Quasi-hydrodynamic Semiconductor Equations

Progress in Nonlinear Differential Equations and Their Applications Volume 41 Editor Haim Brezis Universite Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P.L. Lions, University of Paris IX Jean Mahwin, Universite Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath

Ansgar Jiingel

Quasi-hydrodynamic Semiconductor Equations

Birkhauser Basel . Boston . Berlin

Ansgar Jiingel Fakultat fUr Mathematik und Informatik Universitat Konstanz Universitatsstrasse 10 78457 Konstanz Germany

2000 Mathematics Subject Classification 35K55, 35J60, 35Q99, 76W05

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Jiingel, Ansgar, 1966Quasi-hydrodynamic semiconductor equations / Ansgar Junge\. p. cm. -- (Progress in nonlinear differential equations and their applications; v. 41) Includes bibliographical references and index. ISBN 3764363495 (alk. paper) -- ISBN 0-8176-6349-5 (alk. paper) I. Semiconductors--Mathematical models. 2. Electron transport--Mathematical models. 3. Differential equations, Parabolic. 4. Differential equations, Elliptic. 5. Differential equations, Nonlinear. I. Title. II. Series. QC611.6.E45 J85 2000 537,6'22'0151 18--dc2 1

00-044431

Deutsche Bibliothek Cataloging-in-Publication Data Jiingel, Ansgar: Quasi-hydrodynamic semiconductor equations / Ansgar JUnge\. - Basel; Boston; Berlin: Birkhauser, 2001 (Progress in nonlinear differential equations and their applications; Vo\. 41) ISBN 3-7643-6349-5

ISBN 3-7643-6349-5 Birkhauser Verlag, Basel- Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.

© 2001 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced of chlorine-free pulp. TCF OCJ Printed in Germany ISBN 3-7643-6349-5 987654321

Contents

Preface . . . . . 1 Introduction 1.1 A hierarchy of semiconductor models . 1.2

Quasi-hydrodynamic semiconductor models

IX

1

15

2 Basic Semiconductor Physics 2.1

Homogeneous semiconductors

21

2.2

Inhomogeneous semiconductors

23

3 The Isentropic Drift-diffusion Model 3.1 Derivation of the model . 3.1.1 Semiconductor equations based on Fermi-Dirac statistics . . . . 3.1.2 The isentropic model-scaling . 3.1.3 The convergence result

27 27 30 34

3.2

Existence of transient solutions . 3.2.1 Assumptions and existence result 3.2.2 Proof of the existence result .

36 36 38

3.3

Uniqueness of transient solutions

48

3.4

Localization of vacuum solutions 3.4.1 Main results . 3.4.2 Proofs of the main results 3.4.3 Numerical examples .

62 62

3.5

Numerical approximation . 3.5.1 The mixed finite element discretization in one space dimension. . . . . . . . . . 3.5.2 Numerical examples in one space dimension

65 83

90 90 94

Contents

VI

3.5.3 3.5.4 3.6

4

Current-voltage characteristics . 3.6.1 Numerical current-voltage characteristics .. 3.6.2 High-injection current-voltage characteristics

99 105 112 112 113

The Energy-transport Model Derivation of the model . . . . . . . . . . . . 4.1.1 General non-parabolic band diagrams 4.1.2 A drift-diffusion formulation for the current densities . . . . . . . . 4.1.3 A non-parabolic band approximation. 4.1.4 Parabolic band approximation

127 130 131

4.2

Symmetrization and entropy function

133

4.3

Existence of transient solutions .... 4.3.1 Assumptions and main results 4.3.2 Semidiscretization ....... 4.3.3 Proof of the existence result . .

137 137 139 140

4.4

Long-time behavior of the transient solution.

153

4.5

Regularity and uniqueness . . . . . . . . 4.5.1 Regularity of transient solutions 4.5.2 Uniqueness of transient solutions

156 156 166

4.6

Existence of steady-state solutions

170

4.7

Uniqueness of steady-state solutions

177

4.8

..... Numerical approximation 4.8.1 The mixed finite element discretization in one space dimension . 4.8.2 Numerical results .. ..

180

4.1

5

The mixed finite element discretization in two space dimensions . . . . . . . . . Numerical examples in two space dimensions

119 119

180 185

The Quantum Hydrodynamic Model 5.1

Derivation of the model . . . .

191

5.2

Existence and positivity . . . . 5.2.1 Existence of steady-state solutions 5.2.2 Positivity and non-positivity properties

197 197 206

5.3

Uniqueness of steady-state solutions

209

5.4

A non-existence result . . . . . . . .

212

vii

Contents

5.5

The classical limit 5.5.1 The classical limit of the thermal equilibrium state 5.5.2 The classical limit in the 'subsonic' steady state 5.5.3 Numerical examples . .

217 217 222 235

5.6

Current-voltage characteristics 5.6.1 Scaling of the equations . . . . . . . . . . 5.6.2 Analytical and numerical current-voltage characteristics. . . . . . . . . . . .

238 238

A positivity-preserving numerical scheme . . . . 5.7.1 Semidiscretization in time. . . . . . . . . 5.7.2 Stability bounds and convergence results. 5.7.3 Numerical examples

244 247 253 262

5.7

242

References

267

Index . . .

291

Preface

In the last three decades, the mathematical modeling and simulation of charge transport in semiconductors have become a thriving research area in applied mathematics. Semiconductor device modeling started in the early fifties when the Van Roosbroeck drift-diffusion equations were formulated. These equations constitute the most popular model for the simulation of semiconductors. With the increasing miniaturization of semiconductor devices, one comes closer to the limit of validity of the drift-diffusion equations and a better description of the electrical behavior of the devices is needed. In the last years, various quasi-hydrodynamic or kinetic models have been derived in order to improve the physical description of the semiconductors. The numerical simulation using kinetic equations, however, requires a lot of computing power in real life applications. Therefore, the quasi-hydrodynamic equations seem to represent a reasonable compromise between computational efficiency and an accurate description of the underlying device physics. In this book we concentrate on three different quasi-hydrodynamic models: the isentropic (or degenerate) drift-diffusion equations, the energy-transport model, and the quantum hydrodynamic equations. We present the (formal) derivation of each of the models, analyze the corresponding mathematical problems, discretize the equations, and perform numerical simulations of semiconductor devices showing the particular features of the models. The connections between these models and other semiconductor equations used in the literature are also investigated. Mathematically, the considered models are systems of nonlinear elliptic and parabolic equations, being of degenerate type, strongly coupled and/or with a quadratic growth of the gradient of the solution. We prove the existence and uniqueness of solutions to the (initial) boundary value problems and investigate their qualitative behavior, like positivity of the solutions, space and time localization of so-called vacuum sets (sets where the solution vanishes), longtime asymptotics, and regularity of the solutions. Moreover, we discretize the

x

Preface

equations, using semi-discretization in time and mixed finite element methods, and present numerical examples of modern semiconductor devices like channels of MOS transistors, pn-junction diodes, bipolar transistors, and resonant tunneling diodes. We notice that the considered models also appear in other application areas than semiconductor theory. In general, fluids or gases of charged particles are described. The drift-diffusion equations are connected with models for the flow of immiscible fluids through a porous medium or for electro-diffusion processes in electro-chemistry and biophysics. In fact, in biophysics drift-diffusion models are known as the Nernst-Planck equations. The energy-transport model arises in plasma physics, alloy solidification processes and in nonequilibrium thermodynamics. Finally, models similar to the quantum hydrodynamic equations have been used in superfluidity and thermistor theory. I want to express my gratitude to those who contributed to this work. First of all, I am very indebted to Peter Markowich (Vienna) for his generous support and encouragement. I am grateful to M. Anile (Catania), P. Degond (Toulouse), J. 1. Diaz (Madrid), 1. Gamba (Austin, USA), 1. Gasser (Hamburg), Y. Guo (Providence, USA), M. C. Mariani (Buenos Aires), A. Marrocco (Paris), F. Otto (Bonn), Y.-J. Peng (Clermont-Ferrand), P. Pietra (Pavia), F. Poupaud (Nice), C. Ringhofer (Phoenix, USA), C. Schmeiser (Vienna), C. Schwab (Zurich), Y. Sone (Kyoto), A. Unterreiter (Kaiserslautern), and E. Zuazua (Madrid) for invitations to their institutes and many hours of stimulating discussions. Furthermore, I would like to express my thanks to C. Baiocchi (Pavia), N. Ben Abdallah (Toulouse), F. Brezzi (Pavia), J.A. Carrillo (Granada), C. Cercignani (Milano), L. Caffarelli (Austin, USA), H. Gajewski (Berlin), G. Galiano (Oviedo), C. Gardner (Phoenix, USA), S. Genieys (Lyon), T. Goudon (Nice), J. Hernandez (Madrid), C.D. Levermore (Tucson, USA), J. Naumann (Berlin), B. Perthame (Paris), R. Pinnau (Darmstadt), E. Scholl (Berlin), C.-W. Shu (Providence, USA), J. Soler (Granada), W. Strauss (Providence, USA), G. Toscani (Pavia), L. Tello (Madrid), N. Trudinger (Canberra), and O. Vanbesien (Lille) for valuable discussions and various suggestions on the material. November 2000

Ansgar Jungel

Chapter 1 Introduction

In this chapter a hierarchy of kinetic and quasi-hydrodynamic semiconductor equations is introduced and described. The connections between the various (semi-) classical and quantum models are explained (Section 1.1). Three quasihydrodynamic equations are considered in more detail, and an outline of the contents of this work is presented (Section 1.2).

1.1

A hierarchy of semiconductor models

Roughly speaking, we can divide semiconductor models into two classes: kinetic models and quasi-hydrodynamic (fluid dynamical) models. For each of these classes we can consider (semi-) classical and quantum models (see Fig. 1.1). For instance, the Boltzmann and the quantum Boltzmann equation are kinetic equations, whereas the hydrodynamic, the energy-transport and the drift-diffusion equations are quasi-hydrodynamic models. Fluid dynamical models which take into account quantum effects are, for instance, the quantum hydrodynamic, the quantum energy-transport and the quantum driftdiffusion equations. Other semiconductor models, for instance kinetic models, so-called SHE models or high-field models, can be found in the literature [73, 106, 113, 275, 310, 311], but in this section we only discuss the above mentioned models. (The SHE model will be introduced in Section 4.1.1.) The Boltzmann equation. The Boltzmann equation for Semiconductors models the flow of charge carriers (e.g. electrons) in semiconductor crystals. It describes the temporal evolution of the phase space (position-momentum space) distribution function f = f(x, k, t), where x E JR3 is the position variable, k E B is the wave vector, B denotes the Brillouin zone associated with the underlying crystal lattice [33], and t :::::: 0 is the time. The semi-classical Boltzmann

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations © Springer-Verlag Berlin Heidelberg 2000

Chapter 1. Introduction

2

QUANTUM MODELS

CLASSICAL MODELS: KINETIC

Semiconductor ~oltzmann equation

MODELS

-

- --

model

CD model

:®

:@ I I I I I I I

HYDRO-

CD Isentropic drift-

:@

diffusion model MODELS

Quantum lBoltzmann equation

I I

I I I I I I I

Energy-transport

DYNAMIC

CD

CD: 0/.~ ------:------~----------------

Hydrodynamic

QUASI-

I I I

®

Quantum hydrodynamic model

SchroedingerPoisson system

@ Quantum energytransport model

@ Quantum driftdiffusion model

0 Standard driftdiffusion model

Figure 1.1: A hierarchy of semiconductor models.

equation reads [80, 275, 311]

of

q at +v(k)· \lxf + 1i\lxv. \lkf =

Q(J),

f(x,k,O) = fr(x,k),

x E IR

3

3

x E IR

,

k E B, t > 0,

,

k E B,

(1.1)

subject to periodic boundary conditions on aB. Here, the physical constants are the elementary charge q and the reduced Planck constant ti, v(k) = (ljti)'\hc(k) is the mean electron velocity, c(k) the energy-wave vector function, and V = V(x, t) is the electric potential. The collision operator Q(J) is supposed to model short range interactions of the electrons with crystal impurities, phonons, and electrons (see [275] for some examples). In the parabolic

1.1. A hierarchy of semiconductor models

3

band approximation, the energy band function c(k) can be written as

ti2 c(k) = E c + -2-lkI2 , m* where E c is the conduction band minimum energy and m* is the effective electron mass. The electrostatic potential V is a given function or it is coupled selfconsistently to the Poisson equation cs~V =

q(n - C(x)).

In this equation Cs denotes the semiconductor permittivity and C = C(x) models fixed charged background ions (doping profile). The electron density n = n(x, t) is defined by

n=

hfdk.

The first result on the existence of solutions to the space homogeneous Boltzmann equation has been obtained by Carleman in 1932 [72]. The corresponding £1 theory has been developped by Arkeryd [25]. In [26] solutions close to a space homogeneous solution have been studied. DiPerna and P.L. Lions have proved the first general global existence result for the Boltzmann equation in 1987 [126] (also see the review article of Gerard [178]). Further results can be found in, e.g. [78, 79, 266, 309]. We refer to [27, 80, 278] for reviews of recent mathematical results. Numerically, the Boltzmann equation has been solved by Monte-Carlo methods [12, 13, 202, 250], deterministic particle methods [59, 60, 112, 292] or finite difference schemes [138]. An overview of several numerical techniques is given in [359]. The numerical simulation of the Boltzmann equation (or the BoltzmannPoisson system) requires a lot of computing power in real life applications. Therefore, simpler models which represent a reasonable compromise between the physical accuracy and computational efficiency have to be derived, e.g. fluid dynamical models. One of these fluid dynamical models are the hydrodynamic equations. Hydrodynamic models. The hydrodynamic model is a system of hyperbolicparabolic equations which can be derived from the Boltzmann equation by using a moment method (arrow 1 in Fig. 1.1). The idea of this method is to multiply the Boltzmann equation by 1, v, and ~lvl2 and to integrate over the velocity space. This leads to a system of equations for the first moments of the distribution function. Making an ansatz for the distribution function, a closed system of equations can be obtained. Assuming the parabolic band approximation (see above), we can identify v with k and we set B = 1R3 . For the collision operator QU) a low density

Chapter 1. Introduction

4

collision term is used which is linear in f [275]. The particle density, the current density, and the energy tensor, respectively, are defined (essentially) as the first moments of the distribution function f:

r f(x, v, t)dv, iR3

n

-q E

r

iR3

m* tr 2

=

(1.2)

vfdv,

rv

iR3

(9

vfdv,

where 'tr' is the trace of the tensor v (9 v whose ij-th element is given by As an ansatz for the moment method, the shifted Maxwellian

ViVj'

m* )3/2 (-m*lv-u I2 ) f(x, v, t) = n ( 27fkBT exp 2k T B

can be used since this function belongs to the kernel of QU) [275]. The electron density n, the effective temperature T and the mean velocity u are the free parameters. After inserting the ansatz function, the Boltzmann equation is multiplied by a number of linearly independent functions of velocity and integrated over the velocity space. The result are partial differential equations for the time and space dependent parameters n, T, and u (arrow 1):

on at

- + V'. (nu) kB -au + (V' . u)u + -V'(nT) -

m

m*n

q m*

3

(1.3) (1.4)

-V'V

2 -aT + -TV' . u + u . V'T

at

0,

CT.

(1.5)

The terms on the right-hand sides, stemming from the collision operator, are the relaxation terms. If they are omitted the system (1.3)-(1.5) represents the Euler equations of gas dynamics for a gas of charged particles in an electric field. For the moments (1.2) we get

I n = -qnu, We rewrite the hydrodynamic equations (1.3)-(1.5) in the variables (n, I n , E):

aJn

at

_

!div q

(I n I n (9

n ) _

on _ !divJ at q n qk B V'(nT) m*

+ LnV'v m*

0,

(1.6) (1. 7)

1.1. A hierarchy of semiconductor models

5

-In . \7V + CEo (1.8) These equations can be considered in the whole space with x E 1R3 , t > 0, or in a bounded domain with appropriate boundary conditions [7, 345]. The relaxation terms are written in the relaxation time approximation as (see, e.g., [175])

(1.9) where T p , T w are the momentum and energy relaxation times, respectively, and T L is the lattice temperature. In [58], where the model (1.3)-(1.5) for semiconductors has been introduced, an additional heat conduction term -div (r;,\7T)

°

was added to the left-hand side of (1.8). Here, r;, > is called heat conductivity. In [320] this term was derived from a closure condition in the moment method. Using this heat conduction term, the equation (1.8) transforms into

(1.10) The derivation of the hydrodynamic model from the semiconductor Boltzmann equation has been investigated in, e.g., [12, 13, 320, 350]. The problem of the closure conditions in hydrodynamic equations has been addressed in [12, 14]' using the maximum entropy principle (also see [285]). There is a huge literature for the fluid dynamical limits, in particular Euler and Navier-Stokes equations, from the Boltzmann equation. The first rigorous result for the hydrodynamical limit was carried out by Caflisch [71] and generalized by Lachowicz [252]. The compressible Euler equations have been derived by Nishida [293] and Ukai and Asano [349] based on the work by Grad [183]. The Navier-Stokes equations are derived from the Boltzmann equation in [78, 115]. For new developments, we refer to the works of Levermore et al. [38, 39, 258]. Mathematically, the hydrodynamic equations form a quasilinear hyperbolicparabolic system of balance laws. The solution can become discontinuous in finite time. We refer to [257] for general references on various shock waves phenomena. In the last years, many authors have been examined the existence of solutions to this or to the isentropic model. The isentropic model consists

Chapter 1. Introduction

6

of the first two equations (1.6), (1.7), where the temperature is assumed to depend only on the density:

a> 1,

To> O.

If a = 1 then T is constant, and the corresponding equations are called the isothermal hydrodynamic model. The existence of global weak entropy solutions to the transient isentropic model in one space dimension with zero electric field has been first proved by DiPerna [124, 125] for a = 1 + 2/(2N + 1) where N E N, N > 1, using compensated compactness techniques. Chen and Ding et al. [83, 84, 123] extended the existence result to 1 < a ~ 5/3, whereas P.L. Lions, Perthame and Tadmor [260, 261] gave a proof for all a > 1. The isothermal case a = 1 has been considered by Matsumura and Nishida in [279]. The existence of global solutions to the time-dependent model including the Poisson equation has been proved in [96, 97, 312] for the isothermal case and in [213, 214, 269, 360, 374, 376] for the isentropic case. Degond and Markowich [110, 111]' Gamba [157, 161], Fang and Ito [135, 136] considered the stationary isentropic model in one and two space dimensions. Existence of solutions of the full system of Eqs. (1.6), (1.7), (1.10) in one or several space dimensions has been studied in [8, 371] for the stationary problem and in [361, 370, 378] for the transient equations. For related results for Euler-Poisson systems with geometric structure, see [85, 88]. The numerical discretization and solution of the hydrodynamic model, by using Scharfetter-Gummel type methods (for subsonic transport) or streamline diffusion methods (for transonic flow), is studied in [90, 162, 211, 212, 321, 322]. Other numerical techniques, like the Nessyahu-Tadmor scheme or Gudonov scheme or (essentially non-oscillatory, ENO) shock-capturing algorithms, are used in [87, 137, 164, 210, 267, 318, 336, 374]. Energy-transport models. In applications it turns out that the quotient c 2 = Tp/Tw (see (1.9)) is very small compared to one. Using c as a parameter in the diffusion time scaling (1.11) and letting c --t 0 in the equations (1.6), (1.7), and (1.10), assuming the Wiedemann-Franz law

"'0> 0,

(1.12)

we get the energy-transport model (arrow 2):

an = at _ ~divJ q n

0

,

(1.13)

1.1. A hierarchy of semiconductor models

~~ In =

div Jw = -In

·

7

~V + W(n, T),

(1.14)

L11 (~nn _q~V) + (L 12 _ ~L11) ~T, kBT kBT 2 T ( ~n

q~V)

qJw = L 21 ---:;; - kBT

+

( L 22 3 ) ~T kBT - "2 L21 T'

(1.15) (1.16)

We refer to [175] for details and a justification of the asymptotic limit. Here, U = ~kBnT is the internal energy, W(n, T) = ~nkB(T - TL)/To the energy relaxation term, and the diffusion matrix is given by

L = (L ij ) = /LOnkBT

(~k~T e45 +~/'L:)~BT)2

),

(1.17)

with the mobility constant /Lo = qTo/m*. Usually, Eqs. (1.13)-(1.16) are considered in a bounded domain subject to mixed Dirichlet-Neumann boundary conditions for n, T and initial conditions for n, U (see Section 4.1). The energytransport equations can be obtained formally from the hydrodynamic equations by neglecting all terms including IJn I2 , I n @ I n , and 8tln (see [320]). The model (1.13)-(1.16) can also be derived directly from the Boltzmann equation in the diffusive limit, using the Hilbert expansion method [41, 42, 45, 177]. In this derivation, the dominant scattering mechanisms are assumed to be electron-electron and elastic electron-phonon scattering. The same diffusion equations as above are obtained, but the diffusion coefficients are different. One example is:

L -- (L tJ.. ) -_ /Lon -

(1

~kBT) (k T)2 .

( ) 1.18 B In order to get this diffusion matrix, we assumed the parabolic band approximation, Boltzmann statistics and a special ansatz for the momentum relaxation time (see [107] for details). The corresponding model has been considered by Chen et at. [81]. Different momentum relaxation time approximations give different diffusion matrices (see Section 4.1). Note that both diffusion matrices (1.17) and (1.18) are symmetric and positive definite. Related energy-transport models are derived in [113]. The derivation of the energy-transport model from the Boltzmann equation is explained in more detail in Section 4.1. In the physical and mathematical literature, the energy-transport equations are investigated numerically since several years. They are discretized by using extensions of the Scharfetter-Gummel scheme [23, 338, 339] or by using ENO schemes [208, 210] and solved by using the Gummel method [264,357,358] or the full Newton method [81]. The results are compared to Monte-Carlo simulations [81, 82, 301] or to hydrodynamic simulations [81, 181, 339]. Recently, the equations are numerically solved by using mixed finite elements [107, 277] (see Section 4.8) or by employing compact finite difference schemes [142]. ~kBT

15 4

Chapter 1. Introduction

8

Mathematically, however, the energy-transport equations are analyzed only recently. In [5] a stationary energy-transport system with very special diffusion coefficients (not being of the form (1.15)-(1.16)) has been investigated and the existence of solutions has been proved. Jerome [206] and Griepentrog [184] proved the existence of steady-state solutions of particular systems under restrictive conditions on the data. Some mathematical properties of the energy-transport equations are proved in [4,55]. In [101, 105] the existence and uniqueness of steady-state solutions to the energy-transport system with general uniformly positive definite diffusion matrix have been shown, whereas in the papers [102, 104] the parabolic equations with general diffusion matrix are studied. The existence of global weak solutions could be proved. The question of uniqueness of transient solutions is investigated in [229]. The energy-transport model is studied in Chapter 4. Drift-diffusion models. In gas dynamics, an ideal gas satisfies the gas law = nT, where r denotes the pressure of the gas. In the isentropic case, the temperature (only) depends on the particle density. Then T(n) = T on 2 / 3 holds for particles without spin for adiabatic and hence for isentropic states [98]. Note that the physical unit of To is Kcm 2 • If the diffusion matrix is described by (1.17), we can rewrite the electron current density as r

In

=

f..loC'v(nkBT)- qnV'V),

(1.19)

and we get (1.20) This current relation together with the continuity and Poisson equation

an at

-

1

-divJ q n

cstlV

0,

(1.21 )

q(n - C(x)),

(1.22)

is called the isentropic drift-diffusion model (arrow 3). Mathematically, Eq. (1.21), together with (1.20), is a degenerate parabolic equation. Therefore, socalled vacuum sets {x : n(x, t) = O} may occur locally. Physically, these sets can be interpreted as regions in which the particle density is very small compared to the reference density. The electron density signify the number of electrons in the conduction band of the semiconductor crystal (per space unit). The motion of the holes, or defect electrons, at the top of the valence band also renders a contribution to the current flow in the crystal (see Section 2.1). The hole density is denoted by p, the hole current density by Jp . A positive charge +q is assigned to the holes.

1.1. A hierarchy of semiconductor models

9

Therefore, the evolution of the hole density is described by the equations

ap 1. -at + -dlvJ q P

0,

(1.23) (1.24)

where the Poisson equation now writes Es~V =

q(n - p - C(x)).

(1.25)

The equations (1.20)-(1.21), (1.23)-(1.25) are called the bipolar isentropic driftdiffusion model. Usually, these equations are considered in a bounded domain with mixed Dirichlet-Neumann boundary conditions for n, p, V and initial conditions for n, p (see Section 3.1). There are several derivations of this model. One derivation starts from the hydrodynamic equations. Formally, we obtain (1.20) from (1.7) by assuming isentropic states and by neglecting atJn and the convective term q-1div (In <81 In/n). More rigorously, the isentropic drift-diffusion model can be derived from the isentropic hydrodynamic equations in the zero-relaxation-time limit [86, 199, 216, 217, 232, 233, 234, 255, 256, 268, 273, 288, 315]. Another derivation starts from the relations (1.26)

where the carrier densities are given by Fermi-Dirac statistics [62, 154, 222]: n =

NcFl/2(~~ (
P=

NvFl/2(~~ (
V

+ ~V)).

Here, 0 is called diffusivity. In the high injection case F0~ (y) can be approximated by a constant times y2/3 so that V7
(~~ (
p = Nvexp

(~~ (
V

+ ~v)),

Chapter 1. Introduction

10

we get from (1.26) the (bipolar) standard drift-diffusion model (arrow 4), consisting of the continuity equations (1.21), (1.23), the Poisson equation (1.25) and the current relations

I n = f-Lo(kBTo\7n - qn\7V),

Jp = -f-Lo(kBTo\7p + qp\7V) ,

(1.27)

where the diffusivity is related to the mobility by the Einstein relation Do = UTf-LO,

TTT __ kBTo. q

U'

The physical unit of To is here K. This set of equations was first proposed by Van Roosbroeck in 1950 [319]. The first computational solution was presented in 1964 by Gummel [191] and improved some years later by Scharfetter and Gummel [326]. The developped Scharfetter-Gummel scheme was interpreted as a mixed finite element method in [66, 68]. In the seventies mathematicians started to pay attention to the drift-diffusion model. Mock proved an existence and uniqueness result in 1974 [281]. In the eighties a large amount of papers can be found in the mathematical literature, where the drift-diffusion equations and variants of them are analyzed (see [205, 272, 275, 282] for an overview and references). The modeling aspects are summarized in [194, 333]. Also in the last years the isothermal drift-diffusion model (1.27) and its variants were the subject of various mathematical research both analytically (see, e.g., [2, 131, 132, 133, 134, 145, 151, 167, 230, 231, 233, 289, 290, 314, 327, 328, 355]) and numerically (see, e.g., [152, 176, 195, 197, 206, 207, 283, 324, 362]). The standard drift-diffusion model can also be derived from other equations. For instance, assuming T = const. in the energy-transport model in the formulation (1.19) (for the electron current density), we directly get (1.27). Similar as for the isentropic drift-diffusion model, the standard drift-diffusion equations can be derived from the hydrodynamic model (1.6)-(1.8) (or the energy-transport model) in the zero-relaxation-time limit [175, 216]. Finally, the standard model can be obtained directly from the Boltzmann equation by using the Hilbert expansion method [275, 308]. More general drift-diffusion models than presented here have been derived; see, e.g., [73, 113, 182]. Quantum models. For ultra-small electronic devices in which quantum effects are present, the mathematical semiconductor models have to incorporate the quantum mechanical phenomena. The quantum Boltzmann equation is a kinetic equation describing the evolution of the Wigner function w = w(x, v, t) [193]:

aw +v· \7 w + -()~[V]w q at x m*"

-

w(x, v, 0)

Qn(w),

wo(x, v),

x,v E ffi.3, t > 0,

(1.28) (1.29)

1.1. A hierarchy of semiconductor models

11

Here, Q1i.(w) is the collision operator and V is the effective potential given, in the case of Coulomb interactions, by the Poisson equation (1.22). The electron density and the current density, respectively, are defined by n

=

r w(x, v, t)dv,

lIR3

I n = -q

r vw(x,v, t)dv.

lIR3

The operator (h[V] is a pseudo-differential operator with symbol clef

(8V)1i.(x, "I, t) =

im* ( n T V(X + 2m* "I, t) -

n)

V(X - 2m* "I, t) ,

i.e. B1i.[V]w is defined by

(B1i.[V]w)(x,v, t)

r r (8V)1i.(x,"I,t)w(x,v',t)e(v-v')·7/dv'd"l

_1)3 ( 2n

=

i

lIR3 lIR3

for appropriate functions w. For the mathematical setting of pseudo-differential operators we refer to [343, 346]. In the (semi-)classical limit 0, the collisionless quantum Boltzmann equation, which is called quantum Vlasov equation, becomes (for given V)

n---.

which is similar to the Boltzmann equation (1.1) (arrow 5). We refer to [275, Ch. 1.4] for details on this limit. The quantum Vlasov equation is equivalent to a system of countable many Schrodinger equations for the wave functions 'l/Jk (arrow 6):

n2

ina'l/Jk at 'l/Jk(X, 0)

- 2m*tl'l/Jk- qV'l/Jk, x E IRa, t > 0, 'l/J~(x),

(1.30)

x E 1R3 .

(1.31 )

The existence of solutions to Schrodinger-Poisson systems has been shown in [76,254]. The stationary equations have been studied in [43, 201, 240]. For the mathematical techniques, see [77]. From the quantum Boltzmann equation quantum fluid dynamical models can be obtained. Using a moment method, the (full) quantum hydrodynamic model can be derived (see [163, 171, 173, 174,200,367], arrow 7):

an _ ~div J = at q n aJn at

_

~div (In q

0

(1.32)

,

18)

n

In

) _

qkBV(nT) m*

+ LnV(V +Q) = m*

_

In Tp

,

(1.33)

12

Chapter 1. Introduction

aE

-

at

-

d'IV

(m*

JnIJnl 2q3 n 2

-

= cs~V =

2

5k BTJn + + -2q

-In . \l(V + Q)

I>;

n v

T)

+ GE,

(1.34)

q(n - G(x)),

(1.35)

where E is the energy density given by

the term

Q=~~.;n 2qm*

.;n

is the Bohm quantum correction, and GE is the energy relaxation term defined in (1.9). The heat conduction term -1>;\lT comes from the closure condition. The quantum correction to the energy density was first derived by Wigner [366]. The hydrodynamic formulation of quantum systems has been used already by Madelung in 1927 [265]. Eqs. (1.32)-(1.35) are considered in the whole space JR3 or in a bounded domain with initial conditions for n, I n and E. The question of appropriate boundary conditions is delicate; see [163, 228, 237] for some answers in special situations. Assuming that the particle temperature T is given by the explicit expression T(n) = Ton a - I with a 2 1, we obtain the isentropic/isothermal version of the quantum hydrodynamic equations, where the energy equation can be dropped. In the formal classical limit Ii ----+ 0 we get the hydrodynamic equations (1.6)(1.7) (arrow 8). This limit is justified in [169, 352] for the thermal equilibrium state and in [192] for 'subsonic' solutions. The quantum hydrodynamic model without relaxation terms can also be derived from the system of countable Schrodinger equations (1.30)-(1.31) assuming a closure condition in the energy equation (arrow 9). We refer to Section 5.1 for details of this derivation. The quantum hydrodynamic equations in the thermal equilibrium state I n = 0 are mathematically studied in [64, 169, 298, 352]. The existence and uniqueness of solutions to the stationary model are investigated in [159, 192, 228, 375]. For results on the transient equations, see [168, 170]. Gardner has used this model to simulate numerically resonant tunneling diodes [89, 163, 166], whereas Ferry and Zhou solved numerically a modified set of equations for the simulation of high electron mobility transistors (HEMT; [377]). The full quantum hydrodynamic equations are numerically solved in [302, 306] (also see [367]). A dispersive numerical method is developped in [172]. The quantum hydrodynamic model is analyzed in Chapter 5.

1.1. A hierarchy of semiconductor models

13

It is common to replace the potential V by V +B, where B is a discontinuous function modeling heterostructures. The disadvantage of the above model is that discontinuities in V + B lead to difficulties in the numerical discretization ofEqs. (1.33), (1.34). Gardner and Ringhofer have derived in [165, 166] so-called "smooth" quantum hydrodynamic models where the quantum term involves a smoothed potential. More precisely, they get from the Bloch equation in the Born approximation the equations (1.32)-(1.35) in which the term

ti2 4qm*n

\7Q = - - d i v (n(\7 0 \7) log(n))

in the momentum and energy equation is replaced by

2 qti . ( k n T (\7 0 \7) -) V , 4m*n B

--dlV

with the definition

V(x,,8)

=

1 f!3(,8')2B(,8')3/2 f e- B(!3')(x'-x)2 V (x')dx'd,8'. (1r,82)3/ 2 Jo J.JjP

Here, B(,8') = 2m*,8/((,8 + ,8')(,8 - ,8')ti2) and,8 = l/k BT is the inverse temperature [165]. Moreover, the energy density is now given by

The quantum hydrodynamic equations are obtained in the O(ti2) approximation:

v \7 log(n) if nand I n are varying very slowly such that n can be approximated by exp(- V/ kB T) and T is nearly constant. Then -

1

n(\7 0 \7)V = 3kBTn(\7 0 \7) log(n)

+ O(ti2 )

and we get Eq. (1.33), where the term n\7Q is replaced by (1/3)n\7Q. The factor 1/3 comes from the mixed quantum states ("thermal averaging") and does not appear in the derivation from the Schrodinger-mixed quantum states formulation (see Section 5.1).

14

Chapter 1. Introduction

We can use a diffusion scaling in the full quantum hydrodynamic equations, similar as in the full hydrodynamic model, to obtain the so-called quantum energy-transport model. More precisely, we are using the scaling (1.11), assuming that c 2 = Tp/Tw is a small parameter. Then, letting c ---. 0 in the full hydrodynamic equations, we obtain (arrow 10)

an at

-

In

1. -dlvJ = 0 q n ,

= !-Lo(k B\7(nT) - qn\7(V + Q)),

aE - dlV. (5k _) at 2iB T I n + r;,\7T

where !-Lo

= qTo/m* and

K,

=

=

3 - In . \7(V + Q) - 2To n(T - Td,

r;,O!-Loq-lk~nT. The energy density becomes

3 E = -kBnT 2

h2 24m*

--n~log(n).

Clearly, we can write these equations in the form (1.13)-(1.16) with V replaced by V + Q and the diffusion matrix (1.17). Formally, in the limit h ---.0 we obtain the classical energy-transport equations (arrow 11). The quantum energy-transport equations are yet not studied, neither analytically nor numerically. In the zero-relaxation-time limit the convective term in the isentropic quantum hydrodynamic model can be neglected [227, 303] and we obtain the quantum drift-diffusion equations (arrow 12):

an at

-

1. -dIVJ = 0 q n ,

h2 ) - !-Lo I n =!-Lo (kBTo\7(na ) - qn\7V 2m* n\7 (~vn)

vn '

with the Poisson equation (1.22), considered in a bounded domain with an initial condition for n, mixed Dirichlet-Neumann boundary conditions for n, V and an additional boundary condition for the quantum term (for instance, a homogeneous Dirichlet-Neumann condition for Q; see [237] or Section 5.7). The quantum drift-diffusion model with a = 1 is used to describe the behavior of electrons in the vicinity of strong inversion layers near the gate oxide of MOS transistors [9, 364] or as a quantum correction to the classical driftdiffusion equations [316, 348, 372]. Mathematically, the above equations form a fourth-order parabolic equation which has the important property that the non-negativity of the particle density is preserved [236, 237]. The stationary model has been analyzed in [46] und numerically solved in [305]. The transient

1.2. Quasi-hydrodynamic semiconductor models

15

°

equations are studied analytically and numerically in [236, 237J. In the classical limit Ii, ----; we obtain (formally) the isentropic drift-diffusion equations (arrow 13). Other approaches to the modeling of quantum semiconductors are, for instance, Wigner function methods [28, 248J and density matrix methods [146J.

1.2

Quasi-hydrodynamic semiconductor models

In this work we are concerned with the modeling, mathematical analysis and numerical approximation of three classes of quasi-hydrodynamic models for semiconductors: (i) the isentropic drift-diffusion model, (ii) the energy-transport model, (iii) the quantum hydrodynamic model. The first semiconductor model can be used for devices in high injection situations. It is the simplest model in which an electron temperature is defined, via the ideal gas law. The isentropic drift-diffusion equations extend the standard drift-diffusion model for semiconductors which is widely used in industrial simulations. The second model includes an energy equation for the temperature and is used for hot electron simulations. We expect that this model will be a "standard semiconductor model" for industrial applications in the future since it is simple enough in order to perform efficient numerical simulations but contains important physical features necessary for submicron semiconductor modeling. Finally, the third model is proposed for the simulation of quantum devices, like the resonant tunneling diode. It can be seen as a dispersively regularized version of the hydrodynamic equations. Therefore, the model is very interesting both from an analytical and numerical point of view. For this model, the existence theory and the numerical approximation are not completely understood. These quasi-hydrodynamic models also appear in other application areas than semiconductor theory. Indeed, the isentropic drift-diffusion equations are similar to equations for the flow of immiscible fluids through a porous medium [18, 149J. Drift-diffusion models in general find utility in various electro-diffusion processes in electro-chemistry [34, 91] and biophysics [335J. The drift-diffusion equations are also known as the Nemst-Planck model [37, 130, 299]. Energy-transport models arise in charged particle physics (e.g. plasmas) [108, 109] and in nonequilibrium thermodynamics [249]. They are also used in alloy solidification processes [47, 196J. Finally, the quantum hydrodynamic model describes generally the flow of charged quantum fluids, and very similar model equations have been used in superfluidity [263], superconductivity [140], and thermistor theory [95, 368J. In the zero-relaxation-time limit the

16

Chapter 1. Introduction

quantum hydrodynamic model becomes the quantum drift-diffusion equation for which a simplified version arises in the modeling of interface fluctuations of spin systems [57]. Chapter 3 is devoted to the study of the isentropic drift-diffusion equations. As mentioned in Section 1.1, the model can be derived from transport equations incorporating Fermi-Dirac statistics in high injection situations by using asymptotic analysis. In Section 3.1 results are stated under which assumptions this asymptotic limit can be made rigorously. Mathematically, the isentropic model contains parabolic equations of degenerate type which give rise to interesting qualitative properties of the solutions. The main interest of the Sections 3.2-3.4 is the analysis of this system of equations. In Section 3.2 the existence of global generalized solutions is proven. In the proof, first an approximate nondegenerate problem is solved. A solution of the degenerate equations is found by using a Stampacchia-type truncation method and compactness arguments. Uniqueness of solutions in the class of generalized functions is shown under some additional conditions by using variants of the dual method for quasilinear parabolic equations. In this technique a special test function is employed in the weak formulation of the equations, which solves a backward uniformly parabolic problem (Section 3.3). Due to the degeneracy, solutions may exist for which the carrier densities vanish locally (so-called vacuum solutions). The spatial and temporallocalization of the corresponding vacuum sets is investigated in Section 3.4. We show three properties: (i) finite speed of propagation, i.e. if there are (nontrivial) vacuum sets for t = 0 then there are vacuum sets for small time t > 0, (ii) waiting time, i.e. there is no dilatation of the support of the particle densities for small t > 0, and (iii) formation of vacuum, i.e. there exists a time To > 0 such that there is vacuum for t > To even if the initial densities are strictly positive. The proof of these results is based on local energy methods for free boundary problems. The numerical scheme is explained in Section 3.5. The one-dimensional current relations are discretized by using an exponential fitting mixed finite element method. In Section 3.5.1 we present a discretization of the instationary equations in one space dimension. As an application we simulate a forward and backward biased pn-junction diode. It turns out that there exist (non-trivial) vacuum sets in the backward and slightly forward biased case (Section 3.5.2). Due to the asymptotic limit, the vacuum sets can be physically interpreted as regions in which the carrier densities are very small compared to the reference density of the scaling. Therefore, in these situations, no current flows. For applied biases exceeding a certain

1.2. Quasi-hydrodynamic semiconductor models

17

threshold voltage, there are no stationary vacuum sets and a positive current flows. The mixed finite element method is extended to two space dimensions in Section 3.5.3. Here we solve the stationary equations using a variant of the Gummel iteration procedure. Modeling a pn-junctiondiode and a pnp transistor, the good features of the scheme (mainly, current conservation and good approximation of sharp shapes) are illustrated (Section 3.5.4). Static current-voltage characteristics for diodes are presented in Section 3.6. As mentioned above, for applied voltages smaller than the threshold voltage, no current flows. For larger applied biases, the dependence of the current on the voltage turns out to be polynomial. This is in contrast to the Shockley relation which predicts an exponentional dependence in the isothermal drift-diffusion model. However, it is well known that in real diodes, the curve slows down if the applied bias is large enough. This indicates that we are modeling a diode in the high injection regime.

