHAPTER 4 Recent Analytical Developments in Micromagnetics

356

X3

X3

i, 1

x2

I

hcrit~ (d) 2

X2

Charge-free

FIGURE 4.33 Curling. Hence the magnetostatic contribution, which can be computed as the integral of the product of (4.207) and (4.208) over ~ x (0, s x {0}, scales as follows: tw

2

IVU6ml2dx2 dx3 "" -~-~-.

(4.209)

We see that the sum of exchange contribution (4.206) and magnetostatic contribution (4.209) is minimized for w ~

.

(4.210)

For this choice we indeed obtain hcrit "~" [~,e2 dt ~ ] 2/3 9 Notice that the consistency condition t << w << s precisely turns into the conditions that define Regime III. The hypothesis that the experimentally observed period Wexp of the concertina pattern is the frozen-in length scale w (4.210) of the unstable mode at critical field is tempting. For a quantitative comparison we need a more quantitative version of (4.210). It will be provided in Subsection 4.8.1. Regime IV: Curling This regime is driven by the avoidance of both surface and volume charges. Hence, the model am should be tangential to 0f2 and divergence-free in f2. We make the following choice: am2 - 2(s -1/2 s i n ( - ~ ) cos(E~), c~m3 - 2(s -1/2 } cos(-~) sin(E~). (4.211) The resulting finite perturbations are helicoidal in nature, see Fig. 4.33. The scaling of hcrit is easy to explain: there is no contribution from the magnetostatic energy. In the exchange energy, the contribution from c93c~m2is dominant and scales as (d/t) 2, which yields the scaling of hcrit.

357

4.8 BEYOND LOW-ENERGY STATES

Asymptotic Identification of the Unstable Mode in Regime III Theorem 4.8.1 identifies all scaling regimes by determining the scaling of hcrit, but it does not determine the degenerate subspace. We now present a result from [15], which rigorously identifies the asymptotic degenerate subspace in the new Regime III. As we shall see, it confirms the heuristic ansatz (4.205) from above.

The Scaling The approach of [15] is to identify the asymptotic minimizers (kl,~m2 (x2, x3), c~m3(x2, x3)) by characterizing the F-limit of (4.200). For the notion of F-limit, see Appendix B. This requires a nondimensionalization. (4.210) suggests measuring length anisotropically in the following units

( 11 3

Xl,

X2 -- s X2,

X3 -- tx3,

(4.212)

with the implicit understanding that this also means kl = (d_~s 1/3 /Cl,

02 - f-1~2, dx2 - f, clx2 and so on. In view of the constraint in (4.200) we rescale the infinitesimal perturbation as follows 6m -

(s -1/2 ~m.

In view of Theorem 4.8.1 and hcrit -= 89min HessE ~ the Hessian itself has to be rescaled as

dt t 2/3

HessE ~ =

~

HessE~

We finally introduce the two nondimensional parameters

9-

2t2j3 ( ~-~

(t2)2j3

and

6 "=

and

6 << 1.

d-~

'

(4.213)

which characterize Regime III: << 1

(4.214)

CHAPTER 4 Recent Analytical Developments in Micromagnetics

358

The Result

THEOREM4.8.2. The variational problem of minimizing A

!HessE 0 (m*) (kl ' am ' am) 2

subject to

f(o,e)x(O,t)laml 2 dx2 dx3 -

1 (4.215)

F-converges in the regime (4.214) under weak convergence of (]C1, (5"m2,c~"m3)in R x L2 ((0, 1) x (0, 1))2 to the variational problem of minimizing

fr f0 1(~m2)2 d~:2 + 1/01 (~2 ~m2)2 d~:2

subject to

2 IfCll

f0 1 I~212d~:2

-

1

(4.216)

if(~"m2, c~'m3)isoftheform ((5"ma(x2),0) with c~"m2 E H01((0,1)), and +oo ifthis is not the case.

Since (4.216) can be solved explicitly, one obtains in the original variables: COROLLARY4.8.3. Let hcrit and (kl, (5m2, (5m3) be the minimum and a minimizer of (4.200), respectively. In the regime (4.214), we have

(~2) 2/3 d-t hcrit (~)1/3 1 tg

f(0,e) x (o,t) I~m(x2, x3) -

~

( 11:4/3 )

~ 3 ~

Ikll ~

(2) 2/3

,

(4.217)

,

(4.218)

(sin(re x2/g.), O) 12 dx2 dx3 << 1.

(4.219)

From (4.218) and (4.219) we infer that the unstable subspace asymptotically consists of all perturbations ~m that are indeed of the form (4.205), where the period w = 2rt of oscillation in the infinite direction Xl is asymptotically given by

(d2~2)

w ~

32rt

t

1/3

(4.220)

which specifies (4.210). We return to the hypothesis that the observed period Wexpof the concertina pattern is the frozen-in length scale (4.220) of the unstable mode.

4.8 BEYONDLOW-ENERGY STATES TABLE 4.5 pattern. t d 8 48 60 10 8 8

359

Comparison of predictions and observations of the period of the concertina

s d

Wexp d

Wtheo d

Wtheo Wexp

4000 8000 3600 2000 2800 7000

1200 700 540 500 800 1 700

586 512 279 347 462 851

0.49 0.73 0.52 0.69 0.56 0.50

w

We have compared Wexp to w for eight experiments pictured in [29]. These experiments cover a substantial range of the nondimensional parameters t/d and f,/d, see Table 4.5. We find a deviation by a factor 0.5-0.7, which warrants closer investigation in the future. The somewhat bold hypothesis that the period of the concertina pattern is the frozen-in length scale of the unstable mode could be easily falsified by a weakly nonlinear analysis: if the phase transition at hcrit is of first order (i.e. discontinuous), there is little reason why the unstable mode at hcrit should have an influence on the state for hext > hcrit. If, however, the phase transition is of second order (i.e. continuous) this seems more plausible. In the next subsection we shall argue that the phase transition is mild in the sense that the state is confined to a small neighborhood of the bifurcation point, cf. Corollary 4.8.6.

Asymptotic Identification of the Type of Bifurcation In this section we present [16], in which the type of bifurcation is identified. The factorization (4.199) shows that the unboundedness of the sample in the xl-direction gives rise to a continuous spectrum with a bottom hcrit not separated by a spectral gap. In order to avoid the well-known subtleties related to such a situation, without destroying the translation invariance in Xl, [16] imposes a convenient finite periodicity L in this direction: m(xl q- L, X2, X3) -- m(xl, x2, x3).

This also has the advantage of admitting a well-defined energy functional E (energy over the length of one period):

E(m) - d2 f J(o ,L) x (0,s

IVml 2 dx + l x (0,t)

q-2hext ~(0,L) x (0,(~)x (0,t)

IVUml2 dx

J(0,C) x [~2

ml dx,

(4.221)

360

CHAPTER 4 Recent Analytical Developments in Micromagnetics

~ hcrit FIGURE 4.34 Sub-, trans-, and supercritical bifurcation.

where Um inherits the periodicity of m in Xl. We think of the period L as being large compared with the intrinsic w, cf. (4.220). Classically, one distinguishes between a subcritical, a transcritical and a supercritical bifurcation, see Fig. 4.34 for a sketch. In the subcritical case, there are no local minima close to m" for hext > hcrit. In the transcritical and supercritical cases, there are local minima close to m * for hext > hcrit. But, only in the supercritical case, are these local minima 'shielded' from m u c h lower energy levels further away (we think, for instance, of m = ( - 1 , 0, 0)) by an energy barrier in the following sense: there exists an open ball B around m" in m-space such that inf{ E(m) l m ~ B } < inf{ E(m) l m ~ cgB }

(4.222)

for hext sufficiently close to hcrit. Hence the supercritical type can be assimilated to a second-order phase transition, whereas the two other types correspond to a first-order phase transition: for the supercritical type, both for conservative and dissipative dynamics, the system will stay close to m* for a while as hext increases beyond hcrit, provided that the initial kinetic energy is sufficiently small. For the subcritical and the transcritical type, m will suddenly move far away from m *, at least generically. In this sense, the supercritical type is less dependent on the specific dynamics than the two other types. The traditional approach to find the type of bifurcation is to expand E, in terms of m - m" and hext - hcrit. If there is a cubic term in m - m ", the bifurcation is transcritical. If there is no cubic term but a nondegenerate quartic term, the bifurcation is super- or subcritical depending on whether the quartic term is positive or negative definite. The traditional approach seems rather delicate for our multiscale environment, where one has to combine the asymptotic expansion in m - m ~ with the asymptotics of Regime III. [16] argues that the bifurcation is confined to a potential well in the sense of (4.222), cf. Corollary 4.8.6.

4.8 BEYOND LOW-ENERGY STATES

361

This is done in two steps: 9 In Theorem 4.8.4, one zooms in on m* in the energy landscape at the right scale. The zooming-in is combined with the asymptotic Regime III. Again, this is carried out in the framework of Fconvergence. The F-limit only contains quadratic, cubic and quartic terms and can be seen as a normal form of E near m* in Regime III. 9 In Theorem 4.8.5 it is s h o w n that the F-limit is coercive. Roughly speaking, this means that the F-limit goes to infinity if the rescaled m - m* goes to infinity, provided the rescaled hext - hcrit is sufficiently small. We view this approach as a coarser, but more robust, w a y of identifying the type of bifurcation. It seems better suited to multiscale variational problems. The Scaling Length is nondimensionalized as for Theorem 4.8.2, i.e. (4.212). The new element is the rescaling of the (m2, m3)-components of the magnetization. Notice that for m ~ m* - (1, 0, 0), (m2, m3) can be seen as the finite version of the infinitesimal perturbation (6m2,6m3). Remark that ml -- ~ / 1 - m 2 -- m~ ~ 1-- 89 2 + m 2)

for

m ~ (1, 0, 0).

As for Theorem 4.8.2, we expect that in Regime III, the out-of-plane component is strongly suppressed by the penalization of surface charges so that the above turns into ml ~ 1 - 1 m 2.

(4.223)

The magnetization component m2 will be rescaled in such a w a y as to balance the two contributions to the volume charge distribution V 9m, i.e. c~lml

(4212),(4223) ( t )1/3" 1 2 ~, - ~ 01(~m 2)

= 02m2 (4.212) s

and

This leads to (m2, m3) = 81/2 (fit2, if/3).

(4.224)

The external field is measured in units of the critical field as identified in Theorem 4.8.1. As for the energy (4.221) itself, we subtract E(m*)and normalize appropriately:

(dt) 2/3 hext -

~

~/ext,

E - 2 hext C s t -

(d~_~t2)1/3/~.

(4.225)

CHAPTER 4 Recent Analytical Developments in Micromagnetics

362

We are now left with four nondimensional parameters, ~, b from (4.213), and also L a n d f/ext. We are interested in Regime III close to the critical field. Also, in view of the remark on the spectral gap, we have to assume that the xl-period L is not too large with respect to the intrinsic period w of oscillation. This translates into L ~ 1

and

]/ext ~" 1,

(4.226)

where according to (4.212), L is defined as L

liB Co

We think of L as being large, so that it does not m u c h affect the unstable wavelength ~b = (32~)1/3, cf. (4.220). The Result

THEOREM4.8.4. Fix an M ~, 1. The variational problem of minimizing

F: subject to

Iml2 -- 1 and f(0, L)x (0,1)x (0,1) Ira- m* 12d~ -

g]~/[2 (4.227)

F-converges in the regime described by (4.214) and (4.226) under weak convergence for (r~2, r~3) in L2((0, L) x (0, 1) x (0, 1)) 2 to the variational problem of minimizing ~0 ._ f( 0

,L)x(O,1)

-f/extf(0

if (/h2,

(~lrh2)2 d:~, _ff i f ( 0

,L)x(O,1)

,f~)xl~

rh2d~ '

subjectto

11~11-1/2(_ ~1(1~m^22) + ~2t~/2) ^ i2 d~'

f(o

r~2 d~' - A~//2 (4.228) ,L)x(O,1)

th3) is of the form (/h2(:~:'), 0) and Eo - +oo if it is not of this form.

H e r e I~l1-1/2 denotes the operator with Fourier symbol Ifr

The ex-

pression _ ~1 ( ~1 m^ 22) -t- ~2/h2 is to be understood distributionally, where th2 has been extended by zero from ~ x (0, s to ~ x R. As for Theorem 4.8.2, this imposes edge-pinning, i.e. ~/2 (:~2 -- 1) = t~/2 (:~2 ---0) -- 0 in a weak sense. Let us point out that/~0 interpolates between a linear regime and a wall ^ 1^22) and c92th2 in the regime, depending on which of the two terms -~1 (~m

4.8 BEYONDLOW-ENERGY STATES

363

charge distribution is dominant. 9 For dominant ~2/~/2, E0 is close to the F-limit (4.216) of the Hessian integrated over frequencies/c1: E0 ~" f2y ( /c2 f01 rn2 d:~2 4- 1^ / 0 1 (~2/~/2) 2 d:r2 -- J/eKe ~01 /,~/2 d~ 2) d]Cl. L~ 21kll ,2 9 For dominant _ ~1(~1 -m 2),/~0 is close to a one-dimensional model for small-angle thin-film N6el walls ('ripples') normal to Xl, integrated over ~2:

E0 ~ /01(/0 L(~lm2) 2 dxl q- ~1/oL 11~111/2(1ff/2)12 d:~l

-ext/0L~2 dxl ) d~2.

(4.229)

Indeed, (4.102) with the anisotropy term replaced by the Zeeman term turns into the inner integral of (4.229), when approximated and rescaled according to (4.223), (4.224) and (4.212). Figure 4.35 shows the minimizer r~2 of/~0 for varying strength of the external field hext ([lext - fIcrit q- 0.5, hext - flcrit Jr- 6, flext - flcrit Jr- 10). The function/~/2 is represented by a gray scale plot for better comparison with the experimental picture in Figrue. 4.26. The proportions correspond to a value of s = 0.008. Theorem 4.8.5 shows that/~0 is coercive for L,//ext "~ 1" THEOREM4.8.5. There exists a universal constant c > 0 such that for any fn2 L2((0, L) x (0, 1)) we have r~2 d~' 9 Eo(ffl2) >/ c L-5/3 ( f(o,L)x(O,1 ) if/2 d:~') 14/9 - f/ext ~(0,L)x(0,1) It is not claimed that the exponent 14/9 is optimal; it is just important that it is larger than one. Theorem 4.8.5 states that there cannot be too much cancellation between the two terms in the charge density. This is not so obvious - the normal form/~0 for 'buckling' in micromagnetics thus behaves quite differently from the normal form for buckling in plate

364

CHAPTER 4 Recent Analytical Developments in Micromagnetics

FIGURE 4.35 Minimizers for varying field strengths.

theory, the von Kfirm~in model. Corollary 4.8.6 draws the conclusion that the bifurcation is supercritical in the sense of (4.222): COROLLARY4.8.6. There exists an 1Vl ,,~ 1 such that in the regime (4.214) and (4.226), inf {E(m) I f(0

,L)x(O,1)x(O,1)

I m - m*12d~:~
< inf{E(m)]f(o,L)x(O,1)x(O,1 )

Im - m*l 2 d~: -- ~//2 }.

4.8.2 LOCAL MINIMA AND THERMALLY ACTIVATED SWITCHING

The micromagnetic energy is nonconvex due to the restriction ImI = 1. It can have many local minima, each representing a stable magnetization pattern. As the applied field varies, a ferromagnet exhibits hysteresis, because the choice of local minima is history-dependent.

4.8 BEYOND LOW-ENERGY STATES

365

We have discussed the limiting form of the energy in various thin-film regimes. Mathematically speaking, these limiting forms have been justified only for energy minima, i.e. for ground-state patterns. But, practically speaking, we expect them to describe the accessible local minima as well. Thus, we expect stable magnetization patterns in thin films to converge to local minima of the appropriate asymptotic energy. It is natural to seek a more global understanding of the energy landscape. One motivation is the analysis of thermally activated switching. The Landau-Lifshitz-Gilbert equations of micromagnetic dynamics,

Jt-

-TJ x H - ~sTJ x (J x H)

with

H-

~H: c~l'

can be viewed as a damped Hamiltonian system. More explicitly we have 1

[]-[] -- Hext + Hind + Js [2AAm - Ka(Vq~)].

To account for thermal fluctuations we must add a stochastic term:

Jt--7]x

(

H - V I + c ~ 2Wt

-

jx(jxH)

with

D-kBTT]s '

(4.230)

where kB is Boltzmann's constant, T is temperature, and Wt is white noise in both time and space. After division by Is we obtain the equivalent form

mt=-Tmx

H-VI+~2wt

-eTmx(mx

H).

(4.231)

Models of this type go back at least 4:0 years [92]; for numerical studies in the spatially inhomogeneous setting see, e.g. [93-95]. For finite-dimensional damped Hamiltonian systems the effect of thermal noise is well understood. Micromagnetics is infinite-dimensional, so rigorous analysis is more difficult (and presently open), however we expect the situation to be like the finite-dimensional setting. The system should spend most of its time near stable states, with thermally activated transitions between them. The transitions should moreover have Arrhenius kinetics, i.e. the transition rate between any pair a, b of local minima should have the form kab -- Cab e-AE/kBT (4.232) where AE is the height of the energy barrier, and Cabis a suitable prefactor. This formula is applicable when thermal effects are small and transitions

366

CHAPTER 4 Recent Analytical Developments in Micromagnetics

rare, i.e. when AE/kBT >> 1. The minimum-energy saddle separating the two stable states plays the role of a critical nucleus. Once the saddle point is known, the most likely pathway of the transition is easily determined. The prefactor can also be determined, albeit with more difficulty: it depends on the linearization of the energy near the local minimum and the saddle [96]. Rare events are important, but difficult to sample numerically. For example, we expect memory devices to be stable for years, but we can only simulate thermally activated micromagnetics for a fraction of a second. The Arrhenius formula (4.232) is valuable because it gives access to precisely the regime we cannot sample numerically. The bottom line is this: to understand thermally induced switching we need (i) a list of stable states, and (ii) a list, for each pair of stable states, of the lowest-energy saddle between them. For small magnetic elements the numerically accessible r e g i m e - the list of stable states is relatively short and easy to determine: it suffices to solve the overdamped zerotemperature dynamics with a variety of initial conditions. Finding the lowest-energy saddle is, however, considerably more subtle. Recently, the problem of evaluating energy barriers and finding the associated saddles has seen considerable progress. The nudged elasticband method [97,98] and the string method [99] can be viewed as efficient numerical implementations of the classical 'mountain-pass method' for finding saddle points. Recently, Ren and Vanden-Eijnden applied the string method to analyze the thermally activated switching of Permalloy thin films and cylinders in the absence of an applied magnetic field [100]. One of their examples is a thin film 200nm x 200nm in cross-section and 10nm thick. The relevant local minima are C-states and S-states, each of which can be viewed as having a boundary vortex in one corner. The lowest-energy saddle point separating the C and S states turns out to be a flower state, which can be viewed as having a boundary vortex in the middle of one side (see Figure I of [100]). The string method is currently being applied to finite-temperature hysteresis diagrams, and to predicting the temperature-dependence of the coercive field [101]. -

A P P E N D I X A. M A G N E T O S T A T I C S We collect here a few results on magnetostatics. Existence and uniqueness of hin d satisfying hin d -" - V U

(A.1)

APPENDIX B

367

and [

dR3

hind'Vq9-

-[m. do

Vq0, Vq)~ C~XD(~3),

(A.2)

follow from the Riesz representation theorem. Let X-

{Vu'u

E C o e ( ~ 3 ) , V u E L2(R 3, [~3)}

denote the closure of the space of gradients of smooth functions in the L2 topology. Then X is a closed subset of L2(~3, N3) and hind is the projection of - m E L2(R 3, R3) onto X. The main properties of the map P associating hin d - --Pm with each m are collected in the next proposition. PROPOSITION A.1.

(i) For each m ~ L 2, there exists a unique hind E X satisfying (A.2). (ii) The field hin d is characterized as the best approximation of - m within X

l] - m - hindllL2 = min l] - m - hllL2. h~X

(iii) The linear map P 9 - m ~ hind is an orthogonal projection. In particular

IlhindllL2 --IIPmllL2~llmllL 2.

Classically, the induced field may be written as hin d -- - v A - l d i v m. In Fourier space hin d is obtained by a pointwise projection using the matrix ~ | ~/[~[2, i.e. Jt-(hind)i(~) =

~iCj 1~12.~'mj(~) "

(A.3)

A P P E N D I X B. F - C O N V E R G E N C E B.1 MAIN PROPERTIES In this article we frequently study a sequence I k of energy functionals and we are interested in their limiting behavior as k ~ oe. The limit we are looking for is, however, not the usual (pointwise) limit. Rather, we look for a notion of convergence, which implies that minimizers of I k converge to minimizers of I. F-convergence, introduced in 1974 by DeGiorgi and Franzoni [102,103], is exactly this notion. We first give the formal definition and review some

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

368

key properties of F-convergence. Then we illustrate the notion in an easy, but interesting, one-dimensional problem, in fact a one-dimensional version of the sharp interface problem discussed in Section 4.7.2. Two standard references for F-convergence are [104,105]. An easier introduction is [106] and short overviews appear, e.g. in [107,108]. We consider functionals I k defined on a complete metric space X (e.g. the space L2 equipped with the usual distance given by the L2 norm d(f, g) = I[f - gilL2; there is another metric on L2, which is compatible with weak convergence, when restricted to bounded sets in L2). We say that the functional I is the F-limit of the sequence I k if the following two conditions are satisfied: (i) (lower bound);

for every sequence fk --->f we have lim inf I k (fk) >~I(f); k--,oo

(ii) (attainment of lower bound/recovery sequence); exists a sequence gk such that gk --* f and

(B.1) for everyf there

lim Ik(gk) -- I(f). k--,oo

(B.2)

REMARKB.1. We often consider F-convergence with respect to a continuous parameter (such as the aspect ratio t) rather than a discrete index k. Strictly speaking, we have

It

~ I

as

t ~ 0 if I tk F_~I

as

k ---> cx~,for every sequence tk ---> O.

The main difference with ordinary convergence is that the definition of the limit functional I atf involves not only the values of I k at the point f, but also in a neighborhood off. This is crucial for the link between F-convergence and the convergence of minimizers. The following result (which follows essentially directly from properties (i) and (ii) above) makes this precise. THEOREMB.2. Suppose that I is the F-limit of the sequence I k. Let fk be minimizers ofI k. (i) If fk -~ f then f is a minimizer of I. (ii) If g is a minimizer of I then there exists a sequence gk of minimizers of I k such that gk converges to g. REMARKB.3. Various refinements are possible. First, in (i) it suffices that a subsequence fki of the sequence fk converges to f . Convergence of a sub-sequence can

APPENDIX B

369

often be assured under rather general "soft' conditions on the functionals I k, which does not require detailed knowledge of the minimizers. Second, it suffices that the fk are approximate minimizers of I k, i.e. Ik(fk) < Ik(h) + 1~k for all h ~ X. Then the limit f of (a sub-sequence) is still a minimizer of I. This is particularly useful if we do not know whether I k has a minimizer. In that case we still have a slightly weaker version of part (ii) of the theorem, stating that there exists gk converging to g which are approximate minimizers in the above sense. In applications we are often interested in applying additional external fields. This amounts to adding an additional (usually linear) functional h to the energy functional. The operation is compatible with F-convergence as long as h is continuous. PROPOSITIONB.4. Suppose that h is a continuous functional on X (with respect to the given metric) and

ik F I.

(B.3)

I k + h F_~I + h.

(B.4)

Then

B.2 AN EXAMPLE To illustrate the concept of F-convergence, consider scalar functions f defined on the unit interval (0, 1) and the functionals

lk =

1 f01 f ~ ' ( x )

12+ kq~(f(x)) dx,

(B.5)

where q~ is a double-well potential, with q~(-t-1) = 0 and q0 > 0 otherwise. One can take e.g.

Cp(S) = (S2 -- 1) 2. We will see shortly that the F-limit functional will 'live' mostly on nonsmooth functions f. For this reason it is convenient to extend I k to a space X, which is as large as possible by making it +cx~ outside its natural domain of definition. Note that such an extension does not change the minimizers of I k. We may take X - L1(0, 1) (with the metric given by the usual L1 norm) and set

Ik(f) _ { flf_cx~~_ 0C'(x)]2 if- k(p(f(y)) dx,

iffelse.iSsmooth,

(B.6)

370

CHAPTER 4 Recent Analytical Developments in Micromagnetics

(We could also define Ik(f) by the integral expression for a larger class off, e.g. for all f whose (distributional) derivative belong to L2. One can show that this leads to the same F- limit.) What is the F-limit of I k as k ~ ~ ? The second term in the integral suggests that the F-limit will only be finite on functions that take only the values 4-1. Let f be a function which takes only values 4-1 (and is not identical I or -1). If we approximate f (in Ll) by smooth functions, then the derivatives of the approximations fk must become very large at some points as we improve the approximation. The more jumps between - 1 and 1 the function f makes, the larger we expect the energy Ik(fk) to be. The F-convergence result of Modica and Mortola makes this precise. PROPOSITIONB.5. The functionals I k are F-convergent to I where

I(f)

!

Ao 9 (number of jumps of f), +cx~

if f(x) = +l for all x else,

(B.7)

where Ao = 2

1

q)l/Z(t) dt.

This result illustrates that F-convergence is very different from pointwise convergence. In fact, the functionals I k are finite on smooth functions, while I is +oo on smooth functions (except for the function that is identically 1 or identically -1). On the other hand, I is finite on functions that take only the values 4-1, while I k is +cx~ on those functions (again with trivial exceptions). It is instructive to verify Proposition B.5 using the definition of Fconvergence. To keep things simple we only consider a function f, which has exactly one jump, say fix) = - 1 for x < x0, while fix) = 1 for x > x0. To establish the lower bound in the definition of F-convergence we follow Modica and Mortola and apply the arithmetic-geometric mean inequality to the integrand of I k

! 0C'(x)l2 + kcp(f(x))>/21cpl/2(f(x))f'(x)l k

(B.8)

and we observe that the expression inside the absolute value can be written as a total derivative d H(f(x)), where

H(s) is the antiderivative of r

lo s

q01/2(cr)do"

APPENDIX B

371

Now consider a sequence fk, which converges to f in L1. Then, for almost all values of x, we have fk (x) ~ fix) (more precisely this is true for a sub-sequence, and this is enough to verify (B.1)). Choose a < x0 < b such that convergence holds at a and b. Then d Ik(fk) >~ fab 2 -~xH(fk(x)) dx>~2 [H(fk(b)) - H(fk(a))] 2 [H(1) - H(-1)] = 2A = I(f). This finishes the verification of condition (i) of F-convergence. To check condition (ii) we need to construct a sequence gk that approximates the jump function f and does not 'waste' energy. In particular, we should choose gk such that the geometric-arithmetic inequality (B.8) becomes sharp. This is the case if gkI (X ) -- kq)l/2g k(x)" We define an optimal profile h as the solution of h'(~)

-"

q91/a(h(~)),

lim h(~) = 4-1, ~--,+c~

h(0) = 0.

Then we can take gk(x) = h(k(x - xo)), and we see easily that Ik(gk) 2A = I@. This finishes the proof of Proposition B.5 for the case that f makes a single jump. Iff has multiple jumps at Xl, x2 . . . . then for the lower bound one simply applies the previous argument with points al < Xl < bl, a2 -- bl < x2 < b2, etc. For the upper bound one first constructs good local approximations near each xi as above and then interpolates between these approximations (see, e.g. [107,108] for further details). What can we learn about the minimizers of I k from the F-convergence result? The (global) minimizers of I k are actually quite boring, they are given by the two functions f, which are identically 1 or identically -1. Already a slight modification of I k, however, leads to a rather interesting situation. We simply prescribe the average value off to prevent the trivial solution and to forcer to switch from one well to the other. Specifically, fix 2 (strictly) b e t w e e n - 1 and I and set

jk

Ik +~

if f o r else.

(B.9)

Then, by the same arguments as above, one can show that the functionals

jk have a F-limit J, which agrees with I if the constraint is satisfied and is +oc otherwise. The minimization problem for J is very easy to solve. We need to have at least one jump to satisfy the constraint. More jumps cost additional energy. Hence J has exactly two minimizers, f+ and f_, each of which have exactly one jump (at 1/2 4- 2/2, respectively).

