Effective Field Approach to Phase Transitions And Some Applications to Ferroelectrics (World Scintific Lecture Notes in Physics)

S = -—

=

S0-kBT2n rL X i+i

8K3)

Jo

k2dk

ad (cos 6) -etc + 4np2 cos2 6 [a(T - Tc) + *fc2]

*-(s)(^)r^ /• +1

ad(cos 6>)

(65)

The second integral can then be easily carried out, resulting in

•-*- m^i: tan

cos e/y/{a{T

S0-

.-|1

- Tc) +

y/(a{T - Tc) + =

A;2 dfc 6k2/top2)

5k2/top2)

1^) (£) l ^ fc2d^4^2/(a(T-Tc) + ^ )

x tan - 1 ^ ^ / ( a ^ - T J + Jfc2).

(66)

Landau Theory and Effective Field

77

Approach

The region of interest is T = Tc, where k = 0; therefore, one can make the approximation t a n - 1 y/4np2/(a(T

- Tc) + 5k2) « tan _ 1 (oo)

(67)

and then write „

„

fkBTv\

( a \

/ - — ^ n fkm™

k2 dk y/a{T - Tc) + 6k2

(68)

k2dk • (a(T - Tc) + 5k2)3/2

(69)

and, for the specific heat,

ACp

=

vTdf

~

Cp0

kBT2a2 Sn3/2p J JQ

+

Sufficiently close to Tc, a(T — Tc) becomes much smaller than <5fc^ax, and then p

~ °

p0+

(kBT2a2\{a{T-Tc)/8f/2 ^ 87r3/2p J (a(T - Tc))3/2 {6/a{T-Tc)f/2k2dk

Jo

[1 + (S/a(T -

kBT2a2\ \J^72jJ{s^J

Tc))k2f2

fkBT2a2\ Cp0 +

r J0

rmax

/

1 \

(

1 \ Jo

Cp0+

i

x2dx (1+x2)3/2

V"8^7^; [smj 2 x2dx fXm" _ x dx (l + x2)V2J0 (1 + x 2 ) 3 /

(70)

where x = [S/a(T - Tc)]k.

(71)

We can approximate the two definite integrals in Eq. (70) by means of 1/2

y/S[l

= 1/3,

[In x}*-"™ = In

OtTr

1/2

T-Tr

,

(72)

78

Effective Field Approach to Phase

Transitions

which, substituted into Eq. (70), give

1+ 1

l n/ S

1/2

3 2 fe

1 2

"'ma;*

~ All

T-Tr V Tc

(73)

where (T - T c )/T c < 1, and therefore ln((T - Tc)/Tc) < 0, which implies 1,

'

T-Tc

T

_

T

ln

Using this result and consolidating constant terms together with Cpo into C'p0, we can finally write

AC ~ a + (

kBT2a2

)

In

T-Tr

(74)

in which logarithmic correction terms show up. The calculation is fairly laborious but the final result is a rewarding one. This expression, Eq. (74), can be used to analyze experimental data for the specific heat of ferroelectric triglycine sulfate (TGS). Figure 1.9.2 shows 6C,relative units

(T-Tcl,KFig. 1.9.2. Extra specific heat of ferroelectric triglycine sulfate (TGS) above T c , after subtracting the non-singular part, as a function of ln(T — T c ).

Landau Theory and Effective Field

79

Approach

these d a t a in appropriate form for analysis of t h e logarithmic correction t e r m near a n d above T c . T h e above result indicates t h a t t h e effects of fluctuations become prominent when 5

*->i, ' - ^

s

( £ ) (!)'•

<75>

where fcmax = 7r/6, with b t h e lattice constant in t h e polar axis direction. Equation (75) establishes a criterion t o determine how close t o Tc fluctuations become significant. For T G S , we can see in Fig. 1.9.2 t h a t at A T < 20 K fluctuations become appreciable. Then, using t h e fact t h a t aEc = (47r/C)T c « 1 for T G S a n d t h a t b = 12.6 x 1 0 _ 8 c m , we can get an estimate of the coefficient 6 by means of Eq. (75), ™ - K ( S ) (

322

\{l))

1

V

i.e.,<5«10-16esu.

Vl2.6x 1 0 " V '

(76) y

'

This numerical estimate justifies, for the case of T G S , a t least, neglecting the t e r m 1 / 2 / V in Eq. (41), because, using f3 = 1 / 3 P S Q 3 « 2.8 x 1 0 _ 1 3 e s u for T G S , one can check directly t h a t

J/3P4 « «5(VP)2 « SfcLx-P2 * s(j)

P2

(77)

since substituting numbers on t h e left-hand side of the inequality is several orders of magnitude smaller t h a n t h e right-hand side.

References 1. See, e.g., L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon Press, Oxford, 1980). 2. See, e.g., P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and to Critical Phenomena (John Wiley, New York, 1977). 3. See, e.g., B.A. Strukov and A.P. Levanyuk, Principios de ferroelectricidad (Ed. Universidad Autnoma de Madrid, Madrid, 1988) (Translated from the original in Russian by F. Agullo Rueda). 4. K. Ema, Y. Ikeda, M. Katayama and K. Hamano, J. Phys. Soc. Japan 49B, 181 (1980).

Chapter 1.10

Equation of State and the Scaling Function* The scaling hypothesis has been the starting point for most recent developments in the physics of critical phenomena. 1-5 Several attempts to obtain explicit scaling functions h(x), which specify the corresponding equation of state, have been made, 6 - 9 most of them showing good agreement with selected experimental data, and/or numerical calculations for specific theoretical models. One particularly simple expression10 can be obtained directly from the scaling assumption together with the additional assumption that the critical point $ = (di/dM 1 / | S ) = constant (with k = H/M). This simple expression was shown to be in fair agreement with experimental data for several transitions, 10 ' 11 and with numerical results 10 for the Ising and Heisenberg models. It was later subject to criticism 13 on the grounds that, in general, one should make allowance for $ = $(a;), being x = t/M1^, thus restricting its applicability to regions where $ could be expected to remain very nearly a constant. In the present work, it will be shown that two different approximate expressions for the scaling function near a magnetic critical point can be obtained along paths of constant k or constant t. These two expressions will be shown to describe more accurately h(x) in regions near the critical isotherm (t = 0) and near the H = 0 line. It is useful to start the discussion by considering the mean field model case, which, in spite of lacking the complexities characteristic of most real systems, affords a convenient testing ground. Consider, for instance, an ideal ferromagnetic system whose behavior could be described by the Weiss theory (mean field). Its equation of state *Work previously published under the title "Complementary expressions for the scaling function near a magnetic critical point", J.A. Gonzalo, J. Phys. C: Solid State Phys. 1 3 , 241 (1980). Copyright © 1980 IOP Publishing Ltd. 81

Effective Field Approach to Phase

82

Transitions

can be written as H = (l + t) t a n h - 1 M - M,

(1)

where H = [H]/[H}0, M = [M]/[M]0, and t = (T - Tc)/Tc are dimensionless variables representing the magnetic field, the magnetization, and the temperature difference to the Curie point, respectively. The critical exponents have their classical values /3 = 1/2,5 = 3,7 = 1. Dividing both sides of Eq. (1) by M 3 and developing t a n h - 1 M in a power series, near the critical point, we have H M3

1 t /tanh_1M = —l 1- I M 3 M2 V _1 /tanh M \ 1

n

1

1

.A

1

)

M2

M 3

and taking into account that near the critical point M?
(h = H/M5,

x = t/M1/l3),

(3)

which is consistent with the scaling assumption H = Msh(t/M1/0),

i.e., h = h(x)

or

x = x(h).

(4)

From Eq. (2) it is clear that the scaling assumption may lead to incorrect results for h w 1/3 or x « —1/3, corresponding, respectively, to very small x ox h (i.e., to x or h comparable to M2 or t). In those cases, the terms in parentheses in Eq. (2) cannot be neglected "a priori.'" Also, whenever derivatives of the main variables are evaluated (for instance, % _1 at H = 0, t > 0), approximations like Eq. (3) may lead to incorrect results. Equation (1) can be rewritten in terms of the variables k = H/M, t, and M, in either of the two following forms: t= (l +

fc)(M/tanh_1M)

k= ( l + i ) ( t a n h

-1

- 1,

(5a)

M/M) - 1,

(5b)

from which the special derivatives $ — (d£/dM 1 / /3 )/ c and ip — (dt/dMs~1)t, taken, respectively, at k = constant and t = constant, can be easily

Equation of State and the Scaling

83

Function

computed * = (1 + jfe) [tanh - 1 M - M(\ - M 2 ) - 1 ] / 2 M ( t a n h " 1 M ) 2 , 2

ip = (1 +1) [M(l - M )-

1

1

3

- tanh" M] / 2 M .

(6) (7)

For M 2 <S 1, they become, respectively, $»-I(l +

fc),

^ « |(l+t),

(8) (9)

approaching a constant (with the same absolute value for both cases) when fcCl and t < l , i.e., in the vicinity of the critical point (k = 0, t = 0). Let us turn our attention now to the more general case of scaling with non-classical critical exponents, where, in general, 1/(3 ^ (S — 1) and consequently 7 = 3(5 — 1) ^ 1. We will discuss the possibility of obtaining two expressions for the scaling function obtained taking $ = constant and ip = constant, which should be valid in two neighborhoods of the critical point located near the t = 0 and H = 0 lines, respectively. (i) From k = H/M = Ms~lh(x), one can get, by differentiation at k = k(Ml/^,t) *=(-*>-) \dMW)k

(10) = constant,

=yV™1/0=x-3(6-1)*®w } dk/dt h'{xy

(11) (

}

In a certain close vicinity of the critical point where M1^ -C 1, 0 < k < l , a very weak dependence of $ on M 1 ^ and k may be expected (in analogy with the mean-field case). Hence, taking $ « $o along x = 0, we can integrate Eq. (11), obtaining X [* !£*idx = 3(8 1) [ ^—dx, Jo h(x) 'Jo ^ - * o i.e.,

h(x) h(0)

( x -$ V -*o

0(6-1)

(12)

or taking h(0) = ( - $ 0 ) w _ 1 ) , h(x)=(x-^0fS-1) = (x-%y.

(13)

84

Effective Field Approach to Phase

Transitions

The integration has been performed along lines of constant k with origin in the critical isotherm (x = 0) either toward the low-temperature phase (x < 0) or toward the high-temperature phase (x > 0). Then, we should expect less accurate results as we approach x = 3>o and x « +oo, which are farther away from x = 0, where <&o has been defined. (ii) From t = Ml^x{h),

(14)

by differentiation at t = t(Ms~1, k) = constant, ( dfc \dMs-l)t i

II

n,

\

l

1 dt/dM'-rS-l dt/dk

ill,

i t i i v f

=

1 x{h) (3{8-l)x'{h)'

l

'

In order to proceed as in (i) to take H = 0 as the starting line we need to know whether ip (H = 0) is well behaved for M&~1 M5 + • • •. w h i c h should be valid for HmO, \t\ fixed. One gets I/J = 2as(6 — l)~1M3~s, which remains finite at t < 0 (where M —> Ms) but diverges at t > 0 (where M —> 0). Then one must distinguish between these two cases. Integrating Eq. (15) along lines of constant |i| with origin in h = hi, we obtain, if tp\ does not depend too drastically on h,

Jhi x(h)

(3(6 - 1) Jh.

h-ij>i

i.e.,

x(h)

( h — ip.,

l//3(<5-l)

(16)

For t < 0 (a; < 0), we can take Hi = 0; hence, hi = Hi/M^ = 0 and

_ xl-HiiM. ^

x

~ (6 - l)Ml-

_

r:W-i>

~ (6 - l)(t/x(0))W-V

_ r^_ ~ 6 - l

ls_1} = ( X m

~ V°-

If the upper limit of integration in Eq. (16) remains h -C 1, we should not expect large departures of i[>(h) from ^o- Thus, for t < 0,

-x(h)

(y>o - hyw*-1)

1/7

Equation of State and the Scaling

Function

85

For t > 0 (x > 0), on the other hand, we cannot take Hj = 0 (Mi = 0) as the starting line, since for S > 3 we would have hf =* r " 1 ^ ^ - 1 ) / - ^ - 1 -+ oo and V/" = 2 oo along this line (we note that hf » V>i" as iWj —» 0). It may be argued, however, that taking an arbitrary large hf = constant as the starting line (which can hence be set arbitrarily close to H = 0 at t = 0, but departs increasingly from H = 0 as t increases) one can write •£W._

(ft-^i)

1

i e

0<x=(h-4,+)

h

(17b)

using a;+ = (/i+ — ^ " ) 1 /' ? (' J_ 1 ). It seems reasonable to expect that expression (17a) for t < 0, and the less satisfactory approximation (17b) for t > 0, should work reasonably well for h « 0 and /i « /i^~, respectively, but also that they fail as we move away from H = 0. It was pointed out that the simple expression given by Eq. (13) appears to be in fair agreement with experimental data for various systems (ferromagnetic and non-ferromagnetic) as well as with numerical calculations of h(x) for the Ising and Heisenberg models. However, as previously indicated, one may expect difficulties at and near the H = 0 line, both for t < 0 and t > 0, since o was defined at x = 0, which is as far as it can be from x = $o (for t < 0) and x — +oo (for t > 0), respectively. In particular, for t < 0, x —> <&, Eq. (13) would imply an infinite zerofield susceptibility at the coexistence curve when j3(S — 1) = 7 > 1, i.e., Xk1

=

OH dM

H=0

= [SM'-^x-Qo)™-1) - (6 - l)M*-\x

-

fco)**4-1'-1]^.

(18)

On the other hand, for t > 0, x —• 00, the power series expansion of h(x), defined by Eq. (13) is given by hk{x) = (x- $ o ) W _ 1 ) = x^-V{l

+ [(3(6 - l X - S o ) ] * " 1 + £ } (19)

at variance 12 with the well-known Griffiths expansion 4 h(x) = a^C- 1 ) ] T Hxx2W-V

= xW-V [H, + H2x-2?

+ L]

(20)

except in the leading term. The latter series converges for M < M0(t), thus establishing an upper value for M. Equation (13) then leads to incorrect

86

Effective Field Approach to Phase

Transitions

results at, and presumably very near to, the H = 0 line, as anticipated. Let us now look at the results that are obtained using Eqs. (17a) and (17b) instead, in which ip is defined toward the H = 0 line. For t < 0, x -> -(^o) 1 / / 3 ( < 5 _ 1 ) , from Eq. (17a), dH dM

xr1

H=0

+ 6M*-\-X)W-V}x_{Mims_iy

(21)

The first term on the right-hand side is zero for H = 0, and putting MS-1

=

(Ml/f>)f)(6-l)

=

(_x)-W-l)

(_t)/3(*-l)

(22)

within the second term, we have Xr 1 = ( * - l ) ( - t ) W - 1 ) = r - 1 | t P ,

(23)

which is non-vanishing and shows the correct temperature dependence. For t > 0, on the other hand, we may try using Eq. (17), and argue that as hf is made arbitrarily large, Mi becomes arbitrarily small, hence approaching the if = 0 line. Then, X,-1 = {SM'-^WW-V+tf] x^M5'1^)

(8-l)M*-l{x)W-»}x^Xi

-

- (5 - l)(t)W-V].

Now, taking into account that Mf^hf we get Xt

i _ u\P(s-i) (t)W-i)

(24)

= Hi/Mi K, x t _ 1 as Mi —> 0,

= r;1|tpr.

(25)

Combining Eqs. (23) and (25), we obtain the following ratio of critical amplitudes: r + / r _ = (6-

I).

(26)

It can thus be concluded that, while hk{x) from Eq. (13) and b{x) from Eqs. (17a) and (17b) are only approximations of the scaling function h(x), they show asymptotically correct behavior at the t = 0 and H = 0 lines, respectively, and, together, they appear to give a simple and accurate representation of h{x) (see Fig. 1.10.1). However, since both the scaling assumption and the assumptions that $ = constant and tp = constant are

Equation of State and the Scaling

Function

87

/<0

^

CP t

/>0

m

Fig. 1.10.1. Vicinity of the critical point (CP) in the t-H plane, schematically showing (unshaded) regions where h(x), as given in Eq. (13), is a valid approximation for the scaling function.

asymptotic approximations themselves, an evaluation of the scaling function based upon these assumptions cannot be expected to give an exact representation of h(x) everywhere. Most available sets of scaling provide exact representation of h(x) everywhere. Most available sets of scaling data M(H,T), which tend to have more experimental point for t —> 0 than for H —> 0, are more in consonance with hk(x). A comparison of h(x) = hk(x), the scaling function given by Eq. (13), with recent, very accurate, experimental data on nickel by Oddou et al.13 shows excellent agreement in the entire range of the data. This is shown in Table 1.10.1, where 7 = [In /i]/[ln(l + x)

(27)

is calculated from the experimental values of h = h/ho, and x = x/xo, using ho = 0.228 and XQ = 4.087 (this amounts to a change in normalization parameters [H]o and [M] 0 ). The numerical values of 7 obtained in this way do not deviate appreciably from the experimental value 7 = 1.32, compatible with the experimental values for /? = 0.379 ± 0.003 and S = 4.48 ± 0.09 [i.e., f3(5 - 1) = 1.32 ± 0.05]. Note that these values for (3 and 5 are in excellent agreement with those obtained by means of renormalization group calculations for the three-dimensional Heisenberg model. 14 The numerical values of 7 = In h/ ln(l +2;) are obtained from Ni data 13 ; h = h/ho, where ho = 0.228, directly from the data for x = 0; x = X/XQ,

Effective Field Approach to Phase Table 1.10.1. X

-3.5 -3 -2 -1 0 1 2 3 4 5 6 7 8

Transitions

Critical exponent 7 from h(x) for Ni. h

X

h

7

0.018 0.039 0.096 0.159 0.228 0.303 0.384 0.468 0.561 0.654 0.750 0.846 0.954

-0.856 -0.734 -0.489 -0.245 0 0.245 0.489 0.734 0.979 1.223 1.468 1.713 1.957

0.079 0.171 0.421 0.679 1 1.329 1.684 2.052 2.460 2.868 3.289 3.710 4.184

1.310 1.333 1.289 1.284

— 1.298 1.309 1.306 1.319 1.319 1.318 1.314 1.320

where XQ = 4.087, from best fit to 7 = 1.32 ± 0.05 (experimental) with the highest point x = 8, h = 0.954. This illustrates the good overall fit of the scaling function h(x) = (x + l ) 7 to the best available data for Ni. It may be noted that Eq. (26) predicts a ratio ( r + / r _ ) = (8 — 1) = 3.48 for nickel (isotropic Heisenberg system). Available estimates using the renormalization group approach 15 for the three-dimensional dipolar (n = 1) and Ising models lead to ( r + / r _ ) D i P = 2 and ( r + / r _ ) i s i n g = 5.07, compared with ( r + / r _ ) D i p = 2 and ( r + / r _ ) I s i n g = 3.82 obtained from Eq. (26). It is customary to analyze the interdependence of the scaling function h(x) by means of the parametric representation H = H(0)r05,

M = M(6)r0,

t = t(0)r,

(28)

where, in particular, the linear form of Ho and Lister is specified by H(0) = A{\ - 62),

M(6) = KB,

t{9) = (1 - b262).

(29)

The linear model, i.e., Eq. (29), substituted into the scaling equation H{6)/Ms{6)

= /i[*(6>)/M1//3((9)]

(30)

establishes the form of the function h(x). We can check the consistency of the linear model's fit to the data for Ni as follows: (a) When the system crosses the critical point x = t/M1/13 changes sign from x < 0 (at T < Tc) to x > 0 (at T > Tc) going through x = 0, where

x = t{e)/M1/0(e) = (i-b2e2)/Ke = o, ec = r 1 = 0.7543.

(31)

Equation of State and the Scaling Function

89

(b) At the same time, h(x) = H/M5 goes from h(x) < h(0) to h(x) > h(0) crossing the critical point (see below) from T < Tc to T > Tc. In the linear model, on the other hand, h(x) goes through a maximum at 2 d \ H(9) 1 d A9(l-9 ) d9 M5{9) ~ d9

= 0,

9m=

(j~)

= 0-7969.

(32)

Then for 9 > 9C, up to 9 — 9m, h{x) is still increasing in spite of the fact that already T > Tc, and for 9 = 9m, h(x) decreases instead of increasing, reaching values lower than that corresponding to 9 — 9C. This shows that, while this model produces a good overall fit to the data, it leads to inconsistencies precisely nearest the critical point. Changing the value of 9C toward that of 9m would probably worsen the overall fit to the data. On the other hand, the simpler scaling function given by Eq. (13) gives an overall fit to the data as good as that for the linear model, and a better fit in the close neighborhood of the critical point, being free from the inconsistencies mentioned above. It is fairly obvious that h(x) = H/M goes over from h(x) < h(0) to h{x) > h(0) in crossing the critical point from T < Tc to T > Tc. Indeed, at the H = 0 line (which is the one that crosses the critical point) we have lim

(H/M5)

=0

for t < 0 {x < 0)

(33)

and, for t > 0 (x > 0), hin (H/M*) t 7 i < \\mo(H/M*)tn+v since M n + 1 < M„ for the same H. the critical point (H = 0, t = 0, M dummy variable r, which does not of values H{9) = 0, t(9) = 0, M(9) model.

0
In other words, with the linear model, = 0) can only be reached by setting the enter h(x) or x, equal to zero. The set = 0 is inaccessible for the same for this

References 1. 2. 3. 4. 5.

(34)

B.J. Widom, J. Chem. Phys. 43, 3998 (1965). Domb and D.L. Hunter, Proc. Phys. Soc. 86, 1147 (1965). L.P. Kadanoff, Physics (NY) 2, 263 (1966). R.B. Griffiths, Phys. Rev. 158, 176 (1967). M.E. Fisher, Rep. Prog. Phys. 30, 615 (1967).

90

Effective Field Approach to Phase Transitions

6. M.S. Green, M. Vicentini-Missoni and J.M.H. Levett Sengers, Phys. Rev. Lett. 18, 1113 (1967). 7. A. Arrot and J.E. Noakes, Phys. Rev. Lett. 19, 786 (1967). 8. J. Ho, Phys. Rev. Lett. 26, 1485 (1971). 9. S. Milosevic and H.E. Stanley, Phys. Rev. B 6 , 986, 1002 (1972). 10. J.A. Gonzalo, Phys. Rev. B 8 , 3482 (1973). 11. A. Cammanasio and J.A. Gonzalo, Solid State Commun. 16, 1169 (1975). 12. T.N. Caphart and M.E. Fisher, Phys. Rev. B l l , 1262 (1975). 13. J.L. Oddou, J. Berthier and P. Peretto, Phys. Rev. B17, 222 (1978). 14. C. Domb and M.S. Green, Phase Transitions and Critical Phenomena, Vol. 6 (New York: Academic Press, 1977). 15. A. Aharony and P.C. Hohenberg, Phys. Rev. B 1 3 , 3081 (1976).

Appendix: Effective Field Approach to Superconductors It is obvious that the phenomena of superconductivity and superfluidity are basically quantum-mechanical phenomena which, for a rigorous theoretical description, require the explicit use of quantum-mechanical language. However, it may be of some interest to show that, once the existence of Cooper pairs is assumed, an intuitive and simple effective field approach can also be used for these phase transitions, leading, as shown below, to reasonable estimates of some basic physical parameters (densities of Cooper pairs). (a) Superconductors. Up to this point, we have outlined the formal BCS approach to superconductive transitions. In what follows we will try to introduce an effective field approach to the superconductive transition, using the critical magnetic field as the order parameter. This approach is somewhat unorthodox and admittedly over-simplified, but may be intuitively useful to illustrate in a simple way the features that the superconductive transition has in common with other phase transitions, such as the ferromagnetic transition for instance. The basic concept to describe a superconductive transition is that the of "Cooper pair": two electrons of opposite momenta and opposite spin attract each other due to the combined effect of the polarization of the lattice ions surrounding them and also to a minor extent due to the spinspin interaction. Figure A.l illustrates how the polarization of the lattice ions can give rise to an effective attractive force between two "over-screened" electrons of opposite momenta at a given instant of time. At the instant of capture in the picture, electron (1) is shown with the surrounding ions over-drawn to itself, while electron (2) is shown with the surrounding ions over-drawn away from itself, giving rise to effective charges slightly positive for e(l) and strongly negative for e(2) at that particular instant. At a later instant,

91

92

Effective Field Approach to Phase

o o o o

Transitions

o o o o

O /© £ N O Netjittractive O 9 y ? O O V ? W O * force ~" O ^ ( 4 O O O O O O O O O Fig. A.l. Pictorial description of the electron-phonon-electron attractive interaction between electrons e(l) and e(2) in a superconductor.

the oscillations of the lattice ions surrounding the electrons will bring out a reversal of the sign and the size of the effective charges for e(l) and e(2), the process repeating itself again and again. Thus, schematically, we may picture a Cooper pair as a bound pair of electrons with opposite spins moving around each other in a circular path. This should give rise to a dipolar (orbital) magnetic moment fi, associated with the current loop formed by the two moving electrons. Figure A.2 depicts an "ideal" Cooper pair together with its associated orbital dipole moment. But the weak electron-phonon-electron interaction is capable of overcoming the Coulomb repulsion only for pairs of electrons that are physically far away from each other, so that the polarization of the lattice becomes more effective. This limits the total number of electrons with opposite momenta and spin which can go into pairs to a certain maximum number N of possible bound pairs, such that for these pairs k + fco > kp (where |/CD| is the maximum momentum for the phonon within the Debye sphere

7C:

s*=-s

Fig. A.2. Pictorial representation of Cooper pair with associated dipole moment fi. Dotted arrows indicate the mutual attractive force between electrons e(l) and e(2).

93

Appendix

and kp the maximum momentum for individual electrons within the Fermi sphere) and the electron that has absorbed the phonon can go into an unoccupied state outside the Fermi sphere after absorbing the phonon. (We are talking of Debye sphere and Fermi sphere as approximations to the actual Brillouin zone and Fermi surface, respectively, to make matters simple.) Since in any solid |/CD| -C |&F|, only a small fraction of the total number of electrons within the Fermi sphere can go into pairs under the most favorable conditions, i.e., under T = OK. At higher temperatures, the random thermal motion of the ions in the lattice will tend to break down more and more pairs until at a given critical temperature T = Tc all pairs will be broken and the transition from the superconductive to the normal state will have taken place. In this way, our basic assumptions to try an effective field approach to the superconductive transition are: (i) each Cooper pair carries with it a diamagnetic (orbital) dipole moment fi, which is expected to interact collectively with all other dipole moments so as to minimize the total free energy, giving rise to a macroscopic magnetization; (ii) there is a maximum number N of possible bound Cooper pairs at T = 0 K, and, as the temperature rises, Cooper pairs break down into individual electrons, which, free to move, tend to oppose the macroscopic magnetization of the remaining pairs. Let us write the magnetic field as B = H + 4wM,

(1)

where H would correspond to an external magnetic field and M to the magnetization per unit volume M = Nfi. For H = 0, we have Bs =

4TTMS

=

4TTNSMS

(effective field)

(2)

where Ms is the spontaneous diamagnetic magnetization. There are two possible states for each pair of electrons with \k\ in the range allowed to form Cooper pairs: bound pair state and broken pair state. The respective population densities for broken and bound pairs at a certain T will then be given by TVi = (N/Z) exp(Bsfi/kBT),

(3)

N2 = (N/Z) exp(-B B /VfcBT),

(4)

where the partition function is given by Z = exp(Bsn/kBT)

+ exp(-Bsn/kBT)

and

N = Nt + N2.

94

Effective Field Approach to Phase

Transitions

Subtracting Eq. (4) from Eq. (3), we get N2 - Ni N2 + N!

=

exp(g sM /fc B T) - exp(- J B s/ x/fc B r) exp(Bs(i/kBT)+exp(-Bsfi/kBT)

/5s/x\ \kBT J

(5)

and taking into account that A7r(N2 - Njfi

= BB,

AwNfi = Bs0

(6)

where Bs0 is the spontaneous magnetic field (order parameter) at T = OK, we get the equation of state for the system

S^ t< " h l^J =lanh ^57 1 ' = t.fl.nh I

I = f-.flnh I

P)

where Tc =

5SOMAB

= 4irNn2/kB.

(8)

We may note that the above results imply an energy "gap" at T = 0 K between the bound and broken state of a Cooper pair given by - A 0 = Bs0n = kTc,

i.e., 2A 0 = AkBTc,

(9)

which is in very good agreement with the relationship between the experimentally observed Ao (using infrared absorption spectroscopy) and the experimental value of To, for many type I superconductive transitions. The precise BCS result, somewhat different from the approximate result given in Eq. (7), is 2A 0 « 3.5fcBTc, which is fairly close to the one in Eq. (9). From Eq. (7) it is easy to get the critical exponent for the temperature dependence of the order parameter at T < Tc. This is done, as usual, by expanding the hyperbolic tangent in a power series and keeping only the first two terms for T near and below T c , where the argument is small, resulting in Bs(T)/Bs(0)

« V3[l + (T/Tc)}1/2,

(10)

which gives the effective field critical exponent (3 = 1/2. In a similar way the finite jump in the specific heat per unit volume at T — Tc can be obtained as AC«^VfcB,

(11)

which implies a critical exponent a = 0, in agreement with the experimentally observed behavior.

Appendix

95

Using Eqs. (6) and (8) one can obtain numerical values for /z and N as follows: /x = kBTc/Bs0,

(12)

N = Bs0/iTTfi.

(13)

Here one can substitute the known experimental values of Tc and Bs0 to obtain numerical estimates of \i and N. For instance, using Tc = 7.19 K and Bs0 = 800 gauss for superconductive Pb, one gets // = 1.23 x 10~ 18 (cgsemu units) and iV — 5.12 x 10 19 Cooper pairs/cm 3 , which are reasonable numbers. Table A.l gives numerical values for fi, using Eq. (12), and for N, using Eq. (13), from the experimental values for Tc and Bs0, quoted by Ashcroft and Mermin, 1 corresponding to the bcc superconductive series One can compare the values of N obtained from Tc and Bs0 by means of Eq. (11). For instance, for Nb, AC/Cn{Tc) = 1.9 and Cn(Tc) = 7 T C = (7.56 x 103)(9.26) = 7.0 x 10 4 erg/cm 3 , according to Ref. 1. Then, N = [ l . 9 C n ( T c ) ] / ! * B = 6.42 x 10 20 ,

(14)

which is not in perfect agreement with N — 2.44 x 10 20 , given in Table A.l, but is not badly in disagreement either. Due to the fact that Cooper pairs are thought to be of macroscopic size in comparison with atomic dimensions, we can, in principle, make a classical analysis of the elementary diamagnetic dipole moment g associated with it in terms of current i carried by the two electrons moving at velocity v in a circular path of radius r enclosing an area S as follows:

"-«-Hs?y-(!)"=(!yTable A.l Superconductive parameters for some bcc elements (cgs-emu units). Superconductor V Nb Ta

Tc 5.3 9.26 4.48

Bso 1020 1980 830

/J- =

fceTc/Bst) 18

0.71 x 1 0 0.64 x 1 0 ~ 1 8 0.74 x 1 0 " 1 8

N =

Bso/4irn

1.14 x 10 2 0 2.44 x 10 2 0 0.88 x 10 2 0

<si»

96

Effective Field Approach to Phase

Transitions

where u = v/r is the angular velocity. On the other hand, the energy gap can be written as A 0 = hu

(16)

and, consequently, the radius r and the velocity can be eliminated from Eqs. (15) and (16) as r=[(V(e/c))(/VA0)]1/2

(17)

and v = ur=

(Ao/ft) [(h/(e/c)) ( M / A 0 ) ] V *

(18)

in terms of constants and /i, which are known. For Nb we get, using p = 0.64 x 10 18 and the experimental value for A 0 = 2.42 x 10~ 15 erg, r N b = 4.16 x 10~6cm,

(19)

6

Vm = 9.6 x 10 cm/s.

(20)

The orders of magnitude for r and v are, in principle, not unreasonable, as discussed below. Let us assume that the effective mass of the Cooper pair electrons in the case of superconducting Nb is (m*/m) times larger than the bare electron mass. For instance, for Nb m* « 12m at the normal state (T > T c ), and it is likely that m* be even larger at the superconducting state, where the electrons are "dressed" by neighboring ions as they move, due to the electron-phonon interaction. We may now compute directly the fraction of electrons into the first Brillouin zone gone into Cooper pairs (N/N-QZ), and the value of the Fermi velocity (VF) for Nb, and compare them with the results previously obtained. Assuming a spherical Fermi surface of radius AIF for the electron states, and a thin outer shell of thickness kg for those electrons forming Cooper pairs, we have N

_ 47rfcgfcg

kg

NZB~

fkF

kF

A0hs ^SFf'/h

(

]

which gives (N/NBZ) m terms of observable quantities, such as Ao, m*, £y, and s « ( C n / p ) l / 2 , the longitudinal sound velocity for the phonons. For Nb, using A 0 = 2.42 x 10" 15 erg, m* = 12(9.1 x 10" 2 8 )g, eF = 5.32 eV and

Appendix

97

s = (1.47 x 10 1 2 /8.4) 1 / 2 cm/s, we get ( f )

« 10.1X10-

(22)

which is reasonably close to

(£) »««'»-'

(23)

V v c / Nb

1 22 obtained using N = 2.44 x 1020 from Table A.l and Nc = v- = 5.5 x 10 . The Fermi velocity is given by

vF = (2e F /m*)

(24)

For Nb, this equation gives («F)Nb«3.9xl07cm/s,

(25)

which is somewhat larger, but of the same order of magnitude than 0)Nb « 0.96 x 107 cm/s

(26)

obtained by the simple arguments in Eq. (20). Of course, the effective field approach to superconductivity outlined above is overly simplistic to cope with the complexities of the superconducting phase transition. The semi-quantitative numerical agreements pointed out in this chapter therefore cannot be taken too literally. Nevertheless, it is remarkable that such a simple approach is able to describe in such a semi-quantitative way the features of some simple type I superconductors. (b) Superfluids. Let us try such a necessarily over-simplified description of the superfluid transition from the effective field viewpoint. Two neighboring helium atoms feel a net attractive force (through a Lennard-Jones interaction potential, for instance), which is countered by the centrifugal force associated with their motion around each other (even at OK, because of their zero-point energy) and they can form a stable superfluid Cooper pair. Like in the superconducting case, Cooper pairs can be either in the bound state or in the broken state, which works so as to suppress superfluidity. When a semicircular vibrating wire of radius R is slowly rotated back and forth through superfluid helium well below T c , 2 the wire experiences a very low drag force. As the rotation velocity increases, however, there is a sudden, very large increase of drag force due to pair breaking at a certain

98

Effective Field Approach to Phase

Transitions

characteristic speed of the wire, i>max- The relative speed of two helium atoms at this point may be estimated as vp = (vmax/R)r0,

(27)

where r0 is the nearest-neighbor separation between helium atoms. Since all bound Cooper pairs may be thought of as acting cooperatively to oppose a pair breaking, the energy necessary to break a pair may be assumed to be 2w = {Fs)vp = (NsMvP)vpj

(28)

where Fs is a generalized "effective field," conjugated with the velocity v, given in terms of the density of Cooper pairs present at NS(T) and the momentum of each helium atom within a pair. Prom here we can proceed like in previous sections to get

^

N

^ Ns0

= tanh (J?-)

= tanh (»«MW»*

\kBTj

\

T

£_)

,

iVs0y

(29)

v

;

which determines, in this simplified picture, the temperature dependence of the order parameter (superfluid Cooper pair density). Equation (29) can be written as Ns

^

= t a n (T

c

Ns \

Hr^J'

(30)

where Tc = Ns0Mvl/2kB.

(31)

Thus, we can get for the superfluid energy gap ^ A = Ns0Mv2p = 2kBTc,

(32)

which is analogous to the expression obtained in Chapter 1.4 for the superconducting gap. The maximum number of Cooper pairs in 4 He is given by N = p/2M = 1.09 x 10 22 pairs/cm 3

(p = 0.145g/cm 3 ),

(33)

which can be compared to ATs0 from Eq. (31) using the experimental values for Tc = 2.18K and for vp = (vmax/R)r0, where vmax = 0.9cm/s, R = 0.15cm, and VQ = 2.51 x 10 _ 8 cm may be used, resulting in vp — 1.5 x 10 _ 7 cm/s. 3

Appendix

99

W i t h these d a t a Ns0 = 2kBTc/Mvl

« 0.40 x 10 2 2 p a i r s / c m 3 ,

(34)

which is not far from t h e experimental maximum number of possible Cooper pairs at OK in 4 H e given by Eq. (33).

References 1. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976). 2. J.P. Carney, A.M. Guenault, G.R. Pickett and G.F. Spencer, Phys. Rev. Lett. 62, 3042 (1989). 3. This can be extrapolated from data for noble gas crystals. See table in N.W. Ashcroft and D. Mermin, Solid State Physics, p. 401 (Holt, Rinehart and Winston, New York, 1976).

Part 2

Some Applications to Ferroelectrics: 1970-1991

Chapter 2.1

Behavior at T = Tc of P u r e Ferroelectric Systems with Second Order Phase Transition*

The second order transition in ferroelectric triglycine sulfate (TGS) seems to be a very good test case for the mean field theory. Previous work 1 ' 2 has shown that the behavior of the dielectric constant and the spontaneous polarization is in agreement with the mean field predictions. The present investigation, partially reported in a previous letter, 3 aimed at a more complete analysis of the order-disorder cooperative transition by means of a detailed study of the variation of polarization with electrical field, as well as with the temperature near the critical point. Accurate data of P versus E near Tc allow the determination of critical exponents through log-log graphic representations. In addition, once the two fundamental parameters /? and d, defined by (-PS)TRSTC = const, x [1 — (T/Tcyf and 1 6 (P)T^TC = const, x E / , are determined, the way is open for the search of a "law of corresponding states" in terms of the properly "scaled" variables. The sample preparation and experimental procedure were described in previous communications. The determination of the Curie temperature was done in two different ways. First, a plot of the squared spontaneous polarization (Ps2) vs. temperature was made, which showed an almost perfect linear behavior yielding Tc by extrapolation to Ps2 = 0. Alternatively, the method described by Kouvel and Fisher 4 was used, yielding the same result within experimental accuracy. It has been noted by Reese5 that corrections due to the electrocaloric effect should be considered. While the accurate

*Work previously published under the title "Equation of state for the cooperative transition of triglycine sulfate near T," J.A. Gonzalo, Phys. Rev. B 1, 3125 (1970). Copyright © 1970. The American Physical Society. 103

104

Effective Field Approach to Phase

Transitions

determination of these corrections near Tc is not easy, reasonable estimates indicate that our results would not be substantially altered by them. It may also be noted that perfect compensation of the P vs. E hysteresis loops very near Tc could not be fully achieved with the Sawyer-Tower circuit, possibly due to a field dependence of the conductivity of the crystal. This behavior actually sets limits of AT = Tc - T at +0.22 and -0.04°C within which a reliable determination of P for very small E was not possible. As is well known, an increase of the amplitude of the ac field applied to the sample for displaying the P vs. E curve produces a relatively small increase of the absolute value of the polarization with respect to the corresponding values for our ac amplitude. However, the relative variation of the polarization as a function of temperature was checked for various field amplitudes and it was found to be the same, the absolute values being different only by a constant factor. This effect might be attributed to a consistently partial switching of the ferroelectric domains at low ac amplitudes. The constant ac field amplitude chosen in our case, E = 190V/cm, was relatively low, which helps to keep down the electrocaloric effect in the vicinity of Tc. The experimental results below T c , as described in a previous letter, 3 were shown to yield the value of the critical exponents S and (3, along with four other exponents indicating the field and temperature dependence of both derivatives of the polarization with respect to field and temperature. The experimental values obtained from log-log plots of the data are given in Table 2.1.1 and compared with those calculated from the mean field model, using the expression6 where EQ « 4.4 x 106 V/cm is the saturation internal field and N/u, = 4.3 /iC/cm 2 is the saturation polarization. It is interesting to note that, as it should be expected (see Appendix), the ratio of the two critical exponents relating the same derivative of the free energy to field and temperature is constant. In Table 2.1.2 a summary of data for polarization and field at various temperatures below and above Tc is given. These data were "scaled" to determine p = P/t^6 and e = E/t0S, and plotted using a log-log scale. It can be seen from Fig. 2.1.1 that the scaling of the data is quite good, giving the evidence of the existence of a law of corresponding states. From this, the sequence of critical exponents, found directly and reported in a previous short communication, 3 results in an automatic fashion. What is more important, however, is the fact that this log-log representation, which shows the critical behavior over a wide range of three decades in the reduced field e, shows clearly the asymptotic behavior of the equation of state for

Behavior at T = Tc of Pure Ferroelectric

Systems

105

Table 2.1.1. Experimental critical exponents from TGS compared with mean field theory predictions. Defining relationship

Mean field relationship

Experimental value

(i^teO-eTs

73

= 1/5 = 0.32 ± 0.02

( P ) e = 0 ~ ei*

74 = P = 0.50 ± 0.03

( — | ~e77 V9e/t=0

77 = 0.66 ± 0 . 0 5

dp

; e79

9tJt=o

'

"

-eTio

\dt)e=0

79 = - 0 . 3 3 ± 0.05

"

""

710 = - 0 . 4 5 ± 0 . 1 0

1

73

(P) e =o = ( 3 t ) 1 / 2

74 = / 3 = -

= 5 = 3

77 =

9e / t=o

)e=0

1

(P)t=o = p e ) 1 ^

1-P2)

~e~>s 78 = - i = 0.95 ±0.10

de

Theoretical value

-P2) f^) = £• \9eJe=0 (t-p2) _1 tanh p(l-p2) dp\

—

\dt)t=o 9p\

-tanh_1p(l-p2)

a«/ e =o'

(*- P 2 )

- -

78 = - 1

1 79 = — x

1 71

°

=

"2

Table 2.1.2. Polarization vs. field for T G S from hysteresis loops in the vicinity of the Curie temperature. Below Curie temperature 2

2

Above Curie temperature

P(/iC/cm )

AT ( x l O " °C)

E• ( V / c m )

P(/iC/cm2)

A T ( x l O - 2 °C)

E (V/cm)

0.217 0.217 0.217 0.244 0.244 0.244 0.271 0.271 0.271 0.271 0.298 0.298 0.298 0.298 0.326 0.326 0.326 0.326 0.326

4.2 10.7 17.1 4.2 10.7 17.1 4.2 10.7 17.1 30.1 4.2 10.7 17.1 30.1 4.2 10.7 17.1 30.1 43.0

49.7 27.7 9.3 73.0 46.9 24.5 101.8 72.4 46.4 7.0 137.9 104.5 74.6 25.4 182.7 144.5 104.6 53.8 9.9

0.1085 0.1085 0.1085 0.1085 0.163 0.163 0.163 0.163 0.217 0.217 0.217 0.217 0.298 0.298 0.298 0.298

2.3 8.7 15.2 21.7 2.3 8.7 15.2 21.7 2.3 8.7 15.2 21.7 2.3 8.7 15.2 21.7

12.7 21.1 30.3 40.2 33.1 49.3 60.6 71.9 69.8 88.1 104.3 117.7 131.8 148.7 166.4 186.1

(1 - t) t a n h _ 1 ( P ) -P,E

= E/E0,

t = 1 - (T/Tc),

P = P/Np,.

106

Effective Field Approach to Phase r—I

I I I rl

-I—I—I

Transitions

I I I 11

e. E/h-(T/T c |l^ 8

Fig. 2.1.1. Log-log plot of the scaled polarization vs. scaled electric field for ferroelectric TGS near the Curie temperature. The full line is renormalized mean field equation of state with m = 0.450, n = 0.139.

both small and large e, above and below T c . This asymptotic character is in complete analogy with the observations of Green et al.7 for liquid-vapor transitions in a good number of systems. We have also recently examined very accurate data 8 ' 9 from ferromagnetic transitions, and the asymptotic trend for small and large scaled magnetic fields is seen again to be fully analogous. The asymptotic behavior can be summarized as follows: Below Tc: (la)

pe—o = const., Pe-,00 = const, x e1/5.

(lb)

Above Tc: pe-»o = const, x e,

(2a)

1

(2b)

pe-,oo = const, x e ' .

Behavior at T = T c of Pure Ferroelectric

Systems

107

The implications of these expressions are obvious: (la) means that for E Tc. This equation can be expanded in the following way:

Efi=(p+

i p 3 + ^P5A\ id (P + i p 3 + ^ P 5 A ) - P

1

3 2 l 2 Putting e = eE/t / and p3= +tP/t Ml3 / + 5P 5 (1 ^ ) + A M = HP + gP

Obviously, if t < 1 (for instance, 1.0 x 10" 5 < t < 3.0 x 10" 3 in our experiment), this expression reduces itself to e_ = p f^p2

- 1J

for T < Tc,

(3a)

e+ = P (\p2

+ l)

for T > Tc.

(3b)

To check these equations against the experimental results it has been found necessary to introduce proportionality factors for e and p in the above equations. They become 1 2 2— me- = np If —np

me+ = np ( -n2p2 + 1 j . To introduce these proportionality factors is equivalent to modifying the normalization parameters in such a way that (Nfx)/n replaces (Nfi),

Effective Field Approach to Phase

108

Transitions

and Eo/rn replaces EQ. The best fit to the data is obtained with m = 0.450,

n = 0.139.

Figure 2.1.1 shows a plot of the mean field equation of state in scaling form, along with the experimental data. The agreement is very good except for a few points for T < Tc in the intervening region between small and large e, which fall slightly above the theoretical curve. The estimated experimental errors go from 5 to 1% as p increases and from 10 to 2% as e increases. The realization that the asymptotic behavior specified by Eqs. (la)-(2b) is not only characteristic of our ferroelectric cooperative transition, but also of liquid-vapor and magnetic cooperative transitions, strongly suggesting the convenience of using it along with Widom's homogeneity requirements to specify the equation of state for the system under consideration throughout Tc and in its vicinity. Since the formulation of the homogeneity assumption 10 for the free energy of a cooperative system undergoing a second order phase transition, considerable progress has been made in the understanding of the critical phenomena. 11 Griffiths12 has studied the problem of constructing explicit analytic expressions for the equation of state relating the scaled extensive variable (polarization, magnetization, volume, etc.) to the intensive variable (electric field, magnetic field, pressure, etc.), respectively, for the case of rational critical exponents. Very recently, several empirical 9 ' 13 and parametric 14 expressions have been proposed to fit the equation of state of some real systems. We wish to construct a compact expression of the free energy in a simple way, matching the critical exponents sequence both above Tc and below it, as well as the asymptotic behavior indicated in the preceding paragraph, from the law of corresponding states. Let us assume, following Widom 10 and Griffiths,12 that the free energy about the critical point can be given simultaneously by

F{X,t) =

X^+^'s

^ + a^x^)+a4xWs)

+A

forX1//3,5»<(T>Tc),

h + hf^j+ht^)

(4a)

+A

for t0S > X (T < T c ).

(4b)

Behavior atT = Tc of Pure Ferroelectric

Systems

109

Here, X = X/X0 is the reduced intensive variable and t = 1 — (T/Tc) the reduced temperature. (It may be noted that while the expansion (4b) is very familiar in the literature since the introduction of the homogeneity assumption, relatively less attention has been paid to the complementary expansion (4a); Ho and Litster have made use of the latter in their recent work9 on CrBr3.) These expansions ensure a sequence of critical exponents of the expected form (F)t=0 «

X^)l\

C = (d2F/dt2)t=0

R

S =

X(6+l)/6-(l//36)

(dF/dt)t=o

(5a)

,X(S+l)/5-2(l/0S)^

(F)T=0«^+1),

Y =

t0(6+l)-06

(dF/dX)x=0

(5b)

(d2F/dX% ! = 0 w t«*+i)-2/M,

Yx =

where the "gap" exponents are (1/A) = 1/(36 and A = (36, respectively. The last two expressions of (5b) are easily recognizable as the defining equations for the indices (3 and —7 = (3 — A, respectively. Let us call Y the partial derivative of the free energy with respect to the variable X. Its meaning will be, of course, that of the respective extensive variable in the various cases (polarization, magnetization, volume, etc.). Prom (4a) we obtain OF dX

Y

Xl's

+

Xl's ai + a,2

x

- (36)

dx

t Xl/Ps

a\ + a 2 t 1

x /^

t

a3

+ 3a 3

x1/?6 t

+A

1 36

x //

t \Xl/Ps

(6a) X

&i+26 2

363

w]

+L

(6b)

or, in other words, y

= Y/t" = xlls[Cl + c2(x-1/(jS)

+ c 3 (ar 1 / / 3 5 ) 2 + L]

for x = X/tps y=[di+

d2(x) + d3(x)

2

+ L]

> 1 (T > T c ),

(7)

for x «; 1 (T < T c ).

(8)

Our aim is to get a single expression for y which combines Eqs. (7) and (8) at T < Tc, approaching each of them for the limiting cases of x » 1

110

Effective Field Approach to Phase

Transitions

and x <^C 1. One can write formally y = x1/sip1(x-1/f3d) + %fa(x) for T < Tc 1 for T < Tc y = z / ^ (a;-i//M)

(9) (10)

where ipi and tp2 stand for the factors within brackets in Eqs. (7) and (8). This is our equation of state which already involves the correct sequence of critical exponents throughout the series expansion of ip\ in powers of x-i/ps a n ( j 0 f ^,2 m p 0 w e r s 0 f x. Below Tc, according to Eqs. (4a) and (4b), ipi{x~1/^d)2 should predominate for x 3> 1, and ^(x) for x -C 1. Above T c , it is clear that the spontaneous order ceases to be non-zero for x = 0, so it is reasonable to eliminate the contribution from ip2(x). At this point, the empirical asymptotic behavior indicated in the preceding paragraph should be incorporated. Below T c , for x -C 1, 4>2(x) should approach a constant, and for x 3> 1, ipi(x~1//3S) should also approach a constant, according to (la) and (lb), respectively. Above T c , ipi(x) does not exist and V'i(£~1/'8'5) should approach a value proportional to a;1_(1/'5)forI < 1 , remaining the same as below Tc for x 3> 1. One could try different functional expressions for tp2 and tp\, all of them susceptible to being expanded in the power series of the required form. In principle, a logarithm, a binomial, or an exponential would meet this requirement. However, after testing these three forms against the experimental data, not only for TGS but also for magnetic and liquid-vapor systems, one comes to the conclusion that the logarithm changes too slowly with x and the exponential, on the other hand, too rapidly, in order to satisfy the asymptotic behavior indicated. On this ground, only the binomial forms are left as satisfactory ones. The simplest binomial forms one can think of, meeting the above-mentioned requirements, are

1/0S

ip1(x-

)

= A

1+

fe) /

1

Mx) = B

-I/PS' -P6{l-1/S)

T \ "

if

r\i\

\

5

(11a)

-(l/06)(l-0)

(lib)

The exponents -08(1-1/5) inEq. (11a) and -(l/05)(l-0) inEq. (lib) are the simplest ones that keep the homogeneity of Eq. (9) from x/x\ <S 1 to x/x2 3> 1 through the whole range in x. Also, the former is automatically required by condition (2b).

Behavior at T = TC of Pure Ferroelectric Systems

111

As a check of the "phenomenological" equation of state obtained, the principal critical exponents may be calculated. Below T c , from Eqs. (6)(10), one obtains x < 1, x>i,

x «i,

Y « fo^ » Btp, 1

y « V'IX /'

5

(12) 1

5

« AI / ,

(13)

1 s

1

ay/sx = x ' d^1/dx + (1/5) x^ /^-Vi + t0dil>2/dX = Cit13-^

= di"7'.

(14)

Similarly, above T c , a; < 1, i»l, a; « 1,

y « ^ i ^ 1 / 5 » 0, y » ^ 8Y/8X

1 / < 5

(15) 1 5

(16)

« AX / ,

= X^^/dX -

= C^ "

+ {1/5) X 4

(1/

*

)_

Vi

7

(17)

= C2^ .

The constants that appear in Eqs. (14) and (17) are, respectively, _

A

1-/3

B

A

It is interesting to note that 7 = 7', also supported by available experimental evidence. Figure 2.1.2 shows that the use of Eqs. (11a) and (lib) in the equation of state given by (9) and (10) leads to excellent agreement with the experimental data for ferroelectric TGS, below Tc as well as above Tc in this case, y = p (scaled polarization) and x = e (scaled electric field). Only four dimensionless numerical parameters have been used, their values being A = 2.12,

Xl=ei=

0.907,

B = 1.87,

x2 = e2 = 1.425.

(18)

For various liquid-vapor systems, it was earlier reported by Green et al.7 that the data suggest a scaling law asymptotic behavior as that indicated by Eqs. (1) and (2). Using Green and coworkers' critical exponents, j3 = 0.35 and 5 = 5.0, one could try to fit Eqs. (9) and (10) to the data, given in the chemical potential-density representation, in order to determine A, x\, B, and x2. Since the scattering of the experimental points (collected from many authors on many different systems and temperature intervals) is fairly high, it does not seem to be justified. However, accurate data 1 5 for He 4 are available. By using the tabulated data of Roach, 15 one can

112

Effective Field Approach to Phase

Transitions

• • E/ll-(T/T e l|/3S

Fig. 2.1.2. Log-log plot of the scaled polarization vs. scaled electric field for ferroelectric TGS near T c . Pull line is the phenomenological equation of state with A = 2.12, B = 1.87, xi = 0.907, x2 = 1.425.

calculate the fundamental critical exponents in the pressure-volume representation. Since the transition occurs at a very low temperature, it is not surprising that the asymmetry of the experimental data is considerable. It is convenient to bypass this difficulty by using the following definitions: (^gas — Miq) cx AT13 V{-AP)

(along coexistence curve),

- V(AP) - V(AP) oc AP1/S

(along critical isotherm),

(19) (20)

where V is the volume, T is the temperature, and P is the pressure. In this way, a nicely defined straight line for two decades up to the vicinity of the critical point is obtained in the log-log plot, which yields /?. The analog plot for S is only approximately linear in the last decade up to the vicinity of the critical point, and since the trend suggests an increasing value of S, we extrapolate to the closest value that does not violate Griffith's inequality, taking a' w 0. This results in (3 = 0.411

and

5 = 3.84

(21)

Behavior at T = TC of Pure Ferroelectric

113

Systems

These numerical values are somewhat different from those obtained by Roach 15 and Vicentini-Missoni13 but it should be taken into account that they used the density instead of the volume and neglected the asymmetry. By using the exponents given by Eq. (21), one can scale the data corresponding to several isotherms above and below T c . The results are seen in Fig. 2.1.3(a) where the phenomenological equation of state is represented, with

y= v =

AV/Vc a— Ifi

, and x — p ~

AP/Pc t06

where V = 14.49cc (per gram), Tc = 5.193K, Pc = 1710.0Torr. The equation that best fits the data has been obtained by using the dimensionless

err T l

'•• "I

60 50


4.0 30 '

'"T-« I ' 1 • I T T I ' I

—i—i—i

-»

i

-i-X-r-f'TlJ

-***^^**' '.

*~~^9^

ZlIi-4-fl-"'~0r*

S " 3.43

_ C > - - - ^ H ^ ^ ^ V A 5 ^ r A T (T < Tc 1 AT(T>TC) „ » " " ^ * T > Tc - ~~k~^ \P^

T Zp

-

<>

i i i .)• pivpn—

$' 0.41

O 0.053*K * 0.027 V 0.004

"

• O.OOS'K A 0.027 T 0.071

<]

> 1.0

o.e 0.6

. r„ .1.1

..

'' */ ' / • ^ / 4

i . - j u j ' . hrf.ij.l.M.1. 10°

(Rooch) .

1

. 1 . 1 , l.l.UJ,!

..j

L. .

i . i .

.

i .~ r

p=a£/|i-(T/T c i|^ ( o ) Liquid—vopor

H«

I ' I ' I ' I'l

10

* i

./38

10'

(b) Ferromo«Mtic Ni. Fig. 2.1.3. Log-log plots of scaled quantities for (a) liquid-vapor He 4 , volume versus pressure; (b) ferromagnetic Ni, magnetization versus magnetic field (averaged data). Full lines are phenomenological equations of state with (a) A — 1.92, B = 1.83, x\ = 7.57, x2 = 11.3; and (b) A = 1.18, B - 1.42, x i = 0.334, x2 = 0.360.

114

Effective Field Approach to Phase

Transitions

parameters 4 = 1.92, B = 1.83,

x1=p1=7.57, ar 2 =p2 = H.3.

l

}

An estimate of the experimental errors from Roach's data indicates that they go from 7 to 2% as v increases, and from 20 to 4% as p increases. Finally, accurate data for the ferroparamagnetic transition in Ni, by Kouvel and Comly,8 have also been examined. The result is shown it Fig. 2.1.3(b). Again, the corresponding phenomenological equation of state is plotted with M/M0 V = m=

, and

t0

, x = h=

H/H0 , 6

where M0 = 58.6emu/g, H0 = kTc/fi0 = 15.2 x 10 Oe. The equation that best fits the data is obtained with A = 1.183, B = 1.421,

an = fti = 0.334, x2 = h2 = 0.360.

(

'

Since no tables are given in Ref. 8, we cannot give an estimate of the relative errors. However, they must be small, given the small scattering of points in the graphs. In Table 2.1.3, a comparison is made of the main critical exponents and coefficients for TGS, He 4 , and Ni, along with those of the mean field theory. The principal conclusions of the present work may be stated briefly as follows: 1. The scaled data for P versus E from TGS in the vicinity and at both sides of Tc satisfy very approximately the mean field theory predictions. 2. A phenomenological equation of state has been constructed, based on the homogeneity assumption, which is simpler than previous proposals Table 2.1.3. Comparison of critical exponents and coefficients (see text) for various real systems and the mean field model. System Ferroelectric TGS Liquid-vapor He 4 Ferromagnetic Ni Mean field

0

S

PS

A

B

0.50 0.41 0.38 0.500

3.0 3.8 4.6 3.00

1.50 1.57 1.74 1.500

2.12 1.92 1.18 1.44

1.87 1.83 1.42 1.73

Behavior at T = TC of Pure Ferroelectric

Systems

115

and reflects in a natural way the different asymptotic behavior for lowand high-scaled intensive variables (electric field in the case of TGS). 3. The application of this phenomenological equation of state to representative ferroelectric, liquid-vapor, and ferromagnetic transitions shows fair agreement with the data in all three cases. In the case of TGS, the agreement is excellent and improves that of the mean field theory. All experimental points are within the estimated error limits. In the case of He 4 there is appreciable scattering of points at low p above T c , and at high p below Tc. However, taking into account the experimental uncertainties, the agreement is fair. In the case of Ni, for which very accurate data are available, the agreement is also fair, but some small systematic deviations seem to be present, especially for T > Tc.

Appendix From the inspection of Eqs. (3a) and (3b), it is readily seen that the two critical exponents from the same derivative of the free energy, Fn^m = dn+mF/Xndtm, are related in a simple way. Let us call -jh and 'jk+i as exponents that define the dependence with X and t, respectively. They are 7fe = (6 + 1)/S - m{l/p8)

- n, (A.l)

7fc+i

= 0(6 + 1)/S - n/35 - m,

as obtained by using Eq. (4a) for 7fc and Eq. (4b) for "fk+i- These exponents define relationships F

1

m,n

F

a;XSk(X>t

; l//3«5) ;

^

a-

£7fc+l


»X)

(A.2)

The ratio is then Ik 7fe+i

"

(S + l)/S - m(l/f3S) -- n 0S(5 + 1)/S -nPd-m ~ [35 ~ C ° n S t

(A 3)

'

This general equality is quite useful, and is already implicit it earlier theoretical work (see Fisher's 12 work and references therein). By using Eq. (A.3), one can easily construct equalities relating triads of critical exponents. Let us take, for instance, the four most commonly used exponents, i.e., 5 (critical isotherm), /? (coexistence curve), —7' (compressibility versus temperature), and —a' (specific heat versus temperature). Four triads can

116

Effective Field Approach to Phase

Transitions

be made in the following way: [(1/5) + 1 ] / ( - a ' + 2) = 1/05, [(1/5) - 1}/(-i)

= 1//35,

i.e., a ' + (3(1 + 5) = 2,

(A.4)

i.e.,

(A.5)

7

' +0(1-8)

= O

and eliminating successively (3 and (5 from Eqs. (A.4) and (A.5), a' + 25 - i - 5(a' + 7') = 2,

(A.6)

a' + 7 ' + 2/3 = 2.

(A.7)

Expressions (A.4), (A.5), and (A.7) can be recognized as the equality form of relationships introduced by Griffiths, W i d o m (also referred to as the Kouvel-Rodbell relation), and Fisher-Rooshbrook. By using (A.3), any desired relationship between three arbitrary critical exponents may be easily obtained. T h e series of experimental exponents available from T G S enable a direct experimental check on the constancy of the exponent ratio specified by Eq. (A.3). Table 2.1.3 shows t h a t the critical exponents associated with Fifi = P, F2fi = dP/dE, and F\^ = dP/dT satisfy the expected ratio Ik/lk+i = 3/2.

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

J.A. Gonzalo, Phys. Rev. 144, 662 (1966). P.P. Craig, Phys. Lett. 20, 140 (1966). J.A. Gonzalo, Phys. Rev. Lett. 2 1 , 749 (1968). J.S. Kouvel and M.E. Fisher, Phys. Rev. 136, A1626 (1964). W. Reese (private communication). It should be noted that precisely because of the method we have used to determine T, in spite of the fact that the corrections due to the electrocaloric effect can affect the absolute value of T, the associate correction in (T — T c ) should go to zero as one approaches T c , the temperature at which the hysteresis loops are recorded to practically disappear. Recently, very accurate experiments by E. Nakamura et al. (Proceedings of the Second International Meeting on Ferroelectricity, Kyoto, Japan, 1969), free from electrocaloric effects, establish conclusively the classical values previously reported for the exponents 7 and (3, the former even at temperatures as close to T as T — Tc « 0.01° C. J.A. Gonzalo and J.R. Lopez-Alonso, J. Phys. Chem. Solids 25, 303 (1964). M.S. Green, M. Vicentini-Missoni and J.M.H. Levelt Sengers, Phys. Rev. Lett. 18, 1113 (1967). J.S. Kouvel and J.B. Comly, Phys. Rev. Lett. 20, 1237 (1968). J.T. Ho and J.D. Litster, Phys. Rev. Lett. 22, 603 (1969). B. Widom, J. Chem. Phys. 4 3 , 3898 (1965).

Behavior atT = Tc of Pure Ferroelectric Systems

117

11. See, for example, M.E. Fisher, Rep. Progr. Phys. 30, 615 (1967). 12. R.B. Griffiths, Phys. Rev. 158, 176 (1967). 13. M. Vicentini-Missoni, J.M.H. Levelt Sengers and M.S. Green, Phys. Rev. Lett. 22, 389 (1969). 14. P. Schofield, Phys. Rev. Lett. 22, 606 (1969). 15. P.R. Roach, Phys. Rev. 170, 213 (1968).

Chapter 2.2

Effects of Dipolar Impurities in Small Amounts*

It is well known1 that a-alanine doped triglycine sulfate (TGS-a) crystals have outstanding pyroelectric properties at room temperature and have been widely investigated as ideal materials for pyroelectric detectors. The presence of an a-alanine concentration dependent effective internal bias in these crystals stabilizes their excellent pyroelectric properties in a wide temperature interval from room temperature to well above the transition temperature, Tc « 49°C, of pure TGS, at which its crystal structure changes from P2i to P2i/m. Early work 2-4 on TGS-a implicitly assumed that, since the polarity of this crystal was consistently recovered on going back and forth through T c , the effective internal bias was roughly temperature independent. Later experimental investigation of the temperature dependence of low-frequency TGS-a hysteresis loops in the close vicinity of Tc showed a pronounced temperature dependence of the internal bias, 5 which, from an approximately constant value at T
120

Effective Field Approach to Phase

Transitions

temperature behavior of pure TGS under the effect of external fields of variable intensity) the pronounced dependence of the effective internal bias in TGS-a crystals. Figure 2.2.1 summarizes schematically the different effects of external and internal biases in TGS. Recent work9 analyzing the effect of hydrostatic pressure in pure and a-alanine doped TGS suggests that the remarkable effectiveness of very small amounts (30-100 ppm by weight in the sample) of a-alanine to produce a bias can be understood in terms of local extended defects, made External Field

30

T

55

»

T

55

cz x!0" a

4»

T

SO

c

Fig. 2.2.1. A comparison of hysteresis loops, inverse dielectric constant and specific heat behavior for pure TGS (with various external bias fields) and TGS-a (with various alanine concentrations giving rise to different internal bias fields). Note that in all three cases the effect of an internal bias field near and above T c is weaker than the corresponding effect of a constant external field. The numbers in figures correspond to increasing internal/external bias field magnitudes. Data are schematic plots of actual results from the following sources: (Al) Ref. 6; (A2) Ref. 2; (Bl) and (B2) Ref. 8; (CI) Ref. 3; (C2) this paper.

Effects of Dipolar Impurities

in Small

Amounts

121

up of an a-alanine substituted unit cell and its surrounding mechanically distorted neighbors, which act in such a way as to stabilize the polarity of the crystal in going back and forth through the transition. Its authors do not discuss the temperature dependence of the bias. In the present paper, we report hysteresis loops and dielectric constant observations of the internal bias field in TGS-a crystals with various aalanine concentrations through T c . The temperature dependence of the effective internal bias field (E\,) and the zero field polarization (PQ) can be analyzed within the framework of the statistical theory of ferroelectrics, 10 ' 11 taking duly into account the distinct response of the randomly distributed polar centers associated with molecular units containing a-alanine (in contrast with response of normal, all glycine units) and can be shown to describe correctly the observed temperature dependence. TGS-a single crystals, grown from different water solutions (7-20% molar content), were kindly supplied by F. Jimenez (U. de Sevilla). They were of reasonably large size and good optical quality. From them various plates of area ranging from 0.3 to 2 cm 2 , and thickness ranging from 0.15 to 0.3 mm, were cut. Gold leaf electrodes were attached at the main surfaces of these samples. A set of three samples spanning a relatively wide range of a-alanine concentration (as determined from room temperature direct E\, measurements) were selected for investigation. A digital oscilloscope (Nicolet Explorer I) was used to observe the hysteresis loops in a wide frequency range (0.1-1000Hz). For dielectric constant measurements an ESI Video Bridge model 2100 was used. The driving voltage amplitude for hysteresis loops measurements was 300 V, and 1V for dielectric constant measurements. Constant external voltages up to 600 V for hysteresis loops and up to 45 V for dielectric constant were applied to the samples during the measurements. As it might be expected voltages of opposite polarity showed opposite effect in the crystal dielectric response, i.e., one polarity brings the data on TGS-a closer to those of pure TGS, while the opposite polarity takes them away. The temperature control of the sample holders in the thermal bath was made with a Haake F3, which gives a thermal stability of ±0.005°C. In most cases, however, we used slow thermal heating and cooling rates with periodic temperature recordings. The temperature was measured by means of a chromel-constantan thermocouple and a Keithly 195 digital voltmeter with a resolution of ±0.5 /xV.

122

Effective Field Approach to Phase

Transitions

The dependence of the hysteresis loops shape with the driving field amplitude was first checked at room temperature, and a sufficiently large amplitude (Eo « 2.5 kV/cm) was chosen for the measurements to be taken at T « Tc in such a way that the influence of this parameter, as well as the driving field frequency, were not very pronounced. In order to be able to compare hysteresis loops data with dielectric constant data, both sets of measurements were made at a frequency of 1 kHz. Figure 2.2.2(a) shows a set of data of effective internal bias field E^ as a function of temperature in an interval of ±5°C around Tc for three samples which had room temperature internal bias field of 80 V/cm (#1), 390 V/cm (#2), and 515 V/cm (#3). The room temperature E\> value is known to be roughly proportional to the a-alanine content in the crystals; therefore, these samples spanned a range of nearly one decade in maximum (room «00

4 BO

-£

O

Sample 3

0

Sample 2

V

Sample 1

360

> u

JUC

120

44

46

48 TEMPERATURE

SO

52

6*

(*C)

1.0

\

0 44

46

46 TEMPERATURE

SO

52

54

f'C)

Fig. 2.2.2. Variation of the internal bias field (a) and zero field polarization (b) with temperature near Tc for three samples of increasing alanine content. Data points labeled X correspond to sample 3 for lowering temperature.

Effects of Dipolar Impurities

in Small

123

Amounts

temperature) E\> value. It was observed that there is a certain "thermal hysteresis" between the rising and lowering temperature data. Sufficiently away from Tc both sets of data for decreasing and increasing T coincide very well. This might be, in principle, attributed to a pronounced temperature dependence of the a-alanine polar centers' relaxation time near Tc. A similar, and even more pronounced "thermal hysteresis" effect has been observed in specific heat measurements 7 ' 8 and it has been seen that the rising temperature data approach more closely the successive equilibrium states, as inferred from the fact that, for cooling, if one waits for very long times at a fixed T, the value of E\, approaches asymptotically the corresponding i?b value for heating. This effect is still under investigation and eventually the results would be published elsewhere. Figure 2.2.2(b) shows the zero field polarization dependence with temperature. In this figure we can see clearly the smooth decrease of the polarization through T c , which is due to the presence of a-alanine polar centers in the crystal lattice. Figure 2.2.3 gives a representation of Eb (normalized to its maximum value) versus PQ (normalized to the zero field polarization at T = Tc) obtained also from the same hysteresis loops at various temperatures. Only data for rising temperature are shown. It can be seen that, at least for

| > H ^

y •

&S/

' / •

/ /<=

•

'

Sample 3

<s> Sample 1 ?

Sample 1

-

Theory

•e,o E

1 O

•<

.8 p

1.2

1.6

2

o/Poc

Fig. 2.2.3. Plot of internal bias field normalized to its value, E^/E^MI versus zero field polarization normalized to its critical temperature value, Poc = Po(Tc), for the same three examples of Fig. 2.2.2. The continuous line gives the theoretical relationship obtained from Eq. (15).

124

Effective Field Approach to Phase

Transitions

the range of a-alanine concentration under investigation, the normalized data fall approximately into the same universal curve, within experimental accuracy (about 5% in the worst cases). We may infer from these data that there is no interaction among a-alanine polar centers at the low levels of concentration in the samples, roughly estimated to be in the lOppm range by comparison with previous work.9 In the same figure we have plotted the theoretical curve obtained from a simple statistical theory applicable to a dipolar ferroelectric with a diluted concentration of randomly distributed polar centers. The agreement between the experimental behavior of E^{PQ) and the theoretical relationship, to be discussed below, is excellent. Figure 2.2.4 shows the dielectric constant, s', as a function of temperature for the same set of samples with varying concentration of a-alanine at 1kHz and driving voltage amplitude IV. The frequency dependence between 20 Hz and 20 kHz was relatively small. We also saw in these measurements the same phenomenon of thermal hysteresis between sets of data for rising and lowering temperatures. Let us consider a ferroelectric crystal 10,11 with N normal ferroelectric unit cells, each one possessing a normal dipole moment //, which can be oriented parallel or antiparallel to the effective molecular field: EeS = E + (3P,

I 45

1 47

1 49 TEMPERATURE

i S)

(1)

i 53

I 55

(C )

Fig. 2.2.4. Plot of the dielectric constant versus temperature for the same three examples studied in hysteresis loops measurements. As an inset we can see the plot of the dielectric loses versus temperature near Tc for the same samples.

Effects of Dipolar Impurities in Small Amounts

125

where E is the external field, (3 the mean field coefficient (f3 = 4irTc/C, with C = A T / A e - 1 = Curie constant), and P the polarization per unit volume. At T < Tc, EeK is not zero even for E — 0. In equilibrium or quasi-equilibrium conditions, iVlPl2 =

(2)

N2P21,

where N\ is the number of unit dipoles pointing in the direction favored by the effective local field and N2 the number of unit dipoles pointing in the opposite direction. Taking into account that, P12 = Po ex.p(-Eefffi/kBT),

P21 = Po

exp(EeS^/kBT), (3)

Po = wexp(-Af//fc B T),

where p\2 and P21 are dipole jump probabilities per unit time between asymmetric states for opposite directions, po is the jump probability per unit time between symmetric states, and At/ is the height of the potential barrier between states, and is a characteristic attempt frequency. It is straightforward to get, knowing that Ni + N2 = N,

P=(N1-

N2)n = tanh

(E + f3P)v kBT

— N/j, t a n h

TCE + (3P ~T (3N(i

(4)

where Tc = /3A//z2 /kB- We have here assumed for the sake of simplicity that the dipoles are not deformable, and then that the value is the same for low and high values of Eeff. For E = 0 we have, —r- = tanh Nfj,

II3L T NJJL

(5)

where Ps is the spontaneous polarization, corresponding to E = 0. If we consider now a ferroelectric with a randomly distributed small number of polar centers Na
126

Effective Field Approach to Phase

Transitions

mechanically distorted, neighbors.) Then we can write, Pa

•EeffMa

tanh

= tanh

"/WW

(6)

Taking into account that Eh = f3Pa,

EhM =

(3NA^,

(7)

where EbM is the bias field maximum value and that, for JVa -C N, Tc is still given by Tc = f3N/j?/kB, we can rewrite Eq. (6) as Eb EbM

= tanh

Tc Ma Pp

or =

Ma =

I^tanh-^Eb/EbM]

(8)

On the other hand, for the normal glycine dipolar units under the influence of Eb and using Eq. (4), we have = tanh

T Eh + 0Po]

N/J,

(9)

and making the use of the fact [see Eq. (7)] that Eb 0Nn

Eh JVaMa EhM N fi '

(10)

we can rewrite Eq. (9) as —- = tanh

T Eh Nafi& .T c EbM N n

P0" Nfi_

—- = tanh

T Eh ^VaMa Tci SbM N fj,

Po~ V/x.

or (11)

We can eliminate a, the ratio of polar center dipole moment to normal unit cell dipole moment (which is not necessarily constant), from Eqs. (8)

Effects of Dipolar Impurities

in Small

Amounts

127

[Pol

(12)

and (11) to get —— t a n h - 1 •C/bM

[^1

N P0

Ebu

T —-tanh - 1 To

Taking into account that the change in E\> occurs in the very close vicinity of Tc (see Fig. 2.2.1) where T w Tc, and that there PQ/NH < 1, we can approximate Eq. (12) by Eh

Eh tanh E\bM

^bM

I^Po_ NaNn

H

P^ Nil

Nn

N A /P0 Na3 \NIM

(13)

For T = Tc, we define Ehc = Eh (T = Tc) and P0c = P0 (T = Tc). Then, Ebc/Eh = Ebc/Ebu = 0.5 in all cases, and Po/N/x = P0c/N(j,, changing with the a-alanine concentration in such a way that the larger that concentration is, the larger POC/NJJL becomes. Then, right at the transition temperature, Eq. (13) reads, (0.5) tanh" 1 (0.5)

N_\ ATa3

Nn

(14)

and from Eqs. (13) and (14) we get 1/4

Pv_ PQC

( # - ) tanh" (0.5) t a n h - 1 (0.5)

(15)

This expression, obtained in a straightforward way from simple statistical physics considerations, fits very well the experimental data for all samples examined, as shown in Fig. 2.2.3. A good test for the results obtained from the statistical theory developed in the last paragraph consists in checking whether it is able to give estimates of the alanine concentration in the samples (not in the aqueous solutions from which they were grown, where the concentration is, of course, much higher) in agreement with the chemical analysis determinations previously made by other authors 9 on samples with similar values of internal bias.