In Chapter 4 the energy-transport model is studied. The derivation from the Boltzmann equation by using a Hilbert expansion method is sketched and the diffusion coefficients are computed for parabolic and non-parabolic band diagrams (Section 4.1). The existence of solutions to the parabolic system is investigated in Sections 4.2 and 4.3. The energy-transport model is presented in a more general context of nonequilibrium thermodynamics. In fact, this model is an example of a general diffusion system for a gas consisting of several components of charged particles. Mathematically, the model consists of nonlinear strongly coupled parabolic or (in the stationary case) elliptic equations. One of the main difficulties in the analysis of this problem is the fact that the variable of the time differential operator and the variable of the space differential operator are different and related by an algebraic relation. The existence proof is based on two main ideas: (i) a transformation of variables which symmetrizes the equations and (ii) a priori estimates by using the entropy function (Section 4.2). The method of the proof is a semidiscretization in time which is also of interest from a numerical point of view (Section 4.3). The properties of the entropy function allow to perform the asymptotic limit as the time tends to infinity (Section 4.4). More precisely, we show that the time-dependent variables tend to the variables of the thermal equilibrium state (no current flow) if t --7 00. In Section 4.5 the regularity and uniqueness of the transient solutions are investigated. We prove that there exists a solution whose time derivative is a L 2 function and whose (spatial) gradient lies in the space LP for some p > 2. It is well known that for parabolic systems we cannot expect more regularity in general [215]. The uniqueness of solutions can be shown in a class of functions

18

Chapter 1. Introduction

slightly smaller than the class of weak solutions. In the proof we use a variant of the elliptic dual method. The next two sections are devoted to the existence and uniqueness of steadystate solutions. The existence proof is based on the above mentioned transformation of variables and utilizes the Leray-Schauder fixed point theorem. Uniqueness can only be shown if the boundary data are not far from the thermal equilibrium state. In general, we cannot expect uniqueness for arbitrary data since there exist semiconductor devices having multiple states [363]. In Section 4.8 we present a numerical discretization of the one-dimensional stationary energy-transport model. The equations can be written in driftdiffusion-like form such that an exponential fitting mixed finite element method can be used. The numerical simulation of a ballistic diode shows the behavior of hot electrons. The results for parabolic and non-parabolic band structures are compared to the simulations using the standard drift-diffusion model. The quantum hydrodynamic model is investigated in Chapter 5. In Section 5.1 we present a (formal) derivation of the model from the many-particle Schrodinger equation and we derive physically relevant boundary conditions for the resulting system of degenerate elliptic equations. The existence of solutions to the stationary problem is shown in Section 5.2. We prove that there exists a steady-state solution if the thermal energy is much larger than the electric energy. The main difficulty of the proof is to find a lower bound for the electron density. The existence of a uniform lower bound is related to the regularity of the gradient of the so-called quantum Fermi potential. For a simplified quantum system, which can be derived (formally) for ultra-small devices, a generalized solution is presented for which the particle density vanishes at one point. Moreover, for the one-dimensional stationary model with special boundary conditions, we can give a non-existence result for (weak) solutions. The uniqueness of steady-state solutions follows for sufficiently large scaled Planck constant (or, equivalently, sufficiently large temperature; Section 5.3). For a vanishing Planck constant (which gives the classical hydrodynamic model), there are situations where at least two solutions exist. As the scaled Planck constant tends to zero, we expect that the solution of the quantum model converges to a solution of the corresponding classical model. In Section 5.5.1 we investigate the classical limit of the thermal equilibrium state, whereas in Section 5.5.2 the classical limit of the 'subsonic' state is proved. Numerical simulations show that in the 'transonic' state strong convergence of the physical quantities cannot be expected in the classical limit. The main application of the quantum hydrodynamic model is the simulation of resonant tunneling diodes, where negative differential resistance has to be expected (see Section 2.2). The current-voltage characteristics J(U) of a tunneling diode is investigated in Section 5.6. The characteristic of a reduced

1.2. Quasi-hydrodynamic semiconductor models

19

quantum model, where diffusion and relaxation effects are neglected, show negative differential resistance (i.e. dJ/dU < 0), even in the absence of potential barriers. In the zero-relaxation-time limit, the quantum hydrodynamic model reduces to the quantum drift-diffusion equations. The numerical discretization of the transient model is investigated in Section 5.7. More precisely, a semidiscretization in time is proposed, preserving the positivity of the elctron density, and the convergence of the discrete solutions is proved. Numerical results for a one-dimensional resonant tunneling diode are performed. The static current-voltage characteristic show negative differential resistance in some region. Thus even for the rather simple quantum drift-diffusion model, negative differential resistance effects can be reproduced. The physical reason for these effects is the external potential barrier where the electrons are tunneling through. As seen in Section 5.6, these effects are reinforced by the convective term in the quantum hydrodynamic model. Notations. We end this section with some notations used in this book: Let n E ~d with d 2:: 1 be a bounded domain with boundary E CO,l, = f D nf N , measd-l(fD) > 0, and f N is relatively open in an. The exterior normal unit vector is denoted by 1/. Denote by WS,p(n) and WS,p(n; ~m) the usual Sobolev spaces of real-valued, vector-valued functions, respectively [1]. The norm of WS,p(n) and WS,p(n; ~m) is denoted in both cases by II· Ils,p = II· Ils,p,n. We define V = HJ(n U f N ) = {u E Hl(n) : u = on fD} with dual space V* and duality pairing (-, .). Furthermore, we set WS,P(X) = WS,P(O, T; X) if X is a Banach space and QT = n x (0, T), T > 0. As usual, V' = grad, V'. = div, ~ = divgrad, and at = a/at. Furthermore, s+ = max(O,s) and s- = min(O,s) for s E R We denote by C, Ci positive constants with values varying from occurence to occurence.

an

°

an

Chapter 2 Basic Semiconductor Physics

This chapter presents a short summary of the physics and properties of semiconductors. Only those subjects relevant to the presented mathematical and numerical analysis in the following chapters are included here. We explain basic notions of the structure of homogeneous semiconductors (Section 2.1) and describe the inhomogeneous semiconductor devices pn-junction diode, bipolar transistor, MOSFET, ballistic diode, and resonant tunneling diode (Section 2.2). We refer to the standard textbooks, e.g. [342, 284, 33, 61, 144] for a detailled consideration of solid-state physics and to [36, 69, 129, 139] for new developments in semiconductor physics.

2.1

Homogeneous semiconductors

Any crystallin solid possesses an energy-momentum relationship c(k), where k is the wave vector, called the (energy) band structure of the solid. The upper bands are called the conduction bands; the lower bands are named the valence bands (Fig. 2.1). More precisely, the energy levels c(k) depend on k in a complicated way depending on the semiconductor material. Often the parabolic band approximation is used, i.e. c(k) = E c + (n2 /2m*)lkI 2 for the conduction band and c(k) = E v - (n2 /2m*)lkI2 for the valence band. Here, E c is the conduction band minimum, E v the valence band maximum, n is the reduced Planck constant and m* the effective electron mass. In this approximation, E g = E c - E v is called the energy gap. A solid with an energy gap will be generally nonconducting at the temperature T = 0 K. However, when the temperature is not zero there is a nonvanishing probability that some electrons of the (occupied) valence bands will be thermally excited across the energy gap into the conduction bands. The thermally excited electrons are capable of conducting, and hole-type conduction can occur in the band out of which they have been excited. Indeed, each empty

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations © Springer-Verlag Berlin Heidelberg 2000

22

Chapter 2. Basic Semiconductor Physics

state in the valence band can be considered as being occupied by a particle of positive charge. These fictitious particles are called holes. Solids that are insulators at T = 0 K, but whose energy gaps are of such a size that thermal excitation can lead to observable conducting at temperatures below the melting point, are known as semiconductors. Evidently the distinction between a semiconductor and an insulator is not a sharp one, but roughly speaking in most important semiconductors E g is less than 2eV and frequently as low as 0.5eV (silicon: E g = 1.12eV at T = 300K). (a)

Ee ED

E

conduction band

(b) E

---------

Ev valence band

e

EA Ev

k

E

conduction band

---------

valence band

k

Figure 2.1. Schematic energy band diagram for (a) n-type, (b) p-type semiconductors. If the crystal contains only a very small amount of impurities, one speaks of an intrinsic semiconductor. In the intrinsic case, conduction band electrons can only have come from formerly occupied valence band levels, leaving holes behind them. The density of conduction band electrons n is therefore equal to the density of valence band holes p: def

ni = n = p, where ni is called intrinsic density. The intrinsic density of silicon is ni 1.02 . 10 10 cm- 3 at room temperature. If impurities contribute a significant fraction of the conduction band electrons and/or valence band holes, one speaks of an extrinsic semiconductor. Impurities that contribute to the carrier density of a semiconductor are called donors if they supply additional electrons N D to the conduction band, and acceptors if they supply additional holes NAto (i.e. capture electrons from) the valence band. The doping impurity or doping profile is denoted by C:

If C > 0 holds in the crystal, the solid is called n-type semiconductor, if C < 0, p-type semiconductor. When a semiconductor is doped with donor or acceptor impurities, impurity energy levels are introduced (Fig. 2.1). The donor and acceptor levels lie very close to the boundaries of the forbidden energy region between conduction

2.2. Inhomogeneous semiconductors

23

and valence band. It is far easier thermally to excite an electron into the conduction band from a donor level ED, or a hole into the valence band from an acceptor level EA, than it is to excite an electron across the entire energy gap from valence to conduction band. Therefore the impurities will be a far more important source of carriers than in the intrinsic case. The process of transfering an electron of the conduction band into the lower energetic valence band is called (band-to-band) recombination of electron-hole pairs. Each of the recombination processes releases energy in the form of thermal energy via nonradiative recombination or luminescence via radiative recombination. The inverse process of transfering a valence electron to the conduction band is termed generation of electrons and holes. Each of the generation processes needs energy which can be supplied as thermal energy (phonon fields), optical energy (electromagnetic fields), or electrical energy (electric field). The net recombination-generation rate R is the difference of the recombination rate F and generation rate G: R = F - G. In the Shockley-Read-Hall model the recombination-generation term depends on the electron and hole densities n, p, respectively: R(n,p) = q(n,p)(np - n~),

°

where q(n,p) is a positive function. In thermal equilibrium (no total current flow) the recombination rate equals the generation rate, and R( n, p) = or np = n~ holds.

2.2

Inhomogeneous semiconductors

Depending on the semiconductor material, the doping profile and the geometry, semiconductor devices show a variety of different kinds of electrical behavior. Here we consider the pn-junction diode, the bipolar transistor, the MOSFET, the ballistic diode, and the resonant tunneling diode. These devices are modeled in this book. pn junctions are of great importance both in modern electronic applications and in understanding other semiconductor devices. A pn-junction diode consists of a region where C(x) < 0, the p-region, and a region where C(x) > 0, the n-region, and has two Ohmic contacts rD (see Fig. 2.2a). The remaining part of the boundary r N is insulating. If no potential is applied at the Ohmic contacts, the diode is in thermal equilibrium. Due to the doping profile and the diffusive forces, electrons and holes diffuse from one side of the diode to the other. The resulting transfer of charge builds up an electric field opposites the diffusive currents, until an equilibrium condition is reached. The corresponding potential is called built-in potential. When a potential is applied such that the field in the junction region is reduced (or increased), we say that the diode is forward (reverse) biased. Under forward bias the diffusion current will dominate due to the reduction of

24

Chapter 2. Basic Semiconductor Physics

no

rD p n

n+

n+ n

rN p

rD

(a)

1-----------1

p

p

(c)

(b)

Figure 2.2. Cross sections of the devices: (a) pn diode, (b) pnp transistor, (c) MOSFET.

the electric field. There will be a tendency to return to equilibrium; carriers recombining in steady state will be replaced by those entering from the metallic contacts. Thus a net current flows, opposite to the direction of the field. Electron-hole pairs are thermally generated continuously everywhere in the semiconductor. Under reverse bias the generated pairs will tend to separate, thereby reducing the probability that they will recombine locally and producing reverse currents in the diode. When a sufficiently high field in the reverse direction is applied, the junction "breaks down" and conducts a very large current. The most important mechanism in junction breakdown is avalanche multiplication (or impact ionization). Important informations of a device are contained in the static currentvoltage characteristic, i.e. the relation between contact voltages and current through the contacts under steady conditions. For applied voltages U not too large compared to the thermal voltage UT = 0.025 V at room temperature (low injection case), the experimental results can be in general represented by the Shockley equation Js

> 0,

where the total current J is the sum of the electron and hole current: J = I n + Jp . From the Shockley equation follows that the diode operates like a valve. For "large" applied voltages (for instance, U> lOUT) the growth of the current-voltage characteristic slows down and the above formula is no longer valid in this range (high injection case). The bipolar junction transistor is a device whose performance is based on the interaction of two pn junctions. Each of the three differently doped regions has an Ohmic contact (Fig. 2.2b). There are two possibilities of configurations: the pnp and the npn transistor. The device acts as amplifier if a configuration

25

2.2. Inhomogeneous semiconductors

n (a)

GaAs

-s'"

~

-S'"

c:;)

-s'"

c:;)

:;:

GaAs

(b)

Figure 2.3. Cross sections of the devices: (a) ballistic diode, (b) resonant tunneling diode.

is considered in which the current through the contact of the middle region and the voltages at the contacts of the outer regions are prescribed. The MOSFET (metal-oxide semiconductor field-effect transistor) is a three layer device with two highly doped n+-regions called source and drain and a more weakly doped p-region called bulk (see Fig. 2.2c). A metal electrode (referred to as gate) between drain and source contacts is separated from the semiconductor substrate by a thin oxide layer (denoted by 0 0 in Fig. 2.2c). In general, the device is used as a switch. The MOSFET is a unipolar device, i.e. charge transport is only due to electrons. Applying a significant positive bias at the gate, an n-type inversion layer is formed near the oxide surface. Because the electrons in the inversion layer are restricted to moving in a thin channel close to the surface, their behavior is often similar to a two- or even one-dimensional electron gas. The n+nn+ ballistic diode attempts to simulate the channel of a MOSFET (Fig. 2.3a). Here, the weakly doped n-region, modelling the channel, is between two highly doped n +-regions. The resonant tunneling diode consists of two very thin AlGaAs layers within a central GaAs layer (see Fig. 2.3b) and two heavily doped GaAs n+-regions on the outside (called source and drain). The substrate of the n-channel between the n+-regions is usually weakly doped. Tunnel diodes can be used at very high frequencies as amplifiers or oscillators. These devices are driven by quantum effects. Indeed, to flow from the source to the drain contact, an electron must tunnel through the two potential AlGaAs barriers in series. The probability of this is normally low. The applied bias however alters the position of the bound J

u Figure 2.4: Schematic current-voltage characteristic of a tunneling diode.

26

Chapter 2. Basic Semiconductor Physics

state energy in the central layer with respect to that of the tunneling electrons. This leads to the phenomenon of resonant tunneling. The most important effect of resonant tunneling is the negative differential resistance, i.e. in a certain range of the applied bias U, the current J is decreasing with increasing voltage: dJjdU < 0 (see Fig. 2.4).

Chapter 3 The Isentropic Drift-diffusion Model

In this chapter the isentropic drift-diffusion equations are studied. First, the model is derived from current relations involving Fermi-Dirac statistics (Section 3.1). The existence and uniqueness of solutions to the time-dependent problem are proved in Sections 3.2 and 3.3. Since the model contains parabolic equations of degenerate type, solutions may exist for which the carrier densities vanish locally. These solutions are called vacuum solutions. In Section 3.4 we study the space and time localization of the corresponding vacuum sets. The isentropic drift-diffusion equations are discretized in one and two space dimensions by using an exponentially fitted mixed finite element method. Numerical examples modeling junction diodes and transistors are presented (Section 3.5). Section 3.6 is devoted to the study of the static current-voltage characteristics of diodes.

3.1 3.1.1

Derivation of the model Semiconductor equations based on Fermi-Dirac statistics

The isentropic drift-diffusion model can be derived from a drift-diffusion system incorporating general Fermi-Dirac statistics. We start from a system consisting of the continuity equations for the electron and hole densities n, p, respectively, and of the Poisson equation for the electrostatic potential V:

an 1 ---\1.J at q n ap 1 -+-\1.J at q P €sb.V

=

-R(n,p),

(3.1)

-R(n,p),

(3.2)

q(n - p - C)

inDo

(3.3)

The recombination-generation rate is denoted by R( n, p) and the doping profile characterizing the device under consideration by C = C(x).

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations © Springer-Verlag Berlin Heidelberg 2000

28

Chapter 3. The Isentropic Drift-diffusion Model

One main assumption is that the current densities I n , Jp are given as the product of carrier densities and gradients of the quasi Fermi potentials epn, epp (see, e.g., [62]):

qJ.LnnV'epn,

(3.4)

-qJ.LpPV'epp·

(3.5)

The other main assumption is the relation between the densities and the corresponding quasi-Fermi potentials which is in general given by Fermi-Dirac statistics: n

(3.6)

P =

(3.7)

Here, J.Ln and J.Lp denote the electron and hole mobilities and D n , D p the corresponding diffusion coefficients. Note that for constant mobilities and diffusivities, the current relations can be written equivalently as

(DnnV'F0~

In

q

Jp

-q ( DppV'

(;J -

J.LnnV'v) ,

F0~ (~v) + J.LppV'V )

.

The function F 1 / 2 is the Fermi integral with index 1/2 defined by F

1

2

Y =

/ ()

~ roo .jt dt .,fi Jo 1 + exp(t - y)"

(3.8)

The physical constants are the elementary charge q, the dielectric permittivity lOs of the semiconductor material, the conduction and valence band edge energies E e and E v respectively, and the effective densities of states in the conduction and valence band N e , N v defined by Ne

=

k T) 2( me B 27rfi2

3/2 '

In this definition, kB is the Boltzmann constant, T the (lattice) temperature and me, m v the effective mass of electrons and holes, respectively [56]. n c lR. d (d ~ 1) is the (bounded) semiconductor domain. The equations are supplemented with physically motivated boundary conditions. The boundary an consists of two disjoint subsets r D and r N. The carrier densities and the

29

3.1. Derivation of the model

potential are fixed at f D (Ohmic contacts), whereas fN models the union of insulating boundary segments:

n

= nD, P = PD, V = VD I n . v = Jp • v = V7V . v = 0

(3.9) (3.10)

where v denotes the exterior normal vector of 80, which is assumed to exist a.e. We assume that the densities at time t = 0 are known:

n(O) = nI,

p(O) = PI

in

n.

(3.11)

If the Einstein relations

hold, we get the set of equations (3.1)-(3.7) described in Bonc-Bruevic and Kalashnikov [62] and analyzed mathematically by Gajewski and Groger [154]. A numerical treatment can be found, e.g., in [356]. In low injection situations where the carrier densities differ not much from the corresponding thermal equilibrium densities, we can use Boltzmann statistics since :F1 / 2 (y) '" exp(y) (y --+ -00). Replacing the Fermi integral by the exponential function gives the standard drift-diffusion model (see Section 1.1). In devices under high injection conditions, however, Boltzmann statistics are no longer valid and Fermi-Dirac statistics have to be used [300]. If the densities exceed the corresponding effective densities we can replace :F0~(Y) by 'rJoy2/3 since

:F0~(Y) '" 'rJOy2/3 (y

--+

00),

_(3J7f )2/3

'rJo -

4

'

and the current relations become (for constant mobilities and diffusivities) q

(d

5 3 n V7(n / ) -

-q ( dpV7 (p 5 / 3 ) where

dn =~D 2/3 n,

(3.12)

I-l n nV7V) ,

+ I-lppV7V)

2'rJo d p = ~Dp.

,

(3.13)

5Nc 5Nv With these relations, (3.1)-(3.2) become a coupled system of degenerate parabolic equations. It is well known that for degenerate equations, there may exist solutions which vanish locally. From a physical point of view this is not unreasonable since in semiconductor devices regions with very low carrier densities exist. In this chapter we will make evident that the equations (3.1)-(3.3), (3.12)-(3.13) describe a semiconductor in the high injection regime.

Chapter 3. The Isentropic Drift-diffusion Model

30

In the physical literature, there are different models taking into account high injection effects, assuming applicability of Boltzmann or Fermi-Dirac statistics [280, 287, 297, 307]. Poupaud and Schmeiser [313] have derived a model for degenerate semiconductor material from the Boltzmann equation in the case of high particle densities. Here, Fermi-Dirac statistics are assumed, but the model differs from our degenerate model. Golse and Poupaud [182] also derived a drift-diffusion model incorporating Fermi-Dirac statistics from the Boltzmann equation. This system of equations is similar to (3.1)-(3.7). The current relations (3.12), (3.13) can be interpreted in the language of gas dynamics. We assume that the particles behave - thermodynamically spoken as ideal gas such that the gas law r = nT holds (r denotes the pressure). In the isothermal case T = const. the pressure turns out to be linear: r(n) = n. In the isentropic case, however, the temperature (only) depends on the concentrations. Then T(n) = n 2 / 3 holds for particles without spin for adiabatic and hence for isentropic states [98], which implies r(n) = n 5 / 3 . Therefore the diffusion terms in (3.12), (3.13) can be written as qd n \7r(n) , -qdp \7r(p), respectively. 3.1.2

The isentropic model-scaling

Before presenting the scaling for the system (3.1)-(3.7), we specify the boundary conditions and the model for the recombination-generation rate. The derivation of the boundary conditions and the recombination-generation model is based on a discussion of the thermal equilibrium state. In thermal equilibrium we have vanishing current densities. Hence the quasi-Fermi potentials are spatially constant in n since the carrier densities are positive [154]. The electrostatic potential is determined only up to a constant; we choose this value such that 'Pn

E ) , ="21 ( hi + q g

'Pp

="21

( hi

E +q

g

)

,

where E g = E c - E v is the band gap and hi E lR is a constant. For notational convenience we introduce the chemical potentials

Then the carrier density relations imply Dn

-hn(n) - V J.Ln

E

c +-,

q

D ph (p) -V+-. Ev -J.Lp p q

(3.14)

3.1. Derivation of the model

31

Therefore, the thermal equilibrium densities have to satisfy the compatibility condition p n hi = D hn(n) + D hp(p) f-Ln f-Lp

in n.

(3.15)

The constant hi can be determined by considering an intrinsic semiconductor where C == 0 holds. Then n = p = ni with the intrinsic density ni, a material constant, and n ( Dh Dp ( hi = n nd + - hp nd· f-Ln

f-Lp

The condition (3.15) corresponds to the equilibrium condition np = n; for the standard drift-diffusion model where hn and hp are replaced by the logarithm. Now we are in the position to formulate boundary conditions for Ohmic contacts. As for the standard model we assume that the carrier concentrations at Ohmic contacts are in thermal equilibrium and that the space charge density -n+p+C vanishes. Then the boundary values are the coordinates of the unique point (nD,PD) of intersection of the equilibrium curve (3.15) and the line (3.16)

n-p- C=O.

Clearly, it holds nD,PD > O. The boundary potential VD can be determined by (3.14): D nh ( nD ) f-Ln n

1 (E + E ) + -2q v

D ( ) - - hp PD f-Lp p

C

1 -h· 2 t

1 c 1 + "2(E + E v ) + "2hi' q

(3.17)

For recombination-generation mechanisms we use a class of models compatible with (3.1)-(3.7) in the sense that the recombination-generation term vanishes in thermal equilibrium: (3.18)

where Q is a non-negative bounded function, 8 is increasing with 8(0) = 0, and f is increasing with f(8) ~ 0 as 8 ~ -00. For densities below their equilibrium values, generation occurs faster than recombination, and we get R( n, p) < O. In regions where the carrier concentrations exceed the equilibrium values, recombination occurs faster than generation and R( n, p) > 0 holds. Gajewski and Grager [154] have used the model (3.18) with the special choice 8(x) = x and f(x) = exp(x). For low densities we have F0~(Y) r-.J

Chapter 3. The Isentropic Drift-diffusion Model

32

In(y) (y --+ 0). Then, assuming the Einstein relations Dn = UTMn, Dp = UTMp, where UT = kBT/q, we obtain the well-known Shockley-Read-Hall term

R(n,p)

=

Q(n,p)(np - n;).

(3.19)

Therefore the model (3.18) is compatible with the standard drift-diffusion model in which (3.19) is used. Now we scale the system (3.1)-(3.7) in the case of high injection, where we assume heavily doped devices. This corresponds to degenerate semiconductors in which the Fermi energies F n = qi.pn, F p = -qi.pp lie in the conduction and valence band resp., i.e.

Indeed, this is equivalent to the relations

D n F- 1 (~) kBT » , Mn 1/2 N e q

D pF- 1 Mp

1/2

(L)

N » v

kBT q ,

which implies n » Ne,p» N v . Since at room temperature the effective densities N e , N v are of order 1019 cm- 3 , this corresponds to high injection situations. We define G to be a characteristic value for the doping profile (e.g., G = max lei) and we set N = max(Ne , N v ). Then we introduce the parameter c = (N/G)2/3 which is assumed to be "small", i.e. c « 1. With the reference length L (e.g., the diameter of n), the reference density G, the reference diffusivity V and mobility [j, the reference voltage Uo/c (where Uo = V/ jj), and the reference time T = L 2 c/ [jUo, the scaling is as follows:

n = Gne , p = Gpe, e = Gee, Y = (Uo/c)ye, Di=VD'f, Mi=[jM~ (i=n,p), t=Tt e , x = Lx e, R(n,p) = (G/T)Re(ne,pe). With this scaling the dimensionless scaled system reads

where

he (pe) cF- (NN cpc ) p

=

1

1/2

v

3/ 2

'

33

3.1. Derivation of the model

and CsUo

qC£2c is the scaled Debye length. In applications, the Debye length is "small"; indeed, ).c « 1 if C » N. Note that

where 'fln = (3VifN/4Nc )2/3,

The limit c

-t

'flp = (3VifN/4Nv )2/3.

(3.23)

0 gives formally

~:: - ~. [M~nO~ (~~ h~(nO) apo _ atO

vo) ]

~ . [MOpO~ (D~ Mg hO(pO) + yo)] p

p

(.\0)2 ~ VO where h~(s) = 'flns2/3, h~(s) = 'flps2/3. We have assumed that N° converges to a positive constant. Indeed, the diffusivities and mobilities generally depend on the carrier densities [342], so we assume that the dependence is such that N' remains positive as c - t O. We expect that for c « 1, (nc,p'o, Vc) solves approximately (3.24)-(3.26). Then, using (n c, pC, Vc) as solution for the limiting problem and scaling back to the physical variables and parameters we obtain (for constant diffusivities and mobilities)

an 1 - - ~.J at q n -an + -1 ~ . Jp at q

-

= -

R(n p)

= -

R(n p)

"

"

In = q (Dn~(n5/3) - Mnn~V), Jp = -q (D p~(p5/3) + MpP~V),

(3.27) (3.28)

(3.29)

cs~V=q(n-p-C).

We get the equations (3.1)-(3.7) with F01 replaced by h~, h~ respectively. Furthermore, the system (3.24)-(3.26) can be written as

ano at - M~~ . (~r~(nO) - nO~VO) apo at - M~~ . (~r~(pO)

(3.30)

+ pO~VO)

(3.31)

(>\O)2~VO

(3.32)

34

Chapter 3. The Isentropic Drift-diffusion Model

(2D~1]n/5/L~)(nO)5/3, r~(pO) = (2D~1]p/5/L~)(pO)5/3. In the where r~(nO) following subsection we present a rigorous result on the limit € --t O. Before concluding this subsection we study the boundary conditions in the limit € --t O. The scaled boundary value (nb,pb) in thermal equilibrium is the unique solution of

(3.33) In the limit

€ --t

0, (n~, p~) is the solution of

Therefore n~ = C,

o -- 0 , nD

p~ = 0,

p~ =

-c,

if C > 0 at

rD,

(3.34)

if C < 0 at

rD,

(3.35)

and the boundary densities are only non-negative. Here, Va = Va(x) denotes the applied potential. 3.1.3

The convergence result

The limit explained in the previous subsection can be made rigorous. For this result we need the following assumptions. (HI) 0 C ~d is a bounded Lipschitzian domain (1 :::; d :::; 3) with 80 r D urN, r D n r N = 0, meas(r D ) > 0, and r N is open in 80. (H2) R: [0,00)2 and

--t

R is Lipschitzian uniformly in

lim R(n,p):::; -Pc 'ip ~ 0,

n-O+ where Pc

> 0 if € > 0 and Pc

€,

-R(n,p) :::; cR(1

+ n + p)

lim R(n,p) :::; -Pc 'in ~ 0,

p-O+

= 0 if € = O.

(H3) C E LOO(O); A, Dn , Dp and /L

= /Lp = /Ln are positive constants.

(H4) nD,PD, VD E W1,OO(0); nI,PI E LOO(O); nD,PD, nI,PI ~ if € > 0 and "fe = 0 if € = O.

"fe

with

"fe

>0

35

3.1. Derivation of the model

The condition (H2) is used to show the non-negativity of the carrier densities. For instance, the Shockley-Read-Hall recombination-generation term R(n,p) = np - n~ satisfies (H2). The assumption J-ln = J-lp can be satisfied by rescaling the variables and parameters and is therefore not restrictive. Notice that the boundary functions are defined more generally than in the preceding subsection. For notational convenience we omit the index c in this subsection. The existence of a solution to the system based on Fermi-Dirac statistics (3.20)-(3.22) under the initial and boundary conditions (3.9)-(3.11) has been investigated in [154]. More precisely, it is shown that there exists a solution (n,p, V) E (LOO(QT) n H 1(V*))2 x L 2(H 1) of (3.20)-(3.22), (3.9)-(3.11) with n,p E L 2 (H 1 ) and

0< Cl(c) :S n,p:S C2 < 00 in QT. In addition, if d :S 2 and nI,PI E w1,m(n) for some m > 2 then the solution is unique. These results are proved by Gajewski, Groger and Rehberg for some special recombination-generation rate [154, 187]. Careful reading of the proof shows that condition (H2) is sufficient to conclude the result. We can now state our convergence result. The proof is based on L oo estimates for nand p. Since the technique is similar to the proof of the existence result in Section 3.2 we do not present it here and refer to [222]. Theorem 3.1.1 Let the hypotheses (H1)-{H4) hold. Assume that D~, D~, J-lc , 1/)..€ are bounded uniformly in c, G€, ni, pi are uniformly bounded in Loo, nh, ph, VD are uniformly bounded in L oo nH 1 and R€ is uniformly bounded in L oo . Then there exists a subsequence of (n€, p€, V€) (not relabeled) which converges to a solution (nO,pO, VO) of the limiting problem {3. 30)-{3. 32), (3.9)-{3.11). More precisely,

We have formulated a convergence result but no information on the order of convergence has been obtained. It is possible to derive an O(c)-type convergence result under stronger assumptions on the boundary functions and the recombination-generation rate: (H5) The initial and boundary densities are strictly bounded away from zero: nb,Pb 2 'Yo

> 0 on r D ,

ni,pi 2 'Yo

> 0 in n

and the recombination-generation rate satisfies lim

n->O+

R€(n p) ' n

< +00 Vp 2 0,

R€(n,p) . 11m

p->O+

p

\-I

< +00 vn 2 O.

36

Chapter 3. The Isentropic Drift-diffusion Model

Theorem 3.1.2 Let (H1)-(H5) hold, let assume that

an

=

rD,

an

E

c2+ry

Iinh - n'c - >'°1 IICc - cOll o,2,n + IIRc - ROllo,oo,IR~ C

-

< < < <

(7] > 0) and

CDc, coc, Coc, Coc

for some constant Co > 0 independent of c. Then there exists a solution (nO,pO, Va) of (3.30)-(3.32), (3.9)-(3.11) such that for every solution (nc,pc, Vc) of (3.20)-(3.22), (3.9)-(3.11) it holds

Iincwhere c

nOllo,2,QT

> 0 is

+ Ilpc - pollo,2,QT + 11\7(Vc - VO)llo,2,QT ::::; cVi,

independent of c.

The proof of this theorem uses the dual method for quasilinear parabolic equations and can be found in [222].

3.2 3.2.1

Existence of transient solutions Assumptions and existence result

In this section we show the existence of weak solutions to the scaled isentropic drift-diffusion model (see (3.30)-(3.32))

atn atp -

\7. I n \7. Jp >.2~V

n=nD,

In.v

-Rn(n,p), -Rp(n,p),

n-p-C

I n = fLn(\7rn(n) - n\7V),

(3.36) (3.37) (3.38)

rD x (0, T), r N x (0, T),

(3.39) (3.40) (3.41)

Jp = fLp(\7rp(p) + p\7V) , in QT = n x (0, T),

P=PD,

V=VD \7V . v = 0,

= Jp . v = 0, n(O) = nI, p(O) = PI

in n.

on on

We recall that n, p denote the electron and hole densities and I n , Jp the current densities, respectively; V is the electrostatic potential, R i the recombinationgeneration rate and C the impurity concentration describing spatially fixed charges in the bounded domain n c IR d (d 2: 1). Furthermore, fLn, fLp E IR are the (scaled) mobilities and >.2 > 0 the (scaled) Debye length of the material. The pressure functions are denoted by r n , rp . For the isentropic drift-diffusion model it holds:

a>1.

37

3.2. Existence of transient solutions

In the case of the standard drift-diffusion model the pressure functions are linear: rn(u) = rp(u) = u. Existence and uniqueness of solutions for r(u) = u has been shown by several authors under different assumptions on the drift terms (e.g. nonconstant mobilities), the reaction rate and the boundary conditions [134, 153, 281, 331, 332]. In this section we assume that the pressure functions are nondecreasing and satisfy ri(O) = 0, i = n,p. Since the parabolic equations are nonlinear and (possibly) of degenerate type, existence does not follow from standard theory. First we replace the pressure by a nondegenerate function and cut off the nonlinear functions appropriately. Existence of solutions of this approximate problem can be shown by employing the Schauder fixed point theorem. A Stampacchia-type truncation method yields £00 bounds for the solutions independent of the approximation parameters by using a monotonicity property due to the structure of the drift terms. This is the main step of the proof; in contrast to the results of [218] only VV E £2(0 X (0, T)) is needed. By using further a priori estimates independent of the approximation and compactness arguments related to a result of Dubinskii [128] we obtain a global solution of the degenerate system in 0 x (0, 00). Our notion of solution is a weak one, i.e. n,p E £00, Vrn(n), Vrp(p) E £2. It is well known that because of the degeneracy, solutions exist only in some generalized sense [241]. Our basic hypotheses are the following: (HI) 0 c jRd is a bounded domain, d 2: 1, with ao E CO,l, rD UrN aO,rD nr N = 0, meas(rD) > 0, and r N is relatively open in ao. (H2) ri E C 1 ([0,00)), ri(O) = 0, and ri and r~ are nondecreasing in (0,00), i = n,p.

(H3) R: QT X jR2 ~(.,., n,p) E

jR2 is a quasi-negative Caratheodory function such that £OO(QT), and

---7

(3.42) for (x, t) E QT, n,p 2: 0, CRi 2: 0, i = n,p.

(H4) C E £00(0); 0 :s: nI,PI E £00(0); 0 VD E £00(H 1); J.L = J.Ln = J.Lp, .A> O.

:s:

nD,PD E £OO(QT)

n H 1(QT),

We say that f : QT X jRn ---7 jRm is a Caratheodory function if f(-, z) is measurable for any z E jRn and f(x, t,·) is continuous for almost all (x, t) E QT. The assumption of equal mobilities can be satisfied by scaling the variables and parameters and is therefore not restrictive. We say that a function R : QT x jR2 ---7 jR2 is quasi-negative if for (x, t) E QT, ~(x, t, n,p) :s: 0 for every (n,p) E

38

Chapter 3. The Isentropic Drift-diffusion Model

such that n = 0 and p ~ 0, or p = 0 and n ~ O. For instance, the ShockleyRead-Hall term (3.19) satisfies (H3). The hypothesis on the monotonicity of r~ can be relaxed, see Remark 3.2.4. The tripel (n, p, V) is called generalized solution of (3.36)-(3.41) if (n, p, V) E (L=(QT) nH1 (V*))2 x L 2(H 1 ) is such that ]R2

n,p ~ 0 a.e. in QT, rn(n) - rn(nD), rp(p) - rp(PD) E L 2(V), Vet) - VD(t) E V, nCO) = nI, p(O) = PI a.e. in 0, where V

= HJ(O U r D ), and iffor all ¢

E

L 2(V), 'ljJ E V the equations

r (Otn, ¢)v*,vdt + /kn Jr (V'rn(n) - nV'V) . V'¢dxdt Jo r Rn(n, p)¢dxdt, J r (OtP, ¢)v*,vdt + /kp Jr (V'rp(p) + pV'V) . V'¢dxdt Jo T

Qr

=

-

Qr

T

=

-

r

JQr

Qr

Rp(n, p)¢dxdt,

>,2l V'V(t)· V''ljJdx = - l (n - p - C)'ljJdx are satisfied. Here (', .) v*,v denotes the duality pairing between the dual space

V* and V.

Theorem 3.2.1 Let (H1)-(H4) hold and let T alized solution (n,p, V) of (3.36)-(3.41).

> O. Then there exists a gener-

Proof of the existence result The proof is divided into several steps. First step: approximation by a nondegenerate problem. Set SK = max(O, min(s, K)), riK(s) = ri(sK) + ES for K > 0, Let T > O. First we will solve the system

3.2.2

E

> 0, i = n,p.

Otn -/kn V' . (V'r;K(n) - nKV'V)

-Rn(nK,PK ),

(3.43)

+ PKV'V)

-Rp(nK,PK),

(3.44)

OtP - /kp V' . (V'r~K(n)

>,2~V

n- P- C

in QT

(3.45)

subject to initial and boundary conditions (3.39)-(3.41). To solve this problem we use Schauder's fixed point theorem (see [180, Cor. 11.2]).

39

3.2. Existence of transient solutions

Let (N, P) E L 2(QT)2 and let V(t) = V[N, P](t) E H1(n) be the unique solution of A2~V(t) = N(t) - P(t) - C,

V(t)

=

VD(t)

on

r D , 'VV(t) . II =

on

0

rN.

Then V: (O,T) ---+ H1(n) is (Bochner-)measurable and V E L 2 (H 1). Next we solve the decoupled equations

Otn - J.Ln'V . ((r~K)' (N)'Vn - N K'VV)

=

-

(3.46)

OtP - J.Lp'V . ((r~K)' (P)'Vp + PK'VV)

R.n(NK, PK ),

=

-

Rp(NK, PK )

(3.47)

with initial and boundary conditions (3.39)-(3.41). Since

-J.Ln'V· (NK'VV) - Rn(NK , PK), J.Lp'V . (PK'VV) - Rp(NK' PK)

E

L 2(V*),

standard results on linear evolution equations ensure the existence and uniqueness of a solution n,p E L 2 (H 1) n H1(V*) of (3.46)-(3.47), (3.39)-(3.41) [100, Ch. 18]. Thus the fixed point operator

S: L 2(QT)2 ---+ L 2(QT)2, (N,P) f-t (n,p), is well defined and S(L 2(QT )2) C (L 2(H 1 )nH 1 (V*))2. By using n-nD,p-PD E L 2 (V) as test function in the weak formulation of (3.46)-(3.47) and employing Green's formula (see [35, Ch. 18] for the use of this formula suitable for mixed problems), a standard Gronwall estimate yields

Il n IIL=(£2) + Il n ll£2(Hl) + IlpIIL=(£2) + Ilpll£2(Hl) :s; c(K, T), where c(K, T) > 0 does not depend on N or P. Consequently,

lIotnIlL2(v*)

+ lIotplI£2(v*)

S; c(K, T)

and

sup {IIS(N,P)II£2(Hl)nHl(v*) : (N,P) E L 2(QT)2} < 00. In view of standard compactness results (see, e.g., [259, Ch. 1] or [373, p. 450]) this implies that S(L 2(QT )2) is precompact in L 2(QT)2. To show that S is continuous, let (Ni,p i ) C L 2(QT)2 be a sequence such that (Ni, pi) ---+ (N,P) in L 2(QT) (i ---+ (0). Set (ni,p i ) = S(Ni,p i ). Since S(L 2(QT)2) is relatively compact in L 2(QT)2 and bounded in L 2(H 1 )2 we obtain for a subsequence (not relabeled)

(ni,p i ) ---+ (n,p) (ni,pi) ~ (n,p) (N i , pi) ---+ (N, P)

in L 2(QT),

weakly in L 2(H 1 ), a.e. in QT'

Then (n,p) = S(N, P) follows from standard arguments (see, e.g., [218]) which proves the continuity of S. Now the existence of a solution (n€:,p€:, V€:) E

Chapter 3. The Isentropic Drift-diffusion Model

40

(L 2(H 1) n H1(V*)? x LOO(H 1) is a consequence of Schauder's fixed point the-

orem. Second step: uniform LOO bounds for (nc,pc). To obtain the lower bound we use (nc )- = min(O, n c ) E L 2 (V) as test function in (3.43) (see [347, Thm. 1.56]):

r (n C)-(t)2dx + EJ.Lndx r 1V7(nc )-1 dxdt 2 Jn J :s J.Ln r n1.-V7Vc. V7(n c)-dxdt - r R n (n1.-,p1.-)(n C)-dxdt. J J

~

2

Qt

Qt

Qt

:s

By taking into account n1.- = 0 in {n c O} and the quasi-negativity of Rn we get ~ (n c)-(t)2dx:S 0

In

and thus n C ~ 0 a.e. in QT. Similarly, we have pc To obtain the upper bound set

~

0 a.e. in QT.

k = max{llnIllo,oo,n, IlnDllo,oo,rDx(O,T), I!Plllo,oo,n, IlpDllo,oo,rDX(O,T)},

let K ~ k, q E N, q ~ 1, and define

r
Jo

and consequently

In

it (OtnC,
>

(
q~ 1

In

(n1.- - k)+(t)q+ldx

(d. [221, Section 5]), we get

r

r

_1_ (n1.- - k)+(t)q+1dx + J.Ln E q(n1.- - k)+(q-l)IV7(n1.- - k)+1 2 dxdt q + 1 Jn JQt

r (n1.- - k)+qV7VC . V7n1.-dxdt + J.Lnq r k(n1.- - k)+(q-l)V7Vc . V7n1.-dxdt J - Jr R n (n1.-,p1.-)(n1.- - k)+qdxdt

< J.Lnq

JQt

Qt

Qt

41

3.2. Existence of transient solutions

r + J.Lnk r \7V€· \7(nK - k)+qdxdt J - Jr Rn(nK,PK)(nK - k)+qdxdt

q J.Ln-\7V€· \7(nK - k)+(q+l)dxdt q + 1 JQt Qt

Qt

r

--q- J.Ln (n€ - p€ - C)(nK - k)+(q+l)dxdt q+1)..2JQt k - J.Ln (n€ - p€ - C)(nK - k)+qdxdt )..2

r

JQt

- Jr Rn(nK,PK )(nK - k)+qdxdt,

(3.48)

Qt

where we have used (3.45). We estimate the last term on the right-hand side of (3.48). By employing Young's inequality

aqb < _q_a q+1 + _1_ bq+1 -q+1 q+1

for all a, b 2: 0, q 2: 1,

and by using (3.42) we obtain

- Jr Rn(nK,PK )(nK - k)+qdxdt < CRn r (1 + 2k + (PK - k) + (nK - k))(nK J < -q-2CRn(1 + k) r (nK - k)+(q+l)dxdt q+1 J

(3.49)

Qt

k)+qdxdt

Qt

Qt

CRn + --(1 + 2k)meas(QT) q+1

+ CRn

r ((nK - k)+(q+l) + (PK - k)+(q+l»)dxdt.