372

CHAPTER 4 Recent Analytical Developments in Micromagnetics

The F-convergence result and Theorem B.2 imply that all minimizers of jk must converge to either f+ or f_ (after choosing a sub-sequence). B.3 HIERARCHIES OF F-LIMITS F-convergence reveals limiting features of a family of functionals at a certain energy scale. By using different rescalings one can extract different features of the family of functionals (we have seen that, e.g. in Section 4.4.3). This fact can also be easily illustrated using the example discussed in the last subsection. If we consider the differently scaled functionals

~k = f01 ~-~ ] ~c,(x) 12 +cp0C(x))dx,

(B.10)

then it is not difficult to show that ~k s ~ where

1

I(f) =

f0

q~**(f(x)) dx,

(B.11)

where q~** is the convex envelope of q~ and where we consider F-convergence with respect to weak convergence in L2. As long as q~grows superlinearly at oo, its convex envelope vanishes exactly on the convex hull of the energy wells, i.e. the set [-1, 1]. The set of minimizers of I is very degenerate. Every function taking values in [-1, 1] is a minimizer. The F-limit of the rescaled functionals I k = k~Ik helps to break this degeneracy. Only functions that take only the values 4-1 have finite energy and the limit functional distinguishes among these by the number of jumps.

APPENDIX

Symbol J Is m A Ka

q~ Hind

U

C. F R E Q U E N T L Y

USED

SYMBOLS

Meaning

First appearance

Micromagnetic energy (in J) Magnetization (in Tesla) Saturation magnetization Normalized magnetization J/Js Exchange stiffness (in J/m) Crystalline anisotropy constant (in J/m 3) Anisotropy function Magnetic field induced by J (in A/m) Potential of Hind

(4.1) (4.1) (4.2) (4.3) (4.1) (4.1) (4.1) (4.1) (4.6)

APPENDIX C

Vacuum permeability 4/1: x 10 -7 V s / A m Region occupied by ferromagnetic body C~~ ([~ 3 ) Space of smooth functions, which vanish outside a compact set /C Easy magnetization directions Ka Magnetostatic energy density j2/2//0 E Normalized energy [E/K d (units of volume) Normalized applied field/,t o Hext/Js Hext Normalized induced field Hind U Potential of Hin d Q Quality factor Ka/K d Bloch-line width Av/-A~ d -- dBL I ~-/Ka = E / Q Bloch-wall width Av/-A-~a- d/v/--Q dBW t Thickness of a prism-shaped film t Time l Typical lateral dimension of a film T, Aspect ratio t/l f~' Two-dimensional base of a film Fully nondimensional energy e = E/(12t) e O9 rescaled (nondimensional) three-dimensional domain (0 I Rescaled two-dimensional domain K, Rescaled Bloch-line length/~ = d/l It Rescaled potential Rescaled fields hind, hext Demagnetizing tensor Do9 (/9** Convex envelope of (p Xl Tangential coordinates x I = (Xl, x2) VI In-plane gradient (~1, (~2) 9v Fourier transform mI In-plane magnetization m I = (m I , m2) V Single-layer potential e? Nondimensional two-dimensional energy as a function of the aspect ratio "c Scaled applied field V Rescaled single-layer potential V1• Rotated gradient (-(~2, ~1) C~o (~; [~3) Space of smooth vectorfields vanishing outside a compact subset of Rescaled energy functionals is,Q, is Perf2E Perimeter of a set E BV(~) Functions of bounded variation T Absolute temperature ? Gyromagnetic ratio Damping factor in LLG equations W White noise Boltzmann's constant kB /.to

f~

373

(4.5) (4.1) (4.9) (4.10) (4.13) (4.14) (4.15) (4.16) (4.15) (4.18) (4.19) (4.22) (4.24) Section 4.3 Section 4.8.2 only Section 4.3 (4.26) (4.27) (4.29) (4.28) (4.28) (4.30) (4.31) (4.32), (4.33) (4.36) (4.38) (4.43) (4.43) (4.44), (4.100) (4.45) (4.47) Section 4.4.2 (4.67) (4.71) (4.84) (4.178) (4.187), (4.190) Proposition 4.7.4 Proposition 4.7.4 Section 4.8.2 Section 4.8.2 Section 4.8.2 Section 4.8.2 Section 4.8.2

374

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

ACKNOWLEDGEMENTS We would like to thank Rudi Sch/ifer for the very fruitful collaboration over the last six years. Much of ADS' work was carried out at the Max Planck Institute for Mathematics in the Sciences, Leipzig, where he was leader of a junior research group between 1998 and 2003. ADS and SM were supported by the DFG priority program SPP 1095 'Analysis, modeling and simulation of multiscale problems'. RVK was supported by the NSF through grants DMS-0101439 and DMS-0313744. FO was supported by the DFG research center SFB 611 'Singular phenomena and scaling in mathematical models' at the University of Bonn. We thank Rub6n CanteroAlvarez, J6rg Drwenski, Hans Kn~ipfer, Thomas Roessler, and Jutta Steiner for their most valuable help.

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72. G. Carbou, 'Regularity for critical points of a nonlocal energy', Calc. Var. 5, 409-433 (1997). 73. G. Alberti and S. M~iller, 'Two-scale Young measures for variational problems with multiple scales', Comm. Pure Appl. Math. 54, 761-825 (2001). 74. L.D. Landau and E. Lifshitz, 'On the theory of the dispersion of magnetic permeability in ferromagnetic bodies', Phys. Z. Sowjetunion 8, 153-169 (1935). 75. G. Anzellotti, S. Baldo, and A. Visintin, 'Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism', Appl. Math. Optim. 23, 171-192 (1991). 76. L. Modica and S. Mortola, 'Un esempio di F-convergenza', Boll. Un. Mat. Ital. 14, 285-299 (1977). 77. L. Modica, 'The gradient theory of phase transitions and the minimal interface criterion', Arch. Ration. Mech. Anal. 98, 123-142 (1987). 78. E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkh~iuser, Basel (1984). 79. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992). 80. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, (2000). 81. M. K16man, 'Soft ferromagnets revisited', Phase Transitions 30, 217-232 (1991). 82. F. Bethuel, H. Brezis, and F. H61ein, Ginzburg-Landau Vortices, Birkh~iuser, Basel (1994). 83. M. Struwe, 'On the asymptotic behaviour of the Ginzburg-Landau model in two dimensions', Diff. Int. Eqns. 7, 1613-1624 (1994). 84. M. Struwe, 'Erratum: on the asymptotic behaviour of the GinzburgLandau model in two dimensions', Diff. Int. Eqns. 8, 224 (1995). 85. M. Kurzke, 'Boundary vortices in thin magnetic films', Calc. Var. PDE, in press. 86. A.J. Newell and R.T. Merrill, 'Single-domain critical sizes for coercivity and remanence', J. Geophys. Res.-Sol. Ea. 104 (B1), 617-628 (1999). 87. A. Aharoni, Introduction to the Theory of Ferromagnetism, 2nd edition, Oxford University Press, Oxford, (2000). 88. W.F. Brown, 'Criterion for uniform micromagnetization', Phys. Rev. 105, 1479-1482 (1957).

380

CHAPTER 4 RecentAnalytical Developments in Micromagnetics

89. E.H. Frei, S. Shtrikman, and D. Treves, 'Critical size and nucleation field of ideal ferromagnetic particles', Phys. Rev. 107, 446-455 (1957). 90. A. Aharoni and S. Shtrikman, 'Magnetization curve of the infinite cylinder', Phys. Rev. 109, 1522-1528 (1958). 91. A. Aharoni, 'Magnetization curling', Phys. Stat. Sol. 16 1-42 (1966). 92. W.F. Brown, 'Thermal fluctuations of a single-domain particle', Phys. Rev. 130, 1677-1686 (1963). 93. G. Brown, M.A. Novotny, and P.A. Rikvold, 'Langevin simulation of thermally activated magnetization reversal in nanoscale pillars', Phys. Rev. B 64, 134422 (2001). 94. J.L. Garcia-Palacios and F.J. Lazaro, 'Langevin-dynamics study of the dynamical properties of small magnetic particles', Phys. Rev. B 58, 14937-14958 (1998). 95. R.H. Koch, G. Grinstein, G.A. Keefe, et al., 'Thermally assisted magnetization reversal in submicron-sized magnetic thin films', Phys. Rev. Lett. 84, 5419-5422 (2000). 96. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd edition, Springer, Berlin (2004). 97. G. Henkelman and H. Jonsson, 'Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points', J. Chem. Phys. 113, 9978-9985 (2000). 98. H. Jonsson, G. Mills, and K.W. Jacobsen, 'Nudged elastic band method for finding minimum energy paths of transitions', in: Classical and Quantum Dynamics in Condensed Phase Simulations, p. 385, B.J. Berne, G. Ciccotti and D.F. Coker Eds, World Scientific, Singapore (1998). 99. E. Weinan, W. Ren and E. Vanden-Eijnden, 'String method for the study of rare events', Phys. Rev. B 66, 052301 (2002). 100. E. Weinan, W. Ren and E. Vanden-Eijnden, 'Energy landscape and thermally activated switching of submicron-size ferromagnetic elements', J. Appl. Phys. 93, 2275-2282 (2003). 101. D. Liu, C.J. Garcia-Cervera and E. Weinan, 'Magnetic switching of ferrromagnetic thin films under thermal perturbation', J. Appl. Phys., in press. 102. E. De Giorgi, 'Sulla convergenza di alcune successioni di integrali del tipo dell'area', Rend. Mat. 8, 277-294 (1975). 103. E. De Giorgi and T. Franzoni, 'Su un tipo di convergenza variazionale', Atti Accad. Naz. Lincei Rend. C1. Sci. Fis. Mat. Natur. 82, 842-850 (1975).

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104. G. Dal Maso, An Introduction to F-convergence, Birkh~iuser, Basel (1993). 105. H. Attouch, Variational Convergence for Functions and Operators, Pitman, London (1984). 106. A. Braides, F-Convergence for Beginners, Oxford University Press, Oxford, (2002). 107. G. Alberti, 'Variational models for phase transitions, an approach via F-convergence', in Calculus of Variations and Partial Differential Equations (Pisa, 1996), G. Buttazzo et al. Eds., Springer, Berlin (2000), pp. 95-114. 108. J. Jost and X. Li-Jost, Calculus of Variations, Cambridge University Press, Cambridge, (1998). 109. G. Carbou, 'Thin layers in micromagnetism', Math. Models Methods App1. Sci. 11, 1529-1546 (2001). 110. G. Carbou and P. Fabrie, 'Regular solutions for Landau-Lifshitz equation in a bounded domain', Diff. Int. Eqns. 14, 213-229 (2001). 111. G. Carbou, P. Fabrie, and O. Gu6s, 'Couche limites dans un mod61e de ferromagn6tism6, Comm. Partial Differential Equations 27,1467-1495 (2002). 112. L.A. Privorotskii, Thermodynamics of Domain Structures, Wiley, New York (1976). 113. C. Cartensen and D. Praetorius, Effective simulation of a macroscopic model for stationary micromagnetics, Comput. Methods Appl. Mech. Engrg. 19, no. 2-5, 531-548 (2005). 114. H. Forster, T. Schrefl, R. Dittrich, W. Scholz, and H. Filder, 'Fast Boundary Methods for Computing Demagnetizing Fields', IEEE Trans. Magn. 39(5), 2513-2515 (2003).

CHAPTER 5

Computational Aspects of Micromagnetics Rok Dittrich, Josef Fidler, Dieter Suess, Werner Scholz, and Hermann Forster, Vienna University of Technology, Solid State Physics, Wiedner Haupstr. 8-10, A- 1040 Vienna, Austria Thomas Schrefl, University of Sheffield, Department of Engineering Materials, Mappin Street, Sheffield $1 3JD, UK

Contents 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Introduction Magnetic Memory and Sensors Micromagnetics Examples Langevin Dynamics Energy Barrier: The Path Method Applications Energy Barriers in Magnetic Random Access Memory Elements References

383 384 386 394 405 411 421 424 429

5.1 I N T R O D U C T I O N The development of advanced magneto-electronic materials requires a precise understanding of their magnetic behavior. A prominent example is magnetic recording systems for future high-density information storage, where both recording media and recording heads have to meet certain characteristics. These structures are so small that quantum mechanical effects, like exchange, have to be taken into account. However, they are too large for a pure quantum mechanical description, which would exceed the capabilities of today's ab initio computational models. On this intermediate level, between the macroscopic world and a description with atomic resolution, micromagnetic models provide a useful tool for detailed The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

384

CHAPTER 5 Computational Aspects of Micromagnetics

predictions of the magnetic properties and magnetization processes. The theory of micromagnetism combines Maxwells equations for the magnetic field with an equation of motion describing the time evolution of the magnetization. The local arrangement of the magnetic moments follows from the complex interaction between intrinsic magnetic properties, such as the magnetocrystalline anisotropy, and the physical/chemical microstructure of the material. The finite-element method is a highly flexible tool to describe magnetization processes, since it is possible to incorporate the physical grain structure and intergranular phases. Recent developments of advanced numerical methods allow for the first time the modeling and dynamic simulation of an entire media-head system, including subgrain discretization of the media and a full three-dimensional finite-element mesh of the entire ring head. Moreover, the development of high-performance parallel finite-element micromagnetics packages makes full use of the computational power of novel multiprocessor machines and allows the use of large-scaled complex models [1,2].

5.2 M A G N E T I C M E M O R Y A N D S E N S O R S Every year the number of bits that are crammed onto each square inch of a commercial hard disk almost doubles [3]. Fortunately, this growth rate cannot continue forever because of physical limits. The limit in the case of hard-disk drives is given by an effect called superparamagnetism. It denotes the phenomenon that the magnetic regions on the disk cannot retain their magnetic orientation (=the data) over the lifetime of the product when they become too small. Novel storage media can push this limit some steps further away. For development of novel magnetic structures it is important to have a detailed understanding of the underlying physical phenomena. The computational models provide great freedom in the choice of experimental conditions and in the variation of material parameters. This provides useful hints for artificial structuring of future magnetic materials. In addition to measurements of the remanent magnetization and the coercive field, it is possible to study the details of the magnetization distribution and the magnetization reversal processes, which are difficult to investigate experimentally. An example where micromagnetic simulations are indispensable for the understanding of magnetic processes are the so-called magnetic random access elements (MRAM). Magnetic random access memory has the potential to store more data, access those data faster and use less power than current memory technologies [4]. You hit the power button on your

5.2 MAGNETIC MEMORY AND SENSORS

385

FIGURE 5.1 Array of MRAM cells with current lines. To switch the center cell a current is applied in the two red-colored current lines. The resulting field (the arrow at 45~ is strong enough to switch the magnetization. Half the selected cells feel a weaker field (the horizontal and the vertical arrow) either parallel or perpendicular to the easy axis.

television and it instantly comes to life. But do the same thing with your computer and you have to wait a few minutes while it goes through its boot-up sequence. In order to guarantee the high-speed data access there are no moving parts in the MRAM storage. Instead, the data are accessed through a grid of current lines, where each crossing point carries one bit of data (one MRAM cell) and can be addressed individually (see Fig. 5.1). Another example are magnetic sensor elements. Magnetic sensors require a high sensitivity so that small magnetic fields can be detected. On the other hand, thermal fluctuations, which will lead to thermal noise, should be suppressed. The free layer of a multilayer sensor element is soft magnetic and may have a size well below one micrometer. Thermally induced magnetization processes may cause local or global magnetization rotations that cause the magnetization noise. With decreasing lateral extension of the elements, thermal fluctuations become more pronounced. We use two complementary methods to study such thermal effects. On short time scales we directly integrate the stochastic equation of motion of the system at finite temperature. This gives information about the thermal noise on the time scale of some nanoseconds. For longer time scales, thermal fluctuation can drive the magnetization to switch to another energy minimum. To study the rate of such seldom but important events we apply the n u d g e d elastic-band method. Starting from an initial guess for

386

CHAPTER 5 Computational Aspects of Micromagnetics

the path that connects two local minima of the system, a highly probable path is found by moving the points along the path according to an algorithm that resembles tensioning an elastic band across a mountain. This method then gives the relevant saddle point in between, which determines the probability of switching. The basic structural units of all magnetic systems are nanoscale magnetic particles. The particles exhibit a uniaxial anisotropy either induced by the particle shape or a magnetocrystalline anisotropy. The magnetization of the particle is preferably oriented parallel to the anisotropy axis. Thus, at zero applied field, the particle has two distinct states corresponding to an magnetization angle of r = 0 and q~ = ~. Under the influence of an external field the magnetization may rotate out of its preferred orientation. As long as the external field is small the magnetization rotates reversibly out of its easy direction. In magnetic field sensors the rotation angle is measured by the change of resistivity. In storage elements a high external field is applied that causes the magnetization to rotate by ~. The magnetic particle will remain in the new state and the information is stored, unless it is changed again by an external field or spontaneously switches owing to thermal activation. The calculation of the critical fields that induce magnetization reversal and the calculation of the time required by the magnetization to reach its new equilibrium state is the subject of micromagnetics. The calculation of the probability of the magnetization to switch below the zerotemperature switching fields requires the extension of classical micromagnetics to nonzero temperature.

5.3 M I C R O M A G N E T I C S 5.3.1 BASIC PRINCIPLES

The basic concept of micromagnetism is to replace the atomic magnetic moments by a continuous function of position. In a continuum theory, the local direction of the magnetic moments may be described by the magnetic polarization vector J(r) = ~toM(r) = #om(r)/V.

(5.1)

The magnetic polarization, J, is proportional to the magnetization, M, which is given by the magnetic moment, m, per unit volume, V. /~0 is the magnetic permeability of vacuum. The second principle of micromagnetism treats the magnitude of the magnetization as a function of

5.3 MICROMAGNETICS

387

temperature only. The modulus of the magnetic polarization IJI = Js(T)

(5.2)

is assumed to be a function of temperature, T, and to be independent of the local magnetic field. Thus, the magnetic state of the system can be uniquely described by the direction cosines, bk (r), of the magnetic polarization, J = Jsb. Here k runs from I to 3 and denotes the Cartesian coordinates. In a metastable equilibrium state, bk(r) minimizes the total Gibbs free energy of the system. The contributions to the total magnetic Gibbs free energy are derived from classical electrodynamics, condensed matter physics, and quantum mechanics so that the continuous expressions for the energy describe the interactions of the spins with the external field, the crystal lattice, and the interactions of the spins with one another. The latter consists of longrange magnetostatic interactions and short-range quantum-mechanical exchange interactions. The competitive effects of the micromagnetic energy contributions upon minimization determine the equilibrium distribution of the magnetization. The minimization of the ferromagnetic exchange energy aligns the magnetic moments parallel to each other, whereas the minimization of the magnetostatic energy favors the existence of magnetic domains. The magnetocrystalline anisotropy energy describes the interaction of the magnetization with the crystal lattice. Its minimization orients the magnetization preferably along certain crystallographic directions. The minimization of the Zeeman energy of the magnetization in an external field rotates the magnetization parallel to the applied field. The total magnetic Gibbs free energy, Et, may be written in the following form [5]: Et --

--~Hd" J +

3 A ~ (Vbi) - Ku ( u . b) 2 - Hext" i=1

J

} dV.

(5.3)

In (5.3) the first term of the integrand is the magnetostatic energy density, the second term is the exchange energy density, the third term denotes the magnetocrystalline anisotropy density, and the last term is the Zeeman energy. The integral extends over the total volume of all magnetic particles. The four basic energy contributions to the total energy are shown in Fig. 5.2. A, Ku, and u denote the exchange constant, the uniaxial magnetocrystalline anisotropy constant, and the magnetocrystalline anisotropy direction (easy direction), respectively. The intrinsic magnetic properties A, Ku, and Js and the orientations of the easy axes can be determined experimentally. The intrinsic material properties may vary locally with r. For example, in a

388

CHAPTER 5 Computational Aspects of Micromagnetics

FIGURE 5.2 The four basic energy contributions of the total Gibbs free energy of a magnetic system. The exchange energy tries to align the spins parallel. Thus, it competes with the magnetostatic energy where closure domains are favorable. Additionally, there can be a presence of a magnetocrystalline anisotropy that defines easy directions (low-energy directions) of the magnetization. When a field is applied the Zeeman energy part favors the alignment into the direction of the applied field. granular magnet the crystalline orientation and thus u varies from grain to grain. Hext is the applied field. The demagnetizing field, Hd, follows from magnetic volume charges, V 9J, and magnetic surface charges, n . J (n denotes the unit vector normal to the surface of the magnetic particles). From this starting point there are several methods to study the micromagnetic system described by Eqn. (5.3). A commonly used technique is to optimize the total energy of the system with respect to the direction cosines bk (r). To simulate a hysteresis loop, the applied field is changed by a small step and the energy optimized until equilibrium is reached. Then the field is changed a step further and so on [6]. With this technique one can obtain information about the hysteresis properties, such as coercivity, energy density products, and information on stable and metastable magnetization configurations. However, if the time we are interested in approaches the nanosecond regime, dynamic effects have to be taken into account [7]. Another disadvantage of the energy optimization techniques is that no information is obtained about the reversal mode and the reversal times. For applications in magnetic information storage, the dynamic effects become increasingly important as the areal density increases. The reversal times of the magnetization in media and write heads determine the limits of the read/write speed of data in the recording device [8]. Therefore, micromagnetic dynamics provides a helpful tool to study the limits of such devices, but also provides hints for the design of novel magnetic systems. Thermal effects become increasingly crucial as the size of these structures drastically scales down. Examples are magnetic recording devices

5.3 MICROMAGNETICS

389

and magnetic sensors. State-of-the-art magnetic materials in a hard disk typically consist of thin granular magnetic layers with grain sizes in the range of ~ 8 nm. Thermal effects in such small structures can induce spontaneous reversal of the magnetization and result in a loss of the recorded data. Since these events are very rare, but still important should they occur, it is important to focus on this topic. We apply a so-called nudged elasticband method [9] to numerically predict the probability of such rare events in systems that can be highly complex, with many degrees-of-freedom. 5.3.2 FINITE-ELEMENT DISCRETIZATION

In order to be able to perform numerical calculations and simulations with a computer we first have to formulate the problem in a way that the computer can understand. Our region of interest, the magnetic region, must be expressed as a finite number of evaluation points in space, where each point has a corresponding region surrounding it. The regions are constructed in a way so that there is no overlap between them. The solution of the equations can be evaluated numerically on each of these points. Two ways of discretizing the regions can be used: the finite-differences method simply uses a homogeneous mesh of rectangular prisms. More complex element types are used in the finite-element method. In our simulation package we use tetrahedral finite elements [1]. Surfaces are meshed with triangles. The advantage of the finite-differences method is that it gives a simpler formulation of the discretized equations than the finite-element method. In addition, a fast and powerful algorithm, FFT (fast Fourier transformation), can be applied to solve the time-consuming evaluation of the demagnetizing field [10]. The advantage of the finite-element method, however, is the great flexibility in the modeling of the magnets (see Fig. 5.3). Complex shapes and granular materials can be modeled. The mesh size can be adapted where needed and kept coarse in less-important regions. In many cases this gives a drastic speed-up as compared with a homogeneous mesh [11]. An example is circular soft magnetic dots where the mesh size has to be fine only at the position of the magnetic vortex core [12]. The mesh can also be changed during the simulation itself. During the motion of a domain wall through a magnetic wire, the mesh is adapted while the simulation proceeds. The mesh is fine at the position of the wall and coarse in the rest of the magnet. This gives the impression that a region with fine mesh size moves together with the domain wall (moving mesh) [13]. The numerical treatment starts with the discretization of the total Gibbs free energy Et (see Eqn. (5.3)). The magnetic polarization, J(r), is expanded

CHAPTER 5 Computational Aspects of Micromagnetics

390

FIGURE 5.3 The use of finite elements for the discretization of the magnetic region gives great flexibility in the modeling. Complex shapes and granular materials can be modeled. Moreover, the mesh size can be adapted where needed, and can also be changed during the simulation itself. with piecewise linear basis functions q~i(r). For one component of J we can write N

Jk(r) ~ Jk(r) = Js(r) E bk, iq~i(r)"

(5.4)

i=1

Jk(r) is the finite-element approximation of the kth component of the magnetic polarization. The coefficient bk, i denotes the kth component of the normalized spontaneous polarization (Ibk, il~l) on the node point i. The number of unknowns (bk, i) is three times the number of node points, N, of the finite-element mesh. To perform the time integration, we have to calculate the effective field on every node point of the finite-element mesh. However, we cannot directly use the analytic formula for the effective field, which follows from the negative functional derivative of the total Gibbs energy (5.3) as H e ft --

c~Et

~j

2A 2Ku = ~ V2j + ~ (J. u ) u + H d if- Hex t.

(5.5)

Here A is the exchange constant, Ku is the magnetocrystalline anisotropy constant, u is the unit vector parallel to the anisotropy axis, Hd is the magnetostatic stray field, and Hext is the external applied field. The first term on the right-hand side of Eqn. (5.5) is the exchange field. Its calculation needs the second derivative of the magnetic polarization. Numerically, the second derivatives cannot be calculated directly using linear basis functions.

5.3 MICROMAGNETICS

FIGURE 5.4 The volume example.

391

Vi surrounding

node i shown in a two-dimensional

In addition, the calculation of the stray field, which follows from the gradient of a scalar potential, is crucial. With linear basis functions, the gradient of the potential, which is proportional to the stray field, is only defined within an element but not on the node points. To overcome this problem we start from the total Gibbs energy for a ferromagnetic particle (5.3). The total energy is an integral over the particle volume. For the magnetic polarization J(r) the expansion according to (5.4) is used. The kth component of the effective field on node i is approximated using the box scheme [14] Hk, i eff ~

1

cgEt

Imil Obk, i

(5.6)

,

where Imil denotes the modulus of the magnetic moment on the node point i. It follows from the integral

Imil=lv IJ(r)ldV-fvJs(r)dV i

that

Vi is the surrounding

(5.7)

i

volume (Fig. 5.4) of the node i, such that

E Vj - V

and

V i M Wj --

0

for i ~=j.

(5.8)

J

5.3.3 CALCULATION OF THE STRAY FIELD

The most time-consuming part in the calculation of the effective field is the long-range magnetostatic interaction, which is described by the stray field, lid. The stray field, lid, is obtained from a boundary-value problem AU=

v.j /~0

and

Hd

----VU.

(5.9)

CHAPTER 5 Computational Aspects of Micromagnetics

392

To apply the boundary condition at infinity, a hybrid finite-element boundary-element method [15] is used. No finite elements are needed outside the magnetic particle to solve the boundary-value problem (5.9). This is the advantage of the hybrid finite-element (FE)/boundary-element (BE) method. For the solution of Eqn. (5.9) with the hybrid FE/BE method one Poisson equation with Neumann boundary conditions and one Laplace equation with Dirichlet boundary conditions have to be solved. To obtain the boundary conditions a matrix vector product has to be performed. We split the total magnetic potential U into two parts, U = U1 4- U2. The potential U1 solves the Poisson equation (5.9) inside the magnetic particles with Neumann boundary conditions at the surface of the magnets and it is zero outside the magnets. The potential U2 solves the Laplace equation everywhere in space and shows a jump at the surfaces of the magnets. Thus, U2 is the potential of a dipole sheet at the surfaces of the magnet. After discretization [16] the integral operator may be expressed as a matrix vector product U2 ~- BU1.

(5.10)

Here B is an Ns x Ns matrix relating the surface nodes of the finite-element mesh to each other. Ns is the number of boundary nodes. The vectors U1 and U2 contain the potentials U1 and U2 at all surface nodes. The storage requirement for the matrix B is the bottleneck of the method since B is fully populated. Especially for thin films the method loses efficiency, since most of the nodes are located at the boundary and the boundary-element matrix will exceed available computer memory. To overcome this problem, fast boundary-element methods [17-19] have been applied in micromagnetics

[20]. 5.3.4

DYNAMICS

In a first step we study a single small magnetic particle. If the particle is sufficiently small (smaller than 10 nm) the magnetization within the particle will be uniform. It is therefore sufficient to describe such a particle as one single macroscopic magnetic moment. The dynamics of the magnetization is described by quantum mechanics. Since we are operating on a length scale where many atomistic magnetic moments contribute to the macroscopic description, we can apply the equation for the expectation values. At zero temperature the magnetization dynamics is then described by the Gilbert equation of motion [21]. c3J _ a dJ 0t -lylJ x Heft - ~ J x ~--/,

(5.11)

393

5.3 MICROMAGNETICS

where J is the magnetic polarization vector, Heff is the effective field, and Js is the spontaneous polarization. J is assumed to be a continuous function of space. To obtain the general form 0y 0t

= fit,

y)

(5.12)

for an ordinary differential equation (ODE), we transform Eqn. (5.11) into the mathematically equivalent Landau-Lifshitz-Gilbert (LLG) equation 0J r =

171 1 + ~2J x Heft

171

1 if-0~2Js J x (J x Heft),

(5.13)

with the gyromagnetic ratio

171 = 2.210175 x 105 As' m

(5.14)

and the phenomenological Gilbert damping constant a. The first term on the right-hand side of Eqn. (5.13) describes the precessional motion of the magnetization in the presence of a magnetic field, here the effective field Heft. If taking this term only, the magnetic polarization J would move undamped on a cone around Heft with a precession frequency Heft (see Fig. 5.5 on the left). No energy would be dissipated, and the magnetic system would stay on an equipotential line in the corresponding energy landscape. The second term on the right-hand side of (5.11) is the phenomenologically introduced dissipative term proportional to the generalized velocity. This damping term drives the magnetic polarization J towards the direction of the effective field (see Fig. 5.5). Equilibrium is reached when J and Heft are parallel. The magnetic system has then dissipated its energy and reached the local energy minimum of (5.3). The strength of the damping term is determined by the damping constant ~. In magnetic recording media, typical values for a are 0.02 [22]. This describes a relatively weakly damped system, and a dynamic treatment is important. Taking ~ = 1 describes the high-damping limit of (5.11). Here, similar results are obtained when comparing energy minimization methods with the results obtained by integrating the equation of motion (5.11).

