P e t e r A. M a r k o w i c h C h r i s t i a n A. Ringhofer Christian Schmeiser
Semi. conductor Equations
Springer-Verlag W i e n N e w \ f a r k
Semiconductor Equations P. A. Markowich C. A. Ringhofer C. Schmeiser
Springer-Verlag
Wien New York
Peter A. Markowich Fachbereich Mathematik Technische Universitat, Berlin
Christian A. Ringhofer Department of Mathematics Arizona State University, Tempe, Arizona, USA
Christian Schmeiser Institut fur Angewandte und Numerische Mathematik Technische Universitat, Wien, Austria
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustration, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1990 by Springer-Verlag Wien Typeset by Asco Trade Typesetting Ltd., Hong K o n g Printed in Austria by Novographic, Ing. W. Schmid, A-1238 Wien Printed on acid-free paper
With 33 Figures
Library of Congress Cataloging-in-Publication Data. Markowich, Peier A., 1956Semiconductor equations Peter A. Markowich, Christian Ringhofer, Christian Schmeiser. p. cm. Includes bibliographical references. I S B N 0-387-82157-0 (U.S.) 1. Semiconductors Mathematical models. I. Ringhofer, Christian, 1957- . I I . Schmeiser, Christian, 1958- . III. Title. TK7871.85.M339. 1989. 621.381'52—dc20
ISBN 3-211-82157-0 Springer-Verlag Wien-New York ISBN 0-387-82157-0 Springer-Verlag New York-Wien
Preface
I n recent years the mathematical modeling of charge transport i n semiconductors has become a t h r i v i n g area i n applied mathematics. The drift diffusion equations, which constitute the most popular model for the simulat i o n of the electrical behavior o f semiconductor devices, are by now mathematically quite well understood. As a consequence numerical methods have been developed, which allow for reasonably efficient computer simulations in many cases o f practical relevance. Nowadays, research on the drift diffusion model is o f a highly specialized nature. I t concentrates o n the explorat i o n of possibly more efficient discretization methods (e.g. mixed finite elements, streamline diffusion), on the improvement of the performance o f nonlinear iteration and linear equation solvers, and on three dimensional applications. The ongoing m i n i a t u r i z a t i o n of semiconductor devices has prompted a shift of the focus o f the m o d e l i n g research lately, since the drift diffusion model does not account well for charge transport i n ultra integrated devices. Extensions o f the drift diffusion model (so called h y d r o d y n a m i c models) are under investigation for the modeling o f hot electron effects i n submicron MOS-transistors, and supercomputer technology has made it possible to employ kinetic models (semiclassical Boltzmann-Poisson and WignerPoisson equations) for the simulation of certain highly integrated devices. The focus of this b o o k is the presentation o f the hierarchy o f semiconductor models ranging from kinetic transport equations to the drift diffusion equations. Particular emphasis is given to the derivation of the models and the physical and mathematical assumptions used therefore. We do not go into the mathematical technicalities necessary for a detailed analysis of the models b u t rather sacrifice r i g o u r for the sake of conveying the basic p r o p erties and features of the model equations. The mathematically interested reader is encouraged to consult the references for in-depth investigations of specific subjects. We address applied mathematicians, electrical engineers and solid state physicists. The exposition is accessible to graduate students i n each of the three fields. I n particular, we hope that this b o o k w i l l be useful as a text for advanced graduate courses i n this area and we urge students to w o r k the
Preface
vi
problems, w h i c h can be found at the end o f each chapter, for a deeper penetration o f the material. We are grateful to our colleagues U . Ascher, J. Batt, F. Brezzi, P. Degond, P. Deuflhard, D . Ferry, W . Fichtner, L . Gastaldi, P. Kasperkovitz, N . K l u k s dahl, D . M a r i n i , H . Neunzert, F. Nier, R. O ' M a l l e y , F. Poupaud, T. Seidman, S. Selberherr, H . Steinriick, P. Szmolyan, and T . T a y l o r for many hours of stimulating discussions. I n particular we are indebted to A . A r n o l d , N . Mauser, P. Pietra, and R. WeiB for reading large parts of the manuscript. We thank U . Schweigler for skillfully t y p i n g a p o r t i o n of the manuscript. W e are indebted to the Centre des Mathematiques Appliquees o f the Ecole Polytechnique, Palaiseau, France, and to the I n s t i t u t fur Technische Elekt r o n i k o f the Technische Universitat H a m b u r g - H a r b u r g where a part of the manuscript was written. The first and the t h i r d author acknowledge support from the "Osterreichischer Fonds zur F o r d e r u n g der wissenschaftlichen Forschung" under the grant numbers P6771 and P4919, respectively. The second author acknowledges support by the N a t i o n a l Science F o u n d a t i o n , U S A under grant no. P M S 880-1153. Paris, 1989 Peter A . M a r k o w i c h
Christian A . Ringhofer
Fachbereich M a t h e m a t i k Technische Universitat Berlin StraBe des 17. Juni 136 D-1000 Berlin 12
Department o f Mathematics A r i z o n a State University Tempe, A Z 85287, U S A
Christian Schmeiser Institut fiir Angewandte u n d Numerische M a t h e m a t i k Technische Universitat W i e n Wiedner HauptstraBe 8 - 1 0 A-1040 W i e n , Austria
Contents
Introduction 1 K i n e t i c T r a n s p o r t M o d e l s for Semiconductors 1.1 1.2
1.3
1.4
Introduction 3 The (Semi-)Classical L i o u v i l l e E q u a t i o n 4 Particle Trajectories 5 A Potential Barrier 5 The Transport E q u a t i o n 8 Particle Ensembles 8 The I n i t i a l Value Problem 9 The Classical H a m i l t o n i a n 11 The Semi-Classical L i o u v i l l e E q u a t i o n 12 Magnetic Fields 16 The B o l t z m a n n E q u a t i o n 16 The Vlasov E q u a t i o n 17 The Poisson E q u a t i o n 21 The W h o l e Space Vlasov P r o b l e m 23 Bounded Position D o m a i n s 24 The Semi-Classical Vlasov E q u a t i o n 25 Magetic Fields—The M a x w e l l Equations 26 C o l l i s i o n s — T h e B o l t z m a n n E q u a t i o n 28 The Semi-Classical B o l t z m a n n Equation 30 Conservation and Relaxation 32 L o w Density A p p r o x i m a t i o n 33 The Relaxation T i m e A p p r o x i m a t i o n 34 Polar O p t i c a l Scattering 35 Particle-Particle Interaction 36 The Q u a n t u m L i o u v i l l e E q u a t i o n 36 The Schrodinger E q u a t i o n 37 T u n n e l i n g 38 Particle Ensembles and Density Matrices 40 Wigner Functions 41
3
viii
Contents
The Q u a n t u m Transport E q u a t i o n 42 Pure and M i x e d States 44 The Classical L i m i t 48 N o n n e g a t i v i t y o f Wigner Functions 50 A n Energy-Band Version o f the Q u a n t u m L i o u v i l l e E q u a t i o n 52 1.5 The Q u a n t u m B o l t z m a n n E q u a t i o n 57 Subensemble Density Matrices 58 The Q u a n t u m Vlasov E q u a t i o n 60 The Poisson E q u a t i o n 63 The Q u a n t u m Vlasov E q u a t i o n o n a Bounded Position Domain 65 The Energy-Band Version o f the Q u a n t u m Vlasov E q u a t i o n 65 Collisions 67 1.6 Applications and Extensions 68 M u l t i - V a l l e y M o d e l s 69 Bipolar M o d e l 71 T u n n e l i n g Devices 75 Problems 77 References 80
2 F r o m Kinetic to Fluid Dynamical Models
83
2.1 2.2 2.3
I n t r o d u c t i o n 83 Small M e a n Free P a t h — T h e H i l b e r t Expansion 85 M o m e n t M e t h o d s — T h e H y d r o d y n a m i c M o d e l 90 D e r i v a t i o n of the D r i f t Diffusion M o d e l 91 The H y d r o d y n a m i c M o d e l 92 2.4 Heavy D o p i n g Effects—Fermi-Dirac D i s t r i b u t i o n s 94 2.5 H i g h Field E f f e c t s — M o b i l i t y M o d e l s 95 2.6 Recombination-Generation M o d e l s 98 Problems 101 References 102 3 The Drift Diffusion Equations 3.1 3.2 3.3 3.4
3.5
104
I n t r o d u c t i o n 104 The Stationary D r i f t Diffusion Equations 108 Existence and Uniqueness for the Stationary D r i f t Diffusion Equations 110 F o r w a r d Biased P-N Junctions 116 The E q u i l i b r i u m Case 118 The N o n - E q u i l i b r i u m Case 125 A s y m p t o t i c V a l i d i t y i n the One-Dimensional Case 129 Velocity Saturation Effects—Field Dependent M o b i l i t i e s 130 Reverse Biased P-N Junctions 133 M o d e r a t e l y Reverse Biased P-N Junctions 134
Contents
ix
P-N Junctions U n d e r Extreme Reverse Bias Conditions 134 The One-Dimensional Problem 135 The T w o - D i m e n s i o n a l Case 141 3.6 Stability and C o n d i t i o n i n g for the Stationary Problem 143 3.7 The Transient Problem 148 3.8 The Linearization of the Transient Problem 149 3.9 Existence for the N o n l i n e a r Problem 153 A s y m p t o t i c Expansions for the Transient D r i f t Diffusion Equations 156 3.10 A s y m p t o t i c Expansions on the Diffusion T i m e Scale 157 3.11 Fast T i m e Scale Expansions 162 The Case of a Bounded I n i t i a l Potential 163 Fast T i m e Scale Solutions for General I n i t i a l D a t a 165 Problems 171 References 172 4 Devices 4.1 4.2
4.3 4.4
4.5
4.6
4.7
175
I n t r o d u c t i o n 175 Static Voltage-Current Characteristics 176 P - N Diode 180 The Depletion Region i n T h e r m a l E q u i l i b r i u m 181 Strongly Asymmetric Junctions 185 The Voltage-Current Characteristic Close to T h e r m a l Equilibrium 188 H i g h I n j e c t i o n — A M o d e l Problem 191 Large Reverse Bias 192 Avalanche B r e a k d o w n 195 Punch T h r o u g h 197 B i p o l a r Transistor 198 Current G a i n Close to T h e r m a l E q u i l i b r i u m 199 P I N - D i o d e 202 Thermal Equilibrium 203 Behaviour Close to T h e r m a l E q u i l i b r i u m 206 T h y r i s t o r 208 Characteristic Close to T h e r m a l E q u i l i b r i u m 210 Forward Conduction 212 Break Over Voltage 215 M I S D i o d e 218 Accumulation 221 D e p l e t i o n — W e a k Inversion 222 Strong Inversion 223 MOSFET 225 D e r i v a t i o n of a Simplified M o d e l 228 A Quasi One-Dimensional M o d e l 230 C o m p u t a t i o n of the One-Dimensional Electron Density
231
Contents
X
C o m p u t a t i o n of the Current 233 Gunn Diode 235 B u l k Negative Differential C o n d u c t i v i t y Traveling Waves 238 The G u n n Effect 240 Problems 242 References 243
4.8
Appendix 245 Physical Constants 245 Properties of Si at R o o m Temperature Subject I n d e x
246
245
237
Introduction
Semiconductor device m o d e l i n g spans a wide range of areas i n solid state physics, applied and c o m p u t a t i o n a l mathematics. The involved topics range from the most basic principles o f kinetic transport i n solids over statistical mechanics to complicated bifurcation problems i n the mathematical description o f certain devices and to numerical methods for p a r t i a l differential equations. This b o o k tries to give an overview o f the involved models and their mathematical treatment. I t addresses, on one hand, the engineer and the physicist interested in the mathematical background of semiconductor device modeling. O n the other hand it can be used by the applied mathematician to familiarize himself (herself) w i t h a field which has immediate and technologically relevant applications and gives rise to a whole variety of interesting mathematical problems. The scope of semiconductor device modeling is clearly interdisciplinary. Q u a n t i t a t i v e answers are needed to describe devices and these answers can be obtained from a variety of different physical models. We start from the most basic physical principles for kinetic transport o f charged particles. T h e n we discuss a hierarchy o f simplified model equations c u l m i n a t i n g in the drift diffusion equations w h i c h are the most widely used model today. I n order to make this b o o k accessible to as wide a range o f readers as possible the emphasis has been placed o n concepts, and mathematical details have been replaced by references to the corresponding literature. I n the first Chapter the classical and q u a n t u m mechanical transport models in ensemble phase space and single particle phase space are discussed. F u r t h e r m o r e it is shown how the q u a n t u m mechanical models can be incorporated i n t o the classical transport picture via the so called semiclassical models. The solution o f transport equations i n phase space is a very complex task. Therefore, simplified equations for integral quantities, such as particle and energy densities, are frequently used. These simplified equations are p a r t i a l differential equations i n position space only. The derivation of these equations, i.e. the h y d r o d y n a m i c models and finally the drift diffusion equations, is the subject o f Chapter 2. Chapter 3 is devoted to a
2
Introduction
mathematical discussion o f the drift diffusion equations w h i c h are the underlying model for the b u l k of the simulations performed today. Chapter 4 is concerned w i t h the analysis o f specific device structures. Here the analytical tools developed i n Chapter 3, mainly asymptotic analysis, are used to approximately calculate current flows and to study the qualitative behaviour of voltage-current characteristics. Each Chapter is self contained, m a k i n g it possible to use parts o f the b o o k as a text i n seminars or courses. A t the ends of the Chapters we have collected selections o f problems (referenced i n the text) which should make it easier for the student i n such a course to reflect o n the presented material.
Kinetic Transport Models for Semiconductors
1.1 Introduction In this Chapter we shall derive and discuss transport equations, which model the flow o f charge carriers i n semiconductors. The c o m m o n feature o f these equations is that they describe the evolution o f the phase space ( p o s i t i o n - m o m e n t u m space) density function of the ensemble of negatively charged conduction electrons or, resp., positively charged holes, which are responsible for the current flow i n semiconductor crystals. The kinetic equations are the starting point for the derivation of the drift diffusion semiconductor model (often referred to as the Basic Semiconductor Equations or the van Roosbroeck Equations), which, together w i t h its extensions (hydrodynamic models), constitutes the core o f state-of-the-art semiconductor device simulation programs. This already necessitates a close scrutiny o f kinetic transport models. A n o t h e r reason is provided by the fact that the mathematical assumptions, which allow the derivation of the drift diffusion m o d e l from the kinetic models and which guarantee its validity, are—particularly for highly integrated devices—not satisfied. Thus, kinetic models must be used for the simulation o f such devices. U n t i l recently this approach was generally not taken since the numerical solution o f the kinetic equations requires a lot o f c o m p u t i n g power i n real life applications. However, w i t h the reduction o f cost o f supercomputer technology, which was at least partly p r o m p t e d by efficient V L S I - s i m u l a t i o n , the numerical treatment of kinetic models for semiconductors was facilitated for at least some realistic applications. We expect the trend towards the kinetic equations to continue i n the near future and, thus, we encourage simulation oriented researchers to become acquainted w i t h these models. Principally, the kinetic equations split i n t o q u a n t u m mechanical, semiclassical and classical models. The q u a n t u m mechanical models are based on the many-body Schrodinger equation or, equivalently, on the q u a n t u m Liouville equation obtained from the Schrodinger equation by performing the Wigner transformation. The starting p o i n t for the classical models is the description o f the m o t i o n o f particle ensembles based o n Newton's second law. A probabilistic r e f o r m u l a t i o n o f these canonical equations o f
4
1 Kinetic Transport Models for Semiconductors
m o t i o n immediately gives the classical L i o u v i l l e equation, which describes the evolution of the phase space d i s t r i b u t i o n function of the particle ensemble. The q u a n t u m L i o u v i l l e equation is consistent w i t h its classical counterpart i n the sense that i n the (formal) classical l i m i t h -> 0, where h denotes the Planck constant scaled by 2n, the q u a n t u m L i o u v i l l e equation reduces to the classical L i o u v i l l e equation. The semi-classical L i o u v i l l e equation can be regarded as a modification o f the classical L i o u v i l l e equation, which incorporates the q u a n t u m effects o f the semiconductor crystal lattice via the band-diagram of the material. The L i o u v i l l e equations contain m a n y - b o d y effects i n the sense that they are posed on the usually high-dimensional ensemble phase-space, whose coordinates are the position and m o m e n t u m coordinates of all particles o f the ensemble. The interaction force field, which appears i n these equations, generally depends on all these coordinates. Thus, it is desirable (and, i n fact, necessary i n order to facilitate a numerical solution) to reduce the dimension of the Liouville equations. The procedure for the reduction of the dimension o f the L i o u v i l l e equations is based o n postulating properties o f the interaction force field. T w o cases are usually considered: W h e n only l o n g range forces (like the C o u l o m b force) are considered, then the Vlasov or collisionless B o l t z m a n n equation is obtained i n either the classical or semiclassical f o r m u l a t i o n and the q u a n t u m Vlasov equation i n the q u a n t u m mechanical case. These equations have the form o f single particle L i o u v i l l e equations supplemented by an effective field equation, w h i c h depends on the position space number density of the particles. The effective field equation represents the (averaged) effect o f the many-body physics. If, i n a d d i t i o n to the long range forces, short range forces are included, then the (semi-) classical and, resp., q u a n t u m B o l t z m a n n equations are obtained. These equations contain collision integrals, which model the short range interactions (scatterings) of the particles w i t h each other and/or w i t h their environment. The specific form of the kernel of the collision operator, which is nonlocal i n the m o m e n t u m direction, is determined by the considered short range interaction mechanisms. I n Section 1.2 we discuss the classical and semi-classical L i o u v i l l e equations and i n Section 1.4 their q u a n t u m mechanical counterparts. Section 1.3 is concerned w i t h the classical Vlasov and B o l t z m a n n equations and Section 1.5 w i t h the corresponding q u a n t u m equations. I n Section 1.6 we discuss the applications o f kinetic transport models to semiconductor physics and modeling.
1.2 The (Semi-) Classical Liouville Equation I n this Section we shall derive the basic equation, w h i c h governs the m o t i o n of an ensemble o f charged particles under the action of a d r i v i n g force assuming that the particles obey the laws o f classical mechanics. Since,
5
1.2 The (Semi-)Classical Liouville Equation
usually, it is not possible to obtain enough data to determine the initial state of the ensemble exactly, we shall take a probabilistic p o i n t of view and reformulate the equations for the trajectory of the ensemble as a deterministic equation for the p r o b a b i l i t y density of the ensemble in the positionm o m e n t u m space. This microscopic equation is referred to as classical L i o u v i l l e equation. We start out by considering
Particle
Trajectories
We shall at first analyze the m o t i o n of a single electron i n a vacuum under the action of an electric field E. We associate the position vector x e U and the velocity vector v e U —both assumed to be functions of the time t—with the electron. Then, i n the absence of a magnetic field, the force !F, w h i c h acts on the electron, is given by 3
3
& = -qE
(1.2.1)
(see, e.g., [1.31]). Here q(>0) of the electron is — q. Newton's second law reads:
denotes the elementary
charge, i.e. the charge
J * = mv,
(1.2.2)
where m stands for the mass of the electron and ' " ' denotes differentiation w i t h respect to the time t (v is the acceleration vector). By inserting (1.2.1) i n t o (1.2.2) we obtain the system of ordinary differential equations x = v
(1.2.3)
v=-~E
(1.2.4)
m
for the trajectories of the electron i n the position-velocity space. Together w i t h a given initial state x(t = 0) = x , 0
the system t o r y w(t; x N o t e that, and on the
A Potential
0
v(t = 0) = v
(1.2.5)
0
(1.2.3), (1.2.4) constitutes an initial value p r o b l e m for the trajec, v ) = (x{t), v{t)), w h i c h passes t h r o u g h ( x , v ) at time t = 0. generally, the electric field E depends on the position vector x time t, i.e. E = E(x, t). 0
0
0
Barrier
As an example and for future reference we consider the one-dimensional m o t i o n of an electron across a t h i n and high potential barrier. The static potential V is depicted i n Fig. 1.2.1. The corresponding electric field
1 Kinetic Transport Models for Semiconductors
Fig. 1.2.2 Phase portrait s > 0
1.2 The (Semi-)Classical Liouville Equation
7
E = — V is given by x
0.
1 e
2 '
2 '
— e < x < 0.
0 < x < s
r.
e is a small positive parameter. The equations (1.2.3), (1.2.4) for the trajectories are easily integrated and we o b t a i n the phase p o r t r a i t shown i n Fig. 1.2.2. The t w o thickly d r a w n curves are ' l i m i t i n g ' characteristics. A particle w i t h velocity | v \ < y/l/e cannot cross the barrier, it is reflected. O n l y particles w i t h \ v\ > cross over. As e -> 0 + the barrier becomes thinner and higher, precisely speaking V ——> —(m/q)S(x). The l i m i t i n g phase p o r t r a i t (e = 0) is depicted i n F i g . 1.2.3. A l l particles, no matter h o w big their velocity, are reflected. I n Section 1.4 we shall consider the corresponding q u a n t u m mechanical model, which behaves totally different.
V
X
Fig. 1.2.3 Phase portrait e = 0
1 Kinetic Transport Models for Semiconductors
8
The Transport
Equation
Assume n o w that instead of the precise i n i t i a l position x and i n i t i a l velocity v of the electron we are given the j o i n t p r o b a b i l i t y density f = / ( x , v) of the initial position and velocity of the electron, fj has the properties 0
0
{
fi(x,
fj(x,
v) > 0,
7
v) dx dv = 1,
(1.2.6)
where the integration is performed over the whole (x, f)-space. Then P(B) :=
f,(x,
v) dx dv
is the p r o b a b i l i t y to find the electron i n the subset B of the (x, i;)-space at time t = 0. I t is our goal now to derive a c o n t i n u u m equation for the p r o b a b i l i t y density / = f(x, v, t), w h i c h evolves from f = f(x,v,t = 0). It is reasonable to postulate that / does not change along the trajectories w, i.e. we require l
f(w(t;
x, v), t) = f,(x,
v)
(1.2.7)
for all x, v and for all f ^ 0. Differentiating (1.2.7) w i t h respect to t gives a,/ + x - g r a d j - + > - g r a d j = 0
(1.2.8)
l
and we obtain from (1.2.3), (1.2.4): d.f+
i;-grad
J C
/ - - £ - g r a d , / = 0, m 1
t > 0.
(1.2.9)
This equation is the famous Liouville (or transport) equation governing the evolution of the position-velocity p r o b a b i l i t y density / = f(x, v, t) of the electron i n the electric field E under the assumption that the electron moves according to the laws of classical mechanics. The m o t i o n is assumed to take place w i t h o u t interference from the environment (e.g. the semiconductor crystal lattice) or, equivalently, the electron moves i n a vacuum.
Particle
Ensembles
I n solid state physics one is usually not only concerned w i t h the m o t i o n of a single particle but of an ensemble of interacting particles. F o r the single electron L i o u v i l l e equation (1.2.9) the position vector x and the velocity vector v are i n U (or i n U , d = 1 or 2, i f the m o t i o n can be restricted to a one- or resp., two-dimensional linear manifold). I n the case of an ensemble consisting of M particles, however, the position vector x and the velocity vector v of the ensemble are 3M-dimensional vectors, i.e. x = ( x x ) , v = ( v v ) where x , v e U represent the position and, resp., velocity vector of the i - t h particle of the ensemble. Also, the force field = 3
d
l
3
l
t
M
;
t
5
M
1.2 The (Semi-)Classical Liouville Equation
9
!F ) is a 3 M - d i m e n s i o n a l vector, w h i c h i n general depends o n all 6 M position and velocity coordinates and o n the time t. = J^(x, v, t) denotes the force acting o n the i-th particle. I f all the particles of the ensemble have equal mass m (which we shall assume henceforth) then the trajectories of the ensemble satisfy the system of ordinary differential equations i n the 6 M - d i m e n s i o n a l ensemble positionvelocity space: M
x, = v
t
(1.2.10) i =
1 v = t
m
1 , M . (1.2.11)
-&,
As above, we denote the ensemble trajectory, w h i c h passes t h r o u g h the initial state (x , v ), by w(t; x , v ) = (x(t), v(t)). The classical (ensemble) L i o u v i l l e equation 0
df t
0
0
+ v • gmdj
0
+ -&• m 3M
g r a d j = 0,
(1.2.12)
3M
n o w posed for x e U , v e U is derived from (1.2.10), (1.2.11) as i n the single electron case. Here / = f(x, v, f) denotes the j o i n t position-velocity p r o b a b i l i t y density of the M - p a r t i c l e ensemble at time t, i.e. r f(x, v, t) dx dv P (B, t) M
denotes the p r o b a b i l i t y to find the particle ensemble i n the subset B of the 6 M - d i m e n s i o n a l ensemble position-velocity space at the time t (the preservation of the nonnegativity of / and the conservation of the integral of / over U w i l l be shown below under appropriate assumptions o n the force field iF). The L i o u v i l l e equation (1.2.12) is linear and hyperbolic, its characteristics are the ensemble trajectories satisfying (1.2.10), (1.2.11). I t has to be supplemented by the initial c o n d i t i o n 6M
f(x,v,t
The
Initial
= 0) = f (x,v).
(1.2.13)
I
Value
Problem
We consider the L i o u v i l l e equation (1.2.12) subject to the i n i t i a l c o n d i t i o n (1.2.13) for x e U , v e U . I n order to distinguish between the position and velocity spaces we shall i n the sequel often write x e [ R , v e U . F r o m (1.2.7) we conlude f(x, v,t)>0,xe U , ve U for all t ^ 0 for which a solution exists, i f fj(x, v) ^ 0, x e U^ , v e M™. Thus, the n o n negativity of / is preserved by the evolution process generated by the L i o u v i l l e equation. F o r the following we shall assume that the force field is divergence-free w i t h respect to the velocity: 3M
3M
3M
3 M X
M
3M
3M
1 Kinetic Transport Models for Semiconductors
10
div„^ = 0 ,
x e
3 A/
veM™,
t^O.
(1.2.14)
3 M
We integrate (1.2.12) over l , assume that the solution decays to zero sufficiently fast as | x | -»• oo, \ v\ -* oo and calculate using (1.2.14): / d i v „ ^ dv = 0.
• g r a d , , / dv = — We obtain d_
f(x,
dt
v, t) dv dx = 0
and conclude that the integral of / over the whole position-velocity space is conserved i n time: p
f(x,
v, t) dv dx =
fj(x,
v) dv dx = 1,
r 0. (1.2.15)
The preservation of the nonnegativity of / and the conservation of the whole-space integral (1.2.15), b o t h directly i m p l i e d by the derivation of the Liouville equation and by the assumption (1.2.14) on J*, allow the full probabilistic interpretation of the solution of the initial value problem for the L i o u v i l l e equation (cf. Liouville's Theorem [1.13]). F o r the following the moments of the p r o b a b i l i t y density / w i t h respect to the velocity w i l l be of importance. A t this point we introduce the zeroth order moment f(x,
,(x, t)
(1.2.16)
v, t) dv
and the (negative) first order moment Jcimu(x>t)'=
-Q
(1.2.17)
vf(x,v,t)dv.
The function n = n , ( x , t) is the position p r o b a b i l i t y density of the particle ensemble, i.e. c l a s s
PMJA,
t)
c
ass
"class(*, t) dx
is the p r o b a b i l i t y to find the ensemble in the subset A of the position space R at the time t. n is called classical microscopic particle position density. -'class represents a flux density, it is called classical microscopic particle current density. The conservation property (1.2.15) can now be restated as 3M
c I a s s
class(x, f) dx =
r
w i t
h "class./M = JR3M/J(X, v, t) dv.
"class,/(*) dx,
t^O
(1.2.18)
11
1.2 The (Semi-)Classical Liouville Equation M
By formally integrating the L i o u v i l l e equation (1.2.12) over Uf the conservation law 0
3,"class ~ diV ^class = 0,
we o b t a i n
(1.2.19)
x
which is referred to as macroscopic particle c o n t i n u i t y equation. The solvability of the initial value p r o b l e m (1.2.12), (1.2.13) for the L i o u v i l l e equation is closely related to the global (in time) existence of the characteristics w(t; x, v) for all (x, v) e U^ x Uf, , w h i c h in t u r n is related to the regularity and g r o w t h properties of the force field . I f the maps M
M
w(r; •, • ) : K
x
M
M
x
l R r -»
M
^f ,
t ^ 0
(1.2.20)
are sufficiently smooth and one-to-one, and if f is sufficiently differentiable, then the unique solution / of (1.2.12), (1.2.13) is given by {
f(x,
v, t) = Mw'^t;
x, v)),
x e l f ,
3M
veU ,
t^O.
(1.2.21)
The invertibility requirement of the maps w(f; •, •) excludes the intersection of trajectories ('collisions' of ensembles). M a t h e m a t i c a l l y it prohibits certain strong singularities o f the force field 3F at finite x, v and t. A n L - s e m i g r o u p analysis of the classical L i o u v i l l e equation (1.2.12) for r-independent, static gradient force fields can be found in [1.46]. 2
The Classical
Hamiltonian
We consider the m o t i o n o f an electron ensemble under the action of a velocity-independent force field J = J^(x, t) and denote (as i n the single electron case) the negative force per particle charge by E: 27
E =
.
(1.2.22)
q Assuming that E = E(x, t) is a gradient field E = -grad^F,
(1.2.23)
we can write the t o t a l energy o f the ensemble as sum of the kinetic and potential energies 2
s
tol
m\v\ = ^~-qV(x,t).
W h e n the t o t a l energy s
(1.2.24) tol
is expressed i n terms o f the m o m e n t u m vector
p = mv,
(1.2.25)
then we o b t a i n the classical H a m i l t o n i a n o f the ensemble H{x,p,t)
= ^-qV(x,t). 2m
(1.2.26)
12
1 Kinetic Transport Models for Semiconductors
The equations (1.2.10), (1.2.11) for the ensemble trajectories are n o w equivalent to the so-called canonical equations of m o t i o n (or H a m i l t o n i a n equations): x = grad H
(1.2.27)
p = -grad H
(1.2.28)
p
x
(see, E = For The
e.g., [1.35]). N o t e that H = 0 holds along the trajectories for static fields E{x). I n the static case the energy (1.2.26) is conserved by the m o t i o n . time dependent fields E = E(x, t) we have H = d H. L i o u v i l l e equation expressed in the (x, /^-coordinates takes the form t
df
+ ^-grad f-qE-grad f x
= 0,
3 M
t>0. (1.2.29) The 6 M - d i m e n s i o n a l (x, p)-space is usually referred to as (ensemble) phase space. t
p
xeU™,
peU , p
m
The Semi-Classical
Liouville
Equation
So far the particles were assumed to move w i t h o u t interference from their environment, or equivalently, in a vacuum. I n a semiconductor, however, the ions i n the crystal lattice induce a lattice-periodic potential, w h i c h has a significant effect on the m o t i o n of the charged particles. Since the period of the lattice potential is very small (typically of the order of magnitude 10~ cm), it is necessary to use q u a n t u m mechanics to model its impact o n the transport of charged carriers. F o r precisely this reason the L i o u v i l l e equation, w h i c h has incorporated the q u a n t u m effects of the crystal lattice, is referred to as semi-classical transport equation. We start out w i t h the mathematical set-up of the crystal lattice structure. We denote the (infinite periodic) crystal lattice by 8
E = { i f l j i , + ja
(2)
+ la \i,j,leZ}.
(1.2.30)
(3)
a, a , a are the p r i m i t i v e lattice vectors. The corresponding reciprocal lattice is given by (l)
(2)
(3)
w
L:=
(2)
{ia
+ ja
(3)
+ la \i,j,
I e 1},
(1.2.31) (1)
i2
where the reciprocal p r i m i t i v e lattice vectors a , a \ iJ)
a -a {i)
i3)
a
3
e U
satisfy
= 2n8f.
(1.2.32) 3
A connected subset Z c [ J is called a p r i m i t i v e cell of the lattice L , i f i t satisfies the following t w o conditions: (a) The volume of Z equals \a -(a x a ) | , w h i c h is the volume of the parallelepiped spanned by the p r i m i t i v e lattice vectors of L . (b) IR = [J T Z, where T Z denotes the translate of Z by the lattice vector x. This means that the whole space U is covered by the u n i o n of translates of Z by the lattice vectors. (1)
{2)
(3)
3
xeL
X
X
3
Primitive cells of the reciprocal lattice L are defined accordingly.
1.2 The (Semi-)Classical Liouville Equation
13
Lattice L
The (first) Brillouin zone B is defined as that p r i m i t i v e cell of the reciprocal lattice L , which consists o f those points, which are closer to the origin than to any other p o i n t o f L (see, e.g., [ 1 . 4 ] , [1.31] for details). I t is easy to show that the B r i l l o u i n zone B is p o i n t symmetric to the origin, i.e. k e B iff — k e B holds. Fig. 1.2.4 shows a two-dimensional lattice, its reciprocal lattice and the B r i l l o u i n zone. Consider n o w an electron whose m o t i o n is governed by the potential V generated by the ions located at the points o f the crystal lattice L . Clearly, V is L-periodic, i.e. L
L
VAx + X) = V {x), L
xeUl,
X e L.
(1.2.33)
The steady state energies e of the electron are the spectral values of the Schrodinger equation H \jj L
= #
(1.2.34)
w i t h the q u a n t u m mechanical H a m i l t o n i a n H
L
= - ^ A - q V
L
(1.2.35)
(see Section 1.4 for details). Bloch's Theorem (see, e.g., [ 1 . 4 ] , [1.31]) asserts that the bounded eigenstates I/J can be chosen to have the form of a plane wave e ' times a function w i t h the periodicity of the lattice L : k x
14
1 Kinetic Transport Models for Semiconductors ikx
i/,(x) = e u (x),
x e U
k
(1.2.36)
x
3
u (x + X) = u (x), k
xeU ,
k
XeL,
x
(1.2.37) 3
where, principally, k is an arbitrary (wave) vector i n R . Inserting (1.2.36) i n t o (1.2.34) gives the eigenvalue equation - ^ ( A u
4
+ 2ik-Vu ) k
+ (^\k\
2
- qV (x)^ju L
k
= eu
(1.2.38)
k
subject to the periodicity c o n d i t i o n (1.2.37). F o r given k e U the p r o b l e m (1.2.38), (1.2.37) constitutes a second order self-adjoint elliptic eigenvalue problem posed o n a p r i m i t i v e cell of the crystal lattice L . Thus, we may expect an infinite sequence of eigenpairs e = £ (/c), u (x) = u ,(x), / £ Note that (1.2.36), (1.2.37) can be reformulated as k
;
ikX
3
+ X) = e il/(x),
xeR ,
k
k
XeL.
x
k x
Since e' ' = 1 for all k e L , X e L we conclude that the set of wave functions \jj and the energies e are identical for any t w o wave vectors which differ by a reciprocal lattice vector. Thus, we can assign the indices / e N i n such a way that the energy levels e,(k) and the corresponding wave functions •Ak,;( ) ' ' k,i( ) periodic on the reciprocal lattice L : x
=
e k Xu
x
a
r
e
s,(k + K) = e,(fc), ipk+K.i = h.h
ke B,
KeL,
keB,
KeL,
Ie N leN.
(1.2.39) (1.2.40)
Obviously, no i n f o r m a t i o n is lost when the wave-vector k is constrained to the B r i l l o u i n zone B. F o r a t h o r o u g h mathematical analysis o f the spectral properties of the Schrodinger equation w i t h a periodic potential we refer to [1.47]. The function e, = e,(/c), continuous o n the B r i l l o u i n zone B, is called l-th energy band o f the crystal. The corresponding mean electron velocity is given by P |
(*) = ±gra<Wfc)
(1.2.41)
(see [1.4]). I n practice the lattice potential V is not k n o w n exactly, and therefore a p p r o x i m a t i o n methods have to be used to compute the band diagram {e,(rc)|/c e 8 } , N for a given material. F o r the technologically most i m p o r tant semiconductors the band diagrams can be found i n the literature (see [1.31], [ 1 . 4 ] ) and we shall henceforth assume that the energy bands and, thus, the velocities (1.2.41) are k n o w n functions o f k. Consider n o w the m o t i o n o f an electron ensemble, where all M electrons are l o c a t e d ' i n the same energy band e,, i.e. the wave-function = t/' (x, f) of the /-th electron is represented by a 'linear c o m b i n a t i o n ' of eigenstates i/j ,(x) over ke B: L
E
(i)
k
c (K w
B
t)i// (x)dk, ktl
i=
1 , M .
15
1.2 The (Semi-)Classical Liouville Equation
3
I n the sequel we denote the wave-vector o f the i-th electron by k e U , k = (k , k ) e U , and its position vector by x, e U , x = ( x , , . . . , x ) e U . Also, from now o n we shall d r o p the band index / assuming that we are dealing w i t h a specific band. I t is well k n o w n that in the presence of a d r i v i n g force = <¥{x, k, t), = ..., 3P ), periodic in /c, for i = 1, . . . , M , the semi-classical equations o f m o t i o n i n the (x, /c)-space read (see [1.31], [1.4]): t
3M
x
3
M
M
3M
M
*i = v{k )-\
. i = 1,.... M .
t
> J
fife; = J^
(1.2.42) (1.2.43)
N o t e that band transitions are excluded since the band index is fixed in the equations. I f the d r i v i n g force is independent of the wave vector k, we again write £= E(x, t) for the negative force per particle charge (see (1.2.22)). If, i n addition, the field £ is a gradient field, then, using (1.2.23), we set up the semi-classical electron ensemble H a m i l t o n i a n H(x, p, t) = f
6 (|J
- q V(x, t),
where we set p = ( p . . . , p ). l 5
M
(1.2.44)
Here
p = hk t
(1.2.45)
l
denotes the crystal m o m e n t u m vector of the i - t h electron (see [1.4]). The semi-classical equations o f m o t i o n (1.2.42), (1.2.43) are then equivalent to the H a m i l t o n i a n equations (1.2.27), (1.2.28) corresponding to the semiclassical H a m i l t o n i a n function (1.2.44). The semi-classical electron-ensemble L i o u v i l l e equation reads: M 1 3,f + Z v(k ) • g r a d , , / + - & • g r a d / = 0, i=t n t
k
t > 0,
(1.2.46)
3M
where x e I R , k e B for i = 1, . . . , M . W e impose periodic boundary conditions for k , i = 1 , . . . , M : t
t
f{x,
kj, . . . , k , ..., k , t
M
t) = f(x, / c j , . . . ,
/c;,..., k , M
t),
/c, 6 dB. (1.2.47)
The definitions of the electron ensemble position density and of the electron ensemble current density have to be modified: X
"class, fl( ' 0
X
:
f(x,k,t)dk
—
(1.2.48)
v(k)f(x,k,t)dk
:
"/class,B(- > 0 "
(1.2.49)
B"
where we set t'(/c) : = (v(k ),v(k )). The periodicity of / i n k and the point-symmetry of the B r i l l o u i n zone B i m p l y that the conservation property (1.2.18) and the conservation law x
t
M
16
1 Kinetic Transport Models for Semiconductors
(1.2.19) also h o l d for the semi-classical L i o u v i l l e equation, if the d r i v i n g force J " is divergence-free w i t h respect to k, i.e. i f d i v J * = 0 holds. W h e n the parabolic energy-wave vector relationship h \k\ e(k) = ——, keU (1.2.50) 2m 5
k
2
2
3 M k
for electrons i n a vacuum is used, then we o b t a i n v = p/m = hk/m and the semi-classical L i o u v i l l e equation reduces to its classical counterpart (1.2.12). Magnetic
Fields
A further extension o f the L i o u v i l l e equation is concerned w i t h the inclusion of magnetic field effects. Consider a single classical electron, which moves under the action o f an electric field E and of a magnetic field w i t h i n d u c t i o n vector B . The magnetic field generates a c o n t r i b u t i o n to the d r i v i n g force J^, which is n o w given by (see [1.4]): ind
.<¥ =-q(E
+ v x B ).
(1.2.51)
ind
The corresponding classical single-electron L i o u v i l l e equation reads: 8J+
vgmd f-^-(E x
m
+ v x B
3
i n d
)-grad„/,
t > 0,
(1.2.52)
3
where / = f(x, v, t), x e 1R , v e ER . I n the semi-classical case we o b t a i n dj
+ v(k) • g r a d , / - | ( £ + v(k) x B
i n d
) • g r a d j = 0,
t > 0 (1.2.53)
3
w i t h / = f(x, k, t), x e [R ., ke B, subject to periodic boundary conditions o n dB.
1.3 The Boltzmann Equation F o r an ensemble of many interacting particles there are t w o fundamental difficulties connected w i t h the L i o u v i l l e equation: o Models for the d r i v i n g force, which comprise short range and l o n g range interactions, are not readily available, o The dimension o f the M - p a r t i c l e ensemble phase space is 6 M , w h i c h is very large i n practical applications. Even disregarding the p r o b l e m o f constructing appropriate d r i v i n g forces, the high dimensionality o f the L i o u v i l l e equation, when employed as a semiconductor transport model, is p r o h i b i t i v e for numerical simulations. Consider a VLSI-device w i t h 1 0 c o n d u c t i o n electrons i n the active region. T h e n the L i o u v i l l e equation for the electron ensemble is posed o n the 6 x 10 -dimensonal phase space! 4
4
1.3 The Boltzmann Equation
17
The main goal of the subsequent analysis w i l l be a reduction of the dimension of the L i o u v i l l e equation. This w i l l be accomplished as follows. A t first we shall derive a system o f equations for the position-velocity densities of subensembles consisting of d electrons w i t h d ranging from 1 to M. This system, called the B B G K Y - h i e r a r c h y (from the names Bogoliubov [ 1 . 9 ] , B o r n and Green [1.10], K i r k w o o d [1.30] and Y v o n [1.64]), is obtained by assuming a certain structure o f the interaction field (weak two-particle interactions) and by integrating the L i o u v i l l e equation w i t h respect to the position and velocity coordinates o f M — d particles for d = 1 , . . . , M. Then the formal l i m i t M -» oc is carried out and a particular solution of the hierarchy, determined by a single function o f three position, three velocity coordinates and time, is constructed. This particular solution, based on the assumption that the particles o f a small subensemble move independently of each other, represents the electron number density i n the physical phase space U x Uf,. I t is the solution of the so-called Vlasov equation, which can be considered as an 'aggregated' one-particle L i o u v i l l e equation supplemented by a self-consistent (mean) field relation. The Vlasov equation is a macroscopic equation describing the m o t i o n of a weakly interacting large particle ensemble. Usually, i t is employed to model the C o u l o m b interaction caused by a typical (weak) l o n g range force. 3
However, when charge transport i n a semiconductor is considered on a sufficiently large time scale, then the m o t i o n o f the particles is decisively influenced by strong short range forces, so-called scatterings, or i n a fully classical picture, collisions of particles. F o r the accurate description o f charge transport in semiconductors the short range interactions o f the particles w i t h their environment (crystal lattice) are usually more i m p o r t a n t than short range forces between particles,-which only play a significant role when the particle density is very large. I n order to account for these effects, we shall extend the Vlasov equation and o b t a i n the B o l t z m a n n equation for semiconductors. The B o l t z m a n n equation was derived by L . B o l t z m a n n i n 1872 as a model for the kinetics of gases. Its most distinguished feature is the appearance of a nonlinear and nonlocal 'collision operator', which is responsible for formidable mathematical difficulties i n the analytical and numerical treatment. We shall discuss a modification o f the collision operator, which allows a proper description o f the m o t i o n o f charged particles i n semiconductors.
The
Vlasov
Equation
We consider an ensemble o f M electrons w i t h equal mass, denote—as in the previous Section—the position vector of the ensemble by x = (x ..., x) and the velocity vector by v = ( v v ) , where x e U , v e U are the position and velocity coordinates, resp., o f the i - t h electron. We make the following assumptions: lt
3
l
f
M
t
M
3
t
IS
1 Kinetic Transport Models for Semiconductors
(i) the electrons move in a vacuum, or equivalently, the impact of the semiconductor crystal lattice on the m o t i o n is neglected, (ii) the force field !F acting o n the ensemble is independent of the velocity vector, in particular magnetic field effects are ignored, (iii) the m o t i o n is governed by an external electric field and by two-particle interaction forces. The first t w o assumptions are for simplicity's sake only, they w i l l be discarded o f later on. The t h i r d is crucial for the derivation o f the Vlasov equation. We denote the force field per electron charge by £ = E(x, t) (see (1.2.22)), £ = ( £ E ) , where £ , e U is the field exerted on the i-th electron (per unit charge) and set: 3
M
M
E (x,
t) = E (x
t
en
f) + X Eint(*„ xj).
h
(1.3.1)
7=1 j*i
£ denotes the external electric field and £ the two-particle interaction field. The ansatz (1.3.1) means that the force exerted o n the i-th electron is the sum of the electric field acting on the i-th electron and of the sum of the M — 1 t w o - b o d y forces exerted on the i - t h electron by the other electrons of the ensemble. Moreover, we suppose that the electrons are indistinguishable i n the sense that the interaction force £ = £ ( x , y) is independent o f the electron indices. Also, by the action-reaction law, the force exerted by the i-th electron on the y'-th electron is equal to the negative force exerted by the y'-th electron o n the i - t h electron: e x t
i n t
i n t
Etat(*«> */)
=
E
~
int(Xj,
x„
Xi),
Xj
i n t
3
e U.
(1.3.2)
F o r n o t a t i o n a l convenience we set £ ( x , x) = 0 (we shall later o n consider forces £ ( x , y) w i t h singularities at x = y). The L i o u v i l l e equation for the j o i n t position-velocity density / = f(x , x , it,..., v , t) o f the ensemble then reads: M q M $tf + Z i ' § x,/ Z ex,(x,-, t) • grad / ;=i "i;=i i m
i n t
1
M
M
v
A
-
-
r a d
M
I
£
M
I
£
i n t
(x,,x,)-grad
t ) j
/ = 0.
(1.3.3)
Note that the assumption (1.3.2) implies that the density / is independent of the numbering o f the particles for all times i f it is initially, i.e. /(X] , . . . , x , M
v ,..., l
v , t) = f(x ),..., M
x
n(1
r c ( M )
, v ,..., %{l)
x eRl,
v ,
!',-eR
t
n(M)
3 t
t), (1.3.4)
holds for all permutations n o f { 1 , . . . , M } and for all times t, i f it holds for the initial d a t u m f, = f(t = 0), w h i c h we shall assume henceforth. We n o w set up the j o i n t position-velocity density f of a subensemble consisting of d electrons: w
19
1.3 The Boltzmann Equation d)
f (x
X,, V,
u
v , f) d
f(x ,...,x , 1
t)
V ...,V ,
M
U
M
R3(M-d>
x dx
d+1
... dx
dv
M
...
d+1
dv ,
(1.3.5)
M
w
w i t h 1 < d ^ M — 1. A n equation for f is obtained by integrating the L i o u v i l l e equation (1.3.3) w i t h respect to 3 (M — d) position and velocity coordinates and by assuming that / decays to zero sufficiently fast as | x | -> oo, \v \ —> oo: ;
t
id)
+ 1 V g r a d ^ - ^ X E ,(x„ t) • g r a d , /(d) -
1
Z
x Z
Z E (x int
D I V
e
m —i
i=l
Xj)-gmd fM
h
- ± m
Vi
^
^int(-"-i' X * ) . / *
M
- d )
dx^
dl)^
{
i= l
(1.3.6)
= 0,
where we denoted f l v , v^, t). I n order to f (X\,..., x, X derive (1.3.6) observe that the terms w i t h index i ^ d + 1 i n the sum i n v o l v ing the outer field £ vanish by the divergence theorem. The same holds true for the terms w i t h i ^ d + 1 i n the sum i n v o l v i n g the spatial derivatives and for the terms w i t h i ^ d + 1 i n the double sum i n v o l v i n g the interaction field E . By (1.3.4) each term w i t h 1 ^ i < d gives an identical c o n t r i b u t i o n for each j ^ d represented by the last sum i n (1.3.6). The equations (1.3.6) for 1 < d ^ M — 1 constitute the so-called BogoliubovB o r n - G r e e n - K i r k w o o d - Y v o n ( B B G K Y ) hierarchy for the classical L i o u v i l l e equation (see [1.13]). I n general this system o f equations cannot be solved explicitly, however it is accessible to an asymptotic analysis for M large compared to d, i.e. i n the case o f small subensembles of a large particle ensemble. This is particularly interesting for us since i n semiconductor physics one is usually concerned w i t h extremely large charge carrier ensembles. I n order to be able to carry out the l i m i t M -> oo at least formally, we assume that | £ | is o f the order of magnitude 1 / M for M large, which very reasonably implies that the t o t a l field strength | £ | exerted on each electron remains finite as M -* oc. d + 1
d
d
e x t
im
i n t
;
F o r a fixed subensemble size d the equation (1.3.6) then becomes i n the l i m i t M -> oc: W
SJ
d
+ Z V grad ./< > - * Z E Jx , Xi
e
t
t)-grad /« P |
d
n 0.
J
Mf^EJix^xJdXt
dv* (1.3.7)
20
1 Kinetic Transport Models for Semiconductors
Intuitively, it is reasonable to assume that the electrons o f a subensemble, w h i c h is small compared to the t o t a l number o f electrons, move independently of each other. I n terms of the subensemble p r o b a b i l i t y density f this implies the ansatz: (d)
(d)
f (x ,...,X , 1
V ...,V ,
d
u
P
0 = f l
d
X
(1.3.8)
V
( h
h
;=t
The one-particle density P = / ' (1.3.8) w i t h d = 2: 8,P + v g r a d P - — £ m x
l )
e f f
is obtained from (1.3.7) w i t h d = 1 by using
( x , t) • g r a d „ P = 0, t > 0,
X
6
v e
(1.3.9)
with E {x, ett
t) = £
e x t
( x , t)
+
MP{x*,
v*, t)E (x,
x j dv*
int
t > 0,
dx*,
x e
(1.3.10)
A simple calculation shows that (1.3.8) is a particular solution o f (1.3.7) for arbitrary deN i f P solves (1.3.9), (1.3.10). Equivalently, the solution f , d e N,of the B B G K Y - h i e r a r c h y (1.3.7) can be factored according to (1.3.8), if the i n i t i a l data f (t = 0),d e N, a d m i t such a factorization. We define: (d)
w
F(x,
v, t) = MP(x,
n(x, t) =
(1.3.11)
v, t)
(1.3.12)
F(x, v, t) dv.
The quantities F and n represent the expected electron number densities i n phase space and i n position space resp., i.e. F(x, v, t) is the number o f electrons per unit volume i n an infinitesimal n e i g h b o u r h o o d o f (x, v) at time t and n(x, f) is the number o f electrons per u n i t volume i n an infinitesimal neighbourhood of x at time t. We m u l t i p l y (1.3.9) by M and o b t a i n the so-called Vlasov equation [1.13], [1.33]: d F + v • grad^F - — £ m t
e f f
• g r a d , £ = 0, t
3
3
xeU ,
veU ,
x
E f(x, t{
t) = £
e x t
( x , t) +
n(x*, t)E {x,
t>0,
(1.3.13)
f>0.
(1.3.14)
x*)dx*,
int
3
xeR , x
21
1.3 The Boltzmann Equation
The macroscopic electron current density is given by J = —q
vF dv.
(1.3.15)
The equation (1.3.13) has the form of a single particle Liouville equation. M a n y - b o d y physics enters only t h r o u g h the effective field £ , which i n t u r n depends o n the number density n (self-consistent field modeling). The characteristics e f f
x = v,
i> = - - £ m
e f f
( x , t)
(1.3.16)
are the trajectories of electrons m o v i n g i n the field £ . They are the l i m i t i n g ( x , D )-trajectories of the Liouville equation (1.3.3) as M -* oo. The Pauli principle of solid state physics states that two electrons cannot occupy the same state (x, v) at the same time t (see, e.g., [1.34], [1.31]). Since £ ( x , v, t) can also be interpreted as the existence p r o b a b i l i t y of a particle at the state (x, v) at time t, we expect e f f
;
;
0 sS F(x, v, t) ^ 1,
3
3
xeU ,
veU ,
x
f>0
(1.3.17)
to hold. I t is easy to show by using the characteristics (1.3.16) that upper and lower bounds of solutions of the Vlasov equation are conserved i n time. Thus, i f we require 3
0 sc £ ( x , v, t = 0) ^ 1,
3
xeU ,
veU ,
x
(1.3.18)
then (1.3.17) holds and the Vlasov equation satisfies the Pauli principle (an existence p r o b a b i l i t y £ ( x , v, t) larger than 1 can formally be interpreted as a m u l t i p l e occupancy of the state (x, v) at the time t). The Vlasov equation is nonlinear w i t h a nonlocal nonlinearity of quadratic type. I t provides a macroscopic description of the m o t i o n of many-body systems under the assumption of a weak interaction caused by a long range force (see [1.13], [1.31], [ 1 . 4 ] for various applications). I n particular, it does not account for scatterings of particles generated by strong short range forces and, thus, it only represents a useful model on a time scale m u c h shorter than the mean time between t w o consecutive scattering events.
The Poisson
Equation
The most i m p o r t a n t long range force acting between t w o electrons is the C o u l o m b force modeling the mass-action law (see, e.g. [1.4]). I t is represented by the interaction field E (x,y)=
.— 4TI£ |X -
int
S
~j3 > y\ 3
3
x,yeU ,
x^y.
(1.3.19)
The p e r m i t t i v i t y e accounts for the dispersive effect of the considered host material. s
22
Kinetic Transport Models for Semiconductors
We o b t a i n the corresponding effective field from (1.3.14) £
e f f
( x , f) = £
e x t
( x , t) -
dx
^J *
4ne,
(1.3.20)
A simple c o m p u t a t i o n shows div £„
(1.3.21)
div E„
and rot £
e f f
= rot £
e x t
(1.3.22)
.
If the exterior field is vortex-free rot £ „ , = 0,
(1.3.23)
v
then the effective field is vortex-free, too, and there are potential functions KH> Kxt
s
u
c
n
t n
at
£ ff =
-grad,K
e f f
(1.3.24)
£ x, =
-grad F
e x t
(1.3.25)
e
e
x
hold. Then (1.3.21) can be rewritten as •AK,
•AK,
q
(1.3.26)
n.
The effective potential satisfies a Poisson equation w i t h a right hand side which depends linearly o n the electron number density n. Assume n o w that the external field is generated by ions of charge + q, which are present i n the material. Then, again by Coulomb's law we have 9 47C£«
x — x.
13
'
(1.3.27)
where C(x, t) is the number density of the background ions (in position space at the time t). We calculate div £„
C
(1.3.28)
and AK,
He
(1.3.29)
follows. By inserting i n t o (1.3.26) we o b t a i n the w e l l - k n o w n form of the Poisson equation - e . A Ke f t where
(1.3.30)
1.3 The Boltzmann Equation
23
p = q(C - n)
(1.3.31)
is the charge density of the system consisting o f conduction electrons and positively charged background ions. The Vlasov equation w i t h the C o u l o m b interaction field is usually referred to as Vlasov-Poisson equation.
The Whole
Space
Vlasov
Problem
We consider the Vlasov equation (1.3.12), (1.3.13), (1.3.14) o n the whole position-velocity space [R x [R supplemented by the initial c o n d i t i o n 3
F(x,
v
3
, t = 0) = Fj(x, v),
x eU ,
ve Uf,.
x
(1.3.32)
3
By integrating (1.3.13) over [R we o b t a i n the macroscopic conservation law qd n - d i v J = 0
(1.3.33)
t
assuming that F decays sufficiently fast to zero as 11?| —»- oo. Integration over R gives the conservation o f the t o t a l number of particles 3
nj(x)dx,
n(x, t) dx
t>0
(1.3.34)
if J -» 0 as Ixl -» oo. W e denoted Fj(x,
n ( x ) := 7
v) dv.
N o t e that the macroscopic electron c o n t i n u i t y equation (1.3.33) and the Poisson equation (1.3.30), (1.3.31) do not constitute a 'closed' system of partial differential equations, since a relation for the current density J i n terms o f the potential V and the number density n is not available yet. The derivation of such an equation, which describes the current flow i n semiconductors, is the subject o f Chapter 2. A rigorous mathematical analysis (existence, uniqueness and regularity o f solutions) of the Vlasov equation is beyond the scope of this book. We refer the interested reader to the references [1.21], [1.14], [1.62], [ 1 . 5 ] . Here we only m e n t i o n the basic decoupling approach to the construction of a solution: eff
m
(0)
(0)
m
(i) G i v e n F = F (x, v, t), compute the number density n = n (x, t) from (1.3.12) and the effective field £<$ = £<$(x, t) from (1.3.14). (ii) Set £ = Ef \ i n (1.3.13) and solve the so obtained linear hyperbolic equation subject to the initial c o n d i t i o n (1.3.32). This is done by using that the solution F = F ( x , v, t) is constant along the characteristics e f f
(
111
( 1 )
x = v A
E[V (x, m (
t).
24
1 Kinetic Transport Models for Semiconductors <1
F o r appropriate i n i t i a l data the sequence F \ obtained by iterating (i), (ii), can be shown to converge to a limit function, which is a solution of the initial value p r o b l e m for the Vlasov equation. The mathematical sophistication goes i n t o p r o v i n g bounds for (derivatives of) F and E% which are u n i f o r m in / and which allow the passage to the l i m i t as / -> x . Details can be found in [1.14], [1.62], [ 1 . 5 ] . ( , )
Bounded
Position
Domains
I n semiconductor simulation transport equations are usually solved o n bounded position domains, which represent the device geometry. We n o w consider the Vlasov-Poisson problem (1.3.12), (1.3.13), (1.3.30), (1.3.31) for x e Q, where Q ^ u is a bounded convex d o m a i n . The velocity variable v is still assumed to vary i n all U . Obviously, a boundary c o n d i t i o n for (1.3.13) has to be imposed on those subsets of dQ x U at which the x-characteristics point i n t o Q. These so-called inflow boundaries are given by x
3
3
T_ : = {(x, v)\xedQ,
» e R* v(x)-v < 0 } ,
(1.3.35)
where v(x) denotes the unit o u t w a r d n o r m a l vector to dQ at x e SQ. The outflow segments are T
+
: = {(x, v)\x e 8Q, v e M?„ v(x)- v > 0}.
(1.3.36)
M o s t simply, Dirichlet boundary conditions for the Vlasov equation are imposed on the inflow segments: F(x, v, t) = F {x, v, t),
(x,v)eT_,
D
t > 0
(1.3.37)
w i t h 0 ^ F ^ 1 given. Clearly, the solution o f the one-particle L i o u v i l l e equation (1.3.13) then still satisfies the bounds (1.3.17), i.e. the Pauli principle also holds for the i n i t i a l boundary value p r o b l e m for the Vlasov equation. Moreover, the electron c o n t i n u i t y equation (1.3.33) is still valid, however, instead o f (1.3.34) we now obtain by employing the divergence theorem D
d_
n(x,t)dx=
F \v(x)-v\ds(x) D
F | v ( x ) - v\ds(x)
dv
dt
r
+
dv, (1.3.38)
where s(x) denotes the surface measure o n <9Q. Thus, the t o t a l number o f particles is generally not conserved, its rate of change is the difference o f the i n c o m i n g and the outgoing fluxes. By integrating (1.3.38) w i t h respect to f and using F ^ Owe o b t a i n the estimate: n(x, t) dx ^
n,(x) dx
F \v(x)-v\ds(x)dvdt. D
(1.3.39)
The right-hand side represents a b o u n d for the g r o w t h o f the t o t a l number
1.3 The Boltzmann Equation
25
of electrons caused by the inflow which is determined by the boundary c o n d i t i o n (1.3.37). Also, boundary conditions for the Poisson equation (1.3.30) are required. Usually, mixed N e u m a n n - D i r i c h l e t conditions are imposed. Therefore, we split the boundary cQ i n t o Dirichlet segments 5 Q and N e u m a n n segments ("Q w i t h dQ u dQ = dQ, dQ. n dQ =
. We impose D
v
D
E
e f f
N
D
-v = £ -v 6
V =V e(t
b
N
ondQjv
(1.3.40)
o n dQ ,
(1.3.41)
D
where E is a given vector field on 8Q and V a given real-valued function on cQ. . N e u m a n n segments model insulating device boundaries, artificial b o u n d ary segments introduced i n order to separate the considered device from neighboring devices i n a V L S I - c h i p and semiconductor-oxide interfaces in MOS-devices (see [1.37], [1.51] and Chapter 4 for details). Dirichlet boundaries represent contact segments on w h i c h a bias is applied to the device. We remark that the inflow Dirichlet boundary c o n d i t i o n (1.3.37) i s — f r o m a physical p o i n t of v i e w — n o t fully compatible w i t h the mixed N e u m a n n Dirichlet conditions (1.3.40), (1.3.41) for the Poisson equation. Actually, Dirichlet inflow conditions should only be prescribed at the 'Dirichlet inflow segments' r_r\(dQ x R ) and reflecting boundary conditions o n the ' N e u m a n n inflow segments' T_ n (dQ x R ): b
N
b
D
3
D
3
N
F(x, v, t) = F(x, v - 2v(x) (v(x) • v), t),
3
(x, v) e T_ n (dQ
N
x R )
We refer the interested reader to [1.13] for details. A n L^-semigroup analysis of transport operators on bounded position domains can be found i n [1.15].
The Semi-Classical
Vlasov
Equation
Instead o f t a k i n g the classical L i o u v i l l e equation as basis for the derivation of the Vlasov equation we can also start out from the semi-classical formulat i o n (1.2.46). W h e n the above assumptions on the external and internal fields are made, we o b t a i n by proceeding as i n the classical case: d,F + v(k)grad F x
- ~£
e f f
g r a d £ = 0, k
xeRl, £
e f f
( x , t) = £
e x t
( x , f) +
keB,
t)E {x,
(1.3.42)
x*)dx*,
int
XGR , 3
x
w i t h the electron number density
t > 0,
t>0
(1.3.43)
26
1 Kinetic Transport Models for Semiconductors
F(x,k,t)dk.
n(x, t)
(1.3.44)
As before, B denotes the B r i l l o u i n zone o f the semiconductor crystal and v(k) the electron velocity corresponding to a specific energy band. The position-wave vector number density F is assumed to satisfy periodic boundary conditions i n k: F(x,k,t)
= F(x, -k,t),
xeU ,
keoB,
x
t > 0.
(1.3.45)
The distinguished feature of the semi-classical Vlasov equation is that i t takes i n t o account the (quantum) effects o f the lattice periodic potential generated by the ions o f the semiconductor crystal lattice. The (semi-classical) electron current density is defined by J{x,
t ) = - q \
v(k)F(x,k,t)dk.
(1.3.46)
Clearly, the semi-classical Vlasov-Poisson equation can also be posed on a bounded p o s i t i o n d o m a i n Q. The D i r i c h l e t boundary c o n d i t i o n on the inflow segments then reads: F(x, k, t) = F (x, k, t),
(x,k)er_,
D
t > 0
(1.3.47)
w i t h F given and D
T_ : = {(x, k)\xe8Q,
k e B, v(x) • »(fc) < 0 } .
(1.3.48)
The current c o n t i n u i t y equation and the conservation of the t o t a l number of particles (see the case Q = IR ) holds as i n the classical case. Also, the Pauli principle is satisfied for all times, i f it is satisfied i n i t i a l l y and on the inflow boundaries. 3
Magnetic
Fields—The
Maxwell
Equations
So far we neglected the effects o f magnetic fields i n the derivation o f the classical and semi-classical Vlasov equations. Assume n o w that a magnetic field w i t h effective i n d u c t i o n vector B = B ( x , t) e U exerts influence on the m o t i o n of the electron ensemble, too. Then, by setting B = B i n the classical L i o u v i l l e equation (1.2.53) we o b t a i n the Vlasov equation 3
e f f
e f f
i n d
d F + vgmd F t
x
- ?-(E m
e[f
+ v x B
e f f
) • grad^F = 0,
e f f
(1.3.49)
supplemented by the effective field equation (1.3.14). The electric and magnetic field quantities are not independent, their relationship is governed by the M a x w e l l equations (see, [1.26], [1.51]), w h i c h for an arbitrary m e d i u m read:
27
1.3 The Boltzmann Equation
d i v D = /> divB
(1.3.50)
= 0
i n d
(1.3.51)
rot £ = -d,B
(1.3.52)
rot H — 8 D + J t
(1.3.53)
tot
supplemented by the material equations Z) = £ £
(1.3.54)
= pH.
(1.3.55)
s
B
ind
The three-dimensional field quantities have the following meaning: D: B
i n d
electric displacement vector : magnetic i n d u c t i o n vector
£:
electric field vector
H:
magnetic field vector
J :
total current density vector
tot
u denotes the permeability o f the m e d i u m and, as before, £ the permittivity. We assume that the m e d i u m is isotropic and homogeneous and, thus, p and e are constant positive scalars. We insert (1.3.54) i n t o (1.3.50) and set £ = £ s
s
e f f
£ div £ s
Setting B
ind
c f f
= B
eff
= p.
(1.3.56)
gives
div B
e f f
= 0
rot £
e f f
=
(1.3.57) -c,B
(1.3.58)
eU
and by inserting (1.3.55), (1.3.54) i n t o (1.3.53) we obtain -rot
B
ef(
= e dE s
t
ef(
+ J
t o t
.
(1.3.59)
Assume now that the external field is generated by positively charged ions. Then the equations (1.3.49), (1.3.56)—(1.3.59) constitute a 'closed' system o f partial differential equations when supplemented by (1.3.31), (1.3.12) and by J
tol
= J
ion
+ J,
(1.3.60)
where J is given by (1.3.15) and J denotes the current density caused the flux of the positively charged ions w i t h the number density C = C(x, A mathematical analysis o f the so-called V l a s o v - M a x w e l l system can found i n [1.22]. We remark that a semi-classical V l a s o v - M a x w e l l system can easily be rived by c o m b i n i n g the 'ansatze' o f this and o f the previous paragraph. i o n
by t). be de-
28
1 Kinetic Transport Models for Semiconductors
Collisions—The
Boltzmann
Equation
The Vlasov equation accounts for l o n g range interactions o f particles. Short range interactions, w h i c h occur i n the form o f 'collisions' o f the particles w i t h other particles o f the ensemble and w i t h their environment, were neglected so far. These collisions have the effect that the particles are instantaneously scattered from one state to another in such a way that their velocity vector and, consequently, their m o m e n t u m and wave vectors, change extremely fast, while the change of the position vector takes place slowly. The goal o f the following considerations is to derive an extension of the Vlasov equation, which includes a description o f the l o n g range interactions and a statistical account o f the scattering events. The starting p o i n t for a phenomenological derivation o f this equation, formulated first by L . B o l t z m a n n i n 1 8 7 2 for the description o f n o n e q u i l i b r i u m phenomena i n dilute gases, is the observation that the rate o f change of the number density F = £ ( x , v, t) o f the ensemble due to the convection caused by the effective field £ vanishes along the characteristics ( 1 . 3 . 1 6 ) when collisions are neglected: e f f
= 0 . ^
(1.3.61)
/conv
It is reasonable to postulate that the rate o f change of £ due to convection and the rate of change of £ due to collisions balance: dt\
o
m
(«fe)coii"
(1-3.62)
E x p l i c i t l y , ( 1 . 3 . 6 2 ) reads: a,£ +
l
;-grad
; c
£-^£
m
e f f
-grad ,£ l
= f~-)
\dt
,
J
(1.3.63)
eM
where the effective field £ satisfies ( 1 . 3 . 1 4 ) . The rate P(x, v' -> v, t) o f a particle w i t h position vector x, at the time t, to change its velocity vector v' i n t o v due to a scattering event is assumed to be p r o p o r t i o n a l to the occupation p r o b a b i l i t y £ ( x , v', t) o f the state (x, v') at time t. Also, i n order to account for the Pauli principle, i t is assumed to be p r o p o r t i o n a l to 1 — £ ( x , v, t), w h i c h is the p r o b a b i l i t y that the state (x, v) is not occupied at the time t. We thus set e f f
P(x,
v' -+v,t)
= s(x, v', v)F(x,
v',
-
F{x, v, t)),
(1.3.64)
where s is the so-called scattering rate. M o r e precisely speaking, s(x, v', v) dv' is the transition rate for an electron w i t h position vector x to change its velocity vector v' belonging to the volume element dv' (around v') to v. Clearly, the scattering rate is determined by the physics o f the considered scattering mechanism. Those scattering mechanisms, which are i m p o r t a n t i n semiconductor physics, w i l l be discussed below.
29
1.3 The Boltzmann Equation
The rate of change of the number density £ due to collisions at (x, v, t) is given by the 'sum' of the rates of particles being scattered from all possible states (x, v') i n t o the state (x, v) at the time f minus the sum of the rates of the particles being scattered from the state (x, v) i n t o any possible state (x, v') at the time t: — at
(x, v, t) =
[ P ( x , v' -*v,t)
J
— P(x, v -* v', t ) ] dv'. (1.3.65)
eoU
We insert (1.3.64) i n t o (1.3.65), set
m
,u 661
-={£l»
-
and obtain Q{F)(x,
v, t) =
[s(x, v', v)F'(l
- F) - s(x, v, v')F(l
- £')] dv', (1.3.67)
where we denoted: F = F(x,v,t),
F = F(x,v',t).