128

Effective Field Approach to Phase

Transitions

From Eq. (14) we get

Since we know P 0 c experimentally (see Fig. 2.2.2(b) for T = Tc) and Nfj, = ( P S ) M = 4 / i C / c m 2 = 1.2 x 10 3 esu, we can substitute in Eq. (15) the values of (Poc/Nfi), which are 0.04 for sample # 1 , 0.075 for sample # 2 , and 0.09 for sample # 3 . These lead to values of Na/N ranging from 10~ 4 to 10~ 5 , i.e., 10-100ppm, which is in good agreement with those quoted by Novik et al. for samples with room t e m p e r a t u r e internal bias in the same range as those of our samples.

References 1. M.E. Lines and A. Glass, Principles and Applications of Ferroelectric and Related Materials (Oxford University Press, Oxford, 1980). 2. K.L. Bye, P.W. Whipps and E.T. Keve, Ferroelectrics 4, 253 (1972). 3. K.L. Bye, P.W. Whipps, E.T. Keve and M.R. Josey, Ferroelectrics 7, 179 (1974). 4. E.T. Keve, Phillips Tech. Rev. 35, 247 (1975). 5. J.L. Martinez, A. Cintas, E. Dieguez and J.A. Gonzalo, Ferroelectrics Lett. 44, 221 (1983). 6. J.L. Martinez, J.E. Lorenzo, E. Dieguez and J.A. Gonzalo, Cryst. Lattice Defects Amorph. Mater. (1987). 7. J. del Cerro, Private communication. 8. F. Jimenez, Ph.D. Thesis, Universidad de Sevilla (1987). 9. V.K. Novik, N.D. Gavrilova and G.T. Galstyan, Sov. Phys. Crystallogr. 28, 684 (1983). 10. J.A. Gonzalo and J.R. Lopez-Alonso, J. Phys. Chem. Solids 25, 303 (1964). 11. J.A. Gonzalo, Phys. Rev. B9, 3149 (1974).

Chapter 2.3

Mixed Ferro-Antiferroelectric Systems and Other Mixed Ferroelectric Systems*

Considerable attention has been devoted in recent years to the investigation of phase transitions in mixed crystals with competing interactions and, in particular, to the mixed ferro-antiferroelectric system 1 (RDP)i_! (ADP)^ made up of rubidium and ammonium dihydrogen phosphate. The phase diagram of this system is shown in Fig. 2.3.1. In this chapter, we will investigate first a simple two-sublattice classical model for ferro- and antiferroelectricity, which will be shown to describe consistently the transition for pure RDP and pure ADP. Next, we will consider the behavior to be expected for the phase transition classically, i.e., in the absence of quantum fluctuations of the elementary dipole moments. Finally, we will introduce the appropriate quantum corrections in the expressions for Tc and TN and will show that satisfactory agreement with the observed phase diagrams is obtained. It is well known that crystals isomorphous with potassium dihydrogen phosphate (KDP) often show ferroelectric or antiferroelectric phases at low temperatures. The fact that an antiferroelectric arrangement of dipoles is sometimes realized suggests that a simple two-sublattice model, similar to the one for antiferromagnets, 2 may be useful to describe the phase transition in the KDP family. Since there are four chemical units (Z = 4) per (conventional) tetragonal unit cell in KDP crystals, the system may be thought of as a superposition of two interpenetrating bcc sublattices, each *Work previously published under the title "Quantum effects and competing interactions in crystals of the mixed rubidium and ammonium dihydrogen phosphate system," J.A. Gonzalo, Phys. Rev. B 39 (1989). Copyright © 1989. The American Physical Society. 129

130

Effective Field Approach to Phase

150

Transitions

IROPVx lAOPl*

P£ (Telral

(Af£ JtOrtho)

'Dipolc glass phase reported 0 0.2 RDP

0.4

0.6

0.8

X

1.0 p

AD

Fig. 2.3.1. Phase diagram of (RDP)i_ ; E (ADP)x. Observed, solid line (see Ref. 1); theory (including quantum effects); crosses [Eqs. (22) and (23) with xc = 0.22 and yN = 0.26, respectively; see text].

containing two chemical units. Let us call A and B as these two sublattices, and take into account that we must consider intralattice (AA, BB) and interlattice (AB, BA) interactions. In the mean field approximation, the local effective field acting on unit dipoles at points in lattice A and B, respectively, will be (£ eff )a = £ + aPa + /3Pb,

(1)

(£eff)b - E + f3Pa + aPh,

(2)

where E is the external field, a(AA) = a(BB) and /3(AB) = /3(BA) are mean field coefficients, and Pa and Pb are the pertinent sublattice polarizations per unit volume. Using these effective fields in the standard 3 mean field expression for the polarization as a function of temperature we get

Nfia

tanh

kT

^..^(Ggte),

(3)

(4)

Mixed Ferro-Antiferroelectric

Systems

131

where N is the total number of sublattice sites (or unit cells) per unit volume, and /i a and /^b are the respective sublattice unit dipoles, which in the ordered phase can be pointing parallel (ferroelectric case: \ia = /Ub) or antiparallel (antiferroelectric case: \ia = —/^b) to each other, depending on the geometry of the lattice. Since the two sublattices are chemically identical, we take |/i a | = |/-*b|- From Eqs. (l)-(4), we get E=—

kT P t a n h - 1 - A - - aPa - (3Pb, Ma N/J,a kT

E=—

Pu tanh-1-^-/3Pa-aPb, Mb Nfih

(5) (6)

which, together, define an equation of state for E, P = P a + Pb and T. Let us consider first the ferroelectric case, in which at T < Tc we have a non-zero spontaneous polarization P s = P s a , Psb = 0 for E = 0. Adding Eqs. (5) and (6), and taking into account that t a n h - 1 Z± ± t a n h - 1 Zi = t a n h - 1 ( Z i ± ^2)/(l ± Z\Z-i) and that \i& — /J,\, = /z/2, we get 2£ = 2 - t a n h

\

+ A{Pa/N^iPh/Nfl)-(<*

+ P)(P* + P*)-

W

It is easy to get from here the spontaneous polarization below the transition temperature, but not very far from it, as

P./^«VS(T C /T-I)V» 1 Tc=t±mi.

(8)

The low amplitude (SE —* 5P <€! Nfj,) inverse dielectric constant e - 1 = dE/AndP can be obtained from Eq. (7) with the following results: e-1 = (T-ec)/C+, 2

C+ = 2nNfj, /k,

at T > GC(PS = 0), e c = T
(9)

and e" 1 = (0C - T)/C-,

at T < 6 C (P S = 0); C_ = Nnn2/k.

(10)

Next, we consider the antiferroelectric case. At E = 0, we have now Ps = Psa + Psb = 0, P s a = —Psb = PSs being the sublattice spontaneous

132

Effective Field Approach to Phase

Transitions

polarization, obtained solving 4(Pss/A^) =T_ tanh- 1 {4(P ss /7V/i)/[l + 4(Pss/ArM)2]} TN'

_ N

(a-p)N^ 4fc '

[

'

which has a non-vanishing solution for Pss at T < T N , the antiferroelectric transition temperature. Below TN, and not very far from it, we can get from Eq. (11) Pss/Nfi « (>/3/2)(l - T / T N ) 1 / 2 .

(12)

The low-amplitude expression for £ _ 1 can be obtained from the sum of Eqs. (5) and (6) for 5E -> SP <S Nfj,, resulting in e-1 = ( T - 0 N ) / C + ) C+ =

atT>TN(Pss=0),

2

2irNn /k,

©N - —A

r

,

(13)

and in another, more complicated expression, at T < TN, which leads to

e-1(T^)=s~1(T+)=-/3/S^ £-!(()) = oo,

(i.e., e(0) = 0 ) .

(14)

For RDP (ferroelectric, second order transition) we can get the sum of the sublattice mean field coefficient from Eqs. (8) and (9), and the observed values for 6 C = 147 K and C+ = 4.14 x 103 K, as a + (3 = 87r6 c /C + = 0.89 > 0.

(15)

For ADP (antiferroelectric, first order transition) we get the sum and difference of a and /? from Eqs. (13) and (11), along4 with 0 N = —14K, T N = 148 K, and C+ = 2.67 x 10 3 K, as a +/3 = 87re N /C + = -0.13 < 0, a-(3

= 87TT N /C* + = 1.39 > 0,

(16) (17)

and, therefore, a = 0.63, /? = —0.76. That the two-sublattice mean field relations obtained for KDP-type crystals show a fair degree of internal consistency can be checked in several ways. For instance, for RDP we get n = (kC+/2-!rN)1/2 « 6.2 x 10- 1 8 esu from Eq. (9) and fx = P8B.t/N = 7.0 x 10 _ 1 8 esu from the low-temperature (saturation) value of the spontaneous polarization P s a t « 5.6/xC/cm2 = 5.6 x 3000esu, ./V = ( c a 2 ) - 1 = 2.36 x 10 21 unit cells/cm 3 . For ADP we

Mixed Ferro-Antiferroelectric

Systems

133

can use Eq. (14) to get an independent estimate of j3 = e(T^)/8n. Using e(Tj7) ~ 14 and e(T^) « 30, 4 one gets, respectively, /?_ = —0.55 and j3+ = —1.19. These values average out very close to the one previously obtained from Eqs. (16) and (17). It is of interest to point out that, since a and /? depend on the tetragonality (c/a — 1) of the lattice (they should be zero for a cubic lattice if they are originated by electric dipole-dipole interactions) we are entitled to assume a = a(c/a — 1) and (3 = P{c/a — 1), and we can use the calculated values for ADP, namely C 0 and also a linear increase in TN at x < 1. Since TC(RDP) = 147K, T N (ADP) = 148K, and AT(RDP)' = 2.36 x 10 21 and A^(ADP) = 2.37 x 10 21 are very close to each other, one may expect a similar linear trend of transition temperature with composition from both sides toward zero. Then, classically, from Eqs. (8) and (11), we have kTZ(p) = ^ - N ^ ,

kT*c{x) =

^±IN(1

- x)n2 = fcTc*(0)(l - x), (18)

A:TN(0) = ^ T A ^ 2 ,

kT^y)

= ^-N{\

- y)»2 = fcTN(0)(l - y), (19)

where y = 1 - x, and T*(0), T N (0) are the classical transition temperatures for the pure ferroelectric and antiferroelectric systems. Quantum

134

Effective Field Approach to Phase

Transitions

mechanically, on the other hand, we have huJoc I 2 + (n)Tc j = —~Nn2 J«WON

for x = 0,

f 2 + TN J = ^-J-N^2

for

» = 0.

where V^k^Oc and 1 / 2 ^ O N are the zero point energies and (n) is the average number of energy quanta excited above the ground state at temperature T, given by Planck's relationship. Therefore, huoc/2 tanh[/iwoc/2fcTc(0)] = ^^-Nfj,2 /IWON/2

= kT*{0),

tanh[/iwON/2fcTN(0)] = ^ p - i V > 2 = kT£{0),

where Tc(0) and TN(0) are the quantum mechanical transition temperatures, which reduce to Tc*(0) and T"N(0), respectively, for /iu>Oc/2A;Tc(0) -C 1 and hu)o^/2kT^(0) <S 1. The last two equations yield Tc(0) = ( ^ 0 c / 2 f c ) / t a n h - 1 ( Z c ) ;

Zc = /iwoc/2fcTc*(0),

(20)

T N (0) = (/iw 0N /2fc)/tanh- 1 (Z N );

Z N = hu;ON/2kT^(0),

(21)

and, therefore, we finally get Tc(x) = T c ( 0 ) t a n h - 1 ( Z c ) / t a n h - 1 [ Z c / ( l - a;)],

(22)

T N (0) = r N ( 0 ) t a n h _ 1 ( Z N ) / t a n h - 1 Z N / ( 1 - j/)].

(23)

Since Tc(xc) = 0 for Zc = 1 — a;c and TN(T/N) = 0 for ZN = 1 — 2/N, it is sufficient to know the experimental values for xc and yN to completely specify the phase diagram. For the (RDP) x (ADP)i_ x system, xc ~ 0.22, 2/N ~ 0.26 (see Fig. 2.3.1). The agreement between theory and experiment for this mixed system is excellent, as seen in Fig. 2.3.1. References 1. For a recent review, see, for example, H. Terauchi, Phase Transitions A7, 315 (1986) and references therein. 2. See, for example, A. Dekker, Solid State Physics (Prentice Hall, Englewood Cliffs, NJ, 1962), pp. 484-488. 3. See, for example, J.A. Gonzalo and J.R. Lopez Alonso, J. Phys. Chem. Solids 95, 303 (1964); J.A. Gonzalo, Phys. Rev. B9, 3149 (1974).

Mixed Ferro-Antiferroelectric Systems

135

4. Ferroelectrics and Related Substances, edited by K.H. Hellmege and A.M. Hellwege, Landolt-Berstein, New Series, Group III, Vol. 16 (Springer-Verlag, Berlin, 1981 and 1982). 5. U.T. Hochli, Ferroelectrics 35, 17 (1981); see also K.A. Miiller, Jpn. J. Appl. Phys. (Suppl.) 24, 89 (1985). 6. W. Windsch, H. Braeter and B. Milsch, Ferroelectrics 47, 213 (1983).

Chapter 2.3.1

Comment on "Ferroelectricity in Zinc Cadmium Telluride"

Ferroelectricity in Zn^Cdi-^Te has recently been reported by R. Well et al.1 for Zn content from 4 to 45%. This is very interesting because, for the first time, ferroelectricity is found in materials crystallizing in the zincblende configuration, either of the binary constituents making up the mixed crystals being non-ferroelectric. Attempts by the authors 1 to fit the experimental curves giving the transition temperature as a function of Zn concentration, Tc(x), and the Curie constant as a function of Zn concentration, C(x), were unsuccessful because of the difficulty to match the rapid decrease of Tc(x) for x < 0.04. The aim of this comment is to show, hopefully, that the dipolar, effective field approach, 2 to ferroelectric transitions applied to this system can give a fair description, at least in semi-quantitative terms, of the dependence of transition temperature (Tc) upon concentration (x). The pure CdTe structure is made up of Cd tetrahedra with Te atoms at their center. Substitutional Zn atoms give rise to tetrahedra with 0, 1, 2, 3, and 4 zinc atoms, with different probabilities determined by the average concentration i in a homogeneous sample. Obviously, 0 and 4 zinc atoms produce symmetric tetrahedra with no dipole moment. It may be assumed also that 2 zinc atoms do not give rise to any dipole moment along (111). On the other hand, 1 and 3 zinc atoms produce asymmetric tetrahedra with possible non-zero dipole moments pointing in (111) directions. The probability of realization for tetrahedra with 1 zinc atom is p\x{l — x) 3 , and for tetrahedra with 3 zinc atoms it is p3 = x3(l — x). The effective field2 at a dipole site will be, therefore, Erf = E + frPi + 137

foPs,

(1)

138

Effective Field Approach to Phase

Transitions

where E is the external field, (3\ and P3 are effective field coefficients, and P\ and P3 are polarizations due to tetrahedra with 1 and 3 zinc atoms, respectively. These polarizations are given2 by Pi/Nun P3/N3fi3

=tanh(EeStx1/kBT), =

(2)

tanh(EeSfjL3/kBT),

where Nx = (4/o~ 3 ) x (1 - x) 3 , i\r3 = (4/a~ 3 )x 3 (l - x), and Hi, /j,3 are the respective elementary dipole moments. Combining the relationships for P\ and P3 in Eqs. (2), one gets PiNi/n t a n h - 1

(PI/NKH)

= [(faNtvD/kBT]

+ p3N3^3 tanh" 1 ( A P i + p3P3),

which for T < T c , i.e., P i / A ^ i
(3)


+ foN^DkB = (pNn2/kB){x(l

Tc(x) = (PiNitf

(P3/N3fi3)

- x ) 3 + x 3 (l - x ) } (4)

taking pi = /?2 — P, Mi = M3 = A*- Numerical values for /? = A-KTC/C, where C is the Curie constant, for N = 4/a+ 3 , where 3 a+ 3 = l/2a 3 (CdTe) + a 3 (Z n i Te) = 2.489xl0 2 2 cm 3 , and for (j. = [fc B C/4xiV{x(l-a;) 3 +x 3 (l-a:)}] are easily obtained, and also for x = 0.25, p = 4.1 (close to the Lorentz value 4/3) and fi ss 3.0 x 10~ 18 esu. For other concentrations the g values differ 1

•"-

^

6. E

400

-

1—

*c 3 O

//

200

// "

/ 0.0

0.1

0.2

0.3

0.4

x, Proportion of Zn Fig. 2.3.1.1.

Curie temperature vs. x (Zn proportion).

0.5

Comment

on "Ferroelectricity

in Zinc Cadmium

Telluride"

139

from (3{x = 0.25) by no more t h a n 20% and the // values from fi(x — 0.25) by no more t h a n 40%. Figure 2.3.1.1 shows the calculated Tc(x) according to Eq. (4) with (3Nfi2/kQ « 5.0 x 10 3 consistent with the parameter values above, within the quoted uncertainties, along with experimental d a t a of Well et al.A No errors bars are given for the d a t a in Ref. 1, b u t the overall agreement seems to be fair. It may be noted t h a t the calculated spontaneous polarization for x = 0.10, Ps = ATi/ii = ( 4 / a 3 ) x (1 - x ) 3 w 1.9 x 10 3 esu « 0 . 6 3 / i C / c m 2 , obtained with fj,(x) = 3.6 x 1 0 _ 1 8 e s u , is also compatible with the value observed from hysteresis loops.

References 1. R. Weil, R. Nkum, E. Muranovich and L. Benguigui, Phys. Rev. Lett. 62, 2744 (1989). 2. J.A. Gonzalo, Phys. Rev. B39, 12297 (1989). 3. See, for example, N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976). 4. The vertical scale in Fig. 4 of Ref. 1 should be in /xC/cm 2 (private communication by R. Weil). Note also that the loop is not compensated for excess conductivity.

Chapter 2.4

Relaxation Phenomena Near Tc*

Lithium tantalate is a ferroelectric compound with important technological applications due to its nonlinear optical and photorefractive properties. 1 LiTaOs and its isomorph LiNb03 are high-temperature ferroelectric materials, 2 with critical temperatures of 618 and 1140°C, respectively. Usually, they are ground with a slightly non-stoichiometric composition, 3 ' 4 which can vary in a narrow range around the congruent melting composition. The congruent melting composition gives rise to crystals with a relative abundance of 2.8 of Li + vacancies. An important increase of the dielectric constant is observed at low frequencies in the 400-900°C range. The influence of charged mobile defects (such as Li + vacancies) on dielectric properties has been studied for LiTaC>3,5 LiNb03, 6 and, more recently, some perovskite-type crystals. 7 The transition temperature of LiTaC>3 is inside the temperature range where dispersion on the dielectric constant caused by mobile defects is important, and it is necessary to separate out the ferroelectric contribution in order to study the critical instrinsic relaxation behavior near Tc. The experimental real and imaginary parts of the dielectric constant appear to show the sum of the contribution of mobile defects, which has a 1/LO behavior, and a Debye-like ferroelectric contribution. In the present work, which extends previous e(u) measurements up to 10 MHz, the ferroelectric relaxation time is obtained for LiTaC>3 and it is analyzed in terms of mean field theory predictions.

*Work previously published under the title "Ferroelectric relaxation of LiraO," C. Prieto and J.A. Gonzalo, Solid State Commun. 61, 4373 (1987). Copyright © 1987. Pergamon Press PLC. 141

142

Effective Field Approach to Phase

Transitions

A very good optical quality single crystal of LiTaC-3 was donated by G. Samara, from Sandia Laboratories. The sample was disk shaped, 9.6 mm in diameter and 2.7 mm in thickness. Gold leaf electrodes were attached to the main surfaces perpendicular to the ferroelectric c-axis. Measurements were performed in a high-temperature furnace, which allowed work at variable oxygen pressure. The sample temperature was measured with a Pt/Pt-Rh(13%) thermocouple by a digital microvoltimeter and remained constant within O.IK for each set of data. The capacitance and dielectric losses of the sample were previously measured with an ESI2100 video-bridge from 20Hz to 20kHz at various temperatures near Tc. In order to characterize the temperature-dependent relaxation of e(u>) well above Tc, additional measurements were performed with a HP-4192A at these temperatures from 10 Hz to 10 MHz. The real part of the dielectric constant of LiTa03 is represented in Fig. 2.4.1 for various temperatures, as a function of the frequency. At very low frequencies there are 1/u contributions identified with mobile charged defects motion. 5 The ferroelectric relaxation appears clearly when the extrapolated 1/w contribution is subtracted from the experimental data points. Figure 2.4.2 shows the imaginary part of the dielectric constant corresponding to the same data as in Fig. 2.4.1. It is possible to observe the peaks' characteristic of ferroelectric relaxation in natural scale once the mobile defects contribution has been subtracted.

10'

to2

io*

10* FREQUENCY (Hz)

Fig. 2.4.1. Log-log plot of the real part of the dielectric constant as a function of frequency for LiTaC>3 at three temperatures. Broken lines are ferroelectric contributions (see text).

Relaxation Phenomena

3

143

Near T c

. V

.-ioi

<*>

10s

\

U

X

—

V

£

V

l!

-•'""••

0.8

10'

K \

10' . - - ' • ?

10' 10!

ft*

'\>-. \

/

.

""---..,«v"»-'e 10*

10' FREQUENCY

(Hz)

Fig. 2.4.2. Log-log plots of the imaginary part of the dielectric constant as a function of frequency for LiTaC>3 at three temperatures. Broken lines are ferroelectric contributions; in this case, they have been represented in the left-hand natural scale (see text).

-1.0

log (AT/T) Fig. 2.4.3. Log-log plot of temperature dependence of inverse ferroelectric relaxation time vs. ( A T / T ) .

We can obtain at every temperature the characteristic time from the maximum of the ferroelectric contribution to the imaginary part of the dielectric constant as a function of the frequency, or from the inflection point of the real part as a function of frequency. Figure 2.4.3 is a log-log plot of the inverse relaxation time vs. (AT/T) to show the critical exponent of this dependence, x = (logr _ 1 )/(log A T / T ) . Mean field theory (neglecting fluctuations) gives for this exponent a value

144

Effective Field Approach to Phase

Transitions

x = 1, which is approximately the observed value. The same behavior can be observed if one analyzes the temperature dependence of the loss factor at a fixed frequency.5 The observed relaxation behavior of an order-disorder ferroelectric may be explained using a double-well potential model by means of a mean field approach. The effective field on an elementary ferroelectric dipole is jEeff = E + (3Pd, where Pd is the dipolar contribution to the polarization. In thermal equilibrium, the number of dipoles in both directions and the jump probabilities between the two equilibrium positions are related by -/VlPl2 = ^V2P21-

(1)

The expression for the jump probability is p = vexp(-U/kBT)

exp(P-EeS/kBT)

= pQexp(nEeS/kBT),

(2)

where U is the potential height barrier and fi is the electric dipolar moment. The total number of dipoles is N = N1+N2 and the dipolar polarization can be written as Pd = M-^2 — N\). In thermodynamical equilibrium we have Pd N/i

Nijl-expi-U/kaT)) N1(l + exp(2U/kBT))

_

u(E + pPd) kBT

[)

and for zero external field, Ps = Pd, Nfi

\kBTNij,J

\TNfiJ

(4)

If a high enough external field E is applied in opposition to Ps the reversion of this polarization gives a transitory current per unit area —

= N(pi2-p2i)-Pd(Pi2+P2i)-

(5)

Using dimensionless variables, we have % = -sinh(0(£+p)) -pcosh(0(£+p)), at r

(6)

where

From Eq. (6) it is possible to obtain expressions for the coercive field, switching current, and switching time, as well as the ferroelectric relaxation frequency as a function of temperature. An ac field E = EoeluJt with EQ

Relaxation

Phenomena

Near Tc

145

much smaller than the coercive field will give rise to a small change in spontaneous polarization p = ps + Ap.

(7)

Substituting in Eq. (6), and expanding the hyperbolic tangent, Pa

= tanh 0ps = 0pa-^ (Opsf

(8)

and therefore p s = (3(6»-l)) 1 /2.

(9)

Using Eqs. (7) and (9) and the differential Eq. (6), one gets

At

T

T

\TC-T4ir

where the ferroelectric relaxation time can be identified with Tr = T ( 2 T / T C - T), resulting in

dAp At

1 (c-eo_EoeMiujt)_Ap\

( n )

r r \ A-K

(Here eo is the high-frequency dielectric constant depending only on the atomic polarization P ea -) The solution of this differential equation, omitting the transitory term, which decays rapidly, is fd(*) = T - , £ , ~ £ ° -Eoexp(^t).

(12)

Prom this it is possible to calculate the frequency dependence of the dielectric constant as v E + 47rAPea + 47rAFd / , e - e0 \ .{e - e0)uiTr e =- = = Uo + , . o o - «E E V 1 + w2rr2) 1 + o;2rr2 ' (13) which provides Debye-like relaxation behavior with characteristic ferroelectric time r r , which can be expressed as (rr)'1 = (2AT/T)(veM-U/kBT)).

(14)

Near from T c , the exponential factor may be taken as a constant in comparison with (2AT/T).

146

Effective Field Approach to Phase

Transitions

T h e same calculation is valid for T > Tc, where Ps = 0 and e « C/AT (instead of 2C/AT). T h e LiTaC>3 d a t a shown in Fig. 2.4.3 seem to agree well with the predictions of Eq. (14), obtained using a mean field approximation. Nevertheless, measurements have not been made at temperatures very close to T c , and we cannot conclude t h a t this behavior is valid in the closed vicinity of T c . Note t h a t while L i T a 0 3 is usually considered a displacive ferroelectric, recent works 8 ' 9 indicate t h a t it is more justified to take it as a mixed order-disorder displasive ferroelectric.

References 1. A. Rauber, Current Topics in Material Science (North-Holland, Amsterdam, 1978). 2. A.M. Glass, Phys. Rev. 172, 564 (1968). 3. R.S. Roth, H.S. Parker, W.S. Brower and J.L. Waring, Fast Ion Transport in Solids, Solid State Batteries and Devices (North-Holland, Amsterdam, 1973). 4. L.O. Svaasand, M. Eriksrud, G. Nakken and A.P. Grande, J. Cryst. Growth 22, 230 (1974). 5. C. Prieto, L. Arizmendi, J.A. Gonzalo, F. Jaque and F. Agullo-Lopez, Phys. Rev. B 3 1 , 5483 (1985); Phys. Rev. B33, 7625 (1986). 6. C. Prieto, L. Arizmendi and J.A. Gonzalo, Jap. J. Appl. Phys. 24 (Suppl. 2), 1025 (1985). 7. T. Fernandez-Diaz, C. Prieto, J.L. Martinez, J.A. Gonzalo and M. Aguilar, Ferroelectics 8 1 , 19 (1988). 8. E.J. Samuelsen and A.P. Grande, Z. Physik B 24, 207 (1976). 9. C. Prieto, L. Arizmendi and J.A. Gonzalo, Ferroelectrics 55, 63 (1984).

Chapter 2.5

Polarization Reversal in Ferroelectric Systems* The process of ferroelectric polarization reversal under external field pulses of variable strength has been widely investigated in the last three decades. 1 Among uniaxial ferroelectrics, triglycine sulfate (TGS) and its isomorphs are probably the best studied experimentally. Pulvari and Kuebler2 and Chynoweth 3 did preliminary work investigating the switching behavior at low fields. Fatuzzo and Merz4 and Binggeli and Fatuzzo 5 extended the range of observation up to intermediate and high fields. Further work by Hayashi,6 Jaskiewicz and Przeslawski,7 and Perez et al.s contributed to determine the switching behavior at low and very low fields and questioned the validity of the semiempirical model of Miller and Weinreich9 to describe satisfactorily the observed behavior. Recent renewed interest in time-dependent phenomena in cooperative systems near phase transitions 10 justifies to some extent taking a new look at the process of polarization reversal in ferroelectrics, with the aim of obtaining a unified description of switching behavior from the low- to the high-field regions. Figure 2.5.1 shows the behavior of the inverse switching time of a uniaxial ferroelectric like TGS as a function of field. It depicts, in schematic form, actual data 4 ' 5 for TGS. Taking into account that there is a transition region between the very low field region, where the process is extremely slow, and the intermediate region, where the field dependence becomes clearly linear, we can distinguish three regimes: (3) high-field regime, in which (as discussed below) random switching occurs through the volume

*Work previously published under the title "Ferroelectric switching revisited," M.J. Cebezuelo, J.E. Lorenzo, and J.A. Gonzalo, Ferroelectrics 87, 353 (1988). Copyright © 1988. Gordon and Breach Science Publishers S.A. 147

148

Effective Field Approach to Phase

Transitions

Fig. 2.5.1. Inverse switching time or maximum switching current in a sample of thickness d as a function of pulse height of electric field for a uniaxial ferroelectric like TGS. The high-field regime (3) is identified in this work with bulk random switching, the lowfield regime (2) with switching time limited by sidewise domain wall motion, and very low-field regime (1) characterized by partial switching.

of the sample because the external field is strong enough to switch individual dipoles regardless of whether they are at the boundary of a micro (or macro) domain or not; (2) low-field regime, in which switching occurs at the walls of the initially preexistent microdomains (see below) which grow in size under the influence of the external field at a growth rate limited by the slow sidewise motion of the walls (the forward motion being much faster and the sample thickness sufficiently small); and (1) very low field regime, in which partial switching occurs also at the domain walls (this regime is less interesting and will not be discussed further). Between regimes 2 and 3 there is a transition region labeled 2-3. We may note that the switching time is related to the maximum of the switching current, due to the fact that /"* 1 2PS= / i(t)dt«-*mt8)

i.e., im » 4P s i s " 1

(1)

and, since published data report alternative results on im and t" 1 , we will convert data of im to data on t" 1 and vice versa using Eq. (1). Figure 2.5.1 indicates that the field dependence of i " 1 (or im) in the three regimes

Polarization

Reversal in Ferroelectric

Systems

149

mentioned above can be simply given as: Case (3): im ex i " 1 cc(E-

Ec3),

Ec3 = large.

(2)

Ec2 = very small (-C Ec3).

(3)

Ecl = very small (-C Ec2).

(4)

Case (2): i(xt~l

<x{E-

Ec2),

Case (1): very small im ex t'1 oc (E - Ecl),

We will see later that the threshold field values Ec have the meaning of effective coercive fields for the bulk and domain wall (side wise motion) switchings, respectively. In a dipolar uniaxial ferroelectric, we may assume that each individual dipole can be reoriented, by means of an external field, between two equivalent configurations in which the elementary dipole /J, is pointing in the ± direction of the ferroelectric axis. The dipole is embedded in a local field Eeff = E + (3P, where E is the external field, a mean field coefficient (which may in principle be different for dipoles in the bulk of the sample and dipoles at the domain wall), and P the polarization per unit volume. To switch from one configuration to the other the dipole must overcome a potential barrier AC/ (which may also differ for dipoles in the bulk and in a domain wall, since the unit cells at the domain boundaries may be assumed to be somewhat distorted). The transition probability per dipole per unit time is p± =vexp[-(AUmEeB/fi)kBT]

= p0exp[±EeS(j,/kBT],

(5)

where v is an attempt frequency (corresponding to a low lying lattice model able to push the dipole atoms in the direction going from one configuration to the other), kB is Boltzmann's constant, and T is the temperature. In Eq. (5), p+ corresponds to the probability of jumping in the direction favored by the effective local field and p- to the probability of jumping in the opposite direction. The number of dipoles available for jumping in the favored or opposite direction will be, of course, different if we are considering bulk switching (high fields) or domain wall switching (intermediate or low fields). We will examine the rate equation, first for the bulk switching case [case (3)], which can be done disregarding the domain structure, then

150

Effective Field Approach to Phase

Transitions

for the sidewise domain wall motion switching [case (2)], and finally we will discuss the transition region between high and low fields (2-3).

2.5.1.

High Field — Case (3)

If at a given time there are Na dipoles per unit volume pointing opposite to the field and Nb in the direction of the field, the rate equations can be written down as diVa dt

-NaPab +

diVb dt

N a P a b - -WbPba,

(6)

Nbpha

(7)

where Na + Nh = N, pah = p03 exp[-(E - (33Pd)n/kBT], p03 exp[-(E + p3Pd)^i/kBT}. Then, taking into account that Pd = (Nh — Na)[i, dP d dt

Nuipab

i{

and pha

- Pba.) - Pd(Pab + Pb

N/J, sinh

-(E + (33Pd)fi fcRT

Pdcosh

(E + foPd V knT

(8)

for I/T3 = 2po3 = 2wexp(—AU3/k-oT). We can write down this equation in a more compact form, using Tc = /33Np?/k& and ESQ = P3Nn, as dPf dt

=

j _f T±(E Nfi sinh T T3 \

+ (33Pd E.s0

Pd

Tc(E T

+ /33Pd E.s0

(9)

The solution of this equation yields directly the switching current i(t) = dPd/dt. Since the maximum switching current im = 4PS<71 corresponds to (d 2 Pa/di 2 ) m = 0, it can be easily checked that at i — im we have E.s0

r

--^S

N/i

(10)

and, then making use of the fact that there is a threshold switching field Ec3, corresponding to {dpd/dt)mc = 0, we can get, from Eqs. (9) and (10),

Polarization

for P d = Pdm

=

Reversal in Ferroelectric

151

Systems

Pdmc,

Ec3 = foN^ ttanh a n h "' 1 ! ^ ^ ) '

~ (' V " )

•

(n)

This is the "bulk" coercive field, which is two orders of magnitude larger than the ordinary "hysteresis loops" coercive field measured at low fields and 60 Hz (which is determined by domain wall motion). Direct comparison of Eq. (11) with the data of Binggeli and Fatuzzo 5 for TGS at high fields and various temperatures shows good agreement. The temperature dependence of Ec3(T) for T « Tc is very well fulfilled by the data, and the predicted scale factor Es0 = f3Nfi = (4irTc/C)Ps0 1.2 x 10 4 esu/cm = 3.6 x 10 3 kV/cm is well within a factor of two of the observed one. The maximum switching current denned by Eq. (9) with (d2P<j/di2) = 0 can be expanded in terms of (E — Ec3), taking into account that (E - Ec3)/Es0
exp(-A V /fc B T)(T c /T)(£ - Ec3),

(12)

which describes switching in regime (3) and will later be compared with experimental data. 2.5.2.

Low Field — Case (2)

As we have seen above, bulk switching becomes very ineffective when we lower the strength of the driving field, until it stops for E < Ec3. Before that point, however, another switching mechanism becomes effective. It is well known that TGS crystals grown from water solution (usually at temperatures well above room temperature) suffer a thermal shock when they are taken out of the solution and it is also known that cooling shocks12 give rise to many small spike-shaped microdomains (with polarization opposite to that of the bulk crystal), which form a very regular array with an average distance between neighboring microdomains of 0.01 cm. These microdomains tend to be persistent and seem to be locally fixed, in such a way that moderate external driving fields induce growth or decrease in size, but not elimination of microdomains. It is also well known 12 that at moderate fields, forward domain wall motion proceeds at a faster pace than sidewise motion. Then, in order to write down the rate equation for regime (2), under consideration, we may assume that, right after the beginning of the driving field pulse, there is already formed a set of narrow

Effective Field Approach to Phase

152

Transitions

microdomains running from one main surface to the other, which are growing sidewise until their walls meet those of their nearest neighbors. For simplicity we may assume that the extended microdomains are equidistant parallelepipeds arranged periodically with an average distance 2r m between nearest neighbors. The rate equations are then, for t < tm. dt dNb dt

\{r\

-n4

\ J U

\{r\

+n4

-

U

W J

+ 1 + 1

( < * •

r

\

v&, 1

\Pab

(^ \ r Pah \l>s/

/'r\ / r P \ + n4 7 Ph \KuJ \*>s/ /' r \ / r T \ -n4 7 Pb \v'«/ V&,/

(13) (14)

where n is the number of microdomains per unit surface (then n = 1/2 r ^ ) , u is the lateral distance between neighboring unit cells [for TGS, u — (ac sin/3) 1 / 2 ], d is the sample thickness, b is the unit cell parameter along the ferroelectric axis, pab = po2exp(—(E — /3 2 -P)M/A;BT), pba = P02exp(-(E + f32P)n/kBT). Subtracting Eq. (13) from Eq. (14), multiplying by fi, and taking into account that N = Ns(d/b), we have 1/2

dt

Pd

1/2

sinh

Ps

T2

1

cosh

Ik T

T ~T

E + fcPd E,sO

E + f32Pd E,sO

(15)

where, to evaluate (r/u) in terms of (n/Ns) and the switched polarization Pd, we have used [2(Nh/N)/(n/Ns)]1/2,

(r/u) =

P/Ps = (Nh -

(16)

Na)/N.

(17)

It can be easily shown that making appropriate approximations in Eq. (15) one gets (» m ) 2 «64JV/i

P3P* sO fcPs

1/2

1/2

xv2exy>(-AU2/kBT)

E-Et c2 3Sso

3/2

(18)

which gives the switching behavior in regime (2), dependent on the surface density of preexistent microdomains through (n/Ns).

Polarization

2.5.3.

Reversal

in Ferroelectric

153

Systems

Intermediate Field — Case (2-3)

High-field bulk switching (3) takes place at supersonic velocities, while low-field domain wall switching (2) takes place at subsonic velocities. The switching rate in (2) grows faster as the field increases, and eventually the transition probability p eD will approach the maximum value of one when the domain wall speed approaches the sound velocity, giving rise to a transition region from (im(E))2 to (im(E))3 in which the rate growth of im with E small in comparison with either the subsonic rate (2) or the supersonic one (3).

2.5.4.

Very Low Field — Case (1)

As noted above, this corresponds to incomplete (partial) switching, and sample characteristics, including thickness, should play a major role, making it more difficult for theoretical analysis. It would have been desirable to have a complete set of data on im(E) in the same set of well-characterized samples (as to thickness and density of microdomains in equilibrium) covering the whole range of fields from very low (say E < 0.05kV/cm) to very high fields (say E > 50kV/cm). However, combining the information provided by various authors, 2 - 6 ' 1 2 it appears that existing data provide sufficient support for a coherent picture of ferroelectric switching as given by Eqs. (12) and (18), which encompass the range of maximum switching current (and switching time) from low to very high fields. Table 2.5.1 illustrates this fact, displaying numerical estimates of the basic parameters which determine the switching behavior of TGS for 0.05 < E < 50kV/cm. We take (P s 0 /£ s o) = l//?3 = C/4TTT C = 1 as a common factor in (i m )i_3- The dimensionless factor12 (ji/Ns)1/2 « 10~ 5 corresponding to the fraction of intervening dipoles is of the right order Table 2.5.1. Field

im

High (3) Low (2)*

Eq. (12) Eq. (18)

DF 1 (n/Ns)

1/2

~ 10"

5

v (Hz)

AU (eV)

Ec (kV/cm)

(3 (esu)

t „ ~ 1 . 2 x 1012 (vs/u) ~ 7 x 1 0 1 2

0-10 ~0.10

40 0.01

1 1 0 " 3 to 1 0 " 4

D F = Dipole fraction intervening in switching process (3)—(2); v (Hz) = a t t e m p t frequency; AC/ = P o t e n t i a l barrier height; Ec = threshold switching field; /3 = m e a n field coefficient; @i = ECi/(Pdm)i. For more details see references and m a i n t e x t .

154

Effective Field Approach to Phase Transitions

magnitude. Then assuming suitable a t t e m p t frequency values [I>TO, optic, for regime (3) and VA, short wavelength acoustic, for regime (2)], we get reasonable estimates for t h e potential barrier heights. It may be pointed out, among other things, t h a t A t / from high-frequency relaxation d a t a 1 6 on e(w) at T < Tc gives A t / = 0.06 ev, in approximate agreement with A t / = O.lOev from (z m )3 as a function of field; Ecs(T) agrees well with the thermodynamic (Landau expansion of free energy) coercive fields, along with /? 3 ss 1 « 4wTc/C. Ec2 is two orders of magnitude lower t h a n Ec3, corresponding with the usual hysteresis loop coercive field value at room temperature.

References 1. See, e.g., M.E. Lines and A. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon Press, Oxford, 1977). 2. C.F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1742 (1958). 3. A.G. Chynoweth and W.L. Fledman, J. Phys. Chem. Solids 15, 225 (1960). 4. E. Fatuzzo and W.J. Merz, Phys. Rev. 116, 61 (1959). 5. B. Binggeli and E. Fatuzzo, J. Appl. Phys. 36, 1431 (1965). 6. M. Hayashi, J. Phys. Soc. Japan 33, 616 (1972). 7. A. Jaskiewicz and J. Przeslawski, Phys. Stat. Sol. (A) 56, 365 (1979). 8. R. Perez, E. Toribio, J.A. Gorri and L. Benadero, Ferroelectrics 74, 3 (1987). 9. R.C. Miller and G. Weinreich, Phys. Rev. 117, 1460 (1960). 10. J. Axe, Jpn. J. Appl. Phys. 24 (suppl. 24-2), 46 (1985). 11. See, e.g., A. Dekker, Solid State Physics (Prentice Hall, Englewood Cliffs, NJ, 1962). 12. F. Jona and G. Shirane, Ferroelectrics Crystals (Pergamon Press, New York, 1962) and references therein. 13. H.H. Wieder, J. Appl. Phys. 35, 1224 (1964). 14. G. Luther, Phys. Stat. Sol. A 20, 227 (1973).

Chapter 2.6

Polarization Switching by Domain Wall Motion*

Ferroelectric polarization reversal under external field pulses 1-9 of variable strength has been the subject of extensive research since the 1950s. It may be noted that most previous works 1-9 studied ferroelectric switching at room temperature. In this paper, switching measurements have been undertaken near T c , with the aim of obtaining a description of the field and temperature dependence of the maximum switching current (im) at low fields. We will summarize 11 and complement previous theoretical results for low field switching in ferroelectrics, which will be compared later with our experimental results. It is well known that TGS single crystals grown from water solution (usually at temperatures well above room temperature) suffer a thermal shock when they are taken out of the solution and it is also known that cooling shocks 11 give rise to many small spike-shaped microdomains (see Chapter 2.5). For simplicity, we may assume that the extended microdomains are equidistant parallelepipeds arranged periodically with an average distance 2r m between nearest neighbors. For t < tm, the rate equations then are 11 (similar equations can be obtained for tm < t < ts)

£--[(:H(i)~+-(0(f)-

(1)

*Work previously published under the title "Temperature dependence of polarization reversal in TGS family crystals near T c , " M. de la Pascua, P. Sanchez, J.E. Lorenzo, and J.A. Gonzalo, Ferroelectrics 94, 401 (1989). Copyright © 1989. Gordon and Breach Science Publishers S.A. 155

Effective Field Approach to Phase

156

+n4

dt

Transitions

Pab - n4

u

Pba,,

(2)

where Na and ATb are t h e number of unit cells per unit volume with ± polarization, respectively, n is t h e number of microdomains per unit surface (n = l / 2 r ~ 2 ) , u is t h e lateral distance between neighboring unit cells (for T G S , u = (arcsin/3) 1 ' 2 , d is t h e sample thickness, b is t h e unit cell parameter along the ferroelectric axis, and pab, pb& are t h e transition probabilities (per dipole unit time): Pab=PO2[{E

+

Pba = m[-(E P02 =

02P)lM/kBT\, +

(hPMksT],

£l2exp[-Av2/kBT],

where v2 is a n a t t e m p t frequency a n d /32 t h e domain mean field coefficient. Subtracting Eq. (1) from Eq. (2), multiplying by /x, and taking into account t h a t N = Ns (d/b), we have: 1/2

16 ( n_

dPd dt

NfJL[l

+ - cosh Since Nfj, follows:

Tc(E

+ /32Pd

T

const.

{I + u+ f

(3)

esu 3> 1/2, we can rewrite this equation as 1/2

Pd Ps

1

sinh

T h e maximum value of (dPd/dt) vanishes: d2Pd dt 2

Tc (E + (32Pd T V EsQ

sinh

}

Eso

10->22 x 10

dPd dt

+

1/2

Pd

TCE + 02Pd E> sO

occurs when t h e second derivative -1/2

— sinh

const

(4)

Ps 1/2

T E,sO

cosh

TCE + 02Pd T

Es0

TCE + /3 2 P d E.sO

J dt

(5)

¥Es0

(6)

This implies, at t = t m , tanh

\TcE + foPd E,sO

-2P S

1+

Pdn T

Polarization

Switching by Domain

Wall

Motion

157

and, therefore, -2PS

1/2

I —•— I

dm

1+

= const. I 1 , „

1-

Pdm^\ Tc ft ' Ps J 1 T Es0 PdnO\TC (32~ Ps J ' T Es0

-2R 1

1/2-

(7) On the other hand, as shown below (see Eq. (13))

1+

E - (32PS

dm

(8)

3/?2Ps

Ps

and we can write: 1/2

N» (^j1'

im « 6Av2 exp(-AEyfcT) ( jfj" 1/2

E - (32PS 3-BsO

(^

^

x 2 ,

Tc E - f32Ps T 3Eso

1-4

-. 2 ^ - ! / 2

3ESQ

(9) 2

At low fields [ ( £ - / 3 2 P s ) / 3 £ s 0 ] < 1, where /32PS = Ec2 = Ec is the lowfield regime coercive field and including all constant factors in C (constant for each temperature), we get finally: = C(E -

Ecf'2.

(10)

We show next that Wieder's empirical relation 13 for the asymmetry of the switching current peak can be obtained from the present model. From Eq. (6), we can obtain ^

T

/?2Pdm=tanh-i

+

_22 PK [U1 ++ ^ ^ Z k A Ps )

3Eso

TEs0

(11)

and then E= — ^ o t a n h " 1 J- C.

2 1+

Pdm\ TC(32PS T E.sO

/?2Pdm.

(12)

Taking into account that f32Ps
-2PS

1+

Pdm\ TC(32PS T EtsO

/32Pdra = -3/3 2 P d m - 2ftP s . (13)

158

Effective Field Approach to Phase

Transitions

Dividing by QfoPs' Pdm

(

2PS

V 6/32Psy

l

^E+l,

3

(14)

which is the same result obtained empirically by Wieder 13 Pd

2PS

aE + b,

where the field E is defined with opposite sign. The switching behavior of TGS crystals was studied by the conventional technique in which one measures the current that flows through a series resistors connected to the crystal as a function of time t. First, an electrical square pulse is used to align all the dipoles in one direction, then a second pulse of opposite polarity is used to measure the switching current as a function of time. The quantities of interest are: i m , the maximum switching current; ts, the switching time; and tm, the time corresponding to i = im, which should be larger than the rise time of the applied pulse. For a given temperature the time ts decreases and the maximum switching current im increases with increasing applied field, while the switched charge remains constant. The sample was selected to have unbiased and undistorted hysteresis loops (usually distortions are due to impurities and/or mechanical stresses). The sample thickness was 0.1cm and surface area 0.21cm 2 . Gold foil electrodes were attached directly to both main faces and all the measurements were performed in dry air. Figure 2.6.1 shows maximum switching current versus applied field, at several temperatures. In Fig. 2.6.2(a), im vs. (E — Ec)z/2 has been plotted obtaining a set of straight lines with different slopes corresponding to different temperatures. The data agree reasonably well with the theoretical expression in Eq. (10). Both constants, C and Ec, have been computed by means of a least squares fit method for each temperature. In Fig. 2.6.2(b), ln(i m ) vs. 1/E has been represented, which should give another collection of straight lines, according to the Miller-Weinreich model (i and a have been computed by least squares fits, in the same way). Except at very low fields, they appear to follow approximately straight lines in the semilog representation. The first question that arises is which the model fits better the experimental data. Figure 2.6.2(a) and (b) does not solve clearly this problem. Therefore, we have tried another kind of representation: i vs. C(E — Ec)3/2 (Fig. 2.6.3(a)), and im vs. i 0 0 exp(-Q!/£'). These plots allow us to see the

Polarization

Switching by Domain

Wall

159

Motion

4000 • * a • 4 •

3000'

t t I 'C T « l o *C T O O «C T * « -C T O * «C

« T * 4 Q . S -C 4 T * 4 2 »C « T * 4 3 *C x T=44.S -C

2000-

1000-

0 | * r ^ , « . ''** 0.000 0.375

r

0.750

1.500

1.125

E (W/cm)

Fig. 2.6.1. Maximum switching current i m as a function of pulse height of applied electric field for different temperatures.

* I 4 4 4

3000-

T t 0 «C T«10 -C T s a o «C T = | i »C T t l l *C

4 T E 4 0 . 1 *C 4 T H » -C * T s 4 l «C »r T S 4 4 . S *C

< ^zooo

••• *

1000 *

•

•

•

•

•

0.00

Fig. 2.6.2(a). temperatures.

Maximum

*

0.25

switching

0.50 .3/2 (E-EC)J

current

t.00

0.75

(E -

£c)3/2

for

different

differences considerably better: the fit with the Miller-Weinreich theory departs substantially more from the slope-one straight line than the result of the statistical model given by Eq. (10), at least in this range of low fields.

160

Effective Field Approach to Phase

Transitions

A T= 0 B Ts tD

*iv *\ •

* TsJO 1 Tt2» * T;3S

»• 4* '• *»# *\ n. ••.• * « »-». 0 •t

+ T=«o. J • T»»J

-C

V

*.

V,

3.0 0.0

7.0 (kV'cm)

3.5 1/E

Fig. 2.6.2(b).

Natural logarithm of im vs. 1/E for the same temperatures.

4000

I A Ts 0

'C

e T=io >c « Ts20 « T:J8 * Ts31

3000

HO

-c -c -C

1

1

>

i

* T : < 0 , 5 -C * T r < 2 «c * T 5 4 S «C x T M 6 . J «C

O I °

'

0*

°0' .

• >°. *

2000

e

* *

•

1000

0

1000

'

2000 , . ,

'

3000

'

40

C(E-E C ) Fig. 2.6.3(a). Maximum switching current im vs. C(E — Ec)3/2. C and Ec have been determined independently for each temperature, and the data are plotted in normalized form.

Polarization

r—

Switching by Domain

'

• T

i

—

Wall

Motion

—i—

161

r-

—i

*

3000-

4 0 « « »

Ts 0 *C T = 1 0 "C TiJO ' C T = « «C T=3S ' C

* T = 4 0 . S "C • TS42 «C

oj

* T = <S -C * T=<«5

°°

'C

*

o o •

fl o

*v * <

* o°* • .0 * Q

=k 2 0 0 0 -

3 1000-

0-

1

1000

"-

"T

2000

i

"

-

1

3000

— •>

*000

i.exp(-a/E) Fig. 2.6.3(b). Maximum switching current vs. ioo exp(—a/E). ioo and a have been determined independently for each temperature, and the data are plotted in normalized form.

Figure 2.6.3(b) departs from the expected results at (E — Ec) sa 0 because of incomplete switching. In other words, at low fields the area under the switching curve is less than Ps. This behavior is not surprising. The activation field in the Miller-Weinreich model, a, is nearly ten times larger than Ec calculated with the present model, and does not agree with Pulvari and Kuebler.2 On the other hand, the extrapolated value of Ec from our switching data is proportional to the one measured from hysteresis loops at ±50 Hz for temperatures in an interval of about 20 degrees below Tc. At lower temperatures, 50 Hz hysteresis loops values for Ec are more likely to correspond to the transition between low field switching (domain wall motion at subsonic velocities) and high field switching (random polarization reversal throughout the bulk of the crystal at supersonic velocities). The dependence of maximum switching current of TGS crystals with applied field at several temperatures has been investigated and it has been shown to be in good agreement with the predictions of simple statistical model. The predicted field dependence im(E) « C(T) (E - Ec(T)fl2 fits data at T < Tc better than the usual expression im(E) « i^ exp(—a/E) (Miller-Weinreich) and the predicted temperature dependence of the

162

Effective Field Approach to Phase Transitions

threshold field EC(T) « /32Ps(T) field (subsonic) switching.

is also approximately fulfilled for low

References 1. M.E. Lines and A. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon Press, Oxford, 1977). 2. C.F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1742 (1958). 3. A.G. Chynoweth and W.L. Feldman, J. Phys. Chem. Solids 15, 225 (1960). 4. E. Fatuzzo and W.J. Merz, Phys. Rev. 116, 61 (1959). 5. B. Binggeli and E. Fatuzzo, J. Appl. Phys. 36, 1431 (1965). 6. M. Hayashi, J. Phys. Soc. Japan 33, 616 (1972). 7. A. Jaskiewicz and J. Przeslawski, Phys. Stat. Sol. 56, 365 (1979). 8. R. Perez, E. Toribio, J.A. Gorri and L. Benadero, Ferroelectrics 74, 3 (1987). 9. R.C. Miller and G. Weinreich, Phys. Rev. 117, 1460 (1960). 10. J. Axe, Jpn. J. Appl. Phys. 24 (Suppl. 24-2), 46 (1985). 11. M.J. Cabezuelo, J.E. Lorenzo and J.A. Gonzalo, Ferroelectrics 87, 353 (1988). 12. F. Jona and G. Shirane, Ferroelectrics Crystals (Pergamon Press, New York, 1962). 13. H.H. Wieder, J. Appl. Phys. 35, 1224 (1964).

Chapter 2.7

Switching Current Pulse Shape*

Since the early work 1-6 on ferroelectric switching in crystals of the TGS family, attempts were made to describe the shape of the switching current pulse as a function of time. According to Pulvari and Kuebler, 1 the pulse shape is well described by j(t) = 2PS[^)

exp

2 Um

where Ps is the spontaneous polarization, j is the switching current per unit surface, and t the time required to reach j m a x - On the other hand, Fatuzzo and Merz 3 fit the shape by the difference of two exponentials with the time cosntants Ti and T^ as follows: j(t) = J 0 [exp(-£/T 2 ) - e x p ( - t / T i ) ] , where jo is a constant. Both expressions are of an empirical nature, and the criteria to fit the data on j(t) were: 1. Making sure that the area under the curve is: rta

./o /o

jdt = 2Ps.

2. Insuring that the jmax(t) of the empirical expression occurs at the same time as j m a x of experimental data. *Work previously published under the title "Switching pulse shape in single crystals of the TGS family," M. de la Pascua, G. Sanz, and J.A. Gonzalo, Proceedings of the 7th International Meeting on Ferroelectricity, 1989. Copyright © 1989. Gordon and Breach Science Publishers S.A. 163

164

Effective Field Approach to Phase

Transitions

The expression for j(t) giving rise to a triangular (or smoothed triangular) pulse can be obtained directly from the rate equations for polarization change, assuming a certain distribution of preexisting reverted microdomains. For sufficiently high fields (in the subsonic regime), jit) for t < tm is given by 7

and for tm < t < ts by

i® = — {r~r)' where ts is the switching time, defined as the time required for j to approximately complete the switching process (for sufficiently high fields and correspondingly symmetrical pulses, ts = 2tm). The symmetry of the switching pulse depends strongly on the strength of the applied field: it goes from a considerably asymmetric shape at low fields to an almost symmetric shape at sufficiently high fields.5 Figure 2.7.1 gives j(t) for different applied fields. The statistical model predicts triangular pulse shapes but, of course, does not fit the peak roundings of the experimental data. This triangular shape is due to the assumption of an exactly periodic starting distribution of preexisting microdomains in the sample. For a gradual change in n (density of preexisting microdomains) with position along a given direction of the sample (dn/dx — 0), we can describe j(t) by means of a hyperbola that becomes asymptotically a straight line at both sides of tm:

where JT{tm) is the maximum current density in the triangular fitting. The rounding at t gives rise to a tail at t « 2tm. In Fig. 2.7.2 we have plotted the predicted pulse shape after PulvariKuebler, Fatuzzo-Merz and our model along with the experimental data. In conclusion, the pulse shape predicted taking as a starting point the rate equations for polarization reversal at preexisting microdomain walls fits fairly well the experimental data, better than previous empirical expressions (Pulvari-Kuebler, Fatuzzo-Merz) and at least as well as the more elaborate expression given later by Fatuzzo. 4 We note that our expression for j(t)

Switching Current Pulse Shape

165

TIME (MS) Fig. 2.7.1. Switching pulses at various fields for (a) triglycine selenate (at T = —20°C), (b) triglycine sulfate (at T = 0°C), and (c) triglycine fluorberillate (Ref. 2). The prediction given in Ref. 7 is a triangular pulse.

166

Effective Field Approach to Phase

•

_5>

'

Transitions

ocrciwtxrAi. DATA

'

tin

TIME U s )

•

ocmawEHTAi. OATA

PULVAn-KUEKXR

TIME ( M s) Fig. 2.7.2. Experimental (symmetric) pulse shapes and theoretical fits (Pulvari— Kuebler, Fatuzzo-Merz, this work) for (a) trigylcine selenate (at T = —20°C), (b) triglycine sulfate (at T = 0°C), and (c) triglycine fluorberillate (Ref. 2).

Switching Current Pulse Shape

167

is consistent with the law 8 j m « C(E-EC)3/2, rather t h a n with j m « jo exp(—a/E). At least for the crystals of the T G S family investigated, the power law expression gives a better description of the experimental behavior of j m vs. E t h a n the exponential one in a relatively wide range of E.

References 1. 2. 3. 4. 5. 6. 7. 8.

C.F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1315 (1958). C.F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1742 (1958). E. Fatuzzo and W.J. Merz, Phys. Rev. 116, 61 (1959). E. Fatuzzo, Phys. Rev. 127, 1999 (1962). H.H. Wieder, J. Appl. Phys. 35, 1224 (1964). B. Binggeli and E. Fatuzzo, J. Appl. Phys. 36, 1431 (1965). M.J. Cabezualo, J.E. Lorenzo and J.A. Gonzalo, Ferroelectrics 87, 353 (1988). M. de la Pascua, P. Sanchez, J.E. Lorenzo and J.A. Gonzalo, Ferroelectrics 94, 401 (1989).

Chapter 2.8

Elementary Excitations in Ferroelectrics: Dipole Waves

Low-temperature dipole-wave-like excitations in ferroelectrics, which are analogous in certain respects to ferromagnetic dipolar spin waves (see, for example, Ref. 1), might be expected on the grounds that, slightly above zero temperature, collective-wave-like deviations of the elementary dipoles from maximum alignment would require much less energy than the reversal of single dipoles would. Ordinary acoustic phonons can coexist in a noninteracting way with low-energy spin-wave-like excitations in solids possessing a spontaneous polarization. Consider a complex crystal in which lattice points are occupied by rigidly bound groups of atoms instead of single ions. Suppose that these groups of atoms possess permanent electric dipole moments and that, below a certain transition temperature, there is a non-zero spontaneous polarization and, therefore, an associated spontaneous local field. To find rigorously the ground state and the full spectrum of excited states of this system would demand the solution of the many-body problem in one of its most difficult forms. One may reasonably argue, however, that under favorable conditions the dipole wave excitations can be considered as low-energy internal vibrational modes and then can be treated separately from the normal low-temperature acoustic phonons, which involve vibrations of the tightly bound group of atoms making up a unit dipole. Intuitively, the energy required to produce long-wavelength periodic deviations of the rigid dipoles from perfect alignment in the direction opposite to the field goes to zero as q goes to zero. We shall see below that explicit separate consideration of the long-range dipole-dipole forces *Work previously published under the title "Ferroelectric elementary excitations at low temperatures: dipole waves," J.A. Gonzalo, J. Phys. C: Solid State Phys. 20, 3985 (1987). Copyright © 1987. IOP Publishing Ltd. 169

170

Effective Field Approach to Phase

Transitions

acting on a given elementary dipole gives rise to uq « Ssw(q)q2, which in turn leads to T3'2 contributions to the low-temperature specific heat and a low-temperature change in the spontaneous polarization. Ordinary acoustic phonons give rise independently to a Debye T 3 contribution to the specific heat and to a zero net contribution to the change in spontaneous polarization. (It is not uncommon to find reports in the literature of crystalline systems, e.g., some ammonium compounds, in which certain degrees of freedom are "frozen" at T = 0 and then begin to be evident when the temperature is raised, as vibrations of groups of atoms which were formerly behaving as single point-like masses.) This effect, which shows up as a temperaturedependent effective Debye temperature, is broadly ascribed to anharmonic behavior. However, it is well known that high-energy internal modes of molecular units in a solid are often relatively unaffected by the presence of acoustic modes of very different energies. In the same way, it is prefectly conceivable that the dipole waves, which are considered as very-lowenergy internal librational modes, behave as non-interacting modes with respect to acoustic phonons of energy uiq « Sph{q)q, at least for temperatures low enough for the relevant (excited) wave vectors q to be such that q < Ssw/Sph- This is equivalent to saying that short-range forces may remain near equilibrium while long-range dipole-dipole forces (responsible for the dipole waves) are playing an active role. Of course, a definitive check on the actual existence of excitations, which behave as u>q sa Ssw(q)q2, must wait until systematic neutron scattering data well inside the Brillouin zone are available (at low and high temperatures). In the ground state (T = 0), dipole alignment should be maximum, but not perfect, because of zero-point motion. The situation is partly analogous to that encountered in dipolar (not exchange) ferromagnets. The main difference between ferroelectric and (dipolar) ferromagnetic systems lies in the well-defined character of the atomic magnetic dipole moment in the latter, in contrast with the ill-defined electric dipole moment that makes up the complex primitive unit cell in most ferroelectrics. A previous attempt to attack this problem 2 was clearly insufficient and incomplete, partly because not enough experimental evidence was available. A more systematic calculation is presented here, and its results are compared with available experimental data on TGS. A systematic calculation of basic properties in ferroelectrics at low temperatures is interesting in its own right, even though comparison with experiments is difficult because of the somewhat conflicting results obtained by different researchers (see discussion below).

Elementary

Excitations

in Ferroelectrics: Dipole

Waves

171

To investigate the anomalous low-temperature behavior in uniaxial ferroelectrics, we shall consider only the long-range dipole interactions, since short-range interactions will give rise to normal behavior (Debye behavior of the specific heat), which can be treated separately from the former. A simple dipole-dipole interaction Hamiltonian (H = i?dipoie-dipoie) of a system of N elementary electric dipole moments in a uniaxial ferroelectric crystal under zero external field can be written as „

v-^ l - 3 c o s 2 6 ^ w

{i

l

I

^ w

where #«' is the polar angle between l-V and the polar axis z, parallel to the spontaneous polarization. The dipoles are assumed to be located at regularly spaced fixed points in the crystal lattice and almost perfect alignment is expected in the ferroelectric phase at low temperatures. If we leave aside the short-range forces, the net restoring force due to all neighboring dipoles on a given dipole can be represented adequately by the spontaneous local field Es acting on the dipole. The usual mean field approximation consists in assuming that every dipole in the crystal is under the influence of an effective field Eeg = E + Es = E + (3P, where E is the external field, Es the cooperative "spontaneous" field, and /3 a constant. The last assumption is justified only at a temperature sufficiently lower than the transition temperature Tc, so that lattice distortions are small, the unit cell volume remains almost constant and the value of the order parameter is very close to unity. The behavior of the system may change substantially when appreciable anisotropic thermal expansion and new internal degrees of freedom begin to enter the picture. At any temperature higher than OK, every (n(t) undergoes, under the influence of the spontaneous local field Es, a pseudo-regular precession (see, for example, Ref. 3), which can be thought of as a regular precession around Es accompanied by a small mutation of higher frequency, which might perhaps be associated with an "uncertainty" in the /J,Z component, parallel to Es. We may anticipate that this motion is compatible with a lowtemperature specific heat that is proportional to T 3 / 2 , because we have two kinetic degrees of freedom (motion of the tip of m towards z\\Es, and motion perpendicular to z plus one potential energy degree of freedom, defined by the angle a between \n and Es). The angular momentum Li(t) associated with the motion of m (t) is perpendicular to [i\ = qd and to the

172

Effective Field Approach to Phase

Transitions

velocity of its tip, and it is undergoing a regular precession. Therefore, from Hiixi> = /z2 cos aw

LtLv

(2)

2

= L cos aw

(3)

we obtain, dividing Eqs. (2) and (3), 7 2 = H2/L2 = imni/LiLv

= constant,

(4)

L2/2I = M £ s cos 9.

(5)

where, through the virial theorem, e = 2Tk = 2V,

Tk = V,

Here, Tk is the kinetic energy, V is the potential energy, / = Mr2 is the moment of inertia of the dipole, M is its total mass, and 9 is the angle between /z = qd and Es. For 9 small, and taking into account that within the mean field approximation (see, for example, Ref. 2, and references therein) fiEs = /3Nn2 = kBTc

(6)

we obtain, after substituting Eq. (6) into Eq. (5) and Eq. (5) into Eq. (4), 2 7

w (d/r)2(q2/2MkBT).

(7)

Consequently, we can substitute WW = 72LiLi>

(8)

in Eq. (1), leaving it in terms of angular momenta instead of elementary dipole moments. We can now use directly the formalism of angular momentum quantization (see, for example, Ref. 4) for a system of N quantized processing dipoles to obtain wq = 7 2 /i ] P Jw{l - exp[-iq(l - I']} = qi2h ^ We can use (nq) = [exp(hwq/kBT)

H = H0 + J2 hu)2nq =H0 + ^ q

— 1]

1

Jw sin2 \q(i-i')

(9)

(thermal average) in H to

huq[exp(hujq/kBT)

- l]"1.

(10)

g

(Note that a uniaxial ferroelectric is equivalent, for present purposes, to a spin 1J2 ferromagnet and that in the intervening transformations we

Elementary

Excitations

in Ferroelectrics: Dipole

Waves

173

have neglected interactions between dipole waves.) To calculate the lowtemperature contributions of the dipolar waves to specific heat and spontaneous polarization, it is convenient to rewrite H as follows: H - H0 =

2 2 h •y ha{q)Ja sin

y^hwq{nq) q


= 2 , a(q)kB®Jo sin

qdc 2

qdc

(11)

2

where ^

q(i-n

Jw sir

M) y~] Jw sir Jo

/

J

Jw)

dc

qdc 2 1/3

v,

JV-I/3,

e = /iVEJ»'AB.

(12)

These are simple definitions which, for the time being, involve no approximations. The a(q) are dimensionless parameters, in principle of order unity, d is a characteristic length related to the unit-cell volume (vc = iV _ 1 ), and 6 is a characteristic temperature, related to the strength of the dipolar interaction in the geometry of the crystal lattice. The low-temperature specific heat contribution due to dipole-wave-like excitations is given by ACv

=

df ( ^

nqkuJq

1 d (kBT 8ir3dT

) a ( g ) ( e / r ) sin 2 (gd c /2) f • exp[a( ( ? )(e/T) sm2(qdc/2)}

1

d'q

• (13)

This can be written, using x = qdc/2, as ACV

df

Nk T

{& ° L

ag(@/T) sin 2 x fix2 dx BZ exp[a g (6/T) sin2 a;] - 1

(14)

where, for a given range of x, aq is an effective average of a{q) and ft < 4ir is an effective integration solid angle. Since we are interested mainly in bulk properties, the average takes care of the characteristic directional dependence of u)q. We may assume a cylindrical Brillouin zone (BZ) for a uniaxial

174

Effective Field Approach to Phase

Transitions

ferroelectric ("pill box" shaped for TGS) and then perform an approximate integration in two steps. (i) For 0 < q < <2DI, the central sphere with radius equal to the halfheight of the box, Q = 4w, sin2 a; w x2, and the first part of the integral takes the form / Jo

a 9 (e/:Z> 4 {exp[a g (e/T)a: 2 ] - l } " 1 dx.

(ii) For qr>i < q < qt>2, the part of the cylindrical BZ outside the central sphere, where qo2 is the half-diameter of the box, O = ^(qrti/q), sin2 x « 1 (if the ratio <7D2/(7DI is not abnormally large), and the second part of the integral takes the form XDI / Jo

a(9/T)ir{exp[a g (e/T)] - l } " 1 da;.

Using these rather rough approximations (which are needed to make comparisons below with experimental data) and substituting the appropriate results for the definite integrals, one obtains ACV = B^'2

+ B2(6g/T)2 expt-ef/T),

(15)

where Bt = 0.904ATfc B (2a£e) -3/2 ,

(16)

B2 = (23/2ir2)NkBrcA,

(17) 2

e g = 2 a g e = 2ag/i 7 /fc B .

(18)

Here, a§ is the mean value of aq for 0 < q <
Elementary

Excitations

in Ferroelectrics: Dipole

Waves

175

The low-temperature change in spontaneous polarization from its maximum value at = 0 K is given by AP2 = 5^A|if = ^ 7 A L 2 = 7 / i ^ n i 1

=

=7/i]Tng

1

1

8 ^ JBZ exp(hojq/kBT)-ld

q

q

-

(19)

The same procedure used to evaluate AC leads now to A P 2 = A i T 3 / 2 + A2, e x p ( - 9 g / T ) ,

(20)

A1 = 0.469iV 7 /i(2a^e)" 3 / 2 ,

(21)

A2 = (23/2ir2N>yhrpA)

(22)

e g = 2agO = 2al>h12/kB.

(23)

where

Here, again, QQ and a £ are mean values for the integration intervals 0 < q < <7DI and 0 < q < qc2, respectively. (Note that the approximate integrand in the first interval, which contains x2 in the numerator instead of x4, is slightly different from the previous integrand, while the approximate integral in the second interval is exactly the same as before.) The parameters T p « T c and A have the same meanings as those introduced in calculating AC V . Combining Eqs. (16)-(18) and Eqs. (20)-(23), we obtain A1/B1

= 1.92{ar/aCy3/2>yh/kB,

(24)

A2/B2

= {rP/rc)7h/kB

(25)

= jh/kB,

e g / O g = al/a% = 1; fcB0D = 2aDhj2.

(26)

These simple relationships connect low-temperature spontaneous polarization and specific heat behavior through the ratio ^h/kB « 5^/kB. As mentioned in the introduction, the basic objective of this work is to present a systematic theoretical description of the physical consequences of spin-wave-like excitations in ferroelectric crystals, leaving aside a definitive statement on whether they can be identified with the observed lowtemperature anomalies in some ferroelectrics.5'6 Alternative explanations, such as impurity effects, have been suggested7 for the anomalous T 3 / 2

176

Effective Field Approach to Phase

Transitions

specific heat behavior reported for some ferroelectrics. This latter work, however, does not discuss either the exponential dependence with \/T reported at a slightly higher temperature range, or spontaneous polarization data apparently following the same pattern, 8 which was published shortly afterwards. Further experimental work9 suggested a surface-to-volume ratio of the specific heat T 3//2 term coefficient not confirmed by later work,10 in which an excess observed specific heat contribution behaving as T n , with 1 < n < 2, was also reported. In contrast, very-low-temperature specific heat measurements in various ferroelectrics 11 ' 12 using ultrapure samples seem to support no excess low-temperature contributions, or extremely small ones, above the normal T 3 Debye background. In connection with the last data, it may be pointed out that the only exception among the five crystals studied seems to be TGS, for which data are shown to be reduced by a factor of 10. If one plots the points corresponding to T < I K on the same scale as the data for the other samples and then extrapolates to = OK, the result is a T 3 / 2 contribution slightly lower (by less than 30%) but very similar to that originally reported. 5,6 In summary, while the experimental situation is not yet clear, one may say that at least for TGS, after all the available evidence (from both low-temperature specific heat and spontaneous polarization data) has been considered, it is not unreasonable to assume the presence of a T3'2 contribution followed by an exponential contribution. In the following, we have used data of Lawless5,6 for B\ in TGS (assuming that it is at least within an order magnitude of the true experimental value) as well as the data of Vieira et al.s for A\. Low-temperature specific heat data from ferroelectric TGS single crystals 5,6 (in cgs units per unit volume) give £2 = 9.1xl05,

S i = 12.9,

6 g = 67K,

(27)

and corresponding spontaneous polarization data 8 (in cgs esu units) give A1 = 0.99 x 10" 4 ,

v4 2 =28,

6 g = 66K.