5.3.5 TIME INTEGRATION We use the CVODE package [23] for solving both the LLG equation (5.13) and the differential equation defined by the nudged elastic-band method (see Eqn. (5.37)) in Section (5.6.2). The relative performance of the Adams

CHAPTER 5 Computational Aspects of Micromagnetics

394

(~: 1.0 ""............~

p;ecession

c"-"x

FIGURE 5.5 Left: damped gyromagnetic precession motion of a single magnetic polarization vector J towards the effective magnetic field Heft according to the Gilbert equation of motion. Right: precessional motion towards the effective field for strong and weaker damping.

method, the backward differentiation (BDF) method and the BDF method with preconditioning changes depending on the microstructure, material parameters, and the finite-element mesh. The Adams method seems to be attractive because of the low cost per time step. In contrast to the onestep methods (e.g. Runge-Kutta), multistep methods make use of the past values of the solution. At each time step a nonlinear system of equations has to be solved. The Adams method solves the nonlinear system with functional iteration and thus requires only the evaluation of the right-hand side of (5.13). However, if the problem is stiff, the convergence of the functional iterations is slow. For a stiff problem it is advisable to use an implicit method such as BDF. A detailed description of the use of implicit time integration schemes in micromagnetics is given in [24,25].

5.4 E X A M P L E S 5.4.1 STONER-WOHLFARTH PARTICLE If a magnetic particle is sufficiently small the reversal mode with the smallest coercive field will be uniform rotation of the magnetization. Other reversal modes, which include nonuniform reversal modes, are also solutions of Brown's equations [5] but are not observed in nature, since they have much higher coercivity for very small particle diameter. The magnetization within the particle stays homogeneous, since the exchange energy would be too high in the other reversal modes. To describe a such particle it is thus sufficient to study a single magnetic moment including anisotropy, Zeeman and self-demagnetizing energy.

5.4 EXAMPLES

395

5.4.2 EXAMPLE 1: STATIC HYSTERESIS OF A STONER-WOHLFARTH PARTICLE

The static hysteresis curve of a magnetic particle can be calculated by subsequent minimization of the total Gibbs free energy (5.3) for decreasing external field. At first, the system is in a local energy minimum. The energy landscape changes its shape as the external field changes. At a critical value of the external field, the local m i n i m u m vanishes and the system moves towards a different local minimum. This field is usually called the nucleation field. At the nucleation field, the first irreversible switching event occurs as the external field is decreased [26]. For simple systems it is equivalent to the coercive field. In the following, the magnetization reversal processes is described for a small spherical particle. For a spherical particle, the self-demagnetizing energy becomes independent of the magnetization direction. Thus we only need to take into account the anisotropy energy and Zeeman energy. The energy can be written as a function of the angle, ~, between the easy axis and the magnetization direction: E = K1 sin 2 q0

-

HextJs cos

(5.15)

Figure 5.6 shows the change of the energy landscape for different strengths of the applied external field during a hysteresis loop. If the external field is greater than 2K1/Js, only one m i n i m u m exists. Independent of the initial magnetization state, the final state will always be in this minimum. However, in regions where 2K1 IHextl < Js

(5.16)

more than one m i n i m u m exists and the history of the applied field is important to determine in which state the magnetization will be. If we start with a high positive field, which is successively reduced to zero, and then becomes negative, then the angle ~ stays zero until the field reaches the value of -2K1/Js. At this field strength (see point 4 in Fig. 5.6) the energy m i n i m u m disappears. The magnetization state is now no longer a stable local energy m i n i m u m and will thus switch to another minimum. In this case (a small single-domain particle), only one m i n i m u m exists at this critical field strength and the magnetization will thus reverse into this state and stay in that state also after the field has been turned off again (see point 5 in Fig. 5.6). We now see that the magnetization state is not uniquely determined by the strength of the external field. At zero field, for example, we have two possible states in this example: point I or point 5 in

396

CHAPTER 5 Computational Aspects of Micromagnetics

ff ~

' ,,LA'~2,

'

,

,

0.5

o -0.5 -1

-2

I

-1

,

3"

I

0 1 H~~t[2K / Js]

,

2

FIGURE 5.6 Change of the energy landscape during a hysteresis loop for a Stoner-Wohlfarth particle. The external field is applied antiparallel to the easy axis. The reversal coordinate is the angle between the magnetization and the easy axis.

Fig. 5.6, and it depends also on the history of the applied field in which of the states it will be. The history-dependence of the magnetic state leads to hysteresis. For larger particles the situation can become a lot more complex, since the number of energy minima becomes larger. An example is structured soft magnetic materials, which can have many magnetization states (Cstate, S-state, flower-state, vortex-state, see, for example, [27]). Here, it strongly depends on the history of the applied magnetic fields in which of the states the magnetization will be. Some of these states can even be very difficult to access at all. 5.4.3 EXAMPLE 2: GYROMAGNETIC PRECESSION AND DAMPING IN A STONER-WOHLFARTH PARTICLE

Again we study a single domain particle as in the previous capture, which has been saturated by a positive external field. The easy axis direction is supposed to be aligned along the y-axis. We simplify the problem to two dimensions. The effective field can thus be written as a function of the angle q~, -~ 2K1 ( 0 ) Js ( sin(p ) Heft - - ~ s cos (q)) 1 + Hext - N - /~0 cos q~

(5.17)

5.4 EXAMPLES

397

1

Hext (negative)

2K1 Hext > - Js

2K 1 Hext < - Js

FIGURE 5.7 The magnetization keeps the orientation aligned up until the field becomes smaller than the critical value of -2K1/Js. The bottom left figure shows that the effective field can be antiparallel to the magnetization in a stable state of the magnetization.

The term Js I x (j x Heft) - Js J (J Heft) - Heft J2

(5.18)

in Eqn. (5.13) turns the magnetization into the direction of the effective field. Since J x J = 0, the torque is independent of the component parallel to the magnetization direction. The last term in (5.18) thus does not contribute to the damping term that rotates the magnetic polarization towards the field direction. If the magnetization is exactly parallel to the external field, the torque will be zero and therefore the magnetization will not reverse even for Hext > 2K1/Js. The direction of the magnetization has to be perturbed a little bit in order to generate a torque and to obtain realistic coercive fields. Figure 5.7 shows that the magnetization is turned back into the easy axis direction as long as the field strength is smaller than the nucleation field, and is oriented antiparallel to the anisotropy field. Once this critical field strength is achieved, the magnetization of the particle will reverse, since now the anisotropy field decreases with increasing angle ~p.

398

CHAPTER 5 Computational Aspects of Micromagnetics

5.4.4 EXAMPLE 3: STONER-WOHLFARTH ASTROID

As an example for a Stoner-Wohlfarth particle we study a small anisotropic grain as typically used in perpendicular magnetic recording media. We compare two different sizes for such a grain, but keep the material parameters (K1 -- 300 k J / m 3, A - 10 pJ/m, easy axis aligned along the long axis) constant. The smaller particle has a column length of 15 nm and a diameter of about 10 nm. The reversal mode for the case where the field is aligned parallel to the easy axis is shown in Fig. 5.8. Indeed, the size is so small that the particle reverses by homogeneous rotation of the magnetization and behaves perfectly as a Stoner-Wohlfarth particle. The larger grain has a column length of 80 nm and the same diameter as the smaller one. This size is above the domain-wall width for this material. As expected, the magnetization reversal becomes more complex in this case. First a nucleation of reversed magnetization forms at the ends of the particle. A domain wall then moves through the particle. Note that the reversal starts on both ends. The reason for this is the symmetry of the computational model. Slight differences would lead to a break-up of this symmetry and the reversal would proceed from one of the ends only. The angle dependence of the switching field for the smaller particle behaves as predicted by the Stoner-Wohlfarth theory. The results are shown in the solid asteroid in Fig. 5.9. For the larger particle, one would expect a more complex behavior since the magnetization reversal is highly nonuniform and complex. Still, the angle dependence of the switching field behaves like a perfect Stoner-Wohlfarth asteroid but slightly shifted to higher fields. This shift is explained by the contribution of the self-demagnetizing energy. The demagnetizing factor of the more elongated shaped particle is higher than for the smaller particle. This can be seen as an additional anisotropy contribution to the total anisotropy. Consequently, the switching fields are higher. The still very perfect Stoner-Wohlfarth asteroid in this case can be explained as follows: the switching field is mainly determined by the point where the reversed magnetization forms at the end of the particle. The magnetization in a small fraction of the volume (effective volume) at the end starts to rotate. This small fraction of the volume can now again be seen as a Stoner-Wohlfarth particle, which determines the nucleation field. 5.4.5 EXAMPLE 4: FAST SWITCHING OF SMALL MAGNETIC PARTICLES

In many applications static hysteresis calculations give sufficient information about the magnetic properties of the material. However, in

5.4 EXAMPLES

399

FIGURE 5.8 Magnetization reversal for an anisotropic magnetic grain. (K1 -- 300 kJ/m 3, A = 10 pJ/m, the easy axis is aligned parallel to the long axis). Compared are two different sizes for the columnar length (15 nm and 80 nm). The smaller particle reverses by homogeneous rotation of the magnetization (perfect Stoner-Wohlfarth behavior). The longer grain reverses by nucleation and propagation of transverse domain walls. The color code corresponds to the net magnetization along the easy axis. The external reversing field is applied parallel to the easy axis.

modern applications a theoretical understanding of high-speed switching of, e.g. magnetic grains, becomes increasingly important as the data rate of magnetic recording media approaches the GHz regime [28]. At such high frequencies, gyromagnetic effects and precessional motion dominate the reversal process. He and co-workers [7] found that if the rise time of the pulse field is less than a few nanoseconds and the Gilbert damping constant is one, switching can occur well below the Stoner-Wohlfarth limit (see previous section). Mallinson [29] derived the switching time as a function of the field strength for fields parallel to the anisotropy direction. The switching time decreases with increasing external field. Bauer [30] investigated the possible range of switching at external fields below the Stoner-Wohlfarth limit. These phenomena, which occur at very fast changing fields, are known as fast switching effects. The possibilities of switching small magnetic particles below the Stoner-Wohlfarth limit are explored for columnar grains similar to those shown in Fig. 5.8. Equation [5.13] is solved with the external field applied instantaneously.

400

CHAPTER 5 Computational Aspects of Micromagnetics

FIGURE 5.9 Angular dependence of the switching field strength for an anisotropic magnetic grain (K1 = 300 kJ/m 3, A = 10 pJ/m, the easy axis is aligned parallel to the long axis). Compared are two different sizes for the columnar length (15 nm and 80 nm). Although the magnetization reversal becomes highly nonuniform for the larger grain, still the angle dependence of the switching shows a perfect Stoner-Wohlfarth behavior, however slightly shifted to larger fields due to the larger demagnetizing factor (which can be seen as an additional anisotropy contribution) for the longer particle. The particle shape and the used finite-element mesh is shown on the right.

The average magnetization as function of time is shown in Fig. 5.10. The basal plane of the grain is an irregular pentagon with a diameter of about 13 nm. The easy axis is perpendicular to the basal plane (along the z-axis). After saturating the grain along the +z-axis the remanent state is calculated. An external field is applied instantaneously at an angle fl between the field and the minus z-axis. Depending on the damping constant and the height of the grain (column length lc), the reversal process changes from coherent rotation to inhomogeneous reversal. For a damping constant = 0.02, homogeneous rotation only occurs for column lengths lc smaller than 20 nm. For a column of this length we investigate the influence of the strength of the external field on the switching time. We distinguish between two switching times. We define the theoretical switching time as the m i n i m u m length of a field pulse to switch the particle. However, for longer field pulses, the magnetization may precess back and no switching occurs. A suitable definition of a practical switching time is the critical duration of the field pulse, tp, so that all field pulses longer than tp will switch the particle. Figure 5.10 shows the time evolution of the average magnetization parallel to the z-axis. The magnetization oscillates owing to gyromagnetic precession. This effect is more pronounced with larger

5.4 EXAMPLES

401

'

,

0.5

A A. i~ -

N

l vvv it,

-0.5

I

0.1

0.2 time (ns)

FIGURE 5.10 The average magnetization parallel to the long axis as a function of time for an angle of one between the field and the long axis. The theoretical switching time tth and the practical switching time tp are marked. The inset shows the shape of the columnar grain.

angle ft. The arrows in Fig. 5.10 show the theoretical switching time, tth, and the practical switching time, tp. Figure 5.11 shows the time dependence of the magnetization for different strengths of the external field. At t = 0 an external field of 0.95, 1.2 and 2.8 (2K1/Js) is applied. For small values of the external field the zcomponent does not decrease monotonically but shows oscillations, which correspond to the precession of the magnetization around the effective field. This effect can only be found if the axial symmetry is broken (easy axis does not point exactly parallel to field direction) and an increase of Jz does not necessarily lead to an increase of the energy. With increasing damping constant, the relaxation towards the minimum dominates the motion of the magnetization and the oscillations of the magnetization vanish for every strength of the external field. Figure 5.12 shows the switching time tp as a function of the strength of the external field. It is astonishing that the switching time does not decrease with increasing external field in the whole regime but shows a maximum slightly above the anisotropy field.

CHAPTER 5

402

Computational Aspects of Micromagnetics

0.5

i.,~ N m

-

-0.5

Hex t = 2.8

n

0.5

1

1.5

2

time (ns) FIGURE 5.11 Jz as function of time after different external fields (0.95, 1.2 and 2.8 times 2K1/Js) are instantaneously applied to the saturated state. The column length of the grain is 20 n m (~ = 0.02).

'

I

'

I

'

I

'

po~

3.5 r~ =

:

I

'

I

'

o T=OK

~

, ~ - * c o l u m n a r grain

3

"= 2.5

1.5 _ oo

-1

-1.2

-1.4

-1.6

-1.8

-2

H c x t ( 2 K I / J s)

FIGURE 5.12 The switching time of the columnar grain (solid line), a single magnetic momenent (circles) as a function of the magnitude of the external field (fl = 1~ The switching time does not decrease with increasing external field in the whole regime but shows a m a x i m u m slightly above the anisotropy field. The solid line gives the switching time for a single magnetic moment at a temperature of T = 300 K. Thermal fluctuations do not change the dependence of the switching time on the external field.

5.4 EXAMPLES

403

For a better understanding of this effect we simplify our model. We calculate the time evolution of a single magnetic moment under the action of an external field. Again we assume a uniaxial anisotropy along the zaxis. Initially, the magnetization points parallel to the positive z-axis. Then the external field is applied instantaneously at an a n g l e / / t o the negative z-axis. Figure 5.12 shows that the dependence of the switching time as a function of the field strength is very similar for the granular grain and the single magnetic moment. Thus we assume that the reversal process of a columnar grain can be compared with that of a single magnetic moment. The path which the system follows through the energy landscape may explain the switching behavior more clearly. Figures 5.13 and 5.14 show the path of the magnetization through the energy landscape and the locus of the magnetization just after the application of the field Hext = 0.9(2K1/Js) and Hext = 1.3(2K1/Js), respectively. Before the application of the external field, the system is in equilibrium at Jx = Jy = 0 marked with a dark dot in (A) and (B) in Figs. 5.13 and 5.14. If a field of 1.3(2K1/Js) is applied, the energy and the effective field remains small and it precesses several times around the effective field, with Jz almost one (Fig. 5.14(C)). Since for Jz almost one the torque J x H is small, the relaxation towards the reversed state is slow. A large switching time occurs. Only for fields exceeding 1.1 times the anisotropy field does the switching time decrease. This is due to the linear increase of the torque with the strength of the external field. If the external field is comparable with the anisotropy field, a more complex energy landscape is obtained, because anisotropy energy and Zeeman energy contribute in the same order of magnitude to the total energy. The bold line in the contour plot in Fig. 5.13(B) gives the path of the magnetization for an external field of 0.9(2K1/Js). The precession drives the magnetization away from the effective field (Fig. 5.13(D)), which leads to a high torque. Fast switching occurs when the magnetization does not relax back to the local m i n i m u m shown in Fig. 5.15 (A) but moves towards the local minimum, which corresponds to the reversed state. For an external field Hext~O.89(2K1/Js), the initial motion of the magnetization along a path with nearly constant energy leads to magnetization reversal. The m i n i m u m switching time is 1.4 ns for Hext = 0.89(2K1/Js), whereas a field of Hext > 1.9(2K1/Js) is required to switch within the same time for large external fields. The switching time decreases significantly with larger angle fl between the external field and the particle's easy axis. Figure 5.15 gives the switching time tp for fl = 45 ~ between the field and the negative z-axis. Again the switching time shows a maximum close to the coercive field obtained from the Stoner-Wohlfarth theory. Owing to the gyromagnetic precession,

404

CHAPTER 5 Computational Aspects of Micromagnetics

FIGURE 5.13 Energy landscape and paths of the magnetic polarization towards the switched state when an external field of 0.9 (2K1/Js) is applied. (A) and (B) show the energy landscape and the contour plot in terms of its normalized magnetic polarization, respectively; (C) and (D): locus of the polarization as a function of time. The circles are plotted every 0.005 ns. (D) shows only the first 0.15 ns after applying the external field.

the switching time shows jumps as a function of the field strength. Again a complex energy landscape causes fast switching at low fields. Different fields cause different paths through the energy landscape. It is interesting to note that at some specific fields, for example, at Hext = 0.43(2K1/Js), switching does not occur. The magnetization rotates back towards the local minimum, which corresponds to the nonswitched state after precessing one complete circle. However, at lower fields, switching is possible again provided Hext~O.4(2K1/Js). The observed switching times for fl = 45 ~ are smaller by about a factor of 1/10 as compared with the switching times obtained for fl = 1~ The dependence of the switching time on the strength of the applied field does not change with thermal fluctuations. The solid line in Fig. 5.15

5.5 LANGEVIN DYNAMICS

405

FIGURE 5.14 Energy landscape and paths of the magnetic polarization towards the switched state when an external field of 1.3 (2K1/Js) is applied. (A) and (B) show the energy landscape and the contour plot in terms of its normalized magnetic polarization, respectively; (C) and (D): locus of the polarization as a function of time. The circles are plotted every 0.005 ns. (D) shows only the first 0.5 ns after applying the external field.

was calculated using Langevin dynamics (see Section 5.5) for a temperature of T = 300 K.

5.5 L A N G E V I N D Y N A M I C S With decreasing device dimensions, thermal fluctuations m a y ultimately limit the performance of magnetic materials used for spin-valve sensors, MRAM (magnetic r a n d o m access memory) cells, or magnetic storage media. It therefore becomes increasingly important to introduce thermal effects into the numerical simulations. The previous sections show the finite-element treatment of micromagnetic dynamics at zero temperature. Powerful time integration methods

CHAPTER

406

0.4

I

5

Computational

'

I

A s p e c t s of M i c r o m a g n e t i c s

'

I

...,~"V~/~,.. 0.35 "~

'

I " T=0K -

I -

Z=3OoEi

_

0.3 o

"o 0.25

~

o

0.2 0.15 0.1 I

i

I

-0.5

I

I

-1

- 1.5

Hext (2KI/J s)

FIGURE 5.15 Switching time of a single magnetic moment as a function of the field strength; (circles): fl = 45 ~ T = 0 K; (solid line): fl = 45~ T = 300 K.

can be applied since we are dealing with an ordinary differential equation. To introduce temperature into the dynamics equations, a r a n d o m l y fluctuating field is added to the total effective field. Equation (5.11) then transforms into

3J 0~ -

-171J x

( H e ft q- H t h ) -

a

3J

(5.19)

~s J x ~3---t

We can transform (5.19) into the mathematical equivalent stochastic Landau-Lifshitz-Gilbert (LLG) equation. The stochastic Landau-Lifshitz equation is a system of 3N Langevin equations with multiplicative noise [31]. In the following, we denote the unit vector parallel to J with u.

~3ul = Al(t) + E Blik(t)Hlk (t) ~3t k 1+

1+

0r2 U x H e ft

(5.20)

0r2 U • (U x H e f t )

,

(5.21)

i

BIj --

-I~1~

l

~,ijkUj

I~,1~

1+

l l O~2 ( u i u k -- (~ik),

(5.22)

i

where the indices i, j, k run over the three space directions, and the index I = 1 . . . . . N runs over the computational cells. The drift term, A I, is the

407

5.5 LANGEVIN DYNAMICS

right-hand side of the deterministic Landau-Lifshitz-Gilbert equation. is the Gilbert damping constant and 7 is the gyromagnetic ratio. Hth is the random thermal field. The thermal field is assumed to be a Gaussian random process with the following statistical properties:

(Hlh, i(t) ) -- 0

(5.23)

(H~h,i(t), Hkh,j(t')) -- 2D~ij~tk~(t- t').

(5.24)

The average of the thermal field, taken over different realizations, vanishes in each direction i in space. The thermal field is uncorrelated in time and space. The strength of the thermal fluctuations follows from the fluctuation-dissipation theorem [31] D = ~

~kBT .

13'l#olm;I

(5.25)

In (5.25) m l is the total magnetic moment of the computational cell l (see Fig. 5.4 and Eqn. (5.7)). Numerically, Eqns (5.20) to (5.25) can be solved using a semi-implicit time integration method [32]. The time is divided into a lattice of discrete points tn. The Langevin equation is solved in the time interval tn to tn+l The right-hand side of (5.20) is evalwith the initial condition u li = u l(tn). i uated at the midpoint -in -~ (tn 4- tn+l)/2 -- tn 4- At~2. The update of the direction cosines, uli, is given by Aul . A. ( . ,.

. (U)I-1 --I Ui, 7.l+1 u i ,.

..)At

4- ~

eikl ( . . . . (-if) I-1 , --I Ui, 7.1+1 u i . . . . )AH

h,k

k

(5.26) with the noise integral AH~h,k _ f t n + l H 1 (t)dt th,k 9 Jtn

(5.27)

The magnetization direction at the midpoint of the time interval is

(u)li = (ul(tn) 4- uli(tn+l)/2 - ul(tn) 4- Au/2,

(5.28)

assuming linearization of u ;i within the time interval. The midpoint value, -1 tt i , is given implicitly by Eqns (5.26) and (5.28), which are solved by functional iteration. Numerical tests show that about five iterations are sufficient to gain the required accuracy of 10 -5 . From the midpoint value, the

408

CHAPTER 5 Computational Aspects of Micromagnetics

magnetization at the time tn+l is calculated from (5.28) and then normalized: (5.29) u~li = 2~li _ ul(tn) fil

u/ = ~.

(5.30)

5.5.1 EXAMPLE: SMALL MAGNETIC CO-PARTICLES

In this example the attempt frequency and energy barriers are calculated for Co-nanoparticles. Furthermore, a fit of the calculated relaxation time to the Arrhenius-N6el law is presented. The particle size is assumed to be so small (~ 3 nm) [33], that we can treat the particle as a single magnetic moment. In the experiments by Wernsdorfer [33] a field lower than the zero-temperature switching field was applied to a small Co-nanoparticle. The anisotropy energy used is given by

Eo(m)/V = -Klm2+K2m

2 _ K4 (mx,2 m y2 + mx,2 m z, 2 + my,2 mz,), 2

(5.31)

where K1 and K2 are the anisotropy constants along z and x, the easy and hard magnetization axes, respectively. K4 is the fourth-order anisotropy constant and the (xyz) coordinate system is deduced from (xyz) by a 45 ~ rotation around the z-axis. The crystalline anisotropy constants used are K1 = 220 k J / m 3, K2 - 90 k J / m 3 and K3 = 10 k J / m 3 and the saturation polarization Js is 1.76 T. The particle diameter is 3.18 nm. First, we perform a numerical telegraph noise experiment. At zero applied field we integrate the stochastic Landau-Lifshitz equation (5.20) and monitor the total magnetization. The magnetization switches subterraneously between the two local minima parallel to the easy axis. The relaxation time of the particle is given by 1 - 7- e x p ( E B / ( k B T ) ) .

(5.32)

J0 Here EB is the energy barrier separating the two local minima of (5.31) and )Co is the attempt frequency. The higher the temperature the smaller is the average time r until switching between the two energy states occurs. The energy barrier follows from plotting the relaxation time as function of the inverse temperature. Taking the logarithm of (5.32) gives 1 EB In 9 = l n ~ -} kBT -

~

(5.33)

5.5 LANGEVIN DYNAMICS

409

FIGURE 5.16 Relaxation time for the spontaneous rotation of a small Co-particle at nonzero temperature and zero applied field.

We simulate the thermally induced switching of the particles for different temperatures. Figure 5.17 shows the fit of the numerical data to the Arrhenius-N6el law (Eqn. (5.32)) for two values of the damping constant. From the fit we can derive the attempt frequency (f0 = 1.2 x 101~ Hz) and the energy barrier. The energy barrier agrees well with KIV as expected for uniform rotation. From Fig. 5.16 we can thus see that the damping constant e only affects the prefactor in (5.32) but not the energy barrier (slope of the line). Figure 5.17 presents the results of numerical waiting-time experiments for different applied fields. Now an external field is applied and the time until magnetization reversal occurs, the so-called waiting time, is measured. The external field is well below the switching field at zero temperature. It is varied in the range from p0 H = 0.186 T to/~0 H - 0.264 T. The Gilbert damping constant is c~= 0.1. Again the energy barriers are derived from a fit to the Arrhenius-N6el law (5.32). According to (5.33), the energy barrier is given by the slope of the lines in Fig. 5.17. The energy barrier as a function of the field (Fig. 5.17 on the right) follows the well-known relation [33]"

EB=K1V(1-Next) H0

2

'

(5.34)

where H0 is the intrinsic switching field at T = 0. Figure 5.18 gives a histogram of the waiting times at P(t), which is the probability that the particle has switched by a certain time. The rescaled

410

CHAPTER 5 Computational Aspects of Micromagnetics

FIGURE 5.17 Relaxation time for spontaneous rotation of a small Co-particle at nonzero temperature and zero applied field (on the left). The derived energy barrier is given as a function of the applied field (on the right).

FIGURE 5.18 Waiting-time histogram and probability of not switching.

probability of not switching Pnot = 1 - P(t), is fitted with a single exponential function. Finally, we calculate the coercivity as a function of the temperature. The external field is applied at 0 ~ with respect to the easy axis and at 45 ~ with respect to the easy axis. Again the numerical results are compared with experimental data and theoretical predictions [33]. The coercive field decays with T 1/2 and T 2/3 for an external field applied at 0 ~ and 45 ~ respectively (see Fig. 5.19). The calculated coercive field is independent of the Gilbert damping constant. The above results demonstrate that Langevin dynamics, together with proper numerical methods for stochastic differential equations, can be applied to treat temperature effects in micromagnetics.

5.6 ENERGY BARRIER: THE PATH METHOD

411

FIGURE 5.19 Numerically calculated coercive field as a function of temperature of the Co-nanoparticle. The external field is applied at 0~ and at 45 ~ with respect to the easy axis. The Gilbert damping constant is c~= 1 (top) and c~- 0.1 (bottom).

5.6 E N E R G Y B A R R I E R : T H E P A T H M E T H O D 5.6.1 RARE BUT IMPORTANT EVENTS The previous section s h o w e d the numerical m e t h o d for the simulation of thermal effects on short time scales. The thermal energy kBT is comparable with the energy barrier EB that separates two local energy minima, and thus direct simulations of the escape over the energy barrier using Langevin equation(s) are possible. Unfortunately, this is no longer the case w h e n kBT < < EB, which is the case for most of the interesting problems in magnetic recording applications. The m a i n difficulty in the c o m p u t a t i o n

412

CHAPTER 5 Computational Aspects of Micromagnetics

of such transition processes is caused by the disparity of the time scales. Let us consider a simple gradient system dx

dt

= VV(x).