(1.3.68)
Q is called collision operator and Q(F) collision integral. The B o l t z m a n n equation (1.3.63) then can be w r i t t e n i n the form d F + v• grad^F - — £ m t
e f f
• g r a d „ £ = Q(F), xeRl,
3
veU ,
t>0
(1.3.69)
supplemented by the effective field equation (1.3.14). W h e n the C o u l o m b force is used to model the long range interaction, then (1.3.69), (1.3.14), (1.3.12) is often referred to as Boltzmann-Poisson problem. We remark that the presented derivation of the B o l t z m a n n equation is purely phenomenological. A more rigorous approach for gas-dynamics can be found i n [1.13]. In a d d i t i o n to the nonlinearity caused by the self-consistent field, the collision integral Q(F) introduces another quadratic nonlinearity, which is nonlocal i n the velocity direction. A rigorous mathematical analysis of the Boltzmann equation—even w i t h a given electric field—is extremely complicated and by far beyond the scope of this book. Below, we shall only sketch an existence p r o o f and discuss some qualitative properties, i n particular those w h i c h are i m p o r t a n t for the derivation of the drift diffusion a p p r o x i m a t i o n (see Chapter 2). We refer the reader to the b o o k [1.13] for a wealth of i n f o r m a t i o n o n the (gas-dynamical) B o l t z m a n n equation and for a huge collection of references. Also, for the mathematically oriented reader, we mention the recent paper [1.23], where existence, globally i n time, of solutions of the (gas-dynamical) B o l t z m a n n equation i n the field-free case £ EE 0 is shown. E F F
30
1 Kinetic Transport Models for Semiconductors
The Semi-Classical
Boltzmann
Equation
F o r the modeling o f transport i n semiconductors the B o l t z m a n n equation in the semi-classical f o r m u l a t i o n is usually employed i n order to incorporate the q u a n t u m effects o f the semiconductor crystal lattice as discussed i n Section 1.2. Therefore, we start out w i t h the semi-classical Vlasov equation (1.3.42), (1.3.43) and include the collision effects as above. We obtain: d F + v(k)-grad F t
x
- |£
e f f
• g r a d . F = Q(F), xeU ,
keB,
x
t > 0,
(1.3.70)
where the collision integral Q(F) is given by Q(F)(x,
k, t) [s(x, k', k)F'(l
— F) — s(x, k, k')F(\
- F ) ] dk'.
(1.3.71)
B
We denoted F = F(x,k,t),
F = F(x,k',t).
(1.3.72)
Clearly, the periodic boundary c o n d i t i o n (1.3.45) has to be imposed. Also, the (semi-)classical Boltzmann-Poisson p r o b l e m can be posed o n a bounded position d o m a i n Q c u .. The b o u n d a r y conditions o n <9Q are the same as for the Vlasov-Poisson equation. We impose the i n i t i a l c o n d i t i o n 3
F(x, k,t = 0) = F,(x, k),
(1.3.73)
which is assumed to obey the Pauli principle, i.e. 0
Fj(x, k) ^ 1
(1.3.74)
holds. I n the bounded position d o m a i n case the inflow d a t u m F also has to be between 0 and 1. We shall n o w sketch the existence and uniqueness p r o o f for the B o l t z m a n n Poisson p r o b l e m i n the whole space case as presented i n [1.44]. The p r o o f proceeds by a decoupling iterative approach similar to the existence p r o o f for the Vlasov-Poisson problem. O n l y the collision integral has to be taken care o f accordingly. We set F = 0 and construct a sequence o f approximations { F } as follows. G i v e n F , we compute the number density D
( 0 )
< ( )
( e N o
F" dk B
(l)
and the effective field E% by inserting n kernel (1.3.19). Then we have
i n t o (1.3.14) using the Poission
1.3 The Boltzmann Equation
l+1
31
(M
dF >
(
+ i>(fc)-grad F > - ? £ ' n
t
X
+1
e
) f
f
-grad F"
+ 1 )
T
m
= QvJF" \
F)
(1.3.75)
subject t o the initial c o n d i t i o n (1.3.73) and the periodic boundary c o n d i t i o n on cB, where e
<
LIN
(F '
+1,
(
, F ") (l)
[s(x, k', k)F \l
- F
( I + 1 )
+1)
) - s(x, k, k')F« (\
B
- F<"')] dk'. (1.3.76)
We used the obvious n o t a t i o n w
(
F
= F '\x,
Clearly, Q
lm
Q
L I N
k', t).
can be w r i t t e n as (F"
+ 1 )
( , )
(
, F ) = A<'\1 - F '
+ 1 )
+1
) - B<'>F" >
(1.3.77)
where m
(,)
s(x,k',k)F
A
dk', (1.3.78)
B
w
s{x, k,k')(l
-F )dk'.
The problem (1.3.75) is a linear transport equation, w h i c h can be solved by the method of characteristics o r by semigroup theory. The characteristic form of (1.3.75) reads {l+1)
dF =-=— at
(l)
(,+1)
il)
+ (A" + B )F
=A
(1.3.79)
along the characteristics determined by E%: x = v(k),
hk = -qE%.
(1.3.80)
<1)
< 0
Since s > 0 we conclude A 5s 0 from (1.3.78) i f F ^ 0. Integration o f 11.3.79) gives F ' ^ 0. F o r G := 1- F we obtain from (1.3.79) (
JG
(
m
)
, , + 1 )
, , + 1 )
dt (
+ 1 )
)
(
,
(
+ (/4" + B ' ) G ' (l)
+ 1 ,
(
)
= 5'.
(1.3.81)
( , + 1 )
If F " sc 1 we obtain B ^ 0 and G ^ 0 follows by integrating (1.3.81). Thus, if the initial d a t u m F, satisfies the Pauli principle, all iterates F satisfy the Pauli principle for all times t > 0 and, by passing t o the l i m i t l-> oo, we conclude that the B o l t z m a n n equation conserves the upper b o u n d 1 and the lower b o u n d 0, i.e. the solution F satisfies ( , )
0
f
^ 0 .
It was actually shown i n [1.44] that the sequence { F
(1.3.82) ( , )
}
( e S !
j converges t o the o
32
1 Kinetic Transport Models for Semiconductors
unique solution of the Boltzmann-Poisson p r o b l e m if the transition rate s is sufficiently regular and positive.
Conservation
and
Relaxation
We integrate the collision integral (1.3.71) w i t h respect to the wave-vector k and obtain Q{F)dk
=
[s(x, fe', fe)F(l -
F)
J B J B
- s{x, k, k')F(l
- F ' ) ] dk' dk = 0.
(1.3.83)
This implies the conservation law (1.3.33) and—for the whole space p r o b lem—the conservation of the t o t a l number of electrons (1.3.34). Collision processes neither destroy nor generate particles. Another i m p o r t a n t property of the collision operator is related to the relaxation of the state of the ensemble towards local thermodynamical e q u i l i b r i u m . The so-called principle of detailed balance asserts that the local scattering probabilities vanish for all states (x, k), (x, k') i n thermal e q u i l i b r i u m (see[1.4], [1.8]), i.e. s(x, fe', k)F' i\ e
- F ) = s(x, k, k')F (l e
e
- F' )
(1.3.84)
e
holds, where F denotes the e q u i l i b r i u m number density. I t follows from standard statistical mechanics that F is given by the F e r m i - D i r a c statistics e
e
m
FD
%
(L3 85)
" ^C^)
'
where s(k) is the considered energy band of the semiconductor, s denotes the Fermi-energy, k the B o l t z m a n n constant, T the lattice temperature and F
B
F
»
( L 3
= TTe"
'
8 6 )
(see [1.31]). F r o m (1.3.84) we obtain the following property of the scattering rate s by a simple calculation: s(x, k, k') = e x p ^
( / C
|~
£ ( / c ) r
^ ( x , fe', k).
(1.3.87)
I t was shown i n [1.44] that the c o n d i t i o n (1.3.87) on ,s is sufficient and necessary to guarantee that the n u l l manifold of the collision operator Q consists of F e r m i - D i r a c distributions, i.e. Q(F) = 0,
0 ^ F ^ 1
implies that F is of the form (1.3.85) for some Fermi-energy —oo E (X, i) ^ oo i f (1.3.87) holds. F
(1.3.88) E
F
=
33
1.3 The Boltzmann Equation
The q u a n t i t y (1.3.89)
s(x, k, k') dk'
X(x, k)
is called collision frequency. I t measures the strength of the interaction at the state (x, k) corresponding to the transition rate s. Its reciprocal 1
r ( x , k) :=
(1.3.90)
A(x, k)
is the relaxation time describing the average time between t w o consecutive collisions at (x, k). W e shall see below that T represents the time scale o n which the density F relaxes towards an e q u i l i b r i u m state (1.3.85).
Low Density
Approximation
I n many semiconductor device applications the particle density F is small, i.e. 0 < F ( x , M ) « l
(1.3.91)
holds. Very often the quadratic terms i n the collision operator are ignored i set to zero) i n these cases. Then the so obtained simplified linear collision operator is given by Q {F)(x, L
k, t) =
[s(x,k',k)F
- s(x,k,k')F]dk'.
(1.3.92)
J B
Obviously, Q also satisfies the conservation property L
Q (F)dk
= 0.
L
(1.3.93)
l B
The principle of detailed balance n o w gives s(x, k', k)F' = s(x, k, k')F . e
(1.3.94)
e
I n the context of the l o w density a p p r o x i m a t i o n the F e r m i - D i r a c distribut i o n is usually approximated by the M a x w e l l i a n d i s t r i b u t i o n , w h i c h reads M(k)
= N* exp I -
~ kT
I,
N*
B
{~B
exp
B
]dk (1.3.95)
W i t h F = M(k) we o b t a i n from (1.3.94) e
s(x, fe', fe)M(fe') = s(x, k,
fc')M(fe),
(1.3.96)
which is equivalent to the c o n d i t i o n (1.3.87) obtained for the nonlinear collision operator under the assumption of the F e r m i - D i r a c e q u i l i b r i u m i:stribution.
34
1 Kinetic Transport Models for Semiconductors
We easily conclude that the scattering rate s can now be w r i t t e n as s(x, k, k') = (x, k', k)M(k'),
(1.3.97)
where is symmetric w i t h respect to k and k'\ <j>{x,k,k') = 0(x,k',k).
(1.3.98)
The function cj> is called collision cross-section. It is now easy to show that the null-space of the low-density collision operator Q is spanned by the M a x w e l l i a n : L
Q (F)
= 0^F(x,
L
k, t) = n(x, t)M(k).
(1.3.99)
By using (1.3.97) and (1.3.98) we can n o w write Q in the form L
Q (F)(x, L
k, t)
The Relaxation
(x, k', k)[M(k)F'
Time
- M ( / c ' ) F ] dk'.
(1.3.100)
Approximation
I t is particularly interesting to investigate whether the solutions of the B o l t z m a n n equation converge to an e q u i l i b r i u m state as t -» oo. Since this analysis is very complicated for the nonlinear collision operator Q as well as for the low density a p p r o x i m a t i o n Q we shall carry out another simplification of the collision integral. W h e n the initial d a t u m F is close to a multiple of the M a x w e l l i a n it is natural (and mathematically convenient) to approximate F' in (1.3.100) by n(x, t)M(k'). By using the definition of the relaxation time T given by (1.3.90) we obtain L
2
Q (F)(x, R
k, t) = —-i-(F(x, T(X, k)
k, t) - M(k)n(x,
t)).
(1.3.101)
N o t e that the so-called relaxation time a p p r o x i m a t i o n collision operator Q is linear and local i n the wave vector k. W h e n the characteristics x = x(r), k = k(t), w h i c h satisfy R
x = v(k),
x(t = 0) = x
hk=-qE ,
0
k(t = 0) = k
e[t
0
are introduced, then the B o l t z m a n n equation w i t h the collision operator takes the form —F dt
= —-(F x
- Mn)
Q
R
(1.3.102)
along (x(f), k(t)). By straightforward integration we obtain ' F(x(t),k(t),
t) = e-"*[F (x ,k ) I
0
0
s/r
+ - ' n(x(s), s)M(k(s))e x o
ds
, /
(1.3.103)
1.3 The Boltzmann Equation
35
where—for the sake of s i m p l i c i t y — w e assumed the relaxation time T to be constant. Also, the electron density n = J F dk is assumed to be k n o w n i n 11.3.103). I t is now an easy exercise to show that B
F(x(r), k(t), t) - n(x(t\
t)M(k(t))
~ e'"',
t -> oc
(1.3.104)
holds, i.e. the relaxation time x is the scale o n which F returns to the e q u i l i b r i u m density from the perturbed state F along the characteristics. I n the collisionless case (Vlasov equation) there is no mechanism, which forces the state o f the particle ensemble to relax towards thermodynamical e q u i l i b r i u m i n the large time l i m i t . This is also expressed by the fact that the Vlasov equation is time reversible for static exterior fields while the Boltzmann equation is not. The relaxation behaviour of the solutions of the Boltzmann equation is precisely caused by the effect of collisions. This is represented mathematically by the famous H - T h e o r e m (see [1.13] for the gas-dynamics case and [1.44] for the semiconductor case). 7
Polar
Optical
Scattering
Typically, the transition rates, determined by the physics of the considered scattering process i n the semiconductor crystal are highly nonsmooth functions of k, k' (more precisely speaking they are, i n general, distributions). As a typical example we present the transition rate for the polar optical scattering process m o d e l i n g collisions of electrons w i t h phonons, which are quantized vibrations of the semiconductor crystal lattice (see [1.31] for details on the physics). I t is of the following form (see [1.43]):
s (k, p0
S
y , ((No
k') =
k ]2
+ 1 ) W ) - e(k) +
+ N 5(e(k') 0
hoj ) 0
- e(k) - hco )), 0
(1.3.105)
where ha) is the (constant) energy of a polar optical phonon, N p h o n o n occupation number given by the Bose-Einstein statistics 0
w
is the
0
°
=
M ^ ) - ' r
,
u
i
o
>
6
and 2
q h(o
( 1
0
^
=
1 \
SAX^~I)-
,
1
3
'
1
0
7
)
Here r. denotes the vacuum permittivity, e the high frequency relative p e r m i t t i v i t y and e the l o w frequency relative p e r m i t t i v i t y o f the semiconductor. 8 stands for the D i r a c distribution. 0
a
r
36
1 Kinetic Transport Models for Semiconductors
Particle-Particle
Interaction
The transition rate s, which models particle-particle scatterings depends on the density F itself (see [1.28]) and, consequently, this scattering mechanism introduces another nonlinearity into the B o l t z m a n n equation. The corresp o n d i n g model is of the form [F](x,
V
k,k') H(\k
- k ' \ , \ k - k' \)F (l Q
0
- F )d 3
0
0
ko
E
dk' dk , 0
(1.3.108)
0
where H is nonnegative, F' = F(x, k' , t), F = F(x, k , t), 0
8
ko
= S(k
0
0
+ k ' - k ' - k), 0
0
0
5 = 5(e(k ) E
0
+ e(k')-
e(k' ) 0
s(k)).
When s [ F ] is inserted into the collision integral, it becomes apparent that the corresponding collision operator is nonlinear of fourth order. Particleparticle scattering is only relevant for very high local densities. I t is neglected in most practical situations. We remark that there are various other scattering mechanisms which play a role i n semiconductor physics (see [1.48]). F o r a particular practical application the relevant mechanisms have to be identified, the corresponding scattering rates then have to be added to o b t a i n the total scattering rate modeling all the considered interactions. p p
1.4 The Quantum Liouville Equation I n ultra-integrated semiconductor devices the characteristic length of the active region is usually under 1 /urn. Very often these devices are operated at large applied voltages, which leads to extremely high local electric field strengths. W i t h today's technology electric field peaks o f 10 V / c m are usually reached. I t is w e l l - k n o w n that potential variations of this order o f magnitude lead to q u a n t u m effects, which cannot be properly described by the so far presented classical or semi-classical transport models. O n the other hand, there is a group o f semiconductor devices, whose performance explicitly relies on a quantum mechanical phenomenon, namely the tunneling effect (e.g. the so-called tunnel diode, see [1.57]). 6
F o r thgse reasons and, i n particular, since t o m o r r o w ' s semiconductor technology promises an even higher degree of m i n i a t u r i z a t i o n and integration, it is of great importance to devise transport models capable of describing q u a n t u m phenomena, which are still sufficiently simple to allow for reasonably efficient numerical simulation. I n this Section we shall consider a q u a n t u m transport model based o n Wigner functions. I n t r o d u c e d by E. Wigner i n 1932 as q u a n t u m equivalent of classical particle d i s t r i b u t i o n functions, Wigner functions were closely scrutinized by theoretical physicists but only recently their value for semiconductor simulation was discovered.
1.4 The Quantum Liouville Equation
37
We start the presentation w i t h the basic q u a n t u m mechanical equation o f motion.
The Schrddinger
Equation
I n a q u a n t u m mechanical set-up the m o t i o n of an electron is described by the Schrddinger equation: ihd il/= Hip,
(1.4.1)
t
where the q u a n t u m H a m i l t o n i a n operator H for a single particle i n a potential field E(x,t)=
-gmd V(x,t)
(1.4.2)
- q V ( x , t )
(1.4.3)
x
is given by H = ~ A
x
(see, e.g. [1.34], [1.25]). N o t e that the q u a n t u m H a m i l t o n i a n is obtained from the classical H a m i l tonian function (1.2.26) by inserting the m o m e n t u m operator p = —ih g r a d .
(1-4.4)
x
A solution \jt = i//(x, t) o f the Schrddinger equation is called a wave function. The square of its modulus n
q u
,»:=W
2
(1-4.5)
represents the q u a n t u m mechanical p r o b a b i l i t y density for the position of ;he electron, i.e. the number | i H x , t)\ dx
(1.4.6)
2
IA is the p r o b a b i l i t y o f finding the electron in the subset A of the position space at the time t. Note that ^ \il/\ dx i s c o n s e r v e d by the m o t i o n . M u l t i p l y i n g (1.4.1) by the complex conjugate \jj of the wave function \]/, integrating over Ml and . i k i n g imaginary parts gives: 1 f — \il/\ dx = 0 (1.4.7) dt J R3 3
2
2
assuming that the potential V is real valued (as we shall do henceforth) and that \\i decays sufficiently fast to zero as | x | -* oo. Thus, c
r n
(x, t) dx =
n (x) quanJ
dx,
n
q u a n < /
= l^l
2
(1.4.8)
Kinetic Transport Models for Semiconductors
38
follows, where ij/j is the initial d a t u m for the Schrodinger equation Mx, t = 0) =
fa(x).
(1.4.9)
Clearly, (1.4.8) is the q u a n t u m equivalent o f (1.2.18) i n the single electron case. We calculate the rate o f change o f the p r o b a b i l i t y (1.4.6): 2
\iP\ dx
dt
(ij/d [j/ + \jjd \j/) dx t
t
(\jiH\li - \irH\ji) dx A
ih_
((// Ai/^ — i/>At/>) dx
2m (bars denote complex conjugation)._ Since i/r A i ^ — ij/Ai// = div (ijj x
d
2
\ijj\
dx
grad^i// — ijj grad i/^) we o b t a i n x
div J
(1.4.10)
dx
quan
dt with (1.4.11) By the divergence theorem d
1 \i//\ dx = q
(1.4.12)
2
It
A
V
^quan ' A
holds, where v is the outer unit n o r m a l to dA. Thus, the vector J is called q u a n t u m mechanical electron current density. F r o m (1.4.10) we obtain the conservation law for the one-electron case: A
^r»quan
q u a n
- div J
q u a n
= 0,
(1.4.13)
which is analogous to the classical equation (1.2.19). The eigenvalue p r o b l e m for the Schrodinger equation 2
h
2m
A\p — qV(x)\jj
= B\JJ
(1.4.14)
is obtained by l o o k i n g for time-periodic solutions of the form e x p [ — (i/h)ei] \j/(x). The spejctral values e of (1.4.14) are the possible energies o f the electron (cf. Section 1.2). Tunneling We shall now present the maybe most simple, explicitly solvable model for the tunneling o f a particle t h r o u g h a potential barrier. Therefore we consider
39
1.4 T h e Q u a n t u m L i o u v i l l e E q u a t i o n
the one-dimensional steady state Schrddinger equation (1.4.14) w i t h V(x) = — {m/q)d{x), i.e. we assume that the potential barrier is infinitely high and infinitely thin. I n Section 1.2 we analyzed the m o t i o n o f classical electrons in the same potential field and showed that they are reflected at the barrier x = 0 no matter how large their velocity. As we shall see now their q u a n t u m mechanical behaviour is entirely different. We remark that we proceed analogously to [ 1 . 6 ] in this paragraph. The equation of m o t i o n for electrons w i t h the energy e n o w reads: ,2
• 00 < X <
(1.4.15)
00
w i t h \i = —, s > 0. I t can be shown that (1.4.15) is equivalent to m 2
H
i>xx + e«A = 0 .
(1.4.16)
x ^ 0
.MO+)
(1.4.17)
2
^ ( 0 - ) ) = «A(0)
(1.4.18)
(see Problem 1.20). By solving (1.4.16) we obtain:
• A M
=
a exp
i——-x I + b exp I i
c exp
2s i —— x
x < 0 (1.4.19)
\
+ d exp
i
2e
x > 0
We now assume that a monoenergetic beam of particles, represented by a right m o v i n g wave of the form exp( — iy/2ex/n), is aimed at x = 0 from x = — oo. We then expect a reflected left-moving wave for x < 0 and a transmitted r i g h t - m o v i n g wave for x > 0. This gives a = 1 and d = 0: x J + b exp
exp
i
'2E
x < 0 [1.4.20)
2c c exp
x > 0
—;
The interface conditions (1.4.17), (1.4.18) give b =
fi^jlei 2
— 1
2n e + 1
/
y U / 2 f i ( ^ Z f : + i) v
v
(1.4.21)
2
2n i: + 1
2
R := \b\ is the p r o b a b i l i t y that a particle o f energy e is reflected at the barrier, it is therefore called reflection coefficent. T = | c | is the p r o b a b i l i t y that the particle is transmitted t h r o u g h the barrier, it is called transmission 2
1 Kinetic Transport Models for Semiconductors
40
coefficient. We calculate 2
1
2u e T = - ^ — - . 2p s + 1
R = ^~ 7, 2f.i e + 1 2
2
(1.4.22)
2
Obviously R + T = 1 holds. Also R = 1, T = 0 for e = 0 and R = 0, T = 1 for s = oc. Thus, a particle w i t h zero energy (corresponding to zero velocity) is reflected and a particle w i t h infinite energy (infinite velocity) is transmitted. These are the only energy values for which the classical and the q u a n t u m cases agree. F o r all energy values 0 < e < oo there is a nonzero reflection p r o b a b i l i t y and a nonzero transmission p r o b a b i l i t y . Also, note that i n the classical l i m i t h -* 0, w h i c h implies \i -* 0, we o b t a i n R = 1 and T = 0.
Particle
Ensembles
and Density
Matrices
The m o t i o n o f a particle ensemble consisting o f M electrons is described by the Schrodinger equation (1.4.1) w i t h the M - b o d y H a m i l t o n i a n 2
M
h -—Y K -qV{x ,...,x ,t), 2m ; t l
H=
J
i
l
(1.4.23)
M
where x e IR^ denotes the position vector o f the i-th electron. As in the classical case we shall in the sequel denote the position vector o f the ensemble by x = ( x x ) e Ul . T h e n the q u a n t u m p r o b a b i l i t y ensemble position density n and the q u a n t u m ensemble current density •Jquan defined as i n (1.4.5) and (1.4.11) resp. The conservation law (1.4.13) also holds true for the electron ensemble. I n the conservation property (1.4.8) the integration has to be stretched over U^ . F o r future reference we introduce the density m a t r i x p, corresponding to the wave function \jj of the M - e l e c t r o n ensemble, w h i c h is defined by ;
M
l
5
M
q u a n
a
r
e
M
pir, s, t) = \jj{r, t)\p{s, t),
r.seRf.
(1.4.24)
The diagonal elements represent the ensemble position density p(x, x, t) = n
q u a n
( x , t)
(1.4.25)
and the ensemble current density is given by •/q an(*, U
t) = ^ ( §
r
a
d s
- g r a d ) p ( - , •, t)\ . r
r=s=x
(1.4.26)
Differentiating (1.4.24) w i t h respect to t and using the Schrodinger equation (1.4.1) gives the e v o l u t i o n equation for the density m a t r i x p: ihd p = (H -H )p, t
s
r
(1.4.27)
where H , H stand for the H a m i l t o n i a n acting on the s and, resp., r variable. W i t h the M - b o d y H a m i l t o n i a n (1.4.23) the equation (1.4.27) reads explicitly s
r
41
1.4 The Quantum Liouville Equation
= - ^ ( K P ~ A p ) - q(V(s, t) -
ihd
tP
r
V(r, t))p.
(1.4.28)
This equation is the Heisenberg equation of m o t i o n . We refer those readers, w h o have a deeper interest i n q u a n t u m mechanics to the textbooks [1.25], [1.34]. F o r the mathematically oriented reader interested in analytical results o n the Schrddinger equation (which has been the subject for an intensive mathematical scrutiny), we recommend [1.46].
Wigner
Functions
We shall now reformulate the q u a n t u m equations o f m o t i o n i n kinetic form. Therefore we introduce the change o f coordinates h s = x - — n 2m
h 2m
(1.4.29)
in the density m a t r i x p and set u(x, r\,t) = p\x
+
h
(1.4.30)
—n,x-—r],t].
I n the sequel we shall often consider Fourier transforms o f functions which depend on rj. Since (h/2m)n has the dimension o f length, we conclude that r\ has the dimension o f inverse velocity, and, thus, the dual variable o f rj has the dimension o f velocity. Therefore, we denote it by v and define the Fourier transform (1.4.31)
iv
g(v)e~ "dv
$Fg{n) :=
M
of a function g = g(v), g: Uf -> C. The inverse Fourier transform o f a function h = h{n), h: U* -> C reads M
iv
(2n 3M :
h(n)e "
dr\.
(1.4.32)
R-JM
The Wigner function w, which corresponds to the wave function \j/ (or, equivalently, to the density m a t r i x p given by (1.4.24)) is defined as the inverse Fourier transform o f u w i t h respect to rj: w := ,!F
u
(1.4.33)
or. explicitly: w(x,
V,
t)
(2n)
3M
P[x + —n, »3M V 2m
x
,riv
2m
ri,t)e
dt].
(1.4.34)
It was introduced by E. Wigner i n 1932 (see [1.63]) and, as we shall see below, its construction constitutes a major break-through i n the quest for a kinetic f o r m u l a t i o n o f q u a n t u m transport.
42
1 Kinetic Transport Models for Semiconductors
Using (1.4.25) we immediately derive that the mean value of the Wigner function w w i t h respect to the velocity v is the q u a n t u m electron ensemble position density w(x,v,t)dv,
"quanta *)
(1.4.35)
since u(x, Y\ = 0, t) = # w ( x , r\ = 0, t) = p(x, x, t) holds. Also, we obtain from (1.4.26), (1.4.29): •Jquan
= ~Q g ^ d , , Q | „
= 0
.
(1 .4.36)
By t a k i n g the gradient of (1.4.31) w i t h grad^ ^g(t]) = — HF{vq){n). Thus, •Jquanf-^ t ) =
-q
respect
on
we
vw(x,v,t)dv
conclude
(1.4.37)
follows. The first order moment of the Wigner function vv w i t h respect to the velocity v, m u l t i p l i e d by — q, is the q u a n t u m current density of the electron ensemble. Thus, as far as the zeroth and first order moments are concerned, the Wigner function behaves as the classical particle d i s t r i b u t i o n . However, as w i l l be demonstrated later on, the Wigner function does not necessarily stay nonnegative i n its evolution process. U n l i k e i n the classical case, it can therefore not be interpreted as a p r o b a b i l i t y density. I n the literature it is often referred to as 'quasi-distribution' of particles. F o r precisely this reason Wigner functions were not employed for practical simulations u n t i l recently, when they were rediscovered as the maybe only q u a n t u m transport model for semiconductors w h i c h is accessible to numerical simulations. O n a theoretical physics level, however, Wigner functions have been intensively scrutinized (see [1.58], [1.12], [1.20]).
The Quantum
Transport
Equation
The evolution equation for the Wigner functions is obtained by transforming the Heisenberg equation (1.4.28) for the density m a t r i x p to the (x, r])coordinates given by (1.4.29):
c u + i d i v J g r a d ^ u) + iq— t
—-
n
—
-u
= 0
(1.4.38) and by Fourier transformation d w + f g r a d ^ w + — 0 [ F ] w = 0, m t
f t
xeRf,
M
veUf ,
t > 0.
(1.4.39)
1.4 The Quantum Liouville Equation
43
The operator 9 [V~\ is defined by h
( ^ 0 „ [ F ] w ) ( x , 1, t) V
x
+
t
( = im — ^ -
V
2 m ^ ) -
\
x
~ 7 ^ — -
r
}
,
t
( # w ) (x,n,t)
(1.4.40)
H or, explicitly: ( A [ K ] w ) ( x , f, 0 F
(
X
+
^ ^ ) '
F
X
(
~ 2 ^
( 2 T I ) 3M Hv
l
y
x e ~" "
dv' drj.
f
w(x, i / , t) (1.4.41)
A n operator, whose Fourier transform acts as a m u l t i p l i c a t i o n operator on the Fourier transform o f the function, is called a linear pseudo-differential operator and the m u l t i p l i c a t o r is called the symbol of the pseudo-differential operator. F o r the mathematical analysis o f this type o f operators we refer the reader to [1.52], [ 1 . 5 9 ] , [1.60], [1.61]. Thus, 0 [V~\ is a pseudo-differential operator w i t h the symbol h
v(x (5VMx,
+ ^-n,
n, t) := im-±
—
tj-
vlx-~~r],t ^
— — ' -
(1.4.42)
and the q u a n t u m L i o u v i l l e equation (1.4.39) is a linear pseudo-differential equation. The local term 8 w + t'-grad^ w describes the m o t i o n of free electrons just as i n the classical case, the nonlocal term (q/m) O \_V~\w, which generally couples all velocities and frequencies, models the acceleration by the field t
h
E(x, t)=
- grad
x
V(x, r).
(1.4.43)
It is the q u a n t u m analogue of the term q/m g r a d V • grad . / , which appears in the classical L i o u v i l l e equation (1.2.12). F o r m a l l y , the symbol satisfies x
t
h
(dV) -^igrad V-ri h
(1.4.44)
x
and in the formal l i m i t h -> 0 the equation (1.4.38) reduces to d.u + i d i v J g r a d , u) + ;' — g r a d „ V-riu = 0, m
(1.4.45)
which is the Fourier transformed Liouville equation d w + v • g r a d vv + — grad^ V• g r a d „ w = 0 m t
l
since !F~ {it\u)
x
= grad^J^'u).
(1.4.46)
44
1 Kinetic Transport Models for Semiconductors
Thus, the q u a n t u m L i o u v i l l e equation becomes the classical L i o u v i l l e equat i o n when the so-called classical l i m i t h - » 0 is carried out formally. Later on we shall make this statement mathematically precise. By the above derivation the q u a n t u m L i o u v i l l e equation follows directly from the Schrodinger equation. M a n y - b o d y physics enters t h r o u g h the number of coordinates ( 3 M position and 3 M velocity coordinates for an M-electron ensemble) and t h r o u g h the form o f the many-body potential V. Thus, the q u a n t u m L i o u v i l l e equation is by no means simpler than the many-body Schrodinger equation, actually, the number o f coordinates doubled. As we shall see in the next Section, its advantage is the kinetic form, which is accessible to a one-body a p p r o x i m a t i o n i n w h i c h many-body physics only enters t h r o u g h an averaged potential. Also, the kinetic equation allows a f o r m u l a t i o n on bounded p o s i t i o n domains, subject to (more or less) physically reasonable boundary conditions. This is o f particular importance for the numerical s i m u l a t i o n of semiconductor devices. Very often, the following generic n o t a t i o n for pseudo-differential operators is used: F o r the operator (Ag)(v)
=
iil
{2K
v)
a(rj)g(v')e '- "
,3M
dv' dt]
(1.4.47)
3
R "
w i t h the symbol a = a(n), one writes a (-grad
„ ) g:=Ag.
(1.4.48)
i
Then, the convection operator 6 \V~\ can be expressed as h
v(x
+
t
^2mi , t ) -] v ( x\
g r a d
* 2m\ *
n = (5F)Jx,jgrad ,H
(1.4.49)
0
and the q u a n t u m L i o u v i l l e equation (1.4.39) takes the form / h grad,, \ / h grad,, V 2m; / V Zmi c,w + v • g r a d w + iq —, h v
w = 0. (1.4.50)
Pure and Mixed
States
We consider the whole space p r o b l e m for the 3 M - d i m e n s i o n a l q u a n t u m L i o u v i l l e equation (1.4.50) subject to the initial c o n d i t i o n w(x, v, t = 0) = wj(x, i'),
M
x e U , X
ve
.
(1.4.51)
1.4 The Quantum Liouville Equation
45
The solution o f this initial value p r o b l e m is the Wigner function w(x, v, t)
+
x
=
a^l A ;
')*{"-Si*
' ) ' * "
d
"
<
1
A
5
2
1
for all times t ^ 0 if and only if the initial d a t u m w satisfies ;
for some function i/f = ^ ( x ) , which is the initial wave function o f the state ifr. (1.4.53) is equivalent to the following conditions o n the initial density matrix: 7
(a)
7
^—\n (r,s)
= 0,
Pl
(b)
p,(r, s) = p,(s, r),
(1.4.54)
ores where we set (r, s) : = Fwj{x, n) (1.4.55) evoking the coordinate transformation (1.4.29). N o t e that (1.4.54) (b) is equivalent to w being real valued. If (1.4.52) holds, then the q u a n t u m state o f the electron is fully described by the single wave function i/f = ij/(x, t). I n q u a n t u m physics this is referred to as a pure q u a n t u m state. Clearly, for i n i t i a l data vv , which do not satisfy (1.4.54), the solution w o f the q u a n t u m L i o u v i l l e equation is not o f the pure state form (1.4.52), and, thus, a more general s o l u t i o n representation has to be sought. Let = ij/ (x, t), i/> = i / / ( x , t) be t w o solutions o f the Schrddinger equation. By a simple computation it is immediately verified that the product u' (r, r)(/^ (s, f) solves the Heisenberg equation (1.4.27) and by linearity we conclude that all linear combinations o f such products of the form Pl
7
7
ll)
l)
(2)
(2,
(2|
l)
i
p{r,s,t):=Y Pi V {r,tW \s,t) i.j J
(1.4.56)
i
are solutions of (1.4.27), too. A solution o f the q u a n t u m L i o u v i l l e equation is then obtained by setting r = x + (h/2m)n, s = x — (h/2m)ri and by inverse Fourier transformation. T o solve the initial value p r o b l e m (1.4.50), (1.4.51), the coefficients p and the wave functions \jj at f = 0 have to be adapted to the initial function w . We must require (m)
tj
t
ft('.s)
Efl,^W(j).
=
d-4.57)
i-i 2
This gives a clear indication o n how an L - t h e o r y for the q u a n t u m Liouville equation should be set up. F o r the following we assume 2
3 M
Wj e L (U
X
M
x U* )
(1.4.58)
46
1 Kinetic Transport Models for Semiconductors 2
3M
and choose a complete o r t h o n o r m e d system of L ( I R ) - f u n c t i o n s W ^ / e M - Then {i/4'V)»/'/' ( )};,./ew is complete o r t h o n o r m e d system i n L (R x [ R ) (see, e.g., [1.46]). We compute the initial density m a t r i x p from the initial d a t u m w by using (1.4.55) and expand p, i n t o the Fourier series (1.4.57). We obtain the Fourier coefficients p^: )
2
iM
s
a
3 M
l
7
)
)
p,(r, s ) ^ ' ( r ) ^ ' ( s ) dr ds.
Pij
(1.4.59)
The next step is to solve the Schrodinger equations 1
h -—Ail/2m
ihd i//= t
qV(x,t)\I/,
x e R
3 M
,
t>0
(1.4.60)
3M
^(x, t = 0) = 4if{x),
x e U
(1.4.61)
m
for \j/ = \f/ {x, f) and / e M. Then the solution of the initial value p r o b l e m for the q u a n t u m L i o u v i l l e equation is obtained by employing the coordinate transformation (1.4.29) and by Fourier transformation: v
f
> ) = T^-TW Z (2ny i,ft M
Pu
N
x
J 3« R
"A"' \ + ^~n, V 2m
U)
t) iA / \
(x-^-n,t 2m
iv
xe "dt].
(1.4.62) 2
3 M
M
This solution vv exists for all f ^ 0 (as convergent series i n L ( R x Uf )) if the Schrodinger equation (1.4.60) has a solution globally i n time for all initial data in L ( I R ) . Conditions on the potential V, w h i c h guarantee the global existence of L wave-functions for L initial data can be found i n [1.46], [1.29]. It is easy to show that the solution w remains real valued for all t > 0, i f it is real valued initially (which we shall assume henceforth). F o r such initial data a more convenient solution representation can be obtained. By a simple functional analytic argument (see [1.38]) we conclude the existence of a complete o r t h o n o r m a l system of L (U ) functions such that 2
3 M
2
2
2
k
p,(r,s)=
3M
k
X lJ' \rW \s) fee \
(1.4.63)
holds w i t h ik)
ik)
p,(r, s) (r)(f> (s) dr ds. J3M
v
(1.4.64)
D3M
By proceeding as above we obtain the diagonal representation of the density matrix p(r,s,t)=
X WKr, keN
w
t)<j> (s, t)
and of the solution of the q u a n t u m Liouville equation
(1.4.65)
47
1.4 The Quantum Liouville Equation
x d> (x-^-n,t)e "dn, \ 2m J iv
m
(1.4.66)
(k)
where = (x, t) denotes the solution of the Schrddinger equation (1.4.60) w i t h i n i t i a l d a t u m tj> = {x). We conclude from (1.4.66) that the general L - s o l u t i o n o f the i n i t i a l value problem for the q u a n t u m L i o u v i l l e equation can be w r i t t e n as an infinite sum of Wigner-functions. Thus, the q u a n t u m L i o u v i l l e equation is capable of describing a r b i t r a r y mixed quantum states, w h i c h cannot be represented by a single wave function. It is an easy exercise to show that initially o r t h o n o r m a l wave functions remain o r t h o n o r m a l for all times. Since the functions <j> {x) are o r t h o n o r m a l i n L ( f J ) , the wave functions (x, t) are o r t h o n o r m a l for t > 0 and we o b t a i n from Parseval's inequality {k)
(k)
2
ik)
2
3 M
{k)
IIP(">
'J
f
)llL2(R3«xR3M) =
Since for every function g e Wg\\L2 3«) (R
=
(2TT)
HPiIIl2(R3m*R3M), 2
M
IIg||
L 2 ( R
t > 0.
(1.4.67)
L (Uf, ) 3 M
'
2
3
(1.4.68)
M )
holds (see [1.46]), we conclude from (1.4.66), (1.4.67): W
f
l l ( ' > ') )lli.2(Rj"xR3M) = IIWiH/^RaMxRBM),
f > 0.
(1.4.69)
2
The L - n o r m of the solution of the q u a n t u m L i o u v i l l e equation is timeinvariant. A n analysis o f the steady states o f the q u a n t u m L i o u v i l l e equation, also based on the representation (T.4.62), can be found i n [1.18]. We n o w formally integrate (1.4.66) term by term over [ R , use 3Af
g(v) dv = (^g)(r
= 0)
1
and obtain 2
» a „ ( * , t) = [ w(x, v, t)dv= X t)\ . J R3« k6 M A n o t h e r formal term by term integration, n o w over U ,
(1.4.70)
qU
M
X
n (x, quan
t)dx= (k)
X k
=
gives
% {x)dx.
(1.4.71)
uanJ
2
Here we used | 3m\(f> (x, t)\ dx = 1. (1.4.71) establishes the conservation o f the integral o f tlie q u a n t u m ensemble position density for mixed states. We remark that the m a i n ingredient for the existence of n and for the mathematical justification of the performed term by term integrations is the H
q u a n
1 Kinetic Transport Models for Semiconductors
48
assumption i ^ 0 , V ^ e N , which by (1.4.70) guarantees the nonnegativity of the p o s i t i o n density « (see also [1.38]). W e shall come back to this point later on. If the potential V is continuous i n x and i f w decays sufficiently fast as | x | -» oo, \v\ -> oo, we o b t a i n from (1.4.42): f O iV~\w dv = (8V) (^w)(x, n = 0, t) = 0. (1.4.72) t
q u a n
h
h
JR3M
Thus, integrating the q u a n t u m L i o u v i l l e equation w i t h respect to v gives the conservation law Qd n t
quin
- div J
q u a n
= 0
(1.4.73)
for the q u a n t u m current density. We refer the mathematically oriented reader to the reference [1.38] for a rigorous presentation o f the results o f this paragraph based on a functional analytic framework for the Schrodinger and the q u a n t u m L i o u v i l l e equations. A different approach for bounded potentials can be found i n [1.39].
The Classical
Limit
We consider a quadratic potential o f the form T
V(x, t) = {x A(t)x
+ b(t)-x
+ c(t),
(1.4.74)
where A(t) is a realvalued symmetric 3 M x 3 M - m a t r i x , b(t) a real 3 M - v e c t o r and c(t) e R. The superscript ' T " denotes transposition. E v a l u a t i o n o f the symbol (8V) defined i n (1.4.42) gives h
(5V) (x, h
n, t) = i{A(t)x
+ b(t))-rj
(1.4.75)
and the q u a n t u m L i o u v i l l e equation becomes B.w + i r g r a d w + —(A(t)x m x
+ b(t))-grad,,
w = 0.
(1.4.76)
Thus, i n the case of a quadratic potential (linear field) the q u a n t u m L i o u v i l l e equation and the classical L i o u v i l l e equation are identical. Clearly, this does not h o l d true for more general potentials. However, since (1.4.44) holds for sufficiently smooth potentials, we are led to the conjecture that limits as h -> 0 of solutions o f the q u a n t u m L i o u v i l l e equation are solutions of the classical L i o u v i l l e equation, i f the potential V is sufficiently smooth. Results o f this type were proven i n [ 1 . 3 8 ] , [1.39] by employing functional analytic methods. Here we shall present a more basic technique based on asymptotic expansions. We shall proceed somewhat formally, but a justification of the expansion m e t h o d is possible even w i t h o u t much mathematical sophistication. F o r the sake o f simplicity we consider the one-dimensional m o t i o n o f an electron i n a static potential V = V(x), which is assumed to be infinitely
1.4 T h e Q u a n t u m L i o u v i l l e E q u a t i o n
49
differentiable. Then, by formal power series expansion, we o b t a i n 00
2 k + l
n
2k+l
d V(x)
where we set \JL = h/m. W e are therefore motivated to expand the solution w = w i n powers of p , t o o . We make the ansatz h
2
u\x,t\,t)~
2k
t
u (x, t)n 2k
(1.4.78)
%
h
h
for the Fourier transform u = •!^w . The coefficients u are as yet u n k n o w n . I n order to keep matters as simple as possible we assume that the i n i t i a l d a t u m Wj and, consequently, u = are independent of h. We insert the expansions (1.4.77), (1.4.78) i n t o the Fourier transformed q u a n t u m L i o u v i l l e equation (1.4.38) and o b t a i n by equating coefficients o f equal powers o f JX : 2k
T
2
.
. du
.qdV
0
d t U o
+
l
+
dx~fr,
l
{x)r,U
mdx~ °
ux vr/ u (x, 0
= °'
m ux
^ ^
n, t = 0) = Uj{x, rj)
for k = 0 and d,u + i^-^+ --j-( ) l 2k Su dx drj m dx q n ^ d '^V(x) l
2k
k
=
u (x, 2k
2
r u
2
mS4'(2/+l)!
U 2 k
dx*'"
2
(
' "
1
A
8
0
)
q,t = 0) = 0 1
for k > 0. W e set w of the expansion
= ^~ u
2k
w\x,
x
2k
2k
00 X ik(x, w
v,t)~
and o b t a i n equations for the coefficients w
2k
2k
v, t)n
(1.4.81)
by inverse F o u r i e r transformation o f (1.4.79), (1.4.80). The leading term w satisfies the classical L i o u v i l l e equation dw t
0
^qdVix)^ + —d w m ax
+ vd w x
0
v
0
0
= 0, (1.4.82)
w (x, v, t = 0) = w,(x, v) 0
and the higher order coefficients solve inhomogeneous versions of the classical equation: dw t
2k
+ vd w x
2k
+
q dV(x) „ —d w m dx v
2 k
=
R , 2k
(1.4.83) w (x, 2k
v,t = 0) = 0,
50
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
where the right-hand side R depends o n ^-derivatives of w , . . . , w and o n x-derivatives o f V. The presented expansion procedure is only formal. I t was shown i n [1.53] that approximations of arbitrary high order, say 0(p. ) for reN, are obtained by cutting the expansion (1.4.81) at the index k = r — 1, i f the potential and the initial d a t u m are sufficiently smooth. This a p p r o x i m a t i o n result can easily be extended to more dimensions. F o r weaker convergence results (under less stringent regularity assumptions) we refer to [1.38], [1.39]. So far, the theory for the classical l i m i t o f the q u a n t u m L i o u v i l l e equation in the case of n o n s m o o t h potentials is n o t well developed. However, an asymptotic analysis for a highly irregular potential, namely the onedimensional barrier V(x) = —(m/q)S(x) discussed i n the Paragraph on tunneling, was presented i n [1.53]. I t is shown that the solutions o f the corresponding q u a n t u m L i o u v i l l e equation tend to the classical l i m i t (total reflection, see Section 1.2). The q u a n t u m corrections, e.g. the tunneling current, are o f order h . 2k
0
2k
2
2r
1
T o get a feeling for the 'actual size' o f h, the q u a n t u m L i o u v i l l e equation has to be scaled appropriately (see P r o b l e m 1.29 and Section 1.6). Then it becomes apparent that the 'scaled Planck constant' is indirectly p r o p o r t i o n a l to the square of the characteristic device length. Q u a n t u m effects become more pronounced as the active device length decreases.
Nonnegativity
of Wigner
Functions
A t first we consider a pure q u a n t u m state w i t h wave function L (U ) and Wigner function w = given by 2
i//(-,t)e
M
X
w ^ x , v, t) f
1 (2n)
h \\i x + — n, f i M x - — n, t )e' "dn. 2m 2m v
3M
(1.4.84)
A w e l l - k n o w n result (see [1.27]) asserts that w^, is nonnegative everywhere if and only i f either \jt = 0 or i f \p is the exponential of a quadratic i n x, i.e. i^(x, t) = exp
r
~ x A ( r ) x - a(t)-x
+ P(t)
x e
)3M
t > 0, (1.4.85)
where A(f) is a complex 3 M x 3 M - m a t r i x w i t h a positive definite symmetric real part, a(f) is an arbitrary complex 3 M - v e c t o r and /?(t) e C. Since the p r o o f of this result is instructive we shall present it here for the one-dimensional case. Theorem 1.4.1: Let wfy(x, v)
1 2n
+
h
2 l) l \ tn
l /
x
2m x e
n
i
v
)e " dn, v e
(1.4.86)
1.4 T h e Q u a n t u m
Liouville Equation
51
2
be the Wigner function
of the state ij/ e L (U).
w+(x, v) ^ 0,
x e U,
Then
ve U
x
(1.4.87)
r
holds if and only if either there are complex coefficients such that \\i is given by (a)
i]/(x) = exp
(b)
*j, = 0.
X, a, y with Re X > 0
X ocx + y J,
(1.4.88)
xe
or (1.4.88)
Proof: A t first note that \p = 0 i f and only i f = 0. N o w let \\i = ij/ given by (1.4.88) (a). Using the w e l l - k n o w n formula
ilv
~U/2)x
+ zx ^
2
7
\2n
_
x
•
y
z 2 / 2
\
2n
R e i > 0 , Re
for z e C, we can easily compute the Wigner function (1.4.86). I t is o f the form WA,«. (X,
=:
(1.4.89)
3
„ from (1.4.90)
>o,
f)
7
>0
be
where p is a real p o l y n o m i a l of degree t w o i n x a n d v. T o establish the necessity o f (1.4.88) we assume ^ 0, ip argument shows that 2
0. A simple
*
w ,(x, f ) w l/
-.v
T:
l 2 ; 0
( x , u) dx dv
v
1
27
(1.4.91)
'AW'Ai.z.oW dx
holds for z e C. Since ^ 0, ^ 0 a n d since w is of the form (1.4.90), the left hand side of (1.4.91) is positive. Thus, the right hand side is nonzero for z e C and, consequently, the entire function l
F(z)
-(x*l2)-zx
[j/(x)e
z
0
(1.4.92)
dx
has n o zeros i n C. W e estimate (1.4.92) using (1.4.89): i ^ w i
2
^
! i i / ^
2
T
O
y ^
(
R
e
z
)
(1.4.93)
2
and conclude from Hadamard's Theorem (see, e.g., [1.50]) that F is o f the form F(z) =
a z 2 + b z + c e
.
(1.4.94) a y 2
i h y + c
We set z = - i v and o b t a i n from (1.4.92) that F(iy) = ~ is the Fourier transform o f \]j(x)e~ . Since the only class o f functions whose Fourier transforms are exponentials o f quadratic polynomials are o f that type themselves (see, e.g. [1.46]), we conclude (1.4.88) (a). • e
x212
52
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
The class o f potentials V, which generate wave functions o f the type (1.4.85) can easily be determined by inserting (1.4.85) i n t o the Schrddinger equation. A simple c o m p u t a t i o n shows that V(x, t) is quadratic in x, i.e. it is o f the form (1.4.74). As we already k n o w , the q u a n t u m L i o u v i l l e equation reduces to the classical L i o u v i l l e equation for such potentials and the preservat i o n o f the nonnegativity for a r b i t r a r y initial data is immediate. The more interesting—and quite s t r i k i n g — p a r t o f the result, however, is the necessity of (1.4.88) and, consequently, of the class of quadratic potentials for the nonnegativity o f the Wigner function o f a pure state. The situation for mixed q u a n t u m states, i.e. for arbitrary initial data Wj e L (U x Rf, ) for the q u a n t u m L i o u v i l l e equation is more c o m p l i cated and a necessary c o n d i t i o n for the nonnegativity of the solution w has not been obtained yet. 2
3M
M
A sufficient c o n d i t i o n for the nonnegativity o f the electron position density « , however, can easily be obtained from the representations (1.4.66) and (1.4.70). A t first we remark that the coefficients k are the eigenvalues o f the operator R,: L ( I R ) - L (U ) defined by q u a n
k
2
3 M
2
3M
(r,
(R,f)(r):=
s)f(s) ds
Pl
(1.4.95)
(see [1.38]). The function p , which is the i n i t i a l density m a t r i x o f the mixed state, is called positive semi-definite, i f the integral operator R, is positive semi-definite, i.e. i f X ^ 0, V/c e N . By (1.4.70) the positive semi-definiteness of p, is sufficient to guarantee the nonnegativity o f the electron density n for x e I R , t ^ 0. For potentials more general than the quadratic (1.4.74) we cannot expect an initially nonnegative solution of the q u a n t u m L i o u v i l l e equation to remain nonnegative for all time, t ^ 0. I n spite o f the nonnegativity of the electron position density for positive semi-definite initial density matrices a fully probabilistic interpretation o f the q u a n t u m L i o u v i l l e equation is therefore not possible. This fact was quite a deterrent for the practical use of Wigner functions. O n l y recently they were employed for semiconductor device simulations o u t o f sheer need o f a q u a n t u m transport model, w h i c h is simple enough to allow a reasonably efficient numerical solution. The results obtained are very convincing (see [1.49], [1.32]) and further research on the q u a n t u m L i o u v i l l e equation is ongoing. t
k
3M
q u a n
An Energy-Band
Version
of the Quantum
Liouville
Equation
So far the presented q u a n t u m transport model does not account for the effect of the crystal lattice o n the m o t i o n o f the electrons. F o r semiconductor simulation it is desirable to employ a transport model, which is capable of describing q u a n t u m effects like tunneling and which also contains a description of the crystal lattice structure like the semi-classical L i o u v i l l e equation of Section 1.2. The model presented below was introduced in [ 1 . 1 ] .
53
1.4 T h e Q u a n t u m L i o u v i l l e E q u a t i o n
We consider a single electron in the (fixed) energy band e = e(k), defined for k i n the first B r i l l o u i n zone B o f the semiconductor (and extended periodically to Uf). T h e n the semi-classical H a m i l t o n i a n is given by H(x,
k, t) = e(k) - q V(x, t).
(1.4.96)
N o t e that the position vector x and the crystal m o m e n t u m vector p = hk are canonically conjugate. By the correspondence principle o f q u a n t u m mechanics (see, e.g., [1.40]) the q u a n t u m mechanical H a m i l t o n i a n operator i n wave vector formulation is obtained by substituting the position operator i g r a d for the posit i o n variable x i n the H a m i l t o n i a n function (1.4.96). The corresponding Schrddinger equation (in wave vector formulation) then reads formally: k
ih d,ij/ = [e(k) - qV(i g r a d ) ] &
$ = $(k, t).
k
(1.4.97)
The pseudo-differential operator V(i g r a d ) w i l l be defined below. We assume that the wave function \j/ is periodic i n k w i t h the periodicity of the reciprocal crystal lattice L . Clearly, the reason for this assumption is the periodicity o f the energy band e = s(k). Therefore, we can expand ip i n t o a Fourier series and, hence, its Fourier transform is a discretely defined function on the direct lattice L : k
ikx
t£(M)=
E Hx, t)e~ , XGL 1 \fi{k, t)e dk,
keB
,kx
(1.4.98)
xeL.
(1.4.99)
Here, \B\ denotes the volume o f the B r i l l o u i n zone. We now define the potential operator V(i grad^) by (V(i gmd )f)(k) k
=
~
ixik k)
X
V(x)f(k')e -
dk'
(1.4.100)
B x€ L
for f = f(k), keB. The Schrddinger equation i n space representation is obtained by Fourier transforming (1.4.97) using (1.4.98), (1.4.99): ihdtf
= [t:(-i
gradj
- qV(x, f)]iA,
i//= \j/(x, t),
(1.4.101)
where we denoted (c(-igrad )/)(x)
1
x
X
ik(x x)
e(k)f{x')e -
dk
(1.4.102)
B xe L
for / = f(x), x G L . I n order to define the corresponding Wigner function we set up the density matrix p(r, s, t) = i//(r, t)ij/(s, t),
(r, s) e L x L ,
which satisfies the Heisenberg equation
(1.4.103)
54
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
ih d p = [ / : ( - / grad J -e{-i
grad )
t
r
- qV{s, t) + qV(r,
(r,s)sLxL.
(1.4.104)
Similarly to the vacuum case we introduce the coordinate transformation x +
(1.4.105)
X —
and define the function u(x, v, t) := p\ x + - , x
; 1.4.106)
; t
for x e\L and for all v w h i c h can be represented as a difference of t w o points in L . Since the grid, o n w h i c h u lives, is not rectangular in the (x, v)-space, it is convenient to introduce additional gridpoints such that (x, v) e\L x L and to set u = 0 on these (artificially introduced) points initially. We remark that this only simplifies the n o t a t i o n . I t has no effect on the solution since u remains zero on the additional gridpoints for all times i f it is zero there initially. The Wigner function is then defined as the Fourier transform of u w i t h respect to v: 1 (1.4.107) k e B. X u(x,v)e~ \ xe-L, w (x,k,t)= ik
It satisfies the following pseudo-differential equation 8,w+-
1
1
2i
2i 1
v(x
2i
grad,
g r a d ) + qV\x x
x e - L ,
w = 0
+ — grad
keB,
k
(1.4.108)
which, i n explicit n o t a t i o n , reads: 1
ih
2\B\
elk 2B x'
+
el k
eL/2
V[
X
-V[x
\D\ V6l x e
iv(k-k')
w(x, k') dk'
0. (1.4.109)
The solution vv of (1.4.109) subject to an i n i t i a l c o n d i t i o n w(t = 0) = vv is the quasi-distribution of the electron i n the phase space \L x B. As usual, the electron position density is given by 7
n(x,t)=
w(x,k,t)dk,
xe~L
(1.4.110)
1.4 The Quantum Liouville Equation
55
and the electron current density J(x,
t)
v(k)w(x,k,t)dk. (1.4.111) B The band-diagram q u a n t u m Liouville equation (1.4.108) has t w o properties, which are different from the vacuum q u a n t u m L i o u v i l l e equation (1.4.50). Firstly, the admissible values o f the position vector x are discrete (more precisely, they are restricted to \ times the crystal lattice L) and the /c-vector is restricted to the first B r i l l o u i n zone B o f the crystal. Secondly, the equation (1.4.108) is nonlocal i n the position vector x as well as i n the wave vector k. The nonlocality i n x is introduced by the band-diagram e = e(/c). We remark that the d e r i v a t i o n o f the band diagram q u a n t u m L i o u v i l l e equation is based on the H a m i l t o n i a n operator H
B
= e t - i g r a d j - qV{x, t),
obtained formally from the semi-classical H a m i l t o n i a n function s(k) — qV(x, r) by employing the correspondence principle. I t can be shown that H is an a p p r o x i m a t i o n for the 'full' q u a n t u m H a m i l t o n i a n H — qV, where H , given by (1.2.35), represents the q u a n t u m effects of the periodic lattice potential. The a p p r o x i m a t i o n quality deteriorates i f the m o t i o n of the electron is not confined to the given band e ([1.56]). Obviously, it does not make sense to carry out the classical l i m i t h -> 0 i n (1.4.108). I n order to analyze the relationship o f the full q u a n t u m band diagram transport model (1.4.108) to the semi-classical single electron L i o u v i l l e equation B
L
L
df t
+ ^grad
s(k) • g r a d / + | g r a d
k
x
x
V-grad f
= 0,
k
3
xeU ,
keB,
(1.4.112)
which is subject to periodic b o u n d a r y conditions o n B B , b o t h equations have to be appropriately scaled first. F o l l o w i n g [ 1 . 1 ] we introduce the scaling x x = — , o s
t t = — , ^0
k = Ik, s
s
x
e (K) s
= -e(k),
V (x ) = s
s
1 L = —L, s
B=
IB (1.4.113)
^-V(x)
where we denoted scaled quantities by the index 's\ x is the length of the active region o f the considered semiconductor device and / is a characteristic lattice spacing o f L (for the sake of simplicity we assume that the g r i d - p o i n t distances o f L i n all three directions are of equal order o f magnitude, otherwise we w o u l d have to scale the three components o f k separately). N o t e that the volume o f the B r i l l o u i n zone is o f the order o f magnitude l / / . The scaling factors t , V , e stand for a characteristic observation time, characteristic potential value and m a x i m a l value o f e o n B resp. 0
3
0
0
0
56
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
W i t h this scaling the equation (1.4.108) reads (after d r o p p i n g the index V ) : ae[k
+ — grad,
ae
2i
| x + — grad,
V\x
(1.4.114)
grad,
grad.
2i
w
where the dimensionless constants are given by /
£r,tj £t l
qV t l
0 Q
a. =
o 0
(1.4.115)
fo%Q
WXQ
F o r comparison, the scaled semi-classical L i o u v i l l e equation (1.4.112) is of the form f + a g r a d , s(k) • grad f + b g r a d V • g r a d / = 0. t
x
x
(1.4.116)
k
F o r a typical tunneling diode the following scaling factors may be chosen 8
x
0
= 10- m,
/ = 10^
ls
qV
0
1 0
m,
t =100
1 4
s,
(1.4.117)
1 8
= lO- J,
e = 10- J. 0
T h e n the constants a, a, b are a»0.01,
ax
I,
bxOA.
(1.4.118)
Usually, a is small while a and b are constants o f the order o f magnitude 1. Physically, a small value o f ac means that the characteristic device length is large compared to the crystal lattice spacing. Thus it makes sense to consider the l i m i t of the scaled equation (1.4.114) as a -> 0 while the constants a, b are kept fixed. We remember that the equation (1.4.114) is posed on the phase space xeL
= aL ,
keB
0
(1.4.119)
where the scaled B r i l l o u i n zone B has a volume w h i c h is of the order o f magnitude 1 and L is the crystal lattice scaled by 21. Thus, the lattice spacing o f L is o f the order of magnitude 1. I t is apparent n o w that three different limits have to be carried out simultaneously i n (1.4.114) in order to o b t a i n the scaled semiclassical transport equation (1.4.116): 0
0
(i) a -> 0 ' i n the lattice'. The lattice a L becomes finer as a -> 0 and we expect the discretely defined Wigner functions w to converge to a function defined o n U x B. (ii) a - » 0 ' i n the band operator' 0
x
ai
e ( k + — grad j - e x
grad,
•J.
which tends formally to the differential operator a g r a d e(/c)-grad . t
x
57
1.5 T h e Q u a n t u m B o l t z m a n n E q u a t i o n
(iii) a -> 0 i n the potential operator bi a
v\
x
+ Yt
g T a d k
V[ x
grad, 2i
which tends formally to the differential operator b grad
x
V(x) • grad
k
.
F o r sufficiently smooth energy bands and potentials we then expect the solutions w = w of (1.4.114) to converge to the solution o f (1.4.116). A rigorous mathematical treatment o f the one-dimensional case can be found in [1.54]. For numerical simulations it is desirable to derive an energy band q u a n t u m transport model, which is simpler than (1.4.114) but still capable o f modeling q u a n t u m effects like tunneling. The maybe most i n t r i g u i n g way to achieve this is to perform the limits (i) and (ii) but to leave the potential energy term unchanged. The transport equation obtained in this way then reads x
w, + a g r a d c(/c) • grad.,. w +
bi
k
V\ x
: grad, 2i
w = 0,
[x + — g r a d
x e
k
keB.
(1.4.120)
We remark that the equation (1.4.120) is—even for pure q u a n t u m states— not equivalent to the Schrddinger equation. For a mathematical analysis of (1.4.120) (coupled w i t h a self-consistent potential model, see Sections 1.3 and 1.5) we refer to [1.16], [1.17]. A many-body version o f the energy band q u a n t u m transport model can easily be derived by starting out from the many-body H a m i l t o n i a n (1.2.44). Since the derivation does not give new insights we do not present the details here. We conclude this Section w i t h the remark t h a t — s i m i l a r l y to the classical case—magnetic field effects (and spin effects) can also be taken i n t o account in setting up the q u a n t u m L i o u v i l l e equation. Since the models are highly complicated (particularly when the spin is included) and since they are not employed in practical semiconductor device simulations we only refer to [1.3] for the derivation and mathematical analysis of the electromagnetic q u a n t u m Liouville equation for electrons w i t h spin.
1.5 The Quantum Boltzmann Equation The application of the q u a n t u m L i o u v i l l e equation, which is the q u a n t u m analogue o f the classical L i o u v i l l e equation, to the modeling of many-body systems involves the same problems as i n the classical case. Firstly, realistic models for the many-body potential, which comprise l o n g range and short range interactions, are generally not available. Secondly, the dimension o f the phase space o n which the M - p a r t i c l e q u a n t u m L i o u v i l l e equation is
58
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
posed, equals 6 M , which is by far too large for numerical simulations in practically relevant applications. In this Section we shall derive single particle approximations o f the q u a n t u m Liouville equation, w h i c h contain a self-consistent potential equation to account for the many-body effects. Just as i n the classical case presented i n Section 1.3 we shall at first consider l o n g range forces only and derive the corresponding q u a n t u m Vlasov equation. I t has the form of a single particle q u a n t u m L i o u v i l l e equation supplemented by a Poisson equation for the effective potential, when the particle interaction is modeled by the C o u l o m b force. T h e n we shall discuss short range interactions (scattering events), which lead to the q u a n t u m B o l t z m a n n equation. For the derivation o f the q u a n t u m Vlasov equation we shall proceed similarly to the classical case. We shall set up the q u a n t u m analogue of the B B G K Y - h i e r a r c h y and use the Hartree a p p r o x i m a t i o n to o b t a i n an equat i o n for the one-body density m a t r i x whose F o u r i e r transform w i t h respect to the dual velocity variable is the q u a n t u m Vlasov equation. Short range interactions w i l l be incorporated as i n the classical case, namely by a collision integral operator, which appears o n the right-hand side o f the q u a n t u m B o l t z m a n n equation.
Subensemble
Density
Matrices
We consider an ensemble of M electrons of equal mass m, whose m o t i o n is governed by the Schrodinger equation (1.4.1) w i t h the M - b o d y H a m i l t o n i a n (1.4.23). The ensemble density m a t r i x p is defined by
= il/(r ,...,r ,t)ij/(s ,...,s ,t), 1
M
1
r , s, e R \
M
(1.5.1)
t
where i/r is the wave function of the ensemble. We recall that p satisfies the Heisenberg equation (1.4.27). F o r the following we shall assume that the electrons o f the ensemble are indistinguishable i n the sense that the density m a t r i x remains invariant under any p e r m u t a t i o n o f the r- and s-arguments, i.e. p{r ,...,
r , s , ..., s , t) = p{r (\),
i
M
1
M
n
s
• • •,
S
n ( i ) ' • • • > %(M)> ^) (1.5.2) 3
holds for any p e r m u t a t i o n %• of the set { 1 , . . . , M } and for all r s e U , t ^ 0. The c o n d i t i o n (1.5.2) is satisfied if either the wave function \\i is antisymmetric h
i//(x ...,x , u
M
t) = sgn(7c)^(x
M1)
, ...,x
n m
, t),
t
V T I , V X , , t^O (1.5.3)
or i f i t is symmetric il/(x ,...,x ,t) 1
M
= \l/(x ,...,x ,t), M1)
n(M)
V T I ,
V X „ t ^ 0. (1.5.4)
1.5 T h e Q u a n t u m B o l t z m a n n E q u a t i o n
59
The property (1.5.3) holds for ensembles o f Fermions and (1.5.4) for ensembles of Bosons (see [1.34]). The former represents the P a u l i principle of q u a n t u m mechanics mentioned i n Section 1.3, w h i c h prohibits the double occupancy of states, i.e. the wave functions o f Fermions satisfy il/(x ,...,
x,
1
t) = 0
M
x
if
x
i = j
f°
r
i^j-
(1.5.5)
Since the particles considered i n this b o o k (electrons and holes) are Fermions, we shall assume (1.5.3) and, consequently, (1.5.2) to h o l d henceforth. N o t e that, by the Schrddinger equation, the potential V has to satisfy %
V(x ,...,x ,t)=V(x ,...,x ,t), 1
M
n(l)
VTI,
nm
VX,-,
f^O (1.5.6)
for an ensemble of Fermions. I t is easy to show that the anti-symmetry o f \p and consequently (1.5.2) are conserved i n the e v o l u t i o n process i f (1.5.6) holds. T o model the m o t i o n of subensembles we shall n o w introduce subensemble density matrices. The density m a t r i x corresponding to a subensemble consisting o f d particles is obtained by evaluating the ensemble density m a t r i x p at r = s for i = d + 1 , . . . , M and by integrating w i t h respect to these coordinates: t
{
(i)
p {r ...,r ,s ,...,s ,t) u
d
i
d
'.=
p(ri,...,
r, u d
d + 1
, . . . , u , s ,..., M
s, u
1
d
d + 1
,..., u , M
t)
J R3IM-A
x du ,...,du . d+1
(1.5.7)
M
(d)
The trace o f p represents the q u a n t u m electron position density o f the J-particle subensemble: w
< u a „ ( * i , ...,x ,t)
= p (x
d
...,x ,x ,...,x ,t).
u
d
l
(1.5.8)
d
The subensemble q u a n t u m electron current density is given by x
"^quanC^l i • • • J d> 0 ihq = — (grad , - g r a d , 2m s(d
r
id)
d l
)p
( d ,
(..t)\
(1.5.9)
r < d ) = s t d ) = x ( d )
w
(d)
where we set r = ( r , r ) , s = (s ,..., s ) and x = ( x , x ) . Clearly, the indistinguishability p r o p e r t y (1.5.2) is inherited by the subensemble density matrices, i.e. 1
d
1
d
p* \ r , . . . , r , s . . . , s , i) = p (r (i), w
1
d
l 5
d
a
1
• • •,
f (d)' a
s
s
o(i)i
3
;
0 (1.5.10)
• • • •> o(d)>
holds for all permutations a o f { 1 , . . . , d) and a l l r s e IR , f > 0. h
d
60
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
The Quantum
Vlasov
Equation
As i n the classical case we assume n o w that the potential V is the sum o f an external potential and an internal potential stemming from two-particle interactions: M
V(x ...,x , h
where V
int
\
M
M
f ) = I K ( x „ t) + - t 2 V (x„ i=i i=i j=i
M
ext
iat
xj),
(1.5.11)
is symmetric
V (x ,x )=V {xj,x ), int
l
J
iat
l,j=l,...,N.
l
(1.5.12)
The factor \ i n (1.5.11) is necessary since each particle pair is counted twice in the sum representing the accumulated two-particle interactions. The Heisenberg equation of m o t i o n for the ensemble density m a t r i x p then reads: 2 N
- —
ih d,p=
M
M
£ (KP
2m i=i A
M
E i=i
A
~ r,P) ~ q
F
( ext(%
t) -
V (r„ ext
t))p
M
I W ^ S j ) - ^ , ^ .
(1.5.13)
i i=i j=i We remark that the ensemble is assumed to move i n a vacuum and that magnetic field effects are ignored at this point. We set u, = s = r, for / = d + 1, M i n the equation (1.5.13) and integrate over [ R x ••• x U . Assuming that p decays to zero sufficiently fast as |r,| -> oo, \S[\ -> oo, we o b t a i n by using the definition of the subensemble density m a t r i x p given by (1.5.7) and the indistinguishability property (1.5.10): t
3
3
d+i
M
(d)
i h d
t
P
w = ~ i ( A p ^ - A 2m 1=1 ' S
-
i
t (V (s ;=i ext
t) -
h
r
i
p W )
V (r exl
t))p
h
(d)
d
q(M-d)
X ;=i - V ^ u J l p W d u , ,
(1.5.14)
for 1 ^ d ^ M — 1, where we denoted mi)
p
=
p
w i) +
( r i >
...,r ,u*, ,..., d
Sl
s , u„, t). d
(1-5.15)
The system o f equations (1.5.14) constitutes the q u a n t u m equivalent of the B B G K Y - h i e r a r c h y presented i n Section 1.3. As i n the classical case, i t is not possible to solve the system exactly for finite M , therefore we shall again consider the l i m i t M -+ oo for small subensembles. Then, at least a particular solution can be obtained. Clearly, this l i m i t i n g procedure is reasonable since
1.5 T h e Q u a n t u m B o l t z m a n n E q u a t i o n
61
we are interested in a single particle type a p p r o x i m a t i o n (d = 1) of the q u a n t u m L i o u v i l l e equation for large electron ensembles. Analogously to the classical case, we assume that the two-body interaction potential V is of the order of magnitude 1/M as M -» cc w h i c h implies that the total potential generated by each particle int
M
V
u
J x x
,
M
t) = X VtJx,,
Xj) + V (x , ext
t),
t
l ^ l ^ M
j=i
remains finite as M -» oo. F o r a fixed subensemble size d we obtain by going to the l i m i t M -> oo i n (1.5.14);
-
KextV (r xt
h
0)p
( d )
J {V (s„ int
1=1
«*) -
^„,(r„ uJ ) M p < "
+ 1 )
• (1-5-16)
As i n the classical case we now assume that the particles i n the subensemble move independently from each other (which, again, is reasonable for small subensembles). This is reflected by the so-called Hartree ansatz (see [1.7]): J! R(r„s„t).