(28)

Then, from Eqs. (24), (25), and (26), respectively, we obtain three independent estimates of 7, in cgs esu units: 7

= 5.2xl05(c£/ag)3/2,

7

= 4.0xl0 6 ,

7 = (l/aD)1/2x2xl06.

(29)

The internal consistency of these data is excellent, since the ao and an are dimensionless numbers of the order of unity. Now, let us return to the theoretical expression for 7 in Eq. (7). Substituting in some values, we can check whether this theoretical value for

Elementary

Excitations

in Ferroelectrics: Dipole

Waves

177

7 based on the motion of molecular dipole units is or is not in agreement with the experimental values in Eq. (29). Let us use, again in cgs esu units, q w e = 4.8 x 10 1 0 , N = v'1 = 0.155 X 10 2 2 , d = n/e = Ps0/Ne « 1.7 x 1CT 8 , T PS vl/S « 8.6 x 1(T 8 , M = M m p = 321 x 1.67 x 1 0 " 2 4 and kB Tc = 1.38 x 1 0 " 1 6 x 322. Substituting these values in Eq. (7), we obtain 7caic « 13 x 10 6 ,

(30)

which is in fairly good agreement with 7 [Eq. (29)]. It has been shown t h a t spin-wave-like elementary excitations in ferroelectrics lead to extra low-temperature contributions to the specific heat of the form ACV(T) = B i T 3 / 2 + B 2 ( e g / r ) 2 e x P ( - 6 g / T ) , and low-temperature changes in the spontaneous polarization of the form A P S ( T ) = AlT'il2

+ A2 e x p ( - e g / T ) ,

where the coefficients in b o t h expressions are related to each other by l.92(a%/a$y3/2jh/kB,

A1/B1 = A2/B2

=

lh/kB,

e g / e g = 1, fcBeD = 2a D /i 7 2 . T h e low-temperature d a t a available for T G S are in fair agreement with these expressions.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

J.R. Herringa and M. Marrenga, Physica A 38, 371 (1975). J.A. Gonzalo, Ferroelectrics 20, 159 (1978). A. Sommerfeld, Mechanics (New York: Academic Press, 1964). P.L. Taylor, A Quantum Approach to the Solid State (Prentice-Hall, Englewood Cliffs, NJ, 1970). W.N. Lawless, Phys. Rev. Lett. 36, 478 (1976). W.N. Lawless, Phys. Rev. B 14, 134 (1976). S. Kirkpatrick and C M . Varma, Solid State Commun. 25, 821 (1978). S. Vieira, C. de las Heras and J.A. Gonzalo, Phys. Rev. Lett. 4 1 , 1822 (1978). W.N. Lawless, Phys. Rev. B 23, 2421 (1981). M.C. Foot and A.C. Anderson, Ferroelectrics 62, 11 (1985). H. Grimm, R. Villar and E. Gmelin, Proc. 17th Int. Conf. Low Temperature Physics, ed. U. Eckern et al. (Amsterdam: North-Holland, 1984), p. 1377. R. Villar, E. Gmelin and H. Grimm, private communication (1986).

Chapter 2.9

Low-Temperature Behavior of Ferroelectrics*

In previous work1 it was shown that quantized dipole-wave-like excitations, analogous to dipolar ferromagnetic spin waves, can give a consistent description of the low-temperature specific heat and spontaneous polarization behavior of ferroelectric crystals. The basic features of the dipole-dipole hamiltonian and the main results relative to specific heat and polarization will be summarized. An analysis of available experimental data in the light of these results, previously restricted to triglycine sulfate,1 is extended here to data on other displacive and/or order-disorder ferroelectrics, in particular lithium niobate (LiNbOs), barium titanate (BaTiOs), and sodium nitrate (NaN02). The overall picture emerging from three independent estimates of 7 = \n/L\ (where fi is the unit electric dipole moment and L the angular momentum of the atoms making up the unit dipole in its precession with respect to the local spontaneous electric field) shows a fair degree of consistency for all these crystals, taking into account the experimental uncertainties and the approximations involved in the theoretical calculation. The ground state of a normal, high T c ferroelectric, must show deviations from perfect alignment of the elementary dipoles because of the uncertainty principle. The connection established through 7 between the electric dipole moment vector and the corresponding angular momentum vector allows one to calculate the elementary excitations spectrum of the lattice of interacting dipoles, assuming that these spin-wave-like excitations are much lower in energy than the ordinary acoustic phonons. The quantum

*Work previously published under the title "Low T quantum effects in high T c Ferroelectrics," J.A. Gonzalo, Ferroelectrics 79, 19 (1988). Copyright © 1988. Gordon and Breach Science Publishers S.A. 179

180

Effective Field Approach to Phase

Transitions

calculation is made in a way analogous to the one applicable2 for dipolar ferromagnets. Under these assumptions, the dipole-dipole hamiltonian for a lattice of N dipoles in a uniaxial ferroelectric under zero external field can be written as

\l

ll> \

i

\

J

W

where On, is the polar angle between (l-l') and the polar z axis. Since1 each /ii (dipole moment) is perpendicular to its corresponding L\ (angular momentum), 7 2 = fi2/L2 « constant, and H can be put in terms of the i s . Making use of the standard commutation relations for angular momentum, performing a Holstein-Primakoff transformation of L^ = Lx ± iLy in terms of annihilation (a) and creation (a + ) operators of angular momentum quanta, and changing from local to collective coordinates, one finally gets H = £0 + ] P h[-j2h( Jo - Jq)\a+aq = e0 + ] P hioqnq,

(2)

where Jq = J-q = ^2t Ju> exp{ig(Z — I')} for Bravais lattices and toq = 212h J2 Jw sin2 \q{l - I')

(3)

i

gives the dispersion relation for dipole waves, which for low |g|
exp{-9g/T}

(4)

and (AP s ) d w « AXT^2

+ A2 e x p { - e g / T } ,

(5)

respectively, where Ai/B! A2/B2 ©D/©D

= (1.92)(a 0 3 /aC)- 3 / 2 7 /i/fc B l p

c

= (r /r )7h/kB

= ^h/kB,

= <*o/«o = 1,

fc9D

- 2a D /i 7 2 .

(6) (7) (8)

Low-Temperature

Behavior of Ferroelectrics

181

Table 2.9.1. Crystal LiNb03 BaTi03 NaNC-2 TGS

A

I(cgs

X

1.47 4.2 2.4 1.0

x x x x

u.)

A2 (cgs u.) 0 g (K)

10"4 10"4 10"4 10~ 4

56 28

130 167

—

—

28

66

Bi (cgs u.)

B2 (cgs u.)

28 65 41 13

18 26 16 9

x x x x

eg(K)

Refs.

128 143 81 67

3, 5 3, 5 4,6 3, 5

10 5 10 5 10 5 10 5

Table 2.9.2. Crystal

7

LiNbC-3 BaTi03 NaN02 TGS

x

(aoPi

0.36 0.44 0.40 0.53

/« 0 -T 3 / 2 x x x x

10 6 10 6 10 6 10 6

Eq.

7

Eq.

7 x (a

(6) (6) (6) (6)

4.0 x 10 6 1.4 x 10 6

(7) (7) (7) (7)

2.9 3.3 2.3 2.0

—

4.0 x 10 6

-C\-l/2 D ;

Eq.

10 6 10 6 10 6 10 6

(8) (8) (8) (8)

x x x x

Here a0' , a D ' as well as 7 P ' C , are dimensionless numbers of order unity, G D ' are characteristic temperatures, 7 = \/i/L\, and h and fee are Planck's and Boltzmann's constants. Since the A and B coefficients appearing in Eqs. (4) and (5) can be determined from available d a t a from some ferroelectrics one can make independent estimates of 7 from Eqs. (6) and (8). Table 2.9.1 shows sets of A's and £?'s d a t a from various sources for four oxide and non-oxide ferroelectrics. From these d a t a and Eqs. (6)-(8) one gets three independent estimates of 7, which show a fair degree of internal consistency as shown in Table 2.9.2.

References 1. 2. 3. 4.

J.A. Gonzalo, J. Phys. C 20, 3985 (1987) and references therein. J.R. Herringa and M. Marrenga, Physica A 38, 371 (1975). J.A. Gonzalo, C. de las Heras and S. Vieira, Ferroelectrics 24, 101 (1980). C. de Las Heras, J.A. Gonzalo and S. Vieira, Ferroelectrics 33, 13 (1981). (Note that A has been determined directly from Fig. 2, because the corresponding numerical value inTH appears to be mistaken. Numerical values for A and B are not quoted, since they may be seriously affected by the strong secondary pyroelectric contribution.) 5. W.N. Lawless, Phys. Rev. B 14, 134 (1976). 6. W.N. Lawless, Phys. Rev. B 25, 1730 (1982).

Chapter 2.10

Logarithmic Corrections*

Radium dihydrogen phosphate (RbDP), a well-known ferroelectric isomorph of potassium dihydrogen phosphate (KDP), shows a second order phase transition at Tc « 146 K at ambient pressure. Its dielectric, thermal, structural, and lattice-dynamical properties have been extensively studied. 1-9 The second order character of the transition, in contrast to that of KDP, is generally agreed upon, while at the same time its proximity to a tricritical point is stressed. In their 1969 paper 10 on phase transitions of uniaxial ferroelectrics, Larkin and Khmel'nitskii (LK) predicted logarithmic corrections to classical (Curie-Weiss) behavior of the dielectric constant and other anomalous physical properties near T c , suggesting specifically triglycine sulfate (TGS) and RbDP to test their theory. Logarithmic corrections in the specific heat of TGS have been observed recently by Ema, 11 but no such corrections have been unambiguously observed in the dielectric constant for this crystal as far as we know. Recent dielectric constant measurements by Sandvold and Courtens 12 on tris-sarcosine calcium chloride produced a good fit to an empirical expression including logarithmic corrections compatible with LK theory. No clear crossover to normal Curie-Weiss behavior was discernible above Tc from these data. The samples were cut from a single crystal grown from water solution in the form of plates of 4 x 4 x 0.5 mm 3 with the main surfaces perpendicular to the polar c-axis. Gold-leaf electrodes were directly attached to the main surfaces. The sample holder was located within the evacuated cavity of a cryostat filled with liquid N2. The slow spontaneous heating technique *Work previously published under the title "Dielectric-constant logarithmic correction in R.DH2PO above the Curie point," J.L. Martinez and J.A. Gonzalo, Phys. Rev. B 32, 400 (1985). Copyright © 1985. The American Physical Society. 183

184

Effective Field Approach to Phase

TYansitions

was used in which the rate of heating of the evacuated sample cavity was governed by liquid-nitrogen evaporation at a rate of about 0.5 K/min. At regularly spaced time intervals we measured the capacitance and loss factor (by means of an ESI universal video bridge) and the sample thermocouple temperature (by means of a Fluke microvoltimeter). The relative precision of the capacitance measurements was always better than 0.1% and the relative precision of temperature readings better than 0.02 K. The residual capacitance of the system to be subtracted from capacitance readings was carefully determined using a set of single crystal cleaved plates of NaCl of 0.5 mm thickness and of varying area provided with gold-leaf electrodes, and extrapolating linearly C for A (area) —• 0, obtaining 7.1 pF as residual capacitance. The field amplitude used was 1 V/cm, considered sufficiently low for this kind of measurement. Measurements of sample capacitance and dielectric loss using an ac amplitude of 1V were performed at the following frequencies: 20 Hz, 100 Hz, 1 kHz and 10kHz (all peak values fall within 5% of each other). In all cases the temperature dependence was qualitatively the same: a gradual and relatively small increase in capacitance below Tc followed by a much more pronounced decrease above Tc, which did not quite fit the Curie-Weiss law in the immediate vicinity of Tc. This is illustrated by the gradually decreasing slope in e" 1 vs. T just above the transition temperature shown in Fig. 2.10.1 for 1 kHz, but the same feature was observable for all other frequencies at which measurements were performed. Rounding due to impurity-induced small bias or thermal gradients should show up as a bend in the slope in the opposite direction. Previous work on RbDP by Gladkii and Sidnenko1 and by Peercy and Samara 4 do not show evidence of logarithmic correction.

TIKI

Fig. 2.10.1. Inverse dielectric constant vs. temperature for RDH2PO4 (F = 1 kHz, V = lVrms).

Logarithmic

185

Corrections

On the other hand, our data contain many more points per degree, and the bending of our data points for £ _ 1 vs. T is directly visible in a linear plot only in an interval of about 5K above Tc. Roughly speaking the width of the critical region should be of this order. It may be pointed out that the peak value of the dielectric constant in our data is substantially lower than others 2 - 4 previously reported. This might be due to lower sample quality but in any case impurities and/or defects usually give rise to bending of the e~l{T) curve in the opposite direction. We have tried to fit the behavior of £ _ 1 (T) above the transition to the expression of LK (Ref. 10) which can be written as e~l = (AT/Cni

+ gHATo/C)}-1/3,

(1)

where AT = T — Tc, C = Curie constant above T c , and g and ATo are characteristic parameters of the material undergoing the second order phase transition. Equation (1) can be rewritten as

£

_! _ '

AT C[l+g\n(AT0)}-^

= ^[l-
9 l+5lnATo

- I / O

InT

(2)

To determine the possible presence of the logarithmic correction term in our data the most sensitive parameter by far is T c . To perform a preliminary determination of T c , weighting properly the data closer to the transition, one can use the following procedure: Assuming that a power law e~l = A(AT)1 is approximately fulfilled, one can plot the ratio [£ _ 1 /(d£ _ 1 /dT)] vs. temperature. If the assumption is correct one obtains a linear plot, and from the extrapolation to £ _ 1 = 0 one obtains Tc directly. We followed this procedure with our data, obtaining Tc = 145.90K. Using this Tc with selected data for e~l encompassing an interval of 12 K above the transition (at higher temperatures a trend toward normal Curie-Weiss behavior is discernible) one can easily estimate the other parameters, g' PS —0.860 and C" « 1420. These parameter values were subsequently used as initial values in a computer fit of Eq. (2) in which T c , g' and C" were left free to change, to the full set of data for e _ 1 (T) in the whole temperature interval omitting only the first three points corresponding to nearly flat £ _ 1 very near T c . The parameters thus obtained were Tc = 145.85 K, g' = —0.86, and C = 1425.36. The semi-logarithmic plot in Fig. 2.10.2 shows the good fit of the theoretical expression for £ _ 1 (T) with the experimental values.

186

Effective Field Approach to Phase

Transitions

Fig. 2.10.2. Semi-logarithmic plot of (e - £ 0 ) _ 1 vs. T for R b H 2 P 0 4 . The theoretical curve is the LK expression [Eqs. (1) and (2)] with T c = 145.85, g' = - 0 . 8 6 , C" = 1425.36, and eo = 3.5 (solid curve). Points are experimental data.

At temperatures for which A T > 12 K, the data appear to show a gradual tendency toward Curie-Weiss behavior, which is slightly enhanced if one subtracts from the e value eo ~ 4.5 ±1.5 estimated from room-temperature measurements. Taking into account that the value for the Curie constant C obtained from our data for 12 < AT < 40 K is C = 2.546 x 103 (lower by 13% than some previously reported 1 ) one can obtain from C' = C(l+g S'

In T 0 ) 1 / 3 = 1425.36

= 0 ( 1 + 5 In To)" 1 = -0.86

(3) (4)

the corresponding values for the parameters in the LK expression g m (—0.150) AT 0 ?»237K.

(dimensionless),

(5) (6)

According to this, the value g is small and negative 13 and there would be a change of sign in the product (g' In AT) in Eq. (2) at T = TQ = 237K above T c , which, probably by a fortuitous coincidence, happens to fall very close to the next high-temperature transition in RbDP (from tetragonal to monoclinic), which takes place at T = 86°C. A negative g implies a divergence of e _ 1 above Tc at AT = AT 0 exp(l/
Logarithmic Corrections

187

LK state in their p a p e r 1 0 t h a t Eq. (1) is applicable for displacive-type transitions, whereas e « \T - Tc\~l x In 1 / 3 \T - Tc\ is applicable for o r d e r disorder-type transitions. R b D P is generally considered to show an o r d e r disorder-type transition, like its isomorphs at ambient pressure, but there is also experimental evidence 4 ' 1 4 t h a t it presents an u n d e r d a m p e d soft-mode indicative, at least partially, of a displacive-type character of the transition. In fact, the existence of double potential-well minima separated by energy barriers does not exclude the possibility of relative displacements of these minima, accompanied by changes in the height of the barrier, which could give the transition a mixed (order-disorder) - (displacive) character. Thus, the applicability of Eq. (1) to the R D P transition should not be ruled out entirely on the grounds of its having order-disorder character. T h e mean deviation N

R = (1/iV) 22 \£TLpt - £i"c1alcl/£i"c1alc i=l for the logarithmic correction fit was i?LC = 0.0037, clearly better t h a n the deviation for a power law ( e _ 1 = A ( A T ) 7 i ? P L = 0.0075). A t t e m p t s to fit our d a t a with Curie-Weiss and power laws using background terms linear in t e m p e r a t u r e gave definitely inferior fits to those obtained with a logarithmic correction.

References 1. V.V. Gladkii and E.V. Sidnenko, Fiz. Tverd. Tela (Leningrad) 13, 1642 (1971) [Sov. Phys. — Solid State 13, 1374 (1971)]; see also M. Amin and B.A. Strukov, Fiz. Tverd. Tela 10, 3148 [10, 2498 (1969)]. 2. C. Pierre, J.P. Dufour and M. Remoissenet, Solid State Commun. 9, 1493 (1971). 3. B.A. Strukov, A. Baddur, V.N. Zinenko, A.V. Mishchenko and V.A. Kopstik, Fiz. Tverd. Tela (Leningrad) 15, 1388 [Sov. Phys. — Solid State 15, 939 (1973)]. 4. P S . Peercy and G.A. Samara, Phys. Rev. B 8, 2033 (1933). 5. P. Bastie, J. Lajzerowicz and R.J. Schneider, J. Phys. C 11, 1203 (1978). 6. F. Troussaut, P. Bastie, J. Bornarel and M. Vallade, Ferroelectrics 39, 1053 (1981). 7. P. Schnackenberg, J. Pipman, V.H. Schmidt and G.F. Tuthill, Ferroelectrics 39, 1045 (1981). 8. V.H. Schmidt, P.T. Schnackenberg, A.S. Western, A.G. Baker, C.R. Bacon and W.P. Crummett, Ferroelectrics 24, 275 (1980).

188

Effective Field Approach to Phase Transitions

9. K. Gesi, K. Ozawa, T. Osaka and Y. Makita, J. Phys. Soc. Jpn. 52, 2538 (1983). 10. A.I. Larkin and D.E. Khmel'nitskii, Zh. Eksp. Teor. Fiz. 56, 2087 (1969) [Sov. Phys. — JETP 29, 1123 (1969)]. 11. K. Ema, J. Phys. Soc. Jpn. 52, 2798 (1983). 12. E. Sandvold and E. Courtens, Phys. Rev. B 27, 5660 (1983). 13. The negative value of g is somewhat unexpected. According to LK, however, g = 37 = Tcbv/32n (As 3 ) 1 ' 2 , and, even if all the quantities involved are positive, the square root implies a ± sign compatible with a negative g value. On the other hand, a reasonably good fit can be obtained using C = 2250, a positive g = 0.05, AT 0 = 20 K, and Tc = 145.2 (corresponding to g' = 0.0435 and C' = 2357), which presents a divergence at A T « 9.7 x 10 9 , very far from Tc. The quality of the fit with the set of parameters in an interval of AT w 20 K above T c is, however, definitely inferior to that obtained using a negative g = —0.86 and C" = 1425, as indicated by a mean deviation (R) about ten times larger in the former case. 14. P.S. Peercy, Phys. Rev. Lett. 3 1 , 379 (1973).

Part 3

Some Applications to Ferroelectrics: 1991-1997

Chapter 3.1

Pressure Dependence of the Free Energy Expansion Coefficients in P b T i 0 3 and B a T i 0 3 , and Tricritical Point Behavior* R. Ramirez, M.F. Lapefia, and J.A. Gonzalo Departamento de Fisica Aplicada (C-IV), Universidad Autonoma de Madrid, E-28049 Madrid 34, Spain

Previous data for the pressure dependence of the order parameter for BaTi03 at room temperature (RT) as well as new data for the pressure dependence of the order parameter for PbTi03 at RT have been analyzed within the framework of a phenomenological theory for ferroelectrics. It is concluded that in BaTiC>3 a tricritical point is close but not accessible at RT, while in P b T i 0 3 it is accessible at p = 1.75 x 10 1 0 dyn/cm 2 , allowing a full characterization of the pressure dependence of the freeenergy expansion coefficients.

It is well known 1 ' 2 t h a t rj, the order parameter [either the spontaneous polarization P s or equivalently the unit-cell spontaneous strain z1'2 = (c/a — 1) 1//2 ] of BaTiC>3, undergoes a weakly discontinuous first order transition under high hydrostatic pressure at room t e m p e r a t u r e (RT), which indicates t h a t the crystal is not far from a tricritical point ( T C P ) at RT and p*; 2.2 x 1 0 1 6 d y n / c m 2 . On the other hand, Raman-scattering work 3 on P b T i 0 3 has indicated t h a t it undergoes a continuous secondorder phase transition under hydrostatic pressure at R T . Since the phase transition of this crystal at ambient pressure is known to be first order, *Work previously published in Phys. Rev. 42, 2604 (1990). Copyright © 1990. The American Physical Society. 191

192

Effective Field Approach to Phase

Transitions

this suggests a changeover from first order to second order at a certain intermediate pressure. Recent X-ray diffraction data 4 indicate that at RT and at p*, 1.75 x 10 1 0 dyn/cm 2 ( l P a = 10dyn/cm 2 ; l f r 8 k b a r ) , there is a discontinuity in the slope of the order parameter rj [this time defined as z1/2 = (c/a — 1)1//2] as a function of hydrostatic pressure, which can be interpreted as the changeover from first order to second order behavior. It may also be pointed out that unpublished X-ray diffraction data 5 on atomic position shifts as a function of pressure seem to support this interpretation. Figure 3.1.1 depicts the RT order-parameter pressure dependence of PbTiC>3 and BaTiOs, showing the qualitative difference between both cases. This difference of behavior will be subject to detailed analysis in this work, within the framework of the phenomenological theory of ferroelectricity.6 Let us explicitly include the pressure dependence into the free-energy expansion of the ferroelectric system in terms of even powers of the

(a)

Second order

(b)

Fig. 3.1.1. Order parameter [TJ = z 1 / 2 = (c/a - l ) 1 / 2 ] vs. pressure (p) at room temperature for (a) PbTiC>3, showing discontinuity in slope at p = p c , identified as the tricritical point, and (b) BaTiC>3, showing no such discontinuity at room temperature, but displaying first order transition at p = p*.

Pressure Dependence

of the Free Energy Expansion

Coefficients

193

polarization, F(p) = F0(p) + \X{P)P2

+ \aP)P4

+ \C(P)P6 + L,

(1)

where we assume ,n^(dx(p)\ X(P) = X(O)+[-W)O, n

n

,Tc(0)

P = ^c(o)

to) = e(o) + ( ^ )

,

+ [dTc(p)/dp]oP +

idC(p)/dp)oP>

m

(2)

(3)

P = C(P-PC),

which implies, for a first order transition at p = 0,

«0)<0,

i.e.,c=(2®)o>0,

and C(P),

C(o) = C-

From Eq. (1), taking into account that E =

(4) dF(p)/dP,

E = x(p)P + ttp)P3 + aP)P5+L,

(5)

which, for E = 0, P = Ps, gives O = X ( P ) P + £(P)P S 2 + C(O)P S 4 , and, therefore, p?(p) = ( ^ )

{-^(P) ± [e(p) - 4CX(P)]1/2}

(6)

where only the plus sign within curly brackets has a physical meaning. To carry out the analysis of (i) the PbTi03 case, where we have identified p = p c as the pressure corresponding to the TCP, i.e., to £(pc) = 0, and (ii) the BaTiC-3 case, where p = p* is the pressure at which a first order transition takes place under pressure at RT, we can get the relevant useful

194

Effective Field Approach to Phase

Transitions

relationships as follows. Case (i): p = pc (TCP), f (pc) = 0, Ps2=P

2 s

(Pc)=(^)

1 / 2

,

i - e . , C = ^

(7)

from Eq. (6) for £ = 0, (8)

ifo from Eq. (5) for E = 0, and

sn

-e(o)

^P Jo

Pc

(9)

from Eq. (3). Case (ii): p = p* (first order transition), £(p*) < 0, ^ ( P ' ^ ^ A

£V)=4CX(P*)

(10)

from Eq. (6) for p = p*, where from £(?*) = c ( p * - p c ) = [-4Cx(p*)] 1/2 .

(11)

Combining Eq. (11) with £(0)=c(0-pc),

(12)

we can get p c and c in terms of known quantities. Table 3.1.1 summarizes the relevant parameters for PbTiC-3 and BaTiOs given in the literature. 2 ' 7 Using these data and the relationship given by Eqs. (7)-(9) for P b T i 0 3 and Eqs. (10)-(12) for BaTi0 3 , one can obtain the free-energy expansion coefficients, which have not been previously reported Table 3.1.1. Transition parameters for PbTiOs and BaTiOs under hydrostatic pressure. Numerical data for T c , (dT c /dp)o, C, and (dC/dp)o are from Ref. 2 except the value in parentheses, which is from data in Ref. 7. Crystal

p (dyn/cm 2 )

T c (K)

(dT c (p)/dp) 0 (K cm 2 /dyn)

PbTiO 3 (0) PbTi03(pc) BaTiO 3 (0) BaTiQ 3 (p*)

0 1-75 x 10 1 0 0 2.2 x 10 1 0

773 649 395 287

-7.1 x HT9 — -4.8 x 10"9 —

C (K)

4.1 2.5 1.6 1.3

x 10 5 X 10 5 X 10 5 xlO5

(dT c (p)/dp)o (K cm 2 /dyn) -9.1 x 10"6 — - 1 . 4 x lCT 6 —

Pressure Dependence of the Free Energy Expansion

Coefficients

195

in the case of PbTiC>3 [where we have used a saturation spontaneous polarization value -Pso(O) = 57/xC/cm 2 = 1.71 x 10 5 esu], as well as their pressure dependence, which are given in Table 3.1.2. For BaTiC>3 we have taken P s0 (0) = 25^C/cm 2 = 0.75 x 10 5 esu. The pressure corresponding to the tricritical point of BaTiC>3 can be obtained from Eq. (12) once c has been determined. It turns out to be pc = 2.24 x 10 1 0 dyn/cm 2 , slightly larger and very close to p* = 2.20 x Table 3.1.2. Free-energy expansion coefficients for P b T i 0 3 and B a T i 0 3 under hydrostatic pressure. Crystal

p (dyn/cm 2 )

(d£/dp)o

£ (esu)

C (esu)

(esucm 2 /dyn) PbTiO 3 (0) PbTi03(pc) BaTiO 3 (0) BaTiQ 3 (p*)

-1.27 x 10~ 1 2 0 -6.79 x 1 0 " 1 3 0.09 x 1 0 ~ 1 3

0 1-75 x 10 1 0 0 2.2 x 10 1 0

(a)

6.06 x 1 0 " 2 3 30.4 x 1 0 - 2 3

5.73 5.73 2.33 2.33

x x x x

10-23 10"23 10~26 10"26

0.26

(b) 30 * lo-« 25 •5~

1 s

20

a.

10

PL,

5

15

0

0

0.3 0.S 0.0

1.2 l.S P (GP«)

l.B 2.1

2.4

Fig. 3.1.2. Schematic representation of order parameter n(p,T) = Ps(p,T), for (a) P b T i 0 3 and (b) B a T i 0 3 , showing a qualitative difference stemming from the fact that the T C P falls on a different side of the RT isotherm for either system.

196

Effective Field Approach to Phase

Transitions

1010 dyn/cm 2 . As pointed out above, however, the TCP is not accessible at RT (T = 23°C), but it should be accessible at a slightly lower temperature. Figure 3.1.2 summarizes, in a schematic way, the behavior of the order parameter with pressure and temperature for the ferroelectric perovskites PbTiOs and BaTiC>3. The qualitative difference between the pressure behavior of both systems at RT isotherm for the former and the latter system. We may conclude that a consistent characterization of the pressure dependence of the phase transition under hydrostatic pressure at RT is possible within the simple framework of the phenomenological theory of ferroelectricity. This is worked out under the assumptions implied in the present discussion, i.e., a weak pressure and temperature dependence on the pressure of the coefficients X(P) a n d £(p)Acknowledgments One of us (R.R.) would like to thank R. Nelmes for his hospitality at the University of Edinburgh, Edinburgh, United Kindom, where he became acquainted with the study of pressure effects on phase transitions. We also acknowledge helpful comments from B. Jimenez, from the Consejo Superior de Investigaciones Cientificas, Madrid, Spain. The financial support of Comision Interministerial de Ciencia y Tecnologfa, Madrid, Spain, is also acknowledged. References 1. G.A. Samara, in Advances in High Pressure Research, edited by R.S. Bradley (Academic, New York, 1969), Vol. 3, Chap. 3. 2. G.A. Samara, Ferroelectrics 2, 277 (1971). 3. J.A. Sanjurjo, E. Lopez-Cruz and G. Burns, Phys. Rev. B28, 7260 (1983). 4. R.J. Nelmes and A. Katrusiak, J. Phys. C 19, L725 (1986). 5. R.J. Nelmes, A. Katrusiak, H. Vincent and R. Ramirez (unpublished). 6. F. Jona and G. Shirane, Ferroelectric Crystals (Pergamon, New York, 1962). 7. R. Clarke and L. Benguigui, J. Phys. CIO, 1963 (1977).

Chapter 3.2

Ultrasonic Study of the Ferroelectric Phase Transition in R b D 2 P 0 4 * Z. Hu and C.W. Garland Department of Chemistry and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

J.A. Gonzalo Departamento de Optica y Estructura de la Materia, C-IV Universidad Autonoma de Madrid, 2804 9 Madrid, Spain

The velocity and attenuation of shear elastic waves have been measured in the paraelectric phase of tetragonal R.DD2PO4. The softening of the elastic stiffness C66 can be well described by a Curie-Weiss law. The polarization relaxation time r exhibits conventional critical behavior: T = TO(T — Tc)" 1 . Both TO and T c are very sensitive to deuteration. The values Tc = 229.1 K and T 0 = 2.61 x 1 0 " 1 0 s K were obtained for a RbD2P04 crystal that is 87% deuterated, whereas Tc = 147 K and TO = 0.3 x 10" 1 0 sK have been reported for R b H 2 P 0 4 .

3.2.1.

Introduction

Rubidium dihydrogen phosphate R b H 2 P 0 4 (RDP) and its tetragonal deuterated counterpart RbD2PC>4 ( D R D P ) are isomorphs of the wellknown KH2PO4 (KDP) family. 1 A comparison of the acoustic behavior of R D P and D R D P has been undertaken to increase our understanding of t h e ferroelectric phase transition in K D P - t y p e crystals and t o provide *Work previously published in Ferroelectrics Taylor & Francis Group. 197

112, 217 (1990). Copyright © 2005.

198

Effective Field Approach to Phase

Transitions

useful background data for the orientational glass-forming mixed systems RDP/ADP and DRDP/DADP, where A = NHj. 2 - 3 RDP has a tetragonal structure at room temperature. DRDP can be obtained with either monoclinic or tetragonal structures, depending on the crystal growth conditions. 4 In this paper, we will be concerned only with the tetragonal form of DRDP. Upon cooling, both RDP and DRDP undergo ferroelectric phase transitions from tetragonal (space group 142c!) to orthorhombic (Fdd2). The critical temperature Tc strongly depends on the amount of deuteration, ranging from Tc ~ 147 K for pure RDP 5 to Tc ~ 223 K for a 80% deuterated sample. 6 The ferroelectric transition in RDP is second order. 5 However, a 95% deuterated sample exhibits a well-defined first order discontinuity in the dielectric constant at T\ > Tc.7 This crossover from second order to first order upon deuteration implies that a tricritical point exists at an intermediate deuteration concentration. Ultrasonic and hypersonic shear waves are effective probes for studying ferroelectric transitions in the KDP family.8 Approaching Tc from above, the acoustic attenuation increases and the acoustic velocity decreases due to piezoelectric coupling between the elastic strain and the electric polarization. The acoustic properties of RDP have been investigated by several groups, 9 - 1 1 but DRDP has not been studied previously. Very recently, an impulsive stimulated Brillouin scattering study 5 has demonstrated that the shear elastic behavior of RDP can be completely understood with mean-field theory, in contradiction to earlier specific heat 12 and dielectric 13 measurements, which had indicated the presence of a logarithmic correction term. In this paper, data are reported for a single crystal of DRDP; that is, ~87% deuterated. Analysis of the velocity and attenuation of the elastic shear wave yields the elastic constant CQQ and the polarization relaxation time r as functions of temperature in the paraelectric phase of DRDP. Our results can be well described by mean-field theory. The effect of deuteration on the static and dynamic critical behavior will be discussed and compared with that observed for KDP and KH 3 (Se0 3 ) 2 -

3.2.2.

Experimental

A single crystal of DRDP with tetragonal structure was grown by Interactive Radiation, Inc. The sample had dimensions of ~ 1 x 1 x 1.6 cm 3

Ultrasonic Study of the Ferroelectric Phase Transition

in RbDzPOi

199

and a pair of flat and parallel surfaces normal to the x-axis. The nominal concentration of deuterium was stated to be 97%. From the value of Tc determined from our ultrasonic measurements and the relation between Tc and deuterium concentration given in Ref. 6, we estimate that the actual deuteration of the sample is about 87%. The absolute sound velocity u was measured at intervals of roughly 35 K with an accuracy of ± 1 % using the Papadakis pulse-overlap technique. Relative changes in sound velocities as a function of temperature were determined with high precision using a computerized phase-sensitive detection technique and a MATEC MBS 8000 system. 14 A lithium niobate transducer (10 MHz fundamental frequency, 41° x-cut) was bonded to the sample with Nonaq grease. The acoustic wave of interest is the transverse mode propagating with wave vector parallel to the [100] axis and polarization vector parallel to the [010] axis. This mode is associated with the shear elastic constant c§§ = pu2. We have used p = 2.866g/cm 3 , the room temperature mass density of RDP, 9 for all of our calculations. The effects of deuteration and thermal expansion on the mass density are estimated to be less than 1.5% and have been neglected. At room temperature, the values (in units of 10 9 N/m 2 ) of e n , c 44 , and c 66 are 61.9, 10.13, and 3.14 for our DRDP sample, in comparison with 67.0, 10.27, and 3.59 for a pure RDP crystal. 9 Absolute attenuation values were obtained from a measurement of first and second echo amplitudes when T was far away from Tc. Near Tc, changes in attenuation were measured by following the amplitude of the first echo as a function of temperature. All measurements were confined to the paraelectric phase since the echo pattern disappeared in the ferroelectric phase due to domain-wall scattering. A cooling rate of about 0.1 K/min or slower was used.

3.2.3.

Results and Discussion

Figure 3.2.1 shows the elastic stiffness C66 of DRDP as a function of temperature at various frequencies. As expected, the CQQ softens on cooling toward the ferroelectric transition. The lowest observed CQQ value is 0.635 x 109 N/m 2 at 7\ = 230.28 K, which is about 1.2 K above the extrapolated second order critical temperature Tc. The abrupt disappearance of the echo pattern indicates that the sample undergoes a weak first order phase transition at T\. Values of C66 obtained with three ultrasonic frequencies are in excellent agreement and demonstrate the absence of velocity dispersion.

200

Effective Field Approach to Phase

i

|

•

••

i

i

Transitions

•

i

•

-

3.0

-

j r

2.5 2.0

-

1.5

-

3 3

1.0

-i

-

0.5 n

>

i

i

i

240

260

280

300

T (K)

Fig. 3.2.1. Temperature dependence of the C66 elastic stiffness for 87% deuterated D R D P at 10 (o), 31 (+), and 52MHz (A). The solid line is a Curie-Weiss fit using Eq. (2).

Thus, we consider that these cee values represent the static behavior of the system. As for all KDP-family crystals, the lowest order Landau expansion of the free energy can be expressed as 8 F = F0 + i a ( T - T0)Pi + i c g 6 e | + a36P3e6,

(1)

where the order parameter P3 is the electric polarization along the 2-axis, eg is the elastic strain, Cg6 the bare elastic stiffness, 036 the piezoelectric constant, and To the temperature at which the crystal would undergo ordering in the absence of piezoelectric coupling, a is proportional to the dielectric Curie constant C (a = C/4ir). Prom Eq. (1), one can obtain 8 cm

u

66

T-Tc T-T0'

(2)

with rp _ rp

,

Ca36 4-Trr0 '

(3)

Ultrasonic Study of the Ferroelectric Phase Transition in RbD2P0i

201

where T c is the second order transition temperature at constant (zero) stress. The ce6 elastic stiffness data for DRDP are well described by this mean-field equation as shown in Fig. 3.2.1. The least-squares fitting parameters c^6, T c , and T0 are 3.4 x 10 9 N/m , 229.08K, and 223.97K, which can be compared with 4.2 x 10 9 N/m 2 , 147.0 K, and 140.4 K for RDP. 5 Values of AT = Tc - T0 for RDP varying from 0.2 to 6.6K have been obtained by different techniques (see Ref. 5), and our value of AT = 5.1 K for DRDP is close to AT = 6.6 K for RDP obtained from the stimulated Brillouin data. 5 Published values of the dielectric Curie constant C for DRDP range from 3140 K (Ref. 7) to 3800 K (Ref. 6). Adopting an average value of 3470 K for C, one obtains d36 = 2.5 x 10 4 esu, which is close to the value 036 = 3.0 x 10 4 esu for RDP. 5 Indeed, all the parameters needed to describe the static c6e behavior of DRDP are similar to those for RDP except T c , which is increased ~82 K by deuteration. The same conclusion is obtained from a comparison of c66(T) data for KDP 1 5 and DKDP. 16 The ultrasonic attenuation in DRDP is shown in Fig. 3.2.2 as a function of temperature for three different frequencies. As in other KDP-type

50 |—r

T

40

~30 E

U "^

m T>

"20

10 235

240 T (K)

245

250

0 Fig. 3.2.2. Attenuation a of the C66 acoustic wave vs. temperature at frequencies of 230 10 (o), 31 (+), and 52MHz (A).

202

Effective Field Approach to Phase

Transitions

crystals, there is a substantial increase in the attenuation as Tc is approached. It was not possible to follow the ultrasonic echoes all the way down to the transition for measurements at 31 and 52 MHz due to high attenuation and serious pulse distortion. In the low-frequency regime (LOT
6 6 ~~ c 6 6

2

/

LO2T,

ac = a-a0=

A

\

(4)

zuc 66 where r is the polarization relaxation time at constant stress, u the sound velocity, to = 2irf the angular frequency, and ao the noncritical background attenuation. We have tested the quadratic frequency scaling by plotting (a — ao)/to2 vs. T, as shown in Fig. 3.2.3. The values of ao are small and independent of temperature. The validity of the quadratic frequency dependence for a c is consistent with our elastic stiffness data, which exhibit no sign of frequency dispersion. Since the bare stiffness Cg6 is known from the Curie-Weiss fit to the static CQQ(T) data, Eq. (4) and the data shown in Fig. 3.2.3 will yield the **. J

-i

1

,

,

^3.0

a x

% o

3 1.5

230

235

240 T (K)

245

250

Fig. 3.2.3. Values of (a - a0)/ui2 obtained at 10 (o), 31 (+), and 52 MHz (A) as a function of temperature. The ao values in dB c m - 1 are 2 for 10 MHz, 6 for 31 MHz, and 3 for 5 MHz.

Ultrasonic Study of the Ferroelectric Phase Transition in RbDiPOt,

203

i

^

6

-

-

AJHI"

-

-

-y 230

—

m^ I

235

240 T (K)

I

245

250

Fig. 3.2.4. Polarization relaxation rate 1/r vs. temperature for 87% deuterated DRDP. The solid line is the best fit with Eq. (5).

relaxation time r. Figure 3.2.4 shows that the critical relaxation rate 1/r is a linear function of temperature, as expected for conventional van Hove critical slowing down:

- = T

-{T-Tc).

(5)

T0

The linear fit shown in Fig. 3.2.4 corresponds to To = 2.61 x 10~ 1 0 sK and Tc = 229.1 K. Comparing this T 0 value for DRDP with r 0 = 0.3 x 10~ 10 s K for RDP, 5 we obtain a deuteration amplitude ratio R = T^/T^ = 8.7 for RDP. This ratio is very similar to the values R = 8.3 for KDP 1 5 ' 1 6 and R = 7.5 for the ferroelastic system KH3(Se03)2- 17 For uniaxial ferroelectrics with soft acoustic modes, critical fluctuations can be suppressed by the long-range interactions associated with piezoelectric coupling between electric polarization and elastic strain. In fact, renormalization group calculations 18 have shown that for KDP-type systems the upper marginal dimensionality is 2.5. Therefore, mean-field behavior without logarithmic .corrections is expected, as observed in RDP and DRDP.

204

Effective Field Approach to Phase Transitions

T h e traditional microscopic theory for the ferroelectric phase transition in the K D P family is based on an order-disorder model t h a t considers proton tunneling between two sites of adjacent t e t r a h e d r a (for a review, see Ref. 19). P r o t o n motion plays an important role in triggering this transition, as indicated by the large increase in transition t e m p e r a t u r e upon deuteration. Another mechanism 2 0 for the K D P - t y p e transition has been proposed in which the isotope effect on T c is caused by a geometric distortion of PO4 tetrahedra. R a m a n scattering in R D P and D R D P has given s u p p o r t 2 1 for this new mechanism. Our dynamic result, i.e., a relaxation amplitude for D R D P ~ 9 times larger t h a n t h a t for R D P , may also be explained qualitatively by this model: since the distortion of PO4 t e t r a h e d r a on deuteration results in an increase in the energy barrier for the flip-flop motion of PO4, the polarization relaxation time increases upon deuteration. Our results, however, cannot distinguish which mechanism, proton tunneling or distortion of PO4 tetrahedra, plays the essential role in the ferroelectric phase transition.

Acknowledgments We wish to t h a n k V. Hugo Schmidt for providing the D R D P sample. This work was supported in p a r t by NSF grant DMR-87-10035 and in p a r t by U.S.-Spain Joint Committee for Scientific and Technological Cooperation award CCB8604012.

References 1. K. Gesi, K. Ozawa, T. Osaka and Y. Makita, J. Phys. Soc. Jpn. 52, 2538 (1983). 2. E. Courtens, Hely. Phys. Acta 56, 705 (1983). 3. H. Grimm, K. Parlinski, W. Schweika, E. Courtens and H. Arend, Phys. Rev. B33, 4969 (1986). 4. M. Sumita, T. Osaka and Y. Makita, J. Phys. Soc. Jpn. 50, 154 (1981). 5. L.T. Cheng and K.A. Nelson, Phys. Rev. B37, 3603 (1988). 6. B.A. Strukov, A. Baddur. V.N. Zinenko and A.V. Mishchenko, Sov. Phys. Sol. State 15, 939 (1973). 7. J.L. Martinez and J.A. Gonzalo, Solid State Commun. 52, 1 (1984). 8. E. Litov and C.W. Garland, Ferroelectrics 72, 19 (1987). 9. V. Mnatsakanyan, L.A. Shuvalor, I.S. Zheludev and I.V. Gavrilova, Sov. Phys.-Cyst. 1 1 , 412 (1966).

Ultrasonic Study of the Ferroelectric Phase Transition in RbD^PO^

205

10. M.C. Pierre, J.P. Dufour and M. Remoissenet, Solid State Commun. 9, 1493 (1971). 11. G.P. Singh and B.K. Basu, Phys. Lett. 64A, 425 (1978). 12. M.A. Amin and B.A. Strukov, Sov. Phys. Solid State 10, 2498 (1969). 13. J.L. Martinez and J.A. Gonzalo, Phys. Rev. B32, 400 (1985). 14. P.W. Wallace and C.W. Garland, Rev. Sci. Instrum. 57, 3085 (1986). 15. C.W. Garland and D.B. Novotny, Phys. Rev. 177, 971 (1969). 16. E. Litov and E.A. Llchling, Phys. Rev. Lett. 21, 809 (1968). 17. C.W. Garland, G. Park and I. Tatsuzaki, Phys. Rev. B29, 221 (1984). 18. R.A. Cowley, Phys. Rev. B13, 4877 (1976); R. Folk, H. Iro and F. Schwabl, Z. Phys. B25, 69 (1976). 19. V.H. Schmidt, Ferroelectrics 72, 157 (1987). 20. M. Tokunaga and I. Tatsuzaki, Phase Transitions 4, 97 (1984). 21. J.L. Martinez, J.M. Calleja and J.A. Gonzalo, Solid State Commun. 52, 499 (1984).

Chapter 3.3

New Technique for Investigating Ferroelectric Phase Transitions: The Photoacoustic Effect* Jorge 0 . Tochot, Rafael Ramirez, and J.A. Gonzalo Departamento de Fisica Aplicada C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain Measurements of the photoacoustic effect in properly oriented triglycine sulfate crystals from 30 to 60° C show clearly the ferroelectric phase transition in this crystal manifested by the pronounced thermal expansion anomalies. The same method is generally applicable to similar phase transitions.

Of the large number of properties that can be studied by photoacoustic (PA) techniques, the probing of thermoelastic properties of materials is one of those less explored. 1 ' 2 Acoustic waves generated after the absorption of modulated or pulsed optical radiation, the photoacoustic effect, provide information of sound velocity, specific heat, thermal expansion, thermal conductivity, and other related parameters in a very attractive way. While studies in the gas phase are performed mainly by the use of modulated irradiation and microphone detection, 3 in condensed matter acoustic waves can be efficiently detected by piezoelectric transducers (PZTs) in direct contact with the material, due to the good acoustical impedance matching that they present. PZTs have frequency-dependent responses with sharp resonances related mainly with its physical dimensions. Hundred *Work previously published in Ferroelectrics 59, 14 (1991). Copyright ©1990. Taylor & Francis Group. t Permanent address: Centro de Investigaciones Opticas and Departamento de Fisica, La Plata, Argentina. 207

208

Effective Field Approach to Phase

Transitions

kilohertz resonance frequency is found for PZT transducers but several hundreds of megahertz can be reached with polyvinylidene diflouride piezoelectric films (PVF2). Such ranges of frequencies are easily generated with pulsed laser excitation. 4 The PA signal from a sample may change due to several factors. In the direct coupling method, the acoustic wave is related with the thermal expansion AVth of the irradiated volume VQ. In the more studied isotropic case

AHh = (P/CpP)H, where /3 is the volume expansion coefficient, Cp is the specific heat at constant pressure, p is the density, and H is the heat deposited in the volume VQ. This expansion creates a pressure wave that travels outwards at the velocity of sound. Since the electrical signal generated in the transducer is proportional to pressure, PA signal = K(0/Cpp)(l

- KT^Bb,

where A is the optical absorbance of the sample, EQ is the pulse laser energy, and K is a constant that includes the response properties of the transducer. This formula is valid for a small radius irradiated cylindrical region,1 and for an optically and thermally thick sample. In this way, the PA detection of phase transitions is possible if the thermal parameters change as the experimental conditions change. Triglycine sulfate (TGS) is a well-known ferroelectric, which is being used as a pyroelectric detector. 5 It shows a second order phase transition 6 to the nonpolar high-temperature phase at Tc = 49.3°C. The point group symmetry of the ferroelectric phase is monoclinic 2 and that of the nonpolar phase is 2/m. Below the transition temperature, a spontaneous polarization appears along the monoclinic b axis. Good optical quality, transparent single crystals of TGS can be easily grown from water solution. The acoustic detector and the electrical impedance adapter used in this work were built at Max-Planck Institut fur Strahlenchemie by Braslaysky following an early design of Patel and Tam. 2 PZT is 4 mm in diameter and 4 mm in length. TGS slabs, cut from different crystal, orientations with typical dimensions of 5 x 5 x 1 mm 3 , were coupled to the detector with a drop of vacuum grease. The temperature of the system was stabilized better than 0.5°C. Laser pulses were attenuated by a polarizer to energies below 1 mJ and focused by a 15-cm focal length single cylindrical lens (Fig. 3.3.1(a)). The volume irradiated in the sample was 5 x 0.2 x 1mm 3

New Technique for Investigating

Ferroelectric Phase Transitions

209

PA SIGNAL

(b>

f,o t.a 1.4 WAVELENGTH (j*m)

TIME DElAYC^s}

Fig. 3.3.1. (a) Top view of the experimental setup. CL, cylindrical lens; PZT, piezoelectric transducer; E, energy meter; T, temperature sensor, (b) Optical absorbance of TGS in the near-IR. (c) Time delay between the laser line and the first peak of the PA signal for 6-cut TGS.

(Z, X, Y). The absorption in the near infrared (IR) of these samples allows us to use 1.06-/jm laser pulses from a Nd:YAG laser (Fig. 3.3.1(b)). The crystal attached to the PA detector was mounted on a micrometer translation stage that provided a fine X position. Transducer signals were observed in an oscilloscope, digitalized, and saved in a microcomputer. Small changes along the X direction of the position of the irradiated zone of the crystal, by translation of the whole-system crystal-detector amplifier, produce differences in the delay between the laser line and the arrival of the first acoustic signal to the detector. The sound velocity in TGS

210

Effective Field Approach to Phase

Transitions

can be measured in this way for longitudinal waves propagating parallel to different axis. Figure 3.3.1(c) shows the results for the 6-axis as an example. Time-resolved PA signals from PZT detectors show ringing at the resonance frequency and our attention is in the early part of the PA signal. In this way, we avoid contamination by acoustic reflections in the boundaries of the sample. To check the polarity of the detection system we corn-pared directly the signal obtained below and above the phase transition temperature in a crystal where the 6-axis corresponds to the X direction with the signal obtained when the crystal is replaced by a spectrophotometric quartz cuvette with 0.1 absorbance C0SO4 water solution in it and using a l m J , 532 nm laser pulse. The electrical polarity of the detection system produces negative-going first pulses when the expansion coefficient is positive. The modification of the volume irradiated from the standard cylindrical shape, with its axis in the direction of irradiation to a prism with its larger side perpendicular to both the direction of irradiation and the acoustic-wave propagation, allow us to measure the thermal properties in pure, crystallographic directions. In these conditions, the amplitude of the PA signal can be written as PAS =

KotifiviyCp,

where on is the thermal expansion coefficient along the i direction of the crystal and / is a function of sound velocity in the aforementioned direction Vi. Such a function depends 1 ' 2 on the ratio between the pulse duration of the laser and the transit time of the acoustic wave inside the irradiated zone in our geometrical conditions d{v) — v1/2. By measuring the PA signals in terms of temperature we follow the thermal properties of the sample. Figure 3.3.2 shows several signals at different temperatures for a sample where the experimental X direction corresponds to the crystallographic 6-axis. The change in sign of the acoustical first signal can be clearly seen when the phase transition occurs. The amplitude of the PA signal reflects the big change in the thermal expansion coefficient. This fact can be confirmed by plotting the amplitude of the first PA signal peaks versus temperature. Figure 3.3.3(a) shows the results for a crystal oriented with 6-axis along the X direction. For comparison, we show the relevant thermal expansion coefficient a/,, given by Shibuya and Hoshino,7 below the phase transition temperature and, with more resolution by Ema et al.8 above the phase transition temperature. The PA signal was normalized to fit the thermal expansion coefficient at 60° C.

New Technique for Investigating

Ferroelectric Phase Transitions

211

5 10 TIME (yu-s) Fig. 3.3.2. Time-resolved PA signal of 6-cut TGS at different temperatures. The PA signal at 50° C shows a shoulder at the wing of the first peak; similar "PA phase reversal" has been previously reported [A.C. Tarn and C.K.N. Patel, Appl. Opt. 18, 3348 (1979)].

Figure 3.3.3(b) shows similar results for a sample oriented with the caxis along the X experimental direction. When this sample was rotated and the a± direction (perpendicular to the a-axis) was in the X experimental direction, the PA signal was more complicated due to mixing of waves with different propagation velocities and polarities. The PA signal reflects mainly the changes in the thermal expansion coefficients along the direction perpendicular to both the irradiation direction and the acoustic-wave propagation direction. While the specific heat and the sound velocity also change through the phase transition, in the case of TGS these changes are much smaller than that of the thermal expansion coefficients. The method can be used to directly measure the thermal expansion coefficient. It is essentially different, for instance, from the capacitance method

212

Effective Field Approach to Phase

30

40 50 60 TEMPERATURE <° C)

Transitions

30 40 50 60 TEMPERATURE C O

70

80

Fig. 3.3.3. Amplitudes of PA signal vs. temperature for (a) 6-cut TGS, first peak and (b) c-cut TGS, first (o) and second (•) peaks. The continuous lines are published values for the relevant thermal expansion coefficient (see Refs. 7, 8).

where it is necessary to register, with high accuracy, the changes in thickness and in temperature in order to calculate the derivative dC~1/dT of the capacitance. 9 We have measured sound velocity with axis selection in TGS, a transparent material that allows us to register the time delay of acoustic waves and the irradiation position with high accuracy, unlike the case of nontransparent materials. 10 It can be concluded that the PA effect is a new, sensitive, and convenient tool to investigate many phase transitions with pronounced thermal expansion anomalies.

Acknowledgments One of us (J.O. Tocho) has a sabbatical grant from the Ministerio de Educacion y Ciencia, Spain.

New Technique for Investigating Ferroelectric Phase Transitions

213

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

A.C. Tam, Rev. Mod. Phys. 58, 381 (1986). C.K.N. Patel and A.C. Tam, Rev. Mod. Phys. 53, 517 (1981). H. Vargas and L.C.M. Miranda, Phys. Rep. 161, 43 (1988). S.E. Braslaysky and K. Heihof, Handbook of Organic Photochemistry (CRC, New York, 1989), Vol. 1, p. 327, and references therein. K.L Bye, P.W. Whipps and E.T. Keeve, Ferroelectrics 11, 525 (1988). Landolt-Bernstein, Crystal and Solid State Physics (Springer, New York, 1982), New Series, Group III, Vol. 16b, p. 223. I. Shibuya and S. Hoshino, Jpn. J. Appl. Phys. 1, 249 (1962). K. Ema, M. Katayama, Y. Ikeda and K. Hamano, J. Phys. Soc. Jpn. 46, 347 (1979). H.K. Shiirmann, S. Gillespie, J.D. Gunton and T. Mihailisin, Phys. Lett. 45A, 417 (1973). L. Narrai and W. Volken, Solid State Commun. 70, 223 (1989).

Chapter 3.4

Tricritical Point Behavior and Quadrupole Interactions in Ferroelectrics* Rafael Ramirez, Maria Fe Lapefia and Julio A. Gonzalo Departamento de Fisica Aplicada, C-IV, Universidad Autonoma, 28049 Madrid, Spain

It is shown that the generalization of the effective field expression to take into account in addition to dipolar, quadrupolar, octopolar, and higher order multipolar terms leads to a more general equation of state that describes better the dependence of polarization on temperature and external electric field. The cases of hydrogenated and deuterated members of the TGS family are examined.

It is well known t h a t the usual mean field approach to ferroelectric transitions takes into account dipole-dipole interactions only. T h e effective field is assummed to be given by Eefi = E + f3P,

(1)

where E is t h e external field, (3 a mean field coefficient depending on the geometry of the lattice, and P the polarization. This leads 1 to an equation of s t a t e T e= — tanh_1p-p,

(2)

where e = E/{3NpL, Tc = f3Nfi2/kB, p = P/Nfi, N being the number of elementary dipoles per unit volume, and \i their dipole moment. Equation (2) for e = 0 fixes the behavior of the spontaneous polarization PS(T) *Work previously published in Ferroelectrics 124, 1 (1991). Copyright © 1991. Taylor & Francis Group. 215

216

Effective Field Approach to Phase

Transitions

with temperature in the range 0 T < Tc (it may be noted that at low temperature collective dipole wave-like excitations not taken into account in the above equation of state may play a prominent role) and leads to a continuous (second order) transition with well-defined universal features. In this work, we consider a most general expression of the effective field Eeff = £ + /3P + 7 P 3 .

(3)

It is st right forward to show that in this case the equation of state becomes e = — tanh

l

p - p - gp,

(4)

where g = (7/'(3)(N)j,)2. It is in turn clear that, taking into account the power series expansion of the inverse hyperbolic tangent, , -1

*

3

1

5

(5) p = p + -p + -p5 • 3 5 For the g value going over from g < 1/3 to g > 1/3 we get a changeover from continuous to discontinuous transition, and therefore a tricritical point. tanh

L

6

Fig. 3.4.1. Reduced spontaneous polarization (Ps/Pso) vs. reduced temperature for crystals in the TGS family. Curves are obtained from Eq. (6), and g values indicate strength of quadrupolar interaction.

Tricritical Point Behavior and Quadrupole Interactions

in Ferroelectrics

217

Hysteresis loops spontaneous polarization data for triglicine sulfate,2 selenate, 3 and fluoberillate4 (TGS, TGSe, TGFB) are given in Fig. 3.4.1, together with the theoretical spontaneous polarization curve from Eq. (4), which, taking e — 0, is given by

£ = (l+flrf)[—£T-1,

(6)

J-c Ltanh ps_ with g values g = 0, g = 1/12, g = 1/6, g = 1/4, and g — 1/3. One can see that the behavior of ps(T) for TGS, TGSe, and TGFB can be described much better with g values in the interval 0 < g < 1/3 than with 3 = 0, which corresponds to the case of dipole-dipole interactions only. In Fig. 3.4.2 the same data as in Fig. 3.4.1, in an expanded temperature scale, are plotted along with the corresponding curves defined by Eq. (6) with g values optimized to best fit the data. Since the value of Pso an(A the value of g are not known in advance with sufficient precision, it is convenient to design an unbiased procedure to determine both simultaneously. To this end we may note that the ratio {psi/pS2) corresponding to the two temperatures T\ and T2, close to T c , is independent of PSQ. Taking, for instance, p s l at {T2/Tc) = 0.95 and P s 2 at (T2/Tc) = 0.99, plotting the

Fig. 3.4.2. Same data as in Fig. 3.4.1 in expanded temperature scale near T c ; curves are from Eq. (6) with g values giving optimum fit to data for TGS, TGSe, and T G F B , respectively.

Effective Field Approach to Phase

218

Transitions

Table 3.4.1. Crystal

Tx (K)

g (dimensionless)

P s 0 (^C/cm 2 )

TGS TGSe TGBF DTGS DTGSe

322.4 295.7 346.2 333.0 307.0

0.19 0.32 0.28 0.24 >l/3

4.08 4.55 4.86 2.60 3.50

ratio (psi/ps2) as a function of g at closely spaced Ag intervals in the range 0 < g < 1/3, and inserting experimental ratios (P s i/^s2) = {Psi/Ps2) corresponding to TGS, TGSe, and TGFB one gets directly the corresponding g values, which are, respectively, g = 0.19 (TGS), g = 0.32 (TGSe), and g = 0.28 (TGFB). Then one can automatically determine Pso from Psi/psi and Ps2/Ps2, yielding, with excellent consistency, Pso = 4.08//C/cm (TGS), P s 0 = 4.55 ^C/cm (TGSe), and Ps0 = 4.86/xC/cm (TGFB). Table 3.4.1 summarizes the relevant data, transition temperature, saturation polarization, and g values for crystals in the TGS family including data 5 ' 6 for the deuterated compounds DTGS and DTGSe. The last one shows a first order transition accompanied by thermal hysteresis and, therefore, g > 1/3. It is of some interest to compare the saturation spontaneous polarization data obtained from fits of Eq. (6) as described above to estimates based upon evaluation of iVM using the crystallographic unit cell volume vc, which gives (N = 2vc = 2(abcsen/3) _1 ), Z = 2 being the member of molecular units per unit cell and the Curie constant, C = e(T — Tc) = A-KN^I2/k& (for T>TC). With these data (NkBC/4Tr)1/2.

Ps0 =N^=

(7)

Table 3.4.2. Crystal

C (K)

Ref.

TGS TGSe TGBF DTGS DTGSe

3560 4050 2630 2550 3280

6 7 4 8 7

N (mol. units/cm 3 ) 3.12 2.99 3.16 3.12 2.99

x x x x x

10 2 1 10 2 1 10 2 1 10 2 1 10 2 1

Ref.

P s 0 (^C/cm 2 )

9 10 11

3.7 3.8 3.2 3.1 3.4

— —

THcritical Point Behavior and Quadrupole Interactions

in Ferroelectrics

219

Table 3.4.2 gives the relevant d a t a for the same crystals belonging to the T G S family as in Table 3.4.1, assumming t h a t unit cell volumes for D T G S and D T G S e are not very different from those belonging to the hydrogenated compounds. It can be seen t h a t the saturation polarization values obtained in this way are in general somewhat lower b u t not very different from those given in Table 3.4.1. It can be noted t h a t deuteration increases the effective quadrupolar interaction as well as the transition t e m p e r a t u r e from T G S (g = 0.19) to D T G S (g = 0.24) and from T G S e (g = 0.32) to D T G S e (g > 1/3). This means t h a t at certain fractional deuteration x, which appears t o be located in the range 0 < x < 0.32, the crystal must show tricritical point behavior.

Acknowledgment We wish t o acknowledge the financial support of C I C y T for this work.

References 1. See, for instance, J.A. Gonzalo, Phys. Rev. B l , 3125 (1970) and references therein. 2. S. Tribewasser, IBM J. Res. Dev. 2, 212 (1958). 3. Z. Malek, J. Strajblova, J. Novotny and V. Mecerek, Czech. J. Phys. B 18, 1224 (1968). 4. H.H. Wieder and C. R. Parkeson, J. Phys. Chem. Solids 27, 247 (1966). 5. V.P. Konstantinova, J.M. Silvestrova, L.A. Shuvalov and V.A. Yurin, Bull. Acad. Sci. USSR, Phys. (English Transl.) 1206 (1960). 6. J.A. Gonzalo, Phys. Rev. 144, 662 (1966). 7. K. Gesi, J. Phys. Soc. Jpn. 41, 565 (1976). 8. A. Cammasio and J.A. Gonzalo, Solid State Commun. 16 (1975). 9. E.A. Wood and A.N. Holden, Acta Cryst. 10, 145 (1957). 10. M. Yamada, Masters Thesis, Hokkaido University (1979). 11. F. Warkusz and K. Lukaszewicz, Bull. Acad. Polon. Sic, Ser. Sci. Chim. 2 1 , 271 (1973).

Chapter 3.5

Frequency and Temperature Dependence of Sound Velocity in TGS Near Tc* M. de la Pascua t and Zhibing Hu Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

R. Ramirez, M. Koralewshi*, and J.A. Gonzalo Departamento de Fisica Aplicada, C-IV, Universidad Autonoma de Madrid, 28049-Madrid, Spain

Ultrasonics (10 and 50 MHz) and Brillouin (18.2 Hz and 19.2 Ghz) measurements of the longitudinal sound velocity in TGS propagating along the crystallographic c-axis (_L to the ferroelectric fr-axis) have been performed in the vicinity of the transition temperature Tc = 49.3°C. The data confirm a Debye-type relaxation behavior with a polarization relaxation time of the form T = A(TC — T)~* for T < Tc, which is in fair quantitative agreement with the relaxation time from dielectric measurements at high frequencies.

Triglycine sulfate (TGS) shows a typical second order ferroelectric transition 1 at Tc = 49.3°C. Sound velocities along various crystal directions have been previously measured by ultrasonics 2 and light scattering 3 " 5 techniques at a fixed frequency. In this work, we report high-resolution *Work previously published in Ferroelectrics Letters 14, 91 (1992). Copyright © 1992. Taylor & Francis Group. tOn leave from Dpto. Fisica Aplicada, C-IV, Universidad Autonoma, 28049 Madrid, Spain. *On leave from Universitat im Adama Mickiewicka 60-780 Poznan, Poland. 221

222

Effective Field Approach to Phase

Transitions

ultrasonics data and Brillouin scattering data taken at several frequencies that confirm a Debye-type relaxation behavior and allow a comparison of the anomalous temperature-dependent relaxation time obtained from elastic measurements (ultrasonics and Brillouin) with the corresponding anomalous relaxation time from high-frequency dielectric constant measurements. The samples used in both ultrasonics and Brillouin experiments were properly oriented TGS parallelepipeds, cut from a large TGS single crystal grown from water solution donated to us by Dr. J. Przeslawsky, University of Wroclaw (Poland). The experimental setups for ultrasonics (MIT, Chemistry Department) and Brillouin scattering (UAM, Applied Physics Department) were the same as those used in previous work by the MIT and UAM groups. Figure 3.5.1 shows the ultrasonic longitudinal sound velocity for waves propagating along the c-axis (_L to the ferroelectric 6-axis) in a relatively wide temperature range around Tc. It can be seen that the data for 10 MHz (crosses) and 50 MHz (dots) differ from each other only in a narrow temperature interval of less than one degree below Tc. At T > Tc, the longitudinal sound velocity appears to be temperature as well as frequency independent in the temperature range examined.

4150

4100

jlAtv

+ 10 MHz • SO MHz

^4050 ^4000

\

V

i

3950

3900.

40

45

5

°T['C]55

60

Fig. 3.5.1. Longitudinal sound velocity along the c-axis of a TGS single crystal as a function of temperature at the frequencies 10 MHz (+) and 50 MHz (•) from ultrasonics measurements.

Frequency and Temperature Dependence of Sound Velocity in TGS

223

«fau

«K

4100 o

\*?4050

19.2

-

„«

GBs

» 1&.Z GHz a 14.8 GHz (R'f. 5)

,

*

s° »

.

0,8

39S0

J

""""SO

SO

1

m

r.rn40

I

s0

e0

Fig. 3.5.2. Sound velocity along the c-axis of TGS as a function of temperature at 19.2 GHz (o), 18.2 GHz ( + ) , present work, and at 14.8 GHz ( • ) ; work reported in Ref. 5.

Figure 3.5.2 shows the hypersonic sound velocities obtained by Brillouin scattering with Ar + laser (single mode) lines of wavelength 5145 A (corresponding to 18.2 GHz) and 4880 A (corresponding to 19.2 GHz) along with previous data from Yagi et al. obtained with a He-Ne laser line of wavelength 6328 A (corresponding to 14.8 GHz). While the experimental uncertainty showed by the data points is not negligible (due in part to the fact that the frequency separation is small), it can be seen that the trend of the sound velocity curves with frequencies is as it should be expected, i.e., the curve corresponding to the highest frequency (19.2 GHz) shows the smaller dip with temperature, and the one corresponding to the lowest one (14.8 GHz) shows the largest dip. The widths of the Brillouin peaks and the intensity of the central quasi-elastic peak in our spectra (not shown in Fig. 3.5.1) show basically the same temperature dependence reported by Yagi et al.5 in their investigation at 14.8 GHz. It may be noted that the hypersonic velocity curves of Fig. 3.5.2 are crossed by the ultrasonic velocity curves of Fig. 3.5.1 at about seven degrees below the transition. This should not be so under normal conditions, since at any given temperature the crystal should behave as consistently hardening (showing therefore higher sound velocity) with increasing frequency in a relaxation process. One possible explanation for the observed behavior is that, at the ultrasonic experimental frequencies of 10-50 MHz one is beginning to approach the lower frequency region where piezoelectric resonance effects take place.

224

Effective Field Approach to Phase

Z.5 x 10'

I

Transitions

I

/

2.0

/

1.5

-

1.0

- I

7 /

0.5

i

i

0.05

0.15

0.10

AT = Tc -

T

[K]

Fig. 3.5.3. Inverse elastic relaxation time r 1(T) in the ferroelectric phase of temperature difference to the Curie temperature (T c ).

These effects can give rise to a superimposed apparent hardening of the crystal with lowering frequency (see Fig. 3.5.3). An anomalous elastic behavior obeying Debye-type relaxation characterized by a relaxation time r can be described by C'(w) = C^ C »

-

(Coo -

Coo

—

Co

1 + UJ2T2 C0)tOT

1 + LU2T2

'

(1) (2)

In terms of ultrasonics data the above equations become (3)

F

1 + L02T2

LU

(4)

where p is the density, v(to) the sound velocity, v^ = i>(oo), vp = u(0), and a (a;) the attenuation. On the other hand, in terms of Brillouin shifts, they become 2irAv

2

=P n

27rAt; 00 \

|fc|

5v {2TTAV\

PZ7T — - '

\k\ V |fc|

'

p[(27rAt;0O/|fc|)2 - (2TT At; 0 /|fc|) 2

—

1 + UJ22-T2

p[(2TrAv00/\k\)2-(2TrAv0/\k\)2]ojT 1 + L02T2

(5) (6)

Frequency and Temperature Dependence of Sound Velocity in TGS

225

where Av = v\k\/2ir is the Brillouin shift, and Sv = a\k\/2Av is the peak full-width at half-maximum. In Eqs. (l)-(6), C^ — pv"^ = p(2'wAv0O/\k\)2 should be basically the same at T < Tc and T > Tc, being temperature independent in a certain temperature range around T c . Co = pv2, = p(2nAvo/\k\)2, on the other hand, should be markedly temperature dependent below Tc but almost temperature independent above Tc. Below Tc the temperature dependence of the relaxation time 5 is given by T=

A(T-TC)~

(7)

We can use the ultrasonics data of Fig. 3.5.1 to determine the temperature dependence of T close to Tc. Knowing two sound velocities (corresponding to two different frequencies t>i = ui/2ir and t>2 = U>2J2-K at a given temperature), it is straightforward to get the relaxation time at this temperature by means of Eq. (3) as follows: v~ v lo

v\vr) -V2{V2)

[vl,+V>l)][vlo-V2{^)] (8)

(l+47T2i/jV2)-1 Vo

•v(v2)

(l+47r2i/2r2)-1

since in our case WQO + v(v{), vx + v(v\), Solving Eq. (8) for r one gets

2v0 1/2

VQO-V(VI,T)

v<x>—v(vi,T)

r(T)

1

(S)'

(9) 4TT2I/|

Substituting the data of Fig. 3.5.1 in the useful interval in which v(vi,T) and v(v2,T) are clearly distinguishable (0.03 < (Tc — T) < 0.15) one gets experimental points for r _ 1 ( T ) as shown in Fig. 3.5.3. From these data one gets the coefficient that appears in Eq. (7) A = (4.6 ± 0.3) x 10 - l i

(10)

which is in good agreement with the results previously obtained by Yagi et a/.5 from Brillouin scattering data at 14.8 GHz. From our Brillouin data at 18.2 and 19.2 GHz one can get also the coefficient A taking advantage of the fact that the maximum width dvm observed at a certain temperature T m for a given frequency OJ is given by

226

Effective Field Approach to Phase

Transitions

i.e., T-1=A-1{Tc-Tm)

= 2nAvm.

Eq. (6) as wr = l,

Substituting here the experimental values Av = 19.2 GHz = (Tc - T m ), 5K, one gets ,4 = (4.1 ±0.3) x HT 1 1 ,

(11) LJ/2-K,

(12)

in good agreement with the value previously obtained by Yagi et al.,5 and also with the value deduced from our ultrasonic data, given by Eq. (10). Finally, it is of interest to compare quantitatively the temperature dependence of elastic relaxation (obtained from ultrasonics and Brillouin data) with that of the dielectric relaxation (obtained from high-frequency dielectric constant data) near T c . In principle, they should be the same because in both cases the relaxation process is induced by the fluctuations of the spontaneous polarizations. This can be done analyzing the highfrequency dielectric constant data of Luther and Muser 6 on TGS through the transition. Cole-Cole plots of the imaginary versus the real part of the dielectric constant at several temperatures below T c give nearly perfect semicircles, indicating a well-behaved Debye-type relaxation with a single relaxation time. The dielectric time is given by T-1 = CJT = A-1 (Tc-T),

(13)

where, from the experimental data at AT, 2K, where LJT — 2irvr = 2n x 4.5 x 108 rad/s, one gets A=(

2 T X 4

5 X 1 2

°9);

(7.0 ± 2) x 1 0 "

(14)

The agreement between the result given in Eq. (14) and in Eqs. (10) and (12) is not unsatisfactory, taking into account the experimental uncert a i n t i e s involved.

Acknowledgments We would like to thank Prof. C. Garland for support and valuable criticism in the early stages of this work. The financial support of CICyT is also acknowledged.

Frequency and Temperature Dependence of Sound Velocity in TGS

227

References 1. See Landau-Bornstein, Crystal and Solid State Physics, Vol. 16, pp. 223-236 (Springer Verlag, New York, 1982). 2. E.J. O'Brien and T.A. Litovitz, J. Appl. Phys. 35, 180 (1964). 3. R.W. Gammon and H.Z. Cummins, Phys. Rev. Lett. 17, 193 (1966). 4. T. Yagi, M. Tokunage and J. Tatsuzaki, J. Phys. Soc. Jpn. 37, 1717 (1974). 5. S.T. Yagi and T. Tatsuzaki, Ferroelectrics 52, 67 (1983). 6. G. Luther and H.E. Musser, Naturforsch. A 24, 389 (1969).