(5.35)

V(x) is the potential energy of the system. Without the presence of noise, the system would simply move towards the closest energy minimum of the potential. In the presence of noise however, the system can evolve away from the minimum and the whole configuration space can be explored by this type of stochastic dynamics. The time to escape from the energy minimum grows exponentially with the barrier height, the escape time is inversely proportional to the probability to escape from a local minimum of a barrier with barrier height, Eb.

Pescape cx: exp

(-EB/(kBT))

(5.36)

Now, two different time scales are involved. For a very long period the system stays in the vicinity of the energy minimum. From time to time, the system can overcome the energy barrier and switch to another minimum. These events occur very seldom and are therefore called rare events. We are interested in calculating the probability of such rare events to Occur.

An example where such events are crucial is magnetic storage media. Here data bits are stored in small magnetic elements where the magnetization can only occupy two stable states (energy minima). It is desired that spontaneous switching events between these states occur as seldom as possible, since they lead to the loss of stored data. The desired time scale for the stability is in the range of years and decades. On the other hand, the duration of the switching event itself takes only a few nanoseconds. The dynamics proceeds by long waiting periods around the local minima followed by sudden jumps from one state to another. In the simulation of the dynamics of the system, most of the computation time is spent in resolving unimportant fluctuations around the minima. We would have to wait a for very long time to observe the interesting but rare switching events. This is what is meant by the disparity of time scales. Various methods have been developed to speed up the computation and to overcome this problem [34,35]. When kBT < < EB the task simplifies because it is sufficient to find the lowest saddle point between the two energy minima. The height of the saddle point will then give the relevant energy barrier. The problem of finding the saddle point cannot be solved analytically, except for simple situations. The reason is that at a saddle point neither a maximum nor a

5.6 ENERGY BARRIER: THE PATH METHOD

413

FIGURE 5.20 Thermal switching of the magnetization: fluctuations drive the magnetic system from the stable region A to another stable region B in the energy landscape. The path with the smallest energy barrier is chosen since transition frequency decreases exponentially with the height of the energy barrier. To determine the crossing point with the lowest energy the separating saddle point(s) between the two regions needs to be calculated. An example for two stable regions is the stable states of a magnetic grain in Co-Cr-Pt (Js = 0.5 T, A = 10 pJ/m, K1 = 300 kJ/m 3) perpendicular recording media. At a column length of 80 nm the saddle point is a two-domain state with a transverse domain wall in the center.

m i n i m u m should be achieved. The solution can only be found by calculating the gradient of the energy and inspecting the Hessian matrix that leads to nonlinear equations in the general case. We lack general methods for solving such systems. Instead we will use a path method that sources for highly probable paths between energy minima in high-dimensional energy landscapes. Various methods exist to calculate such paths. We use the so-called n u d g e d elastic-band method (NEB) [36]. Starting from an initial guess for the path that connects two local minima of the system, a highly probable path is found by moving the points along the path according to an algorithm which resembles tensioning an elastic band across a mountain. Using such a method we do not only obtain the saddle point(s) between energy minima, but also get a more global view of the energy landscape. An example for the path over the energy landscape is shown in Fig. 5.20. It gives the two energy minima and the saddle point of an elongated magnetic particle.

414

CHAPTER 5 Computational Aspects of Micromagnetics

5.6.2 THE N U D G E D ELASTIC-BAND METHOD

Transition events over large energy barriers (> 50kBT), as in magnetic recording media, occur on very long time scales (e.g. decades). As discussed in the previous section such events cannot be simulated directly by stochastic time integration methods. To simulate such events we directly search for the most probable path through the energy landscape for such an event. A common feature of path-finding methods is the discrete representation of the path connecting the initial state of the system with its final state. First, we construct a sequence of magnetic states from the initial magnetization state, M(i), to the final magnetization state, M(f). An optimization algorithm is then applied until at any point along the path the gradient of the energy is only pointing along that path. This path is called the minimum energy path, which means that the energy is stationary for any degree-of-freedom perpendicular to the path. The minimum energy path typically represents the path with the greatest statistical weight. From this path, statistical quantities, such as the transition rate for the thermally induced magnetization reversal, can be estimated. Various other methods exist that can calculate either the saddle points (transition state) or the entire minimum energy path for the thermally activated barrier crossing. Most of these methods come from chemical physics and condensed matter physics where thermally activated barrier crossing is of great interest for the calculation of reaction rates. A nice overview of methods for finding saddle points and minimum energy paths in highdimensional systems is given in [37]. Henkelman and J6nsson proposed the nudged elastic-band method to calculate minimum energy paths [36]. We represent a path by a sequence of images. An initial path is assumed that connects the initial magnetization state M(i) = M(1) with the final magnetization state M(f) = M(m) for k runs from I to m. The path is optimal if for any image M(k) the gradient of the energy is only pointing along the path or, in other words, the component of the energy gradient normal to the path, D, is zero. If t denotes the unit tangent vector along the path, a minimum energy path has the following property D(k) = VE(M(k)) - [VE(M(k)). t] t = 0,

for k = 1 . . . . . m.

(5.37)

The optimal path can be found using an iterative scheme. In each iteration step the images move towards lower energy in a direction perpendicular to the path. So the image M(k) is moved into the direction, D(k). Figure 5.21 shows the calculation of D(k). In order to keep an equal distance between successive images a spring force may be introduced [36].

5.6 ENERGY BARRIER: THE PATH METHOD

415

FIGURE 5.21 Schematic representation of the path, which shows how D(k) in Eqn. (5.37) is calculated. A number of images is chosen to discretize the path connecting the two states A and B.

A more detailed description of the method and the implementation in our simulation package is given in [9].

5.6.3 EXAMPLE 1: TWO COUPLED SINGLE-DOMAIN PARTICLES

In order to test our path-finding method we start with a simple test case. We study two single-domain particles with saturation polarization Js, uniaxial anisotropy (K1), and identical volumes (V). The line connecting the particles is parallel to the uniaxial anisotropy direction. The distance between the particles is R. In polar coordinates the magnetization of this two-particle system is described by four angles (01, ~1, 02, ~2)" Although there are four independent variables, only cases where the magnetization vectors of the two particles are in the same plane describe equilibrium states. It is therefore sufficient to study cases where only 01 and 02 vary. The total energy then simplifies to [38]: E/(KIV)

- sin 2 01 -ff sin 2 02 -}- kint [sin 01 sin 02 -- 2 cos 01 COS 02],

(5.38)

416

CHAPTER 5 Computational Aspects of Micromagnetics

with the reduced interaction constant

12v k i n t - 2#0R3K1 9

(5.39)

Thus, EB can be calculated analytically in this case. With this reduction to two dimensions the energy landscape with paths can be plotted, which helps us to verify our method and results. Figure 5.22 shows one possible case of the energy landscape where no external field is applied, with kint = 0.2. In the initial state the magnetization of both particles is parallel (upup). We then connect the initial state with the final state (down-down) with a straight line. This initial path describes the coherent rotation of the magnetization vectors where the two magnetic moments rotate from the initial state to the final state in opposite directions (asymmetric fanning ~01 = -~02). The energy landscape with the initial path and the minimum energy path are shown in Fig. 5.22. The initial path (dotted line) moves over the top of the hill. Applying the algorithm described in the previous section we obtain a minimum energy path as given by the solid line in Fig. 5.22. In the minimum energy path one particle starts rotating. The system passes the first saddle point and reaches a local minimum. The moments are antiparallel. Then the second particle rotates. The system crosses a second saddle point and ends in the final state where the magnetic moments are parallel. It becomes obvious from the energy landscape that there is a second possible minimum energy path, which can be reached from the same initial path. Both paths describe the same process. The particles start to switch in the opposite order in the two paths. Figure 5.23 shows the energy as a function of the arc length of the initial path and the minimum energy path. The path becomes longer and the maximum develops two saddle points and a local energy minimum. 5.6.4 EXAMPLE 2: ANISOTROPIC COLUMNAR GRAIN

In recent years there has been a renewed interest in perpendicular recording, since an improvement of the areal density in longitudinal recording is getting increasingly difficult. The main advantage of perpendicular recording is that the areal density can be increased without reducing the volume of the magnetic bit. A large volume of the magnetic domain can be realized with an increased column length lc (film thickness). For coherent rotation a higher grain volume leads to a higher energy barrier and hence to an improved thermal stability. Thus, twice the exchange length 2R0, which is assumed to be the limit between coherent rotation and inhomogeneous reversal processes, is the upper bound for the column length in perpendicular recording media [39]. For Co-Cr-Pt, a typical perpendicular recording

5.6 ENERGY BARRIER: THE PATH METHOD

417

FIGURE 5.22 Two-dimensional energy landscape representing two anisotropic magnetic moments with weak exchange coupling between them. The dashed line shows the initial guess, and the solid line the resulting minimum energy path obtained from the nudged elastic-band method. Energy minima, maxima and saddle points are indicated in the figure. The contour plot with equipotential lines is shown on the left.

FIGURE 5.23 Energy along the initial path and the minimum energy path for a system of two interacting magnetic moments.

material,

2#oA 2R0- v

Js2

~50nm.

(5.40)

418

CHAPTER 5 Computational Aspects of Micromagnetics

Equation (5.40) is a very rough estimate of the critical column length of the grains. Using the nudged elastic-band method, we can very rigorously calculate the energy barriers and the thermal reversal modes for the thermally induced switching event. In the calculation, we compute the minimum energy path for a uniformly elongated particle with an irregular pentagonal cross-section as a function of the columnar length. The grain geometry with the finite-element mesh is shown in Fig. 5.9. For exchange lengths that are not small compared with the irregularity, the results are independent of the detailed shape of the cross-section. Thus, we will normalize our results with respect to the mean diameter. Fluctuation effects in magnetic materials are often described in terms of magnetic viscosity and activation volumes [40,41]. The magnetic viscosity depends on the change of the energy barrier with the external field dE/dH. Traditionally, the activation volume is defined via the activation energy of a nucleus of reverse magnetization [42], VactJsHf/l~o = kBT. Usually, Vact is referred to as the activation volume and Hf as the fluctuation field. The so-defined activation volume has no direct physical counterpart. The activation volume can be determined experimentally from viscosity measurements or from the sweep rate dependence of the coercive field [41]. For typical elongated particles, as utilized in magnetic-tape recording, this volume is often smaller than the actual particle volume. Such a reduced effective volume is attributed typically to nonuniform reversal processes. For a uniformly magnetized elongated particle the energy barrier for fields less than the zero-temperature switching field is

EB - KeffVeff ( 1 - Hext ) n

(5.41)

This equation is similar to Eqn. (5.34) that holds for small particles where the magnetization reversal mode is uniform rotation. For larger particles, K1 is replaced by an effective anisotropy, and the particle volume is replaced by an effective volume, n = 2 for uniform rotation and an external field parallel to the particle's easy axis. For zero applied field we have E0 = KeffVeff.

(5.42)

A knowledge of the energy barrier E0 and the switching field H0 gives the volume Veff. For a uniformly magnetized particle that reverses by uniform rotation, Veff is equal to the actual particle volume. For nonuniform reversing, Veff is smaller than the total particle volume and n can be different from two. We use finite-element micromagnetics to determine E0 and

H0.

5.6 ENERGYBARRIER: THE PATH METHOD

419

FIGURE 5.24 Left: magnetization states along the initial path and the minimum energy path for an elongated Co-Cr-Pt particle with the easy axis parallel to the long axis. Gray: magnetization up; black: magnetization down. Right: energy along the initial path and the minimum energy path. In the minimum energy path the magnetization reverses by nucleation followed by domain-wall motion.

Using the nudged elastic-band method and the finite-element method to discretize the micromagnetic energy of the columnar grain we first calculate E0, the energy barrier for zero applied field. An example for a computed minimum energy path (MEP) at zero applied external field is shown in Fig. 5.24. The two stable magnetization states are given by the magnetization uniformly oriented along the easy axis of the particle. For the initial guess of the reversal path we simply assume the uniform rotation of the magnetization from one of the states to the other (Fig. 5.24). After applying the nudged elastic-band method we obtain the MEP as shown in the bottom sequence in Fig. 5.24. A nucleation of reversed magnetization forms followed by the motion of a transverse domain wall through the particle. The saddle point, which gives the height of the energy barrier, is a state where the particle has a domain wall exactly in the middle of the column. Owing to the high magnetocrystalline anisotropy, the height of the energy barrier is mainly determined by the domain-wall energy of the two domain states. In the following, analytical estimation we neglect the contribution from the magnetostatic interaction. In the following, F denotes the area of the basal plane of the elongated grain. The domain-wall energy is Ewall - 4F Av'-A-~, while the energy for coherent rotation is Erot - KIV.

420

CHAPTER 5 Computational Aspects of Micromagnetics

Now if the particle length is increased, only V increases while the crosssection, F, remains constant. Thus, there will be a critical length where Ewan < Erot and the nucleation and expansion of a domain wall becomes energetically favorable. The critical length, lcrit becomes Ewall(lcrit) - Erot(/crit)

~

A /crit = 4FFK1 A~/-X~ = 4 ~ Kl1"

(5.43)

Note that the result is independent of the cross-section, F, (or the diameter), and is equal to four times the Bloch parameter vIA~K1 . If the column length is smaller than 4vIA~K1 we expect uniform rotation. WithA = 10 p J / m and K1 = 300 k J / m 3, the critical length becomes ~ 23 nm. This agrees very well with the numerical results obtained from the nudged elastic-band method. Figure 5.25 shows the maximum exchange energy and the effective volume for the thermal reversal of an elongated Co-Cr-Pt particle as a function of the column length. The diameter is 13 nm. Above a size of 20 nm the reversal mode becomes inhomogeneous. Below 20 nm the maximum exchange energy is almost zero. Here the particle reverses by coherent rotation. Above 40 nm the particle is large enough to form a full domain wall. Increasing the length further does not change the maximum exchange energy any more. Once the wall is formed it only moves through the particle. The same is true for the energy barrier, here expressed as an effective switching volume. Above 40 nm the effective volume Veff saturates. It is therefore clear that an further increase of the column length of the grains does not increase the thermal stability in perpendicular recording material. The exchange energy curve in Fig. 5.26 further shows that the reversal mode continuously changes from homogeneous rotation to a domain-wall nucleation. Equation (5.43) predicts a sharp transformation between the modes. In the numerical calculation the saddle-point state for small particle size has the magnetization homogeneous and perpendicular to the easy axis. With increasing size the magnetization state becomes more and more inhomogeneous with a flower-like magnetization state. Finally, above 40 nm a full domain wall can be formed. Above a length approximately equal to the exchange length, the energy barrier / effective volume is constant, being independent of the column length and thus from the total volume. Below the exchange length the energy barrier decreases linearly. The ratio of the particle volume over the effective volume is a measure of the degree of uniformity of the magnetization during thermal switching of the particle. This ratio is 0.3 for the longest column (70 nm). With decreasing particle length the ratio increases and

421

5.7 APPLICATIONS

- 7500 0.5 0.4

Exchange energy of the Saddle point

5000

0.3 .-41-

. . . .

II . . . .

II . . . .

41-----I1

ga 0.2

2500 iI /

O.1

.

-

---20s I

%

I I~4IV

f 40

60

,,=,

-~

columnlength(rim)

I Eex = 0 ~

uniform rotation for l < 20 n m

FIGURE 5.25 Maximum exchange energy and effective volume for the thermal reversal of an elongated Co-Cr-Pt particle as a function of the column length. The diameter is 13 nm. Above a size of 20 nm the reversal mode becomes continuously more and more inhomogeneous. Above ~ 50 nm column length, the size is large enough to form a full domain wall. The maximum exchange energy saturates as a function of the length and has the value of a domain-wall exchange energy.

becomes unity for lengths less than the exchange length. Uniform rotation occurs and the effective volume equals the particle volume.

5.7 A P P L I C A T I O N S In the following, the above theory is applied to simulate the dynamic properties of hard-disk media and the thermal stability of magnetic r a n d o m access memories. 5.7.1 FAST SWITCHING OF TILTED MEDIA The advantages of tilted media in perpendicular recording have been recognized for more than ten years for metal-evaporated tape recording [43]. However, just recently tilted magnetic recording has been proposed for disk recording [44]. Using magnetic media that consist of grains with an

422

CHAPTER 5 Computational Aspects of Micromagnetics

FIGURE 5.26 Saddle-point states and the corresponding energy barrier as a function of the column length. For small particle size the energy barrier increases almost linearly with the length. Above a certain length (~ 40 nm) saturation is reached: here the energy barrier is determined by the energy to form a domain wall. Increasing the length further does not increase the energy barrier further since the domain-wall energy only depends on the cross-section area. At small sizes (< 20 nm) the magnetization reverses by uniform rotation. Between 20-40 nm column length there is a continuous transformation towards the domain wall.

angle of the easy axes and the film plane of optimally 45 ~ yields significant advantages over perpendicular recording. First, the coercive field is smaller by a factor two compared with perpendicular media as shown by the Stoner-Wohlfarth theory [45]. As a consequence, high anisotropy magnetic materials can be utilized, leading to a better thermal stability or a higher aerial density. Second, higher data rates can be realized owing to the high torque that acts on the magnetization. Third, a high signal-tonoise ratio (SNR) can be achieved, since grains with slightly different easy axes have the same switching field in first order. The high sensitivity of the switching field (and hence also of the SNR) on the easy axes' distribution is one of the main physical concerns in perpendicular recording. He and co-workers [7] showed that the Stoner-Wohlfarth theory and dynamic

5.7 APPLICATIONS

423

FIGURE 5.27 Geometry used for the calculation of the switching field with the finite-element model. The left model represents a perpendicular recording media, in the right model the grains are tilted by 45~ with respect to the film normal.

magnetization models based on the Landau-Lifshitz equation (5.11) predict different switching fields for small damping constants and small rise times of the external field. In particular, they showed that switching of single-domain particles is possible for fields below the Stoner-Wohlfarth limit. Thus, using fast field pulses for the writing process, materials with high magnetic anisotropy can be utilized. Interestingly, for external fields of the order of the Stoner-Wohlfarth switching field, the switching time decreases with smaller magnitude of the external field pulse [46,47]. As a consequence, the writing process can be accelerated by decreasing the magnitude of the external field. In Section 5.3.4 we studied fast-switching effects of one single anisotropic magnetic spin and one grain in a perpendicular recording media. In this work we investigate the possibility of fast-switching effects in a more realistic model of a recording layer, which includes both subdiscretization of the grains and intergranular exchange coupling between them. We compare perpendicular recording media with tilted recording media. Finite-element calculations for weakly exchange-coupled and magnetostatically coupled granular media are presented. Both perpendicular and tilted recording media with a weak exchange coupling between the grains are investigated. Assuming an intergranular phase with a thickness of 1 nm and an exchange stiffness of A - 3 x 10 -14 corresponds to a coupling strength of J - 0.3 m J / m 2. The ratio Eexchange/Ean~sotropyof the maximum possible interface coupling energy of one grain ( E e x c h a n g e - 40 nm 2 x 0.3 m J / m 2) and the maximum possible anisotropy energy of one grain (Eanisotropy - 1000 nm 3 x 350 k J / m 3) gives a value of 0.04. This parameter set therefore corresponds to weakly coupled grains. The material parameters within the grains are: A - 10 pJ/m, Ku - 350 k J / m 3, Js - 0.5 T. The geometry of the tilted media and the perpendicular media are shown in Fig. 5.27.

424

CHAPTER 5 Computational Aspects of Micromagnetics

The thickness of the perpendicular media and the tilted media is 12 nm and 8.5 nm, respectively. The average grain diameter is 10 nm for both media. More than 45 000 finite elements are used to provide a finite-element edge length smaller than the exchange length. A normal distribution of the easy axes' directions with a maximum opening angle of 5 ~ is assumed for both media. For the perpendicular media the average easy axis direction is parallel to the film normal. For the tilted media the angle between the average easy axis direction and the film normal is 45 ~. Thus, for both samples the average easy axis direction points parallel to the edges between the grains. The external field is applied 45 ~ off the film normal and parallel to the film normal for the perpendicular media and the tilted media, respectively. The field rise time is 10 ps. The results presented in Fig. 5.28 show that switching is possible for fields smaller than the Stoner-Wohlfarth switching field, which is one half of the anisotropy field

Ku ls

HSW = 0.5Hani = ~ .

(5.44)

Figure 5.28 compares the switching time as a function of the external field strength, for the perpendicular media and the tilted media. Owing to the difference in the demagnetization fields for the two media, the switching times of the two investigated samples significantly differ. It is interesting to note that for small field values the switching time decreases with smaller magnitude of the external field. A similar dependence of the switching time with the field strength has been observed for single-domain particles [46,47]. The finite-element simulations also confirm that the switching time and the switching fields are insensitive to small variations of the easy axes' distribution for the tilted media. A change of the opening angle of the easy axes' distribution from 5 ~ to 1~ changes the switching time by less than 0.1 per cent. Small switching field distributions lead to a high signal-to-noise ratio.

5.8 ENERGY BARRIERS IN M A G N E T I C R A N D O M ACCESS M E M O RY ELEMENTS The energy barrier between stable configurations determines the probability of a thermal switching event. In magnetic random access memory (MRAM) elements the energy barrier is determined by the shape of the element (shape anisotropy) and the induced crystalline anisotropy. This should guarantee a lifetime of a stored bit of about 10 years. We calculate

5.8 ENERGY BARRIERS IN MAGNETIC RANDOM ACCESS MEMORY 0.2

'

I

'

I

'

I

'

I

'

I

'

I

'

I

425

'

]U] tiltedmedia ] 0 perpendicular media 0.15

O~E]

O0 ._ 0.1

-o / ~

9,6

o 0.05

7 ~ [] I

Ooi4 ' 0.5

,

I

0.6

,

I

0.7

,

I

~

I

0.8 0.9 Hext (2 Kl/Js)

~

I

1

i

I 1.1

FIGURE 5.28 Comparison of the switching time for perpendicular recording media and tilted media. For the simulation of the perpendicular media the angle between the external field and the film normal is 45 ~. The external field is applied parallel to the film normal for the tilted media at an angle of 45 ~ with respect to the film plane for the perpendicular media. The external field is given in units of the anisotropy field.

energy barriers for elliptical MRAM elements using the n u d g e d elasticb a n d m e t h o d (NEB) [9]. This m e t h o d finds m i n i m u m energy paths in high-dimensional energy landscapes and is a rigorous w a y to compute the saddle point(s) between local energy minima. Previous studies [48] on thermal effects in MRAMs are based on Langevin dynamics. A field near the switching field m u s t be applied in order to decrease the energy barrier and to observe thermal switching of the magnetization. Using the n u d g e d elastic-band method, we can also calculate energy barriers for zero applied external fields. 5.8.1 M O D E L OF THE FREE DATA LAYER

Our models of the free layer of the M R A M cell are based on experimental w o r k [49], which studies the thermal reversal of elliptical Ni-Fe-Co M R A M elements. We calculate m i n i m u m energy paths (MEP) as a function of the external in-plane field, which is smaller than the zero-temperature switching field. Material parameters of Ni-Fe-Co [50] are used (Js = 1.068 T, A - 10 p J / m , Ku - 510 J / m 3) with the anisotropy axis, u, parallel to the long axis of the element. We vary the size of the elements, starting

426

CHAPTER 5 Computational Aspects of Micromagnetics

PATH 2

tO

3=

FIGURE 5.29 Three minimum-energy paths for the thermal reversal of a 4 nm thin Ni-Fe-Co MRAM element are found between the two stable magnetization states. An opposing field is applied along the easy axis (= long axis of the cell). In paths I and 2 the magnetization stays in-plane, crossing a single barrier. In path 3 a two-step reversal mode is found passing a metastable state (vortex in the center). For an applied field in the range 0-1.2 k A / m path 3 has the lowest barrier. From 1.2 kA/m-6.4 k A / m path I has the lowest barrier and above 6.4 k A / m path 2 has the lowest barrier. The zero-temperature switching field in a quasistatic calculation is 7.5 kA/m.

with an element of 1 120 x 400 n m 2 size and decreasing the size d o w n to 62.5 x 25 n m 2. Aspect ratio (2:5) and thickness (4+ nm) are kept constant. Figure 5.29 shows a top view of the model with three possible paths for a 1 120-nm element. In these calculations the external field is applied parallel to the easy axis (easy-axis field).

5.8.2 REVERSAL MODES AT 1.12/~m SIZE By varying the field strength along the easy axis, three m i n i m u m - e n e r g y paths are found (Fig. 5.29) for a particle size of 1.12 #m. Two of them show a single energy barrier where the magnetization reverses by inhomogeneous rotation in the film plane (path I and path 2 in Fig. 5.29). The third m o d e is a two-step reversal, which has a metastable state where a vortex is in the center of the element (path 3 in Fig. 5.29). A dynamic hysteresis

5.8 ENERGY BARRIERS IN MAGNETIC RANDOM ACCESS MEMORY

427

-~'u ,1.~" path 1 800

path 2 path 3

~, 600

9-r 400

200

0

3.2 4.8 easy axis field (kA/m)

1.6

6.4

FIGURE 5.30 Energy barrier as a function of the easy-axis field strength for 1.12 /~m particle size (see Fig. 5.29). The underlying thermal reversal mode (paths) depend on the strength of the external field. In paths 1 and 2 the magnetization stays in-plane, crossing a single barrier. In path 3, a two-step reversal mode is found passing a metastable state (vortex in the center).

calculation (integration of the Landau-Lifshitz-Gilbert equation using a Gilbert d a m p i n g constant of a = 1) gives a zero-temperature switching field of 7.5 k A / m . The underlying reversal m o d e during the switching driven by an external easy-axis field larger than 7.5 k A / m is similar to the m o d e of path 2 in Fig. 5.29. This is in agreement with the following observation: at a field strength of Happ~ 1.2 k A / m the m o d e of path 3 has the lowest barrier while for Happ~
Figure 5.31(A) and (B) show the size dependence of the energy barrier w h e n an easy-axis field is applied. The extrapolation to the zero barrier height

CHAPTER 5 Computational Aspects of Micromagnetics

428

400

1120 nm 125 nm

800

62.5 nm .~

300 g

600--

_g

,.,,i

_

200 "~ 400

,.O

(D

=

100

O

0t 0

~

0 (A)

12

24

36

easy axis field (kA/m)

48 (B)

'

1 , I , I 150 300 450 area1/2 (rim)

600

FIGURE 5.31 (A) Energy barrier as a function of the easy-axis field strength for different particle sizes. The extrapolation to zero barrier height gives the switching field. (B) Energy barrier at zero external field as a function of the particle size (given as the square root of the surface area). Thickness (4 nm) and aspect ratio (2:5) are fixed as in Fig. 5.29. The energy barrier depends almost linearly on the square root of the area of the MRAM element. (Fig. 5.31(A)) gives a switching field that agrees with the value obtained by the solution of the LLG equation for slowly varying applied fields. Figure 5.32 shows the switching astroid. It gives the switching field for different orientations of the applied field. The applied field direction is given by its components parallel to the long axis of the ellipse (easy axis) and parallel to the short axis of the ellipse (hard-axis). At zero external field the energy barrier depends almost linearly on the square root of the area of the top surface of the element (Fig. 5.31(B)). 5.8.4 STABILITY OF HALF-SELECTED BITS Figure 5.32 shows that the lowest switching fields are obtained w h e n applying the field in-plane at about 45 ~ to the easy axis. At 1.12 #m size of the long axis, this corresponds to a hard-axis and easy-axis field of 3.4 k A / m (both a 3.4 k A / m easy-axis field and a 3.4 k A / m hard-axis field are applied to obtain the 45 ~ in-plane field). To ensure thermal stability, the magnetization must stay stable w h e n applying either a hard- of or an easy-axis field of this strength. At an easy-axis field of 3.4 k A / m , the energy barrier

REFERENCES

429

56 62.5 nm 125 nm 1120 nm

48 ,-, 4O < 32 .2 24 t~ r...,

16

0

8

16 24 32 40 48 easy axis field (kA/m)

56

FIGURE 5.32 Switching astroid obtained by the solution of the LLG equation for three different sizes of a 4 nm thin elliptical MRAM element of aspect ratio 2:5. Each point in-plane represents an applied field with a field component along the easy axis (long axis of the ellipse) and a field component along the hard-axis (short axis of the ellipse). If the point representing the applied field lies inside the astroid no switching occurs. Switching occurs if the point representing the applied field lies on or outside the astroid.

is 75 kB • 300 K (Fig. 5.31), which is high e n o u g h to ensure stability in the range of decades. For hard-axis fields, the situation is found to be less critical. The lowest switching field (Fig. 5.32) is obtained with a superposition of a 11.7 k A / m hard-axis field and an 11.7 k A / m easy-axis field. The energy barrier is about 140 kB x 300 K for a hard-axis field of 11.7 k A / m and 80 kB x 300 K for an easy-axis field of 11.7 k A / m and for a particle size of 125 nm. Also, note that the thermal stability is more critical w h e n the field is applied opposite to the easy axis than w h e n applied perpendicular to it.