(d)
p (r ,...,r ,s ,...,s ,t)= l
d
1
d
(1.5.17)
i=l ( 1 )
We o b t a i n an equation for the one-particle density m a t r i x R := p setting d = 1 i n (1.5.16) and by e m p l o y i n g the ansatz (1.5.17) for d = 2: it, d,R = -^(A R 2m S
- A R) - q(V (s, r
eft
t) -
K (r, eff
r, seU , 3
by
t))R, r>0
(1.5.18)
w i t h the effective potential relation V (x, eU
t) = V Jx, e
t) +
MRix^x^tWJx^Jdx*.
(1.5.19)
It is now an easy exercise to show that a particular solution of (1.5.16) for arbitrary d is given by (1.5.17), i f R satisfies (1.5.18), (1.5.19). We m u l t i p l y (1.5.18) by the total number of particles M , introduce the coordinate transformation r = x + ^~rj, 2m and obtain
s = x - ^ - n 2m
(1.5.20)
62
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
2m
d U + i d i v (grad, U) + iq
(7 = 0,
t
t>0,
(1.5.21)
where we set fj(x, n, t) = MR{r, s, t).
(1.5.22)
Inverse Fourier transformation o f (1.5.21) w i t h respect to n gives 5 , p y + i ; - g r a d , W + - 0 f t [ V ; ] W = O, f f
xeKj,
ceRj,
m
l
where the velocity v is the dual variable oft] and W := ^^ JJ. differential operator 0 is defined i n (1.4.41). We have
f > 0, (1.5.23)
The pseudo-
h
MR(x,
x, t) = U(x, r\ =
0,
t) =
W(x, v, t) dv
and, thus, the effective potential equation (1.5.19) can be rewritten as V {x, e[f
t) = V (x, ext
t)
«(**. *)ViJ.x,
x) #
dx ,
xeRl,
t
t>0,
(1.5.24)
where n(x, t)
W(x, v, t) dv,
t > 0
x e
(1.5.25)
denotes the q u a n t u m electron number density. The macroscopic q u a n t u m current density is given by J(x,
t) = —q
vW(x, v, t) dv,
x e
t > 0.
(1.5.26)
The equation (1.5.23) supplemented by the effective potential relation (1.5.24) is called q u a n t u m (or nuclear) Vlasov equation (see [1.41]). Analogously to the classical case it has the form o f a single-particle q u a n t u m L i o u v i l l e equation. M a n y - b o d y effects o n l y come i n by the equation (1.5.24) for the effective potential, i n which the electron number density n enters. The q u a n t u m Vlasov equation is a nonlinear pseudo-differential equation. The symbol
X
(<5F ),,(x, n, t) = im eff
+
2 ^ '
f
(1.5.27)
63
1.5 T h e Q u a n t u m B o l t z m a n n E q u a t i o n
depends on n by (1.5.24) and, thus, on the solution W itself. The nonlinearity 0/i [^eff] W i °f quadratic nonlocal nature. Since (1.5.23) is a single-particle q u a n t u m L i o u v i l l e equation, the analysis of Section 1.4 applies to the linear p r o b l e m w i t h V given. I n particular, contrary to the classical Vlasov equation, the q u a n t u m Vlasov equation does not preserve the non-negativity o f the solution W. The number density n, however, remains non-negative for all times, i f the initial single-particle density m a t r i x R(t = 0) is positive semi-definite (see Section 1.4). Also, the q u a n t u m Vlasov equation formally converges to the classical Vlasov equation with £ = — g r a d V as h -> 0. The q u a n t u m Vlasov equation models the q u a n t u m mechanical m o t i o n o f a large particle ensemble m o v i n g i n a vacuum under the influence of an exterior potential field t a k i n g i n t o account weak, l o n g range interactions of particles. Thus, as the classical Vlasov equation, it is a transport model useful on a time-scale much shorter than the mean time between t w o consecutive scattering events. C o n t r a r y to the classical Vlasov equation, the q u a n t u m Vlasov equation is capable of modeling the tunneling effect, w h i c h makes it particularly attractive as a t o o l for the numerical simulation of ultraintegrated semiconductor devices. s
e[{
e f f
x
The Poisson
e[{
Equation
T o account for the C o u l o m b interaction we set V {x,y)
3
= - T - r ^ l ' 4?i8 | X - y\
lBt
x,yeU ,
x*y.
(1.5.28)
s
Again, e denotes the p e r m i t t i v i t y of the semiconductor. Obviously s
grad, V
iM
= -£
,
i n t
x ^ y
(1.5.29)
holds, where £ is the C o u l o m b interaction field (1.3.19). V is up to the factor q/e the normalized fundamental solution o f the Laplace equation. The self-consistent potential equation (1.5.24) then reads i n t
int
s
K (x, tt
t) = V (x,
dx*
t)
exl
'
47C6 J 3 S
R
| X-
(1.5.30)
X j
and the Poisson equation -e AF s
e f f
3
= -s AV -qn, s
xeM ,
ext
x
f>0
(1.5.31)
holds. I f the external potential field K is generated by ions o f charge + q present i n the material, then (1.3.27) follows, where C = C(x, t) denotes the number density of the background ions. I n this case we o b t a i n (1.3.29) and, consequently, (1.3.30), (1.3.31). We remark that different effective potential equations, w h i c h also include local values o f the number density n, exist i n the literature, too. They can be seen as attempts to describe m e d i u m range particle interactions. Since these ext
64
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
models are not used i n semiconductor simulations, we shall not discuss them here in detail. The interested reader is referred to [1.7], [1.41]. We supplement the Vlasov equation (1.5.23), (1.5.24), (1.5.25) by the initial condition W(x, , t = 0)=
3
W,(x, v),
v
3
xeU ,
ve U .
(1.5.32)
The main difficulty i n the mathematical analysis o f the q u a n t u m Vlasov equation lies i n the fact t h a t — c o n t r a r y to the classical case—an L ^ t h e o r y for the linear q u a n t u m Liouville equation, which w o u l d guarantee the existence of the number density n, does not exist yet for a sufficiently broad class of potentials. Clearly, the L - t h e o r y is not sufficient since w ( - , - , f ) e L (U x U ) does not i m p l y that n is well-defined. Recently, the existence and uniqueness o f a solution, globally defined i n f, was proven i n the one- and three-dimensional cases by reformulating the q u a n t u m Vlasov equation as system of countably many Schrodinger equations coupled to the self-consistent N e w t o n i a n potential relation (see [1.55] for the onedimensional and [1.11] for the three-dimensional case). This reformulation is based on the equivalence of the linear q u a n t u m Liouville equation to a system o f countably many Schrodinger equations presented i n Section 1.4. The q u a n t u m Vlasov equation (1.5.23), (1.5.24), (1.5.25), (1.5.32) can be w r i t t e n as 2
2
3
X
2
h = -^A 2m
k)
w
ih d,f
(k)
- qV , (k
v
xeU
_ J L f
^
e x t
4jre J I
x
t>0
3
t = 0) = \x), v
x e U,
eff
R J
x
d
x
Jx-xJ
where n is given by (1.4.70): n(x,t)=
k
£
2
X \f \x,t)\ . k
k= l
I n the one-dimensional case the Green's function — l/(47r|x — x j ) has to be substituted by — |x — x j . N o t e that the scalars X and the i n i t i a l data (x) only depend o n the initial Wigner function Wj{x, v) (see Section 1.4 for details). This Schrodinger-Poisson system can be analyzed under reasonable assumptions on X and (x) i n a more straightforward way than the pseudodifferential equation form of the q u a n t u m Vlasov-Poisson system. A n analysis of the one-dimensional steady state Schrodinger-Poisson p r o b lem, posed o n a bounded interval, was presented i n [1.19]. A n existence and uniqueness result for the one-dimensional q u a n t u m Vlasov-Poisson p r o b l e m w i t h periodic boundary conditions i n x can be found i n [1.2]. The spectral properties o f the linearized equation are also discussed i n that paper and the convergence o f the solution to the solution of the corresponding classical p r o b l e m as h -> 0 was proven, too. {k)
k
(k)
k
65
1.5 T h e Q u a n t u m B o l t z m a n n E q u a t i o n
By integrating the q u a n t u m Vlasov equation w i t h respect to the velocity v and by proceeding as for the q u a n t u m L i o u v i l l e equation in Section 1.4 we obtain the conservation law: qd,n
div
(1.5.33)
= 0.
J
Also, if W is sufficiently regular, then the conservation of the total number of particles follows by integrating (1.5.33) w i t h respect to x: n(x,
f)
n ( x ) dx,
dx
7
(1.5.34)
t > 0,
where we set n,(x) = j " 3 W,(x, v) dv. H
The Quantum
Vlasov
Equation
on a Bounded
Position
Domain
As its classical analogue, the q u a n t u m Vlasov equation can also be posed on a bounded position d o m a i n which, i n semiconductor device modeling, represents the device geometry. G i v e n the bounded convex d o m a i n Q c ul, the inflow boundary c o n d i t i o n W(x, v, t) = W (x, v, t), D
(x,t)eL,
t > 0,
(1.5.35)
where the inflow segment T_ is defined i n (1.3.35), can be imposed. Then the Poisson equation (1.5.31) is also posed on Q and supplemented by Dirichlet or mixed D i r i c h l e t - N e u m a n n boundary conditions for K o n 3D. However, due to the nonlocal character of the pseudo-differential operator fyiC^eff] potential V still has to be defined o n the whole position space R . Thus, the solution o f the Poisson equation has to be extended from Q to U i n order to be used as an i n p u t for the Vlasov equation. The p r o b l e m of determining physically reasonable extensions has not been solved satisfactorily yet. I n one-dimensional simulations a piecewise constant continuous extension is n o r m a l l y used. eff
t r i e
e{{
x
3
A disadvantage of the inflow boundary conditions is that they do not exclude the reflection of i n c o m i n g waves. A b s o r b i n g boundary conditions, which provide a better model for O h m i c contacts, were derived in [1.24] by means of the theory of pseudo-differential operators. Analytical results for the linear q u a n t u m transport equation (with given bounded potential) subject to inflow boundary conditions can be found in [1.39].
The Energy-Band
Version
of the Quantum
Vlasov
Equation
The q u a n t u m Vlasov equation presented above does not take i n t o account the impact o f the semiconductor crystal lattice on the m o t i o n o f the particles. I n order to do this the (multi-particle version o f the) energy-band q u a n t u m L i o u v i l l e equation (1.4.108) has to be taken as starting point for the q u a n t u m
i
66
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
B B G K Y-hierarchy. Since the involved calculations are along the lines of the vacuum problem presented above, we d o not give them here, but merely state the result. The q u a n t u m Vlasov equation o n the B r i l l o u i n zone B is just the single particle energy band q u a n t u m Liouville equation (1.4.108) 8 W+ t
£
[
k
+
1 ^ g
1 f a d
*)
_ fi
I
k
~ Yi
g r a d
* j
W = 0,
+
q
V e ( 1
xe]-L,
[
x
+
g
2i keB,
2
r
a
d
f>0, (1.5.36)
where L denotes the crystal lattice and e = s(k) the considered energy band of the semiconductor. N o t e that the (quasi) d i s t r i b u t i o n W = W(x, k, t) is defined for x e \L, k e B and t > 0. The equation for the effective potential is obtained by replacing the integration i n (1.5.24) by a sum over x e jL: Kft(x,
t) = V {x,
t) +
eu
X
n(**.0Knt(*>*«)> xe\L,
f > 0 .
(1.5.37)
Clearly, the number density n is the integral o f W over the B r i l l o u i n zone B n(x, t)
1 xe-L,
W(x, k, t) dk,
t >0
(1.5.38)
and the current density is given by v(k)W(x,
J(x, t) =
k, t) dk,
t >0
x e - L ,
(1.5.39)
w i t h the velocity 1 v(k) = - g r a d
k
e(k).
(1.5.40)
We n o w scale the equations (1.5.36), (1.5.37), (1.5.38) by using (1.4.113) and, additionaly W{x, k, t)
W
X t -,lk,-
1
n(x, t)
t
X
XQ
(1.5.41) Then, after d r o p p i n g the index '5', (1.5.36) reads dW + t
as[k
+ ~ grad,
+ bV [ e{[
x +
ae I k — — g r a d
X
^.grad J-bV k
x e aL , 0
x
e!(
~2i
k e B,
g r a d
"
t>0.
W= 0, (1.5.42)
67
1.5 T h e Q u a n t u m B o l t z m a n n E q u a t i o n
The constants a, b and a are given i n (1.4.115). W h e n the C o u l o m b interact i o n potential (1.5.28) is taken, then the scaled discrete effective potential relation (1.5.37) takes the form: V (x, eff
t) = F ( x , t)-c ext
a 3
X e 3tL
" ( * * , t)-
-,
0
X^X
0
xe<xL , 0
t > 0,
(1.5.43)
with
W h e n the typical numerical values (1.4.117) are taken, then c is of the order of magnitude 1. The scaled number density n is obtained by integrating W over the scaled B r i l l o u i n zone B. A n existence and uniqueness result for the i n i t i a l value problem (1.5.42), (1.5.43), (1.5.38) can be found in [1.16]. We remark that, due to the boundedness of the B r i l l o u i n zone 73, an L - t h e o r y for (1.5.42) is sufficient to guarantee the existence of n, since W(x, •, t) e L {B) implies that n(x, t) is well-defined. As discussed i n Section 1.4 the formal l i m i t of (1.5.42) as a -> 0 is the scaled semi-classical L i o u v i l l e equation (1.4.116) ( w i t h V substituted by F ) . O b viously, i n the limit a - » 0 the sum i n (1.5.43) has to be replaced by the integral and the discrete effective potential relation (1.5.43) becomes (the scaled version of) (1.5.30). A mathematical justification of this semi-classical l i m i t in the one-dimensional case can be found i n [1.54]. I n many applications the exterior potential F has locally large gradients or even jump-discontinuities. Then the tunneling effect becomes i m p o r t a n t and the semi-classical Vlasov equation does not give realistic results. Since, however, the energy band e is a smooth function of k it is even i n these cases reasonable to carry out the partial limits 'a -> 0 i n the g r i d ' and 'a -> 0 i n the pseudo-differential operator i n v o l v i n g e' and to leave the potential energy pseudo-differential operator unchanged. Then the model equation (1.4.120) (with V replaced by V ) supplemented by the 'continuous' effective potential equation (1.5.24) is obtained. A mathematical analysis of this model (with a justification of the partial l i m i t procedure) can be found i n [1.16], [1.17]. We believe that this q u a n t u m transport model is highly appropriate for the simulation of ballistic phenomena i n ultra-integrated semiconductor devices since it allows for a description of the band structure of the crystal and for the modeling of tunneling. Also, from the numerical point of view, i t is significantly simpler than the 'discrete-x' problem (1.5.36), (1.5.37), (1.5.38). 2
2
eff
ext
ef(
Collisions Just as its classical counterpart, the q u a n t u m Vlasov equation is time reversible (for static exterior fields), i.e. it does not contain a mechanism which
68
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
forces the ensemble to relax towards thermodynamical e q u i l i b r i u m i n the large time l i m i t t -> oo. I n order to achieve this relaxation property we have to include the effects o f short range interactions modeled by scattering events of particles. This, however, cannot be achieved by the purely phenomenological approach presented i n Section 1.3 for the (semi-) classical case, since the n o t i o n of characteristics does not make sense for the q u a n t u m Vlasov equation. Principally, t w o different approaches are used for the derivation of the q u a n t u m B o l t z m a n n equation. The first is based on Green's function techniques (see [1.28]) and the second o n the Wigner formalism combined w i t h a modification of the Hartree ansatz(see [1.12]). Since b o t h approaches are highly complicated, we shall not present them here, but merely state the result, which is intuitive when one is familiar w i t h the semi-classical Boltzm a n n equation. The q u a n t u m B o l t z m a n n equation has the form o f an inhomogeneous q u a n t u m Vlasov equation, where the inhomogeneity represents the quant u m collision integral q
d,w + v md g
x
w + -o [v „w m h
=
e
3
xelR ,
Q*(W), 3
veU ,
t>0.
(1.5.45)
The q u a n t u m collision operator Q is nonlocal i n the velocity direction and, except when particle-particle interactions are considered, quadratically nonlinear. The particle-particle scattering q u a n t u m collision operator is nonlinear o f fourth order i n W (see [1.28]). We remark that q u a n t u m scattering operators, which are nonlocal i n the time direction, can also be found i n the literature (see [1.36]). The equation (1.5.45) is supplemented by an effective potential relation o f the form (1.5.24) and by the initial c o n d i t i o n (1.5.32). As i n the classical case the collision operator satisfies the conservation property h
Q,,(W)dv
= 0.
(1.5.46)
Its precise form depends on the considered scattering processes, however, to our knowledge, simulation o f semiconductor devices w i t h physically realistic q u a n t u m scattering operators have not been performed due to the enormous numerical complexity involved. F o r a numerical study of t u n neling devices using the relaxation time a p p r o x i m a t i o n for the scattering operator we refer to [1.32].
1.6 Applications and Extensions I n this Section we shall discuss specific applications of kinetic transport equations to the modeling of semiconductors. I n the course o f this we shall
69
1.6 A p p l i c a t i o n s a n d Extensions
extend the transport models derived i n the previous Sections to cover the particular requirements o f semiconductor device physics. A t first we w i l l present a multi-valley semi-classical transport model, which is of particular importance for the simulation of GaAs (Gallium-Arsenide) devices. T h e n we proceed to discuss bipolar semi-classical models, which constitute the basis for the derivation of the h y d r o d y n a m i c and drift diffusion models in Chapter 2. Finally, we shall summarize the state-of-the-art of q u a n t u m modeling of ultra-integrated semiconductor devices.
Multi-Valley
Models
It is w e l l - k n o w n that the energy-wave vector function e = e(/c) has several m i n i m a for, e.g., the semiconductor GaAs. These m i n i m a , also termed energy-valleys, are separated by energy-shifts and, very often, the band diagram e(k) is approximated by a parabola i n the neighbourhood o f each valley. F o r GaAs three types of valleys have to be distinguished: the l o w energy T-valley and the higher energetic L - and Z-valleys w i t h energy shifts each of the order o f magnitude 0.4 eV (see [1.48] for precise data). F o r the sake o f simplicity we shall for the following neglect the X-valley and only consider a model, which comprises the T- and the L-valleys. This a p p r o x i m a t i o n is justified by the fact that the highest energy X-valley can only be occupied at very high electric field strengths. Also we remark that, due to the symmetry properties of the B r i l l o u i n zone, several L - (and X-) valleys exist. I n the sequel we shall treat them as equivalent. A parabolic band a p p r o x i m a t i o n for the energy-wave vector relation i n the T-valley reads 2
h Biik) = y ^ / m , + k /m 2
2
2
+ k /m ),
k = (k,, k ,
3
2
T
k) 3
where the o r i g i n o f the /c-space has been placed at the location o f the band m i n i m u m and a suitable r o t a t i o n has been performed. The parameters m , m , m are called effective masses. This is m o t i v a t e d by a comparison of the velocity 1
2
3
rad
£
f e
M k ) = \ S f c r ( ) = Hkjm^
T
k /m , 2
k /m )
2
3
3
w i t h the velocity-wave vector relation v = hk/m for electrons in a vacuum. F o r m a l l y , the parabolic band a p p r o x i m a t i o n can be obtained by a scaling of the wave vector w h i c h magnifies the vicinity o f the band m i n i m u m . Accordingly, the boundary of the B r i l l o u i n zone is moved towards infinity and, as an a p p r o x i m a t i o n , B is replaced by U . A p a r t from that, we shall use the c o m m o n , although not rigorously justified, assumption that the effective masses i n the different directions are equal: 3
70
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s m
=
i
m
2
m
=
=
3
m
r
W i t h analogous assumptions for the L-valleys we obtain sAk) =
f
^
,
s (k) = A + ^
,
L
(1.6.1)
where m is the effective mass o f an electron i n the L-valley and A is the energy difference between the bottoms o f the t w o valleys. I t is convenient to split the electron d i s t r i b u t i o n function F i n t o a part corresponding to the T-valley and a part corresponding to the L-valley since the electrons move w i t h i n each valley even under l o w field strengths, while the transfer from the lower valley i n t o a 'higher' one requires the presence of high electric fields. T a k i n g i n t o account the m u l t i p l i c i t y o f the L-valleys we set: L
F(x, k, t) = F (x, k, t) + N F (x, r
L
k, t),
L
(1.6.2)
where N denotes the number o f L-valleys. Each of the d i s t r i b u t i o n functions F , F is assumed to satisfy a B o l t z m a n n equation: L
r
8F t
+ v (k) • g r a d , F
r
r
- \ E
T
e((
L
• g r a d , F = Q (F ) r
r
+ Q . (F ,
r
v L
F ), (1.6.3)
r
L
n
d,F
L
+ v (k)-gmd L
F - f E
x
L
e f f
- g r a d , F = Q (F ) L
L
+ Q , (F ,
L
L r
F ), (1.6.4)
r
L
n
where v (k) = l/h g r a d , e (/c), v (k) = l/h g r a d , e (k) denote the velocities of electrons i n the T- and L-valleys, resp. Q and Q are the intravalley collision operators. They are b o t h o f the f o r m (1.3.67) ( w i t h appropriate T- and L-valley collision rates s and s , resp.). The B o l t z m a n n equations (1.6.3), (1.6.4) are coupled by the intervalley collision integrals Q , (F , F ) and Q , (F , F ), which, i n the l o w density a p p r o x i m a t i o n , are given by: r
r
L
L
r
r
L
L
L r
r L
r
r
L
L
QT.L^T,
F ) = | L
( 5 , ( x , k', k)F' - s L
r
L
r > L
( x , k, k')F )N r
dk',
L
(1.6.5) F
QLAFP,
(s (x, r>L
L
k', fe)F - s (x, r
LS
k, k')F )N L
L
dk',
(1.6.6)
where s (x, k, k') and s (x, k, k') denote the transition rates from the state (x, k) of the T-valley i n t o a state (x, k') of one of the L-valleys and, resp., vice versa. The effective field £ is related to the electron number density rL
Lr
e f f
n = n + T
Nn, L
L
F dk, r
n, =
F dk L
R3 by the equation (1.3.14).
(1.6.7)
71
1.6 A p p l i c a t i o n s a n d Extensions
The intervalley transition rates s Maxwellian M (k) r
= N *exp r
r
L
, s
L
r
can be expressed i n terms o f the
M (k) L
kT
= Atfexp
-
N* =
exp
-
(1.6.8a)
fe T B
B
e (k) kT r
M * 0
exp
fe T
n
dk
B
(L6.8b) by i n t r o d u c i n g the inter-valley cross-sections
r L
s
r > L
( x , fe, fe') = <7 (x, FjL
s (x, fe, fe') =
ff (x,
L>T
Clearly, er
r L
a
L
r
and cr
L r
Ljr
fe, fe,
and
a
L r
:
k')M {k'),
(1.6.9)
fe')M (fe').
(1.6.10)
L
r
are nonnegative. If, i n a d d i t i o n , they satisfy (1.6.11)
( x , fe', fe) = cr ( x , fe, fe') V x ,fe,fe', r L
then the property n Q r . L ^ r . ^ L ) = QL,T(?T,
F ) = 0~(F , F ) L
r
L
=
(M , M ) r
\ + N
L
L
(1.6.12) holds, i.e. the kernel o f the inter-valley collision operator is spanned by the Maxwellians and we are led to expect the pair o f distributions ( F , F ) to relax towards an element of this kernel i n the large time l i m i t f -» oo. F o r the precise structure of the cross-sections er and a and for numerical results of the two-valley model for GaAs we refer to [1.43]. r
r L
Bipolar
L
L T
Model
I n a typical semiconductor the c o n d u c t i o n band is rather scarcely populated. F o r the technologically most relevant semiconductor silicon the intrinsic carrier concentration n at room-temperature is of the order of magnitude 1 0 / c m . M o s t of the electrons are valence electrons, i.e. they are responsible for the chemical c o m p o u n d o f the semiconductor crystal. W h e n the crystal is electrically neutral, then to each c o n d u c t i o n electron there corresponds a 'hole' i n the valence band, to w h i c h the positive charge + q can be assigned. Since the gap between the valence and the c o n d u c t i o n band (usually referred to as the bandgap) is significantly large for semiconductors, quite a l o t o f energy is necessary to transfer electrons from the valence band to the c o n d u c t i o n band. This process is called generation of electron-hole pairs, i.e. an electron is generated i n the c o n d u c t i o n band and a hole i n the valence band. The inverse process, that is the transfer of a c o n d u c t i o n electron i n t o the lower energetic valence band, is termed recombination of electron-hole pairs. O b v i o u s l y , the somewhat artificial i n t r o d u c t i o n of i
n
3
72
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
positively charged holes i n semiconductor physics gives a simple way o f accounting for the valence electrons, whose m o t i o n renders a c o n t r i b u t i o n to the current flow i n the crystal (hole current). F o r more i n f o r m a t i o n o n the basic physical properties o f semiconductors we refer to [1.51], [1.57]. I n the sequel we shall equip quantities, which correspond to electrons, w i t h the index n and quantities which correspond to holes, w i t h the index p. F o r example, F„ n o w stands for the electron d i s t r i b u t i o n and F for the hole distribution. We denote the number densities by p
F
n =
n
dk,
(1.6.14)
F dk p
and the current densities v„(k)F dk,
v (k)F dk,
J=q
n
p
(1.6.15)
p
where v„ and v denote the electron and hole velocities resp. related to the electron and hole band diagrams by v = (\/h)V e , v = —(\/h)V s . The temporal e v o l u t i o n o f the d i s t r i b u t i o n functions F„ and F i s — i n the semi-classical framework—governed by the system o f B o l t z m a n n equations: p
n
dF t
n
dF t
p
+ v„{k) • g r a d F„
£
x
+ v (k)-gmd p
e f f
F + f £
x
p
k
n
p
• g r a d , F„
k
Q„(F„) +
- g r a d , F = Q (F )
e f f
p
p
p
p
I„(F ,F ), 4.6.16) n
+ I (F , p
p
F ). (1.6.17)
n
p
n
Q„ and Q stand for the electron and, resp., hole collision operators. They are supposed to model the short range interactions of the corresponding type of particles w i t h their environment, i.e. w i t h crystal impurities, phonons etc. Mathematically, they are of the f o r m (1.3.67) w i t h transition rates s„ and s resp., determined by the physics o f the considered collision processes. M o s t i m p o r t a n t l y , they satisfy p
p
s„(x, k, k') =
e x
s (x,
e x
p
k, k') =
£
p( " p(
( f e
£ p ( f c
]~
£ T
fc~
)
"~ )'" £ p ( f e )
r
)
( x
5 p ( X
k
' '' k
'
w h i c h leads to the relaxation properties o f Q , n
k )
( L 6
(
''
1
'
6
1 8 )
1
9
)
Q. p
The operators /„, I model a recombination and generation process o f electron-hole pairs. They are given by: p
I (F , n
n
F) = p
[g(x, k', k)(l
F„)(l
- F' ) - r(x, k, k')F F' -\ p
n
p
dk' (1.6.20)
73
1.6 A p p l i c a t i o n s a n d Extensions
I (F„, F ) = P
lg(x, k, fe')(l - F ){1 - F ) - r(x, k', k)F^
p
H
p
JB
dk', (1.6.21)
where g(x, k, k') represents the rate of generation of an electron at the state (x, k) and of a hole at the state (x, k'). r(x, k, k') is the analogous local recombination rate. The expressions (1.6.20), (1.6.21) are derived by accounting procedures similar to the derivation of the single particle scattering integral presented i n Section 1.3. The nonnegative functions g and r are related by: £
r(x, k, k') = e x p ^ "
( f c )
~^
( f c , )
^ ( x , k', k).
(1.6.22)
This equation guarantees that the null-manifold of /„, I consists solely of pairs of F e r m i - D i r a c distributions w i t h the same F e r m i level (see [1.44]). Recombination and generation of carriers balance i n thermal e q u i l i b r i u m . The effective field E enters i n b o t h B o l t z m a n n equations (1.6.16), (1.6.17). The sign of £ i n the hole transport equation (1.6.17) is reversed due to the opposite flow direction of the positively charged holes i n the electric field p
e[[
e f f
E Ke
Obviously, b o t h electrons and holes contribute to the space charge density p. Also, for practically all semiconductor devices, ionized impurities, which mainly determine the performance of the device under consideration are present in the semiconductor crystal. These impurities are implanted i n t o the semiconductor crystal i n the fabrication of the device by a technologically highly complicated process (see [1.51]). We shall denote the so-called i m p u r i t y (or doping) profile by C. I t is given by the difference of the number densities of positively charged d o n o r ions and negatively charged acceptor ions. F o r the following we shall exclude mobile impurities, i.e. we shall assume that C is a function of the position variable x only, i.e. C = C(x). By simply adding up the charges, we obtain the t o t a l charge density p=
-
q
(
n
- p - C ) .
(1.6.23)
W h e n the C o u l o m b interaction is accounted for, we have the following effective field equation: i f ^eff
—
^cxt +
x
x
4tol
where £ represents an exterior electric field acting on the semiconductor device. A mathematical analysis, w h i c h gives a global (in t > 0) existence and uniqueness result for the electron-hole Boltzmann-Poisson system (1.6.16), (1.6.17), (1.6.24) subject to i n i t i a l conditions and periodic boundary conditions o n dB, can be found i n [1.44]. I t is based o n an iteration method, w h i c h is i n spirit similar to the single particle B o l t z m a n n equation method pree x l
74
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
sented i n Section 1.3. We remark that the b i p o l a r p r o b l e m still preserves the upper b o u n d 1 and the lower b o u n d 0 for F„ and F , i.e. 0 ^ F„ ^ 1, 0 ^ F ^ 1 holds for a l l times t > 0 i f i t holds i n i t i a l l y (Pauli principle). F o r real-life simulations o f semiconductor devices the bipolar B o l t z m a n n Poisson p r o b l e m has to be formulated o n a bounded p o s i t i o n d o m a i n Q £ [R> representing the geometry o f the semiconductor device. Then, boundary conditions for the d i s t r i b u t i o n functions F , F have to be prescribed on the inflow segments as discussed i n the previous Sections and the effective field equation (1.6.24) is replaced by the Poisson equation p
p
3
n
-e AF s
e f f
= p,
p
xeQ
(1.6.25)
subject to N e u m a n n - D i r i c h l e t boundary conditions o n 80.. The exterior field then originates from the D i r i c h l e t b o u n d a r y c o n d i t i o n for F , which represents voltages externally applied to the device. The occurance o f recombination-generation o f carriers modifies the conservation laws for the current and for the number o f carriers. By integrating the Boltzmann equations (1.6.16), (1.6.17) over the B r i l l o u i n zone B we obtain eff
qd,n - d i v J„ = -qR
(1.6.26)
qd p + divJ =
(1.6.27)
t
-qR,
p
where R is the recombination-generation rate, which, expressed i n terms of the d i s t r i b u t i o n functions F„, F , reads: p
R
I (F , p
n
F)
I (F„,F )dk.
dk=-
p
n
p
(1.6.28)
B
N o t e that the conservation laws (1.6.26), (1.6.27) are nonlinearly coupled due to recombination-generation processes. The total number of each type o f particles is n o t conserved anymore. Subtracting (1.6.26) from (1.6.27) and using the definition of the charge density (1.6.23) gives the conservation law for the t o t a l current density J defined as the sum o f the electron and hole current densities: J = J + J„.
(1.6.29a)
n
This conservation law reads: d p + div J = 0 .
(1.6.29b)
t
N o t e that we used the assumption d C = 0. F o r the whole space case we o b t a i n the conservation of the charge by integrating (1.6.29b) over U^: t
p{x,t
p(x, t) dx =
= 0)dx,
Vt>0.
(1.6.30)
R3
The modification o f (1.6.30) for the bounded x - d o m a i n case is obvious. A l o w density a p p r o x i m a t i o n of the electron-hole B o l t z m a n n system can be obtained by assuming 0
0scF «l p
(1.6.31)
^5
1.6 A p p l i c a t i o n s a n d Extensions
and by setting all quadratic terms in F„, F , which appear in the collision and recombination-generation integrals, equal to zero. Also, i n many applications, the generation and recombination relaxation times p
T
G
( X ,
g{x, k, k') dk'
k) =
[1.6.32) r(x,
T (X, k) R
k,k')dk'
are large compared to the collision relaxation time. I n these cases the recombination-generation integrals /„, I are usually neglected, which leads to a significantly weaker coupling o f the t w o B o l t z m a n n equations. This approach is physically meaningful in close-to-thermal-equilibrium conditions. p
Tunneling
Devices
Charge transport in semiconductors is collision dominated when the observation time period is significantly larger than the collision relaxat i o n time. Thus, the simulation o f l o w and medium frequency devices (like M O S F E T s , bipolar transistors and thyristors) must be based on mathematical models, which stem from the B o l t z m a n n equation. F o r extremely high frequency devices, however, the interesting time-scale for simulations is usually short and, consequently, the charge transport is m a i n l y ballistic, i.e. collisionsless models can be used. Such devices are very small (the w i d t h of the active region may be of the order of magnitude 20 nm) and they operate under high electric field strengths (very t h i n potential barriers w i t h height 0.3 eV often occur). Therefore, the device operation is driven by q u a n t u m effects and simulations based on a (semi-) classical Vlasov model give totally unrealistic results. F o r such situations the q u a n t u m Vlasov equation, w h i c h models the collisionless q u a n t u m transport of electrons, is well suited. As a typical example we consider the resonant tunneling diode depicted in Fig. 1.6.1. The device has t w o A l G a A s ( a l u m i n i u m - g a l l i u m arsenide) quan-
in
GaAs C= 10 crrT 18
3
t/i
in <
<
I = 60 nm F i g . 1.6.1
GaAs C = l0 crrT 18
3
76
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
Vext
0.3 eV
I = 60 nm — F i g . 1.6.2
turn barriers o f thickness 5 n m separated by a G a A s q u a n t u m well, which has the thickness 5 n m . The barrier and the well are undoped, while the bulk regions outside the barriers are doped w i t h donors of concentration 1 0 c m , i.e. the d o p i n g profile C = 1 0 c m i n the bulk regions and C = 0 in the barriers and i n the well. The q u a n t u m barriers are 0.3 eV high (see Fig. 1.6.2) and b o t h b u l k regions are contacted. The (already somewhat simplified) device geometry suggests a onedimensional q u a n t u m Vlasov-Poisson model (see Section 1.5): 1 8
- 3
1 8
d,W + t ' - g r a d , W + - 0 * [ K m
e f f
- 3
] W = 0,
0 < x
veU,
t>0 (1.6.33)
W{x, v, t = 0) = W,(x, v),
0<x
veM,
(1.6.34)
supplemented by div(e (x) grad V) = q(n - C(x)), s
V(0,t)=V , o
0<x
(1.6.35)
V(l,t)=V ,
(1.6.36)
1
where n denotes the q u a n t u m number density r n =
Wdv.
(1.6.37)
Note that the p e r m i t t i v i t y e now appears 'inside' the divergence operator since it is position dependent due to the t w o different materials o f which the device is made up. The boundary data V , V determine the biasing condition. s
0
x
77
Problems
The effective potential V is obtained by adding the material potential V of Fig. 1.6.2 to the solution V of the Poisson p r o b l e m (1.6.35), (1.6.36) and by extending to U : eU
exi
x
\V ,
x^O
0
Kttfa
t) = < V(x, t) + F ( x ) , ext
0 < x < I.
[V ,
x>l
x
Also, we have to define boundary conditions for the q u a n t u m Vlasov equation at x = 0 and x = /. As discussed i n Section 1.5 the simplest choice is to take inflow Dirichlet data W(0,
v, t) = W (v, t), 0
W(l,v,t)
= W (v,t), 1
v>0,
t>0
(1.6.39)
v<0,
t>0.
(1.6.40)
For the simulation o f ideal contacts i t is more appropriate to choose n o n reflecting boundary conditions (see [1.24] and the discussion i n Section 1.5). For appropriate choices of the initial and boundary data and for numerical results o f simulations of the resonant tunneling diode we refer to [1.32]. These results clearly demonstrate the power o f the q u a n t u m Vlasov-Poisson p r o b l e m in modeling ultra-integrated semiconductor devices o n sufficiently short time-scales. Due to the ongoing m i n i a t u r i z a t i o n o f V L S I structures we expect this research area to become extremely i m p o r t a n t i n the near future. I n particular the inclusion of physically realistic q u a n t u m scattering models, w h i c h w i l l allow simulations o n significantly larger time scales, and the band diagram Wigner-Poisson model presented i n Section 1.5, which incorporates the crystal structure o f the semiconductor, are going to play a major role soon.
Problems 1.1
Solve the i n i t i a l value p r o b l e m (1.2.9), (1.2.13) for a constant electric Field E. D r a w the (x, u)-phase p o r t r a i t o f the characteristics i n the o n e - d i m e n s i o n a l case.
1.2
Show that the Z / - n o r m s , 1 s£ p < oo o f non-negative s o l u t i o n s o f the L i o u v i l l e e q u a t i o n (1.2.9), (1.2.13) are conserved i n the e v o l u t i o n process. Remark: T h e L - n o r m o f / is defined by: p
II/IIip
:=
(J^
1 / 1 " dx
dvj"
,
1 <
p
<
oo
11/11,., : = sup | / ( x , v)\ . X, V p
_
1
Hint: M u l t i p l y (1.2.9) by / a n d integrate over decays sufficiently fast as |.x| ~> oo, \v\ -* oo. 1.3
x Rl assuming t h a t the s o l u t i o n
D r a w the phase p o r t r a i t o f the characteristic e q u a t i o n s (1.2.3), (1.2.4) for the oned i m e n s i o n a l p o t e n t i a l b a r r i e r g i v e n by
78
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s
fo, V(x)
x
0
0<X
=
E>0,
V eU. 0
x ^ e Discuss the cases V > 0, K < 0 a n d consider the l i m i t s e. -> 0, V -> - oo, V -» oo. 0
1.4
0
0
0
L e t H = H(.x, p, f) be a general H a m i l t o n i a n f u n c t i o n . D e r i v e the L i o u v i l l e e q u a t i o n c o r r e s p o n d i n g t o the e q u a t i o n s o f m o t i o n (1.2.27), (1.2.28). S h o w t h a t the L - n o r m o f the s o l u t i o n o f the c o r r e s p o n d i n g i n i t i a l value p r o b l e m is preserved i n t i m e . 2
1.5
Solve the t h r e e - d i m e n s i o n a l L i o u v i l l e e q u a t i o n (1.2.52) (subject t o an i n i t i a l c o n d i t i o n ) for c o n s t a n t electric a n d magnetic
1.6
fields.
a) A r e the c o n s e r v a t i o n laws (1.2.18), (1.2.19) v a l i d for the 'magnetic field p r o b l e m ' (1.2.52)? b) Is n o n n e g a t i v i t y preserved b y (1.2.52)?
1.7
m
m
D e f i n i t i o n : A m a p vv: R
-> R
is called v o l u m e preserving, i f \o\(A)
= vo\(w(A))
for a l l
m
measurable subsets A <= R . P r o v e t h a t the characteristic m a p w(t, -,-): R —* R defined by (1.2.20), (1.2.10), (1.2.11) is v o l u m e preserving for a l l t > 0 i f (1.2.14) holds. P r o v e the analogous result for the case o f a n o n - v a n i s h i n g m a g n e t i c field. T h e f l o w associated w i t h a v o l u m e - p r e s e r v i n g m a p is called i m c o m p r e s s i b l e . C o n c l u d e the c o n s e r v a t i o n o f the L - n o r m s o f the s o l u t i o n s o f the L i o u v i l l e e q u a t i o n f r o m the i n c o m p r e s s i b i l i t y o f the f l o w defined by the characteristic m a p . 6M
6M
p
1.8
L e t / ( x , v) = 5(x - x , v - v ),x e R™,v e R™ be the i n i t i a l d a t u m for the L i o u v i l l e e q u a t i o n . S h o w t h a t the s o l u t i o n o f the i n i t i a l value p r o b l e m is given b y fix, v, t) = <5(x — x(t; x , v ), v — v(t; x , v )). 7
0
0
1.9
0
0
Q
0
0
0
Show t h a t the L i o u v i l l e e q u a t i o n is time-reversible for static force-fields, i.e. i f a s o l u t i o n exists for t > 0, t h e n i t also exists for r < 0 a n d the s o l u t i o n for t < 0 can be c o n s t r u c t e d f r o m its values for t > 0.
1.10 Verify t h a t the f u n c t i o n f o f (1.3.7).
w
g i v e n b y (1.3.8) w i t h P satisfying (1.3.9), (1.3.10) is a s o l u t i o n
1.11 L i n e a r i z e the V l a s o v e q u a t i o n (1.3.13), (1.3.14) at the e q u i l i b r i u m s o l u t i o n F = F (v), £ = 0. T h e so o b t a i n e d p r o b l e m is called r a n d o m phase a p p r o x i m a t i o n . W h i c h physical s i t u a t i o n does i t model? e
e f f
1.12 C o n s i d e r the o n e - d i m e n s i o n a l e q u a t i o n (1.3.13) (xeR ,veR ) with £ = £ (x) given. S h o w t h a t F = F(x, v) is a steady state s o l u t i o n i f a n d o n l y i f F is a f u n c t i o n o f the H a m i l t o n i a n , i.e. i f a n d o n l y i f there is a f u n c t i o n <j>: R -> R such t h a t F(x, v) = (f>(H(x, v)), H(x, v) = mv /2 - qV (x), h o l d s w i t h d/dx V = -E . x
v
e f f
e f f
2
M
M
e f !
Remark: T h i s is used t o c o n s t r u c t steady state s o l u t i o n s o f the c o u p l e d V l a s o v - P o i s s o n p r o b l e m (see [ 1 . 4 2 ] ) . 1.13 C o n s i d e r the V l a s o v e q u a t i o n (1.3.13), (1.3.14) w i t h a s m o o t h i n t e r a c t i o n field £ ( w h i c h m a y be o b t a i n e d by s m o o t h i n g the C o u l o m b field (1.3.19) a b o u t x = y). T a k e F(x, v, t = 0) = S(x — x , v — v ). S h o w t h a t the s o l u t i o n F(x, v, t) has the f o r m o f a ^ - f u n c t i o n centered at a p o i n t (x(t), v(t)) eRl x IR^. D e r i v e the i n i t i a l value p r o b l e m for i n l
0
(x(i),
0
v(t)).
1.14 C a r r y o u t the c a l c u l a t i o n s t o derive the p r o p e r t y (1.3.87) o f the t r a n s i t i o n rate s f r o m the p r i n c i p l e o f detailed balance (1.3.84) b y using the F e r m i - D i r a c statistics (1.3.85), (1.3.86). 1.15 Solve the r e l a x a t i o n t i m e a p p r o x i m a t i o n (1.3.101) for a given c o n s t a n t effective £ = £ , a constant r e l a x a t i o n t i m e t a n d n(x, t) = 1. e f f
field
Problems
79
Hint:
Use the r e p r e s e n t a t i o n (1.3.102) a n d i n v e r t the characteristic m a p (.x , k ) -»
(x(t),
k(t)).
0
0
1.16 F o r m u l a t e the B o l t z m a n n e q u a t i o n w i t h a m a g n e t i c field a n d the B o l t z m a n n - M a x w e l l system. Solve the c o r r e s p o n d i n g r e l a x a t i o n t i m e a p p r o x i m a t i o n for given constant electric a n d m a g n e t i c fields a s s u m i n g n = 1. 1.17 Prove t h a t i n i t i a l l y o r t h o g o n a l wave functions r e m a i n o r t h o g o n a l for a l l times, i.e. conclude
W W / ' W (2)
where t/>'", tp a n d i/>,
<2)
li)
i2)
<J/ (x,t)ij/ (x,t)dx
dx = 0
= 0,
t > 0 ,
are s o l u t i o n s o f the M - p a r t i c l e S c h r d d i n g e r e q u a t i o n w i t h i n i t i a l d a t a
resp.
1.18 L e t the o n e - d i m e n s i o n a l static p o t e n t i a l be g i v e n b y fO, = \
V(x)
x < 0, x > a a > 0.
n
Solve t h e eigenvalue p r o b l e m (1.4.14) f o r the S c h r d d i n g e r e q u a t i o n , i.e. find e e IR a n d i/> = iA(x) e L (U) such t h a t (1.4.14) h o l d s . C o n s i d e r the cases V > 0, V < 0 a n d the l i m i t s V -* oo, V -* — oo. 2
0
0
0
0
1.19 C o m p u t e the density matrices for the eigenstates o f P r o b l e m 1.18. C a l c u l a t e the W i g n e r functions for the l i m i t i n g cases V - > oo, V -* — oo. 0
0
1.20 P r o v e t h a t (1.4.15) is e q u i v a l e n t t o (1.4.16), (1.4.17), (1.4.18). Hint:
M u l t i p l y (1.4.15) b y a C°°-test f u n c t i o n
a n d use d(xW(x)
x > 0. 2
1.21 C o m p u t e the L ( R ) - e i g e n s t a t e s o f (1.4.15), t h e i r density matrices, W i g n e r functions, p a r t i c l e a n d c u r r e n t densities. 1.22 L e t the o n e - d i m e n s i o n a l static p o t e n t i a l be g i v e n b y V(x)
[To,
x > 0
0,
.xs=0.
=
C o m p u t e the reflection a n d t r a n s m i s s i o n coefficients for a m o n o e n e r g e t i c beam o f electrons represented b y a r i g h t - m o v i n g wave. 1.23 D e r i v e the q u a n t u m L i o u v i l l e e q u a t i o n f o r V(x) = —(m/q)S(x),
x e R. S i m p l i f y
the
o p e r a t o r 0 \_V~\ as m u c h as possible. W r i t e d o w n the F o u r i e r t r a n s f o r m e d e q u a t i o n a n d h
f o r m u l a t e i t w i t h o u t <5-functions. Hint:
D e r i v e interface c o n d i t i o n s at the lines x = (h/2m)n,
x = —(h;2m)n
similarly to
P r o b l e m 1.20. 1.24 L e t e , e 0
l
be eigenvalues
eigenfunctions t / ' , i / ' 0
2
1
o f the S c h r d d i n g e r e q u a t i o n (1.4.14) w i t h 3M
e L {U ).
corresponding
P r o v e t h a t i'(e — e^/h is a n eigenvalue o f the q u a n t u m 0
Liouville equation w i t h eigenfunction 4>o (x
+
(x
2m J
\
- ~ n ) e
k
' i
dn.
2m
Remark: I t is s h o w n i n [ 1 . 3 8 ] t h a t the s p e c t r u m o f the q u a n t u m t r a n s p o r t o p e r a t o r T = f g r a d + (qlm)6 [V~\ is the closure o f the set c o n s i s t i n g o f the values i(a — P)/h, q u a n
x
h
80
1 K i n e t i c T r a n s p o r t M o d e l s for S e m i c o n d u c t o r s where a, /? are spectral values o f the c o r r e s p o n d i n g H a m i l t o n i a n o p e r a t o r . T h i s result, together w i t h a m o r e detailed analysis o f the s t r u c t u r e o f the s p e c t r u m was used i n [ 1.18] to characterize the steady states o f the q u a n t u m L i o u v i l l e e q u a t i o n .
1.25 Show that d [K]
maps real valued functions i n t o real valued functions, i f V is real valued.
h
1.26 L e t V = V(x) e R h o l d . S h o w t h a t the pseudo-differential o p e r a t o r d \_V~\ is f o r m a l l y s k e w - a d j o i n t o n L {R x R* ), i.e. p r o v e h
2
M
M
X
f0 tV~\g h
dv dx
g6 [_V-\fdvdx. h
1.27 C o m p u t e the p o t e n t i a l , w h i c h corresponds t o the wave f u n c t i o n (1.4.85). 1.28 C o n s i d e r the i n i t i a l value p r o b l e m for the q u a n t u m L i o u v i l l e e q u a t i o n (1.4.39) w i t h i n i t i a l d a t u m w,(x, v) = 5(x — x , v — v ) for fixed p o i n t s x e R , v e R^ . W h a t are the c o n d i t i o n s o n the p o t e n t i a l such t h a t the s o l u t i o n is g i v e n by w(x, v, t) = 5(x — x(t), v — v(t)\ where (x(t), v(t)) is a c u r v e i n the (x, u)-space w i t h i n i t i a l value ( x , y ) ? Is the given (5-initial d a t u m q u a n t u m m e c h a n i c a l l y admissible ( u n c e r t a i n t y principle)? M
0
0
0
0
X
M
0
0
1.29 C o n s i d e r the q u a n t u m L i o u v i l l e e q u a t i o n (1.4.39) as m o d e l for an u l t r a - i n t e g r a t e d s e m i c o n d u c t o r device o f characteristic l e n g t h / = 1 0 ~ m . I n a t y p i c a l o p e r a t i o n m o d e the p o t e n t i a l is o f the o r d e r o f m a g n i t u d e V = 0.5 e V a n d a t y p i c a l s i m u l a t i o n t i m e scale is T — 1 0 ~ s. Use these values a n d m = 0.6 x 1 0 ~ k g t o scale the q u a n t u m L i o u v i l l e e q u a t i o n b y i n t r o d u c i n g dimensionless variables. Identify the parameter, w h i c h plays the r o l e o f h i n the scaled e q u a t i o n . Is it s m a l l ( c o m p a r e d to 1)? 8
1 4
3 1
References [1.1]
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Semiconductors
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From Kinetic to Fluid Dynamical Models
2
2.1 Introduction Different approaches to the solution o f the kinetic transport models discussed i n Chapter 1 are possible. A l t h o u g h several p r o m i s i n g attempts towards a numerical solution have been undertaken i n the recent past (we only m e n t i o n particle methods [2.10] and spectral methods [2.13]), the application of numerical methods remains to be a formidable task i n general. A p a r t from that, solutions o f the kinetic equations contain i n many cases (e.g. close to equilibrium) a g o o d deal o f redundant information. I n this Chapter fluid dynamical models for semiconductors w i l l be i n t r o duced. They represent a reasonable compromise between the contradictory requirements o f physical accuracy and c o m p u t a t i o n a l efficiency. Their common feature is the fact that the number of independent variables is reduced from seven (3 space + 3 velocity coordinates + time) to four (3 space coordinates + time). The dependent variables can usually be interpreted as averages (moments) of the phase space number density w i t h respect to the velocity. T w o different approaches for the d e r i v a t i o n o f fluid dynamical models from kinetic equations exist. They w i l l be presented i n the Sections 2.2 and 2.3, respectively. The first is a p e r t u r b a t i o n argument. I t exploits the smallness of a dimensionless parameter, namely the scaled mean free path, which appears i n an appropriately scaled version o f the B o l t z m a n n equation. F o r the B o l t z m a n n equation o f gas dynamics an expansion o f the solution i n powers of the mean free path has been introduced by H i l b e r t [ 2 . 9 ] and, accordingly, bears his name. I n the context o f semiconductors, the H i l b e r t expansion has been recently carried out and t h o r o u g h l y analyzed by Poupaud [2.11]. The m e t h o d is presented i n Section 2.2 for a standard bipolar model w i t h the assumptions o f l o w densities and small electric field. I n this case the leading terms i n the expansion are governed by the standard drift diffusion equations for semiconductors which have been derived by van Roosbroeck [2.18] for the first time. A second way for o b t a i n i n g fluid dynamical models are moment methods. Compared to the H i l b e r t expansion, their application requires a good deal
84
2 F r o m K i n e t i c to F l u i d D y n a m i c a l M o d e l s
of physical i n t u i t i o n or a-priori-knowledge about the solution o f the Boltzm a n n equation. Also, the authors are not aware of any rigorous mathematical justification. The m a i n ingredient o f a moment method is an ansatz for the phase space density which prescribes the dependence on the velocity and which contains several parameters depending on position and time. After inserting the ansatz, the B o l t z m a n n equation is multiplied by a number of linearly independent functions o f velocity and integrated over the velocity space. The result are differential equations for the time and space dependent parameters. I n some cases not all integrations can be carried out explicitely. T h e n the terms i n question are usually replaced by phenomenological models. T w o different moment methods are presented in Section 2.3. I n the first one, the ansatz for the phase space density is motivated by the results of Section 2.2. I t leads to a system w h i c h can be reduced to the drift diffusion equations by a perturbation argument. Because o f the choice o f the ansatz all the integrations can be carried out explicitely in this case. The second ansatz [ 2 . 2 ] , usually called shifted Maxwellian, is motivated by the collision term of the B o l t z m a n n equation for m o n a t o m i c gases (see [2.3]). I t leads to a modified version of the Euler equations o f gas dynamics for a gas of charged particles i n an electric field. The difference to the Euler equations is the appearance of relaxation terms. I n general, these cannot be evaluated explicitely. They are usually replaced by relaxation time approximations. The resulting system (possibly including an extra heat conduction term) is referred to as the hydrodynamic model for semiconductors. The m a i n assumptions i n the derivation o f the drift diffusion equations are low carrier densities and small fields. The first assumption can be discarded of i f a nonlinear collision term is used i n the B o l t z m a n n equation. This is necessary when the position space number density is large, w h i c h i n t u r n is to be expected for large d o p i n g concentrations. The H i l b e r t expansion [2.11], which differs considerably from that o f Section 2.2, is carried out i n Section 2.4. The h y d r o d y n a m i c model is usually employed to give an appropriate description of high field phenomena. A different approach for the modeling of high field effects is presented i n Section 2.5. A Hilbert expansion for a rescaled B o l t z m a n n equation [2.12] leads to a hyperbolic drift equation (compare to Section 3.11). Unfortunately, the m o b i l i t y coefficient i n the drift term, which depends on the electric field, cannot be evaluated explicitely. However, very accurate data from measurements are available which can be used for fitting the coefficients i n an ansatz describing the qualitative behaviour. The recombination-generation terms (1.6.20), (1.6.21) describe direct bandband recombination caused by p h o t o n transitions. I n practical situations several other recombination-generation mechanisms are i m p o r t a n t . I n Section 2.6 models for band-trap capture and emission, Auger recombination, and impact ionization are presented.
2.2 Small M e a n Free P a t h — T h e H i l b e r t E x p a n s i o n
85
2.2 Small Mean Free Path—The Hilbert Expansion We consider the bipolar model derived i n Section 1.6: d F„ +
g r a d F - j j i v g r a d , F„ = Q (F )
t
x
n
n
n
+
I (F„,F ) n
p
(2.2.1) d,F
+ v (k)• g r a d Fp + ^E-grad
p
p
x
F = Q (F )
k
p
p
+
p
I (F„,F ) p
p
w i t h the low density approximations (1.3.92): ' Ux,
Q (F ) p
k, k>) ( e x p
r-^j
F' - exp n
= £ t {x, k,fe')(exp (-^P-J
p
( j ^ ] F ) dk' n
^ " exp (•
p
fe r B
(2.2.2) for the collision terms and w i t h the models I (F„, F ) = - [
p
e
fe,
( fs (k') g(x, fe, fe ) I exp I n
I (F , p
n
F) = p
(fc)
(fc,)
fe')(exp( - ~^ )F.F; - s (k)\ " 1 F„F
- l ) dfe'
p
T
p
\ -\\dk (2.2.3)
for the recombination-generation rates. N o t e that " ' " denotes evaluation at fe' as i n Chapter 1. I f we assume that the c o n d u c t i o n electrons are located close to the c o n d u c t i o n band m i n i m u m and the holes close to the valence band m a x i m u m a parabolic band a p p r o x i m a t i o n for the energy-wave vector relations can be used. I t reads (see Section 1.6)
(2.2.4) e (k) p
= E
v
- £ - W ,
where E denotes the c o n d u c t i o n band m i n i m u m , E the valence band m a x i m u m and m and m the effective masses o f resp. electrons and holes. This gives the velocities c
v
n
p
1 h v„(k) = - g r a d e„(k) = —k, n m„ k
(2.2.5) v
1
h
n
m
p
86
2 F r o m Kinetic to Fluid Dynamical Models
I f the effective masses o f electrons and holes are o f the same order o f magnitude, the exponential terms i n (2.2.2) and (2.2.3) suggest the i n t r o d u c t i o n of the reference velocity v = ^Jk T/m . F o r the following the equations (2.2.1) w i l l be w r i t t e n i n terms o f the scaled wave vector and velocities B
v
ns(k )
v
— k,
s
s
s(k )
P
n
—
s
k . s
m
p
A n appropriate scaling o f the collision and recombination-generation terms shows that they are p r o p o r t i o n a l to the reciprocals o f characteristic time constants, which can be interpreted as average relaxation times. W i t h the average velocity v, the relaxation times x and T , corresponding to collisions and recombination-generation respectively, can be w r i t t e n as c
T
= 1 /V,
C
T
C
R
=
R
1 /V, R
where i and i denote the mean free paths between t w o consecutive scattering and, respectively, recombination-generation events. I t is a well k n o w n fact (see e.g. [ 2 . 1 1 ] ) that the relaxation times correspondi n g to the collision terms ( ~ 1 0 ~ s) are m u c h smaller than those of the recombination-generation terms ( ~ 1 0 ~ s). Thus, c
R
1 2
9
i «
i
c
R
2
holds. We denote the r a t i o i /i c
R
by a and introduce a reference length i by Q
a = i /h • c
T h e n a can be interpreted as a scaled version o f the mean free p a t h between t w o scattering events. The choices of the reference time x and the reference field strength U /i complete the scaling. Here the reference voltage U = k T/q is the so called thermal voltage. A t r o o m temperature the reference field strength is o f the order o f 10 V / c m . I n V L S I applications, electric fields can be m u c h larger. This shows that the analysis given below does n o t appropriately account for c o m m o n l y occurring high field effects. The scaled version o f (2.2.1) is given by R
T
T
0
B
2
2
2
a d,F„ + <x{v (k)-gmd n
2
a dF t
p
+ a{v (k)-gmd
x
p
x
F„ - £ - g r a d F„} = Q„(F„) + a I„(F„, k
F + £ - g r a d F } = Q (F ) p
k
p
p
p
2
+ a I (F , p
n
F) p
F ), (2.2.6) p
where the index "s" i n the scaled quantities is n o w o m i t t e d for reasons o f n o t a t i o n a l convenience. The scaled collision and recombination-generation terms have the same form as the unsealed versions (2.2.2) and (2.2.3) w i t h the integration taken over and the exponential terms
2.2 S m a l l M e a n Free P a t h - T h e H i l b e r t E x p a n s i o n
,
fs„(k)\ e x p
Wr)
87
f-e (k) p
a n d
e
x
p
l " ^
replaced by 2
e x p ( £ + |/V| /2)
and
c
exp(^-E
2
+
v
^\k\ /2
respectively, where E and E are the scaled (by k T) conduction band m i n i m u m and valence band m a x i m u m , respectively. The H i l b e r t expansion is an expansion o f solutions of (2.2.6) i n terms o f powers of the scaled mean free path a: c
v
B
F„ = Fno + a^ii + P = po + <*Fpi F
F
+••'•
Equations for the coefficients i n this ansatz are obtained by substitution into (2.2.6) and equating coefficients o f equal powers of a. The equations for the leading order terms Q (F ) n
= Q (F )
n0
P
p0
= 0
have the solutions F
= n(x, t)M„(k),
n0
where M„, M
p
F
p0
= p(x,
t)M (k), p
denote the the scaled Maxwellians (see (1.3.95)) 2
M,(*) = i e x p ( - | * | / 2 ) ,
The constants 2
N = (2nf c
,
N. =
are chosen such that the integrals o f the Maxwellians over the /c-space are equal to one. This implies that the as yet unspecified quantities n(x, f), p(x, t) are scaled position space number densities o f electrons and holes, respectively. Equating coefficients o f a. i n (2.2.6) leads to M„v -{gmd n n
x
M v -(grad, P
p
+nE)
=
Q„(F ),
p - pE) =
Q (F ).
nl
p
pl
The analysis of these equations is facilitated by the following result. I n its statement, several technical assumptions concerning the collision cross sections (j> and are omitted. n
p
88
2 F r o m Kinetic to Fluid Dynamical Models
Lemma (Poupaud [2.11]): A) A necessary solvability of an equation of the form Qipif)
and sufficient
condition
= 9
for
the
(2-2.8)
is gdk
= 0.
(2.2.9)
If (2.2.9) holds, (2.2.8) has a one dimensional linear manifold of solutions the form f = f„ + q M where f„ denotes a particular solution and q a parameter. B) The equations lp
n/p
n/p
lp
Q„(h„) = M„v ,
Q (h )
n
p
=
p
3
have solutions h„(x, k), h (x, h) e R -fi„(x)I
n
of is
Mv p
which
p
v„ ®h dk=
n/p
p
satisfy
< 0,
3
(2.2.10) v ®h dk= p
-p (x)I
p
p
< 0,
3
T
3
where I is the three dimensional unity matrix and a® b = ab , for a,b e U , denotes the tensor product. Furthermore, the j-th component of h„ is an odd function of the j-th component of k and has the form 3
/p
h Jk)
= h(k
nlp
\Pjk\),
p
where \Pjk\ denotes the Euclidian norm of the projection perpendicular to the kfdirection.
of k onto the plane
I n terms o f the scaled current densities J (x,
t) = n„{grad
J (x,
t) = - p ( g r a d
n
n + nE),
x
(2.2.11) p
p
x
p -
pE)
the solution o f (2.2.7) is given by Fnl F
= Jn'KIHn J
<7„M„
+
h
M
P 1 = - p' plP-
+
P
where q„(x, t) and q (x, t) are as yet unspecified. E q u a t i n g coefficients o f oc ,j ^ 2, i n (2.2.6) gives p
j
S F„.j-2
+ Vgrad, F ^
dtF j-2
+ V g
t
- £grad
nJ
P
= Q {F ) p
pJ
+
r
a
d
* F^
+ £-grad
pJ
l (F _ ,F ^ ). p
nJ
2
p
2
F_
t
nJ
t
l
F,,^
89
2.2 S m a l l M e a n Free P a t h — T h e H i l b e r t E x p a n s i o n
Assuming that we k n o w the terms up to the order j — 1, these are equations of the form (2.2.8) for F and F . The solvability c o n d i t i o n (2.2.9) for j = 2 implies nj
pj
— R,
d.n — d i v , J„=
(2.2.12) d p + d i v J = — R, t
x
p
where the position space recombination-generation rate R is given by R = A(x)(np
— nf).
(2.2.13)
The quantities n, and A(x) are defined by 'N N exp{-EJ2), c
^(x) = n
v
- 2
g(x, k, k') dk •S3 R3
dk',
X
where E =• E — E denotes the scaled bandgap o f the semiconductor. The recombination-generation term R has the form o f a mass action law w i t h the reaction rate A(x) and the scaled intrinsic number n . Unsealed versions of (2.2.11) and (2.2.12) are given by the system o f partial differential equations g
c
v
t
J = QP- (U n
n
J=
grad n + nE\
T
~qp {U
P
P
qd,n - d i v J = n
grad p + pE),
T
~qR,
qd p + div J = - q R , t
p
called the drift diffusion equations o f semiconductors. This name originates from the type of dependence o f the current densities o n the carrier densities and the electric field. The current densities are the sums of drift terms (with the mobilities p and p ) and diffusion terms ( w i t h the diffusivities D = p U and D = p U ). The equations n
p
p
p
n
n
T
T
DJp.
= D /p =U
n
p
p
(2.2.15)
T
are k n o w n as the Einstein relations. As before, the unsealed quantities i n (2.2.14) are denoted by the same symbols as their scaled counterparts. I n particular, the scaling introduced above implies the reference value (k Tm ) /h for number densities. Accordingly, the unsealed recombination-generation rate R has the form (2.2.13) w i t h the unsealed intrinsic number given by il2
B
2
(2nk T^m m y B
H i
=
{
n
h
2
3
n
(-E
p
g
J
e
X
P
( 2 ^ T
Another result of the scaling procedure is that the mobilities are inversely p r o p o r t i o n a l to the square roots of the effective masses. The fact that heavy carriers are slower than light ones is responsible for the G u n n effect discussed i n Section 4.8. F o r a self consistent treatment of the electric field, (2.2.14) has to be supplemented by the Poisson equation (1.6.23), (1.6.25). The resulting system, originally due to van Roosbroeck [2.18], is called the basic semiconductor
90
2 F r o m Kinetic to Fluid Dynamical Models
device equations. I t has been the subject of an intensive mathematical, numerical as well as physical scrutiny. A presentation of the most i m p o r t a n t results is the subject of the Chapters 3 and 4. Obviously, the leading terms in the H i l b e r t expansion cannot satisfy general initial and boundary conditions for the B o l t z m a n n equation. I f the prescribed data do not have the form of Maxwellians, initial and boundary layers have to be introduced for the construction of a complete formal a p p r o x i m a t i o n of the solution. This has been carried out by Poupaud [2.11], who also gave a justification for the formal a p p r o x i m a t i o n procedure presented above.
2.3 Moment Methods—The Hydrodynamic Model A second way for the derivation of fluid dynamical models from the Boltzm a n n equation are m o m e n t methods. As mentioned in the i n t r o d u c t i o n they consist of an ansatz for the d i s t r i b u t i o n function and a system of necessary conditions for solutions o f the B o l t z m a n n equation. H o w "close to sufficiency" these conditions are, can i n general only be judged by physical reasoning. I n this Section we consider the classical B o l t z m a n n equation for one type of charge carrier (say electrons): d.F + v• g r a d F — —E-grad„ m
F = Q(F)
x
(2.3.1)
w i t h a l o w density collision term: (f>(x, v, v'){MF'
Q(F)
— M'F)
dv',
where the M a x w e l l i a n is given by ilt/ ^
(
m
/2
Y
f-m\v\*
The ;'-th order moment of the d i s t r i b u t i o n function F is defined as the tensor M of rank /', whose components, which depend o n position and time, are given by 0 )
• V C . . ; U . t)
M
( 0 ,
( x , t)=
• Vj F(x, v, t) dv
for
j ^ 1,
F(x,v,t)dv.
The relevance of the moments is due to the fact that they are related to physical quantities i n a simple way. Examples are:
91
2.3 M o m e n t M e t h o d s — T h e H y d r o d y n a m i c M o d e l
M
( 0 )
(l)
— qM
tn — tr{M ) (2)
n
position space number density,
J
current density,
S"
energy density,
(2.3.2)
where tr denotes the trace (of a matrix). Equations for the moments can be derived by m u l t i p l y i n g the B o l t z m a n n equation by powers o f v and by integrating over the velocity space. This leads to the infinite hierarchy dM
+ d i v , M » = 0,
dM
+ div, M
{0)
( 1
t
ll)
t
( 2 )
(0)
+
-M E
vQ(F)
dv,
m dM
(2)
(2.3.3)
<3
+ div, M > + 2 - M m
t
(
1
)
® E =
v®vQ(F)
dv,
A c c o r d i n g to the physical interpretation o f the moments these equations represent conservation laws. The first one—already discussed in Chapter 1 —represents the conservation of charges. The practical use o f the hierarchy (2.3.3) is limited o n one hand by the fact that all the moments are coupled, such that t r u n c a t i o n of the hierarchy does not give a closed system for a finite number o f the moments. O n the other hand, the terms originating from the collision integral do not depend o n the moments i n a simple way i n general. These difficulties are overcome by m a k i n g an ansatz for the distrib u t i o n function w h i c h a p r i o r i l y fixes its dependence on the velocity. This usually introduces position and time dependent parameters which then are determined by a truncated version o f (2.3.3).