JQt

A similar inequality can be derived for Rpo Hence

- Jr (Rn(nK,PK)(nK - k)+q + Rp(nK,PK)(PK - k)+q)dxdt :S q: + c lot ((nK - k)+(q+l) + (PK - k)+(q+l»)dxdt Qt

1

(3.50)

where C > 0 is, here and in the following, a constant independent of K, c and q with different values at different occurences. Now observe that the inequality

-(n€ - p€)((nK - k)+(q+l) - (PK - k)+(q+l»)

:s 0

(3.51)

Chapter 3. The Isentropic Drift-diffusion Model

42

holds. Add the inequality (3.48) for n E and a similar inequality for pE and use (3.50), (3.51) and J..t = J..tn = J..tP to get _1_ {((ni< - k)+(t)q+l q+ 1

Jn

<

q: + kt 1

c

+ (Pi< -

k)+(t)q+l)dx

((ni< - k)+(q+l)

+ (Pi< -

k)+(q+l))dxdt

- _q_J!:... { (n E - pE - C) ((ni< - k)+(q+l) - (Pi< - k)+(q+l))dxdt q+l)..2JQt

- J..tk { (n E _ pE - C) ((ni< - k)+q - (Pi< - k)+q)dxdt

<

)..2

JQt

1

c

q: + kt

((ni< - k)+(q+l)

+ (Pi< -

k)+(q+l))dxdt,

using Young's inequality again. Gronwall's lemma then implies

t > 0, and finally,

Since the right-hand side of this inequality does not depend on K > 0 and q 2: 1, we can let K --+ 00 and then q --+ 00 to obtain the desired Lco estimate

t > O.

(3.52)

The constants c, A > 0 depend on the data but not on c > O. With this L CO bound we can remove the cut-off function by choosing K > ce AT . We see that (nE,pE, V E) solves

atn E - J..tn \7 . (\7r~(nE) - n E\7V E) atpE - J..tP \7 . (\7r~(pE) + pE\7V E) )..2b.V

E

-Rn(nE ,pE),

(3.53)

_Rp(nE,pE), n E - pE - C in QT

(3.54)

subject to the initial and boundary conditions (3.39)-(3.41). Third step: further a priori estimates for (n E, pE).

(3.55)

We derive a priori estimates for (nE,pE) independent of c > 0 such that a strongly convergent subsequence in L 2(QT) can be extracted. Take n E - nD E

3.2. Existence of transient solutions

43

L 2 (V) as test function in (3.53). Then

r (nC- nD)2(t)dx + r (r;)'(nC)IV'nc 2dxdt 2 In J r (r;)'(nC)V'nc . V'nDdxdt J + r (nC- nD)V'Vc . V'(nc - nD)dxdt J + r nDV'Vc . V'(nC- nD)dxdt- r (nD)t(nc - nD)dxdt J J + ~ r(n] - nD(O))2dx - r Rn(nC,pC)(nc - nD)dxdt 2 In J

~

JLn

I

Q,

JLn

Q,

JLn JLn

Q, Q,

Q,

Q,

h + ... +16 ,

We estimate the right-hand side term by term. Set M = ee AT ~ Iinc Ilo,oo,QT (d. (3.52)) and eM = sup{(r~Y(s) : 0 ~ s ~ M, 0 < c < I}. Then

h

~ ~n

r (r;)'(nC) lV'nc 2dxdt + ~n r lV'nDI2dxdt. J J eM

1

Q,

Q,

From (3.55) and (3.52) follows

h =

r

JLn V'Vc.V'(nC-nD)2dxdt = 2 Q,

J

-

r

JL,n (nC- pC-C)(nC-nD)2dxdt ~ e, 2/\ 2 Q ,

J

where e > 0 is independent of c > O. Since nD(t)(nc - nD)(t) E V, L 2 (O), we can apply Green's formula to estimate h: 13

-JLn

<

r V'nD' V'VC(nC- nD)dxdt-

JQ,

r ~VcnD(nC - nD)dxdt

JQ ,

JLn (11V'nDllo,211V'Vcllo,21Inc - nDllo,oo

+ II~Vcllo,21InDllo,21Inc ~

JLn

nDllo,oo)

e.

Finally, 14 16

~vc(t) E

< 118t nDllo,21lnc - nDllo,2 ~ e, < IIRn(nC,pC)llo,2I1nc - nDllo,2 ~ e.

44

Chapter 3. The Isentropic Drift-diffusion Model

Hence

ll(nc - nD)(t)116,2,n +

r lV7r (n J n

QT

C

)1 2dxdt::;

CM

JrQt (r;)'(n

C

)IV7nC I2dxdt

r (r;)'(n )lV7n I2dxdt J c

c

< c, ::;

c.

(3.56)

QT

Next use ¢ E L 2 (V) in (3.53). Then we conclude from

11

T

(Ot nC , ¢)v*,vdt!

that

::;

IIV7r;(nc)1I0,211V7¢llo,2 + ILn Iinc 110,00 IIV7v cl o,211V7¢llo,2 + IIRn(n c ,pc)llo,211¢110,2

IIOtnCII£2(V*) ::; c.

(3.57)

Analogous bounds can be found for pC. Fourth step: the limit as E; --7 O. Define

1 8

Pi(S)

JrUT)dT,

S

2 0,

{v E LOO(n): v 2 0, Pi(V) E H 1 (n)},

Si

(llV7 Pi(vWdX) 1/2 =

Mi(v)

(l

r~(v)IV7VI2dX)1/2

V E S, i = n,p.

Then, for i = n, p,

(i) V>.

E

[0, 1]' v E Si : Mi(>'v) ::; >'Mi(v),

(ii) {v

E

Si : Mi(v) ::; I} is relatively compact in L 2(n).

r~

is nondecreasing and thus, for>. E [0,1], v E Si,

Indeed,

M i (>'V)2 = >.2l r~(>,v)lV7vI2dx ::; >.2l r~(v)lV7vI2dx = (>'Mi (v))2. The assertion (ii) can be proved similarly as in [259, Ch. 1, Prop. 12.1]. Therefore, taking into account (3.56) and (3.57), the assumptions of Theorem 3.2.2 (to follow) are satisfied for B = L 2 (n), B 1 = V* and we can extract subsequences (not relabeled) such that

n C --7 n,

pC

--7

P

(E;

--7

0)

in L 2(QT) and a.e. in QT.

(3.58)

3.2. Existence of transient solutions

45

Theorem 3.2.2 Let Band B l be Banach spaces and S be a set such that S B C B l and the embedding B "---t B l is continuous. Furthermore, let M : S [0,(0) be such that

--t

c

(i) VA E [0,1], v E S : M(AV) ~ AM(v),

(ii) {v E S: M(v)

~

I} is relatively compact in B.

Finally, set

where 1 < q, s < 00, Cl, C2 compact in Lq(O, T; B).

> O.

Then F C Lq(O, T; B) and F is relatively

Before we prove Theorem 3.2.2 we finish the proof of the existence theorem. From the L OO bound (3.52) for (n£,p£), the bound (3.57) and the compactness of the embedding L2 (0) "---t V*, an application of Aubin's lemma [337] gives (by passing to a subsequence) in CO ([0, T]; V*). Thus n£(O) --t n(O) in V* which implies n(O) = nI. Similarly, p(O) = PI. Since \7V£ ---'" \7V weakly in L 2(QT) and ~ V£ = n£ - p£ - C in L 2(V*) we get ~ V = n - p - C. An elliptic estimate yields

from which we conclude (3.59) It remains to show that (n,p, V) is a solution of (3.36)-(3.41). Since r;(n£) --t rn(n) a.e. in QT (see (3.58)) and r;(n£) is uniformly bounded in L 2 (QT), an application of [259, Ch. 1, Lemma 1.3] gives

weakly in L 2 (QT) and, taking into account (3.56), r~(n£)

(3.60)

---'" rn(n)

From the bound (3.57) we have weakly in L 2 (V*).

(3.61)

46

Chapter 3. The Isentropic Drift-diffusion Model

Analogous results hold for pC. We get for

E L 2 (V)

r (nc\lVc - n\lV) . \ldxdt

JQT

(n Jr QT

<

I

---t

O.

(3.62)

c - n)\lV· \ldxdt! + Ilncllo,ooll\l(Vc - V)llo,211\lllo,2

Finally, the continuity of Rn in (n,p) implies Rn(nc ,pc) Hence, from (3.60)-(3.62), we can let c ---t 0 in

r (8 n c,dxdt-J.Ln Jr Jo

---t

Rn(n,p) in L 2 .

T

t

QT

= -

r Rn(nc,

JQT

nC\lVc.\ldxdt

QT

pC) dxdt,

E L 2 (V),

which gives the weak formulation of (3.36). Similarly, we can treat equation (3.37). This proves the theorem. 0 Now we prove Theorem 3.2.2. A similar theorem was first stated by Dubinskii [128], but he supposed instead of (i): (i)' V>. E JR, v E S : M(>.v) =

1>'IM(v).

To show that the conclusion of the theorem follows from (i), (ii) instead of (i)', (ii), we only need to prove that VT]

> 0 : 3C1/ > 0 : Vu, v

E S :

Ilu - vilB

~

T](M(u) + M(v) + 1) + C1/llu - vllBll

(3.63) since this inequality is the starting point of the proof of Dubinskii and the remaining proof is as in [128]. If the inequality (3.63) is false, there exist T]o > 0 and Un, V n E S such that

(3.64) and M(u n ) ~ M(un)jMn ~ 1, M(v n ) ~ 1 due to assertion (i). From (ii) we conclude that (after extracting a subsequence)

Un

---t

in B

U,

Then (3.64) gives Un - vn This contradicts (3.64).

---t

(n

---t

00).

0 in B 1 and thus u = v, i.e. Un -

vn

---t

0 in B. 0

47

3.2. Existence of transient solutions

Remark 3.2.3 We have shown in Theorem 3.2.1 the global existence of solutions (n,p, V) E (L~c(O, 00; L oo (0)))2 x L~AO, 00; H1(0)) of (3.36)-(3.41). This follows from our estimates and the standard continuation argument. Remark 3.2.4 We discuss some generalizations of (H2)-(H4). (i) The condition (H2) can be relaxed if Ri = 0 (i = n,p). Then we do not need the monotonicity of r~. Indeed, since this property was only needed in the compactness argument for r~(nc) (and r~(pc)) we only have to show that

weakly in L 2 (QT).

(3.65)

Set M = 1 + Ilnllo,oo and rnM(s) = min(rn(M), max(O, rn(s))), s E R Then (3.65) follows from the fact that rnM, considered as functional L 2(QT) ---t (L 2(QT))*, (rnM(v),W) = JrnM(v)wdxdt, is maximal monotone (see [218] for details). (ii) Furthermore, we can replace the assumption on the monotonicity of r~ by the hypothesis that the inverse function r i 1 exists and is (globally) Holder continuous of order () E (0,1). Since the monotonicity condition has been used to prove the strong convergence of n C to n in L 2 ( QT), we have to apply another compactness argument. We use the following result due to Chavent (see [149]):

Let f be Holder continuous of order () E (0,1) with f(O) = 0, and let 0 < s < 1, 1 < p < 00. Then, for all u E WS,P(O),

Lemma 3.2.5

Ilf(u)lllIs,p/lI,n ~ cllull~,p,n, where c > 0 only depends on the Holder norm of f. Applying this result to

where c

f

= r;l and using (3.56) we get

> 0 does not depend on c. Thus n C is uniformly bounded in

L2/lI(WlIs,2/lI)nHl(v*). Since the embedding W lIs ,2/lI(0)

'---t

L 2(0) is compact,

Aubin's lemma [337] implies the existence of a subsequence (not relabeled) such that n C ---t n in L 2 (QT). (iii) It is possible to consider more than two species and nonconstant mobilities f-ti(X, t) using the methods in [218] but we have to assume \7V E Loo(QT) to get a similar global existence result as above.

48

3.3

Chapter 3. The Isentropic Drift-diffusion Model

Uniqueness of transient solutions

Several methods are used in the mathematical literature to prove the uniqueness of weak solutions to quasilinear parabolic equations of the type OtU -

div (\l¢(u) + b(u)e) = f(u),

where e is a prescribed vector field. A successful technique is based on the use of the test function sign+(ul - U2) in the above equation, where sign+(s) = 1 if s > 0 and sign+(s) = 0 if s :S 0, and Ul and U2 are two solutions [6, 147, 148, 149, 295]. The difficulty of this method is to show that the solution has enough regularity in order to define the sign function as an admissible test function. This justification can be carried out, for instance, by doubling the time variable and then passing to the limit in which these variables collapse. This technique, first developped for hyperbolic equations by Kruzhkov [251]' has been applied recently to scalar parabolic equations [148, 149, 295]. These techniques can be applied to degenerate equations. However, the isentropic drift-diffusion equations also possess a nonlocal nonlinearity due to the drift terms (i.e. e = \lV[n,p]), and it can be seen that the mentioned methods cannot be applied directly to the semiconductor problem. For uniformly parabolic semiconductor equations, a monotonicity method has been used which is able to deal with the coupling of the electrostatic potential. This technique is a variant of the usual monotonicity in the sense of Browder-Minty [150, 151]. However, it seems that this method cannot be applied to degenerate parabolic equations. Another successful technique applicable both to uniformly parabolic and degenerate equations is the dual method. In this method a special test function ¢ is employed in the weak formulation, where the function ¢ solves a backward parabolic problem. The method has several variants, cf. [11, 22, 30, 54, 121, 127, 218, 369]. With this technique, the nonlocal nonlinearity and the degeneracy of the isentropic semiconductor equations can be treated, however, not both together. Therefore, we get a uniqueness result for the degenerate equations with given electrostatic potential [218] and for the nondegenerate problem with coupling due to the potential. Furthermore, we can prove uniqueness of solutions if the carrier densities satisfy a regularity condition or if the electrostatic potential satisfies some extra boundary condition (see below). At our knowledge, there is up to now no method which is able to deal with both difficulties (except in the special situations of [22, 118, 120]). The isentropic semiconductor model is formally similar to equations for the flow of two immiscible fluids through a porous medium where uniqueness results in the general case are also yet open problems and only partial results have been found [22, 149].

49

3.3. Uniqueness of transient solutions

Our uniqueness results which are based on duality techniques can be summarized as follows: For strictly positive initial and boundary data, the isentropic model has a unique solution for small time. Indeed, the densities can be proven to be strictly positive, so the problem is uniformly parabolic. Then a test function is chosen which solves a backward parabolic problem (dual method). Employing appropriate a priori estimates for this test function, the uniqueness result can be proved. We have also uniqueness of solutions in the class of functions satisfying

in the case of one space dimension, i.e. n E JR. Numerical results (see Section 3.5) indicate that the densities satisfy this regularity assumption. For the proof we define positive approximations of the initial and boundary data such that the solution of this approximate problem becomes strictly positive. Then the dual method is employed, where the additional regularity assumption is used. For an extension to the multi-dimensional case, see [120]. In the case of the unipolar model (i.e. p = 0), assume that the electrostatic potential satisfies at the Dirichlet boundary the conditions

v=

VD,

\7V· 1/ :2: O.

Then uniqueness of the solution follows. Numerical simulations show that the conditions can be satisfied in special situations. The method of the proof is inspired by Rulla [323]. We only need the hypothesis that ri is nondecreasing. Therefore, the additional assumption can be interpreted as an entropy-type condition since the case ri = const. is possible such that the problem becomes hyperbolic. First we prove the uniqueness of solutions to (3.36)-(3.41) if generate.

ri

is nonde-

Theorem 3.3.1 Let the hypotheses (H1)-(H4) of Section 3.2 hold. Furthermore,

assume that r~ :2: 8 > 0 in [0,(0), an = r D E C2+1) (rJ > 0), R i is locally Lipschitz continuous in (n,p) uniformly in (x, t), and VD E L<X>(W 2 ,q) for q > d. Then the system (3.36)-(3.41) has a unique solution (n,p, V) in the class offunctions (L<X>(QT) nL 2 (H 1 ) nH 1 (V*))2 x L<X>(W 2 ,q).

Proof. Let (nj,pj, Vj) E (L<X>(QT) n L 2(H 1) n H 1(V*))2 x LOO(W 2,q) (q > d, j = 1,2) be two solutions to (3.36)-(3.41). Notice that the regularity for Vj follows from .6.Vj(t) E LOO(n) and the boundary conditions. Since q > d, we have \7Vj E LOO(QT). The continuity of the embedding L 2(V) n H1(V*) ' - t

Chapter 3. The Isentropic Drift-diffusion Model

50

CO ([0, T]; £2(0)) implies that nj,pj E CO ([0, T]; £2(0)). Set n = nI - n2, P = PI - P2, V = VI - V2 and Q = 0 x (kT, (k + 1)7) where kEN and 7 > 0 is to be determined. Let 't/J E Coo(Q), 't/Jlan = 0 be a test function. Then

There exist sequences (A1)), (B1)) A1)

-t

B1)

A

-t

in £2(QT),

c

D(QT) such that

IIA1)llo,oo ~ IIAllo,oo,

in £2( QT) as

J.ln~VI

1] - t

0,

inf{A1):

1]

> O} > 0,

IIB1) 110,00 ~ IIJ.ln~VIilo,oo.

We rewrite (3.66):

- Jr nOt't/Jdxdt- Jr A"t::..'t/Jndxdt- Jr B1). ~'t/Jndxdt + Inr n(t)'t/J(t)dxl(k+l)T Q

k

k ~VI kn2~V, ~'t/Jdxdt k

Q

(A - A1))t::..'t/Jndxdt +

+ J.ln

kT

Q

(J.ln

-

- B1)) . ~'t/Jndxdt

(3.67)

R't/Jdxdt.

We choose as test function the solution 't/J1) E Coo(Q) of the nondegenerate retrograde problem

where f1) E D(QT) is a sequence of smooth functions such that f1) - t n in £2( QT), f1) ~ n in QT. The existence of a unique classical solution 't/J1) follows from standard parabolic theory [253]. Using the maximum principle and Gronwall's inequality we obtain the following standard estimates on 't/J1) (see [218]

3.3. Uniqueness of transient solutions

51

for details):

1'ljJ7)(t)1 ~ Ilf7)llo,oo,o < M, sup II V''ljJ7) (t) 110,2,0 < c(8) Ilf7) 110,2,Q,

(3.69) (3.70)

tE(k'T,(k+l)T)

11'ljJ7) 110,2,Q + IIV''ljJ7) 110,2,Q < I /:i'IjJ7) 110,2,Q <

c(8)JTllf7)110,2,Q, c(8),

(3.71) (3.72)

where c(8) is a positive constant independent of TI, but depending on 8, and where M = Ilnllo,oo,QT' The estimate (3.71) follows from (3.70) and Poincare's inequality. For the estimates (3.70) and (3.72) we need the boundedness of V'V1 in Loo(Q). From (3.67) we get

10 f7)ndxdt

~

in

n(kT)'ljJ7)(kT)dx + MilA - A7) 110,211/:i'IjJ7) 110,2

+ MlllLnV'V1 -

+ LM

in

B7) 110,2 IIV''ljJ7) 110,2 + MlLnIIV'VII0,2 IIV''ljJ7) 110,2

(n + p)I'ljJ7)ldx,

where LM is a Lipschitz constant for

10 f7)ndx

~

M

in

Rn in [O,MF, and, using (3.69)-(3.72),

n(kT)dx + c(8)MIIA - A7)1I0,2

+ c(8)JTMlllLnV'V1 - B7)110,21If7)llo,2 + c(8)JTMIIV'VIl0,21If7)110,2 + c(8)JTLM llf7)110,2(lIn II0,2 + IlpIl0,2)' Letting TI

----+

0 we obtain

10 n2dxdt ~ M in n(kr)dx+ c(8, M)JT(lInll~,2 + Ilpll~,2)'

An analogous inequality holds for p. Adding both inequalities yields

10 (n2 + p2)dxdt ~ M in (n(kr) + p(kr))dx+ c(8, M)JT 10 (n2 + p2)dxdt.

Now choose r > 0 such that JT < 1/c(8, M). If k = 0 then

if k = 1 then

in in

(n2(t)

+ p2(t))dx =

0

1ft E [0, r]j

(n2(t)

+ p2(t))dx =

0

1ft E [r,2r].

Repeating this argument finally gives n(t) =0 and p(t) =0 in n for all tE [O,T]. 0

52

Chapter 3. The Isentropic Drift-diffusion Model

We say that a solution (n, p, V) of (3.36)-(3.41) is a limit solution if it is obtained as (L 2 weak) limit of solutions (n€,p€, V€) E (L=(Qr) n L 2(H l ) n H l (V*))2 x L=(H l ) of (3.53)-(3.55), (3.39)-(3.41). For the first uniqueness result of the degenerate system we need the following propositions.

°

Proposition 3.3.2 Let (H1)-(H4) of Section 3.2 hold, letT> 0, and let (n,p, V) be a limit solution of (3.36)-(3.41). Assume that there exist constants ry, A> such that nD,PD 2: rye-At

rD

on

x (0, T).

(3.73)

Then there exist Ao 2: A and To E (0, T] such that n(t),p(t) 2: rye-Aot

in

n x (0, To).

(3.74)

The constant Ao does not depend on t (but on T). Proof. Let (n€, p€, V€) be approximate solutions of (n, p, V) and let Ao 2: A. Set w = rye-Aot and use (n€ - w)- = min(O, n€ - w) as test function in (3.53). Set M = max {lln€llo,=,QT : E > O} (see (3.52)). Omitting the index E we get

~

Inr(n -

w)-(t)2dx + J-tn E

r

JQ,

Aow(n - w)-dxdt

r r J + r (Ao J r J

+ J-tn

JQ,

where

Cl, C2

Q,

JQ,

(n - w)\7V· \7(n - w)-dxdt

r

w\7V· \7(n - w)-dxdt -

JQ ,

Rn(n,p)(n - w)-dxdt

Q,

Q,

Cl

r

+ J-tn

f).V(n - w)-2dxdt

_J-tn 2

<

r 1\7(n - w)-1 2dxdt

JQ,

J-tnf).V - e

(n - w)-2dxdt +

°

Aot ry

Rn(n,p))w(n - w)-dxdt

r (A J o

C2 -

Q,

e

Aot ry

R)w(n - w)-dxdt,

> depend on J-tn, M and IICllo,= and where R = max{IRn(x, t, n,p)1 : (x, t)

If

Ao -

C2 -

A t-

e

0

E

QT,

Rh 2:

°

Inl, Ipl

:s M}. (3.75)

3.3. Uniqueness of transient solutions

53

for t ~ 0 small and Ao ~ A large enough, we can apply Gronwall's inequality to obtain (n - w)-(t)2dx ~ 0

in

and thus nc(t) ~ w a.e. in n. Letting E ----t 0 gives n(t) ~ w in n x (0, To). Analogously, we get p(t) ~ w in n x (0, To). To prove (3.75) set K = 1 + C2 + Rh and Ao = max(A,2K). Choose o < To ~ T small enough such that

eAoToRh Then, for t

~

To,

~

K.

eAot Rh ~ K ~ Ao - K

o

from which (3.75) follows. This proves the proposition.

In Proposition 3.3.2 we have proved that, starting from (strictly) positive initial data and prescribing positive boundary conditions, the densities n, p remain positive at least for small time (nonvacuum solution). We show now that there exists a nonvacuum solution for all t < 00 if the reaction terms Rn, Rp do not grow too "fast", i.e. if recombination is not "too large". Proposition 3.3.3 Let the hypotheses of Proposition 3.3.2 hold and suppose in addition that

Rn(x, t, n,p) . 11m n->O+ n

~

an,

Rp(x,t,n,p) . 11m p

p->O+

~

ap ,

(3.76)

for all p E [0, M], n E [0, M], respectively, where an, a p 2 0 and M is as in the proof of Proposition 3.3.2. Then there exists Ao ~ A such that n(t),p(t) ~

,e-

Aot

in QT.

(3.77)

(3.77) holds as long as (3.73) and (3.76) are valid. Proof. Let (nc,pc, Vc) and w be as in the proof of Proposition 3.3.2. Omitting the index E we estimate as follows (see the preceding proof):

54

Chapter 3. The Isentropic Drift-diffusion Model

<

r (Ao + r J r J

JQt

C2 -

an)w(n - w)-dxdt

(n - w)-2dxdt

cl

Qt

<

(n - w)-2dxdt,

cl

Qt

if Ao 2 max(A, C2

+ an),

and similarly for p.

o

From the proof of Theorem 3.3.1 and Propositions 3.3.2 and 3.3.3 now follows the first uniqueness result for the degenerate system. Theorem 3.3.4 Let the assumptions (H1)-(H4) of Section 3.2 hold and let T > 0. Furthermore, assume that an = rD, an E c2+'I) (", > 0), ~ is locally Lipschitz continuous in (n,p) uniformly in (x, t) and there exists "IT > Osuch that

Then there exists To E (0, T] such that the system (3.36)-(3.41) is uniquely solvable for t < To. If in addition (3.76) holds then To = T.

For the second uniqueness theorem we need an additional regularity assumption for the carrier densities with respect to the space variable (see Remark 3.3.6). Theorem 3.3.5 Let the assumptions (H1)-(H4) of Section 3.2 hold. Furthermore, let d = 1, rn(s) = rp(s) = sa, s 2 0, with 1 < 0: < 2, and let R i be locally Lipschitz continuous in (n,p) uniformly in (x, t) satisfying (3.76). Then there exists at most one solution (n,p, V) of (3.36)-(3.41) in the class of functions satisfying rn(n), rp(p) E L 2(H 1 ), V E L 2(H 1 ), and

n,p E LOO(W1,1) n H1(V*). Proof. The proof is valid for the multi-dimensional case d > 1 except in one argument. Therefore we present a general proof and make explicit the step where d = 1 is required. First step: approximate solution. Consider the problem (3.36)-(3.41) with initial and boundary data

3.3. Uniqueness of transient solutions

55

There exists a solution (n c, pC, vc) to this problem with data (3.78) satisfying, thanks to Proposition 3.3.3, inQT

°

independent of c. Furthermore, the sequence for some constant c > (nc, pC, vc) converges strongly in (L 2(L 2))2 X L 2(H 1) to a solution (n1' P1, V1) to (3.36)-(3.41). This can be seen as in the proof of Theorem 3.2.1. Let (n2,P2, V2) be another solution to (3.36)-(3.41) such that n2, P2 E Loo(W 1,1) and set n = n C - n2, P = pC - P2, V = vc - V2. Second step: the dual problem. Define

Ac = and let A~, such that

B~ E

1 1

r'(BnC+ (1 - B)n2)dB 2

(3.79)

C1cQ-1,

D( QT) be smooth approximations of A c , J-lnv'V c , respectively,

°

Then, for T E (0, T) and for smooth test functions ¢ with ¢ = on an x (0, T) in the weak formulation of the equations satisfied by nC, n2 respectively, we get

- Jr (a ¢ + A~f:::.¢ + B~ . ¢)ndxdt+ Inrn(T)¢(T)dx c r¢(O)dx - r r (rn(nC) - r n(n2))'\l¢· vdadt In Jo Jan QT

t

1 +1 + J-ln

QT

QT

n2'\lV . '\l¢dxdt

(n(AC -

A~)f:::.¢ + n(J-ln'\lVC - B~) . '\l¢

(3.80)

- (Rn(nC, pC) - Rn(n2,P2))¢)dxdt.

°:s

:s

Now let X E D(n), X 1, and let ¢~ be the unique classical solution of the parabolic backward problem

at ¢ + A~f:::.¢ + B~ . '\l¢ = ° in QTl ¢ = ° on an x (O,T), ¢(T) = X in n.

The existence of a classical solution follows from standard parabolic theory [253]. Note that the problem is uniformly parabolic thanks to (3.79).

56

Chapter 3. The Isentropic Drift-diffusion Model

Third step: a priori estimates for The solution



satisfies the following a priori estimates:

o :S
r

sup

anX(O,T)

(3.81)

r

1\7
C2 E1 -

o

(3.82) (3.83)

.

The constant Cl (E) > 0 does not depend on TJ and C2 > 0 does not depend on E and TJ. The first estimate follows from the maximum principle [253, Thm. 5.2 and 2.1]. By using ~ 0 such that for Xo E there exists a sphere B with center y and radius 00 such that B c IRd\n and Xo E aBo Set

an

an

an

D(x) =

Ix - yl- 00

for x E QT'

S(r)

=

1- e- Kr

1 _ e- K

'

r 2 0,

where K is to be determined later. Let

q(x, t) = ±'lj;.,,(x, t) + S e;

(D(x)+t-t O ) Eo - 1

'

defined in Ne; = {(x, t) E QT : 0 :S D(x) < EO-I, to :S t < to + EO-I}. From :S 1 it can be seen that q 2 0 on the lateral surface and the top of Ne;. Set L = + A~~ + B~· \7. Then Lq < 0 in Ne;; indeed, for (x, t) ENe; we have

o :S 'lj;~

at

Lq =

since S"/S' = -K. From

Ix - yl2 00

in Ne; and (3.79) we obtain

if we choose K > 0 large enough. By the maximum principle (see, e.g., [65]) it follows q 2 0 in Ne;. Since q(xo, to) = 0, q attains a minimum at (xo, to). By

3.3. Uniqueness of transient solutions

57

the maximum principle again, (oq/ov)(xo, to) :S 0 from which we conclude the assertion.

Fourth step: end of the proof. We rewrite the equation (3.80):

l

n(r)xdx

:S E

r
In

+ J.Ln

-1

1

Q,.

Q,.

n2 V'V . V'
(Rn(nE:,pE:) -

Rn(n2,P2))
+ Ilnllo,oo,Q,. IIAE: - A~llo,2,Q,. 11~
The regularity assumption on n2 is needed for the estimate of the third term on the right-hand side:

1 Q,.

n2 V'V . V'
-1 Q,.

V' n 2' V'V
1 n2~
V

< (11V'n211 Loo(£1) IIV'VII L1 (LOO)

+ Il n21ILoo(Loo) II~ VII£1(£1)) 11 0,

~V =

oxxV, and we get, for

IIV'V(t)llo,oo,n :S cllV(t)112,1,n :S c(lln(t)llo,l,n + IIp(t)llo,l,n). Therefore

1 Q,.

n2V'V,

V'
< (1I n21ILoo(W 1 ,1) + Iln21ILoo(Loo))II~VII£1(£1) < c4(ll n ll£1(£1) + Ilpll£1(£1)),

where C4

> 0 depends on the Loo(W1,1) and Loo(L oo ) norms of n2.

58

Chapter 3. The Isentropic Drift-diffusion Model Now let c M = sup{llncllo,oo,QT' Iln2/lo,oo,QT' IIp llo,oo,QT' Ilp21Io,oo,QT : € > O} <

00

and let L M be a Lipschitz constant for Rn in QT x [0, M]2. Then

-1

QT

(Rn(nC, pC) -

Rn(n2,P2))¢~dxdt:=; LM

1 QT

+ Ipl)dxdt.

(Inl

Therefore we get from (3.82)

inrn(T)xdx

:=;

CEO

+ C3€2-a + Cs

+ cI(€)(IIAc -

r (Inl + Ipl)dxdt

i QT

A~llo,2,QT

+ IIJ.kn \lVC -

B~llo,2,QT)'

The constant Cs = max(c4' L M ) does not depend on € or 'T/. We get a similar inequality for P = pc - P2 and 'l/J E V(n), 0 :=; 'l/J :=; 1. Hence, after adding the inequalities for n(T)X and p(T)'l/J and letting 'T/ --+ 0, we get

r(n(T)X + p(T)'l/J)dx:=;

k

CEO

+ C3€2-a + Cs

r

i~

(Inl

+ Ipl)dxdt.

The strong convergence of n C and pc to ni and PI, respectively, in L 2(QT ) implies n(T) --+ (ni - n2)(T), p(T) --+ (PI - P2)(T) as € --+ 0, for a.e. T E (0, T). Observing that a < 2, we obtain after letting € --+ 0

inr ((ni -

n2)(T)X + (PI - P2)(T)'l/J)dx :=; Cs

irQT (ini -

n21

+ IPI - P21)dxdt.

Now let Xm, 'l/Jm E V(n) be sequences converging for m --+ 00 to the characteristic functions of {(ni - n2)(T) 2: O}, {(PI - P2)(T) 2: O}, respectively (in the L 2 topology). Then

inr ((ni -

n2)+(T) + (PI - P2)+(T))dx:=; Cs

inr ((n2 -

nI)+(T) + (P2 - PI)+(T))dx:=; Cs

r (ini -

i QT

n21

+ IPI -

P21)dxdt.

Changing the role of n C, n2 and pC, P2, i.e. defining n = n2 - n C and P = P2 _ pc , we arrive at the estimate

r (ini - n21 + IPI - p21)dxdt,

i QT

which implies

inr (I(ni -

n2)(T)1 + I(PI - P2)(T)I)dx :=; 2csl (ini - n21 QT

+ IPI

- p21)dxdt.

We conclude finally with Gronwall's lemma that (nI-n2)(T) = 0, (PI-P2)(T) = o for a.e. T E (0, T). D

3.3. Uniqueness of transient solutions

59

Remark 3.3.6 By using n~-a - (nb)2-a, p~-a - (pb)2-a as test functions in Eqs. (3.53), (3.54), respectively (with nb, pb defined in (3.78)), it is not difficult to see that we get the estimate n e , Pc E L 2 (H 1 ) uniformly in c > if 1 < a < 2. Thus, after letting c --t 0, we obtain

°

We conjecture that in the case of one space dimension and 1 < a < 2, it even holds n, p E Loo(W1,I). This conjecture is confirmed by numerical simulations (cf. Section 3.5). For the last uniqueness result we need an additional condition for the electric field \7V on the Dirichlet boundary. Theorem 3.3.7 Let the assumptions (H1)-(H4) of Section 3.2 hold. Furthermore, let r n be (only) nondecreasing in [0,00), let

for u, v

~

0, (x, t) E Qr, and finally, let

\7V . v

~

0

on fD x (0, T)

(3.84)

for every solution (n, V) of the unipolar problem Otn - \7 . (\7r n (n) - n\7V) >.2b.V

-Rn(n), n - C in Qr

(3.85) (3.86)

subject to the initial and boundary conditions (3.39)-(3.41) forn and V. Then there exists at most one solution (n, V) of the unipolar problem in the class of functions satisfying n E LOO(Qr) n H1(V*), rn(n) E L 2(H 1), V E LOO(W 2,OO). Proof. Let (nl, VI) and (n2, V2) be two solutions to the above unipolar problem and set n = ni - n2, V = VI - V2. Using an appropriate test function ¢ in (3.85) and taking the difference of the equations for nl and n2, we obtain

r (Ot n , ¢)dt + Jr \7(rn (nt} - r n (n2)) . \7¢dxdt

Jo

1 -1 Qr

Qr

nl \7V . \7 ¢dxdt +

Qr

1 Qr

n \7V2 . \7 ¢dxdt

(Rn(nt} - Rn(n2))¢dxdt.

(3.87)

60

Chapter 3. The Isentropic Drift-diffusion Model

We choose ¢(t) E H1(n) as the unique solution of -b.¢(t) = n(t) in n,

¢(t) = 0

on

rD,

V¢(t) . II = 0

on r N

for a.e. t E (0, T). Then ¢ E H1(H 1) and, since b.¢ = _,X2b.V in Qn

iT

1

2

211V¢( T) Ilo,2,n,

(8t n, ¢)dt

,XllvVllo,2,Q,. <

IIV¢1I0,2,Q,..

Thus (3.87) implies

~

2

Inr IV¢(TWdx + JrQ,. (Tn(nl) 1

nlVV . V¢dxdt

-1 Q,.

Q,.

-1

Q,.

Tn(n2))(nl - n2)dxdt

(3.88)

b.¢VV2 · \7¢dxdt

(Rn(nl) - Rn(n2))¢dxdt.

Set M = Ilnlllo,oo,QT. The first term on the right-hand side of (3.88) is estimated as follows:

For the last term on the right-hand side of (3.88) we use the assumption on Rn, the Lipschitz continuity of Tn and Poincare's inequality:

1 Q,.

~

(Rn(nd - Rn(n2))¢dxdt C2(E)

1 Q,.

2 1\7¢1 dxdt + EC31 (Tn(nd - Tn(n2))(nl - n2)dxdt, Q,.

for every E > O. For the second term on the right-hand side of (3.88) we need the assumption (3.84). Formally, we get

- Inrb.¢\7V2 . \7¢dx Inr\7(\7V2 . \7¢) . \7¢dx - Janr (\7V2 · \7¢)V¢· IIdCT. =

Since ¢ = 0 on Thus

rD,

\7¢ lies in the direction of ±II on {x E

rD : V¢(x) i- O}.

- Janr (\7V2 . \7¢)\7¢. IIdCT = - Janr \7V2 · 1I1\7¢[2dCT = - Jrr VV2 · 1I1\7¢1 2dCT. D

3.3. Uniqueness of transient solutions

61

Denoting by H(V2) the Hessian matrix of V2, we further get

in

\7(\7V2 . \7¢) . \7¢dx

and therefore, using (3.84),

-in ~¢\7V2'

\7¢dx

=

in

(\7¢)H(V2)(\7¢)T dx -

~

- ~2 irD r \7V2 . vl\7¢1 2d<7

< c(V2)

in

in ~V21\7¢12dx

2 1\7¢1 dx.

The constant c(V2) > 0 only depends on the L OO (W 2,OO) norm of V2. The estimate can be made rigorously by using an approximation argument (see [323]). For this approximation, we only need the regularity assumptions \7¢(t), ~¢(t) E L 2(0) and V E L oo (W 2,OO). With the above estimates we can write (3.88) as

~

2

inr 1\7¢(TWdx + (1 :::; c(€)

Choosing

€

1 QT

€C3)

irQT (rn(nl) -

rn (n2))(nl - n2)dxdt

1\7¢1 2dxdt.

> 0 small enough and applying Gronwall's lemma, we obtain for

This implies n

= 0 and

V

= 0 in

T

E

(0, T).

o

QT'

Remark 3.3.8 (i) We can show the uniqueness of solutions (n, p, V) to the bipolar problem (3.36)-(3.41) in the class of functions indicated in Theorem 3.3.7 under the assumptions

for (x, t) E QT, U, V, fL, v 2=: 0, i = n, p, and

\7V· v = 0

on f

D

x (0, T).