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CHAPTER 5 Computational Aspects of Micromagnetics

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18. S. Kurz, O. Rain and S. Rjasanow, 'The adaptive cross-approximation technique for the 3-d boundary-element method', IEEE Trans. Magn. 38, 421-424 (2002). 19. L. Grasedyck and W. Hackbusch, 'Construction and arithmetics of hmatrices', Computing 70(4), 295-334 (2003). 20. H. Forster, T. Schrefl, R. Dittrich, W. Scholz, and J. Fidler, 'Fast boundary methods for magnetostatic interactions in micromagnetics', IEEE Trans. Magn. 39(5), 2513-2515 (2003). 21. T. L. Gilbert, 'A Lagrangian formulation of gyromagnetic equation of the magnetization field', Phys. Rev. 100, 1243 (1955). 22. N. Inaba, Y. Uesaka, A. Nakamura, M. Futamoto, Y. Sugita, and S. Narishige, 'Damping constants of Co-Cr-Ta and Co-Cr-Pt thin films', IEEE Trans. Magn. 33(5 Part 1), 2989-2991 (1997). 23. S.D. Cohen andA. C. Hindmarsh, 'CVODE, a stiff/nonstiffODE solver in C', Computers in Physics 10(2), 138-143 (1996). 24. B. Yang and D. R. Fredkin, 'Dynamical micromagnetics by the finite element method', IEEE Trans. Magn. 34(6), 3842-3852 (1998). 25. D. Suess, V. Tsiantos, T. Schrefl, et al., 'Time resolved micromagnetics using a preconditioned finite element method', J. Magn. Magn. Mater. 248, 298-311 (2002). 26. M. E. Schabes, 'Micromagnetic theory of non-uniform magnetization processes in magnetic recording particles', J. Magn. Magn. Mater. 95, 249-288 (1991). 27. R. Hertel and H. Kronmuller, 'Micromagnetic simulation of the domain structure of a flat rectangular permalloy prism', J. Appl. Phys. 85(8 Part 2B), 6190-6192 (1999). 28. D. Weller and A. Moser, 'Thermal effect limits in ultra high-density magnetic recording', IEEE Trans. Magn. 35, 4423-4439 (1999). 29. J.C. Mallinson, 'Damped gyromagnetic switching', IEEE Trans. Magn. 36, 1976-1981 (2000). 30. M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Phys. Rev. B 61, 3410-3416 (2000). 31. J. L. Garcia-Palacios, F. J. Lazaro, 'Langevin-dynamics study of the dynamical properties of small magnetic particles', Phys. Rev. B. 58, 14937-14958 (1998). 32. P. D. Drummond and I. K. Mortimer, 'Computer simulations of multiplicative stochastic differential equations', J. Comput. Phys. 93, 144-170 (1991).

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33. W. Wernsdorfer, 'Classical and quantum magnetization reversal studied in nanometer-sized particles and clusters', in Advances in Chemical Physics, Volume 118, I. Prigogine, S.A. Rice (Eds), John Wiley and Sons, New York (2001). 34. A. F. Voter, F. Montalenti, and T. C. Germann, 'Extending the time scale in atomistic simulation of materials', Annu. Rev. Mater. Res. 32, 321-346 (2002). 35. J. Xue and R. H. Victora, 'Micromagnetic predictions for thermally assisted reversal over long time scales', J. Appl. Phys. Lett. 77, 34323434 (2000). 36. G. Henkelman and H. J6nsson, 'Improved tangent estimate in the NEB method for finding minimum energy paths', J. Chem. Phys. 113, 99789985 (2000). 37. G. Henkelman, G. J6hannesson, and H. J6nsson, 'Methods for finding saddle points and minimum energy paths', in Progress on Theoretical Chemistry and Physics, S. D. Schwartz Ed., Kluwer Academic Publishers, Boston, 269-300 (2000). 38. W. Chen, S. Zhang, and H. N. Bertram, 'Energy barriers for thermal reversal of interacting single domain particles', J. Appl. Phys. 71(11), 5579-5584 (1992). 39. H. J. Richter, 'Recent advances in the recording physics of thin-film media', J. Phys. D: Appl. Phys. 32, 147-168 (1999). 40. R. Street and J. C. Woolley, 'A study of magnetic viscosity,' Proc. Phys. Soc. A 62, 562-572 (1949). 41. R. Street and S. D. Brown, 'Magnetic viscosity, fluctuations, field, and activation energies (invited)', J. Appl. Phys. 76(10), 6386-6390 (1994). 42. A. Lyberatos and R. W. Chantrell, 'Thermally activated magnetization reversal in the micromagnetic approach', J. Phys. D: Appl. Phys. 29(9), 2332-2342 (1996). 43. H.J. Richter, 'An approach to recording on tilted media', IEEE Trans. Magn. 29, 2258-2265 (1993). 44. Y. Y. Zou, J. P. Wang, C. H. Hee, and T. C. Chong, 'Tilted media in a perpendicular recording system for high areal density recording', Appl. Phys. Lett. 82(50), 2473-2475 (2003). 45. E. C. Stoner and E. P. Wohlfarth, 'A mechanism of magnetic hystersis in heterogeneous alloys', Phil. Trans. R. Soc. 240, 599-642 (1948). 46. D. Suess, T. Schrefl, W. Scholz, and J. Fidler, 'Fast switching of small magnetic particles', J. Magn. Magn. Mater. 242, 426-429 (2002).

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47. K. Z. Gao, E. Boerner, and N. H. Bertram, 'Fast switching in a singledomain particle under sub-Stoner-Wohlfarth switching fields', Appl. Phys. Lett. 81(21), 4008-4010 (2002). 48. R.H. Koch, G. Grinstein, G. A. Keefe, Y. Lu, P. L. Trouilloud, and W. J. Gallagher, 'Thermally assisted magnetization reversal in submicronsized magnetic thin films', Phys. Rev. Lett. 84, 5419-5422 (2000). 49. N. D. Rizzo, M. DeHerrera, J. Janesky, B. Engel, J. Slaughter, and S. Tehrani, 'Thermally activated magnetization reversal in submicron magnetic tunnel junctions for magnetoresistive random access memory', Appl. Phys. Lett. 80, 2335-2337 (2002). 50. S. Ingvarsson, G. Xiao, S. S. P. Parkin, and W. J. Gallagher, 'Thickness-dependent magnetic properties of Ni81Fe 19, Co90Fe 10 and Ni65Fe15Co20 thin films', J. Magn. Magn. Mater. 251, 202-206 (2002).

CHAPTER 6

Nonlinear Magnetization Dynamics. Switching and Relaxation Phenomena Giorgio Bertotti, Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada delle Cacce 91, 10135 Torino, Italy Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA Claudio Serpico, Dipartimento di Ingegneria Elettrica, Universita degli Studi di Napofi 'Federico I1" via Claudio 21, 80125 Napoli, Italy

Contents 6.1 6.2 6.3 6.4 6.5

Introduction Landau-Lifshitz-Gilbert (LLG) Equation Geometrical and Analytical Techniques for LLG Equation in Uniformly Magnetized Bodies Magnetization Switching and Relaxation Spin-Torque-Driven Magnetization Dynamics Acknowledgements Appendix A. LLG Equation in Different Systems of Coordinates Appendix B. Poincar6-Melnikov Method Appendix C. Classification of Bifurcations References

435 438 462 488 524 549 549 552 557 561

6.1 I N T R O D U C T I O N The theoretical study of spin dynamics governed by the Landau-Lifshitz equation has been the focus of considerable research for many years [1-7]. Traditionally, this study has been driven by the ferromagnetic resonance problems [8-11]. In those problems, the main part of magnetization is pinned by a strong constant magnetic field, while only a small transverse component of magnetization executes resonance motions caused by radio-frequency fields. These small magnetization motions are studied by The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

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CHAPTER 6 Nonlinear Magnetization Dynamics

linearizing the Landau-Lifshitz equation and, for this reason, the literature on magnetization dynamics is mostly concerned with the linearized Landau-Lifshitz equation. Recently, new directions of research have emerged that deal with large motions of magnetization and that require the analysis of the nonlinear Landau-Lifshitz equation. These new areas of research are the fast precessional switching [12-15] of magnetization in thin films and magnetization dynamics induced by spin-polarized current injection [16-21]. The research on fast precessional magnetization switching has been motivated by promising technological applications in the area of magnetic data storage as well as by the pure scientific quest for ultra-fast magnetization reversals. Usually, magnetization reversals are produced by applying magnetic fields oriented almost antiparallel with respect to initial magnetization directions. These reversals are driven by relaxation (damping) processes and, for this reason, they can be termed as 'damping' switchings [22]. They are realized through numerous precessional oscillations and, as a result, they are relatively slow. A new mode of magnetization switching has recently emerged. This mode exploits fast precessional magnetization dynamics and, for this reason, it is coined as 'precessional' switching. Precessional switching phenomena have been mostly studied experimentally and through numerical solution of the Landau-Lifshitz equation. It has been found that this switching is very sensitive to shapes and durations of magnetic field pulses, and the seemingly stochastic nature of precessional switching has been observed for some ranges of pulse durations [23]. No comprehensive analytical and qualitative study of precessional switching has been performed and the very notion of precessional switching has not been defined in precise terms based on phase portraits of nonlinear magnetization dynamics. Magnetization dynamics induced by spin-polarized current injection is of considerable theoretical and technological interest as well. It has been experimentally observed that spin-polarized current injection may result in switchings or microwave oscillations of magnetization [20,21]. Extensive efforts have been devoted to the explanation of these experimental results in view of the new physics invoived and the promising applications to new types of current-controlled memory cells [24] and microwave sources [17,20]. However, the precise nature of magnetization dynamics when spin-polarized currents and external magnetic fields are simultaneously present has not yet been fully grasped. The reason is that spin-transfer driven magnetization dynamics is an essentially nonlinear phenomenon of considerable complexity and it must be treated as such. This suggests that the appropriate framework for the study of magnetization dynamics induced by spin-polarized current injection is the nonlinear dynamical

6.1 INTRODUCTION

437

system theory [25,26] and, particularly, its branch known as the bifurcation theory [27]. The intent of this chapter is to systematically apply the methods and concepts of nonlinear dynamical system theory to the analysis of large magnetization motions governed by the Landau-Lifshitz equation. Precessional switchings of magnetization and magnetization dynamics driven by spin-polarized current injection are treated as very important applications. However, many results obtained for those applications are of considerable interest in their own right. Accordingly, the chapter is organized as follows. Sections 6.2 and 6.3 deal with the theoretical aspects of Landau-Lifshitz dynamics. The phenomenological derivation of the Landau-Lifshitz equation as a dynamic constitutive relation consistent with micromagnetics constraints is provided. Then, Landau-Lifshitz dynamics is treated as a nonlinear dynamical system defined on a sphere. The phase portraits of this dynamical system, its noncanonical Hamiltonian structure, and the nature of dissipation are discussed. A novel numerical integration technique that preserves the magnitude of magnetization is presented. Special attention is paid to the conservative Landau-Lifshitz dynamics and detailed analytical expressions are derived for this dynamics in terms of elliptic-type functions and elliptic-type integrals. These expressions are very instrumental in the analysis of precessional switching and spin-polarized current-induced dynamics of magnetization. Sections 6.4 and 6.5 deal with precessional switchings of magnetization and spin-polarized current-induced phenomena. Precessional switchings of magnetization for longitudinal and perpendicular media are discussed for rectangular field pulses, and expressions for critical fields and pulse durations are derived. It is demonstrated that the seemingly stochastic nature of precessional switching in longitudinal media is related to the entanglement of separatrices in the high-energy region of the phase portrait and can be completely avoided by the appropriate choice of pulse durations. An 'inverse-problem' approach is developed for precessional switchings of magnetization driven by nonrectangular magnetic-field pulses. This inverse-problem approach is purely algebraic in nature; it is applicable to longitudinal and perpendicular media and it leads to expressions for field pulses that guarantee desired switchings. A novel approach to the analysis of magnetization relaxation to equilibrium is presented. This approach is based on the averaging technique and results in a differential equation for the free energy. The significance of this approach is that the free energy is a 'slow' variable whose dynamics is not obscured by precessional oscillations and, for this reason, reveals the actual rate of relaxation to equilibrium. Finally, the magnetization dynamics driven by spin-polarized current injection is discussed. It is demonstrated that this dynamics can be treated

438

CHAPTER 6 Nonlinear Magnetization Dynamics

as a perturbation of the conservative dynamics caused by intrinsic damping and spin transfer. By using this central idea as well as the bifurcation theory for nonlinear dynamical systems, complete stability diagrams are obtained for the case when injected spin-polarized currents and external magnetic fields are simultaneously present. Quantitative predictions are derived for the critical values of switching currents and magnetic fields, for amplitudes and frequencies of magnetization self-oscillations as well as for the conditions for hysteretic transitions between self-oscillations and metastable states of magnetization.

6.2 L A N D A U - L I F S H I T Z - G I L B E R T

(LLG) E Q U A T I O N

6.2.1 LANDAU-LIFSHITZ EQUATION AND ITS CONNECTION WITH MICROMAGNETICS

Micromagnetics Most studies of nonlinear magnetization dynamics in ferromagnetic media are based on the dynamic generalization of micromagnetics [28-32]. For this reason, it is appropriate to start this chapter with a brief discussion of the micromagnetic description of ferromagnetic bodies subject to classical electromagnetic fields. Micromagnetics is a continuum theory, which is highly nonlinear in nature, and effects on rather different spatial scales are involved, namely, short-range exchange forces and long-range magnetostatic effects. The state of the ferromagnetic body is described by the differentiable vector field M(r) representing the local magnetization at every point inside the body. For ferromagnets well below their Curie point, at the smallest spatial scale compatible with the continuum hypothesis, the strong exchange interaction prevails over all other types of force. This fact is taken into account by imposing the following fundamental constraint: [M(r)[ = Ms(T),

(6.1)

which means that the magnitude of the local magnetization vector at each point inside the body is equal to the saturation magnetization Ms (T) at the given temperature T. The direction of M(r) is in general nonuniform and, at equilibrium, is determined by minimizing an appropriate Gibbs-Landau free-energy functional GL(M(.); Ha, T) dependent on the applied magnetic field Ha, and the temperature T. In the sequel we will omit the dependence of GL and Ms on T, since the temperature will always be assumed to be uniform in space and constant in time.

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

439

The minimization of the free-energy functional under the constraint given by Eqn. (6.1), leads to the so-called Brown's equation: M x Heff -- 0,

(6.2)

where the effective field Hef f is defined as (minus) the variational derivative of the free-energy functional with respect to the magnetization vector field. Equation (6.2) expresses the fact that the local torque exerted on the magnetization by the effective field must be zero at equilibrium [28,29]. It is important to underline that the component of Heff along M plays no role, as a direct consequence of the constraint expressed by Eqn. (6.1). The effective field Heff takes into account all the information about the geometry and the magnetic properties of the system. When magnetoelastic effects are neglected, it can be written as: Heff - H a ( t ) + H M if- HAN if- HEX,

(6.3)

where Ha(t) is the applied field, FlEx is the exchange field, HAN is the crystal anisotropy field and H M is the magnetostatic field produced by the magnetization vector field M(r). The first two terms in Eqn. (6.3) represent the Maxwellian part of the effective field: Ha(t) is produced by external sources whereas H M is determined by solving the magnetostatic Maxwell equations: curl n M ~- 0, div H M -- - d i v M, (6.4) subject to the appropriate interface conditions at the body surface. The term nAN is a purely local term that depends on the magnetization vector at the point considered and on some structural parameter characterizing the local anisotropy. Finally, the exchange field is given by FlEX -- (2A/#0Ms2)V2M, where A is the exchange stiffness constant (#0 is the vacuum permeability) [31], and the Laplacian term phenomenologically describes exchange forces that tend to oppose magnetization nonuniformities. In addition to Maxwellian interface conditions, another boundary condition comes from the minimization of the micromagnetic free energy at the surface of the body. In its simplest form, this condition consists of the requirement of zero normal derivative of M at the body surface: 8M -- 0. 8n

(6.5)

This condition is expected when no surface anisotropy is present. Surface anisotropy gives rise to pinning of the magnetization at the body surface, a fact that can substantially alter the response of the body to time-dependent

440

CHAPTER 6 Nonlinear Magnetization Dynamics

external fields. In particular, nonuniform magnetization modes may appear even under spatially uniform driving fields. These effects will not be considered in this chapter. They should be treated by appropriate generalizations of the methods presented in the following sections. The fact that magnetostatic equations, Eqns (6.4), are used implies that we are in fact neglecting any propagation effect, which means that the electromagnetic wavelength must be much larger than the linear dimensions of the body under study. Finally, it is worth remarking that no eddy current contribution has been included in Eqn. (6.3), which is a valid approximation for nonmetallic materials only. Magnetization dynamics in metallic systems leads to highly nontrivial complications, which will not be discussed.

Dynamical Generalization of Brown's Equations When M x Heft # 0, the system is not at equilibrium and will evolve according to some appropriate dynamic equation. The most used dynamic equation is the Landau-Lifshitz equation, originally proposed by Landau and Lifshitz [1] for the description of domain-wall dynamics. This equation is based on the idea that in a ferromagnetic body the effective field Heff will induce a precession of the local magnetization M(r, t) of the form: 8M = - y M x Heft, 8t

(6.6)

where y > 0 is the absolute value of the gyromagnetic ratio. Equation (6.6) gives rise to motions in which the value of the saturation magnetization Ms = IMI is preserved, because M . c3M/St = 0 by construction. In this respect, Eqn. (6.6) is consistent with the fundamental micromagnetic constraint expressed by Eqn. (6.1). On the other hand, this equation cannot describe any approach to equilibrium, because dGL/dt = -Heft 9c3M/c3t = 0, which means that the dynamics is indeed nondissipative. Energy relaxation mechanisms can be taken into account by additional phenomenological terms, chosen through heuristic considerations. In their original paper, Landau and Lifshitz introduced a contribution to cgM/cqt proportional to the Heff component perpendicular to the magnetization:

~gM 0t

= --~L

(

M x Heff q-~?LMs Heff-

M. Heff) Ms

'

(6.7)

where YL is a gyromagnetic-type constant and ~L accounts for damping effects. Owing to the phenomenological nature of Eqn. (6.7), the constant YL is not necessarily coincident with the gyromagnetic ratio ?. The compatibility of Eqn. (6.7) with Eqn. (6.6) is guaranteed provided that YL = Y for

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

441

- 0. The rationale behind Eqn. (6.7) is that Heff identifies the direction in M space of steepest energy decrease, so this is the natural direction for magnetization relaxation. However, the magnetization magnitude must be preserved as well, which means that only the Heft component perpendicular to M may contribute to ~gM/Ot. This component coincides with the vector - M x (M x Heff)/M 2, so Eqn. (6.7) can be written in the equivalent form: 0M 0t = --TL M x Heft

~Ms ?LM x (M x Heff).

(6.8)

This is the form in which the Landau-Lifshitz equation is ordinarily written in the recent literature. The normalization used is such that the dissipation constant ~ is a dimensionless quantity. Its value is always very small, of the order of 10-4-10 -3 in garnets or of the order of 10 -2 in Permalloy. The parameter ?L is usually assumed to be equal to the gyromagnetic ratio 7. The Landau-Lifshitz equation can be derived and justified on the basis of a different line of reasoning, which reveals its universal character [5]. In this regard, let us look for the simplest dynamic equation consistent with the fundamental micromagnetic requirements, that is, the preservation of the magnetization magnitude and the validity of Brown's equation at equilibrium. As a first step, we restrict our choice to relaxation equations of the form: 0M -- V(M, Heft, r, t). (6.9) 0t The function V is a vector that can always be decomposed along three mutually orthogonal directions arbitrarily chosen. As the direction of M plays a particular role in our problem (no magnetization change can occur along M itself), we choose M, M x a, and M x (M x a) as base vectors, where a is an u n k n o w n vector to be determined. Equation (6.9) becomes: ~M ~t

=ClM+c2Mxa+c3Mx

(M x a).

(6.10)

Let us discuss how the micromagnetic constraints affect Eqn. (6.10). By taking the scalar product of both sides of the equation with M, we find that 01MI2/0t = 2ClIMI 2. Therefore cl -= 0, because the magnetization magnitude must be preserved. The two remaining terms must be consistent with Brown's equations, that is, the equilibrium condition OM/Ot = 0 must be equivalent to M x Heff = 0. The two terms under consideration represent vectorial components along mutually orthogonal directions, so they must be individually equal to zero for OM/~gtto be zero. Thus the equilibrium condition is M x a = 0. This condition will be equivalent to Brown's equations if a = Heft. When use is made of these results, Eqn. (6.10) is

442

CHAPTER 6 Nonlinear Magnetization Dynamics

reduced to the Landau-Lifshitz equation, Eqn. (6.8), with c2 = --?L and C3 =

--~ ~L/Ms.

The previous argument applies to any dynamic equation of the form of Eqn. (6.9) that one might decide to use, provided the equation preserves ]MI and is consistent with static micromagnetics. This suggests that differences among different forms of dynamic equation can always be removed by appropriate renormalization of the coefficients 7L and 0~. In this sense, the Landau-Lifshitz equation has a universal character. On the other hand, there is no a priori reason for assuming that the coefficients ?L and ~ should be constant. In general they are expected to be certain functions of the state of the system. It is only for the sake of simplicity that these terms are assumed to be constant parameters in most studies of magnetization dynamics.

Landau-Lifshitz and Gilbert Forms of the Dynamical Equation In addition to the Landau-Lifshitz equation, another equation widely used in the description of magnetization dynamics is the one proposed by Gilbert [2,3]: ~9M ~ 0M 0t = -?c M x Heff + MssM x 0--t-"

(6.11)

Also in this case ~Gcan be in general different from ? provided that 7c = ? for % = 0. Equation (6.11) deserves special attention, because it can be derived from a suitable Lagrangian formulation of magnetization dynamics and a Rayleigh dissipation function [33]. Writing Eqn. (6.11) in the form:

~M ( % 0M) 0t = -TG M x H e f f 7cMs Ot

(6.12)

shows that the Gilbert form is equivalent to subtracting from the effective field a viscous-like term proportional to the time derivative of magnetization. Equation (6.11) is mathematically equivalent to Eqn. (6.8). The equivalence is readily proven by applying the vector product operator, M x ..., to both sides of Eqn. (6.8). This calculation shows that the two equations are equivalent provided that: % = ~ - ~'

?G - ?L(1 + 0r

(6.13)

The mathematical equivalence of Eqns (6.8) and (6.11) might have been expected, because Eqn. (6.11) is itself consistent with the micromagnetic

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

443

constraints ]M(t)] = Ms and M x Heft = 0 at equilibrium, and, as we just demonstrated, this type of dynamics can always be described by Eqn. (6.8). However, Eqns (6.8) and (6.11) are not physically equivalent, because they have different precessional and damping terms and the two gyromagnetic parameters 7Land 7c are not coincident. This raises the interesting question of which is the equation where precession and damping are properly separated. This question acquires particular relevance whenever a is expected to be a nontrivial function of the state of the system. In fact, this issue is part of the more general problem of what are the limits under which the phenomenological introduction of damping is acceptable and is in agreement with microscopic models of spin dynamics [5,34]. In the rest of this chapter, we will describe magnetization dynamics by means of Eqn. (6.11). We will take 7c = 7 and we will denote % simply as a. We will refer to this equation as the Landau-Lifshitz-Gilbert (LLG) equation. 6.2.2 LLG DYNAMICS FOR UNIFORMLY MAGNETIZED BODIES

The Landau-Lifshitz-Gilbert equation, Eqn. (6.11), defines a highly nonlinear dynamical system that may exhibit very complex features, such as nonlinear resonances, quasiperiodicity, chaos, turbulence-like dynamics, and nonlinear wave propagations [7]. It is quite remarkable that such a complicated evolution problem may admit exact spatially uniform solutions under certain assumptions about the geometric and physical properties of the system. Spatially uniform solutions are important for several reasons. First, magnetic storage technologies and spintronics are moving toward the design of increasingly smaller devices with dimensions in the nanometer range. At this spatial scale, exchange forces strongly penalize magnetization nonuniformities and therefore spatially uniform magnetization processes are expected to be the main modes of evolution. Second, the uniform-mode theory provides a basis for the study of more complex situations where magnetization dynamics is not spatially uniform. In fact, nonuniform magnetization configurations typically arise in small systems from uniform magnetization states through instability processes. These mechanisms can be analyzed by using perturbation theory around spatially uniform solutions. Finally, spatially uniform magnetization dynamics deserves special attention because it is the simplest nontrivial case of nonlinear magnetization dynamics. This case should always be the starting point for the study of nonuniform dynamics, in order to discriminate which phenomena can be ascribed to the presence of magnetization nonuniformities and which can still be explained by nonlinear spatially uniform

444

CHAPTER 6 Nonlinear Magnetization Dynamics

FIGURE 6.1 Ellipsoidalferromagnetic body and Cartesian reference frame. magnetization dynamics. In this sense, the study of nonlinear magnetization dynamics presented in subsequent sections can be viewed as the dynamic generalization of the classical Stoner-Wohlfarth model [31,35], which is the starting point for most of the static descriptions of small magnetic systems.

Spatially Uniform Solutions Spatially uniform solutions of the LLG equation exist under the following conditions: 1. the ferromagnetic body is of ellipsoidal shape (see Fig. 6.1); 2. no surface anisotropy is present, so Eqn. (6.5) is valid at the body surface; 3. the parameters (e.g. anisotropy constants, anisotropy axis direction) that characterize the local anisotropy field HAN are spatially uniform; 4. the applied field Ha is spatially uniform; 5. the initial distribution of magnetization is spatially uniform. The first condition guarantees that internal demagnetizing fields will be spatially uniform whenever the magnetization is. Indeed, the solution of magnetostatic Maxwell equations (Eqns (6.4)) in uniformly magnetized ellipsoidal bodies is spatially uniform and can be expressed in terms of the demagnetizing tensor. By choosing a system of unit vectors (ex, ey, ez) along the principal axes of the ellipsoid (see Fig. 6.1), one obtains: H M (M)

= - N x M x e x - NyMyey - NzMzez,

(6.14)

where Nx, Ny, Nz are the demagnetizing factors and Nx + Ny + Nz = 1. The second condition is necessary to prevent the pinning of magnetization at the body surface, which could produce spatial magnetization nonuniformities. The third condition guarantees that the anisotropy field HAN is

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

445

spatially uniform when M is spatially uniform. We will limit our discussion to the case of uniaxial anisotropy, where: HAN (m) -- HAN (eAN 9m ) eAN.

(6.15)

In this equation, m - M / M s , HANis an anisotropy parameter, and eAN is the unit vector along the easy axis direction. Under these conditions, the effective field associated with a spatially uniform magnetization state is itself spatially uniform and can be written as:

Heft(M, t) - Ha(t) + HAN (M) - N x M x e x - N y M y e y - NzMzez.

(6.16)

This result is tantamount of the fact that the LLG equation admits spatially uniform solutions. Indeed, consider the initial-value problem for Eqn. (6.11) under the condition of spatially uniform initial distribution: M(r, t - 0) - M0.

(6.17)

The spatially uniform vector field M(t) is the solution of this problem if it is a solution of the ordinary differential equation: dM

~

dt

Ms

M x

dM dt

= - 7 M x Heft(M, t),

(6.18)

under the initial condition M(t - 0) - M0. We underline the fact that the solution of the LLG initial-value problem is unique, so it necessarily coincides with M(t). Once we have recognized this important property of LLG magnetization dynamics in ellipsoidal particles, the issue of stability of spatially uniform modes naturally arises. The question is whether small nonuniform perturbations of the initial uniform magnetization distribution will increase during the time evolution or will remain small. The quantitative theory for studying this issue has not been completely established yet. The problem has been investigated in detail for certain particular cases such as nucleation theory and spin-wave analysis [29,30]. Nonuniform perturbation modes and spin-wave parametric instabilities under rotating fields are analyzed in some detail in Chapter 7 of this volume of 'The Science of Hysteresis'. In this chapter, we will limit ourselves to the study of stability with respect to uniform perturbations of the nonlinear dynamical system described by Eqn. (6.18).