Derivation
of the Drift
Diffusion
Model
The first moment m e t h o d presented here is m o t i v a t e d by the results o f the H i l b e r t expansion. W i t h the particular solution h(x, v) of the equation Q(h) =
vM,
whose properties are given i n the L e m m a of the preceding Section, we make the ansatz F(x, v, t) = n(x, t)M(v)
+
1
J(x, t)-h(x,v),
H(x)k T B
where p.(x) satisfies v (x) h(x, v) dv = JR3
—p(x)U I . T 3
(2.3.4)
92
2 F r o m Kinetic to Fluid Dynamical Models
Straightforward integration gives the relations for the moments of (2.3.4) i0)
M
(l)
= n,
-qM
= J,
M
, 2 )
=
n ^ / m
3
and a comparison w i t h (2.3.2) shows that the choice of the symbols n and J in (2.3.4) is justified. The energy density is given by £ =
-k Tn. B
The first t w o equations i n (2.3.3) i m p l y qd,n — d i v , J = 0, um
(2.3.5) c,J + qp(U
g r a d , n + nE) = J.
r
q The factor pm/q m u l t i p l y i n g the time derivative of the current density is the current density relaxation time. I t is usually assumed that this relaxation time is small compared to characteristic time constants i n the drift diffusion a p p r o x i m a t i o n (2.2.14) (see [2.16]). Thus, the term ~(pm/q)d J i n (2.3.5) is neglected and a unipolar drift diffusion model is obtained from (2.3.5). t
The Hydrodynamic
Model
A different ansatz for the d i s t r i b u t i o n function is motivated by the collision term for a dilute gas of r i g i d spheres (see [ 2 . 3 ] ) . F o r this case the n u l l manifold of the collision term is five dimensional and its elements can be written as m
A
M
(
V'
2
f-m\v-v\ \ 2
.....
where n, T and the three components of v are the free parameters (see [2.3, pp. 78ff]). A d i s t r i b u t i o n function of the form (2.3.6) is called displaced (or shifted) Maxwellian. Here, (2.3.6) can be used as an ansatz for a moment method w i t h the parameters depending o n position and time, n, T , and v can be interpreted as number density, effective temperature, and mean velocity, respectively. Since an effective temperature different from the lattice temperature is allowed, it is plausible that certain high field effects are taken i n t o account by (2.3.6). F o r the moments of (2.3.6) we have e
e
w
M
= n,
M
( 1 )
= nU,
M'
2 1
= n (v v +
~~I?J ,
which implies that the energy density can be w r i t t e n as the sum of a kinetic and a thermal c o n t r i b u t i o n : 2
tn\v\ 3, -Y~+2 B e k
T
^
93
2.3 M o m e n t M e t h o d s - - T h e H y d r o d y n a m i c M o d e l
F o r the determination o f the u n k n o w n s the first t w o equations and the trace of the t h i r d equation i n (2.3.3) are used. Straightforward but lengthy computations lead to the system, usually referred to as the h y d r o d y n a m i c semiconductor model: d,n + div(nl') = 0, k q + — g r a d ( « T ) + — E = {d v) ,
d v + (v-grad)F t
e
mn
t
m
c
n
,
7
(z.j./j
2 d T + - T div v + v grad T = (d t
e
e
e
T),
t
e
c
where we denoted dv _ dv _ dv - + v — + v — , v = (v v , v ). cx ox ox I f the terms on the right-hand sides, stemming from the collision terms, are omitted, then (2.3.7) are the Euler equations of gas dynamics for a gas o f charged particles i n an electric field. A weakness o f the ansatz, when applied to the semiconductor problem, is displayed by exactly these terms. They are given by 1 f (d v) vQ(F) dv, (vgrad)v
_ = v
l
T
2
1
t
3
u
2
2
3
3
c
m = 3/c n
(o,T )
e c
2
\v\ Q(F)
2m d v - — v 3k n
vQ(F)dv.
B
B
I n general, it is impossible to o b t a i n the dependence o f the integrals on the parameters explicitely. F o r the purpose o f simulation the collision terms are often replaced by relaxation time approximations. We refer the reader to [2.1] for a model which seems to meet w i t h approval i n the literature. The problems at the end o f this section shed light on the mathematical properties o f the h y d r o d y n a m i c semiconductor model. I n [ 2 . 2 ] , where the model (2.3.7) i n the context o f semiconductors has been introduced, an additional heat conduction term i d i v ( x grad T ) 3k n e
B
was added to the left hand side of the temperature c o n t i n u i t y equation. Here, y. denotes the heat conductivity o f the electron gas. The type o f the differential equations i n (2.3.7) changes at the transition from subsonic flow to supersonic flow. I n the supersonic regime the occurance of electron shock waves is possible. The interested reader can find a brief discussion of the nonlinear wave structure in [ 2 . 5 ] . A different model w i t h an account for energy flow has been proposed in [ 2 . 7 ] . I n [ 2 . 8 ] a simplified version has been derived by a perturbation argument, which can be interpreted as a modification of the drift diffusion model. Its special appeal lies i n the fact that high field effects are modelled in a way compatible w i t h experiment.
94
2 F r o m Kinetic to Fluid Dynamical Models
2.4 Heavy Doping Effects—Fermi-Dirac Distributions I n this Section we consider cases where the d i s t r i b u t i o n function is not necessarily small compared to one. Therefore a nonlinear collision term has to be used i n the B o l t z m a n n equation. A scaled version of the classical unipolar model (2.3.1) reads a dF
+ tx(vgrad F
2
t
- E• g r a d „ F) = Q(F),
x
(2.4.1)
where the collision integral is given by <j>(x, v, v'){MF'{\
— F) — M'F(\
- F')) dv'
(see Section 1.3) w i t h the M a x w e l l i a n exp(-M /2).
M(v) = (2ny
312
2
As i n Section 2.2, a denotes the scaled mean free path and we introduce a power series expansion of F in terms of a: F = F + aF + • • • . 0
1
The equation Q(F ) 0
= 0
implies [2.11] that the leading term is a Fermi-Dirac F
F (\v\ /2-Q>), 2
=
0
distribution:
D
where F
M
=
r
h
holds and O = (x, t) is the F e r m i energy (see Chapter 1). Equating coefficients of cc i n (2.4.1) gives F ( l - F )v(gmd 0
0
x
O + E) = L(
(2.4.2)
where L{Q>) is the Frechet derivative of Q evaluated at F : 0
{x, v, v')(M(f'(l
- F ) 0
F' f) 0
-M'(f{\-F' )-FJ'))dv'. 0
A result [2.11], which is i n the spirit of the L e m m a in Section 2.2, states that an equation of the form L(
(2.4.3)
has a solution i f and only i f gdv = 0
(2.4.4)
2.5 High Field Effects—Mobility Models
95
holds and that the n u l l space of L ( O ) is spanned by F ( l — F ) . I n [2.11] it is shown that the equation 0
L(
-
0
0
F) 0
has a solution w i t h the property that the m a t r i x v® h dv
n(
is positive definite. The solution of (2.4.2) can be w r i t t e n as F, = ( g r a d ,
0
2
Equating coefficients of a i n (2.4.1) gives an equation of the form (2.4.3) for F . The solvability c o n d i t i o n (2.4.4) implies 2
d,n-di\J
= 0,
(2.4.5)
where the scaled electron density and current density are given by n(O)
F dv,
J = FI( + E).
0
E q u a t i o n (2.4.5) is a nonlinear parabolic equation for the F e r m i energy <5>. Since the electric field can be expressed in terms of the electrostatic potential as E = —grad F t h e current density is p r o p o r t i o n a l to the gradient of the quasi-Fermi potential q>„ =
div(n( ) = 0 . n
The perturbation argument leading to the fluid dynamical model (2.4.5) has been justified i n [2.11]. A practical application w o u l d be facilitated by some, at least qualitative, knowledge of the dependence of the m a t r i x IT on its argument. A t present, the authors are not aware of results in that direction.
2.5 High Field Effects—Mobility Models As mentioned i n Section 2.2 the validity of the H i l b e r t expansions presented so far is restricted to the case of small electric fields. A totally new situation occurs i f the scaled electric field is large, say of the order of magnitude of x " . The appropriately rescaled B o l t z m a n n equation then reads 1
ad F t
+ xv • g r a d , F — F - g r a d , F = Q(F),
(2.5.1)
where we also introduced a time scale faster than the one used i n Section 2.2. F o r the collision term we choose a (linear) l o w density a p p r o x i m a t i o n . The leading term of the H i l b e r t expansion satisfies -£-grad .F l
0
= <2(F ). 0
(2.5.2)
This equation does not o n l y occur i n the context of the H i l b e r t expansion
46
2 F r o m Kinetic to F l u i d Dynamical Models
but is, by itself, of physical interest as a model for stationary, homogeneous situations. I t is well k n o w n that i t does not have an integrable solution i n general, w h i c h w o u l d be necessary for the definition of the position space number density n. The nonexistence of such a solution is called runaway phenomenon. F o r a mathematical analysis of related questions we refer the reader to [ 2 . 4 ] . The occurance of runaway depends o n the collision frequency X(v) =
dv'.
The following result can be found i n [2.12]: L e m m a : A necessary solution of (2.5.2) is
condition
for the existence
of a positive,
integrable
X(sE) ds = oo, i.e. the collision frequency field.
does not decay too fast in the direction of the electric
F o r the following we make the assumption that (2.5.2) has a positive solution M (v) w h i c h satisfies E
M (v)
dv = 1.
E
Then the leading term i n the H i l b e r t expansion has the form F
0
= n(x,
t)M (v). E
Equating coefficients of a i n (2.5.1) gives d (nM ) t
+ div(vM n)
E
E
- E-grad, F = Q(F ). 1
1
Integration i n the ^-direction implies d,n + div(?J(£)n) = 0, w i t h the average velocity defined by v(E) =
J
vM
E
R
dv.
3
(2.5.3) is a hyperbolic equation for n. Thus, n m i g h t have discontinuities. A similar situation occurs i n gas dynamics where the Euler equations allow for shocks. These shocks are eliminated by i n t r o d u c i n g viscosity terms and, thus, considering the Navier Stokes equations, w h i c h can be derived from the B o l t z m a n n equation of gas dynamics by the Chapman-Enskog method. A similar approach [2.12] leads to a diffusion term of order a i n the present situation: d n + di\(v(E)n t
— aD(E) grad n) = 0,
where D(E) is the (positive definite) diffusivity tensor.
97
2.5 H i g h Field E f f e c t s — M o b i l i t y M o d e l s
It is easy to see that U(0) = 0,
g r a d v(0) = -Hoh
<
£
(2.5.4)
0
holds for the average velocity, where the l o w field m o b i l i t y / i can be computed as i n Section 2.2. Unfortunately, it is impossible to obtain the dependence of v on the field explicitely. F o r simulation purposes it is common to use an ansatz fitted to experimental results. As a first step, it is certainly reasonable to write v i n the form 0
v(E) =
-n{\E\)E
w i t h /<(0) = n . Here (2.5.4) is taken i n t o account and it is assumed that the direction of the average velocity is given by the direction of the field. Experiments show the effect of velocity saturation at large electric fields: 0
l i m \v(E)\
=
v . sM
|E|->oc
A model for the m o b i l i t y w h i c h shows this behaviour has been derived in [2.7] and [ 2 . 8 ] : 2^o 1 +
x
(2.5.5)
/ l +(2/. |£|/t; 0
s a t
)
2
Several other models w h i c h are similar to the above are used (see [2.16] for an overview and references). I n numerical simulations it is c o m m o n to use a drift diffusion model w i t h a m o b i l i t y like i n (2.5.5) and to compute the diffusivity from the Einstein relations (2.2.15). However, it is likely that i n reality the Einstein relations are violated for high electric fields. I n particular, the diffusivity cannot be expected to decay for large electric fields. The transferred electron effect in two-valley semiconductors (e.g. GaAs) w i t h large effective mass of the electrons i n the upper valley leads to a n o n m o n o tone velocity-field relation (see Fig. 2.5.1). I t has already been mentioned
V
v
E a! F i g . 2.5.1 V e l o c i t y vs. field for (a) Si, (b) G a A s
E b)
98
2 F r o m Kinetic to Fluid Dynamical Models
that large effective mass means l o w m o b i l i t y . This explains the effect on the velocity-field relation because for higher electric fields the electron density i n the upper band increases (see [2.17] for details). I n these cases (2.5.5), w h i c h is an acceptable model for Si and Ge, has to be changed accordingly.
2.6 Recombination-Generation Models The recombination-generation rate R = A(np -
nf)
derived i n Section 2.2 is a model for direct band-band recombination caused by p h o t o n transitions. I t is well k n o w n that other recombination-generation mechanisms are m u c h more i m p o r t a n t i n semiconductor devices. I n this Section we discuss three such mechanisms w h i c h are usually taken i n t o account i n semiconductor device modelling. This w i l l , however, n o t be done on the level of the B o l t z m a n n equation. Instead, models w i l l be presented which can be used directly i n the fluid dynamical equations. The first mechanism to be considered is Auger recombination. T w o different processes are shown schematically i n F i g . 2.6.1: a) Electron capture: A n electron moves from the c o n d u c t i o n band to the valence band and recombines w i t h an hole there. Its energy is transferred to another electron i n the c o n d u c t i o n band. b) Hole capture: A n electron moves from the c o n d u c t i o n band to the valence band and recombines w i t h an hole there. Its energy is transferred to another hole i n the valence band. The processes acting i n the opposite directions: c) electron emission, d) hole emission are also possible. The rates o f these processes i n position space are modelled by mass action laws. W i t h the l o w density assumption they are given by
conduction band
1
v a l e n c e band l)
t)
Fig. 2.6.1 A u g e r r e c o m b i n a t i o n
99
2.6 R e c o m b i n a t i o n - G e n e r a t i o n M o d e l s conduction band
t r a p level
v a l e n c e band a)
b)
Fig. 2.6.2 B a n d - t r a p c a p t u r e
a) b) c) d)
2
C„n p, C np , C n, C , 2
p
n
pP
The principle of detailed balance (see Section 1.3) states that each process balances its counterpart i n thermal e q u i l i b r i u m . The thermal e q u i l i b r i u m c o n d i t i o n np = nf derived i n Section 2.2 implies C
2
C„ = n C „
2
=
nC
The t o t a l Auger recombination-generation rate can then be w r i t t e n as K a u = (C„n +
(2.6.1)
C p){np p
Next we consider band-trap capture and emission. These are i m p o r t a n t in the presence o f impurities which generate additional energy (trap) levels i n the forbidden band. The processes shown i n Fig. 2.6.2 are a) Electron capture: A n electron moves from the conduction band to an unoccupied trap. b) Hole capture: A n electron moves from an occupied trap to the valence band. A n hole disappears. Again, the processes i n the opposite direction are also possible. The rates are given by a) C n(N n ), b) C pn , c) Cn, d) Q ( i V - n ) , a
tI
tr
b
c
lr
u
tr
t r
where N denotes the density of traps and n the density of occupied traps. The densities n, p, and n satisfy the differential equations lr
tr
lr
d n - - div J„ = C n t
c
d,p -
1
a
div J = C {N p
cl,n = C n(N tr
- C n(N
tI
a
lt
d
tI
tr
- n) tt
- n ) - Qn tr
t r
-
n ), tr
(2.6.2)
C pn , b
+ C„pn
tT
tI
- C (N d
tI
-
n ). tt
100
2 F r o m Kinetic to Fluid Dynamical Models
Since the impurities are assumed to have fixed positions, the trapped electrons do not contribute to the current flow. I n the classical theory of band-trap transitions, due to Shockley, Read [2.15], and H a l l [ 2 . 6 ] , it was assumed that the relaxation of n towards e q u i l i b r i u m happens m u c h faster than the relaxation of n and p. A l t h o u g h the authors are not aware of a rigorous j u s t i f i c a t i o n of this assumption i t w i l l be adopted here because the resulting recombination-generation m o d e l has been generally accepted. I f a moderate time scale is considered, this assumption justifies setting d n = 0 i n (2.6.2). F r o m the resulting algebraic equation n can be computed: tI
t
u
tI
Cn +
Q
a
Cn + C + Cp + C a
c
b
d
This implies for the Shockley-Read-Hall (SRH) recombination-generation rate: np-n p 1
x (n
l
+ n j + T„(p +
p
'
where the densities n and p are given by i
ni = C /C , c
1
p
a
C /C
=
1
d
b
and the carrier life times T„ and x by p
T„ = (C N y , a
t[
r
l
p
=
( C M '
1
.
The requirement that the recombination-generation rate vanishes i n thermal e q u i l i b r i u m implies n p = nf. A more detailed analysis shows that n and p depend on the l o c a t i o n o f the trap level [2.17]. I n particular, 1
1
x
1
holds i f the trap level is i n the middle o f the forbidden band. Strictly speaking, the above considerations for Auger and S R H recombinat i o n are o n l y v a l i d close to thermal e q u i l i b r i u m and for small electric fields. This observation is o f particular importance for the generation process corresponding to Auger r e c o m b i n a t i o n at high electric fields. A n effect called impact ionization w h i c h cannot be modelled by (2.6.1) is observed. A phenomenological description is p r o v i d e d by the c o m m o n l y used m o d e l (see [2.16]) R =-*„\J \/q-a \ n
a
p
J \/q,
(2.6 A)
p
where the i o n i z a t i o n rates <x„ and a are strongly field dependent. A simple choice is the so called lucky drift model [2.14]: p
a„ = a * e x p ( - £ f 7 | E | ) ,
<x = o £ e x p ( - £ p
d p
7|£|),
(2.6.5)
101
Problems A
where a* and oc^ are m a x i m a l i o n i z a t i o n rates, and £ " " and E" field strengths.
are critical
Problems 2.1 D e r i v e the f o l l o w i n g e q u a t i o n for the rate o f change o f the energy: d S
dt
m = ~ 2
JEdx
dx
f r o m (2.3.2), (2.3.3). Assume t h a t M
1
2
\v\ Q(F)
3
)
dvdx
vanishes as | x | - » oo.
2.2 D e f i n i t i o n : A flow is called i r r o t a t i o n a l i f its velocity vector satisfies c u r l v = 0. Simplify the velocity e q u a t i o n o f the h y d r o d y n a m i c m o d e l (2.3.7) assuming that the flow o f the electron gas is i r r o t a t i o n a l . Hint: P r o v e ( f J g r a d ) ? = \ g r a d (|t>| ) — v x c u r l v. 2
2.3 D e f i n i t i o n : A flow is called incompressible i f its velocity vector satisfies d i v v = 0. Simplify the h y d r o d y n a m i c m o d e l (2.3.7) assuming t h a t the flow o f the electron gas is incompressible. T a k e the r e l a x a t i o n m o d e l for the temperature: (d,T )
e c
(T e
=
T)
x
,
T
where T > 0 denotes the (constant) lattice t e m p e r a t u r e a n d x > 0 the (constant) t e m perature r e l a x a t i o n t i m e . Solve the electron c o n t i n u i t y e q u a t i o n a n d the t e m p e r a t u r e e q u a t i o n (in terms o f the v e l o c i t y field v) w i t h the i n i t i a l d a t a T
n(x,t T (x, e
= 0) = n,(x),
x e U \
t = 0) = TAx),
x e R \
2.4 C o n s i d e r the steady state h y d r o d y n a m i c m o d e l (2.3.7), i.e. set d,n = 0, 8,v = 0, d,T = 0. Assume infinitely fast t e m p e r a t u r e r e l a x a t i o n ( 0 is an a r b i t r a r y constant, is a s o l u t i o n o f the t e m p e r a t u r e e q u a t i o n . Remark: p = k nT is called pressure o f the electron gas a n d a r e l a t i o n o f the f o r m p = p(n) is called e q u a t i o n o f state. T h u s , under the above assumptions, the electron gas has the e q u a t i o n o f state p = Kk n . e
2ji
e
B
c
e
e
sli
B
2.5 C o n s i d e r the o n e - d i m e n s i o n a l steady state h y d r o d y n a m i c m o d e l (2.3.7) w i t h an e q u a t i o n of state p = p(n) (see P r o b l e m 2.4) a n d w i t h the velocity r e l a x a t i o n t e r m v (B,v) =
,
c
where the velocity r e l a x a t i o n t i m e x , is positive. P r o v e t h a t the n o n l i n e a r c u r r e n t dependent drift-diffusion e q u a t i o n c
2
J = qn
m J 1 - — + -p(«) q n q T
+nE
holds w i t h the electron m o b i l i t y jx = x^qjm. Hint: N o t e that J = const, holds i n the o n e - d i m e n s i o n a l steady state case! Remark: W e conclude t h a t the "classical" drift-diffusion m o d e l (2.2.14) is o b t a i n e d f r o m the h y d r o d y n a m i c m o d e l (at least i n the steady state o n e - d i m e n s i o n a l case w i t h velocity
102
2 F r o m Kinetic to Fluid Dynamical Models 2
r e l a x a t i o n ) by neglecting terms o f o r d e r J (small c u r r e n t l i m i t ) a n d using the e q u a t i o n o f state p(n) = (U q)n (linear pressure-density relationship). r
2.6 D e f i n i t i o n : (i) A flow is called subsonic, i f its velocity field satisfies \v\ <
Jp'irijm.
i) is called soundspeed o f the flow. (ii) L e t 4
F:
R
-> R,
F = F(r, q, u, x)
be a f u n c t i o n . T h e e q u a t i o n F(u ,
u , u, x) = 0
xx
x
is called (everywhere) e l l i p t i c i f cFjdr
> 0 on
4
R.
Prove t h a t the drift-diffusion e q u a t i o n o f P r o b l e m 2.5 together w i t h the c o n s e r v a t i o n e q u a t i o n J = 0 leads t o an e l l i p t i c e q u a t i o n for the density n i f a n d o n l y i f the flow is subsonic. x
References [2.1] [2.2] [2.3] [2.4]
[2.5]
[2.6] [2.7] [2.8] [2.9] [2.10] [2.11] [2.12]
[2.13] [2.14] [2.15]
G . Baccarani, M . R. W o r d e m a n : A n I n v e s t i g a t i o n o f Steady-State V e l o c i t y O v e r s h o o t Effects i n Si a n d G a A s Devices. S o l i d State Electr. 28, 4 0 7 - 4 1 6 (1985). K . Blotekjaer: T r a n s p o r t E q u a t i o n s for Electrons i n T w o - V a l l e y S e m i c o n d u c t o r s . I E E E T r a n s . E l e c t r o n . Devices ED-17, 3 8 - 4 7 (1970). C. C e r c i g n a n i : T h e B o l t z m a n n E q u a t i o n a n d Its A p p l i c a t i o n s . S p r i n g e r - V e r l a g , N e w Y o r k (1988). G . F r o s a l i : F u n c t i o n a l - A n a l y t i c Techniques i n the S t u d y o f T i m e - D e p e n d e n t E l e c t r o n Swarms i n W e a k l y I o n i z e d Gases. P r e p r i n t , I s t i t u t o d i M a t e m a t i c a , U n i v e r s i t a d i A n c o n a (1988). C. L . G a r d n e r , J. W . Jerome, D . J. Rose: N u m e r i c a l M e t h o d s for the H y d r o d y n a m i c Device M o d e l : Subsonic F l o w . Presented at the m e e t i n g o n M a t h e m a t i s c h c M o d e l l i e r u n g u n d S i m u l a t i o n elektrischer Schaltungen, O b e r w o l f a c h (1988). R. N . H a l l : E l e c t r o n - H o l e R e c o m b i n a t i o n i n G e r m a n i u m . Physical Review 87, 387 (1952). W . H a n s e n , M . M i u r a - M a t t a u s c h : T h e H o t - E l e c t r o n P r o b l e m i n S m a l l Semic o n d u c t o r Devices. J. A p p l . Phys. 80, 6 5 0 - 6 5 6 (1986). W . H a n s e n , C. Schmeiser: H o t E l e c t r o n T r a n s p o r t i n Semiconductors. Z A M P 40, 4 4 0 - 4 5 5 (1989). D . H i l b e r t : M a t h . A n n . 72, 562 (1912). B. N i c l o t , P. D e g o n d , F . P o u p a u d : D e t e r m i n i s t i c Particle S i m u l a t i o n s o f the B o l t z m a n n T r a n s p o r t E q u a t i o n o f Semiconductors. J. C o m p . Phys. 78, 3 1 3 - 3 5 0 ( 1 9 8 8 ) . F. P o u p a u d : Etude M a t h e m a t i q u e et S i m u l a t i o n s N u m e r i q u e s de Quelques E q u a tions de B o l t z m a n n . Thesis, U n i v . Paris 6 (1988). F. P o u p a u d : R u n a w a y P h e n o m e n a a n d F l u i d A p p r o x i m a t i o n U n d e r S t r o n g Electric F i e l d i n S e m i c o n d u c t o r T h e o r y . M a n u s c r i p t , L a b o r a t o i r e de M a t h e m a t i q u e s , U n i v e r s i t e de N i c e (1988). C. A . Ringhofer: A Spectral M e t h o d for N u m e r i c a l S i m u l a t i o n o f Q u a n t u m T u n n e l i n g P h e n o m e n a . S I A M J. N u m e r . A n a l . (1989) (to appear). W . Shockley: P r o b l e m s Related t o p-n J u n c t i o n s i n S i l i c o n . S o l i d State Electr. 2, 35-67(1961). W . Shockley, W . T . Read: Statistics o f the R e c o m b i n a t i o n s o f Holes a n d Electrons. Physical Review 87, 8 3 5 - 8 4 2 (1952).
References
103
[ 2 . 1 6 ] S. Selberherr: Analysis a n d S i m u l a t i o n o f S e m i c o n d u c t o r Devices. S p r i n g e r - V e r l a g , W i e n - N e w Y o r k (1984). [ 2 . 1 7 ] S. M . Sze: Physics o f S e m i c o n d u c t o r Devices, 2 n d edn. J o h n W i l e y & Sons, N e w Y o r k (1981). [ 2 . 1 8 ] W . V . v a n Roosbroeck: T h e o r y o f F l o w o f Electrons a n d H o l e s i n G e r m a n i u m a n d O t h e r S e m i c o n d u c t o r s . Bell Syst. T e c h n . J. 29, 5 6 0 - 6 0 7 (1950).
The Drift Diffusion Equations
3.1 Introduction The drift diffusion equations are the most widely used model to describe semiconductor devices today. The b u l k o f the literature on mathematical models for device simulation is concerned w i t h this nonlinear system o f partial differential equations and numerical software for its solution is commonplace at practically every research facility i n the field. F r o m an engineering p o i n t of view, the interest i n the drift diffusion model is to replace as m u c h laboratory testing as possible by numerical s i m u l a t i o n i n order to minimize costs. T o this end, it is i m p o r t a n t that computations can be performed i n a reasonable amount o f time. This implies that the involved mathematical models cannot be t o o complicated, such as, for instance, the higher dimensional transport equations described i n Chapter 1. F o r the current state o f technology the drift diffusion equations seem to represent a reasonable compromise between c o m p u t a t i o n a l efficiency and an accurate description of the underlying device physics. Therefore transport equations are used mainly to compute data for the model parameters i n the drift diffusion equations i n the engineering environment. I t should be pointed out, however, that, w i t h the increased m i n i a t u r i z a t i o n o f semiconductor devices, one comes closer and closer to the limits o f validity o f the drift diffusion equations, even i n an i n d u s t r i a l environment. The reason for this is, o n one hand, that i n ever smaller devices the assumption that the free carriers can be modelled as a c o n t i n u u m becomes invalid. O n the other hand the drift diffusion equations are derived t h r o u g h a l i m i t process where the mean free path of a particle tends to zero. T h r o u g h m i n i a t u r i z a t i o n and the use o f materials other than silicon this mean free path becomes larger and larger i n comparison to the size o f the device. I n a d d i t i o n , q u a n t u m mechanical effects start to play a more and more i m p o r t a n t role i n novel device structures. F o r these reasons, and because of the r a p i d increase i n available c o m p u t i n g power, transport equations w i l l be used more and more i n device simulation i n the future. B u t even then the drift diffusion equations w i l l remain an i m p o r t a n t t o o l since the microscopic effects not described by them occur only locally. Thus, the most likely approach w i l l be to use more
105
3.1 I n t r o d u c t i o n
sophisticated models only locally, for instance i n the channel of a M O S transistor (see Chapter 4), and to use the drift diffusion equations in the parts of the device where they suffice to describe the physics. I n this Chapter we w i l l discuss the analytical properties of the drift diffusion equations. We w i l l m a i n l y be interested in the structure of their solutions. O f course this structure w i l l strongly depend on the underlying geometry; i.e. on the device under consideration. We w i l l , however, not discuss specific devices i n this Chapter but leave their discussion to Chapter 4. So, we w i l l consider only P-N j u n c t i o n s and w i l l not concern ourselves w i t h how different types o f devices can be made up by configurations of these P-N junctions. We consider the system o f p a r t i a l differential equations a) div(« grad V) = q(n — p — C) b) div J„ = q(8 n + R) t
c) div J = q(-d,p-R)
(3.1.1)
p
d) J„ = q(D„ grad n - fi n grad V) n
e) J = q(-D p
p
grad p - p. p grad V), p
where V denotes the electric potential ( — grad V is the electric field.), n and p are the concentrations o f free carriers of negative and positive charge, called electrons and holes, and J and J are the densities of the electron and the hole current respectively. D , D , p. and \i are the diffusion coefficients and mobilities o f electrons and holes respectively, e is the p e r m i t t i v i t y constant whose approximate value i n silicon is 1 0 ~ As V " c n T . q is the elementary charge whose value is approximately 1 0 ~ As. We assume the device geometry to be given by a d o m a i n Q _\ U w i t h d = 1, 2 or 3. Physically d = 3 holds, o f course. F o r many devices, however, it suffices to consider t w o dimensional models (d = 2) since their extension in one dimension is m u c h larger than in the other t w o . Even one dimensional models are sometimes used today. The b o u n d a r y <9Q o f the d o m a i n Q is assumed to consist o f a D i r i c h l e t part dfl and a N e u m a n n part dCl : n
p
n
p
n
p
12
1
1
19
d
D
<9Q = c f i
D
u <9Qjv ,
N
dQ
D
n cQ
N
= { }.
(3.1.2)
The Dirichlet part dQ. o f the boundary corresponds to Ohmic contacts. There the potential V and the concentrations n and p are prescribed. The boundary values are derived from the following considerations. A t O h m i c contacts the space charge, given by the right-hand side of (3.1.1)a) vanishes. So D
n - p - C = 0
for
xedQ
D
(3.1.3)
holds. F u r t h e r m o r e the system is i n thermal e q u i l i b r i u m there, which is expressed by the relation np = nf
for
x e
dQ . D
(3.1.4)
106
3 The Drift Diffusion Equations 1 0
3
n is the intrinsic density (; 1 0 c m i n silicon at r o o m temperature). Moreover, the quasi F e r m i levels <j>„ and <j) , given by i
p
V - U In
n Jh
(3.1.5)
= V + U In
b)
T
T
assume the values o f the applied voltage at O h m i c contacts. Here U denotes the thermal voltage which, at r o o m temperature, is roughly 0.025 V . F r o m the conditions (3.1.3)—(3.1.5) the b o u n d a r y values for V, n and p can be uniquely determined. Inserting (3.1.4) i n t o (3.1.3) gives one quadratic equat i o n for n and p each, w h i c h have unique positive solutions given by T
b)
n(x, t) =
n (x)
HC(x)
+ VC(x)
p(x, t) =
p (x)
\(-C(x)
+ JC(x)
for
D
D
2
2
+ 4n, )
2
+ 4nf)
xe8Q . D
(3.1.6)
(3.1.5) gives the b o u n d a r y values for the potential V: c)
V(x, t) = V (x, t) = U(x, t) + D
V (x) bl
U In
V (x) bi
for
T
x e
d£l . D
U(x, t) denotes the applied potential. So, differences i n U(x,t) between different segments o f 8Q. correspond to the applied bias between these contacts. Note, that (3.1.4) immediately implies that <j> equals <j> at O h m i c contacts. The N e u m a n n parts dQ o f the boundary model insulating or artificial surfaces. Thus a zero current flow and a zero electric field in the n o r m a l direction are prescribed. D
n
p
N
dV a)
(x, t) (:= grad V-v) = 0
cv b)
J„(x, t)-v
0,
c)
J (x, t) • v
0
p
(3.1.7) for
x e <3Q
A
I n this Chapter v w i l l always denote the unit o u t w a r d n o r m a l vector on the boundary cQ. I n a d d i t i o n the concentrations o f the free carriers n and p at time r = 0 are prescribed. n(x, 0) = n'(x),
p(x, 0) = p'(x)
for
x e Q
(3.1.8)
holds and the complete i n i t i a l b o u n d a r y value p r o b l e m is given by the equations (3.1.1), the boundary conditions (3.1.6), (3.1.7) and the i n i t i a l conditions (3.1.8). This setting is not sufficient to describe devices like M O S transistors which, i n a d d i t i o n , contain an oxide layer attached to the semiconductor. I n the oxide a different set of equations holds and interface conditions are given at the semiconductor oxide interface. We w i l l leave the discussion of this case to Chapter 4.
3.1 I n t r o d u c t i o n
107
Various models for the recombination rate R i n (3.1.1)b)c) can be found in the literature (see [3.34]). I n this Chapter we w i l l , for the sake o f simplicity, only consider the Shockley Read H a l l term which is o f the form z (n p
+
Hi) +
T (p + n ) ' t
n
Here, again, n denotes the intrinsinc density. T„ and z are the lifetimes o f electrons and holes respectively. Other models, such as the Auger- or the impact i o n i z a t i o n model can be found in Chapter 2. We w i l l always assume the mobilities and the diffusion coefficients to satisfy the Einstein relations t
p
D„ = n„U ,
D = pU,
T
p
p
(3.1.10)
T
where U ( = 0.025 V ) is the thermal voltage. There is a variety of models for the mobilities fi and u . They can be grouped i n t o t w o different categories which have to be treated i n different ways analytically. I n one case they are simply functions o f position. I n the other case they are modelled as dependent o n the electric field —grad V in order to represent so called velocity saturation effects. The field dependent m o b i l i t y models w i l l be discussed i n detail i n the corresponding Section of this Chapter. We refer the reader to [3.34] for a discussion of different m o b i l i t y models. In this Chapter we w i l l restrict ourselves to the analysis of P-N junctions. The term P-N j u n c t i o n refers to the sign change o f the d o p i n g concentration C(x) in (3.1.1)a). This d o p i n g concentration is produced by diffusion of different materials i n t o the silicon crystal and by i m p l a n t a t i o n w i t h an i o n beam. This produces a preconcentration o f ions i n the crystal which is modelled by the function C(x). So C(x) = C ( x ) — C_(x) holds where C and C are the concentrations o f negative and positive ions respectively. Where the preconcentration o f negative ions predominates i n Q, C(x) < 0 holds and these subregions o f Q are called F-regions. Similarly, i n the N-region, where the preconcentration o f positive ions dominates, C(x) > 0 holds. The boundaries between the N- and the F-regions, i.e. the manifolds where C(x) changes its sign, are called P-A/junctions. F o r the simplest P-N j u n c t i o n device, the P-N diode, the geometrical configuration i n the t w o dimensional case is depicted in F i g . 3.1.1. T
n
p
+
+
3fi
D 2
Fig. 3.1.1 P-N
diode
108
3 The Drift Diffusion Equations
The function o f a P-N j u n c t i o n diode is, roughly speaking, that o f a valve. I f a potential difference of an appropriate sign, that means a difference in the values of U(x, f) i n (3.1.6)c) between the Dirichlet boundary parts dQ and cQ. in F i g . 3.1.1 is applied to the P-N j u n c t i o n , a so called depletion region w i l l form around the P-N j u n c t i o n . There, very few free carriers w i l l exist and n = 0 and p = 0 w i l l hold i n (3.1.1), inside the depletion region. The depletion region w i l l effectively act as an insulator and no, or very little, current w i l l flow. I f the potential difference is applied the other way a r o u n d the depletion region w i l l vanish and current w i l l flow. This is the simplest type of semiconductor device and w i l l be used i n this Chapter to explain the analytical features of the drift diffusion equations. D1
D2
The m a m difficulties i n the treatment of the drift diffusion equations are, on one hand, their nonlinear nature and, on the other hand, the extreme differences i n the magnitude of the involved quantities. These differences lead to an almost singular type of behaviour o f solutions of the drift diffusion equations. Solutions of the drift diffusion equations w i l l exhibit an extreme layer behaviour i n the spatial as well as in the time direction and w i l l , therefore, be amenable to singular perturbation analysis. I n this Chapter we w i l l analyze the drift diffusion equations by means o f such an asymptotic analysis. This w i l l provide an understanding o f the types of mechanisms which are i m p o r t a n t i n different subregions o f the device d o m a i n Q and on different time scales. Different approaches have to be used depending on the biasing conditions and the geometries under consideration. We w i l l keep the discussion i n rather broad terms i n order for the results to be applicable to as wide a range o f geometrical configurations, and therefore as wide a range of devices, as possible. Excerpts of the analytical machinery developed in this Chapter w i l l be used to treat specific devices in Chapter 4.
3.2 The Stationary Drift Diffusion Equations The majority o f the analytical and c o m p u t a t i o n a l w o r k on the drift diffusion equations so far has been concerned w i t h the stationary p r o b l e m . This means that the drift diffusion equations (3.1.1) are considered at a steady state and that the time derivatives c n and d p i n (3.1.1)b)c) are neglected. So we consider the p r o b l e m t
t
a)
div(e grad V) = q(n — p — C)
b)
div J = qR
c)
div J = -qR
d)
J„ = q{D grad n - p n grad V)
e)
J = q(~D
n
(3.2.1)
p
n
p
n
p
grad p - p p grad V), p
together w i t h the boundary conditions
3.2 T h e S t a t i o n a r y D r i f t D i f f u s i o n
Equations
109
a)
n(x) = n (x) = ±(C(x) + JC(x)
+
b)
p(x) = p (x) = i ( - C(x) + ^ C ( x )
c)
F(x) = K ( x ) = C/(x) +
2
D
D
u
=
ln(^^j
bi
for
d)
^ ( x ) ( : = g r a d V-v) = 0 dv
e)
J„(x) • v = 0,
f)
xe5Q
7 (x) • v = 0 p
+ 4B?)
V (x)
D
V {x)
2
4nf)
(3.2.2)
D
for
x 6 5fi . N
F r o m a practical point o f view, the interest i n the stationary drift diffusion equations lies i n the dependence o f the current densities J and J on the applied bias, the d o p i n g concentration C(x) and the geometry. F o r instance, one tries, t h r o u g h v a r i a t i o n of these parameters, to minimize the so called leakage current. T h a t is the small current still flowing t h r o u g h a reverse biased P-N j u n c t i o n . While the tools to treat such a complicated problem, as the drift diffusion equations in higher dimensions, have to be computational, a great deal of insight can be gained by analysis. The development of numerical methods for the drift diffusion equations benefits greatly from the analytical understanding o f the solutions, and might even be impossible w i t h o u t it. I n the following Sections we are interested in four basic questions about the stationary drift diffusion equations: n
1. Does a solution exist, and, i f yes, h o w smooth is it? 2. W h a t is the structure of solutions of the stationary drift equations? 3. W h a t are the stability properties of these solutions?
p
diffusion
Question 1 is o f a general mathematical interest. I t turns out that solutions exist and lie in the function spaces one w o u l d expect them to. Question 2 is o f extreme importance for the development of numerical methods. As mentioned earlier, solutions o f the drift diffusion equations exhibit layer structure. T h a t means that they have large gradients locally, usually near the P - N junctions. The performance o f numerical methods can be i m p r o v e d a great deal if they are adapted to this layer behaviour. As had to be expected, the structure of solutions to the drift diffusion equations looks quite differently for forward and reverse biased P-N junctions. Thus, these t w o cases are treated separately i n the subsequent Sections. The stability properties of the stationary drift diffusion equations are extremely dependent on the geometry and not analyzed satisfactorily at this point. We w i l l present the available results in Section 3.6. There it turns out that the c o n d i t i o n i n g of the drift diffusion equations is acceptable if every N- and P-region contains a contact. I f this is not the case, i.e. i f so called floating regions are present, the stability properties can worsen drastically.
110
3 The Drift Diffusion
Equations
3.3 Existence and Uniqueness for the Stationary Drift Diffusion Equations Before analyzing the structure and the properties of solutions to the equations obtained by the steady state drift diffusion a p p r o x i m a t i o n i n the subsequent Chapters, we w i l l first briefly discuss the existence of these solutions. F r o m a practical p o i n t of view one w o u l d prefer existence results based on the implicit function theorem. Such results w o u l d provide inform a t i o n about the n o r m of the inverse of the linearized problem. They w o u l d show the existence of an isolated solution and, more i m p o r t a n t l y , give an indication of the c o n d i t i o n i n g of the p r o b l e m and of the convergence p r o p erties of Newton's method. The c o n d i t i o n i n g of the steady state drift diffusion equations has been investigated i n [ 3 . 3 ] and various papers in the literature ( [ 3 . 5 ] , [3.17], [3.28], [ 3 . 2 9 ] ) deal w i t h the convergence of iterative methods. However, they do not treat the whole coupled system of equations or they assume the existence of a suitable a p r i o r i b o u n d o n the inverse of the linearization. A t the end of this Section we w i l l briefly discuss these results. The o n l y existence results available for the coupled p r o b l e m and arbitrarily large bias are not constructive since the arguments are based on the Schauder Fixed P o i n t Theorem i n one way or the other. I n this Chapter we w i l l give an example for an existence theorem for a simplified case (Theorem (3.3.16)) i n order to outline the basic approach. We refer the reader to the literature ( [ 3 . 4 ] , [3.10], [3.19], [ 3 . 2 4 ] ) for results i n more general cases. Since the scaling is of no particular importance for the purposes of this Section we w i l l treat the drift diffusion equations i n an unsealed form for the moment. So we consider the system a)
8 AV=
b)
div J = qR,
d)
d i v J = -qR,
— C{x))
q(n — p
c)
n
J„ = q(D„ grad « — p„n grad V) e)
p
J = q(-D p
(3.3.1)
grad p - p p grad V).
p
p
Equations (3.3.1) have the disadvantage of containing the convection terms — n grad V and — p grad V w h i c h p r o h i b i t the use of the m a x i m u m principle in a simple way. I f the Einstein relations D =
Up
n
T
D
}
p
=
(3.3.2)
Up T
p
can be assumed, w i t h U the thermal voltage (see [3.35]), it is beneficial to change from the concentrations n and p to the so called S l o t b o o m variables u and v given by T
a)
y,VT
n = n e u,
b)
i
p = n e -v/u ;
T
V.
(3.3.3)
The current relations then become a)
riVT
J„ = qU n pi„e T
grad u,
t
(3.3.4) b)
vlv
J = -qU n p e~ T p
T
i
p
grad v.
3.3 Existence a n d Uniqueness for the S t a t i o n a r y D r i f t Diffusion
111
Equations
After inserting the current densities J„ and J i n t o the c o n t i n u i t y equations (3.3.l)b)d) one obtains the elliptic system p
a)
VIUr
e AV = q n ^ ' a V!V
b)
Un
c)
Un
T
div{u e >
t
T
- e~ v)
qC(x)
grad u) = R
n
ViU
dW(u e- '
t
-
(3.3.5)
grad v) = R.
p
I n this form the c o n t i n u i t y equations (3.3.5)b)c) are self adjoint. I n the S l o t b o o m variables u and v the boundary conditions (3.2.2)d)e)f) at artificial or insulating surfaces become pure N e u m a n n conditions dV
du
dv
0.
dv
(3.3.6)
F o r O h m i c contacts we obtain from (3.2.2)a)b)c) V\fa =V \ a , B
D f
l
v
u\
B
= u\ ,
enD
v\g
D eiiB
IVT
1
v
n o
= v\ , D SsiD
(3.3.7)
!Vr
w i t h u = nj e~ " n and v = n^ e " p . We remark that, since n and p represent physical concentrations, the Slotboom variables u and v have to remain positive. Existence theorems for the problem (3.3.5)—(3.3.7) usually employ the Schauder Fixed Point Theorem. The construction of the fixed point map depends on the form of the recombination rate, the mobilities, the geometry and so on. T o outline the basic idea we w i l l give an existence p r o o f following the approach in [3.19] but we w i l l use some simplifying assumptions. We w i l l consider the Shockley Read H a l l recombination term only. So after changing variables to (F, u, v) the recombination rate R i n (3.3.5) is o f the form D
D
„ ni
v
v
x (e > 'u p
D
D
uv - 1 + 1) + x (
n
[ X X
+ 1) •
v
*>
We assume that the mobilities //„ and p. are uniformly bounded functions of position o n l y and that p
0 < \x
n
n„(x) ^ p , n
0 < p. < n (x) p
p
< p
p
V.veQ
(3.3.9)
hold. F u r t h e r m o r e we w i l l take the boundary cQ and the boundary data V , u , and v i n (3.3.7) to be as smooth as necessary. We reiterate that existence results for more general situations can be found in the literature (see [3.19, Chapter 3 ] ) . F o r instance the form of the recombination rate can be taken so as to include the Auger recombination term w i t h o u t any additional problems. The boundedness of the mobilities away from zero is a more severe restriction. I t excludes the field dependent mobilities used to model velocity saturation effects. A c o n d i t i o n of the form (3.3.9) is necessary, however, to guarantee the u n i f o r m ellipticity of the continuity equations. Therefore most existence proofs do assume an a p r i o r i b o u n d o n the mobilities even when modeling them as dependent on the D
D
D
112
3 The Drift Diffusion Equations
field —grad V. O n the other hand the inclusion o f an oxide layer, where Laplace's equation has to be solved for V, does not pose a major p r o b l e m . The fixed point m a p is constructed such that its evaluation only involves the solution o f semilinear or linear scalar b o u n d a r y value problems. Let G be given by G(u , v ) = (u , where (u , f ) is computed from (u , v ) as follows. 0
0
l
1
1
0
0
Step 1: Solve Poisson's equation v
r,AV
+ qnAe ^u
- e
Q
_ K / p
' » ) - qC{x) = 0 0
(3.3.10)
dV
0,
a7 for V = V . L
Step 2: Solve v,,UT
U
a)
div(p e
T
grad u)
n
- 1
UVr
+ T<e ^u
= 0
+ 1) + x {e -Vi[U
v
0
T
n
(3.3.11)
1)
du = 0,
b)
\SC1
D
= uD\dn
D
for u = u . 1
Step 3: Solve div(p„e
a)
V i I V t
1
+ x (e ^u v
p
grad v)
+ 1) + T {e-™*v
0
H
0
+ 1) (3.3.12)
dv
0,
b)
v D\Pa
D
oil,
for v By solving the b o u n d a r y value problems (3.3.10)—(3.3.12) we mean a solution in the usual weak sense (see [3.12]). O b v i o u s l y a fixed p o i n t o f the nonlinear operator G is a weak s o l u t i o n o f the coupled p r o b l e m (3.3.5)—(3.3.7). The existence o f such a fixed p o i n t is established by showing that the map G is completely continuous and by a p p l y i n g the Schauder Fixed Point Theorem. O f course, for this approach one has to choose an appropriate space for defining G. W e w i l l leave this question for later (for Theorem (3.3.16)) and first convince ourselves that the m a p G is well defined; that means that the involved b o u n d a r y value problems are uniquely solvable. A l l three problems (3.3.10)—(3.3.12) can be w r i t t e n i n the general f o r m — div(a(x) grad w) + f(x, w) = 0,
(3.3.13)
dw
a7
x e Q
=
0,
w \ d a
D
=
w
D
\ g n
D
,
3.3 Existence a n d Uniqueness for the S t a t i o n a r y D r i f t Diffusion E q u a t i o n s
113
where w takes the place of V, u and v respectively. The coefficient a(x) in (3.3.13) is either the constant e or equal to p. e or u c ' ' . I n any case it is uniformly bounded away from zero if JX„ and p. are and i f V is bounded, which makes the semilinear equation (3.3.13) uniformly elliptic. f(x, w) is a monotone increasing function of w i n all three cases (3.3.10) (3.3.12) if u and v are positive. I n (3.3.11) and (3.3.12) / is linear in w. The existence of a unique solution of semilinear partial differential equations of the type as in (3.3.13) is, under certain assumptions, a standard result in the theory of elliptic partial differential equations. We w i l l state i n L e m m a 3.3.14 such a result i n a form suitable for our purposes and refer the reader to [3.19] for the proof. The result of L e m m a 3.3.14 requires that the coefficient a(x) i n (3.3.13) is in the space L * ( Q ) , i.e. that a(x) is bounded uniformly i n Q. The solution w(x) w i l l lie in the intersection of the spaces L * (Q) and H (Q). H (Q) is the space of functions w h i c h are square integrable and whose gradient is square integrable as well. So VilVl
1
n
(
n
p
x
0
0
1
2
(w(x)
2
+ \Vw(x)\ )dx
1
< oo
n holds. Lemma 3.3.14: Let the following assumptions hold: A l ) The function f(x, w) is monotonically increasing in w for all x e Q. A2) a(x) e L (Q) and a(x) ^ a > 0 holds for some constant a. A3) There exist functions g(w) and g(w) such that g(w) ^ f(x, w) ^ g(w) hold V x g Q, V w. A4) There exist solutions w and w of g{w) = 0 and g(w>) = 0. Then there exists a unique solution w ofthe problem (33.13) in H (Q.) n L ( Q ) . This solution satisfies X
1
X
w ^ w(x) ^ w w = m i n < i n f w , w >, (dn ~ J D
D
w = max < sup w , w >. ( ? n J D
(3.3.15)
D
The p r o o f can be found in [3.19]. Using L e m m a 3.3.14 we can now, by showing that the map G is well defined and completely continuous, employ the Schauder theorem to establish the existence of a weak solution to (3.3.5). Theorem 3.3.16: Let K ^ 1 be a constant -^u (x),v (x)^K D
satisfying
Vxe5Q .
D
D
Then the problem v,v
a)
e AV = qn (e 'u
b)
C7j.fi,. div(n„e T
{
vlu
vlu
- e' ^v) grad u) = R
-
qC(x)
114
3 T h e D r i f t Diffusion
c)
Un T
t
V / U l
div(g e p
dV
grad v) = R dv
du
d)
0
cv
1
has a solution estimate
(V*, u*, v*) e (H ( f t ) n L ° ° ( Q ) )
1
^ K
M(X),
^
K
3
wh/'c/?
satisfies
the L ' -
i«Q,
2
m i n ( i n f F , l / In D
Equations
CC + ( C + 4 n
r
<
V(x) (3.3.19)
V(x) ^ max I sup K , <J In D
r
2\l/2> (C + ( C + 4nf) 2
2tt:
in Q
where C < C(x) ^ C fto/ds. Proof: First we choose an appropriate space for the fixed point map G. Let be defined by 2
(u, v) e L ( Q ) : — K
u, v < K a.e. in Q } ,
(3.3.20)
2
where L ( Q ) is the space of square integrable functions; i.e. the space of functions (u, v) for w h i c h 2
|(w(x), f ( x ) ) | dx < oo holds. We show that G maps Jf i n t o itself and is completely continuous. Given (u , v ) e A", by virtue of L e m m a 3.3.14, there exists a solution V of (3.3.10). g and g can be chosen as 0
0
l
(3.3.21)
[L viu _ -viu e
T
Ke
T
-v/u,
vlu
g(V) =
n q[Ke i
qC.
K
Solving g(V) = 0 and g( V) = 0 gives V = U In T
V = U In T
K 2n
2
(C + (C
+
2
2Kn
(C + (C
v2
4nf)
+ 4n]2\l/2
(3.3.22)
3.3 Existence a n d Uniqueness for the S t a t i o n a r y D r i f t D i f f u s i o n
Equations
115
A p p l y i n g L e m m a 3.3.14 to equation (3.3.11) we use Ku §W
= e
- 1
\
YIU
T
V U
f- ' r e
M - i r + i) + M-ir-
+ 1
where V < V^x) < K holds, and obtain u = l/K. u = K which implies - ^
M
l
Analogously we obtain
( x K K .
(3.3.24)
In the same way we obtain l/K ^ v (x) ^ K. Thus, G maps Jf i n t o itself. The c o n t i n u i t y of Jf is a simple consequence of the well posedness of uniformly elliptic boundary value problems. O n the other hand the continuous dependence of u and v o n the data of the corresponding boundary value problems implies L
x
x
l l " l l l l . 2 . n + K l l l . 2 , Q < F(||« |l2.n» K I I 2 ,n> 0
u
II dII
i,2,n>
i,2.n) (3.3.25)
II^dII
for some positive and continuous function F. Here, the symbols || • || I I ' lli,2.n denote the norms i n L (Q) and H^Q). So
2
and
n
2
1/2
ll/l 2,n
l/WI
l/lli.2.n =
(l/WI
2
dx
\ 1/2 2
2
+ |V/(x)| )Jx
holds. Thus, l i " ! | i i , , o + ll illi,2,n ^ const holds for all ( M , V ) i n Jf. The Rellich K o n d r a c h o v Theorem (see [ 3 . 1 ] ) n o w assures that G(..V) is precompact in ( L ( Q ) ) . This, together w i t h the c o n t i n u i t y of G, gives complete continuity and the Schauder Fixed P o i n t Theorem (see [3.12]) assures the existence of a fixed point of G which is a solution of (3.3.5)—(3.3.7). • u
2
2
0
0
2
Theorem 3.3.16 serves as a typical example of an existence result for the steady state drift diffusion problem. Various other results of this type treat more complicated geometries or parameter models, which affects the structure of the fixed point map and introduces additional technical complications. Such results can be found i n [3.16], [3.10], [3.18] or [3.19]. G l o b a l uniqueness of the solution of (3.3.5)—(3.3.7) cannot be expected i n the general case since there are devices, such as thyristors, whose performance is based explicitely on the existence of multiple solutions (see Chapter 4). One can, however, obtain a uniqueness result in the case that the applied bias, and therefore the current densities J„ and J , are sufficiently small. I n the case of thermal e q u i l i b r i u m (no voltage applied, J = J = 0) the system (3.3.5) reduces to the scalar p r o b l e m p
n
p
116
3 T h e D r i f t Diffusion E q u a t i o n s e
AV
= n q(e
E
v)
{
E
-
qC(x) (3.3.26)
V, where u = u and v = v are constant. One can show easily the isolatedness of the solution V o f (3.3.26) and estimate the n o r m of the inverse of the linearization of (3.3.5) at the s o l u t i o n (V , u , v ). The implicit function theorem then implies a unique solution of (3.3.5)—(3.3.7) for sufficiently small voltages and (consequently) current densities. E
D
E
D
E
E
E
E
3.4 Forward Biased P - N Junctions We now t u r n to analyzing the structure and the quantitative properties o f solutions of the drift diffusion equations (3.2.1). The m a i n t o o l for this analysis is singular p e r t u r b a t i o n theory. I t is well k n o w n that the solutions of the drift diffusion equations behave differently i n different subregions o f the device. F o r instance, steep gradients occur locally across P-iV j u n c t i o n s and i n n a r r o w regions underneath semiconductor-oxide interfaces, i.e. i n the channel of a M O S transistor (see Chapter 4). The basic idea o f singular perturbation analysis is to replace the Basic Semiconductor Equations locally, i n different regions o f the device, by simpler problems whose solutions contain all the essential qualitative features o f the original solution. These approximations are then used to gain insights into the behaviour o f the solution which could not be achieved by l o o k i n g at the structurally more complicated original system. Since we restrict ourselves i n this Chapter to simple P-N junctions, and leave more complicated devices w i t h more than one j u n c t i o n to Chapter 4, we only have to distinguish between t w o basic situations. I n the case o f a reverse biased P-N j u n c t i o n one observes the formation o f a depletion region w i t h no, or very few, free carriers, so n = p = 0 i n (3.2.l)a) holds. This region acts effectively as an insulator and only a very small current, the so called leakage current, flows. I f the potential difference is applied the other way around, i.e. i f a forward bias is applied, the depletion region vanishes. The free carriers tend to neutralize the d o p i n g concentration and current flows. So p — n + C = 0 w i l l h o l d except i n n a r r o w layers a r o u n d the P-N j u n c t i o n s where large electric fields and concentration gradients w i l l occur. These t w o situations require t w o different types of p e r t u r b a t i o n analysis. I n this Section we w i l l concentrate on the forward bias case where, except i n the above mentioned layer regions, the space charge q(p — n + C) is very small. We start by bringing the drift diffusion equations (3.2.1) into an appropriate scaled and dimensionless form. Suppose the geometry under consideration has a characteristic length scale (for instance the diameter) L . We use the scaling x =
Lx
s
(3.4.1)
3.4 F o r w a r d Biased P-N
117
Junctions
for the position variable x. x , the scaled position variable, is then at most of order 0 ( 1 ) and dimensionless. F o r the dependent variables in (3.2.1) we use the following scaling which has turned out to be the most useful for the singular perturbation analysis of forward biased P-N junctions (see [3.21], [3.18]) a) V = U V , b) n = Cn , c) p = Cp , s
T
d)
s
s
J„ = —^—J„ ,
e)
s
J
s
=—-—J ,
p
Ps
where the subscript s denotes the scaled and dimensionless variable. U in (3.4.2) is the thermal voltage which, at r o o m temperature, has a value of 0.0259 V and C is the m a x i m a l absolute value of the d o p i n g concentration C(x). F o r V L S I applications typical values of C range from 1 0 c m " to 1 0 c m ~ . ft i n (3.4.2)d)e) is a characteristic value for the mobilities //„ and H and is, for silicon, usually of the order of 1000 c m V s . We w i l l only consider situations i n this Section where we can assume the Einstein relations T
1 5
2 1
3
3
2
- 1
-1
p
D„=U fi , T
D =U .„
n
P
(3.4.3)
TP
to hold. Thus we set P-n = Wn »
U
P-p = PPp .
s
A, = TPPn
s
D
,
s
p = U flU ^ . (3.4.4) r
p
U s i n g this scaling, the drift diffusion equations (3.2.1) become a)
/} AV = n -
b)
div J„ = R ,
d)
div J
s
s
s
P s
P
s
C (x) s
c)
s
= - R
-
J„ = fi s
,
s
e)
ns
J
P s
(grad n - n grad F ) s
s
(3.4.5)
s
= p. ( - grad p - p grad F ) Pa
s
s
s
w i t h C(x) = CC (x). R i n (3.4.5)b)d) is the appropriately scaled recombination rate. I f the Shockley Read H a l l term is taken as a model for the recombination rate, R is of the form s
s
s
R
* = ! (n Ps
s
4
n Ps - <5 + 5 ) + t„ (p
.
—r—n5T• + S)
s
2
s
<--^ 3 4 6
2
s
The boundary conditions (3.2.2) read i n scaled form a)
F (.x) = V (x) = V (x) + v
K , (x) = In
Ds
s
1 ;2
26
V (x) bU
-(c (x) + V Q W
2
s
4
+ 2
^ 4
b)
« (x) = n „ ( x ) = |(C (x) + JC (x)
c)
p (x) = p ( x ) = i ( - C ( x ) + V C ( x )
d)
-=-*(x) = 0 ,
s
s
s
Ds
t I
J (x) • v = 0 Ps
+ 4«5 )
s
s
s
2
4
+ 4«5 )
for
x e dQ
J ( x ) v = 0, n
for
x e dQ . Ns
(3.4.7)
118
3 The Drift Diffusion Equations
F r o m here on we w i l l o m i t the subscript s for n o t a t i o n a l convenience. The parameters X i n (3.4.5)a) and 8 i n (3.4.7) are i - i - ^ r . qCL J
* - ( ± r . \C
2
K
.3.4.8,
X is the scaled m i n i m a l n o r m e d Debye length o f the device (see [3.35]). X w i l l act as a singular p e r t u r b a t i o n parameter in the forward bias case as well as in the reverse bias case i n the next Section. However, for reasons that w i l l be explained i n the corresponding Section, a different form o f scaling has to be used for P- Adjunctions under extreme reverse biasing conditions.
The
Equilibrium
Case
The derivation o f an a p p r o x i m a t i o n to the s o l u t i o n o f a given p r o b l e m via singular p e r t u r b a t i o n analysis follows a certain recipe. The steps i n this recipe become technically more involved the more structurally complex the original p r o b l e m is. As a matter of fact, for a nonlinear system of p a r t i a l differential equations, such as the drift diffusion equations, we w i l l i n general not be able to carry o u t some o f these steps. So i t is for instance still an open problem to show that the asymptotic expansion derived i n the next Section actually approximates the solution o f (3.4.5) except i n some special cases. This does n o t diminish the value o f the singular p e r t u r b a t i o n approach since one can always resort to numerical computations to 'convince' oneself o f the validity of the expansions. I n order to demonstrate the basic techniques involved i n singular p e r t u r b a t i o n analysis, and to give a flavor of the type of results obtained, we w i l l consider the case o f a P-N j u n c t i o n i n thermal e q u i l i b r i u m first, using the scaling (3.4.2) for the forward bias case. This has the advantage that parts o f the system (3.4.5) can be integrated exactly and (3.4.5) reduces to a nonlinear Poisson equation for the potential V. W e w i l l also restrict ourselves to the t w o dimensional case. So we assume that the device occupies a region Q £ U w i t h a D i r i c h l e t boundary o Q and a N e u m a n n boundary dQ (dQ = 8Q u dQ , dQ. n d€l = { } ) . The drift diffusion equations can be reduced to a nonlinear Poisson equation w i t h a certain set o f boundary data corresponding to zero applied bias. This set is given by 2
D
N
D
N
2
a)
V(x) = V = In
(C(x) + ^C(x)
b)
n(x) = n (x)
= \(C{x)
c)
p(x) = p (x)
= i ( - C(x) + yJC{xf
d)
^ W
bi
D
D
J(x)v
= 0, = 0
D
2
+ ^C(x)
J (x)-v
= 0,
for
xedQ .
n
N
+
N
+ 48*) 4
48 ) A
+ 48 )
for
x e dQ
D
(3.4.9)
3.4 F o r w a r d Biased P-N
119
Junctions
So U = 0 i n (3.4.7) holds. Here, again Q denotes the d o m a i n o f the device and dQ and dQ denote the D i r i c h l e t and N e u m a n n parts o f the boundary, v is the unit o u t w a r d n o r m a l vector o n the d o m a i n b o u n d a r y dQ. A solution n, p, J„ and J o f the c o n t i n u i t y equations and current relations (3.4.5)b)-e) is given by D
N
p
2
2
= S
v
Se.
J„ = 0,
J = 0 .
(3.4.10)
(3.4.10) is a solution o f the c o n t i n u i t y equations as l o n g as there is no generation-recombination i n thermal e q u i l i b r i u m , i.e. as l o n g as R = 0 holds whenever np = 5 . This c o n d i t i o n is satisfied for the Shockley Read H a l l term (3.4.6). The Poisson equation then reduces to A
2
2
a)
X AV=
b)
V\an =
V (x)U
B
dV c)
57 en*
— i
v
S (e
bi
D
-C(x) 1
In
:
28
2
4
(C(x) + ^C(x)
+ 4<5 )
(3.4.11)
= 0.
To keep matters simple, we w i l l take Q to be a rectangle w i t h dQ. consisting of the t w o vertical sides and dQ. o f the t w o h o r i z o n t a l sides. The P-N j u n c t i o n shall be given by a curve F = {(x, y) = (X(s), Y(s))} which intersects dCl i n the t w o points ( A ^ s J , Y{s )) and (X(s ), Y(s )). The situation is depicted schematically i n F i g . 3.4.1. We assume the P-N j u n c t i o n to be abrupt. So C(x) has a j u m p discontinuity at r and is as smooth as we like (say constant) away from T. We can assume w i t h o u t any loss of generality that the tangent vector (dX/ds, dY/ds) =: (X, Y) o n T is normalized. Also, to a v o i d technical difficulties, we assume that T intersects dQ. i n a right angle and w i t h o u t curvature. So D
N
N
t
2
2
N
X(s)
+
Y(s)
1
X(s )
=
X( )
Y(s )
2
2
Vs (3.4.12)
t
Si
t
= 0,
1,2
holds.
F i g . 3.4.1 S i m p l i f i e d g e o m e t r y
120
3 The Drift Diffusion
Equations 4
X i n (3.4.1 l)a) w i l l be very small in practice. So, for instance, for L = 10 cm and C = 1 0 cm in (3.4.2) X = O ( 1 0 ) holds. I f the term A V in (3.4.1 l)a) is of moderate size this implies that the right hand side of (3.4.1 l)a), the space charge, is small (of order X ). I f the scaling (3.4.2) was correct t\V w i l l not be large except, maybe, in small subregions (layers) where V has a steep gradient. I t is therefore reasonable to assume that, away from these layers, V will be approximated by V , the solution of the zero space charge approximation 1 9
3
_ 3
1
0
2
0 = 2S s i n h ( F ) - C(x).
(3.4.13)
0
(3.4.13) is a simple problem since the differential equation (3.4.1 l)a) has been reduced to an algebraic relation. I n the language of singular perturbation theory (3.4.13) is called the reduced problem and V is called the outer, or reduced, solution (see [3.25]). We use the index 0 for the outer solution since V w i l l be the zeroth order term of an asymptotic expansion in powers of / at the end of this Section. Solving (3.4.13) we obtain 0
0
C(x) + V C ( x )
V (x) 0
2
4
+ A3
(3.4.14)
2
2<5
V (x) also satisfies the boundary conditions (3.4. l l ) b ) c ) assuming that dC/dr\ = 0 holds. O n the other hand, since the d o p i n g profile C(x) is discontinuous at T, so is V . Because of the regularizing effect of the Laplace operator we w o u l d expect the solution V to be continuous, and so V is probably not a good a p p r o x i m a t i o n to V i n the vicinity of V. The reason for this is that our original premise, namely that div(grad V) is of moderate size, is not valid there. I n order to investigate the solution in a neighbourh o o d of T we employ a local coordinate transformation. F o r a point (x, y) close to T let s denote the parameter value of the nearest curve point (X(s), Y(s)) and Q the perpendicular distance of (x, y) to V divided by X as shown i n Fig. 3.4.2 (£ > 0 o n one side of T and £ < 0 on the other side). Using the normalized tangent and n o r m a l vectors (X, Y) and (Y, —X), the transformation (x, y) <->(c, s) is given by 0
SilN
0
0
x = X(s) + Xc Y(s) (3.4.15) y=
Y(s)-XcX(s).
This coordinate transformation is one to one as l o n g as \X£\ is sufficiently small, i.e. i n a neighborhood of T. I n the (£, s) variables (3.4.11 )a) reads d\V
2
= 2S s i n h ( F ) - C + O(X).
(3.4.16)
I n order to obtain an a p p r o x i m a t i o n to V, which is valid in a neighbourhood of T as well, we w i l l derive the layer term V w h i c h is a function of the 0
3.4 F o r w a r d Biased P-N
Junctions
121
y
(x,y)
X
F i g . 3.4.2 L o c a l c o o r d i n a t e s
transformed variables (£, s). So we approximate V by F ( £ , s) inside the layer. So, letting k go to zero, we o b t a i n 0
2
d V (^, 0
sinh(F (£,
2
s) = 28
0
s)) - C(X(s)
(3.4.17)
+ , Y(s) + ) .
C(X(s) +, Y(s) + ) i n (3.4.17) means the corresponding onesided l i m i t of the function C at the P-N j u n c t i o n T. E q u a t i o n (3.4.17) is called the layer equation i n the language of singular p e r t u r b a t i o n theory and V is called the inner s o l u t i o n (see [3.25]). The layer equation is n o w also simpler than the original equation since it is only a family of o r d i n a r y differential equations i n £ parameterized by s. F o r £ -> +oo the layer term V should match w i t h the outer solution V . So we require that 0
0
0
P (oo, s) = V(X(s) 0
+ , Y(s) + ) ,
F ( - o o , 5:)= V(X(s)-,
Y(s)-) (3.4.18)
0
holds. Let us first check whether V can be really used as a layer term i n (3.4.16), i.e. whether there is a solutive V of (3.4.19) which converges to V for Q to + o o . E q u a t i o n (3.4.17) can be integrated i n terms o f the inverse o f a m o n o t o n e function. M u l t i p l y i n g b o t h sides of (3.4.17) by d^V and integrating once gives 0
0
0
0
^V (c, 0
2
2
s) = 2d cosh(F (c, s)) 0
-C(X(s)-, U\V (Q, 0
2
V
0
s)
0
A-(s),
for
A {s),
for
c<0
s) = 2<5 cosh(F (c,.s)) 0
-C(X(s)
If
Y{s)-)V (Z,
2
converges to
V
0
+ , Y(s) +
for c to
)? (Z,s) 0
± o c , V (±oo, 0
+
s) = V (X(s)±, 0
c>0. (3.4.19) Y(s)±),
122
3 The Drift Diffusion
Equations
<9 f ( + oo, s) = 0 holds. This gives for A.{s) and A+(s) i n (3.4.19) 4
0
2
A (s)
= 28 cosh(F (X(s) + , Y(s) + )
+
0
-C(X(s)
+ , Y(s) + )V (X(s)
+ , Y(s) + )
0
(3.4.20) 2
A_(s)
= 28 c o s h ( F ( X ( s ) - , T ( s ) - ) 0
-C(X(s)-,
Y(s)-)V (X(s)-,
Y(s)-).
0
Moreover, the sign of the j u m p discontinuity of V determines the sign a of d,V . 0
0
a(s) : = s i g n ( 5 K ( & s)) = s i g n [ F ( X ( s ) + , Y(s) + ) 4
-
0
0
V (X(s)-,
(3.4.21)
T(s)-)]
0
holds. So by t a k i n g square roots we obtain J2tr(s)
= d&{Z,
2
s){25
c o s h ( f ( £ , s)) 0
- C(X(s)±,
Y(s)±)V {Z,
(3.4.22)
112
s) -
0
A (s)}- . ±
Integrating (3.4.22) w i t h respect to i, n o w gives a)
F (V (£,s),s)
= /2aZ
for
c; > 0
b)
F.(f ^,s),s)
= [fiat;
for
f < 0
c)
F+(z, s)
+
0
y
0
(3.4.23)
dy 2
B( ) v S
d)
^
2
cosh(y) — C y — A + {s) T
A
B(s)=V (0,s) o
C(X(s)-,
(s)-A_(s)
+
Y(s)-)
- C(X(s)
+ , Y(s) + )
where F and denote the onesided limits at Y. B(s) i n (3.4.23)d) has been computed by setting £ = 0 i n (3.4.19). Since the integrand i n (3.4.23)c) is positive F+(z, s) is a monotone function of z and therefore the equations (3.4.23)a)b) are solvable for all £. So the layer p r o b l e m (3.4.17) has a solution V w h i c h converges exponentially towards V for £ to + o o . T o obtain a uniform O(k) a p p r o x i m a t i o n of the solution V we define the composite expansion K (see [ 3 . 2 5 ] ) by 0 T
0
0
C
0
f V (x, y) + [P (& s) 0
0
for
0
m ^ )
f > 0
(3.4.24)
V (x, y) + [ ? „ ( £ , s) 0
for
V (X(s) + , Y(s) +
K (X(s)-, 0
y( )-)]^(^) S
£ > 0.