62

Chapter 3. The Isentropic Drift-diffusion Model

The second condition allows to estimate the term J~'l/J\7V2' \7'l/Jdxdt,where 'l/J is the solution of -~'l/J = PI -P2 under homogeneous mixed boundary conditions, as in the preceding proof. (ii) From numerical simulations of a one-dimensional forward biased pn diode (see Section 3.5), it can be seen that the condition (3.84) on the sign of \7V . II on the boundary is satisfied if the applied potential is large enough. (iii) In the one-dimensional case d = 1, the electrostatic potential satisfies the regularity V E L oo (W 2 ,00). Indeed, since n - C E LOO(QT) we obtain VEL 00 ( QT ). Furthermore, we have immediately VEL 00 ( QT) and

oxx

3.4 Localization of vacuum solutions 3.4.1 Main results The isentropic drift-diffusion model contains parabolic equations of degenerate type so that solutions may exist for which the carrier densities n, P vanish locally. These solutions are called vacuum solutions and the corresponding sets {n = O} and {p = O} are called vacuum sets. We are interested in the existence of (non-trivial) vacuum sets and the space localization of vacuum solutions. Our results can be summarized as follows: 1. Finite speed of propagation. If there are vacuum sets initially then there are vacuum sets for small t > 0:

meas{ n(O) = O} > 0, meas{p(O) = O} > 0 ~ meas{n(t) = O} > 0, meas{p(t) = O} > O. This property shows that the speed of propagation of the support of n and p is finite. 2. Waiting time. Under some structure condition on R(n,p) and some "flatness" condition on nI there is no dilatation of the initial support:

{n(O) = O} C {n(t) = O}

for small t

> O.

3. Formation of vacuum. Under some structure condition on R( n, p) there exists a To > 0 such that there is vacuum for t > To: meas{ n(t) = O} > O. It is well known that the speed of propagation of the solution to uniformly parabolic equations is infinite. However, for degenerate parabolic equations (of

63

3.4. Localization of vacuum solutions

porous media type b.u O ), the speed of propagation is finite [241], which shows some hyperbolic behavior of the solutions. The third result states that there is vacuum (under some conditions) even if the initial densities are strictly positive. The proof of these results to be formulated precisely below is based on local energy methods for free boundary problems. The idea of these methods is to introduce an energy functional (usually given by the norm in the natural energy spaces associated to the equations) and to derive a (differential) inequality for the energy functional. From this inequality the desired qualitative properties of the solutions can be deduced. The energy methods have two principal features. First, they are local methods, i.e. they operate in subsets of the corresponding domain without need of global informations like boundary conditions or boundedness of the domain. Secondly, they have a very general setting, allowing to consider, for instance, problems in any space dimension or with coefficients depending on the space or time variable. The energy method we use does not need any monotonicity assumption on the nonlinear functions and it requires no comparison principle. The method has been introduced by Antontsev [15] and developped by J. I. Dfaz and Veron in [122] and by Antontsev, J. I. Dfaz and Shmarev in [16, 17, 18, 19, 20, 117] for parabolic equations of degenerate type. The energy methods have been extended to equations of arbitrary order [48] and have been applied to equations or systems of equations [49, 119, 155, 156, 221, 247]. We also refer to [19, 21] for an overview of the existing literature. We now turn to the precise formulation of the above localization results. The last result is only valid if the local energy of the density is small enough. The local energy Dn(P) of n in a domain P C QT = n x (0, T) is defined by

where

t1vnO(x, T)1 2 dxdT,

t

n(x, T)o+ f3 dxdT,

sup sE(i,t)

r

Jpn{T=S}

n(x, s)o+ldx,

with i, t > 0, and f3 E (0,1) is a constant to be precised below. Throughout this section we assume that s 2 0,

a>

1,

64

Chapter 3. The Isentropic Drift-diffusion Model

R(n,p) = Rn(n,p) = Rp(n,p) for n,p 20, /-Ln = /-Lp = 1, and that there exists a solution (n, p, V) to (3.36)-(3.41) satisfying n,p E LOO(QT) n H 1(V*),

r(n), r(p) E L 2 (H 1),

V E L OO (H 1).

The existence of a solution with these regularity properties is shown in Section 3.2. We have the following theorems (also see Fig. 3.1).

Theorem 3.4.1 (Finite speed of propagation) Let Xo En, 0 < Po < dist(xo, an) and T > O. Assume that nI = 0,

PI = 0

and R(u, v)(uO:

+ vO:)

2

-h:R( uo:+

1 + vO:+!)

for all u, v 2 0

(3.89)

with h:R 2 0 hold. Then there exist T 1 > 0 and a non-increasing function P satisfying p(T) > 0, 0::; T < T 1, and p(O) = Po such that n(x, t) = 0,

p(x, t) = 0

for a.e. x E Bp(t)(xo), t E (0, Td.

For the next theorems we need a stronger condition on R(n,p):

R( u, v) 2 bu{3

for all u, v 2 0,

b > 0, a

+ {3 < 2.

(3.90)

Theorem 3.4.2 (Waiting time) Let Xo E n, 0 < Po < P1 < dist(xo, an) and T> O. Assume that (3.90) and

r

} Bp(xo)

n~+!

(3.91)

::; co(p - pon-

for 0 < p < P1 hold, where co > 0 and "(=

d(a - 1) + 2(a + 1) >1. a-I

(Recall that d 2 1 is the space dimension.) Then there exist (0, T) such that if co ::; 101 then

101

> 0 and T2

E

n(x, t) = 0 Theorem 3.4.3 (Formation of vacuum) Let Xo E nand T > O. Assume that (3.90) holds. Then there exist M > 0, T 3 E (O,T), and "(,10 E (0,1) such that if Dn(QT) ::; M then

n(x, t) = 0 where p(t) = "((t - T 3 )c.

for a.e. x E Bp(t)(xo), t E (T3 , T),

3.4. Localization of vacuum solutions

65

t

t

t T

Tz

n=O r

r

r

Figure 3.1: Localization of the vacuum sets. The proofs of these theorems are presented in Section 3.4.2. The difficulties in proving the above results are due to the coupling of the equations (3.36)(3.38) and in particular, due to the drift terms div(nVV), -div(pVV). Indeed, the electric field - VV induces (or prevents) a flow of electrons or holes in some direction influencing the support of the carrier densities. The condition (3.90) is almost optimal in the following sense. Let R( u, v) ~ bu{3 for all u, v 2: satisfying a > 1 and a + f3 > 2, and let the initial and boundary densities be strictly positive in n, n x (0,00), respectively. Then, choosing f3 2: 1 (thus a + f3 > 2), there exists a solution (n,p, V) to (3.36)(3.41) satisfying n(t) > in n, 0< t < 00.

°

°

This result follows as in the proof of Proposition 3.3.3. Hence, in this situation, no vacuum occurs. No results are available in the limit case a+f3 = 2 (however, see [20]). The three localization results are illustrated by numerical examples in one space dimension in Section 3.4.3. For the discretization we use an exponentially fitted mixed finite element method as in [220]. Modeling a one-dimensional forward biased pn-junction diode, the presented properties can be verified. 3.4.2 Proofs of the main results For the proofs of Theorems 3.4.1-3.4.3 we have to estimate the local energies in the domain

P

=

{(x, T) E ~d

X

[0, (0) : Ix - xol ~ r(T), T E (i, tn,

where i, t E [0, T]' i < t, Xo E n, and r E C 1 (i, t). In this subsection, r always denotes a radius (function). Since the pressure function r(s) is taken to be sa and does not appear in this subsection, there should be no confusion of the meaning of r. The lateral surface of P is given by

8zP={(X,T): Ix-xol=r(T), TE(i,tn,

Chapter 3. The Isentropic Drift-diffusion Model

66

and the outer unit normal v

V

where

ex

x

= (Vx , vr ) of P has the components

}1 + r'(T)2'

=

V

r

-r'(T)

= -Jr:l=+::::::::::::r'~(T=;):;;:2'

is the unit vector in the direction of V x ' We choose the parameters r(T) as follows:

t and the function

(i) Theorem 3.4.1: P is a truncated cone with r(T) = p-MT, 0 < o < T < t and M > O. (ii) Theorem 3.4.2: P is a cylinder Bp(xo) x (0, T) with 0 0< T < T.

E

t,

< p :S Po,

< p :S Po and

(iii) Theorem 3.4.3: P is a paraboloid with r(T) = "f(T - t)J.L, t < T < T and "f, J.L E (0,1). The proof of the localization results is based on two technical lemmas. The first lemma is a local integration by parts: Lemma 3.4.4 Assume that P C QT. Then for almost all holds

r(V'nO< -

}p

<

nV'V) . V'nO
_1_

a

r

+ I} pn{r=i}

__ 1_ a +1

r

}alP

r (V'nO< -

kiP

t, t

E

[0, T], t < t, it

nV'V) . vxnO
r

n(t)o<+ldx_ _ 1_ n(t)o<+ldx a + 1 }pn{r=t}

no<+lvrdadT_

r

}p

(3.92)

R(n,p)nO
By using spherical coordinates and Fubini's theorem, it can be seen that the integrals over 81P are well defined for almost all r (T). A similar inequality holds for the hole density p. Proof. We only give a sketch of the proof since it is a straightforward extension of the proof of Lemma 3.1 given in [122]. Define, for p, T E [0,00), x E ]Rd, l < t < T, h > 0, and k, mEN, the following truncation and approximation functions

{ {

0 -m(p - r(T)) 1

~m(lx - xol,T)

o

if r(T) :S p < 00 if r(T) - Ijm :S p :S r(T) ifO:Sp:Sr(T)-ljm, if Ix - xol < r(T) else,

67

3.4. Localization of vacuum solutions

if 0 ~ r ~ i, t ~ r if ~ - l/k ::? r ~ t

0

-k(r -_t) { k(r-t)

~k(r)

< 00

ift~r~t+l/k

ifi+l/k~r~t-l/k,

1

h-l J;+h nc>(x,s)ds if Ix - xol < r(r) o else.

{

Then Wk,m,h(X, r) = ~k(r)'lj;m(x, r)Th(x,r) is an admissible test function in (3.36) satisfying w(x, t) = w(x, i) = O. Thus

iT In

(\7nC> - n\7V) . \7wk,m,hdxdr

iT In 8W~,;:,h -iT In

=

n

dxdr

(3.93)

R(n,p)Wk,m,hdxdr.

The left-hand side can be written in spherical coordinates (f, w) with center Xo as

iT In

(\7nC> - n\7V) . \7Wk,m,hdxdr

-

T

io

+

~k(r)m

T

io

~k(r)

lr(1')

1

1

r(1')-l/m

ir(1') 0

Sd-l

fw (\7nC> - n\7V)· -1_IThfd-ldwdfdr Sd-l rw

'lj;mh-1

l 1'+ (\7nC> h

n\7V) . \7nC>(s)ds

l'

,

X fd-l dwdfdr

where Sd-l is the sphere of radius 1 with center Xo and x - Xo = fw. Letting first k ~ 00, then h ~ 0 and m ~ 00, we get by arguing as in [122] lim {T {(\7nc> _ n\7V) . \7wk,m,hdxdr k,h,mJo

In

t{

Jt JSd-l

+ (

(\7nC> - n\7V) . ~nC>fd-l

(1') {

Jt Jo

{ (\7nC> kIP

JSd-l

Iwl_r=r(1')

dwdr

(\7nC> _ n\7V) . \7nC>fd-1dwdfdr

n\7V) . vxnC>dadr+ {(\7nc> - n\7V) . \7nC>dxdr.

Jp

68

Chapter 3. The Isentropic Drift-diffusion Model

Now we turn to the right-hand side of (3.93). It holds T n 8W~,h,m dxdT = -k mpmThdxdT

r Inr

Jo

it

r

+ k (+l/k Jt

+

mpmThdxdT

JBr(T) (xo)

T

r ~k(T)m r

Jo

+

r

t-l/k J Br(T)(XO)

T

r'(T)nThdxdT

J{r(r)-l/m
rT ~k(T) r

Jo

J Br(T)(XO)

n(x, T)VJm(X, T)h- 1(nC>(x, T + h) - nC>(x, T))dxdT.

Performing the limit k --+ 00 gives T hm n 8Wkmh 8" dxdT k---+oo 0 n T

.li

-r + r J

n(x, t)VJm(x, t)Th(X, t)dx

JB r (,) (xo)

n(x, i)VJm(x, i)Th(X, i)dx

Br(i.) (xo)

+

tm r

Jt

r'(T)nThdxdT

J{r(r)-l/m
r

+ (

n(x,T)VJm(X,T)h-1(n(x,T+h)C>-n(x,T)C»dxdT

Jt J Br(T) (xo)

h+···+h After letting h lim

--+

(h

0 and then m

+ h) =

--+ 00,

r

-

J B r (,) (xo)

h---+O,m---+oo

n(x, t)c>+ldx +

and lim

h---+O,m---+oo

h

we obtain

= -

r

Jetp

Thanks to the convexity of the function x

r

n(x, i)c>+ldx

J Br(i) (xo)

vrnc>+ldadT.

f-t

x1+1/c> the inequality

x1/c>(y - x) :::; a : 1 (yl+l/c> - x1+1/c» holds for all x, y 2: 0, and therefore

14

:::;

t r

_a_ a + 1 Jt

J Br(T)(XO)

VJm(x, T)h- 1(n(x, T + h)c>+l - n(x, T)c>+l)dxdT.

69

3.4. Localization of vacuum solutions

The right-hand side converges in the limit h

Hence

.ll T

hm

k,h,m

0

n

0, m

n aWkhm a" dxdr

< -10:

--+

+1

1

r

pn{r=i}

r

-

--+ 00

to

1

n(t)Odx - -1n(t)Odx 0: + 1 pn{r=t}

__ 1_ nO+1vrdadr. 0: + 1 lOIP

Finally, it holds

o

This proves the lemma. The second technical tool is an interpolation-trace lemma. Lemma 3.4.5 Let B = BR(XO) C lRd be a ball of radius R and let u E W1'P(B) with 1 < p < 00. Then

> 0 and center Xo (3.94)

where Co

> 0 is independent of u and R, and 1 ~ s 1 < q < p(d - 1) d-p ,

l
~

p, ifp

< d,

1

~

q, r

< 00

ifp = d,

1

~

q, r

~ 00

ifp> d,

and the exponents are given by

e=

pqd-r(d-l) (d ) d E (0,1), qp +r - r

-

8 = ps + d(p - s) > 1. ps

The proof can be found in [19]. In the case q = p and s = r the lemma is proved in [122].

Chapter 3. The Isentropic Drift-diffusion Model

70

Proof of Theorem 3.4.1. Using local elliptic regularity theory (cf., e.g., [180]) and noting that n,p E LOO(Bpo(xo)), we see that \7V E LOO(Bpo(xo) x (0, T)). Let M = II\7Vllo,oo,B po (xo)X(O,T), c E (0, Po), tl = c/2M,

and consider the cone p = P(p, t) = {(x, T) : x E Br(xo), T E (0,

tn,

where p E (c, po], t E (0, tl), and r = r(p, T) = P - MT. For almost all p and T it holds

r\7V. \7n o+ldxdT __ a_ r .6.Vno+1dxdT a+l }p + ~ r (\7V. vx)no+1dadT, a + }etp ~

a+ l}p

1

and therefore, we conclude with Lemma 3.4.4

r

r

_1_ n(t)o+ldx + l\7n o l 2 dxdT a + 1 )pn{r=t} }p

<

r

(3.95)

r

_1_ n(O)o+ldx + (\7n o . vx)nOdadT a + 1 )pn{r=O} }etp __ 1_

r (v

a+l}et p

-L

r

+ \7V. vx)no+ldadT _

r

_ a_ .6.Vno+1dxdT a+l}p

R(n,p)nOdxdT

h+···+1s · Since n(O) vanishes in Bpo(xo), we have h = O. For the estimate of 12 observe that in spherical coordinates with center Xo (cf. [20, 122])

8En

(

8p P,T

Hence

)

3.4. Localization of vacuum solutions

71

We use the interpolation-trace Lemma 3.4.5 with p = q = 2 and r = s = 1+1/ a:

where

e=

d(a - 1) + (a + 1) E (0,1), d(a - 1) + 2(a + 1)

8 = 2(a + 1) + d(a - 1) > 1. 2(a + 1)

(3.97)

By the definition of r, we have

Thus, applying Holder's inequality with exponent

c5 max(l, (2/c)8), we obtain

l/e

and setting

K1

I IInQII~,2,8BrdT::; I (1IV'nQII~,2,Br + IlnQI16,l+1/Q,BJ81InQII~~~~:;Q,BrdT t

t

2K1

< 2K , (J,'lllInallhB.dT + J,'llnalli'l+l/a,B.dT)' x

(J,'lI na lli"+l/a,B.dT) 1-6

< 2K1t l - 8(En(p, t) + t1bn(po, tdQ-1)/(Q+l)bn(p, t)) 8bn (p, t)2Q(1-8)/(Q+1), where

This yields Iln Qllo,2,8I P

< K 2t(1-8)/2(En (p, t) + bn(p, t))8/2b n (p, t)Q(1-8)/(Q+l) < K 2t(1-8)/2(En (p,t) +bn(p,t)y',

where Ki = 2K1 max(l, t1bn(po, tl)(Q-l)/(Q+l)) and /-L =

We conclude

e

a

2 + a + 1 (1 -

e) E (1/2,1).

Chapter 3. The Isentropic Drift-diffusion Model

72

Thanks to the special structure of r = r(p, r) and the definition of M, we have M + V'V· ex l/r + V'V . l/x = > 0, \.11 +M2 so that 13

:s O. Furthermore,

where K 3 = a~lllD. Vllo,oo,QT' For p we get an analogous inequality to (3.95) and similar estimates involving the local energies E p and bp defined by

Therefore we have the estimate

r

_1_ (n(t)a+1 0: + 1 }Br(xo)

< K,,(l-Oj/2 ( + K3

L

+ p(t)a+1)dx +

r

}p

(IV'n a

2

1

+ IV'pa l2)dxdr

(J~n) 1/2 (En + bn )" + (&:;) 1/2 (E

(na+l

+ pa+l)dxdr -

L

p

+ bp j ")

R(n,p)(na + pa)dxdr,

where

K 42 = 2K1 max

(1 t b (p , 1 n

0, t 1 )(a-1)/(a+l) , t 1 bp (p 0, t 1 )(a-1)/(a+1») .

Employing the assumption on R( n, p) gives _1_ 0:

r

+ 1 } Br(xo)

(n(t)a+l+p(t)a+l)dx+En(p,t)+Ep(p,t) 8E

< 2K4 t(1-()/2 ( 8pn + + (K3

+ /'i,R)

L

8J

8E) 1/2

(En

+ Ep + bn + bp)(p, t)11-

(n a+1 + pa+1 )dxdr

8 (En + E p)) 1/2( En + E p + bn + bp)( p, t )11< 2K4 t (1-()/2 ( 8p + tKs(bn + bp)(p, t),

73

3.4. Localization of vacuum solutions

with K 5 = K 3 + /'i,R. Since the right-hand side of the above inequality is nondecreasing in t, we can write

(b n + bp + En

:S

+ Ep)(p, t)

8 2(a + 1)K4 t(1-0)/2 ( 8p (En

+ Ep)) 1/2 (b n + bp + En + Ep)(p, t)/-L

+ (a + 1)K5 t(b n + bp)(p, t). Choosing t

< t2

= min

(h, (2(a + 1)K5 )-1),

(bn + bp + En + Ep)(p, t)

we get

:S 4(" + 1)K,t(1-'1/2 ( X

(b n + bp + En

~ (En + E p)) '1'

+ Ep)(p, t)/-L

(3.98)

and

where K 6 = 16(a+ I? Kl. Integrating this differential inequality for (En + E p ) in (p, Po) gives (note that J-L > 1/2)

(En

+ Ep)(p, t)2/-L-1 :S (En + Ep)(Po, t)2/-L-1 -

Let

K 6 1t O- 1(po - p).

p(t) = Po - K 6 t 1- O(En + Ep)(Po, t)2/-L- 1.

Then p(O) = Po and p is non-increasing. Choose T 1 E (0, t2) such that p(T1) > c. Then, for t E (0, T 1) and P E (c, p(t)],

(En

+ Ep)(p, t)2/-L-1 < (En + Ep)(p(t), t)2/-L-1 < (En + Ep)(Po, t)2/-L-1 -

K 6 1t O- 1(po - p(t)) = 0.

Thus (see (3.98)), for P = p(t),

n(x, t)

=

p(x, t)

The conclusion follows.

=

°

for a.e. t E (0, T 1), x E

Bp(t) (xo).

D

The proof of Theorem 3.4.3 contains an estimate used in the proof of Theorem 3.4.2 and is therefore given before.

74

Chapter 3. The Isentropic Drift-diffusion Model

Proof of Theorem 3.4.3. We take the paraboloid

P = P(t) = {(x, T) : x

E

Br(xo), T E (t, Tn,

°

where t E (0, T), r = r(T, t) = ,,/(T-t)'\ ,,/, c E (0,1). Choose,,/ > small enough such that 2,,/max(1,T) ~ dist(xo,aO). Then r(T,t) ~ ,,/TJ-L ~ dist(xo,aO)j2 and Br(T,t) (xo) C w for some domain wee 0, for t E (0, T) and T E (t, T). We get from (3.95)

_1_ a

r

+ 1 }pn{T=T} < _1_

rlV'nQ dxdT n(tr~+ldx + r }alP

n(Tr~+ldx +

r

l

2

}p

(V'nQ • IIx)nOtdadT

+ 1 )pn{T=t} - ~1 r (liT + V'V . IIx )n Ot +1 dadT - ~

a

a+ }alP

-l

r ~VnQ+ldxdT

a+1}p

R(n,p)nOtdxdT

h+···+h. Since measd-l (P n {T proceed as follows:

= t}) = 0, it holds h = 0. For the estimate of 12 we

Taking into account IlIxl proof of Theorem 3.4.1)

_ dEn (t)

dt

we obtain

~

1 and (with spherical coordinates (r,w) as in the

3.4. Localization of vacuum solutions

75

Using the interpolation-trace inequality (3.96), the boundary integral can be estimated by 'Yf

r

J&lP

<

I~r ,-1\vx ln 2°do-dT ~ ut

C51T IT - tl x

(lIV'n

< K , (t) ( x

O

1 e -

rT IT - tl 1- r e

Jt

J&Br(XO)

n2°do-dT

max(I, r- 280 )

IIO,2,B r (xO)

+ IinoIIO,l+1/o,B

r

(xO)

f8 11no 11~~~~:}o,Br(xo)dT

1,T IIl1n Ili",B,(Xo)dT + 1,T Iln lIi,1 +1/u,B,(x,) dT) , U

U

(1,T IlnUlli ,,+1/U,B,(x,)dT) '-',

where we have used Holder's inequality with exponent I/O, the inequality (a + b)2 ~ 2(a2 + b2) and the definition K 1(t) = 2ch- 288 sup max (IT

- tI 1- e, IT _ tI1-e-2e88).

TE(t,T)

The constant K 1 (t) is finite if we choose f ~ 1/(1 +280). Furthermore, K 1 (t) K~ ~f 2ch- 288 max(T 1-e, T1-e-2e88). Therefore

fi(tl:nvxln'UdndT)

<

'I'

K,(LIlln I'
x


'u/(u+1)dT

)'1'

[1,T (in nU+1dx)'u/(U+l)dTr-')/2

< K 2(En(t) + (T - t)bn(m, T)(o-l)/(o+l)b n (r, t)) 8/2 X

(T - t)(1-8)/2bn (r, t)O(1-8)/(o+1)

< K 2(T - t)(1-8)/2 max (1, (T _ t)bn(m, T)(O-l)/(O+1)) 8/2

+ bn(r, t))o/2bn (r, t)O(1-8)/(0+1) K 3 (En (t) + bn(r, t))'.t, X

<

(En(t)

~

Chapter 3. The Isentropic Drift-diffusion Model

76

where

r

bn(t) = bn(r, t) =

sup n(T)o+ldx, TE(t,T) JBr(r,t) (xo) m = ')'max(l, T), J.L = 0/2 + a(l- o)/(a + 1) E (1/2,1), and

K 3 = K 2 T(1-6)/2 max ( 1, Tb n (m , T)(o-l)/(O+l)

6/2 )

.

We conclude

K3

(

h :S.JYE -

dE (t))1/2 ;t (En(t)

+ bn(r, t))/l.

Now we estimate the integral h. Here we need the assumption on R(n,p). Since Do V E £00 (QT), we get, by elliptic theory, the interior regularity V E £00(0, T; w 2 ,q(w)) for all q < 00. (Recall that w cc n.) Hence, M = IIV'V/lvX>(O,T;CO(w» is finite. Then T h < _1_ (/lITI + IV'VI.llIx l)n o+1dadT a + 1 Jt J8B r(xo) T < (1 + M) nO+1dadT Jt J8B r(xo)

rr rr

(1

+ M) iT Ilnoll~~~1/o,8Br(xo)dT.

(3.99)

Let>. E (1 + (3/a, 2/a); since a + (3 < 2 by assumption (3.90), the interval is non-empty. We apply the interpolation-trace lemma 3.4.5 with q = 1 + l/a, p = 2, S = 1 + (3/a and r = >.: Ilnollo,1+1/o,8Br(xo)

:S

co(IIV' no llo,2,B r (xo)

+ r-81Inollo,1+I3/o,Br(xo)Y~

x II nO I/~:>'~Br(xo)'

where

+ 1 - >.a) + >'a da(2->')+2>.a

(j = ~ d(a

a+1

E

(0,1),

+ d(a - (3) > 1. 2(a+(3)

15 = 2(a + (3)

We use Holder's inequality with exponent Q = (1 - (3)/(1 the last norm:

+a

-

a>.) > 1 for

3.4. Localization of vacuum solutions

where

2c~+1!O max

77

(1, r(T, t)-88 C0+1)!O)

[ (1

X max 1,

f3

)OCO-f3)(O+1)!C20 CO+f3))]

n(T)o+ dx

,

Br(xo)

and 1/1 =

B(a+1) 2a

+

(1-B)(a+1) aAQ < 1,

1/2 =

For future reference we note that, since aA 1/1

+ 1/2 =

(1 - B)(a + 1) aAQ' > O.

(3.100)

< 2,

ad(2 - A) + aA + 2 ad(2 _ A) + aA + aA

> 1.

Integrating the above estimate for n o +1 over (t, T) gives

(3.101)

Chapter 3. The Isentropic Drift-diffusion Model

78

Since VI < 1, we can employ Holder's inequality with exponent Ilvl to get (T - tt2bn (r, t)V2

(i

T

t

K 4(T)I/(I-vt}dT

)1-1'1

1

x (En(t) + C n (t)t . Recall that C n (t) = if

Choosing

€

(3.102)

Jp n Ot +{3dxdT. The integral involving K 4(T) is well defined

iT

r(T, t)-89(Ot+l)/(0t(I- V1»dT <

00.

< a(l- vl)I(80(a + 1)) it holds _€

ZO(a + 1) >

-1

a(l- vt}

,

and the integral converges. Thus it holds K 5 ~f

(i t

T

K 4(T)I/(I-vt}dT

)1-1'1

< 00,

where K 5 depends also on the L Ot +{3 norm of n in w x (0, T). Using Young's inequality with exponent Ilvl gives: (E + C )V1bv1(V1+V2-1) . b(1-vt}(V1+ V2) n

n

n

n

< vl(En + Cn)b~1+V2-1 + (1- Vt}b~1+V2 < (En + Cn)b~1+V2-1 + b~1+V2

=

°

bV1+V2-1(E n n

+ Cn + bn·)

Recall that VI + V2 - 1 > (see (3.101)). This estimate and the inequality (3.102) concludes the estimate of 13 (see (3.99)):

h

< (1 + M)

r

Jetp

n Ot +1dadT

< K 5(1 + M)(T - tt2bn (r, ttl +1'2- 1(En(t) + Cn(t) + bn(r, t)). More generally, we have proven the following result: Let P be given by P = {(X,T) :

Ix - xol ::; r(T),

Then it holds

~LP nOtHdadT

< c(t - it2 max X

T E (i,tn·

[1, (h r(T)-V3 dT) 1-1'1]

b~1+V2-1(En

t

+ C n + bn ),

(3.103)

3.4. Localization of vacuum solutions

79

where c > 0 only depends on the L 00 (0, T; CO (w)) norm of \7V and the U"+!3(w x (0, T)) norm of n, on 0, 0:, j3 and d. Furthermore, VI and V2 are given by (3.100), and V3

=

80(0: + 1) . 0:(1 - VI)

We need this result in the proof of Theorem 3.4.2. It remains to estimate the integrals 14 and Is: 14

+ Is :::;

IID.Vllo,oo,wX(O,T)

:::;

L

n a+1dxdT - "'R

(3.104)

L

n a+!3dxdT

K 6 (T - t)bn(r, t) - "'RCn(t),

where K 6 = IID.Vllo,oo,wX(O,T)' Therefore, we have shown that

r

_1_ n(T)a+ldx 0: + 1 }Br(T,t)(XO)

<

# K

(

dE dtn (t)

+

r l\7n a }p

l

2

dxdT + "'R

)1/2 (En(t) + bn(t))JL

r n a+!3dxdT }p

+ K 5 (1 + M)(T - tt2bn (r, ttl+V2-1(En(t) + Cn(t) + bn(r, t)) + K 6 (T - t)bn(r, t). Since the right-hand side of this inequality is non-decreasing in T, we can replace the left-hand side by 1

0: + 1 bn(t) + En(t)

+ "'RCn(t).

Then, taking c > 0 small enough, setting t* = T - c, and using bn(r, t) :::; K, where K is the global energy, we get for t E (t*, T),

~ ( ,,: 1bn(t) + En(t) + KROn(t»)

: :; # K

(

dE dtn (t)

)1/2 (En(t) + Cn(t) + bn(t))JL.

Thus

where the constant

K2 ~ 2K3 >0 7 j'YEmin(l, "'R, (0: + 1)-1)

(3.105)

80

Chapter 3. The Isentropic Drift-diffusion Model

is independent of t. Integrating this differential inequality in (0, t) with t E (t*,T) gives E n (t)2f.L- I < E n (0)2f.L- I -

-

-

t

K

t*

7

< K 2f.L- I - -K < 7

°

if K 2f.L- I :S t* / K 7 . Recall that J.L > 1/2. We conclude E n(t)2f.L-I = for t E (t*, T),

°

and (see (3.105)), for some T2 E (t*,T), n(x, T) =

°

for a.e. Ix -

xol

:S ,(T - T2)f.L, T E (T2,T).

This proves the theorem.

D

Proof of Theorem 3.4.2. We consider the cylinder P = P(p, t) = Bp(xo) x (0, t),

with P E (€, PI), t E (0, T), and € E (0, po). Taking into account hypotheses (3.90) and (3.91), we get from (3.95):

r

r

VT

=

r

_1_ n(t)o+ldx+ lV'nQl2dxdT + "'R nQ+{3dxdT + 1 JBp(xo) Jp Jp

a

~1

<

r

a + JBp(xo) __ 1_

(p -

Po):~dx + t

t r

a+1Jo JaBp(Xo) h+"'+h

r

Jo JaBp(Xo)

°

and the (3.106)

(V'nQ . vx)nQdcrdT

(V'V. vx)nQ+ldcrdT_ _ a_

r ~VnQ+IdxdT

a+1Jp

As in the proof of Theorem 3.4.1 we get the estimate

h

where

<

K~ = 2c6 max(l, €-28) max(l, Tbn(po,T)(Q-I)/(Q+I))

and J.L E (1/2,1) (see the proof of Theorem 3.4.1 for the definitions of Co > and 8 > 0). From (3.103) we conclude

h

< M <

r nQ+IdcrdT

JalP Mctl/2 max(l, (T€-1/3)1-1/1 )bn(po,T)'-'1+1/2- I (En + en + bn)(p,t),

°

81

3.4. Localization of vacuum solutions where V2, V3

M = II\7VllvXl(O,T;CO(Bpo(xo)))'c> 0 does not depend on P or t, and VI, > 0 are given by (3.100) and (3.104). Note that VI < 1 and VI + V2 > 1.

Thus

h ::; K 3 t V2 (En + Cn + bn)(p, t),

where K 3 = M cmax(l, (Tc- V3 )1-Vl )bn(po, T)Vl +v2- 1 . Finally, the integral h is estimated by

with

K 4

= II~Vllo,oo,Bpo(xo)x(O,T). Therefore we obtain from (3.106)

where K s = meas(Bpo(xo)). Since the right-hand side is non-decreasing in t, we can replace the left-hand side by 1

--lbn(P, t)

a+

+ En(p, t) + "'RCn(p, t).

Choosing t > 0 small enough, we get

(b n + En

+ Cn)(p, t)

::;

2K6 K sco(p - po)~

+ 2K6 K 2t(1-0)/2 (8~n) 1/2 (En + bn)(p, ty", where Ki 1 = min(l, "'R, (a+l)-l). By Young's inequality with exponent 1/ J-L > 1 we get

(1 - J-L)(b n + En

+ Cn)(p, t)

::;

2K6Ksco(p - Po)~ 8E ) 1/2(1-J.L) + (1 - J-L)K7 ( 8pn ,

where K 7 = (2K6K 2T(1-O)/2)1/(1-J.L). Therefore, setting Kg = (2Ks K 6/(1 -

J-L))2(1-J.L) ,

En(p, t)2(1-J.L)

::;

Kgc~(1-J.L) (p _ po)~')'(1-J.L) + Ki(l-J.L) 8~n (p, t),

where p E (c,pt}. Now we can apply the following lemma (cf. [17,18,20]):

82

Chapter 3. The Isentropic Drift-diffusion Model

°

Lemma 3.4.6 Let TJ E (0,1), co, Po, 8 > 0, ~ c < Po, and let ¢ E CO([c, Po + 8] x [0, T]) be a non-negative function, non-decreasing in t and satisfying ¢(Po +

8,0) =

°

and ¢(p, t)T/

~ K~~ (p, t) + co(p -

po)1(1-T/)

°

for P E [c, Po + 8], t E [0, T]. Then there exist C1 > and t* E (0, T) such that if co < C1 then ¢(Po, t) = for t E (0, t*).

°

We finish the proof of the theorem before proving the above lemma. Since TJ ~f 2(1 - J.L) < 1 and 2'Y(1 - J.L) = TJ/(1 - TJ), the assumptions of the lemma are satisfied and we conclude the existence of C1 > and T 2 E (0, T) such that for all co E (0, cd

°

°

that means, n(x, t) = for x E B po (xo), t E (0, T 2 ). This proves the theorem. 0 It remains to prove Lemma 3.4.6. For this, define the function z(p) = A(pPO)l/(l-T/), for p 2 Po, with A > 0. Then z(p) solves

z(p)T/ z(po where t* >

+ 8)

°

K~; (p) + co(p > ¢(po + 8, t*),

PO)T//(l-T/) ,

P E [po, Po

+ 8),

will be specified later, if the conditions K

AT/ = - - A + co I-TJ

and

A 2 8- 1/(1-T/)¢(po

+ 8, t*)

are satisfied. This is possible if the function K

f(x) = xT/ - - - x - co, I-TJ

x 2 0,

has a root in the interval I ~f [8- 1/(1-T/)¢(po + 8, t*), 00). Now, f has a maximum at X m = (K/TJ(1 - TJ))1/(T/-1) with value

)T//(T/-1) K f(x m ) = (1 - TJ) ( TJ(1 _ TJ) - co·

°

Since TJ < 1, the value f(x m ) is positive for sufficiently small co > 0. The function f is strictly concave and satisfies f(O) = -co < and f(x) ----7 -00 as x ----7 00. Therefore, f has two roots in (0,00). At least one of the two roots lies in the interval I if we choose t* > small enough (noting that ¢(po + 8,0) = 0).

°

83

3.4. Localization of vacuum solutions

The conclusion of the lemma follows by a monotonicity argument. Indeed, it holds


in [Po, Po + 8) x [0, t*). Multiplying this inequality with sign+(
(
+ 8, t) - z(Po + 8))+.

Since z(Po + 8) 2
0:::; (
= 0,

o

hence the conclusion.

3.4.3

Numerical examples

We present numerical examples which illustrate the properties of the vacuum sets {n = o} and {p = o} proved in the previous subsection, namely (i) finite speed of propagation, (ii) waiting time, and (iii) formation of vacuum. The isentropic drift-diffusion model in one space dimension reads:

atn - ax (ax (n 5 / 3 ) - naxV) atp - ax (ax (p 5 / 3 ) + paxV) A2 axx V

-R(n,p), -R(n,p), n-p-C

in (0,1) x (0, T)

with initial and boundary conditions

n(O, t) = no, p(O, t) = Po, V(O, t) = Yo, n(l, t) = nI, p(l, t) = PI, V(l, t) = VI, n(x,O) = nI(x), p(x,O) = PI(X), x E (0,1). The equations are in dimensionless form (see Sections 3.1.2 and 3.5.2 for details on the scaling). The doping profile is given by

°

C(x) = {-I if < x < 0.7 +1 if 0.7 < x < 1. We take the squared Debye length A2 = 1.6.10- 3 . This choice of parameter and functions corresponds to a silicon pn-junction diode of length L = 1O- 3 cm with the moderate doping concentration ICI = 10 15 cm- 3 (see Section 3.5.2). The above system is numerically solved by using an exponentially fitted mixed finite element method for the discretization with respect to the space

Chapter 3. The Isentropic Drift-diffusion Model

84

variable and an explicit Euler method for the discretization with respect to the time variable. The numerical discretization is explained in Section 3.5.1. Example 1. We take the boundary values

no = 0, nI

=

1,

Po = 1, PI

= 0,

Vo = -2.5, VI

= -3.75.

°

In semiconductor simulations, the boundary values are usually chosen such that (i) the total space charge -n+p+C vanishes at the Ohmic contacts x = and x = 1, (ii) the boundary densities are in thermal equilibrium, and the boundary potential is the superposition of the thermal equilibrium value and the applied potential (see Section 3.1.2). The above values satisfy these conditions with an applied voltage of U = 1.0 V. Thus we are modeling a forward biased pn diode. We neglect in this example recombination-generation effects: R(n,p) = 0 (see Examples 2 and 3 for non-vanishing R(n,p)). The initial densities are shown in Fig. 3.2 and 3.3. In Fig. 3.2 the temporal evolution of the hole density p is depicted. Initially, there is a vacuum region for p consisting of the interval [0.2,1.0]. For increasing t > 0, the vacuum set becomes smaller and finally, it becomes trivial (i.e. only p(l, t) = 0) after some time. A similar behavior can be observed for the electron density in Fig. 3.3. This shows the finite speed of propagation of the support of p and n (see Theorem 3.4.1). In Fig. 3.4 the electron density is shown for small values of time. Here, the vacuum set for n becomes larger for small time (t = 0.01), and later (t = 0.05), the size of the vacuum set is decreasing. This illustrates the waiting time property even in the absence of recombination-generation effects (see Theorem 3.4.2). Example 2. We use the same boundary values as in the first example but different initial functions (see Fig. 3.5). In this example we study the effects of the recombination-generation term. We choose:

R(n,p) = cR(np)f3. From Fig. 3.5, where CR = 0, we see that the vacuum set for the electrons is trivial (i.e. only n(O, t) = 0) for sufficiently large time. At t = 4.0 the electron density has almost reached the stationary state. If CR = 1 and (3 = 0.2 there are (non-trivial) vacuum sets for n for sufficiently large time, e.g. t = 0.7 (see Fig. 3.6). This example shows the property of formation of vacuum due to the presence of a strong recombination term (see Theorem 3.4.3). Note that the assumption 0: + (3 < 2 is satisfied. (The hole density is positive in the region where n = 0.) Choosing CR = 1 and (3 = 0.5, there are no vacuum solutions for n for t 2:: 0.7 (Fig. 3.7). In this situation the condition 0: + (3 < 2 is not satisfied.

3.4. Localization of vacuum solutions

85

Example 3. For the last example we choose initial conditions such that n(x, 0) and p(x, 0) are strictly positive. Furthermore we use different boundary values than in Examples 1 and 2, namely no=l, nI

Vo=-H~-I),

Po=2,

= 2, PI = 1,

-H ~ - 1) -

Vo =

~5.

Here the total space charge vanishes, but the boundary functions are not in thermal equilibrium (cf. Section 3.1). Again, for vanishing recombinationgeneration, the vacuum sets {n(t) = O} are empty for all t 2: 0 (Fig. 3.8). If CR = 5 and f3 = 0.1, there exists to > 0 such that the vacuum sets for n(t) have positive measure for all t larger than to (Fig. 3.9). In this situation, it holds a + f3 < 2. In Fig. 3.10 the electron density in the case CR = 5 and f3 = 0.6 is presented. The condition a + f3 < 2 is not satisfied, the vacuum sets are empty for all time. For t = 2, the solution is close to the steady state. However, if we choose CR = 10 and f3 = 0.6, there are vacuum solutions for t 2: 2 (Fig. 3.11), although it holds a + f3 > 2. The recombination effects are so strong that vacuum occurs. This does not contradict the non-vacuum result mentioned in Section 3.4.1 since this result is only valid for f3 2: 1.

p rT-----------,r------..,,~

1.10 1.00

.---=-::..:..:.::"'::.............

0.90

"\

f-

\

i

I

0.40

\

0.30

\.

0.20

\\

\\

i

:

:.

0.00 .......... I 0.00

~

~,

'"

'"

\

0.10

j';2:0-·

'.

\ ~

i

0.50

i;oT"

~ \ ~ :~ :

\

t

0.60

-~

,,

\

\'0

\

0.80 0.70

,

-_.. .......