LLG Equation in Dimensionless Form It is useful and insightful to rewrite the LLG equation in dimensionless form. There are two natural scales in the problem. Fields and magneti-

CHAPTER 6 Nonlinear Magnetization Dynamics

446

zation are naturally m e a s u r e d in units of Ms. Accordingly, we introduce the dimensionless vectors m - M / M s and heft = Heft~Ms. On the other hand, a natural time scale is provided by (TMs) -1. Accordingly, we normalize time as t ---> 7Ms t. In Permalloy, with 7 = 2.2 x 105A-lms -1 and Ms = 8 x 10SAm -1, one finds 7Ms = 176 GHz, that is, (TMs) -1 ~ 6 ps. In dimensionless form, Eqn. (6.18) can be written as follows: dm

dt

czm x

dm

dt

-- - m x heft(m, t).

(6.19)

The normalized effective field heft is given by: heft(m, t) = Heft~Ms = ha(t) 4- h M(m) 4- bAN(m),

(6.20)

where ha(t) = Ha(t)/Ms, h M (m) = H M(M)/Ms, and hAN (m) = HAN (M)/Ms. According to Eqns (6.14) and (6.15), one has: h M(m) = - N x m x e x - Nymyey - Nzmzez,

(6.21)

bAN (m) = /~AN(CAN " m)eAN'

(6.22)

where NAN --- NAN/Ms is the normalized anisotropy constant. The s u m of the two fields h M(m) and hAN(m) is a linear function of m which can be expressed as: h M(m) 4- bAN (m) ~---- D . m,

(6.23)

where D is a symmetric matrix whose components are given by:

Dij = Ni~ij - /~ANeAN,/eAN,j.

(6.24)

Subscripts i, j indicate x, y, z components, eAN,X, eAN,Y, eAN,Za r e the direction cosines of eAN, and cS/jis Kronecker's symbol. The symmetric matrix D can be diagonalized by expressing its components in the Cartesian reference frame associated with its eigenvectors ex,, ey,, ez,. In this reference frame, heft = (hax' - Dx, mx, ) ex, 4- (hay, - Dy, my, ) ey, 4- (haz' - Dz, mz, ) ez,, where hax', hay', haz' are the components of ha along ex,, ey,, ez,, and Dx,, Dy,, Dz, are the eigenvalues of D. In the sequel, the less general case will be discussed where the easy-axis direction CAN is aligned with one of the principal axes of the ellipsoid, i.e. with one of the unit vectors ex, ey, ez. In this case: h e f t - (hax - Dxmx)ex + (hay - Dymy)ey + (haz - Dzmz)ez,

(6.25)

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

447

where the parameters Dx, Dy, Dz include the effects of both demagnetizing fields and crystal anisotropy fields. This case is simplified by the fact that the eigenvectors of D are along the principal axes of the ellipsoid. Although this is not a strictly necessary assumption, it has the advantage of making the interpretation of various results much more transparent, owing to the fact that m and ha are expressed in terms of their components along the magnetic body geometrical axes ex, ey, ez. The results reported in the rest of this chapter can be extented to the general case by using an appropriate change of reference frame. In the sequel, we will also assume that the parameters Dx, Dy, Dz are ordered as follows:

Dx<~Dy<~Dz,

(6.26)

which can always be achieved by an appropriate permutation of the Cartesian axes. Equation (6.26) indicates that the axis ex is the easy axis of the magnetic particle. Under the assumptions made, the free energy associated with the magnetic body is expressed in normalized form as: 1 2 (M,9 Ha) 1 Dxm2+ -~Dymy 1 2q_~Dzm gL(In; ha) - GL#oM2 V _- -~ z - haxmx-haymy-hazmz,

(6.27) where V is the volume of the body and GL(M; Ha) is the micromagnetic free energy. The effective field is related to the energy through the expression: heft - - m--c3 ~ ' g ) (m" ha).

(6.28)

As a final remark, we briefly mention some useful alternative forms of the normalized LLG equation. First, we observe that Eqn. (6.19) can be rewritten as: dm ( dm) dt = - i n x heft(m, t) - c~--~- , (6.29) which is the normalized version of Eqn. (6.12). By using Eqns (6.8) and (6.13), the last equation can be transformed into the Landau-Lifshitz form: din dt

1 1 + ~2 m x heft(m, t)

1 4-c~2 m x ( m x heft(m, t)).

(6.30)

Both Eqns (6.29) and (6.30) can be written in the form: dm = -m x ~, dt

(6.31)

CHAPTER 6 Nonlinear Magnetization Dynamics

448

where the vector ~ is equal to: dm

9-~G -- heff - a d---T

(6.32)

in the case of Eqn. (6.29), or: 1

- 1 + 0r2 [heft if-0r

x heft]

(6.33)

in the case of Eqn. (6.30). Energy Balance in LLG Dynamics The free energy of the magnetic body undergoes a certain time evolution during the magnetization motion. The equation describing this time evolution is readily obtained from the LLG equation written in the form: d m / d t = - m x "/'r (Eqns (6.31) and (6.32)). Dot multiplication of both sides of the equation by 9-~c yields 9-~c 9d m / d t = 0. Then, by using the fact that heff -= -eggL~cgin, one arrives at the energy balance equation: d

dm 2

-~gL (m; ha) -- --~X - - ~

-- m .

dha d--T"

(6.34)

The two terms on the right-hand side respectively represent the power dissipated because of intrinsic damping and the power supplied through the action of the time-varying applied field. In connection with this last term, it is worth mentioning that in magnetization switching under pulsed fields, the applied field rise-time can be often considered, as a first approximation, negligible with respect to the magnetization response characteristic time. In these cases, the analysis can be greatly simplified by assuming that the applied field has a discontinuity at the pulse start time, say t - to. At that moment the time derivative d m / d t will be discontinuous as well, as can be seen from Eqn. (6.19). The magnetization evolution, on the other hand, will remain continuous since a finite-amplitude discontinuity in d m / d t cannot produce any discontinuity in m(t) itself. In the presence of field discontinuities, Eqn. (6.34) is to be interpreted in the generalized sense, since dha/dt will exhibit a Dirac-delta singularity localized at t - to. One concludes that at t = to the free energy instantaneously changes by the finite amount Ag - -m(t0) 9 Aha, where Aha is the jump in the applied field. When the applied field is constant, the energy balance equation is reduced to the simpler form:

dgL dt-

_

--~

dm

--~-

2

(6.35) '

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

449

which has the following important consequences. 9 The dynamics of the system is such that the free energy g~ (re(t); ha) is always a decreasing function of time. 9 The time derivative of & (re(t); ha) is zero only when d m / d t = 0, namely, when the system reaches equilibrium points. Since equilibrium points are stationary points for the energy, the dynamics is directed toward energy minima. 9 No self-oscillations, i.e. periodic solutions, are admissible. Indeed, the necessary condition for the existence of a nonconstant periodic solution is that gL(re(t); ha) is a periodic, and thus nonmonotone, function of time. This requirement is in contradiction to the monotone time decrease predicted by Eqn. (6.35).

6.2.3 LLG DYNAMICS ON THE UNIT SPHERE Conservation of M a g n e t i z a t i o n A m p l i t u d e It was previously mentioned that LLG dynamics (Eqn. (6.19)) always preserves the magnitude of the magnetization. In other terms, the quadratic form m2 4- m~ 4- m2 is an integral of motion of Eqn. (6.19)" m 2x 4- m y2 4- m 2 z

_ 1.

(6.36)

Therefore, Eqn. (6.19) defines a dynamical system evolving on the surface of the unit sphere. The evolution of this dynamical system proceeds along the lines of the vector field: v(m, ~, t) = - ~ m

1 4- 0r2

x heff(m, t) - ~ m 1 4- 0r2

x (m x heff(m, t)), (6.37)

given by the right-hand side of Eqn. (6.30). These lines are tangential to the unit sphere for all m, c~and t. In the interesting particular case where the applied field is constant (dha/dt = 0), the dynamical system defined by Eqns (6.19) or (6.30) is autonomous, since the effective field does not depend explicitly on time anymore. Accordingly, the vector field v(m, a, t) is stationary (v = v(m, a)). The fact that the evolution of magnetization is driven by a stationary vector field tangential to the unit sphere has some remarkable consequences of topological nature. 9 There must exist equilibrium states for the system (points where v = 0), because a stationary vector field on the sphere has always singular points [25]. In general, singular points of a vector field

450

CHAPTER 6 Nonlinear Magnetization Dynamics can be either zeros or poles where the magnitude of the vector field tends to infinity. However, the LLG vector field v is continuous and, consequently, it cannot have poles. 9 The number of equilibria is at least two and is even under all circumstances. This conclusion is derived from the Poincar6 index theorem [25], which asserts that the number of nodes, foci and centers minus the number of saddles of any autonomous dynamics on the sphere is equal to two. The distinctive qualitative features of saddles, nodes, foci, and centers are shown in Fig. 6.2(a)-(d) (see [25,26,36] for more details on the classifications of equilibria in nonlinear dynamical systems). 9 Chaos is precluded, because the phase space is two-dimensional [26,36]. This is the consequence of the generalized version of the Poincar6-Bendixson theorem, which states that on two-dimensional manifolds the only possible steady states are either stationary states associated with equilibria or self-oscillations (periodic steady states) associated with limit cycles of the dynamics. The distinctive features of limit cycles are shown in Fig. 6.3.

When the vector field v(m, a, t) is not stationary, LLG dynamics becomes much more complicated, and quasi-periodic or chaotic solutions may appear [37]. In this chapter we will not encounter these cases since we will restrict our analysis to transient phenomena in which the field is either constant or becomes constant after a certain time has elapsed. When the applied field becomes constant, the energy balance expressed by Eqn. (6.35) precludes the appearance of self-oscillations in the system dynamics: the magnetization will always evolve toward one of the stable equilibria. In the case of coexisting stable equilibria, what will be the final equilibrium state is determined by the applied field history and by the initial magnetization state. Limit cycles and self-oscillations play a important role in spin-torque-driven magnetization dynamics, where the energy balance is no longer given by Eqn. (6.35), and consequently the appearance of limit cycles is no longer ruled out. This interesting case will be discussed in Section 6.5. There is a special case of nontransient variations of the applied field in which magnetization dynamics can nevertheless be reduced to autonomous form. This is the case where the magnetic body has uniaxial symmetry and the applied field is time-harmonic and circularly polarized. By taking advantage of the symmetry of the problem, one can introduce a reference frame in which the applied field is constant and the dynamics is reduced to an autonomous dynamical system on the unit sphere [38]. This case is treated in detail in Chapter 7 of this volume of 'The Science of Hysteresis'.

451

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

r (a)

(b)

(c)

1 (d)

FIGURE 6.2 Qualitative sketches of magnetization dynamics on small portions of the unit sphere in neighborhoods of equilibrium points of different nature. (a) Saddle equilibrium; (b) stable node (the sketch for the unstable node can be obtained by inverting the arrows); (c) stable focus (the sketch for the unstable focus can be obtained by inverting the arrows); (d) center (this kind of equilibrium is generally present only in conservative (undamped) systems).

(a)

(b)

FIGURE 6.3 Schematic representation of limit cycles. (a) Stable limit cycle (a, i.e. attractive) surrounding unstable focus; (b) unstable limit cycle (r, i.e. repulsive) surrounding stable focus.

Spherical and Stereographic Coordinate Representations The evolution of magnetization on the unit sphere can be described by using appropriate two-dimensional coordinate systems. In this section, we introduce two systems that will be often used in the sequel: spherical and stereographic coordinates. In the spherical coordinate system, the position of the vector m on the unit sphere is identified by the angle 0 that m forms with the axis ez and the angle ~b that the projection of m on the plane (ex, ey) forms with the axis ex (see Fig. 6.4(a)). The connection between Cartesian and spherical coordinates is expressed by the following formulas: mx = sin 0 cos ~b,

my = sin 0 sin ~b,

mz = cos 0.

(6.38)

The vector quantities in both sides of Eqn. (6.19) are tangent to the sphere ImI = 1 at the point corresponding to m. Let us choose as basis vectors

CHAPTER 6 Nonlinear Magnetization Dynamics

452

FIGURE 6.4 Spherical and stereographic representation of magnetization state. Z is the unit sphere, n is the (x, y)-plane and C is the unit circle on n. (a) In the spherical coordinate representation, the point m on Z is identified by the angles 0 (angle between m and ez) and ~ (angle between the projection of m on the plane and ex). (b) In the stereographic representation, the point m on Z is projected into the point w of the plane n by finding the intersection between n and the line connecting m to the south pole S of the unit sphere Z. in this tangent plane the unit vectors eo and er associated with the polar coordinates 0 and ~b. The unit vectors (eo, er m) form a positively oriented triple (m x eo = er It can be verified that: dm dO ~_~ dt = dt eo + sin 0 er

(6.39)

In addition, the component of the effective field (heff(m, t))o,4) tangential to the unit sphere at the point m is given by: (heft(m, t))0,r =

-

cggL 0--Oe0

sin 0 cg~ber = heff, oe 0 + heff'r162

(6.40)

where the function gr (0, ~; ha) in Eqn. (6.40) is the function obtained by using the spherical-coordinate representation of m (Eqn. (6.38)) in Eqn. (6.27), while heff,0 and heff,~ are the projections of the effective field along the unit vectors e0 and er (complete expressions for these quantities are given in Appendix A). By using Eqns (6.39) and (6.40), and by projecting both sides of Eqn. (6.19) along e0 and er the following system of two coupled ordinary differential equations is obtained: dO ~_~ d---}-+ ~ sin 0 - heft,4~ dO 0 -a-G; + sin Cll:

1 0gL sin 0 c~b'

- -heff, 0 -

~0

(6.41)

(6.42)

453

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

Spherical coordinates are quite a natural reference system to represent magnetization evolution on the sphere. However, the ensuing description of magnetization dynamics is not always completely transparent in the vicinity of the sphere poles where the angle ~bis undefined. For this reason, it is useful to introduce the stereographic projection of the unit sphere on the plane (Fig. 6.4(b)). The stereographic coordinates Wx and wy are connected to the Cartesian coordinates by the following relations:

mx Wx - 1 4- mz 2Wx m x = 1 + IWI2'

wy =

my , 1 4- mz

2wy my - 1 + IWI2'

(6.43)

1 - IW] 2 mz = 1 + Iwl2'

(6.44)

where w - Wxex 4- wyey. The main properties of this transformation are the following: angles between intersecting lines are preserved (conformal transformation) [39]; m = +ez is projected into the origin of the plane; the north hemisphere (mz > 0) is projected inside the unit circle; the south hemisphere (mz < 0) is projected outside the unit circle; m = -ez is projected into infinity. The LLG equation can be written in stereographic coordinates in the following form (the proof is reported in Appendix A):

dwx d w y - ( 1 + IwI2) 2 heff,wy, d--T 4- ~ ~ 2 dwx d---~ -- -

dw v dt

(6.45)

(1+ ]wI2)2 2

(6.46)

heff'wx '

where heff,wx = -OXL/ C~Wx,heff,wy = -6~gL/ C~Wy,and the function gL(Wx, Wy ; ha) is obtained by substituting Eqns (6.44) into Eqn. (6.27).

Geometrical Interpretation of Damping Equations (6.41) and (6.42) can be written in terms of the e0 and e~ components of dm/dt (see Eqn. (6.39)) as follows:

(1 -~

~)( 1

dO/dt )_( sin0ddp/dt

heff,$ ) -heff, 0 "

(6.47)

By inverting the matrix in the left-hand side of Eqn. (6.47), one obtains:

1(1 tlhe t

dO/dt ) _ sin 0 ddp/dt 1 + a2

+a

1

-heff, 0

"

CHAPTER 6 Nonlinear Magnetization Dynamics

454

This equation admits an interesting geometrical interpretation. Let us introduce the angle 6 defined as tan 6 - a. Equation (6.48) can be rewritten as:

( dO/dt ) sin0dqb/dt =

1 v / i + 0r

(cos~-sin~)( + sin 6 COS6

heff,~ ). -heff,0

(6.49)

In the case of conservative dynamics (a = 0), Eqn. (6.49) is reduced to:

dO/dt (sinOdc/~/dt)-( heff'c/~_hef 0 ).f,

(6.50,

We see that the dissipative dynamics (Eqn. (6.49)) is obtained from the conservative one by a counterclockwise rotation through the angle 6 tan -1 a and a subsequent rescaling by the factor 1/v/] + a 2. This remarkable property can be expressed in general vectorial terms by using the vector field v(m, a, t) given by Eqn. (6.37). In terms of v(m, ~, t), the LLG equation is expressed as: dm = v(m, a, t). dt

(6.51)

Let v0(m, t) = v(m, a = 0, t) be the vector field corresponding to the undamped dynamics. Then, Eqn. (6.49) implies the following relation: 1 v(m, :r t) = ~ R(m, ~). v0(m, t), v/1 +0r 2

(6.52)

where the matrix R(m, ~) represents a rotation through the angle ~ around the axis identified by the magnetization unit vector m. In the extreme cases of slightly dissipative (a ~ 0) and strongly dissipative (a --, cx~) systems, Eqn. (6.52) yields the following results. 9

~ --~ 0. In this case, to the first order in ~, v(m, ~, t) ~ R(m, ~) 9 v0(m, t), i.e. the effect of damping is a small rotation through the angle a of the undamped vector field. This fact has direct consequences on the nature of equilibrium points. Let us consider the particular case of stationary applied fields (v and v0 do not depend on t). When = 0 the system is conservative and for this reason it can have only two types of equilibria: saddles (see Fig. 6.2(a)) and centers (see Fig. 6.2(d)). It can be verified that a small counterclockwise rotation of the vector field transforms centers into foci (see Fig. 6.2(c)), while saddle equilibria are only slightly distorted. This is expressed in bifurcation theory by saying that saddle equilibria are structurally stable with respect to generic perturbations of the differential equation.

455

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

9 ~ ~ cx~. In this case, to the first order in l/a, v(m, a, t) ~ (1/~)R(m, ~/2) 9v0(m, t), which means that magnetization dynamics is governed by a differential equation of the type: d m / d t = --(1/a)[eoCggL/ 00 + e~(1/sin O)CggL/Cgq5]. This equation shows that the magnetization evolves along the lines of steepest energy descent toward energy minima. Equilibrium points are generally either nodes (see Fig. 6.2(b)) or saddles. Indeed, one can verify by comparing Fig. 6.2(d) and Fig. 6.2(b) that a ~/2 rotation of the vector field generates, in the neighborhood of a center, a qualitative dynamics corresponding to a node. A similar discussion can be carried out in the case of stereographic coordinates. By following the same line of reasoning leading to Eqn. (6.49), one obtains the equation: d(wx)_

1

(1 + Iw12) 2

heff,wy ) -heff,wx ' (6.53) which has the same geometrical interpretation as Eqn. (6.49). The geometrical properties of the dynamics are conserved as the direct consequence of the conformal nature of the stereographic projection. dt

wy

v/1 if_ a2

2

t

COS +sin

-sin~ cos ~

t(

6.2.4 MATHEMATICAL STRUCTURE OF LLG EQUATION

Hamiltonian Structure of Conservative LLG Dynamics Let us start our discussion by considering the mathematical structure of the LLG equation in the conservative case, i.e. for ~ = 0. The equation can be written in the following form: dm OgL eggL dt = m x 0m - A(m). ~m'

(6.54)

where A(. ) is the matrix such that A(u) 9w = u x w for all vectors u and w. Equation (6.54) does not exhibit the typical Hamiltonian form since it involves an odd number of state variables, contrary to usual Hamiltonian systems where state variables (coordinates and associated conjugate momenta) are always even in number. Nevertheless, in modern analytical mechanics [40] Hamiltonian systems are defined in more general terms as systems admitting a representation in terms of Poisson brackets. In the case of Eqn. (6.54), we can introduce the 'rigid-body' Poisson bracket defined as~ {f, h / - - m -

0f x 9A(m)- ~ , 0m cgm ~gm cgm

(6.55)

CHAPTER 6 Nonlinear Magnetization Dynamics

456

wheref(m) and h(m) are two generic scalar functions of m. By using Eqn. (6.55), Eqn. (6.54) can be written in the Hamiltonian form:

dmx dt = {mx, gL },

dmy dt -- {my, gL },

dmz dt = {mz, gL },

(6.56)

in which the function gL can be interpreted as the Hamiltonian of the system. It can be proved that the bracket {f, h} defined by Eqn. (6.55) satisfies the four fundamental properties of Poisson brackets: (i) bilinearity (i.e. linearity with respect to f and h separately); (ii) antisymmetry ({f, h} = -{h,f}); (iii) Leibniz rule: {f, hi} = l{f, h} + h{f, l}; and (iv) the Jacobi identity: {{f, h}, l} + {{l,f}, h} + {{h, l},f} = 0. By using these properties, it can be proved that the total time derivative of a generic function f(m, t) along the trajectories of Eqn. (6.54) can be expressed as:

df _ {f, gL} + COl dt 0t"

(6.57)

From this formula it follows that a function f that does not explicitly depend on time is an integral of motion if and only if its Poisson bracket with the Hamiltonian is zero, i.e. {f, gL } = 0. In particular, the antisymmetry property of Poisson brackets implies that {gL, gL } = 0, that is: d

0

~-~gL(m; ha) = -~gL (m; ha).

(6.58)

Therefore, the energy gL is an integral of motion in all cases where the applied field is constant. There are two main differences between the Poisson brackets defined by Eqn. (6.55) and the Poisson bracket usually defined for Hamiltonian systems in terms of generalized coordinates and conjugate momenta. The first difference is that Eqn. (6.55) is explicitly dependent on the state of the system m, because the antisymmetric matrix A depends on m. The second difference is that there exist integrals of motion C(m), which satisfy the condition {C, h} = 0 for any arbitrarily chosen scalar function h. Functions C(m) fulfilling this condition are usually referred to as Casimir integrals. They have a rather different nature in comparison with usual integrals of motion in Hamiltonian dynamics. Usually, a function I(m) is an integral of motion if the Poisson bracket with the Hamiltonian is zero. This is usually connected to the presence of symmetries in the Hamiltonian with respect to certain transformation groups (Noether's theorem). The Casimir integral has a much stronger property that is not related to the form of the Hamiltonian but only to the geometrical structure of the dynamics. In the

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

457

case of conservative LLG dynamics (Eqn. (6.54)) the Casimir integral is Iml 2. Indeed, for any scalar function h(m)" {Iml2 , h } - 2 m .

Oh ) _ 0. m x c3m

(6.59)

Therefore, the preservation of the magnetization magnitude can be interpreted as the presence of a Casimir integral associated with the Poisson bracket defined by Eqn. (6.55).

LLG Dissipation Structure Starting from conservative LLG dynamics (Eqn. (6.54)), it is possible to introduce dissipative effects without affecting the preservation of magnetization magnitude by adding a 'dissipation field' hdis to the effective field h e ft - -

--63gL/Om: dm dt

-- --m

• (heft if-hdis).

(6.60)

The field hdi s takes into account dissipative processes and it may depend on dm/dt, m, ha, and other physical parameters of the system. In order to represent dissipative processes, hdis has to fulfill the following inequality [41]: dm hdi s 9 -~ ~0. (6.61) Therefore, in the case of constant field, d g L / d t - hdis.dm/dt~<0, i.e. the free energy gL is a decreasing function of time. This is consistent with the idea that dissipative forces produce a relaxation dynamics toward free-energy minima. In addition, we assume that dissipative processes act only when magnetization is varying and therefore do not affect equilibria. This means that hdi s ----0 when d m / d t - 0. The mathematical form of h d i s c a n be specified by using the Rayleigh dissipation function approach. This approach usually appears in the framework of Lagrangian formalism and it consists of introducing additional viscous friction forces linearly depending on generalized velocities (i.e. time derivative of the generalized coordinates). These viscous forces can be expressed in terms of the Rayleigh dissipation function 7r through the equation: dm 07~ h d i s - - A d i s - dt = c~rh' (6.62) where: ~, -- 11il-adi

s 9 rh,

(6.63)

458

CHAPTER 6 Nonlinear Magnetization Dynamics

rh = dm/dt, and Adis is a symmetric positive definite matrix that does not depend on rh but may depend on m, ha, and other physical parameters associated with the magnetic system. Then, the dissipative LLG equation is given by: dm (0g L ~T4) ( din) m x ~m + ~-~ = m x hef f A d i s . dT . dt

(6.64)

By dot multiplying both sides of the equation by (Sgr/Sm + 8T4/Srh), we end up with the following energy balance equation: dgL -- --27~ + toga dt-~'

(6.65)

which in the case of constant applied field is reduced to: dgL = -27"4. dt

(6.66)

This equation clarifies the physical meaning of the Rayleigh dissipation function: 2/4 represents the rate of decrease of the free energy due to dissipative processes that drive the system toward equilibria. The Gilbert form of magnetization dynamics (see Section 6.2.1.3) corresponds to a special choice for T4: dm

T4--~ ~

2

,

(6.67)

which amounts to choosing the simple diagonal form: Adis = aI (I is the identity matrix).

Geometrical Techniques for Numerical Integration of LLG Equation When it is desirable to analyze LLG dynamics without making any simplifying assumption, the only available general quantitative method of analysis is numerical integration. Numerical integration of the LLG equation is indeed widely used in micromagnetics [42,43]. However, existing general discretization techniques (e.g. linear multistep, explicit Runge-Kutta methods) usually corrupt the intrinsic properties of the dynamics such as the preservation of magnetization magnitude, the structure of dissipative processes, or the Hamiltonian structure in the limit of zero damping. In this section, we discuss special techniques designed to reproduce the fundamental properties of LLG dynamics in numerical integration. In order to develop a numerical integration scheme for Eqn. (6.19), we consider the temporal mesh consisting of the sequence of time instants

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

459

tl, t2, t3 . . . . , which, for the sake of simplicity, are taken uniformly spaced, i.e. tk+l -- tk = At. A numerical time-stepping scheme is a map: m (k+l) --

{~k(At,m (k), m (k-l) . . . . )

k = 1, 2 . . . . .

(6.68)

which generates a sequence of approximate values m (k) of the u n k n o w n solution re(t) of the LLG equation at the time instants tk. The first point to be addressed is whether there are techniques that preserve the magnetization magnitude in numerical integration. In the study of this problem, we will make use of the general form of the LLG equation given by Eqn. (6.31). The conservation of Iml in the numerical integration can be achieved by using the following midpoint rule formulas: dm

d---t-(tk+ At/2)

--

m ( t k + l ) -- m ( t k )

At

()

+ O At 2 ,

(6.69)

m(tk + At/2) = m(tk+l) + m(tk) + 49 /" ( A2) t 2

(6.70)

where we use the notation O(At 2) to denote the terms of the second order of smallness with respect to At. By substituting Eqns (6.69) and (6.70) into Eqn. (6.31), we obtain: m ( t k + l ) - m(tk) At

=--(m(tk+l)+m(tk))x'l~(tk+At/2)+O(At2) " 2

(6.71)

To arrive at the final form of the scheme, we have to express 7-r + At~2) in terms of values of 7-r at the mesh points. This can be done with different techniques that will be specified in the sequel. In any case, we will arrive at some expression of the form: 7-~(tk + At/2) -- Hk (m(tk+l), m(tk), m(tk_l) . . . . ) + (..9(At2),

(6.72)

where I-Ik(') is a function that provides a second-order approximation to 7-r + At~2) in terms of m (k+l), m (k), m (k-l), etc. By using Eqns (6.71) and (6.72), we can devise the following second-order-accuracy time-stepping scheme: re(k+1) --m(k) - - - - 2

(re(k+1) q-

x 7-~

,

(6.73)

where: (6.74)

460

CHAPTER 6 Nonlinear Magnetization Dynamics

The main property of the general numerical scheme expressed by Eqns (6.73)-(6.74) is that it preserves the magnetization magnitude. Indeed, by computing the scalar products of both sides of Eqn. (6.73) with the vector m(k+l) + re(k), we obtain: (m (k+l) + m(k)) 9 (m (k+O -- m (k)) --O,

(6.75)

which is equivalent to:

I m (k+l) 12=1 m (k) 12-- const

(6.76)

It is remarkable that this conservation property is valid regardless of the choice of the function Hk used in Eqn. (6.72) to approximate the value of 9-~(tk + At/2). The conservation of Iml is also valid regardless of which expression of 9-~ is used, i.e. it holds both for ~ = 9"~G(see Eqn. (6.32)) and for ~ = 9"~L(see Eqn. (6.33)). The numerical scheme described by Eqn. (6.73) is always implicit in nature since it contains m(k+l) on both sides of the equation. However, if we use explicit extrapolation formulas in Eqn. (6.74), namely, functions Hk that do not depend on m(k+l) but only on m at previous steps, then the determination of m(k+l) in Eqn. (6.73) requires only the solution of a linear equation. In this case, it is convenient to rewrite the finite-difference scheme as follows:

m (k+l) -}- ~-~tm(k+l) x "]'~(k+l) -- m (k) -- Atm(k) x "J~(k+l) 2

(6.77)

The last formula can be interpreted as a linear vector equation for m (k+l) (or as three linear simultaneous equations for the Cartesian components of m(k+l)). A very effective explicit second-order extrapolation formula can be devised assuming ~ = "H,L(m, t) (see Eqn. (6.33)), and using the following Adams' extrapolation: 3

1

( )

9-~(tk + At/2) - ~'H,L (m(tk), tk)- -~9~ (m(tk-1), tk-1) + 0 At2 , (6.78) which leads to the following value of the field 7~(k+l) in Eqn. (6.73)" 9.~(k+89 -- -~ 3 9"~L (m (k) , tk)_1 -~ 9"{~ (m(k_l) , t k_ l ).