I n order to not having to deal w i t h exponentially small terms at the b o u n d ary dQ, the function <j> serves to eliminate the layer term away from Y. I t satisfies 0 0
<j) £ C ,
(f>(r) = 1
for
|r| < 1,
^(r) = 0
for
|r| > 2. (3.4.25)
3.4 F o r w a r d Biased P-N
123
Junctions
The composite zero order a p p r o x i m a t i o n K ' then satisfies the differential equations together w i t h the boundary conditions up to terms of order 0(A) (see Problem 3.4). So far we have only derived necessary conditions for the terms i n our asymptotic a p p r o x i m a t i o n . The question which arises is o f course whether, and i f yes i n what sense, the composite a p p r o x i m a t i o n VQ does approximate the solution V. I n this simple case this question can be answered precisely by using the m a x i m u m principle (see [3.12]). I f we define the remainder Q by 0
0
Qo = V-Vj,
(3.4.26)
we derive the f o l l o w i n g remainder equation for Q
0
2
X AQ
= F ( x , y, Qo) - *G(x, y)
0
<2oU> = o, D
= 0
dv 2
F(x, y, Q ) = 25 0
(3.4.27) (3.4.28)
c
2
sinh(K (x, y) + Q ) - 28 0
0
c
sinh(F (x, y ) ) . (3.4.29) 0
The function G(x, y) i n (3.4.27) is o f a complicated form i n v o l v i n g the lower order partial derivatives o f V w.r.t. c and s w h i c h arise from the coordinate transformation (x, y) <-•(£, s). However, G(x, y) is a bounded function (see Problem 3.1). (3.4.27) implies that the derived a p p r o x i m a t i o n , at least, solves the differential equation up to terms o f order O(X). Since F ( x , y, Q ) is m o n o t o n i c a l l y increasing i n Q , it follows from the m a x i m u m principle, that the p o i n t (x, y), where Q attains its m a x i m u m Q, either lies on the Dirichlet boundary 8Q or that 0
0
0
0
D
F(x, y, Q) - /.G(x, y ) ^ 0
(3.4.30)
holds. Since F is m o n o t o n e i n Q , Q ^ Q holds, where Q is the solution of 0
F ( x , y , 0 - / G ( x , y ) = O.
(3.4.31)
Thus maxjj Q (x, y) = 0(A) holds. m i n < 2 ( x , y) = 0(A) and thus Q
In
the
same
way
one
shows
a
m a x \ Q ( x , y ) \ = 0(2.) n
(3.4.32)
0
holds and VQ is an 0(A) a p p r o x i m a t i o n to V. Since the remainder term Q is of order 0().) it is reasonable to assume that, for X = 10~ for instance, w h i c h corresponds to a device diameter of 1 /; i n silicon and a m a x i m a l d o p i n g C of roughly 1 0 cm , VQ is a good a p p r o x i m a t i o n to the solution V. I f even better approximations are desired these can be obtained by further expansion i n powers o f X. We set, formally, 0
3
1 9
V=
V(x,y,X)x
3
£
N
VZ(x,y)X .
(3.4.33)
(3.4.33) is an asymptotic expansion and the sum o n the right hand side w i l l
124
3 The Drift Diffusion Equations
in general not converge. I f the sum is truncated after the Nth term the remainder Q w i l l be o f order 0(X ). F , V , V and Q were already derived above. The higher order terms are determined by inserting (3.4.33) i n t o the equation (3.4.1 l ) a ) and expanding i n powers o f A. This gives N
C
N
a)
0
F
1
=
0
0
0
0 2
b)
A F „ _ = 28 c o s h ( F ) F „ + F„(V ,V „ ),
c)
dV
2
0
0
2
2
n
= 28 cosh(F )V 0
n
n
+ G (V ,..., n
n = 2,3...
x
0
V„, V ,..., 0
n = 1,2,... V„{x, y) + for d)
s)
V (X(s) n
+ , Y{s) +
)-]^(y/k)
£ > 0
v: v„(x, y) + for
MZ,s)
V (X{s)-,Y(s)-)My/k) a
£ > 0 . (3.4.34)
and so on. So, a p p r o x i m a t i o n s of any order c o u l d be obtained theoretically. Note, however, that V i n (3.4.34)b) is a function o f AK„__ . Thus, w i t h each a d d i t i o n a l t e r m i n the expansion regularity is lost^iepending on the geometry of Q and T. I f V is smooth enough so that V , V„ are reasonably well defined, it can be shown by using the same m a x i m u m principle argument as above that the remainder t e r m Q is o f order X i n the L ^ - n o r m . If we reconsider the procedure to o b t a i n an asymptotic expansion o f the solution of the e q u i l i b r i u m problem (3.4.11), the steps involved are n
2
0
0
N
N
A) Determine the reduced equation by setting X equal to zero. B) Check w h i c h parts of the p r o b l e m (i.e. b o u n d a r y conditions, c o n t i n u i t y requirements etc.) can be satisfied w i t h the outer s o l u t i o n obtained i n Step A . C) A d d a layer correction term to the outer s o l u t i o n where the outer solution is apparently insufficient (in this case i n the vicinity o f the P-N j u n c t i o n T). D ) Show that the so derived reduced p r o b l e m and the layer p r o b l e m actually have solutions w i t h the desired properties (regularity, decay etc.). E) Show, that the composite expansion actually is an asymptotic approxi m a t i o n to the solution. I n the case o f the e q u i l i b r i u m p r o b l e m , when everything can be reduced to a single differential equation, we c o u l d actually carry out all the five steps above. However, when currents are present, and one has to deal w i t h the whole coupled system (3.4.5), this w i l l i n general not be the case. One can always set the p e r t u r b a t i o n parameter to zero and o b t a i n a reduced equation. So Steps A and B above are usually no p r o b l e m . Also the d e r i v a t i o n of the layer equations i n Step C is possible i f one finds the right coordinate
3.4 F o r w a r d Biased P-N
125
Junctions
transformation. I n the case of the steady state Basic Semiconductor Equations this is no problem either. T o find the solution o f the reduced problem is more tricky. I n the e q u i l i b r i u m case the reduced problem consists of the transcendental equation (3.4.13) which obviously had a unique solution. I n the non e q u i l i b r i u m case the reduced problem is still a system o f nonlinear partial differential equations. The layer equations usually degenerate i n t o ordinary differential equations, and so finding solutions to them is easier. T o show i n Step E that the derived expansion actually is an asymptotic a p p r o x i m a t i o n of the solution is generally a very difficult task for systems of nonlinear partial differential equations. Since the whole approach consists of finding an approximate solution which satisfies the boundary value p r o b l e m up to small terms, Step E involves the estimation of the inverse o f the involved nonlinear differential operator. U s i n g the implicit function theorem this task can be reduced to estimating the inverse o f the linearization at the solution. However, even to show that this linearization has an inverse whose n o r m is bounded uniformly i n the p e r t u r b a t i o n parameter /., is usually impossible for complicated systems of partial differential equations, except i n special cases.
The Non-Equilibrium
Case
When the boundary data V i n (3.4.9) are not given by the built-in-potential V the densities n and p i n (3.4.10) do not satisfy the boundary conditions and therefore do not constitute a solution o f the boundary value problem. So, since we cannot simply integrate o u t the c o n t i n u i t y equations and the current relations, the singular p e r t u r b a t i o n analysis has to be carried out on the whole coupled system (3.4.5). As pointed out before, the derivation of the reduced equations and the layer equations is straightforward i f one follows the Steps A C above. T o prove that the asymptotic expansions actually constitute an a p p r o x i m a t i o n o f the solution is not possible in general except in special cases such as i n one dimension. The a p p r o x i m a t i o n property has been verified, however, numerically (see [3.21]). As it already was the case for the existence and uniqueness results in Section 3.3, it is more convenient to write the drift diffusion equations i n the Slotboom variables V, u and v o f (3.3.3). Using the scaling introduced at the beginning of this Section, they are given by D
bl
2
v
2
u = S ne' ,
v
v = b~ pe .
(3.4.35)
The transformed p r o b l e m n o w reads 2
2
a)
/ AV
b)
d i v J„ = R,
c)
div J =
d)
V\ =V \ ,
p
aaB
v
= S (e u
D dfh
- R ,
— e~ v) — C(x) v
v
J = u„e
grad u
n
v
J = -n e~ p
grad v
p
u\ea = u \ , B
D ?ilo
= v\ , D esiB
(3.4.36)
3 The Drift Diffusion Equations
126
dv
du
dv
~dv~
dv dn„
dv
f)
u (x) =
e-
g)
V (x) = U(x) + In
e)
D
V M
d)
,
U(x)
r (x) =
e
D
' 1
D
U(x)
0.
2
(C(x) + ^C(x)
2
2S '
+ V (x),
A
+ 4S
xeBQ,
bi
A g a i n U(x) denotes the applied potential at the contacts. Note, that u(x) = v(x) = 1 holds i n the e q u i l i b r i u m case discussed above. I n the n o n equilibr i u m case the form o f the mobilities j . i and i impacts the form of the asymptotic expansions significantly. The t w o basic cases to be distinguished are the presence and the absence o f velocity saturation effects. If, i n the presence o f velocity saturation, fi„ and \i behave like l / | g r a d V\ for large values o f the electric field —grad V this implies that inside the layers the mobilities are o f order 0(A). We w i l l leave this case for later on and consider the absence o f velocity saturation first. So we simply assume for the moment that u„ and p. depend on the position x only (and not o n X). Since we w i l l not derive the higher order terms i n the asymptotic expansion we w i l l d r o p the subscript 0 for the zero order term from now on. So, for I -> 0, the reduced p r o b l e m for the outer solution vv = (V, u, v, J , J ) is given by n
h p
p
p
n
2
v
v
a)
0 = b (e u-
e~ v)-
b)
div J = R,
C(x) v
J„ = fi e
n
n
div J„
R
e)
grad u v
-p. e
= uD\en ' D
~dv
du dv
Ba„
dv N
D I (flp
o,
dv
dn
(3.4.37)
grad v
p
d) dV
p
and it is understood that vv = (V, u, v, J„, J ) denotes the zero order term o f the outer solution. R i n (3.4.37)b)c) denotes the r e c o m b i n a t i o n rate evaluated at the reduced s o l u t i o n vv. (The form o f the recombination term is not i m p o r t a n t for the moment.) Firstly we observe that we have three b o u n d a r y conditions at each part of the b o u n d a r y while we have only t w o second order differential equations (3.4.37)b) and (3.4.37)c). However, since the boundary conditions at O h m i c contacts are derived from the c o n d i t i o n o f vanishing space charge, they are consistent w i t h the zero space charge a p p r o x i m a t i o n (3.4.37)a). I n other words, i f we solve (3.4.37)a) for V and insert i n t o (3.4.37)b) and (3.4.37)c), we o b t a i n the boundary value p r o b l e m p
a)
div J = R,
b)
div J
c)
n
R,
n
L
D\cn„ •
71
01
J = i^e *'"'
grad u
n
JT =
-^-^'""^gradTJ v D\?n
D
(3.4.38)
3.4 F o r w a r d Biased P-N
Junctions
du d)
dv
dv
= 0.
dv 5n»
Sili,
127
2
e)
C(x) + Jc~{x)
V{x, u, v) = In
+
4
45 uv
2
25 U
for u and v. V = V(x, u, v) then satisfies the boundary c o n d i t i o n (3.4.37)d)e) automatically i f dC/dv\ = 0 holds. (3.4.38) constitutes the reduced p r o b lem. Again, as i n the e q u i l i b r i u m case, the necessity for a layer term arises from the fact that V, defined by (3.4.38)e), is discontinuous at the P-N j u n c t i o n Y because of the discontinuity o f C(x) there. Performing the same local coordinate transformation (3.4.15) as in the e q u i l i b r i u m case, we derive the layer term w = (V, u, v, J„, J ) which is a function of the layer variables c and s i n (3.4.15). A g a i n we change from the (x, y) variables to the (£, s) variables. This yields, for A -> 0, dilD
p
a)
2
dV
2
b) c)
V(i s)
v
d [_e - u(Z,
d^J -n p
s)
s) - e ^ v(£,
= 0,
0 = ^ (s)e^%a(t
= 0,
0=
s)] - C+(s) (3.4.39)
s)
+
f
-^(s)e- ^%m,s).
T
Here n = (Y{s), —X(s)) denotes the n o r m a l vector on the P-N j u n c t i o n T. In (3.4.39) / + ( s ) denotes the appropriate one sided limit of the function / at T. So f (s) = f(X(s) + , Y(s) + ) for £ > 0 and f_{s) = f(X(s)-, Y(s)-) for I < 0 holds. Since the layer terms have to match the outer solution w at c = + o o , we immediately o b t a i n +
a)
u(c, s) = u(X(s),
b)
s)-n(s)
c)
Y(s)),
6tf, s) = v(X(s),
= J (X(s),
Y(s))-n(s),
= J (X(s),
Y(s))-n(s).
n
J (Z,s)-n(s) p
p
Y(s)) (3.4.40)
A layer term occurs i n the potential V and i n the tangential components of the current densities J and J only, while u, v,J„-n and J • ~n are continuous across Y. The layer equation is given by n
a) b)
d\v 2
dV
2
v
p
s)
S le ^ u(s) =
p
,-V(i,s)
0(S)]
2
8 [e
C (s)
for
Q > 0
C-(s)
for
Z
+
where «(s) stands for u(X(s), Y(s)) and so on. As in the e q u i l i b r i u m problem, we require V to match the outer solution V at <J = +oo. So we require t/(OO,
s) = V(X(s)
+ , Y(s) + ) ,
V(-co,
s) = V(X(s)-,
Y(s) (3.4.42)
After solving the reduced p r o b l e m and the layer p r o b l e m (3.4.41)-(3.4.42) the tangential components o f the layer current densities are given by
128
3 The Drift Diffusion Equations
a)
J =
b)
J =
v
t(s)p. (s)e d u(s)
n
n±
s
(3.4.43)
v
-7(s)n {s)e- d v(s),
p
p±
1
t(s) = (X(s),
s
Y(s))
Since the layer p r o b l e m (3.4.41)-(3.4.42) a r o u n d the P-N j u n c t i o n is o f the same form as i n the e q u i l i b r i u m case the same technique can be applied to show that (3.4.41)-(3.4.42) has a s o l u t i o n V w h i c h converges m o n o t o n i c a l l y towards the one sided l i m i t of the outer s o l u t i o n V for £ to + GO . I f no velocity saturation effects are considered, and the mobilities p, and \i are simply functions o f position, bounded u n i f o r m l y away from zero, the same approach as i n Section 3.3 can be used to show the existence of a s o l u t i o n o f the reduced p r o b l e m (3.4.38). I f the r e c o m b i n a t i o n rate is given by the scaled Shockley Read H a l l t e r m n
p
uv — 1 ,„ . . i7 v , (3.4.44 z (e u + 1) + x (e- v + 1) then the corresponding fixed p o i n t m a p G is given by G(u , v ) = ( u v) where u and v satisfy the linear b o u n d a r y value problems R =
v
p
V
n
0
x
0
l 9
±
x
a)
Uv t
F
d i v [ / i „ e ° grad u ] t
v
p
b) c
)
Vl
div I/i e p
u
D
— D\?CI
D
du
1
d)
~8v~
cQ.v
d7
>
0
e)
n
n
1
v
l
+ 1)
0
—1 v
rJe^u,
+ 1) + T (e' 'v n
0
+ 1) (3.4.45)
u
i\aa
— nl?n >
D
D
0. 2
F = ln
+ 1) + r (e- °v uv
grad i ^ ] =
U
i \dn
—1
0
v
z (e °u
4
C(x) + JC{x)
+ 4<5 ti u " 0
0
2
2S u
n
C(x) + y C ( x ) f)
n = l n
2<5
2
+ 4<3S t; " 1
0
2 Ml
We refer the reader to [3.19] for a more detailed existence p r o o f for the reduced p r o b l e m . As i n the e q u i l i b r i u m case (3.4.24), the zero order c o m posite expansion is given by w(x, y) + [w(£, s) -
w(X(s)for
w(x, y) + [vv(£, s) -
w(X(s)
I > 0,
(3.4.46)
Y(.s)-)]^(v^) for
£ < 0,
where the vector w denotes (V, u, t>, J„, J ). The d e r i v a t i o n o f the higher order terms i n the asymptotic expansion is again straightforward (see P r o b l e m 3.5). p
3.4 f o r w a r d Biased P-N
Asymptotic
Junctions
Validity
129
in the One-Dimensional
Case
T o show that the obtained a p p r o x i m a t i o n is asymptotically valid, i.e. to show that the solution o f (3.4.36) converges i n some sense to for X -» 0, is, i n full generality, a yet unresolved problem. Ideally, one w o u l d like an explicit estimate of the difference between the exact solution and the asymptotic a p p r o x i m a t i o n obtained above. Unfortunately, such estimates are available only i n special cases (no generation terms, one dimensional problems etc.). I n more than one dimension it has been established that the solution of the full p r o b l e m (3.4.36) converges to WQ as X tends to zero, i n the absence o f recombination and generation terms (see [3.15]). However, this result says n o t h i n g about the rate o f convergence, and therefore about the a p p r o x i m a t i o n q u a l i t y for finite X. Results w h i c h do also provide convergence rates are only available i n the one dimensional case. They can be found i n c.f. [ 3 . 2 ] and [3.23]. I f we consider the one dimensional model o f a P-N j u n c t i o n the system (3.4.36) reduces to 2
2
dV
a)
X
b)
l J„
2
v
~ ^ = 5 {e u dx
(3A47)
d J =-K>
Tx
C(x)
= R,
x
c)
v
- e~ v)-
^ dv v
J =~P- e-
P
P
P
T
x
,
where x is the position variable w h i c h varies, after scaling, between x = — 1 and x = 1. The boundary conditions i n the one dimensional case are of the form V ( x )
(x)
= ~
V(x)
= V (x) = U(x) + I n
u
e
,
v(x) =
D
= U (x) + V (x), bi
U(x)
e
2
2
23 for
(C(x) + V W
+ 43"
x = - 1
x = 1.
and
(3.4.48)
I n [3.23] an abrupt P - N j u n c t i o n i n the center of the device is assumed. So C(x) i n (3.4.47)a) has a j u m p discontinuity at x = 0 and C(x) > 0 for x > 0 and C(x) < 0 for x < 0 holds. Also the absence of recombination-generation effects is assumed; so R = 0 holds. The authors show that i f the boundary potential V varies w i t h i n a certain range then D
max
\V
—
VQ \ +
[-l.i]
c
max \u — u \ + max \v 0
i-t.i]
^ c o n s t A c521 5 |TK/ m( l ) _2
D
—
u
oi
[-l.i] T F/ D t(_- 1\ \) _1_ + i1|| 33 „e o0 .. 5=( 1. /L(,1i ))-- L - i ) , (
(3.4.49)
holds. Since the p r o o f is technically quite involved it w i l l be o m i t t e d here. It should be pointed out, however, that the p r o o f makes explicit use of the
3 The Drift Diffusion Equations
130
fact, that i n the one dimensional case J„ and J are constant i n the absence of recombination-generation. Thus i t is unlikely that it can easily be generalized to higher dimensions. A l t h o u g h this is unsatisfactory for practical purposes, the result contains an interesting feature. I f we choose typical values for the device parameters, say 1CT cm for the length L o f the device and 1 0 c m " for the m a x i m u m d o p i n g concentration, and calculate the resulting values for X and 3 i n silicon, the bounds o n the applied potential difference, which have to be assumed i n order for the estimate (3.4.49) to be valid i m p l y that the applied bias satisfies p
4
1 9
3
-0.8
V ^
U \U(\) T
-
U(-1)]
^
0.2 V .
(3.4.50)
I n this setting a negative potential difference corresponds to forward bias and a positive one to reverse bias. F o r such a diode —0.8 V represents quite a large forward bias. O n the other hand one w o u l d expect the drift diffusion equations to give a reasonable description of the device for reverse bias values of up to a couple o f volts. So this is consistent w i t h the original premise of this Section to give an asymptotic analysis for forward biased P-N junctions. I t also is an i n d i c a t i o n that i n general the validity of the above derived approximations w i l l break d o w n for some moderate values of the reverse bias. Thus, the reverse bias case w i l l be treated by a different k i n d o f asymptotic analysis i n Section 3.5.
Velocity
Saturation
Effects—Field
Dependent
Mobilities
I t is a well k n o w n fact o f device physics, that the p r o p o r t i o n a l i t y o f the drift velocities p„ grad V and — p gmd V to the electric field —grad V only holds at moderate field strengths. D u e to carrier heating, the velocities saturate for high electric fields, i.e. p
lim
\p grad V\ = v„
and
n
lim
|grad V\-*cc
\p grad V\ = v p
p
Igrad K|-»co
(3.4.51) holds, where v„ and v are the saturation velocities. I n order to reflect this property, the mobilities p„ and p, are modeled as dependent on the electric field such that the saturation relation (3.4.51) holds. One way to do this is to choose the mobilities as p
p
M
"
^
v„ + / I J g r a d V\ '
^
^
v + /J |grad V\ ' p
p
(3.4.52) where p and p are field independent, m o b i l i t y models (see [3.34]). I f the original drift velocities p grad V are small compared to v (3.4.52) represents only a small p e r t u r b a t i o n o f the original model and n
p
n
Pn ~ Pn i
Pp ~ P~p
p
n p
(3.4.53)
3.4 F o r w a r d Biased P-N
Junctions
131 s
holds. O n the other hand the new drift velocities j.i grad V and — p.* grad V satisfy the saturation relation (3.4.51). O f course there is a multitude o f possible formulas for p. and up which w o u l d do the same j o b . We refer the reader to [3.34] for other formulas used i n practice. I t should be pointed out, however, that the form of p and p is based o n no other physical consideration than the saturation relation (3.4.51), and different models are obtained only by fitting experimental data. I f we use the scaling of this Section together w i t h (3.4.52) we o b t a i n n
s
n
s
s
n
2
2
v
a)
? AV
b)
div J„ — R,
v
= 8 (e u
- e~ v) s
p
C(x)
v
J„ = n „e
grad u (3.4.54)
c)
div J =-R,
d)
A -
s
v
J = -n e~
p
p
grad v
p
+. ^nlgrad V\' .
A"
— <x + ^" | g*r a d V\ ' p
p
w i t h a„ , the scaled saturation velocities, given by p
^n
= ~ r , n„u
*
=
pP
J
h
(3-4-55)
T T -
nU
T
p
T
Inside the layer, where grad V w i l l be large ( = 0(1/1)), this w i l l obviously change the structure of the solution and o f the singular perturbation approx i m a t i o n . I n a d d i t i o n , as we w i l l see, the different layer behaviour also impacts the reduced p r o b l e m and therefore the t o t a l current flow t h r o u g h the P-N j u n c t i o n . The reduced equations remain the same, however, w i t h jx and Hp i n (3.4.37) replaced by n and p . F o l l o w i n g the expansion procedure gives, for the layer equations s
n
n
2
f{
s)
a)
d\ V = 5 [e ^ u(ct,
b)
d J {Z,s)-n(s)
c)
J -n
i
n
f
= Ac-e^
s)
s) - e~ ^ v{^
= 0,
a
p
s
)
d J (£,s)-n(s) i
p
s)] = 0
-C (s) ±
(3.4.56)
cW,s)
w i t h n(s) = {Y(s), —X(s)) as the normalized n o r m a l vector on the j u n c t i o n T. Again, we o b t a i n a) b)
J -n " J -n n
p
=
J (X(s),Y(s))-n n
1 =
P-N
(3.4.57) J (X(s),Y(s))-n. p
The difference to the unsaturated case comes from the current relations (3.4.56)c)d). F r o m (3.4.56)c) we o b t a i n
3 The Drift Diffusion
132
a)
Equations
v
d^u = e [J„(s) - n(s)] — ad^V y. n
b)
f
d v = e [ 7 ( s ) - n ( s ) ] — ad.V a £
(3.4.58)
p
p
c)
rj = s i g n [ ^ F ( i f , s ) ] .
Here Jj,(s) denotes J^(X(s), Y(s)). I f F is monotone i n c; for fixed s then a is independent of £ and (3.4.58) can be integrated. This gives v
u = -e- \_J„(s)-n(s)]
—
(3.4.59) 0 = -^[J ( )-n( )]-
S
S
w i t h integration constants ,4„ and / f . Because of the matching conditions u(±ao) = u(X(s)±, Y(s)±), i ; ( ± o o ) = v(X(s) + , Y{s)±) the integration constants are given by p
a)
A (s) = u (s) n
+
v
+
-(J (s)-n(s))e~ ^ n
««
(3.4.60) b)
v
A (s) = v (s) + - (J (s) • (X. p
+
n(s))e ^
p
v
v_(s) +
-(J (s)-n(s))e ^. p
Here u + (s) denote the onesided limits u(X(s) + , Y(s)±) and so on. So the layer p r o b l e m is given by (3.4.56)a) together w i t h the boundary conditions I/(oo, s) = V(X(s)
+ , Y(s) + ) ,
V(-oo,
s) = V(X(s)-,
Y(s)-). (3.4.61)
One can again show, as i n the e q u i l i b r i u m case, that the solution V exists and is indeed monotone for fixed s. So er i n (3.4.58) is really independent of £. The difference in the reduced p r o b l e m comes from the fact that^ now a layer term arises i n the u and v variables as well. This implies that V, u and TJwill be discontinuous across the P - N j u n c t i o n Y. Thus the reduced problem can not be formulated as i n (3.4.38) but i n a d d i t i o n interface conditions have to be applied for u, v across Y. F r o m (3.4.60) we o b t a i n that the functions v
a)
it +
e~ (J„-n(s))^~^
b)
v +
e (J -n(s))
v
(T
p
3.5 Reverse Biased P-N
Junctions
133
remain continuous. Thus the reduced p r o b l e m is o f the form a) b)
s
div J„ = R, div J =
grad u
n
- R ,
p
V{x u
J„ = p e - ^ s
V(x a B)
J = - n e- ' ' p
grad v
p
for (x, y) e Q together w i t h the boundary conditions d)
"lan = " J a n . .
„ e)
8u —
D
dv = —
v\
dilB
=
v\ , D dni>
>
(3.4.63)
=0
on <9Q, and the interface conditions which say that the terms f)
v
u + e~ (J„-n(s))°^,
<x(s) = s i g n ( F +
V_)
n g)
v +
h)
J„ • n(s)
i)
J • n(s)
v
e (J -n(s)) p
p
remain continuous across F. Since the reduced p r o b l e m has a different form, the effect of the field dependent mobilities is not only noticable locally, inside the layers, but also globally i n the outer solution. I t w i l l therefore influence the current flow t h r o u g h the P-N j u n c t i o n although p and p w i l l be close to p and fi for the outer solution. Note, that, since the effect o f the field dependent m o b i l i ties on the outer solution occurs t h r o u g h the interface conditions (3.4.63)f )g), this effect w i l l be the stronger the larger the j u m p in V at the P-iV j u n c t i o n r , and therefore 8^ V, is. So the locally large electric fields at the P - N junctions influence the solution globally. s
s
n
p
n
p
3.5 Reverse Biased P - N Junctions We now t u r n to the study o f P - N j u n c t i o n s under reverse biasing conditions. In the reverse bias case the singular p e r t u r b a t i o n scaling, introduced i n the previous Section, describes the solution of the drift diffusion equations only up to a moderate value o f the reverse bias. The reason for this is that, i f a potential difference of the appropriate sign is applied to the P-N j u n c t i o n , one observes the f o r m a t i o n o f a depletion region i n the semiconductor where no free carriers are present. This region acts as an insulator and, ideally, no current flows. I n Section 3.4 equation (3.4.5) implied that the space charge p = —n + p + C(x) is o f order ) } except in n a r r o w layer regions. Therefore, if that scaling is used, the depletion region has to lie entirely w i t h i n the P-N
134
3 The Drift Diffusion Equations
j u n c t i o n layer. O n the other hand the limits of validity o f this asymptotic analysis which we have briefly discussed i n the previous Paragraph (see (3.4.50)), indicate that the a p p r o x i m a t i o n breaks d o w n anyway for large reverse bias values. Thus, for large reverse bias, the layer region becomes so bloated that the scaling (3.4.2) has to be modified. We w i l l at first analyze the formation of the depletion region for moderate reverse bias values, using the scaling (3.4.2). This can be done by an a d d i t i o n a l asymptotic analysis of the layer term V i n (3.4.41) for 5 -» 0.
Moderately
Reverse
Biased
P-N
Junctions
I f we compute the layer term V i n (3.4.41) i n the same way as i n the e q u i l i b r i u m case, we o b t a i n K i n terms o f the inverse o f a m o n o t o n e function. a)
F (V(Ls),s)
b)
F_(V(Z,s),s)
+
'
F {V,s)
c)
= ^flol;
for
{ ^ 0
= ^2at
for
£ < 0
y
(3.5.1)
dz
T
2
z
z
K o) '3 {e u{s) (
2
v
= d {e ^ u(s)
d)
+ e- v(s))
sJ
s)
+ e"
FT<s
- C+(s)z - A + (s)
%(s)) -
C (s)V (s) T
T
A_(s)-A (s) +
V(0)
C {s)-C.(s)' +
For fixed V # V+ and 3 -» 0, F ( F , s) w i l l tend to F (V, T
F
T n
(F,5)=
To
s), given by
*
(3.5.2) z)
Solving F ( F , s)
lac, yields
T
1
C (F T
T
- F(0))
(3.5.3)
which gives d\ V = —C and therefore n = p ^ 0 inside the layer. The limit 3 -> 0 i n F is not uniform for Fclose to F . This raises the general question whether performing an a d d i t i o n a l asymptotic analysis for small 3 in the layer equations is exact, i.e. whether the t w o limits / -> 0 and 3 -> 0 commute. We w i l l leave the discussion o f the interdependence of these t w o limits to Chapter 4. A t this p o i n t the precise transition mechanism between the forward bias case and the situation under extreme reverse biasing conditions, treated i n the next Section, is not understood completely. T
P-N
Junctions
Under
?
Extreme
Reverse
Bias
Conditions
I f we increase the reverse bias further, the depletion region, and w i t h it the P-N j u n c t i o n layer, w i l l cover a significant part o f the device and the approx-
3.5 Reverse Biased P-N
Junctions
135
i m a t i o n , derived i n the previous Section, breaks d o w n . F o r P-N junctions under extreme reverse biasing conditions an asymptotic analysis based on a different scaling can be performed. This scaling was first introduced i n [ 3 . 6 ] and is given by 2
qL C
Cn ,
K,
CPs
s
(3.5.4) qp. U C
qjl
T
UC T
~L
'
The scaled boundary value p r o b l e m then becomes a) b)
div J„ =
c)
div J
d)
Ks\?n„
e) f)
=
Ps
2
p (X
2
X J„
grad n - n grad V )
n
s
2
-X
2
A J„
R> s
Us +
n
Kijen
s
grad p
s
s\dn = n'D \d(l T
s
D
D
>
da. = 0
dv n = i(Q(x) +
VQ(.x)
s
C (x)
2
+
+ , 'CXA
2
s
s
p grad V ) s
s
=
Ps\dn
D
Pb'Jdiii, (3.5.5)
4
4«5 ), 4
+ 45
2
U(x)e5 g)
2
qL n
t
n,.(x) =
h)
^ qL
. n n
C,(x) + V Q ( x )
2
+ 48*
2
2S
i
qh C
2
C
I n this scaling the depletion region, i.e. the region where AV = — C holds, is not necessarily small. So, i n particular so called punch t h r o u g h effects can be considered. (3.5.5) is, however, a scaling for very large bias values, since in order to keep U (x) i n (3.5.5)d) o f order 0(1), the unsealed bias U(x) has to grow w i t h the m a x i m a l d o p i n g concentration C. The analysis o f the drift diffusion equations i n the scaling (3.5.4) leads to the solution o f free boundary problems w i t h the edges of the depletion region as the free boundaries. We w i l l first treat the one dimensional case, where these edges are just points, and leave the discussion of the results k n o w n for the higher dimensional cases for later on. s
The One-Dimensional
Problem
The asymptotic analysis for extreme reverse bias is technically more complicated than for the forward bias case. The solution o f the reduced
136
3 T h e D r i f t Diffusion E q u a t i o n s
p r o b l e m behaves differently i n different subregions of the device, namely inside the depletion region and outside. The layers occur n o w not directly at the P - N j u n c t i o n but at the edges o f the depletion region. So, i n a d d i t i o n to finding the reduced solution, one has to determine the location o f these edges. I n general this w i l l result i n the reduced p r o b l e m becoming a free boundary problem. We w i l l consider the one dimensional case first. I n this case, the determination o f the boundary o f the depletion region becomes particularly simple since it consists o n l y o f t w o points. Aside from being a free boundary p r o b l e m the reduced p r o b l e m is also structurally more complex since, as we w i l l see, setting the perturbation parameter X to zero i n (3.5.5) yields an underdetermined problem. I n the language o f singular p e r t u r b a t i o n theory (3.5.5) is called a 'singular' singular p e r t u r b a t i o n p r o b l e m (see [3.31]). I n the one dimensional case we assume that, after scaling, the semiconductor is positioned i n the interval [ 0 , 1]. The one dimensional equivalent to (3.5.5) is then o f the f o r m a)
V" = n — p — C(x)
b)
J' = R,
c)
j ; = -R,
d)
V(x) = V (x) = U(x) + V {x),
X J„
2
= n„(X n'
2
N
XJ 2
-
nV)
2
= p. (-X p'
P
>
for
x e [ 0 , 1]
-pV)
p
(3.5.6) D
X
P( )
e)
f°
X
= PD( ) 2
n
D
n{x) =
bi
r
x = 0
and
4
= \(C + JC
+ 4S ),
p
D
n {x), D
x = 1
= \{-C
2
+ ^C
4
+ 4«5 ).
Here ' denotes differentiation w i t h respect to x. The l o c a t i o n of the P-N j u n c t i o n is at some p o i n t y e (0, 1). So, again employing the simplification of an abrupt P-N j u n c t i o n , we assume that C(x) has a j u m p discontinuity at x = y and is as smooth as we like elsewhere. Since this analysis shall h o l d for large reverse bias, we have to agree o n a sign convention for the d o p i n g profile C and the bias. So we assume that sign(C(x)) = sign(x - y)
(3.5.7)
holds. A large reverse bias then corresponds to V (i) — V (0) > 0. L e t t i n g X go to zero i n (3.5.6) gives the reduced p r o b l e m D
a)
V" = n - p -
b)
J ; = R,
c)
J ; = - R ,
o=-Li pv'
d)
V(x)=V (x),
n(x) =
C(x)
0 =
D
for
-n„nV'
x e [0, 1]
p
D
p(x) = p (x)
for
(3.5.8)
n (x), D
x = 0
and
x = 1.
The equations (3.5.8) allow t w o possible solutions: Case 1:
V' # 0,
Case 2:
V' = 0.
D
n = 0,
p = 0.
3.5 Reverse Biased P-N
Junctions
137
I n the first case, inside the depletion region, V" = — C(x) w i l l hold. I n the second case n and p cannot be determined immediately from the equations (3.5.8). Since n = p = 0 contradicts the boundary conditions, and we do not expect the depletion region to form at the contact, we derive the following problem for V V = 0 V"(x) =
C(x)
for
0 ^ x ^
for
^ X ^
for
V' = 0
x
2
(3.5.9)
x
2
< x ^ 1,
where x and x , the endpoints of the depletion region, have yet to be determined. I f the concentrations n and p stay bounded for X -> 0, then so do the derivatives of V and no zeroth order layer term can form i n the potential variable. Thus, the reduced potential V and its derivative have to remain continuous across x and x . Since V = 0 holds for 0 ^ x < x we have {
2
1
V(x )=V (0), i
2
l 5
P ' ( x ) = 0.
D
(3.5.10)
1
Inside the depletion region V is then given by V(x) = V (0)
(3.5.11)
C(n) dt] dC-
D
Since V = 0 holds i n [ x , 1] we have 2
V(x )=V (l), 2
V'(x )
D
(3.5.12)
= 0.
2
M a t c h i n g (3.5.11) w i t h the c o n d i t i o n (3.5.12) yields "C C{n) dt] d£, a) V {0)-V {1) D
D
(3.5.13) b)
*2
0
C(w) dn.
Let us first convince ourselves that (3.5.13) has a solution ( x x ). A family of solutions ( x ( 0 , x ( f ) ) of (3.5.13)b) alone, which depends continuously differentiably on the parameter t, has to satisfy l 5
x
2
2
(3.5.14)
C(x (f))x (f)-C(x (0)x (f)^0. 2
2
1
1
I f we add the equation x = 1 and the initial conditions x ( 0 ) = x ( 0 ) = y to (3.5.14) we obtain a system of ordinary differential equations whose solution satisfies (3.5.13)b). M o r e o v e r x ^ 0 and x ^ 0 holds and so x stays to the left of the P-N j u n c t i o n y and x stays to the right of y. I f we define f(t) by 2
x
1
2
2
1
2
f(t)
C{t]) dt] di
=
(3.5.15)
x,(t) Jx,(r)
then f(0) = 0 holds and we have to solve f(t) = V (0) - V (l) obtain a solution of (3.5.13). D i f f e r e n t i a t i n g / , we see that D
D
in order to
138
3 The Drift Diffusion
C(x )(x
f'{t)=-C(x )x {x l
1
2
1
Since, because of (3.5.14), x obtain f'(t)K
2
—x
(3.5.16)
2
^ 1, and therefore x
t
Equations
2
—x
^ f holds, we
x
(3.5.17)
-rmin{C(x)}. x^y
So f(t) is monotone from (—oo, 0 ] onto ( — oo, 0 ] and for reverse bias, that means for V (0) — F ( 1 ) < 0 , (3.5.13) w i l l have a unique solution. After finding x and x , the reduced potential V is uniquely determined by equation (3.5.9) together w i t h the boundary conditions (3.5.6)d) and the requirement that V and the reduced field V remain continuous. So far, n, p, J and J are k n o w n o n l y w i t h i n the depletion region [ x x ] where, because of (3.5.8), D
D
{
n
2
p
l 5
a)
/T(x) = p(x) = 0
b)
J„(x) = J (x ) n
J (x) p
+
l
=
2
(3.5.18)
Jp(x )-
Ro(0
l
for
dC
!
X
< X
2
holds. Here R {x) denotes the recombina^tion rate R evaluated at n = p = 0. Outside the depletion region, where V = 0, the reduced equations are insufficient to determine n and p. The only i n f o r m a t i o n about the reduced carrier concentrations is given by the c o n d i t i o n of vanishing space charge 0
n - p - C
= 0
for
x e [0, x ) u ( x , 1 ] . t
(3.5.19)
2
I n this sense (3.5.6) is a singular singular perturbation problem (see [3.31]). The standard approach to deal w i t h such a situation is to find a transformat i o n of the dependent variables w h i c h reduces the problem to a regular singular perturbation problem. I n our case such a transformation has been given i n [3.30] and is of the form (n, p)<-^{p, w), a)
p = n — p,
w = np
b)
n = i ((pp ++ V 7 > +
(3.5.20) 2
4 w
>,
P = U-P
+ VP
2
+
4
w
) -
In the transformed variables the equations (3.5.6) read aj
V" = p — C
b)
J'
— R,
w
1 -Jp 2
c)
n
•>; =
1
R _
+ 4w _Pn
1
V Pp. ~2
^ _
(3.5.21)
P
_P
n
P\ P
d) e)
X*p'
V
+ 4w V + X-
J„ Pn
p(x) for
C(x) = 0
wX and
5\ 1.
J„
+P
for
P
V(x)
V (x) D
x e ( 0 , 1)
3.5 Reverse Biased P-N
Junctions
139
Outside the depletion region, where V = 0 holds, the reduced equations i n the transformed variables are given by a)
V = o,
c)
K =
d)
w'
b)
p = C -R
R, 1
~ 2
for
2
/p
V
(3.5.22) 1
+ 4w Pn
~2
Pp.
P
.Pn
Pp_
x e [ 0 , X j ) u ( x , 1] 2
subject to the boundary conditions 4
w(x) = <5 ,
F(x) = K (x)
for
D
x = 0
and
x = 1.
(3.5.23)
Since the right-hand side o f (3.5.22) stays bounded, w cannot exhibit layer behaviour. This gives the conditions o n the reduced solution at x = x and x = x,. 1
a)
w(Xi) =
b)
J„(x )
w(x )
0
2
0
J (x ) p
(3.5.24)
R (C)
2
= J (x )
2
p
l
+
*o(C)
For ^ = 0 the p r o b l e m (3.5.22)^(3.5.24) has the trivial solution. One can show w i t h a perturbation argument a r o u n d this trivial solution that for 3 sufficiently small (3.5.22)-(3.5.24) has a unique solution (see [3.30]). Thus we have completely determined the solution of the reduced p r o b l e m and can now t u r n to the calculation o f the layer terms. Under strong reverse bias conditions the layers do not occur at the P-N j u n c t i o n y, as i n the forward bias case, but at X j and x , the edges o f the depletion region. F r o m (3.5.20) we calculate 2
2
n(x)
p(x)
=
=
| ( C ( x ) + JC(x)
+ 4w]
0 i(-C(x) + 'C(x) v
2
for
x e [0, x ) u ( x , 1 ] ,
for
xe(x,,x ),
1
2
2
(3.5.25)
+ 4vv;
for
x e [0, x ) u ( x , 1 ] , x
2
0
for xe(X[,x ). Since w ( x ) = vv(x ) = 0 holds, n is discontinuous at x = x , where C > 0 holds, and p is discontinuous at x = x , where C < 0 holds and the corresponding layer terms are needed at the boundaries o f the depletion region. The layer equations involve the higher order terms in the expansion. A t the right edge o f the depletion region (x = x ) we introduce the fast layer variable c = (x — x )//„. E x p a n d i n g the Poisson equation we see that, close to x = x , the potential V is of the form 2
x
2
2
t
2
2
2
V(t)
= V(x
2
+ It)
2
+ k $(c)
3
+ 0(/. ),
(3.5.26)
140
3 The Drift Diffusion
Equations
where <j> denotes the second order layerJerm. (3.5.26) implies that, close to x_= x , the field (l/X)d^V is given by [_V'{x + / £ ) + ).d (j)(c,)~\. Expanding V'(x + /.<;) around /. = 0 gives 2
2
i
2
2
,A [-£C(x ) +
+ 0(P)
2
^
V
i
a
=
^ % m
+ 0 ^ )
for
£<0,
for
£>0.
( 1 5
-
2 7 )
Inserting this i n t o the equations and letting /, go to zero while keeping c fixed gives the following layer equations and matching conditions: \n — p a)
bU
[n — p — C(x ) 2
b) c)
^ J „ = 0,
£ < 0
for
i > 0
d J, = 0 e
C
d i
for
+
8
n = \ % - / ^ ^ [no^ -p[-^C(x ) + ^ ] dp -pd^<j) %
d) e)
2
r
l° \ 0 for £ < 0
t
<^(oo) = (j>(—oo) = 0,
for
(3.5.28)
L, > 0
n(oo) = C ( x ) ,
n(—oo) = 0,
2
p(oo) = p(—oo) = 0. Again, the equations for the carrier concentrations can be integrated exactly and we obtain a)
n
C(x )e^
2 / 2 , C (
2
2
* >
C(x )e* 2
b)
for
Q<0
for
i;>0
(3.5.29)
p = 0.
Thus the only boundary value p r o b l e m to solve for the layer terms is the equation (3.5.28)a) together w i t h the boundary conditions (3.5.28)e). Using the same methods as i n the previous Section, one can show that this boundary value problem has a unique solution (see Problem 3.6). A similar expansion has to be performed a r o u n d the left edge o f the depletion region, where p has a layer term p(rj) which depends o n the layer variable r\ = (x — x )j/.. This layer p r o b l e m is of the form x
a)
—p
for
Y\ >
— p — C(x,)
for
n < 0
0 (3 5 30)
b)
p =
C( )e-^"
2l2)ClXl)
for
rj>0
C(xi)e"^
for
t]<0.
Xl
I n order to o b t a i n a u n i f o r m 0(1) a p p r o x i m a t i o n o f the solution one can again derive the composite expansion (V , n , p , J , J ) given by c
c
c
c
c
n
p
3.5 Reverse Biased P-N c
a)
141
Junctions 2
c
V = V + 0{}. ),
h)
c
=
J =J n
{n(x) + m)
+ 0(A) + 0(A)
2
c
'
(p(x) + p(n) + C( )
=
+ 0(A)
Xl
P
J = J + 0(A) p
for
" \n(x) + n(c) - C(x )
C
l
+ 0(A),
n
[ p ( x ) + p(w) + 0 ( 2 )
for
p
0 ^ x ^
x
x ^ x
^ 1
2
2
for
0 < x ^ x,
for
x , < v sc 1.
The derivation of higher order approximations is straight forward and is left to the reader (see Problem 3.7). The asymptotic validity o f (3.5.31) can again be shown in special cases. However the k n o w n results here are o f a different flavor than i n the forward bias case (see [ 3 . 6 ] , [ 3 . 7 ] , [ 3 . 8 ] , [3.9]). They are based on compactness arguments and do not give any convergence rates. So they only say that, for A -> 0, the solutions converge to the reduced solution (V, n, p, J„, Jp) derived above. Since the one-dimensional case is in no way particular, as far as asymptotic validity is concerned, we w i l l defer discussion of these results to the next paragraph, where we w i l l deal w i t h the two-dimensional case.
The
Two-Dimensional
Case
I n more than one dimension the approach to finding an asymptotic approx i m a t i o n o f the solution follows the same pattern as i n the one dimensional case. However, the solution o f the resulting free boundary problem is much more difficult since the boundary of the depletion region is a curve i n the t w o dimensional case and a surface in the three-dimensional case. F o r the sake of simplicity we w i l l restrict ourselves to the case o f t w o dimensions and leave generalizations to the three-dimensional case to the reader. (They are straight forward.) We w i l l use the same assumptions on the device geometry as i n the forward biased case. So the device occupies again a region Q a U w i t h a boundary dQ. = dQ u cQ where the boundary conditions 2
D
n
V\dn — ^olaou'
\dn
B
b)
B
N
—
n
D\dti i D
Pirn,, — P Dim,,
(3532)
dV dv an*
hold. Setting A to zero in the equations (3.5.5) gives the reduced problem
— C
a)
AV = n — p
b)
0 = ngradF,
d)
div J„ = R,
c) e)
0=-pgradF
div J = p
(3.5.33)
—R.
Jn and J have to be determined from the higher order terms in the expansion. (3.5.33) suggests that the device d o m a i n Q splits into a region where grad V = 0 (V = constant) holds, and the depletion region where n = p = 0 holds (see F i g . 3.5.1). p
142
In Q
3 The Drift Diffusion
D £
Equations
, i.e. i n the depletion region, the solution is given by n = p = 0,
A F = - C
for
xefl
(3.5.34)
D £
and outside the reduced solution is given by a)
g r a d F = 0,
c)
p= (-C
2
b)
n = \(C + JC
+ 4w), (3.5.35)
2
+ JC
2
+ Aw),
x e Q j u Q2
•
where vv is the transformed variable according to (3.5.20). w is then given as the solution of the two-dimensional equivalent of (3.5.22) (see P r o b l e m 3.8). I n order for this approach to be feasible, we have to assume that the boundary data V are piecewise constant on dQ so as not to contradict (3.5.35)a). So we assume D
V
D
= U
D
for
1
x e dQ ,
V = U
Di
D
for
2
x e dQ . Di
(3.5.36)
F o r the same reason as i n the one-dimensional case the a p p r o x i m a t i o n to the potential V does not have a zero order layer term, and so V and grad V are continuous across X and X , the boundaries of the depletion region. This gives the reduced problem for V: 1
2
a)
AV = —C
b)
V\
Xl
= U
grad V- \
V\
Xi
= U,
gradF-v |
c)
lt
X
x e Vl
2
V=\J
for
for
DE
=0,
Xi
2
xeQ,,
Q ,
X 2
(3.5.37)
= 0
V = U
for
2
xeQ , 2
where Vj and v are the unit n o r m a l vectors on X and X respectively. The asymptotic validity of the derived a p p r o x i m a t i o n has been shown in [ 3 . 7 ] and [3.8] for the one-dimensional case and i n [ 3 . 9 ] for t w o and three dimensions. A t least for the higher-dimensional cases the available results are of a very weak type. They only say, that the solutions of the full p r o b l e m (3.5.5) converge to the solution of the reduced p r o b l e m (3.5.34)—(3.5.37) i n the L°° weak star sense. This means, that for any test function <j> e L ' ( Q ) 2
1
2
3.6 S t a b i l i t y a n d C o n d i t i o n i n g for the S t a t i o n a r y P r o b l e m
V dx
Vdx,
ri(/> dx
143
n dx, (3.5.38)
p dx
p(/> dx
holds. I n [ 3 . 9 ] this result has been shown for a one carrier model (that means C(x) < 0 everywhere and n = 0 everywhere) and for the special case 3 = 0. I n this case the smooth variable vv vanishes identically and p = 1 holds outside the depletion region.
3.6 Stability and Conditioning for the Stationary Problem W h e n solving any k i n d of equation one of the most i m p o r t a n t questions from a practical point of view is that of stability and conditioning. These terms refer to the sensitivity of the computed solution o n the data. I n our case the input data are given by the boundary data on the Dirichlet part of the boundary, the d o p i n g concentration and the geometry of the device. The question of stability and c o n d i t i o n i n g becomes particularly i m p o r t a n t when the drift diffusion equations are solved numerically. Errors, introduced by a numerical solution, can usually be analyzed by regarding the computed solution as the exact solution of a perturbed problem. Stability hinges o n an estimate of the inverse of the linearization of the involved differential operator. As pointed out earlier such an estimate, w h i c h is independent of the perturbation parameter A, is not available i n the general case. I n this Paragraph we w i l l give estimates of approximations to the inverse of the linearized operator and briefly sketch h o w the performance of iterative solution methods for the nonlinear p r o b l e m can be influenced by using different sets of dependent variables. I f we formulate the p r o b l e m to be solved as F(z) = g,
(3.6.1)
where z is the solution and g denotes the data, we consider the perturbed problem F(z')
= g'.
(3.6.2)
The terms stability and c o n d i t i o n i n g now refer to the absolute and relative effect of the perturbation i n g o n the solution z. So one looks for constants K and K , s u c h that c
a)
\z-z'\\^K \\g-g'\ s
(3.6.3) \z - z ' l l
. „
\g - g'
b) holds, where || • || is some suitable n o r m . K is called the c o n d i t i o n number and K the stability b o u n d . Even i f the data of the p r o b l e m are k n o w n c
s
144
3 The Drift Diffusion Equations
precisely the stability and c o n d i t i o n i n g o f a p r o b l e m are of great significance since errors made in numerical computations can usually be estimated t h r o u g h backward error analysis (see [3.38]). I f we solve (3.6.1) numerically the operator F is replaced by some approximate, discrete operator F and F(z) = g
(3.6.4)
is solved instead. (Usually F also includes the effect of roundoff errors.) The concept o f backward error analysis now implies that z, the solution o f the discretized equation w i t h the exact data, can be interpreted as the solution o f the exact equation w i t h perturbed data. T h a t means that there exists a small p e r t u r b a t i o n Sg of the data g, such that F(z) = g + Sg
(3.6.5)
holds. So errors in numerically obtained solutions can be measured in terms of the stability bounds. I n t u r n it can be said that problems, where K and K , are too large, are practically very difficult to solve. They are called i l l posed problems. Stability and c o n d i t i o n i n g are, by virtue o f the mean value theorem, closely related to the linearization F' of F since c
s
Sg = F'(z)(z
- z) + o(\\z - z\\)
(3.6.6)
holds. O n the other hand the c o n d i t i o n i n g of the linearized operator F' is also of great importance for the performance o f iterative methods for the solution of the drift diffusion equations since they are usually based o n a linearization technique. The most prominent of these methods is Newton's method, which is of the form a)
w
= w + dw
b)
F'(w )dw
k+i
k
(3.6.7) =
k
-F(w ), k
where w and w , denote the old and the new iterates and (3.6.7)b) has to be solved for the increment dw i n each iteration. I f we apply Newton's method to the drift diffusion equations we o b t a i n the system k
t +
I
;. A* 2
a a
21
a
2l
-1 \
a dR/dn 22
3 1
= div(n n n
k
dV\
dR/dp «3 3
grad * ) ,
1
ffA
=
dn
(3.6.8)
h
[dp j a
3 ]
= di\(-n p p
k
grad * ) ,
dR « 2 2 = div [ p „ ( - g r a d * + * grad K J ] + — , on PR "33
=
div[/ip(-grad
* - * grad V )~\ + — , k
where * is a placeholder symbol. We have used the forward bias scaling of Section 3.4 here since it is o f greater practical significance (because it is
3.6 S t a b i l i t y a n d C o n d i t i o n i n g for the S t a t i o n a r y P r o b l e m
145
relevant for a much larger bias range). Also, for the sake of simplicity, i t is assumed that the mobilities ii and p. do not depend on V, n and p and that the recombination rate R does not depend o n V. The current densities J and J have been eliminated by inserting them i n t o the c o n t i n u i t y equations. O f course (3.6.8) is the linearization of the continuous problem and has to be replaced by the jacobian of the corresponding difference operator when solving the drift diffusion equations numerically. Reasonable stability bounds (which are independent of k) for the coupled system (3.6.8) are not available yet. One can however, i n lieu of such a rigorous approach, carry out a 'decoupled' analysis along the lines of [ 3 . 3 ] . F o r this purpose i t is beneficial to perform a variable transformation i n (3.6.8). This transform a t i o n is given by n
p
n
p
J
jdV\
ldV\ dn
= T
T =
y
0 0\ 1 0
1 n
(3.6.9)
\-p o \ ) Note, that the transformation (3.6.9) is the linearization of the nonlinear transformation to the S l o t b o o m variables n = e u, p v. The transformed equations then read VP]
2
idV\ J
1 /
1a
lfi\ ,
y l
z
J =\
fl = n
«21
^a
1 2
-A A
3 1
-1
1
n
"22
dR/dn
dR/dp |
(3.6.10)
"33
* + K + p) k
dR
= div(-JBfc *) + ( — "31 = d i v ( - J , *) Pk
dR \Pk^ dp
+
8R\
dR dn dR *
d i v [ / u „ ( - g r a d * + * grad V )~] + k
'22
d i v [ / ( ( - g r a d * - * grad F J ] + p
dn 8R * dp
and the update is given by the formula V
= V + dV,
p
= ( 1 +dV)p
k+l
k+{
k
k
n
k+1
= (1 + dV)n
k
+ y,
(3.6.11)
+ z.
Note, that, i f the drift diffusion equations are w r i t t e n in the Slotboom variables (F, u, v) and Newton's method is applied to the system (3.4.36), up to a transformation the same linear system has to be solved for the increments (dV, du, dv). I n this case (dV, du, dv) are related to (dV, y, z) via v
y = e *du,
v
z = e~ «dv.
(3.6.12)
Thus the difference between applying Newton's method to the system (3.4.5)
146
3 The Drift Diffusion
Equations
and (3.4.36) is only in the updating strategies, w h i c h are of the form a)
V
= V + dV,
k+1
n
k
= (1 + dV)n
k+l
+ y,
k
Pk+i = (1 — dV)p
+ z
k
(3.6.13) b)
V
k+X
v
= V + dV,
e ^u
k
iV
k + 1
Vk+1
v
= e (e «u dV
e~ v
+ y),
k
y
= e~ (e- -v
k+1
k
+ z),
respectively (see Problem 3.2). I n practice the m a t r i x J i n (3.6.10) is frequently replaced by a, i n some sense, simpler m a t r i x i n what is called approximate N e w t o n methods (see [ 3 . 5 ] ) . The most widely used approximate N e w t o n method consists of neglecting the subdiagonal terms i n (3.6.10) and solving the system a G
I,
y
1
n
-1
0
o
\*
(3.6.14)
0
0
«3 3
in each iteration, where the diagonal blocks a are the same as i n (3.6.10). I f this approach is used together w i t h the update (3.6.13)b) one obtains a version of the so called G u m m e l method (see [3.14]). Clearly, neglecting the subdiagonal terms i n (3.6.10) assumes small currents and recombination rates. We refer the reader to [ 3 . 5 ] , [3.17], [3.28] and [3.29] for a convergence analysis and various acceleration techniques for G u m m e l type methods. The iteration (3.6.14) has the advantage that only three scalar linear differential equations have to be solved at each step instead of a coupled system. F o r each of these separate equations stability bounds are easily obtainable. So, for instance, for the (1, 1) block of (3.6.14) a boundary value problem of the form u
a)
-PAdV
b)
dV\
eno
+ (n + p)dV = 0,
= f (3.6.15)
3d V —
0 0O»
has to be solved. ( W i t h o u t loss of generality we can assume that the last iterate satisfies the boundary conditions and therefore the boundary conditions for the increment are homogeneous.) D i v i d i n g (3.6.15)a) by (n + p) and applying the m a x i m u m principle yields max \dV\ ^ max n + p n n
(3.6.16)
which suggests a r o w scaling of the first r o w of the N e w t o n equation (3.6.10) by (n + p). After this row scaling the stability b o u n d for (3.6.15) equals one. So the Poisson equation is generally well conditioned. The c o n d i t i o n i n g of the linearized c o n t i n u i t y equations is a more subtle problem since here the stability bounds depend o n the type of the device. I f we consider the (2, 2) block of (3.6.14) i n the case R = 0, a boundary value
147
3.6 S t a b i l i t y a n d C o n d i t i o n i n g for the S t a t i o n a r y P r o b l e m
problem of the form d i v [ > ( - g r a d y + y grad V )~] = g n
k
(3.6.17) y Ian, = 0 •
( - grad y + y grad V ) • v | k
a n j s
= 0 Vk
has to be solved. E m p l o y i n g the transformation y = e Vk
a)
— d'\v(u„e
du, we obtain
grad du) = g (3.6.18) 0.
A p p l i c a t i o n of the m a x i m u m principle (see [ 3 . 1 2 ] ) gives that max\du\^Ke~^max\g\, n a
V := min V n
(3.6.19)
k
holds, where the constant K depends only on the geometry of Q and the m i n i m u m value of u (see Problem 3.3). Transforming back i n t o the y variable yields n
y
max \y\ ^ Ke ~^ n
max \g\, n
V:=maxV . o .
(3.6.20)
k
So the stability b o u n d for the linearized electron c o n t i n u i t y equation depends exponentially on the potential difference. Remember that, at O h m i c contacts, n
a
, = U + 1»
C
+
+
f;
4
^)
(3.6.21,
holds. I f we consider, for instance a device w i t h a contacted n- and a contacted p- region and a piecewise constant d o p i n g profile (C = C > 0 in the n-region and C = — C_ < 0 i n the p- region), where a potential of U„ and U is applied, respectively, we obtain +
p
e
v - v
^
s - 4
e
u „ - u „
c
+
c
_
(3.6.22)
I f we use scaling factors corresponding to a 1/i device geometry and a m a x i m a l d o p i n g concentration of 1 0 the right-hand side of (3.6.22) is of the order of 1 0 in thermal e q u i l i b r i u m (U = U ) and becomes even larger in the reverse bias case. The question which arises immediately is whether the b o u n d (3.6.20) on the stability constant is sharp. Such high stability bounds are actually observed for complicated devices in numerical calculations (see [3.19]), but not for a simple P-N j u n c t i o n diode. Generally i t can be said that the bound (3.6.20) is attained in the presence of so called floating regions. These are p- or n-regions w i t h o u t an O h m i c contact (see [3.34]). One can, at least in the one-dimensional case, show that a stability estimate of the form 1 9
1 8
n
9 y ^ K max max n n n n
p
(3.6.23)
holds w i t h a moderate constant K, i f every p- and n- region is contacted.
148
3 The Drift Diffusion Equations
We refer the reader to [ 3 . 3 ] for further details on the c o n d i t i o n i n g and stability o f the drift diffusion equations and of its effect o n the performance of iterative methods.
3.7 The Transient Problem The remaining part o f this Chapter is concerned w i t h the analysis o f the transient drift diffusion equations. So, we consider the p r o b l e m a)
div(e grad V) = q(n — p — C)
b)
d i v J„ = q(d n + R),
d)
J„ = q(D„ grad n - p„n grad V)
e)
J = q( — D grad p — p p grad V)
f)
n{x,0)
g)
V(x, t) = V (x, f ) ,
p
p
= n\x), D
p(x, t) = p (x, t) D
h)
ev — (.x, 0 = 0, dv
div J = q( — d p — R)
c)
t
p
t
for
p
p(x,0)
=
I
p (x)
n(x, t) = n (x,
t),
D
for J - v = 0,
xeQ,t>0
x e dQ
D
J-v = 0
for
xe8Q , N
where the coefficients E and q etc. have the same meaning as i n the stationary case. F r o m a practical p o i n t o f view the interest i n the transient drift diffusion equations is o f a different nature than that i n the steady state problem. I n the steady state p r o b l e m one is interested p r i m a r i l y i n the voltage current characteristics, that is i n the current flow as a function of the applied potential difference. W h e n considering the transient p r o b l e m one is i n terested i n the time required by the solution to reach a steady state. As far as numerical simulations are concerned, it is also o f interest to calculate the transient response o f several coupled devices constituing, say, a F l i p - F l o p . The structure of the transient solution is obviously more complex because of the a d d i t i o n a l dimension time. N u m e r i c a l calculations are more expensive for the same reason. O n the other hand, the transient p r o b l e m is, in some sense, easier to deal w i t h analytically. F o r instance, while the steady state p r o b l e m admits multiple solutions, a feature which is exploited technically i n c.f. a thyristor (see Chapter 4), the s o l u t i o n to the transient p r o b l e m w i l l , i n general, be unique. Existence results for the steady state drift diffusion equations have to rely o n the Schauder fixed p o i n t principle, while those for the transient p r o b l e m can employ c o n t r a c t i o n arguments, at least locally i n time. Generally speaking, there are t w o types o f transient analysis, one 'close' to a steady state and one 'far' from it. I n the first case one is interested i n what happens, i f a small p e r t u r b a t i o n is introduced i n a stationary i n i t i a l state.
149
3.8 T h e L i n e a r i z a t i o n o f the T r a n s i e n t P r o b l e m
Thus, one considers the problem 8,w = F(w),
w(t = 0) = vv* + ef,
(3.7.2)
where F, i n our case, stands for the nonlinear differential operator corresponding to the steady state p r o b l e m and vv = (n, p). vv* is the solution of the stationary problem, so F(w*) = 0 holds, ef is a small perturbation, so /' is some function and i: is a small parameter. I f we set vv = w* + ez, z satisfies 8 z = F'(w*)z t
+ 0(e),
z(t = 0) = / .
(3.7.3)
Thus, in first order, the behaviour of the solution vv can be determined by analyzing the linearized p r o b l e m 8 z = F'(w*)z,
z(f = 0) = / .
t
(3.7.4)
I n particular, it can be deduced that, if the solution z o f (3.7.4) decays to zero as f -» co, w* is a dynamically stable steady state. We will treat the linearized problem (3.7.4) i n the next Paragraph. O f course, this approach is only valid i f the perturbation sf is sufficiently small and, in this sense, is rather an extension of the steady state analysis. I f the behaviour of transient solutions is of interest over longer periods of time, in particular if a P-N j u n c t i o n is switched from forward to reverse bias, the full, nonlinear p r o b l e m (3.7.1) has to be dealt w i t h . Here the analysis follows the same pattern as i n the stationary case. Once again, the m a i n instrument to understand the behaviour of the solution of (3.7.1) w i l l be asymptotic analysis.
3.8 The Linearization of the Transient Problem I f we employ the forward bias scaling of Section 3.4 we obtain the system 2
a)
X &V=n-p-C
b)
L TT^ t
(3.8.1)
2
8 n
= div J„-
R,
c)
J = Li (grad n - n grad V) n
n
U Ll T
1} d) —— 8 p = - d i v J - R, e) J = ^ ( —grad p — p grad V). U fi A l t h o u g h the scaling is of no particular importance for the purpose of this Section we use the scaling of the stationary forward bias case for the time being, f i n (3.8.1) is the unsealed time. (3.8.1) suggests the scaling t
T
t = T%h,
(3.8.2)
where t is the scaled time variable. Since ft is the scaling factor of the electron and hole mobilities u„ and p , and D„ = V u holds, the timescale given by (3.8.2) corresponds to the diffusion timescale of carriers. W i t h this scaling s
p
p
T
np
150
3 The Drift Diffusion Equations
(3.8.1) reads 2
a)
X AV = n — p — C
b)
d,n = d i v J - R,
d)
d p = — d i v J — R,
c)
n
ts
J„ = ^ „ ( g r a d n — n grad V) e)
p
J = p ( - grad p p
p
(3.8.3)
grad V).
F r o m here on we w i l l o m i t the subscript s. I f we linearize the operator F i n (3.7.2), corresponding to the steady state problem, a r o u n d any s o l u t i o n w = (n, p) along the direction z = (u, v) we o b t a i n the linearized operator
—
div /„ — R u n
F'(w)z
— div /
— Ru n
Rv p
—
R„v
p
/„ = p (grad n
I
p
p
(3.8.4)
u — u grad V — n grad (f>)
= p ( - g r a d v - v grad V - p grad ), p
where V and
2
A
(3.8.5) )} A<j) = u — v together w i t h homogeneous b o u n d a r y conditions for . There are four m a i n questions w h i c h are o f interest i n connection w i t h the transient linearized problem 1. Does the linearized p r o b l e m (3.7.4) have a unique solution? 2. Is the linearization stable; i.e. i f we consider the i n i t i a l b o u n d a r y value problem d z = F'(w)z t
+ g,
Z
( t = 0) = / ,
(3.8.6)
can z be bounded i n terms o f / and g7 3. Does the s o l u t i o n z o f (3.8.6) w i t h a homogeneous r i g h t - h a n d side g = 0 tend to zero for t -» oo? 4. D o the eigenvalues o f F ' have negative real parts? Clearly questions 3 and 4 have to do w i t h the dynamic stability o f a steady state, i f w is chosen as the stationary s o l u t i o n o f F(w) = 0. The answers to questions 1 and 2 tell whether linearization is justified at all. O f course we cannot expect an unqualified 'yes' as the answer to questions 3 and 4 since we k n o w that there exist d y n a m i c a l l y unstable steady states to the drift diffusion equations (see Chapter 4). W e w i l l , however show that steady states, up to a certain a m o u n t of current flow and r e c o m b i n a t i o n , are stable. The answer to all four questions is based on an energy estimate; i.e. an estimate of the form ^ X < z , z > ,
(3.8.7)
where < •, • > denotes a suitable scalar product. This scalar p r o d u c t is o f a
3.8 T h e L i n e a r i z a t i o n of the T r a n s i e n t P r o b l e m
151
p r o b l e m specific form and has been used i n [3.24] for the linearization a r o u n d e q u i l i b r i u m (J„ = J = 0) and i n [3.22] for the linearization a r o u n d a general s o l u t i o n of the transient or steady state problem. Since, i n order to deal w i t h the eigenvalue p r o b l e m , we w i l l have to consider complex valued eigenfunctions we w i l l define the scalar product for complex valued functions. p
=
u
v
Definition 3.8.8: Let z = (u , v ) and z ( n i) be complex valued functions. T h e n we define the scalar product < • , •> and the n o r m || • || i n the following way. First we solve 1
)} A
2
1
t
2
(3.8.9)
u-, — v-,
together w i t h the b o u n d a r y conditions (x) = 0
for
2
xedQ ,
^(x) = 0
D
for
x e
8Q . N
(3.8.10)
dv Then < z
l 5
z > is given by 2
On
z >
n
2
(3.8.11)
^2 J + »1 ( — +2 )dx.
N o t e that the bars denote complex conjugation here. We leave it to the reader to show that < • , •> really constitutes a scalar product (see Problem (3.9)). The energy estimate is now given by L e m m a 3.8.12: Let z = (u, v) be a complex valued function. Re ^
-£
+ holds for all c e [ 0 , 1) with a, ,
K
j
=
2
Li n\gmd
a\
Kj
R
n
1 - £
+ K
Kj and K
Then
+ u_p|grad p T dx
(3.8.13)
given by
R
(3.8.14)
P = - + <j>, P + max
X
2\™
R, K
R
= - max | «p + 11 max 2 n n 1
and <j> the solution of X Acj) = u — v together with the homogeneous conditions (3.8.10).
boundary
The p r o o f o f this energy estimate is technically quite complicated and can be found i n [3.22]. W e w i l l o m i t the p r o o f here and instead discuss its implications o n the questions 1-4 above.
152
3 The Drift Diffusion
Equations
The estimate (3.8.13) immediately implies the stability of (3.8.6). I f we define the n o r m ||z|| by ||z(i)|| = V ,
(3.8.15)
and take z = (u, v) to be real, we obtain d \\z\\ =
Re.
t
(3.8.16)
Using the energy estimate (3.8.13) and the Cauchy-Schwartz inequality, this gives d,\\z(t)\\ <(Kj
+ K )\\z(t)\\
+ \\g(t)\\
R
(3.8.17)
and, by G r o n w a l l ' s inequality lK +K
v
\\z(t)\\ ^ e ' * (ll/H
(K
+ J
+K
)s
e- > * \\g(s)\\ds).
T o show that the solution z decays for sufficiently small current flow and recombination rate and homogeneous g = 0 we estimate in terms of jn(yU„M|grad oc\ + p, p\gmd fi\ ) dx. R e w r i t i n g , one obtains 2
2
p
=|
n\oL + \ + p\fi — <j>\ +(v — u)<j> dx 2
ua + vpdx = j 2
2
« | a + ^ | + p | / ? - r > | + P | g r a d \ dx
^ const^||a||
2 2 ( n )
+ P||
2 2 ( n )
2
+
(3.8.18)
2
(n + pM
+ P|grad
\ dxj.
Rewriting the Poisson equation i n terms of a and jS we obtain I
2
A = (n + p)(j> + not - pp.