\

'

- - JI ' - -

0.50

-'--' :-

1.00

Figure 3.2: Example 1: hole density, R(n,p) =

x

o.

86

Chapter 3. The Isentropic Drift-diffusion Model D -----_._-_.~---

'--""TO~

, ,-.

1.10

..':

#

:

,' ..

1.00

I,

(

'

0.90

~

.~ •

0.80

.

t .. 1 I 1

0.70 0.60

O.SO

,

r,;o:or i;oT'-

j.;'2:0'·

I

I

.~

0.40

.:" "

0.30

,

"

I'

0.20 0.10 0.00

...

" ',' ",1:

I

",

-----

........... ."

0.00

j

j

1.00

O.SO

Figure 3.3: Example 1: electron density, R(n,p) = O.

n

1=0.0

1.00 0.90

"

0.80

i;;Q.bT i;(l:/\r'

/--,:>?

j,';cfos"

/'

" :' ,, ,..

..,' ., . 1/ .,,. ! I .,,. 1/ ·· Ii

0.70

,

I

0.60

,

,:

I

O.SO 0.40 0.30

," ,: :

'

I

0.20

:

I

0.10

,

0.00 0.60

0.70

j

: : "j

0.80

0.90

1.00

Figure 3.4: Example 1: electron density, R(n,p) = O.

87

3.4. Localization of vacuum solutions n .----r"---r--~--~

1=0.0 1=0.'-

i;oT"

j,;;(o'

1.10 1.00 0.90 0.80 0.70 0.60 O.SO

0.40 0.30 0.20 0.10 0.00 0.00

0.20

0.40

0.80

0.60

1.00

Figure 3.5: Example 2: electron density, R(n,p) = O.

n

1=0.0

,;;or

1.20

i;o.r

1.10

j~~o-

1.00

i

i;

!;

0.90

i.

0.80

il

'I

0.70

1;

Ii

0.60

il

0.50

~--)j

.

0.40

,.,.~'

0.30 0.20 0.10 0.00

..",

/

/

j

0.00

.' , " , .'.' , ,

....:::.~:::::~:~, . ' 0.20

0.40

0.60

0.80

1.00

Figure 3.6: Example 2: electron density, R(n, p)

It

= (np)0.2.

88

Chapter 3. The Isentropic Drift-diffusion Model D

........----,-----.---..------,...-----.-.j;;;ij'J) 1.20

1,,(l.1

i;oT'

1.10

t~~o'

1.00 0.90 0.80 0.70 0.60

0.50 0.40 0.30

I

0.20 0.10

/'

L~ . :·::

0.00

0.00

0.20

0.40

0.60

0.80

1.00

x

Figure 3.7: Example 2: electron density, R(n,p) = (np)O.5.

n

r - r - - - - - - - - - - r - - - - - - - - " 1=0.0 j;;['J

2.20

i;o:is"

2.00

t~2~O-'

1.80 1.60

1.40 1.20 1.00

"-...

0.60

:. / . ..

--

~'.:.:.:.:.:::..

0.80

-

0.4Of-0.20 0.00 0.00

I

I

O.SO

1.00

x

Figure 3.8: Example 3: electron density, R(n,p) = O.

89

3.4. Localization of vacuum solutions n

- - - - , . - , 1=0.0

2.00

~­

i~~HI­

1.80

i;2~0-'

1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.50

x

1.00

Figure 3.9: Example 3: electron density, R(n,p) = 5(np)O.1.

n r-r------,----------y-o~

2.00

~ ~:n-­

1.80

(,;20-'

1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 ......... 0.00

--'0.50

--'--' 1.00

x

Figure 3.10: Example 3: electron density, R(n,p) = 5(np)O.6.

90

Chapter 3. The Isentropic Drift-diffusion Model n ,.,..-------r------.~~

2.00

,1

//

1.80 /"

1.40

., ,',

!

/

I

il,

1.20

.

/'1 I

1.60

,,;0.02 i~]8-'

1';2~O-·

,:

~'

1/,'

1.00 0.80 0.60 0.40 0.20 0.00 0.00

O.SO

1.00

It

Figure 3.11: Example 3: electron density, R(n,p) = 10(np)o.6.

3.5 3.5.1

Numerical approximation The mixed finite element discretization in one space dimension

In this subsection we derive a discretization scheme for the equations (3.36)(3.41) in one space dimension. Scharfetter and Gummel [326] proposed to treat the equations of the standard drift-diffusion model as differential equations in nand p with I n , Jp , and Vx assumed constant between mesh points. In [24] the mixed exponential fitting method is adapted to a unipolar advectiondominated drift-diffusion model with nonlinear pressure. Mixed finite element methods are described, e.g., in [63, 66, 68, 93, 317]. The basic idea of the method is to write the continuity equations of the standard model as (In)x = (e V wx)x, (Jp)x = -(e- V zx)x through the change of variables w = ne- v , z = pe v . We shall adapt this method to the nonlinear bipolar model. We assume throughout this section that

and we neglect recombination-generation effects, that means we set R( n, p) = 0 (see Section 3.6.1 for nonvanishing recombination). Moreover, we assume that the boundary data are time-independent. For convenience, we set J.Ln = J.Lp = 1.

3.5. Numerical approximation

91

Time discretization. The problem (3.36)-(3.41) is solved via the explicit Euler method as follows. Let tj = jk, j 2: 0, k > 0, and set

Given nO, pO verifying the boundary conditions (3.39) solve for j 2: 0:

n

HI _

j

k n - (((nj)",)x - nj(Vj)x)x

pJ+lk- pJ - (((pl)"')x ),2(Vj)xx

=

+ pl(vj)x)x n j - pl- C

° ° in

(3.107) (3.108)

n=

(O,L)

(3.109)

subject to the boundary conditions (3.39). The explicit Euler method has the disadvantage that k must be chosen very "small" compared to the mesh size h (more precisely k/h2 ~ 1/2), but it is simple to implement and sufficient for our purpose. Space discretization. As mentioned above we discretize the equations (3.36)(3.37) via the exponential fitting mixed finite element method. The Poisson equation (3.38) can be discretized using standard finite differences. Let Xi = ih, where h = L/N, i 2: 0, h > 0, and set Ii = (Xi-I, Xi), i = 1, ... , N, and

In the following we restrict our attention to the hole density in (3.108). Equation (3.107) can be discretized analogously. We will discretize (3.110)

where r'(p) = ap",-1 and!k = (pJ+l - pJ)/k. The index j is here and in the following omitted. The discretization is done in several steps. First step: approximation of r' . We approximate r' (p) by a piecewise constant function r defined by

(3.111)

The current Jp in (3.110) is then approximated by Jp

c::::'

-(riPx

+ pYx)

in h i = 1, ... ,N.

92

Chapter 3. The Isentropic Drift-diffusion Model

Second step: change of variables. Using the local Slotboom variables Z

= pe V / ri

I i,

l'n

the expression for the current becomes

J p ~ -(rie-V/ri zx)

in h

If ri = 0 the current vanishes; thus we assume in the following ri > O. Eq. (3.110) can be rewritten in terms of the variables Jp and z as:

Find (z, Jp ) E L 2 (1)

X

HI (1) such that

r-:-IeV/riJ 0, t P + z X '" -

(3.112)

(Jp)x=-h

(3.113)

inh

Here V is given (from (3.109)) such that Vx is constant in h i = 1, ... , N. Multiplying (3.112) and (3.113) by test functions and integrating over all intervals gives the following formulation:

(J LJ ~

L...J

Ii

i=l

t

N

~l

~

J =-L J

!....eV/riJ Tdx r. p

Ii

ZTx dx

+ [pev/riTlxi

)

~0

Xi-l'

(3.114)

N

(Jp)x
h
(3.115)

Jp(Xi - 0)) = O.

(3.116)

~l

~

N-I

L q(Xi) (Jp(Xi + 0) i=l

The equation (3.116) describes the continuity of Jp at the interval end points. Here T is a piecewise regular function,
Xh Wh Ah,PD

{TEL 2 (I): T(x)=aix+biinli, i=l, ... ,N}, {
We approximate

Jp ~ J; E Xh,

Z ~ zh E Wh,

P ~ ph E Ah,PD'

V E Xh.

93

3.5. Numerical approximation

The discrete system then consists of three equations for the three unknowns J;, zh and ph: Find

J; E Xh, zh E Wh, ph E Ah,PD such that

i=l

iIi r.

t (r ~eV/riJ;Tdx

r

-

iIi

zhTxdx +

[PheV/riT]~~_l)

= 0, (3.117)

L J(J;)x¢dx = - L 1!k¢dx, N

~l

N

~

~l

(3.118)

~

N-l

L q(xd (J;(Xi + 0) -

(3.119)

J;(Xi - 0)) = 0,

i=l

for T E Xh, ¢ E Wh, q E Ah,o. Eqs. (3.117)-(3.119) can be recognized as the one-dimensional lowest order Raviart-Thomas mixed scheme with hybridization [317,24]. Last step: the final scheme. Set Ji

= J;(Xi),

Pi

= ph(Xi)'

Zi

= zh(Xi)'

Vi = V(Xi),

i

= 0, ... , N.

We will approximate the integrals in (3.117) and (3.118). Choose ¢ = 1 in ¢ = 0 elsewhere. Then we get from (3.118) J;(Xi) - J;(xi-d

=

h

r (J;)x dx = - iIir /kdx.

iIi

Approximating the integral on the right-hand side by -h!k(Xi-l), we see that +1 . P;-l -k P;-l + ~h (Jip,t. _ Ji. ) = o. p,.-l We approximate J; ~ J i in Ii (see Section 4.8.1 for another idea); then, since Vx is constant in h the integral in (3.117) can be computed if we take T = 1 in Ii, T = 0 elsewhere. Thus from

r ~eV/riJidx + PieVi/ri - Pi_levi-dri

iIi ri

= 0

it follows after elementary computations Ji -- -

(Vi -

Vi-I cot h 2

Vi-I) Pi - Pi-l -Pi Pi-l Vi-I -+-Vi --2ri h 2 h

Vi -

if ri > 0 (3.120)

and Ji = 0 if ri = O. The discretization (3.120) can be seen as a nonlinear Scharfetter-Gummel scheme [326, 66], the nonlinearity being taken care of by the coefficient rio

Chapter 3. The Isentropic Drift-diffusion Model

94

vg, vt E

°

The final scheme is the following: Given the boundary conditions ~, rJo, piN' nl, n~ 2 (j > 0) and initial conditions (p?)~o, (n?)~o solve for j 2 0, 1 :S i :S N - 1: j+l

ni

-

k '+1

j

ni

1 (

h

-

j

j

'

2

A

if ri

V:~1 +

° °

2v:h

j

-

2

~N+l,

)

0,

(3.121)

Jj )

0,

(3.122)

I n ,i+l - In,i

j PI k- PI + h1 (Jp,i+l -

E

p,i

V:~1

(3.123)

> 0, ri > respectively, and

if ri = 0, ri =

J~,i = 0,

J~,i = 0,

respectively.

3.5.2 Numerical examples in one space dimension As example we present the numerical simulation of a silicon pn-junction diode. A pn diode consists of two differently doped regions, the p-region where the preconcentration of negative ions dominate: C(x) < 0, and the n-region where the concentration of positive ions dominate: C(x) > (see Section 2.2). We solve numerically a scaled version of the drift-diffusion model (3.27)-(3.29) which includes the physical parameters, in one space dimension. The semiconductor domain is given by the interval n = (0, L), where L > 0. The diode is defined by the doping profile, the physical parameters and the boundary conditions. We assume:

°

(i) The permittivity

€s

is constant throughout the device.

(ii) The diffusivities D = D n = D p and the mobilities J.L = J.Ln = J.Lp are constant throughout the semiconductor. (iii) The boundary densities nD, PD are equal to the thermal equilibrium values, and VD is the superposition of the thermal equilibrium boundary potential and the applied potential Va (see (3.34)-(3.35)). At the cathode a forward bias U > is applied, i.e. Va(O) = and Va(L) = -U.

°

°

3.5. Numerical approximation

Parameter q Cs

J.L

L Lo Co

Vb

95

Physical meaning elementary charge permittivity constant mobility constant device length length of p- region doping concentration barrier potential

Numerical value 10 -HI As 10- 12 Asy- 1cm- 1 103 cm2y-1 s -1 1O- 3 cm 0.7.10- 3 cm 1015 cm- 3 0.8Y

Table 3.1: Physical parameters (iv) The doping profile C is given with abrupt junction at x = L o:

C(x)

=

{-co C1

~f 0 < x < L o If L o < x < L,

where the absolute values of the doping in the n-region and the p-region are equal: Co = C1 > O. Whereas the first hypothesis is generally accepted for isotropic and homogeneous materials [342, Ch. 1], the second assumption is only valid in an average sense, but simplifies the computation of the diffusivity (see below). The hypothesis that the doping of the n-region and the p-region are equal is only used in the numerical simulations. The subsequent considerations do not need this condition. The numerical values of the physical parameters are given in Table 3.1. According to (3.34)-(3.35) the boundary data are given by

nD(O) = 0, PD(O) = Co, nD(L) =

C1,

PD(L) = 0,

5 D 2/3 VD(O) = -2j;C1 ,

VD(L) =

~ ~ C~/3 -

(3.124)

U,

(3.125)

The behavior of the diode presented below is similar for moderate or heavy doping. We take a moderate doping (see Table 3.1) which reduces the numerical difficulties in solving the Poisson equation (the scaled Debye length A being not too small). This choice does not contradict our assumptions made for the asymptotic analysis (see Section 3.1) where C » N c , C» N v is assumed. This condition is satisfied in the case of small (lattice) temperature T such that N c = const.T 3 / 2 and N v = const.T 3 / 2 are smaller than ICI. The electrons and holes thus behave like fluid particles. It remains to compute the diffusion coefficient in (3.27), (3.28). In the standard drift-diffusion model (with linear pressure) the diffusivities are derived from the Einstein relations (3.126)

Chapter 3. The Isentropic Drift-diffusion Model

96

where UT = kBT/q is the thermal voltage and k B the Boltzmann constant. These expressions follow from the equations (3.27)~(3.28) for rn(s) = rp(s) = s in the thermal equilibrium case I n = Jp = 0 and the Maxwellian form of the equilibrium densities [33, Ch. 26]. Since the equilibrium densities are not given as exponential functions of the electrostatic potential in our model, we cannot expect that the above relations hold for the isentropic drift-diffusion model. We compute the diffusion constant by the following consideration. The barrier or diffusion potential Vb (also called built-in potential; see Section 2.2) can be obtained from physical measurements and is given by Vb = VD(L) - VD(O) if U = O. From (3.124)-(3.125) follows in the thermal equilibrium U = 0: 2 Vb D = 5" 2/3 2/3/1-· (3.127) Co + C1 If we consider pressure functions rn(s) = rp(s) = sa with 0: > 1 and take into account the intrinsic density ni (see Section 3.1.2), it can be seen that the relation (3.127) becomes D

_0:-1 '" .... Coa-I

a -

Vb

+ Ca-I 1

(3.128)

2n a-l/1i

(see [220]). This expression is consistent with the Einstein relation in the sense that (3.128) coincides with (3.126) as 0: --+ 1. Indeed, since -0:- (a-l Co - Ca-I 1 0:-1

+ 2n·a-I) I

--+ 1n -COCI 2~

as 0:--+ 1

and Vb = UT In(cocdnr) if 0: = 1 [342, Ch. 2.3.1], we get aso:--+1. We note that in Section 3.1.2 we have used a scaling that gives (3.127) instead of (3.128) since ni « N e , N v . Now we bring the equations (3.27)-(3.29) into an appropriate scaled dimensionless form. The scaling is as follows (d. [220,271,274]):

where

Vb

UO=S'

L2 r--- /1-Uo '

Defining the scaled Debye length as in Section 3.1.2 by ..\2 = €sUO/qL2CO and keeping the same notation of all variables and functions, we get from (3.27)~ (3.29):

-R(n,p),

(3.129)

3.5. Numerical approximation

97

atp - ax (ax (p5/3)

+ paxV) ,\2axx v

-R(n,p), n-p-C

(3.130) (3.131)

subject to the initial and boundary conditions

n(O, t) = 0, p(O, t) = 1, V(O, t) = -5/2, n(l, t) = 1, p(l, t) = 0, V(O, t) = 5/2 - U, n(x,O) = nI(X), p(x,O) = PI(X), x E (0,1), t > 0.

°

Here, U > is the scaled applied voltage. This system of equations is solved numerically via the mixed finite element method described in the previous subsection. In the following we restrict ourself to examples of steady-state solutions, obtained by letting t ----; 00 numerically. Time-dependent solutions are presented in Section 304.3. We neglect recombination-generation effects (see Section 3.6.1 for nonvanishing recombination rates). In Fig. 3.12 the stationary carrier densities and the electrostatic potential for different applied voltages are shown. The semiconductor interval is decomposed uniformly with 50 nodes. The solution of the thermal equilibrium state is presented in Fig. 3.12a. We see that there are nontrivial vacuum sets

{n

=

O}

=

{p

[O,x n ],

=

O}

=

[xp , 1]

for some X n , x p E (0,1). It can be shown analytically by using maximum principle arguments that the densities are exactly zero (and not only numerically zero) in the indicated intervals [239]. In Fig. 3.12b the solution for the reverse (unsealed) bias U = -0.8V is depicted. The sizes of the vacuum sets are larger than in thermal equilibrium and no current flows. There are also (non-trivial) vacuum sets if a slightly forward bias U = O.35V is applied (Fig. 3.12c), and again no current flows, whereas for U = 1.0V the vacuum sets are trivial (Fig. 3.12d). In thermal equilibrium the sum of the hole diffusion current -ax (p5/3) and the drift current -pax V is zero. When U < 0 we expect that for t > 0 the (absolute value of the) electric field laxV(x,t)1 is larger than laxV(x,O)1 for x near the junction (see Fig. 3.12a and 3.12b). Thus drift current dominates and the holes flow in the direction of the anode x = until reaching the steady state. Therefore, there is vacuum for all t > in the reverse bias case. Numerically, it turns out that vacuum occurs for all time if U E (-00, Uth) where Uth c:::: OAV is called threshold voltage. In Section 3.6 we show that a positive current flows if U > Uth.

°

°

98

Chapter 3. The Isentropic Drift-diffusion Model

(a)

lr=====:::::--r::::===-t

(b) lr=====::::::::------::::===1 p

n

p

0.8

0.8

0.6

0.6

0.4

0.4 0.2

0.2

p

n

n

0l.:====:::::::;:====::;:=:::::=;~~=~ o 0.2 0.4 0.6 0.8 1

:~;~I===II o

0.2

0.4

0.6

0.8

p

04==:=:==::;::=;:=:;:::::======.1 o 0.2 0.4 0.6 0.8 1

1

(c)

(d)

lr======::::----:::==1 p 0.8 0.6 0.4 0.2

n

p

04==:=:==:::::::=::=::~:;::::;==:J

o

0.2

~i:l

o

0.2

0.4

0.4

0.6

0.8

1

11 :~::rSJSI

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Figure 3.12: Steady state electron and hole densities and electrostatic potential.

99

3.5. Numerical approximation

3.5.3 The mixed finite element discretization in two space dimensions For the numerical descretization of the isentropic model in two space dimensions we have to extend the mixed finite element method described in Section 3.5.1. In this subsection we consider a scaled version of the steady-state equations obtained from (3.27)-(3.29). The scaling is the same as in Section 3.5.2. Discretization of the continuity equations. In the following we describe in detail the discretization of the hole density p. The equation for the electron density n is discretized in the same way, with the obvious changes due to the different sign of the diffusion term. The subscript p in the hole current density will be dropped. We assume that V is a given piecewise linear function, stemming from the discretization of the Poisson equation. Moreover, for the sake of simplicity, the domain n is assumed to be a bounded polygon in JR2. The mixed exponential fitting scheme introduced in the linear case (see [66, 67, 68]) can be sketched as follows: (i) transformation of the problem by means of the Slotboom variable to a symmetric form; (ii) discretization of the symmetric form with mixed finite elements; (iii) suitable discrete change of variable to rewrite the equation in terms of the physical variable p. Due to the presence of the nonlinear term r(p), a Slotboom variable does not exist in the present case. Then, we do not discretize directly the continuity equation but a suitable approximation. We observe that \?rp(p) = r~(p)\?p and we approximate r~(p) by a piecewise constant function r ' . The equation to be solved is then J = -r'\?p- p\?V, div J = 0 in n. (3.132)

More precisely, let {1hh be a regular family of decompositions of n into triangles T (see [92]). Given a piecewise linear function p 2: 0 on a triangulation 1h, we define r'IT ~f r~(p(bT))'

(3.133)

where bT denotes the barycenter of the triangle T. The choice of the function p depends on the global iterative procedure, and an explicit choice will be made below. In order to describe the scheme let us assume first r'IT > O. Then, in the triangle T a "local" Slotboom variable is defined by p = e- vlr' p

in T,

(3.134)

and the approximate current becomes

J = _r'e- vlr'\?p

in T.

(3.135)

Since J = 0 where p = 0, we define J = 0 in T if r'IT = O. Notice that r'IT = 0 if and only if p == 0 in T, since p is linear in T and nonnegative. Equation

Chapter 3. The Isentropic Drift-diffusion Model

100

J as

(3.132) can then be written in terms of the variables P and

J=

-r'e-V/r'\lp

in T

ifr!z, > 0,

(3.136)

J= 0

in T

if r!z, = 0,

(3.137)

div J = 0

in T for all T E 7h,

(3.138)

J E H(div; 0),

(3.139)

P = ev/-' r PD on rD,

J. // = 0 on rN, (3.140) where H(div; 0) = {T E (L 2(0))2 : divT E L 2(0)}. We recall that the property J E H(div; 0) is equivalent to the requirement that the normal component is continuous at the interelement boundary [63]. In order to derive the mixed finite element discretization of (3.136)-(3.140), for every triangulation 7h, let us introduce the set of triangles where r'IT > 0: Dh

=

{T

E

7h : r'IT > O}

C

7h.

Clearly, D h == 7h if r' > 0 in the whole domain O. Denote by £h the set of edges of Dh. According to [317], we define the following set of polynomial vectors (Raviart-Thomas space): RT(T) = {T = (T1,T2) : T1 = a

+ f3X1'

T2 = 'Y + f3X2' a, f3, 'Y E ~},

and the following finite dimensional spaces Xh {T E (L 2(D h ))2 : TIT E RT(T) for all T E Dh}, Wh {4> E L 2(D h) : 4>IT E Po for all T E Dhl,

Ah,E"

{fL E L 2(£h) : for all e E

fLle

E Po for all e E £h,

rh },

1

(fL -

~)ds =

0

where ~ is any function in L 2 (r D ), P k is the space of polynomials of maximal order k, and rh = {e E £h : e C rD} c rD· We consider the following approximation of equations (3.136)-(3.140):

1

Find Jh E Xh, Ph E Wh, Ph E Ah,PD such that:

L (r ;,ev/r'Jh'TdX- r phdivTdx+

TE'Dh

JT

JT

8T

ev/r'PhT.//ds) =0 for all

T

E Xh,

(3.141)

L rdiv Jh4>dx = 0 for all 4>

E

W h,

(3.142)

L r Jh · //fLds = 0 for all fL

E

Ah,o,

(3.143)

TE'D h JT

TE'D h J8T

3.5. Numerical approximation

101

The scheme is completed by defining Jh = 0, Ph = 0, Ph = 0 outside the domain 'Ph. The function Jh is an approximation of the current j (and in turn of J), Ph is an approximation of P, and Ph is an approximation of P at the interelement boundaries, as proved in [29J. The first equation is a discrete weak version of (3.135), where integration by parts and summation over all T E 'Dh have been used together with the inverse of the transformation (3.134) on the edges. The second equation is a discrete weak version of div j = O. Since Jh· v is constant on the edges of £h, the third equation imposes that the normal component of Jh at the interelement boundaries is continuous. Moreover, it implies Jh . v = 0 on the boundary edges of'Dh not included in rho Hence, the properties Jh E H(div; 0) and Jh . v = 0 on r N are guaranteed. Due to the L 2-regularity of Xh and Wh, JhlT and PhlT see only the triangle T. Thus, the algebraic system associated to (3.141)-(3.143) can be simplified, using the so-called "static condensation" procedure [143J. Eq. (3.142) implies that, for every T E 'Ph, we have div Jhl T = 0 and consequently, JhlT is a constant vector, which we denote by JT. Taking now in the equation (3.141) 7 = (1,0) and 7 = (0,1) in T (7 = 0 elsewhere), we obtain (3.144)

where pk is the constant value of Ph on the edge ek and v(k) is the outward normal to ek. Then, inserting the value of JT given by (3.144) in (3.143), we are left with an algebraic system in the unknown Ph. More precisely, for the basis functions J.L in Ah,o, we make the obvious choice J.L = 1 on one edge and J.L = 0 on the others. From (3.144) in (3.143), one can easily compute the contributions of the triangle T to the global stiffness matrix T _ -'( (i) (j») S ij - r n .n

Ie

J

I-I J.

eV/r'ds

IT eV/r'dx ej

'

i,j = 1,2,3,

(3.145)

where we set n(k) = Ieklv(k). The integrals appearing in (3.145) can be computed exactly, since V is linear in each triangle. Remark 3.5.1 Notice that n(i) . n(j)

T

L ij

=

ITI

i,j = 1,2,3,

is the elementary stiffness matrix corresponding to a PI non-conforming finite element discretization of the Laplace operator (see below for a precise definition). From (3.145) we see that the elementary stiffness matrix corresponding

Chapter 3. The Isentropic Drift-diffusion Model

102

to the scheme (3.141)-(3.143) is obtained from the "Laplace" matrix multiplied by column by the strictly positive coefficient

The correction coefficient takes care of the drift part of the operator and it adjusts automatically from a pure diffusive to a strongly advection dominated regime. An easy observation is that, in the case V constant, the matrix (3.145) reduces to the Laplace matrix L~. We refer to [66, 67] for an exhaustive discussion in the small diffusion case. Iterative scheme. Several iterative procedures for solving the coupled system, in the case of the classical linear drift-diffusion problem, can be found in the literature (see, e.g., [206, 245] and the references therein). Here, we present a modification of the so-called Gummel method. The Gummel iteration procedure can be regarded as an approximate Newton method, where the Jacobian of the system is replaced by a diagonal matrix. It is well known that a very basic construction of the approximate Jacobian (such as taking the diagonal of the Jacobian itself) produces a nonconverging algorithm. The idea of the Gummel scheme is to incorporate information about the strong coupling of the unknowns into the Poisson equation and then construct the iteration matrix by taking the diagonal of the Jacobian of the modified system. Such a method has the advantage that only three decoupled linear systems have to be solved at each step instead of a coupled system. The electrostatic potential V and the charge densities nand p are related, through the quasi-Fermi potentials

= h(n) - V,


=

h(p) + V,

where h is the enthalpy defined by h' (s) = r'(s)/ s which implies, for r( s) 0: > 1, h(s) = _O:-s<>-1, s 2 0 0:-1

= s<>,

(see Section 3.1.2). Now, the above relation can be written as

n = g(V +
p = g(-V +
where g( s) is the generalized inverse of the enthalpy function h(s):

g(s) = { ~-l(s)

ifO<s
(3.146)

3.5. Numerical approximation

103

Then, the Poisson equation reduces to (3.147)

Differentiation with respect to V implies that at each step a problem of the form -,\2tl(8V) + [g'(- V

+
8V = 0

V (8V) . v = 0

on r D,

on

rN ,

in 0,

has to be solved, where

g'(8)

g(8)/r'(g(8))

= {

Since in our applications r(p) = pO, with 1 2-0

g'(-V +
a

if 0 if 8

< 8 < 00 ~

O.

< a < 2, we have

g'(V +
2-0

=~. a

Then, the Gummel iteration procedure can be described as follows. We assign p(O) 2: 0, n(O) 2: 0 (verifying the boundary conditions), and compute V(O) as solution of the Poisson equation ,\2 tl V(O) = n(O) - p(O) - C in 0, V(O) = VD on r D , VV(O) . v = 0 on r N . Then, for each k 2: 0, we define the following iterative procedure. Given p(k) 2: 0, n(k) 2: 0, V(k),

=

Step 1. Define V(k+l)

V(k)

-,\2tl(8V) + 8V Step 2. Define

+ 8V,

with 8V such that

(a- 1 [p(k)]2-0 + a- 1 [n(k)j2-o)8V

+ p(k) - n(k) + C in 0, on r D, V (8V) . v = 0 on r N .

=

tl V(k)

=

0

p = p(k) and 1'~ according to (3.133).

Step 3. Find p(k+l) such that

div J(k+l) = p p(k+l) Step

4.

0 , lk+l) = -1"p Vp(k+l) - p(k+l)VV(k+1 ) p

= PD on rD,

J~k+l) . v

= 0 on r N .

in 0 ,

(3.148) (3.149)

Project p(k+l) on the cone of vectors with nonnegative components.

Step 5. Define

n=

n(k) and 1'~ according to (3.133).

Chapter 3. The Isentropic Drift-diffusion Model

104 Step

6. Find n(k+I) such that div J~k+I)

= 0,

J~k+I)

n(k+I) = nD

Step

= T~ Vn(k+I) -

on rD,

n(k+I)VV(k+I)

J~k+l) . v = 0

on r N

.

in 0, (3.150) (3.151)

7. Project n(k+I) on the cone of vectors with nonnegative components.

Step 1 is performed by means of a PI non-conforming finite element method, that is a Galerkin scheme where the finite element space is defined by Xh,TJ = {v E L2 (0) : VIT E PI(T), T E 7", v is continuous at the midpoints of the internal edges, v = 'rl at the midpoints of the edges of r D }, with 'rl a given function in CO(rD). Clearly, 'rl is replaced by VD in the trial space and by o in the test space. Step 3 and Step 6 are solved via the mixed exponential fitting scheme presented above. We point out that, in the drift-diffusion system with linear diffusion, p(k+I) 2: 0 (n(k+l) 2: 0) would be a consequence of the monotonicity property of the matrix associated to the problem, as shown in [66]. Here such a property is not guaranteed, although all the numerical tests performed so far provided a nonnegative numerical solution at each step of the iterative procedure. After convergence of the iterative scheme, the current densities Jp and I n are computed using formula (3.144). We stress once more the fact that the use of the mixed formulation guarantees conservation of the current densities (i.e. the jump of the normal component of the current at the interelement is zero). It is well known that a Gummel-type iteration scheme is very sensitive to the choice of the initial guess p(O), n(O), in particular far from thermal equilibrium and for small Debye lengths. We have chosen to couple the procedure with a continuation in the parameter A. Alternatively, a continuation in the applied bias can be used.

Remark 3.5.2 Some remarks on the computational aspects of the numerical scheme are in order. In the code the positivity set 'Dh used in the mixed finite element discretization has been defined by 'D h = {T E 7" : T'lT> 8}, with 8 = 10- 8 for all the tests reported in the next subsection. Indeed, the definition of the numerical free boundary depends on the behavior of the solution in the neighborhood of the free boundary (see [294] and the references therein). Our choice of 8 has been confirmed by the numerical experiments. As pointed out in [67], the matrix corresponding to the discretization of the continuity equations can have columns with very small coefficients. From the expression of the elementary matrix defined in (3.145), following [67], it can be seen that this can happen, for the hole continuity equation, when the potential V has a local minimum or, for the electron continuity equation, when

3.5. Numerical approximation

105

the potential has a local maximum. In order to avoid instabilities, the following correction on V has been used in the code. At each iteration, after Step 1 in the iteration procedure, if l/i(k+l) at the i-th node is smaller (larger, respectively) than all the other values at the neighbouring nodes, then the value of l/i(k+l) is set equal to the minimum (maximum, respectively) value of V(k+l) at the neighbouring nodes. The continuation in the parameter A is performed as follows. We start with A ~ 10- 1 and any initial guess, then A is slowly decreased using as p(O), n(O) the solution just computed for the previous A. When the problem is solved for varying applied voltages, a continuation in the applied voltage is then used (keeping A fixed). 3.5.4

Numerical examples in two space dimensions

The tests presented in this subsection are intended to illustrate the good features of the numerical scheme described above. We consider (i) a reverse biased pn diode, (ii) a forward biased diode, and (iii) a pnp transistor. As in the previous subsections, we neglect recombination-generation effects and the semiconductor material is silicon. In all three examples, the device is assumed to be a square of size 1O- 3 cm x 1O- 3 cm and the mobilities are constant throughout the domain with value fLn = fLp = 103 cm 2 /Vs. The derivation of the diffusivity D = D n = D p is slightly different here, compared to the computation in Section 3.5.2. Assuming that the built-in potential Vb is given by the value found in the isothermal case, the diffusion coefficient reads

(3oUT

D=~fL, Cm

(3 - ~ In Co + In C1 o - 5 Co2/3 + c 2/3 1

-

2ln ni 2n 2/3 ' i

where Cm = max (co , C1), Co > 0 and -C1 < 0 are the constants of the doping profiles at the contact of an n-region, p-region, respectively, ni = 10 10 cm- 3 is the intrinsic density, and UT = 0.025 V is the thermal voltage (see [235] for details). The first set of numerical tests is devoted to the simulation of the pn diode. The geometry of the device is shown is Fig. 3.13, decomposed into triangles with the nonuniform mesh, slightly refined about the junction, shown in Fig. 3.14. The number of nodes (mid-points of triangles) is 680. First we consider a strongly reverse biased diode with an applied voltage of U = -2.0V, defined as the difference of the applied potential at the contacts of the n-region, p-region, respectively. A moderate doping profile C = _10 15 cm- 3 in the p-region and C = 10 15 cm- 3 in the n-region (abrupt junction) is chosen. The (unsealed) built-in potential is Vb = 0.576 V. The corresponding electron

106

Chapter 3. The Isentropic Drift-diffusion Model

and hole densities are depicted in Fig. 3.15, where nontrivial vacuum sets near the Ohmic contacts are formed. In the so-called depletion region, about the junction, both the electron and hole densities vanish. The total current through the contacts is zero here. In the case of a forward biased diode, the vacuum sets can be trivial, i.e. the densities only vanish at the contact. The computations are carried out with a doping profile C = -10 16 cm- 3 in the p-region and C = 10 16 cm- 3 in the n-region, and with an applied voltage U = 0.8 V. Here, the built-in potential is equal to Vb = 0.691 V. The carrier densities are shown in Fig. 3.16. The values of the (unsealed) maximal densities are maxp = 1.44· 10 16 cm- 3 and maxn = 1.45· 10 16 cm- 3 . The current (density) flow from the anode to the cathode is reported in Fig. 3.17. In this situation, a nonvanishing total current density can be observed at the contacts. In Fig. 3.18 the level curves of the unsealed electrostatic potential are depicted expressed in Volt. In the second numerical test set we model a pnp transistor with the geometry given in Fig. 3.19, decomposed with the non-uniform mesh with 903 nodes of Fig. 3.20. The doping profile is C = _10 15 cm- 3 in the p-regions and C = 10 15 cm- 3 in the n-region. With the emitter potential UE = 0 as reference point we choose the collector potential Uc = -2.0 V and the base potential UB = -1.0V. Then the emitter-base voltage is UEB = +1.0V and the basecollector voltage UBC = -1.0 V, i.e. the emitter-base junction is forward biased and the base-collector junction is reverse biased. Fig. 3.21 shows the electron and hole densities. The maximal value of the (unsealed) electron density is max n = 1.44.1015 cm- 3 ; the maximal value of the (unsealed) hole density is max p = 1.46 . 10 15 em -3. The total current is depicted in Fig. 3.22. The base current is comparable to the collector current which means that the transistor is in the high injection regime. For a well-designed device, working in the ideal (active) region [342, Ch. 3.2], the base current should be much smaller than the collector current. However, in the high injection regime, it is well known that the base current can be comparable to the collector current.

3.5. Numerical approximation

p

_________________

107

I I I I I I I I I I I I I I I I I 1

N

Figure 3.13: Geometry of the pn diode.

Figure 3.14: Mesh for the pn diode.

108

Chapter 3. The Isentropic Drift-diffusion Model

Figure 3.15: (a) Electron and (b) hole density in the reverse biased diode.

Figure 3.16: (a) Electron and (b) hole density in the forward biased diode.

3.5. Numerical approximation

109

\

,

I

Figure 3.17: Current density in the forward biased diode.

[===

--------

O.407E.OO

Figure 3.18: Level curves (in V) of the electrostatic potential (U = O.8V).

Chapter 3. The Isentropic Drift-diffusion Model

110

Emitter

Base

p

N

p

Collector Figure 3.19: Geometry of the pnp transistor.

Figure 3.20: Mesh for the transistor.

3.5. Numerical approximation

111

Figure 3.21: (a) Electron and (b) hole density in the transistor.

1

Figure 3.22: Current density in the transistor.

112

3.6 3.6.1

Chapter 3. The Isentropic Drift-diffusion Model

Current-voltage characteristics of diodes Numerical current-voltage characteristics

The current-voltage characteristic of the silicon diode described in Section 3.5.2 from the degenerate model is depicted in Fig. 3.23. Due to the presence of vacuum sets, i.e. regions with vanishing carrier densities, the total current vanishes for small applied voltages U E [0, OAV]. For U> OAV, a positive current flows. The maximal voltage where no current flows is called threshold voltage Uth ~ OAV. The threshold voltage can be computed analytically [239] and is equal to Uth ~ 0.3998V. In Fig. 3.24 the characteristic from the standard model with corresponding parameters is shown. Also in this model, the current is "small" below a certain bias and is much larger for "large" voltages. More precise information can be obtained using a logarithmic scale (Fig. 3.25). For biases U < 0.5V the current depends exponentially on the voltage, which is referred to as the well-known Shockley relation: Js

> O.

For larger applied biases U > 0.5V the growth of the curve slows down. This behavior also appears in characteristics of real diodes [325]. Notice that in the standard model the threshold voltage, i.e. the bias where the behavior of the characteristic changes, cannot be computed exactly. From the double-logarithmic representation (Fig. 3.26) we see that in the degenerate model, the dependence of the current on the voltage is approximately polynomial: where (3 changes with U. We distinguish three regions (this division should be seen as in a numerical sense since it has no mathematical justification): For applied biases close to Uth we get (3 = 1.5 ± 0.1 (U E [OAV, OA4V]). This is in good agreement with analytical computations where J'" (U - Uth) 1.5 as U --t Uth [239]. For larger voltages U E [OA4V,1.8V] we have (3 = 1.81 ± 0.02. For larger biases (3 starts decreasing and we get (3 = 0.80 ± 0.03 for U E [lOV, 50V]. In this region the current densities reach values of 104 Acm -2 which can be obtained for real diodes only under special conditions. Here, the current is mainly given by the drift terms and the characteristics of the two models are expected to be similar in this high injection region (see Fig. 3.26). For vanishing recombination-generation the steady-state electron and hole current densities are constant and thus, the occurence of vacuum regions for both types of carriers implies that the total current vanishes. This cannot be expected if recombination-generation effects are accounted for. We expect that in a layer around the pn-junction,where both carrier densities are positive,

3.6. Current-voltage characteristics

113

recombination-generation effects are responsible for non-vanishing current densities. In Fig. 3.28 the characteristic for a diode including the recombinationgeneration term (3.152) is shown. Note that R(n,p) satisfies the condition (H3) of Section 3.2.1 and vanishes if n, p are thermal equilibrium densities (see Section 3.1.2). Since R(n,p) is always non-negative only recombination effects are modeled. After an appropriate scaling of the constant hi (see (3.15)) we can see that

is another admissible model which gives similar numerical results as (3.152) if hi « 1 (see [239]). For small applied biases the current of the model including recombination is larger than the current where recombination is neglected which is due to the recombination current near the junction (cr. [342]). However, for larger biases, the recombination effects cause a reduction of the carrier densities such that the total current becomes smaller compared to the case without recombination (Fig. 3.28). 3.6.2

High-injection current-voltage characteristics

We derive static current-voltage relations of a unipolar diode modeled by the degenerate equations under high injection conditions. We consider a onedimensional semiconductor device with two Ohmic contacts. The unipolar steady-state equations read (cf. (3.30), (3.32))

(In)x = 0, I n = Dn (n 5 / 3 )x - J.lnnVx , ,\2Vxx = n - C in (0,1)

(3.153) (3.154)

subject to the boundary conditions

n(O) = nD,O, V(O) = 0,

n(l) = nD,l, V(l)=Vb -U.

(3.155)

As in Section 3.5, U is the applied voltage and Vb the built-in potential. For notational convenience the reference point for the potential is chosen such that V(O) = 0. We consider two different scalings. First rescale the equations (3.153)-(3.155) by introducing (3.156)

Chapter 3. The Isentropic Drift-diffusion Model

114

with 8 > O. Substituting (3.156) in (3.153)-(3.155) and letting formally 8 we obtain the limiting problem for 'lj;: jx = 0,

=

j

-J.lnP'lj;x, >?'lj;xx = P in (0,1),

'lj;(1) = -u.

'lj;(0) = 0,

~

0,

(3.157) (3.158)

Set E = -'lj;x. Using maximum principle arguments it can be seen that E 2: 0 in [0,1]. Thus j is a non-negative constant. The problem (3.157)-(3.158) can be solved by integrating. We obtain _

u - 'lj;(0)

_

_

~

2

'lj;(1)- 3 J.lnA.

