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 = 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
. 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 £ 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 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 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
"
= 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, 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
( 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 L
( x , fe, fe') = ,
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
(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)
+ 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() 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(| ) — 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 (