(3.8.19)
M u l t i p l y i n g b o t h sides by
2
2
( l | g r a d \ + (n + p) ) dx ^ c o n s t ( | | a | | , L
(n)
+
\\P\\ )U\\ LHa)
LHa)
n UWmct)
< const(||a||
L2(n)
C o m b i n i n g (3.8.18)-(3.8.20) gives ^ const(||a|| 2 + L
(n)
+ ||)S||
WPW^f
t2(n)
3
8
2 0
).
< - - )
•
(3-8.21)
2
2
Since the L - n o r m of a function can be bounded in terms of the L - n o r m of its gradient i f the function is sufficiently smooth-and vanishes on at least a part of the d o m a i n boundary <9Q (see [3.1]), there exists a constant K such that ^ K
(/u„n|grad a|
2
2
+ ^ p | g r a d >3\ ) dx p
(3.8.22)
3.9 Existence for the N o n l i n e a r P r o b l e m
153
holds. Thus, it can be deduced from (3.8.13) that Re ^ ( ( k
+ y^Kj]
r
& ~ e
2
(p„rc|grad a\
2
+ / i „ p | g r a d p ] ) dx
holds. I f Kj and K are sufficiently small the right hand side can be made negative by an appropriate choice of e and, using the same argument as above, l i m , ^ ||z(f)|| = 0 holds for g = 0. T o estimate the eigenvalues of F'(w), (3.8.13) can be used immediately. I f a complex eigenfunction z satisfies R
coz = F'(w)z
(3.8.24)
m u l t i p l i c a t i o n by z yields (Reco) = Re .
(3.8.25)
Setting c = 0 i n (3.8.13), we obtain Reco ^ Kj + K . Again, using the same argument as before, the real parts of all eigenvalues of F'(w) are negative i f Kj and K are sufficiently small. I n [3.22] a version of L e m m a 3.8.12 is used to show that the operator F'(w) generates an analytic semigroup on L ( Q ) equipped w i t h the n o r m || • ||. Thus the existence of a solution to the p r o b l e m (3.8.6) is guaranteed. So the answer to all four questions asked in the beginning is a c o n d i t i o n a l 'yes'. There exists a solution to the linearized problem (3.7.4). This solution is stable in the sense of the estimate (3.8.17). The solution of the linearized problem w i t h homogeneous boundary conditions tends to zero for t -» oo for sufficiently small current flow and recombination. R
R
2
3.9 Existence for the Nonlinear Problem I n this Section we w i l l show the existence of a solution to the transient drift diffusion equations (3.8.3). As mentioned previously, the existence of a solution can, at least for a sufficiently small time interval, be established by a contraction argument. There are various papers in the literature establishing existence for various models for the mobilities and the recombination rates (see c.f. [3.11], [3.32], [3.33]). I t should be noted that the presence of velocity saturation effects complicates existence proofs considerably if the Einstein relations (a)
D = U i, n
Tf n
(b)
D =U i„ p
Tf
(3.9.1)
are kept. I n this case the differential operator loses its u n i f o r m parabolicity for large values of the electric field — grad I " and the fixed p o i n t map used below w i l l not necessarily be well defined. The existence p r o o f for the transient problem consists essentially of t w o
154
3 The Drift Diffusion Equations
steps. First, short time existence is established by a c o n t r a c t i o n argument. Then a p r i o r i estimates are used to show that this solution can be continued a r b i t r a r i l y i n time. O t h e r approaches, treating more sophisticated models (i.e. avalanche generation), use a Schauder type fixed p o i n t argument directly (see c.f. [3.32]). We w i l l follow the lines o f [3.11] but, for the sake o f simplicity, we w i l l n o t consider velocity saturation effects. Let us first consider the existence o f a solution for a sufficiently short time interval. We define the fixed p o i n t m a p F by F(n, p) = (u, v) i n the following way. Step 1: G i v e n n and p solve the Poisson equation 2
I
AV = n - p -
C(x) (3.9.2)
dV v w = v
D
,
-
= 0 oil,
for V. Step 2: Solve a)
d,u = d i v [ / i „ ( g r a d u — n grad V ) ] — R(n, p)
b)
d,v = d i v [ / < ( g r a d v + p grad K ) ] — R(n, p)
c)
u(t = 0) = n ,
d)
v
t e (0, T)
p
du
7
(t = 0) = p',
lf-«„ v\
s a
D>
=p ,
0
Jv dv
(3.9.3)
0
D
da,
for u and v. For the fixed p o i n t argument we w i l l use the following n o r m : (n,p)\\
max { | | n ( / ) | |
=
L2(Q)
+ IIPWIILW
r
1/2
+
(3.9.4)
I f ||(n, p)\\ is finite then standard existence results (see [ 3 . 3 7 ] ) for linear parabolic p a r t i a l differential equations i m p l y that (3.9.3) has a unique solut i o n (u, v). W e n o w show that the m a p F is contractive for a sufficiently small time interval (0, T) o n the set M := a
{(n, p): \\(n, p)\
(3.9.5)
a}.
For the difference (du, Sv) = F(n , x
p ) - F(n , t
2
p) 2
we o b t a i n the i n i t i a l b o u n d a r y value p r o b l e m
(3.9.6)
155
3.9 Existence for the N o n l i n e a r P r o b l e m
a)
d du = d i v [ p „ ( g r a d Su — n grad V + n grad V )~\ — oR
b)
d,8v = d i v [ ^ ( g r a d <5u + p , grad V — p grad V Y\ — dR
c)
Su(t = 0) = 0,
d)
t
x
x
p
1
2
2
2
ddu <5"lan = 0> D
dv(t = 0) = 0,
5v\
1
rfi.v
ddv
= 0,
enB
(3.9.7)
0
7
1
e)
2
0
7
SR := R(n , p ) — R(n , p ). x
x
2
2
M u l t i p l y i n g (3.9.7)a) b y Su, integrating by parts w i t h respect t o x a n d integrating w i t h respect t o t from t = 0 t o t = T yields 1 rr H„||grad Su(s)\\ ds \Su(t)\
+
L2{n)
(3.9.8)
[ p „ ( n grad V — n grad F ) - g r a d du — 5Rdu~\ dx ds. 1
o
x
2
2
J
We estimate the right-hand side o f (3.9.8) by 1
r
T
\\du(t)\\ ma) +
p j g r a d 5u(s)\\
LHn)
l(\\5n grad V \\
^ const
a
Su\\ 2
• ||grad
+ K grad
lAim
SV\\ ) LHil)
(\\on\\ 2
+
L in)
ds
L iC1)
+ WMmnMSuhwlds, (3.9.9) where dn denotes (n — n ) etc. The right-hand side o f (3.9.9) can be estimated further using the definition o f M , the Sobolev imbedding theorem and the G a g l i a r d o - N i r e n b e r g inequality (see c.f. [3.37]) ||flf|| r ^ const ||flf|lir Q) 11^11,1(0) ^c(e)\\g\\ + e\\g\\ (3.9.10) x
2
a
L
(n)
(
LHn)
HHa)
l
for any 8 > 0, r ^ 2 and any H (Q) function g w i t h s = d(j — j) and d being the dimension o f the p r o b l e m . We o b t a i n ||<5ngrad V \\ 2
+ \\n grad dV\\
LHn)
x
^ const[a(\\5n|| •(c(e)||5n|| 2 L
Lr(n)
(n)
2
s; c(a, e)(\\dn\\
+ ||c5p|| , ) + ( | | n L
+ \\Sp\\hm+
(n)
+ e||<5n|| i )]||grad H
+ ||<5p||
L2m
(3.9.11)
L2(n)
(n)
2 2(Q(
) +
2
||
+
L 2 ( Q )
IIP IILW 2
du\\ 2 L
in)
2
e(\\Sn\\
HHn)
II^UHMQ))-
Inserting (3.9.11) into (3.9.9) and using the inequality \ab\ ^ ~ a 2e
2
2
+^b , 2
a,b,se
s > 0
(3.9.12)
156
3 The Drift Diffusion Equations
yields 1 2
0
nJgrad5u{s)\\ 2 ds L
(c(fl, £)(\\5n\\
^ const
(3.9.13)
ia)
+
2
L2in)
\\5p\\
2
)
L2(n)
o + e(l!<5«ll '(n) +
+ H^lliW)
II^PIIHMII)
H
2
< m a x { r c ( a , e), fi} ||(<5n, t)p)|| + e
d
s
l<5w(s)llH'
A similar inequality holds for dv and, by choosing s and T sufficiently small, F can be made contractive i n M ; i.e. 0
< ill(Sn,8p)||,
V ^ p J ^ p ^ e M ,
(3.9.14)
holds. Because o f the Banach fixed p o i n t theorem there exists exactly one fixed p o i n t (n*, p*) w i t h (n*, p*) = F(n*, p*) i n M . This fixed p o i n t is the unique, weak solution of (3.8.3) i n (0, T). In order to show the existence of a global solution for a r b i t r a r y time intervals [0, T], it is necessary to show, i n a d d i t i o n , an a p r i o r i estimate on the solution o f the form a
(\\n(s)\\
\n(t)\\ma) + \\p(t)\\mn) +
+ | | p ( s ) | | „ i ) ds ^ c(t) (3.9.15)
HHil)
(n)
o
for the solution (n, p), where c(t) is a bounded function. The p r o o f o f this a p r i o r i b o u n d involves a quite complicated argument based o n a p r o b l e m specific L y a p u n o v function. We refer the reader to [3.11] for the details. Using (3.9.15), the existence of a solution can then be established for a r b i trarily large time intervals.
Asymptotic
Expansions
for the Transient
Drift
Diffusion
Equations
As i n the steady state case, a great deal of insight i n t o the structure of the solutions o f the transient p r o b l e m can be gained by asymptotic analysis. The structure o f solutions o f the transient drift diffusion equations is quite a bit more complicated than that o f stationary equations. I n a d d i t i o n to all the structural complexities already present i n the steady state case, one is confronted w i t h different time scales whose occurence depends o n the form of the i n i t i a l data. I n this Section we w i l l first derive an asymptotic a p p r o x i m a t i o n (for small k ) o n the time scale given by the scaling (3.8.2). We w i l l refer to this time scale as the 'slow' or diffusion time scale i n the future. There we w i l l observe the same spatial structure as i n the stationary problem; i.e. layers a r o u n d the P-N j u n c t i o n s etc. However, for this a p p r o x i m a t i o n to be valid, it w i l l be necessary that the i n i t i a l and boundary data satisfy certain conditions. I f a steady state solution is taken 2
3.10 A s y m p t o t i c Expansions o n the D i f f u s i o n T i m e Scale
157
as initial d a t u m , which usually is the case, and i f the applied boundary potential varies o n l y o n the diffusion time scale, then we can conclude from the results o f Section 3.4 that these conditions are satisfied. So the fast time scale only occurs because o f perturbations i n steady state initial conditions or because o f rapid changes i n the externally applied bias. Perturbations i n the initial conditions have to be considered as soon as the transient drift diffusion equations are solved numerically. I t turns out that the structure of the temporal layer solutions depends strongly on the type o f perturbation introduced.
3.10 Asymptotic Expansions on the Diffusion Time Scale We w i l l first derive the zero order term o f the asymptotic expansion of the solution o f the transient drift diffusion equations o n the time scale given by (3.8.2). This time scale corresponds to the diffusion of carriers since the diffusion coefficients D„ and D i n (3.7.1) have been scaled to order 0(1). As we w i l l see, the structure o f the zero order term o n this time scale is not essentially different from that o f the stationary p r o b l e m in the sense that the reduced problem (3.4.37) n o w simply evolves i n time. The spatial layers remain located at the P-N junctions and w i l l only widen or n a r r o w according to the applied bias. p
Existence o f a solution to the transient reduced p r o b l e m could, i n principle, be shown by the same methods as for the full p r o b l e m i n Section 3.9. However, we w i l l sketch a different type o f existence p r o o f at the end o f this Section i n order to reflect o n the structure o f the solution of the transient reduced p r o b l e m . This w i l l express the fact that, for certain types o f devices such as diodes or bipolar transistors, the free electrons w i l l actually m a i n l y populate the A/-region and the holes w i l l populate the P-region under moderate biasing conditions. Setting the p e r t u r b a t i o n parameter I = 0 i n (3.8.3) we o b t a i n the reduced equations 0 = n
b)
d,n = div J„ -
c)
d p=
d)
J = - M g r a d P + P grad V)
e)
J„ = / U g r a d n - n grad V),
t
—
p
— C(x)
a)
-div J
R p
-R
(3.10.1)
p
which i m p l y vanishing space charge everywhere. Again, as i n the stationary case. (3.10.1)a) is consistent w i t h the Dirichlet boundary conditions since, i f only O h m i c contacts are present, n
D
— p — C(x) = 0 D
for
x e
cQ
D
(3.10.2)
3 The Drift Diffusion
158
Equations
holds. The reduced equations (3.10.1) have the following interpretation. W h i l e the full transient drift diffusion equations state the conservation o f the t o t a l current J = J„ + JP-X2
grad (d V),
(3.10.3)
t
the displacement current —X2 grad(5 V) can be neglected away from spatial and temporal layers, and only the drift diffusion current J + J is conserved. The spatial layers on the slow time scale occur near the P - N j u n c t i o n where, because o f steep gradients or discontinuities i n the d o p i n g profile C(x), the reduced concentrations n and p are discontinuous. The spatial layer terms, and the corresponding layer equations, are o f the same form as for the stationary problem. F o r instance i n t w o space dimensions the layer terms are functions o f the form (
n
T
w = w(Z,s,t),
w = (V, n, p, J , J ) , n
p
(3.10.4)
p
where (£, s) are again given by the same local coordinate transformation near the P-N j u n c t i o n as i n the stationary case, s is the curve parameter of the P-N j u n c t i o n and Q is the perpendicular distance of the p o i n t (x, y) to the P-N j u n c t i o n divided by X. C a r r y i n g out this variable transformation and inserting the layer term (3.10.4) i n t o the equations gives a)
d\V = n - p - C(s) + O(X)
b)
d^n-nd^V
d)
8 p + pd V= i
= O(X), 0(X),
i
c)
^ ( J „ • n) = O(X)
e)
d (J -n) i
(3.10.5)
= 0(X).
p
Letting X go to zero, and after integration o f the current relations and the continuity equations, one obtains, similar to the stationary case 9
a)
»({, s, t) = A (s,
t ) e
b)
P(£, s, t) = A (s,
t ) e -
c)
JM, s, t) = J e ^ - ^ '
d)
J (Z,s,t)
e)
d\9
n
p
V
^ 9
^
)
(3.10.6)
n
= J *<-*<.*.'>^>
p
= n - p -
where A and A n
p
p
C,
are given by
a)
A„(s, f) = n(X(s),
Y(s), t j e - ^ w - ' w - "
b)
A (s, t) = p(X(s),
Y(s),
(3.10.7) p
t
)e
v l X ( s )
-
r { s M
,
and the interface conditions at the P-N j u n c t i o n are given again by the requirement that y
ne~ ,
v
pe ,
J 'n, n
J 'n p
(3.10.8)
are continuous across the P-N j u n c t i o n . Equations (3.10.1) can be reformulated as a system o f one elliptic coupled to one parabolic equation. Differentiating (3.10.1)a) w i t h respect to time and inserting from
3.10 A s y m p t o t i c E x p a n s i o n s o n the D i f f u s i o n T i m e Scale
159
(3.10.1)b),c) gives div(J„ + J,)
= 0.
(3.10.9)
So the system (3.10.1) can be rewritten as
a) div(J„ + Jp) = 0 b)
d p = -div
J
t
p
- R
(3.10.10)
c)
J„ =
d)
Jp = P [ —grad p — p grad
C)gradF]
p [grad(p + C ) - ( p + n
F].
P
After inserting for Jn and Jp from (3.10.10)c),d) (3.10.10)a) is an elliptic equation for F while (3.10.10)b) is a parabolic equation for p. Thus, the reduced p r o b l e m is given by the equations (3.10.10) together w i t h boundary conditions a)
p(x,t)
b)
dp JL[x, t) = 0, dv
F(x, t) = V (x, t)
= p (x), D
for
D
dV — (x, i) = 0, dv
,xecfi , D
f>0 (3.10.11)
for
xedQ ,
t > 0,
N
the interface conditions (3.10.8) and an i n i t i a l c o n d i t i o n for p. n(x, t) is then given a posteriori by n(x, t) = p(x, t) + C(x)
(3.10.12)
and satisfies automatically the b o u n d a r y conditions n(x, 0 = n (x) D
(= p {x) + C(x)) D
for
xedQ , D
t>0. (3.10.13)
One feature of the solutions o f the drift diffusion equations, that has not yet been expressed t h r o u g h our asymptotic analysis, is that, away from P-N junctions, the carrier densities n and p w i l l be quite small i n the P- and N-region, respectively (see [3.35]. I n the case, when an A^-region is adjacent only to P-regions, and when all the N- and P-regions are contacted this fact can be explained by an asymptotic analysis using the built i n potential. This a d d i t i o n a l asymptotic analysis has been used i n [3.26], [3.27] to show the existence o f a solution to the reduced p r o b l e m (3.10.10), (3.10.11), (3.10.8). We assume that the device d o m a i n Q is the disjoint u n i o n o f K P- and Af-regions whose b o u n d a r y contains exactly one contact each. K
a)
Q={jQj,
5Q .n5Q # { }, J
j=l,...,K
D
J = 1
b)
sign(C(x)) = const
(3.10.14) inQy,
j=\....,K.
Using the f o r m of the boundary data V and p in our scaling, we have D
D
(see [3.19] or Section 3.1)
160
3 The Drift Diffusion
a)
V (x, t) = Vj(t) + V (x) D
for
hi
2
C(x)
b)
VJx)
= \n
c)
p (x)
=
d)
n (x)
= p (x)
Equations
Z
+ JC(x)
+ 4<5
2
28
(3.10.15) D
D
2
8 txp(-V (x)),
x e dQ,
bi
+
D
C(x).
Here Vj is the externally applied potential, so the difference between the Vj is the applied bias and V is the so called built in potential, which is due only to the d o p i n g (see [3.19]). N o t e that V = 0 holds i n the case of an undoped semiconductor. 3 i n (3.10.15) is given by bi
bi
2
S
=
(3.10.16)
max | C(x) xeil
and w i l l be quite small i n practice. We have neglected the effect of <5, so far, since it appears only logarithmically i n the boundary conditions (3.10.11) and V w i l l be of moderate size, even for small <5. I n order to analyze the structure of solutions of the reduced problem, we employ the variable transformation V -* <j>, p -» u, n -» C(x) + u, given by bi
A
a)
V(x, t) = 3 (t>(x, t) + V (x) + Vj(t)
for
bi
4
b)
p(x, t) = d u(x,
c)
J„ = <5 /„,
4
2
t) + 8 exp(-V (x))
x e Q,
for
bi
xeQ
(3.10.17)
4
d)
J = <5 / . p
P
Note, that this transformation implies automatically n — p — C = 0 everywhere. Inserting (3.10.17) i n t o the equations (3.10.1) yields a)
div(/„ + I ) = 0
b)
d u = —div I — S
p
t
p
4
grad u — w(grad V + <5 grad <j>) bi
2
•C + JC
4
+ 43
grad <j>
(3.10.18)
4
D
>
I
P = PP
-grad u — u(grad V + <5 grad <j>) bi
2
C + JC
4
+ 4<5
grad
S i n (3.10.18)b) is the transformed recombination rate. I f R is given by the Shockley Read H a l l recombination term, we have A
a)
R =
np-d
— , n + p + 23
2
b)
uJC
S = A
4
2
+ 4<5 +
28 u + v C
8*u 4
1
+ 4 t ) + 28
3.10 A s y m p t o t i c Expansions o n the D i f f u s i o n T i m e Scale
161
The boundary conditions (3.10.11) become a)
u(x, t) = 0
for
x e <3Q ,
b)
du — (x, 0 = 0 dv
for
c)
{x, 0 = 0
for
x e
dQ ,
d)
dj>_ ^-(x, dv
for
xe
dQ ,
D
xedQ
w
(3.10.20)
0=
0
D
N
and for the interface conditions we o b t a i n that the functions a)
[ m ( C + ^C
b)
[ u ( - C + JC
c)
l ~n
2
+ 43*) + 2 ] exp(Vj + 3*) 2
+ 43*) + 2 ] e x p ( - V, - 3*)
and
n
d)
(3.10.21)
l ~n p
are continuous across P-N junctions. I n (3.10.21) Vj takes on different values on different sides o f the P - N j u n c t i o n (in the different subdomains Clj). I f we now assume that (like c.f. i n a M O S F E T , see Chapter 4) an N - r e g i o n is adjacent only to P-regions, and vice versa, the problem degenerates i n t o a linear p r o b l e m as 3 -> 0. This linear p r o b l e m is given by a)
div(/„° + 7°) = 0
b)
c,u° = - d i v 7° - S°
c)
/ „ ° = P „ [ g r a d u° - u° grad V - ( ( - C + \C\)/2)
grad
d)
/ ° = P [ - g r a d w° - u° grad F
grad
(3.10.22) hi
P
- ( ( C + \C\)/2)
6 I
together w i t h the boundary conditions (3.10.20). The interface conditions then become the conditions that a)
7„-7f,
b)
I -n
(3.10.23)
p
are continuous across P-N junctions and that a)
u°(x, t)C(x)
+ 1 = sxp(V (t)
-
k
for
Vj(t))
x e cQj n cQ
if
k
C(x) > 0
or b)
-u°(x,
in (3.10.24)
t)C(x)
+ 1 = exp{Vj(t) for
-
V {t)) k
x e cQj n c Q
k
if
C(x) < 0
in
Qj
holds. Here u°{x, t) and C ( x ) are to be understood as the one sided limits o f u° and C i n Q - at the P - N j u n c t i o n cQj n r Q (see Problem 3.10). Note, that the p r o b l e m (3.10.22), (3.10.23), (3.10.24) is linear. Standard arguments can be applied to establish the existence of a solution (see [3.26], [3.27]). I f the appropriate spaces are chosen (see [3.26]) the existence of a 7
fc
162
3 The Drift Diffusion
Equations
solution to the reduced problem (3.10.10) (3.10.11) can be established by a perturbation argument for small S.
3.11 Fast Time Scale Expansions The slow time scale expansions, derived i n the previous paragraph, i m p l y that certain conditions for the inner and outer solution h o l d for all time. These are given by a)
d i v ( X ( x , t) + JJx, t)) = 0, Vt > 0 . _ - . A = fie*~* , c) p = pe^~*. p
b)
(3.11.1)
Therefore, i n order for the a p p r o x i m a t i o n derived in the previous paragraph to hold, the initial data n and p' have to be such that (3.11.1) is satisfied at t = 0. As we have seen i n Section 3.4, this is the case when n and p' are solutions o f the stationary problem. I n this and in the next paragraph we w i l l analyze the structure o f the solutions i f this is not the case. The m o t i v a t i o n for this analysis is twofold and we w i l l distinguish between t w o principal cases. Firstly we are interested i n steep, or i n the extreme case discontinuous, changes i n the applied bias. So n' and p' w i l l be solutions o f the steady state problem l
1
a)
2
SS
!
A AV
b)
div J„
c)
div
= n -p'
=
d)
J„
ss
e)
J
s s
f)
V (x)=
- C
= R
ss
—R 1
ss
= p „ ( g r a d n - n' grad
V)
= p . ( - g r a d p - p' grad
(3.11.2) p
1
ss
V)
p
ss
s
V* (x),
p'{x)
n'(x) =
= p (x)
for
D
SS
n (x), D
x e dQ
D
1
1
8V dn dv g) ^ - = 0 , ~ = 0, °4= 0 for xeda . dv cv ov We then solve the transient initial value p r o b l e m (3.7.1) together w i t h the boundary conditions H
V(x,
t) = V (x, t)
for
D
xecQ , D
(3.11.3)
s
where V (x, 0) # V^ {x) holds. Obviously, i n this case, V(x, 0) w i l l not equal V (x) because of the different boundary values at the Dirichlet boundary dQ . Therefore D
ss
D
div(J„ + J ) = 0
(3.11.4)
p
w i l l also not h o l d at t = 0. I n the second case we w i l l consider the solution o f the transient problem for general initial functions n and p'. The fundamental difference between the 1
163
3.11 Fast T i m e Scale Expansions
two cases is that in the first case V, J„ and J stay bounded at f = 0 for X - » 0 , while i n the second case V(x, t = 0) = X~ A~ (n — p' — C) w i l l be u n i formly of order X~ , and so w i l l be J and J at t = 0. This case can be interpreted as a r a n d o m p e r t u r b a t i o n of the initial data caused by either external radiation or roundoff errors introduced by a numerical solution. A n alternative interpretation w o u l d be to regard, after a rescaling of the potential V, the resulting problem as the transient drift diffusion equations under extreme reverse biasing conditions. p
2
1
I
2
n
The
Case of a Bounded
Initial
p
Potential l
1
We w i l l first consider the case when the initial data n and p are such that the potential V and the current densities J„ and J stay bounded for X -» 0. So, again restricting ourselves to the two-dimensional case, we assume the same configuration as for the steady state problem i n Section 3.4 p
a)
n' = n'(x,y)
b)
p = p (x,y)
1
I
+ n'^,s)
+ O(X)
+ p'(Ls)
+
0(X),
where (x, y)<-• (
2
l
l
a)
X AV
= n'- p - C
b)
V'(x) = V (x, 0)
for
D
dV' ——(x) = 0 dv and J „ and J , given by
for
x£dQ ,
(3.11.6)
D
x e dQ
N
!
p
a)
1
1
Jl = / U g r a d n - n grad V)
b)
(3 117)
.//^(-gradp'-p'gradF') ;
stay bounded as X goes to zero. N o w J„ and J w i l l , i n general, not satisfy (3.11.4) w h i c h makes a correction on a faster time scale necessary. I n [3.26] it has been shown that the only possible time scale, in the absence of velocity saturation effects, is of the form p
2
t = t/X
(3.11.8)
(see also Problem 3.11). Performing this change of variables, we obtain a)
2
X AV
= 2
h - p - C
b)
d n = X (div
b)
c p = ; . ( - d i v J -R)
d)
J„ = A'„(grad n - n grad V)
e)
J = u ( - grad p - p grad V),
T
J„ -
R)
2
t
p
p
n
(3.11.9)
164
3 The Drift Diffusion
Equations
w h e r e ' ' denotes the dependent variables on the fast time scale. Since J J stay bounded as X -> 0 we obtain
n
and
p
n(x, y, T) = n'ix, y),
p(x, y, x) = p'(x, y).
(3.11.10)
The fast time scale equation for the potential F i s obtained by differentiating (3.11.9)a) w i t h respect to T. This yields d AV = div(J„ + J ) , t
(3.11.11)
p
where J and J are n o w given by (3.11.9) w i t h the concentrations n and p replaced by n' and p . The fast time scale equation is of the form n
p
1
8 AV= X
-di\[_(p n'
1
+ p p')
n
grad V~\ + div(p„ grad n -u
p
p
grad
1
p ).
(3.11.12) Thus, V can satisfy the i n i t i a l conditions and the boundary conditions r
a)
V(x, 0) =
V (x)
b)
V(x, x) = V (x, t)
for
D
dV — (x, x) = 0
for
xedQ ,
(3 n 13)
D
x e
dQ . N
I n [3.26] i t is shown that the fast time scale p r o b l e m (3.11.12) (3.11.13) has a unique solution V. A simple energy estimate (see again [3.26]) shows that the fast time scale potential decays towards a steady state solution V , J " , satisfying m
a) b)
d i v ( J ^ + Jp°) = 0 ~ ' , , „ j * =p (gradn -n gradF )
c)
= p ( - g r a d p - p grad
l
/
(3.11.14)
c 0
n
7
7
p
V ). m
The corresponding current densities J ^ and n o w satisfy the c o n d i t i o n (3.11.4) and so the fast time scale solution can be matched to the slow time solution, derived i n (3.10.1)—(3.10.6), i n the usual way (see [3.25]). So, i n the case of a bounded initial potential, the o n l y correction o n a faster time scale takes place i n the potential V. The physical reason for this is that the timescale given by (3.11.8) corresponds to the dielectric relaxation time. By definition this is the time scale on which the electric potential V adjusts to a new charge d i s t r i b u t i o n . The carriers move o n a m u c h slower time scale and therefore h = n and p = p holds i n (3.11.10). 1
7
The above analysis can be used to describe the effect of r a p i d changes in the applied bias. I f we start from a steady state solution, given by (3.11.2), and change the applied bias at O h m i c contacts o n the dielectric relaxation time scale, i.e. i f V(x,t)=
V (x,jj\ D
for
xedQ
D
(3.11.15)
holds, we can describe the behaviour of the solution on the fast time scale by corrections i n the potential and i n the drift current densities alone. We set
3.11 Fast T i m e Scale E x p a n s i o n s
a)
165
ss
V(x, t) = V (x)
+ (x, T)
ss
b)
J (x, t) = J (x)
c)
J {x, f) = J | ( x ) + /u p (.v) grad ^ ( x , T)
n
- iiy(x)
n
s
grad (x, x)
(3.11.16)
7
p
p
2
ss
s
w i t h T = t/X and V , J% and J initial-boundary value problem
s p
given by (3.11.2).
/
a)
d A = - d i v [ ( / i „ n + p. p')
b)
Mx, x) = V (x, x) -
z
grad
p
V (x, 0)
D
for
D
f\ x e dQ
D
(3.11.17)
T) = 0
c)
— (x, dv
d)
<j>{x, 0) = 0.
for
x e dQ
N
In this way the effect of, for instance, a steep ramp in the applied bias can be easily described (see Problem 3.13). The situation is somewhat more complicated i f velocity saturation effects are considered; i.e. if the mobilities u„ and fi i n (3.11.9) depend on the electric field —grad V as well. I n this case an intermediate time scale of the form a = t/X is present on which also the concentrations n and p can evolve inside the spatial layer regions. The presence of this intermediate time scale has, however, no effect o n the over-all picture; that means on the solutions away from spatial layer regions. We refer the reader to [3.27] for a detailed analysis. p
Fast
Time Scale
Solutions
for General
Initial
Data
I f arbitrary functions are prescribed as initial data for n' and p' in (3.1.1) severe complications are introduced i n t o the preceeding asymptotic analysis. M o s t notably, the resulting potential at time f = 0, which satisfies a)
X
b)
V(x, 0) = V (x, 0)
2
AV = n' - p' - C D
dV —(x,0) = 0 dv
for
for
x e cQ . D
(3 H 18)
xedQ
N
2
w i l l become of order 0 ( / ~ ) , uniformly in Q. as /. tends to zero and so w i l l the current densities J„ and J at t = 0. One w o u l d hope that, after initially being very large, V, J and J w o u l d settle d o w n to 0(1), so that they can be matched to the diffusion time scale solution derived before. Unfortunately, no general results, which are valid in more than on space dimension, are available to this end. As we w i l l see, the resulting fast time scale equations are o f mixed elliptic-hyperbolic type and this makes the analysis o f their behaviour for large x highly complicated. I t could be argued that this complication is somewhat artificial. I t arises from the fact that n' and p i n p
n
p
1
166
3 The Drift Diffusion Equations
(3.1.1) are regarded as i n i t i a l data and V(x, 0) is given a posteriori by the solution o f the Poisson equation. I f one w o u l d regard the i n i t i a l potential and one carrier density as i n i t i a l d a t u m instead, the fast time scale expansion derived i n the previous paragraph w o u l d completely suffice. The results o f this paragraph actually suggest that one should proceed i n this manner, when solving the transient drift diffusion equations numerically since otherwise the resulting equation represent a differential algebraic system o f index 2 (see [3.13] for the definition of the index o f a differential algebraic system). The case of large (0(X~ )) i n i t i a l potential is, however, of a d d i t i o n a l interest since it w i l l also describe the transient behaviour of the drift diffusion equations under extremely large bias. 2
1
Inserting the potential V, given by (3.11.18), and the i n i t i a l functions n and p i n t o the c o n t i n u i t y equations and current relations (3.8.3)b)-e) at time t = 0 shows that the time derivatives o f n and p are o f order 0{X~ ) at t = 0. This implies again the fast time scale variable to be o f the f o r m 1
2
t x =
¥-
Also, because o f the size o f the potential at time t = 0, V, J„ and J have to be rescaled. We set p
V{x,t)
J (x, p
= «*0-,
T) =
(3.11.19)
,
p 2
J„(x,
T) =
.
This yields the equations a)
A = h — p — C
b)
3 n = d i v ( / J - X R,
2
c)
t
2
dj=
-div(I )-A R p
P 2
c)
/„ = p. (X
d)
I = H ( - X grad p - p grad )
n
(3.11.20)
grad n - n grad <j>) 2
p
P
subject to the i n i t i a l and b o u n d a r y conditions a)
fi(x, 0) = n'ix),
b)
2
c)
(j)[x, T) = X V (x,
d)
S-(x, dv
for
D
T) = 0
p(x, 0) =
for
p'(x) x 6 dCl
D
(3.11.21)
xedQ . N
2
Because of the rescaling the small parameter X does n o t appear i n the Poisson equation anymore, it appears as small diffusion coefficient i n the current relations. Also, by (3.11.20) a) the effect of recombination-generation is 0(X ) on the dielectric relaxation time scale. N o t e , that the p r o b l e m (3.11.20)—(3.11.21) is the transient equivalent o f the steady state p r o b l e m (3.5.5) under extreme reverse biasing conditions. Thus, 2
3.11 Fast T i m e Scale E x p a n s i o n s
167
the p r o b l e m (3.11.20)—(3.11.21) has t w o interpretations. I t can either be interpreted as the transient p r o b l e m under moderate biasing conditions w i t h general i n i t i a l data n and p . I n this case (j)(x, x) w i l l be o f order 0(X ) at the D i r i c h l e t boundary dQ . O r i t can be interpreted as the transient problem under extreme reverse biasing conditions, i n which case (x, T) w i l l be of order 0 ( 1 ) at dQ . F o r X -> 0(zero diffusion l i m i t in (3.11.20)) we o b t a i n the mixed elliptic-hyperbolic system 1
1
2
D
D
a)
A ^ ° = n° - p° - C
b)
B n° = - d i v ( n ° grad °),
c)
d p° = d i v ( p ° grad °)
(3.11.22)
r
T
subject to the initial and boundary conditions 0
a)
fi (x, 0) = n\x),
c)
°(x,x) = (/>%(x)
d)
C
-f-( ,x) dv
= 0
x
b)
p°(x, 0) =
for
xedQ
p\x) (3.11.23)
D
for
xedQ
N
w i t h D(X) = l i m _, X V (x, Ax), i.e. <^(x) = 0 holds for moderate biasing conditions and ^,{x) = 0(1) holds for extreme biasing conditions. A n additional boundary c o n d i t i o n has to be imposed on n° whenever the characteristic directions o f (3.11.22)b) p o i n t i n w a r d from the boundary, i.e. wherever 2
;
0
D
grad^°-v<0
(3.11.24)
holds. (Here, as always, v denotes the unit o u t w a r d n o r m a l vector on the boundary <3Q.) Similarly, a boundary c o n d i t i o n has to be used for p° wherever grad (f>° • v > 0 holds. F o r the one dimensional case, a p r o o f that the solution (<j>, n, p) of the fast time scale p r o b l e m actually converges to the solution o f (3.11.22)—(3.11.23) in the weak sense as A -> 0, can be found in [3.20]. T o analyze the solution o f (3.11.22)—(3.11.23) is extremely t r i c k y since the parts of the d o m a i n boundary r Q , where a boundary c o n d i t i o n on the concentrations n° and p° is required, w i l l depend on the solution <j>° itself. Several special cases have been investigated i n [3.36] and we w i l l present here only the simplest one of them i n order to give the reader some idea of the situation. A standard device to reduce the drift diffusion equations to the simplest possible case is to assume a one-dimensional semiconductor w i t h a piecewise constant and antisymmetric d o p i n g concentration (see [3.21]). I n this case (3.11.20) reduces to 2
a)
d i) = n — p — C
b)
c\n = c I
d)
X N
2
/„ = X 8 n x
2
- A R, - ndj,
c) e)
2
d p = -c I t
I
- AR
x p
2
P
= -X c p x
-
(3.11.25) pcj.
168
3 The Drift Diffusion Equations
(For the sake of simplicity we have taken the mobilities to be constant, and therefore scaled to 1, here.) The semiconductor occupies the region Q = ( — 1 , 1). The piecewise constant d o p i n g concentration C(x) satisfies n C
h
\ (
X
)
=
for
l
1
1
- 1
for
<0, (
O O ^ l .
3
-
U
'
6
)
Thus, the P - N j u n c t i o n is located at x = 0. This setting allows the reduction of the drift diffusion equations via the symmetry (see [3.21]) x) = p(x, T),
a)
n(-x,
b)
c)
{ — x, T) = — (f>(x, T)
/ „ ( - x , x) = I„(x, for
T),
(31127)
x e (0, 1)
(see also P r o b l e m 3.14). T o simplify matters even further Szmolyan i n [3.36] assumes that the hole concentration p is neglegibly small i n the N - r e g i o n ; i.e. he assumes that p(x,x)
= 0
for
xe(0,
1)
(3.11.28)
holds. F o r X -> 0, this leads to the u n i p o l a r p r o b l e m a)
d (j> = ff° - 1
b)
d n° = -d (n°dj°)\
c)
n°(x,0) =
d)
«°(0,T) = 0,
e)
«°(1,T) = 1
f)
A 0 , t ) = 0,
g)
^°(1,t) =
2
x
0
z
f
x
°
r
x
e
(
(
U
)
(3.11.29)
n'(x),
^ .
The boundary conditions (3.11.29)d) and f) arise from the symmetry ass u m p t i o n (3.11.27). ^o = 0 w o u l d h o l d if the transient problem is considered under 'moderate' biasing conditions. I f an extreme bias is applied i n the sense of Section 3.5 (V = 0(X~ ) i n (3.11.21)c)) a value ^ 0 is possible. We introduce the electric field 2
0
D
D
£ ( x , t ) = -dJ°(x,T)
(3.11.30)
as a new variable. Differentiating (3.11.29)a) w i t h respect to x and integration gives a)
d E - Ed E
b)
d n° - Ed n°
t
T
x
x
= -E
+ A(x)
= n°(l - n°)
(3.11.31)
c) A(x) = E(l, x) + d E(l,x). B o t h equations (3.11.3 l ) a ) and (3.11.3 l ) b ) , have n o w the same characteristics x ( x , s, T) satisfying t
a)
3_x(x, s, x) = —E(x, s) s
(3.11.32)
b) x ( x , T, T) = X. If we denote by n and E the values o f n° and E along these characteristics; i.e. a)
E(x, s, T) = £ ( x ( x , s, T), S)
b)
n(x, s, T) = n ° ( x ( x , s, T), S),
(3.11.33)
3.11 Fast T i m e Scale E x p a n s i o n s
169
x(0,0,T)
x(l,0,T)
F i g . 3.11.1
we o b t a i n a)
d E = -E
b)
d h =• n(\
s
s
+ A(s) (3.11.34)
— h).
The values of £ ( x , x) and n ° ( x , x) at the starting p o i n t o f the characteristic curve are the initial values for the o r d i n a r y differential equations (3.11.34). In particular, the values o f n° and E at T = 0 and the values of n° at inflow boundaries are k n o w n . I f we assume that there is no flux of holes from the P-region into the A<-region at the i n i t i a l time x = 0, this means (see [3.36]) a)
£(0,0) > 0 ,
b)
(3.11.35)
£(1,0) < 0 .
The assumption (3.11.35) implies that the characteristic curves x(0, s, 0) and x ( l , s, 0), starting at time x = 0 at the boundaries x = 0 and x = 1, p o i n t inward. Consequently boundary values for w°(0, T) and n ° ( l , T) are prescribed initially. So, the situation is as depicted i n Fig. 3.11.1. The characteristics x(0, s, 0) and x ( l , s, 0) split the d o m a i n (0, 1) x (0, T) i n the three subdomains G , G and G. I t follows from the equation (3.11.34)b) that the boundary values are transported into the d o m a i n i n G and G . Therefore n° = 0 holds i n G and H° = 1 holds i n G . This fact, and equation (3.11.29)a) i m p l y 0
x
0
0
E(x, x) =
{ ( £ ( 0 , T)
for
[E(l
for
x)
x
:
T) e G , (x, T) e G [ . !x.
0
(3.11.36)
Using the differentiated version o f Poisson's equation once more, one easily obtains a)
£ ( 0 , T)
= £ ( 1 , T) +
b)
y(t) =
E(l,s)ds,
y(r) + a (3.11.37) c)
a = £(0,0)-£(0,1).
3 The Drift Diffusion
170
Equations
0
The solution £ and it i n the middle subdomain G can now be determined by straightforward integration along the characteristics. One obtains s
a)
x ( x , s, 0) = x + ( £ ( 1 , 0) - E(x, 0))(1 - e~ ) -
b)
n(x,s,0)=
M
X
)
(3.11.38)
n (x)(l — e
)+ e
7
c)
y(s)
s
£ ( x , s, 0) = ( £ ( x , 0) - £ ( 1 , 0))e-
+ £ ( 1 , s)
w i t h y(s) defined as i n (3.11.37)b). I n [3.36] it is shown that, i f the applied potential % satisfies (f>% ^ a /2, the characteristics x(0, s, 0 ) a n d x ( l , s, 0) remain i n the interval (0, 1) for alls > 0, and that the function y(x) i n (3.11.37)b) has a l i m i t for t -* oo. l i m y ( r ) = y(oo) = - a (3.11.39) 2
t-»oo
holds. Thus, using (3.11.38)a), we obtain a)
l i m x(0, s, 0) =
Jl<jP
D
(3.11.40)
l i m x ( l , s, 0) = 1 + a + , y 2 < ^ • /
s-*oo
Using (3.11.38)b)c), we obtain the steady state solution for £ and n°. a)
b)
n (x, oo)
£ ( x , oo) =
fO
for 0^x^^/277,
1
for , x — ,y 2 ^ 0
X v
/ 2 X ^ x sc 1, (3.11.41) for
0 ^ x ^ 7 2 ^ ,
for
Jlfa
< x ^ 1.
F o r a moderate value of the applied bias, that means for ^ = 0, n° has a j u m p discontinuity at x = 0, the P-N j u n c t i o n , and the boundary x = 0 becomes a characteristic. I n any event n° and £ (and therefore also ) converge to a steady state solution satisfying a)
n°(x, o c ) - C ( x ) = 0,
b)
d I„ = 0.
(3.11.42)
x
This steady state solution can then be matched to the slow (diffusion-) time scale solution derived before. F o r a large value of the applied bias (V = 0(/.~ ) i n (3.11.18)b)), w h i c h implies (j>% ^ 0, the layer moves away from the P-iV j u n c t i o n towards the edge of the depletion region at x = /2<j> f . A t least for this simple model problem, the corresponding limit solution for x -» oc is identical to the stationary solution derived for extremely large reverse bias values i n Section 3.5 (see Problem 3.16). D
2
(
s
)
Unfortunately, this type of analysis breaks d o w n i f more than one carrier is present since it heavily relies on the fact that the equations for the field £ and the concentration n° have the same characteristics (see P r o b l e m 3.15). The above example, and the other special cases analyzed i n [3.36], lead one to expect that the solutions of the fast time scale problem under extreme biasing conditions i n general tends to a solution of the corresponding
171
Problems
stationary p r o b l e m for the fast time variable t -» oo, as it is the case for moderate biasing conditions. Note, that, i f (3.11.29) is regarded as the transient p r o b l e m w i t h general initial data but under moderate biasing conditions (i.e. % = 0), the l i m i t i n g solution <j>(x, x = oo) vanishes identically. This suggests that, for large x, the scaling (3.11.19) has to be reversed again and the fast time scale solution c o u l d be matched to the slow time scale solution derived before. The precise mechanism for this transition from the fast to the slow time scale is, however, not k n o w n yet. So, as for the stationary problem, the asymptotic analysis of the drift diffusion equations i n the reverse bias case is m u c h more complicated than in the forward bias case. The reason is that i n the reverse bias case the reduced p r o b l e m is of mixed elliptic-hyperbolic type, leading to a free boundary value p r o b l e m i n the stationary case. I n general, it can be said that a great deal of a d d i t i o n a l analysis is needed for the drift diffusion equations i n the extreme reverse bias scaling. I t should be pointed out, however, that the value of singular p e r t u r b a t i o n analysis lies i n the general understanding o f the qualitative behaviour o f solutions o f the drift diffusion equations and that singular p e r t u r b a t i o n analysis cannot replace numerical calculations techniques. F o r particular applications the asymptotics has to be adjusted to the device geometry, the models for the mobilities and the recombination rate etc., all o f which w i l l influence the actual solution significantly. Therefore it does not seem reasonable to analyze the general drift diffusion equations i n too m u c h detail w i t h o u t specifying the geometry and the model parameters. F o r this reason we w i l l , i n the next Chapter, t u r n to the study of particular devices where we apply the tools and results developed i n this Chapter for the general problem.
Problems 3.1
D e r i v e the r e m a i n d e r t e r m G(.v, v) i n (3.4.27) a n d s h o w t h a t it is o f o r d e r 0{/.) u n i f o r m l y i n 12.
3.2
Suppose F(w) = 0 is solved by N e w t o n ' s m e t h o d . S h o w that i f the variable t r a n s f o r m a t i o n w = G(s) is used a n d N e w t o n ' s m e t h o d is a p p l i e d to the t r a n s f o r m e d e q u a t i o n the l i n e a r i z a t i o n o f the new u p d a t i n g strategy coincides w i t h the o l d strategy. W h a t does this m e a n for the local convergence rates?
3.3
Use the m a x i m u m p r i n c i p l e t o show (3.6.19).
3.4
S h o w t h a t V£ i n (3.4.24) satisfies the N e u m a n n b o u n d a r y c o n d i t i o n s at d£l holds.
3.5
D e r i v e the b o u n d a r y value p r o b l e m s defining the first-order a p p r o x i m a t i o n i n (3.4.46).
3.6
S h o w t h a t the b o u n d a r y value p r o b l e m consisting o f the e q u a t i o n (3.5.28(a) a n d the b o u n d a r y c o n d i t i o n s (3.5.28)e) has a s o l u t i o n . Hint: Use the same a p p r o a c h as used w h e n s h o w i n g the existence o f a s o l u t i o n t o the drift diffusion equations i n Section 3.3: i.e. find an a p p r o p r i a t e fixed p o i n t m a p .
3.7
D e r i v e the 0 ( / . ) a n d the 0 ( / . ) terms i n the a s y m p t o t i c e x p a n s i o n for the s t r o n g l y reverse biased one d i m e n s i o n a l case (3.5.31).
N
2
i f (3.4.12)
172
3 The Drift Diffusion
Equations
3.8
D e r i v e the b o u n d a r y value p r o b l e m for the s m o o t h v a r i a b l e w, given by the t r a n s f o r m a t i o n (3.5.20) i n the t w o d i m e n s i o n a l case. U n d e r w h a t c o n d i t i o n s is this b o u n d a r y value p r o b l e m u n i f o r m l y elliptic?
3.9
S h o w t h a t the p r o d u c t defined i n (3.8.11) satisfies a l l the requirements o f a scalar product.
3.10 H o w d o the terms ( i / ^ U ) — ^j(t)) i n (3.10.24) behave for 8 -> 0. W h a t does this mean for f o r w a r d a n d reverse biased P-N j u n c t i o n s ? 3.11 S h o w t h a t a w a y f r o m s p a t i a l layers ( t h a t means w h e n the spatial derivatives stay b o u n d e d ) the o n l y fast t i m e scale, w h i c h does n o t y i e l d a t r i v i a l s o l u t i o n , is o f the f o r m (3.11.8). 2
3.12 D e r i v e the 0 ( / . ) a n d 0(/. ) e q u a t i o n (3.11.9).
terms i n the a s y m p t o t i c e x p a n s i o n o f the fast t i m e scale
3.13 Assume a one d i m e n s i o n a l P-N d i o d e m o d e l w i t h a piecewise c o n s t a n t a n t i s y m m e t r i c d o p i n g c o n c e n t r a t i o n , where after scaling the d i o d e is located i n the i n t e r v a l [ — 1, 1] (like i n (3.11.25)—(3.11.27)). Assume the a p p l i e d bias is varied l i n e a r l y f r o m 0 V t o — 0.5 V f o r w a r d bias i n the scaled t i m e i n t e r v a l f r o m t = 0 t o t = )}. Use the a p p r o a c h i n (3.11.16)—(3.11.17) t o c o m p u t e the c u r r e n t at the end o f the s w i t c h i n g p e r i o d approximately. 3.14 Verify t h a t the s y m m e t r y 'ansatz' (3.11.27) is consistent w i t h the b o u n d a r y c o n d i t i o n s (3.11.21). 3.15 D e r i v e the characteristic equations c o r r e s p o n d i n g t o (3.11.32) a n d (3.11.34) i n the b i p o l a r case. T h a t means w h e n p ^ 0 i n (3.11.29)a). 3.16 Verify t h a t the s o l u t i o n (3.5.18), (3.5.25) o f the s t a t i o n a r y p r o b l e m under extreme reverse b i a s i n g c o n d i t i o n s reduces t o the l i m i t s o l u t i o n (3.11.41), for t -> go, o f the transient p r o b l e m i f a u n i p o l a r one d i m e n s i o n a l device w i t h a n t i s y m m e t r i c d o p i n g is considered.
References [3.1] [3.2]
R. A . A d a m s : Sobolev Spaces. A c a d e m i c Press, N e w Y o r k (1975). F . A l a b e a u : A S i n g u l a r P e r t u r b a t i o n Analysis o f the S e m i c o n d u c t o r Device a n d the E l e c t r o c h e m i s t r y E q u a t i o n s . R e p o r t , I n s t i t u t N a t i o n a l de Recherche en I n f o r m a t i q u e et en A u t o m a t i q u e , Paris (1984). [3.3] U . Ascher, P. A . M a r k o w i c h , C. Schmeiser, H . S t e i n r i i c k , R. WeiC: C o n d i t i o n i n g o f the Steady State S e m i c o n d u c t o r D e v i c e P r o b l e m . S I A M J. A p p l . M a t h . 49, 1 6 5 - 1 8 5 (1989). [3.4] R. E. B a n k , et al.: A n a l y t i c a l a n d N u m e r i c a l Aspects o f S e m i c o n d u c t o r Device M o d e l l i n g . R e p o r t 82-11274-2, Bell L a b o r a t o r i e s , M u r r a y H i l l (1982). [3.5] R. E. B a n k , D . J. Rose: G l o b a l A p p r o x i m a t e N e w t o n M e t h o d s . N u m e r i s c h e M a t h e m a t i k 37, 2 7 9 - 2 9 5 (1981). [3.6] F . Brezzi: T h e o r e t i c a l a n d N u m e r i c a l P r o b l e m s i n Reverse Biased S e m i c o n d u c t o r Devices. Proc. 7th I n t e r n . Conf. o n C o m p u t . M e t h . i n A p p l . Sci. a n d Engng., I N R I A , Paris (1985). [3.7] F . Brezzi, A . C. S. Capelo, L . G a s t a l d i : A S i n g u l a r P e r t u r b a t i o n Analysis o f Reverse Biased S e m i c o n d u c t o r D i o d e s . S I A M J. M a t h . A n a l . 20, 3 7 2 - 3 8 7 (1989). [3.8] F . Brezzi, L . G a s t a l d i : M a t h e m a t i c a l Properties o f O n e - D i m e n s i o n a l Semiconductors. M a t . A p l . C o m p . 5, 1 2 3 - 1 3 7 (1986). [3.9] L . Caffarelli. A . F r i e d m a n : A S i n g u l a r P e r t u r b a t i o n P r o b l e m for S e m i c o n d u c t o r s . B o l l e t i n o U . M . I . 7 - B (7), 4 0 9 - 4 2 1 (1987). [ 3 . 1 0 ] H . G a j e w s k i : O n the Existence o f Steady-State C a r r i e r D i s t r i b u t i o n s i n Semic o n d u c t o r s . Z A M M (to appear).
References
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[ 3 . 1 1 ] H . G a j e w s k i : O n Existence, Uniqueness a n d A s y m p t o t i c B e h a v i o r o f Solutions o f the Basic E q u a t i o n s for C a r r i e r T r a n s p o r t i n S e m i c o n d u c t o r s . Z A M M 65, 101-108 (1985). [ 3 . 1 2 ] D . G i l b a r g , N . S. T r u d i n g e r : E l l i p t i c P a r t i a l Differential E q u a t i o n s o f Second O r d e r , 2 n d edn. Springer, B e r l i n (1984). [ 3 . 1 3 ] E. G r i e p e n t r o g , R. M a e r z : Differential A l g e b r a i c E q u a t i o n s a n d T h e i r N u m e r i c a l T r e a t m e n t . Teubner, L e i p z i g (1986). [ 3 . 1 4 ] H . K . G u m m e l : A Self-Consistent I t e r a t i v e Scheme for O n e - D i m e n s i o n a l Steady State T r a n s i s t o r C a l c u l a t i o n s . I E E E T r a n s . E l e c t r o n . Devices 11, 4 5 5 - 4 6 5 (1964). [ 3 . 1 5 ] J. H e n r i , B . L o u r o : S i n g u l a r P e r t u r b a t i o n T h e o r y A p p l i e d to the E l e c t r o c h e m i s t r y E q u a t i o n s i n the Case o f E l e c t r o n e u t r a l i t y . N o n l i n e a r Analysis T M A 13, 7 8 7 - 8 0 1 (1989). [ 3 . 1 6 ] J. W . Jerome: Consistency o f S e m i c o n d u c t o r M o d e l l i n g : A n Existence/Stability Analysis for the S t a t i o n a r y V a n R o o s b r o e c k System. S I A M J. A p p l . M a t h . 45, 565-590(1985). [ 3 . 1 7 ] T . K e r k h o v e n : A P r o o f o f the Convergence o f G u m m e l ' s M e t h o d for Realistic Device Geometries. S I A M J. N u m . A n a l . 23, 1 1 2 1 - 1 1 3 7 (1986). [ 3 . 1 8 ] P. M a r k o w i c h : A S i n g u l a r P e r t u r b a t i o n A n a l y s i s o f the F u n d a m e n t a l S e m i c o n d u c t o r Device E q u a t i o n s . S I A M J. A p p l . M a t h . 44, 8 9 6 - 9 2 8 (1984). [ 3 . 1 9 ] P. M a r k o w i c h : T h e S t a t i o n a r y S e m i c o n d u c t o r Device E q u a t i o n s . Springer, W i e n N e w Y o r k (1986). [ 3 . 2 0 ] P. M a r k o w i c h , P. S z m o l y a n : A System o f C o n v e c t i o n - D i f f u s i o n E q u a t i o n s w i t h S m a l l D i f f u s i o n Coefficient A r i s i n g i n S e m i c o n d u c t o r Physics. J. Diff. E q u . 81, 2 3 4 - 2 5 4 (1989). [ 3 . 2 1 ] P. M a r k o w i c h , C. Ringhofer: A S i n g u l a r l y P e r t u r b e d B o u n d a r y V a l u e P r o b l e m M o d e l l i n g a S e m i c o n d u c t o r Device. S I A M J. A p p l . M a t h . 44, 2 3 1 - 2 5 6 (1984). [ 3 . 2 2 ] P. M a r k o w i c h , C. Ringhofer: S t a b i l i t y o f the L i n e a r i z e d T r a n s i e n t S e m i c o n d u c t o r Device E q u a t i o n s . Z A M M 67, 3 1 9 - 3 3 2 (1987). [ 3 . 2 3 ] P. M a r k o w i c h , C. Schmeiser: U n i f o r m A s y m p t o t i c Representations o f the Basic S e m i c o n d u c t o r Device E q u a t i o n s . I M A J. A p p l M a t h . 36, 4 3 - 5 7 (1986). [ 3 . 2 4 ] M . S. M o c k : Analysis o f M a t h e m a t i c a l M o d e l s o f S e m i c o n d u c t o r Devices. Boole Press, D u b l i n (1983). [ 3 . 2 5 ] R. E. O ' M a l l e y j r . : I n t r o d u c t i o n to S i n g u l a r P e r t u r b a t i o n s . A c a d e m i c Press, N e w Y o r k (1974). [ 3 . 2 6 ] C. Ringhofer: A n A s y m p t o t i c Analysis o f a T r a n s i e n t P-N J u n c t i o n M o d e l . S I A M J. A p p l . M a t h . 47, 6 2 4 - 6 4 2 (1987). [ 3 . 2 7 ] C. Ringhofer: A S i n g u l a r P e r t u r b a t i o n Analysis for the T r a n s i e n t S e m i c o n d u c t o r Device E q u a t i o n s i n O n e Space D i m e n s i o n . I M A J. A p p l . M a t h . 39, 1 7 - 3 2 (1987). [ 3 . 2 8 ] C. Ringhofer, C. Schmeiser: A n A p p r o x i m a t e N e w t o n M e t h o d for the S o l u t i o n o f the Basic S e m i c o n d u c t o r D e v i c e E q u a t i o n s . S I A M J. N u m . A n a l . 26, 5 0 7 - 5 1 6 (1989). [ 3 . 2 9 ] C. R i n g h o f e r , C. Schmeiser: A M o d i f i e d G u m m e l M e t h o d for the Basic Semic o n d u c t o r Device E q u a t i o n s . I E E E T r a n s . C A D 7. 2 5 1 - 2 5 3 (1988). [ 3 . 3 0 ] C. Schmeiser: O n S t r o n g l y Reverse Biased S e m i c o n d u c t o r D i o d e s . S I A M J. A p p l . M a t h . (1989) (to appear). [ 3 . 3 1 ] C. Schmeiser, R. Weiss: A s y m p t o t i c Analvsis o f S i n g u l a r S i n g u l a r l y P e r t u r b e d B o u n d a r y V a l u e P r o b l e m s . S I A M J. M a t h . A n a l . 17. 5 6 0 - 5 7 9 (1986). [ 3 . 3 2 ] T . Seidman, G . T r o i a n i e l l o : T i m e D e p e n d e n t S o l u t i o n o f a N o n l i n e a r System A r i s i n g i n S e m i c o n d u c t o r T h e o r y . N o n l i n e a r Analysis T . M . A . 9, 1137-1157 (1985). [ 3 . 3 3 ] T . Seidman, G . T r o i a n i e l l o : T i m e D e p e n d e n t S o l u t i o n o f a N o n l i n e a r System A r i s i n g i n S e m i c o n d u c t o r T h e o r y , I I : Boundedness a n d P e r i o d i c i t y . N o n l i n e a r Analysis T . M . A . 10, 4 9 1 - 5 0 2 ( 1 9 8 6 ) . [ 3 . 3 4 ] S. Selberherr: Analysis a n d S i m u l a t i o n o f S e m i c o n d u c t o r Devices. S p r i n g e r - V e r l a g , W i e n N e w Y o r k (1984). [ 3 . 3 5 ] S. M . Sze: Physics o f S e m i c o n d u c t o r Devices. 2 n d edn. J o h n W i l e y & Sons, N e w Y o r k (1981).
174
3 The Drift Diffusion
Equations
[ 3 . 3 6 ] P. S z m o l y a n : I n i t i a l Transients o f S o l u t i o n s o f the S e m i c o n d u c t o r Device E q u a t i o n s . N o n l i n e a r A n a l y s i s T M A (to appear). [ 3 . 3 7 ] H . T r i e b e l : I n t e r p o l a t i o n T h e o r y , F u n c t i o n Spaces, Differential O p e r a t o r s . V e r l a g der Wissenschaften, B e r l i n (1973). [ 3 . 3 8 ] J. H . W i l k i n s o n : R o u n d i n g E r r o r s i n A l g e b r a i c Processes. Prentice H a l l , N e w Jersey (1963).
Devices
4.1 Introduction Depending o n the semiconductor material, the d o p i n g profile and on the geometry, semiconductor devices show a variety o f different kinds of electrical behaviour. This Chapter is concerned w i t h an analysis o f the performance of the practically most i m p o r t a n t devices. The starting point is a mathematical m o d e l w h i c h is general enough to include the considered physical effects. Obviously there is not only one choice, and it is t e m p t i n g to take a model which is as simple as possible. However, this involves the danger of neglecting i m p o r t a n t detail. O n the other hand, a very complex m o d e l w h i c h represents a detailed picture o f the physical w o r l d , possibly contains a good deal of superfluous information. Examples o f such complicated models are the transport equations i n Chapter 1 if the mean free path is very small compared to the length o f the active region of a device. M e t h o d s for the systematic simplification of these models belong to the basic parts o f the t o o l k i t of the applied mathematician [ 4 . 7 ] . W h e n the model has been chosen, an appropriate scaling is carried out and the relevant dimensionless parameters are identified (such as the scaled mean free path). After these preparatory steps, the formal machinery of perturbat i o n theory is a t o o l for the systematic simplification o f the problem. F o r the examples given i n this Chapter the simplification goes far enough such that i n f o r m a t i o n about the relations between a few observable quantities can be obtained by analytical methods. This systematic approach leads to mathematically justified results which i n many cases confirm established text b o o k formulas [4.22]. I n the original derivation o f these formulas the necessary simplifications are justified by physical arguments. The power of the formal approach is demonstrated by the fact that some of the results go significantly beyond the classical analysis. Striking examples can be found i n the Sections 4.3 and 4.5 o n the bipolar transistor and the thyristor. F o r all the devices considered here, the smallness o f the scaled mean free path is a valid assumption. The analysis w i l l therefore be based on the drift diffusion equations w i t h the appropriate models for the mobilities and
176
4 Devices
the recombination-generation rate. A second assumption which holds t h r o u g h o u t this Chapter is that the scaled m i n i m a l Debye length /. and the scaled intrinsic number 5 are small. These parameters have been defined i n Chapter 3, where the role of the singular p e r t u r b a t i o n parameter X is analyzed i n detail. This analysis sheds light o n the solution structure but does not simplify the p r o b l e m sufficiently to o b t a i n explicit information. Since we still have the small parameter S at our disposal, additional simplifications are possible. In a complete mathematical model for a semiconductor device, boundary conditions reflect the interaction o f the device w i t h the circuit which it is imbedded i n . I n the following we shall mostly be interested i n dc operating conditions. This means that the times between t w o switching events are long enough for quasi steady states to be reached. I n this case most o f the i m p o r tant i n f o r m a t i o n is contained i n the static voltage-current characteristics, i.e. the relation between contact voltages and currents t h r o u g h the contacts under steady conditions. O n l y in one case (the G u n n diode, Section 4.8) a time dependent problem w i l l be considered. This is due to a lack of m a t u r i t y of the theory o f those semiconductor devices, whose performance is based on dynamic effects. A unified account o f a variety o f the possible dynamic behaviour o f semiconductors can be found i n [4.16]. 1
2
Static
Voltage-Current
Characteristics
k
Suppose that Q c U , k = 1,2 or 3, represents the geometry of a semiconductor device and that a stationary operating p o i n t is described by the driftdiffusion equations 2
X AV = n — p — C, J„ = /x„(grad n - n grad V),
div J = R,
J = - p ( g r a d p + p grad V),
d i v J = —R
p
(4.1.1)
n
p
p
for x e Q. N o t e that (4.1.1) is already i n scaled form. F o r the scaling see Section 3.4. We assume that the device has m O h m i c contacts r , . . . , r _ j and I + I — m oxide regions T , ...,F, attached to its boundary dQ (I") £ dQ., i = 0 , . . . , /). A t the O h m i c contacts we have the boundary conditions 0
m
m
n - p - C at
r,,
= 0,
np = 3 \
V = V —U bi
t
(4.1.2)
i = 0 , . . . , m — 1,
where the b u i l t - i n potential V is given by bi
2
V
bi
= areasinh(C/2<5 ).
Since we are free to choose a reference point for the potential we take U = 0. W i t h o u t going i n t o the details of the description o f the oxide regions at the moment we only m e n t i o n that they lead to the a d d i t i o n a l applied voltages U, U[ and that current flow i n t o the oxide is not possible. Thus, the 0
m
177
4.1 I n t r o d u c t i o n
solutions o f (4.1.1) depend on / applied voltages U , leaving the device t h r o u g h the contact T, is given by
U,. The current /,
x
/, = j "
(J„ + J )-vds,
j = 0, . . . , m - l ,
p
where v denotes the unit o u t w a r d n o r m a l and s the (k — l)-dimensional Lebesgue measure. A static voltage-current characteristic is given by an equation of the form F(I ,U ,...,U )=0 i
1
(4.1.3)
l
representing an /-dimensional surface i n the (/ + l)-dimensional (/,-, U , t/,)-space. Since the solution of the voltage driven p r o b l e m (prescribed Uj, j = 1 , . . . , /) cannot be expected to be unique in general (see e.g. Section 4.5), it m i g h t be impossible to solve (4.1.3) for I . A complete description o f the stationary behaviour is given by an /-dimensional surface in the (I + m — 1)dimensional ( / , , . . . , / _ , U . . . , (7,)-space. Since the total current density J„ + J is divergence free no i n f o r m a t i o n is lost by not considering / which can be computed from ±
t
m
t
u
p
0
m-l
I
/, = o.
i=0
As a first step i n the analysis of voltage-current characteristics we consider the situation close to thermal equilibrium. This means that the applied voltages and, therefore, also the currents t h r o u g h the contacts are small i n absolute value. O u r approach is a perturbation analysis which takes into account the smallness o f b o t h dimensionless parameters X and 3. As i n Section 3.4 we introduce the S l o t b o o m variables [4.19] by the transformation 2
v
n = 3 e u,
2
p =
y
8 e~ v.
The variables u and v are related to the (scaled) quasi F e r m i levels ip and % t>y n
p
>
u = e~' ",
v = e" ".
The O h m i c contact boundary conditions (4.1.2) i n terms of u and v read U i
Ui
u = e , v = e~ , V=V -U atr , i = 0 , . . . , m - 1. Since in thermal e q u i l i b r i u m u = v = \ holds (see Section 3.4), u and v are well scaled as l o n g as the applied voltages are not large compared to the thermal voltage (which is our reference quantity for voltages). Close to thermal e q u i l i b r i u m large electric fields are only expected in small parts o f the device. Thus, position dependent m o b i l i t y models and the Shockley-Read-Hall recombination-generation term A __ np-8 x (n + S ) + z (p + 8 ) bi
i
;
A
2
p
2
n
are certainly sufficient for describing the relevant effects (see Chapter 2).
178
4 Devices
F r o m the formula for the b u i l t - i n potential we see that V is o f the order of magnitude o f In 8~ . The (in practice n o t very restrictive) assumption X In S~ « 1 justifies the use of the zero space charge a p p r o x i m a t i o n w h i c h amounts to replacing X by zero i n (4.1.1). I f the zero space charge assumpt i o n is w r i t t e n i n terms o f the S l o t b o o m variables i t can be used to compute the potential i n terms of u and v. Thus, it remains to consider the equations 2
2
2
2
2
J„ = n„
C + V C + 48*uv - grad u, 2u
d i v J„ = R, (4.1.4)
J
P
-C
= -H
+ J C
+ 4S*uv
2
P
A
grad v,
T
J
div J = p
?
—R.
As discussed i n Chapter 3 there is an i m p o r t a n t difference between the Slotboom variables and the carrier densities. I n the t e r m i n o l o g y of singular p e r t u r b a t i o n theory, u and v are slow variables whereas n and p are fast variables. This means that the derivatives of n and p are locally unbounded as X -> 0 and that they converge pointwise to discontinuous functions. O n the other hand, the derivatives of u and v are uniformly bounded as X -> 0. This implies that their limits are continuous. The above expressions contain the approximations 2
C + JC
2
+ 48*uv
A
- C + JC
+ 48 uv
(4.1.5)
for the carrier densities. F o r small values o f 8 they i m p l y 4
4
n = C + 0(8 ),
p = 0{8 )
i n rc-regions (C > 0)
and
(4.1.6) 4
p = - C + 0(8 ),
4
n = 0(8 )
i n p-regions (C < 0).
Remark: Strictly speaking, the L a n d a u order symbols 0 and o (see e.g. [ 4 . 1 ] for a definition) i n (4.1.6) require the supplement "as 8 -> 0". Here and i n the following the meaning of the order symbols is determined by phrases like "for small values o f <5" instead. N o t e that the conclusion (4.1.6) relies on the assumption that the scaling o f u and v is correct (i.e. u, v = 0(1)), which is not justified a p r i o r i l y . Nevertheless, this type o f argument w i l l be used repeatedly i n this Chapter. I f i t leads to consistent approximations of solutions, this is considered as an—at least heuristic—justification. (4.1.4) and (4.1.6) i m p l y that J„ is 0(<5 ) i n p-regions and J is 0{8 ) i n n-regions. Disregarding situations where an n- or p-region has more t h a n one contact we conclude that the current densities are 0((5 ) t h r o u g h o u t Q. This is m o t i v a t e d by the following argument: Consider an rc-region w i t h one O h m i c contact. T h e n the electron current t h r o u g h the adjacent PNjunctions is 0(<5 ). The same holds for the recombination-generation rate and, by the divergence theorem, for the current t h r o u g h the O h m i c contact. This implies that the electron current density is 0(8 ) t h r o u g h o u t the n4
4
p
4
4
4
4.1 I n t r o d u c t i o n
179
region. The same argument w i t h the obvious changes holds for the hole current density i n p-regions. A p p r o p r i a t e l y scaled current densities are introduced by 4
4
J = dJ , n
J = SJ .
ns
p
(4.1.7)
ps
W i t h these assumptions we are left w i t h the problem, now expressed in terms of the rescaled current densities C + V C + 4S uv oJ„ = Hn ~ -gradu, lu uv — 1 div J„ = x {n + S ) + T„(p + 8 Y 2
A
2
2
p
(4.1.8) -C + yC
2
A
+ 4S uv
uv — 1
divJ
p
r (n p
2
2
+ S ) + z„(p + S )
subject to the boundary conditions V i
Vi
u = e , du
dv
dv
dv
^- = ^
v = e~
at
F,
/ = 0, ...,m— m
=0
at
1, (4.1.9)
l
~
dQ =
dSl\[jr .