_2_

[( J.lnA. J

01/3

2

+

E(O) 2) 3/2 _ E(O) 3] (jl/3 ) (jl/3)'

Under high applied bias, 'lj; is nearly linear and E(O) ':::' u holds (see Fig. 3.27 for the bipolar model). Using this approximation in the above current-voltage relation we get an implicit equation in j for given u, which can be approximated by u ':::' cl/ 2 for some c > 0, i.e.

In

rv

U2

(U ~ 00).

(3.159)

This relation is called Mott's law [62]. Now, if we rescale (3.153)-(3.155) by introducing

P = 8n, the limiting problem 8 jx

~

= 0,

'lj;

= 82 V,

j

= 83 I n , u = 82 U,

0 reads: j

=

-J.lnP'lj;x, 'lj;xx = 0

'lj;(0) = 0,

in (0,1),

(3.160)

'lj;(1) = -u.

Thus E = u in (0,1). Furthermore, jx = 0 implies P = Po = const. in (0,1) (assuming j i- 0). Integrating the second equation of (3.160) gives j = J.lnPou, or I n rv U (U ~ 00). (3.161) This can be recognized as Ohm's law [62]. We interpret (3.159) as the currentvoltage relation for "large" applied voltages and (3.161) as the current-voltage relation for "very large" applied voltages. In the bipolar model similar relations hold as shown in the preceding subsection. It should be noted that (3.159) and (3.161) can also be obtained from the standard drift-diffusion model in high injection situations (see [62, 300]). Indeed, in the proposed scaling, the drift term dominates the diffusion term and the current in the limiting problem is mainly given by the electric field. However, the standard model is derived under low injection conditions and strictly speaking, the limit U ~ 00 seems to be somewhat artificial in this model.

3.6. Current-voltage characteristics

115

J/IE-6A ~---'-··--""I----r----,.----..

400.00 350.00 300.00 250.00 200.00 150.00 100.00 50.00 0.00

0.00

Figure 3.23.

0.50

1.00

1.50

2.00

2.50

UN

Current-voltage characteristic of a pn-diode (computed from the degenerate model).

JIlE-3 A

1.40 1.30 1.20

1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30

1

Ji

0.20

0.10 0.00

~ L - _ - - l_ _-L_ _..l-

0.00

Figure 3.24.

0.50

1.00

1.50

L

I

2.00

2.50

UN

Current-voltage characteristic of a pn-diode (computed from the standard model).

Chapter 3. The Isentropic Drift-diffusion Model

116

-.-',---,---.,-----,..-, a;-ri\;sJ)-

le+03 I e+02

....,....••./ •..__.•...•__................•........•.

.'

le+Ol

le+OO le'()l le.()2 le'()3 le-04 le'()5 '--_-'-_ _-'-_ _-'-_ _...J-_ _--.l..J 0.50 1.00 1.50 2.00 2.50

Figure 3.25.

U

Current-voltage characteristics of a pn-diode from the standard model (a = 1) and from the degenerate model (a = 5/3) (J in A/cm 2 , U in V).

1e-Hl4

.-..---,.---...,.----,.------r-1 a;;ra: 513'-

3

le+03

le+02

le+Ol

3

le+OO

le·OI le-02

L..L

le-02

Figure 3.26.

-'--_ _- L .

le-Ol

le+OO

'--

le~1

1

J..:l

U,U.lh

le~2

Current-voltage characteristics of a pn-diode from the standard model (a = 1) and from the degenerate model (a = 5/3) in double-logarithmic scale (J in A/cm 2 , U in V).

117

3.6. Current-voltage characteristics

(a)

Carrier concentralions (U = to.OY)

B.p

3.50

3.00

0.00

(b)

Y

1.00

0.50

x

=

Polalliat (U to.OY) 0.00 F T - - - - - · - , - - - - - - - - - - ; - ,... poleDiill -1.00 -2.00 -3.00 -4.00 -5.00 .. -6.00 -7.00 -8.00

-9'ool--10.00

0.00

Figure 3.27.

X

_I

0.50

1.00

(a) Carrier concentrations and (b) electrostatic potential versus position (n, p in 10 15 cm- 3 , V in Volt, X in 1O- 3 cm).

118

Chapter 3. The Isentropic Drift-diffusion Model

~

ROO

1e+04

3

le+03 3

le+02 3

le+Ol 3

Ie-+OO

/

3

le.{)\

3 le'{)2

3 _/

/'/ /

/

i

,..-"

\e'{)l

Figure 3.28.

le+OO

le+{)I

u

Current-voltage characteristics of a pn-diode with vanishing and nonvanishing recombination-generation rate in double-logarithmic scale (J in A/cm 2 , U in V).

Chapter 4 The Energy-transport Model

This chapter is concerned with the analysis and numerical approximation of the energy-transport model. Some ideas of the derivation of this model are given in Section 4.1. The energy-transport model is a special case of systems arising in nonequilibrium thermodynamics. In Section 4.2 we define the entropy function of this general system and transform the equations to a symmetrized problem via the dual entropy variables. The existence of transient solutions is shown (Section 4.3) and the long-time behavior of the solutions to the thermal equilibrium state is studied (Section 4.4). Section 4.5 is devoted to the proof of some regularity properties and of a uniqueness result. The existence and uniqueness of steady-state solutions is investigated in Sections 4.6 and 4.7. Finally, the equations are discretized by using mixed finite elements and the numerical solution for a ballistic diode is presented (Section 4.8).

4.1 4.1.1

Derivation of the model General non-parabolic band diagrams

The energy-transport model can be derived from the Boltzmann equation in the diffusive limit. In this section we give a sketch of the derivation. Details and the mathematical analysis can be found in [41, 42, 45, 177]. We consider the electron distribution function f(x, k, t) of electrons in the conduction band of a semiconductor. The position variable is denoted by x E JR3 , k E B is the wave vector, where B denotes the Brillouin zone associated with the underlying crystal lattice [33], and t 2: 0 is the time. For an ensemble of electrons subject to collisions, the scaled Boltzmann transport equation rules the evolution of f according to [80, 275] (4.1)

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations © Springer-Verlag Berlin Heidelberg 2000

120

Chapter 4. The Energy-transport Model

where C is the kinetic energy, V the electrostatic potential, and Q(f) the collision operator. The coupling of V to f through Poisson's equation needs not to be considered such that V is treated as a given function (also see below). Assume that there are collisions due to lattice defects and electron-electron binary collisions. The three main classes of lattice defects are impurities, acoustical and optical phonons. Then we write the collision operator as

where Qimp stands for the collisions with impurities, Qph for collisions with phonons, and Qe for binary electron-electron collisions. In typical hot-electron situations, the relative gain or loss 0: 2 of the electrons during a phonon collision is small, i.e. 0: 2 = cph/cO « 1, where cph, co are the orders of magnitude of the phonon energy and kinetic energy, respectively. By expanding the phonon collision operator Q ph in powers of 0: 2 , we get (at first order) Since Qph,O is an elastic operator, it can be gathered with the impurity collision operator Qimp: For the electron-electron collision operator we write (4.2)

where V e measures the relative strength of Qe. We assume that V e is of order 1, i.e. the order of magnitude of the elastic collision operator and the electronelectron collision operator are comparable (cf. the discussion below). Now we consider a diffusion scaling with parameter 0::

which leads to the following scaled Boltzmann equation (writing t for t l and x for Xl):

For deriving the diffusion limit we perform a Hilbert expansion of

r = fo + o:il +

0:

2

12 + ....

f

=

p::x:

4.1. Derivation of the model

121

Inserting this expansion in the Boltzmann equation (4.3), we get, by identifying equal powers of a,

(Qe (DQe(fo)

+ Qo)(fo) + Qo)(fd

0,

(DQe(fo)

+ Qo)(12)

8~0 + "V kc:(k) . "V xIt + "V xV . "V kit - Q~h,1 (fo)

(4.4)

'hc:(k)· "Vxfo

+ "VxV· "Vkfo,

(4.5)

1 2 - "2- D Qe(fo)(It,Jd,

(4.6)

where DQe(fo) and D 2Qe(fo) denote the first and second derivative of Qo with respect to fo, respectively. It can be shown (see [42] for details) that (4.4) holds if and only if there exist variables f-l and T such that

fo(k)

=

Fp"T(k)

°

clef (

=

ry + exp

c:(k) - f-l)-1 T '

(4.7)

where the parameter ry > measures the level of degeneracy of the electron gas. If ry --+ 0, classical non-degenerate statistics apply. The variable f-l can be interpreted to be the chemical potential of the electrons, and T is the electron temperature. In order to solve (4.5) we introduce the operator

L F = DQe(F)

+ Qo,

defined on the space (L 2(B), (', 'IF), where F = Fp"T' and the scalar product is defined by

(f,gIF

=

r

dk

JB fg F(l- ryF)'

In [42] it is proved that (4.5) admits a unique solution

It where 'l/J1' 'l/J2 : B

--+

= -

-r

f-l - "VxV) 'l/J1 ( "V xT

It

in span(l,c:)l- and

+ (1) "V xT 'l/J2'

1R3 are the unique solutions in span(l,c:)l- of

L F'l/J1 L F'l/J2

=

=

-("Vkc:)F(l - ryF), -C:("Vkc:)F(l - ryF).

Finally, the equation (4.6) is solvable if and only if f-l(x, t) and T(x, t) satisfy the following diffusion system:

:t

n(f-l, T) + div J1 = 0,

~E(f-l,T)

+divJ2 = "VxV· J1 + W(f-l,T),

(4.8) (4.9)

122

Chapter 4. The Energy-transport Model

(4.10) (4.11) where n(J.L,T), E(J.L, T) are the electron density and the density of the internal energy, respectively, defined by E(J.L, T) =

4~3

L

c(k)F/-"T(k)dk.

In a parabolic band structure (i.e. c(k) = const.lkI 2 ), the relations for nand E can be approximated by Boltzmann statistics with B = ~3: (4.12) The carrier flux density is denoted by h, and J2 is the energy flux density or heat flux. Furthermore, L ij are diffusion matrices in ~3x3 given by (4.13) with the tensor product (J ~ g)ij = figj. The diffusion matrix L = (L ij ) is positive definite if T > 0 and -00 < J.L < 00, and it satisfies L ij = L'[j expressing Onsager's relation [188, 249]. The source term W(J.L, T) is given by

and it holds with the lattice temperature TL, which shows that W(J.L, T) is a temperature relaxation term. The equations (4.8)-(4.11) are called the energy-transport model for semiconductors. There are several drawbacks to this approach. First, in the above derivation the electron-electron collision operator is assumed to be of the same order of magnitude as the elastic operator (Le. V e = 0(1)), which is questionable. Another drawback is that the diffusion coefficients are not explicit. To overcome these problems, we can proceed as follows. We assume that Qe is of the order of a 2 , i.e. V e = a 2 v e (cf. (4.2)) or

123

4.1. Derivation of the model

Again we suppose that 0: 2 is small compared to one, and we use the same diffusion scaling as above to get the scaled Boltzmann equation

Performing a Hilbert expansion of f = r", we obtain a diffusion equation for a certain function F, called the spherical harmonic expansion model (see [41] for details):

of

oj

N(c)-+\7 at x .J+\7x V ·oc-

J(x,c, t)

Sc«F),

(4.15)

-L(x,c)(\7xF+\7xV~~),

(4.16)

where N(c) is the density of states of energy c and L(x, c) is a diffusion matrix. Going back to (4.14), the collision term SC«F) is written as

where S;h(F) and Se(F) are defined by the integrals

S~(F)(e) =

1

c- 1 (e)

Q~,l(F)dNe(k).

(4.17) Here, dNe(k) is the Euclidian surface element on the manifold c- 1 (e), weighted by 1/I\7c(k)l, which is related to N(e) by (see [41])

N(e) =

1

c- 1 (e)

dNe(k).

We choose the time unit such that 1/0:2 = 1, and we assume that the electronelectron collision operator veS':(F) is dominant, setting v e = 1/f3 with f3« 1. This amounts to assume that, for a distribution function which is constant on the energy surfaces, the energy loss due to phonon collisions occurs on a longer scale than the thermalization by electron-electron collisions. This hypothesis is valid in hot-electron situations. Therefore we can write

Again, we use a Hilbert expansion of F = Ff3 and J = Jf3 in (4.15)-(4.16). As final result we get the system of diffusion equations (4.8)-(4.11) with diffusion matrices which are differently defined than (4.13) (see [41]).

124

Chapter 4. The Energy-transport Model

Boltzmann equation f(x, k, t)

a?={3---->O

a? ----> 0

Spherical harmonic expansion model F(x, c, t)

(3---->O

Energy-transport model p,(x, t), T(x, t)

Figure 4.1: Diffusion limits for the Boltzmann equation. One advantage of this approach is that the diffusion matrix L = (L ij ) can be computed explicitely in some situations (see below). Thus both presented approaches give the same equations with different diffusion coefficients. The main difference between the two approaches is the order of magnitude of the electron-electron collision operator Q e' In the first approach this operator is assumed to be of order 1, i.e. I/e = a 2 /(3= 0(1), and thus it is comparable to the elastic collision operator, whereas in the second approach Qe is considered of order a 2 , i.e. I/e = O(a 2 ). The two diffusion limits are summarized in Fig. 4.1. In order to get more explicit expressions for the coefficients L ij in terms of n, T (or IL, T), we have to impose some physical assumptions: (HI) The energy-band diagram c: of the semiconductor crystal is spherically symmetric and a strictly monotone function of the modulus k = Ikl of the wave vector k. Therefore, the Brillouin zone equals ~3 and c: : ~ ----t ~,

k

I---->

c:(k).

(H2) A momentum relaxation time can be defined by (3

> -2,

cPo > 0,

(4.18)

where N(c:) = 47rk 2 /1c:'(k)1 is the density of states of energy c: = c:(k) [41, (III.31)], cPo is the transition matrix constant, and No is the phonon occupation number [44, Sec. 4]. (H3) The electron density n and the internal energy E are given by nondegenerate Boltzmann statistics. The assumptions (Hl)-(H2) are imposed in order to get simpler expressions for the variables. In the physical literature, the values (3 = 0 [81] and (3 = 1/2 [264] have been used in the case of parabolic band structure (see Section 4.1.4). The non-degeneracy assumption (H3) is valid for semiconductor devices with a doping concentration which is below 1019 cm -3. Almost all devices in practical applications satisfy this condition.

125

4.1. Derivation of the model

Under these assumptions, the diffusion coefficients are given by

where

471" 2 d(c:) = 3T(e)le'(k)lk and e = e(k)

(see [41, (IV.l7), (III.33)]). We refer to [41] for more general expressions for the diffusion coefficients under weaker assumptions. Notice that due to the special structure of the diffusion matrix, we can interpret the diffusion coefficients to be scalar functions (instead of IR. 3x3 matrices). By assumption (H3), we have for the electron density and internal energy, respectively [41, (IV.16)]:

1

00

n

n(J.L,T) = eJ.'/T

E

E(J.L, T) = eJ.'/T

1

e- c / T N(e)de,

(4.20)

ee-e/T N(e)de.

(4.21)

00

Let ')'(e) = k2 be the inverted e(k) relation. Then N(e) = 271",),(e)1/2,),'(e) and, using (4.18),

871" ,),(e)3/2 4 ')'(e) d(c:) = 3 T(e) ')"(e) = 3¢o(2No + 1) e{3')"(e)2' which yields

or with

(T 2. + J.) L 'J·· -- T i +j-{3-1 eJ.'/Tp{3,

roo

_ 4 P{3(T, f) - 3¢o(2No + 1) Jo

£-(3-2

U

')'(Tu) -u ')"(TU)2 e duo

The electron density and the internal energy read (see (4.20), (4.21))

1 1

00

n

271" eJ.'/T

E

271" eJ.'/T

00

')'(e) 1/2,),' (e)e-e/T dc, q(e)1/2,),'(e)e-e/T de

(4.22)

126

Chapter 4. The Energy-transport Model

or

n = Te/J-/TQ(T,O),

(4.23)

with

Thus we get

The energy relaxation term is given by

w=

1

00

8 1 (e(/J--e)/T)E:dE:,

where 8 1 is the phonon collision operator [41, (IV.18)]. In the Fokker-Planck approximation, we can write this operator as (see [329])

8 1 (e(/J--e)/T) = :E: {8(E:) [

(1 +

To :E:) e(/J--e)/T] } ,

where 8(E:) = ¢oE:f3 N(E:)2 and To = 1 is the (scaled) ambient temperature. With the definition of 8(E:), the above expression can be simplified: W

=

1

(1 -; )

00

_e/J-/T

8(E:)e-e/T

¢oe/J-/T Tf3(To - T)

1 1 00

47r2¢oe/J-/T Tf3(To - T)

Introducing

1

00

Rf3(T) =

dE:

uf3 N(Tu)2e- U du 00

"((Tuh'(Tu)2uf3 e- U du.

"((Tuh'(Tu?uf3 e- u du,

(4.24)

the energy relaxation term can be written as

W _ ~ n(To - T) - 2

Tf3(T)

,

(4.25)

with the temperature-dependent relaxation time (4.26)

4.1. Derivation of the model

127

Usually, the electrostatic potential is coupled selfconsistently to the Poisson equation A2~V = n - C(x), (4.27) where C(x) represents the doping profile. The equations (4.8)-(4.11), (4.27) for (f-l, T, V) or (n, T, V) have to be solved in the (bounded) semiconductor domain 0 C ~d (d ~ 3). The equations are to be supplemented with appropriate initial and boundary conditions. The boundary 80 of the domain 0 consists of two disjoint subsets f D and f N. The chemical potential f-l, the electrostatic potential V, and the temperature Tare fixed at f D, whereas f N models the union of insulating boundary segments (zero outflow); f-l=f-lD,

T=TD,

(4.28)

V=VD

J 1 . 1/ = h . 1/ = V'V . 1/ = 0

(4.29)

where 1/ denotes the exterior normal vector of 80. Due to the relation (4.12) between nand f-l, T, the carrier density n instead of f-l can be prescribed at fD. Thus the boundary condition n = nD,

T = T D, V = VD

on f

D,

is equivalent to (4.28). The initial condition is given by in O.

(4.30)

4.1.2 A drift-diffusion formulation for the current densities A remarkable observation is that the current densities J1 and J2 can be written in a drift-diffusion formulation of the type

V'V

-V'g1(n,T)

+ g1(n,T)T'

-V'g2(n,T)

+ g2(n,T)T.

(4.31)

V'V

(4.32)

(Here and in the following, the gradient V' always means differentiation with respect to the space variable.) Indeed, in the general case the current densities are given by (see (4.7) and (4.16)) Ji

=

-1=

d(c) ( V'e(/-L-c)/T

+ V'V ~ e(/-L-c)/T) ci - 1 dc,

i

= 1,2.

(4.33)

This relation holds true under weak assumptions (see [41] for details) and in particular under the assumptions (H1)-(H3) of Section 4.1.1.

128

Chapter 4. The Energy-transport Model

From (4.33) we get Ji = - V

= d(c)e(/lo-c)/Tc . 1 dc + -VV1= d(c)e(/lo-c)/Tc .

1

t-

o

t-

T

1 dc

0

which equals (4.31), (4.32), respectively, setting 91

=

1=

92

d(c)e(/lo-c)/Tdc,

=

1=

d(c)e(/lo-c)/Tcdc.

The functions 91 and 92 can be computed in terms of nand T, under the assumptions (H1)-(H3) of Section 4.1.1. Indeed, by (4.19), we get 91 = L l1 and 92 = L 2 1, and using (4.22) and (4.23), we can write _ P{3(T,2) -(3 91(n,T) - Q(T, 0) T n,

or

(4.34)

91(n,T) = J.L~1)(T)Tn,

92(n,T) = J.L~2)(T)T2n

(4.35)

with the temperature-dependent mobilities (i)(T) = P{3(T, i + 1) T- 1-{3 ,; = 1,2. J.L{3 Q(T, 0) ,.

(4.36)

We can write the stationary energy-transport model in the drift-diffusion formulation either in the variables n, T and V or in the variables 91, 92 and V. In both cases only the current density relations change. In the former case we have

-V(J.L~1)(T)Tn) + J.L~1)(T)nVV, -V(J.L~2)(T)T2n)

+ J.L~2)(T)TnVV,

(4.37) (4.38)

and in the latter case Ji = -V9i

+ T(

9i )VV, 91,92

i = 1,2.

The electron density is given in terms of 91 and 92 by, see (4.34),

_ Q(T(91, 92), 0)

(

n (91,92 ) - P{3(T(91,92),2)T 91,92

){3

91·

(4.39)

The energy relaxation term in the variables 91 and 92 writes now (recall that To = 1) (4.40)

4.1. Derivation of the model

129

In order to compute the electron temperature in terms of gl and g2, we have to invert the following function (see (4.34)): (4.41) This is possible if the derivative of f is positive for all T > O. The following lemma shows that this is true if the diffusion matrix (L ij ) is positive definite. Now, this property has to be satisfied in order to get a well-posed mathematical problem. Lemma 4.1.1 Let the hypotheses

(Hl)~(H3)

hold. Then

(4.42) Proof. Using the relation TP~(T,£ - 1)

= P(3(T,£) - (£ - f3 - 2)P(3(T,£ -

1),

which can be proved by integration by parts, we obtain

Then, from the formulas

det(L ij ) = e2MITT4-2(3[p(3(T, 4)P(3(T, 2) - P(3(T, 3)2] and n = Q(T,O)Te M1T (see (4.22) and (4.23)), it follows D

For later reference, we rewrite the complete energy-transport model in the (g1' g2, V) formulation: divJ1

0,

divJ2

J1 · \lV + W,

(4.43) (4.44)

g1

J1

-\lg1

+ T \lV,

J2

-\lg2

+ T \lV,

)?~V

g2

n - C(x)

in

n,

(4.45) (4.46) (4.47)

Chapter 4. The Energy-transport Model

130

subject to the mixed Dirichlet-Neumann boundary conditions

g1 = gD,1, g2 = gD,2, V = VD J1 . V = h . v = V'V . v = 0

(4.48) (4.49)

where we have set gD,i = gi(nD, T D), i = 1,2. The functions nand W depend on g1 and g2 according to (4.39) and (4.40), respectively. The dependence of T on g1 and g2 is given by the non-linear equation (4.41).

4.1.3 A non-parabolic band approximation The non-parabolic band structure in the sense of Kane [242] is defined as follows: (H4) Let the energy

£(

k) satisfy £(1

+ 0:£) =

k2 2m*

-.

The constant m* is the (scaled) effective electron mass given by m* = m OkB To/n2 k,/;, where mo is the unsealed effective mass, ko is a typical wave vector, and 0: > 0 is the (scaled) non-parabolicity parameter. Notice that we get a parabolic band diagram if 0: = O. The assumption (H4) implies "f(Tu) = 2m*Tu(1 + o:Tu) , and introducing the functions

P;3(o:T,£)

(Xl

Jo

1

1 + o:Tu £-;3-1 -ud (1 + 20:Tu)2 u e u,

00

q(o:T, £)

(1

+ o:Tu)1/2(1 + 20:Tu)u 1/ 2+£e-

U

du,

we can rewrite P;3 and Q as (see Section 4.1.1) 2

3
E - q(o:T, 1) Tn - q(o:T, O) , where N(T) = 27r(2m*)3/2q(o:T, 0). For the mobilities (4.36) we get the expressions (i)(T) _ P;3(o:T,i + I)T-1/2-;3 /-L;3 - /-Lo q(o:T, 0) , i = 1,2.

4.1. Derivation of the model

131

Here, the mobility constant /-Lo is given by

/-Lo = (31r¢>0(2No + 1)m*(2m*)3/2) -1. The diffusion matrix L

= (L ij ) then reads

Furthermore, introducing

1

00

rj3(o.T) =

(1

+ o.Tu) (1 + 2o.Tu) 2ul+j3e- u du,

we obtain (see (4.24))

and the energy relaxation time (4.26) becomes where

TO

= (21r¢>0(2m*)3/2)

-1 .

Notice that the function rj3 is in fact a polynomial:

rj3(o.T) = The symbol

rCB + 2) + 5r(,6 + 3)o.T + 8r(,6 + 4) (o.T)2 + 4r(,6 + 5)(o.T)3. r

denotes the Gamma function defined by

1

00

r(s)

=

u s - 1 e- U du,

s > 0.

(Here we use the hypothesis ,6 > -2.) Finally, the energy relaxation term (4.40) can be rewritten as

4.1.4 Parabolic band approximation

In the parabolic band approximation case (a = 0) the above expressions simplify. Since q(O, 0) = r(3/2) = y'7f/2 and q(O, 1) = r(5/2) = 3y'7f/4, we get for the electron density and the internal energy the well-known relations

3 E= -Tn 2 .

132

Chapter 4. The Energy-transport Model

In order to compute the mobilities and the energy relaxation time, we have to specify the parameter {J. In the literature the values {J = 1/2 (used by Chen et at., d. [81]) and {J = 0 (used by Lyumkis et at., d. [264]) have been employed. First let (J = 1/2. Then P1/2(0, 2) = V7i/2 and P1/2(0, 3) = T1/2(0) = 3V7i/4

and therefore,

(1) (T) /-L1/2

= /-Lo

T- 1 ,

(2) ( )

/-L1/2 T

=

3 -1 2"/-Lo T ,

Hence, we get the same current density relations and the same energy relaxation term as Chen et at. in [81]:

/-Lo('ln - f'lV),

~ /-Lo ( 'l (nT) -

n'lV) ,

3 n(To - T) 2 TO

w

The energy-transport model with the above relations will be called the Chen model. The diffusion matrix in terms of n, Treads:

L = /-Lon

(~~ 1J~2)'

When (J = 0, we have po(O, 2) = TO(O) = 1, Po(O, 3) = 2 and

(1) (T) = 2/-Lo T-1/2 V7i '

/-Lo

(2)(T)

/-Lo

=

4/-L°T- 1/ 2 V7i '

3V7i 1/2 To(T) = -4-T OT ,

so that the current densities and the energy relaxation term become

2/-Lo ('l(nT 1/ 2) - ~'lV) V7i T1/2'

~ ('l(nT 3 / 2) -

nT 1/ 2'lV),

2 n(To - T) V7i ToT1/2 .

w

The energy-transport equations with these expressions will be called the Lyumkis model. For this model, the diffusion matrix equals _

~

L - V7i/-LOnT

1/2

(1

2T) 2T 6T 2 .

Other choices for the diffusion coefficients can be found in, e.g., [339, 23].

4.2. Symmetrization and entropy function

133

We conclude this section with a remark on the choice of the parameters. In order to determine the energy-transport model completely, the parameters 0:, {3, cPo, No and ko have to be chosen. The mobility constant /10 depends on cPo, No and ko (the dependence on ko comes in via m*), and the constant TO depends on cPo and ko. Instead of choosing the parameters cPo, No and ko, we prescribe /10 and TO whose values (depending on the semiconductor material) can be derived from physical experiments.

4.2

Symmetrization and entropy function

In this and the following sections we consider a slightly more general energytransport model than proposed in Section 4.1 for two reasons. First, the general model can be found in many applications of transport theory of charged particles, e.g. in semiconductor theory [41, 224], in electro-chemistry [91, 116], and alloy solidification processes [47, 196], where several components of charged particles are considered. Secondly, the analysis of the multi-component system is the same as for the energy-transport model (4.8)-(4.11), where only one type of charged particles (i.e. electrons) are considered. The general system also includes the bipolar energy-transport model for electrons and holes, if the temperatures of both components can be described by one temperature function. Thus we consider the following system of equations for the flow of n components of a fluid or gas of charged particles with particle density Pi and charge ei (per particle) for the i-th component:

at '

,

~P'(U) + div J.

:t Pn+ (U) + div I n+ 1

Wi(u, V),

i = 1, ...

,n,

(4.50)

n W n+1 (u, V) - Lek\7V. Jk' k=l

1

n in x (O,T), - L ekPk(u) - C(x) k=l n+l - L Lik(U, V)(\7uk - ekUn+l \7V), i = 1, ... , n k=l

(4.51)

n

~V

Ji

(4.52)

+ 1,

(4.53)

under mixed Dirichlet-Neumann boundary and initial conditions: U=UD,

Ji

.V

V=VD = \7V . v = 0

p(u(O)) = p(UO)

r D x (0, T), on r N x (0, T), in n. on

(4.54) i = 1, ... , n

+ 1,

(4.55) (4.56)

134

Chapter 4. The Energy-transport Model

where v denotes the exterior normal vector of an and, as described in the previous section, an = r D UrN. Here, Ji denotes the corresponding particle flux density (i = 1, ... , n), Pn+1 is the internal energy density and In+l the energy flux density or heat flux. We have introduced the entropy variables

U=(U1"",U n+l),

ui=J.LdT

(i=l, ... ,n)

and

u n+l=-l/T,

where J.Li is the chemical potential of the i-th component and T the temperature. The diffusion matrix L = (Lik) is symmetric and positive definite, which are consequences from Onsager's principle and the second law of thermodynamics [249]. The particle densities Pi and the internal energy density Pn+1 are assumed to depend on U: Pi = Pi(U), i = 1, ... , n + 1. The main assumption on P = (P1, ... , Pn+1) is that

P is strongly monotone

and

3X: P = V' uX·

(4.57)

For the energy-transport model under Boltzmann statistics it holds (see (4.12))

p(u) =

( - U21)

3/2

Ul

e

(

1 )'

(3/2)U2

and X(u) = (-U2)-3/2 exp(ud satisfies V'uX = p. Finally, the functions Wi are source terms, and C(x) models fixed charged background particles. Note that the system (4.50)-(4.56) corresponds for n = 1 and e1 = -1 to (4.8)-(4.11), (4.27)-(4.30). The system (4.50)-(4.56) describes generally the evolution of a particle ensemble, influenced by diffusive, thermal and electrical effects, and is used in nonequilibrium thermodynamics [188, 249]. Mathematically, there arise two main difficulties in the analysis of (4.50)(4.56). First, the variable of the time differential operator, i.e. P, and the variable of the space differential operator, i.e. u, are different and related by the algebraic relation P = p(u). These types of equations with V = 0 have been studied in, e.g., [6, 141,243]. Secondly, the equations form a system of strongly coupled parabolic equations and maximum principle arguments cannot be used in general to derive L OO bounds for the variables Ui. Now, the term Jk . V'V on the right-hand side of (4.51) contains the term LikUn+11V'VI2 which is quadratic in the gradient. Note that, due to the mixed boundary conditions (4.54)-(4.55), we cannot expect to get "regular" solutions, and generally, only IV'V\2 E L 1 (n) holds. But then, Un+l not being in L=(n), the term LikUn+lIV'VI2 is not defined. The key of the proof of the existence of solutions to the steady-state or timedependent equations as well as the long-time behavior is (i) to use another set of variables which symmetrizes the problem, and (ii) to obtain a priori estimates by using the entropy function (see below). These two ideas are connected;

135

4.2. Symmetrization and entropy function

indeed, the existence of a symmetric formulation of the system is equivalent to the existence of an entropy function (see, e.g., [103, 244] for details).

(i) Transformation of variables. Define the dual entropy variables, or electro-chemical potentials,

Wi=Ui-eiUn+IV

(4.58)

(i=l, ... ,n),

and set W = (WI,.'" W n+1)' To simplify the notation, we set e en+l) = (el, ... ,en, 0) and define

= (el,""

n

e .~

=

L ek~k

for ~ E IRn +l .

k=l

Then the problem (4.50)-(4.56) is equivalent to

%tbi(W, V) +divIi

~ bn + l (w, V) + div I n + l

~V

Qi(W,V),

i=l, ... ,n,

Qn+1(W, V)

+ Ve' Q(w, V)

BV + ate. b(w, V),

-e· b(w, V) - C(x)

in

(4.60)

n x (O,T),

(4.61)

n+l

- L Dik(W, V)\7wk,

(4.59)

i = 1, ... , n

+ 1,

(4.62)

k=l

subject to the initial and boundary conditions on rD x (0, T), on r N x (0, T), i = 1, in n,

W=WD, V=VD Ii . v = \7V . v = 0 b(w(O), V(O)) = b(wO, yo)

(4.63) 00"

where

bi(w, V) bn+l(w, V) Qi(W, V)

Pi(U), i=l,oo.,n, Pn+1 (u) + Ve . p( u), Wi(u,V), i=1, ... ,n+1, UD - eUD,n+1 VD, U - eUn+1 VO ,

°

°

n + 1,

(4.64) (4.65)

136

Chapter 4. The Energy-transport Model

VO is the solution of ~VO -e' p(UO) - C(x) under the mixed boundary conditions VO = VD(O) on rD and v'V° . v = 0 on r N , and finally, the new diffusion coefficients are given by D ik

Lik,

i,k=I, ... ,n, n

D n +1,i

= L i ,n+l + V L n

L n+1,n+1

+ 2V L

k=l

i

ekLik,

k=l

ek L n+1,k

+V

= 1, ... ,n, n

2

L

eiekLik.

i,k=l

The equivalence of (4.50)-(4.53) and (4.59)-(4.62) can by shown by elementary computations. Strictly speaking, the two problems are only equivalent if the corresponding solutions are regular enough. The used notion of solution as well as the needed assumptions on the nonlinear functions are stated in the following section. The diffusion matrix D = (Dik) is symmetric and positive definite. Indeed, it holds D = p T LP, where P = (Pik) is the regular matrix defined by

Pik

=

I {

ei V

o

if i = k E {I, ... ,n + I} if k = n + 1, i E {I, ... , n} else.

Thus, if L is symmetric, positive definite, then D is symmetric, positive definite, too. We note that the transformation (4.58) is well known in nonequilibrium thermodynamics [188, §53] and is also used in the standard drift-diffusion model for semiconductors (i.e. T = const.), where Wi is called quasi Fermi potential (cf. Section 3.1.1). Recently, this transformation appears naturally in the derivation of an energy-transport model for semiconductor heterostructures [113]. The symmetrization property of the transformation (4.58) has also been observed by Albinus [3] in the case of the energy-transport model. (ii) Entropy function. To derive a priori estimates we use the entropy function

where UD,n+l = const. < 0 and p = \7X (see (4.57)). In Section 4.4 we show that S(t) is non-negative and satisfies the so-called entropy inequality (if WD =

4.3. Existence of transient solutions

137

const.):

(4.66) and for some c > 0. (If '\lwD does not vanish then the right-hand side of (4.66) has to be replaced by S(t l ) + C(WD)') Introducing the entropy density s(t) by S(t) = s(t)dx, the inequality (4.66) can also be written (formally) as

In

d

n+l

n+l

d: + div L h(Wk - WD,k) L h· '\l(Wk - WD,k) :S 0, =

k=l

k=l

since the matrix D is symmetric, positive definite. Therefore we can interpret Lk h(Wk - WD,k) as the entropy current and Lk h . '\l(Wk - WD,k) as the entropy production rate with the thermodynamic fluxes Ii and the generalized thermodynamic forces '\l(Wk - WD,k) [249]. Furthermore, if V = 0, the entropy density relates the extensive variables PI, ... , Pn+l and the intensive variables UI,· .. , un+! by fJS/fJPi = Ui - UD,i for i = 1, ... , n + 1. Mathematically, s(t) can be interpreted as the Legendre transform of X.

4.3 4.3.1

Existence of transient solutions Assumptions and main results

This section is devoted to the proof of the existence of solutions to the diffusion system (4.50)-(4.56) of Section 4.2. We impose the following hypotheses: (HI) 0 C JRd (1 :S d :S 3) is a bounded domain with Lipschitzian boundary fJO = r D urN, rD n r N = 0, measd-l(rD) > 0, and r N is open in fJO. (H2) P = (PI, ... , Pn+d E WI,oo (JR n+ l ; JRn+l) is strongly monotone and a gradient, i.e.

(p(U) - p(v)) . (u - v) 2:

colu - vl 2

for U, v E JRn+!,

where co> 0, and there exists a (convex) function X E CI(JRn+!;JR) such that P = '\luX· (H3) L = (Lik) E LOO(Qr xJRn+1 xJR; JR(n+l)x(n+I») is a CaratModory function (see Section 3.2.1) and a symmetric, uniformly positive definite matrix.

Chapter 4. The Energy-transport Model

138

(H4) Wi : QT X JRn+1 x JR satisfying

---t

JR (i = 1, ... ,n + 1) are Caratheodory functions

n

2)Wk (u, V) - Wk(U, V))(Uk - Uk) :S 0, k=l

W n+1(X, t,u, V)(Un+l - u) :s 0, :S c(l + lui + IVI), i = 1, ... , n + 1,

e· W(U, V) = 0,

IWi(x, t, u, V)I

u, V,

°

with u < and e = (el,'" ,en,O). l (H5) UD E CO([O,T];H (O;JRn+1)) n H l (0,T;L 2(0;JR n+1)), VD E CO([O,T]; LOO(O)) n Hl(O, T; Wl,po(O)) with Po = 2 if d = 1, Po > 2 if d = 2 and Po = 3 if d = 3; uO = (u~"",u~+1) E Loo(O;JRn+l) n H l (O;JRn+1); C E LOO(O), el, ... , en E JR, and UD,n+1 = U. for all u, V,

In the papers [6, 141] a hypothesis similar to (H2) has been used. It is satisfied in the energy-transport model for semiconductors (see above) if the particle density and the temperature are bounded. Assumption (H3) follows from basic physical principles, as explained in Sections 4.1 and 4.2. The monotonicity condition for W n + l in assumption (H4) means that W n + l is a relaxation term. For the energy-transport model for semiconductors, this condition is verified as can be proven rigorously [41, 42]. We assume that the temperature Un+l is constant at the contacts, i.e. we neglect surface thermal effects. The term W n + l relaxes to the constant temperature at the boundary (assumption (H4)). In the case of the bipolar energy-transport model for semiconductors (Le. n = 2 and PI, P2 are the electron and hole density, respectively), usually the Shockley-Read-Hall term Wl(u) = W 2(u) = -(Pl(U)p2(U) - nt)/(TlPl(U) + T2P2(u)+1) with n~, Tl, T2 > is used as source term [275, 333]. Since el = -1, e2 = +1, we get el WI + e2W2 = 0. Furthermore, Wi is monotone in the sense of (H4). Therefore, the hypothesis (H4) is satisfied in this case. Now we can state the existence theorems:

°

Theorem 4.3.1 Let (Hl)-(H5) hold and let T

> 0. If d :S 2 there exists P > 2

and a solution (u, V) of (4.50)-(4.56) with U -

UD E L 2 (V n +1) n CO([O, T]; L 2 (0; JRn+l )), p(u) E H l ((v*)n+1) n L 2(QT; JRn+1) , V - VD E LOO(V) n Loo(Wl,P).

(4.67) (4.68)

(4.69)

= 3 and 80 = rD E Cl,l. Then there exists a solution (u, V) of (4.50)-(4.56) such that (4.67)-(4.69) hold and, moreover, V E LOO(QT) n L OO (W l .3 ).

Theorem 4.3.2 Let (Hl)-(H5) hold. Let d

°

We recall that V = {u E Hl(O) : U = on rD}. Notice that our regularity results are sufficient to conclude that the product \7V . Ji is integrable in QT'

4.3. Existence of transient solutions

139

4.3.2 Semidiscretization The proof of Theorems 4.3.1 and 4.3.2 is based on a semidiscretization in time (cf. [204,243]). This method is also of interest from a numerical point of view. We introduce partitions = to < tl < ... < tN = T of [0, T] such that def

< 1 and

hj = tj - tj-I def

h = . max hj J=I,... ,N

--t

°

°

(N

--t

sup

(0),

max

N>oj=2,... ,N

(J!L + hj - I

j

h hj

I

)

< 00.

(4.70) For instance, uniform partitions satisfy (4.70). We denote by CN(O,TjX) (for a Banach space X) the space of all functions u : (0, T] --t X which are constant on (tj-I, tj], and we set u(t) = u j for t E (tj-I, tj], j = 1, ... , N. Let u}f), V~N) E CN(O,TjHI(O)) be defined by

ub

Vb

1:~1 UD(T)dT,

def

(u}f))j = hjl

def

(V~N))j = hjl ljt~l VD(T)dT

for j = 1, ... ,N. Furthermore, define the shift operator (TN : CN(O, Tj L2(0)) CN(O, Tj L 2(0)) by ((TNU(N))j

= uj - I ,

j

--t

= 1, ... ,N,

and the linear interpolation of u(N) E CN(O, Tj L 2(0)) by ij,(N)(x, t) = hjl(tj- t)(uj - u j - I ) + uj

for x E 0, t E (tj-I, tj],

for j = 1, ... , N, where uo is the initial value defined in (H5). Then, thanks to (4.70) (see [204, Lemma 5.2.5]), u}f) u-(N) D

V~N)

UD

in L 2 (H I ),

~

UD

in H I (L 2 ),

--t

VD

in HI(H I )

--t

~

as N

--t 00.

The time discretization of (4.50)-(4.56) is j j hjl(Pi(u ) - Pi(u - I )) j j I hjl(Pn+l(U ) - Pn+1 (u - ))

+ div J! =

W i (u j , vj), (4.71)

i = 1, ... , n, + divJ~+1 = W n+1 (uj , vj) n

-L

k=1

ek vvj .