Chapter 3.6

Dipolar and Higher Order Interactions in Ferroelectric TSCC* Gines Lifante and Julio A. Gonzalo Departamento de Fisica Aplicada, C-IV, Universidad Autonoma de Madrid, 28049 Madrid,

Spain

Wolfgang Windsch Fachbereich Physik, Universitat Leipzig, 0-7010 Leipzig, Germany

Relevant data on Tris-sarcosine calcium chloride (TSCC), including temperature-dependent data of spontaneous polarization ( P = 0.27 /iC/ cm 2 ), dielectric permittivity (Curie constant at T > Tc, C = 59K, and Curie temperature Tc = 125.4 K), and specific heat (AC P = 0.27 x 10 6 erg/Kcm 3 , specific heat jump at T = T c , after discounting for logarithmic corrections), have been analyzed within the framework of the effective field approach to ferroelectric transitions, resulting in N = 1.01 x 10 21 dipoles/cm 3 and fj, = 0.80 x 10~ 1 8 esucm, in good agreement with microscopic observations. Consideration of quadrupolar (<7 =£ 0) and octopolar (h ^ 0) contributions to the effective field improve the theoretical fit to (P/Pso) vs. (T/Tc) data.

T h e ferroelectric to paraelectric transition at about T C ,130K, in Trissarcosine calcium c h l o r i d e 1 - 3 (TSCC) has features of an order-disorder 4 , 5 transition as well as of a displacive 6 - 8 transition, which is not uncommon among phase transitions in uniaxial ferroelectrics. T h e crystal s y m m e t r y 9 a t room t e m p e r a t u r e is orthorhombic D\h {Pnma) with four formula units per *Work previously published in Ferroelectrics 135, 277 (1992). Copyright © 1992. Taylor & Francis Group. 229

230

Effective Field Approach to Phase

Transitions

unit cell and becomes orthorhombic C\v {Pn2\a) below Tc. The structural data point out to the existence of two molecular groups per unit cell including each a zwitter ion, which could be interpreted as evidence for two elementary dipolar groups per unit cell, or one dipolar group for every two formula units. In this work, we will analyze available spontaneous polarization, 1 dielectric permittivity, 1 and specific heat 10 data on TSCC within the framework of the effective field approach 11 to ferroelectric transitions to get the relevant parameters /3 (effective field coefficient), /J, (elementary dipole moment), and N (numbers of elementary dipole moments per unit volume) from macroscopic data alone and we will compare them with other structural and microscopic information on these quantities. As is well known, the effective field (or molecular field) approach assumes, in the case of ferroelectrics,11 an effective field acting on each individual dipole (/i) EeS = E + /3P,

(1)

where E denotes the external field, j3 an effective coefficient of proportionality, depending on the crystal structure, and P the polarization. The term (5P involves the dipolar contributions only to the effective field. It will be seen later that higher order contributions can be considered, resulting in a better description of the order parameter dependence on the temperature. The simple effective field assumption leads in a straightforward manner to the equation of state P_

=- {Mw)}-

where Nfi = Poo is the saturation polarization and Tc = f3Nfi2lkQ the transition temperature. Prom this equation of state a Curie-Weiss law for the dielectric susceptibility follows immediately e = e0 + ^

,

C

=

^

forT>Tc.

(3)

Makita's data 1 on spontaneous polarization imply a low-temperature (saturation) polarization P s 0 = Nil = 0.27 ^C/cm 2 = 810esu/cm 2 .

(4)

The same author's data on dielectric permittivity along the ferroelectric axis indicate that the temperature dependence of this physical quantity for

Dipolar and Higher Order Interactions

in Ferroelectric TSCC

231

T > Tc is well fitted by eb-£bo = ^ ^ r

(Tc = 125.4 K),

(5)

where mostly the electronic contribution is £b0

= 5.4(T > T c ),

(6)

the Curie constant is C = 4 ^

= 59K.

KB

From these data alone we get (3 = ^

11

= 26.7,

/x=Mii=0.80xlO-

(8) 18

esu,

(9)

P"sO

and N = — = 1.01 x 10 21 dipoles/cm 3 .

(10)

The last numerical value can be immediately compared with the number of formula units per unit volume, which is given by Zv~l = Z{abc)~l = 2.44 x 10 21 formula units/cm 3

(11)

obtained using 1 a = 9.156(10)1, b = 17.460(5) A, and c = 10.265(5) A. This is compatible, within experimental error, with the assumption that there are two dipole units per unit cell in this crystal. The jump in specific heat 10 at T = Tc is given by AC P = 20 J/Kmol = 0.815 x 106 erg/cm 3 ,

(12)

where use has been made of the fact that the molecular weight of TSCC is M = 377 and its density p = 1.53g/cm 3 . However, the specific heat peak shows clearly that there is an upwards curvature in Cp(T) for increasing T just below Tc, indicative of the need for logarithmic corrections if one wants to compare with the purely effective field jump AC P (eff. field) = ^NkB-

(13)

232

Effective Field Approach to Phase

Transitions

We have estimated the approximately constant slope of AC P (T) well below T c , where logarithmic corrections are presumably negligible and we have obtained AC P (eff. field);

0.27 x 106 erg/cm 3 .

(14)

Substituting numerical values from Eqs. (10) and (14), as well as Boltzmann's constant k-g = 1.38 x 10~ 16 erg/K, we get from Eq. (13) N;

1.3 x 10 21 dipoles/cm 3 ,

(15)

in fair agreement with the value obtained in Eq. (10), taking into account the uncertainties involved. The results thus obtained for /i (dipole moment) and for the number of elementary dipoles per unit cell can be compared with published ENDOR data 12 in TSCC, which seem to support in a quantitative manner the values obtained in Eqs. (9) and (10). ENDOR investigations on manganese doped TSCC revealed a tilting of the sarcosine zwitter ions type II, which are located in a general position in case of transition to the ferroelectric phase. No positional changes could be detected on the sarcosine zwitter ions of type I lying in the (010) mirror planes above T c . The tilting within the molecular planes [(011) and (011)] of the sarcosine type II is not connected with the spontaneous polarization. The tilt perpendicular to the molecular planes of sarcosine type II is temperature dependent. It proceeds for sarc 6 and sarc 2 as well as sarc 3 and sarc 5 in the opposite direction so that the mirror plane is lost. A tilt angle of 3.0° ± 0.5° was measured. The increase of the angle with decreasing temperature indicates a displacement process responsible for the observed spontaneous polarization. With the help of CNDO calculations a dipole moment of 2.7 x 10~ 29 C m (= 8.28 x 10~~18esu) was determined for the sarcosine zwitter ion. Together with the tilting angle of 3° a spontaneous polarization of 0.33 /iC cm" 2 (= 1000esu/cm 2 ) along (010) was calculated, which is in good agreement with the experimental values. The sarcosine zwitter ions 1, 3, and 5 and 4, 2, and 6 are lying in different planes along (100). All are connected by a strong chemical bonding to the calcium ion, and the oxygens of the carboxyl group constitute a deformed octahedron. For each set of three sarcosine zwitter ions as well as for the calcium-(sarcosine) 6 complex one can combine the observed tilt to obtain

Dipolar and Higher Order Interactions

in Ferroelectric TSCC

233

the spontaneous polarization. In the first case, this would mean that there are four polar units in the unit cell, whereas in the second case only two polar units would be present. The calculations given here confirm the earlier statement, which was based on EPR as well on Raman investigations, that the Ca(sarc)6 complex is the polar entity in TSCC. In the case of uniaxial ferroelectrics of the TGS family it has been shown recently 13 that assuming an effective field EeS = E + f3P + ^P3+5P5,

(16)

which includes quadrupolar ( 7 ^ 0 ) , octupolar (5^0), etc., terms beyond the purely dipolar (j3 / 0) term, allows a much better description of the temperature dependence of the spontaneous polarization. Using Eeg from Eq. (16) the expression for the spontaneous polarization becomes P —— = tanh

1+

(17)

*'j|)+A(l

where g = (7//3)(iV/i)2, h = (S/f3)(N^,)4:, etc. This can be written as

Tt^-Hp./N»)_1=g(gy+h(g_

Tc

(Ps/Nfi)

\NfiJ

\Nn

+L

(18)

Available experimental data 1 of (PS/N/J.) vs. (T/Tc) for TSCC have been analyzed, and Fig. 3.6.1 shows that the agreement between experimental points and the theoretical expression, Eq. (18), improves from the purely dipolar (/? ^ 0, g = 0, h = 0) to the dipolar plus quadrupolar (/? ^ 0, g 7^ 0, h = 0) to the dipolar plus quadrupolar plus octupolar expression (/? ^ 0, g ^ 0, h ^ 0). It may be noted that the best fit is obtained with g, 0.19 (<7TC = 1/3 for a tricritical point), and h = 0.15 (hue = 1/5 for a higher order multicritical point). In conclusion, we may say that a clear improvement in the description of the temperature dependence of the order parameter (Ps) in TSCC is obtained including quadrupolar and octupolar terms in the local effective field at the dipole sites. It may be noted that a quadrupolar contribution involves a field gradient, which may give rise to some "stretching" of the elementary dipole moment, thus incorporating, at least partly, displacive features in the physical description of the transition.

234

Effective Field Approach to Phase

1.0

-

- r ^ - *

Transitions

'1nr;Ss£CLfi v

• 2en- 0.6 a.

0.4

-

g = 0 g = .215 9 =

0.2

1

""0.0

0.2

.

.19

_i

0.4

.„,

6

\

a.

\

%

\

TSCC

\ 1 M

h = 0 h = 0 h = .15

\\n

>

L-

0.6

0.8

i—

9

(

1

1.0

T/Tc Fig. 3.6.1. Reduced spontaneous polarization (Ps/N[i) vs. reduced temperature (T/Tc). Experimental points are from Makita (Ref. 1). Theoretical curves are given by Eq. (18) with the successive sets of g and h values: (1) g = 0, h = 0; (2) g = 0.215, h = 0; (3) g = 0.19, h = 0.15.

Acknowledgment We wish t o acknowledge financial support for this work from C I C y T (Comision Interministerial p a r a Ciencia y Tecnologia). One of us (W. Windsch) is indebted to the Spanish M E C (Ministerio de Educacion y Ciencia) for supporting a three-month stay a t UAM, Madrid.

References 1. Y. Makita, J. Phys. Soc. Jpn. 20, 2073 (1965). 2. M. Fujimoto, Cz. Pawlaczyk and I.G. Unruh, Phil. Mag. 60, 919 (1989). 3. J. Petzelt, A. A. Volkov, Yu. G. Goncharov, J. Albers and A. Klopperpieper, Solid State Commun. 73, 5 (1990). 4. T. Chen and G. Schaack, J. Phys. C 17, 3801 (1984); ibid. 17, 3821 (1984). 5. V.H. Schmidt, Solid State Commun. 35, 649 (1980). 6. S.D. Prokhorova, G.A. Smolensky, I.G. Siny, E.G. Kuzminov, V.D. Mikvabia and H. Arndt, Ferroelectrics 25, 629 (1980). 7. G.E. Feldkamp, J.F. Scott and W. Windsch, Ferroelectrics 39, 1163 (1981). 8. G.V. Kolzov, A.A. Volkov, J.F. Scott, G.E. Feldkamp and J. Petzelt, Phys. Rev. B28, 255 (1983). 9. T. Ashida, S. Bando and M. Kakudo, Acta Cryst. B28, 1560 (1972). 10. T. Matsuo, M. Manson and S. Sunner, Acta Chem. Scand. A33, 781 (1979).

Dipolar and Higher Order Interactions in Ferroelectric TSCC

235

11. See, for instance, Julio A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). 12. H. Metz, R. Bottcher and W. Windsch, Chern. Phys. Lett. 182, 132 (1991). 13. R. Ramirez, M.F. la Pena and J.A. Gonzalo, Ferroelectrics (in press).

Chapter 3.7

Thermal Hysteresis and Quadrupole Interactions in Ferroelectric Transitions* Julio A. Gonzalo, Rafael Ramirez and Gines Lifante Departamento de Fisica de Materiales, C-IV Universidad Autonoma de Madrid, 28049-Madrid, Spain

Marcell Koralewski Institute

of Physics, Adam Mickiewicz 60-780 Poznan, Poland

University,

It is shown that a self-consistent calculation of the maximum amount of thermal hysteresis accompanying discontinous ferroelectric transitions can be made if one takes into account the quadrupolar contribution to the effective ferroelectric field. Comparison of calculated and observed thermal hysteresis in several ferroelectrics belonging to different families shows fair agreement.

T h e phenomenon of thermal hysteresis in discontinous phase transitions is a ubiquitous one, not only in ferroelectrics but also in other systems undergoing a variety of phase transitions. Little theoretical attention has been put, as far we know, in calculating the amount of thermal hysteresis, and in correlating its value with t h a t of other physical parameters characterizing the phase transition. Our purpose in this work is to show, within the framework of a generalized effective field approach, t h a t it is possible to correlate succesfully the amount of thermal hysteresis with the relative strength of the quadrupolar interaction contributing to the effective field. *Work previously published in Ferroelectrics Letters 15, 9 (1993). Copyright © 1993. Taylor & Francis Group. 237

238

Effective Field Approach to Phase

Transitions

It is well known that the observed thermal hysteresis may depend on the particular experimental conditions, such as heating and cooling rates, and on the particular experimental technique used. We are concerned here only with the "ideal" (maximum) thermal hysteresis under quasi-equilibrium conditions. The effective field approach 1 to ferroelectric phase transitions, fully analogous to the Weiss2 theory of ferromagnetism, is certainly the simplest possible one, and in its simplest version (Eeg = E + f3P), which takes into account only dipole interactions, is capable of describing fairly well the main features of continous ferroelectric transitions. 3 In this work, we will use a more general expression EeS = E + 0P + 7 P 3 + SP5L

(1)

for the effective field, in terms of an external field E and odd powers of the polarization P, which correspond successively to contributions dipolar, quadrupolar, octupolar, etc. The coefficients /?, 7, and S may be expected to depend on the geometry of the lattice and on the spatial charge distribution within the unit cell. The equation of state 1 is given by

-^=tanh(^V Nfi

\kBT

(2) w

J'

where P is the polarization, Pso = N[i is the low-temperature saturation polarization, N being the number of elementary dipoles per unit volume and fi the electric dipole moment, kB is Boltzmann's constant, and T the temperature. Keeping terms only up to the quadrupolar contribution in Eeg given by Eq. (1) one can rewritte Eq. (2) as T e= — tanh" 1 p-p-gp3,

(3)

J-c

where e = E/PN/i, p = P/N/i, 9 = {-y/f3)(N[i)2, and Tc = pN(j,2/kB is the Curie temperature. For e = 0, p = ps (spontaneous polarization), Eq. (3) is conveniently rewritten as

7r(1+5Ps)t^^'

P s =

w

(4)

Here, g < | gives rise to continous (second order) transitions, while g > \ gives rise to discontinuous (first order) ones. Since we are concerned

Thermal Hysteresis and Quadrupole Interactions

in Ferroelectric Transitions

0.8

-

0.6

i z r-—— ^.^__— ~-—^ ^ "~~-^-^^

^ ^ ^ ^

0.4

^

•

-

^

.

^ ^ ^ ^\ ^\. N. -t> \ 3=1/3 \ \

0.2

"

239

^--^9=0.50

N^ g=0.45 >v g=0.40

\

/

// / ' ^"" , •;'_'' t s

s '

/

s* J - '

i

+

9 1

0.0 0.98

1.00

1.02

T/T c Fig. 3.7.1. Reduced spontaneous polarization (Ps/Pso) for g values above g = | .

vs. reduced temperature

(T/Tc)

here with discontinous transitions, for which thermal hysteresis may be expected, we consider the case g > | . Figure 3.7.1 shows p = {Ps/Ps0) as a function of (T/Tc) from Eq. (4) for several g values just above g = 5. An increasing thermal hysteresis for increasing g between Tc and a certain temperature T* is noticeable. Figure 3.7.2 gives the discontinuous jump in spontaneous polarization p* = (P*/Ps0) occurring at T = T* as well as the reduced thermal hysteresis ATCcal

H

(5)

Tc. obtained from Eq. (4) as shown below, as a function of g. The expression of (T*/Tc) as a function of p* can be easily obtained as follows. Equation (4) establishes an explicit relation for (T/Tc) as a function of ps = (Ps/.Psrj) and the transition temperature T* for rising T corresponds to a minimum of (T/Tc) as a function of ps. Therefore, d(T/T c ) dp s

2gp*s T=T*

L(l+3(Ps*) 2 )

1

(T*/Tc) +

[(1 + 9Pf)P;(l - gpf)}-1 ( 0

0.

T*

(6)

240

Effective Field Approach to Phase

Transitions

0.6

0.4 P>*

0.2

0.0 0.3

g Fig. 3.7.2. Spontaneous polarization jump (P*/Pso) and thermal hysteresis (AT/TC) as a function of g in the interval ~ < g < | ; g = (a//3)(N/j,)2 gives a measure of the quadrupolar interaction contribution to the effective field.

Simplifying and eliminating (T*/Tc) we get ( ^ )

= ( i + f l ( p : ) 2 ) ( 3 f l ( p : ) 2 + i)

(7)

Then substituing g in terms of p* from Eq. (4), T*\

Tj

_1 = 2 3 ^tanh Ps\ V Pt )

1

(l-Ps*)2J

(8)

which gives T* in terms of p* or vice versa. It may be noted that, for sufficiently small p*, Eq. (8) further reduces to 1

l-i(rf)

(9)

Equations (8) and (9) establish a conection between the ammount of thermal hysteresis accompanying a ferroelectric phase transition and the value of the discontinuous jump in spontaneous polarization occurring in the same transition. As might be expected, the amount of thermal hysteresis vanishes as p* approaches to zero. For 0 < g < | one has a continuous (second order) transition, such as the one found3 in the undeuterated crystals of the triglycine sulfate family TGS (sulfate) and TGFB (fmoberillate). For g one has a continuous, tricritical point transition, which seems

Thermal Hysteresis and Quadrupole Interactions

in Ferroelectric Transitions

241

Table 3.7.1. Crystal

T c (K)

T* (K)

P s c (/iC/cm 2 )

Pso (/iC/cm 2 )

Refs.

DTGSe KDP BaTi03

307.0 122.56 374.3

307.8 122.70 390.4

1.5 2.2 15.5

3.5 5 27.0

4,5 6,7 9, 10

Crystal

A T o b s (K) a

A T o b s (K) b

Refs.

AT c a l (K)

DTGSe KDP BaTi03

0.6 3.7

0.8 0.14 1.6

5 8,6 11,9

0.6 0.25 3.2

a b

From unit cell parameters as determined by X-ray work. Prom dielectric constant measurements.

to be very close to TGSe (selenate). For g > | , as we have seen, one has discontinuous (first order) phase transitions. They are realized in some deuterated crystals of the triglycine sulfate family, such as DTGSe, as well as in most crystals of the potassium dihydrogen phosphate family (KDP) and the perovskite family. Table 3.7.1 gives relevant data to characterize the thermal hysteresis of the ferroelectric transition in DTGSe, KDP, and BaTiO-3. To determine AT c a | we have proceeded as follows. First, from the known value of P s = P s c at T = Tc (somewhat below T*, which is the transition temperature for rising temperature) and the saturation spontaneous polarization, we have obtained psc = Psc/Pso- From 'taBh^ftc _ J 1 ps J pic

(1Q)

[deduced from Eq. (4) for T = Tc], we have obtained a numerical value for g. Then, from the functional relationship between g and p*, depicted in Fig. 3.7.2, a numerical value for p* has been finally obtained, which, when substituted in Eq. (8), leads directly to a calculated value for the thermal hysteresis AT ca i. We have to do this instead of directly using p* experimental values because the spontaneous polarization is changing rapidly as T approaches T*, and available data are rarely accurate. Ar o bs (u.c.p.) from the temperature dependence of the unit cell parameters as given by X-ray investigations, and ATODS(£), from dielectric constant data for rising and lowering temperatures, agree reasonably well with AT ca i for the crystals in Table 3.7.1. It must be noted that, in going through the thermal hyateresis temperature interval, the crystal is in succesive

242

Effective Field Approach to Phase

Transitions

metastable states and this may easily result in somewhat inaccurate numerical values for AT 0 b s - In this respect the usual larger values given by unit cell parameters measurements seem to be more reliable t h a n those given by dielectric constant a n d / o r hysteresis loops measurements. We may conclude t h a t if the relative strength of quadrupolar contributions to the effective field in a ferroelectric are such t h a t g = (a/P)(N/j,)2 > | one can expect appreciable thermal hysteresis in ferroelectric transitions as in fact observed. T h e relevance of quadrupolarlike interactions in ferromagnets 1 2 and ferroelectrics 1 3 has been previously recognized. It should not be ruled out altogether t h a t higher order (octupolar, etc.) interactions play a role at the phase transition. This would complicate the calculation of the amount of thermal hysteresis and it will be left for further work.

Acknowledgments We wish to acknowledge the financial support of C I C y T for this work. One of us (M.K.) would also like to acknowledge the D G I C y T .

References 1. J. A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). 2. P. Weiss, J. Phys. 6, 667 (1907). 3. R. Ramirez, M. Fe Lapefia and J.A. Gonzalo, Ferroelectrics 124, 1 (1992). 4. Z. Malek, J. Strajblova, J. Novotny and V. Merecek, Czech. J. Phys. B18, 1224 (1968). 5. K. Gesi, J. Phys. Soc. Jpn. 41, 565 (1976). 6. B.A. Strukov, A. Baddur, V.A. Koptsik and LA. Velichko, Fiz. Tverd. Tela 14, 1034 (1972); Sov. Phys. Solid State 14, 885 (1972). 7. G.A. Samara, Ferroelectrics 5, 25 (1973). 8. J. Kobayashi, Y. Uesu, I. Mizutani and Y. Enomoto, Phys. Stat. Sol. (a) 3, 63 (1970). 9. J.A. Gonzalo and J.M. Rivera, Ferroelectrics 2, 31 (1971); J.M. Rivera and J.A. Gonzalo, Rev. Mexicana de Fisica 19, 297 (1970). 10. S.H. Wemple, M. di Domenico Jr. and I. Camlibel, J. Phys. Chem. Solids 29, 1797 (1968). 11. R.G. Rhodes, Acta Crystallogr. 2, 417 (1949). 12. H.H. Chen and RM. Levy, Phys. Rev. B7, 4267 (1973). 13. Q.Z. Kai, Z.J. Bo and W.C. Lei, Ferroelectrics 101, 159 (1990).

Chapter 3.8

Specific Heat and Quadrupole Interactions in Uniaxial Ferroelectrics* Beatriz Noheda, Gines Lifante and Julio A. Gonzalo Departamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

The inclusion of quadrupolar and higher order terms into the effective field in second order transition in uniaxial ferroelectrics is shown to describe well the jump in specific heat at T, Tc, for crystals of the TGS family. It is shown that the numerical values for the coefficient g = (•y/p)(Nfi)2, corresponding to the quadrupolar interaction term, obtained (a) from specific heat data and (b) from spontaneous polarization data (both at T, T c ) are in good agreement with each other for crystals of the TGS family.

It has been previously reported 1 t h a t introducing an effective field Ees = E + f3P + jP3 + 5P5 • • • , which implicitly includes quadrupolar ( 7 P 3 ) , octopolar {SP5), and higher order terms in addition to the usual dipolar {(3P) term, leads to an improved description of the phase transition 2 in uniaxial ferroelectrics. T h e resulting equation of state is given by e = (^-Jtanh^p-{l+gp2

+ hp4 + ---)p,

(1)

where e = E/f3N(i is t h e dimensionless normalized field, T c = fiNfj? t h e transition (Curie) t e m p e r a t u r e , p = P/Nfi the normalized polarization, and *Work previously published in Ferroelectrics Letters 15, 109 (1993). Copyright © 1993. Taylor & Francis Group. 243

244

Effective Field Approach to Phase

Transitions

g = ('j/(3)(Nfi)2, h = (S/f3)(Nfi)4, etc., N being the number of elementary dipole moments per unit volume and fi their electric dipole moment. For E = 0, the temperature dependence of the spontaneous polarization is given [Eq. (1)] by ^ = r^^(l+gP2s+hpl+L) Jc tanh ps

= -J^(l+g>P*). tann ps

(2)

Near the transition temperature (T < T c ), this leads to p2; 3(1 — T/Tc) for g' = 0 and

^d^1"!)

for

^^°-

(3)

This result will be used below to determine the behavior of the ferroelectric specific heat jump from T < Tc to T > Tc, which will allow one to get an independent determination of g from specific heat data, to be compared with a previous determination of g from spontaneous polarization data. To get the appropriate expression for the specific heat per unit volume at T < Tc we may write down2 the ferroelectric internal energy per unit volume in terms of the spontaneous polarization Q(T) = - i / ? P 2 ( T ) ,

(4)

which allows one to get the differential of entropy

ts^yjfi,

( 5)

and, consequently, the ferroelectric specific heat per unit volume

^m-r(f)T-i«J»rt'(^). Substituting p2s (T < Tc) from Eq. (3) and using Tc = j3Nn2/kB

ACp(Tc) = - i / W

2

(« we get

^ ^ ) =I ^ ^ B ,

(7)

at T < Tc and, of course, ACp(Tc)=0,

p 2 ( T > T c ) = 0, at T >Tc.

(8)

Specific Heat and Quadrupole Interactions

in Uniaxial Ferroelectrics

245

Therefore, the relative importance of g is automatically reflected in the specific heat jump at Tc, which is 3 (AC p ) m a x = (l)-iVfcB for g = 0 (dipolar interactions only)

(9)

and becomes 3 (AC p ) m a x -> (OO)-ATA;B

1 for g = - (tricritical point).

(10)

Let us investigate the case of uniaxial ferroelectrics of the TGS family, in particular triglycine sulfate (TGS), triglycine fmoberilate (TGFB), and triglycine selenate (TGSe). To get the corresponding numerical values for the quadrupolar interaction coefficient g the only information needed is the experimental value of the jump in specific heat per unit volume at T = T c , i.e., (AC p ) m a x in erg/cm 3 K, and the number of dipole moment per unit volume N = Z/vc (with Z = 2, the number of formula units per unit cell, and vc = abcsma, the monoclinic unit cell volume in terms of the lattice parameters a, 6, c and the angle a, as given by X-ray diffraction data 3 ). Relevant specific heat data for TGS, 4 " 6 TGFB, 7 and TGSe, 8 - 9 including both a.c. calorimetry and adiabatic calorimetry measurements, are given in Table 3.8.1, where the g(ACp) value from Eq. (7) is given by 1 ff=

3

Nk¥ 2(AC„

(11)

The last column in Table 3.8.1 gives the g(Ps) value obtained from the temperature dependence of the spontaneous polarization of TGS, 1 0 , 1 1 TGFB, 1 2 and TGSe 13 in the close vicinity of T c . These values are basically confirmed by the g(ACp) values obtained from specific heat data. A trend may be noted from TGS, which at ambient pressure is farthest from the tricritical point (g = | ) , to TGSe, close to it at the same pressure. Table 3.8.1. Crystal Tc (K)

(ACp)max N (ACp)max/(§JVKB) (erg/cm 3 K) (dipoles/cm 3 K)

TGS TGFB TGSe

0.618 x 10 7 1.35 x 10 7 1.93 x 107

322.2 346.2 295.3

1.56 x 10 22 1.58 x 10 22 1.50 x 10 22

1.19 4.18 6.2.2

g(ACp)

g(Ps)

0.16 0.25 0.28

0.17 0.27 0.32

246

Effective Field Approach to Phase

Transitions

tA cm '-5 Vi N KB 5.0

2.5

0 0.010

0.00 5

0.00

(T c -T)Ac — " Fig. 3.8.1. Plot of reduced specific heat vs. reduced temperature for uniaxial ferroelectric crystals of the TGS family.

Figure 3.8.1 depicts the normalized ferroelectric specific heat data from T G S , 4 - 6 TGFB, 7 and TGSe 8 ' 9 in a narrow vicinity of T c . The TGS data indicate that (AC p ) m a x is only slightly larger than (3/2)./V&B, the value corresponding to g = 0 (almost negligible contribution from quadrupolar or higher order interactions). On the other hand, the TGSe data show that (AC p ) m a x is about six times larger than (3/2)N1CB- This means that the thermal energy release or absorption by the crystal is going through the order-disorder phase transition taking place in a much shorter temperature interval in the latter than in the former case, which might be some relevance in connection with the working of devices making use of the rate of change of spontaneous polarization near T c . Figure 3.8.2 shows experimental data of g' = g + ftp2 + • • • , as a function of the squared reduced spontaneous polarization, Ps2 = (Ps/Nfi)2, for TGS, 10 - 11 TGFB, 1 2 and TGSe 13 obtained using Eq. (2). It may be seen that the TGS data, available in an extended range of temperature below T c , and therefore going back to p 2 values close to one, show that higher order multipolar contribution become non-negligible at low temperatures. The g(Ps) value for TGSe from these data is perhaps too close to the tricritical value g = \ and is somewhat larger that the g(ACp) value given above. More detailed and precise measurements of spontaneous polarization for this crystal, especially in the close vicinity of T c , would be certainly welcome.

Specific Heat and Quadrupole Interactions

in Uniaxial Ferroelectrics

247

0.7 5 T6Se:Pso=4.14 ptC/em2,Tc = 295.VK T6FB: Pso- 4.49 /iC/cm 2 , Tc = 34„6.2°K TGS : Pso=4.28;uC/cm2,Tc = 321.20K 0.50

i 0.2 5

T6S

•-CH>oW^n^o^^ o o

°%o 0

0.50

1.00

Fig. 3.8.2. Plot of g' determined from experimental data on spontaneous polarization at given temperatures using Eq. (2) as a function of squared spontaneous polarization.

It may pointed out that the inclusion of quadrupolar and higher order terms in the effective field responsible for the ferroelectric phase transition improves the theoretical fit to temperature-dependent spontaneous polarization 1 ' 14 in some second order transition uniaxial ferroelectrics, describes semiquantitatively the amount of thermal hysteresis 15 in various typical first order transition ferroelectrics, allows one to get a fair quantitative prediction of the composition at which a mixed ferroelectric/nonferroelectric system approaches the tricritical point, 16 and, as shown in this work, produces a good correction between thermal data (specific heat) and dielectric data (spontaneous polarization) in the vicinity of a ferroelectric transition. All these results put together make a good case for the actual applicability of the generalized effective field concept to ferroelectric phase transitions.

Acknowledgment Financial support from CITyT is gratefully acknowledged.

References 1. R. Ramirez, M.F. Lapena and J.A. Gonzalo, Ferroelectrics 124, 1 (1991) (we note some fairly obvious typing mistakes in this paper).

248

Effective Field Approach to Phase Transitions

2. J. A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991), pp. 56-63. 3. See, for instance, Landolt-Bornstein, Group III, Vol. 16, Ferroelectrics and Related Substances (Non-oxides) (Springer-Verlag, Heidelberg, 1982). 4. R. Tobon and J.E. Gordon, Ferroelectrics 17, 409 (1977). 5. S.A. Taraskin, B.A. Strukov and V.A. Meleshina, Sov. Phys. Solid State 12, 1089 (1970). 6. W. Reese and L. May, Ferroelectrics 4, 65 (1972). 7. V.A. Chernenko and I.N. Polanov, Sov. Phys. Solid State 2 1 , 1059 (1979). 8. B.A. Strukov, S.A. Taraskin and V.A. Koptsik, Sov. Phys. Crystallogr. 13, 541 (1968). 9. K. Ema, K. Harnano, K. Kurihara and T. Hatta, J. Phys. Soc. Jpn. 43, 1954 (1977). 10. S. Triebwasser, IBM J. Res. Dev. 2, 212 (1958). 11. A.G. Chynoweth, Phys. Rev. 117, 1235 (1960). 12. H.H. Wieder and C.R. Parkerson, J. Phys. Chem. Solids 27, 247 (1966). 13. Z. Malek, J. Strajblova, J. Novotny and V. Mecerek, Czech. J. Phys. 18, 1224 (1968). 14. G. Lifante, J.A. Gonzalo and W. Windsch, Ferroelectrics 135, 277 (1992). 15. J.A. Gonzalo, R. Ramirez, G. Lifante and M. Koralewski, Ferroelectrics Lett. 15, 9 (1993). 16. G. Lifante, J.A. Gonzalo and W. Windsch (to be published).

Chapter 3.9

Field-Dependent Temperature Shift of the Dielectric Losses Peak in TGS* Tomas Iglesias, Beatriz Noheda, Gines Lifante and Julio A. Gonzalo Departamento de Fisica de Materiales, C-IV Universidad Autonoma de Madrid, 28049 Madrid, Spain

Dielectric constant and loss factor measurements have been performed in triglycine sulfate single crystals at the vicinity of the Curie temperature ( r c = 48.5° C) as a function of a.c. driving field (20 < v < 106 Hz) in a wide range of field amplitude (50 > Eo > 0.15 V/cm). Well-defined shifts of loss factor peak (Z) max ), increasing monotonously with the field, not previously reported as far as we know, were observed and analyzed in terms of the energy dissipated within the vanishing hysteresis loops at T « Tc corresponding to the Eo « EC(T) coercive field.

Triglycine sulfate is a prototypical example of second order transition uniaxial ferroelectric. 1 Its dielectric properties, 2 including the dielectric loss factor, 3 ' 4 have been widely investigated, but no systematic study of the field dependence of these properties has been previously performed. Such a study may help to clarify some controversial 5 ' 6 previous results. Properly oriented single crystal plates, with the main surfaces perpendicular to the ferroelectric 6-axis, were cut from a large single crystal of good optical quality grown from water solution in the s t a n d a r d way. Gold leaf electrodes were attached to the main surfaces. While samples of various thicknesses were examined, a conveniently thick cleaved sample of thickness d « 0.2 cm was used for t h e measurements reported here. To check

*Work previously published in Ferroelectrics Letters 17, 3/4 (1994). Copyright © 1994. Taylor &c Francis Group. 249

250

Effective Field Approach to Phase

Transitions

the quality of the sample, hysteresis loops were observed through T = Tc on the screen of a Nicolet 310 digital oscilloscope, which showed negligible asymmetry. Capacitance and losses factor were measured with an automatic Hewlett Packard Universal Bridge 4284 A with high accuracy. The temperature of the sample, within the sample holder immersed in a large-capacity water bath controlled by a Haake F-3, was measured by a chromel-alumel thermocouple and recorded automatically by means of a Keithley 196 system DMIM microvoltimeter. Figure 3.9.1 shows the losses factor D = tan 5, along with inverse dielectric constant, as a function of temperature in the vicinity of Tc = 48.56°C at a frequency v = 1kHz for the following a.c. field amplitudes: EQ = 50,15,5,1.5, 0.5V/cm. It can be clearly seen that (a) D m a x is unusually large for large fields and decreases sharply for small fields down to values about 20 times smaller; (b) there is a shift in temperature AT m = T(DmeiX) — Tc, which is also very substantial for large fields and becomes practically vanishing for the lowest fields. Figure 3.9.2 is a blow up of the region closest to Tc in Fig. 3.9.1, showing in more detail the behavior for the three lowest field amplitudes. Table 3.9.1 summarizes the main results contained in the data of Figs. 3.9.1 and 3.9.2 and shows, in the last column, that in spite of the large variations in AT m and EQ for the family of curves depicting D(T), which span almost two orders of magnitude, the ratio ATm/Eo(sin 5)1/2 remains very approximately constant. These results can be analyzed in terms of the energy dissipated at T < Tc within the vanishing hysteresis loops corresponding to coercive field values, EC(T), approximately matching the a.c. field value spanned by EQ in V/cm. The energy dissipated per unit volume and unit time by a dielectric characterized by a complex dielectric constant e(u>) = e'{ui) — ie"(u>), where s"(to)/e'(LO) = sin S, tan 5 = D, is given by 7 W = \^Ets"{u)

= \~Ele'{^)

sin 6.

(1)

On the other hand, the energy dissipated per unit volume and time within a hysteresis loop in a ferroelectric material is W = ~PP?,

(2)

where (3 = Es/PsAirTc/C is the effective mean field coefficient8 and Ps the spontaneous polarization per unit volume at the corresponding temperature. Therefore, we can set, for a ferroelectric near a second order phase

Field-Dependent

Temperature Shift of the Dielectric Losses Peak in TGS

1.0

1

1

1

1 KHz

0.8 -o U O 0

o

.' ° «

0.6 -

fc< w 0.4 v

v

•A - *

0

B

- «° C

CO

w

.

*

O

id

251

0

O

<*

»•

o

IT

0.2

".

»

O

DJ

° oo B

V j * — * * * * * * " "

"""o

,

-

H

•"•BffiB&SBf^L 1 —

-•'

1

i

' " ^

•

£

10~

» 50 V/cm * 1.6 V/cm • 15 V/cm • 0.5 V/cro • 5 V/cm • 0.15 V/cm

^

"*\

!

, \ V

8

fv&

*J

<J CD

o w u

„

\."\5

-

V\ "A v\ *• "a

^r

1 -

'

\

$

> i

46

44

50

48

52

Temperature ( C) Fig. 3.9.1. Triglycine sulfate losses factor (D) and inverse dielectric constant (e a function of temperature for various a.c. field amplitudes.

1

) as

transition, with e'(cj) fulfilling the Curie-Weiss law, '1 Wm&x

C

sin 5 n

~ 8TT ° V 2 T, - T,,

a 2 H^3/3f 8TT

V

P2

T

c

1

T

s0)

J


-1 c j

(3)

252

Effective Field Approach to Phase

0,20

P

1

r-

r

—1

°A

1

:

1 KHz

A

0.15 --

Transitions

*

A

U

o

A

1->

V

0.10

* r O

«J

V

0

o

w CO m m O

»

0

0

•«

0.05 -

°0 » ••

* A° v .»« o o g

,

• *» •• 0.00 ,-4 10

.

1

••

i

' •

• * • 0 • •

w C

a o

l

l

«

!

• SO V/cm * 1-5 V/cm » IS V/cro • 0.5 V/em « 5 V/cm • 0.1S V/cm

.

• * * * •

a

**

a

**

-

a

J-. 4)

?> d 48.2

>

1

i

1

1

48.3

48.4

48.5

48.6

48.7

48.8

Temperature (°C) Fig. 3.9.2. Triglycine sulfate D and e~l vs. T as in Fig. 3.9.1 in the close vicinity of Tc to show in more detail the curves corresponding to the lowest field amplitudes.

where W from Eqs. (1) and (3) is used and a(w) is defined as a dimensionless coefficient of proportionality, equal or less than unity, which may be expected to be somewhat frequency dependent, and PS(T) = V3Ps0 x (1 - T/Tc) 1 / 2 has been substituted.

Field-Dependent

Temperature Shift of the Dielectric Losses Peak in TGS

253

Table 3.9.1. Losses factor of TGS near T c as a function of a.c. field (1 kHz). E0 (V/cm) 50

15 5 1.5 0.5

A T m (°C)

Dmax

sin<5m

2.52 0.94 0.26 0.04 0.01

0.87 0.78 0.47 0.18 0.06 0.04

0.66 0.62 0.43 0.18 0.06 0.04

—

0.15

ATm/E0(sm5n 11/2 0.062 0.079 0.076 0.068 0.085

—

Equation (3) leads to AT m = AEoisinS^2,

AT m = Tc - T m ,

(4)

where A_(27r/3)

1 2

a{uj)

/

fy \kB

(5)

after substituting Ps0 = Nfi, C = A-K^L2 /k-Q (Curie constant), and Tc = fUN^/k-Q (Curie temperature) in terms of microscopic quantities into Eq. (3). The data for AT m and i?o(sin S^)1/2 are plotted in log-log form in Fig. 3.9.3, and the linear fit through the experimental points with slope equal to one implies, ,4 o b s « 0.057,

(6)

while, from Eq. (5), using fi = (fcBC/47rA'')1/2 = 1.58 x 10~ 18 esu cm,8 we get (27T/3)1/2 / 1 . 5 8 x l 0 " 1 8 \

0.0165

,.

nn

Then a(to) is the order of and less than unity, as expected, and the validity of Eq. (4) to describe the observed behavior is confirmed. It may be concluded: (a) that a novel effect, the temperature shift with field amplitude in the losses factor peak near a second order ferroelectric, has been experimentally established; (b) the calculated linear behavior of AT m (temperature shift) with E0 (field amplitude) is fulfilled by the present data for TGS in a remarkable range, spanning more than two orders of magnitude.

254

Effective Field Approach to Phase Transitions

m (°C)

,o»

ICT1

10" 2

J

10"'

10°

EQ (sin

L

10*

10*

1/z

5m) (volt/cm)

Fig. 3.9.3. Log-log plot or temperature shift AT m = Tc - T(D m a x ) as a function of field amplitude times (sin5m) 1 / 2 where <5m = t a n - 1 ( D m a x ) . The straight line through the experimental points is (AT m ) c a | using Bq. (4). Further work at still lower a.c. field amplitudes and various frequencies is under way.

Acknowledgments We acknowledge the financial support of C I C y T (Spanish Scientific and Technological Research Commission) under Materials Research G r a n t t o investigate ferroelectric phase transitions.

References 1. See, for instance, F. Jona and G. Shirane, Ferroelectric Crystals (Macmillan, New York, 1962). 2. See, for instance, Landolt-Bornstein, Group III; vol. 16, Ferroelectrics and Related Substances (Non-Oxides) (Springer-Verlag, Heidlberg, 1982). 3. R.M. Hill and S.K. Ichiki, Phys. Rev. 132, 1603 (1963). 4. R. Garcia and J.A. Gonzalo, Phys. Rev. Lett. 50, 1501 (1983). 5. E. Courtens. Phys. Rev. Lett. 5 1 , 844 (1983). 6. J. Kortzler, Phys. Rev. Lett. 5 1 , 1811 (1983). 7. See, for instance, A.J. Dekker, Solid State Physics (Prentice Hall, Englewood Cliffs, NJ, 1962). 8. See, for instance, J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991).

Chapter 3.10

Discontinuity and Quasitricritical Behavior near Tc in Ferroelectric Triglycine Selenate* Tomas Iglesias, Beatriz Noheda, Gines Lifante, and Julio A. Gonzalo Departamento de Fisica de Materiales, C-IV Universidad Autonoma de Madrid, 28049 Madrid, Spain

Marcel Koralewski Institute

of Physics, Adam Mickievichz 60-780 Poznan, Poland

University,

Precise dielectric constant and hysteresis loops measurements at very low heating and cooling rates (<~0.5°C/h) near Tc « 22.0CC in single crystals of triglycine selenate reveal previously unobserved discontinuities and a very small thermal hysteresis (AT « 0.05°C), indicative of a first order transition very close to a tricritical point. The set of tricritical exponents / 3 « i , J « 5 , 7 « 1 , borne out by present data, and a « | (from Ema's specific heat data) describe very well the observed behavior in the vicinity of the phase transition.

Triglycine selenate 1 [ ( N H 2 C H 2 O O O H ) 3 H 2 S e 0 4 ] , hereafter TGSe, is a wellknown ferroelectric material pertaining to the triglycine sulfate (TGS) family t h a t undergoes a typical order-disorder transition at about T c s=s 22° C; the space group of the lower-temperature phase being P2\ and t h a t of the higher-temperature phase P2\/m. T h e main features of the phase transition in T G S e are like those of other members of the same ferroelectric family and, *Work previously published in Phys. Rev. B 50, 10307 (1994). Copyright © 1994. The American Physical Society. 255

256

Effective Field Approach to Phase

Transitions

up to the present, it has been generally considered as a second order, 2 or continuous, phase transition. Some prominent characteristics of the TGSe transition make it worth further scrutiny. Its specific heat peak 3,4 is about six times larger than the specific heat peak of TGS, which may be taken as an indication that the transition is close to first order, if not first order. Also, in contrast to TGS, a moderate amount of deuteration 5 induces a clear discontinuity at the transition, accompanied by sizable thermal hysteresis. The fact that both deuteration and hydrostatic pressure are known to influence the character of the transition has led Okada and Suzuki6 to investigate a possible tetracritical point in this crystal, which, according to them, could be attained at p « 2.3kbar and x = 0.38 (deuterium concentration). In this work, we report precise dielectric constant and hysteresis loop measurements, performed at very low heating and cooling rates (~0.5°C/h) in the vicinity of the phase transition, from which we conclude that the phase transition of TGSe is discontinuous (first order) at ambient pressure and zero deuteration, with a small thermal hysteresis A T « 0.05°C. In addition, we are able to show that tricritical exponents, (3 « | , 8 « 5, and 7 « 1, obtained from our data, together with the specific heat exponent a « | , from Ema's data, 3 characterize the order-parameter behavior of the system in a certain range of temperature below Tc. The samples were parallelepipeds with surface area (perpendicular to the fr-axis) ranging from 6 to 16 mm 2 , and thickness (along the fe-axis) ranging from 2 to 6 mm, cut from a good optical quality TGSe single crystal grown from aqueous solution. Gold leaf electrodes were attached directly to the surfaces perpendicular to the ferroelectric 6-axis. The sample holder was well sealed within a thick-walled copper container and was immersed in a 15-1 temperature-controlled oil bath regulated by a Hake model F-3, which operated in both the slow-heating and the slow-cooling modes at a rate of about 0.5°C/h. The temperature was measured by means of a chromel-alumel thermocouple with a Keithley digital microvoltmeter model 196 system DMM. Capacitance (C) and loss factor (D) were measured at frequencies between 1kHz and 1MHz by a Hewlett-Packard precision LCR meter model HP-4284-A with very good accuracy and stability. From these data both the real and the imaginary parts of the dielectric constant could be obtained. As discussed below, the very high loss factor (of the order of unity) observable at the peak, just below the capacitance maximum, should be interpreted in terms of energy dissipation within the very low coercive-field hysteresis loops at T near and below T c , driven by the small (~2 V/cm) radiofrequency signal (in fact the high-£> values disappear

Discontinuity

and Quasitricritical

Behavior Near T c

257

at T > T c ), rather than in terms of intrinsic Debye relaxation behavior, which would take place only at much higher frequencies (~GHz). Hysteresis loops were observed by means of a simple Sawyer-Tower circuit with phase compensation at a frequency of 10 Hz in a Nicolet model NIC-310 digital oscilloscope with high resolution (4000 points per loop). For reading the polarization vs. field data from the loops it was important to determine precisely the true center, and this was done both by displaying the digital information on the monitor of a desktop IBM PC computer and by recording the trace of the loop with a precision Hewlett-Packard plotter. Capacitance and hysteresis loop measurements could be taken instantaneously every second, and the resolution in temperature could be as high as 10~ 4o C from point to point. Some of the capacitance and loss data were taken with this resolution, but no such high resolution is needed for hysteresis loop data and, on the other hand, managing 4000 points per loop per second would become impractical. It may be noted that using relatively thick samples (~6 mm along the fr-axis) improves the reproducibility of the data. Figure 3.10.1 shows the dielectric constant (e) and loss factor (D = s"/e') for a relatively thick sample (4.5 mm) both during heating and cooling the sample slowly at about 0.5°C/h. It may be noticed that at temperatures above the transition the heating and cooling data points fall on top of one another. A clear discontinuity in the heating curve of e(T) at T ss 22.16°C corresponding to the peak value can be observed, accompanied by a small thermal hysteresis of about AT « 0.045°C. At T sa 21.16°C, 1°C below the peak, another less pronounced discontinuity shows up in the heating curve accompanied by a thermal hysteresis of about AT « 0.10°C. Discontinuities appear also in the curves of D(T) at the same two temperatures, but the peak value of D(T) is located at the lower one, which is displaced by about 1°C with respect to the peak value of e(T). D(T) drops discontinuously to a near-zero value just above the peak value of s(T). The peak values of D are about 0.5 and 0.7 for heating and cooling, respectively. Since these high values can be expected 7 for normal Debye behavior only near the relaxation frequency (w ~ 2ir x 1 0 9 s - 1 ) , where Anax ~ wr ~ 1, provided that e(w = 0) » E(LO = co), we may conclude that they are due to energy dissipation within the low-coercive-field hysteresis loops driven by the low 1-kHz amplitude of the bridge oscillator at temperatures below and very close to the transition temperature. If this is the case, one would expect D to be proportional (for Ec < Eo, the driving field amplitude) to the area enclosed by the loops, roughly proportional to the spontaneous polarization after reaching the maximum, and near zero

258

Effective Field Approach to Phase

Transitions

Temperature ( C) Fig. 3.10.1. Dielectric constant (e) and dissipation factor (D) of TGSe as a function of temperature ( / = 1kHz) [heating (•) and cooling (o)].

after the transition to the paraelectric phase is accomplished. The fact that D drops discontinuously to zero at the transition may be taken as an indication that the spontaneous polarization also drops discontinuously to zero at the transition. Subsequent measurements of spontaneous polarization vs. temperature through the transition confirmed this interpretation and showed that the ratio of T>max at AT = 1°C below the transition to D just below transition is roughly the same as the ratio of P S 2 (AT = 1) to P S 2 (AT = 0) = (P*) 2 - Measurements on good-quality single crystals of triglycine sulfate under identical conditions showed no noticeable discontinuity in a large peak with e m a x = e(T « Tc) « 2.8 x 105.

Discontinuity

14

and Quasitricritical

16

18

20

•T

o

22

259

Behavior Near Tc

24

26

28

30

,

o

2.5

-

O o

2.0

" o

\

1.5

CJ

4.

1.0

\

•• •

0.5

00

t

•

1

Temperature (°C) Fig. 3.10.2. Inverse dielectric constant ( e - 1 ) and spontaneous polarization (P s ) of TGSe as a function of temperature [heating (•) and cooling (o)].

Figure 3.10.2 gives the inverse of the dielectric constant £ _ 1 (T), and the spontaneous polarization PS(T), measured at 10 Hz as a function of temperature in the vicinity of the transition. It can be seen that the spontaneous polarization loops suddenly to almost zero at about the same temperature as the transition discontinuity in e _ 1 (T). Furthermore, the ratio of the slopes of £ - 1 ( T ) vs. T below and above the transition [(1/C_)/(1/C+) « 4.5] is close to the value of 4 corresponding to tricritical point behavior and considerably higher than the value of 2 pertaining to a standard critical point. The behavior of the spontaneous polarization PS(T), obtained from hysteresis loops under slow cooling, indicates a discontinuous drop at

260

Effective Field Approach to Phase

Transitions

about the same temperature as the discontinuity in the dielectric constant. It may be noted that due, at least, in part, to imperfect phase compensation of the hysteresis loops some amount of rounding at the transition is almost unavoidable. The estimated discontinuous jump in spontaneous polarization from these data is P* ss 0.90/xC/cm 2 . The saturation spontaneous polarization Ps0 = Nfj, for TGSe has not been measured directly, as far as we know, but a fair estimate can be obtained taking into account that P s 0 « 4.20/xC/cm 2 for TGS, 8 and that the ratios N{TGSe)/N(TGS)

= NSe/Ns

« 1.56 x 10 22 /1.5 x 10 22 ,

and M(TGSe)MTGS) =

[(NSeCs)/(NsCSe)}1/2,

where Cs e = 4050 K and Cs = 3650 K are the respective Curie constants, C = 4nfi2/kB- Using these data 2 we get Pso ~ 4.50/xC/cm 2 , slightly larger than the value Pso ~ 4.14/zC/cm 2 used in Ref. 4. Then we can get an estimate of the maximum thermal hysteresis AT m a x to be expected in terms of p* = P*/Ps0 « 0.20 from9 T*/Tc = 1 + AT m a x /T c =[1 - ( p : ) 4 / 5 ] - x = 1.00032,

(1)

where T* = 295.16 K = 22.03°C is the (first order) transition temperature and Tc = 295.06 K = 21.93°C the Curie temperature. The resultant value for ATmax = 0.10°C is compatible with the observed thermal hysteresis AT = 0.045°C. The small discontinuities for TGSe in e(T) and PS(T) at the transition and the barely observable thermal hysteresis just mentioned suggest that quasitricritical behavior may be expected just below the transition temperature in this crystal. In this case the tricritical exponents for the spontaneous polarization, the critical isotherm, the inverse dielectric constant, and the specific heat would be given by psoc(AT)", e-1«(AT)T>

eocp5,

/3=];

7 = 1;

4

ACpocAT~a,

S = 5, 1 a = -,

(2)

instead of by (3 = 5, S = 3, 7 = 1, a = 0, which are the usual critical exponents for a first order transition. These results can easily be obtained from the equation of state relating field (e), polarization (p), and temperature (T/Tc), in dimensionless units, e = (T/Tc - \)p + bp3 + cp5L

(T
p < 1),

(3)

Discontinuity

and Quasitricritical

Behavior Near T c

261

10*

Critical exponents for TGSe 0=Ln(Ps)/Ln(AT) a = Ln(AC p )/Ln(AT) 10'

E o

\ o

a. en Q_

10°

P|=Ps(T*)

/

T
i
10"' AT=TC-T

io°

io'

(°C)

Fig. 3.10.3. Log-log plots of spontaneous polarization (this work) [heating (•) and cooling (o)] and ferroelectric specific heat [Ema's data (Ref. 3)], as a function of A T = Tc—T (Ema's T c corresponds to our T*). Numerical values for the pertinent exponents /3 and a are given. Spontaneous polarization data for T > Tc but T < T* (corresponding to A T < 0) do not appear in this plot.

for the cases 6 = 0 (or b 0 (b » cp2) (normal critical point). Figure 3.10.3 depicts, in log-log scales, the temperature dependence of our data of spontaneous polarization for TGSe with Tc = 21.93°C. The value obtained for the exponent (3 — 0.26 ± 0.02 is very close to the tricritical value (3 = j . In the same graph the specific-heat data for TGSe of Ema 3 are plotted directly in a log-log scale. The ferroelectric contribution to the specific heat, after subtracting the background (corresponding to nonferroelectric contributions), given in Fig. 9 of Ema's paper 3 in arbitrary units has been normalized by dividing ACP by the maximum value ACpm, corresponding presumably to A T = T* - T — 0°C (with T* corresponding to Ema's T c ). The value for a = 0.50 ± 0.02, obtained neglecting the first few points with AT < 0.5CC, is in excellent agreement also with the tricritical-point value a = \. Finally, Fig. 3.10.4 shows the field dependence of the polarization at a temperature T = 21.92°C just below Tc < T* = Tc + AT = 22.03°C.

262

Effective Field Approach to Phase

Transitions

Critical e x p o n e n t s for TGSe C\2

1/6=0.23

£ O

±0.03

(T * T c )

o OH P *=P s ('r .

;

)V

• '

.

10

l

100

E(V/cm) Fig. 3.10.4. Log-log plot of polarization ( F ) vs. field (E) at T = 21.92°C, close to the temperature corresponding to the tricritical isotherm, giving the exponent 1/5.

Again, the exponent value 1/5 = 0.23 ± 0.03 is very close to the tricritical value 1/5 = 1/5. The error quoted for 1/5 corresponds to 40 < E < 150V/cm, since deviations toward 1/5 = 0 for E —» 0 are expected for a discontinuous transition, and also toward 1/(5 > 1/5 for E —> Emax due to imperfect phase compensation of the hysteresis loop, originally compensated at T > Tc. 1/5 has been taken, therefore, as the asymptotic value away from E = 0, neglecting rounding effects in the loops. The exponent 7 corresponding to e _ 1 (T) is not shown explicitly but it is clear from the asymptotic linear behavior of e _ 1 above and below Tc that its value is very close to 7 = 1, while possible logarithmic corrections very near Tc are not excluded. As should be expected the observed exponents fulfill very well Widom's equality, /?(<J-1)«7,

(4)

(3(5 + 1) « 2 - a.

(5)

and the Griffiths equality

In summary, the results reported here clearly indicate that the ferroelectric paraelectric transition of TGSe is first order and very close to tricritical. It may be noted that the early literature on other important ferroelectric crystals, like potassium dihydrogen phosphate and BaTiC>3, presented the respective transitions as second order transitions, and only later did more

Discontinuity

and Quasitricritical

Behavior Near Tc

263

precise measurements establish their first order character. Related previous work on the T G S e phase t r a n s i t i o n 1 0 - 1 4 did not substantiate either the discontinuity at T = 20.0°C or a clear quasitricritical behavior. A mean field approach is applicable to the case of T G S e (and other uniaxial ferroelectrics) because long-range dipolar forces suppress the polarization fluctuations.

Acknowledgments We would like to t h a n k A. Czernecka (Poznan) for providing us with some good samples of TGSe, and the C I C y T (Spanish Research Agency) for financial support (Grant no. MAT90-0304).

References 1. See, for example, F. Jona and G. Shirane, Ferroelectric Crystals (Pergamon, New York, 1962); B.T. Matthias, C.E. Miller and J. Remeika, Phys. Rev. 104, 849 (1956). 2. See, for example, Landolt-Bornstein, New Series, Group III, Vol. 16b, Ferroelectrics and Related Substances: Non-oxides (Springer-Verlag, Berlin, 1982). 3. K. Ema, J. Phys. Soc. Jpn. 52, 2798 (1983). 4. B. Noheda, G. Lifante and J.A. Gonzalo, Ferroelectrics Lett. 15, 109 (1993). 5. K. Gesi, J. Phys. Soc. Jpn. 41, 1437 (1976). 6. K. Okada and I. Suzuki, Ferroelectrics 39, 1205 (1981). 7. G. Luther, C. Birkenheier, G. Brosowski, H.E. Miiser and H.E. Petersson, Ferroelectrics 8, 569 (1974). 8. LA. G. Chynoweth, Phys. Rev. 117, 1235 (1960). 9. J.A. Gonzalo, R. Ramirez, G. Lifante and M. Koralewski, Ferroelectrics Lett. 15, 9 (1993). 10. K. Gesi and K. Ozawa, J. Phys. Soc. Jpn. 40, 599 (1976). 11. H. Yamashita, I. Todo and I. Tatsuzaki, J. Phys. Soc. Jpn. 44, 1261 (1978). 12. H. Yamashita, Y. Takeuchi and I. Tatsuzaki, J. Phys. Soc. Jpn. 49, 1894 (1980). 13. Y Takeuchi, H. Yamashita and I. Tatsuzaki, J. Phys. Soc. Jpn. 50, 9 (1981). 14. Y. Takeuchi, H. Yamashita and I. Tatsuzaki, J. Phys. Soc. Jpn. 50, 2022 (1981).

Chapter 3.11

Scaling Equation of State for Ferroelectric Triglycine Selenate at T « Tc* T. Iglesias, B. Noheda, B. Gallego, J.R. Fernandez del Castillo G. Lifante and J.A. Gonzalo Departamento Universidad Autonoma

de Fisica, de Materiales, de Madrid, 28049 Madrid,

Spain

Digital data of polarization vs. field on triglycine selenate at closely spaced temperature intervals (AT ss 0.015) in the vicinity of the quasitricritical point of triglycine selenate have been collected. These data fulfill very well the scaling equation of state e+ = rnp + (l/5)p 5 (where e_ and e+ correspond to T < Tc and T > Tc, respectively), where p = ( P s / P s 0 ) / { 1 - T/Tcy'A and e = {E/Ea0)/{1 - T/Tc}5'* are the scaling variables, with tricritical exponents /3 = 1/4, <5 = 5.

Scaling theory, as is well known, was first introduced by W i d o m 1 to describe the behavior of simple fluids near critical points, and later extended t o describe the scaling equation of s t a t e of fluids2 and other systems including ferromagnetic, order-disorder alloys, ferroelectrics, etc. Not many investigations 3 ' 4 of the scaling equation of s t a t e in ferroelectric systems have been reported, in p a r t due to the fact t h a t truly continuous second order transitions in ferroelectrics are not very frequent. No experimental investigation of the scaling equation of s t a t e of a ferroelectric near a tricritical point 5 has been previously reported, as far as we know. In this chapter, we report digital hysteresis loop d a t a in the vicinity of the ferro-paraelectric transition of triglycine selenate (TGSe), which happens to be very close 6 to a tricritical point. T h e phase transition, *Work previously published in Europhysics E D P Sciences.

Letters 28, 91 (1994). Copyright © 1994.

265

266

Effective Field Approach to Phase

Transitions

occurring for undeuterated TGSe at about 22°C, shows a minute discontinuity in the dielectric constant accompanied by a small thermal hysteresis of about 0.05°C, which indicates the proximity to the tricritical point. Tricritical exponents /3 = log A P s /log AT « 1/4, 6 = (log AE/ log A P ) T c « 5, a = logACp/logAT « 1/2, are observed, which differ from the classical critical exponents (5 = 1/2, 5 = 3, a = 0 reported for the transition in isomorphous triglycine sulfate (TGS). 3 The main objective of the present work was to check whether the corresponding scaling equation of state (with tricritical point exponents) describes satisfactorily the observed behavior, using digital hysteresis loop data, which are expected to define the transition much better than manually obtained hysteresis loop data. The samples were gold-electroded plates cut with main faces perpendicular to the ferroelectric 6-axis from a large-optical-quality TGSe single crystal grown from water solution. For the particular crystal for which data are shown below the area was 10.5 mm 2 and the thickness 4.7 mm. The sample holder was well sealed within a thick-walled copper container immersed in a controlled-temperature oil bath regulated by a Haake, model F-3, using slow heating and cooling rates (~0.5°C/h). The temperature was measured by means of a chromel-alumel thermocouple with a Keithley digital microvoltimeter, model 196. Hysteresis loops in the temperature interval 294.5 < T < 296.3 K were observed and recorded by means of a Nicolet, model NIC-310, digital oscilloscope connecting the sample to a Sawyer-Tower circuit with phase compensation. For each loop, at every temperature, 4000 points were recorded, thus ensuring high resolution in the definition of the P vs. E curve. The frequency of the driving field was 10 Hz and the amplitude 425 V/cm. The true center of the loops was determined precisely by means of a simple computer program, 7 which ensured symmetric ±PS and symmetric ±EC values simultaneously, Ps and Ec being the spontaneous polarization and the coercive field, respectively. Using a generalized effective field8 £ eff = E + (3P + 7 P 3 + SP5L,

(1)

where E is the external field and P the polarization, in which successively higher order terms in the polarization can be attributed to dipolar {(3P), quadrupolar (7P 3 ), octopolar {SP5) etc., contributions, the equation of state for a uniaxial ferroelectric can be written as F1 T ^ - = -tanh-1

+

4

P — p x 4

.^^te 1

!™ ( Nn " £,'• L

(2)

Scaling Equation of State for Ferroelectric Triglycine Selenate

267

where PNfi = Eso, is the saturation spontaneous field and N/J, = Pso the saturation spontaneous polarization, N being the number of elementary dipoles per unit volume and fi the elementary dipole moment. In terms of reduced variables e = E/Es0, p = P/Pso, this equation becomes e = — tanh" 1 p - [1 + gp2 + hp4L}p,

(3)

where g^j(Nn)2,

h=^(N^L.

(4)

Expanding the inverse hyperbolic tangent in series for p
T.-1h(m-')'p+(iT.-h)'L-

(5)

which for g « 1/3, fc
P {T/Tc - 1}V4

1 (T 5 \T<

P {T/Tc - l}i/4

T5

(6)

or em«mp+(l/5)p5,

(7)

where e m = e/{l - T/Tc}5/4 and mp = mp/{l - T/Tcy/4 correspond, respectively, to T < Tc (ferroelectric phase) and T > Tc (paraelectric phase). It will be shown below that Eq. (7), which is the tricritical point (TCP) scaling equation of state for uniaxial ferroelectrics, describes quite well the behavior of TGSe in the vicinity of T = TTCP • Figure 3.11.1 shows a plot of the fourth power of the spontaneous polarization (P s in yuC/cm2) as a function of temperature in the vicinity of the transition at closely spaced temperature intervals. Imperfect phase compensation of the hysteresis loops, which is unavoidable, because the conductivity of the crystal is temperature dependent at T « Tc, gives rise to a small residual contribution at temperatures just below and just above T c . Thus, points with P < PTes (residual polarization) should not be considered to describe the true scaling behavior. These data allow one to make a first estimate of Tc = 295.818K and support the TCP exponent value (3 = 1/4 for TGSe. The small discontinuity expected for Pa(T) at T = T* (first order transition temperature) is rounded off in this representation. Figure 3.11.2 gives a plot of the fifth power of the polarization (also in /LiC/cm2) as a function of field (in V/cm) for various temperatures in

268

Effective Field Approach to Phase

Transitions

294.2 294.6 295.0 295.4 295.8

296.2

Fig. 3.11.1. Spontaneous polarization up to the fourth power for TGSe in the vicinity of Tc ss 295.810 K. Due to unavoidable imperfect phase compensation of the hysteresis loops, a small residual contribution at T close to T c can be observed.

0.20

: •

0.16

-

1

z

-—>

:

u O

:

PH

0.08 r*

'a o-ia

•>•« . . ...

.^

j * .

%o°

»,

I 0.04 r s* ,>»

^^-Yi>•

sK'BJilnT'TTTii 11

0.00

100

hi

200 •ECV/cm)

> 11111 11 u . L i

300

400

Fig. 3.11.2. Polarization up to the fifth power vs. field at T = 295.749, 295.786, 295.810, 295.830, 295.848, 295.872 K in the vicinity of Tc m 295.810 K. A small residual contribution, corresponding to that shown in Fig. 3.11.1, is present at T close to T c .

Scaling Equation of State for Ferroelectric THglycine Selenate

269

close vicinity of T c . The small residual contribution at T « Tc mentioned before is also apparent, causing finite intercepts at E = 0 for the straight lines approximating the relationship between P 5 and E. These data confirm indirectly the previous estimate of Tc and support the TCP exponent value 8 » 5 for TGSe. The scaled digital data from the full set of P vs. E curves in close vicinity of Tc (below as well as above the phase transition) are given in Fig. 3.11.3. It can be seen that these data fulfill quite well the behavior predicted by Eq. (7) giving an excellent fit to both branches of the scaling equation of state. As mentioned above, points with small P < Pres (residual polarization) are expected to depart from the scaling behavior, and have been omitted from the plot. It is clear that data for about 50 closely spaced isotherms (Tn+i -Tn < 0.03°C) from T = 294.49 to 296.31 K collapse fairly well on top of each other in spite of the large variation in absolute polarization value, more than two orders in magnitude. The T > Tc branch shows systematic departures toward the low-field side for each isotherm from the theoretical curve due to incompletely suppressed residual polarization, but the envelope of sets of points belonging to different isotherms defines quite

1.80 Inp 0.80

-0.20

-1.20

-2.20 -8.80

-4.80

-0.80 In e

3.20

7.20

Fig. 3.11.3. Scaling equation of state for TGSe at T « T c , /3 = 1/4, 6 = 5. Experimental points for T < Tc and T > T c are obtained from digital hysteresis loops in the interval 294.49 < T < 296.31 K. The continuous line is the tricritical scaling equation of state e m = rap + ( 1 / 5 ) J 3 5 , with - for T < Tc and + for T > Tc. Here, p = ( P s / P s 0 ) / { 1 T/Tc}1/4 and e = (E/Es0)/{1-T/Tc}5/4 are the scaled field and the scaled polarization, respectively. Points with E < 1/2 E m a x were used.

270

Effective Field Approach to Phase

Transitions

well the general behavior in agreement with the theoretical curve. In order to optimize the fit of Eq. (7) to the data, computer fits with Tc changing by very small amounts (0.002 K) were made within certain intervals of the original estimate of T T from the d a t a in Fig. 3.11.1. T h e best fit was obtained for Tc = 295.810K, with P s 0 = 5.60/xC/cm 2 and Es0 = 40.5 V / c m . T h e set of d a t a points with the highest p and e values correspond t o an isotherm very close and just above the critical isotherm. T h e T < Tc branch shows very good collapsing of d a t a toward low values (describing the behavior of the spontaneous polarization away from T c ) but shows a distinct, albeit small, systematic deviation from t h e tricritical point equation of s t a t e at higher e values, probably related to the slightly discontinuous character of the transition in TGSe, which is already of first order, 6 as indicated more clearly by dielectric constant data. It may be pointed out t h a t Eq. (7) properly speaking does not contain adjustable parameters. T h e value of Tc is refined around a fixed transition t e m p e r a t u r e given by various experimental observations. T h e value of P s o, corresponding to the expected low-temperature spontaneous polarization value, is also slightly refined from t h a t previously estimated, 6 combining dielectric d a t a for T G S e and T G S . And, finally, t h e value of Eso = (3Pso, where (3 = A-KTC/C is the dipolar mean field coefficient (not to be confused with the spontaneous polarization critical exponent /3, which is j3 ~ 1/4 for T C P behavior), is also fixed in terms of the transition temperature, t h e Curie constant, and the low-temperature spontaneous polarization.

Acknowledgments We acknowledge the financial support of CICyT, grant PB93-1253, and the Comunidad de Madrid, grant AE00138/94, for this work.

References 1. B. Widom, J. Chem. Phys. 3, 3898 (1965). 2. M.S. Green, M. Vicentini-Missoni and J.M.H. Levelt Sengers, Phys. Rev. Lett. 18, 1113 (1967). 3. J.A. Gonzalo, Phys. Rev. Lett. 2 1 , 749 (1968); Phys. Rev. B 1, 31225 (1970). 4. R. Ramirez, C. Prieto, J.L. Martinez and J.A. Gonzalo, Phase Transitions 9, 253 (1987). 5. For an in-depth theoretical discussion, see, e.g., J.D. Lawrie and S. Sarbach, Theory of Tricritical Points, in Phase Transitions and Critical Phenomena,

Scaling Equation of State for Ferroelectric Triglycine Selenate

TJX

edited by C. Domb and J.L. Lebowitz, Vol. 9 (Academic Press, New York, 1984). 6. T. Iglesias, B. Noheda, G. Lifante and J.A. Gonzalo, Phys. Rev. B. (to be published). 7. B. Gallego, J.R. Fernandez del Castillo, T. Iglesias, B. Noheda, G. Lifante and J.A. Gonzalo, ISMPC-Physics and Computers (to be published). 8. R. Ramirez, M. Fe Lapen and J.A. Gonzalo, Ferroelectrics 124, 1 (1991).

Chapter 3.12

Composition Dependence of the Ferroelectric-Paraelectric Transition in the Mixed System PbZri^Ti^Oa* Beatriz Noheda, Noe Cereceda, Tomas Iglesias, Gines Lifante and Julio A. Gonzalo Depariamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Hui Ting Chen and Yong Ling Wang Shanghai

Institute of Ceramics, Chinese Academy of Sciences, 1295 Ding Xi Road, Shanghai 200050, China

The dielectric constant and losses of good-quality ceramic samples of Nb-doped ( l w t % N b 2 0 5 ) PbZn-^TixOs about the ferroelectricparaelectric transition for 0.03 < x < 0.09 have been measured with the aim of investigating the behavior of the thermal hysteresis with composition and the possible existence of tricritical points in this mixed system. The trend of the data, analyzed in terms of a generalized effective field theory (EeB = E + (3P + 7 P 3 H ), indicates that two tricritical points, at x = 0.26 and x = 0.51, are implied by the observed behavior at lower x.

T h e mixed system P b ( Z r i _ a , T i a ; ) 0 3 , abbreviated P Z T , in which ferroelectric and antiferroelectric transitions were reported by Shirane and Suzuki, 1 ' 2 has been widely investigated because of its outstanding piezoelectric and pyroelectric properties. While the basic features of the ferroelectric phase diagram have been well known for a long time, little attention has been paid, as far as we know, t o the composition dependence of the character *Work previously published in Phys. Rev. B 51, 22 (1995). Copyright © 1995. The American Physical Society. 273

274

Effective Field Approach to Phase

Transitions

(first or second order) of the ferroelectric-paraelectric transition. The character of the transition, and the possible existence of one or more tricritical points separating composition regions with discontinuous first order transitions and continuous second order transitions can be investigated, in good-quality ceramic samples of varying composition, by means of simple dielectric constant and dielectric loss measurements. A decreasing trend with changing composition in the transition thermal hysteresis, or better in the difference between transition temperature (T*) and extrapolated Curie temperature (T c ), is a clear indication that the system is approaching a tricritical point. We report dielectric measurements in Nb-doped (lwt% Nb2C>5) PZT ceramics for 0.03 < x < 0.09, prepared and sintered at 1340°C for 1.5 h at the laboratories of the Shanghai Institute of Ceramics, and for x = 0.22, kindly provided by Dr. W. Wersing (Siemens AG). A straightforward analysis of the data using a generalized effective field approach, in which the standard molecular field Eeg = E + (3P is generalized3 to include higher order terms, i.e., Eeg = E + f3P + 7 P 3 + • • •, leads to estimates of tricritical compositions at x = 0.26 and 0.51, between which continuous (second order) ferroelectric-paraelectric transitions are expected, the transitions 4 ' 5 corresponding to both ends, x = 0 and 1, of the phase diagram being clearly discontinuous (first order). Interest in the ferroelectric properties of mixed lead zirconate/titanate ceramics has been boosted recently by the prospects of using PZT thin films in nonvolatile memories.6 The samples were thin disks (thickness ~ 1 mm, diameter ~ 1 cm) cut from sintered and hot-pressed cylindrical samples with compositions x = 0.03, 0.06, 0.08, and 0.09, doped with 1 wt% N b 2 0 5 to improve the physical properties of the ceramics. A high-temperature furnace built at the Crystal Growth Laboratory of the UAM, Madrid, controlled by a Eurotherm Temperature Controller/Programmer with high-resolution ramp (model 903P/IS/HDV), was used. The samples were thermally treated by keeping them for several hours at temperatures of about 400° C before the measurements, and the slow heating and cooling rates employed were of the order of 20°C/h between room temperature and 400°C. The capacitance and dissipation factor were measured during the heating and cooling runs at regular intervals separated by about 0.1°C by means of an automatic Hewlett-Packard Bridge Precision LCR Meter (model 4284A) with an accuracy better than one part in 104 at a frequency of 1 kHz with a field amplitude of 8.3 V/cm. Figure 3.12.1 depicts the inverse dielectric constant e"x vs. temperature for 0.03 < x < 0.09. The slope of e" 1 vs. T above the transition

Ferroelectric-Paraelectric

••'•••

1

'

Transition in the Mixed System PbZr\^xTixOz

1

. • • -

\

jJ I .