N
t
i=
0
I n (4.1.8) we dropped the index s of the rescaled current densities. N o w we make use of the smallness of <5. Solutions of (4.1.8), (4.1.9) are approximated by letting 5 tend to zero which gives grad u = 0,
div
Ll U p
\
A
—— grad v = V c j
UV — 1
—-
i n w-regions
TC P
and
(4.1.10)
, \
1
• • / Pn ' div I —
grad v = 0,
UV
— 1
grad it) = Y J c f
r e
P" §
l o n s
-
O u r a p p r o x i m a t i o n amounts to replacing the quasi F e r m i level of the majority carriers by a constant. Then the other quasi F e r m i level can be computed by solving a linear elliptic equation. Consider the p r o b l e m of determining v in an /i-region Q. . Assuming the constant value of u as given we observe that v — l/u solves a homogeneous linear equation. The boundary of Q_ splits i n t o N e u m a n n segments dQ n dfijy and Dirichlet segments S S which consist of PA -junctions adjacent to Q and (possibly) a contact. The boundary conditions at O h m i c contacts and the fact that v is constant in p-regions i m p l y that v takes constant values v v along S , S . respectively. I n t r o d u c i n g the functions q> ,..., q> by solving +
7
+
l
f
p
+
u
t
p
p
±
p
180
4 Devices
div
grad v
7
I
J!
p
•
(4.1.11)
1
^
A
we can determine v i n terms of the cp/s: v
+
= ~
i
( J-VJVJv
As a consequence we obtain A ^ (up, - 1) 4. = - L — § 7=1
r a d
C
Since the cpj do not depend o n the applied voltages they only have to be computed once for a given device. I n the same way formulas for u and J„ i n p-regions can be obtained. These ideas w i l l be used i n the following Sections for the c o m p u t a t i o n of voltage-current characteristics. They w i l l lead to explicit formulas i f the functions cpj for a l l n- and p-regions of the device are assumed to be k n o w n .
4.2 P-N Diode The importance of understanding the electrical behaviour of P - N junctions is twofold. O n one hand the P - N j u n c t i o n diode itself is a device w i t h a wide variety of applications and o n the other hand the performance of more complicated devices (such as the bipolar transistor and the thyristor) is based on the interaction of P - N junctions. I n this Section we consider steady states of a P - N diode w i t h O h m i c contacts. The mathematical model consists of the steady state drift diffusion equations (4.1.1). The choice of models for the mobilities and the recombinationgeneration rate R w i l l depend on the effects we want to take into account. We consider (4.1.1) in the d o m a i n Q 0), and the P - N j u n c t i o n T (see Fig. 4.2.1). F o r simplification we assume an abrupt P - N j u n c t i o n , i.e. the d o p i n g profile C has j u m p discontinuities along T. A t the contacts r £ 8Q n 8Q. and ^ c n dQ the boundary conditions (4.1.2) are satisfied w i t h U = 0 and {7, = U where U is the applied voltage. A l o n g the insulating (or artificial) part of the boundary 8Q = 8Q\(r u T , ) homogeneous N e u m a n n boundary conditions for V, n and p are prescribed. I n this Section the singular perturbation analysis of the Sections 3.4 and 3.5 is extended by further simplifications of the a p p r o x i m a t i n g reduced and layer problems. These additional simplifications are carried out by exploiting the smallness of the scaled intrinsic number 5 = n-JC. The solution of the simplified problems allows to obtain explicit information o n the geomek
+
o
+
0
N
2
o
4.2 P - N D i o d e
181
r
r, F i g . 4.2.1 T w o - d i m e n s i o n a l cross section o f a P - N d i o d e
try of the depletion region and on the voltage-current characteristics close to thermal e q u i l i b r i u m . A d d i t i o n a l i n f o r m a t i o n on the voltage-current characteristics i n high injection and large reverse bias situations (in particular the critical voltages causing punch-through and avalanche breakdown) w i l l be obtained by considering the simplified one-dimensional model n = (o,i),
r
0
= {o},
r
1
=
{i},
r = {x } 0
(4.2.1)
(4.2.2)
The Depletion
Region
in Thermal
Equilibrium
I f the potential d r o p across a P - N j u n c t i o n is large compared to the thermal voltage a certain region around the P - N j u n c t i o n is depleted of charge carriers. As discussed i n Section 3.5 this effect is especially i m p o r t a n t for reverse biased P - N junctions. I n thermal e q u i l i b r i u m a depletion region occurs i f the b u i l t - i n potential is large enough w h i c h always is the case i f a significantly large d o p i n g concentration is assumed. Recalling the results from Section 3.4 for the thermal e q u i l i b r i u m problem )} AV=
2
2d sinh V— C, (4.2.3)
182
4 Devices
the solution is approximated by the solution V(x) =
V (x) bi
of the reduced equation 2
0 = 2b sinh V — C away from T whereas close to T i t can be approximated by a layer solution V(s,0
= hmV(s,AO.
(4-2.4) k
1
I n (4.2.4), V is w r i t t e n i n terms of the local coordinates (s, r) e U ~ x U along r w i t h the tangential components s and the n o r m a l component r (r > 0 in Q ) . £ = r/X is a fast variable and V is a solution o f the layer p r o b l e m +
2
df V = 2d sinh V — C 1 F(s, - oo) = V (s, 0 - ) ,
. F(s, oo) = V (s,0 + ) ,
bi
(4.2.5)
bi
where the n o t a t i o n -C_(s) = C ( s , 0 - ) C(s, 0 :=
C (s)
= C(s,0 + )
+
for
^ < 0,
for
£>()
was introduced. A detailed discussion o f the formalism leading to (4.2.5) is given i n Section 3.4. Since the tangential component s o n l y appears as a parameter, (4.2.5) is essentially a one-dimensional p r o b l e m . A n approximate solution can be obtained by exploiting the smallness o f <5 for significantly large d o p i n g concentrations. We rewrite the b o u n d a r y conditions at <J = + oo as 2
2
A
V~(s, oo) = In 5~ + In C + 0(S ), . V(s, - oo) = - I n S' - In C + 0(S ). +
2
(4.2.6)
A
This shows that V is badly scaled because its b o u n d a r y values at + oo become unbounded as b -> 0. Instead of b we introduce the small parameter y = (In b ~ ) ~ and the scaling 2
2
2
l
W = yV of the dependent variable. The new reference value U /y T
= U ln(C/n ) T
;
for voltages is of the order of magnitude of the b u i l t - i n voltage. F o r small values o f 7 it is large compared to the thermal voltage. After substitution o f W i n (4.2.5) the factor 1/y appears i n front o f the second derivative. I t can be eliminated by the rescaling n= of the independent variable. The resulting p r o b l e m reads
4.2 P - N D i o d e
183
-W-
dW 2
y 1 + y In C
W{oo)
+
)
-
(4.2.7)
c,
(4.2.8)
+ TST,
-1 - y l n
W(-oo)
1
(4.2.9)
+ TST,
where T S T stands for franscendentally small terms, i.e. terms w h i c h are 0(e~ ) w i t h a positive constant c. I n general, the symbol " T S T " denotes terms w h i c h are small compared to any power of the small parameter considered. Since the derivative of the right-hand side of (4.2.7) w i t h respect to W is positive, the m a x i m u m principle (see [4.13]) can be applied. L e m m a 3.3.14 implies the estimates cly
- 1 + 0(y) ^ W ^ 1 +
0(y)
w h i c h i n t u r n i m p l y the boundedness of the carrier densities e x p [ ( i y — l ) / y ] and exp [ ( — I T — l ) / y ] u n i f o r m l y i n y. By the differential equation (4.2.7) d W is also u n i f o r m l y bounded, w h i c h can be used for a justification of the l i m i t i n g process y -> 0 i n (4.2.7), (4.2.8), (4.2.9) (see [ 4 . 2 ] ) . G o i n g to the l i m i t we obviously have 2
-1 ^
1,
W(-oo)
= - 1 ,
d W + C ^ 0
for
W < 1,
d W + C ^ 0
for
W >
2
2
W(oo) = 1, (4.2.10)
-1,
and W is continuously differentiable by the boundedness of d W. The p r o b l e m (4.2.10) is a standard double obstacle problem. This term originates from an interpretation of W as the displacement of a string under the action of a lateral load C w h i c h lies between t w o obstacles represented by the bounds 1 and — 1 i n our situation. I t can be shown that (4.2.10) has a unique solution by r e w r i t i n g i t as a variational inequality [ 4 . 2 ] . The above treatment outlines the p r o o f of a convergence result w h i c h is well k n o w n , because (4.2.7), (4.2.8), (4.2.9) is a standard penalisation of (4.2.10) (see [4.2]). The solution of (4.2.10) splits Q i n t o three subdomains: the coincidence sets where W = ± 1 holds and the noncoincidence set where — 1 < W < 1 and d W + C = 0 holds. Leaving the solution of (4.2.10) to the reader (see Problem 4.2) we only state the result that the noncoincidence set is given by the ^-interval 2
2
(-2/ /C_(C_/C v
+
+ 1), 2 / / C V ( C _ C v
+ 1))
(4.2.11)
/
w i t h the length 2 1 / C + 1/C_ . N o t e that the l i m i t i n g carrier densities V
+
vanish i n the noncoincidence set of (4.2.10) w h i c h means that the noncoin-
184
4 Devices
F i g . 4.2.2 T h e d e p l e t i o n r e g i o n i n t h e r m a l e q u i l i b r i u m
cidence set is depleted o f charge carriers. A n interval o f the form (4.2.11) is obtained for every point along the P - N j u n c t i o n T. The depletion region is, thus, given by the u n i o n o f the n-intervals (4.2.11) taken along the values o f the tangential coordinate s along T (see F i g . 4.2.2). W i t h the a p p r o x i m a t i o n U = 2U /y o f the unsealed b u i l t - i n voltage, the unsealed w i d t h o f the depletion region is given by bi
T
Remark: Here and i n the sequel was denote scaled and unsealed quantities by the same symbols. Formulas i n terms of unsealed quantities are explicitely announced in the text in order to a v o i d misinterpretations. The above expression for the w i d t h o f the depletion region can be found i n textbooks on semiconductor device physics (see e.g. [4.22]). There i t is obtained by the a p r i o r i assumption that a depleted region exists, and that zero space charge prevails i n the rest o f the device. We conclude this Paragraph w i t h a mathematical remark. The results have been obtained by the l i m i t i n g process X-*0 followed by 3 -> 0 applied to the thermal e q u i l i b r i u m p r o b l e m (4.2.3). We p o i n t out that the limits commute. By i n t r o d u c i n g W i n (4.2.3) and letting 8 -> 0 we arrive at a m u l t i dimensional double obstacle p r o b l e m which contains the singular perturbat i o n parameter X/yfy. A n analysis o f this singularly perturbed obstacle problem can be found i n [4.15]. There it is shown that its solution can be approximated by solving the one-dimensional obstacle p r o b l e m (4.2.10) i n the j u n c t i o n layer. A p a r t from these t w o approaches an analysis where X and 3 tend to zero simultaneously is also possible (see [4.11]).
185
4.2 P - N D i o d e
Strongly
Asymmetric
Junctions
The performance of most semiconductor devices relies o n the interaction of strongly asymmetric P - N junctions, i.e. junctions w i t h d o p i n g levels of different orders of magnitude in the n- and p-regions. I n this Paragraph an analysis of the j u n c t i o n layer p r o b l e m (4.2.7), (4.2.8), (4.2.9) for strongly asymmetric junctions is given. The presentation is based o n the w o r k of W a r d [4.25] w h o also pointed out the similarity of the p r o b l e m to a M I S diode (see Section 4.6) i n strong inversion. I n this paragraph, advanced concepts of singular perturbation theory are introduced. They are also i m p o r t a n t for the analysis i n the Sections 4.6 and 4.7 o n the M I S diode and the M O S F E T . +
I n the case of an N P j u n c t i o n C » C_ holds. The depletion region (4.2.11) obtained i n the preceding Paragraph spreads mostly i n t o the p-region. Concentrating o n the potential i n the p-region, we assume that the d o p i n g concentration i n the p-region has been chosen as reference value i n the scaling. This implies C_ = 1. N o t e that this is i n contrast to the scaling philosophy adopted i n the biggest part of this b o o k where the d o p i n g profile is scaled to the m a x i m a l value 1. I t is n o w justified since the m a i n interest lies i n an analysis of the region w i t h lower doping. The p r o b l e m (4.2.7), (4.2.8), (4.2.9) w i l l again be analyzed by letting y tend to zero. The asymmetry of the j u n c t i o n is taken i n t o account by keeping the q u a n t i t y +
A : = y In C
+
fixed as y - » 0 (A d o p i n g concentration of 1 0 / c m i n the n-region and 1 0 c m i n the p-region for Si at r o o m temperature leads to A ~ 1). The first step is the c o m p u t a t i o n of the value of the potential at the j u n c t i o n n = 0. M u l t i p l i c a t i o n of (4.2.7) by d„ W and integration gives 1 8
1 4
3
3
2
(5, W) /2
(W— = y exp I —
1\ /— W — 1 \ I + y exp I - CW + k (4.2.12)
with Aly
_ (e (A
+ 1 - y) + T S T
~ [ 1 - 7 + TST
for
r\ > 0,
for
n<0.
The values of k have been determined from (4.2.12) evaluated at n = ± oo and (4.2.8), (4.2.9). Requiring c o n t i n u i t y of the derivative of W at rj = 0, the value of W at the j u n c t i o n W(0) = A + 1 - y + T S T
(4.2.13)
can be computed. Since this value is equal to I F ( x ) up to 0(y), most of the variation of the potential takes place i n the p-region as expected. There, (4.2.12) can be rewritten as
186
4 Devices
W V 2 7 exp
(W
-
1\
V
/
/
+
exp
/ - W V
)'
1\ /
+ W + 1 - y + TST . (4.2.14)
Since the i n i t i a l value (4.2.13) is larger than 1, the derivative of W at n = 0 becomes unbounded as y - > 0 , and (4.2.14) is singularly perturbed. The asymptotic analysis below w i l l show that the solution has m u l t i p l e layers. A very t h i n i n i t i a l layer is followed by a transition layer describing the behaviour o f the solution close to W = 1. This transition layer connects the initial layer to a depletion region where W is between — 1 and 1. The edge of the depletion region is a free boundary where W takes the value — 1 which is asymptotically equal to its value at n = - co. Initially, the solution is expected to vary fast. A n a p p r o x i m a t i o n depends on a layer variable 1 a
=
W)
where 8(y) = o ( l ) holds. A differential equation for the a p p r o x i m a t i o n is obtained by letting y -» 0 i n the transformed differential equation (4.2.14). However, for nonlinear equations the l i m i t i n general depends o n the values w h i c h the solution takes. F o r (4.2.14) i n particular, the relative orders of magnitude o f the terms under the square root depend on the value o f W. I t is, for example, possible to generate different situations by choosing different values W i n 0
W=W
+ yy
0
(4.2.15)
where the new variable y has been introduced. The choices o f S(y) and o f W are governed by an heuristic principle: The limiting equation should contain as many terms as possible. A l i m i t i n g equation which satisfies this requirement is called a significant degeneration (see [ 4 . 1 ] , sometimes also distinguished limit [4.5]). The initial c o n d i t i o n (4.2.13) requires the choice W = A + 1 and, thus, 0
0
W {a) in
= A + 1 + yy(a)
for the initial a p p r o x i m a t i o n W . W i t h this choice, the heuristic principle in
implies 5(y) =
exp( — A/2y).
W i t h the fast variable
'1 we o b t a i n the degeneration yl2
d„y = J2e ,
y(0) = - 1
w i t h the solution y = — 2 ln(y/e
—
a/yjl).
(4.2.16)
4.2 P - N D i o d e
187
The a p p r o x i m a t i o n (4.2.16) has been obtained by d r o p p i n g all the terms under the square root i n (4.2.14) except the first. Obviously, this term only dominates as long as W is larger than one. The transition o f W t h r o u g h values close t o 1 is governed by a significant degeneration o f (4.2.14), obtained by setting 1
W (x) = 1 + / In y " + y z ( r ) ,
x = rj/y
tr
and letting y tend t o zero: d z = J2{e* z
+ 2).
(4.2.17)
The independent variable x is fast compared t o the original variable n but slow compared t o the i n i t i a l layer variable a. The solutions of (4.2.17) are given by 2
z = - l n ( | s i n h ( r + c)),
(4.2.18)
where c is a constant o f integration. This constant is determined by a procedure called matching, w h i c h has already been introduced i n Section 3.4. The underlying idea is that the domains o f validity o f the a p p r o x i m a tions W and W overlap. I n its simplest form, matching w o u l d amount t o equating the l i m i t o f W as a -> — GO t o the value o f W at x = 0. However, the l i m i t o f W does n o t exist and we have to proceed differently. R e w r i t i n g W i n terms o f the slower variable x gives in
lT
in
u
in
in
A 2y
W (xJ^e ' ) in
1
2
~ 1 + y In y " - y l n ( r / 2 ) .
(4.2.19)
Obviously, this coincides w i t h W for small values o f x i f we set c = 0 i n (4.2.18). The transistion layer a p p r o x i m a t i o n W takes W from values larger than 1 to values smaller than 1. F o r W between 1 a n d — 1, the carrier densities vanish as y -> 0. The a p p r o x i m a t i o n W (ri) satisfies the l i m i t i n g equation tr
tT
depl
W e
P
. = V2(PF
d e p l
+ 1)
(4.2.20) 2
w i t h the general solution l F i = — 1 + (f?/>/2 + c) . The free constant c can again be computed by matching. A straightforward c o m p u t a t i o n gives d e p
l i m W (n/y)= tr
1 + 2r\.
(4.2.21)
This agrees for small rj w i t h W
depl
^dep,=
- 1 +(n +
i f we set c =
v
2. Thus, VT
dep]
is given by
2
2) /2.
Since the solution of (4.2.14) is obviously monotone, l F looses its validity at the p o i n t rj = — 2, w h i c h represents the edge of the depletion region. The m o n o t o n i c i t y also implies that for rj < — 2 the potential can be a p p r o x i mated b y the constant value — 1 . This is a continuously differentiable extension o f the depletion layer s o l u t i o n W and a singular solution o f the differential equation (4.2.20). F o r the carrier densities we o b t a i n that i n a t h i n region adjacent t o the P - N j u n c t i o n the electron density is m u c h larger than the hole density. This t h i n d e p l
depl
188
4 Devices
layer is followed by a depletion region w i t h a sharp edge. Outside the depletion region the hole density is approximately equal to the d o p i n g concentration and the electron density is much smaller (compare to (4.1.6)).
The Voltage-Current
Characteristic
Close
to Thermal
Equilibrium
The ideal diode operates like a perfect valve. The t o t a l current t h r o u g h the device is zero i n reverse bias (i.e. for negative values of applied voltage U) and grows w i t h the applied voltage i n forward bias (see Fig. 4.2.3). I n this Paragraph we shall demonstrate t h a t — a t least close to thermal e q u i l i b r i u m —the P - N diode approximately exhibits this behaviour. F o r the P - N diode problem, (4.1.10) implies in
a
+
(4.2.22)
= 1 . u - 1 Q
m
,
«|r =l,
u
u\
0
r
= e , (4.2.23)
u
e
div
grad . A = ^ = ^ -
in
Q , +
c| =l,
=
r
u
e~ .
(4.2.24) The solutions of (4.2.23) and (4.2.24) are given by u
u\ _
= (e
»j
= (1 - e " >
n
0 +
-
Dcp, + 1,
J J Q . = ( c
u
2
+ e~ ,
J\
p a+
D
- 1 ) ^ 7 grad
= (1 - e
u
^ ,
) ^ grad q> , 2
where the functions cp, and q> are the unique solutions of the problems 2
I
u
F i g . 4.2.3 I d e a l d i o d e characteristic
4.2 P - N D i o d e
189
Pn grad (pj ic.
div
in
Q_,
TjC| = 1,
dv an n an
= 0
v
and div
grad
^
2
in
2
Q , +
0.
dv an n an.
2
v
The current / t h r o u g h the device i n the direction from r to r is obtained by c o m p u t i n g the current across the P - N j u n c t i o n T. I f v denotes the unit n o r m a l vector along V oriented i n t o Q , we have 0
t
+
/
(J„ +
u
(e
J )vds p
" grad
- 1)
grad cp I • v ds. 2
The first term on the right-hand side can be rewritten as • v ds = an.
2
T7^(p
x
n
I grad
M
using integration by parts. A n analogous c o m p u t a t i o n for the second term finally leads to a voltage-current characteristic of the form u
I = I (e -
(4.2.25)
1),
s
which approximately exhibits the behaviour described at the beginning of this Paragraph. Under reverse bias (U < 0) the current does not vanish but it is bounded below by the leakage current 1
+
| ^
n
p
| g r a d < p
2
|
2
+ ^ U v
(see Fig. 4.2.4). O u r result (4.2.25) is a scaled version of the famous Shockley equation [4.18] which i n terms of unsealed variables reads / = I,(e u/v
T
1).
I n the simplified one-dimensional case (4.2.1), (4.2.2) the problems for q and cp can be solved explicitely (see Problem 4.4) and the unsealed saturation x
2
4 Devices
190
F i g . 4.2.4 V o l t a g e - c u r r e n t characteristic
current is then given by /. = qV^rnf
+ ?
L
coth ( - 7 ^ = )
£
coth
(4^)).
(4.2.26)
where L denotes the unsealed length o f the device. F o r devices, where the lengths x and L — x o f the p- and n-regions are large compared to the diffusion lengths y/p T U and y/g. r U , the coth-terms can be replaced by 1 which leads to the saturation current of the classical Shockley equation. I n the case of a very short device (i.e. the arguments of the coth-terms are small) (4.2.26) reduces to 0
0
n
h = q U
T
n
T
n f ( - ^ + . \x C.
p
p
T
) (L-x )C+J
T
0
0
for the unsealed saturation current. This formula shows that for very short devices the characteristic close to thermal e q u i l i b r i u m is essentially independent of recombination-generation effects. I n the derivation of the Shockley equation we used the assumptions that the zero space charge a p p r o x i m a t i o n holds ( w i t h the exception o f a t h i n layer) and that the error made i n the a p p r o x i m a t i o n (4.2.22), (4.2.23), (4.2.24) remains small. The first assumption is violated i n strong reverse bias when the w i d t h o f the depletion region is not small compared to the length of the device. The second assumption is satisfied as long as the (scaled) applied voltage U is small enough for e to be small compared to <5~ , i.e. i f U is small compared to the b u i l t - i n potential. I f this is not the case the assumption that the charge carrier densities are close to their e q u i l i b r i u m values does not h o l d any more. This situation is called high injection. v
4
191
4.2 P - N D i o d e
High
Injection—A
Model
Problem
I n [4.10] the solution of the zero space charge p r o b l e m for a simple model device, the one-dimensional symmetric diode, was computed explicitely. As no further approximations are necessary i n this case the resulting currentvoltage characteristic remains v a l i d i n high injection. The necessary computations are outlined below. Suppose that the center o f the device is represented by the p o i n t x = 0, that the d o p i n g profile is piecewise constant and an o d d function. We take constant mobilities and neglect the effects o f recombination-generation. Symmetry arguments allow the ansatz V(-x)=V(x), JJPn
n(-x)
= p(x),
= Jp/Hp = J/(P-n + lip)
J,
w h i c h i n t u r n allows to compute the solution by considering only half of the device (say, the n-region). Thus, the p r o b l e m is posed on the scaled n-region (0, 1) and the b o u n d a r y conditions V(0) = 0,
n(0) = p(0)
at x = 0 follow from the symmetry ansatz. As expected the solution o f the resulting p r o b l e m has a b o u n d a r y layer at zero and the solution of the zero space charge a p p r o x i m a t i o n satisfies 0 = n — p — 1, 1 = n'-nV, 1=
(4.2.27)
-P'-PV
subject to the b o u n d a r y conditions V(0) - In n(0) = - V(0) - In p(0), (4.2.28) V(l)
= V (\) bi
U/2,
p(l) =
+ J \ + 43
where the constant 1 i n the reduced Poisson equation is the scaled d o p i n g concentration and 2
V (l)
= areasinh(l/2c) )
bi
holds. The boundary c o n d i t i o n at x = 0 can be interpreted as the symmetry c o n d i t i o n (p (0) = —(p (0) for the reduced quasi F e r m i potentials. After integration o f (4.2.27) subject to the boundary conditions at x = 1 the carrier densities and the potential can be expressed i n terms o f / : n
p
n = p+ v
1 =^(1 + J l + z
V = V bi
A
48
+ 4/(1 - x ) ) ,
v
(4 2 29) 4
U/2 + , / ! + 4b - + 4 / ( 1 - x) -
A
V
T + 45
.
The voltage-current characteristic i n implicit form is obtained by substitut i o n of (4.2.29) i n t o the b o u n d a r y c o n d i t i o n at x = 0:
192
4 Devices
F i g . 4.2.5 F o r w a r d bias characteristic ( l o g a r i t h m i c scale)
^ =
n
-ini±^^S yrT4^-yrT4F.
i
+
4
V ^
+
4
1
(4.2.30)
4
W i t h I = 0(c) ) this equation includes the Shockley equation as the l i m i t i n g case S -> 0. I n a high injection situation it predicts a quadratic dependence of the current o n the applied voltage which, i n unsealed variables, takes the form 4
qC(g.
+ g. )
n
p
8LU
T
This equation illustrates the result that the g r o w t h o f the voltage-current characteristic slows d o w n i n high injection, which is well k n o w from experiments and numerical computations (see F i g . 4.2.5). The solution (4.2.29) also shows that the charge carrier densities grow linearly w i t h the applied bias and that a significant potential d r o p takes place t h r o u g h o u t the device as opposed to low injection.
Large
Reverse
Bias
The Shockley equation looses its validity as soon as the depletion region covers a significant part of the device. This is the case i f the scaled reverse bias — U is 0 ( / t ~ ) . As discussed i n Chapter 3 a scaling different from that in (4.1.1) has to be used i n this situation. I n order to be able to obtain 2
4.2 P - N D i o d e
193
explicit i n f o r m a t i o n o n the voltage-current characteristic we restrict ourselves to the one-dimensional situation (4.2.1), (4.2.2). W i t h the transformation 2
cp = X (V -
V (0)) u
(4.1.1) becomes q>" = n — p — C, 2
2
X J„ = u (X n'
-tup'),
n
2
2
XJ
= -n (? p'
P
J„ = R,
+ p
p
(4.2.31) J' =-R p
subject to the boundary conditions 9(0) = 0 ,
p ( l ) = «Pi.
«(0) - p(0) + C
= n(l) - p(l) - C
n(0)p(0) = n ( l ) p ( l ) = 2
where , = X (V {\) Read-Hall t e r m
-
bi
n
R
= 0,
+
(4.2.32)
5\
V (0) - U) > 0 holds. F o r i? we take the Shockleybi
p
5
A
~ T (n + S ) + T (p + 2
p
n
2
d )'
As i n Section 3.5 an approximate solution is obtained by going to the limit / -> 0 in (4.2.31), (4.2.32). The reduced equations q>" = n — p — C, tup' = pep' = 0 i m p l y that the device splits i n t o a depletion region (n = p = 0) and a zero space charge region (n — p — C = 0). The reduced potential q> is a solution of the double obstacle problem Q^q>^
^(1) =
for
!,
^" + C > 0
for
(p > 0
^ , (4.2.33)
w i t h the noncoincidence set (see Problem 4.6) (x x ) r
r
= \x
0
-
/
2
2(
x
"
o + u
Pi V C ( l + C /C_) +
+
(4.2.34) which represents the depletion region. I t has the length
x - x, = V2(p,(l/C r
+
+ 1/C_) .
(4.2.35)
The l i m i t i n g p r o b l e m i n (0, x,) u ( x , 1) is completed by (see Chapter 3) r
194
4 Devices
0 = n —p
—
C,
T
(np)' = -±p-- S-n, Pn P-p P
T
p
(
n
(4.2.36)
+
d
2
)
+
T
n
{
p
+
5
2
)
subject to the auxiliary conditions (np)(0) = (np)(\) J (x ) n
- J( )
r
= 5\
p
4-
+
= 0,
+ 1/C_) ,
(4.2.37)
T
s2 = - — — ^ 2 ^ ( 1 / 0 + + 1/C.) . ~r
- J( )
r
r
—~ V 2 ^ ( l / C
T
J (x )
= (np)(x )
3
=
n Xl
(np)(x,)
p Xl
The fact that np is a slow variable, w h i c h does not have a j u m p across the edges of the depletion region, explains the equations for np at x, and x . The right hand sides of the last t w o equations are caused by generation effects in the depletion region and can be obtained by integrating the differential equations of the current densities. F o r r 5 = 0(4.2.36), (4.2.37) obviously has the solution = J„ = J = O.This motivates the rescaling 5 w = np, d I„ = J„, 8 I = J . The transformed p r o b l e m is approximated by going to the l i m i t S -» 0 w h i c h implies for the charge carrier densities: r
2
p
2
2
2
p
p
2
n = max(0, C ) ,
p = max(0, — C)
in
(0, x,) u ( x , 1). r
The variables w, I„, I solve the problem p
I C_/p
in
(0, x ) ,
I C /p
in
( x , 1),
N
w
=
P
n
+
p
(
r
CW/T„C_
r = - v = [W/T C n p 1
1
P
+
in
(0,
x,),
in
(x , r
1),
subject to the auxiliary conditions w(0) = w ( x ) = w(x ) = w ( l ) = 0, (
I„(x ) - I { ) r
n Xl
r
=
— /2
r„ + I (x ) p
r
- I (x,) p
= _ _ ^
T „
p
1
1
+ 1/C_),
V
( 1 / C
+
+ 1/C_).
The solution of this p r o b l e m only involves the integration of linear differential equations w i t h constant coefficients and is left to the reader. A typical charge carrier d i s t r i b u t i o n is depicted i n F i g . 4.2.6. The only source o f current flow is generation i n the depletion region and
195
4.2 P - N D i o d e
n
'I x,
1
'\ xr
x0
1 x
F i g . 4.2.6 C h a r g e carrier d i s t r i b u t i o n i n s t r o n g reverse bias
the approximate current-voltage characteristic reads J = -?—J'2 {\IC l
T „
+
+ 1/C_).
(4.2.38)
+
I n terms o f unsealed quantities this gives J = -Jh— /2e q{VC y
s
+
+ l/C_)(-t/),
which is a standard textbook formula [4.22]. I t states that the leakage current caused by generation i n the depletion region is p r o p o r t i o n a l to the w i d t h of the depletion region and, thus, to the square r o o t of the applied bias. Avalanche
Breakdown
I n large reverse bias impact ionization might cause j u n c t i o n breakdown. This phenomenon can be explained by analysing (4.2.31), (4.2.32) w i t h the generation t e r m
*--*«p(-^)i*i-«i«n'(-^f)wThis is a scaled version of the impact ionization model (2.6.4), (2.6.5). The factor X i n the scaled critical field strengths XE„, XE has been introduced because XE and X.E are o f the order of magnitude of the scaled Debye length X i n typical cases. The dimensionless parameters E , x„ are assumed to take moderate values. F o r X -* 0 the reduced potential is that of the preceding Paragraph and, obviously, the depletion region is the same, too. I n t r o d u c i n g p
n
n
4
w = l i m (np
d -*0
4
/„ = l i m ( J „ < T ) , 4
d ^0
l
p
p
p
= lim 4
d ^0
4
(J S~ ) p
196
4 Devices
the problem in the zero space charge regions (0, x,) and ( x , 1) becomes r
I„C-/n„ w =
•I C /n p
+
p
in
(0, x ) ,
in
( x , 1),
(
r
/ ; = / ; = 0,
(4.2.39)
vv(0) = w ( l ) = 1,
w(x,) = w ( x ) = 0. r
F r o m (4.2.39) we obtain / I
"
l < 0
x
- '
_ Jf!L_
)
/ • p l ( x
x,C_'
"
1 )
(l-x )C ' r
+
Whereas the ionization rates vanish i n the zero space charge regions as X -» 0, they take their m a x i m a l values a and oe w i t h i n the depletion region. As long as the current densities /„ and I remain negative the differential equations n
p
p
hold i n the depletion region (x„ x ). conditions r
/ „ ( * . ) = - - £ - ,
ipMp y
x,C_ '
W h e n subjected
h
-
"
to the
boundary
-
(1 - x ) C + r
this system can be solved explicitely giving the following equation for the current I = I„ + I : p
x iXp
x
)
(a e ' ~ ''
Xr(Xp
—
!X
)
ae ~ " n
a - a„ p
Pn „x,(a„-oc„) _ x,(a -a„) rpPp xA* -x„) x,C_ " (1 — x )C p
c
r
_
p
(4.2.40)
+
The current blows up when the term i n the brackets o n the left hand side of (4.2.40) vanishes. This occurs for x - x, = — - — ln(a /a„). r
(4.2.41)
p
F r o m (4.2.41) and the formula (4.2.35) for the length of the depletion region the breakdown voltage
i
n
/
r
\
1 / r
J—^—
2(1/C+ + 1 / C _ ) V « - a„ P
ln(« /a„)) /
2
P
can be computed. I n terms of unsealed variables this gives
N o t e that i n the case a„ = a = a the c o n d i t i o n (4.2.41) becomes p
(4.2.42)
4.2 P - N D i o d e
197
oe(x — x,) = 1, r
which is a special case o f the classical c o n d i t i o n [4.22]: B r e a k d o w n occurs as soon as the integral o f the i o n i z a t i o n rate over the depletion region takes the value 1. E q u a t i o n (4.2.40) shows that the current also becomes unbounded i f one of the boundaries of the depletion region reaches a contact, i.e. i f x, = 0 or x = 1 holds. The results of the preceding Paragraph show that this happens for the critical voltages r
2
| x C _ ( l + C_/C ) +
or
2
|(1 - x ) C ( l + C /C_) 0
+
+
respectively. This effect is called punch through. There are three possible reasons for j u n c t i o n breakdown: Avalanche generat i o n and punch t h r o u g h at the left or right contact. B r e a k d o w n occurs as soon as one of the t w o critical voltages given above or the avalanche b r e a k d o w n voltage (4.2.42) is exceeded. N a t u r a l l y , only the smallest of the three values is o f practical interest. A n existence result by M a r k o w i c h [ 4 . 8 ] for (4.2.31), (4.2.32) w i t h a simplified version of the i o n i z a t i o n rates shows that for every value o f the applied potential a stationary s o l u t i o n w i t h finite current exists. Thus, the fact that the current obtained by our analysis tends to infinity at a critical voltage does not necessarily mean that the same holds for the s o l u t i o n of the full p r o b l e m (4.2.31), (4.2.32). I t only tells us that we reached the l i m i t o f the validity o f our asymptotic analysis. The following Paragraph w i l l show that, by rescaling and c a r r y i n g out a different p e r t u r b a t i o n procedure, the cont i n u a t i o n o f the current-voltage characteristic after punch t h r o u g h can be computed. I n reality, however, the sharp increase i n the current causes thermal effects which are not taken i n t o account by our model and w h i c h m i g h t very well be the actual reason for b r e a k d o w n .
Punch
Through
F o r simplicity we consider the model p r o b l e m w h i c h we dealt w i t h i n the Paragraph o n high injection. As we are interested i n the situation after punch through, we introduce the rescaled current density 2
; = x J/(p
n
+ Li ) p
as suggested above. W i t h the scaling for large reverse bias the p r o b l e m now reads
= n — p — 1,
/ = )}ri - ncp', / = -)}p'
- pep'
subject to the b o u n d a r y conditions
(4.2.43)
198
4 Devices
(4.2.44)
"(0) = P(0), 4
«(l) = p ( l ) + 1 = 1 ( 1
(Pl,
+ 4<5 ).
The analysis o f the Paragraph "Large reverse bias" implies that the part o f the depletion region o n the rc-side is given by the interval (0, yjltpy )• Thus, punch t h r o u g h occurs at the applied bias cp = \ and, therefore, we assume cp > \ for the following. By m u l t i p l y i n g the Poisson equation by
1
cp'icp" + 1) = 0 for the reduced potential follows. Since cp is a slow variable it is expected to satisfy the original boundary conditions after going to the l i m i t /. -> 0. Thus, we o b t a i n
The reduced charge carrier densities are given by n = p=
-I/q>'
= -l/((p
x
+ i - x ) .
The stability of boundary layer equations at x = 1 for n and p depend on the sign of cp'(l). Since 0 holds, the layer equation for n is stable while that for p is unstable. This means that p cannot have a boundary layer at x = 1. The requirement that p takes the prescribed boundary value at x = 1 gives the approximate voltage-current characteristic / = - ( ^
4
i ) i ( - i + y i + 4<5 )~
-
Upi
After punch through, the characteristic can be extended linearly. The order of magnitude o f the currents is larger than before by the factor A ' . I n terms of unsealed quantities the above equation reads 2
2
The punch t h r o u g h voltage is given by U = —qL C/s a p p r o x i m a t i o n for the characteristic is valid for U < U . pt
s
and the above
pt
4.3 Bipolar Transistor The bipolar transistor is a device whose performance is based on the interaction o f t w o P - N junctions. Thus, there are t w o possibilities: A P N P and an N P N - c o n f i g u r a t i o n . We restrict o u r discussion to PNP-transistors because the resulting theory carries over to the NPN-case w i t h the obvious changes. Each of the three differently doped regions has an O h m i c contact. This means that an appropriate model has to be at least two-dimensional. Thus,
F i g . 4.3.1 T w o - d i m e n s i o n a l cross section o f a b i p o l a r t r a n s i s t o r
k
the device is represented by a d o m a i n Q £ U w i t h k = 2 or 3. The middle (in our case n-) region is usually called base region (Q ) whereas the t w o outer (p-) regions are the emitter and resp. collector regions (Q and Q ). The contacts corresponding to these regions are denoted by Y £ dQ n dQ, q = B, E, C, the emitter j u n c t i o n by Y and the collector j u n c t i o n by r (see F i g . 4.3.1). F o r a three terminal device the possible steady states constitute a t w o parameter manifold (see Section 4.1). A n appropriate choice of parameters which depends on the circuit configuration permits a convenient interpretation of the results. Here we concentrate o n the so called common-emitter configuration where usually the current t h r o u g h the base contact and the collector-emitter voltage are prescribed. The relevant o u t p u t quantity i n this situation is the collector current. I n the common-emitter configuration the transistor acts as an amplifier. F o r significant values of the collector-emitter voltage the common-emitter current gain B
E
c
q
EB
SJc '
q
B C
dI
B
is large. Here I is the collector current and I the base current leaving the device. c
Current
Gain Close
B
to Thermal
Equilibrium
We consider the drift-diffusion equations (4.1.1) subject to the boundary conditions
200
4 Devices
(n-
p - C)|
(V-V )\ bi
(V-
= 0,
Tt
V )\ = bi
4
= 0,
F j ! u r i > u r c
n p l r r „ u r = £U
(V-
c
V )\ = bi
~U ,
TB
BE
-U
Tc
CE
where U and U denote the base-emitter and collector-emitter voltages, respectively. The simplified equations (4.1.10) i m p l y the following representations for the S l o t b o o m variables: BE
CE
L
(e '»* -
emitter
u
V
l)pj + 1
1
v
base
v E
e"
(1 - e ' )(p
2
v E
v
(e '
collector
- e ")(p
+
A
+ (e
UcE
v E
- e ' )q>
3
+ e
U
E
*
-Uc,
VcE
e
e
where the functions (p
4
(J„ + J.) • v ds
u
= (e »* - 1) _ ( v„-u e
a
- ^ y grad _
P P
grad
cp
2
grad q> -v ds 3
for the emitter current and
(P =
3
«pi = o
94 = 0 lp, =
1
lp = 1 ip =0 2
3
92=0 (p =1 3
= 1
F i g . 4.3.2 B o u n d a r y c o n d i t i o n s t o r the
4.3 B i p o l a r T r a n s i s t o r
201
V
(Jn + Jp)-
ds
JC
1)
v
-(e >
grad q> • v ds 2
1)
Pn grad
C
4
R
for the collector current. I n the above equations C = C | and similar definitions for C and C hold, v denotes the u n i t n o r m a l vector along the junctions p o i n t i n g i n the direction from the emitter to the collector. The base current is given as the difference I = I — I . I n the c o m m o n emitter configuration we are interested i n prescribing U and I instead of U . Fortunately the above formula for I turns out to be a one-to-one relation between I and U . The o u t p u t characteristic is obtained by c o m p u t i n g U i n terms of I and substitution i n the equation for I : E
B
n £
c
B
E
c
CE
B
BE
B
B
BE
BE
B
c
a
+ a exp( - U )
3
4
a +a l
3
C £
+ (flj - a e x p ( - l /
2
(4.3.1)
a exp(l/ )
CE
2
C £
))-
a-, + a. + a
exp(-U )
4
CE
and U
BE
I
= In
B
+ a
+ a
3
4
(4.3.2)
+ a exp( — U A
c
w i t h the parameters grad cp - v ds,
a, =
2
Pn
grad
a-, =
3
\C \
grad <j» ) • v ds, 4
r
a-, =
grad
\CA
Pn grad (/> • v ds \C \
grad cp -v ds. 2
PP.
4
r
grad
r„ C
B
Since the dependence of I o n 1 is linear, the common-emitter current gain only depends on the collector-emitter voltage, i.e. / i = p{U ). F o r collectoremitter voltages w h i c h are significantly larger than the thermal voltage (which is our reference q u a n t i t y for voltages), it can be approximated by c
B
CE
)8(oo) = aja . 3
(4.3.3)
A n application of the m a x i m u m principle implies that 0 <
4 Devices
202
property is sufficient for determining the signs o f the n o r m a l derivatives along the contacts and PN-junctions. I f this knowledge is taken into account, it turns out that all the parameters a, are sums o f positive quantities. Thus, j8(co) is large as long as b o t h terms which sum up to a are small compared to a . F o r the first term this certainly holds i f the d o p i n g i n the emitter region is large compared to that i n the base region. The second requirement 3
1
grad (p 'V ds «
(4.3.4)
grad cp -v ds
2
2
essentially refers to the flow of m i n o r i t y carriers i n the base region o f the device. I n the simplified case of a constant d o p i n g profile i n Q , the above c o n d i t i o n only depends on the geometry of the base region. The function cp describes a situation where a certain current carried by m i n o r i t y carriers (in our case holes) is injected into the base region t h r o u g h the emitter j u n c t i o n . It is required that this current leaves the base region essentially t h r o u g h the collector j u n c t i o n w i t h o u t causing a significant m i n o r i t y carrier current t h r o u g h the base contact. The classical analysis o f bipolar transistors uses a one-dimensional model. This obviously raises the question o f h o w to model the influence of the base contact. Since the usual O h m i c contact boundary conditions cannot be applied i n this situation, simplifying assumptions have to be made. The major assumption turns out to be that the current t h r o u g h the base contact is entirely caused be majority carriers. The one-dimensional analysis can therefore only produce reliable results for devices which satisfy the requirement (4.3.4). B
2
The formula (4.3.2) shows that U tends to a positive l i m i t i n g value as U -^ oo. This shows that the emitter j u n c t i o n is forward biased w i t h a voltage which remains bounded as U -> oc whereas the collector j u n c t i o n is reverse biased w i t h a voltage g r o w i n g w i t h U . As i n the case o f the PN-diode we expect a depletion region of significant w i d t h a r o u n d r w h i c h has not been taken i n t o account by the above analysis. The influence of the widening of the depletion region on the o u t p u t characteristics o f bipolar transistors is usually referred to as the Early effect i n the literature (see e.g. [4.22]). Similarly to the case o f a reverse biased diode it can be shown that the collector current does not saturate for large collectoremitter voltages as opposed to the above result (4.3.1). A second l i m i t a t i o n for the above theory are high injection situations where the collector current becomes large enough for o u r scaling to be invalid. This leads to a decrease of the current gain which is c o m m o n l y referred to as the Webster effect (see [4.22]). BE
CE
CE
CE
B C
4.4 PIN-Diode The P I N - d i o d e is a device where an n- and a p-region are separated by an intrinsic region ((-region) w i t h a very l o w concentration of ionized i m -
203
4.4 P I N - D i o d e
purities. I t behaves essentially like a P - N diode but has some peculiar additional features. Its practical importance is due to a higher conductivity in the forward bias regime and a larger breakdown voltage compared to PN-diodes. The first of these properties w i l l discussed below, for the second we refer to the literature (see e.g. [4.12]).
Thermal
Equilibrium
I f the P I N - d i o d e is represented by the d o m a i n Q d o p i n g profile satisfies <0
in
Q_,
C(xH = 0
in
Q .
> 0
in
Q
\ k = 1, 2 or 3, the
0
+
where Q_, Q , fi are disjoint, simply connected subdomains o f Q w i t h 0
+
Q_ u Q u Q 0
+
= Q,
Q_ n Q
+
= { }.
We consider the idealized situation o f a vanishing d o p i n g concentration in the /-region Q . The d o p i n g profile is supposed to have j u m p s along the piand w'-junctions and the p- and n-regions have O h m i c contacts 0
r _ £ dQ n 3 Q „ ,
r
+
c f f i n
8Q
+
(see F i g . 4.4.1).
F i g . 4.4.1 Cross section o f the P I N - d i o d e
4 Devices
204
We consider the thermal e q u i l i b r i u m p r o b l e m 2
)} AV = 2S sinh V — C
in
Q,
dV (V-
K )| . w
r
u r
=0, 2
where SQjy = c Q \ ( r _ u T ) and V = areasinh(C/2<5 ) holds. As i n the case of the P - N diode approximations of the solution can be obtained by exploiting the smallness o f X and S. I n t r o d u c i n g the small parameter y = (In r 5 ) and the rescaled variable W = yV we expect W to be uniformly bounded i n X and S. The new p r o b l e m reads (compare to (4.2.7), (4.2.8), 4.2.9)) +
- 2
2
X AW = exp
bi
- 1
' W - l \
exp
f - W - Y
(4.4.1)
C,
7
7
8W 1 + 0(y),
W\ _ r
W\r
= 1 + 0(7),
w i t h the new parameter X = X/sJy w h i c h we also assume to be small. F o r P - N diodes we noted that the limits X -* 0 and y -*• 0 i n the p r o b l e m corresponding to (4.4.1) commute. This statement is w r o n g for the P I N - d i o d e . L e t t i n g X -> 0 i n (4.4.1) we obtain the reduced solution - 1 + 0(7) W =
0
yV =\ bi
1 + 0(7)
in
Q_ ,
in
(! ,
in
Q .
0
+
It has j u m p discontinuities at the j u n c t i o n s which are smoothed by appropriate layer terms. The layer equation i n the /-region reads W1\ f-W dj W = exp I - — - I — exp y
j
\
- 1 7
where Q is the fast variable. A n estimate for the thickness of the layer can be obtained from the linearization o f this equation at the stationary point W = 0. The result is the characteristic length ^fy exp(l/2y) i n terms of the variable c. I f we return to the original length scale the layer thickness is of the order of magnitude of Xyjy exp(l/27) = X/S which can be interpreted as the scaled intrinsic Debye length. O u r approach is justified i f the layer thickness is small which is equivalent to smallness of the intrinsic Debye length compared to the w i d t h o f the /-region or, equivalently, to the relation
4.4 P I N - D i o d e
X«
205
5
in terms of our original parameters. I f on the other hand S « X holds ( P I N - d i o d e w i t h short /-region) we a p p r o x i mate the solution of (4.4.1) by first letting y tend to zero. Arguments as i n the case of the P N - d i o d e show that the result is the double obstacle p r o b l e m - 1 ^ W < 1,
W\
= - 1,
T
W\
T
2
for
W < 1,
2
for
W > -1
X AW + C < 0 X AW + C $s 0
= 1, (4.4.2)
w h i c h can be rewritten as the f o l l o w i n g singularly perturbed variational inequality: F i n d We K w i t h 2
X
I grad W- grad(
C(x)((/) - VF)dx, <2
VcpeK, (4.4.3)
where l
K = {(peH (0):
cp\ _ = - 1 , ^ | r
r +
= 1, - 1 < ^ < 1}
is a closed, convex subset of the space H ' ( Q ) of square integrable functions w i t h a square integrable gradient. Since the d o p i n g profile vanishes in the /-region, the formal l i m i t of the v a r i a t i o n a l inequality as / -> 0 only consists of contributions from the p- and n-regions. The formal l i m i t i n g problem does not have a unique solution. The set of solutions is given by X = {
= 1,
a
We expect the l i m i t of W as /. -> 0 to be an element of xF o r i// e # the l i m i t i n g variational inequality C(x){cp - \jt)dx, VcpeK, (4.4.4) a holds. Replacing (p by \p in (4.4.3) and by I F i n (4.4.4) and adding up the resulting inequalities gives 0 >
grad I F g r a d f i / / -
W) dx ^ 0,
V^e/.
(4.4.5)
m
Since we require W to be an element of x the limit / . - > ( ) the l i m i t i n g solution can be determined uniquely from the variational inequality (4.4.5). A justification of this formal procedure can be found [ 4 . 6 ] . The solution of (4.4.5) satisfies W] _ = - 1 , n
H/|
n
= 1 ,
A1F = 0
in
Q
0
(4.4.6)
which differs strongly from the l i m i t i n g solution W for the case X « S above. F o r a short P I N - d i o d e the zero space charge a p p r o x i m a t i o n does not h o l d w i t h i n the /-region.
206
4 Devices
Behaviour
Close to Thermal
Equilibrium
We cannot apply the considerations from the beginning of this Chapter to the P I N - d i o d e because the ('-region requires special care. F o r simplicity we restrict o u r attention to a one-dimensional P I N - d i o d e w i t h l o n g f-region (/. « 8) and constant mobilities. The geometry is given by n = (0, 1), with
Q_=(0,x ),
Q = ( x , 1)
1
0 < x
< x
x
+
(4.4.7)
2
< 1.
2
The zero space charge a p p r o x i m a t i o n (A = 0 i n (4.1.1)) leads to the following relations between the carrier densities and the S l o t b o o m variables u and v in the /-region: 2
n = p = d y/uv
in
(x
x ).
l 5
2
F o r the Shockley-Read-Hall recombination-generation term this implies 2
R =
8 (uv -
— T (JUV
+
P
-
2
1) '-^=
1) + X (JuV
8L uv - 1) = — ^ +
n
x
1)
n +
in
*p
2
W i t h i n the /-region R is 0(5 ) w h i c h is i n contrast to n- or p-regions where R is 0(8 ). This implies that properly rescaled current densities are i n t r o duced by A
J„
=d J,
J = 8
ns
p
J
ps
A
2
instead of (4.1.7) where a factor o was used instead of 5 . This fact demonstrates the higher c o n d u c t i v i t y of the P I N - d i o d e mentioned i n the introduct i o n o f this Section. After substitution of the rescaled current densities i n (4.1.4) we let 8 -> 0 and o b t a i n v' = 0,
J = 0
i n the p-region (0, x j ,
u = 0,
J = 0
in the n-region ( x , 1 ) ,
n
p
2
J„ = p J~vfuu'',
J
n
p
'uv — 1
J'=
—J' = — D
=
-Upjujvv',
i n the /-region ( x , , x ) . 2
T
+
T
W i t h U denoting the applied voltage we have the boundary conditions M
(0) = i?(0) = 1,
u(l) = e , u
»(1) =
which imply v =
1
in
(0, x ) x
and u
u = e
W i t h vv : = yjuv
in
u
e~ ,
(x ,1). 2
a straightforward c o m p u t a t i o n gives
207
4.4 P I N - D i o d e
in
J
W' = JJUn ~ p/Hp
(x
1 ;
x ) 2
and, thus, w —1 w
in
=
(x,,x )
T„
with
2
+ t
!//*„ +
r
\/n„ (4.4.8)
where L denotes the ambipolar diffusion length (see [4.22]). The solution o f (4.4.8) is u
s i n h ( x — x)/L 2
w = 1 + ( w ( x , ) - 1)
a
sinh(x -
x,)/L
2
sinh(x — x , ) / L
+ ( w ( x ) - 1) 2
sinh(x —
a
a
(4.4.9)
x )/L
2
l
a
and the values o f the Slotboom-variables at the j u n c t i o n s are 2
2
u ( x j = w(x,) ,
D(X ) = w(x ) c 2
2
_ c /
.
A system o f algebraic equations for w ( x ) and w ( x ) is obtained by n o t i n g that t
_
i
J pw n
u
2
v
n
holds. Integrating these equations gives 2 In w ( x j - U =
*
x
x
HnW . .x, T„ +
2
X 2
1/ =
2 In w ( x ,
w- 1
U
*2
- P
V V
.
x
ds dx,
X
p
w- 1 Tn
ds dx.
+ ?p
A l t h o u g h the integrations on the right hand sides can be carried out explicitely the above system does not allow for an explicit solution for vv(x,) and w ( x ) . However, it is amenable to an asymptotic analysis as U ->• ± x which shows that 2
lim
w(xj) =
lim w(x
0
2
t/->-oo 1 2
holds and that w ( x ) and w ( x ) are 0{e ' ) density is given by x
2
1
w
J„(x
2
T„
+
?
as U -> oo. Since the total current
dx
R
we arrive at the conclusion that the voltage-current characteristic of the P I N - d i o d e has the same qualitative behaviour as that of the P-N-diode. The reverse bias saturation current can be computed by substituting w ( x , ) = w ( x ) = 0 i n (4.4.9) and evaluating the above integral. I n the forward bias 2
208
4 Devices
situation we have the result J =
u 2
0(e ' ).
Remark: Compare this result to the P-N-diode where the current grows like e. u
4.5 Thyristor Thyristors are devices w i t h four differently doped regions. Their practical importance is due to the existence o f m u l t i p l e steady state solutions under certain biasing conditions. I n particular it is possible to switch between an O F F state w i t h very l o w current and a conducting O N state. I n this Section we discuss the static voltage-current characteristic of a t w o terminal device, the so called Shockley-diode (Fig. 4.5.1a). The S-shaped forward bias characteristic is depicted qualitatively i n Figure 4.5.1b. I t consists o f three parts which are separated by the critical points corresponding to the holding voltage U and the break over voltage U . The currents on the lowest (forward blocking) branch are small leakage currents. The points o n the middle branch between U and U correspond to solutions which are dynamically unstable in the voltage driven case. Solutions on the upper (conduction) branch are characterized by significant current flow. Note, that for all applied voltages between U and U t w o dynamically stable steady state solutions exist. h
br
br
h
h
br
The qualitative behaviour o f the characteristic has t r a d i t i o n a l l y been analyzed by replacing the thyristor by t w o bipolar transistors where the base region of one transistor is identified w i t h the collector region of the other, and vice versa. O n l y quite recently Rubinstein [4.14] and Steinriick [4.20], [4.21] explained the structure o f the characteristic by applying p e r t u r b a t i o n methods and bifurcation theory to the drift-diffusion model.
r,
r
2
a) F i g . 4.5.1 (a) Cross section o f the Shockley d i o d e , (b) v o l t a g e - c u r r e n t characteristic
b)
209
4.5 T h y r i s t o r
i
I
Ih
Im
u
a)
2
X U
b)
F i g . 4.5.2 T h y r i s t o r characteristic for (a) small voltage scaling (b) large voltage scaling
The key to the analysis is the observation that different scalings have to be used for o b t a i n i n g different parts of the characteristic i n order to single out the d o m i n a n t effects. T w o observations are i m p o r t a n t i n this context. Firstly, the currents o n the b l o c k i n g branch are several orders smaller than on the middle branch. Second, the break over voltage usually assumes very large values compared to the h o l d i n g voltage which is of the order o f magnitude o f the b u i l t - i n voltage. This leads to the conclusion that a small-voltage-scaling m i g h t result i n an a p p r o x i m a t i o n of the characteristic which has t w o saturation currents corresponding to the blocking and the middle branch (Fig. 4.5.2 a). Since the currents o n these t w o branches have different orders o f magnitude, they are obtained separately by considering different scalings. Finally, the situation close to the break over voltage can be analyzed by considering the large-voltage-scaling which was already used for P-N-diodes under large reverse bias. This should provide a piece o f the characteristic which connects the t w o parts discussed above (Fig. 4.5.2 b). I n the following Paragraph the lower branch for small voltages is considered which can be analyzed for a multi-dimensional model. Here the same ideas which have been used i n the Sections on diodes and bipolar transistors for the situation close to thermal e q u i l i b r i u m are applied. Since a four layer device can be obtained by a small p e r t u r b a t i o n o f the d o p i n g profile o f a P I N - d i o d e , it is clear that n o t every four layer device has a characteristic like that depicted i n F i g . 4.5.1 b. I t w i l l be demonstrated that there are t w o possibilities. I n one case the behaviour expected of a thyristor is obtained: A n a p p r o x i m a t i o n for the lower branch of the characteristic saturates for large voltages. I n the second case the characteristic grows exponentially like that of a diode. O n l y i n the former situation the device has a chance to show the performance w h i c h we expect o f a thyristor. The distinction between the t w o cases is determined by the sign of a parameter which depends on the geometry, the d o p i n g profile and the recombination-generation rate. F o r discussing the situation along the middle and conducting branches we rescale the currents and restrict ourselves to a one-dimensional model. I f the
210
4 Devices
above mentioned parameter takes small values and has the appropriate sign the existence o f a saturation current corresponding to the m i d d l e branch can be shown if an a d d i t i o n a l c o n d i t i o n on the device parameters is satified. It w i l l also be demonstrated how the critical point at the h o l d i n g voltage can be computed in this case. The part of the characteristic connecting the lower and middle branches is essentially governed by t w o physical effects. One is the widening of the depleton region a r o u n d the middle P - N - j u n c t i o n o n the characteristic, the second is impact ionization. O n l y the first effect is taken i n t o account in the final Paragraph o f this Section where an a p p r o x i m a t i o n for the break over voltage is computed. This result can i n general only be expected to be qualitatively correct because there is strong evidence that the influence of impact ionization on the value of the break over voltage cannot be neglected (see [4.22]).
Characteristic
Close
to Thermal
Equilibrium
The aim o f this Paragraph is to derive an a p p r o x i m a t i o n of the lower branch of the characteristic of a thyristor close to thermal e q u i l i b r i u m . The equations (4.1.10) for the S l o t b o o m variables w i l l be used w i t h the simplifying assumption o f vanishing recombination-generation effects. I t does not cause mathematical problems to include Shockley-Read-Hall recombination, only the resulting calculations w o u l d be m u c h more involved (see Problem 4.9). We denote the differently doped regions o f the device by Q , , Q and assume Q, and fi to be p-regions and Q > d ^ 4 to be n-regions. The j u n c t i o n separating Q, and Q , is denoted by r . The regions Q, and Q have the O h m i c contacts T and T , respectively (see Fig. 4.5.1 a). By the approximate equations (4.1.10) the S l o t b o o m variable u corresponding to electrons is constant in Q and Q whereas v is constant in Q, and Q : 4
ar
3
2
;
+ 1
0
2
i>|
Ql
= l ,
4
v
u\
Qi
4
4
= e ,
3
w
v\
ih
= e- ,
v
u\
n4
= e
(4.5.1)
holds, where U denotes the applied voltage and the values of the constants V and W are as yet u n k n o w n . The S l o t b o o m variables corresponding to the m i n o r i t y carriers can be expressed in terms of the functions cp i = 1 , . . . , 4 defined on £2, as solutions o f problems o f the form (4.1.11) w i t h h
%lr,_, = ° .
I f V and W are considered as given u and v are completely determined by (4.5.1) and v
"In, = 1 +(e u\
a3
= e
v
+
l)(p
lt
v
v
(e -e )q> , 3
vC
1 +( w w + e= e
1)
(4.5.2) 4 •
The simplicity of these formulas is due to the assumption o f vanishing recombination-generation.
4.5 T h y r i s t o r
211
It remains to determine the values o f V and W. N o t e that the current t h r o u g h the device is equal to the current across any o f the surfaces r . By the zero recombination assumption the same holds for the electron and hole currents taken separately. The conditions that the electron current across Y is equal to that across r and that the hole current across Y is equal to that across T lead to the equations ;
x
3
2
4
v
w
(e v
v
v
-\)K =e~ (e
- e )
x
w
e (e~ -
u
\)K
u
i
,
W
= e (e-
2
K
-e~ )K , 4
where the K , are given by K
t = \
P»( )/IC| grad
i=l,...,4
P
w i t h v being the unit n o r m a l vector along T, p o i n t i n g o u t w a r d o f Q,. The m a x i m u m principle implies that (p takes its m a x i m a l value 1 along T, which in t u r n implies that K , is positive, K can be interpreted as a measure for the c o n d u c t i v i t y o f Q, for m i n o r i t y carriers. The t o t a l current t h r o u g h the device is given by t
t
v
I = (e
u
-
w
1 ) , + (e K
-
1)K . 4
v
n
E l i m i n a t i o n o f e from (4.5.3) leads to a quadratic equation for e~ w h i c h has a unique positive solution. The asymptotic behaviour as U tends to infinity depends on the sign o f the parameter A := K K X
4
— KK 2
3
.
(4.5.4)
F o r negative values o f A, W tends to a positive l i m i t i n g value as U -> oo whereas V grows like U. Thus, the current grows as e i n this situation w h i c h is not what we expect from a thyristor. A possible reason for A < 0 is low d o p i n g i n the middle regions compared to the outer regions w h i c h corresponds to a device w h i c h is close to a P I N - d i o d e . I f A is positive W grows as U and V tends to a l i m i t i n g value as U -> oo. The current saturates i n this case and we are led to the conclusion that A > 0 is a necessary c o n d i t i o n for a four layer device having a t h y r i s t o r characteristic. O u r results also show that the potential drops across the (forward biased) outer j u n c t i o n s Y and T remain bounded whereas that across the (reversed biased) j u n c t o n T grows as U. v
x
3
2
The reverse bias characteristic is analyzed by considering the asymptotic behaviour o f the s o l u t i o n as U -* — x . I t turns out that the current saturates at a negative value. A g a i n the voltage d r o p is concentrated to one P - N j u n c t i o n . I f the parameter B := K K X
2
—
K K^ 3
is positive V and W tend to negative l i m i t i n g values as C -» — x . This implies that the voltage d r o p across the j u n c t i o n T grows w i t h U. F o r negative values of B b o t h V and W tend to — x linearly i n U. I n this case the biggest part o f the voltage d r o p takes place at Y . 3
x
212
4 Devices
Forward
Conduction
I n this Paragraph an a p p r o x i m a t i o n o f the voltage current characteristic o f a thyristor i n a neighbourhood o f the h o l d i n g current w i l l be derived. A one-dimensional model w i t h vanishing recombination-generation rate is considered for simplicity, but recombination rates of the Shockley-ReadH a l l as well as Auger type could be easily included (see [4.20]). O u r approach is based o n an observation concerning the current controlled problem. F r o m F i g . 4.5.2 we conclude that this p r o b l e m is quite i l l conditioned. However, the analysis below w i l l show that only the determination of the potential is critical, whereas the carrier densities depend on the current in a smooth way. I t is possible to take advantage o f this fact by decoupling the current controlled p r o b l e m and using T a y l o r expansions i n terms o f the current for a p p r o x i m a t i n g the carrier densities. I n a second step the potential w i l l be computed leading to an a p p r o x i m a t i o n o f the voltage-current characteristic i n the form U = U(I). U n d e r the assumption that the parameter A, which has been defined i n the preceding Paragraph, is positive and an additional c o n d i t i o n o n the device is satisfied, U(I) has the expected behaviour. I t is defined for / > I , which is the saturation current o f the middle branch of the characteristic, and takes its m i n i m u m at I = I which is the h o l d i n g current. m
h
I n order to keep the calculations as simple as possible, the d o p i n g profile is assumed to be piecewise constant and the mobilities are assumed to be constant. The d o m a i n Q o f the preceding Paragraph corresponds to the interval (0, 1), and the surfaces T , . . . , T to the points 0
0 = x
< ••• < x
0
4
4
= 1.
(4.5.5)
The d o p i n g profile is given by -C, C
2
in
(x ,x,),
in
(x,,x ),
in
(x ,x ),
in
(x ,x ).
0
2
C(x) = <|
(4.5.6) -C
3
C
4
2
3
3
4
The zero space charge a p p r o x i m a t i o n reads 0 = n — p — C, J„ = p„(n' - nV), J= P
-p (p'
+
p
(4.5.7) pV)
w i t h the j u m p conditions [Tlx,
= P
n
" L , = - D n p] ., x
i=l,2,3
which can be interpreted as the c o n d i t i o n that the quasi F e r m i levels do not have j u m p s across the junctions. Replacing <5 i n the O h m i c contact b o u n d 4
4.5 T h y r i s t o r
213
ary conditions by zero the m i n o r i t y carrier densities vanish i n thermal e q u i l i b r i u m . This motivates the i n t r o d u c t i o n o f a new variable vv by setting np = /vv where / = J„ + J is the t o t a l current. This leads to the representation p
2
n = \(C
+ JC
2
+ 4/w),
p = i ( - C + JC
+ 4/w)
for the carrier densities. The current densities are rescaled by replacing J„ and J by 1J„ and IJ , respectively. By e l i m i n a t i o n of the potential from (4.5.7) the p r o b l e m p
p
vv' = ^ - ( - C + V C
2
w(0) = w ( l ) = 0 ,
J + J
+ 4/w) - - ^ - ( C + V ^ n
p
2
+
4 / w
4
)>
5
8
< - - )
= l
for vv is obtained. N o t e that the t r i v i a l solution corresponding to / = 0 has been eliminated by our choice o f scaling. The p r o b l e m for the determination of the potential reads J
r
_ Jj^±Jdh_ JC + 4/w
=
I
f
o
r
x
/
i=l,2,3,
x
2
V(0) = V (0),
2
m
hi
x
= [ l n ( C + JC
i
+ 4/w)],,,
i = 1,2,3. (4.5.9)
The fact that vv and V can be computed consecutively from (4.5.8), (4.5.9) is the decoupling mentioned above. The solution o f (4.5.8) can be a p p r o x i mated by T a y l o r expansion i n terms o f / . F o r the leading terms w , J„ , J the differential equation i n (4.5.8) reduces to 0
— J C/Li n0
W
n
for
C < 0,
for
O O .
0
p0
N
JpoC/fip
The solution can be w r i t t e n i n terms o f the quantities K , defined in the preceding Paragraph w h i c h are given by Un
. _
C i ( X i - *o) ' k
C (x 2
Pn
=
3
=
4
C (x -x )' 3
3
ih
2
-
2
x )' ±
Pp Q(x -x )' 4
3
in the one-dimensional case. w is piecewise linear and takes the values 0
w (xi) = 0
K
3
( K
2
+
K )/2, 4
w (x )= - / l / I , 0
(4.5.10)
2
* v ( x ) = K (K + K )fL at the junctions. A is defined i n the preceding Paragraph and 0
3
2
X
3
214
4 Devices
holds. The leading coefficients for the current densities are
•AO i
=
K
1
K
3 (
K
2
^4)/^'
+
-^pO
=
^2^4(^1 +
(4.5.11)
3)ft^-
K
W i t h the assumption that A is positive (which has been found to be necessary for a thyristor) w (x ) is negative w h i c h obviously is unacceptable for an a p p r o x i m a t i o n of the product of the carrier densities. F o r w t a k i n g negative values the p r o b l e m (4.5.9) for the potential does not have a solution. Let us reconsider the result we are a i m i n g at. The p r o b l e m is expected not to have a solution for values of the current between the saturation currents of the middle and the b l o c k i n g branch. Since the saturation current of the blocking branch is 0(8*) and the a p p r o x i m a t i o n 8 = 0 was used i n this Paragraph, the b l o c k i n g branch reduces to / = 0. Thus, i t is to be expected that there is no solution for currents below a certain threshold. This is i n agreement w i t h our results so far. 0
2
4
A further analysis requires the c o m p u t a t i o n of higher order terms i n the T a y l o r expansion of w. The first order t e r m w solves a linear p r o b l e m just as w , but w i t h different inhomogeneities. I n the following only its value w , ( x ) at the middle j u n c t i o n w i l l be used. O u r second assumption o n the device (besides A > 0) is that this value is positive. The a p p r o x i m a t i o n t
0
2
vv(x )~vv (x ) + /vv (x ) 2
0
2
1
(4.5.12)
2
implies that the p r o b l e m has a solution i f I > L = -w (x )/w (x ) 0
2
1
2
holds. This argument is justified as l o n g as I is small enough for the T a y l o r p o l y n o m i a l (4.5.12) to be a g o o d a p p r o x i m a t i o n at / = I . This c o n d i t i o n is satisfied i f the parameter A is sufficiently small, i.e. i f the device is close to the critical case A = 0 where the shape of the characteristic changes qualitatively. W i t h the boundary c o n d i t i o n m
m
K ( l ) = V {\) - U bi
for the potential at the right contact the voltage current characteristic is obtained by integrating (4.5.9): U = U
bi
JjHn
+ I
2
JC J 0
+
J
plPp
dx -
+ 4/w
X IMC
+ y c
2
+ 4/w)],,
i= l
where U = V (Y) — V (0) denotes the b u i l t - i n voltage. U s i n g the T a y l o r expansions computed above this equation can be simplified considerably. The integral w i l l be replaced by bi
bi
bi
and the arguments of the logarithms by
4.5 T h y r i s t o r
215
2C
for
C > 0,
2Iw /\C\
for
C < 0
and
i = 1,3,
2/(w + /wJ/ICI
for
C<0
and
i = 2.