Jk'

(4.72)

140

Chapter 4. The Energy-transport Model ~Vi = -e· p(ui ) - C(x)

n+1 J[ = - L

in f2,

(4.73)

Lik(Ui , Vi) (\7u{ - eku~+1 \7Vi),

k=1

i = 1, ... , n

+ 1,

(4.74)

with boundary conditions

u i -- u iD' Vi = VDi Jit . v = \7Vi . v =

on

°

rD,

(4.75) (4.76)

For given p(un) and VO (computed from ~ VO = e . p(un) - C with mixed boundary conditions), Eqs. (4.71)-(4.76) define recursively (u i ,Vi). Using the discrete transformation j

2: 0,

(4.77)

the above discrete system is (formally) equivalent to

hjl(bi(Wi , Vi) - bi(Wi - l , Vi-I)) +div Ii = Qi(W i , Vi),

i = 1, ...

h;t(bn+I(Wi , Vi) - bn+I(Wi - l , Vi-I)) + div I~+1 Qn+1 (wi, Vi) ~Vi

+ Vi e . Q(wi, Vi) + hjl (Vi -e.b(wi,Vi)-C(x)

(4.78) (4.79)

- Vi-I)e . b(Wi-I, Vi-I),

inf2,

n+1 - LDik(Wi, Vi)\7w{,

,n,

i = 1, ...

(4.80)

,n+ 1,

(4.81)

k=1

subject to the boundary conditions

wi=wb, Iit . v

Vi=VJy

= \7Vi . v =

°

rD, on r N ,

on

(4.82) (4.83)

where wiD = u iD - euiD,n+1 ViD E H I (f2.JRn+1) in view of (H5) . ' For given (wi-I, Vi-I), the problem (4.78)-(4.83) is a system of strongly coupled elliptic equations for (wi, Vi). If (wi, Vi) E H I (f2;JRn+l) x LOO(f2) is such that Vi E W I ,P(f2) with p > 2 if d ::; 2 and p = 3 if d = 3, then u i E H I (f2;JRn+1), and the problems (4.78)-(4.83) and (4.71)-(4.76) are equivalent. 4.3.3

Proof of the existence result

To prove Theorems 4.3.1 and 4.3.2 we proceed as follows. First we solve recursively (4.78)-(4.83) to get functions w(N) E CN(O, T; HI) and V(N) E

4.3. Existence of transient solutions

141

CN(O, T; HI) where w(N) = (wI, ... , w N ) and V CN ) = (vI, ... , V N ). Then, setting u CN ) = wCN) - eW~)1 V CN ) E CN(O, T; L 2 ), we prove that (u CN ), VCN)) solves (4.71)-(4.76) for j = 1, ... , N. Using compactness arguments we let

N

to get a solution to the continuous problem (4.50)~(4.56). The proof of the existence of solutions to (4.78)-(4.83) is based on estimates on the discrete entropy function ---> 00

sj

~f S(uj , vj)

=

in

j j (p(u ). (u -

~UD,n+1

-

in

ub) - (X(uj ) -

X(ub)))dx

V~Wdx.

1V'(vj -

Notice that, since X is convex and UD,n+1 = U < 0, we have sj 2': 0 for all

j 2': O.

The following lemma provides a priori estimates for the existence of solutions to the discrete system (4.78)-(4.83) as well as for the proof of Theorems 4.3.1 and 4.3.2. Lemma 4.3.3 Let (wj , vj) E H 1(O;ffi.n+ 1) x (H 1(O) nLOO(O)) be a solutionto

(4.78)-(4.83). Then j Ilu llo,2,n Ilu CN )IIL=C£2)

+ IIvjI11,2,n + hjllwjlll,2,n < + IIVCN)IIL=CH1) + Ilw CN )II£2CH1) <

where CI > 0 is independent of (w j , V j ) and (w(N), V(N)) and N.

C2

> 0 is

CI,

(4.84)

C2,

(4.85)

independent of

Proof. Using the convexity of X, i.e. X(u j ) - X(u j -

l

) -

V'uX(uj - I ). (u j - u j - 1)

2': 0,

we can estimate as follows: sj - Sj-1

<

in

(V'uX(uj ) - V'uX(uj - I )). (u j - ub)dx

in + in (+ in ( -u

V'(vj - V j j p(u -

U

-

l

).

l

) .

V'(vj -

(ub - ub-

l

)

V~)dx + x(ub) -

\I(Vj-, - Vir')· \I(Vj, - Vj,-')

~I\I(Vj, - Vj,-')I') dx

K 1 +···+K4 .

1 X(ub- ))dx

Chapter 4. The Energy-transport Model

142

First we estimate the integrals K 1 and K 2 . Since (wj , V j ) solves (4.78), (4.80), we obtain

r I~(u{ - Ub,k)dx j + hj t r Qk(W , Vj)(u{ - Ub,k)dx k=l in

-h/t div k=l in

+

j r (Pn+l(U ) - Pn+l(Uj-l))(U~+l - ub n+l)dx in '

-u

in

e· (p(u j ) - p(uj-1))(vj -

V~)dx.

(The integral involving div I~ has to be understood as the dual product between V* and V.) Taking into account (4.79), we get K1+ K2

=

r I~(wt - Wb,k)dx

-hj t div k=l in

r divI~ek(Vj(w~+l-WD,n+l)

-hjt k=lin

+WD,n+l(Vj-V~))dX+hjtrQk(wj,vj)(u{-ub,k)dx

in -in

+

k=l in

j j (bn+1(w , vj) - bn+1(w -1, Vj-l))(W~+l - wD,n+d dx (e. (p(u j ) - p(uj-1))vj

+e·p(uj-1)(vj - Vj-l))(W~+l -WD,n+l)dx - U

in

e· (p(u j ) - p(uj-1))(vj -

V~)dx

n+l -hj

L rdiv I~(w{ - Wb,k)dx

k=l in

rdivI~ek(Vj(w~+l-WD,n+l) n+l j +WD,n+l(vj - V~))dx + hj L rQk(W , vj)(u{ - Ub,k)dx k=lin -hjt k=lin

143

4.3. Existence of transient solutions

Vje'Q(wj,vj)(W~+l -wD,n+l)dx

+hj i

p(Uj-l))Vj(W~+l -

- i e . (p(u j ) - Ui

WD,n+l)dx

e· (p(u j ) - p(Uj-1))(vj - Vjy)dx.

Using (4.78) again and the assumption (H4), we obtain K 1+ K2 =

n+l -hj

L rdiv I~(w~ - Wb,k)dx

k=lin

- hju i

j j Q(w , V ). e(Vj - Vjy)dx

r Qk(wj,vj)(u~-ub,k)dx

+hjI: k=lin n+l I~ . \7(w~ - Wb,k)dx hj k=l in j - hju i W(u , vj)· e(Vj - vjy)dx

L r

+ hj + hj

< -hj

I: inr I: inr

j (Wk(u , vj) - Wk(ub,

k=l

Wk(ub,

k=l n+l

L

k,i=l

i

vjy)(u~ -

vjy))(u~ -

Ub,k)dx

Ub,k)dx

j Dki(W , vj)\7w~ . \7(w~ - Wb,k)dx

+chj i1uj -ubI2dx+c(c,ub,vjy)hj,

employing Young's inequality, where c of D, we obtain n+l

> O. Taking into account the symmetry

j r Dki(W , vj)\7w~. \7(w~ - Wb,k)dx k,i=l in

L

144

Chapter 4. The Energy-transport Model

1 n+1

r

L in Dkl(W j , vj)\7w~ . \7w~dx

2

k,l=l

+~

n+l

L ~ Dkl(W j , V j )\7wb,l' \7wb,k dx .

k,l=l

By Stampacchia's estimate (see [340]), we get, since d :S 3,

IIVj 110,00,n :S c(l + Iluj 110,2,n). Since the matrix D is positive definite, there exists 8 (uj ) > 0 such that n+l

L

j Dik(U , Vj)~i~k 2: 8(uj)I~12

for all ~ E

jRn+l.

(4.86)

i,k=l

Indeed, 8 depends on the LOO(n) norm of vj, by the definition of D, and thus on the L 2 (n) norm of u j , by the above elliptic estimate. Taking into account \7wD,n+1 = 0 due to (H5) and Dik = Lik for all i, k = 1, ... , n by the definition of D, we get further

This implies

Now we estimate the remaining integrals K 3 and K 4 . Recalling that h j < 1, we employ Young's inequality with E > 0 to get K3

=

-

~ ((p(u j - 1) -

- x(ub)

p(ub- 1 )). (ub - ub- 1) + P(ub- 1 ). (ub - ub- 1 )

+ X(ub-1))dx

< Ehj - 1 ~ !u j - 1 - ub-112dx + c(E)hj~l ~ IUb - ub-112dx + c(ub- 1 )

4.3. Existence of transient solutions and

~

K4

ch j - 1

in

145

in 1\7(V~

V~-lWdx + c(C)hj!l

1\7(Vj-1 -

V~-lWdx.

-

We proceed by estimating m

sm - SO = Z)Sj - Sj-1) j=O 1 m n+1 m < -hj 18(uj)I\7W~12dX + c h j luj - ubl 2dx 2 j=l k=l n j=O n

L L

t,

+E

+ c(c)

hj !nIV(V;- Vbl!2dx+ c( E)

f

j=l

l

L

h j 1 11\7(V~ n

t,

hj' !nlub - uj) 'I'dx

f

V~-lWdx + c(c)

j=l

hj,

where we have used the fact that hj /h j _ 1 is bounded independently of j and N (see (4.70)). Since

t, in hj 1

1 (Iub - ub- 12 + 1\7(V~ -

afj(N) 2

~ I ;: ~

aV(N)

t2(L2)

+ II

c

~

V~-lW)dx

2

t2(Hl)

independently of N (see Section 4.3.2), we get finally

<

sm - SO

1

-2

. f;hj n+1 ~ In 8(uj)l\7w~12dx + f; hj In lu

+c

m

m

t, in hj

c

1\7(vj -

.. J -

ubl 2dx

V~)12dx + c(c, UD, VD,T).

On the other hand, since p is strongly monotone,

in 1 1

sm =

X

2:

Co 2

(p(u m ) - p(su m + (1 - s)ui)))

(u m - (sum

+ (1 -

s)uD)) ~dx - !UD,n+1 1-

s

2

r lum - ui)1 2dx + !Iul r 1\7(Vm - VE'Wdx. 2 in

in

inr 1\7(V

m - VE'Wdx

146

Chapter 4. The Energy-transport Model

The above two estimates imply

Applying the maximum over m = 1, ... , N yields

2Co I U (N) -U D(N) 112£=(£2) + 211_111 u V (N) 1

N

n+l

J=O

k=l

+ 2 L c5(u j )hj < SO + €llu(N)

-

- VD(N) 1 £=(V) 2

L ilV'w112dx

u~) 11£2(£2)

This implies, for sufficiently small

(4.87)

n

€

+ €IIV(N) -

vit)11£2(v) + c{€).

> 0,

where c> 0 only depends on the given data. In particular, j = 1, ... ,N.

It can be seen that therefore

for j = 1, ... , N.

c5( u j ) 2': 150 > 0 Hence we conclude from (4.87)

Ilw(N) Ili2(Hl) :S C

and

with c > 0 only depending on the data. This finishes the proof.

o

We are now able to prove the existence of solutions to the discrete problem (4.78)-(4.83). Lemma 4.3.4 Let (wj - 1,Vj-l) E Hl(o;~n+l) X (H1(0) n £00(0)) be given. Then there exists a solution (w j , vj) E Hl(o;~n+l) X (H1(0) n Loo(O)) to

(4·78)-(4.83). Proof. The proof is based on the Leray-Schauder fixed point theorem. To define the fixed point operator, let U E £2(0; ~n+l) and let V E H1(0) be the unique solution of

L\V = -e' p(u) - C(x) in 0,

V

= Vb on r D , V'V· v = 0 on rN.

4.3. Existence of transient solutions

147

From Stampacchia's estimate for elliptic equations [340] follows that V E Loo(o.). Thus W ~f u- VUn+le E L2 (o.;lRn+l). Now consider the linear system

for w:

n+l -div L

k=l

Dik(W, V)\7wk

n+l -div L

Dn+l,k(W, V)\7wk

k=l

a( - hjl (bn+l (w, V) - bn+1(w j - 1, Vj-l))

+ Qn+l(w, V) + Ve· Q(w, V) + hjl(V - Vj-l)e. p(u j - 1)),

where i = 1, ... , n and a E [0,1], subject to the boundary conditions

n+l w=awDonf D ,

LDik(W,V)\7wk·v=OonfN, i=l, ... ,n+1.

k=l

By Lax-Milgram's theorem, there exists a unique solution w E H1(o.; lRn+1), since the right-hand side of this elliptic problem lies in L 2 (o.; lRn +1 ). Finally, define u = w + VWn+le E L 2 (o.;lRn+1). Hence the fixed point operator S : L 2 (o.;lRn +1 ) x [0,1] ----+ L 2 (o.;lRn +1 ), (u,a) f---+ U, is well-defined. It holds S(U, 0) = for all U E L 2 (o.;lRn+l). Every fixed point u of S with a = 1 solves (4.78)-(4.83). Indeed, let S(u, 1) = u then we only have to show that W = W. Now, Wn+l = Un+l = Wn+l and Wi = Ui - VWn+le = Wi for i = 1, ... ,no Therefore, W = U - V un+l e. To show that S is a compact operator, let U E L 2 (o.; lRn+l) be a fixed point of S with a E [0, 1]. Similarly as in the proof of Lemma 4.3.3 we get the estimate

°

where c>

°

Ilullo,2,n + Ilwlll,2,n + 11V111,2,n ::::: c,

is independent of u, w, V, and a. Thus

II\7ullo,3/2

< c(ll\7wllo,2 + II\7Vllo,21Iwn+lllo,6 + 1IVIIo,611\7wn+lllo,2) < c(llwlll,2 + 11V111,21IWn+l11 1,2) < c,

using H1(o.) <.......t L 6 (o.). Standard arguments show that S is continuous and compact, noting that the embedding W 1,3/2(o.) <.......t L 2 (o.) is compact (for d ::::: 3). Leray-Schauder's theorem gives the desired result. 0

148

Chapter 4. The Energy-transport Model

Corollary 4.3.5 Let (u j - 1, V j - 1) E H 1(n; IRn+l) x (H 1(n) n LOO(n)) be given. Then there exists a solution (u j , vj) E H 1 (n;lR n+l) x (H 1(n) n £00(0,)) to

(4· 71)-(4. 76).

The corollary is a consequence of the following observations: Taking into account the estimate (4.85) for u(N), the regularity result for elliptic problems of [186, 291], and Stampacchia's elliptic estimate [340]

IlV 110,00,0 ~ c(1 + Iluj 110,2,0), j

we conclude: Lemma 4.3.6 There exists p

> 2 such that (4.88)

where c > 0 is independent of N. Furthermore, p = 3 if d = 3 and 80, = r D C 1 ,1.

E

We proceed by proving further a priori estimates independent of N. From (4.85) and (4.88) follows: Lemma 4.3.7 It holds

Ilu(N) 11£2(Hl)

< c,

IIJt) 11£2(£2)

< c,

(4.89) i = 1, ... ,n + 1,

(4.90)

where Ji(N) E CN(O, T; £2) is such that Ji(N) = J1 on (tj-1, tj], j = 1, ... , N, and c > 0 is independent of N. Proof. We have for i = 1, ... , n

IIV'ut) 11£2(£2)

~

IIV'w~N) 11£2(L2) + cllw~~ 11£2(Lq) IIV'V(N) IIL''''(LP)

+ cllV'w~~II£2(£2)IIV(N)IIL=(L=)' where p > 2 and q = 2p/(p - 2) if d ~ 2, and p = 3, q = 6 if d = 3. Since the embedding H 1(n) '---+ Lq(n) is continuous, we obtain (4.89). To prove (4.90), we use that £ is bounded in QT X IRn +1 x lR:

II Ji(N) 11£2(L2) ~ c( where i = 1, ... , n

n+1

L IIV'u~N) 11£2(£2) + Ilu~;~T)111£2(Lq) IIV'V(N) IIL=(LP») ~ c, k=1

+ 1 and p and q are as above.

(4.91) 0

149

4.3. Existence of transient solutions

Our final a priori estimates are necessary for the compactness argument. For this, define the linear interpolation of p(u(N)) by jj(N) (x, t) = hjl(tj - t)(p(u j ) - p(u j - l )) + p(u j ) for x E 0, t E (tj-l, tjl, and j = 1, ... , N. Recall that h = maxj hj and that aN is the shift operator defined by (aNu(N))j = u j - l for j = 1, ... , N (see Section 4.3.2). Lemma 4.3.8 It holds for a positive constant c independent of N: Ilu(N) - aNu(N) Ili2(£2)

II T

< ch,

II £2(V*) + Ilpi(N)II£2(Hl)

op· (N)

<

(4.92) i = 1, ... ,n + 1.

C,

(4.93)

Proof. Use u j - u j - l - (ub - ub- l ) as test function in the weak formulation of (4.71)-(4.72) and use the monotonicity of p (see (H2)) to get j j cohjl llu j - u j - 112dx:S hjl l (p(u j ) - p(u - l )). (u - uj-l)dx hjl l (p(u j ) - p(u j - l )). (ub - u1)l)dx

+

rL in

n+l

Jk . 'l(u{- U{-l - (ub k "

k=l

+l

W(u j , yj). (u j - u j - l - (ub - ub-l))dx

- inr L ek'lyj . Jk(U~+1 k=l n

<

ub-~))dx

~ hjl llu j n+l

j 12 u - 1 dx

u~~~ -

(ub n+l "

+ Chjl llub

+ L(lIJkllo,2 + IIWk(u j , yj)llo,2)(llu j k=l

n

+cL

k=l

-

Ub-~+l))dx

1 - ub- 12dx

j I - u - III,2

j II'l yj llo,pIIJkllo,2(llu - uj-11Io,q

+ lI ub -

+ Ilub -

I ub- III,2)

ub-11Io,q)

with p and q as in the proof of Lemma 4.3.7. Summing this inequality over j from 1 to N yields:

150

Chapter 4. The Energy-transport Model

< c( 1 + IIJ(N) 11£2(£2) + Ilu(N) 11£2(£2) + IIV(N) 11£2(£2)

+ II VV (N)IIUX'(Wl'I')IIJ(N)II£2(£2l) (1Iu( Nl ll£2(Hl) + Ilugv)II£2(Hl) a-(N)

+ II

< c,

~

t2(£2)

using Lemmas 4.3.6 and 4.3.7. This proves (4.92). To prove (4.93) observe that for i = 1, ... ,n + 1,

-(N)112£2(£2) II VPi

by Lemma 4.3.7. Since for i = 1, ... , n

we get immediately for i = 1, ...

+ 1,

,n,

N

Lhj 11Ip(uj ) - p(uj-1)11~* j=l

< Iidiv Ji(N) 1112(v*) + IIWi(u(N), V(N))1112(£2) :'S

c.

J!

For i = n+ 1, observe that VV j · is bounded in L 2 (L 2 p/(p+2 l ) independently of N with p as in Lemma 4.3.6 and that L 2 p/(p+2) '----t V*, and therefore

n

+ c L Ilek VV(N) < c. This proves the lemma.

.

Jk

N

) 11£2(£21'/(1'+2))

k=l

o

Proof of Theorems 4.3.1 and 4·3.2. First step: strong convergence of u(N). We choose partitions of [0, T] satisfying (4.70). From Lemma 4.3.8 follows that for all i = 1, ... , n + 1 the sequences

4.3. Existence of transient solutions

151

are bounded in L 2 (H 1) n H1(V*) and therefore relatively compact in L 2 (Qr) (by Aubin's lemma, see [204, 337]). Taking into account Lemma 4.3.7, there exists a subsequence of N, not relabeled, such that (Pi(N))N

U(N) ~ U p(N) ~ r p(N) - t r

weakly in L 2 (H 1 ), weakly in L 2 (H 1) and in H1((v*)n+l), in L 2 (Qr;lR n+ 1 ) (N - t (0).

(4.94) (4.95) (4.96)

Now we identify r with p(u) by using a monotonicity argument. From Lemma 4.3.8 we conclude

thus, p being uniformly Lipschitz continuous,

Recalling that

for x E

n, t

E (tj-l, tj], and j = 1, ... , N, we conclude

which implies p(U(N))

-t

(4.97)

r

It holds for all v E L 2 (Qr;lR n+l)

1 QT

Letting N

- t 00

(p(u(N)) - p(v)) . (u(N) - v)dxdt ~

o.

gives, taking into account (4.94) and (4.97),

1 QT

(r - p(v)) . (u - v)dxdt ~ 0,

and hence r = p(u) by the strong monotonicity of p. Since, for N

- t 00,

we get finally (4.98)

152

Chapter 4. The Energy-transport Model

Second step: convergence of V(N) and JCN). By Lemma 4.3.3 the sequence (V(N)) is uniformly bounded in L 2 (H 1 ), i.e. there exists a subsequence '(not relabeled) such that (4.99) and therefore ~V=-e·p(u)-C,

since V~N) --; VD in L 2(H 1) (see Section 4.3.2). This result implies, together with (4.98), V(N) --; V in L 2(H 1). (4.100) The strong convergence of u(N) and V(N) in L 2(L 2) implies Lidu(N), V(N)) --; Lik(U, V) in L 2(L 2). From (4.88) and (4.89) we conclude that U~:~)lV'V(N) --; Un+1V'V in L1(L 1). Hence, since Lik(U, V) E L OO (Q7)' J(N)-, J weakly in L 1(Q7; ~n+1), where J = (J1 , ... , In+d and

n+l Ji = -

L Lik(U, V) (V'Uk -

i = 1, ... , n

ekUn+1 V'V),

k+1

+ 1.

In view of the bound (4.90) this yields (for a subsequence) (4.101)

Third step: the limit N --; is

1° 7

j{)Pi(N)) v' v dt -

\-{)-,¢

t

'

00

in the equations. The weak formulation of (4.71)

1 QT

-1

J i(N) . V'¢dxdt -

Wi (N) U ,V (N)) ¢dx dt,

QT

for ¢ E L 2 (V), where i = 1, ... , n. Using (4.95), (4.98)-(4.101), we can let N --; 00 to obtain the weak formulation of (4.50). In order to get the weak formulation of (4.51) we observe that for ¢ E L 2(V) n LOO(L OO )

(N--;oo), using (4.100) and (4.101). Finally, U - UD E L 2(V n+1) and p(u(O)) = p(uO) in D, since the bound (4.93) implies p(u) E CO([O, T]; L 2(D; ~n+l)) [373, Ch. 23], and p(N)(t = 0) = p(UO). Furthermore, by the strong monotonicity, U E CO([0,T];L2(D;~n+l)). The theorems are proved. 0

153

4.4. Long-time behavior of the transient solution

4.4 Long-time behavior of the transient solution The solution constructed in the previous section converges to the thermal equilibrium state as the time tends to infinity, if the boundary data are in thermal equilibrium. The thermal equilibrium state is defined by Ji = 0, i = 1, ... ,n+ 1, which is equivalent to \lWi = 0 for i = 1, ... , n + 1. In particular, the temperature T = -1/wn +l is constant in O. We assume: n

(H6)

L Wk(UD, VD) = O. k=l

(H7) UD E H1(O), VD E HI (O)n£<Xl(O), \lwD

=

0, and ~VD

=

-e,p(uD)-C.

The first assumption is needed to show that the entropy function Set) introduced in Section 4.2 is non-increasing. In the case of the bipolar energytransport model for semiconductors, it holds W1(UD, VD) = W 2 (UD, VD) = 0, so the assumption (H6) is satisfied for this model. The second assumption means that the boundary data is time-independent and in thermal equilibrium. Setting Ck = WD,k = const., the differential equation for VD is equivalent to ~VD = -e· p(Ck + ekuVD) - C. Using mixed boundary conditions for VD as in Section 4.3.2, we see that this problem is uniquely solvable in H1(O) since p is monotone and u < O.

Theorem 4.4.1 Let (H1)~(H7) (see Section 4.3.1) hold. Then there exist c, /.L > o such that for t > 0

Ilu(t) - uDllo,2,n + 1!V(t) - VD II1,2,n :::; ce-J.Lt, where (u, V) is the solution constructed in Section 4.3. Furthermore, Set) ;:::: 0 is non-increasing and S (t) ---t 0 as t ---t 00. Proof. In the proof we use ideas of [154]. First, introduce the entropy density

set)

~f p(u(t))· (u(t) -

UD) - (x(u(t)) - X(UD)) -

~UD,n+lI\l(V(t) -

VDW.

In view of assumption (H2) of Section 4.3.1 and UD,n+l = U < 0 it holds

set) set) where

Cl, C2

> O.

> cl(lu(t) - uDI 2 + 1\l(V(t) - VDW), < c2(lu(t) - uDI 2 + 1\l(V(t) - VD)1 2),

Set

So(t) = So(u(t))

~f

l

(4.102) (4.103)

(p(u(t)) . (u(t) - UD) - (X(u(t)) - X(uD)))dx.

154

Chapter 4. The Energy-transport Model

Using methods of convex analysis as in [154, p. 15] and the convergence results of the previous section, it can be seen that for a.e. 0 :s: TI < T2 < 00 (4.104) This result also follows directly from the convexity of X. Indeed, let 0 = to < tl < ... < tN = T be a partition of [0, T] and let £, m be such that TI E (te-l' tel and T2 E (tm-b t m ]. Then

1 t

Tn

t(

a _(N)

n+l "

L

(

(N)

aPk ,Uk

t

k=l

f 1

(p(uj )

)

*

V ,v

dt

p(uj - l )). (uj

-

-

uD)dx

n

j=HI m

>

-UD,k

2: (So(u

j

) -

So(uj - l ))

j=HI

So(um )

-

so(ue)

and thus

Letting N -+ 00 (thus h -+ 0) gives (4.104). Now we estimate similarly as in the beginning of the proof of Lemma 4.3.3. Employing (4.50)-(4.53), we get for all J.L ~ 0,

1 T

o

n+l

a

eJ.Lt(2:(-at Pk(U),Uk-UDk) ' v*,v

l V'~~

k=l

- UD,n+1

. V'(V - VD)dx + J.L

r eJ.Lt (- t (div Jk'Uk - UD,k)V* ,v

io

k=l

n

+ 2:(Wk(u, V),Uk -

UD,k)V*,V

k=l

a + + ( ~,Un+l pn

l

)

UD,n+1 V*,v

l

s(t)dx)dt

155

4.4. Long-time behavior of the transient solution n 8 - UD ,n+l 'L" ek/\ -8t Pk(U), V - VD) v· V k=l '

+ J.L

L

s(t)dx)dt

r e/-L Inr (I: h· 'V(Wk - WD,k) Jo k=l t

n+l

+L

k=l

<

Wk(U, V)(Uk - UD,k) + J.Ls(t))dxdt

1.'e"' in (-

.(V(t))

+ c2J.Llu-

uDI

2

~ IV(w. - WD,.)]'

+ c2J.L1'V(V -

(4.105)

2 VD)1 )dxdt,

where we have used (H4), (H6), (4.103), and (4.86) for the last inequality. If J.L = 0 we get, using (4.102), Ilu(T) - uDII0,2,n + IIV(T) - VDIh,2,n ::; c,

with c > 0 independent of T > O. Therefore, p(u) E £00(0,00; £2(0)) and, by Stampacchia's elliptic estimate, V E £00(0,00; £00(0)). Hence there exists 80 > 0 such that 8(V(t)) 2 80 . For J.L > 0 we obtain from (4.105) e/-LT Cl

L

(IU(T) - uDI 2 + I'V(V(T) - VD)1 2)dx ::; eW 8(T)

< 8(0) +

r e/-L r(- 8 I: j'V(Wk - WD,kW + c2J.Llu- uDI 2 Jo In k=l t

+ c2J.L1'V(V -

0

VD )1 2)dxdt.

Now, if we can show that

r (Iu - uDI 2 + I'V(V - VD)1 2)dx ::; r L 1'V(Wk - WD,k)1 2dx, In In k=l n+l

C3

(4.106)

then the theorem follows after choosing J.L ::; 80/(c2c3). To prove (4.106) we use (H2), (H7), and (4.52):

~ Llu2

2 2 uDI dx + CO L1w - wDI dx

L

(p(u) - p(UD)) . (w - wD)dx

156

Chapter 4. The Energy-transport Model

>

>

in -in

Co

Co

2

lu - uDI 2dx - u

e· (p(U) - p(UD))(V - VD)dx

e· (p(u) - p(UD))(Un+1 - u)Vdx

rIu -

in

- c(V)

°

in

in

uDI 2dx - U

r1\7(V -

in

VD )1 2dx

IWn+1 - WD,n+11 2dx,

where c(V) > depends on the LOO(n x (0,00)) norm, of V. Employing Poincare's inequality and observing u < 0, we obtain (4.106). D The long-time behavior of (weak) solutions to parabolic equations and systems using the entropy function as a Lyapunov functional has also been studied in [74, 75, 114,296] where exponential or algebraic decay rates (depending on 0. bounded or 0. = ffi.d) have been obtained.

4.5 4.5.1

Regularity and uniqueness of transient solutions Regularity of transient solutions

For the uniqueness theorem we need some regularity results which are provided in this subsection. More precisely, we show that there exists a number q > 2 and a solution (u, V) of the time-dependent problem (4.50)-(4.56) satisfying u E LOO(H1) n H 1(L 2 ), \7u E Lq(Qr), and V E LOO(W1,00). The following assumptions are imposed: (HI) 0. C ffi.d (1 :::; d :::; 3) is a bounded domain with boundary an = rDUr N E C1,1, r Dn rN = 0, measd_l(rD) > 0, and r N is open and closed in an, Le. r N coincides with connected components of an. (H2) p = (Pl,oo.,Pn+1) E W 1,00(ffi.n+1;ffi.n+1) is strongly monotone and a gradient, i.e. there exists a constant Co > such that (p( u) - p( v)) . (u - v) 2: colu - vl 2 for all u,v E ffi.n+l, and there exists a function X E C 1(ffi.n+l) such that \7 uX = p.

°

(H3) L = (Lik ) with L ik E W1,T(n) n LOO(n), where r = 2 if d = 1, r > 2 if d = 2, and r = 3 if d = 3, is a symmetric uniformly positive definite matrix satisfying, for some A, A> 0,

AI~12:::;

n+l

L

i,k=l

Lik(X)~i~k :::; AI~12

for ~ E ffi.n+1.

4.5. Regularity and uniqueness (H4) Wi : QT X IRn +1 x IR satisfying

-t

157 IR (i = 1, ... , n + 1) are Caratheodory functions

n

~)Wk(U, V) - Wk(U, V))(Uk - Uk) ::; 0, k=l

e· W(U, V) = 0,

IWi(x, t, u, V)I ::; c(1

for all u, V, '11, V, with

u)::; 0, 1, ... , n + 1,

Wn+l(x,t,u,V) (Un+l -

+ lui + IVI),

u < 0 and e =

i =

(el,'" ,en,O).

(H5) UD E HI(O,T; H I (n;lRn + I )), VD E CO([O,T]; L=(n)) n L=(O,T; W 2,p(n)) n HI(O,T; WI,po(n)) with Po = 2 if d = 1, Po > 2 if d = 2, Po = 3 if d = 3, and P > d, uO E L=(n; IRn +l ) n HI(n; IRn +I ), UD,n+1 = U = const. We discuss the assumptions. The condition that rN is relatively open and closed means that the intersection of D and r N is empty. This condition is needed to conclude that the electric field -VV(t) is an essentially bounded function. For general mixed boundary conditions, this regularity may fail at points where the Dirichlet and Neumann boundary segments meet. Indeed, for general "smooth" boundaries, it is well known that we can only expect the regularity VV(t) E W I ,4/3-c(n) for all € > 0 [354]. For polygonal domains in 1R 2 , we get VV(t) E WI,S(n) '---+ L=(n) for some s > 2 if the angle between a Dirichlet and Neumann boundary side is smaller than 7r /2 (see [185]). The monotonicity condition for Wn+l in assumption (H4) means that W n+ l is a "relaxation" term. For the energy-transport model for semiconductors, this condition is verified (see Section 4.1). Furthermore, for the bipolar energytransport model where n = 2, the source terms are monotone and satisfy el WI + e2W2 = 0 (see [104] for details). Therefore, the hypothesis (H4) is satisfied in this case. Finally, we assume in (H5) that the temperature un+l is constant at the contacts, i.e. we neglect surface thermal effects. Our proof only works if the second order operator is linear, i.e. we need to assume that L ik only depends on the space variable (and not on u). Alt and Luckhaus proved in [6] the regularity p(u) E H I (L 2 ) for solutions to systems of the type

r

i = 1, ... ,n + 1,

which corresponds to (4.50)-(4.56) in the case of vanishing electrostatic potential V = O. However, their assumptions exclude the case ai(u, Vu) = L-k L ikVUk·

Our first regularity result is as follows:

Chapter 4. The Energy-transport Model

158

Theorem 4.5.1 Let the hypotheses (H1)-(H5) hold. Then there exists a solution (u, V) to (4.50)-(4.56) satisfying

where p > d (see (H5)). In particular, \7V E LOO(QT)' At first sight, it may appear advantageously to use the formulation (4.59)(4.65) in the dual entropy variables (w, V) to prove the above regularity result, since the gradient terms Jk' \7V and div (U n+l \7V) disappear. However, in this formulation the problem becomes quasilinear with a diffusion matrix depending on V(x, t) (see Section 4.2). Using our technique of proof, it can be seen that we would need LOO(QT) estimates for aV/at which are not available. Therefore, we use the formulation in the (primal) entropy variables (u, V). For systems of elliptic or parabolic equations, it is well known that there does not exist a regularity theory as for single elliptic or parabolic equations. For instance, there exist examples of linear elliptic systems whose weak solution is not bounded if d 2 3 [179, Ch. II]. Counterexamples to the regularity of weak solutions to parabolic systems can be found in, e.g., [215, 341].

Proof of Theorem 4·5.1. First step: an approximate solution. In Section 4.3 it is shown that there exists an approximate solution (U(N), V(N) to the problem

a-(N)

~ +d' J(N)

at a-(N) Pn+l at

IV

t

+ d'IV J(N) n+l

W(U(N) V(N) t ,

i = 1, ...

,

,n,

(4.107)

n

Wn+l(U(N) , V(N) - LekJt)· V(N), (4.108) k=l n

D.v(N)

L ekPk(u(N) - C(x), k=l n+l 'LJ " Lik ()( (N) - ekUn+l (N) \7V (N) , X \7u k k=l i = 1, ... ,n+ 1,

(4.109)

(4.110)

subject to the boundary and initial conditions

U(N) = ut;),

V(N) = VbN )

p(u(N)(-, 0» = p(uO)

on

an x (0, T),

in 0,

(4.111) (4.112)

4.5. Regularity and uniqueness

159

(see Corollary 4.3.5). To simplify the presentation we choose uniform partitions = tj - tj-1. Recall that u(N)(t) = u j , V(N)(t) = vj if t E (tj-1, tj] and

h

= h- 1(tj

jj(N)(t)

j j - t)(p(u ) - p(u - 1))

+ p(uj )

for t E (tj-1, tjl, j = 1, ... ,N. The boundary data u}f) and V~N) are defined in Section 4.2. They satisfy [204]

u}f) --+ UD

in H 1 (H 1 ),

(4.113)

V~N)

in L 2 (W 2 ,p).

(4.114)

--+

VD

From the results in Section 4.3 follows that there exists a subsequence of (u(N), V(N») (not relabeled) which converges to a solution (u, V) to (4.50)(4.56) in the following sense: U(N) --+ U

jj(N)

--+

V(N)

in L 2 (L 2 ),

p(u)

in L 2 (L 2 ),

V

in L 2 (H 1 )

--+

as N

--+ 00,

and the following a priori bounds hold: Ilu(N) 11£<"'(£2)

°

+ Ilu(N) 11£2(Hl) + I Ji(N) 11£2(£2) + Iljj~N) IIHl(V*) + IIV(N) IIL=(L=) ::; c,

(4.115)

where c > denotes here and in the following a constant independent of N. If we can show that u(N) and V(N) are uniformly bounded in L OO (H 1) n H 1(L 2 ), L 00 (W 2,P), respectively, the conclusion of the theorem follows after letting N --+ 00.

Second step: an estimatefor V'V(N). Thanks to elliptic regularity theory and Sobolevembeddings (see, e.g., [180, 347]) it holds, since p > d, IIV'V(N)IIL=(L=)

<

cllV(N)IIL=(w2 ,P)::; c(1

::;

c(1

+ IIp(u(N))IIL=(LP))

+ Ilu(N)IIL=(LP»),

where we have used (4.95) and the hypothesis (H2). Employing the GagliardoNirenberg inequality [373, p. 1034] we get

j j Ilu Ilo,p,n ::; cllu Ilt2,nll uj 116~~n,

j = 1, ... ,N,

where () E [0,1] if d = 1 and () = d(p - 2)j2p if d = 2,3. It holds () E (0,1) if and only if p < 2dj(d - 2). Thus we have to choose p E (d, 2dj(d - 2)) which is possible if d ::; 3. Young's inequality yields for every E >

Iluj 1I0,p,n ::; Elluj 111,2,n + C(E) Iluj Ilo,2,n,

°

160

Chapter 4. The Energy-transport Model

such that we obtain for every 8 > 0 (4.116)

taking into account the uniform a priori estimates. The constant c(8) > 0 is independent of N. Third step: estimatesfor u(N). Let m E {I, ... ,N} and use (a/at) (u(N) uc;:») E L 2(L 2) as test function in the weak formulation of (4.107)-(4.108):

1

Qt m

a-eN)

a

at

at

_P- . _(u(N) _ u(N»)dxdt-

1L

n+l

Qt m

k=l

- J(r

Qt m

D

n+l a r '" J(N) . V'-(u(N) J( k at k ~

Qt m k=l a W k(u(N), V(N») at (u~N) - u¥:,k)dxdt

~

(N) (N) a (-(N) ~ ek Jk . V'V at Un+ 1 k=l

-

u(N»)dxdt D,k

(4.117)

-(N) ) UD,n+l dxdt.

We estimate both sides term by term:

where we have used the condition (H2) and (4.113). The second integral on the left-hand side of (4.117) is splitted into several integrals:

4.5. Regularity and uniqueness

161

For the estimate of the integral K 1 we use the facts that Lik is symmetric and does not depend on u(N) or V(N):

The integral K 2 is estimated by employing the assumption (H3):

L 1] m

K2

t·

t, ljt~l

j=1 tj-l

>

>.

~h 2

Jr

QtTn

1L k

n+1

h-2 (tj - t)

1

n k,i'=1

(tj - t)

~ \7u(N)

at

I

2 1

Lk[(x)\7(U{ - U{-1) . \7(u~ 2

%t \7u(N) 1 dxdt

dxdt.

For every c > 0 we have, using (4.113),

u~-1)dxdt

162

Chapter 4. The Energy-transport Model

For the last integral K 4 we employ Green's formula:

r L J

n+l

K4

=

Qt =

k,e=l

ek(V'Lke(x), V'V(N)u~~)l +Lke(x)V'u~~)l' V'V(N)

+ Lke(x)u(N) ~V(N»)~(:U(N) at e n+l >

_Co

8

1

I-u-I a-eN)

Qt=

at n+l

2

-:u(N»)dxdt D,e

1

dxdt-c

Qt=

a-eN)

I~I at

2

dxdt

ktm k~l (IV'LkeI21V'V(N)12Iu~~12 + ILkeI21V'u~)1121V'V(N)12

-

C

+

ILkeI2Iu~~)1121~V(N)12)dxdt.

The last integral is estimated as follows: Let q = r/(r - 2) (see condition (H3) for the definition of r). Then the embedding H 1 (O) <-t L 2q(O) is continuous and, by Holder's inequality,

1

Qt m

IV' LkeI21V'V(N)12Iu~)112dxdt

< IIV'Lkellioo(£r) IIV'V(N) Ilioo(LOO) Ilu~)1I1i2(L2q) < cIILkellioo(Wl,r) IIV'V(N) Ilioo(LOO) Ilu~~ Ili2(Hl) < 811V'u(N)llioo(o,t=;£2) + c(8), using (4.116). Furthermore, we have

1 ILkeI21V'u~)1121V'V(N)12dxdt Qt m

< IILkelliOO(LOO) IIV'u~~)11Ii2(£2) IIV'v(N) lIiOO(LOO) ~ 811V'u(N)llioo(o,t=;£2) + c(8). In this estimate, the boundedness of V'V(N) in LOO(LOO) is needed. Finally, by Holder's inequality,

1 ILkeI2Iu~)1121~V(N)12dxdt Qt=

~ IILkelliOO(LOO) Ilu~~ IIIOO(£2 P!(P-2» IIV(N) Ili2(W2,P)'

4.5. Regularity and uniqueness

163

Similar as in the second step of the proof we obtain IIV(N) III2(W2.P)

< c(1 + Ilu(N) III2(LP») < c(1 + Ilu(N) III~(Hl) Ilu(N) Ili~(i~»),

and, in view of the uniform estimates of u(N) in L 2(H 1) and L oo (L2), c.