•

I

I

I

I-

1

j 1

-

-

• . . . L ,...

225 Tl-C!

1

/ j-

1

/ -

•

\ I

•

/

/

1

i 1 1

•

: 1

j

j

1

/

• •

V

•

•

250 TCO

/-

i / . \ / ' / -

/

V

225

'j,

j

j •

j /

.J

• . • -

i

• \ ;

•

-

\

.

i

-

•

1

•

•

2S0

1

•

\

!

'

•

•

-1 i 225 T(f)

— i —

,»-o.f»!

\

i

•

, 1, - i — 2S0

r-

1

t

J

•

• -

j ! \ < \

•

-

-" • :

•

I

•

i

'

}k>M \

j

275

-• '

-~i 256

275

ICC)

Fig. 3.12.1. Inverse dielectric constant e~x vs. temperature near the ferroelectricparaelectric transition for Pb(Zri_ x Tia;)03 doped with Nb ( l w t % N b 2 0 s ) for various compositions in the range 0.03 < x < 0.09.

temperature T* is not equally well defined for all four compositions. This leads, in some cases, to deviations from the ideal Curie-law behavior due to differences in the compositional inhomogeneities, which causes uncertainties in the Curie temperature Tc extrapolated at e _ 1 —> 0. In all cases the slope was determined at an interval of about 20°C above T*, taking into account numerically the derivative d e _ 1 / d T and avoiding the rounded region usually observed immediately above T*, which increases for increasing x. Figure 3.12.2 shows dielectric loss data D = tan 5 vs. temperature for the same samples. It may be noted that, for x = 0.03, no clearly defined peaks are observed, but well-defined discontinuous jumps mark the phase transition. The transition temperature T* for increasing T is therefore well defined in every case by the dielectric loss data as well as by the dielectric constant data. Table 3.12.1 gives results for T*, T c , and AT = T* - Tc for PZT compositions with 0.03 < x < 0.09 and lwt% Nb20s as well as 4 ' 5 for P b Z r 0 3 (x = 0),PbTiO 3 (x = 1), and x = 0.22, for later use. It can be seen that the trend of AT is decreasing with increasing x from x = 0.03 to 0.22, as an estimate of AT has been obtained from the difference between ^X-Dmax) for increasing and decreasing temperature. These results can be analyzed by means of a generalized effective field theory in which the standard expression for the effective field3 in terms of

276

Effective Field Approach to Phase

Transitions

r — j — I — »=0.03

n-0.06

-

\

0,10

-•• •• -

•

•

/

0.08

'

/'

'

1

0.04

0.02

•

•

*

J

•

•

t

V

•

J~

...... In,

250

22S

•

•

*

- •y •

•

225 T(»C)

Fig. 3.12.2. Fig. 3.12.1.

;

\=0.0<>

•

•

1

i

;/

i — i —

•• -

•

•

i—

i

0.06

•

'•'"

j-O.OS

•

V__:

•

-

1\ '

J \

; 1——'

\

v_x

.....»_..-J__

2SO

225

250

250

275

•rrc>

Dielectric loss factor D vs. temperature for the same samples as in

Table 3.12.1. Transition temperatures and Curie temperatures for P b ( Z r i _ : c T i : r ) 0 3 as a function of x. x 0 0.03 0.06 0.08 0.09 0.22 1.00

Nb (%)

* 1 1 1 1

* *

T* (K)

T c (K)

503 502.2 505.5 517.4 531.7 571.7 763

460 486.9 496.5 510.0 524.8

A T (K)

720

43.0 15.3 9.0 7.4 6.9 [1.7] 43

* = negligible. A T estimated from loss peaks.

the polarization, EeS = E + 0P,

(1)

where (3 = 4TTTC/C is a dimensionless coefficient depending on the geometry of the dipole lattice, is substituted by 7 EeS = E + (3P + 7 P 3 + L.

(2)

With this substitution the equation of state relating P, E, and T,

Nn

tanh

kBT

(3)

Ferroelectric-Paraelectric

Transition in the Mixed System PbZri-xTix03

277

where N is the number of dipoles per unit volume, JJL the elementary dipole moment, and KB Boltzmann's constant, leads directly to / P \'

l + WMNny^—j

TTTT— — \ 7^r \ t a n h

N/MJ

L

(4) here g = (•j/(3)(Nfj,)2 is a second dimensionless coefficient that plays a decisive role in determining the character of the transition, g being 1/3 for a tricritical point (TCP) transition. For a pure system the Curie temperature is given by pNfi2 Tc = (5) KB

For a mixed system such as PZT, the composition-dependent Curie temperature Tc(x) can be given in a first approximation 8 in terms of the Curie temperatures of the components, (Tc)z -= '-/c(O) = TC{1)

(TC)T

PTNT^T

=

-

(6)

KB

(7)

KB

Tc(x) ~-= Tc(0)

PTNT^

( ! - * ) + ( .PzNztfz (8)

where the ratio pTNTfi2T\ 2

JzNzfi zJ

_TC(1)

Tc(0)'

In our case, we will assume that this ratio is not too sensitive to small amounts of doping, like 1 wt% N ^ O s . Similarly, the dimensionless coefficient g, which determines the phase transition character, and is given for a pure compound by

g = (1/(3)(Ntf

(9)

can be given for a mixed system in terms of the coefficients g of the components, (g)z = 5(0) = hz/Pz)(Nzvz)2, G?)T = 5(1) =

(1T/PT)(NTIJ-T)2,

(10) (11)

278

Effective Field Approach to Phase

Transitions

as 9{x) = 9(0)

5(1)

[

]

+

\IZ/(3Z){NZ»Z

(1T/PT\

(NTHT\2

W / W

\Nznz)

2

[

X)

(12)

where the product of ratios 1T/PT

_lz/Pz_

£(1) 5(0)'

.Nzuz.

and we again expect that this product is not affected substantially by small amounts of doping with Nb. As shown in previous work,9 within the framework of a generalized effective field approach, the dimensionless quantities T*/Tc = 1 + AT/T C , g, and p* = P s (T*)/P s (0) are related among themselves by the relationships 3 tanh i„*p, Ps

T * t a n h _ 1ip^ ;*

(l-fe*)2)"1

(13)

1

(14)

(Ps*)2-

Numerical values for T*/Tc and g as a function of p* are given in Table 3.12.2. Figure 3.12.3 gives AT/TC as a function of g in an extended range obtained by means of Eqs. (13) and (14). For g < 1/3, i.e., below the TCP the transition is second order and no thermal hysteresis occurs under proper Table 3.12.2. Numerical values for {T*/Tc) and g as a function of p* = P s ( T * ) / P s ( 0 ) . Ps

(T'/Tc), Eq. (13)

g, Eq. (14)

0 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

1 1.0036 1.0066 1.0114 1.0191 1.0314 1.0508 1.0803 1.1387

1/3 0.3910 0.4131 0.4415 0.4783 0.5276 0.5944 0.6829 0.8387

Ferroelectric-Paraelectric 0.10

Transition in the Mixed System PbZr\-xTixOz

•-J—T

I

I

I

I

1--1

•

r-—•

•

279

J

0.W

0.06

0.04

0.02

0.00 0.00

0.20

0.40 g

0.W

O.SO

Fig. 3.12.3. Plot of (AT/Tc), where A T = T* (transition temperature) - T c (Curie temperature), vs. g = (7//3)(iV^) 2 (higher order effective field coefficient) after Eqs. (13) and (14).

thermal equilibrium conditions. For g > 0.4, AT/T C increases fairly rapidly. This relationship between AT/TC and g will be used later to calculate AT for comparison with (AT%bs as a function of x. To analyze our PZT results we begin by fitting Eq. (8) to the data in Table 3.12.1 for 0.03 < x < 0.09. We get Tc(x) = TC(0)[(1 - x) + Ax],

A = 1.67, Tc(0) = 476.9K.

(15)

This results in Tc(0) and T c (l) = Tc(0)A values that can be compared with the Curie temperatures for the pure materials (no Nb) reported by other authors. The ratio 4 - 5 (TC)T/(TC)Z ^ 1.56 is not far from A = 1.67 as anticipated. Next, we determine the relevant numerical parameters for Eq. (12). A reasonable fit is obtained for g(x) = s (0)[(l - xf + Bx2},

5 = 1.46, g{0) = 0.534.

(16)

Here, B and g(0) have been estimated from the set of numerical values of g(x) corresponding to 0.03 < x < 0.09, using the AT/T C values given in Table 3.12.1 and the graph in Fig. 3.12.3. It may be noted that g(l) = g{0)B = 0.779 so obtained is larger but not 7 far from (g)T = 0.62 at T = T*, independently determined for the pure titanate according to

280

Effective Field Approach to Phase

Transitions

Table 3.12.1. On the other hand, g(0) = 0.534 is somewhat smaller than (g)z = 0.72 at T = T* for the pure zirconate after Table 3.12.1. The possibility that Nb doping slightly pushes some of the factors determining g for the titanate and the zirconate in opposite directions is not to be expected, a priori, but is not inconceivable. Figure 3.12.4 shows the calculated curve for AT using Eq. (15) for Tc(x) and Eq. (16) for g{x) and the functional relationship between AT/TC and g plotted in Fig. 3.12.3, together with (AT) o b s for x = 0.03, 0.06, 0.08, and 0.09 (lwt% Nb20 5 doped), and for x = 0.22. A first tricritical point at ( £ T C P ) I = 0.26 is indicated by the calculated curve, and a second one at ( ^ T P T ) 2 — 0.51 (not shown in the graph) is also deduced from the same equations for higher x values. Thus, we may conclude that a mixed ferroelectric system like PZT can accommodate more than one tricritical point in a wide enough range of compositions. Recent theoretical work10 on ferroelectric solid solutions suggests

16

14

12

10

1 • 4

: _i

o 0.00

0.05

0.10

0.15 X

i

i_

0.20

0.25

0.30

Fig. 3.12.4. AT = T* -T vs. x for P b ( Z r i _ a : T i a ; ) 0 3 . The curve is calculated from Eqs. (15) and (16) using the plot of (AT/T C ) vs. g from Fig. 3.12.3, the solid circles are experimental points for 0.03 < x < 0.09 (doped with 1 wt% Nb 2 Os) and 0 is for x = 0.22 (estimated from the shift in D m a x for heating and cooling runs). It is worth noticing that the experimental point for x = 0.03 appears to be further from the theoretical value than the rest. This may be due to the fact that Tc(x) has a slightly different behavior in the low-x region 0 < x < 0.03 (Ref. 8).

Ferroelectric-Paraelectric

Transition

in the Mixed System PbZr\^xTixOz

281

t h a t a detailed investigation of the lattice deformation through the transition may be useful to check the phase transition character of P Z T solid solutions at x near the expected trictrical point compositions.

Acknowledgments We would like to t h a n k Gen Shirane for many informative discussions on structural aspects of this P Z T system, and Wolfram Wesing for providing us the sample with x = 0.22. T h e financial support of I B E R D R O L A (through Grant no. I N D E S / 9 4 ) and C I C y T (through Grant no. PB93-1253/94) is gratefully acknowledged.

References 1. G. Shirane and K. Suzuki, J. Phys. Soc. Jpn. 7, 333 (1952). 2. E. Swaguchi, J. Phys. Soc. Jpn. 8, 615 (1953). 3. J.A. Gonzalo, Effective Field Approach to Ferroelectric Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). 4. Z. Ujma and J. Handerek, Acta Phys. Pol. A 53, 665 (1978). 5. G.A. Samara, Ferroelectrics 2, 277 (1971). 6. S.L. Swartz and V.E. Wood, Condens. Matter News 1, 5 (1992); J.F. Scott, J. Kammerdiner, M. Parris, S. Traynor, V. Ottenbacher, A. Shawabkeh and W.F. Oliver, J. Appl. Phys. 64, 787 (1988). 7. G. Lifante, J.A. Gonzalo and W. Windsch, Ferroelectrics 146, 107 (1993). 8. B. Jaffe, W.R. Cook and H. Jaffe, Piezoelectric Ceramics (Academic, New York, 1971), p. 136, Fig. 7.1. 9. J.A. Gonzalo, R. Ramirez, G. Lifante and M. Koralewski, Ferroelectrics Lett. 15, 9 (1993). 10. W. Windsch and H. Braeter, Acta Phys. Pol. 86, 393 (1994).

Chapter 3.13

Observations of Two Ferroelectric Response Times in TGSe at T < Tc* J. Prezeslawski*, J.R. Fernandez del Castillo, T. Iglesias and J.A. Gonzalo

^Institute

Departamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain of Experimental Physics, University of Wroclaw,

Poland

G. Shirane Department

of Physics, Brookhaven National Upton, New York 11973, USA

Laboratory,

Time-dependent dielectric constant measurements in ferroelectric triglycine selenate reveal an existence of two characteristic response times with distinct temperature dependences near T c . Possible reasons connected with this behavior are considered.

Accurate time-dependent dielectric constant measurements have been performed in good-quality single crystals of ferroelectric triglycine selenate with the aim of investigating whether or not more t h a n one characteristic time, possibly related t o the double correlation length observed in several magnetic and structural transitions, can be observed. 1 Dielectric permittivity d a t a were collected by means of a H P Precision L C R Meter 4384A connected to a computer (EQ = 1 V/cm, / m e a s = 1 kHz). Samples were very slowly cooled t h r o u g h Tc, a n d t h e n kept for a long period at constant t e m p e r a t u r e .

*Work previously published in Ferroelectrics Taylor & Francis Group. 283

186, 329 (1996). Copyright © 1996.

284

Effective Field Approach to Phase

Transitions

Figure 3.13.1 shows a typical plot of e * as a function of temperature and indicates a pronounced change of e~l with time at the final fixed temperature T = 22.307°C (AT = TC-T = 0.035°C). Figure 3.13.2 presents Aei and temperature as a function of time for data of Fig. 3.13.1.

e " 1 vs. T for TGSe.

Fig. 3.13.1.

1E+5

'

I

i

'

i

i

l

i

'

i

I

' i ' > I ' '

23 TfC)

1E+4

Fig. 3.13.2.

-i

15000

20000

i

i

i

22

i_

25000 thlM(S)

30000

35000

e and temperature vs. time for data of Fig. 3.13.1.

Observations

of Two Ferroelectric Response Times in TGSe

285

Figure 3.13.3 gives Aei = (e — £too) vs. time (same data as above) in semilog scale from which a T\ is determined: Aei = Aeie~*/ T l , with the result n ( A T = 0.035°C) = 14584s. Figure 3.13.4 shows Aeo = (e — £i) vs. time (again the same data) in semilog scale (after subtracting the previous contribution), from which a TQ i i • i i i r | i i i j | i—n

Ae1

\

M "

T i =14684s \

1E+4

i | i i i

AT = 0.035 °C

—

_

\.

-

^

^^^"^

1 1 1 t I

1 1 1 i

12000

Fig. 3.13.3.

1 i

16000

1 1 1 i

20000 time(s)

1 1 \

I

» I

24000

.i.i.fT' 28000

Aei = (e — £too) vs. time data as in Fig. 3.13.1.

•T-T'J'MI

Ae

I T [ 1 1 I J \ IT

1 * | 1 1 1 1

1' *

\

o

\

t„=2081s \

1E+3 .

~

AT=O.O35OC

_

\

a.-L.) i I i i 1 i l l i i.J L t x-j—I...I. )• r i 1 . i

12000

Fig. 3.13.4.

16000

20000 24000 time (s)

28000

Aeo = (e - £i) vs. time data as in Fig. 3.13.1.

286

Effective Field Approach to Phase

Transitions

is obtained through: Ae 0 = A e 0 i e - t / r 0 , resulting in r 0 (AT = 0.035°C) = 2081s. Similar data for AT = 0.065, 0.18, and 0.31°C were collected and analyzed. Figure 3.13.5 displays graphically the data in Table 3.13.1. The data indicate two distinct response times r^"1 ~ (AT)" 0 , where VQ « 0.5 ± 0.1 and Tj -1 ~ (AT)" 1 , where Vi « 1.0 ± 0.1, with different magnitudes and temperature dependences. It may be noted that the estimated error bars become considerable away from T c . It is well known that noticeable changes of permittivity at a constant temperature below Tc are related to the gradual tendency to attain the equilibrium state of the domain structure. 2 This evolution of the domain structure has been described by a two-step mechanism. 3,4 The first step has been observed as the shrinkage of small domains to the expense of larger domains of opposite polarity. The second one consists of the decelerated formation of 1

" i

" 1

r

, Jr'"<

H-"

: A

V

r f „=c(AT) => " v„«0.5 + 0.1 i, »c (AI)V1

,'

s

v^l.O ± 0.1 •

,L

*

* ,,l

1E-4

Fig. 3.13.5.

1E-3

1

!

>

1

1

1 1

A T / T c

1

1E-2

Temperature dependence of two response times.

Table 3.13.1. Results for T, and TO at various 0.035°C < A T < 0.31°C. A T (°C)

TO (s)

TI (s)

0.035 0.065 0.18 0.31

2081 1624 1094 1120

14584 8577 3811 12665

Observations

of Two Ferroelectric Response Times in TGSe

287

At[s]

e 1500

. -

X #

tOO kHz

1000

X * X

B

A

* x•

D • 250 °

50 kHz

j ^ .

>/^H,

25 kHz

500 Xi» '

1000 Fig. 3.13.6.

i"^

2000

i

^a

I 3000

100 500

I 4000

•

7400

0

17500

+

32400

A

44400

,

Time evolution of Cole-Cole plots at a fixed temperature ( A T = 0.08° C).

a more and more periodic domain structure in the crystal. The rate of the evolution process decreases at temperatures far below T c . In order to compare our response times with time dependence of the reciprocal domain width (density of domains), we have analyzed data reported for a TGS crystal. 5 The comparison is, however, nondirect because of the differences in the characteristics of the transition for TGS and TGSe. We have estimated the two relaxation time values as about 1011 and 7530 s, respectively, taking into account data collected at constant temperature 5 (AT = 0.4°C). The time evolution of Cole-Cole plots can be observed also for TGSe crystals at a fixed temperature below Tc. Our data (Fig. 3.13.6), which were collected at AT = 0.08°C (E0 = 0.5V/cm), served for an evaluation of relaxation time values of a dispersion strength Ae = (e s — e ^ ) . We have obtained two values, 1613 and 9906 s. It may be noted that both values would fit reasonably well with TO and TI in Table 3.13.1. We think that this is the first attempt to investigate the possible existence of two scales in a characteristic parameter (time or length) at T « Tc in an uniaxial ferroelectric. Triglycine selenate was choosen for investigations because its Tc is close to room temperature and because the rather large specific heat anomaly 6 indicates a substantial contribution from fluctuations near the transition temperature. Acknowledgments We acknowledge financial support from CICyT, Grant no. PB 93-1253, and CAM, Grant no. AE 00138/94. One of us (J.P.) wishes to thank DGICyT for financial support for a Sabbatical at UAM, SAB 94-0084.

288

Effective Field Approach to Phase

Transitions

References 1. K. Hirota, G. Shirane, P.M. Gehring and C.F. Majkrzak, Phys. Rev. B49, 17, 11967 (1994). 2. J. Fousek, Czech. J. Phys, B15, 412 (1965); F. Gilletta, Phys. Stat. Sol. (a) 11, 721 (1972). 3. A. Jaskiewicz and W. Gruszczynska, Ferroelectrics 23 173 (1980). 4. N. Tomita, H. Orihara and Y. Ishibashi, J. Phys. Soc. Jpn. 58, 1190 (1989). 5. N. Nakatani, Jap. J. Appl. Phys. 24, L528 (1985). 6. K. Ema, K. Hamano, K. Kurihara and I. Hatta, J. Phys. Soc. Jpn. 43, 1954 (1977); K. Ema, J. Phys. Soc. Jpn. 52, 2798 (1983).

Chapter 3.14

Equation of State for Pressure and Temperature Induced Transition in Ferroelectric Telluric Acid Ammonium Phosphate* Jose R. Fernandez del Castillo, Janusz Przeslawskr', and Julio A. Gonzalo Departamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

A careful experimental investigation of the ferroelectric-paraelectric transition in uniaxial ferroelectric telluric acid ammonium phosphate induced by pressure and by temperature in the vicinity of the phase transition has been carried out. Dielectric constant and hysteresis loop data allow the determination of the combined equation of state as a function of both temperature and pressure, relating accurately polarization, field, pressure, and temperature in the neighborhood of the transition.

Solid-state pressure-induced phase transitions are similar to t e m p e r a t u r e induced phase transitions in many respects. T h e analogy between the role played by hydrostatic pressure in the former case and the role played by thermal energy in t h e latter can be exploited to obtain a compact equation of state describing simultaneously the pressure and t e m p e r a t u r e dependence of the relevant pair of conjugated variables (polarization and electric field in a ferroelectric). To make a proper quantitative investigation of the equation of s t a t e in the vicinity of the transition, accurate d a t a , in terms of the hydrostatic pressure as well as in terms of t e m p e r a t u r e , are needed. *Work previously published in Phys. Rev. B: Rapid Commun. 53, R2903 (1996). Copyright © 1996. The American Physical Society. tOn leave from the Institute of Experimental Physics, University of Wroclaw, 50-205 Wroclaw, Poland. 289

290

Effective Field Approach to Phase

Transitions

Uniaxial ferroelectric telluric acid ammonium phosphate 1 8 (TAAP), chemical formula |Te(OH) 6 | • |2NH 4 H 2 P0 4 | • |(NH 4 )2P0 4 |, is a good candidate for investigation, because it has (for ambient pressure, 1 bar) a second order phase transition 2 to the paraelectric state at Tc(0) sa 320K, conveniently above room temperature, and also because it possesses a negative pressure coefficient,4'5 dTc(p)/dp « 23.58°C/kbar, which allows investigation of pressure-induced transitions at constant temperatures between room temperature and 320 K. This crystal is monoclinic at room temperature (Pn, Z = 2). At the ferroelectric transition the change in space group is P2/n <-> Pn. The main feature of the atomic arrangement is the presence of two distinct anionic groups: TeC-6 and P 0 4 . Two types of planes perpendicular to the [101] direction can be distinguished. In the first type, one can find P 0 4 groups and NH 4 0„ polyhedra, while in the second one, one has TeC>6 groups in addition to P 0 4 groupsand NH 4 0„ polyhedra. 6 Four types of hydrogen bonds, N-H-O, linking P 0 4 tetrahedra are present. All of them are asymmetric and the protons are ordered in the ferroelectric phase. 7 This ordering was confirmed also by the increase of Tc upon deuteration and Raman studies, 8 which showed that the groups of Te06, P 0 4 , and NH 4 do not undergo changes that could trigger the phase transition. In this work, we present and analyze high-resolution data (Ap ~ 1.5 bar steps, in an interval of about 3kbar; A T sa 0.01°C steps, in intervals of about 5°C) on the dielectric constant and hysteresis loops for TAAP single crystals, undertaken with the aim of determining a combined pressure and temperature equation of state. The samples were small platelets, 3 x 2 x 1.2 mm 3 in size, cut from a larger single crystal grown from water solution, coated with silver paste electrodes deposited on the main surfaces. The sample holder, in which pressure and temperature could be accurately controlled, as mentioned before, was an adapted LC10 Unipress cell. Measurements of dielectric constant and loss factor were performed with an HP Precission LCR Meter 4384A {Emeas — 1 V/cm, / = 1kHz). Automatic data, together with temperature and pressure readings, were collected by means of a desk-top computer. A Diamant-Drenck-Pepinsky bridge and a NICOLET 3 10 digital oscilloscope were used to do the digitalized hysteresis loops measurements (.Emeas = 1.8kV/cm, / = 50Hz). A computer program was used to correct for internal bias in the hysteresis loops. Figure 3.14.1 shows dielectric constant data as a function of pressure through the transition at room temperature. The data can be fitted well with a Curie-Weiss-type law with pressure.

Equation of State for Transition in Ferroelectric

TAAP

291

5 x 10- 4

4 x 10"

6.4

p (kbar) Fig. 3.14.1. Inverse dielectric constant e - 1 as a function of hydrostatic pressure at T = 293 K for TAAP. Straight line indicates the Curie-Weiss behavior at p > pc = 5.13 kbar (note that vertical scale starts at e _ 1 = 4 x 1 0 - 4 ) .

Figure 3.14.2 gives similar dielectric constant data as a function of temperature through the transition at several pressures, which also follow well a Curie-Weiss law. It may be noted that £ m a x becomes smaller for lower transition temperatures. Figures 3.14.3(a) and (b) displays normalized square spontaneous polarization as a function of temperature (a) at ambient pressure (lbar), and pressure (b) at constant temperature T = 306.65 K. It may be noted that, because of the lack of perfect compensation in the hysteresis loops, Tc and p c are slightly overestimated. Two sets of hysteresis loop data, one at constant pressure (ambient pressure, i.e., lbar) and another at constant temperature (T = 306.65K, i.e., near and below the ambient pressure Curie temperature) are represented in Fig. 3.14.4 in scaled form. They fit well a combined equation of state, discussed below, which is easily deduced from basic considerations. It must be noted that we are not aware of previous descriptions of observed pressure and temperature behavior near a ferroelectric transition by means of a single equation of state. In an order-disorder ferroelectric phase the existence of microscopic dipole moments reversible under the action of an external electric field can

292

Effective Field Approach to Phase

Transitions

5 x 1(T4-

s-1 6 x 10-4-

4x 1(T4-

2x 10" 4 1 - 5.68 2 - 5.07 3 - 4.32

5 - 3.60 6 - 3.45 7 - 2.57

4 - 4.13 |

28

'

I

I

|

8 - 1.31 I

I

I

|

I

I

32 36 T(°C)

I

|

40

44

Fig. 3.14.2. Isobars of the temperature dependence of the inverse dielectric constant e~l for TAAP.

be associated with the presence of a double potential minimum along the polar axis of the unit cell for specific constituent ions. In the paraelectric, i.e., fully disordered, phase the jump probability per unit cell and unit time of those specific ions between the two symmetric minima is usually given by 7r(0) = ve~^lkaT, where v « k^Qr^/h is the attempt frequency and the potential barrier height between the two minima. If the barrier height is increased by a small amount d(fi, the jump probability should decrease in such a way that d7r0(^) = - ( l / ^ ) 7 r 0 d ^

(1)

and then ir0() = 7r 0 (0)e"

(2)

Here, * can be given in terms of the critical barrier height at which the ferroelectric-paraelectric transition, induced either by rising temperature or rising hydrostatic pressure, takes place. In a transition induced by temperature only (i.e., under ambient pressure) we have 9 <J>eS(Tc,0)=(3Nfi2=kBTc,

(3)

where Tc is the transition temperature and Eaoft = /3Nfi2 is the electrostatic energy of the spontaneous field acting on the unit dipole fi. In a transition

Equation of State for Transition

in Ferroelectric

TAAP

293

Fig. 3.14.3. Normalized spontaneous polarization square (Ps/Pso)2 vs. pressure at T = 306.65 K (a), and vs. temperature at ambient pressure ( l b a r ) (b). P s o = 2.12/iC/cm 2 is the saturation spontaneous polarization.

induced by pressure only (at 0 K) also 4>es{0,Pc) = (3N/J2 = Avcpc

(4)

where pc is the critical hydrostatic pressure and Au c the associated change in unit cell volume, vc, needed to bring about the pressure-induced transition.

294

Effective Field Approach to Phase

3

O Pd = Pd < e ' a t const - ,em P- ( T = 306.65 K ) • pd = pd ( e ) at const, pressure (1 bar).

-

2

Transitions

PrJ = P d /

1

-(T/Tc+p/pc)|1/2

S yf

y S

-

Jf 1 —

^F

-

Jw

0 —

/f

_

:

-1 —

_

Pso= 2 1 2 (xC/cm2 £^=9.90 10 5V/cm

/I

I I | I I I I

-4

/

e = e/|l-(Tn" c +p/p c )| 3 / 2

/I

1 I I | I I I I | I I I I I I I I I | I 1 I 1 | I I I I

-2

0

2

A

6

4

8

10

Ine Fig. 3.14.4. Simultaneous scaling equation of state for the pressure-induced and the temperature-induced ferroelectric transition in TAAP. The continuous line is the theoretical scaling equation of state e = ± p a + \p\, with a — sign for T < Tc and p < pc, and a + sign for T > Tc and p > pc. Tc (p = l b a r ) = 317.25K, p c (T = OK) = 76.87kbar; Pso = 2.12/iC/cm the saturation spontaneous polarization, and Eao = 9.90 X 10 5 V/cm, the saturation spontaneous field, give the best fit to the theoretical equation of state.

Consequently, in the general case of T < Tc, p < pc, we may take 0* = ^eff(T,p)=/3A r /i 2 (^ r + ^ .-'c

(5)

Pc

This expression is used in the following to get in a straightforward manner a combined pressure/temperature equation of state for the ferroelectric transition. In equilibrium under zero external field at the paraelectric phase, where the double potential minimum is symmetric, the number of dipoles per unit volume pointing in the positive direction (A/2) and in the negative direction (A^i) are related by A ^ ^ i = N1T12, and since iri2 = 7T2i = TTO, one has JVi = N2 = N/2. In general, however, i.e., under an effective field EeS = E + f3Pd + 7 P d 3 + 6 P%,

(6)

one has 7T2i=?ro(0)e-^+/3Fd+LW^, 7T12 = 7 r 0 ( 0 ) e ( £ + ^ + L ^ \

(7)

Equation of State for Transition in Ferroelectric

TAAP

295

L)

(8)

which, in equilibrium (N2iT2i = Nynn) leads to (N2 N ^ = t ^ (N2 + N1)li

Nfi

{

E

+

^

+

»

and, using Eq. (5), to the combined equation of state -1

or E

T Tc

P_

tanh"1 ( A

'E + pPtV pNfx ,

j3Pd

PZ +

(9)

PH4 + L

Pc

(10) which can be written in dimensionless form as Tc

pc

tanh

l

pd - p d ( l + gj + hpAd + L)

(11)

where p is the pressure and pd is the reduced dipolar polarization and must not be confused. Tc = Tc(p = 0) is the transition temperature at p = 0, and pc = pc(T = 0) is the transition pressure at T = 0 K. For a continuous second order transition, 10 close to the ordinary critical point (pd
e = ±Pd + gPJL where 3/2'

1 - \TC + it) Pd

+

1/2-

(13)

te £)

As shown in Fig. 3.14.4, Eqs. (12) and (13) describe simultaneously the pressure-induced and the temperature-induced ferroelectric transition of TAAP simply and fairly accurately. From pc and Tc we estimated the unit cell volume change associated with the phase transition as Au c = k^Tjpc = 0.565 A (the actual unit cell volume is vc = 922.731 ). Table 3.14.1 illustrates the good fit of data for Tc(p) and pc(T) from Figs. 3.14.1 and 3.14.2 to Eq. (9), in which (T/Tc+p/pc) play the same

296

Effective Field Approach to Phase

Transitions

Table 3.14.1. H = heating; C = cooling; D = decreasing of pressure. Tc = Tc(j> = 0) = 317.25 K and pc = pc(T = 0) = 76.87kbar were used for all data. Tc{p ^ 0) is the actual transition temperature for a specific pressure p ^ 0; pc(T ^ 0) is likewise the actual transition pressure for a specific temperature T ^ 0. Tc(P) (K)

p (kbar)

299.56 299.56 302.00 30.90 304.36 304.22 304.61 304.30 305.58 305.42 307.28 307.14 308.12 307.94 310.98 310.83 315.35 315.30

H C H C H C H C H C H C H C H C H C

D

Tc(p)/Tc

(P/Pc)

5.688 5.688 5.071 5.071 4.452 4.327 4.452 4.452 4.074 4.137 3.601 3.601 3.394 3.452 2.556 2.578 1.310 1.315

0.944 0.944 0.951 0.951 0.959 0.958 0.960 0.959 0.963 0.962 0.968 0.968 0.971 0.970 0.980 0.979 0.994 0.993

0.074 0.074 0.066 0.066 0.058 0.056 0.058 0.058 0.053 0.054 0.047 0.047 0.044 0.045 0.033 0.033 0.017 0.017

T(K)

Pc(T)

(T/Tc)

293.15

5.13

role as T/Tc

0.924

Pc(T)/Pc

{[Tc(p)/Tc]+(P/Pc)} 1.018 1.018 1.017 1.017 1.017 1.015 1.018 1.017 1.016 1.016 1.015 1.015 1.015 1.015 1.013 1.013 1.011 1.010 {(T/Tc)+\p(T)/Pc}}

0.067

in a transition at zero pressure and as p/pc

0.991

in a transition a t

zero t e m p e r a t u r e .

Acknowledgments We acknowledge t h e financial s u p p o r t of C I C y T (Grant no. PB93-1253). One of us (J.P.) t h a n k s D G I C y T for financial s u p p o r t for a sabbatical (SAB94-0084) at t h e Ferroelectric Materials Laboratory, U A M .

References 1. R.F. Weinland and H. Prause, Z. Anorg. Chem. 28, 49 (1901). 2. S. Guillot Gauthier, J.C. Peuzin, M. Olivier and G. Rolland, Ferroelectrics 52, 293 (1984).

Equation of State for Transition in Ferroelectric TAAP

297

3. R. Sobiestianskas, J. Grigas and Z. Czapla, Phase Transitions. 37, 157 (1992). 4. M.N. Shashikala, H.L. Bhat and P.S. Narayanan, J. Phys. Condens. Matter 2, 5403 (1990). 5. S. Haussiihl and Y.F. Nicolau, Z. Phys. B 6 1 , 85 (1985). 6. M.T. Averbuch-Pouchot and A. Durif, Ferroelectrics 52, 271 (1984). 7. J. Gaillard, J. Gloux, P. Gloux, B. Lamotte and G. Rins, Ferroelectrics 54, 81 (1984). 8. M.N. Shashikala, B. Raghunata Chary, H.L. Bhat and P.S. Narayanan, J. Raman Spectrosc. 20, 351 (1989). 9. See, e.g., J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). 10. T. Iglesias, B. Noheda, B. Gallego, J.R. Fernandez del Castillo, G. Lifante and J.A. Gonzalo, Europhys. Lett. 28, 91 (1994).

Chapter 3.15

Neutron Diffraction Investigation of the FRL-FKH Transition in Nb-Doped PbZri-^Ti^Oa with x = 0.035* Beatriz Noheda, Tomas Iglesias, Noe Cereceda and Julio A. Gonzalo Departamento de Fisica de Materiales, Universidad Autonoma, 28049 Madrid, Spain

Hui-Ting Cien and Yong Ling Wang Shanghai Institute of Ceramics, Chinese Academy of Sciences, 1295 Ding Xi Road, Shanghai 200050, China

David E. Cox and Gen Shirane Department

of Physics, Brookhaven National Upton, NY 11973-5000, USA

Laboratory,

Neutron powder diffraction data have been collected in order to characterize the F R L - ^ R H transition of Nb-doped, Zr-rich PZT. Data for the first few diffraction peaks were obtained in the temperature range 270 < T < 350 K in order to make a detailed characterization of the .FRL-JRH transition, which takes place for this composition at TUT = 323 K. The temperature dependence of the Pb and Zr/Ti(Nb) atomic shifts, and the rotation and distortion of the oxygen octahedra through the transition was derived. The results are compared with published data for PZT with x = 0.10, and indicate that (a) the change in Zr/Ti(Nb) shift through TLH becomes more pronounced, and (b) the distortion and tilt of the oxygen octahedra below TLH become more important for x = 0.035 than for x = 0.10. It has been concluded that the increment in the variation of the spontaneous polarization through

*Work previously published in Ferroelectrics 184, 251 (1996). Copyright © 1996. Taylor & Francis Group. 299

300

Effective Field Approach to Phase

Transitions

TLH with increasing Zr amount is mainly caused by the increase of the Zr/Ti(Nb) displacement in the low-temperature rhombohedral phase.

3.15.1.

Introduction

Ferroelectric ceramics of PZT with high Zr contents show striking electromechanical and pyroelectric properties, 1 which make them useful for technological applications. These ceramics are often doped with small amounts of Nb to reduce aging effects, and to increase the resistivity and the dielectric constant. 2 ' 3 The structural investigation of Nb-doped ceramic samples may at first seem to be an added complication, but it should be emphasized that the incorporation of Nb improves the homogeneity and the mechanical properties of the ceramic material, and enhances the electrical resistivity, facilitating the dielectric measurements through the transition undertaken in parallel with the structural work. The basic features of the phase diagram of PZT solid solutions were established by Shirane et al.4 and Barnett. 6 Michel et al.7 subsequently made an X-ray and neutron powder diffraction study in which they established the space groups of the two rhombohedral phases of PZT, as R3c for the FRL phase with x = 0.1 (two formula units per unit cell) and as R3m for the FRH phase with x = 0.42. Later, Glazer et al.8 carried out further structural work on PZT (x = 0.10) and characterized the FRL phase in terms of the tilting of oxygen octahedra 9 and the accompanying expansion or contraction of oxygen triangles. In the present work, we have analyzed neutron powder diffraction data on PbZri_a;Ti303 with x = 0.035. The aim of this work is to make a more detailed characterization of the shifts in the atomic positions through the transition and to compare them with the data of Glazer et al.8 for an undoped sample with x = 0.10.

3.15.2.

Experimental

The sample was a sintered and hot pressed cylinder of PZT doped with 1.5% by weight of Nb 2 Os about 1cm in height and 1cm in diameter, having a nominal composition Pb 1 _j / / 2 (Zri_ x Ti x )i_j / Nbj / 03 with x = 0.035 and y = 0.039, prepared at the Shanghai Institute of Ceramics. Constituent oxides were ground, pressed into pellets, and fired at 1346°C for 2 h. The neutron diffraction runs were carried out on one of the triple-axis diffractometers

Neutron Diffraction

Investigation

in Nb-Doped PbZr\~-xTixOz

301

at Brookhaven National Laboratory, New York, at intervals of 10 K over the temperature range 270-350 K. Data were collected for the four (F R H ) and five (FRL) lowest angle peaks at nine different temperatures in this temperature region, which spans the F R L - F R H transition (T LH ). Since the ceramic sample was large, and no special orientation is to be expected for this cubic/pseudocubic material, few reflections should, in principle, be sufficient to determine properly the changes in atomic positions, taking into account that neutron data are relatively free from systematic errors. The neutron energy was 14.7 meV. The transition temperature TLH — 323 K was previously determined from dielectric constant and thermal expansion measurements. 1 °

3.15.3.

Results and Discussion

The behavior of the atomic shifts in the rhombohedral phases (aRL = 5.85 A, a R L = 59°42') and FRU (a R H = 4.07A,a R H = 89°39') for the first four peaks, (100), (110), (111), and (200), together with the superstructure (3/2,1/2,1/2) peak at T < TLH (pseudocubic notation) was determined in the following way. The unit vectors of the F R L phase (i?3c), ai,a2,a3, with lengths ar = \/2a, where a is the pseudocubic unit cell parameter, and rhombohedral angle aRL = 59°42', were used throughout. The oxygen triangles were allowed to tilt as a whole perpendicular to (111) for TTH, while the tilt angle was set equal to zero for TLH- The values reported by Glazer8 were taken as initial values for the Pb shifts, s, the Zr/Ti (Nb) shifts, t, the O3 tilts perpendicular to (111), e, and the O3 expansion/contraction within the plane perpendicular to (111), d. In the FRH phase we had three unknown parameters (s, t, d) and four peaks at each temperature. In the Fn phase we had, on the other hand, four unknown parameters (s, t, e, d) and an additional fifth peak, the superstructure peak (3/2,1/2,1/2), which is essentially dependent only on and vanishes at TLH- The atomic positions at each temperature were refined via an iterative manual procedure by minimizing R = Yl I^obs — -fcail/ X^obs- The final R values were smaller than 1.5%. The scattering lenghts used were 6A = 4.928 (6pb = 0.940, other dopants: Ca, vacant), bB = 0.695 (bZr = 0.716, bTi = -0.330, 6 Nb = 0.705, other dopants: Hf), be = 0.578 (60 = 0.580, vacant) all in 10~ 12 cm units. Isotropical temperature factors were assumed at each temperature. The rhombohedral fractional positional parameters used in the R3c phase (FRX) are the same as those used by Megaw and Darlington. 11

302

Effective Field Approach to Phase

Transitions

b s O—fer-

4~i

Q,

^

•

t

-©^"O-—©

a) •a §

i

-•

•"

0.000 •

O -d

0.016 •

•

e

'£ o.oio —

b)

©—e—o—Q—-e35^£) o.ooo-

T(K) Fig. 3.15.1. Fractional shifts s and t for P b and Zr/Ti (Nb) along to the rhombohedral axis [111] (a) and oxygen octahedra tilt, e, and distortion, d (b), as a function of temperature through the -FRL~' ? RH, transition for Pbzi_y/2('Z'fl-xrrix)i-yNby03 (x = 0.035,y = 0.039). Dashed lines are from data by Glazer et al.s for the composition x = 0.10, with T L H = 353 K.

Figure 3.15.1(a) shows the fractional shifts s and t for Pb, Zr/Ti (Nb) with respect to the length Ch = \fvia of the long rhombohedral body diagonal and Fig. 3.15.1(b) the tilt e = (tan w)/4\/3, where is the tilt angle around [111], and the O3 triangle distortion (2d = ±<5(0)/a r ), for the nine temperatures, six below TLH and three above TLH- These results show that the Pb and Zr/Ti(Nb) atomic shifts along [111] decrease by about 18 and 44%, respectively, above the transition, which implies a substantial change in spontaneous polarization, and a high pyroelectric coefficient around the phase transition. The oxygen triangle tilt angle decreases fairly abruptly from about 5° to zero at the same temperature, as expected. Table 3.15.1 summarizes these results and compares the s, t, d, e, and R values obtained with those given by Glazer et al.8 for a PZT sample with higher Ti content {x = 0.10).

Neutron Diffraction

Investigation

in Nb-Doped PbZr\-xTixOz

303

Table 3.15.1. Refined atomic positional parameters between 270 and 350 K for Nb-doped P b Z r i - ^ T i x O s (z = 0.035) containing 1.5% N b 2 O s by weight. F RL (iJ3c)

Glazer8

FRH(R3m)

T(K)

270

280

290

300

310

s t -d e

0.0318 0.0151 0.0028 0.0147

0.0322 0.0148 0.0027 0.0147

0.0312 0.0132 0.0027 0.0145

0.0324 0.0148 0.0027 0.0141

0.0325 0.0148 0.0027 0.0138

320

330

340

350

0.0320 0.0266 0.0263 0.0258 0.0158 0.0080 0.0081 0.0084 0.0028 0.0011 0.0011 0.0011 0.0121

298 0.0316 0.0114 0.0029 0.0137

Our analysis of the temperature dependence of the intensities of a few neutron powder diffraction peaks from Nb-doped PbZri_ a : Ti x 03 through the -FRL^-FRH transition indicates that the shifts of the Pb, Zr/Ti(Nb) ions and the tilt and distortion of the O3 triangles show small but significant changes with respect to those previously reported by Glazer et al.8 for a PZT sample with higher Ti content (x = 0.10). The Pb shift (s) is somewhat higher in our case in the FRL phase due to the longer octahedron edge, as was pointed out by Megaw et al.11 The oxygen octahedra tilt (e) is higher in our case, and the Zr/Ti(Nb) shifts (t) are substantially higher, which is not in agreement with strict packing considerations. 11 It may be concluded that the increase in AP S through TL-H with Zr content 1 is mainly due to the increase of the Zr/Ti shifts in the FRL-

Acknowledgments We acknowledge partial financial support from IBERDROLA (INDES/94) and CICyT through grant PB93-1253. Work at Brookhaven is supported by the US Department of Energy, Division of Materials Sciences, under contract DE-ACO2-76CH00016. References 1. 2. 3. 4.

M. Bernard, R. Briot and G. Grange, J. Solid State Chem. 31, 369 (1980). R. Gerson and H. Jaffe, J. Phys. Chem. Solids 24, 979 (1963). R. Gerson, J. Appl. Phys. 31, 188 (1960). G. Shirane, K. Suzuki and A. Takeda, J. Phys. Soc. Jpn. 7, 12 (1952); E. Sawaguchi, J. Phys. Soc. Jpn. 8, 615 (1953). 5. G. Shirane and A. Takeda, J. Phys. Soc. Jpn. 1, 5 (1952). 6. H. Barnett, J. Appl. Phys. 33, 1606 (1962).

304

Effective Field Approach to Phase

Transitions

7. C. Michel, J.M. Moreau, G.D. Achenbach, R. Gerson and W.J. James, Solid State Comm. 7, 865 (1969). 8. A.M. Glazer, S.A. Mabud and R. Clarke, Acta Cryst. B34, 1060 (1978). 9. A.M. Glazer, Acta Cryst. B 2 8 , 3384 (1972). 10. B. Noheda, T. Iglesias, G. Lifante, J.A. Gonzalo, J. de Frutos, H.T. Chen, Y.L. Wang and G. Shirane, Electroceramics IV (Aachen Sept. 5-7, 1994). 11. H. Megaw and C.N.W. Darlington, Acta Cryst. A 3 1 , 161 (1975).

Chapter 3.16

0 3 Tilt and Pb/(Zr/Ti) Displacement Order Parameters in Zr-Rich P b Z r i ^ T i ^ O s from 20 to 500 K* Noe Cereceda, Beatriz Noheda, Tomas Iglesias, Jose R. Fernandez-del-Castillo and Julio A. Gonzalo Universidad

Autonoma

de Madrid, 28049 Madrid,

Spain

Ning Duan and Yong Ling Wang Shanghai Institute of Ceramics, Chinese Academy of Sciences, 1295 Ding Xi Road, Shanghai 200050, China

David E. Cox and Gen Shirane Department

of Physics, Brookhaven National Upton, New York 11973-5000

Laboratory,

Neutron diffraction and dielectric data have been collected from a polycrystalline sample of PbZri-xTi^Os (a; = 0.035) to determine the temperature dependence of the two-order parameters associated with the oxygen octahedral tilt and the Pb/(Zr/Ti) displacement, respectively, through the sequence of phase transitions FRL-FRH-PC present in this mixed system. The weak coupling between tilt and polarization (displacements) is satisfactorily described within the framework of a two-order-parameters statistical theory in which the respective effective fields involve two higher order terms in addition to a linear term.

•Work previously published in Phys. Rev. B 55, 9 (1997). Copyright © 1997. The American Physical Society.

305

306

3.16.1.

Effective Field Approach to Phase

Transitions

Introduction

Perovskite structures, which are cubic at high temperatures, frequently display a rich variety of phase transitions at lower temperatures 1 that usually involve cation shifts and/or oxygen octahedra tilts, with respect to the high-symmetry phase. These shifts and tilts are basically independent, in the sense that one can appear in the absence of the other, but they can couple weakly to each other. The mixed perovskite system PbZri-^Ti^Os (generally denoted PZT) is especially interesting, 2 not only because it can be prepared over the entire compositional range (0 < x < 1), with a variety of distorted structures at room temperature, e.g., tetragonal {FT), zerotilt rhombohedral high-temperature (.FRH), nonzero-tilt rhombohedral lowtemperature (JFRL), and antiferroelectric orthorhombic {AFo) phases, but also because 3 of its important technological applications. The basic features of the phase diagram of PZT solid solutions were established by Shirane et al.2 Barnett 4 pointed out the existence of an additional low-temperature phase change for Zr-rich compositions, later characterized as the F R L - ^ R H transition. Michel et al.5 determined the space groups of the two rhombohedral phases as R3c (FRL) and R3m (FRR) by means of X-ray and neutron powder diffraction at room temperature using two compositions, x = 0.10 (-FRX, w ith two formula units per unit cell, involving oxygen octahedra tilts) and x = 0.42 (-FRH, with one formula unit per unit cell and zero tilt). In summary, the sequence of phase transitions in Zr-rich PZT is FRL (rhombohedral R3c) —> FRH (rhombohedral R3m) —> Pc (cubic). Later, Glazer et al.6 made a further neutron powder study on PZT with x = 0.10, characterizing the FRL phase in terms of the tilting of oxygen octahedra 7 and the expansion or contraction of the octahedra triangles neighboring the shifted B cation (Zr/Ti). Glazer7 has also made a general classification of tilted octahedra in perovskites. He pointed out that the cation displacements, which are directly related to the ferroelectric character of the perovskites, have only a small effect on the lattice parameters and that, generally, the overall symmetries follow those of the tilts. Research has been done to describe the possible relations between the atomic displacements in these materials. Megaw and Darlington 8 described the perovskite structures in terms of four structural parameters, classifying them by their space groups to allow comparisons. They found no correlation between the cation displacements and the octahedra tilts in the space groups R3c and R3m, to which rhombohedral PZT belongs. A study of such

Oz Tilt and Pb/(Zr/Ti)

Displacement

Order Parameters

307

a correlation, if it exists, is important to determine which parameters are the cause of the deformations and which parameters are only consequences. Nevertheless, even if displacement and tilt are independent, they may be coupled, mainly because of the shared-corner linkage between octahedra, which is required by the packing. We have recently reported some preliminary neutron diffraction measurements 9 on a ceramic sample of PZT with x = 0.035 over a limited temperature region around the phase transition between the two ferroelectric rhombohedral phases, FRL and FRH, which occurs at a temperature TLH ~ 323 K. This work allowed a detailed description of the structural parameters that characterize the transition, namely the cation displacements along the [111] axis, s (Pb) and t (Zr, Ti), the distortion of the oxygen octahedra d, and a parameter e, which describes the rotation of the oxygen octahedra around the [111] axis with a tilt angle, w, in the (111) plane such that tan UJ = \/3e. 6 We have also reported some measurements on the dielectric response of Zr-rich PZT (Ref. 10) in the temperature region around the rhombohedral-cubic phase transition, -FRH--PC> which occurs at « 5 1 0 K and is shown to be first order. The weak coupling between tilt and displacements has been recently studied by Dai et al.11 in PZT (x = 0.35) by means of dielectric constant, hysteresis loops, and dilatometric and electron diffraction techniques. They proposed that the inability of the oxygen octahedra to rotate coherently within a rigid lattice generates random internal stresses, which constrain the polarization. They found doubled hysteresis loops for this composition, which appear to be relaxed with La and Nb doping. To analyze in depth the coupling between tilt and displacements, it is very convenient to use neutron diffraction data, especially to measure the tilt angle, which is difficult to determine by other techniques. In the present work, we investigate the temperature dependence of the two-order parameters associated with the octahedral tilting and the cation displacements over a wide temperature range spanning both the -FRL--FRH and -FRH-PC transitions, with special emphasis on the weak coupling between polarization, directly associated with the cation shifts, and tilt in a Nb-doped PbZri-^TirOs ceramic sample with x = 0.035. Nb doping is important, among other things, to reduce the electrical conductivity and to facilitate dielectric constant and polarization measurements. The squared structure factor of the first superstructure peak, 1/2(311), which can be expressed as | F | 2 = 8[sin2(47re)[l + cos(247rd)] « 16 sin2(47re)

308

Effective Field Approach to Phase

Transitions

is, in practice, directly proportional to the squared rotation parameter e 2 . This allows us to define the temperature dependence of the tilt order parameter as Vs(T)

= [IRLCTV/RUO)] 1 / 2 ,

where /RL is the integrated intensity of the superstructure 1/2(311) Bragg reflection. We have accurately measured J]S(T) between 20 and 375K by neutron diffraction. Dielectric constant data, which show a small anomaly at TLH, and pyroelectric charge measurements through TLH were used to characterize the small decrease in spontaneous polarization that accompanies the •FRL~-FRH transition. Measurements of hysteresis loops, which were previously used to characterize the behavior of the polarization order parameter Ps(T) = Ps(T)/Ps(0) at the T R H ^ P C (ferroelectric-paraelectric) transition, were attempted at T < TLH, but were unsuccessful because of the high values of the coercive field for the sample in this temperature range. The combined neutron diffraction and dielectric data were used for a theoretical analysis of the weak tilt-polarization coupling. Prior work by Halemane et aL 12 ' 13 made use of Landau's theory to describe the simultaneous temperature dependence of tilt and polarization in PZT with a different composition, x = 0.10, but no detailed information on the temperature dependence of the tilt order parameter was available at this time. We describe below an analysis of our data for PZT with x = 0.035 by means of a simple two-order-parameters statistical theory, in which the effective fields14 for both tilt and polarization involve linear, cubic, and fifth power terms in the conjugate variable. In spite of its simplicity, this theoretical approach produces a simple relationship between tilt and polarization, which is borne out quantitatively by the experimental data.

3.16.2.

Experiment

The samples were high-quality polycrystalline ceramics prepared at the Shanghai Institute of Ceramics, with nominal composition Pbi_ 3/ /2(Zri_ a: Ti a ;)i_ 3/ Nbj / 03, with x = 0.035 and y = 0.039, confirmed by chemical analysis. The constituent oxides were ground, pressed into pellets, and fired at 1350°C for 2 h. Cylinders of about 1 cm in height and 1 cm in diameter were sintered and hot pressed for the neutron diffraction runs, and thin plates of about 1 mm in thickness and 1 cm in diameter were used for the dielectric measurements. Data were collected with neutrons

O3 Tilt and Pb/(Zr/Ti)

Displacement

Order Parameters

309

of 14.7 meV energy for several low-angle peaks, including the superstructure peak 1/2(311) (Ref. 9) at one of the triple-axis diffractometers at the Brookhaven National Laboratory HFBR between 20K and TLH — 323 K. The dielectric constant data, capacitance and dissipation factor, were measured at regular intervals of about 0.1 K by means of an automatic HewlettPackard Precision LCR Meter (Model 4284A) with a field amplitude of 8.3 V/cm to an accuracy better than one part in 104 at a frequency of 1 kHz. The rate of temperature change was ~ 20 K/h for both heating and cooling runs. The pyroelectric charge released through the FRL — -FRH transition was measured with a Keithley Electrometer (Model 610C), and measurements of hysteresis loops data were made with a modified Diamant-Pepinsky-Drenck circuit with a Nicolet Digital Scope (Model NIC-310). The temperature for the electrical measurements was measured in all cases by a chromel-alumel thermocouple with a Keithley Digital Multimeter (Model 196). 3.16.3.

Results

Figure 3.16.1 depicts the tilt order parameter rjs{T) obtained directly from the neutron diffraction data. It can be seen that rjs(T) decreases gradually 1

1

'

1

'

i

'

i

'

1

'

1

•

1

'

1

i

—•— " * — • -

r

~~---
^

-

\

• T

T

LH

• I

0

1

50

l

1

100

.

1

150

:___L

ZOO

u___J

250

,

1

300

ii

350

400

T(K) Fig. 3.16.1. Tilt order parameter vs. temperature obtained from neutron diffraction measurements on a ceramic sample of Nb-doped P b Z r i - ^ T i ^ O s with x = 0.035. The tilt order parameter is denned as rja(T) = [ / R L ( T ' ) / 7 R L ( 0 ) ] 1 / 2 , where 7R.L corresponds to the integrated intensity of the first superstructure peak (hkl) = 1/2(311) of the rhombohedral unit cell. Dashed line indicates calculated metastable region between lowtemperature and high-temperature phases. The theoretical curves (full and dashed lines) are calculated from Eq. (7) with x = 0.

310

Effective Field Approach to Phase

Transitions

with increasing temperature, reaches a value of 7? S (T L H) = 0.59, and then drops fairly abruptly to nearly zero for T > T L H — 323 K. As shown below, a good fit of the data for rjs(T) between 20 K < T < T LH is obtained with an extrapolated Curie temperature T ct = 178.2 K, much lower than the transition temperature TLH, and indicative of the pronounced first order character of the transition. The fit is done by varying the tilt saturation value, 7ys(0) = 7?so, and the effective Curie temperature, T ct , from estimated initial values to optimize the agreement between available experimental data and the equation of state arrived at in Section 3.16.4. It may be noted that, in our case, T?S(0) « rjs (20 K) is well defined beforehand, while T ct < TLH = 323 K (transition temperature) is not, because it is inaccessible experimentally. In Section 3.16.5, the fitting process is described in more detail. Figure 3.16.2(a) shows the inverse dielectric constant as a function of temperature over a wide range, from slightly below TLH = 323 K to well above Tpp = 509.6 K, the ferroelectric-paraelectric transition temperature. In this case, the extrapolated Curie temperature, Tc = 489.9 K, is relatively close to the transition temperature. A small anomaly in e _ 1 (T) at T >= TLH marks the onset of the tilting of the oxygen octahedra. Figure 3.16.2(b) presents the behavior of the spontaneous polarization (displacement order parameter) Ps as a function of temperature over the same wide range as in Fig. 3.16.2(a). Data of the hysteresis loops for T > TLH in the region of Tpp are combined with pyroelectric charge measurements made around TLH- They indicate, as expected, a small change in spontaneous polarization at the onset of the tilting transition. It may be noted that the numerical values for Tc and Tpp are not identical to those in Fig. 3.16.2(a) but appear slightly shifted toward lower temperatures. This may be due to the fact that in the measurements of the hysteresis loops, extra heating of the sample under the relatively high driving field is known to take place. The thermocouple, which is not in good thermal contact with the sample, may register a temperature closer to that within the furnace than that of the sample. The data, read automatically from the digital scope, include extra polarization just above the true transition temperature. As is well known, at first order discontinuous transitions in ferroelectric perovskites, the single hysteresis loops evolve toward double loops, which, when imperfectly compensated due to the high conductivity of the sample, may give rise to tails in the apparent spontaneous polarization above the transition temperature, whose precise value may become blurred. It should be pointed out that the estimated ratio between the true (single crystal)

O3 Tilt and Pb/(Zr/Ti)

Displacement

Order

Parameters

311

3.0x1 0 3 2.5X10"3 2.0x1cr3 ~„ 1.5X10'3 1.0x10J 5.0x10"* 0.0 35 30 25

• f *> 10

0 250

300

350

400

450

500

550

600

T(K) Fig. 3.16.2. (a) Inverse dielectric constant vs. temperature (heating and cooling) for Nbdoped P b Z r i _ a ; T i x 0 3 (x = 0.035) showing the - F R L - ^ R H transition associated with the O3 tilt and the Fjm-Pc transition to the cubic paraelectric phase, (b) Spontaneous polarization vs. temperature for Nb-doped P b Z r i - ^ T i x O s (x = 0.035). Since hysteresis loops data show an increasingly large coercive field as T decreases and approaches T L H , making measurements near this temperature impossible, pyroelectric charge measurements were made. It may be noted that in ceramic rhombohedral perovskites the saturation polarization is substantially lower than in single crystals, the ratio being P s (s.c.) RS 1.15 x Ps (ceramic) (see text). Dashed line indicates calculated metastable region. The theoretical curves (full and dashed lines) are calculated from Eq. (4) with e = 0.

polarization and the apparent (ceramic) polarization is about 1.15,15 which has been used to correct the data shown in Fig. 3.16.2(b). 3.16.4.

Theoretical Analysis

To analyze the phase transition sequence ^RL(?? S

> 0;p s > 0; Aps > 0) -* FRB{r)s = 0;p s > 0; Ap s = 0) -* Pcds =ps = Ap s = 0)

312

Effective Field Approach to Phase

Transitions

we first examine separately the polarization order parameter (ps) and the tilt order parameter (r]s) using a common order-disorder statistical approach, and then we investigate the weak coupling between both order parameters, which gives rise to the increase in the polarization order parameter (Aps) apparent at T < T LH [see Fig. 3.16.2(b)]. 3.16.4.1.

Polarization

If there are N interacting elementary dipoles per unit volume in the solid, of which N2 are pointing one way and Ni in the opposite way, (Ni+N2) = N, in thermal equilibrium we have N2(kBeD/h)e-^/kBTe~E^^kBT

=

N1(kBeu/h)e-^kBTe-E^lx/kBT,

(1) where (/cB©D/^)e * d / fcBT is the jump probability per unit time per unit dipole, n, for an effective field Ees = 0, and fa is the height of the energy barrier between the two potential minima corresponding to the two possible orientations (Eeg = 0). The net dipolar polarization (practically identical to the total polarization, P) would then be Pd = (N2 -N1)n

= Nn tanh(£ eff /Vfc B T),

(2)

where fi, as before, is the elementary dipole moment per unit cell. The effective field may be expanded in powers of the polarization, taking into account that, for an external field E — 0, Eeg(P(\) = —Eeg(—Pd). Thus, EeS = E + pPd+1P$

+ 5Pi + ---,

(3)

where /3, 7, and 5 are constant (i.e., temperature independent) coefficients. Using the dimensionless variables, e = E/(3N(j,, p = Pd/N/j, = P/Nn, and substituting T c = (3Nfj?/kB, 9 = (-y/p)N2fi2, h = ( 7 / / ? ) i V V , we obtain the equation of state e = (T/Tc) t a n l T 1 p - p(l + gp2 + hp4 + L),

(4)

which specifies completely the temperature dependence of the spontaneous (e = 0) order parameter p(0) = pB. (This dimensionless field e should not be confused, obviously, with the structural parameter e, which describes the O3 rotation.)

O3 Tilt and Pb/(Zr/Ti)

Displacement

Order

Parameters

313

The temperature dependence of the inverse dielectric constant can also be obtained easily from the equation of state, Eq. (4), as e _ 1 (T) — (T c /C)(de/dp). 3.16.4.2.

Tilt

Similarly, we can try to describe the temperature dependence of the tilt order parameter in the FRL ferroelectric-antiferrodistortive phase, in which the unit cell is doubled, using a statistical approach as follows. If there are N' interactive unit cells per unit volume, consisting of N[ with the two oxygen octahedra within the unit cell tilted in the sequence (+u>, —ui), and A^j with the two oxygen octahedra tilted in the opposite sequence (—IV,+UJ), the net "staggered" tilt per unit volume 6t of the pseudocubic three-dimensional arrangement of cells is 0t = {N'2 -

N[)2\LO\

= N'2\u\ tanh(|X eff \2\co\/kBT),

(5)

where

\xeS\ = x + f3'et + 1'e3t+6'e5t

(6)

is the generalized (torsional) field and |w| is the absolute value of the rotation angle of a single oxygen octahedron in the unit cell (the other oxygen octahedron within the unit cell will have, automatically, a rotation angle with the same value in the opposite direction). Because the FRL phase presents an antiferrodistortive deformation with respect to the higher temperature rhombohedral phase, FRH, Xeg is an effective staggered field (torsional field in our case), conjugate with the staggered tilt strain. The equation of state in dimensionless variables is x= ( — )

tanh

~ l y - r t 1 + 9tV2 + htr?4 + L),

(7)

where x = X/(5'N'2\u)\; r\ = #t/7V'2|a;|, which can also be interpreted as {e(T))/(e(0K)}, due to the fact that e = tanw/4V3 « co/4,^3; T ct = P'N'(2\Lo\)2/kB, gt = ( 7 '//?')(2H) 2 and ht = (5'/p')N'i(2\uj\)i. Here, T ct < TLH is the effective Curie temperature for the transition involving O3 tilting, which is different from Tc < Tpp, the Curie temperature for the ferroparaelectric transition involving Pb and Zr/Ti displacement or order-disorder orientations. The spontaneous (x = 0) tilt order parameter 77(0) = TJS = e s /e s o is therefore given by Eq. (7) with x = 0. Note that e can be interpreted, in

314

Effective Field Approach to Phase

Transitions

general, as a common tilt for all unit cells (displacive transition) at any given temperature, or as an average tilt (e) for a statistical distribution of the tilts through the lattice (order-disorder transition). Very often, transitions have a mixed displacive/order-disorder character that should, to some extent, be taken into account by a generalized effective field that includes terms with higher order powers, as in Eq. (6). 3.16.4.3.

Polarization-Tilt

Coupling

Let us assume that there is a weak coupling between tilt and polarization, or, in other words, that a unit cell tilted in one specific direction favors atomic displacements in one of the two opposite directions perpendicular to the plane of the tilt. In this case, the total interaction energy to be substituted into Eq. (2) is Wa + Wt = Ees + Xeff2uj, instead of W^ = Eefffi only, and we therefore have

ps + Aps = tanh

wd

wt]

kBT

kBT

tanh

\ w« 1

+ tanh

kBT

1 + t a nh

£T]

tanh

&}-. [&

Taking into account that for E = 0, tanh[Wd/fcBT'] tanh[Wt/fcB?1 = rjs, we obtain directly from Eq. (8) Ap s = ( l + p s 7 ? s ) - 1 ( l - P s ) ' 7 s ,

(8)

ps and

(9)

which relates the increase in polarization to the tilt in a very simple way. The prefactor (1 + PsVs)^1 varies smoothly from 0.5 at T = OK to 1.0 at T = TLH- The second factor (1 — pi) is zero at T = OK and is still much less than unity at T = T L H if TLH is substantially lower than TFP as in the present case. Figure 3.16.3 shows the temperature dependence of the spontaneous polarization and the tilt, in excellent agreement with the observed behavior.

3.16.5.

Discussion and Conclusions

As mentioned above, the temperature dependence of the tilt order parameter, rjs(T), and the polarization order parameter, ps{T), are well described by Eqs. (7) and (4) with x = 0 and e = 0, respectively. In Figs. 3.16.1, 3.16.2(b), and 3.16.3, which display Eqs. (7) and (4), full line indicates equilibrium states (heating) and dashed line corresponds to ideal metastable

O3 Tilt and Pb/(Zr/Ti)

Displacement

Order

Parameters

315

T(K) Fig. 3.16.3. Calculated temperature dependence of the tilt and polarization order parameters for P Z T (x = 0.035) indicating that the extra polarization associated with the tilt Aps is well described, below TLH by Ps = (1 +PsVs)~1 (1 — Ps )*?s, Eq. (9). See text for details on the fitting procedure for TJS(T) and ps(T). Dashed lines indicate the calculated metastable region between low-temperature and high-temperature phases. The dotted line for ps(T) below TLH is the calculated polarization in the absence of tilt.

behavior (cooling). The fitting procedure was the following: the experimental values for Vs(T) = [/ RL (T)// RL (0)] 1 /2 and Ps(T) = P s (T)/P s (0), with initial values for 7RL(0), Tct and P s (0), Tc chosen as discussed below, were substituted, respectively, into t_

(^)tanh-17b(r)/»h(T)-l 9t + htVs

St r_

(10)

(^)tanh-1Ps(T)/ps(T)-l P2s(T)

•9 + hpi

(11)

Here, the experimental values for g't, defined by the actual value of the tilt at a given temperature, are plotted vs. r]2, using as normalized parameters / R L ( 0 ) and T ct . If Eqs. (7) and (4) describe correctly the observed behavior, g[ vs. rj% and g' vs. p2 should result in linear plots, giving automatically (
Effective Field Approach to Phase

316

Transitions

between experimental and calculated values. The values of I R L ( O K ) , T ct needed to obtain rjs(T) and T/Tct, and Ps0,Tc to get PS(T) and T/Tc in Eqs. (10) and (11), are not known experimentally, so the fitting procedure is carried out by changing them to get the best linear fit. The initial values are chosen from the experimental data knowing that J R L ( O K ) « I R L ( 2 0 K ) , T ct < T LH = 323K, P s 0 > Ps (430K), and Tc < TFP = 501.0K. Then, varying T ct and / R L ( 0 ) [fixed in practice because the neutron data include points for 7 R L ( 2 0 K ) KS TRL(OK) with very small statistical error], we proceed to minimize the least-square error of the fit to get a final Tct [see Fig. 3.16.4(a)]. Likewise, experimental values for g', defined by the actual value of the spontaneous polarization at a given temperature, are plotted vs. Pg, using as normalized parameters initial estimates for Pso and Tc. Varying again (within narrow limits) Tc and Pso and optimizing the linear fit, we get final values for Ps0 and Tc [see Fig. 3.16.4(b)]. This procedure leads to R < 10" 4 for I R L ( 2 0 K ) / I R L ( 0 K ) = 0.985 and T ct = 178.2 K with estimated uncertainties of the order of 1%, and to R < 10" 4 for P s 0 = 33.0/uC/cm 2 , Tc = 480.0K (with estimated uncertainties of the order of 5%). These fits resulted in linear plots giving gt = 5.1 ± 0.1, ht = —5.8 ± 0.2 for the tilt order parameter, and g = 0.62 ± 0.25, h = —(O.^IJ;^) for the polarization order parameter. It should be noted that the set of experimental data for TJS(T) covered the range 0.06 < T / T L H < 1-00; while the available set for ps(T) covered only

-4-

•

•

g't from experimental date o( .],(T) Eq.(10]

0.4-

2

— - linear lit: g',= 5.1 - S.8 n,

-80.0

0.3 0.2

0.4

0.6 1,

0.8

1.0

0.2

g' from experimental data of p,(T). £q,(11)

— l i n e a r ft g-= 0 * 3 - 0.15p,J 1

0.3

.

1

0.4

1

1

0.5

1

1

0.6

r

1

0.7

r

0.

P.'

Fig. 3.16.4. (a) Plot of (gQexp = /t[T/T ct ?7 s 2 ], where Tct and / R L ( 0 ) are adjusted by a least-squares fit to get the best linear dependence of g[ vs. r£. IRL(T) are the actual measured values of temperature and integrated intensity. It may be noted that [/RL(0)] ' comes out very close to [i R L (20i<:)] 1 / 2 . (b) Plot of (g() ex p = f(T/Tc,p2s = Ps/Pso) vs. p 2 , where T c and Pso are similarly adjusted. In this case P s o is still somewhat larger than P s (T ~ 450 K), which is the maximum value actually measured. Points corresponding to T > Tpp (transition temperature) have been omitted.

O3 Tilt and Pb/(Zr/Ti)

Displacement

Order Parameters

317

0.85 < T/Tpp < 1.00. This is the main reason for the larger uncertainties in the latter. The occurrence of nonvanishing values gt > 1/3, ht < 0, and g > 1/3, h < 0 in the expressions for 7ys(T) and ps(T) implies a first order (discontinuous) character for the transitions, in agreement with the observed behavior. Physical meaning can be attributed to the effective field coefficients /3 t , 7t(st), St(ht) and /?, 7(5), 5(h) in Eqs. (6) and (3), considering the effective field expressions as multipolar expansions. These expansions include successive dipole-like (long range), quadrupole-like (short range), octupole-like (shorter range) terms, summing up the contributions over the whole lattice. However, detailed calculations of this kind in rhombohedral perovskites are nontrivial. The temperature dependence of the weak peak in e(T) at T « TLH can be described only in a semiquantitative manner within the theoretical approach used here to describe ps(T) at the TRL~TRH transition. The presently available information on the trend of AJ> S (TLH) with composition (x) for x = 0.035,9a; = 0.10,6 and x — 0.40 (Ref. 16) indicates a tendency to smooth out the discontinuity with increasing x. This behavior might also be masked by the increasing compositional inhomogeneity of the samples. It would be interesting to be able to predict theoretically the composition dependence of T L H ( ^ ) and A P S ( T L H ( ^ ) ) in terms of gt(x) and ht(x), but this is not possible at present. In summary, the temperature dependence of the tilt order parameter of PZT with x = 0.035, previously investigated in a narrow range near TLH, 9 has been determined by means of neutron diffraction in the whole temperature range from T = 20 K to TLH = 323 K. The associated polarization change PS{T) a t T « TLH has been determined. It may be concluded that the simple two-order-parameters statistical theory outlined in Section 3.16.4 accounts well for the coupling between tilt and polarization determined by neutron diffraction and dielectric measurements, especially in view of the fact that the data were obtained not from single crystals, but from ceramic samples.

Acknowledgments Financial support from CICyT (Grant no. PB93-1253/94), Iberdrola (Grant no. INDES 94/95), and Comunidad de Madrid (Grant no. AE00138-94) is gratefully acknowledged. Work at Brookhaven is supported by the U.S.

318

Effective Field Approach to Phase Transitions

D e p a r t m e n t of Energy, Division of Materials Sciences, under Contract no. DE-AC02-76CH00016.

References 1. See, for instance, Ferroelectric and Related Substances: Oxides, LandoltBornstein, New Series, Group 3, Vol. 16, Pt. a (Springer-Verlag, Berlin, 1981). 2. G. Shirane, K. Suzuki and A. Takeda. J. Phys. Soc. Jpn. 7, 12 (1952). 3. J.C. Burfoot and G.W. Taylor, Polar Dielectrics and their Applications (Macmillan, London, 1979). 4. H. Barnett, J. Appl. Phys. 33, 1606 (1962). 5. C. Michel, J.M. Moreau, G.D. Achenbach, R. Gerson and W.J. James, Solid State Commun. 7, 865 (1969). 6. A.M. Glazer, S.A. Mabud and R. Clarke, Acta Crystallogr. 5 34, 1060 (1978). 7. A.M. Glazer, Acta Crystallogr. B 28, 3384 (1972). 8. H.D. Megaw and C.N.W. Darlington, Acta Crystallogr. A 3 1 , 161 (1975). 9. B. Noheda, T. Iglesias, N. Cereceda, J.A. Gonzalo, H.T. Chen, Y.L. Wang, D.E. Cox and G. Shirane, Ferroelectrics 184, 251 (1996). 10. B. Noheda, N. Cereceda, T. Iglesias, G. Lifante, J.A. Gonzalo, H.T. Chen and Y.L. Wang, Phys Rev. B 5 1 , 16 388 (1995). 11. X. Dai, J.-F. Lie and D. Viehland, J. Appl. Phys. 77, 3354 (1995). 12. T.R. Halemane, M.J. Haun, L.E. Cross and R.E. Newnham, Ferroelectrics 62, 149 (1985). 13. T.R. Halemane, M.J. Haun, L.E. Cross and R.E. Newnham, Ferroelectrics 70, 153 (1986). 14. J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). 15. D. Berlincourt and H.A. Krueger, J. Appl. Phys. 30, 1804 (1959). 16. A. Amin, R.E. Newnham, L.E. Cross and D.E. Cox, J. Solid State Chem. 37, 248 (1971).