(6.79)

This scheme is very fast and very simple to implement. It is also very convenient for the analysis of spatially nonuniform processes, where the

6.2 LANDAU-LIFSHITZ-GILBERT (LLG) EQUATION

461

extrapolation formula given by Eqn. (6.79) can be used at each spatial location by using the values of m (k) and 7-~(k) at the same spatial location at previous time instants. In this way, by using the two-step nature of the scheme, it is possible to achieve a complete spatial decoupling in nonuniform micromagnetic analysis. This two-step scheme presents a minor difficulty for the calculation of 7-~(89 at the first step. This difficulty can be circumvented by using the second-order Runge-Kutta method to compute M(1) needed for the calculation of 7-~(1). Another interesting scheme can be obtained by using the expression 7"r = 9-~c (m, t) (see Eqn. (6.32)), and using the following midpoint formula:

"l.~(tk + At/2)--D. [ m(tk+l)2+ m(tk) 1 + ha(tk + At/2) _ocm(tk+l) -- m(tk)

At

+ O(At2),

(6.80)

which leads to the following expression for the field 7-t(k+l) in Eqn. (6.73):

,}.~(k+l) _ D. Im(k+l)q -m(k)] 2

+ha(tk+At/2)-a

m(k+l)-m(k) At

. (6.81)

By multiplying both sides of Eqn. (6.73) by 7-~(k+l) we find that 7-~(k+1) 9 (m (k+l) -- m(k))/At -- O. Then, by using the symmetry of the matrix D and the midpoint approximation of ha(tk + At/2), namely:

ha(tk + At/2) --

ha(tk+l) + ha(tk) 2

At2 d2ha 4 dt 2

+ Ra(tk, At), (6.82) tk+At/2

where Ra(tk, At) -- O(At 4) is the remainder of the above midpoint expansion, one can readily derive the following equation:

(k+ ) _ R(Lk)

m(k+l)At--m(k) 12-- (re(k+1)2+ m(k)) " (h}k+l)At- h~k))

At At2 d2ha 4 dt 2 +Ra (tk, At).

m(k+ 1) _m(k)

tk+At/2

At

m(k+ 1) _ m (k) At '

(6.83)

CHAPTER 6 Nonlinear Magnetization Dynamics

462

where we have denoted ha(tk) as hff ), while g(Lk) -- gL (re(k); ha(k))" Equation (6.83) shows that, for generic time-varying ha(t), the discretized dynamics reproduces the energy balance (see Eqn. (6.34)) with second-order accuracy. If the applied field ha(t) is piecewise linear with respect to time in each time interval [tk, tk+l], then the last two terms in the right-hand side of Eqn. (6.83) vanish and the energy balance Eqn. (6.34) is exactly reproduced in its midpoint time-discretized version. More importantly, in the case of constant applied field, the last three terms in the right-hand side of Eqn. (6.83) vanish and the energy balance is reduced to the following form: (k+l) _ g(Lk)

At

m ( k + l ) _ re(k) 2

= -~

At

.

(6.84)

Equation (6.84) has very important consequences. First, it shows that the midpoint rule is an unconditionally stable algorithm that reproduces the relaxation behavior in LLG discretized dynamics for any choice of the time step. In addition, for a - 0 the energy is exactly preserved regardless of the time step. This property indicates that for slightly dissipative systems the midpoint rule will tend to reproduce correctly the most important part in the LLG dynamics, i.e. the precessional magnetization motion.

6.3 G E O M E T R I C A L A N D A N A L Y T I C A L T E C H N I Q U E S FOR LLG EQUATION IN UNIFORMLY M A G N E T I Z E D BODIES

6.3.1 EQUILIBRIA General Considerations The first step in the analysis of magnetization dynamics is the determination of the equilibrium points and the study of their stability. In this chapter we are mostly interested in transient problems in which the external field becomes constant after a certain time interval has passed. Although there are special cases where the system can have equilibrium points even if the external field is varying (e.g. both magnetization and external field along the easy axis), we will focus our attention only on the study of equilibria in the case of constant applied field. Indeed, this is the case most relevant to switching and relaxation problems. Equilibrium points are found by solving the equation: m x heff - O ,

(6.85)

463

6.3 GEOMETRICALAND ANALYTICAL TECHNIQUES

under the constraint Im[ = 1. Due to the viscous-like nature of damping in LLG dynamics, the equilibrium points are not affected by the value of and thus the problem of finding equilibria can be formulated in terms of the free energy only. If we take into account the constraint Iml = 1, Eqn. (6.85) can be rewritten in spherical coordinates as two scalar equations:

0gL = 0, a0

0gL = 0, a4

(6.86)

which means that LLG equilibria are always critical points of the free energy gL(m; ha) with m restricted to vary on the unit sphere. In this respect, equilibria can be classified as energy maxima, energy minima and energy saddles. If the magnetic body is invariant with respect to the angle ~b, the second equation in Eqn. (6.86) can be ignored. This is the case analyzed in the Stoner-Wohlfarth model [35], which permits one to give a complete description of the equilibria in the case of magnetic systems with uniaxial symmetry. For more general magnetic systems, it is necessary to consider some appropriate extension of the Stoner-Wohlfarth theory. A rather general approach is to solve the following system of equations: heft(m) - 2m,

Im[ 2 - 1,

(6.87)

which is equivalent to four scalar equations for the four unknowns represented by 2 and the three components of m. This approach was proposed in [44] and leads to the determination of two, four, or six equilibrium points, depending on the value and the orientation of the applied field. In the sequel, we will present a different approach for the determination of equilibria under the assumption that one of the components of the applied field is zero. Although it is less general than the method based on Eqn. (6.87), this approach has the advantage that the solution can be worked out in full analytical detail and can be translated into geometrical constructions that generalize the Stoner-Wohlfarth model.

Equilibria When One Component of the Applied Field is Zero In many cases relevant to applications, one can assume that the applied field ha lies in one of the coordinate planes, namely, that one of the Cartesian components of ha is zero. Under these conditions, the analysis of equilibria can be substantially simplified. In the following, we will assume that haz -- 0, but the theory can be extended to the other cases by an appropriate reorientation of the Cartesian axes. In the case haz -- 0, by using the fact that m2z - 1 - m 2 - m ~ , m z can be eliminated from the expression of the

CHAPTER 6 Nonlinear Magnetization Dynamics

464

free energy, Eqn. (6.27). This leads to the following formula: Dz - D x rE(mx _ ax)2 if- k2 (my - a y ) 2 - P~l-', 2

gL(m; h a ) =

(6.88)

where: a x

~

~

hax ~ Dz-Dx'

hay ay=-Dz_Dy 2

2

p2 = ax + ay

, k2 _ Dz - Dy - Dz-Dx'

Dz .

+ ~

(6.89)

(6.90)

Dz - Dx

Equation (6.88) is written in a particularly convenient form because it already includes the c o n s t r a i n t Iml 2 = 1. In addition, as is discussed in the sequel, it permits one to readily recognize the nature of equilibria, namely, whether they are energy maxima, minima or saddles. In order to determine LLG equilibria, let us consider the components of Eqn. (6.85) along the three Cartesian axes: (Dz-Dy)mz(my-ay) -

(Dz

-

(6.91)

-0,

Dx) mz (rex -

ax) -

O,

(Dz - Dx) my (mx - ax) - (Dz - Dy) mx (my - ay) - O.

(6.92) (6.93)

Due to the form of Eqns (6.91)-(6.93), it is convenient to divide equilibria into two families: equilibria with mz r 0 and equilibria with mz = O. 9 Equilibrium points with mz r O. In this case Eqns (6.91)-(6.92) immediately lead to mx = ax and my - ay. The value of mz can be then determined from the constraint Iml = 1. This leads to two equilibrium points: mx = ax, my - ay, mz - + v / 1 -

a2 - @,

(6.94)

which are symmetric with respect to the (mx, my)-plane. Notice that solutions of this type are possible only when the field lies inside the region confined by: (6.95) a x2 + ay2 = 1, that is: h2x (Dz - DK) 2

+

h2

- 1.

(6.96)

(Dz - Dy) 2

Equation (6.88) shows that these equilibria are always energy maxima.

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

465

9 Equilibrium points with mz = 0. This class of equilibria is characterized by the condition m 2 + m~ = 1. Thus, their positions on the unit circle of the (mx, my)-plane can be described by the angle coordinate ~b0: mx = cos ~b0, my = sin ~b0. The angle q50 associated with the equilibrium points is obtained by solving Eqn. (6.93) rewritten in the form: (Dy - Dx) sin ~b0 cos (P0 4- hax sin ~b0 - hay cos ~b0 - 0.

(6.97)

In order to interpret the meaning of Eqn. (6.97), it is useful to express the system free energy gL (m; ha) (see Eqn. (6.27)) for mz = 0 in terms of the angle ~b0: gL (mz -- O, qbO" h a ) - - 12D x + ~ l (Dy - Dx) sin 2 ~ 0 -hax cos ~b0 - hay sin ~P0.

(6.98)

This energy expression is identical to the one considered in the Stoner-Wohlfarth model [35], with hAN = Dy - Dx

(6.99)

playing the role of anisotropy field. Equation (6.97) coincides with the equation C~gL/CgC/)o -- 0. It can be inferred from this fact that all the results known for the Stoner-Wohlfarth model can be extended to the equilibrium points characterized by mz = 0. In particular, one immediately concludes that there will be either four equilibrium points, two of which will be energy minima, or two equilibrium points, one of which will be an energy minimum. The bifurcation line where the number of energy minima changes from one to two or vice versa is given by the astroid curve in the (hax, hay) control plane shown in Fig. 6.5:

h2/3 h2/3 ax 4-'~ay

--

h2/3 '~AN"

(6.100)

In conclusion, there are (at most) four qualitative different situations, separated by the two bifurcation lines given by Eqns (6.96) and (6.100) (see Fig. 6.5). 9 Region A, six equilibrium points: two minima, two maxima, two saddles; 9 Region B1, four equilibrium points: one minimum, two maxima, one saddle;

466

CHAPTER 6 Nonlinear Magnetization Dynamics hay hANoi Oz-Oy

hay

1"2min B 1max 211saddle

Dz-Dy

haN

hax

hax

B1

C [I ~nx

~

rmin~ ['L1max

hAN=Dy-Dx

(a)

l,i

(b)

FIGURE 6.5 Bifurcation lines in the (hax, hay) control plane (Eqns (6.96) and (6.100)). (a) Dz - Dy < Dy - Dx; (b) Dz - Dy > Dy - Dx.

9 Region B2, four equilibrium points: two minima, one maximum, one saddle; 9 Region C, two equilibrium points: one minimum, one maximum. In this classification, the number of saddles has been indirectly determined on the basis of the Poincar6 index theorem. Let us notice that for certain values of the parameters Dx, Dy and Dz the region B2 may not exist. The condition for the existence of region B2 is that hAN -- Dy - Dx > Dz - Dy. The classification of equilibria in terms of the free energy is also instrumental in the understanding of their stability properties under constant field. According to Eqn. (6.35), if the system is not at equilibrium, magnetization motion is such that the free energy decreases in time for ~ > 0 and remains constant for ~ = 0. Thus, for a > 0, the dynamics will always bring magnetization away from energy maxima and toward energy minima. On the other hand, for c~ = 0 the magnetization will keep on precessing around an energy m i n i m u m or m a x i m u m if it is initially close to it. We conclude that for a > 0 energy maxima are unstable nodes or foci, and energy minima are stable nodes or foci of the dynamics. Conversely, for a = 0 energy maxima and minima are all centers of the dynamics. 6.3.2 GEOMETRICAL STUDY OF CONSERVATIVE DYNAMICS

General Considerations Exact analytical solutions of LLG dynamics exist for a few special cases only; in fact there is no general solution for the nonconservative dynamics. However, in most experimental situations, as well as in magnetic storage and spintronics applications, the damping constant ~ is a very small quantity. This means that, on a relatively short time scale, the magnetization

6.3 GEOMETRICALAND ANALYTICAL TECHNIQUES

467

FIGURE 6.6 Conservative LLG trajectories for ha = 0. s: energy minima; u: energy maxima; d: energy saddles. (a) Trajectories on the unit sphere; (b) stereographic projection. dynamics is expected to be very close to the undamped dynamics. This circumstance suggests approaching the problem by starting with a detailed analysis of the conservative LLG dynamics, which can be exactly solved by using appropriate analytical techniques. The availability of exact solutions for the conservative LLG equation permits one to treat the dissipative dynamics by perturbation methods in which the zero-order term is given by the conservative dynamic solution. This is the main line of reasoning that we will apply throughout the rest of this chapter. The starting point for the analytical solution of the conservative dynamics is the fact that when a = 0 magnetization trajectories are constrained by two integrals of motion: m x2 4- my2 4. m z2__ 1 ,

(6.101)

Dx 2 Dy m2 Dz 2 Y + - 2 -mz - haxmx - haymy - hazmz - got (6.102) gL (m; ha) -- --~-mx + 2

where go is the energy of the initial condition. Thus, magnetization trajectories can be obtained by intersecting the unit sphere described by Eqn. (6.101) with the ellipsoid described by Eqn. (6.102). An example is shown in Fig. 6.6(a) for the case ha - 0. In Fig. 6.6(b), the same trajectories are represented on the plane by means of a stereographic projection (see Fig. 6.4(b)).

CHAPTER 6 Nonlinear Magnetization Dynamics

468

Unit-Circle Representation and Classification of LLG Phase Portraits We will start our investigation of conservative magnetization dynamics by analyzing the problem from the geometrical point of view. We will make the same assumption made in Section 6.3.1, namely, that one of the Cartesian components of ha is zero. In particular, we consider the case when haz = 0. In this case the free energy, Eqn. (6.27), takes the form given by (6.88), while the LLG equation can be written as follows: dmx d t = (Dz - Dy) mz (my - ay), dmy = - (Dz - Dx) mz (rex - ax),

dt dmz dt

= (Dz - D x ) m y (mx - ax) - (Dz - D y ) m x (my - ay).

(6.103) (6.104) (6.105)

During the magnetization motion the energy is conserved and it is equal to the value go associated with the initial condition. By taking into account Eqn. (6.88) and setting the energy equal to go we end up with the equation: (mx - ax) 2 -}- k 2 (my - ay) 2 - p2,

(6.106)

where: p2

2 2 2 Dz - 2g0 = a x + k ay + Dz - D~"

(6.107)

Equation (6.106) shows that the projection of magnetization trajectories on the (mx, my)-plane is described by a family of self-similar elliptic curves with aspect ratio k, all centered at the point (ax, ay). The family of ellipses is generated by varying the free-energy value go within an appropriate interval. Indeed, for a given value of ha, the constant go controls the size of the ellipses through Eqn. (6.107). It must be underlined that the only admissible portions of the ellipses are those lying inside the circle m x2 + my2 ~<1. Notice that mz does not appear in Eqn. (6.106). In addition, by having a closer look at Eqns (6.103)-(6.105), we observe that, for a given point (mx, my), a change in the sign of mz produces a change of the sign of dm/dt. In other terms, the mirror transformation (mx, my, mz) --~ (mx, my, - m z ) produces the transformation d m / d t ~ - d m / d t . From this symmetry property, one can infer that Eqn. (6.106) actually corresponds to two trajectories or two parts of the same trajectory, with positive and negative mz, yielding identical projections onto the (mx, my)-plane. Based on the previous analysis, one can introduce a useful unit-circle representation of the phase portrait of the dynamics inside the circle

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

my

$2

_ _ --. dl","

\\\\ I-'1

c" i ,, ,x..\ ,

'

,,

,,

' , , ,,

x

i .~rS 1

--... ---

71-

S

.

",,

..\\\\\\\\\\

/

,,

\1 .

myt~ ../-" ,

t/I/P

469

"-

: .

.

',mx

'

.

I

,

,,

\\\\-...

t ,

, i

I

',, i

l

,, i

\\

'

'

'

'

'

tI, " , \ /

'

,I /I

' '

,I ii

' '

'

.,/

1

(a)

(b)

F I G U R E 6.7 Unit-circle representation for two different values of ha. Magnetization trajectories are obtained from the intersection of the ellipses given by Eqn. (6.106) with the unit circle m2 + m~
mx2 + m~ - 1, two examples of which are s h o w n in Fig. 6.7. Although this representation involves only two components of magnetization, information on the third component mz can be immediately obtained by using the s y m m e t r y property discussed above and by taking into account that mz2 - 1 - (m 2 4- m~). For example, mz - +1 w h e n (mx, my) - (0, 0) while mz = 0 w h e n (rex, my) lies on the unit circle. Information about the sign of mz can be obtained by analyzing the unit-circle representation in the light of the mirror s y m m e t r y property previously discussed. Two examples of unit-circle representations are shown in Fig. 6.7. In order to clarify the interpretation of this representation, it is useful to discuss this figure in some detail. The ellipses that do not intersect the unit circle (see trajectory L1 in Fig. 6.7(a)) represent two trajectories in the opposite hemispheres that have exactly the same projection on the (rex, my)plane. In addition, the above two trajectories have opposite senses of rotation. These types of trajectory exist only in the case a 2 + a~ < I and they

470

CHAPTER 6 Nonlinear Magnetization Dynamics

always encircle an energy maximum located at the point (ax, ay). In order to understand how to interpret the unit-circle representation of ellipses that intersect the unit circle, let us consider the special case of ellipse L2 depicted in Fig. 6.7(a). Let us suppose that the motion on the ellipse L2 starts from point 1 and the sign of mz is such that the direction of motion is toward point 2. When the magnetization reaches point 2, mz crosses the zero value and changes sign, the motion continues in the opposite hemisphere and the projection of the trajectory on the (mx, my)-plane retraces the curve L2 but in the opposite direction, from 2 to 1. The critical case separating the two classes of ellipses described above is represented by those ellipses that are tangent to the unit circle. In these cases, at the point of tangency we have that the gradient of the free energy, i.e. the effective field heft , which is normal by definition to the constant-energy ellipsoid described by Eqn. (6.102), is aligned with m. The parallelism of heft and m implies that at the tangency point there is an equilibrium. There are three possible cases of tangency. Examples of these three different cases are the trajectories F1, F2 in Fig. 6.7(a) (tangency from the inside), L3 in Fig. 6.7(a) (tangency from outside with the same convexity), and L4 in Fig. 6.7(b) (tangency from outside with opposite convexity). If one takes into account that an increase of energy go produces a decrease of the size of the ellipse (see Eqn. (6.107)), it is possible to arrive at the following conclusions: tangency from the inside corresponds to saddle points; tangency from the outside with the same convexity corresponds to energy minima; and tangency from the outside with opposite convexity corresponds to energy maxima. The case of ellipses tangent from inside to the unit circle is very important. As previously discussed, this point of tangency is a saddle; the corresponding tangent ellipse is a homoclinic trajectory, i.e. a trajectory starting and ending at a saddle equilibrium, or a heteroclinic trajectory, i.e. a trajectory starting from a saddle equilibrium and ending at a different saddle equilibrium. Trajectories, homoclinic and heteroclinic, passing through saddles are also referred to as separatrices, because they create a natural partition of the phase portrait into different 'central regions'. In order to clarify this notion, let us consider the sterographic-plane phase portrait in Fig. 6.8. A closer inspection of this figure reveals that the phase portrait is divided into five regions separated by the saddle trajectories F1 and F2: two regions associated with the two energy minima Sl and s2; two regions associated with the two energy maxima Ul and u2; and one region (indicated with I) between the two homoclinic trajectories F1 and F2, which does not contain equilibria. These regions are 'central regions' in the sense that they are filled by a continuum of closed trajectories and in this respect they resemble the behavior of a dynamical system around center equilibria.

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

Wy

471

F1

Wx

F3

Stereographic-plane representation of phase portrait. Stereographic coordinates are: Wx = m x / ( 1 + mz), Wy = my~(1 + mz). s: energy minima; u: energy maxima; d: energy saddles. Bold lines F 1 and F2 denote separatrices. The dashed line is the unit circle. The symbol I labels the region between the two homoclinic structures F 1 and F2. The line F3, which starts and ends at infinity, separates trajectories rotating counterclockwise around the homoclinic trajectory F2 from those rotating clockwise around u2. This special property of F3 is an artifact of the stereographic projection and is connected to the singularity of this projection for mz = -1. On the unit sphere, the trajectory F3 is similar to all other trajectories encircling u2. FIGURE 6.8

Some general conclusions about phase portraits can be d r a w n from Fig. 6.8. First, phase portraits m a y be partitioned by saddle trajectories into more than one central region. The qualitative features of the dynamics are defined once a list of the central regions and of the equilibrium points is specified. To this end we will introduce the following notation: energy maxima are indicated with u, energy minima with s, and saddles with d. Each central region will be indicated by the symbol (). If a central region contains an energy m i n i m u m or m a x i m u m , then we will write (s) or (u), respectively. In addition, since in generic situations (i.e. situations with no particular symmetries or fulfillment of bifurcation conditions), a saddle point is always associated with two homoclinic trajectories, we will use the notation d( )( ) to indicate the saddle and the two central regions associated with its homoclinic trajectories. It is remarkable that by this notation all the fundamental properties of the dynamics can be specified in terms of a short

CHAPTER 6 Nonlinear Magnetization Dynamics

472

. /IUv all_ _L_"

my

F2min A|2mad~tleS

_

F1 rain le

B, |2 max

c

m;

m~ C

{(s),d(u) 2 }

{dl(S1) (s2),d2(u)2,() } (a)

(b)

my

my

[-2 min d Bg.| 1max - ~ L 1saddle

~

-

~

F 1 min Llmax /

$2

s1

S

~

m

4 C

u {(u),d(s 1)(s2) } (c)

{(SU)} (d)

FIGURE 6.9 Classification of phase portraits by means of unit-circle representation for different regions in the control plane (hax, hay). In particular, unit-circle representations are shown for applied field values in regions A (a), B1 (b), B2(c), C (d), defined in Fig. 6.5(a), (b). string. For example, the phase portrait shown in Fig. 6.8 corresponds to the string {dl(Sl)(S2), d2(ul)(U2), ( )}. The empty parentheses ( ) correspond to the central region I, which does not contain equilibria. This notation can be further extended to indicate the presence of particular symmetries in the phase portrait when one of the field components is zero. In particular, in the case haz = 0, central regions (), which exhibit mirror symmetry with respect to the plane (rex, my), will be indicated by: ( )2 (see Fig. 6.9(a), (b)). With the help of this notation and of the unit-circle representation, one can arrive at a full classification of the different conservative phase portraits. In Fig. 6.9(a)-(d), unit-circle representations corresponding to the four regions A, B1, B2, C defined in Section 6.3.1 (Fig. 6.5) are shown.

6.3 GEOMETRICALAND ANALYTICAL TECHNIQUES my

473

my

s

(a)

{d2(u)2(s+)(s-) }

(b)

{d2(u)2(s+)(s-) }

(c)

{d-(u)2(s+) }

FIGURE 6.10 Unit-circle representation and corresponding string description of phase portraits under x-directed field. (a) Zero-field case characterized by the existence of symmetric heteroclinic trajectories; (b) portrait under nonzero field when heteroclinic trajectories are still present; (c) portrait under larger field when heteroclinic trajectories have been transformed into homoclinic ones. Figure 6.10 presents the highly symmetric case where the field is aligned along the x-axis. This case is interesting because it exhibits the presence of heteroclinic trajectories. We will use the special notation d2( )2( )(), without commas, to indicate the two saddles connected by heteroclinic trajectories and the four associated central energy regions. 6.3.3 ANALYTICAL STUDY OF CONSERVATIVE DYNAMICS The starting point for the analytical solution of the conservative dynamics is Eqn. (6.106), which shows that the projection of the magnetization trajectory onto the ( m x , my)-plane is an arc of ellipse. Therefore, the trajectory can be expressed in the parametric form: m y - a y + ~ s Pi n u ,

m x - ax - p c o s u,

(6.108)

where the connection between the parametric variable u and time is to be determined. Equation (6.108) implies that the remaining magnetization component m z will be given by: mz -

+ V / 1 - (ax - p cos u) 2

-

(ay + ( p / k ) s i n U) 2.

(6.109)

By inserting Eqn. (6.108) into Eqn. (6.104) one finds: du dt

= k (Dz - Dx) mz.

(6.110)

474

CHAPTER 6 Nonlinear Magnetization Dynamics

Since mz depends on u only, one obtains the desired equation for u(t) by using the method of separation of variables: du

= k (Dz - Dx) dt,

(6.111)

~/1 -- (ax -- p cos u)2 - (ay 4- ( p / k ) s i n U)2

where the '4-' signs are no longer indicated because the last equation is to be construed as an equation for multi-branch analytical functions in the complex plane. Equation (6.111) is solvable in terms of elliptic integrals. This can be shown by carrying out the substitution w = tan(u/2), which transforms Eqn. (6.111) into the equation: dw k ~/P4(w) - 2 (Dz - D x ) d t ,

(6.112)

where P4 (w) represents the fourth-order polynomial: P 4 ( w ) - (1 + w2) 2 - lax(1 + w2) - p(1 - w2)] 2 - [ay(1 + w2) + ( 2 p / k ) w ] 2 . (6.113) Equation (6.112) is precisely of the form permitting integration in terms of elliptic integrals and elliptic functions [45,46]. However, the integration is based on the knowledge of the roots of the polynomial P4 (w), which are generally given by rather complicated formulas.

S o l u t i o n U n d e r Zero F i e l d Before attempting the study of Eqn. (6.111) in all its complexity, it is useful to get some first insight into the properties of conservative magnetization dynamics by considering the particular case of zero applied field. This case is simple enough to be solved without excessive technical complications. Under zero field, Eqns (6.103)-(6.105) become formally identical to Euler equations for the angular momentum of a rigid body [47]. The magnetization components (Eqns (6.108) and (6.109)) can be expressed as functions of w = sin u only, since ax = ay = 0 (see Eqn. (6.89)). One finds: mx = mpv/1

where:

-

w 2,

my - -~w,

mz = •

kj1 - p2

w0 = - -

kip

w2

-- W 2,

(6.114)

(6.115)

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

475

k'2 - 1 - k2, and p2 is linearly related to the system energy through Eqn. (6.107), which for zero field is reduced to: (6.116)

p2 = Dz - 2go. Dz - Dx

By making the change of variable w = sin u in Eqn. (6.111), one obtains the following equation: dw

= D~ dt,

(6.117)

V/(1 - w 2) (1 - k2HW2) where: kH -

1

w0

=

klp

1

k V/1 _ p2 '

f ~ = kv/1 _ p2 (Dz - Dx).

(6.118) (6.119)

The solution of Eqn. (6.117) is the Jacobi elliptic function [45]" w = sn (D.n t, kH).

(6.120)

By inserting Eqn. (6.120) into Eqns (6.114) and by using Jacobi elliptic function transformation rules one obtains the magnetization components: mx = :Fp cn ( f ~ t, k~ ),

(6.121)

my = P; sn (f~t, kH) ,

(6.122)

mz - q-~/1 -

p2

dn (D.Ht, kn).