0
0
The resulting a p p r o x i m a t i o n for the voltage-current characteristic reads U = a/ + In
+ l/ w {x ) 0
w
+ In
—— QC
+ 7w (x )
2
1
2
. 4
This formula shows that 7 can be interpreted as the saturation current on the middle branch. I n the l i m i t I -»• I the voltage tends to infinity. The expression m
m
2
I„ = ( - a w ( x ) + V « w ( x ) 0
2
0
2
2
-
4aw (x )vv (x ))/2aw (x ) 0
2
1
2
1
2
for the h o l d i n g current can be found by solving the equation >
= o.
The above procedure can be justified a posteriori i f the computed approxim a t i o n for the h o l d i n g current is small. A mathematical justification by means o f bifurcation theory can be found i n [4.20]. F o r larger values of the current the full p r o b l e m (4.5.8), (4.5.9) has to be solved. I n [4.14] it was shown that the differential equations i n (4.5.8), (4.5.9) can be integrated explicitely reducing the p r o b l e m to a set o f algebraic equations. F o r a more general model including recombination-generation effects the p r o b l e m was solved numerically i n [4.20] resulting i n very satisfactory approximations of the characteristics.
Break
Over
Voltage
The a i m o f this Paragraph is to show the existence and to compute an a p p r o x i m a t i o n o f a break over voltage. A branch o f the characteristic which connects the t w o branches discussed i n the preceding Paragraphs w i l l be constructed. As mentioned above the break over voltage is large compared to the thermal voltage (the reference voltage used u n t i l now). Thus, the potential has to be rescaled. Since the m a i n part of the potential d r o p occurs at the middle j u n c t i o n , a depletion region of significant w i d t h around that j u n c t i o n is to be expected. As i n the case of the reverse biased diode the edges of this depletion region can be obtained by solving a free boundary problem, arising as a singular l i m i t of the rescaled drift diffusion equations. We consider a one-dimensional device for which the drift-diffusion model w i t h a rescaled potential q> = / (V — F (0)) reads 2
q>" 2
A J„ 2
* Jp
= n — p — C, 2
= u (/. ri n
2
= -P (/ P' P
-
tup'), + P
fcl
4 Devices
216
where (p satisfies the boundary conditions
q>(l)=-q> :=X (U -U). 2
l
u
F o r the discussion of the l i m i t X -» 0 we refer to the Paragraph o n reverse biased diodes. I n this l i m i t the device splits i n t o depletion and zero space charge regions. The analysis above suggests that a depletion region occurs a r o u n d x . A more rigorous justification of this p r o p o s i t i o n by the method of matched asymptotic expansions can be found i n [4.21]. A straightforward c o m p u t a t i o n leads to a l i m i t i n g potential which varies quadratically w i t h i n the depletion region (x,, x ) and is piecewise constant outside of this interval. The edges o f the depletion region are given by 2
r
2C cp 3
z
'
vc (c 2
2
_
l
/
2C (p 2
X — X 2 "T v c ( c
+ c )'
r
3
3
2
1
+ c ) 3
This result can be justified by the m e t h o d o f matched asymptotic expansions as l o n g as x, > x , and x < x holds, i.e. for voltages small enough such that punch t h r o u g h does not occur. F o r the c o m p u t a t i o n o f the carrier densities we introduce the variable w as above. Since w vanishes w i t h i n the depletion region it solves the problem r
w
J. = ^ L ( - C 2n„
in
3
+ yJC
+ 4/w)
2
-
v
- M C 2n
+ y C
2
+
4/w)
p
(0, l ) \ ( x „ x ) ,
(4.5.13)
r
w(0) = w ( l ) = 0,
w(x ) = w(x ), (
J„ + J = 1
r
p
subject to w(x,) = 0.
(4.5.14)
The reason for separating the c o n d i t i o n (4.5.14) from the rest o f the problem is that (4.5.13) is a modified version o f (4.5.8). I t reduces to (4.5.8) for (p = 0. F o r given current / and voltage
x
0
w {xi) 0
=
2
-
A/2,
where A and i n £ are defined like A and Z w i t h the parameters K and K replaced by 2
ff
_
Ih
C^Xt-Xi)'
_
3
P-n C (x -x )' 3
3
r
Since i n the preceding Paragraph the first order term evaluated at x was assumed to be positive, w ^ x , ) is positive i f cp is not too large. As an a p p r o x i m a t i o n o f the characteristic, (4.5.14) yields 2
x
4.5 T h y r i s t o r
217
/ = -WofoJ/Wifa).
(4.5.15)
A
By setting 5 = 0 the b l o c k i n g branch o f the characteristic was reduced to 1 = 0. This solution was eliminated from (4.5.13) by the specific scaling used. Thus, (4.5.15) can only be an a p p r o x i m a t i o n o f the middle branch. F o r <j0j = 0 the current takes the value I o f the saturation current of the middle branch as expected. The break-over voltage is determined by the intersect i o n p o i n t o f the middle branch (4.5.15) and the lower branch / = 0 o f the characteristic. Consequently, the break over voltage
lb
w h i c h can be rewritten as (4.5.16) where z = C (x — x ) = C (x over voltage is given by 2
2
;
zl
(Pn = (C
2
+
3
— x ) holds. A n expression for the break
r
2
C )/2C C . 3
2
3
The variable z is p r o p o r t i o n a l to the w i d t h o f the depletion region w h i c h implies that punch t h r o u g h does n o t occur as l o n g as
holds. The smaller solution o f (4.5.16) is given by
2 \K
K
2
3
V
\
K
2
3/
K
KK t
A
Im
F i g . 4.5.3 A p p r o x i m a t e v o l t a g e - c u r r e n t characteristic
218
4 Devices
w h i c h can be estimated from above: Pp
Pn
mm
Pp Pn
By c o m b i n i n g our results an a p p r o x i m a t i o n o f the characteristic is given by the segments o f (4.5.15) and 7 = 0 w h i c h lie between
lb
4.6 MIS Diode The M I S ( M e t a l i n s u l a t o r Semiconductor) diode has applications as a capacitor w i t h voltage dependent capacitance. I n the context o f this b o o k an understanding o f the performance o f this device is an i m p o r t a n t preparatory step for the analysis o f the M O S F E T (see the following Section). The M I S diode consists o f a uniformly doped piece o f semiconductor coated w i t h a t h i n layer o f insulating material which carries a metal contact called the gate (see F i g . 4.6.1). The semiconductor w h i c h has an O h m i c contact, is assumed to be o f p-type (The analysis o f this Section also applies to M I S diodes w i t h rc-type semiconductor after the obvious changes). I n order to simplify the presentation we assume that the insulator is free o f charges and that no trapped charges at the interface between the semiconductor and the insulator occur (See [4.22] for a justification o f these assumptions). Because o f the simple device geometry a one-dimensional model is certainly sufficient for the analysis o f the relevant effects. Since no current flow t h r o u g h the insulator is possible the M I S diode is always i n thermal equilibrium.
N or P
I F i g . 4.6.1 Cross section o f a M I S d i o d e
219
4.6 M I S D i o d e
The uniform concentration o f acceptors i n the semiconductor is introduced as a reference value for the scaling of the d o p i n g profile and the thickness of the semiconductor as the reference length. Thus, the Poisson equation for the scaled potential V 2
X V"
2
v
2
= Se
v
— d e~
+ 1
holds i n the interval (0, 1) representing the semiconductor part of the device. A t the b o t t o m o f the device we impose the O h m i c contact boundary condition
The insulator is located i n the interval ( — d, 0) where d denotes the scaled thickness o f the insulating layer. Since no charges are present i n the insulator the Laplace equation V" = 0 holds i n ( — d, 0). A t the interface between semiconductor and insulator c o n t i n u i t y o f the potential and the electric displacement is required: K(0 + ) = V(0-),
e V'(0 + ) = £ s
i n s
F(0-),
where £ and e denote the permittivities o f the semiconductor and the insulator, respectively. A t the gate contact the potential is prescribed by s
ins
V(-d)=V
+ U.
bi
I n the case o f an ideal M I S diode the contact voltage is given by U. A n ideal M I S diode is characterized by a vanishing metal-semiconductor w o r k function difference cp (see [4.22] for details). I n realistic cases the contact voltage is given by U + cp . The first step i n the analysis is the c o m p u t a t i o n o f the potential and the electric field i n the insulator: ms
ms
V(x) = V(0) + (V(0) - V bi
V'(x) = (V(0)-V -U)/d
U)x/d, in
bi
(-4,0).
This result is substituted i n the interface conditions at x = 0: —
V'(0)=V(0)-V -U, u
^ins
which reduces the p r o b l e m to a boundary value p r o b l e m on the interval (0, 1) w i t h a mixed boundary c o n d i t i o n at x = 0. By i n t r o d u c i n g the change of variables W = yV,
q = 2
—where y = (In ( 5 " )
- 1
xJy/X
is a small parameter—the p r o b l e m is transformed to
220
4 Devices
a^=exp(^^j-exp( ,xd W(0) e
= W(0) - yV
bi
^j +
^
- U,
WUfi/k)
l
in
(0,jy~/k),
= yV .
(4.6.1)
bi
The new parameters i n (4.6.1) are defined by a = —^r-,
U =
yU.
I n Section 4.2 i t was shown that the scaled w i d t h o f the depletion region of a P - N - j u n c t i o n i n thermal e q u i l i b r i u m is 0(k/y/y). I f the thickness o f the insulator is o f that order o f magnitude and the semiconductor is m u c h thicker, kjyfy is a small parameter b u t a is 0(1). F o r significant d o p i n g levels, 7 is also a small parameter. The scaling o f the potential has been chosen such that the scaled b u i l t - i n potential is 0(1): yV =
- 1 +TST.
bi
I t w i l l be assumed that U is also 0 ( 1 ) w h i c h means that the applied voltage is of the order of magnitude o f the b u i l t - i n potential. The solution o f (4.6.1) w i l l be approximated by letting tend to zero. The a p p r o x i m a t i n g p r o b l e m is posed o n the infinite interval (0, oo). After m u l t i p l i c a t i o n by d W the differential equation can be integrated, w h i c h gives
kjyfy
4
(w = 7 exp ( —
(d Wf/2 {
— 1\ / —W — 1\ — J + 7 exp I J + W + k,
(4.6.2)
where the constant o f integration k is computed by evaluating (4.6.2) at £ = oo (where, obviously, d^W = 0 holds): k=
1-
7
+ TST.
Since for U = 0 the solution is constant and equal to the value yV prescribed at infinity, we expect the s o l u t i o n to be decreasing (increasing) for U > 0 (U < 0). This observation determines the sign when t a k i n g the square r o o t o f (4.6.2): bi
r
d^W/y/2 = -sign((7)
I (W - 1 \ (-Wy exp I j + y exp I
1\ I + W+ k . (4.6.3)
E v a l u a t i o n of (4.6.3) at £ = . 0and substitution for d% W(0) (by using the i n i t i a l condition) leads t o the equation (W(0) - yV U)/c^/2 u
=
-**U>J,
exp
(m=l)
+
y exp
(=Z&=±) +
^ (4.6.4)
for 1F(0). I t is easy to show that this equation is uniquely solvable (see [4.9]).
221
4.6 M I S D i o d e
This reduces the p r o b l e m to an initial value p r o b l e m for the first order equation (4.6.3). Because of the similarity of the differential equations (4.6.3) and (4.2.14) parts of the analysis below are very similar to that of the Paragraph "Strongly asymmetric j u n c t i o n s " of Section 4.2. I n particular, we refer the reader to the detailed explanation of the methods of singular perturbation theory there. The aim of our analysis is to obtain an a p p r o x i m a t i o n for the charge (n — p — C) dx i n the semiconductor, where the integrand (the space charge) is given by the right hand side of (4.6.1). The capacitance of the M I S diode is defined by C = dQ/dU. The case (7 = 0, where the potential W is constant and equal to the b u i l t - i n potential ( 1), is c o m m o n l y referred to as flat band condition. The charge Q vanishes i n this situation. The density of the m a j o r i t y carriers, i.e. the holes, is approximately equal to the d o p i n g concentration and the electron density is m u c h smaller. F o r negative U the potential is smaller than i n the flat band case in the vicinity of the interface and, accordingly, an accumulation of holes occurs. The situation where U is positive but small enough for W(0) to be between — 1 and 0, is called depletion because b o t h carrier densities are so small that the space charge is essentially equal to the density of the fixed charges. This is also true for IF(0) between 0 and 1 but i n this case the additional phenomenon occurs that the m i n o r i t y carrier density at the interface is larger than the m a j o r i t y carrier density. Therefore this situation is called weak inversion. Finally, the case W(0) > 1 is referred to as strong inversion.
Accumulation The equation (4.6.4) cannot be solved explicitely, but i t is amenable to an asymptotic analysis as y - > 0 . F o r the case of accumulation we make the ansatz W
=
—
1 — y In y
- 1
+
yy.
Substitution i n (4.6.4) and letting y -» 0 gives an equation for y(0) w i t h the solution: 2
2
y(0) = l n ( 2 a / ( 7 ) . Similarly an initial layer equation for y i n terms of the fast variable x = £/y is determined from (4.6.3):
The solution y = 2 l n ( T / / 2 — Oy/2/U) of the initial value p r o b l e m can be used to compute an a p p r o x i m a t i o n for the charge by n o t i n g that i n the initial layer the space charge is dominated by the hole density N
222
4 Devices
-2
w h i c h implies that i n the case o f accumulation the (unsealed) capacitance can be approximated by the insulator capacitance
The a p p r o x i m a t i o n o f the potential computed above does not converge to the e q u i l i b r i u m value as the fast variable T tends to infinity. This disturbing situation can be eliminated by i n t r o d u c i n g a transition layer for values of W close to — 1 (see Problem 4.11). The transition layer solution tends to — 1 as the corresponding layer variable tends to infinity and it can be matched to the inner layer solution. The c o n t r i b u t i o n o f the transition layer to the charge is small compared to the computed a p p r o x i m a t i o n , and therfore neglected.
Depletion—Weak
Inversion
W h e n W-^O) e (— 1, 1) holds, b o t h exponential terms (4.6.4) can be neglected. The resulting equation has the solution 2
W(0) = - 1 + (yfot
+ 2U -
2
ot) /2
w h i c h is i n ( — 1, 1) i f Ue{0,
2 + 2a).
The upper b o u n d marks the l i m i t between weak and strong inversion. I n terms of unsealed variables i t is given by U = 2V
bi
+ 2 2
where C7 = e qC/C is a reference voltage and C denotes the concentrat i o n of acceptors used for scaling the d o p i n g profile. The b u i l t - i n potential is approximated by V = U /y = U ln(C/n,) i n the above equation. Returning to scaled variables, the onset o f weak inversion is determined by requiring W(0) = 0, w h i c h leads to the voltage ref
s
ns
bi
(7=1
+
T
T
oty/2.
Since W e a n be expected to be between — 1 and 1, the p r o b l e m (4.6.1) reduces to a double obstacle p r o b l e m i n the l i m i t y -*• 0 (compare to the thermal
4.6 M I S D i o d e
223
e q u i l i b r i u m p r o b l e m (4.2.7), (4.2.8), (4.2.9) for the P - N diode). The l i m i t i n g solution is given by | - l + ( f - ^ )
2
/ 2
for
1-1
for
£>£ , d
where the w i d t h o f the depletion region is 2
Zd = Jot
+ 2U - a .
F o r later reference we note that the depletion w i d t h at the onset o f strong inversion (for U = 2 + 2a) is 2. The c o m p u t a t i o n o f the charge reduces to the m u l t i p l i c a t i o n o f the acceptor concentration by the depletion w i d t h . I n terms o f unsealed variables i t is given by
e = c LuVi + 2i7[/ -i). ins
ref
The capacitance depends on the voltage i n this regime and is given by C =
C JJl+2U/U , i
tet
w h i c h reduces to the insulator capacitance for (7 = 0.
Strong
Inversion
F o r analyzing the case o f strong inversion we make the ansatz 1
W
inv
= 1 + y i n y^
+ yz
for the potential i n an inversion layer. A n equation for z(0) is obtained by going to the l i m i t y -» 0 i n (4.6.4). I t has the solution z(0) = l n
( 1 7 - 2 - 2a)(<7 - 2 + 2a) — . 2
Obviously, we only consider voltages larger than the above obtained threshold 2 + 2a for strong inversion. W i t h the fast variable x = £/y, the layer equation
d z= x
z
- V 2 ( e + 2)
is obtained from (4.6.3). The solution o f the i n i t i a l value p r o b l e m for z is given by z = ln(2 sinh
( T + c))
with
1 , U-2 + 2a c = - In — —. 2 (7 — 2 — 2a
Since the solution decreases rapidly (as a function of the fast variable T ) , strong inversion only takes place w i t h i n an inversion layer of thickness 0(y). Outside this layer, there is a depletion region where the potential satisfies the reduced equation
4 Devices
224
The general solution rT
d e p l
= a + H + e/2
can be matched t o the inversion layer solution W terms of the slow variable £:
inv
W
inv
by expressing I4^ i n nv
= l - 2 f + o(l).
M a t c h i n g by equating coefficients implies a = l,b = —2 and, thus, W = depl
2
- 1 +(c-2) /2.
Similarly t o above this solution is valid u n t i l l T takes the value — 1 at £ = 2 which is the depletion w i d t h . F o r £ > 2, W is approximately equal t o — 1. N o t e that the depletion w i d t h is equal t o its m a x i m a l value obtained i n the case o f weak inversion and does not depend o n the applied voltage any more. F o r the c o m p u t a t i o n o f the charge, the contributions from the depletion and inversion layers have t o be added. W i t h i n the inversion layer the electron concentration d e p l
v
z
-^—\ = -e y ) y
=
2
y
sinh
- 2
( T + c)
dominates and integration gives the inversion layer c o n t r i b u t i o n Q total charge:
inv
2
1
Qi„v = 4 / y V y ( e < - D " = -^7=(U
o f the
-2-2a).
Xyjy A d d i n g the depletion layer charge we end up w i t h the unsealed t o t a l charge Q = CaU-2V ). As i n the case of accumulation, this is a linear relation between the voltage and the charge which means that the capacitance is independent o f the voltage and equal t o the insulator capacitance C . O u r results are summarized i n Fig. 4.6.2. Depending o n the value o f U different cases occur. W e have bi
i n s
r
accumulation
f°
depletion
for
0 < U < 1+ a /2,
weak inversion
for
1 + a.y/2 < U < 2 + 2 a ,
strong inversion
f°
r
U <0, x
2 + 2a < U.
We have seen that the capacitance o f the M I S - d i o d e depends o n the voltage. I n the accumulation and strong inversion regimes i t is approximately constant and equal t o the insulator capacitance. I n depletion and weak inver-
4.7 M O S F E T
225
Q
C
ns
a)
b)
Fig. 4.6.2 (a) C h a r g e a n d (b) capacitance vs. voltage
sion i t is a decreasing function of U and takes its m i n i m u m value close to the threshold voltage separating weak and strong inversion. The j u m p discontinuity i n the computed approximated for the capacitance indicates fast v a r i a t i o n close to U = 2 + 2a, w h i c h we have not analyzed i n detail.
4.7 MOSFET The M O S F E T ( M e t a l Oxide Semiconductor Field Effect Transistor) is one of the most i m p o r t a n t semiconductor devices. I n general, it is used as a switch. Its importance is due to the fact that no power is consumed by the switching (as opposed to the bipolar transistor). The M O S F E T is a unipolar device, i.e. charge transport is due to only one type of charge carrier. Since the m o b i l i t y o f the electrons is usually higher than that o f the holes, most M O S F E T s are so called n-channel devices (this term w i l l be explained below). A l t h o u g h the following discussion w i l l be restricted to this group of devices the results carry over to p-channel devices after obvious changes. L i k e the bipolar transistor, the M O S F E T is a three layer device w i t h t w o highly doped n-regions called source and drain and a p-region called bulk w i t h lower doping. Source and d r a i n always have O h m i c contacts whereas the bulk m i g h t be a floating region (no contact), e.g. i n the case of S O I (Silicon On /nsulator) technology. Between the source and d r a i n contacts the semiconductor material is coated w i t h a t h i n layer of oxide w i t h a metal contact, the so called gate (see F i g . 4.7.1). The analysis of the preceding Section applies to the situation close to the semiconductor oxide interface BC away from the endpoints B and C. A p p l i cation o f a sufficiently large voltage at the gate generates an inversion layer in the so-called channel, i.e. the part of the bulk region close to the interface. The fact that the electron density dominates the hole density i n this region has led to the term n-channel. This n-channel is able to carry a significant
226
4 Devices
1
bulk
F i g . 4.7.1 Cross section o f a simplified M O S F E T g e o m e t r y
current from source to d r a i n as soon as there is a potential difference between these t w o contacts. Thus, the source-drain current can be switched o n and off by a p p l y i n g different voltages to the gate. A n i m p o r t a n t parameter for the performance o f the M O S F E T is the channel length L (the length o f the segment BC). Long channel devices are characterized by the requirement that the Debye length i n the channel is significantly smaller than the channel length. This implies that the depletion layers corresponding to the source and d r a i n j u n c t i o n s are well separated, and p e r t u r b a t i o n arguments lead to locally one-dimensional problems. A n a l y t i c formulas for the static characteristics of l o n g channel M O S F E T s have been obtained by models o f different degrees of sophistication (see [4.22] for an overview). Recently, W a r d [4.25] analyzed l o n g channel M O S F E T s by using methods of asymptotic analysis. The following presentation is based on his w o r k ( [ 4 . 2 5 ] , [4.26]). W e p o i n t out that no c o m parable results for short channel devices are available because they are intrinsically d o m i n a t e d by two-dimensional behaviour. O u r mathematical model is a two-dimensional version of the scaled stationary drift-diffusion equations w r i t t e n i n terms of the quasi F e r m i potentials: 2
2
y
2
X A F = b e -^ J„=
2
p
v
-g„S e -«» 2
J
v
= -g d e^p
v
v
- 8 e v-
-
C,
grad cp ,
d i v J „ = 0,
grad cp ,
d i v J = 0.
n
p
(4.7.1)
p
These equations are posed on the rectangle AEFD (see F i g . 4.7.1) and a coordinate system w i t h the o r i g i n i n the p o i n t B has been introduced. The validity o f the two-dimensional model is restricted to devices w i t h sufficient
4.7 M O S F E T
227
channel width, i.e. the w i d t h of the active region o f the device i n the direction perpendicular to the x-y-plane. I n (4.7.1) the effects o f recombination and generation are neglected and the mobilities are assumed to be constant. A l t h o u g h these assumptions cannot be justified i n general the simplified model produces qualitatively correct voltage-current characteristics. I n the scaling, the channel length L (the length of the segment BC i n Fig. 4.7.1) has been chosen as the characteristic length. The d o p i n g concentration has been scaled by the d o p i n g in the bulk region which we assume to be constant. This is a rather stringent assumption which does not h o l d for many practical situations. However, as demonstrated i n [4.25] the principal ideas o f the analysis o f this Section can be carried over to the case o f a d o p i n g profile w h i c h varies in the x-direction (see Fig. 4.7.1). The analysis below w i l l show that the performance o f the M O S F E T can be explained by concentrating o n the p-region. This has motivated our choice of the scaling where the m a x i m a l d o p i n g concentration is not scaled to 1 as usual. The parameter X denotes the scaled Debye length. I n the following, smallness o f X w i l l be assumed w h i c h means, that we are dealing w i t h l o n g channel devices. The rectangle BCJI represents the oxide w h i c h we asume to be free of charges such that the Laplace equation AV=
0
holds there. The quasi F e r m i potentials satisfy D i r i c h l e t boundary conditions along the contact segments AB, CD, and EF. We have =
on
AB,
= %=
on
CD,
on
EF.
P
=
B
The voltages are referenced w i t h respect to the source, i.e. U is the d r a i n source voltage and U the bulk-source voltage. A l o n g the artificial b o u n d aries AE and DF and along the interface BC homogeneous N e u m a n n conditions for the quasi F e r m i potentials h o l d . So currents can leave or enter the rectangle under consideration AEFD only t h r o u g h the source, drain, and bulk contacts. The potential satisfies the usual O h m i c contact boundary conditions at source, bulk, and d r a i n as well as homogeneous N e u m a n n conditions along the artificial boundary segments AE, DF, BE and CJ. A l o n g the interface, c o n t i n u i t y of the potential and the vertical component o f the electric displacement are required: D
B
F(0-,
v ) = F ( 0 + ,>-),
e 8 V{0-,y) ox
x
= e d V(0 s
x
+ ,y),
where e and s are the permittivities o f the oxide and the semiconductor, respectively. A t the gate contact I J the boundary c o n d i t i o n ox
V{-d,
s
y) = VJO, y) + U
G
4 Devices
228
holds, where d denotes the oxide thickness and U is related to the gatesource voltage as explained i n the preceding Section. G
Derivation
of a Simplified
Model
The aim of this Paragraph is to reduce the model to a boundary value problem posed on the smaller rectangle BGHC by i n t r o d u c i n g several simplifying assumptions. A singular p e r t u r b a t i o n analysis w i t h X -> 0 combined w i t h the arguments from the beginning o f this Chapter leads to approximations o f the quasi F e r m i level corresponding to the m a j o r i t y carriers by a constant, i n each pand n-region. Thus, as a first simplification we set (p = 0
i n the source region,
(p =U
i n the d r a i n r e g i o n ,
n
n
D
(p = U p
(4.7.2)
i n the b u l k r e g i o n .
B
N e x t we consider the potential i n the oxide, and assume that the oxide thickness is small compared to the channel length, i.e. d « 1. By i n t r o d u c i n g the independent variable £ = x/d the insulator region is transformed to a square and the potential satisfies dV + ddV = 0. 2
2
2
I n the l i m i t d -> 0 the potential i n the insulator is the solution o f a onedimensional problem. The resulting a p p r o x i m a t i o n for the potential violates the N e u m a n n conditions at the artificial boundary segments. However, it is easy to see that O(d)-boundary layer correctors are sufficient as a remedy. Thus, the a p p r o x i m a t i o n obtained by solving the one-dimensional equation d V = 0 is uniformly valid i n the oxide region. As i n the case o f the M I S diode it leads to a mixed boundary c o n d i t i o n at the interface: 2
£
A
_ _tj
dxV=V
Vbi
at
G
BC
Finally, the p r o b l e m is reduced to the rectangle BGHC. Since the hole quasi F e r m i potential is assumed to be constant i n this region we only consider the Poisson equation and the electron c o n t i n u i t y equation. W i t h the rescaled variables W = yV,
0„ = y
Z = xl~X
H
(X =
Xj fy) s
we have 2
2
2
8 W + Xo
W = exp (
d (ft, exp (^~®"~ t
W
1
~ ^
- ) d
) - exp
2
t
H
y
(ft
~ ™ ~ *) + 1.
exp ( - ~ ^
~
1
d
(4.7.3)
4.7 M O S F E T
229
where U and, i n the sequel, U and U denote rescaled versions of the applied voltages. N o t e that the approximate b u i l t - i n potential U /y = U l n ( C / n ) is the scaling factor for voltages i n (4.7.3). By restricting our attention to the smaller rectangle BGHC we ignore the electron current i n the rest of the device and totally ignore the hole current. The critique of the simplified model given below also applies to these simplifications. The p r o b l e m is completed by prescribing boundary conditions. I n terms of the rescaled quantities the mixed boundary c o n d i t i o n at the interface BC reads B
D
G
T
T
;
= W — yV
xdtW
bi
-
U
G
at
BC
(4.7.4)
where % is defined as i n the preceding Section by ed y- = —; s
.
The analysis below w i l l show that the values of the potential along the pn-junctions BG and CH do not affect the results, and so we leave them unspecified. I n order to avoid nonuniformities along the artificial boundary GH, zero space charge is required there: W = - 1 + U - y l n ^ l + ^ 1 + 4 exp B
n
® -
2
^
at GH (4.7.5)
Recalling the results for the M I S diode, this c o n d i t i o n is satisfactory as l o n g as the edge of the depletion region stays away from GH. A quantitative f o r m u l a t i o n of this requirement w i l l be given below. Prescribing boundary conditions for the electron quasi F e r m i level is a more subtle problem. Starting w i t h the easy parts, we pose a homogeneous N e u m a n n c o n d i t i o n along the interface and the following Dirichlet conditions at the pn-junctions: 0>„ = 0
at
BG,
®„ = U
D
at
CH,
(4.7.6)
which are motivated by (4.7.2). The previous discussion fails to give a handle on the behavior of <&„ at the artificial boundary GH. Therefore a boundary c o n d i t i o n is used which reflects favourable operating conditions. The above description of the device behaviour raises the expectation that current flow only takes place i n the direction tangential to the interface. This motivates the assumption that no current flow occurs across GH, resulting in the boundary c o n d i t i o n c\*„ = 0
at
BC and GH.
(4.7.7)
The results of numerical simulations [4.17, Chap. 9 ] . Show that these assumptions are not necessarily justified. I n [4.17] a parasitic effect is analyzed which can be explained by interpreting the M O S F E T as a bipolar transistor
230
4 Devices
where the source, bulk, and d r a i n are identified w i t h emitter, base, and collector, respectively. Depending o n the details o f the geometry and the d o p i n g profile a small b u l k current can be the reason for a significant current from source to d r a i n w i t h o u t an inversion layer being present. As the numerical results show, this current m i g h t very well have a significant component i n the direction perpendicular to GH. A c c o r d i n g to these observations, certain parasitic effects are a p r i o r i l y precluded by the conditions (4.7.7). Nonetheless, the model (4.7.3)-(4.7.7) w i l l be sufficient for the comp u t a t i o n o f qualitatively correct device characteristics. The t o t a l current from source to d r a i n w i l l be approximated by the electron current across the P-Adjunction BG (or CH).
A Quasi One-Dimensional
Model
I n this Paragraph the smallness o f the parameter 1 w i l l be exploited for a further simplification o f the model (4.7.3)-(4.7.7). As 1 -> 0 the Poisson equation reduces to the o r d i n a r y differential equation
S }
_ exp (
W
W
- * - -
1
) - exp (
V
- - ^ -
>
) +
1
•
H.7.8)
Assuming (D„ to be given, this equation subject to the b o u n d a r y conditions (4.7.4), (4.7.5) constitutes a one-dimensional b o u n d a r y value p r o b l e m for each value of y, w h i c h has a unique solution. As mentioned above the a p p r o x i m a t i o n o f W computed i n this way is independent o f the boundary values at the P - N junctions. Since layer corrections along BG and CH do not significantly affect the final results for the current, they w i l l not be considered here. The asymptotic analysis of the c o n t i n u i t y equation is less straightforward. The reduced p r o b l e m consisting of the differential equation 8, (p
exp
n
8 ^
= 0
and homogeneous N e u m a n n conditions does n o t have a unique solution. It only allows the conclusion that the quasi F e r m i level is independent o f £, i.e. <J>„ = ®„(y). A singular p e r t u r b a t i o n p r o b l e m of this k i n d has been dealt w i t h i n [4.5, Section 4.3] where a formal a p p r o x i m a t i o n o f the s o l u t i o n is derived. A justification for the formal approach can be found i n [ 4 . 3 ] . A d d i t i o n a l i n f o r m a t i o n o n the l i m i t i n g s o l u t i o n can be obtained by integrating the original differential equation i n (4.7.3) i n the ^-direction. U s i n g the boundary conditions, the result is J 8 ( p „ exp ( ^ " y " " y
where
1
)
^
=
0
denotes the Rvalue corresponding to the b o u n d a r y GH. D e n o t i n g
4.7
231
MOSFET
the solution of ( 4 . 7 . 8 ) for given „ by W ( 0 „ ) , we introduce the dimensional electron density
one-
exp(
N(OJ
Since, i n the l i m i t /. -> 0 , <J>„ only depends o n y, the l i m i t of the above equation can be w r i t t e n as d (Li N(Q> )d <S>„) = 0. y
n
n
y
Subject to ( 4 . 7 . 6 ) , this is a one-dimensional b o u n d a r y value p r o b l e m i n the y-direction. Since its f o r m u l a t i o n involves the solution of a p r o b l e m i n the c-direction it m i g h t be called quasi one-dimensional. The current from source to d r a i n is equal to the current across an a r b i t r a r y vertical cross section of the rectangle BGHC and is given by I = Li N( . n
y
(4.7.9)
n
Since the only y-dependence i n the p r o b l e m for the potential originates from
I = Pn
Computation
(4.7.10)
n
of the One-Dimensional
Electron
Density
N o t i n g the similarity of the one-dimensional p r o b l e m for the potential to the M I S diode problem, we expect that the results for the M I S diode essentially carry over to the present situation. The only difference between the t w o problems is the occurence of the parameters O and U i n (4.7.8). Leaving the c o m p u t a t i o n a l details to the reader, we only summarize the results. I n general, M O S F E T s are not operated i n the accumulation regime. Thus, we restrict ourselves to the case U > U . Depletion or weak inversion occurs for values o f U i n the interval n
G
B
B
G
[ l / , 2 +
fl
l/ )].
b
(4.7.11)
B
The c o n d i t i o n U < 2 which is necessary for the validity o f this analysis corresponds to requiring that a not too large forward bias is applied to the source-bulk j u n c t i o n . The onset of strong inversion corresponds to the right end of the above interval or to the c o n d i t i o n W(0) = 1 + d>„. The approximate solution i n the case of depletion or weak inversion is determined by B
W(0) W(c)
= U G
= {
1+ a
2
l + U + (ZB
•l + U
B
2
- aJa
+ 2(U
G
2
Q ) l2
w
d
for
—
for
U ), B
5 < Q
I -
l"'
(4.7.12)
q>Q , d
232
4 Devices
where the depletion w i d t h is given by a
C = V
2
+
a
2(U -U )-x. G
B
N o t e that the potential is independent of the electron quasi F e r m i level i n this case. The m a x i m a l depletion w i d t h , occurring at the onset of strong inversion, is ^2(2 + „— U ). B
I n strong inversion the potential i n the inversion layer is approximated by 1
W
= 1 + 0>„ + y In y-
im
+ yz
with
z = In [(2 + (D„ -
2
U ) s i n h " (y B
1 + (O -
U )/2 ± + c
n
B
(4.7.13) and the constant of integration c being determined from the i n i t i a l c o n d i t i o n 2
z(0) = ln((2 +
2
L / ) / 2 a - 2 -
B
I n the depletion layer the potential is given by (4.7.12) w i t h the depletion w i d t h <; replaced by the m a x i m a l depletion w i d t h given above. Assuming the channel close to the d r a i n to be i n the strong inversion regime, we arrive at the c o n d i t i o n d
'2(2 + U D
U) < ? B
for the validity of our analysis. This inequality means that the m a x i m a l depletion w i d t h along the channel is smaller than the depth of the source and d r a i n regions. I n the depletion/weak inversion regime the electron density is given by /W(0)
- d>„ -
1- ^
2
d
+ £ /2
exp A n a p p r o x i m a t i o n of AT(<J>„) can be computed by d r o p p i n g the quadratic term and replacing t* by oo i n the integration: = f
exp
r
°
>
-
(
4
.
7
,
4
,
I n strong inversion A(<J)„) is the sum of contributions from the inversion layer and the depletion region. The electron density i n the inversion layer is "inv =
l
z
l~ e
w i t h z given i n (4.7.13). Integrating this and adding the depletion layer c o n t r i b u t i o n gives N(
G
— 2 — <&„)/« - 72(2
+ y/j2(2
+ Q> -U ). n
B
+
U) B
(4.7.15)
4.7 M O S F E T
233
Computation
of the
Current
Depending o n the biasing situation, different cases occur. I f the gate voltage U is below the threshold voltage G
U, = 2 + 7.^2(2 -
U) B
depletion/weak inversion prevails t h r o u g h o u t the channel. F o r the c o m p u tation of the current the one-dimensional electron density given by (4.7.14) can be used:
,_*g„p(H&zi) ,
- e - ^ .
(
I n this so called subthreshold region the current saturates for large drainsource voltages at a value which is transcendentally small i n terms of y. The threshold voltage and the subthreshold characteristic i n terms o f unsealed quantities are given by U = 2V t
j
+
bi
=
j2U (2V -U ), tet
HeU n Q LCx n
s
T
_
m
bi
B
-v /u ^
e
B
T
d
where t /
r e f
is defined as i n the preceding Section,
1 "so = n exp ^ — (U - V + U t
G
bi
tet
- ^U JU r
+ 2U -
ref
G
2U )) B
denotes the surface electron concentration close to the source and x
= Jr,J(qC)(^U
d
+ 2U - 2U -
te(
G
B
JU ) re(
is the depletion w i d t h . N o t e that U = U, implies n = C, i.e. the threshold voltage marks the onset o f strong inversion close t o the source. F o r U > U, t w o different possibilities have to be accounted for. I n the case G
so
G
U
G
> 2 + U + a /2(2 + U D
N
D
- U ),
(4.7.16)
B
called non-saturation region, the whole channel is i n strong inversion. Thus, the formula (4.7.15) for N(Q>„) applies and the current is given by 1
/ = P„( (U - 2 - U /2)U G
+
2
D
2
2
^ ( 2 - U f' B
D
- h l l ( 2 + l] D
+ 7 ^ 2 ( 2 +U D
3
fJ ) '
2
B
U ) - 7^2(2 B
U) B
(4.7.17) w h i c h — i n terms of unsealed variables—reads
234
4 Devices
I =
j-CJU -2V -U /2)U G
hi
D
- ^V^qC(^(2V
D
+U
bi
- U /2{2V +U -U ) Ts
bi
D
- U )^ -
D
B
+ U ^2(2V
B
T
2
-^{2V
bi
-
2
U f> B
- U )^j .
bi
B
Considering the dependence o f the current on the d r a i n voltage for a fixed gate voltage U > U the formula (4.7.17) holds as l o n g as G
U
D
t
= U — 2+ a — 2
Dsal
2U - 2U
G
G
B
is satisfied. The saturation voltage is determined by assuming equality i n (4.7.16). F o r larger values o f U a phenomenon called pinch-off occurs. A transition from strong inversion to weak inversion takes place at the pinchoff point where the quasi F e r m i level takes the value U I n this case the one-dimensional electron density is given by (4.7.15) for 0 < „ < f J and by (4.7.14) for U < <£•„ < U . The current is given by D
DsiiV
D s a t
Dsat
/ =7
sat
+^
D
exp (
W
i
0
)
^
l
\
e
- ^
-
e-°")
which is essentially equal to the saturation current 7 obtained by substituting U = C / i n (4.7.17). D u e to this behavior o f the characteristic the set of operating points defined by sat
D
Dsat
F i g u r e 4.7.2 C u r r e n t vs. d r a i n voltage for different U
G
4.8 G u n n D i o d e
U
235
< 2 + U + aV'2(2 + U -
G
D
U)
D
B
is called the saturation region. A n a p p r o x i m a t i o n of the pinch-off p o i n t y* can be computed from (4.7.9) by integration from y = 0 to y = y*: y* =
hJi-
Since i n the saturation region the current is only insignificantly larger than 7 we o b t a i n sat
1 - y* « 1. The distance o f the pinch-off p o i n t to the d r a i n is very small compared to the channel length.
4.8 Gunn Diode The G u n n diode is an i m p o r t a n t microwave device. Its performance is based on the transferred-electron effect described i n Chapter 2 which is responsible for a n o n m o n o t o n i c velocity-field relation. A typical device consists o f a homogeneously doped piece o f a semiconductor whose energy-band structure supports the transferred-electron effect (e.g. gallium arsenide (GaAs) or i n d i u m phosphide (InP)). This Section is concerned w i t h an explanation o f the Gunn effect [ 4 . 4 ] : A microwave o u t p u t can be generated by a p p l y i n g a large enough constant voltage to an n-type piece of G a A s or I n P . The presentation w i l l mostly be based o n the w o r k of Szmolyan [4.23], [4.24] who put the classical analysis (see [4.22] for references) o n a mathematically sound basis. The results o f the final Paragraph are new. Consider a homogeneously doped piece of semiconductor o f length L w i t h constant d o n o r concentration C. A one-dimensional unipolar model is given by the differential equations scE s
x
= q(n -
d,n = d (Dd n x
x
C), -
nv v(E/E )), SM
T
where E denotes the negative electric field, D is the diffusivity and the qualitative behaviour o f the velocity v v(E/E ) is given i n F i g . 4.8.1 b which also explains the meaning o f the saturation velocity v and the threshold field E . The graph of the scaled function v goes t h r o u g h the origin, has a m a x i m u m at 1 and saturates at v = 1 for large arguments. F o r negative values o f the argument, v is defined by o d d extension. The differential equations hold for x i n the interval (0, L ) , representing the device. Since large fields are to be expected, the Einstein relation between the diffusivity and the m o b i l i t y is dropped here (see Chapter 2). I n a d d i t i o n the field dependence of the diffusivity w i l l be ignored for simplicity. However, most of the arguments below go t h r o u g h i f the field dependence is such that the diffusivity is bounded from above and away from zero (see [4.23], [4.24]), a l t h o u g h the computations are more involved. sat
T
sat
T
236
4 Devices
E
E
0
T
a)
b)
F i g . 4.8.1 V e l o c i t y vs. field for (a) Si a n d (b) G a A s
O h m i c contacts are modelled by the boundary conditions n(0) = n(L) = C and the application o f a voltage U is described by an integral c o n d i t i o n for the field: Edx=U. The p r o b l e m f o r m u l a t i o n w o u l d have to be completed by imposing initial conditions for the electron density. However, o u r m a i n interest w i l l lie i n the study o f special solutions o f the differential equations rather than i n the general initial value problem. A scaling is introduced where the device length L and the characteristic time L/u are the reference quantities for length and time. Carrier densities, electric fields, and voltages are scaled by C, E , and E L, respectively. The scaled p r o b l e m reads s a t
T
T
2 5 £ = n - 1, 2
v
d n = d (yd n t
x
-
x
nv(Ej), E dx = U,
n(0) = n ( l ) = 1,
(4.8.1)
where £E qCL' S
T
y
D
are the square of the scaled Debye length and the relative strength of diffusive and convective terms. N o t e , that the same symbols have been used for scaled and unsealed quantities. Considering typical values for the material dependent parameters (see [4.22]), a device w i t h a length of 10 u m or more, and a d o p i n g concentration
237
4.8 G u n n D i o d e 1 5
3
2
of about 1 0 c m ~ , b o t h X and y are small parameters of the same order of magnitude. Assuming the ratio oc = y/X to take moderate values, only one small parameter appears in the differential equations: 2
2
XdE x
= n — 1, (4.8.2)
d,n = d (X ad n — x
x
nv(E)).
I n the following t w o i m p o r t a n t properties of the system (4.8.1), (4.8.2) are discussed: The loss of stability of homogeneous steady states due to bulk negative differential conductivity ( N D C ) and the existence of traveling wave solutions. A c o m b i n a t i o n of these properties w i l l be used for the asymptotic analysis of the G u n n effect.
Bulk
Negative
Differential
Conductivity
Consider a piece of semiconductor w i t h homogeneous carrier density. I n this case the current density is given by J = nv(E) and the bulk differential dJ/dE
=
conductivity
by
nv'(E).
Whereas i n Si dJ/dE is always positive, bulk N D C occurs i n GaAs and I n P . A p a r t from the transferred-electron effect, other physical mechanisms can be responsible for b u l k N D C . We only mention impact ionization induced bulk N D C w h i c h is used i n another microwave device, the I M P A T T [impact ionization avalanche transit time) diode (see [4.22]). F o r a detailed discussion of bulk NDC-effects caused by recombination-generation we refer to [4.16]. N o t e that a global form of N D C has been observed in Section 4.5 in connection w i t h the middle branch of the voltage-current characteristic of a thyristor. As opposed to bulk N D C , which is due to microscopic material properties, this effect is caused by the interaction of P-N junctions. C a r r y i n g out the differentiation i n the right hand side of the c o n t i n u i t y equation leads to
2
2
c,n = X ccd n
— c nv(E) x
n —1 — nv'(E)~-p— , 2
which shows (by the smallness of X ) that for values of n away from the e q u i l i b r i u m value 1, the dynamics of the system are dominated by the ordinary differential equation dn = t
—nv'(E)—^—•
Obviously the stability of the e q u i l i b r i u m solution n = 1 is determined by the sign of v'(E) w i t h stability for v'(E) > 0. This heuristic argument has been
238
4 Devices
made rigorous i n [4.24]. A stationary solution o f (4.8.1), (4.8.2) is given by n = 1,
E = U
which is called the trivial solution from n o w on. The stability o f this solution was examined i n [4.24] by linearization. I t can be shown that for U > 1 (which implies v'(U) < 0) and X small enough the t r i v i a l solution is unstable whereas it is stable for U ^ 1. F u r t h e r m o r e a stable n o n t r i v i a l solution bifurcates from the t r i v i a l solution at the critical voltage where the t r i v i a l solution looses its stability. 2
Traveling
Waves
A n i m p o r t a n t property o f the equations (4.8.2) is the existence o f traveling wave solutions, i.e. solutions which only depend on x — v t where v denotes the velocity o f the wave. These solutions are strictly valid o n l y for an idealized device o f infinite length. However, since the active region o f the solutions w i l l be shown to be very small w i t h fast decay at ± oo, they can be used i n the singular p e r t u r b a t i o n analysis o f the following Paragraph as layer terms in a m o v i n g internal layer. The smallness o f the active region is reflected i n the choice o f the variable 0
0
Assuming n and E to be functions o f s only, the differential equations can be w r i t t e n as d,E = -v d n 0
s
(4.8.3)
n - l , = d (ad n s
-
s
nv(E)).
O n l y solutions which converge as s -» ± oo can be used as layer terms. Besides, it w i l l be shown i n the following Paragraph that the limits of E as s -» ± oc have to be equal. Thus, integration o f the second equation gives ccd n = n(v{E) - v ) + v s
Q
0
(4.8.4)
v(E ) x
where £ is the c o m m o n l i m i t of E as s tends to + x . The further analysis proceeds by studying the phase portraits of (4.8.3), (4.8.4) for various choices of v and E . The above requirements i m p l y that we are l o o k i n g for a homoclinic o r b i t o f the system w i t h respect to a stationary p o i n t (n, E) = (1, E ). I n [4.23] it is shown that such a solution can o n l y exist i f v = v(E ) holds, which means that the velocity o f the wave is equal to the drift velocity of the electrons at infinity. W i t h this assumption the equations read x
0
x
x
0
ad n = n(v(E) s
oa
— v ). 0
The number o f stationary points of (4.8.5) is equal to the number o f solutions
239
4.8 G u n n D i o d e
of the equation v(E) = v . F o r v ^ 1 there is only one stationary point which implies that a homoclinic o r b i t cannot exist. Therefore we assume that v lies between 1 and v(\) from n o w on. I n this case there are t w o stationary solutions £ < 1 < E . A stability analysis shows that the point (1, Ey) is a saddle and that the eigenvalues o f the Jacobian of the right hand side o f (4.8.5) evaluated at (1, E ) are imaginary. Separation of variables and integration gives 0
0
0
t
2
2
oc(n — I n n
—
(v(y) - v ) dy
1)
(4.8.6)
0
which can be used for d r a w i n g a picture o f the phase p o r t r a i t o f (4.8.5), F i g . 4.8.2. For £ = F , , (4.8.6) describes the stable and unstable manifolds of the stationary p o i n t (1, £ , ) . I t is easily seen that the part of the curve w i t h F ^ F , is closed which means that the stable and unstable manifolds meet and a homoclinic orbit exists. O n this o r b i t the m a x i m a l value F of the field satisfies the equation r e f
m a x
(v(y)
-
v ) dy = 0 0
which is k n o w n under the name equal area rule (see Fig. 4.8.3). For £ between £ , and F , (4.8.6) is the equation o f a closed curve a r o u n d (1, E ) corresponding to a periodic solution. Thus, the stationary point (1, £ ) is a center. The homoclinic orbit is the traveling wave solution we have been l o o k i n g r e f
2
2
m a x
4 Devices
240
V
v
0
E
F i g . 4.8.3 E q u a l area rule
for. The qualitive shape of the wave can be determined from Fig. 4.8.2. The field has the form o f a single pulse whereas the electron density forms a dipole w i t h a depleted region (n < 1) followed by a region o f accumulation (n > 1).
The Gunn
Effect
It was observed in [ 4 . 4 ] that for sufficiently large applied voltages small perturbations o f the homogeneous steady state grow, until a stable configu r a t i o n (called domain) is reached which then travels t h r o u g h the semiconductor w i t h o u t changing its form. The electric field outside the d o m a i n is lower than the threshold field E but it takes values in the region of bulk N D C inside the d o m a i n . As soon as the d o m a i n leaves the device the electric field grows to a value above E t h r o u g h o u t the device and a new d o m a i n is built. The a i m o f this Paragraph is to relate the shape and velocity of the d o m a i n to the applied voltage. A p p l y i n g the methods o f singular perturbation theory to (4.8.2), we t r y to o b t a i n a solution which can be approximated by a solution o f the reduced equations T
T
n - 1 = 0,
c v(E) x
= 0
away from layers. We are interested i n the case that the only layer is given by a traveling wave. The above equations w o u l d allow for a j u m p o f £ across the wave if the values at the left and at the right give the same velocity. This w o u l d i m p l y that the integral of F changes w i t h time because the
241
4.8 G u n n D i o d e
integral of E changes as the wave travels t h r o u g h the device and the integral of the c o n t r i b u t i o n from the wave is constant. This contradicts the integral c o n d i t i o n in (4.8.1). Thus, E has to be constant and the travelling wave is given by the homoclinic orbit constructed above. E lies between E and 1 where v(E ) = 1 holds (see F i g . 4.8.1 b). The integral c o n d i t i o n on the field implies E = U because the w i d t h of the d o m a i n is 0 ( i ) . Since we are interested i n applied voltages larger than 1 this w o u l d make a d o m a i n solution impossible. The reason for this p r o b l e m is that the c o n t r i b u t i o n o f the d o m a i n to the integral of E is too small. Therefore we shall t r y to construct a wider d o m a i n w i t h larger values of the electric field. By the equal area rule large values of the m a x i m a l field implythat the velocity of the wave is close to the saturation velocity (see Fig. 4.8.3). This i n t u r n implies that E is close to E . These observations motivate the following transformations i n the traveling wave problem: 0
0
2
0
E = E^ + e/X,
E^ = £
+ Xe,
0
=
a
Xs,
where e and e remain to be determined. Substitution in (4.8.6) gives n — In n — 1 ~
(v(y) - 1 - Xv'(E )e) 0
dy
a 1
(y(y) — I) dy — - v'(E )ee a 0
IE
= A — Bee,
0
where we assumed that the i m p r o p e r integral o n the right-hand side converges. This is an assumption on the speed o f convergence of the velocity to its saturation value as the field tends to infinity. Assuming knowledge of e the rescaled field e i n the d o m a i n can be computed i n terms o f n: l
e = -{Be)~ {n
- In n - 1 -
A) l
and takes the m a x i m a l value A(Be)~ . O b v i o u s l y this equation is valid as long as e remains positive which holds for n between the zeros 0 < n < 1 < ;?! of the right-hand side. I n t r o d u c i n g the transformation i n the differential equation for n implies 2
Xac n n
~ n{v(E
x
+ e/X) - 1 - Xv'(E )e) 0
~ n(o(X) -
Xv'(E )e). 0
I n the l i m i t / - » 0 we o b t a i n c„n =
—Ben
w i t h the solution n = exp( — Beo), where a different choice for the constant o f integration corresponds to a shift in the c-direction which obviously does not change the results. This solution is valid for a between the values a and er where n takes the values n and n , respectively. The construction o f the asymptotic form of the traveling wave solution w o u l d be completed by considering layers i n neighbourhoods x
2
2
x
242
4 Devices
of fjj and G w h i c h smooth the j u m p s i n the electron density from 1 to / i , and from n to 1, respectively. I t remains to determine the value o f e from the integral c o n d i t i o n on the electric field. A s y m p t o t i c a l l y the integral is given by 2
2
£ +
1
e da = E
Q
0
+ KJe
with K = B
2
\n
2
— n , — In — \ \ l n ( n " i ) + A + \ 2
Since e has to be positive, we o b t a i n e =
^K/(U-E ). 0
Summarizing the results o f this Paragraph we note that the dipole formed by the electrons has sharp boundaries represented by <7, and a . The velocity of the wave is close to the saturation velocity and, thus, together w i t h the microwave frequency essentially independent from the applied voltage. 2
Problems 4.1
Instead o f the S R H - t e r m consider a m o r e general r e c o m b i n a t i o n - g e n e r a t i o n m o d e l o f the f o r m 4
R = Q(n, p, x)(np - <5 )
with
Q > 0
i n (4.1.4). T h i s includes the b a n d - b a n d r e c o m b i n a t i o n t e r m (2.2.13) a n d the A u g e r t e r m (2.6.1). C a r r y over the discussion o f the c l o s e - t o - e q u i l i b r i u m case t o this m o d e l . 4.2
Solve the d o u b l e obstacle p r o b l e m (4.2.10) by p a t c h i n g together s o l u t i o n pieces w i t h W = — \, c*W + C = 0 a n d W = 1, respectively, such t h a t W is c o n t i n u o u s l y differentiable. C o n v i n c e y o u r s e l f that the s o l u t i o n is u n i q u e . Verify (4.2.11).
4.3
a) Verify the results i n the P a r a g r a p h " S t r o n g l y a s y m m e t r i c j u n c t i o n s " o f Section 4.2. b) Singular p e r t u r b a t i o n t h e o r y leads t o different a p p r o x i m a t i o n s o f a f u n c t i o n i n different regions. Here these a p p r o x i m a t i o n s are t ^ J c r ) , W {x) a n d W . {r\). T h e question arises i f an a p p r o x i m a t i o n can be f o u n d w h i c h is u n i f o r m l y v a l i d i n the full r e g i o n o f interest. I n general the answer is positive. C o n s i d e r the example u
f{x,
c
e) = cos(e~* + x )
with
e. « 1.
A w a y f r o m the b o u n d a r y layer at x = 0 the a p p r o x i m a t i o n fix)
= cos x
is v a l i d , whereas
f(c)
-X = cos(e * ) ,
{ =
x R
a p p r o x i m a t e s / w i t h i n the layer. T h e m a t c h i n g c o n d i t i o n / ( 0 ) = l i m fic)
= 1
if pl
References
243
holds. A u n i f o r m l y v a l i d a p p r o x i m a t i o n f o r / ' c a n be o b t a i n e d by a d d i n g the i n d i v i d u a l a p p r o x i m a t i o n s / a n d / a n d s u b t r a c t i n g t h e i r " c o m m o n p a r t " w h i c h is 1 i n o u r example. T h u s , we have x
f(x,
s) ~ cos(e~
£
) + cos x — 1.
Use these ideas a n d the c o m m o n parts (4.2.19), (4.2.21) for o b t a i n i n g an a p p r o x i m a t i o n of W w h i c h is u n i f o r m l y v a l i d i n the ^ - i n t e r v a l [ — 2, 0 ] . 4.4
a) Verify the f o r m u l a (4.2.26) for the s a t u r a t i o n c u r r e n t by e x p l i c i t e l y s o l v i n g the p r o b l e m s for tp a n d
4.5
2
a) Verify (4.2.29). b) Rescale the c u r r e n t i n (4.2.30) by / = S T. O b t a i n the Shockley e q u a t i o n by l e t t i n g S -> 0 i n the e q u a t i o n for T. 4
4
4.6
a) Solve the d o u b l e obstacle p r o b l e m (4.2.33) a n d o b t a i n the n o n c o i n c i d e n c e set (4.2.34). b) C o m p u t e vv, /„ a n d I a n d verify the f o r m u l a (4.2.38) for the characteristic.
4.7
C o m p u t e the c o m m o n - e m i t t e r c u r r e n t g a i n (4.3.3) o f a b i p o l a r transistor u n d e r the f o l l o w i n g , s i m p l i f y i n g assumptions: a) A c c o r d i n g to (4.3.4), the second t e r m i n the f o r m u l a for a can be neglected. b) I n the c o m p u t a t i o n o f the base contact can be i g n o r e d . d) T h e m o b i l i t i e s are c o n s t a n t a n d the d o p i n g p r o f i l e is piecewise constant.
p
3
2
2
4.8
Solve the d o u b l e obstacle p r o b l e m (4.4.2) for a s h o r t P I N - d i o d e i n t h e r m a l e q u i l i b r i u m i n the o n e - d i m e n s i o n a l case (4.4.7) a n d w i t h the a s s u m p t i o n o f a piecewise c o n s t a n t d o p i n g profile. Verify t h a t (4.4.6) h o l d s i n the l i m i t i n g case k -> 0.
4.9
a) A n a l y z e the t h y r i s t o r close t o t h e r m a l e q u i l i b r i u m c o n s i d e r i n g the Shockley-ReadH a l l t e r m for r e c o m b i n a t i o n - g e n e r a t i o n . b) I n the case o f the o n e - d i m e n s i o n a l m o d e l (4.5.5), (4.5.6) c o m p u t e the constant A i n (4.5.4).
4.10 Verify (4.5.10), (4.5.11) b y s o l v i n g the p r o b l e m for w , J „ , 0
0
J. p0
4.11 F o r the M I S diode p r o b l e m (4.6.3), (4.6.4) i n the a c c u m u l a t i o n regime an a p p r o x i m a t i o n of the p o t e n t i a l has been o b t a i n e d i n Section 4.6. H o w e v e r , this a p p r o x i m a t i o n c a n n o t be u n i f o r m l y v a l i d because it does n o t converge as £ -> oo. a) I n t r o d u c e a t r a n s i t i o n layer s o l u t i o n o f the f o r m
^
r
a
n
» = -1
+yz(a),
a =
^ -
w h i c h connects the i n i t i a l a p p r o x i m a t i o n t o the prescribed value for I V a t c = x . b) C o n s t r u c t a u n i f o r m l y v a l i d a p p r o x i m a t i o n for the p o t e n t i a l by the m e t h o d i n t r o duced i n P r o b l e m 4.3. c) S h o w t h a t the c o n t r i b u t i o n o f W t o the t o t a l charge is small c o m p a r e d to the a p p r o x i m a t i o n (4.6.5). trans
References [4.1] [4.2]
W . Eckhaus: A s y m p t o t i c A n a l y s i s o f Singular P e r t u r b a t i o n s . N o r t h - H o l l a n d , A m sterdam (1979). A . F r i e d m a n : V a r i a t i o n a l Principles a n d F r e e - B o u n d a r y Problems. J o h n W i l e y & Sons, N e w Y o r k (1982).
244 [4.3] [4.4] [4.5] [4.6] [4.7] [4.8] [4.9] [4.10] [4.11] [4.12] [4.13] [4.14] [4.15] [4.16] [4.17] [4.18] [4.19] [4.20] [4.21] [4.22] [4.23] [4.24] [4.25] [4.26]
4 Devices P. G r a n d i t s , C. Schmeiser: A M i x e d B V P for F l o w i n S t r o n g l y A n i s o t r o p i c M e d i a . A p p l i c a b l e A n a l y s i s (to appear). J. B. G u n n : M i c r o w a v e O s c i l l a t i o n s o f C u r r e n t i n I I I - V Semiconductors. S o l i d State C o m m . 7,88 (1963). J. K e v o r k i a n , J. D . Cole: P e r t u r b a t i o n M e t h o d s i n A p p l i e d M a t h e m a t i c s . Springer, N e w Y o r k (1981). D . K i n d e r l e h r e r , G . Stampacchia: A n I n t r o d u c t i o n t o V a r i a t i o n a l Inequalities a n d T h e i r A p p l i c a t i o n s . A c a d e m i c Press, N e w Y o r k (1980). C. C. L i n , L . A . Segel, M a t h e m a t i c s A p p l i e d t o D e t e r m i n i s t i c P r o b l e m s i n the N a t u r a l Sciences. M a c m i l l a n , N e w Y o r k (1974). P. A . M a r k o w i c h : A N o n l i n e a r Eigenvalue P r o b l e m M o d e l l i n g the A v a l a n c h e Effect in S e m i c o n d u c t o r D i o d e s . S I A M J. M a t h . A n a l . 16, 1268-1283 (1985). P. A . M a r k o w i c h , C. A . Ringhofer, C. Schmeiser: A n A s y m p t o t i c A n a l y s i s o f O n e D i m e n s i o n a l S e m i c o n d u c t o r Device M o d e l s . I M A J. A p p l . M a t h . 37, 1-24 (1986). P. A . M a r k o w i c h , C. Schmeiser: U n i f o r m A s y m p t o t i c R e p r e s e n t a t i o n o f S o l u t i o n s o f the Basic S e m i c o n d u c t o r Device E q u a t i o n s . I M A J. A p p l . M a t h . 36, 4 3 - 5 7 (1986). C. P. Please: A n Analysis o f S e m i c o n d u c t o r P-N Junctions. I M A J. A p p l . M a t h . 28, 301-318(1982). A . Porst: H a l b l e i t e r . Siemens A G , B e r l i n - M u n c h e n (1973). M . H . P r o t t e r , H . F. Weinberger: M a x i m u m Principles i n D i f f e r e n t i a l E q u a t i o n s . Prentice H a l l , E n g l e w o o d Cliffs, N J (1967). I . R u b i n s t e i n : M u l t i p l e Steady States i n O n e - D i m e n s i o n a l Electrodiffusion w i t h L o c a l E l e c t r o n e u t r a l i t y . S I A M J. A p p l . M a t h . 47, 1076-1093 (1987). C. Schmeiser: A Singular P e r t u r b a t i o n A n a l y s i s o f Reverse Biased PN Junctions. S I A M J. M a t h . A n a l . 21 (1990). E. Scholl: N o n e q u i l i b r i u m Phase T r a n s i t i o n s i n Semiconductors. Springer, B e r l i n (1987). S. Selberherr: Analysis a n d S i m u l a t i o n o f S e m i c o n d u c t o r Devices. Springer, W i e n N e w Y o r k (1984). W . Shockley: T h e T h e o r y o f p-n J u n c t i o n s i n S e m i c o n d u c t o r s a n d p-n J u n c t i o n Transistors. Bell Syst. Tech. J. 28, 435 (1949). J. W . S l o t b o o m : I t e r a t i v e Scheme for 1- a n d 2 - D i m e n s i o n a l D . C . - T r a n s i s t o r S i m u l a t i o n . E l e c t r o n . L e t t . 5, 6 7 7 - 6 7 8 (1969). H . S t e i n r i i c k : A B i f u r c a t i o n A n a l y s i s o f the Steady State S e m i c o n d u c t o r D e v i c e E q u a t i o n s . S I A M J. A p p l . M a t h . 49, 1102-1121 (1989). H . S t e i n r i i c k : A s y m p t o t i c A n a l y s i s o f the C u r r e n t - V o l t a g e C u r v e o f a P N P N Semic o n d u c t o r Device. I M A J. A p p l . M a t h . (1989) (to appear). S. M . Sze: Physics o f S e m i c o n d u c t o r Devices. J o h n W i l e y & Sons, N e w Y o r k (1969). P. S z m o l y a n : T r a v e l i n g Waves i n G a A s - S e m i c o n d u c t o r s . Physica D (1989) (to appear). P. S z m o l y a n : A n A s y m p t o t i c A n a l y s i s o f the G u n n Effect. P r e p r i n t , I M A , U n i v . o f M i n n e s o t a (1989). M . J. W a r d : A s y m p t o t i c M e t h o d s i n S e m i c o n d u c t o r D e v i c e M o d e l i n g . Thesis, C a l i f o r n i a Inst, o f T e c h n . (1988). M . J. W a r d , D . S. C o h e n , F. M . O d e h : A s y m p t o t i c M e t h o d s for M O S F E T M o d e l i n g . P r e p r i n t , C a l i f o r n i a Inst, o f T e c h n . (1988).
Append
Physical Constants Symbol
Quantity B o l t z m a n n constant Electron rest mass Electron v o l t Elementary charge P e r m i t t i v i t y i n vacuum Reduced Planck constant
Value 1.38 x 1 ( T V A s / K 0.91 x 1 0 ~ k g 1.6 x 1 0 ~ V A s 1.6 x 1 0 " As 8.85 x 1 0 " A s V " c m " 1.05 x 1 0 " V A s 2 3
^B
3 0
m eV 0
19
1 9
1 4
h
1
3 4
1
2
Properties of Si at Room Temperature Permittivity: e = 11.9e Bandgap: E = 1.12 eV L o w field mobilities: LI„ = 1500 c m V " s" , jx = 450 c m V Typical values for recombination-generation parameters: C„ = 2.8 x 1 0 " cm /s, C = 9.9 x 1 0 " c m / s s
0
g
2
1
1
2
p
3 1
6
3 2
p
T„ =
6
10~ S, 6
T
p
= 1 0
_
5
S
1
6
1
a* = 1 0 c m " , a* = 2 x 1 0 c m " £ ^ = 1.66 x 1 0 V / c m , Ef = 2 x 10 V / c m rit
6
x
6
6
- 1
s"
1
Subject Index
Acceptor 73 Accumulation 221 Ambipolar diffusion length 207 Avalanche breakdown 195 Band diagram 14 Bandgap 89 Base 199 Basic semiconductor device equations 89, 90 B B G K Y hierarchy 19 Bipolar transistor 198 Bloch's Theorem 13 Boltzmann constant 32 Boltzmann equation 28 — , semi-classical 30 Bose-Einstein statistics 35 Bosons 59 Break over voltage 208 Brillouin zone 13 Built-in potential 176 Bulk 225 Bulk differential conductivity 237
Continuity equation 11 Correspondence principle 53 Coulomb force 21 Crystal lattice 12 Current density 10 Current gain 199 Debye length 118 Density matrix 40 Depletion 221 Depletion region 108, 181 Dielectric relaxation time 164 Diffusion time scale 157 Displacement current 158 Distinguished limit 186 Domain 240 Donor 73 Doping profile 73 Drain 225 Drift diffusion equations 89
Capacitance 221 Carrier life times 100 Channel 225 — length 226 — , width 227 Charge density 73 Coincidence sets 183 Collector 199 Collision cross-section 34 Collision frequency 33 Collision integral 29 Collision operator 29 Collisions 28
Early effect 202 Effective mass 69 Einstein relations 89 Electric displacement 27 Electric field 27 Electron capture 98 Electron emission 98 Elementary charge 5 Emitter 199 Energy band 14 Energy density 91 Energy flow 93 Energy-valleys 69 Equal area rule 239 Euler equations 93
Common-emitter configuration 199 Composite expansion 122 Conduction band 71
Fast variables 178 Fermi-Dirac statistics 32
Subject I n d e x Fermi-energy 32 Fermions 59 Flat band condition 221 Floating region 109,147 Fourier transform 41 Gate 218, 225 Generation 71 Gummel method 146 Gunn effect 235 Hamiltonian, classical 11 — , quantum mechanical 13 — , semi-classical 15 Hamiltonian equations 12 Hartree ansatz 61 Heisenberg equation 41 High injection 191 Hilbert expansion 87 Holding current 215 Holding voltage 208 Hole 71 Hole capture 98 Hole emission 98 H-Theorem 35 Hydrodynamic model 93 Impact ionization 100 Inflow boundaries 24 Inner solution 121 Intrinsic number 89 Intrinsic region 202 Inversion, strong 221 —, weak 221 Inversion layer 223 Landau order symbols 178 Layer 120 — , equation 121 — , term 120 Leakage current 109,189 Liouville equation 8 — , semi-classical 12 Liouville's Theorem 10 Lucky drift model 100 Magnetic field 27 Magnetic induction 27 Matching 187 Maxwell equations 26 Maxwellian, displaced 92 — shifted 92 Maxwellian distribution 33 Mean free path 86 M I S diode 218
247 Moment 10 Moment methods 90 Momentum vector 11 M O S F E T 225 Negative differential conductivity 237 Newton's method 144 — , approximate 146 Noncoincidence set 183 Non-saturation region 233 Obstacle problem 183 Ohmic contacts 105 Parabolic band approximation 69 Particle ensemble 8 Pauli principle 21 Permeability 27 Permittivity 21,27 Phase space 12 Phonons 35 Pinch-off 234 — , point 234 PIN-diode 202 P-N diode 107 P-N junctions 107 — , abrupt 119 Poisson equation 21 Polar optical scattering 35 Primitive cell 12 Principle of detailed balance 32 Pseudo-differential operator 43 Punch through 197 Quantum Liouville equation 36, 93 Quasi-Fermi potential 95 Reciprocal lattice 12 Recombination 71 — , Auger 98 — , band-band 98 — , Shockley-Read-Hall 100 Reduced problem 120 Reduced solution 120 Relaxation 32 Relaxation time 33 Resonant tunneling diode 75 Runaway phenomenon 96 Saturation region 235 Scattering rate 28 Schrddinger equation 13, 37 Shockley-diode 208 Shockley equation 189 Significant degeneration 186
Subject I n d '
248 Singular perturbation 118 Slotboom variables 110 Slow variables 178 Source 225 Space charge 105 States, mixed 44 —, pure 44 Subsonic flow 93 Subthreshold region 233 Supersonic flow 93 Thermal voltage 86 Thermodynamical equilibrium 32 Threshold voltage 233 Thyristor 208 Transferred electron effect 97 Trap level 99
Tunneling 36, 38 Valence band 71 Variational inequality 205 Velocity saturation 97 Vlasov equation 17 —, semi-classical 25 Voltage-current characteristics Volume preserving map 78
176
Wave function 37 Webster effect 202 Wigner function 41 Work-function difference 219 Zero space charge approximation 120, 126
This book contains the first unified account of the currently used mathematical models for charge transport in semiconductor devices. It is focussed on a presentation of a hierarchy of models ranging from kinetic quantum transport equations to the classical drift diffusion equations. Particular emphasis is given to the derivation of the models, an analysis of the solution structure and an explanation of the most important devices. The relations between the different models and the physical assumptions needed for their respective validity are clarified. The book addresses applied mathematicians, electrical engineers and solid state physicists. It is accessible to graduate students in each of the three fields, since mathematical details are replaced by references to the literature to a large extent It provides a reference text for researchers in the field as well as a text for graduate courses and seminars.
ISBN 3-211-82157-0 ISBN 0-387 82157-0