Chapter 3.17

Dielectric C h a r a c t e r i z a t i o n of t h e P h a s e Transitions in Pb 1 _ 2 / / 2 (Zr 1 _ i E Ti a ; ) 1 _ y Nb 2 / 03 (0.03 < x < 0.04,0.025 < y < 0.05)* Ning Duan, Noe Cereceda, Beatriz Noheda and Julio A. Gonzalo Departamento de Fisica de Materiales, C -IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

The Curie temperature and the ideal thermal hysteresis of P b 1 _ v / 2 (Zri_ a; Ti a: )i_j / Nbj / 03 (0.03
3.17.1.

Introduction

Solid solutions of lead zirconate t i t a n a t e P b ( Z r i _ x T i x ) 0 3 (denoted P Z T ) are ferroelectric ceramic materials with perovskite structure which have important practical applications in high-e capacitors, infrared pyroelectric *Work previously published in J. Appl. Phys. 82, (2) (1997). Copyright © 1997. American Institute of Physics. 319

320

Effective Field Approach to Phase

Transitions

detectors, piezoelectric devices, and ferroelectric memories, using their outstanding ferroelectric, piezoelectric, and pyroelectric properties. Early studies were focused on the morphotropic phase boundary (MPB), where the piezoelectric electromechanical coupling factor is very high and shows anomalous behavior. In recent years, efforts have been devoted to the Zr-rich PZT range of compositions, due to the abundant phase boundaries between phases, including orthorhombic antiferroelectric (A0), high-temperature rhombohedral ferroelectric ( F R H ) , low-temperature rhombohedral ferroelectric (JPRL), and paraelectric (Pc) phases. 1-4 In particular, the rhombohedral ferroelectric region has attracted considerable interest. The -FRL-^RH phase transition was first discovered by Barnett in 1962.5 The structures of the FRH and FRL phases were later reported by Glazer and Mabud. 6 In these phases one finds rotations and distortions of the oxygen octahedron (FRL phase) and only distortions (FRH phase), respectively, in addition to the cation shifts. Later, several studies in the FRL and the FRH structures were performed.7~9 These results indicated that the increase of the spontaneous polarization near the F R L - F R H phase transition is related to the O3 tilt coupling to cation shifts along the (111) axis, and depends on the composition (Ti content) and the dopant content. Along with the structural research, other physical properties at the FRL-FRH phase transition have also been investigated. 10 It was found that the F R L - F R H transition has the following features: (a) the transition temperature is close to room temperature; (b) there are small variations in dielectric constant through the transformation; (c) there is a considerable pyroelectric effect during the transition (AP « 2 fiC/cm , the pyroelectric coefficient being ~ 1 0 - 7 / i C / c m 2 °C, higher than the usual value, which is around 10~& fiC/cm2 °C). Applications based on the above outstanding properties of the F R L - F R H phase transition are, for example, direct energy conversion from heat to electricity through the FRL-.FRH phase transition 11 ' 12 and its use in infrared detectors. Recently, Zr-rich PZT thin films with F R L - F R H phase transitions have been reported to possess prospective potentialities in the fields of IR detectors, energy converters, and uncooled imaging systems. 13 On the other hand, the use of different dopants allows the enhancement of specific properties and therefore results in wider applications. In the case of Nb doping, for instance, substitution of a small amount of Nb 5 + for Zr 4 + (or Ti 4 + ) increases the bulk resistivity and reduces aging effects, because of the difference in chemical valence between N b 5 + and Zr 4 + , which creates a considerable amount of lead vacancies. 14-17 The phase diagram of

Dielectric Characterization

of the Phase

Transitions

321

Nb-doped Zr-rich PZT has been precisely measured. 18 " 20 The results show that small amounts of Nb dopant shift the ferro-para phase boundary down, and displace the F R L - F R H phase boundary to the left. In this investigation, the composition dependence of the -FR.L_-PC and FRL-FRK transition temperatures and the thermal hysteresis that accompanies the transitions have been investigated in a set of compositions with x in the range of (0.03, 0.04) and y in the range of (0.02, 0.05). The results have been analyzed in terms of the generalized effective theory used previously to study the composition dependence of T(x) for samples with a fixed small amount of Nb (y = 0.026). It may be noted that an important motivation for this work was to investigate the changes in the transition temperature of Zr-rich PZT with varying content of Ti and Nb, since the latter substantially enhances the resistivity of the material, which has important consequences in connection with its use as an energy converter of thermal into electrical energy. A larger internal resistance results in larger power output being obtainable. The range of Nb content has been chosen to get FRL^-PRH transitions in a narrow range not much higher than room temperature. 3.17.2.

Experiment

The samples were prepared by the conventional ceramic process in Shanghai Institute of Ceramics, China. They were calcined at 850° C for 2h and sintered at 1340°C for 2h, then polished into thin disks (thickness of 1mm, diameter of 1.49 cm) for later use. Dielectric constant data, capacitance, and dissipation factor, were measured at regular temperature intervals separated by about 0.1°C, the temperature being controlled by an Eurotherm Temperature Controller/Programmer with a high-resolution ramp (Model 903P/IS/HDV) by means of an automatic Hewlett-Packard Precision LCR Meter (Model 4248A) with an accuracy of better than one part in 10° at a frequency of 1kHz. The field amplitude was 8.0V/cm. The heating and cooling rates were of the order of 20°C/h, between room temperature and 300°C.

3.17.3.

Results and Discussion

Figure 3.17.1 shows the behavior of the inverse dielectric constant with temperature for different compositions around the FRL — Pc transition. From

322

Effective Field Approach to Phase

Transitions

(100-x)/x/y O 97.0/3.0/2.6 • 97.0/3.0/3.9 A 97.0/3.0/5.0

o.o-

T

8.0-

O D 6.

96.5/3.5/2.6 96.5/3.5/3.9 96.5/3.5/5.0

O D A

96.0/4.0/2.6 96.0/4.0/3.9 96.0/4.0/5.0

f

200

240

320 I ' l O

Fig. 3.17.1. Dielectric behavior vs. temperature for various P b i _ y / 2 ( ^ i - x T ^ i x ) i - y NbyC>3 compositions with 0.03 < 0.04 and 0.02 < 0.05 for the F R H - P c phase transition.

these data the ferroparaelectric phase transition temperature, T*, and the Curie temperature, T c , can be obtained directly. The temperature at which e _ 1 begins to decrease nonlinearly upon heating is denned as T*. The slope of e~l vs. T above T* was fitted in a narrow temperature range near T*. The Curie temperature, T c , is subsequently obtained extrapolating linearly £ _ 1 (T) —> 0. It can be seen that Tc and T* decrease with the increase of Nb content, y. The experimental results for T*, T c , and AT = T* - Tc, corresponding to different niobium contents, are presented in Table 3.17.1 along with the dependence on Ti content. This notation for the extrapolated Curie-Weiss temperature (Tc) and the transition temperature (T* > T c ) is the same as that used in our previously published work 21 but differs from that used by other authors, which is To and T c , respectively, for the Curie temperature and the transition temperature.

Dielectric Characterization

of the Phase

323

Transitions

Table 3.17.1. Experimental data for T*, T c , AT (all in K) for different P b ! _ a / 2 ( Z r 1 _ x T i x ) 1 _ H N b „ 0 3 (0.03 < x < 0.04,0.026 < y < 0.05) compositions. (100 -

y (mol%)

x)/x

3.9

2.6

97/3 96.5/3.5 96/4

5.0

T*

Tc

AT

T*

Tc

AT

T*

Tc

AT

503.5 501.7 502.8

489.9 491.2 493.9

13.6 10.5 8.9

487.8 489.3 494.5

473.3 475.1 478.4

14.5 14.2 16.1

478.1 484.8 492.5

458.7 470.1 479.7

19.4 14.7 12.8

We may note that there is some diffuse character in the transition peaks, increasing with the Nb content, which may be attributable to small inhomogeneities in composition. 22-24 However, measurements at different frequencies below 1 MHz show that the peak temperature is not seriously affected by the change in frequency, which may be taken as an indication that the diffuse character is not very pronounced in our samples. It allows us to determine the transition and Curie temperatures within a moderate error range. Estimated asymmetric error bars, which vary from composition to composition and are larger when the Nb content is large, are given in the figures. Figure 3.17.2 shows the dielectric constant and losses factor for the -FRL-^RH transition. The temperature corresponding to the dielectric constant and the losses factor peak value is well defined as the low-high transition temperature, TLH, but the corresponding -FRL-JFRH "Curie temperature," TCT, associated with a hypothetic Curie-Weiss behavior at T > TLH J cannot be obtained directly from our data. Taking into account that TLH increases with the increase of Nb and Ti content, TCT could be expected to increase with increasing Nb and Ti content, but this point needs further study. The behaviors described in Fig. 3.17.1 and Table 3.17.1 can be analyzed in the framework of the generalized effective field theory, 25 using an effective field Eeff = E + (3P + 7 P 3 + 5P5L,

(1)

where E is the external field, P the polarization, and /?, 7, 5 are constants, i.e., temperature-independent coefficients. With the substitution of the dimensionless variables e = E//3Nfi, normalized field; p = P/Nfi, normalized polarization; Tc = BNfi2/he; 2 2 4 g = (S//3)N fi ; and h = (<5//3)iVV > where N is the dipoles number per

324

Effective Field Approach to Phase

* 750

W.,5/3.5/i

«

W..5/3.V3«>

<•

• K . J . * 5/2 6

450

20

40

W)

8ft

100

12(1

VK120

^

y

t-.M).

550

Transitions

<>0

9K!/i

'

or-mo

j

»

97:5/2.6

/

/

/ /

X> J^ 40

•

SO

UKI

U0

20

40

c.O

Ml

UK)

120

120

20

40

00

SO

100

120

<• % M / S 0.04

"

W4/3'»

%.5/3.V3.9

*

%/4/2.ii

96.5/3 5/>fc

H A^ H 0.02

T \\V ^*&®

11.01-

iKKl20

40

(.0

SO

100

120

T(C)

20

40

(.0

80

100

T(Q

T(C)

Fig. 3.17.2. Dielectric behavior and losses factor vs. temperature for the same composition as in Fig. 3.17.1 at the •FR.L _ ^ ? RH phase transition.

unit volume, fx the dipole moment per unit cell, andfeethe Boltzmann constant, we get the following equation of state for a pure system: e = — tanh lp - p(l + gp2 + hp4 H

).

-t c

(2)

For e = 0, p = ps (spontaneous polarization), the above equation is rewritten as Tc

l+gpj tanh

1

+ hp4 ps/ps

(3)

If we consider a mixed system, like Nb-doped PZT, g and h become composition dependent, as we will discuss below. Thus, Eq. (3) can be used to describe the composition dependence of the FRH~-PC phase transition temperature. 3.17.3.1.

Composition Dependence Phase Transition

of the

FRU-PC

We have pointed out, just before Eq. (2), that Tc for a ferroelectric compound has a linear dependence on the number of dipoles per unit volume,

Dielectric Characterization

of the Phase

Transitions

325

N. So, for a mixed system, a linear dependence of Tc on the molar content, x or y, of the components can be supposed, if we consider the parameters involved, {&} and {/J-i}, to be approximately constant in relatively narrow ranges of x and y. The compositions that we have investigated in this work are so close to each other that this constancy can be reasonably assumed. Tc is, then, expressed by means of linear relationships in x and y. As a first approximation, 21 Tc(x,y) =Tc(0,y)[{l-x)

+ Axx],

y = constant,

(4)

Tc(x,y)=Tc(x,y)[(l-y)

+ Ayy],

x = constant,

(5)

where x and y are fixed values of x and y, respectively. The coefficient Ax is given by Ax = T c (l, 0)/T c (0,0) = .j3TNT^l'(3ZNZH2Z where the subscripts T and Z correspond to pure PbTiC>2 and PbZrC>3, respectively. In such a way we are sure to obtain the Tc of the pure compounds in the extremes of the phase diagram (x = 0,1). No such simple relation can be written for Ay, due to the reasons commented upon below. However, Ay can be understood as a fitting parameter analogous to Ax. In Fig. 3.17.3(a), the experimental values of Tc(x,y) vs. x are represented by different symbols along with the linear least-squared fits, plotted as full lines. It may be noted that the slopes are very similar for y = 0.026 and y = 0.039, which indicates that the coefficient Ax is almost composition independent for y < 0.04. Table 3.17.2 gives numerical values for Ax extracted from the experimental data fits. For Ay we cannot confidently use Eq. (5) in the whole composition range (i.e., 0 < y < 1) because it is known that PbNb2C>6 has the tungsten-bronze structure, which is quite different from the perovskite structure. Above the transition temperature, its paraelectric phase is not cubic but tetragonal. This means that Eq. (5) has no meaning in the high Nb range. However, we may expect that small Nb constants (< 4mol%, in our case) do not substantially change the perovskite structure, and, consequently, that we can use Eq. (5) to describe the dependence of Tc(x, y) vs. y in the low Nb constant range. The numerical values for Ay obtained by fitting the experimental data are also given in Table 3.17.2, where we find this coefficient is not too composition dependent for y < 0.04. The experimental data for Tc(x, y) vs. y and their linear least-squares fits are plotted in Fig. 3.17.3(b) by means of different symbols and full lines, respectively. If we take into consideration the effects of x and y, simultaneously, on Tc, the general expression for Tc(x,y) is given, combining Eqs. (4) and (5), by Tc(x, y) = T c (0,0) • [(1 - x) + Axx][(1 - y) + Ayy}.

(6)

326

Effective Field Approach to Phase

(a)

500

i

1

'

Transitions

"•

*-

i

T

. - • • ' •

460 -

>

• • *

• • • • • • ' ! ' " " " " '

i

^0.026 ,= 0.039 y=0.050 •

Tc(x,y)=Tc(0,y)!(1-x)+AJ<x] 0.035

0.040

(b) Tc(x,y)=Tc{x,0)[(1-y)+Ayy]

h460

• • *

x= 0.030 x= 0.035 x= 0.040

0.02

Fig. 3.17.3. Plot of the Curie temperature Tc vs. x (a) and y (b). The full lines are best fits to Eqs. (4) and (5), which are combined into Eq. (6). Points with different symbols are experimental data.

Table 3.17.2. The calculated values for Ax and Ay Tc(x,y)=Tc(0,0)x [(l-x) + Ax[(l-y) + Ayy]. y (mol %)

x (mol %) 3.0 3.5 4.0

in

1.84 2.12 4.78

2.6 3.9 5.0

-3.27 -3.36 -3.44

Dielectric Characterization

of the Phase

327

Transitions

We may conclude that Eq. (6) describes fairly well the x and y dependence of the Curie temperature for Nb-doped Zr-rich PZT. On the other hand, we have defined, just before Eq. (2), the dimensionless parameter g(x,y), which determines the more or less pronounced first order character of the transition, 26 and is dependent on N2, the squared number of dipoles per unit volume. Then, in an analogous way to that for Tc(x,y), we get g(x,y) = g(0,y)[(l-x)2+Bxx2},

y = constant,

(7)

x = constant,

(8)

g(x, y) = 5 (0,0) [(1 - xf + Bxx2] [(1 - yf + Bvy2}.

(9)

2

g(x,y) = g{x,0)[{l-y)

2

+ Byy ],

and

Note that x and y are small in all of our cases, so the higher order terms involving xy, x2y, xy2, and x2y2 in the expansion of Eq. (9) are considerably smaller than the terms in x and y, unless the coefficients Bx and By are unexpectedly large. Neglecting terms of order higher than second order in x and/or y we get g(x, y) 9* 5 (0,0) [l - 2(x + y) + 4xy + (Bx + l)x 2 + (By + l)y2},

(10)

which indicates that, as a first approximation, g(x,y), unlike Tc(x,y), should show a linear dependence on (x+y) in the range of our investigation. Similarly, for parameter h, we have h(x, y) = M0,0) [(1 - x) 4 + Cxx4} [(1 - yf + Cyy4}, 2

2

h(x, y) =* h(0,0) [1 - 4(x + y) + 16xy + 6(x + y )],

(11) (12)

which indicates that, as a first approximation, h(x, y) should show a linear dependence on (x + y) for x, y
Effective Field Approach to Phase

0.00

0,05

0 15

0-10

Transitions

0.20

y

Fig. 3.17.4. Calculated composition dependence of the normalized spontaneous polarization, p* = P*/Pso, at the -FRH^-PC phase transition temperature: (a) p* vs. x for several y values; (b) p* vs. y for several x values. Dashed lines are indicative because the y value is beyond our expected range of approximation.

in higher AT/TC at loss x values, where our theoretical curves depart most from the data. Substituting Eqs. (10) and (12) into Eq. (3) at the FRH to P Q transition temperature (T*) we get the relationship between (x,y) and p*, plotted in Fig. 3.17.4(a) and (b). Also, we can get numerically the calculated (x, y) dependence of AT/TC, which is plotted in Fig. 3.17.5(a) and (b). Full lines correspond to 5(0,0) = 0.59 and /i(0.0) = -0.12, Bx = 1.04, and By = -1.70 for y = 0.026, 0.039, 0.050 and x = 0, 0.05, 0.10, respectively. Also plotted are our experimental data and additional experimental data from Ref. 21. 3.17.3.2.

Comments

on the FRL--FRH Phase

Transition

As we have mentioned before a small amount of Nb content can change significantly the -FRL--FRH transition behavior and other characteristic

Dielectric Characterization

of the Phase

Transitions

(a)

329

"T~ D y= 0 026 O y= 0.039 A y= 0 050 • ref pi](y=C0?6: calculated.

t °,C

(b)

i

—y-

0.05-

<

>

D O A

!

I iJ

0 04-

0 03-

•

Vt
0 02-

x~0 x=0.0f>

^ \ / 0.01-

0 00-

x= 0.030 >.-- 0.035 x- 0.040 let [21] calculated

*^v

x=0 1C

•-'
—,—.— -,

r-^f—^i—^.

— I — 025

Fig. 3.17.5. Plot of AT/Tc vs. (x, y) from Eqs. (3), (10), and (12) for different compositions (lines) together with the experimental d a t a and those from Ref. 21: (a) AT/TC vs. x for different y values; (b) AT/TC vs. y for different x values. Dashed lines are indicative because the y value is beyond our expected range of approximation.

properties such as the dielectric constant, the spontaneous polarization jump (AP S ), and the phase transition order. It is known that the spontaneous polarization discontinuity at the -FRL--FRH phase transition for 95/5 type PZT ferroelectric ceramics arises due to the cation (Pb and Zr/Ti/Nb ions) shifts along the (111) direction, because of the coupling effect between oxygen's octahedra tilt (T?S), and the spontaneous polarization (P s ). The spontaneous polarization discontinuity can be affected by the Ti and Nb content. What we are interested in is how the composition, x and y, affects the APS, and the transition temperature.

330

Effective Field Approach to Phase

Transitions

Based on the generalized effective field approach, the spontaneous polarization change can be shown to be a function of ps (normalized spontaneous polarization) and rjs (normalized spontaneous tilt angle) as follows27: Ap s = ( l + p s 7 7 s ) - 1 ( l - p s 2 ) ? ? s .

(13)

In this equation, ps and rjs are related to temperature through 28 T_ = 1+gpl + hpj Tc

tanh_1ps/ps

and

J^_ = l+gtr£ + htVit TCT

(15)

t a n h - 1 T]s/r]s

respectively. In principle, T c , T C T, g, h, gt, and ht could be given in terms of the composition, x and y, in a similar way to that discussed above for the FRH-PC transition. So, Eq. (13) could, in theory, describe the composition dependence of the spontaneous polarization discontinuity. We lack, however, enough experimental information to get it numerically. Sufficiently detailed measurements of AP S for different compositions have not yet been performed. This is left for further study. At present we have only dielectric constant data, shown in Fig. 3.17.2, for different compositions. From them we can see that increasing Ti or Nb content increases, as is well known, the temperature at which the anomaly of the dielectric constant corresponding to the F R L - ^ R H transition appears. An increase in the diffuse character with the increase of Nb content can also be seen. In Fig. 3.17.6, the FR.L_-FRH transition temperatures are plotted versus (x + y), with x being the Ti content and y the Nb content, for the compositions investigated in this work. The experimental data from Ref. 21, which correspond toy = 0.026, and the phase boundary line for pure PZT 2 9 (y = 0) have also been included. It can be seen that the F R L ^ ^ R H phase boundary is almost unaffected by the substitution of Nb(y) instead of Ti(cc), for small amounts of y (< 5mol%). This dependence supports the idea 20 that the Nb 5 + ions substitute Zr 4 + ions and that they are not located at interstitial positions. 19 This is also consistent with the observed 27 weak coupling between O3 tilt (responsible for the -FRL-JFRH transition) and the polarization.

Dielectric Characterization

I

ouu —

I

' p

£

200-

I

'

1

c

\

mperat i

Transitions

1

\

I

01

of the Phase

1

100-

F

y (% mol) 2.6

A

3.9

•

5.0

o

2.6 (ref.21)

V

0-

RH

I

4

y 1

(ref.29) -

^0 J

\

i-

•

•

0

\ \

331

^

J

F

RL

1

8 x+y (%mol)

1

12

'

Fig. 3.17.6. The effects of Nb contents on the F R L _ - F R H phase transition temperature TLH- Different symbols correspond to different Nb contents and coincide with Jaffe's data without Nb content (full lines). This means that the F R L - F R H transition temperature for small Nb content (in our case < 5 mol%) is well defined by the curve corresponding to pure PZT.

3.17.4.

Summary

In summary, in the range of small amount of Ti(z) and Nb(y) content, the Curie temperature of the Nb-doped PZT system can be described fairly well within the framework of an effective field approach for a combination of Ti and Nb content in the range x < 1, y < 1. The thermal hysteresis at the F R R - P C transition can be described by means of higher order coefficients (g,h,...), determinant of the phase transition character. The experimental errors in the determination of AT are considerable due to the diffuse character of the transition. The behavior of the dielectric constant as a function of Ti(x) and Nb(y) at the F R L - F R H transition has been characterized experimentally in the narrow range 0.03 < x < 0.04, 0.026 < y < 0.05, and the data indicate that the anomaly at T L H becomes smoother and shifts toward higher temperature as the Nb content increases. The amount of shift in the phase boundary

332

Effective Field Approach to Phase Transitions

due to Nb doping (y) is equivalent to t h a t due to Ti(x) substitution, a fact t h a t is related t o t h e secondary role of cation displacements in this transition. Our results provides useful information for choosing compositions for specific applications.

Acknowledgments We wish to acknowledge financial support from C I C y T (Grant no. P B 9 3 1253/94), Iberdrola (Grant no. INDES 94/95), Comunidad de Madrid (Grant no. AE00138-94), and C E A O (UAM).

References 1. K. Roleder and J. Handerek, Phase Transitions 2, 285 (1982). 2. Y.L. Wang, Z.M. Cheng, Y.-R. Sun and X.-H. Dai, Physics B150, 168 (1988). 3. M.J. Haun, E. Furman, S.J. Jang and L.E. Cross, Ferroelectrics 99, 13, 27, 45, 55, 63 (1989). 4. V.A. Isupov, Ferroelectrics 143, 109 (1993). 5. H.M. Barnett, J. Appl. Phys. 33, 1606 (1962). 6. A.M. Glazer and S.A. Mabud, Acta Crystallogr. Sec. B 34, 1060 (1978). 7. X.H. Dai, J.F. Li and D. Vieland, J. Appl. Phys. 77, 3354 (1995). 8. D. Viehland, J.F. Li, X.H. Dai and Z. Xu, J. Phys. Chem. Solids 57, 1545 (1996). 9. B. Noheda, T. Iglesias, N. Cereceda, J.A. Gonzalo, H.T. Chen, Y.L. Wang, D.E. Cox and G. Shirane, Ferroelectrics 184, 251 (1996). 10. X. Dai and Y.-L. Wang, Phys. Status Solidi A 124, 435 (1991). 11. J.A. Gonzalo, Y.L. Wang, B. Noheda, G. Lifante and M. Koralewski, Ferroelectrics 153, 347 (1994). 12. D. Xunhu and Y.L. Wang, Ferroelectrics 109, 253 (1990). 13. R.C. Buchanan, J. Huang and J.E. Sundeen, Electroceramics V, Aveiro, Portugal, 2-4 September 1996 (unpublished), pp. 309. 14. R. Gerson and H. Jaffe, J. Phys. Chem. Solids 24, 979 (1963). 15. S. Takahashi, Ferroelectrics 41, 143 (1982). 16. M. Hand, G.E. Kugel, J. Handerek and Z. Ujma, Ferroelectrics 135, 101 (1992). 17. W.L. Warren, J. Robertson, D.B. Dimos, B.A. Tuttle and D.M. Smyth, Ferroelectrics 153, 303 (1994). 18. Y.L. Wang, New Ceram. 11, 87 (1990) (in Japanese). 19. L. Benguigui, J. Solid State Chem. 3, 381 (1971). 20. Z. Ujma, D. Dmytrow and M. Pawelczyk, Ferroelectrics 120, 211 (1991). 21. B. Noheda, N. Cereceda, T. Iglesias, G. Lifante, J.A. Gonzalo, H.T. Chen, Y.L. Wang, D.E. Cox and G. Shirane, Phys. Rev. B 51, 16388 (1995).

Dielectric Characterization of the Phase Transitions 22. 23. 24. 25. 26. 27. 28. 29.

333

M. Yokosuka and M. Marutake, Jpn. J. Appl. Phys. 1 25, 981 (1986). V.A. Isupov, Ferroelectrics 143, 109 (1993). A.A. Bokov, Solid State Commun. 90, 687 (1994). J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). N. Cereceda, N. Duan, B. Noheda and J.A. Gonzalo, in Ref. 13, p. 193. N. Cereceda, B. Noheda, T. Iglesias, J.R. Fernandez-del-Castillo, J.A. Gonzalo, N. Duan and Y.L. Wang, Phys. Rev. B 55, 6174 (1997). N. Duan, N. Cereceda, B. Noheda and J.A. Gonzalo, Ferroelectrics Lett. Sect. 22, 27 (1996). B. Jaffe, W.R. Cook Jr. and H. Jaffe, Piezoelectric Ceramics (Academic, London, 1971).

Part 4

Some Applications to Ferroelectrics: 1998-2005

Chapter 4.1

Scaling and Metastable Behavior in Uniaxial Ferroelectrics* J.R. Fernandez del Castillo, B. Noheda, N. Cereceda and J.A. Gonzalo Department of Material Physics, C-IV, Autonomous University of Madrid, 2804-9 Madrid, Spain

T. Iglesias Department

of Physics, Brookhaven National New York 11973, USA

Laboratory,

Upton,

J. Przeslawski Institute

of Experimental Physics, University of Max Born Sr. 9, 50-205, Poland

Wroclaw,

Improved experimental resolution and computer-aided data analysis of hysteresis loops at T ~ Tc in uniaxial ferroelectrics triglycene sulfate (ordinary critical point) and triglycine selenate (quasitricritical point) show that scaling holds in a wide range of scaled fields spanning many orders of magnitude, well beyond the asymptotic region (\T — Tc\ —> 0), and that the behavior of metastable experimental points (E < 0, P > 0) approaches the theoretical branch of the respective scaling equation as T->TC.

4.1.1.

Introduction

As is well known, the revival of experimental and theoretical interest in cooperative phenomena and continuous phase transitions, especially in "Work previously published in Phys. Rev. B 52, (2) (1997). Copyright © 1997. The American Physical Society. 337

338

Effective Field Approach to Phase

Transitions

liquid-vapor, order-disorder alloys, and magnetic systems, has stimulated a vast amount of work during the last three decades. 1 This work includes more accurate sets of measurements close to the transitions than in previous works, as well as the development of new theoretical concepts, such as scaling, multicritical points, and the renormalization group theory. Ferroelectric phase transitions, 2 in particular transitions in uniaxial order-disorder ferroelectrics like triglycine sulfate (TGS), have been shown3 many years ago to present classical exponents (ft = 1/2, S = 3, 7 = 1, a = 0) and to obey a classical (mean-field) equation of state. A prominent characteristic of this kind of transition is the role long-range dipolar interactions play in them. More recently,4 another uniaxial ferroelectric of the TGS family, triglycine selenate (TGSe), was found to exhibit classical tricritical exponents (ft = 1/4, S = 5, 7 = 1, a = 1/2) and to follow a quasitricritical equation of state in which the tricritical exponents were used instead of the former classical critical exponents. In this work, we undertake, under substantially improved 3 ' 4 experimental conditions (automated data acquisition and analysis), digital resolution in P, E, and T, a study of the equation of state of both uniaxial ferroelectrics, TGS (ordinary critical point) and TGSe (quasitricritical point). This have been done in order to (a) elucidate possible systematic deviations from asymptotic scaling, (b) investigate the metastability 5 region at T < Tc, and (c) to detect the presence, or lack of it, of logarithmic corrections 6 of pure scaling in the pattern displayed by the data. In this work, we present data that are substantially improved with respect to those given in previously published work. 3,4 The improvements include higher accuracy, a broader temperature range examined, and a much closer set of points giving the P vs. E behavior at T very close to T c , both above and below it. These improvements are important to investigate the evolution of scaled data away from the close vicinity of Tc, where asymptotic scaling was previously detected, and allow us to explore the metastable branch (E < 0, P > 0) of the P vs. E data, not previously investigated.

4.1.2.

Experiment

Samples of different thicknesses and areas were cut from good optical quality single crystals of TGS and TGSe grown from water solution. Symmetric and easily saturated loops were obtained with plates cleaved perpendicular to the ferroelectric 6-axis and electrodes with goldleaf at the main

Scaling and Metastable Behavior in Uniaxial Ferroelectrics

339

surfaces, which were about 0.5 cm 2 in surface and 1 mm in thickness. The measurements were performed on a TGS sample of dimension 0.500 cm 2 x 0.140 cm and a TGSe sample of dimension 0.105 cm 2 x 0.044 cm (electrode area x thickness, respectively). The temperature of the sample was controlled using a temperature controller (Unipan 680) capable of producing very slow linear heating and cooling ramps ( ~ l K / h in our case). Hysteresis loops were obtained using a Diament-Drench-Pepinsky 7 (DDP) circuit, with phase compensation only through change of auxiliary resistance. The loops were recorded in a relatively wide temperature interval encompassing Tc for both crystals [TC(TGS) = (321.470±0.005)K, T c (TGSe) = (294.683 ± 0.005) K] (calculated from the fit to the scaling equation of state), using a digital oscilloscope (Nicolet NIC-310). For each loop, 4000 points were recorded at each temperature, which insured high resolution of the P vs. E, hysteresis loops below T c and nonlinear curves above Tc. It may be noted that special precautions must be taken to get good quality, symmetric, hysteresis loops. Phase compensation, achieved by varying the auxiliary external resistance in the DDP circuit, must be performed as close as possible to T c , because the resistivity of the crystal is temperature dependent through the transition. In this way, the undesirable effects on PS(T) and EC(T) due to under-compensation or overcompensation can be minimized. The small asymmetry present in most loops, resulting from the small bias due to inhomogeneous distribution of charged impurities, should be taken care of, first, by annealing the sample for 20 h at T = 70°C, well above T c , and, second, by shifting the center of the loop to the true center of symmetry. This was achieved with a simple iterative computer program to ensure that \+Ec\ = \— Ec\, \+Ps\ = \—Ps\ with Ps and Es the spontaneous polarization and the coercive field, respectively. Also important is the choice of frequency for the driving field, which should be low (to approach equilibrium conditions) but not too low in order to avoid excess ionic conduction. We have used a frequency around 50 Hz for most measurements. The final, but very important factor, is the unavoidable thermal gradient from the lower to upper surface of the sample that must be minimized. The estimation of this gradient for our experimental setup gave the value of ~0.01K/cm. 4.1.3.

Results

Figures 4.1.1(a) and 4.1.2(a) show typical hysteresis loops for TGS and TGSe in the vicinity of the transition. Thicker lines point which part from

340

Effective Field Approach to Phase

I

'

I

'

I

'

I

'

I

-1.0 -A8 -0.5 -0.3 0.0 0.3 E(V'on)

(a)

'

Transitions

I

05

'

I

'

0.8 "W3

2.5-

o- 1.0-

(b)

-0.2

0.0

0.4 E(VJcm)

0.2

0.6

0.8* 1»3

Fig. 4.1.1. (a) Digital hysteresis loop for TGS just below the transition. Thicker line shows the points from each loop that are shown in (b) and are used as raw data in what follows, (b) P (polarization) vs. E (field) at various temperatures close to T c for triglycine sulfate (TGS) at 307 < T < 323 K. Only 10% of the experimental points at each temperature are shown. Only one-fourth of the recorded temperatures are shown for clarity of presentation. Vertical lines indicate intervening regions in which the data are little affected either by rounding effects or imperfect phase compensation (see text).

each loop is taken to plot Figs. 4.1.1(b) and 4.1.2(b), which show the set of P{E) below and above T c , for TGS and TGSe, respectively. The P(E) values for E < 0, down to the inflection point in P vs. E correspond to metastable states. The curves corresponding to T « Tc were determined, as shown below (Figs. 4.1.3 and 4.1.4), from P3(E) for TGS and P5(E) for TGSe. This allowed us to identify the critical and the tricritical isotherm,

Scaling and Metastable Behavior in Uniaxial Ferroelectrics

I -2

'

I ' 1 0 2 E

341

««»

I ;;#**'• •

(H)

i ' i ' 3 4 e (V / c m )

7X10 3

Fig. 4.1.2. (a) Digital hysteresis loop for TGSe just below the transition. Thicker line shows again the points from each loop that are shown in (b) and are used as raw data in what follows, (b) P (polarization) vs. E (field) at various temperatures close to Tc for triglycine selenate (TGSe) at 284 < T < 298 K. Only 10% of the experimental points at each temperature are shown. Only one-fourth of the recorded temperatures are shown for clarity of presentation. Vertical lines indicate intervening regions in which data are little affected either by rounding effects or imperfect phase compensation (see text).

respectively, through the observation of the straight lines which passes through the origin (E = 0, P = 0). In our opinion, such plots are important since they give a direct, unmanipulated experimental hysteresis loop. In Figs. 4.1.3 and 4.1.4, we present P3(E) for TGS and P3(E) for TGSe, respectively, which result in straight tines for E values higher than certain temperature-dependent threshold, and allow the identification of the

342

Effective Field Approach to Phase

0

Transitions

100 200300400500600 700 800900 1000 E(V/cm)

Fig. 4.1.3. Normalized polarization up to the third power (p 3 ) vs. field (E) at various temperatures (for 321 < T < 322.1 K) close to T c , defining the ordinary critical isotherm (T = Tc = 321.470 K). The actual temperature corresponding to the experimental data ( P vs. E) closest to T c is indicated. Vertical lines show the range of fields used for the linear fit of the critical isotherm.

0.12

I ' I ' I ' ) ' I ' I 0

1000 2000 3000 4000 5000 6000 7000 8000 EfWcm)

Fig. 4.1.4. Normalized polarization up to the fifth power ( P 5 ) vs. field (E) at various temperatures (283.2 < T < 298K) close to Tc, defining the tricritical isotherm (T = T c = 294.683 K). The actual temperature corresponding to the experimental data ( P vs. E) closest to T c is indicated. Vertical lines show again the range of fields used for the linear fit of the critical isotherm.

Scaling and Metastable Behavior in Uniaxial Ferroelectrics

343

critical and tricritical isotherms, respectively, which pass through the origin (E = 0,P = 0). In Figs. 4.1.5 and 4.1.6 plot of Ps2 vs. T for TGS and Ps4 vs. T for TGSe, respectively, gave a linear dependence, as it is to be expected for ordinary critical and tricritical (or quasitricritical) behavior. Nevertheless, the exact determination of Tc becomes somewhat more problematic in this 3-i

TGS Te(Ps 2 ) = 321.331K E = 928.5 V/cm f = 50Hz

Fig. 4.1.5.

Squared spontaneous polarization (P s 2 ) vs. T for TGS.

300 TGSe Tc(Ps 4 )= 295.282 K

E = 7272.7 V/cm f=50Hz

300 T(K)

Fig. 4.1.6.

Fourth power of the spontaneous polarization (P s 4 ) vs. T for TGSe.

344

Effective Field Approach to Phase

Transitions

case, because deviation from linear behavior due to the lack of perfect phase compensation in the loops. 4.1.4.

Equations of State

The equation of state for an uniaxial ferroelectric, which should follow a classical Landau behavior asymptotically, 3 can be extended using a generalized effective field expanded in terms of odd powers of the polarization EeS = E + 0fP + jtP3

+ SfP5 + • • • ,

(1)

where E is the external field, P the polarization, and (3{, 7f, <5f, are constant coefficients depending only on the geometry of the crystal lattice and the charge distribution within a unit cell.4 For a ferroelectric crystal with N unit dipoles (/i) statistically oriented along the ferroelectric axis, with N\ in the direction of the field and N2 in the opposite direction (N1+N2 = N), the polarization is given3 by P=(N2-N1)^

= N^tanh(j^j,

(2)

where fee is Boltzmann's constant and T the temperature. From this relationship it is straightforward to get the equation of state in terms of reduced variables (e = E/Es0, Es0 = /?iV>; p = P/Ps0, Pso = M) as T

e = — t a n h " 1 ^ ) - (1 + gp2 + hpA + • • • )p,

(3) Pi

Pi

1

and, expanding tanh p in powers of p, which is especially useful for a subsequent investigation of the asymptotic equation (e < 1, p C 1, (T Tc)/Tc
I

+

+

k

'- ir' {li-'y {lrc- y

+

--

(4)

This is the general equation of state for a dipolar uniaxial ferroelectric, valid at the vicinity of ordinary as well as quasitricritical points. It is analogous to the equation derived from Landau's theory, 5 but not identical because it contains specific temperature dependences of the coefficients of the successive powers of the polarization.

Scaling and Metastable Behavior in Uniaxial Ferroelectrics

345

For TGS (ordinary critical point, g < 1/3), Eq. (4) can be written in the scaling form as ±p +

(G

T

$C[(T-Tc)/Tc,p]

where e = e / | T - T c / T c | 3 / 2 and p = p/\T -Tc/T^l2 ables, and

1 (T~TC

$C[(T-Tc)/Tc,p] =

3 V Tc

±

\p

(5)

are the scaled vari-

I'9

T-Tr

1 (T-T< ~5 Tc.

(6)

In Eq. (5), the ± signs correspond to the T > Tc branch and the T < Tc branch, respectively, and <J>C = [(T/Tc)Tc,p\, which becomes important only for T below and away from T c , changes from $ c [0,p] —> 0 at T « Tc to $c[—1,1] —» const at T w 0, always for e ^C 1. It may be noted that ESQ = PN/x is in practice much larger than the breakdown field, and therefore the behavior at e « 1 is not relevant experimentally. For TGSe (quasitricritical point, g « 1/3), on the other hand, the term in p 3 disappears, and Eq. (4) should be written in scaling form as ±P-

1 T 57;

<M(T-TC)/TC)P]U5,

where e = e/|(T - T c )/T c | 5 / 4 , p = p/\(T - Tc)/Tc\l'\ $tc[(T-Tc)/rC)p]

"1T-T C ± [5 Tc ~h 1 /r-Tc\i 2 p" 7 \ Tc ) \

(7)

and

[(H «}•

/T-Tc (8)

Note that scaled variables and $ tc [(2 1 - Tc)/Tc,p] are now defined differently. Again, in Eq. (7) the ± signs correspond to the T > Tc branch and T < Tc branch, respectively, and &tc[(T — Tc)/Tc,p], which also becomes important only for T < Tc and toward the low-temperature region, goes from $ c [0,1] -> 0 at T « Tc to $ c [ ( - l ) , p ] - • const, at T RS 0, always for e < 1.

346

Effective Field Approach to Phase

Transitions

Fig. 4.1.7. Scaled data l n p vs. l n e at T < Tc, including metastable data {E < 0, P > 0) and at T > T c , for TGS (307 < T < 323K), where e = e / | ( T - T c ) / T c | 3 / 2 and p s p/\(T — TcVTc] 1 ' 2 are the scaled field and the scaled polarization, respectively. The continuous curve is the asymptotic equation of state [Eq. (9)] for an ordinary critical point. The dashed line shows the expected asymptotic behavior as T —> OK. Note that experimental data collapse on the asymptotic equation of state in a wide range (more than ten orders of magnitude in scaled field and four orders in scaled polarization).

Figure 4.1.7 gives lnp vs. lne for TGS, together with the asymptotic equation of state corresponding to an ordinary critical point e = ±p+(^-g\p3

(g = 0.24 ±0.05).

(9)

It can be seen that for T 0, p —> 1 (P s —> Pso)> and p goes to unity, which results in lap going to zero. For T > Tc, the fit to the asymptotic equation (9) is almost perfect in the whole experimental range. It may be noted that the set of experimental points further up in the graph corresponds to a particular temperature extremely close to T c , and that the nearest sets at both sides are substantially away form Tc, in comparison. A sequence of sets at more closely spaced temperatures would have closed the gap between neighboring set of isothermal points in the graph.

Scaling and Metastable Behavior in Uniaxial Ferroelectrics

347

Fig. 4.1.8. Scaled data l n p vs. lne, at T < Tc, including metastable data (E < 0, P > 0) and at T > T c for TGSe (284 < T < 298K), where e = e/\(T - T c ) / T c | 3 / 2 and p = p / | ( T — TcJ/Tcl 1 ' 2 are the corresponding scaled field and scaled polarization, respectively. The continuous curve is the asymptotic equation of state [Eq. (10)] for a tricritical point. The dashed line shows the expected asymptotic behavior as T —> OK. Again, the asymptotic equation is valid in a wide range of field and polarization.

Likewise, Fig. 4.1.8 gives, for TGSe, lnp vs. lne, wherep and e are now defined in the way appropriate for tricritical point behavior, and shows the asymptotic equation of state corresponding to a tricritical point. e = ±p + \\-h)f

{h = 0.14 ±0.05).

(10)

Also in this case the agreement between theory and experiment is good and scaling holds very well for all our data obtained for T > Tc. For the data obtained below T c , the deviation from the asymptotic equation is smaller than in the previous case, as expected. Note that experimental data in both cases (TGS and TGSe) collapse on the asymptotic equation of state in a wide range (more that six orders of magnitude). The coefficient of p5 changes again gradually toward one as p goes to unity with T —> 0.

4.1.5.

Metastable Behavior

Figures 4.1.7 (TGS) and 4.1.8 (TGSe) also include points corresponding to the metastable behavior (E < 0, P > 0) shown in the loops (see

348

Effective Field Approach to Phase

Transitions

Figs. 4.1.1 and 4.1.2). It is well known that, due to the forward and sidewise motion of domain walls,8 the ferroelectric coercive field, which determines the metastability region, is orders of magnitude lower than the ideal (thermodynamic) coercive field. It may therefore be expected that, as Tc is approached from below, the "contrast" between domains decreases, the effective field at the domain boundaries approaches asymptotically to the ideal (bulk) effective field value. If this is so, we may expect that the scaled data corresponding to the metastable portions of the loops approach the metastable branches of the scaling equations, ~e = -p+(^-g)p3

forTGS,

(11)

-e = -p + (\-hjp5

forTGSc,

(12)

which are represented as ln|p| vs. ln|e| in Figs. 4.1.7 and 4.1.8, returning from the left and going down. It can be seen that the shapes of the curves defined by the data are similar to those defined by Eqs. (11) and (12), and that there is a clear tendency in both sets of data to move toward the metastable branches of respective scaling equations.

4.1.6.

Discussion

We may conclude that scaling holds in uniaxial ferroelectric TGS (TGSe) at points substantially further away from critical (tricritical) point than previous investigations. 3 ' 4 Deviations from asymptotic scaling for T < Tc take place, gradually, toward the expected behavior when T —> T c , in accordance with analytic expression, given by Eqs. (5) and (7), obtained within a generalized effective field approach. For points in metastable regions (E < 0, P > 0) the data show a clear tendency, both for TGS and TGSe, to approach the metastable branches of the corresponding scaling equations (solid line in Figs. 4.1.7 and 4.1.8). No logarithmic corrections 6 are clearly visible in our scaled data. Such corrections (predicted a long time ago9) are important for uniaxial ferroelectrics but have not yet been fully elucidated experimentally by hysteresis loop measurements. This might be taken to imply that the role of fluctuations is less important in uniaxial ferroelectrics at T sa Tc than that of other nonlinear electric or elastic interactions in the crystals. Around Tc ± 1K the estimated error of polarization for a driving field amplitude is less than 3%,

Scaling and Metastable Behavior in Uniaxial Ferroelectrics Table 4.1.1. Crystal

Tc(K)

Scaling constant for uniaxial ferroelectrics TGS and TGSe. C(K)

Eso/Pso (uC/cm2)

TGS TGSe

321.47 294.68

349

3650 4050

4.2 4.5

4TTTC

(°C)

9

h

0.24 «l/3

0.14

(10 6 V/cm) 4.41 2.89

1.16 0.76

1.10 0.91

which sets an upper limit for possible logarithmic corrections. Nevertheless, as it was previously mentioned it is quite difficult to achieve perfect phase compensated loops at T ~ T c . To the best of our knowledge this is the first experimental work exploring the metastable behavior at T w T c in uniaxial ferroelectrics. Table 4.1.1 summarizes scaling d a t a for T G S and TGSe. T h e respective Curie constants are from previous dielectric constant measurements. 3 , 1 0 , 1 1 T h e consistency between /3 = ES0/PSQ and /3 = (AirTc)/C is much better t h a n in the previous works. 3 ' 4

Acknowledgments We acknowledge the financial support of C I C y T (Grant no. PB93-1253) and the Comunidad de Madrid, Project no. AE00138/94. One of us (J.P.) t h a n k s D G I C y T for financial support for a sabbatical (SAB94-0084) at the Ferroelectric Materials Laboratory, UAM.

References 1. See, for example, H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971). 2. See, for example, N. Goldenfeld, Lectures on Phases Transitions and the Renormalization Group (Addison-Wesley, Reading, MA, 1992). 3. See, for example, F. Jona and G. Shirane, Ferroelectric Crystals (Pergamon, New York, 1962). 4. J.A. Gonzalo, Phys. Rev. B 1, 3125 (1970). 5. T. Iglesias, B. Noheda, B. Gallego, J.R. Fernandez del Castillo, G. Lifante and J.A. Gonzalo, Europhys. Lett. 28, 91 (1994), and references therein. 6. See, for example, L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd edn. (Pergamon, Oxford, 1980). 7. A. Aharony and M.E. Fisher, Phys. Rev. B 8, 3342 (1973). 8. H. Diamant, K. Drenek and R. Pepinsky, Rev. Sci. Instrum. 28, 30 (1957).

350

Effective Field Approach to Phase Transitions

9. M.J. Cabezuelo, J.E. Lorenzo and J.A. Gonzalo, Ferroelectrics 87, 353 (1988). 10. A.I. Larkin and D.E. Khmelnitskii, Zh. Eksp. Teor. Fiz. 56, 2087 (1969) [Sov. Phys. JETP 29, 1123 (1969)]. 11. T. Iglesias, B. Noheda, G. Lifante and J.A. Gonzalo, Phys. Rev. B 50, 10307 (1994).

Chapter 4.2

Energy Conversion with Zr-Rich Lead Zirconate/Titanate Ceramics* Ning Duan, Noe Cereceda, Beatriz Noheda, Jose Raul Fernandez-del-Castillo and Julio A. Gonzalo Departatnento de Fisica de Materiales, Universidad Autonoma de Madrid, 28049 Madrid, Spain

The conversion efficiency and power output to convert thermal energy to electrical energy by means of a ferroelectric-ferroelectric phase transition has been investigated. The material used was the ceramic mixed system lead zirconate/titanate (PZT) with a high Zr-enriched composition (Zr/Ti = 97/3%). We have studied the performance dependence on sample thickness and working frequency, the main relevant parameters for a given PZT composition. The observed optimum values of efficiency and power output are analyzed in terms of the two main characteristic relaxation times of the process, the thermal time (sample thickness dependent) and the electric time (thickness independent). Possible ways of improving the overall performance are discussed.

Direct thermal t o electric energy conversion using Zr-rich lead zirc o n a t e / t i t a n a t e (PZT) ferroelectric ceramic plates by thermal cycling at the phase transition between t h e two rhombohedral ferroelectric phases, FRL (low-temperature phase) and F R H (high-temperature phase), has been investigated by several a u t h o r s . 1 " 5 In these works, the advantages of using these kinds of phase transitions for energy conversion applications have been already pointed out, in comparison with other better known ferroelectric transitions. 6 ~ 8 Ferroelectric energy conversion with poled P Z T , through the fRH-i 7 RL first order transition occurs between two phases with nonzero *Work previously published in Appl. Phys. Lett. 77, 13 (1997). Copyright © 1997. American Institute of Physics. 351

352

Effective Field Approach to Phase

Transitions

polarization, and is the only demonstrated case in which no external fields are needed, unlike cases in which the ferroelectric-paraelectric (F-P)7 or the ferroelectric-antiferroelectric (F-AF)6 transitions are used, requiring external fields for cyclic operation. In these investigations, a polarized sample subject to thermal cycling is connected to an external load. By heating and cooling the sample around the -FRH-^RL phase transition temperature (7LH); positive and negative free charges are produced at the electrodes of the ceramic plate, due to the temperature induced change in spontaneous polarization, which is measured as a voltage signal through the load resistance. It is well known that not all thermal energy transmitted to the ferroelectric plate is converted into electricity, apart from the thermodynamic limitation, r\ < rjc — A T / T m a x , because some of the thermal energy is wasted in mechanical expansion/contraction of the material, heating metallic plates in contact with the ceramic, and other processes.9 The actual efficiency and the power output could be, however, considerably improved in several ways, including the design of materials especially adequate for this purpose, combining a small dielectric constant and a large polarization change at the transition, and the so-called "multistage" operation. 10 It consists in using a set of plates with changing composition and profiting from the wasted energy from one sample for heating the next sample with a slightly lower transition temperature. In this way, a set of PZT plates with gradually changing compositions {x\ < X2 < ••• < xn), having transition temperatures T L H ( ^ I ) < T\Jft{x2) < • • • < Tu$(xn), could be used. First, however, a better understanding of the process itself taking place during conversion in our system (PZT plate + experiment setup) is necessary. In the present work, we have investigated the dependence with the sample thickness and the frequency of the output power and the efficiency of plates with a single Zr-rich PZT composition. The experimental apparatus used is the same as in Ref. 4, but several improvements were made and the dynamic range was extended to lower frequencies. The heating source was an IR lamp and the temperature oscillations were created by a chopper. The center and the width of the temperature window could be reliably controlled by changing the lamp intensity as well as the frequency of the chopper, driven by a low-speed motor (range from3mHz to 0.11 Hz). The composition of the thin plates used in the present work was PZT 97/3 + 3.9mol%ofNb 2 0 5 , which has a transition temperature T LH « 49°C, with a polarization change of about A P « 2/xC/m 2 .

Energy Conversion

with Zr-Rich Lead Zirconate/Titanate

Ceramics

353

The specific output power is given by energy/cycle ^out —

-,

IV J —

,

\L)

volume v where energy/cycle is the total output energy per cycle, / is the frequency, and v is the sample volume. The efficiency of the conversion process can be obtained through V

=

^ut = Ein

CIVdt „ IoAtlVdt fi:::vpvCvdT~vpvCvAT>

(2) {)

where pv is the density, C p the average specific heat per unit mass over the working temperature window, AT = T m a x — T m ; n , and At = 1 / / is the time for one cycle. For a given lamp intensity, the temperature interval, AT, can be decreased by increasing the chopping frequency in such a way as to fix it to the place where the maximum change of polarization occurs, increasing, in this way, the efficiency [see Eq. (2)], which must be low enough to allow most of the charge to be, first, liberated at the electrodes after the heating/cooling period and, then, released through the load resistance. Here, the two characteristic times of the process 10 are the electrical relaxation time (r e = pee = RC, where R is the optimized load resistance, equal to the sample resistance, and C is the sample capacitance), depending only on the material resistivity, pe, and its dielectric constant, e, and the thermal relaxation time (-rth = pvCpd2/kth), which depends on the density pv, the average specific heat Cp, over the working temperature window the thermal conductivity kth, and the thickness of the sample d. Then, for a given material, the optimum working frequency can be tuned to the sample thickness. In principle, the lower the r t h the higher the working frequency and the efficiency. However, we know that VoUt = Q/C = APsd/e, where Q is the charge liberated at the surface of the sample, C the capacitance of the sample, APS the change in spontaneous polarization, d the thickness, and e the dielectric constant. Note that PZT around TLH has a conveniently very low s(T) value in comparison with e(T) near the Curie point, contributing to a higher output voltage. We can expect, as it was found, that, for the same composition, V^,ut increases linearly with the sample thickness. The total characteristic time of the process can be estimated from the full-width at half-height of the voltage peak versus time obtained with a frequency low enough to observe the full-peak real shape. These values

354

Effective Field Approach to Phase

• 0-

Transitions

observed T fitting: t= 25 d 2 + 6.4 T

e+Xr

8_

_

^

^ •

6-

V^r 4 - V0.0

i '" 0.1

0.2

0.3

r

'

~T

0.4

d (mm) Fig. 4.2.1. Total optimum characteristic time observed vs. sample thickness. The full line represents the best fit r t o tal = 6.4 + 25d 2 (T to t a i in seconds and d in mm) to the data. The dotted line marks the level of all other contributions to the characteristic time (independent of d), including the electrical characteristic time, Te.

are represented vs. thickness in Fig. 4.2.1. The only expected thickness dependence comes through the contribution of the thermal characteristic time, proportional to the squared thickness (rth oc d2). Then, we can easily extract from Fig. 4.2.1 the observed value of r t o t — Tth, fitting the experimental data to Ttot = Ad2 + B (A = 25s/mm 2 , B = 6.4s), where the first term represents Tth and the second term represents the contributions of the other characteristic times of the process (independent of d). This value (6.4 s) is much higher than the expected electrical timer (re = pe « 0.8 s). We call residual time, r r , to the additional contributions to the total relaxation time, after discounting r e , which are mainly related to the indirect heating of the sample and to the thermal inertia of the whole system (setup + sample). Figure 4.2.2 gives the pout as a function of frequency for three different thickness (0.1, 0.2, and 0.4 mm) in the range of 3 < / < 110 mHz. It is shown that the optimum pout and efficiency correspond to a frequency of 0.04 Hz and a thickness of (0.2 ± 0.1) mm. Under these conditions, we have obtained a peak voltage of ±83 V, an energy density per cycle of 7.5mJ/cm 3 , a specific output power of 0.75mW/cm 3 , and an efficiency given by m = T]/r]c = 0.01, r/c being the Carnot efficiency. From the data in Fig. 4.2.1, we have obtained a r t h = 25, which gives a r t h = I s for a 0.2 mm thickness, of the same order as r e , as can be expected near optimum conditions.

Energy Conversion with Zr-Rich Lead Zirconate/Titanate Ceramics 1.0

( • •

•

'

1

1

'

A

/

*

> * \_ \

355

1

0.4mm 0.2mm 0.1mm

» \

0.5-

0L,

i 0.0 0.00

•

A =

* **"*&. •

i

0.02

'

(

l

0.04

0.06

'

l

0.08

'

—

i

—

0.10

•

—

i

0.12

f(Hz) Fig. 4.2.2. Specific power output vs. frequency for three different thicknesses.

There exists room for further improvement of the o u t p u t power and the efficiency through a synchronized discharge operation. In it, charge is accumulated u p to the m a x i m u m (in an open circuit), t h e n the circuit is closed, and the charge is released t h r o u g h a proper load resistance, just before the chopper changes to another half cycle. Preliminary results show a m a x i m u m o u t p u t voltage 1.6 times t h a t of the value obtained by means of a spontaneous discharge operation. This figure can be improved varying the ratio open circuit t i m e / s h o r t circuit time. In conclusion, it is perhaps somewhat p r e m a t u r e to make a final evaluation of ferroelectric materials as an energy converter, in view of the as yet relatively low efficiencies found. We note, however, t h a t a better knowledge about the conversion process in Zr-rich P Z T samples has improved by a factor of 10 the efficiency of the specific power o u t p u t . This fact may be indicative t h a t further improvements are possible by modifying the operation mode (multistage/multicomposition operation, synchronized discharge operation) and, perhaps more important, by identifying materials with larger A P S , accompanied by moderate A T and low e ( T ) .

Acknowledgments T h e authors are grateful to Eliaz Rodriguez and the Spanish C I C y T (Grant no. PB93-1253) and I B E R D R O L A .

356

Effective Field Approach to Phase Transitions

References 1. J.Y. Lian, Masters Thesis, Shanghai Institute of Ceramics, Shanghai, China, 1985. 2. X.H. Dai and Y.L. Wang, Ferroelectrics 107, 253 (1990); X.H. Dai, Ph.D. Thesis, Shanghai Institute of Ceramics, 1990. 3. Noe Cereceda, Masters Thesis, Universidad Autonoma de Madrid, Madrid, Spain, 1996. 4. J.A. Gonzalo, Y.L. Wang, B. Noheda, G. Lifante and M. Koridewski, Ferroelectrics 153, 347 (1994). 5. B. Noheda, Ph.D. Thesis, Universidad Autonoma de Madrid, Madrid, Spain, 1996. 6. R. Briot, P. Gotthard and M. Troccaz, Silic. Ind. 2, 33 (1982). 7. R.B Olsen, J. Energy 6, 91 (1982); Ferroelectrics 40, 17 (1982). 8. W.H. Clingeman and R.G. Moore, J. Appl. Phys. 32, 675 (1961). 9. A. Van der Ziel, J. Appl. Phys. 45, 675 (1974). 10. J.A. Gonzalo, Ferroelectrics 11, 423 (1976).

Chapter 4.3

Microscopic Characterization of Low-Field Switching in Ferroelectric TGS* Carmen Arago, J.R. Fernandez del Castillo, Beatriz Noheda and Julio A. Gonzalo Departamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Low-field ( < l k V / c m ) switching has been investigated for thin plates (0.06 < d < 0.15 cm) of triglycine sulfate by means of field pulses with a linear rise time. The dependence of the maximum current density, jm, with the field value Em at which this maximum occurs is j m = B i ( £ m - . E c w i ) 3 / 2 , for 0.3 < Em < 0.5kV/cm, and j m ~ B 2 ( £ m - . E c w 2 ) , for 0.5 < -Em < l.OkV/cm. This is satisfactorily explained taking into account the dominant role played by sidewise and forward domain wall motion, respectively.

4.3.1.

Introduction

Ferroelectric switching in triglycine sulfate (TGS) using external field pulses has been investigated extensively by various authors 1 ~ 4 since the late 1940s. While the high-field behavior was satisfactorily explained 5 by random bulk switching of the individual dipoles, the low-field behavior was only tentatively explored in terms of forward and sidewise domain wall motion driven by the action of the switching fields with rectangular pulses. More recently, the switching kinetics, especially in ferroelectric ceramics with prospective applications for computer memories, have been investigated in "Work previously published in J. Appl. American Institute of Physics.

Phys.

357

84, 7 (1998). Copyright © 1998.

358

Effective Field Approach to Phase

Transitions

great detail. 6 ^ 8 The range of fields investigated in PZT thin films is high (200-1000 kV/cm) and the switching time dependence of systems such as the two-dimensional Ising model 7 and the finite size Kolmogorov-Avrami 8 model were also investigated. 4.3.2.

Experimental

In this work, we are just concerned with low fields, below 1 kV/cm. A Hewlett-Packard generator, model 33120A, has been used to apply alternatively negative and positive rectangular field pulses, E{t), which are amplified with a Kepco bipolar amplifier, model BOP 1000M, capable of producing, pulses of ±1000V and ±40 mA. As rectangular pulses are gradually amplified they get an almost linear initial variable rise time, as shown in Fig. 4.3.1, and we may consider the applied field as E(t) = (E0/t0)t for the initial relevant part of the pulse. The switching current density, j(t), is also shown in Fig. 4.3.1. As this curve can be approximated to an isosceles triangle, 9 a simple relation between the maximum current density, jm = j(tm), and the switching time, ts, can be obtained at a fixed temperature as / Jo

m
ts

4^ 3m

t(ms) Fig. 4.3.1. Plot of a positive field pulse, E(t), after amplification, and the corresponding j(t), indicating the meaning of Em = -E(t m ) and j m = j(tm).

Microscopic

Characterization

of Low-Field Switching in Ferroelectric TGS

359

where Ps is the spontaneous polarization. Both j m = j(tm) and Em = E(tm) have been observed and measured from a Hewlett-Packard oscilloscope, model 54603B. Three samples were cut from single TGS crystals, grown from water solution. Sample dimensions were di = 0.15cm, A\ = 1.1cm2; d% = 0.08 cm, A2 = 0.36 cm 2 ; and d3 = 0.06 cm, A3 = 0.64 cm 2 . The ferroelectric axis was parallel to the direction of the applied field. All the experiments were performed at RT (T ^ 20°C).

4.3.3.

R e s u l t s a n d Discussion

Figure 4.3.2 shows the resulting curve j m vs. Em for the three samples. The observed data display the expected low-field behavior, which is very similar qualitatively to that observed with squared and shorter pulses by Fatuzzo and Merz 3 in another range of field values. In the initial regime (I), the maximum current density, jm(Em), grows faster than linearly with Em from a threshold domain wall switching coercive field, Ecw, to E^, E^ being the field corresponding to the transition from regime I to regime II. In the subsequent region (II), jm(Em) grows linearly with Em, i.e., for Em > ££j, up to a much higher field, at which a change to bulk random switching eventually would take place.

40-

• o

•

0.15cm 1.1 cm2 0.08 cm 0.36 cm2 0.06 cm 0.64 cm2

o 0

30-

0

o E E

my

20-

10-

00.2

m

,"o« "^

H«J*

0

o •

•

• • (II)

•

(1)

,

.

1

0.6

Em(kV/cm) Fig. 4.3.2. Maximum switching current, j m , vs. field, Em, for three TGS sample plates, showing the different behaviors at regimes I and II.

360

Effective Field Approach to Phase

Transitions

To describe microscopically the behavior depicted in Fig. 4.3.2, we assume that domain wall switching proceeds from small remnant counter polarized domains at one surface and that both forward and sidewise motion are involved. We assume as well that the sidewise domain wall velocity, u(swm) = Vi(E), has a field dependence different from that of the forward domain wall velocity, u(fwm) = V2{E). At low fields above the threshold Ecvr (known to be much smaller than Ecb, the random bulk coercive field), the tips of preexisting microdomains are expected to arrive first at the opposite surface. The lateral growth of the counter polarized domains is then expected to proceed more slowly (v\ < t^)At a somewhat higher field, E^, the tips of the growing microdomains arrive quickly at the opposite surface at about the same time as the lateral walls of neighboring microdomains (which are considered to be spaced at roughly regular intervals 10 ) meet each other. From E^ upwards the sidewise domain wall motion is expected to became so fast that the lateral growth keeps pace with the forward motion of the broad tip, causing the switching process to be controlled by the forward domain wall motion that proceeds at a lower rate (v\ > 1*2). In the following we make this argument quantitative, starting from the general rate equations that give (dPd/dt) in response to a field E = (Eo/to)t, and we show that good consistency with the observed low-field switching behavior is obtained. dA^ being the number of dipoles per unit of volume that contributes to the switching in a time interval dt, the general rate equation is given by 5

dt

T

[v

^

a;

\T

Es0

J

Here, (JVb + Na) = N is the total number of dipoles per unit of volume in the whole crystal, 1/r is the transition probability for E —> 0 at T > Tc (Curie temperature), E is the external field, (3P& the polarization field, and Es0 = /3Nfj, the saturation field, with /3 = 47rTc/C, the mean field coefficient, C the Curie constant, and /J, the elementary dipole moment. Since at low fields all switching processes take place at domain walls no screening effects need to be considered. It must be noted that, for domain wall switching, only the two monolayers of unit cells in contact at the wall will participate in the switching process. So, we call A^ w the number of dipoles per unit volume susceptible

Microscopic

Characterization

of Low-Field

Switching

in Ferroelectric

TGS

361

of being aligned with the field, and iVaw the number of dipoles per unit volume already aligned. Therefore, iVbw + ^ a w = Nw will be much smaller than N = l/vc, vc = abcsin$ being the volume of the unit cell. Here, a — 9.15 A, b = 12.69 A, and c = 5.73 A are the dimensions of unit cell10 and /? = 105°, the monoclinic angle. Particularizing the general rate equation for domain wall switching is all we need to do to get the switching current time evolution. Then, considering quasi-cylindrical domain walls surrounding the initially grown up microdomains (from the bottom to the top of the crystal plate), n 2n[f(t) + s]d' V the density of dipoles of the outer monolayer to be aligned by the field, is only slightly larger than iV,bw

N

Jv

—

aw —

V

~2nf(t)d~ sb

the density of dipoles of the inner already switched monolayer. We have used in these expressions n = A/2ir(fm)2 as the number of prepolarized nuclei in the surface area A of the plate, f being the maximum radius of the growing domains. V is the total volume of the sample (V = Ad), d is the thickness of the sample, and s = (ac sin /3) 1 / 2 a kind of average dimension of the unit cell perpendicular to the ferroelectric axis. Therefore, considering that r(t) ^> s, JV W = JV,bw

NK

2n

V

2-Kf(t)d sb

iV,bw

and taking into account that Em + 0wPdn tanh x = x, we can write iVibw

•NK

Nhv/ - Na

tanh

Tr. E„

Pw*>dn

E.s0

NK

n

/2TRA

V\~)


which is sufficiently larger than one for domain wall switching /3W -C 1 (as it is shown later), and allows us to neglect the second term in Eq. (1). Substituting iVw and multiplying this equation by fi, we get

m =dPdft) di

JVW(*)M

sinh/^+/wn E.s0

(2)

To get Eq. (2) in final form we need to put Nw(t) in terms of Pd(t). In an interval dt the net increase in the number of dipoles switched by the

362

Effective Field Approach to Phase

Transitions

field is n 27rdrSr V vr '

6NW = and, consequently,

(3) <

*

-

$

&

)

'

»

•

Integrating Eq. (3) from t = 0 to t = t (t
7

r > «

= -PdW-(-Ps) =

P s ( l - ^

which connects f(t) with Pd(i) and allows us to put down the factor 7Vw(t)/x as 2n2nd

fif(t)

=

4M

TrnATX^/P^172' i46

Nfi 1/2

«"- tr-o-t

1/2

(4)

where M is a time-independent factor. The switching current, dP<±/dt, is obtained in final form substituting Eq. (4) into Eq. (2) as

m_^_^_my^m±gm).

(5)

Making use of the fact that j m [ £ m ( t m ) ] corresponds to the maximum switching current for t — tm, Em = Em(tm) = (Eo/to)tm, which implies 'd2Pd dr2

— Uj

clTj X — t u i j

the time derivative of Eq. (5) leads directly to 1

Pd,

1/2

-,1/2

( l/2)(dP d /dt)m(l/P B )

^(TcEm+pwPdm\ J_ \T Es0

B) m^m,

Eso

J

(6)

Microscopic Characterization of Low-Field Switching in Ferroelectric TGS

363

which substituted into Eq. (5) for P
jm(Em)

T

(T, \T

-1/2

(l/2)(imim)/Ps (Em + Pwjmtm)/Es0_

1/2

Tc Em + /? w P d T Eso

(7)

We can distinguish two regimes in the behavior of (I) Em
jm(E„

1/2

ip\ M (% T \T ) \2f3wJ

jm(Em)

= Bi(Em

fN^2fEm-Ecwl^/2 \PS

E.sO

— -E cw i) ' ,

(8)

where (3wPdm has been identified with Ecv,i, the domain wall coercive field for this regime. This is the field dependence, jm(Em), found experimentally in Fig. 4.3.2 for Em < E*m. (II) Em » /3wjmtm resulting in jm(Em)

M T

Er,

E,c w 2

B2 {Em — Ecv,2)

(9)

-^sO

now with the same field dependence found experimentally in Fig. 4.3.2 for Em > -EmIn Fig. 4.3.3 the experimental data for one of the samples are shown together with their fits to the Eqs. (8) and (9). From these we get E^ = 0.5kV/cm, By = 199.83(mA/cm 2 )(kV/cm)- 3 / 2 , Ecvrl = 0.32kV/cm, and B2 = 55.79 (mA/cm 2 )(kV/cm) and Ecvl2 = 0.22kV/cm, respectively. This allows an estimation of a domain wall mean field coefficient, as given, by /3W = E*J{jmtm) = E*J{2PS) = 0.0001, which is dimensionless in esu CGS units. This is much smaller than the bulk effective mean field coefficient for TGS, 0 = 4irTc/C = 1.1, as might have been expected, because the cooperative effect of the unit dipoles on one side of the wall compensates to a large extent that of the unit dipoles on the other side. We have used the corresponding basic parameters (see Ref. 9, p. 62, Table 6.1) for TGS crystal in order to determine the frequencies T\ and T2 for the microscopic jump probabilities at both regimes and we find they are «10 1 3 Hz, which is of the order of low-lying optical frequencies in TGS.

364

Effective Field Approach to Phase

Transitions

40d = 0.08 cm

o

experimental data

s = 0 36 om2

35-

fits

30)m " % ( E m" E e*2>

25-

£ 20-

E

1510-

Jm = 8 1< E m- S cwl)

f

5-

0 -

1 0,2

•

rt s^ 1 • 1 0.3 0.4

!•'

|

0.5

'1

|

1

0.6

|

0.7

>

1

OB

'

l

0.9

•

1

Em(kV/cm) Fig. 4.3.3. Experimental data as in Fig. 4.3.2 for the sample with thickness 0.08 cm. The continuous curves represent the fits obtained with the functional dependencies of jm = Bi(Em - E d ) 3 / 2 , for Em < Em, and j m = B2(Em - Ecw2) for Em > Em.

The energy barriers for switching are in both cases small as might be expected. We may note as well that in the expression of B\ and £?2 there is a weak dependence on the factor (n/A)1/2, through the factor M, which may point out the relevance of the density of preexisting microdomains at the surface of the crystal, related to the growing conditions of a given sample. This would explain the differences in Fig. 4.3.2 for the three samples studied. In subsequent measurements we have also observed that several factors as temperature, humidity, and even the process of measure itself may produce differences in the location of E^, but the general behavior for the two regimes remains unchanged. Summarizing, we have seen that a simple microscopic description of low-field switching in TGS, showing a distinct change of regime in the field dependence of the maximum switching current, is satisfactorily carried out starting from the general rate equation, using common sense microscopic considerations. The resulting expressions for jm(Em) predict well the observed behavior, and a specific temperature dependence, which will be the object of future works. It may be noted that this approach may be useful to study low-field switching in ferroelectric thin films.

Microscopic

Characterization

of Low-Field Switching in Ferroelectric TGS

365

Acknowledgments T h e authors t h a n k Elias Rodriguez for his help with the experimental setup, and J. Przeslawski, B. Mroz, and N. Cereceda for helpful suggestions. Financial support from C I C y T , Grant no. PB96-0037, is gratefully acknowledged.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

C.F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1742 (1958). A.G. Chynoweth and W.L. Feldman, J. Phys. Chem. Solids 15, 225 (1960). E. Fatuzzo and W.J. Merz, Phys. Rev. 116, 61 (1959). B. Binggeli and E. Fatuzzo, J. Appl. Phys. 36, 1431 (1965). M.J. Cabezuelo, J.E. Lorenzo and J.A. Gonzalo, Ferroelectrics 87, 353 (1988). J.F. Scott, L. Kammerdiner, M. Parris, S. Traynor, V. Ottenbacher, A. Shawabkeh and W.F. Oliver, J. Appl. Phys. 64, 787 (1988). H.M. Duiker and P.D. Beale, Phys. Rev. B 4 1 , 490 (1990). H. Orihara and Y. Ishibashi, J. Phys. Soc. Jpn. 61, 1919 (1992). J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications in Ferroelectrics (World Scientific, Singapore, 1991), pp. 175-199. F. Jona and G. Shirane, Ferroelectric Crystals (Pergamon, New York, 1962), pp. 48-55.

Chapter 4.4

Brillouin Scattering Studies of Ferroelectric Triglycine Selenate Sound Velocity Versus Uniaxial Pressure at T ~ T c * J.R. Fernandez-del-Castillo and J.A. Gonzalo Departamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain B . M r o z a n d Z. Tylczynski Institute of Physics, A. Mickiewicz Umultowska 85, 61-614 Poznan,

University, Poland

Brillouin spectroscopy was used to study the influence of uniaxial pressure ab, applied along the b polar direction, on transverse polarization fluctuations in ferroelectric TGSe single crystals. The experiments were performed with ab varying from 0 to 8 MPa at temperatures from 285 to 305 K. It was found that increasing pressure shifts the phase transition temperature toward lower values, producing a slowing down of the relaxation process at T ~ TQ. It was also possible to induce the ferroelectric to paraelectric phase transition at a constant temperature by varying the uniaxial pressure.

4.4.1.