(6.123)

One finds two sets of solutions, depending on the choice made for the sign of mz. T h e corresponding sign of mx is determined by using Eqn. (6.103). Equations (6.121)-(6.123) depend on the energy go of the conservative motion through the parameter p defined by Eqn. (6.116). This parameter has an intuitive geometrical meaning. As shown by Eqn. (6.106), it measures the size of the ellipse along which the magnetization motion takes place. The energy of the system takes values in the interval (see Eqn. (6.102)): Dx <~go<
CHAPTER 6 Nonlinear Magnetization Dynamics

476

my dl p=k

sa mx

C

FIGURE 6.11 Unit-circle representation of phase portrait under zero field, s: energy minima; u: energy maxima; d: energy saddles. Labels H and L denote high-energy and low-energy regions, respectively. They are separated by doubly-connected separatrices F. shown in Fig. 6.11. The energy maxima correspond to p = 0, where the constant-energy ellipse given by Eqn. (6.106) is reduced to a point, whereas p = I represents the energy minima, where the ellipse is tangent to the unit circle from outside. The heteroclinic separatrix trajectories (bold lines in Fig. 6.11) divide the portrait into the high-energy (H) and low-energy (L) regions. 9 Region H. This region actually consists of two separate regions symmetric with respect to the (mx, my)-plane. The system energy varies in the interval Dy/2Kgo~Dz/2, that is: 0~p2~k 2.

(6.125)

According to Eqn. (6.118), 0KkH~1 in this region, so one immediately concludes from Eqns (6.121)-(6.123) that the magnetization motion is a precessional motion about the hard z-axis. By taking into account the properties of Jacobi elliptic functions, one obtains for the precession period: Tgo=

4K(kH) D~H ,

(6.126)

where K(kH) represents the complete elliptic integral of the first kind [45,46].

477

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

9 Region L. This region consists of two regions symmetric with respect to the (my, mz)-plane, in which the energy varies in the interval Dx /2 <~go<~Dy /2. Therefore: k2 <~p2~<1,

(6.127)

which implies l~
(6.128)

my - ~ ~/1 - p2 sn ({)Lt, kL),

(6.129)

mz - 4-~/1 - p2 cn ( q t, kL) ,

(6.130)

9,L = kHf ~ = k'p (Dz - Dx),

(6.131)

1 k kL --k-~H = k, p ~/1 - p2

(6.132)

where:

It can be verified that 0~
(6.133)

In the following two sections we discuss in some detail the solution of Eqn. (6.111) under nonzero field in two cases of particular physical interest, namely, when the field is perpendicular or parallel to the easy axis of the system.

Solution for Field Perpendicular to the Easy Axis This case, where hax = O, hay ~ 0, is of particular importance for the interpretation of precessional switching phenomena discussed in Section 6.4. To avoid ambiguities, we will assume hay :> 0, i.e. ay < 0 (see Eqn. (6.89)). Let us introduce the variable w = sin u. In terms of w, the magnetization components (Eqns (6.108) and (6.109)) are expressed as: p k'p mx - :Fpv/1 - w 2, my - ay + -~w, mz - +--ff-v/(W - w _ ) ( w + - w),

(6.134)

CHAPTER 6 Nonlinear Magnetization Dynamics

478

where:

kay w+-

k ~

P k' 2 4- ~

k2a~ 1 _ p2 q - ~ k 2, .

(6.135)

By introducing the transformation w --- sin u in Eqn. (6.111), one obtains: dw

= k'p (Dz - Dx) dt.

~/(1 - w 2) (w - w _ ) ( w +

(6.136)

- w)

The solution of Eqn. (6.136) depends on how the four roots of the denominator are ordered for different system energies. The unit-circle representation of the phase portrait of the dynamics for ax = 0, ay < 0 is shown in Fig. 6.12 for the particular case where layl < 1 (see Section 6.3.2 for the general discussion of the portrait modifications when lay l > 1). Since we are going to discuss the properties of w• it is useful to recall that, according to Eqn. (6.134), w is a linear function of my and that the condition w = w• is equivalent to mz - 0, which means that it identifies trajectory points located on the boundary of the unit circle. The presence of the field breaks the heteroclinic connections present in Fig. 6.11 into two pairs of homoclinic trajectories (bold lines in Fig. 6.12). These trajectories divide the portrait into high-energy (H), intermediate-energy (I), and low-energy (L) regions. 9 Region H. This region exists only if lay l < 1. It consists of two separate regions symmetric with respect to the (mx, my)-plane, in which the parameter p varies in the interval: 0~p2~k 2 (1 -lay[) 2 ,

(6.137)

and the root ordering is w_ ~ - 1 < 1 < w+. The variable w varies in the interval (-1, 1) during the magnetization motion. When p = 0 (energy maxima), w_ --. - ~ , w+ ~ + ~ . Conversely, w_ = - 1 , w+ > I when p = k(1 -layi). 9 Region I. This is a single region symmetric with respect to both the (mx, my)- and (my, mz)-planes. The parameter p varies in the interval: k2 (1 -layl) 2 ~pa~k2 (1 + ]ayl)2 , (6.138) and the root ordering is - 1 K w _ < 1Kw+. The variable w varies in the interval (w_, 1) during the magnetization motion. 9 Region L. This region exists only if lay l < k'2/k 2. It consists of two regions symmetric with respect to the (my, mz)-plane, in which the

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

479

my~ F2

s2

s1

m~ C

c

F1

J2

FIGURE 6.12 Unit-circle representation of phase portrait when field is applied along positive y-axis, s: energy minima; u: energy maxima; d: energy saddles. Labels H, I and L denote high-energy, intermediate-energy, and low-energy regions, respectively. Lines F 1 and F 2 are homoclinic trajectories.

p a r a m e t e r p varies in the interval:

k2 (1 + lay[) 2 <~p2~<1 q-

k2@ k12 ,

(6.139)

and the root ordering is - 1 < w_<,w+<~l. The variable w varies in the interval (w_, w+) d u r i n g the magnetization motion. W h e n p2 ._ k 2 (1 + lay]) 2, w_ > - 1 , w+ = 1, whereas - 1 < w_ = w+ < 1 (energy minima) w h e n p2 _ 1 + k2@/k'2. Let us first discuss regions H and L. Region I will require a slightly different treatment.

Solution for Region H.

In order to reduce the left-hand side of Eqn. (6.136) to a form expressible in terms of Jacobi elliptic functions it is useful to carry out a transformation leaving the +1 roots of the d e n o m i n a t o r u n c h a n g e d and bringing the remaining two roots to opposite values. This can be obtained b y the following Moebius transformation with fixed points at +1 [39]: =

w- q ,

1-qw

w -

~+q l+q~"

(6.140)

CHAPTER 6 Nonlinear Magnetization Dynamics

480

The parameter q has to be chosen in such a way that ~+ + ~_ - 0, where ~+ - (w+ - q ) / ( 1 - q w + ) . This leads to the equation q2 _ 2sq + 1 = 0, where: s=

1 +w+w_ w++w_

.

(6.141)

Thus, the above condition is fulfilled w h e n q - s + v/s 2 - 1. These values are real only w h e n s 2 - 1 ~ 0 . It can be verified that s 2 - 1 = ( w 2 + - 1 ) ( w 2 - 1 ) / (w+ + w_) 2, so s 2 - 1~0 w h e n the w+ roots are both outside or both inside the interval ( - 1, 1). This is precisely the root ordering existing in regions H and L, respectively. The M6ebius transformation given by Eqn. (6.140) has a determinant equal to I - q2. Therefore, of the two acceptable q values, it is convenient to use the one for which I - q2 > 0, because this will preserve the order of the polynomial roots w h e n making the change of variable from w to ~. This value is: ( ~ 1) q-s 11-~ . (6.142) Let us apply the transformation given by Eqn. (6.140) to Eqn. (6.136). One finds: d~ = f ~ dt, (6.143) ~ ( 1 - ~ 2 ) ( 1 - k2 ~2) where: =

G

(1 qw)2 (1 qw)2 --

w+-q

-- k'p

~l

w_ -q

,

(6.144)

-q 2

k2 _ q2 (Dz - DK).

(6.145)

The argument of the square root in Eqn. (6.145) is certainly positive. In fact, it can be verified that (1 - q2)/(k2 - q2) _ (q _ w _ ) ( w + - q)/(1 - q2). The latter quantity is positive because w_ < q < w+ as a consequence of the order-preserving property of the w --+ ~ transformation. Indeed, w = q is transformed into ~ = 0 by Eqn. (6.140). By integrating Eqn. (6.143) one obtains a solution for ~ in terms of the Jacobi elliptic function [45,46]: - sn (f~t, kH).

(6.146)

By using Eqns (6.146) and (6.140) in Eqn. (6.134) one obtains the complete solution for the magnetization components in Region H: mx - :FP ~/1 - q2

CnH 1 +qsn H'

(6.147)

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

my--ay+;

P q+snH l+qsn H

(6.148)

-_92921

mz - +k'p V/lk - q2 ~ k 21

481

+' dq ns nH H

(6.149)

where snH, cnH, and dn H denote the corresponding Jacobi elliptic functions of argument (f~t, kH). According to Eqn. (6.144), 0 < kH < 1, since k2 = 1/~ 2 and ~+ > 1. The presence of the 4- signs reflects the fact that region H is made up of two parts symmetric with respect to the (mx, my)-plane (see Fig. 6.12), covering the same energy interval. The precession period is Tg0 = 4K(kH)/f ~.

Solution for Region L. Equations (6.147)-(6.149) are valid for region L too. The only important difference is that k2 > 1, because now ~+ < 1. The solution can be reduced to elliptic functions with modulus smaller than 1 by using k n o w n Jacobi function identities [45,46]. The final result is" /

mx - :FP ~/1 - q2

dn L 1+

P q + kL snL m y - - a y + - ~ 1 + qkL snL

mz - +k'p v/1 k

- 92

(6.150)

qkL s n L '

~ 1 - 92 kL cnL 1 - q 2 k 2 1 + q k L sn t '

(6.151) (6.152)

where k2 -- 1/k2. The symbols sn L, cnL, and dn L denote the corresponding elliptic functions of argument (f~ t, kL), where q - kHf~. The precession period is T30 - 4 K (kL)/f~.

Solution for Region I. In this case, the reduction of the left-hand side of Eqn. (6.136) to canonical elliptic form can be achieved in two steps. First, the o r d e r i n g - 1 < w_ < 1 < w+ is transformed into w*_ < - 1 < 1 < w+* by a transformation leaving I unchanged and transforming w_ into - 1 . This intermediate transformation is: 1-w w * - l - 2 1 _ w ~_ ,

w-l+

1 ~ ( 1 - w_ ) (w* - l) 9

(6.153)

In terms of w*, Eqn. (6.136) takes the form: dw* ~/(1 - w . 2 ) (w* - w * ) ( w ~ _

= c k'p (Dz - Dx) dt, - w*)

(6.154)

482

CHAPTER 6 Nonlinear Magnetization Dynamics

where c = (1 - w_)/2 and: w* - 1 w

4 l-w_

1 -w+ w+* - 1 - 2 ~l - w _

~

9

(6.155)

Equation (6.154), with w* < -1 < 1 < w+, can be solved by the method previously used for region H. By introducing the notations: s * _1 + _ w+w_ * ,

(6.156)

+ w*_ '

q*-s*

( j 1) 1-

l-s- ~

,

(6.157)

one can immediately write the solution in the form: ~* = sn (f~ t, k~),

(6.158)

where: k2(1-q*w~-) W~_

m

q*

2

=

(1- q'w*) 2 w* q* m

~l-q .2 -ck'p

k/2 q. 2 ( D z - D x ) .

(6.159) (6.160)

The connection between ~* and the magnetization components is more complicated than before, because of the intermediate transformation from w to w*. After algebraic manipulations analogous to those carried out for region H and an appropriate change of the time origin, the following analytical expressions are obtained: m x - - C p (1--q*)~~+q._q* ~/(dni-/~dni -cnI)q*cni(1+ dni) cn (f~ t/2, kx) ,

(6.161) p cp(1-q*) my -- ay q- - --

k

cpk'(1 +q*) / l - q *

k

dni+cni , dni - q* cnx

(6.162)

~/(dni +kicni) ( l + d n i ) ki, sn(f~t/2,/~) mz = k ~ kI + q* a n / - q* cni cln i~t/-2: ~-)' (6.163) where k/2 _ 1 -k~2. The symbols sni, cni, and dni denote the corresponding elliptic functions of argument (f~ t, k~).

483

6.3 GEOMETRICAL AND ANALYTICAL TECHNIQUES

S o l u t i o n for F i e l d A l i g n e d A l o n g t h e E a s y A x i s This case, where hax ~ O, hay = 0, will be instrumental in the analysis of magnetization relaxation discussed in Section 6.4 and in the interpretation of spin-torque phenomena given in Section 6.5. To avoid ambiguities, we will assume that hax > O, i.e. ax < 0 (see Eqn. (6.89)). Let us introduce the variable w = - cos u. In terms of w, the magnetization components (Eqns (6.108) and (6.109)) are expressed as:

mx - ax + w, m y - 4 - p-~ v / 1 - w 2 ~

m z = 4- ~.~- v / ( w -

w_)(w

-

w+),

(6.164) where:

kaax 1 ~p ( a2~ w+ - k,2p 4- --k'p 2 _ k2 1 - k, 2 j .

(6.165)

We will see that the argument of the square root can become negative under certain conditions, that is, w+ can become complex conjugate. By introducing the transformation w = - c o s u in Eqn. (6.111), one obtains: dw

= k'p (Dz - Dx) dt.

(6.166)

~/(1 - w 2) (w - w_)(w - w+) The solution of Eqn. (6.166) depends on the nature of the roots w+ that may be real or complex conjugate. When the roots are real, the nature of their ordering with respect to • affects the solution as well. The unit-circle representation of the phase portrait of the dynamics for ax < 0, ay - 0 is shown in Fig. 6.13 for the particular case where laxl < k'2. Contrary to what happens when the field is applied along the y-axis (see Fig. 6.12), the presence of the field does not break the heteroclinic connections present in Fig. 6.11, so the portrait is divided into high-energy (H) and low-energy (L) regions similar to those present under zero field. 9 Region H. This region exists only if lax l < 1. It consists of two separate regions symmetric with respect to the (mx, my)-plane, in which the parameter p varies in different intervals, depending on whether region L_ exists or not: 0~p 2~k 2 ( a12-) ~ 0~p2~ (1 -laxl) 2

iflaxl~k'2 if laxl ~k '2.

(6.167)

The roots w+ are complex conjugate if p2 < k2 (1 - a2/k '2) (see Eqn. (6.165)), otherwise they are real and follow the ordering w_ < w+

CHAPTER 6 Nonlinear Magnetization Dynamics

484

my dl

F

L_ S

S1

m~

d2

FIGURE 6.13 Unit-circle representation of phase portrait when field is applied along the positive x-axis, s: energy minima; u: energy maxima; d: energy saddles. Labels H and L+ denote high-energy and low-energy regions, respectively. They are separated by doubly-connected separatrices F.

< - 1 < 1. The variable w varies in the interval (-1, 1) during the magnetization motion. 9 Region L+. This is the low-energy region around the energy minimum at mx = 1. Under positive field (i.e. ax < 0) this region is always present. The parameter p varies in different intervals, depending on whether region L_ exists or not:

k2 ( 1 - - ~ a 2 ) ~ p a ~ ( l + l a x l ) 2 (1 -lax]) 2 ~p2~ (1 + iaxi)2

if laxl~k '2

if laxl/>k '2.

(6.168)

The corresponding root ordering is - 1 < w_ < w+ < 1 if p2 < (1 - iaxl)2, otherwise w_ < - 1 < w+ < 1. The variable w oscillates in the interval (w+, 1) during the magnetization motion. 9 Region L_. This is the low-energy region around the energy mini m u m at mx = -1. Under positive field (i.e. ax < 0) this region exists only if lax I~k '2. The parameter p varies in the interval: k 2 l - k - -a2 ~ ) ~p2~ (1 -laxI) 2

(6.169)

6.3 GEOMETRICALAND ANALYTICAL TECHNIQUES

485

and the corresponding root ordering i s - 1 < w_ < w+ < 1. The variable w oscillates in the interval (-1, w_) during the magnetization motion. The analytical solution for the magnetization motion in the various energy regions can be obtained by applying the same strategy used in the previous section for the case of field transverse to the easy axis. Refer to that section for the definition and discussion of the various parameters used. Case 1. When the w+ roots follow the ordering - 1 < w_ < w+ < 1 (region L_; region L+ for p2 < (1 -laxl) 2) or when they are complex conjugate (region H for p2 < k 2 (1 - a2/k'2)) the solution is obtained by applying to Eqn. (6.166) the transformation Eqn. (6.140). This leads to:

d~

= :Ff~ dt,

~(1- ~2)(~2_k~2)

(6.170)

where the m signs respectively refer to regions L+ and L_, while:

k~2

=

(w+-q)

2

1 -qw+

-

( w--q ) 2 , 1 -qw_

1_q2 q

- k'p

1 - qak[2 (Dz - Dx),

(6.171)

(6.172)

and q is defined by Eqns (6.141), (6.142), and (6.165). The solution of Eqn. (6.170) is: = + d n ( q t, kt), (6.173) where k2 - 1 - k~2. 9 Regions L+ and L_. By using Eqns (6.173) and (6.140) in Eqn. (6.164), one obtains for the magnetization components: q + drk

(6.174)

mx - ax + p l + q d n L'

P ~/1 - q2

my = --~

kL srk

1 + q dn L '

mz - + k ' p v / 1 - q2 ~ 1 - q 2 k 1 - q2k~

kL crk

2 1 4- q dn L '

(6.175)

(6.176)

486

CHAPTER 6 Nonlinear Magnetization Dynamics where the 4- signs respectively refer to region L+ and region L_, whereas snL, cnL, and dn L denote the corresponding elliptic functions of argument (ilLt, kL). The precession period is Tg0 = 4 K(kL)/ilL. 9 Region H. Equations (6.174)-(6.176) are also valid for region H. However, Eqn. (6.171) shows that kL2 < 0 when w+ are complex conjugate, so k2 > 1. One can use the transformation rules obeyed by Jacobi elliptic functions [45,46] to reduce the solution to a form involving functions with a modulus smaller than 1. The result, after taking proper account of sign consistencies, is: q + cnn

(6.177)

q2

(6.178)

mx = a x + p l + q c n n ' P ~/

my=-}-;

mz

1-

snn l+qcn n'

_ :Fk'p v/1 - 92 /

k

1 - 92 d______~__~ Vk2 q- qakH2 1 + q cn n '

(6.179)

where k2 -- 1/k2, kH2 = 1 - k 2, and the two signs refer to the two parts with opposite values of mz of which region H consists. The symbols snn, cnn, and dn n denote the corresponding elliptic functions of argument (f~at, kn), where f~ - kL q . The precession period is Tg0 = 4 K(kn)/f~a. Case 2. When the w• roots follow the ordering w_ < - 1

< w+ < 1 (region L+ for p2 > (1 -laxl) 2) or w_ < w+ < - 1 < 1 (region H for e 2 > k 2 (1 - a2/k'2)) the solution is obtained by applying to Eqn. (6.166) the sequence of transformations used in the previous section for Region I. By first applying the intermediate transformation given by Eqn. (6.153) one obtains: dw*

~/(1 -

w .2)

(w* - w*)(w*

= c k'p (Dz - Dx) dt,

(6.180)

- w~_)

w h e r e - 1 < w*_ < w+* < l for Region L+ and - 1 < w+* < W*_ < l f o r Region H. Equation (6.180) can be solved by the method used for Case 1. Because of the different ordering of w* and w~_ in the two regions, different results are obtained for the magnetization motion. 9 Region L+. 1 - dn" mx - ax + p - c p (1 - q*) 1 + q'dn" '

(6.181)

6.3 GEOMETRICALAND ANALYTICAL TECHNIQUES cp(1 - q * )

487

my =

--

1 + q* /dn* + k*' k* sn* 1 - q ' k * ' v lq-dn--; l + q * d n * '

(6.182)

mZ

ck'p(l+q*)~ k

l-q, ~l+dn* k*cn* (6.183) 1 + q'k*' dn* + k*' 1 + q*dn*"

9 Region H. 1 -dn* mx - ax + p - c p (1 - q*) 1 + q*dn* ' my=+-

cp(1-q*) k

l+q*

(6.184) 1

1 + q'k*' ~(dn* + k*')(1 + an*)

k*2 sn*cn* x 1 4-q'dn" ' c k'p (1 4- q*) mz

-- T

(6.185) ~/ l - q * ~/(dn* 4-k*')(1 + a n * ) V 1 - q'k*'

1

x I + q'dn""

(6.186)

The two signs correspond to the two parts with opposite mz of which region H consists. In the previous expressions, q" is defined by Eqns (6.155)-(6.157), and (6.165), while: k,,2

--t

w+*--9* ) 2 1 - q* w~_

(W*- - 9* ) 2 1 - q* w*

'

(6.187)

and k" 2 = 1_k,,2. The symbols sn', cn', and dn" denote the corresponding elliptic functions of argument (fF t, k*), where:

fF = k'p

1 - q,2 1 - q*2k*'2 (Dz - Dx).

(6.188)

The product sn*cn* in Eqn. (6.185) oscillates at the same frequency as dn*. This is true also for the other magnetization components, which means that the period of the precessional motion is Tg0 = 4 K ( k ' ) / ~ * in region L+ but Tg0 = 2K(k*)/fF in region H.

488

CHAPTER 6 Nonlinear Magnetization Dynamics

6.4 M A G N E T I Z A T I O N

SWITCHING AND RELAXATION

Magnetization switching of magnetic particles and thin films is one of the fundamental issues in applied magnetism. This topic has received considerable attention in connection with the enormous advances in magnetic storage technologies and spintronics [13,22]. Materials and devices used in these areas (e.g. hard-disk magnetic recording materials, magnetic reading sensors such as MR and GMR heads, magnetic RAM elements, etc.), have typically nanoscale dimensions and operate in the GHz range. The theoretical understanding of magnetization dynamics and switching under these conditions is of great relevance to these technologies. Traditionally, the study of magnetization reversal has been carried out by using the classical Stoner-Wohlfarth model [35]. However, in the frequency range relevant to magnetic storage applications, magnetization precession has a crucial influence on magnetization dynamics and therefore has to be appropriately taken into account. In this section, we will study magnetization switching processes driven by an external applied field by using the LandauLifshitz-Gilbert (LLG) equation. In most studies of this subject the relevant equations are solved numerically. In contrast, our approach is based on systematic and rigorous analysis of the LLG equation by using the methods of nonlinear dynamical system theory. These techniques have been outlined in the previous sections of this chapter and turn out to be very effective in the analysis of magnetization switching. In particular, these techniques yield detailed information about the parameters that enter in the design of devices (e.g. critical fields for switching, pulse duration, frequency of precessional motion, and relaxation time in 'ringing' processes). 6.4.1 SWITCHING UNDER RECTANGULAR FIELD PULSES

In this section, we present a complete analysis of the precessional switching process in the case of rectangular applied field pulses. Under usual experimental conditions, this kind of switching is realized in nanoscale thin-film elements [13] by the following steps (see Fig. 6.14). The magnetization is initially along the film easy axis and a field approximately orthogonal to the easy axis is applied in the film plane. The field produces a torque that tilts the magnetization out of the plane. This results in a strong vertical demagnetizing field that yields an additional torque that forces the magnetization to precess in the plane of the film away from its initial position. Magnetization reversal is realized by switching the field off when the magnetization is close to its reversed orientation. After the field is switched off, the magnetization starts to relax due to dissipation and finally reaches the reversed equilibrium state.

6.4 MAGNETIZATION SWITCHING AND RELAXATION

489

FIGURE6.14 Different stages of precessional switching. (a) The torque exerted by the applied field tilts the magnetization from the position m (0) to the out-of-plane position m (1). (b) The resulting vertical demagnetizing field hM produces an additional torque that moves the magnetization from the position m (1) to the position m (2) toward the reversed state -ex.

Switching processes realized in magnetic storage devices are based on other principles. The external field is applied in the direction opposite to the initial magnetization and the reversal is achieved by lowering the free energy of the system. In this process the role of dissipation is crucial and, for this reason, this kind of switching is referred to as damping switching [22]. In contrast, precessional switching is accomplished by controlling the magnetization precession within a time interval so short that the role of dissipation is negligible. This type of switching can be considerably faster and it may require lower applied fields in comparison with traditional switching. However, the switching is realized only if the field pulse duration is accurately controlled in such a way that the field is switched off precisely when the magnetization is close to its reversed orientation.

Precessional Switching: Phase-Portrait Description The basic properties of the precessional switching process are revealed by the investigation of its phase portrait. The case when no field is applied is shown in Fig. 6.15(a). The phase portrait is characterized by six equilibrium points: Sl, s2 (energy minima); Ul, u2 = +cx~ (energy maxima); dl, d2 (saddles). Heteroclinic trajectories connect one saddle to the other. All trajectories, except the heteroclinic ones, enclose one equilibrium point. The shaded regions indicate the low-energy central regions (potential wells) around the energy minima located at m = ex and m = -ex. The remaining white regions are the high-energy central regions around the energy maxima Ul and u2. These shaded and white regions will often be

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CHAPTER 6 Nonlinear Magnetization Dynamics

FIGURE 6.15 Stereographic-plane representation of phase portraits for conservative LLG dynamics. The dashed line represents the unit circle. (a) ha = 0; (b)

ha = hayey (hay > 0).

superimposed to subsequent diagrams in order to give an intuitive idea of the expected dynamics after the field is switched off. According to Fig. 6.15(a), no magnetization switching is possible under zero field, because there is no trajectory connecting the state m -- ex to the shaded region around m = -ex. The phase-portrait modifications consequent to the application of the field along the positive y direction are shown in Fig. 6.15(b). The equilibrium points are displaced to new positions and the heteroclinic trajectories are broken into homoclinic trajectories, that is, closed trajectories starting from a saddle and coming back to the same saddle. A so-called homoclinic structure consisting of a pair of homoclinic trajectories is formed for each saddle. For example, the homoclinic trajectories 1~1 and F2 form the homoclinic structure associated with dl. In this situation, a new type of magnetization trajectory appears in the phase portrait, namely, trajectories that enclose the homoclinic structures (e.g. curve L in Fig. 6.15(b)). It is along this type of trajectory that magnetization can move from one shaded region to the other and, thus, realize the switching. Indeed, if m is initially in the position m = ex in Fig. 6.15(b), the switching is realized if the field is switched off in the time interval during which magnetization is between points I and 2 on the trajectory L. When the field is switched off, the magnetization remains trapped in the potential well around the reversed state m = -ex (see Fig. 6.15(a)). Subsequent relaxation due to the dissipative nature of the dynamics will bring the magnetization to the final equilibrium state m = -ex. It is evident from this picture that switching becomes possible only when the applied field is large enough that the homoclinic

6.4 MAGNETIZATION SWITCHING AND RELAXATION

491

FIGURE 6.16 Stereographic-plane representation of phase portraits of dissipative LLG dynamics for ha = 0. (a) Numerically computed basins of attraction of stable focus Sl (white region) and stable focus s2 (shaded region); (b) qualitative sketch of dissipative dynamics (shaded regions represent the central regions of the corresponding conservative dynamics (see Fig. 6.15)). trajectory F1 does not enclose anymore the initial state m -- ex. It will be shown below that the critical field at which F1 passes through the point m = ex is hay -- ( D y - D x ) / 2 . A field pulse of amplitude exceeding this threshold will guarantee switching, provided the field-pulse duration is properly tuned. If the magnetic field is switched off when the magnetization is not between points I and 2 but in the high-energy regions of the phase portrait, the result of the subsequent relaxation to equilibrium is practically uncertain. This is due to the very convoluted and closely entangled nature exhibited in these regions by the trajectories of the dissipative dynamics leading to different final equilibrium states Sl or s2 (see Fig. 6.16). Indeed, one realizes that there exist two basins of attraction in the dynamics: the basin of initial states which will end up in Sl and that of initial states which will end up in s2. The high-energy regions of the phase portrait are very fine mixtures of these two basins of attraction, the smaller the damping constant a, the finer the mixture. This fine entanglement may lead to a seemingly stochastic nature of precessional switching if the field is switched off when magnetization is in the high-energy regions, because in that case the final equilibrium that will be reached is strongly dependent on the exact initial condition. This seemingly stochastic nature of precessional switching could partly explain various experimental results presented in the literature. In Fig. 6.17, the experimental results obtained in [23] are shown. The outcome of the experiment is characterized statistically by representing the 'probability' of switching versus the applied field-pulse duration.

CHAPTER 6 Nonlinear Magnetization Dynamics

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