Introduction

Analysis of Brillouin spectra from ferroelectric crystals undergoing o r d e r disorder transition is very most attractive method to study t h e so-called •Work previously published in Physica B 262, 433 (1999). Copyright © 1999. Elsevier B.V. 367

368

Effective Field Approach to Phase

Transitions

transverse fluctuations of the order parameter, i.e., the coupling of elastic properties with spontaneous polarization. Since the acoustic mode is piezoelectrically coupled to these fluctuations, the strain related to the phonon propagation induces local changes of polarization, which in turn affect the sound velocity and the attenuation. Thus, the information about polarization relaxation time can be easily obtained from the anomalous changes of the Brillouin line shift Av and its true width T going through the transition temperature. Several papers have been published on Brillouin scattering studies of relaxation processes in various order-disorder ferroelectric systems like TGS, 1 ' 2 TGSe, 3 RBHSO4,4 P b H P 0 4 , 5 and NH4HSO4.6 The common features of the relevant results at the paraelectric to ferroelectric phase transition are the steps down in the phonon velocities propagating in the directions perpendicular to spontaneous polarization and associated changes in maxima of the line width. The relaxation behavior associated with Ps could then be well described within the Landau-Khalatnikov theory 7 giving a linear temperature dependence of the inverse relaxation time, r = A(Tc-T)^ 1 , with the slopes A of the order o f l O ~ n s K . Most of the pressure studies on ferroelectric systems have been made with hydrostatic pressure, which allows a much higher pressure applied without cracking the sample. The results obtained for different orderdisorder ferroelectrics undergoing continuous phase transitions can be summarized as follows: (i) the transition temperature depends linearly on the applied pressure; (ii) the reciprocal dielectric constant at the paraelectric phase follows the Curie-Weiss law and the Curie constant is almost pressure independent; and (iii) close to the phase transition the square of spontaneous polarization is a linear function of both temperature and pressure. 8 The pressure dependence of the phase transition temperature (die/dp) is usually of the order of a few K/kbar and its sign depends on the signs and relative values of contributions (dTc/dau where £) an = p for i = 1-3) given by the uniaxial stress applied along the main crystallographic directions. The influence of uniaxial pressure on the elastic properties of TGS has been studied by Ramirez et al. using the Brillouin scattering technique. 9 Koralewski et al.,10 Stankowska et al.,11 and Hamed et al.12 reported on the changes of dielectric properties of TGSe and DTGSe crystals under the action of uniaxial pressure. In this work, we present the results of combined uniaxial pressure and temperature in the Brillouin scattering spectra of ferroelectric (NH 2 CH2COOH) 3 H 2 Se04 (TGSe). At about 295 K TGSe undergoes a

Brillouin Scattering Studies of Ferroelectric Triglycine Selenate

369

phase transition within the monoclinic system from the centrosymmetric prototype point group 2/m to the ferroelectric 2 one. The experiments have been performed with the uniaxial pressure (at,) and temperature T varying from 0 to 8MPa, and from 285 to 305 K, respectively. The studies presented here were undertaken: (i) to determine the influence of the uniaxial pressure on the temperature behavior of velocity of the phonon coupled to the order parameter; (ii) to investigate the temperature dependence of polarization relaxation time for different values of uniaxial pressure 0-5; and (iii) to verify the hypothesis regarding the possibility of shifting the system from ferroelectric phase to paraelectric phase at constant temperature by applying a uniaxial pressure. We believe we have accomplished the above objectives.

4.4.2.

Experimental Procedure

Single crystals of TGSe were grown at the paraelectric phase (305 K) from the water solution of the initial components at the stoichiometric proportion. Colorless and good-optical quality crystals were obtained. Since our experiment was focused on the critical behavior of the quasilongitudinal phonon propagating along the c crystallographic direction, for which the elastic anomaly (due to the transverse fluctuation of Ps) is most pronounced, 3 the scattering geometry was the following. The sample was illuminated along the direction of the bisectors of a*_ and c-axis. The scattered light was observed at 90° to this direction and the uniaxial pressure o\, was applied along ferroelectric direction b. All surfaces of the samples of typical size 6 x 7 x 7 mm 3 were optically polished. The flatness of the samples faces, especially those perpendicular to the b direction, was checked with the Newton's rings method. It should be noted that TGSe is a very delicate crystal to study under applied uniaxial pressure. Any departure from perfect parallelism of the surfaces perpendicular to the b direction led to sample cracking. The incident light provided by a stabilized single-mode argon-ion laser (Spectra Physics, Model 2020) operating at 514.5 nm was polarized perpendicularly to the scattering plane. The scattered light was analyzed at 90° using a piezoelectrically scanned triple pass Fabry-Perot interferometer (Burleigh Model RC-110) with free spectral ranges (FSR) 44 and 24 GHz. With a normal finesse of about 45 the instrumental line width Fi did not exceed 0.55GHz (for the smaller FSR). Any drifts due to the

370

Effective Field Approach to Phase

Transitions

laser, electronics, or the Fabry-Perot were automatically compensated by a PC-based data acquisition and control system (FPDACS-MUN, Canada). 13 The relation between the observed Brillouin line shift Av and the sound velocity v is given by the Brillouin equation: Av = \v(nf +nj - mns cos @)~1/2,

(1)

where A is the wavelength of the incident light, rii and ns are refractive indices for the incident and scattered light, respectively, and 0 is the scattering angle. The uniaxial pressure and temperature dependence of the Brillouin scattering spectra of TGSe were collected using a vacuum-insulated container constructed for this purpose. The container consists of a heating-cooling holder inside which the sample was placed between two metal pistons. The uniaxial pressure was applied by squeezing an accurately calibrated spring. To avoid sample cracking a thin (15 /xm) metalized mylar foil was placed between the sample and pressing pistons. The estimate error of the applied stress was less than 3%. The temperature of the sample was regulated with a stability of ±0.02 K using an Oxford (model ITC 4) temperature controller. The thermocouple was glued to the sample using the thermoconductive paste.

4.4.3.

Results and Discussion

In Fig. 4.4.1, we present the temperature dependence of the Brillouin line shift Av proportional to the quasi-longitudinal phonon velocity W[ooi] and the related changes of the true line width T obtained after deconvolution with the instrumental line width (Fig. 4.4.1(b)). 14 A step-like anomaly was observed when cooling the sample from the paraelectric phase to the ferroelectric phase. The uniaxial pressureCTJ,was shifting TQ toward lower temperatures, as expected. However, the shift of the phase transition temperature was found to be —0.15K/MPa, which is bigger than those reported in Refs. 10 and 11 (—0.09K/MPa). In addition, a well-pronounced decreasing of the step in velocity with increasing pressure was observed. The inset in the Fig. 4.4.1(a) demonstrates that, for a^ of the order of 20MPa, this step anomaly should decrease linearly to zero. The results presented the Fig. 4.4.1 allowed us to estimate the temperature changes of the polarization relaxation time r for different uniaxial pressures.

Brillouin Scattering Studies of Ferroelectric Triglycine

,,i as

,

, 280

,

! S6

,

, 309

,

Selenate

( 90S

T(K)

Fig. 4.4.1. Temperature dependence of Brillouin line shifts related to the quasilongitudinal phonon [0 0 1] for different constant values of uniaxial pressure (a) and related changes of the line width (b). The vertical scale in the inset is in GHz. All lines are guided on eye.

Assuming that a single relaxation time, r, is appropriate, it can be determined by combining two equations, the first of which expresses15 the dependence of the sound velocity, v, on frequency, to,

»2=^ + ( ^ - ^ V ( i + u V ) - 1 .

(2)

Here, vx is a limiting high-frequency velocity and VQ is a sound velocity not affected by the transition. The second equation is a corresponding expression for sound attenuation, a = (2«)-3(^-VoVr(l+^2r2)-1.

(3)

The relaxation time r is then given by r=(2a)-lv-3(v2-v20),

(4)

Effective Field Approach to Phase

372

l ,, U J . 1 u.,. | i... 1 .a.,

S;

TT,M,1*''^'"r'-""« •

Transitions

!

•

*•

/

4

5 3

o

•

rX

&2 V

1

i

JV / // tf aJy JP

jy

Q a^ =

0 M?«, A » (1 jfifttaSO) 1 (H > s K

o

s s = 3;JMP4. A - (3,75*0.25} K M ' s K

9

a = &9Mfti, A-=<2.aM&30) t4r»aK

10

(Tc-T) (K) Fig. 4.4.2. Temperature dependencies of the inverse relaxation time, in the ferroelectric phase, for different values of uniaxial pressure.

and can be calculated from the true width of the Brillouin line, which is related to a by the relation: T = av. The obtained temperature dependencies of the inverse relaxation times T " 1 are given in Fig. 4.4.2. The observed changes follow the Landau-Khalatnikov expression r = A(TQ — T ) _ 1 with slopes A increasing for rising stresses. This indicates the role of the uniaxial pressure applied along the spontaneous polarization in slowing down the relaxation process. Attempts were undertaken to check the possibility of passing the ferroelectric phase transition at a fixed temperature, varying only the uniaxial pressure <7{,. The temperature of the sample was set at 293.20K, i.e., almost 2° below T o Brillouin spectra were then collected for several rising values of
Brillouin Scattering Studies of Ferroelectric Triglycine

1W»

r

T

l

' "r"'-'"-,Lr~T • i

A

1

•

•

•

16,76

- 0,7

/ ° 5f* 1Am

-

o/

373

1 — • - 0,8

S ' •

w

Selenate

0,8

.

/I

*• o,s j£f«*

& IT*

-"•o

t/

"saw

\°

\

J

•\&m

:

- fta

H

0.2

T = 293.2 K ji™* T Z

3

4

5

MR

6

7

8

Oil

S

Fig. 4.4.3. Brillouin line shift and its width versus uniaxial pressure at constant temperature (293.2 K). Lines are guided on eye.

1

%.*

'

1.Z

• /

"

1.0

/

k" *

•

r

.

/

/

„,.-'

*" flL*

• •

y

-

• u

r

•

jI

:

• *

i

_ _ i

a

'

•

'

•

-afMPaji

Fig. AAA. The uniaxial pressure dependence of inverse relaxation time in the ferroelectric phase.

374

Effective Field Approach to Phase

Transitions

the rather poor accuracy, it may be said that r at constant temperature under an uniaxial pressure (as the active thermodynamic force) follows the rule: r = A(ac -a)'1, with the slope A= (2.9 ± 0.3)10 - 1 1 sMPa. 4.4.4.

Conclusions

The results presented above lead to the following conclusions. The data regarding the phonon velocity W[ooi] and its attenuation obtained by the Brillouin scattering technique for different values of the uniaxial pressure, <7b, indicate a strong coupling to the spontaneous polarization. This is manifested (for er;, = 0) by the step-like change of i>[ooi] versus temperature and accompanied by a distinct maximum of the Brillouin line width at TQ . The same temperature dependencies obtained for o\, ^ 0 are less pronounced: the step down in velocity becomes smaller with increasing uniaxial pressure. With the assumption that these changes are proportional to <Jb, it is estimated that for <7b of the order of 20 MPa the step change of velocity at the actual phase transition temperature should become undetectable. The temperature dependencies of the relaxation time r of the transverse polarization fluctuations were found to be sensitive to the constant uniaxial pressure applied along the polarization direction. The observed changes can be described within the Landau-Khalatnikov theory giving r = A(TQ — T ) _ 1 , with the slopes A proportional to the applied ov Additionally, it was demonstrated that, by applying uniaxial pressures at constant temperature, the crystal studied undergoes the phase transition from the ferroelectric phase to the paraelectric phase. Acknowledgments We wish to acknowledge financial support from MEC through grant PB960037. One of the authors (B.M.) wishes to thank the Spanish Ministry of Education and Culture for supporting him with sabbatical grant (SAB950460). We are indebted to Prof. M.J. Clouter for his help in developing the hardware and the kind donation of software of FPDACS. Thanks are due to A. Karayev for the samples preparation. References 1. W. Gammon and H.Z. Cummins, Phys. Rev. Lett. 17, 193 (1966). 2. Luspin and J.P. Hauret, Ferroelectrics 15, 43 (1977).

Brillouin Scattering Studies of Ferroelectric Triglycine Selenate

375

3. T. Yagi, M. Tokunaga and I. Tatsuzaki, J. Phys. Soc. Jpn. 40, 1659 (1076). 4. Y. Tsujimi, T. Yagi, H. Yamashita and I. Tatsuzaki, J. Phys. Soc. Jpn. 50, 184 (1981). 5. B. Lavrencic, M. Copic, M. Zgonik and J. Petzelt, Ferroelectrics 2 1 , 325 (1978). 6. Hikita and T. Ikeda, J. Phys. Soc. Jpn. 42, 351 (1977). 7. D. Landau and J.M. Khalatnikov, Sov. Phys. Dokl. 96, 469 (1954). 8. Jona and G. Shirane, Ferroelectric Crystals (Pergamon, London, 1962). 9. R. Ramirez, C. Prieto and J. Gonzalo, Ferroelectrics 100, 181 (1989). 10. J. Koralewski, Stankowska, T. Iglesias and J. Gonzalo, J. Phys.: Condens. Matters, 4079 (1996). 11. J. Stankowska, I. Polovinko and J. Stankowski, Ferroelectrics 2 1 , 529 (1978). 12. E. Hamed, Phase Transitions 38, 43 (1992). 13. M.J. Clouter, Data Acquisition and Control System for Fabry-Perot Interferometry, User's Manual (Version 2.02, 1997). Memorial University of Newfoundland, St. John's, A1B 3X7 Canada. 14. A. Pinov, S.J. Candau, J.T. la Macchia and T.A. Litovitz, J. Acoust. Soc. Am. 43, 131 (1968). 15. R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics (North-Holland, New York).

Chapter 4.5

Piezoelectric Resonance Investigation of Zr-Rich P Z T at Room Temperature* N. Cereceda, B. Noheda, J.R. Fdez.-del-Castillo and J.A. Gonzalo Departamento de Fisica de Materiales, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain

J. De Frutos ETSI

Telecomunicacion (UPM), 28040 Madrid,

Ciudad Spain

Universitaria,

A.M. Gonzalez EUIT

Telecomunicacion (UPM), Ctra. 28031 Madrid, Spain

Valencia,

We study the piezoelectric resonances in poled PZT ceramics by means of a microscopic model. It connects the microscopic vibrations of the ionic units, cooperatively producing the piezoelectric effect, with the macroscopic piezoelectric parameters. The behavior at the resonance is well described in a wide range of frequencies, allowing the calculation of piezoelectric constants and the electromechanical coupling factors in terms of microscopic quantities.

*Work previously published in J. Eur. Ceram. Soc. 19, 1259 (1999). Copyright © 1999. Elsevier B.V. 377

378

4.5.1.

Effective Field Approach to Phase

Transitions

Introduction

The use of piezoelectric ceramics is quite extended, because of their reliability in producing detectors, actuators, and different useful devices.1 However, this cooperative effect is still not completely understood. Early attempts made by Schrodinger, Larmor, and Born to use modern lattice dynamics in the description of piezoelectricity were unequal to the task of predicting anything better than a rough approach to the order of magnitude of the piezoelectric effect in the simplest structures. 2 Piezoelectricity is a property appearing in all 20 non-centrosymetric crystal classes, 3,4 and then includes a lot of types of materials, among ferroelectrics. The use of modern calculation techniques allowed some authors to obtain valuable first-principle results for the piezoelectric parameters in some simple compounds. 5,6 In perovskites the calculations based on LAPW-LDA provide some explanations about the origin of ferroelectricity and piezoelectricity in these materials. 7 Recently, some authors presented calculations of piezoelectric properties in pure and mixed perovskites. 8 ' 9 All these calculations are not entirely based on first principles, because they always use some input from experiments, such as the lattice constant, which is critical to get stable structures. Escaping from first-principles and ab initio calculations, we propose a microscopic approach to the piezoelectric resonances. The model described in this work is based on a classical forced and dumped oscillator and allows the expression of the piezoelectric coefficients and the electromechanical coupling factors in terms of microscopic quantities. The material investigated in this work is the well-known PZT: a lead zirconate-lead titanate solid solution (PbZri_ x Ti x 03). In the Zr-rich region of compositions it presents a ferroelectric phase with rhombohedral geometry below the ferroelectric transition temperature. This phase is divided into two other ferroelectric regions (FRL and FRH for low- and hightemperature phases, respectively), in which the positions of cations and oxygen octahedra play a fundamental role. 10,11 The transition temperature (ILH) between them varies from room temperature up to 150°C, depending on composition.4

4.5.2.

Experimental

We have characterized the piezoelectric resonances in PZT 96.3/3.5 + lwt.% ND2O5 ceramic samples. They were sintered at the Shanghai

Piezoelectric

Resonance Investigation

of Zr-Rich

PZT

379

Institute of Ceramics, Shanghai (China), by ordinary methods. The samples were silver-electroded and poled by applying 2kV/cm during 10 min at 125°C in silicon oil medium. The dielectric response was determined by means of a Hewlett-Packard HP4194A-LF impedance analyzer, and characterized using a specially designed software. 12,13 By this way we collected data for the elastic constants, piezoelectric coefficients, and electromechanical coupling factors in complex form. 4.5.3.

Theoretical Model

The model simplifies the material assuming an arrangement of ions placed into alternating layers of positive and negative charges along the ferroelectric axis. In this approach, it is assumed that positive charges (+2) correspond to the P b 2 + ions and the negative charges (—2) correspond to the [(Zr/Ti)0 3 ] 2 - packed ions. We assume the motion of these layers to be described by a forceddumped oscillator, where k(d)x± is the restoring force, M±~f(d)dx±/dt is the viscous force, and qEes(d) exp{—icut} is the effective electric force, driven by the external applied field, EQ exp{—iwt}. The net displacement, x = x+ — X-, connects with the extension of the material, caused by, let us say, an accordion effect. The thickness dependence is included in the description, according to the principle that piezoelectricity is a macroscopic cooperative phenomenon, with the boundary conditions playing a fundamental role. The complex dielectric response is *

£ =£cd + - ^

./i

JT

. .

:—,

(1)

where LjQk(d)/M is the characteristic frequency of the system, A(d) is the amplitude of the resonance, 7(d) is the corresponding dumping factor, and M is the effective mass, defined as 1/M — 1/M+ + 1/M_. Only these thickness-dependent parameters are needed to characterize the resonance, as shown below. The "clamped" dielectric constant, £ccj, does not depend on sample thickness. 4.5.4.

Results and Conclusions

As is known, disk geometry possesses thickness extensional and radial mode vibrations. Figure 4.5.1 represents the dielectric response found for PZT

380

Effective Field Approach to Phase

5 f^*,

Transitions

e MHz

Fig. 4.5.1. Thickness mode resonance in the P Z T 96.5/3.5 + lwt.% ND2O5 sample. e' and e" represents real and imaginary parts of Eq. (1). The fitting parameters are indicated in the plot.

96.5/3.5 + lwt.% ND2O5 ceramic during the thickness mode resonance in a sample with d = 0.4 mm thickness and 2r = 1cm diameter. Points and crosses represent the experimental points for real (e') and imaginary (e") parts, respectively. Lines represent the corresponding theoretical fits, according to Eq. (1). As can be seen, each fit reproduces adequately the experimental behavior in a wide range of frequencies. We studied both thickness and radial mode resonances for samples with different sizes. Figure 4.5.2 reproduces the dependence of the parameters wo, A, and 7 versus thickness for the longitudinal mode resonance. A similar study has been made for radial mode resonance. Table 4.5.1 summarizes the results. The behavior found for the characteristic frequency, uio(d), agrees with the expected one, because it is determined by the boundary conditions. As can be observed from Fig. 4.5.1, there exists a "jump" in the dielectric constant, Ae, which is not size-dependent and can be defined as Ae' = e'(tj —> 0) — e'(u> —> 00) = A(d)/u>o{d). This jump is related to the electromechanical coupling factor as follows: at the thickness mode resonance we expect e'(0) = £33 and £^d(oo) = £| 3 , so we may write, according

Piezoelectric

Resonance Investigation

of Zr-Rich

PZT

381

l/rf,tnm"' Fig. 4.5.2.

Thickness dependence of the fitting parameters A, Wod, and 7. Table 4.5.1. Dependence of the fitting parameters with thickness (d) and radius (r). Thickness mode 2

7 A

ocl/d ocl/d2 ocl/d2

Radial mode oc 1/r 2 ocl/r4 ex 1/r 2

to Meitzler 14.

*? =

Ae £(0)

Ae Ae + £ c d '

(2)

which is the thickness electromechanical coupling factor. We may interpret this as follows. Extrapolating to the case in which the atomic chains consist of only two unit cells (d = 2a), there exists a microscopic elastic constant, K^, and an associated characteristic frequency, wo(2a). Then, for uJo{d) we may write u20(d) =

(Kd/M)(2a/d)2,

(3)

382

Effective Field Approach to Phase Transitions

where a is the lattice parameter. In our samples we found that wo(2a)/27r = 2.8 x 1012 Hz, which is a reasonable value for a low-lying purely ionic mode resonance. For the total viscous force we consider that the cooperative effect during the resonance leads to a sharing of the atomic viscous force and to a total viscous force parameter expressed by 7(rf)=rd(2a/d)

2

,

(4)

where Td = 7(2a) is constant. On the other hand, A(d) oc l/
A{d) _ A-wNg2 EeBJcQ/E W - -273T u%{d)- ~—~nM

,,fiwl(d) • (A\

W

In a centrosymmetric (non-piezoelectric) ionic crystal, the ratio (Eeg/E) is given by Ae/3, combining Lorentz's and Claussious-Mossotti's relations. 15 In a piezoelectric crystal the strength of Ees(d), driven by the external field E, is shared by (n+n~) = [(d/2)/a\2 charged pairs. The effective electric field Eefi results in EeS(d)/E=^[2a/d}2.

(6)

From the above discussion, we may get the piezoelectric coefficient d^z as follows: d33

5d/d _ x/2a _ 1 = —ET = o ~E~ ~ E ~~2

= —£T

{q/M)(EeS(d)/E) ..,2/JN ' aujl{d)

V)

and using Eqs. (5) and (6) we get

*-K^)^-

(8)

By using the values q = 2e, a = 4 A, Ka = 4 x 104 (dyn/cm) and Ae d = 30, we get d33 = 101 x 1 0 - 1 2 C/N. This value is somewhat smaller than I//133 = 320 x 10~ 12 C/N, used to estimate it, but it lies in an expected value in accordance with the values in the literature. The results and the above theoretical development are also applicable to radial mode piezoelectric resonances. Figure 4.5.3 represents the results found for the dielectric response through the radial mode resonance. As can be observed, this resonance is narrower than the thickness one.

Piezoelectric

Resonance Investigation •

|"

of Zr-Rich PZT

383

HUlflM

1400

600 400

S/

1200

mJ2n

m i i i m l

1000

200

i

800

0

r

-200 •400

600 400 /2,t

J2a_JL.'»<-

200

-600 Hmilllililll)ll1HIIII>|IIIHIIIIIIIItllllllltll[llllr

THllllllll|IIIIIMIHlllllllllllll|llllllllllllllllllllllll|IIIMIIIIIIIIIIIIIHIIII

250

200

300

350

f=W2n, kHz Fig. 4.5.3.

Dielectric response, radial-mode piezoelectric resonance.

In order to calculate the piezoelectric coefficient d^i we use now the values KT = 4 x 104 dyn/cm and Ae r = 5, where the sub-index "r" indicates "radial." Then, by using Eq. (8) applied to this resonance, we get ofai = 11.6 x 1 0 - 1 2 C/N, which is in good agreement with the observed one during the standard characterization. The planar electromechanical coupling factor, kt, can be easily calculated by again using Meitzler criterion, 14 using now e'cd = e'r(oo) = e^3, £'r(0) = ej 3 , and the relation eg3 = ej 3 (l - fct2). Table 4.5.2 shows the final results obtained in comparison with the previously done macroscopic characterization.

Table 4.5.2. Comparison of the results obtained after analysis with the "standard" method and with the "microscopic" approach.

Macrosc Microsc

^33


kt

[<320]* 101

11.5 11.6

0.35 0.35

0.14 0.12

*Not directly determined. Estimated using ^33 < I//133. Parameters ^33 and 1^31 are given in units of xHT12C/N.

384

Effective Field Approach to Phase

Transitions

According to all these results, there is a simple description of the piezoelectric resonance in t e r m s of microscopic quantities, establishing connection between the usual interpretation (macroscopic) and the fundamentals of the material. In principle, this microscopic approach could be used directly in other piezoelectric perovskites, and possibly in other piezoelectric materials. Our description works quite well not only at frequencies close to the resonances, but also far from them.

Acknowledgments T h e authors wish to acknowledge the financial support from C I C y T (PB960037), the Spanish Committee for Scientific and Technological Research, and N A T O (CGR-0037). References 1. J.C. Burfoot and G.W. Taylor, Polar Dielectrics and their Applications (Macmillan, London, 1979). 2. W.G. Cady, Piezoelectricity. An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals (Dover, New York, 1964). 3. W.P. Mason, Crystal Physics and Interaction Processes (Academic Press, New York, 1966). 4. B. Jaffe, W.R. Cook and H. Jaffe, Piezoelectric Ceramics (Academic Press, New York, 1971). 5. A.D. Corso, M. Posternack, R. Resta and A. Baldereschi, Phys. Rev B 50(15), 10715 (1994). 6. F. Bernardini, V.D. Fiorentini and D. Vanderbilt, Phys. Rev. B 56, R10024 (1997). 7. R.E. Cohen, Nature 358, 136-138 (1992). 8. G. Saghi-Szabo, R.E. Cohen and H. Krakauer, Ferroelectrics 194, 350 (1997). 9. E. Cockayne and K. Rabe, Phys. Rev. B 57(22), R13973 (1998). 10. A.M. Glazer, S.A. Mabud and R. Clarke, Acta Crystallogr. B34, 1060 (1978). 11. B. Noheda, T. Iglesias, N. Cereceda, J.A. Gonzalo, H.T. Chen, Y.L. Wang, D.E. Cox and G. Shirane, Ferroelectrics 184, 251-255 (1996). 12. A.M. Gonzalez and C. Alemany, J. Phys. D: Appl. Phys. 29, 2476-2482 (1996). 13. C. Alemany, L. Pardo, B. Jimenez, F. Carmona, J. Mendiola and A.M. Gonzalez, J. Phys. D: Appl. Phys. 27, 148-155 (1994). 14. A.H. Meitzler, H.M. Tiersten Jr. and H.F. Tiersten, IEEE Trans. Sonics Ultrasonics SU20, 233-239 (1973). 15. A.J. Dekker, Solid State Physics (Prentice-Hall, Englewood Cliffs, NJ, 1962).

Chapter 4.6

Transition Temperature Dependence in Perovskite Ceramics as a Function of Grain Size* M.I. Marques, N. Cereceda and J.A. Gonzalo Departamento de Fisica de Materiales, C-IV Universidad Antonoma de Madrid, 28049 Madrid,

Spain

Simple effective field arguments assuming a nonisotropic layer of thickness Lc covering the ferroelectric bulk grain of perovskite ceramics are used to determine an expression for the transition temperature TC(L) as a function of grain linear size L. The surface to volume ratio increases as L decreases, reducing the value of TC(L). Experimental data for BaTiC>3 ceramics as a function of grain size are described reasonably well by the expression of TC(L) obtained this way.

4.6.1.

Introduction

Grain size effects on the physical properties of ferroelectric ceramics have been investigated by several a u t h o r s . 1 - 5 Results on B a T i 0 3 ceramics show t h a t the ferroelectric (cubic-tetragonal) transition is suppressed for grain sizes below about 0.09 /an. As the grain size decreases and the surface to volume ratio of the grain decreases, one may expect t h a t in the ferroelectric phase the unit cells near the surface order nonisotropically with respect to the unit cells in the bulk of the grain. This may give rise to a weakening of the overall cooperative interaction of the dipoles within the grain, resulting in a lowering of transition t e m p e r a t u r e .

*Work previously published in Ferroelectrics Letters 25, 103 (1999). Copyright © 1999. Taylor & Francis Group. 385

386

Effective Field Approach to Phase

Transitions

In a recent review by Niepce 3 on permittivity of fine-grained BaTiC-3 ceramics, data by Henning 3 indicate that TC{L) ~ 25°C (298.15K) for Lc ~ 0.8 fim. On the other hand, data by Uchino et al.4 show that TC(L) ~ 20°C (293.15 K) for Lc ~ 0.095 /xm. The qualitative behavior of TC(L) in both sets of data is the same, but the critical grain size to bring down Tc to room temperature is different. There are possible reasons for this discrepancy, but we will use the data of Uchino et al.4 later for numerical comparison with our calculated TC(L), based on simple effective field arguments, as shown below. 4.6.2.

Theoretical Approach

For a single crystal or a large single grain, effective field arguments 6 lead to the identification of kBTc, the thermal energy per unit dipole, with ESQIJL, the interaction energy of the local spontaneous field at T = 0 K with the unit electric dipole per unit cell. Therefore, kBTc » Es0ii « pNv2,

(1)

where Eso = (3Pso = /3N(i, /? = 4irTc/C being the effective field coefficient that is given in terms of the observed transition temperature, T c , and the observed Curie constant, C ( e _ 1 ~ (T-Tc)/C at T not far from T c ). Consider a ceramic with cubic-like grains in which a ferroelectric bulk central cube of volume (L — L c ) 3 surrounded by an nonisotropic surface layer of volume [6(L - Lc)2{Lc/2) +12 (L - L c ) (Lc/2)2 + 8(L c /2) 3 ] (see Fig. 4.6.1). For a cubic-like grain of arbitrary linear size L and surface layer of thickness Lc, instead of Eq. (1), we would have kBTc(L,Lc)

=

kBTc(oo,Lc) 3LC

, ^ ,„ 3LCC Peb 2 ^ /?(Msi//i)2 . /3(/WA0 ,

(L-Lc)

+

+

{L-Lcf

&

1(MCC/M) 2

{L-Lcf (2)

with L » Lc, Eq. (2) results in Tc(L,Lc)«Tc(oo,Lc),

(3)

with L -> Lc and (/3cc//?)(/xcc//i)2 < 1, Eq. (2) gives

T c (L,L c )«T c (oo,L c )

/?cc/

7

2n


(4)

Transition

Temperature Dependence in Perovskite

Ceramics

387

(L-Lc)LcLc/4 bar

T • ^ - • w > ';-^Frw.

(L-Lc)(L-Lc)Lc/2 Layer

L-Lc

T

LcLcLc /8 comer cube Fig. 4.6.1. Cubic-like perovskite grain (projection on plane) made up of a central bulk ferroelectric region of volume (L — Lc)3, with unit cell dipole moment fi, surrounded by surface nonisotropic layers with unit cell dipoles (—fisi) (as the one shown in white), edge bars, with unit cell dipoles (±/z e b) (as the one shown in light gray), and corner cubes, with unit cell dipoles (±/i C c) (as the one shown in black). Note that there are six surface layers of volume (L — Lc)2Lc/2, 12 edge bars of volume (L — Lc)Ll/A, and eight corner cubes of volume L;?/8. Effective field coefficients for the three zones are /3 sl , f3eb and /3CC, respectively.

as expected. With arbitrary L in the interval Lc < L < oo we can compare Eq. (2) with data for fine-grained BaTiOa ceramics by Uchino et al.4

4.6.3.

Results

Figure 4.6.2 shows a fit of Eq. (2) to these data in semilog representation. The parameter values for a best fit are given by

T c (oo,L c )=T c (oo)

(/V/3)(/WM)

2

(/W/?)(/W/i)2 (/W/?)(/WM) 2

401.72 ± 1.7 (K) 0.095 (/im) 1.018 ±0.016 0.98 ±0.012 0.8488 ± 0.0064

The fit is reasonably good and the values for (/Usi//x), (/ieb/A*), and (/i cc //i) seem to be of order unity.

388

Effective Field Approach to Phase

Transitions

420 . 400 360 .

T c (° K) ™ 340 -

s

320 300 .

2ao 1

0.1

L(Mfn) Fig. 4.6.2. ceramics.

Fit of TC(L, Lc), Eq. (2), to experimental data by Uchino et al.4 on BaTiOs

Work on Monte Carlo simulations of fine grain effects on the transition t e m p e r a t u r e of three-dimensional finite-size Ising systems is currently investigated for comparison with experimental data.

Acknowledgments Financial support from C I C y T (PB96-0037) is acknowledged.

References 1. 2. 3. 4. 5. 6.

K. Kinoshita and A. Yamaji, J. Appl Phys. 47, 371 (1976). G. Arlt, D. Hennings and G. de With, J. Appl. Phys. 58, 1619 (1985). D. Hennings, Int. J. High Technol. Ceram. 3, 91 (1987). K. Uchino, E. Sadanaga and T. Hirose, J. Am. Ceram. Soc. 72, 1555 (1989). J.P. Niepce, Electrocerarmics 4, p. 29, Aachen, September 5-7 (1994). See, for instance, J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991).

Chapter 4.7

A Monoclinic Ferroelectric P h a s e in t h e Pb(Zr 1 _ a ; Ti a ; )03 Solid Solution* B. Noheda, D.E. Cox and G. Shirane Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA

J.A. Gonzalo Dept. Fisica de Materiales, U.A.M., 28049 Madrid, Spain

Cantoblanco,

L.E. Cross and S.-E. Park Materials Research Laboratory, The Pennsylvania State University, Pennsylvania 16802-4800, USA

A previously unreported ferroelectric phase has been discovered in a highly homogeneous sample of PbZro.52Tio.4gO3 by high-resolution synchrotron X-ray powder diffraction measurements. At ambient temperature the sample has tetragonal symmetry (o t = 4.037 A, c t = 4.138 A), and transforms below ~250 K into a phase that, unexpectedly, has monoclinic symmetry (a m = 5.717 A, bm = 5.703 A, c m = 4.143 A, j3 = 90.53° at 20 K). The intensity data strongly indicate that the polar axis lies in the monoclinic ac plane close to the pseudocubic [111] direction, which would be an example of the species m3m(12)A2Fm predicted on symmetry grounds by Shuvalov.

*Work previously published in Appl. Phys. Lett. 74, 14 (1999). Copyright © 1999. American Institute of Physics. 389

390

Effective Field Approach to Phase

Transitions

The solid solution system Pb(Zri_ x Ti x )03 (PZT) has a complex phase diagram containing a number of materials that exhibit useful ferroelectric and piezoelectric properties. In particular, compositions near the morphotropic phase boundary (MPB) around x « 0.45-0.5 have attracted considerable interest for many years due to their high piezoelectric response. The PZT phase diagram, as now accepted, was determined by Jaffe et al.1 (Fig. 4.7.1), and nearly four decades of study on the physical and structural properties of PZT have made possible the development of a phenomenological theory to explain the stability of the phases and the properties of PZT over the entire range of the phase diagram. 2 Nevertheless, many questions still remain. It is generally accepted that for ferroelectric compositions with rhombohedral and tetragonal symmetry on the two sides of the MPB the polar axis are [111] and [001], respectively. However, very recently, based on neutron powder diffraction data for several ferroelectric rhombohedral compositions in the range of x « 0.12-0.40, Corker et al.s have proposed a model in which random (100) Pb displacements are superimposed on those along the [111] polar axis, which allows a much better structure refinement than achieved with a normal long-range order model incorporating anisotropic temperature factors. The MPB, which separates rhombohedral Zr-rich from tetragonal Tirich PZT, is nearly vertical along the temperature scale, and many X-ray diffraction studies have been reported over this region.4~9 This boundary is not well defined since it appears to be associated with a phase coexistence

o

to t o

pbZrO, Fig. 4.7.1.

»

«6 «5 So TO s o s o Molt % PbTlOj

too PfrTK),

P Z T phase diagram after Jaffe et al.1

A Monoclinic Ferroelectric Phase in the Pb(Zr\-xTix)C>2,

Solid Solution

391

region whose width depends on the compositional homogeneity and on the sample processing conditions. 7 ~ 10 Cao and Crosse 11 have modeled the width of this region based on the free energy differences between the tetragonal and rhombohedral phases, obtaining an inverse dependence with particle size in a polycrystalline sample. In the present work, we have utilized high-resolution synchrotron X-ray powder techniques to study the structure of a composition close to the MPB (x = 0.48) as a function of temperature, and we report the observation of a low-temperature monoclinic phase in the PZT system. A PZT composition with x — 0.48 was prepared by conventional solidstate reaction techniques using appropriate amounts of reagent-grade powders of lead carbonate, zirconium oxide, and titanium oxide, with chemical purities better than 99%. Pellets were pressed and heated to 1250°C at a ramp rate of 10°C/min, and held at this temperature for 2 h. During sintering, PbZrC-3 was used as a lead source in the crucible to minimize volatilization of lead. The product was found to be a single phase within a detection limit of <2% by conventional X-ray techniques. High-resolution synchrotron X-ray powder diffraction measurements were made at beam line XIA at the Brookhaven National Synchrotron Light Source. An incident beam of wavelength 0.6896 A from a G e ( l l l ) double-crystal monochromator was used in combination with a Ge(220) crystal and scintillation detector in the diffraction path. The resulting instrumental resolution is about 0.01° on the 29 scale, an order of magnitude better than that of a laboratory instrument. For measurements above room temperature, the pellet was mounted on a BN sample pedestal inside a wire-wound BN tube furnace. The furnace temperature scale was calibrated with a sample of CaF2. The accuracy of the temperature was estimated to be within 10 K, and the temperature stability was ~2 K. For measurements below room temperature, the pellet was mounted on a Cu sample holder and loaded in a closed-cycle He cryostat, with an estimated temperature accuracy of 1K and stability of 0.1 K. Coupled 9-29 scans were performed over selected angular regions with a 29 step interval of 0.005 or 0.01° depending on the peak widths. The sample was rocked 1-2° during data collection to improve powder averaging. The diffracted intensities were normalized with respect to the incident beam monitor. The evolution of the (111) and (220) reflections is shown as a function of temperature in Fig. 4.7.2. At the highest temperature reached (736 K), the material is cubic and the sharpness of the peaks [full-width at half-maximum (FWHM) ~0.02°] demonstrates the excellent quality of

392

Effective Field Approach to Phase

Transitions 10ao

(a)

<111)c

(220) c

X=0.48

T-736 Kf

500 0

(b) T=300K

ioo^T

JM_

o

e o

I50Z •H U T

0

_

500 T=20K

"

i S m f i 17",0 " 29 (deg)

*^*wn.i

0 150

27.3 ' 27",6 27.9 2B (deg)

28.2°

Fig. 4.7.2. Evolution of the pseudocubic reflections (111) and (220) for x = 0.48 from T = 736 (a) to 20 K (e).

the sample [Fig. 4.7.2(a)]. From a Williamson-Hall plot, 12 we estimate a particle size of ~0.7/xm and a Ad/d of about 3 x 10~ 4 , corresponding to a compositional inhomogeneity, Ax, better than ±0.5%. On cooling, the symmetry changes to tetragonal at ~660 K and remains tetragonal down to 300 K. There is no sign of any rhombohedral component as shown by the absence of a Bragg peak at the rhombohedral (200) position. However, some of the tetragonal peaks broaden as the temperature is lowered, especially the (111) and (202) reflections [Fig. 4.7.2(b)]. Below room temperature these reflections become distinctly asymmetric, and at 210 K the latter is clearly split into two roughly equal peaks, while the (111) peak has a low angle shoulder [Fig. 4.7.2(c)]. As the temperature is lowered further the splitting increases [Fig. 4.7.2(d)] until at 20 K the (202) is well resolved into two peaks, and the (111) is seen to consist of a central stronger peak with weaker shoulders on both low- and high-angle sides [Fig. 4.7.2(e)]. The positions and intensity ratios of the peaks are very well described by a monoclinic cell in which am and bm lie along the tetragonal [110] and [110] directions (a m « 6m R^ a t \/2), and c m is close to the [001] axis (c m « c t ), as illustrated in Fig. 4.7.3. The monoclinic cell has bm as the unique axis, and the angle between am and c m is ~90.5° at 20 K.

A Monoclinic Ferroelectric Phase in the Pb(Zr\-xTix)Oz

Fig. 4.7.3.

Solid Solution

393

Tetragonal and monoclinic unit cell representations.

Fig. 4.7.4. Lattice parameters of PZT with a composition x = 0.48 as a function of temperature.

The lattice parameters are plotted in Fig. 4.7.4 over the entire temperature range. At the low-temperature phase transition, am is slightly elongated with respect to tetragonal aty/2, whereas bm « at\/2 continues to decrease as the temperature is lowered, and c m « c t appears to reach a broad maximum around the transition. A direct phase transition from a tetragonal to monoclinic phase is rather uncommon, and the existence of the latter is likely to be a direct consequence of the proximity of the MPB. Consequently, one might expect the

394

Effective Field Approach to Phase

Transitions

monoclinic phase to exist over a relatively narrow composition region. However, because of the asymmetrical peak broadening, which occurs between 300 and 210 K, we cannot altogether rule out the possibility of an orthorrombic phase in this region, although we feel this is unlikely. In the tetragonal region of PZT the space group is Pimm, with the polar axis along [001], while in the rhombohedral region, with space group R3m, the polar axis is along the pseudocubic [111]. The most plausible space group for the new monoclinic phase is Cm, which is a subgroup of PAmm and R3m and allows the polar axis to lie anywhere between the [001] and the [111] axes. A preliminary check of the peak intensities indicates that the cation displacements lie close to the monoclinic [201] direction, i.e., close to the rhombohedral [111] axis. If this is the case, monoclinic PZT would be the first example of the ferroelectric species with P% — Py ^ P%, P%, p2 p2 -^ (^ m3rn(12)A2Fm predicted from symmetry by Shuvalov.13 A more detailed investigation of the structure is currently in progress, and the extent of the monoclinic region will be investigated with additional samples containing slightly different amounts of Ti.

Acknowledgments We thank E. Sawaguchi for his efforts trying to locate his 1953 sample for this work, and Evagelia Moshopoulou for her helpful comments. Support by NATO (R.C.G.0037), Spanish CICyT (PB96-0037), and U.S. Department of Energy, Division of Materials Science (Contract no. DE-AC0298CH10886) is also acknowledged. References 1. B. Jaffe, W.R. Cook and H. Jaffe, Piezoelectric Ceramics (Academic, London, 1971), p. 136. 2. M.J. Haun, E. Furman, S.J. Jang and L.E. Cross, Ferroelectrics 99, 13 (1989). 3. D.L. Corker, A.M. Glazer, R.W. Whatmore, A. Stallard and F. Fauth, J. Phys.: Condens. Matter 10, 6251 (1998). 4. G. Shirane and K. Suzuki, J. Phys. Soc. Jpn. 7, 333 (1952). 5. E. Sawaguchi, J. Phys. Soc. Jpn. 8, 615 (1953). 6. P. Ari-Gur and L. Benguigui, Solid State Commun. 15, 1077 (1974). 7. K. Kakewaga, O. Matsunaga, T. Kato and Y. Sasaki, J. Am. Ceram. Soc. 78, 1071 (1995). 8. J.C. Fernandes, D.A. Hall, M.R. Cockburn and G.N. Greaves, Nucl. lustrum. Meth. Phys. Res. B 97, 137 (1995).

A Monoclinic Ferroelectric Phase in the Pb(Zr\-xTix)03

Solid Solution

395

9. M. Hammer, C. Montry, A. Endriss and Michel J. Hoffmann, J. Am. Ceram. Soc. 8 1 , 721 (1998). 10. A.P. Wilkinson, J. Xu, S. Pattanaik and J.L. Billinge, Chem. Mater. 10, 3611 (1998). 11. W. Cao and L.E. Cross, Phys. Rev. B 47, 4825 (1993). 12. G.K. Williamson and W.H. Hall, Acta Metall. 1, 22 (1953). 13. L.A. Shuvalov, J. Phys. Soc. Jpn. 28, 38 (1970).

Chapter 4.8

Tetragonal-to-Monoclinic P h a s e Transition in Ferroelectric Perovskite: T h e s t r u c t u r e of PbZro.52Tio.4sO3* B. Nohedat and J.A. Gonzalo Departamento

de Fisica de Materiales, UAM, 28049 Madrid, Spain

Cantoblanco,

L.E. Cross, R. Guo and S.-E. Park Materials

Research Laboratory, Pennsylvania

The Pennsylvania State 16802-4800, USA

University,

D.E. Cox and G. Shirane Department

of Physics, Brookhaven National Laboratory, New York 11973-5000, USA

Upton,

The perovskite-like ferroelectric system PbZri-^Ti^Oa (PZT) has a nearly vertical morphotropic phase boundary (MPB) around x = 0.45-0.50. Recent synchrotron X-ray powder diffraction measurements by Noheda et al. [Appl. Phys. Lett. 74, 2059 (1999)] have revealed a monoclinic phase between the previously established tetragonal and rhombohedral regions. In the present work, we describe a Rietveld analysis of the detailed structure of the tetragonal and monoclinic PZT phases on a sample with x = 0.48 for which the lattice parameters are, respectively, at = 4.044 A, c t = 4.138 A, at 325 K, and am = 5.721 A, bm = 5.708 A, c m = 4.138 A, (3 = 90.496°, at 20 K. In the tetragonal phase, the shifts of the atoms along the polar [001] direction are similar to those in PbTiC>3

*Work previously published in Phys. Rev. B 74, 14 (2000). Copyright © 2000. The American Physical Society. f Visiting scientist at Brookhaven National Laboratory.

397

398

Effective Field Approach to Phase Transitions but the refinement indicates that there are, in addition, local disordered shifts of the Pb atoms of ~0.2 A perpendicular to the polar axis. The monoclinic structure can be viewed as a condensation along one of the (110) directions of the local displacements present in the tetragonal phase. It equally well corresponds to a freezing-out of the local displacements along one of the (100) directions recently reported by Corker et al. [J. Phys.: Condens. Matter 10, 6251 (1998)] for rhombohedral PZT. The monoclinic structure therefore provides a microscopic picture of the MPB region in which one of the "locally" monoclinic phases in the "average" rhombohedral or tetragonal structures freezes out, and thus represents a bridge between these two phases.

4.8.1.

Introduction

Perovskite-like oxides have been at the center of research on ferroelectric and piezoelectric materials for the past 50 years because of their simple cubic structure at high temperatures and t h e variety of high symmetry phases with polar states found at lower temperatures. Among these materials the ferroelectric P b Z r i - x T i ^ O a (PZT) solid solutions have attracted special attention since they exhibit an unusual phase b o u n d a r y t h a t divides regions with rhombohedral and tetragonal structures, called the morphotropic phase b o u n d a r y (MPB) by Jaffe et al.1 Materials in this region exhibit a very high piezoelectric response, and it has been conjectured t h a t these two features are intrinsically related. T h e simplicity of the perovskite structure is in part responsible for the considerable progress made recently in the determination of t h e basic structural properties a n d stability of phases of some important perovskite oxides, based on ab initio calculations (see, e.g., Refs. 2-9). Recently, such calculations have also been used to investigate solid solutions and, in particular, P Z T , where the effective Hamiltonian includes b o t h structural and compositional degrees of freedom. 1 0 - 1 2 T h e P Z T phase diagram of Jaffe et al.,1 which covers only temperatures above 300 K, has been accepted as the basic characterization of t h e P Z T solid solution. T h e ferroelectric region of the phase diagram consists mainly of two different regions: t h e Zr-rich rhombohedral region ( F R ) , which contains two phases with space groups R3m and R3c, and t h e Ti-rich tetragonal region ( F T ) , with space group PAmm.13 T h e two regions are separated by a b o u n d a r y t h a t is nearly independent of t e m p e r a t u r e , the M P B mentioned above, which lies at a composition close t o x = 0.47. Many structural studies have been reported around t h e M P B , since t h e early 1950s, when

Tetragonal-to-Monoclinic

Phase Transition

in Ferroelectric Perovskite

399

these solid solutions were first studied, 13 ' 14 since the high piezoelectric figure of merit that makes PZT so extraordinary is closely associated with this line. 1 ' 15 The difficulty in obtaining good single crystals in this region, and the characteristics of the boundary itself, make good compositional homogeneity essential if single-phase ceramic materials are to be obtained. Because of this, the MPB is frequently reported as a region of phase coexistence whose width depends on the sample processing conditions. 16 ~ 19 Recently, another feature of the morphotropic phase boundary has been revealed by the discovery of a ferroelectric monoclinic phase (FM) in the PbZri_3;Ti x 03 ceramic system. 20 Prom a synchrotron X-ray powder diffraction study of a composition with x = 0.48, a tetragonal-to-monoclinic phase transition was discovered at ~300K. The monoclinic unit cell is such that a m and bm lay along the tetragonal [110] and [110] directions (a m « bm ~ at\/2), and c m deviates slightly from the [001] direction (c m w c t ). 2 1 The space group is Cm, and the temperature dependence of the monoclinic angle (3 gives immediately the evolution of the order parameter for the tetragonalmonoclinic (.FT--FM) transition. The polar axis of the monoclinic cell can in principle be directed along any direction within the ac mirror plane, making necessary a detailed structural study to determine its direction. In the present work, we present such a detailed structure determination of the monoclinic phase at 20 K and the tetragonal phase at 325 K in PZT with x = 0.48. The results show that the polarization in the monoclinic plane lies along a direction between the pseudocubic [001]c and [ l l l ] c directions, corresponding to the first example of a species with P2 = Py ^ JR 2 . A tentative phase diagram is presented in Fig. 4.8.1, which includes data for the x = 0.48 composition together with those of the recently studied x = 0.50 composition. 22 The most striking finding, however, is that the monoclinic cation displacements found here correspond to one of the three locally disordered sites reported by Corker et al.23 for rhombohedral compositions in the region x = 0.1-0.4, and thus provide a microscopic model of the rhombohedral-to-monoclinic phase transition. This, together with the fact that the space group of the new phase, Cm, is a subgroup of both P4mm and R3m, suggests that FM represents an intermediate phase connecting the well-known F T and F R ZT phases.

4.8.2.

Experimental

A PZT sample with x = 0.48 was prepared by conventional solid-state reaction techniques using appropriate amounts of reagent-grade powders of

400

Effective Field Approach to Phase

Transitions

Fig. 4.8.1. Preliminary modification of the PZT phase diagram. The data of Jaffe et al.1 are plotted as open circles. The i " r — F M and PQ—FT transition temperatures for x = 0.48 and x = 0.50 are plotted as solid symbols. The F^-Fyi transition for x = 0.50 is reported in Ref. 22.

lead carbonate, zirconium oxide, and titanium oxide, with chemical purities better than 99.9%. Pellets were pressed and heated to 1250°C at a ramp rate of 10°C/min, held at this temperature in a covered crucible for 2h, and furnace cooled. During sintering, PbZrC-3 was used as a lead source in the crucible to minimize volatilization of lead. High-resolution synchrotron X-ray powder diffraction measurements were made at beam line X7A at the Brookhaven National Synchrotron Light Source. In the first set of measurements, an incident beam of wavelength 0.6896 A from a Ge (111) double-crystal monochromator was used in combination with a Ge (220) crystal and scintillation detector in the diffraction path. The resulting instrumental resolution is about 0.01° on the 29 scale, an order of magnitude better than that of a laboratory instrument. Data were collected from a disk in symmetric flat-plate reflection geometry over selected angular regions in the temperature range 20-736 K. Coupled 9-29 scans were performed over selected angular regions with a 29

Tetragonal-to-Monoclinic

Phase Transition in Ferroelectric Perovskite

401

step interval of 0.01°. The sample was rocked 1-2° during data collection to improve powder averaging. Measurements above room temperature were performed with the disk mounted on a BN sample pedestal inside a wire-wound BN tube furnace. The furnace temperature was measured with a thermocouple mounted just below the pedestal and the temperature scale calibrated with a sample of CaF2. The accuracy of the temperature in the furnace is estimated to be within 10 K, and the temperature stability about 2K. For low-temperature measurements, the pellet was mounted on a Cu sample pedestal and loaded in a closed-cycle He cryostat, which has an estimated temperature accuracy of 2K and stability better than 0.1K. The diffracted intensities were normalized with respect to the incident beam monitor. For the second set of measurements aimed at the detailed determination of the structure, a linear position-sensitive detector was mounted on the 28 arm of the diffractometer instead of the crystal analyzer, and a wavelength of 0.7062 A was used. This configuration gives greatly enhanced counting rates that make it feasible to collect accurate data from very narrow-diameter capillary samples in Debye-Scherrer geometry, with the advantage that systematic errors due to preferred orientation or texture effects are largely eliminated. A small piece of the sintered disk was carefully crushed and sealed into a 0.2 mm diameter glass capillary. The latter was loaded into a closed-cycle cryostat, and extended data sets were collected at 20 and 325 K while the sample was rocked over a 10° range. With this geometry the instrumental resolution is about 0.03° on the 28 scale. Because lead is highly absorbing, the data were corrected for absorption effects24 based on an approximate value of /xr = 1.4 determined from the weight and dimensions of the sample.

4.8.3.

Phase Transitions

The evolution of the lattice parameters with temperature was briefly summarized in Ref. 20, and a more complete analysis is presented below. The results of the full structure analysis are described later. A transition from the cubic to the tetragonal phase was observed at ~660K, in agreement with the phase diagram shown in Fig. 4.8.1. The measurements made on the pellet in the cubic phase at 736 K demonstrate the excellent quality of the sample, which exhibits diffraction peaks with full-widths at half-maximum (FWHM) ranging from 0.01 to 0.03° as shown

402

Effective Field Approach to Phase p

Transitions

*> Zr o. 5 JU°3 (no

8.0

16.7

6.0

16.8 16.9 M (deg)

!

17.0

17.1

*o

g4.0 0

2

200

400

600 T(K)

2.02Ad/d«5.810 rcos(8)=0.7l(f+5.8 10*sina

0.0 0.0

0.1

0.2 sin e

0.3

0.4

Fig. 4.8.2. The Williamson-Hall plot for P Z T {x = 0.48) derived from the measured diffraction peak widths in the cubic phase (T = 736 K). Particle size and microstrain are estimated from a linear fit (solid line). The plot for the (111) reflection in the cubic phase demonstrates the excellent quality of the ceramic sample (peak width ~0.02°). The plot of Ad/d vs. temperature is also shown as an inset.

for the (111) reflection plotted as the upper-right inset in Fig. 4.8.2. The FWHM's (r) for several peaks were determined from least-squares fits to a pseudo-Voigt function with the appropriate corrections for asymmetry effects,25 and corrected for instrumental resolution. The corrected values are shown in Fig. 4.8.2 in the form of a Williamson-Hall plot 26 r cos 6 = X/L + 2{Ad/d) sin 6,

(1)

where A is the wavelength and L is the mean crystallite size. From the slope of a linear fit to the data, the distribution of d spacings, Ad/d, is estimated to be ~ 3 x 104, corresponding to a compositional inhomogeneity Ax of less than ±0.003. From the intercept of the line on the ordinate axis the mean crystallite size is estimated to be ~ 1 /j,m. A tetragonal-to-monoclinic phase transition in PZT with x = 0.48 was recently reported by Noheda et al.20 Additional data have been obtained

Tetragonal-to-MonocUnic

Phase Transition

in Ferroelectric Perovskite

403

™>Zi0S2Ti0MO3 •

i — • -

<4.15

H 84.054 • * — c

m

0.50

-•—a m /V2 •*— b _/V2

0.25

a,

o

0.00 200

400 T(K)

600

£

800

Fig. 4.8.3. Lattice parameters vs. temperature for PZT (x = 0.48) over the whole range of temperatures from 20 to 750 K showing the evolution from the monoclinic phase to the cubic phase via the tetragonal phase.

near the phase transition around 300 K, which have allowed a better determination of the phase transition to be made, as shown by the evolution of the lattice parameters as a function of temperature in Fig. 4.8.3. The tetragonal strain ct/at increases as the temperature decreases from the Curie point (T « 660K), to a value of 1.0247 at 300K, below which peak splittings characteristic of a monoclinic phase with a m sw bm « aty/2, (3 ^ 90°, are observed (Fig. 4.8.3). As the temperature continues to decrease down to 20 K, am (which is defined to lie along the [110] tetragonal direction) increases very slightly, and bm (which lies along the [110] tetragonal direction) decreases. The c m lattice parameter reaches a broad maximum value of 4.144 A between 240 and 210 K and then reaches a shallow minimum value of 4.137 A at 60 K. Over the same temperature region there is a striking variation of Ad/d determined from Williamson-Hall plots at various temperatures, as shown in the upper-left inset in Fig. 4.8.2. Ad/d increases rapidly as the temperature approaches the FT-Fu transition at 300 K, in a similar fashion to the tetragonal strain, and then decreases

404

Effective Field Approach to Phase

Transitions

rapidly below this temperature in the monoclinic region. Thus, the microstrain responsible for the large increase in Ad/d is an important feature of the phase transition, which may be associated with the development of local monoclinic order, and is very likely responsible for the large electromechanical response of PZT close to the MPB. 1 The deviation of the monoclinic angle /3 from 90° is an order parameter of the FT~FM transition, and its evolution with temperature is also depicted in Fig. 4.8.3. This phase transition presents a special problem due to the steepness of the phase boundary (the MPB in Fig. 4.8.1). As shown in the previous section, the compositional fluctuations are quite small in these ceramic samples (x « ±0.003) but, even in this case, the nature of the MPB implies an associated temperature uncertainty of AT « 100 K. There is, therefore, a rather wide range of transition temperatures instead of a single well-defined transition, so that the order parameter is smeared out as a function of temperature around the phase change, thereby concealing the nature of the transition. Scans over the (220)c region for several different temperatures are plotted in Fig. 4.8.4, which shows the evolution of phases from the cubic phase at 687K (upper-left plot) to the monoclinic phase at 20K (lower-right plot), passing through the tetragonal phase at intermediate temperatures. With decreasing temperature, the tetragonal phase appears at ~660K and the development of the tetragonal distortion can be observed on the left side of the figure from the splitting of the (202)t and (220)t reflections. On the right side of the figure, the evolution of the monoclinic phase, which appears below ~300K, is shown by the splitting into the (222) m , (222) m , (400) m , and (040) m monoclinic reflections. It is quite evident from Fig. 4.8.4. that the (202)t peak is much broader than the neighboring (220)t peak, for example, and this "anisotropic" peak broadening is a general feature of the diffraction data for both phases. Another feature of the patterns is the presence of additional diffuse scattering between neighboring peaks, which is particularly evident between tetragonal (00/) and (/i00) pairs, and the corresponding monoclinic (00/) and (hhO) pairs.

4.8.4.

Structure Determination

A detailed analysis of the 325 K tetragonal and 20 K monoclinic structures of PbZro.52Tio.48O3 was carried out by Rietveld refinement using the GSAS program package. 27 The pseudo-Voigt peak shape function option

Tetragonal-to-Monoclinic

27.6

Phase Transition in Ferroelectric Perovskite

28.0

27.6

405

28.0

26 (deg) Fig. 4.8.4. Temperature evolution of the pseudocubic (220) peak from the cubic (top left) to the monoclinic (bottom right) phase.

was chosen25 and the background was estimated by linear interpolation between fixed values. An important feature of the refinements was the need to allow for the anisotropic peak broadening mentioned above. This was accomplished by the use of the recently incorporated generalized model for anisotropic peak broadening proposed by Stephens, 28 which is based on a distribution of lattice parameters. It was also necessary to take into account some additional diffuse scattering by modeling with a second, cubic, phase with broad, predominately Gaussian, peaks. A similar strategy has been adopted by Muller et al.29 in a recent study of PbHfo.4Tio.6O3. Although in principle this could represent a fraction of untransformed cubic phase, we suspect that the diffuse scattering is associated with locally disordered

406

Effective Field Approach to Phase Transitions

regions in the vicinity of domain walls. The refinements were carried out with the atoms assigned fully ionized scattering factors. 4.8.4.1.

Tetragonal

Structure

at 325 K

At 325 K, the data show tetragonal symmetry similar to that of P b T i 0 3 . This tetragonal structure has the space group P4mm with Pb in 1(a) sites at (0,0,2); Zr/Ti and 0(1) in 1(6) sites at (1/2,1/2, z); and 0(2) in 2(c) sites at (1/2,0, z). For the refinement we adopt the same convention as that used in Refs. 30 and 31 for P b T i 0 3 , with Pb fixed at (0,0,0). However, instead of thinking in terms of shifts of the other atoms with respect to this origin, it is more physically intuitive to consider displacements of Pb and Zr/Ti from the center of the distorted oxygen cuboctahedra and octahedra, respectively. We shall take this approach in the subsequent discussion. The refinement was first carried out with individual isotropic (C/iso) temperature factors assigned. Although a reasonably satisfactory fit was obtained {RF? = 8.9%), Uiso for O(l) was slightly negative and Uiso for Pb was very large, 0.026 A 2 , much larger than UiSO for the other atoms. Similarly, high values for Pb(t/i so ) in Pb-based perovskites are well known in the literature, and are usually ascribed to local disordered displacements, which may be either static or dynamic. Refinement with anisotropic temperature factors 32 (f/n and C/33) assigned to Pb (Table 4.8.1, model I) gave an improved fit (.RF 2 = 6.1%) with Uu(— U22) considerably larger than [733 (0.032 and 0.013 A 2 , respectively), corresponding to large displacements perpendicular to the polar [001] axis. A further refinement based on local displacements of the Pb from the 1(a) site to the 4(d) sites at (x,x,0), with isotropic temperature factors assigned to all the atoms, gave a small improvement in the fit (RF2 = 6.0%) with x; 0.033, corresponding to local shifts along the (110) axes, and a much more reasonable temperature factor (Table 4.8.1, model II). In order to check that high correlations between the temperature factor and local displacements were not biasing the result of this refinement, we have applied a commonly used procedure consisting of a series of refinements based on model II in which Pb displacements along (110) were fixed but all the other parameters were varied. 34,35 Figure 4.8.5 shows unambiguously that there is well-defined minimum in the R factor for a displacement of about 0.19 A, consistent with the result in Table 4.8.1. A similar minimum was obtained for shifts along (110) directions with a slightly higher R factor. Thus, in addition to a shift of 0.48 A for Pb along the polar [001] axis toward four of its 0(2) neighbors, similar to that in

Table 4.8.1. Structure refinement results for tetragonal PbZro.52Tio.4sO3 at parameters at = 4.0460(1) A, ct = 4.1394(1) A. Fractional occupancies N for a P b in model II, where A^ = 0.25. Agreement factors, .Rwp, Rp2, and x 2 , are de Model I, anisotropic lead temperature factors

Pb Zr/Ti O(l) 0(2) JXwp

RF2 2

x

X

y

z

[/(A2)

0

0

0

0.5 0.5 0.5

0.5 0.5 0

C/11 = 0.0319(4) U33 = 0.0127(4) Uiso = 0.0052(6) Uiso = 0.0061(34) Uiso = 0.0198(30)

0.4517(7) -0.1027(28) 0.3785(24) 4.00% 6.11% 11.4

Model II, y

X

0.0328(5)

0.0328

0.5 0.5 0.5

0.5 0.5 0

408

Effective Field Approach to Phase

Transitions

~T

<110> local shifts <100> local shifts

4.4

4.3

i£ 4.2 (A

4.1

4.0

T= 325 K tetragonal phase

0.0

0.1

0.2 Lead shifts (A)

0.3

Fig. 4.8.5. Agreement factor Rv,p as a function of P b displacements for refinements with fixed values of x along tetragonal (110) and (100) directions as described in text. The well-defined minimum at £~0.19A confirms the result listed in Table 4.8.1 for model II.

PbTi03, 3 0 , 3 1 ' 3 6 there is a strong indication of substantial local shifts of ~0.2A perpendicular to this axis. The Zr/Ti displacement is 0.27 A along the polar axis, once again similar to the Ti shift in PbTiC-3. Attempts to model local displacements along (110) directions for the Zr/Ti atoms were unsuccessful due to the large correlations between these shifts and the temperature factor. Further attempts to refine the z parameters of the Zr and Ti atoms independently, as Corker et al. were able to do, 23 were likewise unsuccessful, presumably because the scattering contrast for X-rays is much less than for neutrons. From the values of the atomic coordinates listed in Table 4.8.1, it can be inferred that the oxygen octahedra are somewhat more distorted than in PbTiC-3, the 0(2) atoms being displaced 0.08 A toward the 0(2) plane above. The cation displacements are slightly larger than those recently reported by Wilkinson et al.37 for samples close to the MPB containing a mixture of rhombohedral and tetragonal phases, and in excellent agreement with the theoretical values obtained by Bellaiche and Vanderbilt 38 for PZT with x = 0.50 from first-principles calculations. As far as we are

Tetragonal-to-Monoclinic

Phase Transition in Ferroelectric Perovskite

409

Table 4.8.2. Selected Z r / T i - O and P b - O bond lengths in the tetragonal and monoclinic structures. Models I and II refer to the refinements with anisotropic temperature factors and local (110) displacements for Pb, respectively (see Table 4.8.1). The standard errors in the bond lengths are - 0 . 0 1 A. Bond lengths Tetragonal

Monoclinic

Model I

Model II

Zr/Ti-0(2)

1.85 x 1 2.29 x 1 2.05 x 4

1.85 x 1 2.29 x 1 2.05 x 4

Pb-O(l)

2.89 x 4

Pb-0(2)

2.56 x 4

2.90 2.71 2.67 2.46

Zr/Ti-0(1)

(A)

x x x x

2 1 2 2

1.87 2.28 2.13 1.96 2.90 2.60 2.64 2.46

x 1 X1 x 2 x 2 x 2 x 1 x 2 x 2

aware no other structural analysis of PZT compositions in the tetragonal region has been reported in the literature. Selected bond distances for the two models are shown in Table 4.8.2. For model I, Zr/Ti has short and long bonds with 0(1) of 1.85 and 2.29 A, respectively, and four intermediate-length 0(2) bonds of 2.05 A. There are four intermediate-length Pb-O(l) bonds of 2.89A, four short Pb-0(2) bonds of 2.56 A, and four much larger Pb-0(2) distances of 3.27 A. For model II, the Z r / T i - 0 distances are the same, but the P b - 0 distances change significantly. A Pb atom in one of the four equivalent (x, x, 0) sites in Table 4.8.1 now has a highly distorted coordination, consisting of two short and two intermediate Pb-0(2) bonds of 2.46 and 2.67 A, and one slightly longer Pb-O(l) bond of 2.71 A (Table 4.8.2). The tendency of P b 2 + , which has a lone sp electron pair, to form short covalent bonds with a few neighboring oxygens is well documented in the literature. 23 ' 39 ~ 41 The observed and calculated diffraction profiles and the difference plot are shown in Fig. 4.8.6 for a selected 26 range between 7° and 34° (upper figure). The short vertical markers represent the calculated peak positions. The upper and lower sets of markers correspond to the cubic and tetragonal phases, respectively. We note that although agreement between the observed and the calculated profiles is considerably better when the diffuse scattering is modeled with a cubic phase, the refined values of the atomic

410

Effective Field Approach to Phase

Transitions

10

r

i 4H

?—MrH—fhttfrt *

>

H

"

•»

»——•

:

I4

1, IXLLi -—t

-*—-t-

-tote 1—

—i—

10

20

30 26 (deg.)

Fig. 4.8.6. Observed and calculated diffraction profiles from the Rietveld refinement of the tetragonal (top) and monoclinic (bottom) phases of P Z T (x = 0.48) at 325 and 20 K, respectively. The difference plots are shown below, and the short vertical markers represent the peak positions (the upper set corresponds to the cubic phase as discussed in t h e text). The insets in each figure highlight the differences between the tetragonal and the monoclinic phases for the pseudocubic (110) reflection, and illustrate the high resolution needed in order to characterize the monoclinic phase.

coordinates are not significantly affected by the inclusion of this phase. The anisotropic peak broadening was found to be satisfactorily described by two of the four parameters in the generalized model for tetragonal asymmetry. 28 4.8.4.2.

Monoclinic

Structure

at 20 K

As discussed above, the diffraction data at 20 K can be indexed unambiguously on the basis of a monoclinic cell with the space group Cm. In this case, Pb, Zr/Ti, and 0(1) are in 2(a) sites at (x, 0, z), and 0(2) in 4(6) sites at (x, y, z). Individual isotropic temperature factors were assigned, and Pb was fixed at (0, 0, 0). For monoclinic symmetry, the generalized expression for anisotropic peak broadening contains nine parameters, but

Tetragonal-to-Monoclinic

Phase Transition in Ferroelectric Perovskite

411

when all of these were allowed to vary the refinement was slightly unstable and did not completely converge. After several tests in which some of the less significant values were fixed at zero, satisfactory convergence was obtained with three parameters (i? wp = 0.036, \2 — 11.5). During these tests, there was essentially no change in the refined values of the atomic coordinates. A small improvement in the fit was obtained when anisotropic temperature factors were assigned to Pb (i? wp = 0.033, x 2 = 9.2). The final results are listed in Table 4.8.3, and the profile fit and difference plot are shown in the lower part of Fig. 4.8.6. From an inspection of the results in Tables 4.8.1 and 4.8.3, it can be seen that the displacements of the Pb and Zr/Ti atoms along [001] are very similar to those in the tetragonal phase at 325 K, about 0.53 and 0.24 A, respectively. However, in the monoclinic phase at 20 K, there are also significant shifts of these atoms along the monoclinic [100], i.e., pseudocubic [110], toward their 0(2) neighbors in adjacent pseudocubic (110) planes, of about 0.24 and 0.11 A, respectively, which corresponds to a rotation of the polar axis from [001] toward pseudocubic [111]. The Pb shifts are also qualitatively consistent with the local shifts of Pb along the tetragonal (110) axes inferred from the results of model II in Table 4.8.1, i.e., about 0.2 A. Thus, it seems very plausible that the monoclinic phase results from the condensation of the local P b displacements in the tetragonal phase along one of the (110) directions. Some selected bond distances are listed in Table 4.8.2. The Zr/Ti-0(1) distances are much the same as in the tetragonal structure, but the two sets of Zr/Ti-0(2) distances are significantly different, 1.96 and 2.13 A, compared to the single set at 2.04 A in the tetragonal structure. Except for Table 4.8.3. Structure refinement at 20 K, space group Cm, lattice . 5.70957(14) A, cm = 4.13651(14) R w p = 3.26%, flF2 = 4.36%, x 2 =

Pb Zr/Ti O(l) 0(2) a

results for monoclinic PbZro.52Tio.4sO3 parameters a m — 5.72204(15) A, bm — A, /3 = 90.498(1)°. Agreement factors 9-3.

Xm

Vm

•^m

C/iso (A 2 )

0 0.5230(6) 0.5515(23) 0.2880(18)

0 0 0 0.2434(20)

0 0.4492(4) -0.0994(24) 0.3729(17)

0.0139 a 0.0011(5) 0.0035(28) 0.0123(22)

For P b , [/j so is the equivalent isotropic value calculated from the refined anisotropic values [Un = 0.0253(7) A 2 , U22 = 0.0106(6) A 2 , t/33 = 0.0059(3) A 2 , U13 = 0.0052(4) A 2 ] .

412

Effective Field Approach to Phase

Transitions

a shortening in the Pb-O(l) distance from 2.71 to 2.60 A, the Pb environment is quite similar to that in the tetragonal phase, with two close 0(2) neighbors at 2.46 A, and two at 2.64 A.

4.8.5.

Discussion

In the previous section, we have shown that the low-temperature monoclinic structure of PbZro.52Tio.4sO3 is derived from the tetragonal structure by shifts of the Pb and Zr/Ti atoms along the tetragonal [110] axis. We attribute this phase transition to the condensation of local (110) shifts of Pb, which are present in the tetragonal phase along one of the four (110) directions. In the context of this monoclinic structure it is instructive to consider the structural model for rhombohedral PZT compositions with x = 0.08-0.38 recently reported by Corker et al.23 on the basis of neutron powder diffraction data collected at room temperature. In this study and also an earlier study 42 of a sample with x = 0.1, it was found that satisfactory refinements could only be achieved with anisotropic temperature factors, and that the thermal ellipsoid for Pb had the form of a disk perpendicular to the pseudocubic [111] axis. This highly unrealistic situation led them to propose a physically much more plausible model involving local displacements for the Pb atoms of about 0.25 A perpendicular to the [111] axis and a much smaller and more isotropic thermal ellipsoid was obtained. Evidence for local shifts of Pb atoms in PZT ceramics has also been demonstrated by pair-distribution function analysis by Teslic and coworkers.39 We now consider the refined values of the Pb atom positions with local displacements for rhombohedral PZT listed in Table IV of Ref. 23. With the use of the appropriate transformation matrices, it is straightforward to show that these shifts correspond to displacements of 0.2-0.25 A along the direction of the monoclinic [100] axis, similar to what is actually observed for x = 0.48. It thus seems equally plausible that the monoclinic phase can also result from the condensation of local displacements perpendicular to the [111] axis. The monoclinic structure can thus be pictured as providing a "bridge" between the rhombohedral and tetragonal structures in the region of the MPB. This is illustrated in Table 4.8.4, which compares the results for PZT with x — 0.48 obtained in the present study with earlier results 43 for rhombohedral PZT with x = 0.40 expressed in terms of the monoclinic cell.44 For x = 0.48, the atomic coordinates for Zr/Ti, 0(1), and 0(2) are listed for

Tetragonal-to-Monoclinic

Phase Transition

in Ferroelectric Perovskite

413

Table 4.8.4. Comparison of refined values of atomic coordinates in the monoclinic phase with the corresponding values in the tetragonal and rhombohedral phases for both the "ideal" structures and those with local shifts, as discussed in text. Tetragonal, x = 0.48, 325 K Ideal ^Zr/Ti ZZr/Ti ^O(l) x

O(2) 2/0(2) z O(2)

am(A) 6m (A) Cm (A) /5(°) a b

Local shifts8,

0.500 0.451 0.500 -0.103 0.250 0.250 0.379

0.530 0.451 0.530 -0.103 0.280 0.250 0.379 5.722 5.722 4.139 90.0

Monoclinic, x = 0.48, 20 K As refined 0.523 0.449 0.551 -0.099 0.288 0.243 0.373 5.722 5.710 4.137 90.50

Rhombohedral (Ref. 43), x = 0.40, :295 K Local shifts' 3 0.520 0.420 0.547 0.093 0.290 0.257 0.393

Ideal 0.540 0.460 0.567 -0.053 0.310 0.257 0.433

5.787 5.755 4.081 90.45

Tetragonal local shifts of (0.03, 0.03, 0). Hexagonal local shifts of (-0.02, 0.02, 0).

the "ideal" tetragonal structure (model I) and for a similar structure with local shifts of (0.03, 0.03, 0) in the first two columns, and for the monoclinic structure in the third column. The last two columns describe the rhombohedral structure for x = 0.40 assuming local shifts of (—0.02, 0.02, 0) along the hexagonal axes and the as refined "ideal" structure, respectively. It is clear that the condensation of these local shifts gives a very plausible description of the monoclinic structure in both cases. It is also interesting to note the behavior of the corresponding lattice parameters; metrically the monoclinic cell is very similar to the tetragonal cell except for the monoclinic angle, which is close to that of the rhombohedral cell. Evidence for a tetragonal-to-monoclinic transition in the ferroelectric material PbFeo.5Nbo.5O3 has also been reported by Bonny et al.45 from single crystal and synchrotron X-ray powder diffraction measurements. The latter data show a cubic-tetragonal transition at ~376 K, and a second transition at ^355 K. Although the resolution was not sufficient to reveal any systematic splitting of the peaks, it was concluded that the data were consistent with a very weak monoclinic distortion of the pseudorhombohedral unit cell. In a recent neutron and X-ray powder study, Lampis et al.46 have shown that Rietveld refinement gives better agreement for the monoclinic structure at 80 and 250 K than for the rhombohedral one. The resulting

414

Effective Field Approach to Phase

(a)

(b)

Transitions

(c)

Fig. 4.8.7. Schematic illustration of the tetragonal (a), monoclinic (b), and rhombohedral (c) distortions of the perovskite unit cell projected on the pseudocubic (110) plane. The solid circles represent the observed shifts with respect to the ideal cubic structure. The light gray circles represent the four locally disordered (100) shifts in the tetragonal structure (a) and the three locally disordered shifts in the rhombohedral structure (c) described by Corker et al.23 The heavy dashed arrows represent the freezing-out of one of these shifts to give the monoclinic observed structure. Note that the resultant shifts in the rhombohedral structure can be viewed as a combination of a [111] shift with local (100) shifts, as indicated by the light gray arrows.

monoclinic distortion is very weak, and the large thermal factor obtained for Pb is indicative of a high degree of disorder. The relationships between the PZT rhombohedral, tetragonal, and monoclinic structures are also shown schematically in Fig. 4.8.7, in which the displacements of the Pb atom are shown projected on the pseudocubic (110) mirror plane. The four locally disordered (110) shifts postulated in the present work for the tetragonal phase are shown superimposed on the [001] shift at the left [Fig. 4.8.7(a)] and the three locally disordered (100) shifts proposed by Corker et al.23 for the rhombohedral phase are shown superimposed on the [111] shift at the right [Fig. 4.8.7(c)]. It can be seen that both the condensation of the [110] shift in the tetragonal phase and the condensation of the [001] shift in the rhombohedral phase leads to the observed monoclinic shift shown at the center [Fig. 4.8.7(b)]. We note that although Corker et al. discuss their results in terms of (100) shifts and a [111] shift smaller than that predicted in the usual refinement procedure, they can be equally well described by a combination of shifts perpendicular to the [111] axis and the [111] shift actually obtained in the refinement, as is evident from [Fig. 4.8.7(c)]. We conclude, therefore, that the FM phase establishes a connection between the PZT phases at both sides of the MPB through the common symmetry element, the mirror plane, and suggest that there is not really a morphotropic phase boundary, but rather a "morphotropic phase,"

Tetragonal-to-Monoclinic

Phase Transition in Ferroelectric Perovskite

415

connecting the F T and F R phases of PZT. In the monoclinic phase the difference vector between the positive and negative centers of charge defines the polar axis, whose orientation, in contrast to the case of the F T and F R phases, cannot be determined on symmetry grounds alone. According to this, from the results shown in Table 4.8.3, the polar axis in the monoclinic phase is found to be tilted about 24° from the [001] axis toward the [111] axis. This structure represents the first example of a ferroelectric material with P 2 — Py ^ P 2 , (Px, Py, Pz) being the Cartesian components of the polarization vector. This class corresponds to the so-called m3m(12)A2Fm type predicted by Shuvalov.47 It is possible that this new phase is one of the rare examples of a two-dimensional ferroelectric48 in which the unit cell dipole moment switches within a plane containing the polar axis, upon application of an electric field. This FM phase has important implications; for example, it might explain the well-known shifts of the anomalies of many physical properties with respect to the MPB and thus help in understanding the physical properties in this region, of great interest from the applications point of view.1 It has been found that the maximum values of 1^33 for rhombohedral PZT with x = 0.40 are not obtained for samples polarized along the [111] direction but along a direction close to the perovskite [001] direction. 49 This points out the intrinsic importance of the [111] direction in perovskites, whatever the distortion present, and is consistent with Corker et al.'s model for the rhombohedral phase, 23 and the idea of the rhombohedral-tetragonal transition through a monoclinic phase. It is also to be expected that other systems with morphotropic phase boundaries between two nonsymmetry-related phases (e.g., other perovskites or tungsten-bronze mixed systems) may show similar intermediate phases. In fact, an indication of symmetry lowering at the MPB of the PZNPT system has been recently reported by Pujishiro et al.50 From a different point of view, a monoclinic ferroelectric perovskite also represents a new challenge for first-principles theorists, until now used to dealing only with tetragonal, rhombohedral, and orthorhombic perovskites. A structural analysis of several other PZT compositions with x = 0.42-0.51 is currently in progress in order to determine the new PZT phase diagram more precisely. In the preliminary version shown in Fig. 4.8.1 we have included data for a sample with x = 0.50 made under slightly different conditions 22 at the Institute of Ceramic and Glass (ICG) in Madrid, together with the data described in the present work for a sample with x — 0.48 synthesized in the Materials Research Laboratory at the

416

Effective Field Approach to Phase Transitions

Pennsylvania State University (PSU). As can be seen the results for these two compositions show consistent behavior, and demonstrate t h a t the F M ~ FT phase b o u n d a r y lies along the M P B of Jaffe et al. Preliminary results for a sample from P S U with x = 0.47 show unequivocally t h a t the monoclinic features are present at 300 K. However, measurements on an ICG sample with the same nominal composition do not show monoclinicity unambiguously, but instead a rather complex poorly defined region from 300 to 400 K between the rhombohedral and tetragonal phases. 2 2 T h e extension of the monoclinic region and the location of the F R - F M phase b o u n d a r y are still somewhat undefined, although it is clear t h a t the monoclinic region has a narrower composition range as the t e m p e r a t u r e increases. T h e existence of a quadruple point in the P Z T phase diagram is an interesting possibility.

Acknowledgments We wish to gratefully acknowledge B. Jones for the excellent quality of the x = 0.48 sample used in this work, and we t h a n k L. Bellaiche, A.M. Glazer, E. Moshopoulou, C. Moure, and E. Sawaguchi for their helpful comments. Support by NATO (Grant no. R.C.G. 970037), the Spanish C I C y T (Project no. PB96-0037), and the U.S. Department of Energy, Division of Materials Sciences (Contract no. DEAC02-98CH10886) is also acknowledged.

References 1. B. Jaffe, W.R. Cook and H. Jaffe, Piezoelectric Ceramics (Academic Press, London, 1971). 2. R.E. Cohen, Nature (London) 358, 136 (1992). 3. R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 49, 5828 (1994). 4. W. Zhong, D. Vanderbilt and K. Rabe, Phys. Rev. Lett. 73, 1861 (1994); Phys. Rev. B 52, 6301 (1995). 5. W. Zhong and D. Vanderbilt, Phys. Rev. Lett. 74, 2587 (1995). 6. A. Garcia and D. Vanderbilt, Phys. Rev. B 54, 3817 (1996). 7. K.M. Rabe and U.W. Waghmare, Phys. Rev. B 55, 6161 (1997). 8. K.M. Rabe and E. Cockayne, in First-Principles Calculations for Ferroelectrics, AIP Conf. Proc. No. 436, edited by R. Cohen (AIP, New York, 1998), p. 61. 9. Ph. Ghosez, E. Cokayne, U.V. Waghmare and K.M. Rabe, Phys. Rev. B 60, 836 (1999). 10. L. Bellaiche, J. Padilla and D. Vanderbilt, Phys. Rev. B 59, 1834 (1999).

Tetragonal-to-Monoclinic Phase Transition in Ferroelectric Perovskite

AY!

11. L. Bellaiche, J. Padilla and D. Vanderbilt, First-Principles Calculations for Ferroelectrics: 5th Williamsburg Workshop, edited by R. Cohen (AIP, Woodbury, 1998), p. 11. 12. G. Soghi-Szabo and R.E. Cohen, Ferroelectrics 194, 287 (1997). 13. G. Shirane and K. Suzuki, J. Phys. Soc. Jpn. 7, 333 (1952). 14. E. Sawaguchi, J. Phys. Soc. Jpn. 8, 615 (1953). 15. Y. Xu, Ferroelectric Materials and their Applications (North Holland, Amsterdam, 1991). 16. K. Kakewaga, O. Matsunaga, T. Kato and Y. Sasaki, J. Am. Ceram. Soc. 78, 1071 (1995). 17. J.C. Fernandes, D.A. Hall, M.R. Cockburn and G.N. Greaves, Nucl. Instrum. Meth. Phys. Res. B 97, 137 (1995). 18. M. Hammer, C. Monty, A. Endriss and M.J. Hoffmann, J. Am. Ceram. Soc. 8 1 , 721 (1998). 19. W. Cao and L.E. Cross, Phys. Rev. B 47, 4825 (1993). 20. B. Noheda, D.E. Cox, G. Shirane, J.A. Gonzalo, L.E. Cross and S.-E. Park, Appl. Phys. Lett. 74, 2059 (1999). 21. The [110] and [110] directions are chosen so that the monoclinic angle (5 > 90° to conform with usual crystallographic convention. 22. B. Noheda, J.A. Gonzalo, A.C. Caballero, C. Moure, D.E. Cox and G. Shirane, cond-mat/9907286, Ferroelectrics (to be published). 23. D.L. Corker, A.M. Glazer, R.W. Whatmore, A. Stallard and F. Fauth, J. Phys.: Condens. Matter 10, 6251 (1998). 24. C.W. Dwiggins Jr., Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 3 1 , 146 (1975). 25. L.W. Finger, D.E. Cox and A.P. Jephcoat, J. Appl. Crystallogr. 27, 892 (1994). 26. G.K. Williamson and W.H. Hall, Acta Metall. 1, 22 (1953). 27. A.C. Larson and R.B. Von Dreele (unpublished). 28. P.W. Stephens, J. Appl. Crystallogr. 32, 281 (1999). 29. C. Muller, J.-L. Baudour, V. Madigou, F. Bouree, J.-M. Kiat, C. Favotto and M. Roubin, Acta Crystallogr., Sect. B: Struct. Sci. 55, 8 (1999). 30. A.M. Glazer and S.A. Mabud, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 34, 1065 (1978). 31. R.J. Nelmes and W.F. Kuhs, Solid State Commun. 54, 721 (1985). 32. The structure factor correction is defined in terms of the anisotropic Uij thermal factors as exp{—\2ir2(^2i,juija*a'j)]}, a* being the lattice vectors of the reciprocal unit cell. 33. L.B. McCusker, R.B. von Dreele, D.E. Cox, D. Louer and P. Scardi, J. Appl. Crystallogr. 32, 36 (1999). 34. K. Itoh, L.Z. Zeng, E. Nakamura and N. Mishima, Ferroelectrics 63, 29 (1985). 35. C. Malibert, B. Dkhil, J.M. Kiat, D. Durand, J.F. Berar and A. Spasojevic-de Bire, J. Phys.: Condens. Matter 9, 7485 (1997). 36. G. Shirane, R. Pepinski and B.C. Frazer, Acta Crystallogr. 9, 131 (1956).

418

Effective Field Approach to Phase Transitions

37. A.P. Wilkinson, J. Xu, S. Pattanaik and J.L. Billinge, Chem. Mater. 10, 3611 (1998). 38. L. Bellaiche and D. Vanderbilt, Phys. Rev. Lett. 83, 1347 (1999). 39. S. Teslic, T. Egami and D. Viehland, J. Phys. Chem. Solids 57, 1537 (1996); Ferroelectrics 194, 271 (1997). 40. D.L. Corker, A.M. Glazer, W. Kaminsky, R.W. Whatmore, J. Dec and K. Roleder, Acta Crystallogr., Sect. B: Struct. Sci. 54, 18 (1998). 41. S. Teslic and T. Egami, Acta Crystallogr., Sect. B: Struct. Sci. 54, 750 (1998). 42. A.M. Glazer, S.A. Mabud and R. Clarke, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 34, 1060 (1978). 43. A. Amin, R.E. Newnham, L.E. Cross and D.E. Cox, J. Solid State Chem. 37, 248 (1981). 44. The rhombohedral unit cell can be expressed in terms of a monoclinic one by am = 2a r cos(a/2)4, 6m = 2a r sin(a/2), c m = a r , f3 = 180° — 4>, where cos0 = 1 — 2sin 2 (a/2)/cos(a/2) and aT and a are the R3m cell parameters. Note that o r in Ref. 43 refers to the doubled cell. 45. V. Bonny, M. Bonin, P. Sciau, K.J. Schenk and G. Chapuis, Solid State Commun. 102, 347 (1997). 46. N. Lampis, P. Sciau and A.G. Lehmann, J. Phys.: Condens. Matter 11, 3489 (1999). 47. L.A. Shuvalov, J. Phys. Soc. Jpn. 28, 38 (1970). 48. S.C. Abrahams and E.T. Keve, Ferroelectrics 2, 129 (1971). 49. X.-H. Du, J. Zheng, U. Belegundu and K. Uchino, Appl. Phys. Lett. 72, 2421 (1998). 50. K. Fujishiro, R. Vlokh, Y. Uesu, Y. Yamada, J.-M. Kiat, B. Dkhil and Y. Yamashita, Ferroelectrics (to be published).

Chapter 4.9

Evolution of the Ferroelectric Transition Character of Partially Deuterated Triglycine Selenate* Carmen Arago and Julio A. Gonzalo Departamento de Fisica de Materiales Universidad Autonoma, 28049 Madrid,

C-IV, Spain

Automatically recorded hysteresis loops and dielectric constant data on pure and partially deuterated triglycine selenate and sulfate are used to characterize the transition evolution from quasi-tricritical to clearly first order character. A quantitative assessment of the transition's displacive degree is made through the first anharmonic contribution to the effective field (g), obtained from hysteresis loop data. This characterization is complemented with the assessment by means of a generalized Rhodes-Wohlfarth parameter R = (NkBC/AwPso)1 involving the Curie constant C and the saturation spontaneous polarization Pso as done previously by other authors.

4.9.1.

Introduction

First-principles local density approximation calculations 1 on ferroelectric perovskites have shown t h a t cubic/tetragonal phase transitions in some of these compounds, at first thought to be of the displacive kind, can be well described as order/disorder-type transitions. On t h e other hand, generalized Rhodes-Wohlfarth empirical parameters, 2 ~ 4 analogous to those used to characterize magnetic phase transitions, have been proposed to assess quantitatively the displacive versus order/disorder degree of ferroelectric transitions. *Work previously published in Journal of Physics: Condensed Matter 12, 3737 (2000). Copyright © 2000. Institute of Physics. 419

420

Effective Field Approach to Phase

Transitions

In this work, we focus our attention on transitions in hydrogenated and deuterated crystals of triglycine selenate (TGSe)i_ x (DTGSe)a; and triglycine sulfate (TGS)i_ x (DTGS)a;, which are paradigmatic examples of order/disorder transitions but are not free from typical features of displacive systems such as the discontinuous, first order transition observed in DTGSe. We will see that in these ferroelectrics (whose structure is so complicated as to preclude ab initio calculations to substantiate the mixed displacive order/disorder character) it is still possible to quantitatively characterize the evolution of the displacive component involved in the phase transition. The order/disorder features of a transition may be ascertained through measurements of the transition entropy and the specific heat jump at T c . The displacive characteristics are usually (if it is at all feasible) seen by measuring underdamped soft modes in the vicinity of the transition by means of infrared, Raman, or neutron scattering spectroscopies. Recently, Dalai et al,b have pointed out that diffraction work and NMR data provide convincing evidence that order/disorder and displacive features coexist in H-bonded ferroelectric transitions. Characteristics of mixed order/disorderdisplacive phase transitions can be analyzed within the framework of a generalized effective field approach in which the effective field is given by EeS = E + (3PA + 7 P d 3 + dP%L. The basic differences between a purely order/disorder and a purely displacive transition, if such extreme cases are possible4 at all among ferroelectric transitions, can be visualized in the following way. In an order/disorder transition an ion (atom) or a group of ions (atoms) in the unit cell may occupy either one of two "off center" (shifted) positions between neighboring ions (atoms). These two positions correspond to two potential well minima separated by a relatively large energy barrier at the mid-point between them. In the paraelectric phase the two potential minima are equivalent, and they are equally populated. In the ferroelectric phase the cooperative (dipolar) interaction between cells makes the two wells become asymmetric and a spontaneous polarization sets in. Even at the order/disorder transition, if the two potential minima are deep enough and their separation is well kept, harmonic oscillations around the equilibrium position may be well defined and the normal modes may remain "hard." On the other hand, in a displacive transition, corresponding to the case of a shallow well separated by a weak potential barrier, the separation between the minima (at which the potential may require additional anharmonic terms to be well specified) would evolve from the low-temperature ferroelectric phase toward the

Evolution

of the Ferroelectric Transition

Character

421

transition point and beyond, into the paraelectric phase. At these higher temperatures we would finally have one single effective minimum and the anharmonic oscillations around equilibrium could result in "soft" modes. Depending on the relative sign of higher order terms in the local potential, the dipole moment in the paraelectric phase (T > Tc) could be either larger or slightly smaller than in the ferroelectric phase (T S OK). Triglycine selenate, belonging to the triglycine sulfate family, is an interesting uniaxial ferroelectric whose transition is very close to a tricritical point. 6 - 8 Increasing deuteration changes the transition's features giving rise to a clearly discontinuous transition with a thermal hysteresis AT = 0.8°C for x= 0.96, almost full deuteration. 9 The availability of accurate dielectric and thermal data near the Curie temperature for both TGSe and TGS is important to assess quantitatively the evolution with deuteration of the degree of displacive character of the phase transition.

4.9.2.

Experiment

The samples of TGSe and TGS were prepared at the Institute of Physics, Adam Mickiewicz University, Poznan (Poland). They were cut from single crystals grown from aqueous solution and their shape was almost paralleepipedic. The thickness of these samples was between 0.6 and 1.0 mm and their areas went from 3 to 25 mm 2 . Hysteresis loops were measured at a frequency of 50 Hz with a DDP bridge and they were observed with a Nicolet NIC-310 digital oscilloscope. The temperature was carefully controlled by means of an Unipan 680 controller that allowed very slow heating/cooling ramps maintaining an accuracy of 0.01 K. The actual temperature inside the sample holder was measured with a type-T thermocouple connected to a Keithley 196 voltmeter. The sample holder was well isolated and refrigerated by liquid nitrogen. The whole measuring process was fully automated, which resulted in a real improvement of the data acquisition. In fact, 4000 points were recorded for each loop at temperature intervals of 0.25 K and even lower (0.1K) as the transition temperature was approached. After data recording, an iterative computer program was used to analyze the loops and to obtain the spontaneous polarization Ps at each temperature. Dielectric constant measurements were performed with an HP 428A precision LCR meter at a low voltage of 1V and a frequency of 1 kHz. Also these measurements were automatically programed and recorded.

422

4.9.3.

Effective Field Approach to Phase

Transitions

Results and Discussion

Figure 4.9.1 gives the spontaneous polarization Ps against the temperature T for (a) (TGSe) 1 _ x (DTGSe) x , x = 0, 0.35, 0.80; and (b) (TGS)i_ :E (DTGS) x , x = 0, 0.54, 0.77. It can be seen that the decrease of spontaneous polarization approaching the transition temperature is more pronounced for TGSe than for TGS, as corresponds to quasi-critical behavior, and that for DTGSe the data appear to suggest a small discontinuity at the transition. Assuming an effective field of the form10 EeS = E + pPd[l + g(Pd/Nfi)2

+

h(Pd/N^L],

• TGSe1xDTGSex

(1)

x=0 x = 0 .35

0

•

X = 0.80

T(°C)

2 5-

~

•

TGS,J)TGSr

^ ^ ^ * W 2.0-

x= 0 "

* x=a.n

^ < « ^£^ * «*>„ S w^ s & .

> . ^ ^Su

1.5-

^ V^ «*%>o ^ v \

1.0-

% \

0.5-

% :<>

*A

«

0.0-

A

° 1

•

1

1

•

1

'

1

'

1

'

T(°C) Fig. 4.9.1. Spontaneous polarization Pso against temperature T for different deuteration percentages in (a) triglycine selenate and (b) triglycine sulfate.

Evolution of the Ferroelectric Transition

Character

423

where E is the external applied field, Pa is the dipolar polarization, N the number of dipoles per unit volume g, h a dimensionless coefficient, and // the elementary dipolar moment, the dipoles could be either rigid [fi(T) = ^i(O) = fi(Tc) = constant] or deformable. In this latter case there are two possibilities, /z(T)///(0) < 1 and /j,(T)/n(0) > l, depending on the signs of higher order terms in the local potential. From the equation of state

|r'-(ff)

(2)

in terms of reduced variables e = E//3N/1, p = Pd/Nfi, we obtain

e=Zrtanh-1p-p{l+gp2

+ hp*L)

(3)

-*-C

and with p = ps (the reduced spontaneous polarization) for e = 0, we have

g.ipl)sg+hp^mi^2im_^

(4)

We note that each hysteresis loop provides a spontaneous polarization PS{T), which is normalized, ps(T) = P S (T)/P s 0 , with P s 0 = Nfj,, to be introduced in Eq. (4). Figure 4.9.2 gives g' against ps for (a) (TGSe)i_ a; (DTGSe) :c and (b) (TGS)i_a;(DTGS) x for the different molar fractions. As the plots can be linearly fitted we can determine g as the value at which each fitting line crosses the g' axis. The coefficient g, representing the strength of the quadrupolar interaction, 11-13 is a good indicator for the type of transition of the ferroelectric crystal. In fact, g < 1/3 for a second order transition, g > 1/3 for a first order transition, and g = 1/3 for a tricritical point. We had checked in an earlier work 14 that increasing deuteration in TGS determines an evolution of the transition to a tricritical point but staying still far from it. In a similar way for TGSe, the first order character of the ferroelectric transition becomes more evident 9 as the percentage of deuteration grows. As the transition becomes more first order the g value grows, 13 reflecting likely changes in the local potential. An entirely different parameter, R = /ic/A*o> analogous to the RhodesWohlfarth parameter for magnetic systems as previously mentioned, has been proposed 3 ' 4 to characterize the ferroelectric transition type. Here, nc is defined as the individual dipolar moment in the paraelectric phase, obtained

424

Effective Field Approach to Phase

Transitions

g'=g + hpj

9'

TGSe1.pTGSex

• x=0
•—|—i—i—•—|—i—i—i—i—•—i—i—i—i—i—i—i—i

00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

09

1.0

g' = g + hpf

9 • « &

TGSUlpTGSx

x=0 x=0.54 x=0.77

i—|—•—|—,—i—i—i—i—|—i—i—i—i—•—i—i—i—i

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 4.9.2. Linear plot of g' against normalized spontaneous polarization pi for each deuteration percentage in (a) triglycine selenate, and (b) triglycine sulfate.

from the Curie-Weiss constant, C = e(T — Tc), with e the low-field and lowfrequency dielectric constant, and fio is taken as the saturation individual dipolar moment in the ferroelectric phase (T = OK), given by PSQ/N. SO we can write

R=

{NkBC/4nP^2.

It is evident that in an order/disorder ferroelectric, with rigid dipoles approximately equal in the paraelectric phase and at saturation (T = OK), the value of R must be near one. As the displacive character of the transition becomes more and more pronounced one expects that going through Tc involves not only disordering the preferentially aligned dipoles but also

Evolution of the Ferroelectric Transition

Character

425

releasing their strain (related to the spontaneous field and to the anharmonic potential terms). It may be then expected that the generalized Rhodes-Wohlfarth parameter becomes larger than one. Table 4.9.1 shows the values obtained from hysteresis loops and dielectric constant measurements for the six different samples. Figure 4.9.3 gives R as a function of g for pure and deuterated TGS and pure and deuterated TGSe. It can be seen that there is a linear correlation between both parameters. In the plot we have noted g = 1/3 (which indicates the crossover between continuous and discontinuous transitions) and R = 1 (which is the borderline between the dipole behavior upon crossing the transition). The above data show clearly that R grows when g grows and that R > 1 favors the displacive character of the transition, which becomes more likely to be first order (g > 1/3). Table 4.9.1.

(TGSe)i_ a ; (DTGSe) : I

Dielectric measurements and resulting parameters. T c (K)

AT(K)

Ps0(MCcm-2)

295.16 297.79 303.3 322.18 326.15 330.4

0.1 0.2 0.5

4.5 4.3 4 4.2 3.9 3.6

0 0.35 0.80

(TGSJi-^DTGS),

0 0.54 0.77

— — —

o • R=1

R

-H+

R

4166 4000 3703 3225 2941 2702

0.87 0.89 0.92 0.82 0.84 0.87

(TGS^JDTGS),, (TGSe^JDTGSe),,

+.

i

0.30 0.33 0.35 0.21 0.23 0.25

C(K)

L_

9 Fig. 4.9.3. Plot of Rhodes—Wohlfarth parameter R against g for triglycine selenate and triglycine sulfate with different deuteration percentages. Each point can be identified from the values recorded in Table 4.9.1.

426

Effective Field Approach to Phase Transitions

We may then conclude t h a t the parameter g, which can be easily obtained from hysteresis loop d a t a for most ferroelectric crystals, gives a good characterization of the transition in uniaxial, basically order/disorder ferroelectrics (0 < g < 1/3) as well as in typically displacive ferroelectrics (g = 1) like most pseudocubic ferroelectric perovskites. Cases like LiTaOs, which can be considered a highly distorted perovskite, are good examples of the coexistence of pronounced order/disorder and displacive features. T h e case of Sn2P2S6, with a second order transition at T = 337K, is another example of a transition t h a t can be said to be at the o r d e r / d i s o r d e r displacive crossover. 1 5

Acknowledgments Financial support from D G I C y T , through grant no. PB96-0037, is gratefully acknowledged. We would like to t h a n k G. Shirane and J. M. PerezM a t o for helpful discussions and the latter for making available a preprint of his work on the subject prior to publication. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

W. Zhong, D. Vanderbilt and K.M. Rabe, Phys. Rev. Lett. 26, 1861 (1994). M. Tokunaga, J. Phys. Soc Jpn. 57, 4275 (1988). Y. Onodera and N. Sawashima, J. Phys. Soc. Jpn. 60, 1247 (1991). J.M. Perez-Mato, S. Ivantchev, A. Garcia and I. Etxebarria, Proc. 9th Eur. Meeting on Ferroelectricity, Prague (1999). N. Dalai, A. Bussman-Holder and K.H. Michel, Proc. 9th Eur. Meeting on Ferroelectricity, Prague (1999). T. Iglesias, B. Noheda, G. Lifante, J.A. Gonzalo and M. Koralewski, Phys. Rev. 5 50, 10307 (1994). T. Iglesias, B. Noheda, B. Gallego, J.R. Fdez del Castillo, G. Lifante and J.A. Gonzalo, Europhys. Lett. 28, 91 (1994). B. Fugiel and M. Mierzwa, Phys. Rev. B 57, 777 (1998). K. Gesi, J. Phvs. Soc. Jpn. 4 1 , 565 (1976). J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). J.A. Gonzalo, R. Ramirez, G. Lifante and M. Koralewski, Fermelectr. Lett. 15, 9 (1993). B. Noheda, G. Lifante and J.A. Gonzalo, Fermelectr. Lett. 15, 109 (1993). B. Noheda, G. Lifante and J.A. Gonzalo, Fermelectr. Lett. 17, 25 (1994). C. Arago, B. Noheda and J.A. Gonzalo, Prot. 9th Eur. Meeting on Fermelectricity, Prague (1999). J. Hlinka, T. Janssen and V. Dvorak, J. Phys.: Condens. Matter, 11, 3209 (1999).

Chapter 4.10

Composition Dependence of Transition Temperature in Mixed Ferroelectric-Ferroelectric Systems' Carmen Arago, Manuel I. Marques and Julio A. Gonzalo Departamento de Fisica de Materiales, C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

The composition dependence of the transition temperature Tc(x) for mixed crystals of triglycine sulfate (TGS) and deuterated triglycine sulfate (DTGS) has been investigated experimentally and analyzed theoretically. A similar investigation has been carried out for triglycine selenate (TGSe) and deuterated triglycine selenate (DTGSe). The experimental results agree well with previous data by Brezina (TGS/DTGS) and Gesi (TGSe/DTGSe). Classically, we would expect a linear dependence of temperature Tc(x) = Tc(0) + Ax with composition x. However, Tc(x) is clearly below the straight line, especially at x ~ 0.5. We show that an effective field approach with quantum effects can justify well this lowering of Tc(a;).The experimental data are also compared with classical and quantum Monte Carlo simulations for mixed Ising systems. It is shown that there is a good qualitative agreement with the observed behavior in mixed crystals of the TGS family.

It is a well-known fact t h a t the critical t e m p e r a t u r e Tc for a ferroelectric transition depends on the material's composition. Mixed systems, ferroelectric-antiferroelectric 1 as well as ferroelectric-paraelectric, 2 ' 3 have been analyzed earlier in order to explain theoretically the experimentally

"Work previously published in Phys. Rev. B 62, 13 (2000). Copyright © 2000. The American Physical Society. 427

428

Effective Field Approach to Phase

Transitions

observed transition temperature behavior. In the above-mentioned systems a quantum effective field approach can explain the deviations of Tc(x) from the expected classical results using just mean field considerations. In the present work, we study partially deuterated triglycine sulfate ( T G S ) I _ ; C ( D T G S ) : E and partially deuterated triglycine selenate (TGSe)i_ a: (DTGSe) a ; crystals considered as special cases of ferroelectricferroelectric mixed systems with a small change (about 4%) in transition temperature between Tc(0) and T c (l). Experimental data from earlier works 4 - 7 show that the x dependence of Tc is clearly nonlinear. The lowering of the experimental curve below the classically predicted straight line is, in fact, very small when compared with the absolute values of the critical temperatures. As noted below the experimental error in our Tc measurements is ±0.1 K. In addition it must be noted that in Ref. 4 there is a substantial scattering of the data points in the experimental results, probably caused by the difficulty to evaluate the actual composition of each sample which very likely are not spatially homogeneous. Entropy arguments, analogous to Vegard's rule for the melting temperature vs. composition in alloys, might result in a nonlinearity of Tc(x). We will see below that our Monte Carlo simulations with Ising mixed systems do not support large enough deviations to explain the observed behavior in our case. The (TGSh-^DTGS);, and (TGSe)i_ x (DTGSe) x crystals have been prepared at the Institute of Physics, Adam Mickiewicz University, Poznan (Poland), with different nominal deuteration percentages. Their shape was paralelepipedic with thickness between 0.6 and 1.6 mm and areas around 3-4mm 2 . Dielectric measurements, both hysteresis loops from which we obtain PS(T) and dielectric constant e(T), have been performed in order to determine the critical temperature Tc for each composition. Hysteresis loops were obtained by using a DDP bridge and were observed with a Nicolet 310 oscilloscope. The data recording process was fully automated. Dielectric constant measures were performed with a Hewlett-Packard LCR meter 4284A applying a voltage of 1V and a frequency of 1 kHz. The data were automatically recorded. The sample holder was a copper cylinder with three concentric compartments. The inner compartment contained the sample in contact with the electrodes and the thermocouple whose signal was measured by a Keitley voltmeter model 196. In the intermediate cylinder were located the electric connections to a temperature controller, Unipan type 680, with an accuracy of ±0.1 K. The outside cylinder was used for refrigeration.

Composition

Dependence of Transition

429

Temperature

A

Gesi's data (TGSeJ^DTGSe),, Fitting curve for Gesi's data A This work Quantum effects • Monte Carlo data with quantum effects - ° — Clasical Monte Carlo data & classical approach.

35

35

30

30

25

25

o o

20

-"

1—

0.2

0.0

—I— 0.4

—I— 0.6

20 1.0

—I— 0.8

Deuteration molar fraction x Fig. 4.10.1. Deuteration (molar fraction) x dependence of transition temperature Tc for mixed crystals of triglycine sulfate ( T G S ) I _ : E ( D T G S ) X . 70

70 A

65

A •

Brezina's data (TGS)1x(DTGS)x Fitting curve for Brezina's data This work Quantum effects Monte Carlo data with quantum effects Clasical Monte Carlo data & classical approach j r ^ - / A A

it*

55

45 0.0

0.2

0.4

0.6

0.8

1.0

Deuteration molar fraction x Fig. 4.10.2. Deuteration (molar fraction) x dependence of transition temperature T c for mixed crystals of triglycine selenate (TGSe)i_ x (DTGSe) : E .

Experimental results 5 ' 7 are reproduced in Figs. 4.10.1 and 4.10.2. We have attempted a theoretical analysis similar to the one made before with other mixed systems. In our case, we can think of the ferroelectric-ferroelectric system as a mixture of two different ferroelectrics, with well-defined individual dipolar moments /in (for the pure case) and /^D

430

Effective Field Approach to Phase

Transitions

(for the completely deuterated case). In the classical approach, the transition temperature for each composition would be T ; ( x ) = T c * ( 0 ) ( l - x ) + Tc*(l)*,

(1)

T* being according to the standard effective field theory 8 (3N»2/kB,

Tc* =

where (3 is the common mean field ccoefncient, N the common number of dipoles per unit volume, and kB Boltzman's constant. The energy can be written quantum mechanically as hcoo Q + (n) T o ) = kBTc* and it should be equated to the thermal energy kBT*. Using Planck's distribution law in thermal equilibrium i

(n)n =

o(haj0/kBTc)

_ J'

we have 1 2

1

kBT*(x)

e[hw0(x)/kBTc(x)]

_ I ~

hiU0(x)

This equation can be rewritten as 1 tanh[/iwo(a;)/2A;BTc(a;)]

=

2kBT*{x) hwo(x)

(2)

and from here we obtain T (x) = c[ '

hw0(x)/2kB ta,nh-1[hLO0(x)/2kBT*(x)]'

,g,

Taking Eq. (1) into Eq. (3) and identifying aeff(x)

(4)

= zk^W)

we get finally a

rp I \ _ rp*(Q\ c(X)

-

c[

eff\%)

)

1

~tanh- [aeS(x)/{(l-x)

/g\

+ [T*(l)/T*(OM]'

Composition

Dependence of Transition

Temperature

431

which gives a relationship between the experimentally observed "quantum transition temperature" Tc(x) for each composition and the "classic transition temperatures" Tc*(0) and Tc*(l) for pure and completely deuterated compositions. Classical and quantum transition temperatures are scarcely different for both extreme compositions x = 0 and x = 1. aefi(x) defined by Eq. (4) gives the degree of the "quantum effect" as a function of composition for the partially deuterated systems. This x dependence can be associated to a certain WQ(X), which may be taken to mean that the zero point quantum energy changes slightly with composition. This change becomes a maximum when the deuterated molar fraction is x ~ 0.5 and a minimum for x = 0 and x = 1 (undiluted systems). The interaction potential for identical dipoles (/XH or fir>) is perfectly periodic. On the other hand, the interaction potential between different dipoles (/ZH and /XD) is aperiodic and aleatory. This difference may be expected to show up in the interatomic potential determining the zero point energy and therefore the frequency OJQ. Later, we will see that the observed behavior of Tc{x) can be taken care of through an expression for Awo(a;) reflecting the aperiodic character of the system for x = 0.5. The function Tc(x) given by Eq. (5) has been plotted at Figs. 4.10.1 and 4.10.2 for the mixed TGS and mixed TGSe systems, respectively. As can be seen it agrees fairly well with the experimental data in both cases. At this point, we can think of quantum effects as a good candidate to explain the deviations from the expected linear behavior. It is not clear whether the linear approximation is suitable to describe accurately the composition behavior in a purely classical scenario. Even if mean field approximation describes well the critical behavior of long-range ferroelectrics we do not know if it takes properly into account the effect of short-range interactions between deuterated and nondeuterated dipoles. Thus, we have considered a mixed Ising system with two species, each with a different value of the critical temperature. The Ising model is more appropriate for uniaxial ferroelectrics (such as crystals of the TGS) than other models (e.g., Heisenberg) but we should note that in a ferroelectric the true interactions are not nearest neighbors only, but long range as is well known. To simulate a change on transition temperature similar to the one experimentally observed in (TGS)i_3;(DTGS) x , we have chosen the same experimental ratio between the critical temperatures TC*(0)/TC*(1) « 0.96, Tc*(l) being the critical temperature of the deuterated Ising system with an interaction constant J = 1 and T* (0) the critical temperature of the pure

432

Effective Field Approach to Phase

Transitions

component whose interaction constant J' = 0.96. So this gives rise to the following interactions: deuterated-deuterated equal to 1, deuterated-pure equal to (0.96) 1 / 2 , and pure-pure equal to 0.96. First, we localize the value of the critical temperatures for different values of the concentration x. We determine this value as the point where the susceptibility reaches its maximum. Nowadays, one of the best numerical methods to determine critical temperatures by Monte Carlo simulations is the histogram reweighting method, first introduced by Ferrenberg and Swendsen. 9 ' 10 This method accumulates information about energy histograms at a single temperature to determine the value of a certain property, such as the susceptibility, at another point of the phase diagram. As we are expected to be very close to the critical temperature we have used the Wolff's single cluster algorithm, 11 ' 12 which is most convenient near the critical temperature. Simulations were performed for systems with 1000 spins and periodic boundary conditions at fixed temperatures linearly extrapolated between the values Tc*(0) and T*(l). More than 106 Monte Carlo steps (MCS) were performed for each value of the concentration (x) leaving 105 MCS at the beginning for thermalization. Five different realizations of the randomness were considered for each concentration, resulting in very close values of the critical temperature. We have used a canonical distribution of deuterated dipoles: this means that the concentration of nondeuterated dipoles is kept constant. The deviations from linearity cannot be observed because they are inside of our error estimation, which is around 0.01% smaller than the point size. We may say, however, that there is a tendency for the critical temperature to go slightly below the extrapolated linear value. In fact, when the difference between extreme critical temperatures is made larger (by about 33%), we are able to detect a decrease of 0.3% at x ~ 0.5 from the linear behavior in our numerical simulations. In any case, even with a large difference between the extreme critical temperatures, we are unable to explain the relatively large value, around 1.7% obtained experimentally in our ferroelectric-ferroelectric mixed systems. So we are justified in taking into account possible quantum effects. We consider 1 ' 3 the existence of a zero point energy, fkjJo{x)/2, where LOQ(X) may be x dependent as noted before. From the point of view of the Monte Carlo algorithm it is equivalent to introduce 13 an effective "quantum transition temperature," which may be obtained from Eq. (2). Taking into account that the maximum deviation AWQ(X) of zero point energy seems to

Composition Dependence of Transition Temperature

433

occur at x ~ 0.5 we write Au;o(z) Aw 0 (0.5)

••4x(l-x),

(6)

which implies Aw o (0) = 0 and Awo(l) = 0. It must be noted t h a t we have Chosen [Awo(0.5)] M onte Carlo — [Awo(0.5)] E xperimental-

A possible justification for this x dependence of Awo might be t h a t for x = 0 and i = l w e get something similar to perfect periodicity in the lattice, with a well-defined zero point energy above the occupied potential minima, while at x = 0.5 the aperiodicity is large and we may expect the largest changes in the zero point energy, with respect to a perfect periodic system. Results from Monte Carlo simulations, for classic and q u a n t u m approaches, are also displayed at Figs. 4.10.1 and 4.10.2 for ( T G S ) i _ a : ( D T G S ) a ; and ( T G S e ) i _ x ( D T G S e ) a ; respectively. It can be easily observed t h a t t h e quant u m effects d a t a match b o t h experimental d a t a and theoretical calculations. We can conclude t h a t the q u a n t u m effect approach, which has proved to fit anomalous behavior in other mixed systems, could justify the experimental d a t a of the triglycine sulfate family. Also, we have shown t h a t in Ising systems thermodynamics arguments cannot explain the decrease of Tc(x) from a linear behavior, as in our partially deuterated crystals. Obviously, q u a n t u m effects are more likely to describe well large deviations as those observed in mixed systems made up of very different components.

Acknowledgments Financial support from D G I C y T (Grant no. PB96-0037) is gratefully acknowledged.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

J.A. Gonzalo, Phys. Rev. B 39, 12297 (1988). W. Windsch, Ferroelectrics 47, 231 (1983). J.A. Gonzalo, Ferroelectrics 168, 1 (1995). B. Brezina and F. Smutny, Czech. J. Phys. 18, 393 (1968). C. Arago, B. Noheda and J.A. Gonzalo, Ferroelectrics 238, 1 (2000). K. Gesi, J. Phys. Soc. Jpn. 4 1 , 565 (1976). C. Arago and J.A. Gonzalo, J. Phys.: Condens. Matter 12, 3737 (2000). J.A. Gonzalo, Effective Field Approach (World Scientific, Singapore, 1991). A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988).

434

10. 11. 12. 13.

Effective Field Approach to Phase

Transitions

A.M. Ferrenberg a n d D.P. L a n d a u , Phys. Rev. B 4 4 , 5081 (1991). U. Wolff, Phys. Rev. Lett. 6 2 , 361 (1989). J.S. W a n g a n d R . H . Swendsen, Physica A 1 6 7 , 565 (1990). J . Dec a n d W . K l e e m a n n , Solid State Commun. 1 0 6 , 695 (1998).

Chapter 4.11

Temperature Dependence of Mode Griineisen Parameters in Ferroelectric Perovskites at T = Tc* Carmen Arago, Jorge Garcia and Julio A. Gonzalo Departamento de Fisica de Materiales C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Jose de Frutos Escuela

Departamento de Fisica Aplicada, Tecnica Superior de Ingenieriade Telecomunicacion, Ciudad Universiraria s/n, 28040 Madrid, Spain

An approximate expression for the temperature dependence of the Griineisen parameter 7 T O ( T ) for ferroelectric perovskites is obtained, which implies a Curie-Weiss behavior in the vicinity of the transition temperature. While no temperature-dependent data on 7TO (T) itself is available, indirect estimates based upon the frequency and volume temperature dependence seem to give reasonable results.

T h e mode Griineisen parameter is defined 1 as dw/u) 7 =

dV/V

where duj is the frequency shift of the mode frequency w and dV is the variation of the equilibrium volume V, produced in b o t h cases by changes in pressure. In earlier works, 2 ' 3 the transverse optic Griineisen parameter TToCO has been related to the frequency ratio x = WLO/WTO in several "Work previously published in Phase Transitions 76, 3 (2002). Copyright © 2002. Taylor & Francis Group. 435

436

Effective Field Approach to Phase

Transitions

binary cubic crystals through a rather simple expression Ax2 + B, where the coefficients A and B change for each family of binary compounds. For a diatomic pseudocubic crystal, the equations of ionic motion can be written as mu = F-

e.Eiocai.

(1)

If we substitute the ionic displacement at the r site u(r) = uoe _lw * and divide both members by u(r), Eq. (1) transforms for transverse and longitudinal modes into 2

4ne2 = / - y —, ,

-mwTO 2

,

,

8 7 I

(2)

"e2

/ON

- m w L O = / + y —.

(3)

Subtracting Eq. (2) from Eq. (3) we obtain e2

W L O - < 4 O = 4 T T — =Q2,

which in terms of x =

(4)

can be written as

WLO/^TO

x2 - 1 =

ft2/<4o-

(5)

Differentiating Eq. (4) we obtain 2wLo<9wLO -

2WTO<9WTO

jLv_2n _o^

= I 47r—— 1 ( — — I == -fl

and then, dividing by WTO^WTO and introducing mode Griineisen parameters according to their definition . \7TO/

"TO \ 7 T o /

,-{x2-l),

. V7TO/

(6)

7LO-1/2

1 / ^ 2 1 x 27, ) 2 = —z =*• 7TO = 7LO + TTO - 1 / 2 aH V /

Equation (6) is similar to Wakamura's expression, with the coefficients A = 7LO — 1/2 and B = 1/2, respectively.

Temperature Dependence of Mode Griineisen

Parameters

437

Taking into account the Lyddane-Sachs-Teller relationship involving the high- and low-frequency dielectric constant and the corresponding longitudinal and transverse frequencies we can write 7LO ~ 1/2 ^ 7TO

1 \ eo(T)

£QQ

- 1/2

7TO

e0(T)

7LO

+

1

Here we can make use of the Curie-Weiss law for the low-frequency dielectric constant eo(T) =

C

T>TC,

T-Tc

and then we can obtain a similar expression for 7TO 7TO

7LO

- 2)

C/£oo

T-Tc

+

T>TC,

(7)

which gives a Curie-Weiss law like temperature dependence of the transverse optical Griineisen parameter. Figure 4.11.1 shows schematically the behavior of the inverse of longitudinal and transverse optical Griineisen parameters. Note that the corresponding expression for 7 T O ( T ' < Tc) should contain AC instead of C (for a first order transition) and (Tc - T) in the denominator.

y-l 'LO

0.05

r

\y' \ 'TO

0.01 -

' 100

115

T(°C)

Fig. 4.11.1. Schematic behavior of the inverse Griineisen parameters for BaTiC>3. It must be noted that as the longitudinal optical parameter is temperature independent, the inverse of the transverse optical parameter behaves as a Curie-Weiss like law.

438

Effective Field Approach to Phase

Transitions

T h e formal similarity between Eqs. (6) and (7) is a result of the Lyddanne-Sachs-Teller relation, which results in

which is due to the t e m p e r a t u r e dependence of UITO, as it should be expected for the soft mode t h a t is a signature of the first order ferroelectric transition. If we approximate a perovskite-like BaTiOa as a binary compound 4 formed by two ionic groups, B a + + and TiO^~, we can evaluate, after d a t a from several authors, 5 ~ 9 an approximated value of 7 x 0 — 20, and a sharp discontinuity of the transverse optical Griineisen parameter between T c (the Curie t e m p e r a t u r e obtained from e~l(T) for T > Tc) and Tg (the critical t e m p e r a t u r e at which the discontinuity at e _ 1 ( T ) occurs), 7 x 0 ( ^ 0 ) — 100, which is similar to the behavior of the inverse of the dielectric constant, e _ 1 ( T ) , at the first order ferroelectric transitions. We may conclude t h a t anomalous Curie-Weiss behavior in the temperat u r e dependence of the Griineisen parameter 7 x 0 (T) must be expected near ferroelectric or structural phase transitions characterized by soft modes.

Acknowledgments We gratefully acknowledge the D G C I C y T for the G r a n t BFM2000-032.

References 1. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976). 2. K. Wakamura, Phys. Lett. A 118(8), 419 (1986). 3. T. Fernandez-Diaz and J.A. Gonzalo, Phys. Lett. A 123(7), 349 (1987). 4. J.A. Gonzalo, J. De Frutos and J. Garcia, Solid State Spectroscopies (World Scientific, Singapore, 2002). 5. H.E. Kay and P. Vousden, Philos. Mag. 40, 1019 (1949). 6. G. Shirane and S. Hoshino, J. Phys. Soc. Jpn. 7, 5 (1952). 7. C.J. Johnson, Appl. Phys. Lett. 7(8), 221 (1965). 8. J.A. Gonzalo and J.M. Rivera, Ferroelectrics 2, 31-35 (1971). 9. A. Scalabrin, A.S. Chaves, D.S. Shim and S. Porto, Phys. Stat. Sol. (b) 79, 731 (1977).

Chapter 4.12

Quantum Tunneling Versus Zero Point Energy in Double-Well Potential Model for Ferroelectric Phase Transitions* C.L. Wang1'4, C. Arago*, J. Garcia* and J.A. Gonzalo* ^Department of Physics, Shandong University, Jinan 250100, People's Republic of China ^Departamento de Fisica de Materiales, Facultad de Ciencias C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain

The Ising model in a transverse field, which involves quantum tunneling in double potential wells, has been used for many years to describe ferroelectric phase transitions. An alternative model previously published takes into account the effect of a non-negligible zero-point energy in the double potential wells at the phase transition. A comparison of the respective Curie temperatures, spontaneous polarizations, and susceptibilities shows that both models are nearly equivalent. Quantitative differences of the two methods are apparent at low temperature where the saturation polarization value with non-negligible zero-point energy is higher than the same quantity with the Ising model in a transverse field.

4.12.1.

Introduction

T h e Ising model in a transverse field was first employed by de Gennes 1 to describe order-disorder-type ferroelectrics. T h e model assumes

"Work previously published in Physica A 308, 337 (2002). Copyright © 2002. Elsevier B.V. 439

440

Effective Field Approach to Phase

Transitions

ferroelectrics to consist of pseudo-spins with interactions in a transverse field. By including multi-spin interactions, this model can be used to describe first order phase transitions in ferroelectrics.2 The Ising model in a transverse field has also been applied successfully to study surface and size effects in ferroelectrics. The influence of the surface on the Curie temperature and on the excitation spectrum has been obtained in semiinfinite systems. 3 ' 4 The dependence of Curie temperature on film thickness as well as on surface variation of exchange constant and transverse field have been studied under mean field theory 5 ' 6 and effective field theory. 7 ' 8 Later, this model was extended to calculate polarization and susceptibility,9 as well as dynamical property 10 and other properties of ferroelectric films. In recent years, the Ising model in a transverse field has been used to investigate the phase transition properties in ferroelectric superlattices, 11 pyroelectric properties of ferroelectric multilayers, 12 and phase transition properties of ferroelectric particles. 13 On the other hand, the effective field approach, which is analogous to Weiss' theory for phase transition in ferromagnetism, is another theoretical method to describe phase transition properties in ferroelectrics and other systems such as alloys and superconductors. 14 The inclusion of quadrupolar and higher order terms into the effective field in second order transitions in uniaxial ferroelectrics is shown to describe well the jump in specific heat at the Curie temperature of the triglycine sulfate family crystals. 15 The thermal hysteresis accompanying discontinuous ferroelectric transition can be estimated if one takes into account the quadrupolar contribution to the effective field. The calculated values are in fair agreement with observed thermal hysteresis in several ferroelectrics belonging to different families.16 A general equation of state using the effective field approach has been obtained for pressure- and temperature-induced transitions in the ferroelectric triglycine selenate. 17 By including the zero-point energy, a simple quantum effective field approach has been developed to explain the Curie temperature in mixed ferroelectric system of rubidium/ammonium dihydrogen (RDP-ADP) 1 8 and tris-sarcosine calcium chloride/bromide (TSCC-TSCB). 19 Recently, this quantum effective approach has successfully explained 20 the Curie temperature deviations from those to be expected from the classical limit in triglycine sulfate/deuterated, triglycine sulfate (TGS-DTGS), and triglycine selenate/deuterated triglycine selenate (TGSe-DTGSe) mixed crystals. 21 ' 22

Double-Well

4.12.2.

Potential Model for Ferroelectric Phase Transitions

441

Effective Field Approach

The effective field approach to ferroelectric phase transitions is certainly the simplest possible way to study phase transitions. In its simplest version (Eeff = E + PP), which takes into account only dipole interactions, it is capable of describing fairly well the main features of continuous ferroelectric phase transitions. A more general expression of effective field is £ eff = E + pP + 7 P 3 + SP5 + A,

(1)

given in terms of an external field E and odd powers of the polarization P, which correspond successively to dipolar, quadrupolar, octupolar contributions, etc. The coefficients /3, 7, 6 may be expected to depend on the geometry of the lattice and on the spatial charge distribution within the unit cell. The equation of state is given by Gonzalo 14

P = J v, t a n h (fgf),

(2)

where N is the number of elementary dipoles per unit volume and /i is the electric dipole moment, fee is the Boltzmann constant, and T is the absolute temperature. From this expression we can easily see that the lowtemperature saturated polarization is Pso = Nfi. For simplicity, keeping terms only up to the dipole contribution in Eefi in Eq. (1), one can rewrite Eq. (2) as

Pso = N/j, is the spontaneous polarization at low temperature. Taking into account that as T approaches Tc from below at E = 0, P = Ps approaches zero, as a result the Curie temperature (4)

The equation of state given in Eq. (3) in implicit form can be made explicit as

E

=ir,-*ri£)-',F-

(5)

442

Effective Field Approach to Phase

Transitions

The dielectric constant can be obtained easily from the above expression as follows: ~-l

1

'dE'

4ir

9P.

kBT

N

47T (7V M ) 2 -

E^0

p P2

(6)

4-7T'

For temperatures below the Curie temperature, i.e., at the ferroelectric phase, the above equation can be rewritten in the vicinity of Curie temperature as £-

1

= - ^ ( T - T

c

)

= -§(T-Tc),

T
(7)

Here, C = 4 x N2n/ks is the Curie-Weiss constant. When the temperature is higher than the Curie temperature, at the paraelectric phase (Ps = 0), we have

'-, = -5^?-S-£ (r - r ->-

TiT

(8)

<-

Quantum mechanically, the zero-point energy should be included, so we have hco0(^ + (n)T^=kBT;,

(9)

where T* refers to the classical Curie temperature in Ref. 19: (n) Tc = [exp(/iw0/fcBrc) - l ] " 1

(10)

is the thermal average of the number of energy quanta excited in each unit dipole at temperature T = Tc, above the zero-point energy level EQ = l/2fkoo, or Einstein energy.23 From Eqs. (9) and (10), we can have knT* = B

c

h

^ll

tanh(/iw0/2A;BTc)'

(11) V

;

From the above we can see that the zero-point energy rescales the real temperature to T* or the quantum temperature scale. 24 ' 25 Therefore, the

Double- Well Potential Model for Ferroelectric Phase Transitions

443

quantum Curie temperature can be obtained from Eq. (3) as

H/2

=

(12)

or more explicitly in the form

— w^

tanh- 1 (/iw 0 /2fc B ^At 2 )

.

(13)

Therefore, the dielectric constant at the paraelectric phase [Eq. (8)] can be rewritten as e -i =

_^9

I

A

(14)

At zero temperature, £-

1

( T = 0) = ^ ( i ^

0

- ^

2

) ,

(15)

which means that if the zero-point energy is large enough, there will be no phase transition to the ferroelectric phase. Equation (14) is equivalent to the famous Barret formula,26 which is used to explain the low-temperature behavior of quantum ferroelectrics, such as SrTi03 and CaTiC>3 and their solid solutions.

4.12.3.

Ising Model in a Transverse Field

The starting point of the Ising model in a transverse field (IMTF) is the double-well potential in hydrogen bonds in ferroelectric such as the potassium dihydrogen phosphate. The spin-1/^ Hamiltonian IMTF in the general form27 is

H = -J2 ViS* -\Y,

J s s

^ i J - 2» E w -

(16)

where the first and third sums are over sites i and the second is over sites i and j . The first term involves the transverse field O,, which is related to the tunneling frequency. Sx and Sz are spin-1/^ operators, Sx describes the tunneling effect, and Sz is related to the polarization. The third term accounts for the coupling to an external electric field E{\ / / i s the dipole

444

Effective Field Approach to Phase

Transitions

moment per Ising spin. With the usual notation, (...) denotes a thermal average. The mean field equation derived from Eq. (16) is

where

Hi = H(ni,0,J2Jij(S-)+2fiE^

(18)

is the molecular field acting on spin at in the ith site. The polarization Pi on site i is proportional to (S1?), namely Pi = 2/x(5f) and in the absence of an applied field, Eqs. (17) and (18) give for this average (sz) w

_

S j Msjp + 2/x£ 2[n? + (j«(s;> + 2AtE)2]

[n? + {J13{SD + 2»E)2} 1 2fcBr

tanhf

1/2

\ /• (19)

In a uniform bulk material all fli are equal and in the nearest-neighborexchange approximation all exchange constants J^ are equal to J. Therefore, the spontaneous polarization is related to (Sz) at zero external field in the form 2[n2 + (n0J(Sz))2}1/2

2+ 2 .tanhp y 2kf » H, { T

(20)

B

where no is the number of nearest neighbors. At temperatures close to the Curie temperature, (5?) approaches zero, and the mean field value of the Curie temperature Tc is found by solving

kBTc =

P^

.

(21)

tanh" 1 (2fi/n 0 J) The dielectric susceptibility can be found from Eqs. (19) and (20) as £ = (l/4n)(dP/dE). For the ferroelectric phase e -i =

I

I

WTTN[I2

/((S 2 ))

where the function f((Sf)) and has the form

f((Sz))

2

2

^L-

(22) V

167T7V> 2 '

;

is introduced to simplify the above equation,

2

[ft + (n 0 J(S )) ]

ft2 no J

(l-4(S2)2)(n0J)2-4ft2 4kBT

.2'

-(sy

(23)

Double-Well

Potential Model for Ferroelectric Phase Transitions

445

and for paraelectric phase the dielectric constant becomes _x

=•-1

20.

1

:

2

167rNfi ta,nh(n/2kBT)

4.12.4.

UQJ

s^l

WTTN/J,2'

(2A) y

'

Discussion and Comparison

First, we compare the expressions of the Curie temperatures obtained by both the methods. If we ignore the tunneling term (i.e., the transverse field) in the IMTF, the expression of Curie temperature [Eq. (21)] reduces to kBTc = n0J/A.

(25)

By comparison with Eq. (4) obtained in the effective field approach, we can have the following relation for the parameters in the two models: 0Nfi2 = n0 J/4.

(26)

Therefore, /3 corresponds to the dipole-dipole interaction J in the IMTF. If the zero-point energy is included in the effective field approach, the expression to determine the Curie temperature is Eq. (11) or (12). By comparison with Eq. (21) from IMTF, we can see that the tunneling frequency is equivalent to the zero-point energy in the effective field theory, i.e., r)oj0 = n.

(27)

Equations (25) and (26) can be further confirmed by the dielectric constant at paraelectric phase [see Eq. (17) from effective field approach and Eq. (24) from IMTF]. In Fig. 4.12.1, we present the Curie temperature from the two models for comparison at different zero-point energies or tunneling frequencies. Figure 4.12.1(a) is for effective field approach, and Fig. 4.12.1(b) for IMTF. From the figure we easily see that the Curie temperatures are exactly the same, even the critical behavior near the critical value of zero-point energy or critical frequency, marked by arrows in the figures. However, the two theoretical approaches do not always have exactly the same corresponding expressions. An example is the polarization at low temperature. From effective field theory, as temperature approaches zero, (n) approaches zero; however, the energy does not tend to zero but to

446

Effective Field Approach to Phase

Transitions

Fig. 4.12.1. Transition temperature as a function of zero-point energy (a) and as a function of tunneling frequency (b).

l/2rjuJo, then the low-temperature spontaneous polarization is given by P = iV

M

tanh(^,

(28)

which is not the classical limit Nfx. It means that at zero temperature there are still fluctuations, which prevent all dipole moments from aligning completely in the same direction. From the above we can get the critical value for the zero-point energy as T]LOC =

2N/3/I2

(29)

From Eq. (20) of IMTF at temperatures close to zero temperature we have a spin average given by

(Sz)o =

\Vl-CM/n0J)2,

(30)

which is not equal to 1/2, the classical limit. It means that not all spins are pointing in the same direction. Also, the critical value obtained from the

Double- Well Potential Model for Ferroelectric Phase Transitions

447

0.5

w V CM

0.3

j

"o.O

0.1

.

i

02

i

1

i

0.3

1

0.4

1

I

1

0.6

x = ha> lANu.2^ or n/n J o o

Fig. 4.12.2. The ground-state polarization from Ising model in transverse field and effective field approach with zero-point energy.

above expression is 2QC = n0J.

(31)

From Eqs. (29) and (31), we can see that the critical values are the same if we recall expressions (25) and (26). The physics meaning behind the above two equations are the same. The zero-temperature polarization at different zero-point energies or tunneling is shown in Fig. 4.12.2 for comparison. The difference vanishes when there is no zero-point energy or tunneling frequency, and also near the critical value of the zero-point energy. The larger difference appears around the middle of the critical value. In order to give a comparison in the whole temperature range, the temperature dependence of polarization and dielectric constant are shown in Fig. 4.12.2. To enhance the difference, we choose fko/4Nfi2P = ^1/TIQJ = 0.35 in both diagrams. The difference in saturation polarization is largest at zero temperature (Fig. 4.12.3(a)) and as the temperature increases, the difference becomes smaller. The behavior of the polarization near the critical temperature is identical for both models. For the dielectric constants, as shown in Fig. 4.12.3(b), the overall difference seems to be a little smaller than that of polarization in Fig. 4.12.3(a), and there is no difference at the paraelectric phase. 4.12.5.

Conclusion

From the comparison, we can see that when the zero-point energy is included in the effective field approach, the corresponding equations become

448

Effective Field Approach to Phase

0.20

="" ^

0.15

&

0.10

EFA+

IMTF

I . I .

0 10

Transitions

x=0.35

0.05 •

0.00 0.0

i

1

0.5

i

1.0

1.5

TTTc

(a) 1.4 •

1.2 yEFA+ 1.0 •

IMTF

i::

x=0.35

0.4 0.2 0.0 0.0

(b)

\^~ 0.5

1.0

i

I

1.5

2.0

TTTc

Fig. 4.12.3. The temperature dependence of polarization (a) and dielectric constant (b) at hui/iNn2!3 = a/no J = 0.35.

the same or very similar to the equations from Ising model in a transverse field under mean field approximation. The effective field approach can be used to describe phase transitions in ferroelectrics with classical behavior as well as with quantum effects. The tunneling frequency in the Ising model in a transverse field is then analogous to the zero-point energy. The results from the two models can be said to be nearly equivalent, except the quantitative differences at low temperatures, when the zero-point energy or the tunneling frequency are away from the critical value.

Acknowledgments Financial supports from DGICyT through Grant no. PB96-0037 is gratefully acknowledged. C.L. Wang is financially supported by China Scholarship Council to visit Universidad Autonoma de Madrid.

Double- Well Potential Model for Ferroelectric Phase Transitions

449

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

P.G. Gennes, Solid State Commun. 1, 132 (1963). C.L. Wang, Z.K. Qin and D.L. Lin, Phys. Rev. B 40, 680 (1989). M.G. Cottam, D.R. Tilley and B. Zeks, J. Phys. C 17, 1793 (1984). I. Tamura, E.F. Sarmento and T. Kaneyoshi, J. Phys. (717, 3207 (1984). C.L. Wang, W.L. Zhong and P.L. Zhang, J. Phys.: Condens. Matter 3, 4743 (1992). H.K. Sy, J. Phys.: Condens. Matter 5, 1213 (1993). X.Z. Wang and Y. Zhao, Physica A 193, 133 (1993). E.F. Sarmento and J.W. Tucker, J. Magn. Magn. Mater. 118, 133 (1993). C.L. Wang, S.R.P. Smith and D.R. Tilley, J. Phys.: Condens. Matter 6, 9633 (1994). C.L. Wang and S.R.R Smith, J. Phys.: Condens. Matter 8, 3075 (1996). C.L. Wang and S.R.R Smith, J. Kor. Phys. Soc. 32, S382 (1998). Y. Xin, C.L. Wang, W.L. Zhong and P.L. Zhang, Solid State Commun. 110, 265 (1999). C.L. Wang, Y. Xin, X.S. Wang and W.L. Zhong, Phys. Rev. B 62, 11423 (2000). J.A. Gonzalo, Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (World Scientific, Singapore, 1991). B. Noheda, G. Lifante and J.A. Gonzalo, Ferroelectrics 15, 109 (1993). J.A. Gonzalo, R. Ramirez, G. Lifante and M. Koralewski, Ferroelectrics 15 9 (1993). J.R. Fernandez del Castillo, J.A. Gonzalo and J. Przeslawski, Physica B 292, 23 (2000). J.A. Gonzalo, Phys. Rev. B 39, 12297 (1989). J.A. Gonzalo, Ferroelectrics 168, 1 (1995). C. Arago, M.I. Marques and J.A. Gonzalo, Phys. Rev. B 62, 8561 (2000). C. Arago and J.A. Gonzalo, J. Phys.: Condens. Matter 12, 3737 (2000). C. Arago and J.A. Gonzalo, Ferroelectrics Lett. 27, 83 (2000). E.K.H. Salje, B. Wruk and S. Marais, Ferroelectrics 124, 185 (1991). J. Dec and W. Kleemann, Solid State Commun. 106, 695 (1998). W. Kleemann, J. Dec and B. Westwanski, Phys. Rev. B 58, 8985 (1989). J.H. Barrett, Phys. Rev. 86, 118 (1952). R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics (North-Holland, Amsterdam, 1974).

Index

Aizu, K., 4, 62 alloys quenching, 38 solubility, 37 Andrews, T., 4, 11

critical exponents, 9, 17-20, 24-27, 33, 64, 68-72, 82, 83, 88, 94, 103-105, 108-112, 114-116, 143, 260, 266, 270, 338 isotherm, 18, 26, 48, 49, 71, 72, 81, 84, 107, 115, 260, 270, 342 magnetic field of superconductors, 33 Curie constant, 21, 50, 125, 137, 138, 185, 186, 200, 201, 218, 229, 231, 253, 260, 270, 349, 360, 368, 386, 419 Curie, P., 4

/3-brass, 39, 43 Bardeen-Cooper-Schrieffer (BCS) hamiltonian, 32 Bethe, H., 43 Bogolubov, N., 4 Bose condensation, 53, 54 Bose-Einstein statistics, 53, 54 Bragg-Williams theory, 38, 42-44 Brillouin zone, 93, 96, 170, 173

Debye frequency, 33, 257 sphere, 92, 93 temperature, 3, 33, 34, 170 deuteration effects, 421 dielectric constant, 6, 45, 47, 49, 103, 119-122, 124, 131, 141-143, 145, 183-185, 198, 222, 226, 241, 242, 250, 251, 256-260, 266, 270, 273-275, 283, 289-292, 300, 301, 307-311, 313, 319-321, 323, 329-331, 349, 352, 353, 368, 379, 380, 421, 424, 425, 428, 437, 438, 442, 443, 445, 447, 448 dielectric losses, 249 dipole moment ferroelectric, 61, 62, 124

charge density waves, 34 chemical potential, 33, 54, 73, 111 coexistence curve, 17, 25, 85, 107, 112, 115 composition dependence, 273 compressibility, 18, 19, 115 conjugated variables, 4, 289, 308 continuity phase transition, 3, 5, 62, 68-70, 191, 241, 256, 337, 368, 441 Cooper pair, 4, 32, 34, 53, 56, 59, 91-99 cooperative phenomena discovery, 4 first successful theory, 4 451

452

Effective Field Approach to Phase Transitions

ferromagnetic, 28 of cooper pair, 92, 93, 95 dipole waves (ferroelectric) contribution to change in P s , 170, 171 contribution to specific heat, 170, 171 in some materials, 176 dipole-dipole hamiltonian, 171, 179, 180 domains ferroelectric, 104 ferromagnetic, 62 effect of impurities impurity concentration, 119 internal bias, 119 effect of impurity modified equation of state, 131 effective field in ferroelectrics, 103 in liquid-vapor systems, 7 in superconductors, 31 in superfiuids, 97 inferromagnets, 21 effective mass, 96, 379 elastic compliance, 64, 65 electron-electron interaction, 31, 32 electron-phonon interaction, 32-34, 63, 96 elementary excitations in ferroelectrics, 169 in superfiuids, 97 energy conversion, 351 energy gap, 31, 33, 59, 94, 96, 98 equation of state of ferromagnets, 23 of liquid-vapor systems, 11 excitons, 34 x - FRH transition in PZT, 299 Fermi surface, 93, 96 Fermi velocity, 96, 97 Fermi-Dirac distribution function, 33 ferroelasticity, 61, 62

ferroelectric materials, 141, 250, 255, 296, 319, 349, 355, 398, 413, 415 ferroelectric response times, 283 ferroelectricity, 61 in Zn^Cdi-^Te, 137 ferromagnetic materials hard and soft, 21 uniaxial dipolar, 30 Fisher-Rooshbrook equality, 116 free energy in Landau theory, 67 of liquid-vapor systems, 12 of order-disorder alloys, 4 frequency dependence, 221 generalized (P, T) equation of state, 289 Gibbs free energy, 75 grain size effects, 385 Griffiths equality, 262 Guggenheim, E.A., 7 Heisenberg, W., 29, 81, 85, 87, 88, 431 Helium ( 4 He), 53, 54, 56, 98, 99 homogeneity assumption, 108, 109, 114 hysteresis loops ferroelastic, 61 ferroelectric, 45 ferromagnetic, 21 ideal gas, 11, 13, 16 Keesom, A., 4 Keesom, W., 4 Landau, L. D., 67 lattice parameter, 3, 6, 51, 245, 306, 382, 393, 397, 401, 403, 405, 407, 411, 413 lattice polarization, 48 law of corresponding states, 7, 8, 15, 16, 103, 104, 108 Lennard-Jones potential, 97

453

Index logarithmic correction, 50, 64, 78, 79, 183-185, 187, 198, 203, 229, 231, 232, 262, 338, 348, 349 Lorentz factor, 29, 48 low field switching, 360

renormalization group, 50, 87, 88, 203, 338 resonating valence bonds, 34

quantum tunneling, 439 quasitricritical behavior, 255, 260, 263

scaling and metastable behavior, 337 scaling assumption, 81, 82, 86 scaling equation of state, 265-267, 269 solid solutions interstitial, 37 substitutional, 37 specific heat, 3 in ferroelectrics, 49, 172 in ferromagnetics, 27 in ferromagnets, 173 in liquid-vapor systems, 19 in superconductors, 47 specific heat and quadrupole interactions, 243 spin density waves, 34 spontaneous magnetization, 5, 8, 21-25, 27, 70, 93 spontaneous polarization, 27, 46-48, 50, 70, 103, 125, 131, 132, 139, 145, 163, 169-171, 173-177, 179, 180, 191, 195, 208, 215-218, 226, 229, 230, 232-234, 238-241, 243-247, 250, 257-261, 266-268, 270, 291, 293, 294, 299, 302, 308, 310, 311, 314, 316, 320, 324, 328-330, 339, 343, 352, 353, 359, 368, 372, 374, 420-424, 439, 441, 444, 446 spontaneous strain, 61, 62, 64, 65, 70, 191 superconductivity, 53, 56, 91, 97 superconductive materials metallic, 34 oxide, 31, 34, 36 superfluidity, 4, 53, 54, 56, 59, 91, 97 susceptibility dielectric, 6, 230, 444 magnetic, 26 symmetry breaking, 5

real gas, 11, 15, 18 relaxation (near T c ), 123, 141, 226

Tamman, G., 4, 37 thermal hysteresis, 237-242

Miiller, K. A. and Bednorz, G. J., 34 magnetic susceptibility, 26 mixed ferro-antiferroelectric systems, 129, 133, 427 mixed ferroelectric, 427 mode Griineisen parameter, 435, 436 Onnes, K., 4 Onsager's solution, 8 order parameters dimensionality, 5 fluctuations, 67 order-disorder transition in alloys, 4, 37, 42, 43 ordinary critical point, 68, 70, 71, 295, 337, 338, 345, 346 over-screening, 91 partition function, 12, 23, 48, 93 photoacoustic effect, 207 piezoelectric resonances, 377, 378, 382-384 piezoelectricity, 378, 379 Planck's distribution law, 54, 430 polarization reversal coercive field, 46, 144 pulse shape, 163 switching current, high field, 147 switching current, low field, 147 pressure dependence, 191, 192, 195, 196 pyroelectricity, 46, 119, 181, 208, 273, 300, 302, 308-311, 319, 320, 440 PZT monoclinic phase, 394, 397

454

Effective Field Approach to Phase Transitions

tilts and displacements in PZT, 307 transition entropy ferroelectric, 50, 420 ferromagnetic, 27 transition heat ferroelectric, 43 ferromagnetic, 27 transition temperature, 5 in ferroelastics, 61, 62, 65 in ferroelectric, 50 of ferromagnetic, 21 of order-disorder alloys, 41 of superconductors, 31 of superfluids, 55 transition temperature (mixed ferro-antiferroelectric systems) classical, 133 quantum mechanical, 134 transitions in complex ferroelectric ceramics, 329

tricritical point, 5, 68-70, 72, 183, 191, 192, 195, 198, 215, 216, 219, 233, 240, 245, 247, 259, 265-267, 270, 273, 274, 277, 280, 337, 347, 421, 423 two sublattice model antiferroelectric (afe) case, 131 ferroelectric (fe) case, 131 mixed (fe-afe) case, 129 uniaxial pressure effects, 367 Valasek, J., 4, 45 Van der Waals, J. D., 11 virial theorem, 172 Weiss, P., 4, 21 Widom's equality, 262

Effective Field Approach to Phase Transitions and Some Applications to Ferroeiectrics (2nd Edition) This book begins by introducing the effective field approach, the simplest approach to phase transitions. It provides an intuitive approximation to the physics of such diverse phenomena as liquid-vapor transitions, ferromagnetism, superconductivity, order-disorder in alloys, ferroelectricity, superfluidity and ferroelasticity. The connection between the effective field approach and Landau's theory is stressed. The main coverage is devoted to specific applications of the effective field concept to ferroelectric systems, both hydrogen bonded ferroeiectrics, like those in the TGS family, and oxide ferroeiectrics, like pure and mixed perovskites.

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