IIv(N) III2(W2,P) ::;

Employing the Gagliardo-Nirenberg inequality [373, p. 1034] we find (N) 11 2 Loo (£2p/(p-2) I un+l

where

J1, =

::; C

I un+ (N) 1 2 J.L I (N) 11 2 (1-J.L) 1 LOO(Hl) un+ 1 Loo (£2)'

d(p+2)j4p < 1, and therefore, for any 8> 0, by Young's inequality,

1 ILk£12Iu~;~)1121~V(N)12dxdt::; 811\7u(N)IIIoo(£2) + Qt",

c(8).

The last inequality implies, together with the other above estimates, K4

2:

Co ( -8 lc

I

Qt",

au(N) f it

2

1

2 c811\7u (N) IILoo(o,t",;£2) -

dxdt -

c(8).

Now we turn to the terms on the right-hand side of (4.117):

1L

n+l

a

Wk(u(N), v(N») at

Qt", k=l

~ lei ~t Qt", (

<

+C <

a-(N)

1

Qt",

2 I

(u~N) -

dxdt + C

u<;'k)dxdt

lei (

Qt",

a-(N)

~

2 1

dxdt

(1 + [u(N) 12 + IV(N) 12)dxdt au(N) 12 dxdt+c,

81Qt ", f i t Co

(

1

using (4.115) and (4.113). Finally, again employing (4.115) and (4.116),

~ J(N). nV(N)~(-(N) _ -(N) )d dt lcQt", k=l L.. ek k v at Un+! uD,n+l X

_ (

::;

Co

8

1

Qt",

au(N) 2

I~ I dxdt + at

c

1I Qt",

au(N) 2 D,n+l dxdt at

I

164

Chapter 4. The Energy-transport Model

Hence we get from (4.117):

This implies -Co

8

1

1afj(N) 1 -!)

Qr

vt

2

dxdt

+ (>../2 -

c: - co) sup

tE(O;r)

1 n

lV'u(N)(t)1 2 dx::; c(c:,o).

Therefore, choosing c: > 0 and 0 > 0 small enough, we obtain

1 Qr

afj(N)

--!)

I

vt

1

2

dxdt

+

sup

tE(O,r)

1 n

lV'u(N)(t)1 2 dx ::; c,

where c> 0 does not depend on N. The conclusion of the theorem follows. 0 Using a regularity result for elliptic systems due to [186, 291], we can improve the regularity of the transient solution with respect to the space variable: Theorem 4.5.2 Let the hypotheses (Hl)-(H5) hold and let UD E L 2 (W 1 ,qo) for some qo > 2. Then there exist q E (2, qo] and a solution (u, V) of (4·50)-(4·56)

such that V'u E Lq(Qr). Moreover, u E CO(Qr) if d = 1 and u E £8(Qr) for all s < 00 if d = 2.

Proof. As in the proof of Theorem 4.5.1 we show that the sequence of approximate solutions (u(N») derived in Section 4.3 is uniformly bounded in the space Lq(W1,q). We rewrite the problem (4.107)-(4.108) in the dual entropy variables (w(N), V(N») (see Section 4.2): n+l

div

L Dik(X, vj)V'wL

in

n,

k=l

on

i = 1, ... , n

an,

j

+ 1,

= 1, ... , N,

4.5. Regularity and uniqueness

where

165

11 is defined by

aPi-(N) ( tj) at

a-(N)

Pn+l

at

j Vj) _ W(u t,

(t·) J

'f' = 1, ... , n,

1 ~

a-(N)( .) -W.(uj Vj)-Vje. P tJ

at

t,

ifi=n+1.

Thanks to the a priori estimate (4.115), the diffusion matrix (Dik) is uniformly positive definite, and there exist 80,81 > 0 independent of j and N such that: for all ~ E ~n+1 .

i,k=l We claim that Ii(N) = Ul, ... ,I f ) is uniformly bounded in L 2 (L 2 ). Indeed, thanks to Theorem 4.5.1 and assumption (H4) it holds a-(N)

IIIi(N) 11£2(L2)

S

C( 1 + II ~t

t2(£2) +

Ilu(N) 11£2(£2)

+

IIV(N) 11£2(£2))

Ilu(N) IIL2(£2)

+

IIV(N) 11£2(£2)

SC

if i = 1, ... , n,

III~~lll£2(£2) <

C(l + II

a-(N)

P~+l II ut £2(£2)

+

+ IIV(N) IIL=(L=) II ap~N) II )sC ut £2(£2)

if i = n

+ 1.

The regularity result in [186, 291] now implies the existence of s E (2, qo] such that E W 1,S(0) for all i, j. Furthermore, II wj I11,s,n S c(l + III1110,2,n + Ilwj I11,2,n),

wI

which yields Ilw(N)II£2(W1 ,S)

S

c(l

+ III(N)II£2(L2) + Ilw(N)IIL2(Hl») S

c,

using Lemma 4.3.3, where c > 0 is independent of N. Therefore, letting N ----+ we get w E L 2 (W 1 ,s). It follows u E L 2 (W 1 ,s). Furthermore, since u lies in the space L OO (H 1), by Theorem 4.5.1, we get for q = s/2 + 1 > 2: 00,

iT

IIV'uIIZ,q,ndt

<

iT IIV'ullo,2,nIIV'ull~::ndt

S

IIV'uIIL=(£2)IIV'ull1.~2(Ls)

< IluIIL=(Hl) Ilull1.~2(Wl'S) < 00, which implies V'u E Lq(QT) (takings S 4). Weconcludefromu(N) E LOO(H 1 )n H 1 (L 2 ) and Aubin's Lemma [337] that u E CO ([0, T]; H 1 (0)) '---> CO(QT) if d = 1. The last regularity result for d = 2 follows from u E LOO(H 1) and the Sobolev embedding H 1 (0) '---> £8(0) for all s < 00. 0

166

Chapter 4. The Energy-transport Model

Remark 4.5.3 (i) It is not difficult to see that we even get the regularity \lu E £<'(Lq) with (J = 2q/(q - 1) < 4. (ii) The continuity of u in Qr implies that the temperature T = -1/un +1 is

strictly bounded away from zero in Q r' Moreover, if the particle density is given by n = T 3 / 2 eU1 , we obtain n > 0 in Qr' Therefore, in one space dimension we conclude the strict positivity of the physical variables nand T. 4.5.2

Uniqueness of transient solutions

The uniqueness of solutions to equations or systems of the type Otp(u) - diva(x, t, u, \lu)

=

f(x, t, u)

subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions and initial conditions is a delicate problem. In the case where p( u) is a scalar-valued function, the uniqueness of solutions satisfying Otp(u) E L I (Qr) is shown in [6]. The function p( u) is only assumed to be monotone. Recently, the uniqueness of weak solutions is obtained in [295] by using the method of time doubling (cf. Section 3.3). For vector-valued functions p(u), there exists to our knowledge no general result. In [6] the uniqueness in the class of weak solutions could be proved for linear elliptic operators, i.e. a(x, t, u,p) = A(x, t)p. For the system (4.50)-(4.56) from nonequilibrium thermodynamics, we encounter additional difficulties due to the gradient terms and the electrostatic potential. As in the regularity theorems of Section 4.5.1, it seems difficult to prove the uniqueness of solutions to the system in the dual entropy formulation since the diffusion matrix then depends on V(x, t). We prove uniqueness in the class of functions satisfying u E Loo(W I ,3/2), V E Loo(WI,OO). These regularity properties are necessary to treat the gradient terms involving the electrostatic potential. Note that it is shown in Section 4.5.1 that there exists a solution (u, V) with the above regularity. We have to assume as in [6] that the diffusion matrix does not depend on u. The uniqueness proof is performed by using an elliptic dual method (see [6]), i.e. we take a test function in the weak formulation of the equations, which satisfies an elliptic problem. Theorem 4.5.4 Let the hypotheses (H1)-(H5) of Section 4.5.1 hold. Furthermore, let Wi(x, t, u, V) be globally Lipschitz continuous in (u, V) uniformly in (x, t) for i = 1, ... , n + 1. Then there exists a unique solution (u, V) of (4.50)(4.56) in the class of functions satisfying u E Loo(W I ,3/2), V E LOO(WI,OO). Proof. Let (u\ VI), (u 2 , V 2 ) be two solutions to (4.50)-(4.56) satisfying the above regularity properties. Set r = p(u l ) - p(u 2 ) E L 2 ((v*)n+1). By Lax-

167

4.5. Regularity and uniqueness

Milgram's lemma, there exists Y E L 2(V n+l ) such that for all c/> E L 2(V n+l)

1 Q.,.

n+1 L Lkt(X)"VYk . "Vc/>t dxdt = k,t=1

rr

in

(4.118)

(f, c/»dt,

0

where (-,.) is the usual duality pairing between (v*)n+l and V n+l . Taking the difference of the equations satisfied by (ul, yl) and (u2 , y2), respectively, we get, for t E (0, T),

n+1

r(atf, y)dt - iQtk=1 r L(J~ - J~) . "VYkdxdt io t

=

1 -1 t ek(J~ Qt

n+1 L(Wk(Ul, yl) - Wk(u 2, y2))Yk dxdt k=1

Qtk=1

(4.119)

J~ . "Vy2)Yn+l dxdt .

. "Vyl -

In [6] the following inequality is shown:

for a.e. t E (0, T), which implies, thanks to the uniform positive definiteness of (L kt ), t (4.120) l"Vy(tWdx. (atf, y)dt 2 ~ io 2 Furthermore, in view of (4.118) and assumption (H2) we have

r

1L

inr

n+1

Qt

k,t=1

Lk£(X)"V(U} - ui) . "VYk dxdt

r (p(u p(u2)) . (u r luI - u 2 2dxdt. i Qt

=

i Qt

>

Co

l

) _

l

-

u 2)dxdt

1

Hence, we can rewrite (4.119) using this estimate together with (4.120):

~

inr l"Vy(tWdx + Co irQt luI - u

:; 1 t Qt

k,t=1

212dxdt

ekLkt(X)(U;+1 "Vyl -

U~+I "Vy2) . "VYt dxdt

168

Chapter 4. The Energy-transport Model

n+1

r L(Wk(U I , yl) - W k (U 2, y2))Yk dxdt (4.121) J k=1 - Jr k,l=1 t ekLkl(x)(V'U} . V'yl - V'u~ . V'y2)Yn+Idxdt + r t ekelLkl(x)(U~+IIV'yI12 - u~+lIV'y212)Yn+Idxdt J k,l=1 +

Qt

Qt

Qt

Now we estimate the integrals K I , .•. , K 4 . Since it holds for d:S 3

11V'(yl - y2)11£2(L6) :S clIVI - y211£2(H2) :S cllp(u l ) - p(u 2 )11£2(£2)

(4.122)

:S cilu l - u211£2(£2), we get, using Holder's inequality,

KIlt Qt k,l=1

<

ekLkl(x)((U~+l - U~+l)V'yl + u~+l V'(yl -

y2)) . V'Yldxdt

cllu~+1 - u~+III£2(£2) IIV'yIlluX>(LOO) IIV'YII£2(£2)

+ cllu~+lIILOO(£3) 11V'(yl - y2) 1I£2(L6) IIV'YIIL2(£2)

< cllu~+1 - u~+lII£2(£2) IIV'YII£2(£2) < E:

r luI - u 2 2dxdt + c(E:) jlV'YI 2dxdt,

JQt

1

Qt

where E: > 0 and c(c) > 0 depends on the L=(O) norm of Lkl, the L=(L=) norm of V'Y\ and the L=(L3) norm of U;+I' Note that for d :S 3 the space W I ,3/2(0) is continuously embedded into L3(0). The Lipschitz continuity of Wi and Poincare's inequality imply

K2 <

c

r (luI - u21+ IV I - y 21)IYldxdt

JQt

< c(llu l - u211£2(£2) + 11V'(yl - y2)11£2(£2)) IIV'YII£2(£2) < E:

r luI - u 2 2dxdt + c(c) Jr IV'YI 2dxdt.

JQt

1

Qt

169

4.5. Regularity and uniqueness For the estimate of K 3 we use Green's formula: K3

rt

=

JQt k,l=1

ek (V Lkl(X) . (VyIu} -

Vy2U~)Vn+l

+ Lkf(X)(U}ilyl - u~ily2)Yn+1 + Lkl(X)(U}Vyl- u~Vy2) . VVn+l)dxdt n

< c

L

k,l=1

(IIV LklIILoo(£r)IIVn+III£2(£S) IIVyIIILoo(Loo) Ilu} -

u~II£2(£2)

+ IIV LklllLoo(£r) IIVn+lII£2(LS) Ilu~ IILoo(L3)IIV(yl - y2) 11£2(L6) + IILkfIILoo(Loo)IIVn+IiI£2(L6) Ilu} - u~ IIL2(£2) IlilyIIILoo(£3) + IILkfIILoo(Loo)IIYn+lIIL2(L6) Ilu~IILOO(£3) Ilil(yl - y2)11£2(£2) + IILklIILOO(LOO)IIVVn+IiI£2(£2) Ilu} - u~II£2(£2) IIVyIIILOO(LOO) + IILklIILoo(Loo)IIVVn+III£2(£2) Ilu~IILOO(L3) IIV(yl - y2) 11£2(L6)), where r 2: 2 is defined in hypothesis (H3) and s = 2r/(r - 2). Observing that HI(O) ~ £5(0) and

IlilyIIILOO(£3) :S c(l

+ IluIIILOO(£3)) :S c,

we conclude, employing (4.122),

where c(e:) > 0 depends on Lkf, the L oo (L3 ) norms of u l and u2, and on the Loo(L oo ) norm of Vyl. Finally, using again (4.122), K4

=

1t QT

k,l=1

((U~+l - U~+l)IVyI12

ekelLkl(X)

+ U~+l V(yl + y2) . V(yl

- y2) )Vn+ldxdt

< cllu~+1 - u~+III£2(L2) IIVyIlliOO(LOO) IIYn+III£2(£2)

+ cllu~+IIILOO(£2) IIV(yl + y2)IILOO(LOO) IIV(yl x IIVn+lII£2(L3) < cilu l - u211£2(£2)llvll£2(Hl)

< e:

r lUI - u2 2dxdt + c(e:) r IVvl2dxdt. J J 1

Qt

Qt

y2) 11£2(L6)

Chapter 4. The Energy-transport Model

170

Therefore, we get from (4.121), for a.e. t E (0, r),

~ 2

rl\7y(tWdx + (co - 4c) Jr lUI - u In Q,

2 2 1

dxdt

~ c(c)

r I\7YI dxdt. J 2

Q,

Choosing c > 0 small enough and employing Gronwall's lemma, we conclude

in

which implies u l

-

l\7y(t)1 2 dx = 0

for a.e. t E (0, r),

u 2 = 0 and VI - V 2 = 0 a.e. in Q7"

o

From Theorem 4.5.4 we cannot exclude the existence of solutions in a larger class of functions. However, in one space dimension, we can prove that the solution constructed in the proof of Theorem 4.5.1 as the limit of approximate solutions is the unique solution of the system: Corollary 4.5.5 Let the assumptions of Theorem 4.5.4 hold and let d = 1. Then there exists a unique weak solution (u, V) to (4.50)-(4.56). This solution satisfies u E Loo(H I ) n H I (L 2 ) and V E Loo(WI,oo).

Proof. Let (u 1, VI) be an arbitrary weak solution to (4.50)-(4.56). Furthermore, let (u 2 , V 2 ) be the solution satisfying u 2 E Loo(H I ) whose existence is assured by Theorem 4.5.1. Since d = 1 it holds for i = 1,2,

II\7ViIILoo(LOO) ~ cllV i IILOO(H2) ~ c(1 + IluiIILoo(£2») < 00. We can proceed as in the proof of Theorem 4.5.4 since the constants c(€) only depend on the Loo(Loo) norms on \7V I , i = 1,2, and on the L oo (L 3 ) norm of u 2 , except in the estimate of K 3 . Therefore, it remains to consider the term

r Lk£(X)Yn+I(U} - ui)t::.VIdxdt

JQ ,

< IILHIILoo(Loo) IIYn+1II£2(Loo) Ilu} - ui 11£2(£2) lit::.VIIILoo(£2) ~ cllu} - uill£2(£2) II\7Yn+1II£2(£2), which can now be estimated as in the proof of Theorem 4.5.4. The conclusion of the corollary follows. 0

4.6 Existence of steady-state solutions In this section, we prove the existence of solutions to the steady-state system: div J i div I n+1

t::.V

Wi(u, V), i = 1, ... ,n, n ekJk . \7V + W n+1(u, V), k=1 "l(u, V) - C(x),

-L

(4.123) (4.124) (4.125)

171

4.6. Existence of steady-state solutions n

+ ekUn+l'VV) -

- L Lik(U, V) ('VUk

Ji

Li,n+l (U, V)'VUn+l

k=l in 0, i=I, ... ,n+1.

Ui = UD,i, V = VD Ji 'lI='VV'lI=O

(4.126)

on rD, onr N ,

(4.127)

i=I, ... ,n+l,

(4.128)

We assume that the "total" particle density is given by the general function ",(u, V). As explained in Section 4.2, the key of the proof is the formulation of (4.123)-(4.128) in the dual entropy variables (4.58). Then the above system becomes: div Ii div In+l ~V

Qi(W,V), i=I, ... ,n, Qn+l(W, V) - Ve' Qk(W, V), N(w, V) - C(x),

(4.129) (4.130) (4.131)

n+l (4.132)

- L Dik(W, V)'VWk, k=l

i

Wi

WD,i, V = VD Ii . 1I = 'VV . 1I = 0 =

= 1, ... ,n + 1, (4.133)

on rD, on

rN,

i = 1, ... , n

+ 1,

(4.134)

where h Qi and Dik are defined as in Section 4.2, and N(w, V) = ",(u, V). Our main hypotheses are as follows:

c JR.d (d ~ 1) is a bounded domain with Lipschitzian boundary rD urN, r D n r N = 0, measd-l(rD) > 0, and rN is open in ao.

(HI) 0

ao =

(H2) (L ik ) E Loo(O x JR.n+l x JR.; JR.(n+l)x(n+l») is a Caratheodory function (see Section 3.2) and a symmetric uniformly positive definite (n+ 1) x (n+ 1) matrix. (H3) Wi : 0

X

JR.n+l x JR.

-t

JR. are Caratheodory functions satisfying

n

n

L(Wk(u, V) - Wk(U, V))(Uk - Uk) ~ 0,

k=l

\Wi(x, t, u, V)I ~ c(1 + lui for all u, V, '11,

LekWk(U, V) = 0,

k=l Wn+l(x,t,u, V) (Un+l - u) ~ 0,

V,

with

+ IVI),

i = 1, ... , n

+ 1,

u < O.

(H4) ", E Loo(O x JR.n+l x JR.) is a Caratheodory function; C E Loo(O), el, ... , en E JR.; UD,i, VD E Hl(O) n Loo(O), and UD,n+l = U.

172

Chapter 4. The Energy-transport Model

In hypothesis (HI), also domains with C O,l corners are included. The assumption (H2) follows from basic physical principles, as explained in Section 4.3.1. The monotonicity condition for W n +1 in assumption (H3) means that W n + 1 is a "relaxation" term. For the energy-transport model for semiconductors, this condition is verified as can be proven rigorously [41]. In this example, it holds W 1 (u, V) = O. In the case of the bipolar semiconductor model (see Section 4.3.2), this hypothesis is also satisfied since e1 WI + e2 W2 = O. We assume that the temperature Un +1 is constant at the contacts, i.e. we neglect surface thermal effects. Physically, we expect that the "total" particle density is bounded, so hypothesis (H4) is reasonable. For a more general assumption on TJ see Remark 4.6.4. Before we can state the existence result, we have to explain the notion of solution. We say that (w, V) is a weak solution of (4.129)-(4.134) if w - WD E vn+1, V - VD E V n Loo(O), and (4.129)-(4.131) are satisfied in the (usual) weak sense. The tupel (u, V) is called quasi-weak solution of (4.123)-(4.128) if u - UD E W~,l(O U fN;lR n+1) n L 2 (0;lR n+1), V - VD E V n Loo(O), and the equations (4.123), (4.125) and div I n + 1 = -div (

n

n

k=l

k=l

L ek V Jk) - V L ek Wk + W +1 n

(4.135)

are satisfied in the weak sense. The equation (4.135) follows from (4.124) after inserting (4.123) into (4.124). Note that V Jk lies in L 1 (0), so (4.135) can be interpreted as an equation in (W~,oo(O U f N ;lRn+ 1 ))*, the dual of W~,oo(O U fN; IR n+ 1 ). It follows from these definitions that, given a weak solution (w, V) of (4.129)-(4.134), the tupel (u, V) = (w+eVw n +1' V) is a quasi-weak solution of (4.123)-(4.128). Conversely, every quasi-weak solution of (4.123)-(4.128) satisfying u E H 1 (0) n Loo(O) defines a weak solution to (4.129)-(4.134). If (u, V) is regular such that Jk . VV can be defined, then (u, V) is a weak solution of (4.123)-(4.128) in the usual sense. The main result is the following theorem. Theorem 4.6.1 Under the assumptions (H1)-(H4), there exists a quasi-weak solution (u, V) to (4.123)-(4.128). Moreover, u E W 1 ,P(0; IR n +1 ) with p d/(d - 1) if d 2: 3, p = 2 - E for all E > 0 if d = 2 and p = 2 if d = 1. In view of the above remarks, it is clear that the theorem is a consequence of the following proposition. Proposition 4.6.2 Under the assumptions (H1)-(H4), there exists a weak solution (w, V) of (4.129)-(4.134).

4.6. Existence of steady-state solutions

173

Proof. The proof is based on Leray-Schauder's fixed point theorem. Set S = £2(0; IRn+l) x £2(0) and let (z, U) E S, a E [0,1]. Since the function N(·, z(-), U(·)) is bounded, there exists a unique solution V E H 1 (0) of a(N(x, z(x), U(x)) - G(x)) in 0, aVD on r D , V'V· v = 0 on rN.

~V =

V

=

The boundedness of N implies

1IVIIo,oo,n :s; G1 ,

(4.136)

where G1 > 0 only depends on G, N, 0, and VD. Now solve the linear problem n+1

- div ( L Dik(X, z, V)V'Wk) k=l

= aQi(x, z, V),

i

= 1, ... ,n + 1,

n+1

W = aWD on

r N , L Dik(X, z, V)V'Wk . v =

0 on

k=l

rN ·

(4.137) (4.138)

By Lax-Milgram's theorem, there exists a unique solution wE H 1 (0; IRn+l) of (4.137)-(4.138). Indeed, the right-hand side of (4.137) is a £2(0) function of x (i.e. Qi(-, z('), V(-)) E £2(0)), due to assumption (H3), and WD E H 1 (0; IR n +1 ), due to hypothesis (H4). Thus the system (4.137)-(4.138) can be formulated in the weak sense and Lax-Milgram's theorem applies. Hence, the fixed point operator T : S x [0,1] --t S, (z, U, a) f-+ (w, V), is well-defined. It holds T(z, U,O) = (0,0) for all (z, U) E S. We show that there exists a bound for all fixed points of T. Let (w, V) E HI (0; IRn+l) X HI (0) be such a fixed point (i.e. a solution to (4.129)-(4.134) if a = 1) and use W - aWD as test function in (4.137): n+1

L { Dik(X, W, V)V'Wk . V'wi dx i,k=l in n+1

= a L l Dik(X, w, V)V'Wk . V'WD,i dx i,k=l n

(4.139)

n+1

+L

k=l

( aQk(x,w, V)(Wk - aWD,k)dx.

in

It is shown in Section 4.3.1 that (Dik) is symmetric positive definite, and there exists 8(V) > 0 such that n+1

L Dik~k~i 2: 8(V)I~12 i,k=l

for all ~ E IR n +1 .

Chapter 4. The Energy-transport Model

174

Taking into account the Loo bound (4.136), there exists 00 > 0 such that O(V) 2: 00. Applying Young's inequality to (4.139) and using the assumption (H3), we can therefore write

;0o L inrD;kl\7wD,iI 2dx 2

n+l

i,k=l

+

t inr

aWk(Wk - aWD,k)dx

in ;; I: inr k=l

+a

<

Wn+1(Wn+l - aWD,n+l)dx

o i,k=l

+a

(4.140)

D;kl\7wD,iI 2dx

t inr

Wk(aUD,k, VD,k)(Uk - aUD,k)

in

k=l

+a

Wn+1(u, V) (Wn+l - aWD,n+d dx .

From assumptions (H3)-(H4) follows that

in

Wn+1(Wn+l - aWD,n+d dx

=

in

Wn+l(x,u, V) (un+l - u)dx

+ (1- a) < c(l +

<

in

Wn+l(x,u, V)udx

in (lui +

o n+l

r

IVl)dx)

: {; in [\7wkI 2dx + C(OO)'

Thus we get from (4.140), after using Poincare's and Young's inequalities,

where Cl > 0 depends on the Loo norms of V and the H 1 norm of WD, but not on W or a. Hence, T( w, V, a) is bounded in S. Standard arguments show that T is continuous and compact, noting that the embedding H1(n) '----7 £2(0,) is compact. Now, Leray-Schauder's theorem gives the desired result. 0

4.6.

175

Existence of steady-state solutions

Remark 4.6.3 For smooth data we get weak solutions (u, V) of (4.123)-(4.128) for d ~ 3. Indeed, let an E 3 , I'D nI' N E C 3 , and VD E W1,00(n). From [334] follows that V E w1,q(n) for all q < 4. Thus

c

where r = min(2, 2dq/(2d + dq - 2q)). For d ~ 3, we get r = 2, hence u E H1(n; IRn +l) and Ji . \7V E p(n) for 1 < P < 4/3. Thus the product Ji . \7V is defined and (4.123)-(4.128) can be interpreted in the weak sense. Remark 4.6.4 We can relax the condition that the total density 1]( u, V) is

bounded by assuming that the source terms are strongly monotone (see (4.141)). Indeed, we get the following result: Let the assumptions (H1)-(H4) hold, 1](x,u, V) = 1](x,u) only being a Caratheodory function (see Section 3.2), and let n

:L)Wk(u, V) - Wk(u, V))(Uk - Uk) ~ -clu - uI 2

(4.141)

k=l

for all u, V, U, V, where c > O. Then there exists a quasi-weak solution (u, V) to (4.123)-(4.128). Note that we do not get immediately an Loo norm for V, since 1](u) is not bounded a priori, and the proof of Proposition 4.6.2 cannot be applied directly. To prove the result we proceed as follows. For the application of the fixed point theorem, we have to find an estimate for the solution of (4.137)-(4.138) (for z = w). Use w - crWD as test function in (4.137):

L 1Dik(X, w, V)\7Wk . 'V(Wi - crwD,i)dx

n+l

i,k=l

n

L r crQk(X, w, V)(Wk -

n+l

=

Since \7wD,n+l

k=lJn

crwD,k)dx.

= 0 and Dik = L ik for i, k = 1, ... ,n, we get

Chapter 4. The Energy-transport Model

176

>

> Furthermore, the assumptions (H3) and (4.141) yield

L r (TQk(X,W, V)(Wk -

n+1 k=1

In

(TWD,k)dx

L r (T(Wk(u, V) - Wk((TUD, VD))(Uk - (TUD,k)dx

n+1 k=1

+

In

t.; i

n+1

< -(TC

<

(TWk((TUD, VD)(Uk - (TUD,k)dx

r Iu - (TuDl 2dx + (T L r Wk((TUD, VD)(Uk - (TUD,k)dx In In

-~(T

n+1

i

k=1

Iu -

(TUD!2dx

+ C(WD, VD)(T.

Therefore we obtain

We conclude that there exists an £2 bound for U from which we get an £00 bound for V, by Stampacchia's elliptic estimate. Hence there exists Do > 0 such that D(V) ~ Do > O. This gives an HI bound for w. Now the proof of the existence result is as the proof of Proposition 4.6.2.

4.7. Uniqueness of steady-state solutions

4.7

177

Uniqueness of steady-state solutions

For the uniqueness result, we need additional assumptions. The essential assumption is to require boundary data not far from the thermal equilibrium state. This state can be defined by Ji = 0 for i = 1, ... , n + 1, i.e. no currents flow. This implies Ii = 0 for i = 1, ... , n + 1 and, since (Dik) is invertible, VWi = 0 or Wi = const. in 0 for i = 1, ... , n + 1. Therefore, we have to require that IVWil are small enough in some norm. In general, we cannot expect uniqueness for arbitrary data since, for example for the drift-diffusion semiconductor model (n = 1), there exist semiconductor devices having multiple states [246,363]. The precise assumptions are as follows: (H5) d::; 2. (H6) (u, V)

O.

f--t

Lik(X,U, V) is Lipschitz continuous in jRn+l x jR uniformly in

(H7) Wi = 0 for i = 1, ... , nand W n+ 1 is Lipschitz continuous in Un+l uniformly in all other variables satisfying for all u, V, V

u,

(Wn+l(u, V) - Wn+1(u, V))(un+l - un+d ::; O.

(H8) (w, V) f--t N(x, w, V) is Lipschitz continuous in W uniformly in (x, V) and non-decreasing in V; UD,i, VD E w1,q(0) for some q > 2 (i = 1, ... , n+1).

Whereas the assumptions (H6), (H8) are natural for uniqueness results, the hypotheses (H5) and (H7) are more restrictive. The condition (H5) is connected with the regularity of the solutions to (4.123)-(4.128). If the solution is more regular then uniqueness also holds for d = 3 (see [224]). From assumptions (H3) and (H7) follows that for all u, V (4.142)

For the energy-transport model, we have N(w, V) = (-1/w2)3/2 exp(wl W2V), and since W2 = -liT < 0, the function V f--t N(w, V) is non-decreasing (assumption (H8)). Theorem 4.7.1 Let the assumptions (H1)-(H4) of Section 4.3.1 and (H5)-(H8) hold. Then there exists € > 0 such that if n+l

L IIVWD,kllo,q,n ::;

€,

k=l

there exists a unique weak solution (u, V) of (4.123)-(4.128) satisfying Ui, V E H1(0) n L=(O), i = 1, ... , n + 1.

178

Chapter 4. The Energy-transport Model

Proof. We first prove the existence of a weak solution to (4.123)-(4.128). From Proposition 4.6.2 follows the existence of a solution (w, V) E HI(O;lR n+l ) x (HI(O) nLoo(O)) of (4.129)-(4.134). Taking into account the assumption (H8) we conclude from the regularity result in [186, 291] (also see [224]) the existence of qo E (2,q) such that Wi, V E wl,qo(O) (i = 1, ... ,n + 1). Furthermore, the a priori estimate n+1

n+1

k=1

k=1

L IIVWkllo,qo,n :S c(IIQn+I(W, V)llo,2,n + L IIVWD,kllo,q,n)

(4.143)

holds. Since d :S 2, this regularity result implies W E Loo(O; lRn+l) and u E wl,qo(O;lRn+I ). Thus u E HI(O;lRn+l) n Loo(O;lRn+l) and (u, V) is a weak solution of (4.123)-(4.128). We prove now the uniqueness in the class of weak solutions. In this class of functions the problems (4.123)-(4.128) and (4.129)-(4.134) are equivalent. Thus it is sufficient to prove uniqueness of solutions to (4.129)-(4.134). Let (wI, VI), (w 2,V 2) be two weak solutions to (4.129)-(4.134). Using wi - WD (i = 1,2) as test functions in (4.129) and VI - V 2 as test function in (4.131), we get the estimates

n+1

f; llVW112dX

I lIV(V - V 2 )1 2 dx and hence

< CI

n+1

L r IVWD,kI2dx, k=1 in

i = 1,2,

(4.144)

< - l (N(w l , V 2) - N(w 2,V 2))(V I - V 2)dx, n+1

r IV(V I - V 2Wdx :S CI L r IWk - W~12dx, in k=lin

(4.145)

where CI > 0 depends on the data. Using WI - w 2 as test function in (4.129) gives

n+1 0o [ ; llV(Wk -

< -

L 1(Dlk - D;k)VW;' V(Wk - w~)dx

n+1

i,k=1 n

+l < -

w~Wdx

(Qn+I(W\ VI) - Qn+I(W 2, V2))(W~+l -

n+1

L r(Dlk -

i,k=1 in

D;k)VW; . V(Wk - wDdx,

w~+I)dx

179

4.7. Uniqueness of steady-state solutions

where Dfk = Dik(X,W m , vm), m = 1,2. We have used the facts that (Dik) is symmetric, positive definite (see Section 4.2) and that Qn+1(w, V) = Wn+I(u, V) is monotone due to condition (H7). By Holder's inequality, applied to qo and qb = 2qo/(qo - 2), we obtain further

For d :::; 2 the embedding HI(O,) '---+ LS(O,) is continuous for all s < fore, noting that in view of (4.142) and (4.144)

00.

There-

n+I :::; C

L IIV'WD,kllo,2

k=I

and using the Lipschitz continuity of Dik and (4.143)-(4.145), we get

The constant Ca > 0 depends on the data but not on w D (or €). For sufficiently small € > 0 we get

This implies

WI

= w 2 a.e. in 0, and, using (4.145),

VI

= V 2•

o

180

4.8 4.8.1

Chapter 4. The Energy-transport Model

Numerical approximation The mixed finite element discretization in one space dimension

This section is devoted to the numerical discretization of the stationary energytransport model in one space dimension. First we scale the equations and transform them into a drift-diffusion-like form. The discretization of the scaled equations is then performed by using mixed finite elements. The energy-transport equations (4.37)-(4.38), (4.43)-(4.44) including the physical parameters read divJ1

0,

J1 • \7V + W(n, kBT),

divJ2

-\7(J.L~l)(kBT)kBTn)+ qJ.L~l)(kBT)n\7V,

J1

-\7(J.L~2)(kBT)(kBT)2n)

qh

+ qJ.L~2)(kBT)kBTn\7V.

We bring these equations into a scaled and dimensionless form. Let C m be the maximal value of the doping profile, L the diameter of the device, and UT = kBTo/q the thermal voltage. Using the scaling

n----+Cmn,

J1

----+

C----+CmC, x ----+ Lx, T ----+ (qJ.LoUTCm / L)J1 ,

T----+ToT, V----+UTV, (L 2/ J.LOUT )T, J 2 ----+ (qJ.LoUj.Cm / L)J2,

we get the system (also see (4.43)-(4.47)) (4.146)

0, I n · \7V + W(n, T),

(4.147)

91

-\791

+ T \7V,

(4.148)

-\792

+ 92 T \7V,

(4.149)

n-C, where 91

=

J.L~l)(qUTT) J.Lo

nT,

91

(4.150)

=

J.L~2)(qUTT) J.Lo

2

nT ,

W(n, T) = -~n(T - 1)/Tf3(T), and ),2 = Es UT /qCmL 2 denotes the scaled Debye length. The variables 91 and 92 are introduced in (4.35). Notice that we have used the same notations for the scaled and unsealed variables. The above equations are supplemented, in one space dimension, by Dirichlet conditions for n, T, and V (see Section 4.8.2).

4.8. Numerical approximation

181

We turn to the discretization of (4.146)-(4.150) which is performed in several steps. In the following we describe in detail the discretization of the onedimensional energy flux continuity equations (4.147), (4.149) by means of an exponential fitting mixed finite element method. The discretization of equations (4.146), (4.148) is similar but simpler (since the zero-th order term and the right-hand side of (4.146) are zero). The Poisson equation (4.150) is discretized with a PI finite element scheme. Consequently, in the following V denotes a piecewise linear function and Vx its (piecewise constant) derivative. The exponential fitting mixed finite element method introduced for the drift-diffusion continuity equation (d. [66, 67, 68, 270]) can be sketched as follows: (i) transformation of the problem by means of the Slotboom variable to a symmetric form; (ii) discretization of the symmetric form with mixed finite elements (consequently, the flux is introduced as independent variable); (iii) suitable discrete change of variable to rewrite the equations in terms of the original variables 92. Due to the non-constant electron temperature, a Slotboom variable does not exist in the present case. As starting point of the discretization scheme we define a "local" Slotboom variable, assuming that the temperature is a prescribed piecewise constant function defined in the global iteration process. We refer to the end of the section for an explicit choice of the procedure. A related idea has been used in [235] for the discretization of the non-linear drift-diffusion continuity equation (see Section 3.5). More precisely, introduce a partition 0 = Xo < Xl < ... < XN = 1 of (0,1) and set Ii = (Xi-I, xd, hi = Xi - Xi-1 for i = 1, ... , N , and h = maXi hi. We denote by T the piecewise constant approximation of the temperature (see (4.172) for the precise definition). The equations to be solved are then

-(92)x + 92 Vx/T, J 1 Vx + c19l, where we set

_

(4.151) (4.152)

3

Cg =

2TTf3(T)J.L~g) (T) ,

for f = 1,2, and, for simplicity of notation, we denote the variables again by Jg, 9g, for f = 1,2. In each interval Ii, "local" Slotboom variables are introduced by

Y2

= e- v / T 92

l'n

I i,

(4.153)

and equations (4.151) and (4.152) are written in the interval Ii as:

e- V / T h (h)x +

+ (Y2)x = 0, T C2 eV/ Y2

= J1Vx + c191'

(4.154) (4.155)

182

Chapter 4. The Energy-transport Model

A similar idea for the transformation of the energy-transport equations has been used in [225]. Jerome and Shu [206, 209] have employed a slightly different Slotboom transformation by introducing ¢(x) = Vx(s)jT(s)ds. To define the mixed finite element scheme we follow [270], where a monotonic scheme for the two-dimensional current continuity equation in the presence of a zero-th order term has been developed. The finite dimensional space for the flux variable contains functions of £2(0), which are in each interval polynomials of the form <7(x) = ai + biPi(X), with ai, bi constant and Pi(x) a second order polynomial uniquely defined in the interval Ii as follows. Let P(x) be the second order polynomial with the following properties:

I;

1

1 \

P(x)dx = 0, P(o) = 0, P(I) = 1,

that is, P(x) = 3x 2 - 2x. Moreover, it holds We define Pi(x) (depending on V) by

(4.156)

I; P'(x)dx = 1,10 P(x)2dx = 1

2 15'

Pi(x)

= -P(Xh~X)

if i min = i-I,

(4.157)

Pi(x)

=

if imin = i,

(4.158)

P(X-~:-l)

where i min is the point of minimum of the potential V(x) in the interval h We shall denote by Vmin its minimum value. Notice that the minimum is always attained at one end point of the interval, since V is linear in h If V(x) is constant in h we define Pi(x) = P(X-~:-l). Let us introduce the following finite dimensional spaces:

+ biPi(X)

Xh

{<7 E £2(0) : <7(x) = ai

Wh

{~ E £2(0) : ~ is constant in

Ah,x

in h i = 1, ... , N}, h i = 1, ... , N},

{q is defined at the nodes xo, ... , XN , q(xo) = X(O), q(XN) = x(1)}·

The mixed-hybrid approximation of equations (4.151)-(4.152) is then: Find J~ E X h , g~ E W h , g~ E A h,9D,2' such that:

t. (-1. t. (1.

1. B,g~uxdx [e- g~uC,) + 1. "2g~edx ~ t.1. + ",~)edx,

A,J;udx +

(J;)xedx

-

)

v/'f

(JfVx

= 0,

(4.159)

(4.160)

N

L[qJ~]~:_l i=1

=

0,

(4.161)

4.8. Numerical approximation

183

for all (j E X h , ~ E Wh, q E Ah,o. Jf E Xh is the approximation of the current density J1, gq E Wh is the piecewise constant approximation of gl, stemming from the discretization of the current continuity equation (see (4.169)-(4.171) below). In the first equation A and B denote the piecewise constant functions (approximations of e- V / T ) defined in each interval Ii by

2.-1

A t. ~f h.

't

xi

e -V(s)/Td s,

i = 1, ... ,N,

Xi-l

. /1' , == e -v:m'l.n B i clef

i = 1, ... ,N.

J!l is an approximation of the energy flux J 2, g~ is a piecewise constant approximation of g2 and gq is an approximation of g2 at the nodes (see [29, 270]). The first equation is obtained from a weak version of (4.154), using integration by parts and summation over all Ii together with the inverse of the Slotboom transformation (4.153). Notice that the discrete inverse transformation is not the same for the variables g~ and gq. We refer to [270] for a detailed discussion on the need of different approximations of the exponential function due to (possibly) large value of Vx. The second equation is a discrete weak version of (4.152), obtained from (4.155) where eV / T is approximated by B-1 and the discrete inverse Slotboom transformation for g~ is used. The third equation implies the continuity of J!l at the nodes. The variables J!l and g~ can be eliminated a priori by static condensation, leading to a final algebraic system in the variables gq only. We write J!l E X h as (4.162) for some constants Jg,i, Ji,i' i = 1, ... , N. Set g2,i = g~IIil g2,i gq(xd, Vi = V(Xi) and qi = q(Xi)' Taking (j E X h such that (j = 1 in Ii and (j = 0 elsewhere in equation (4.159) gives o . = _e-VdTg . + e-Vi-I/T g . h.A-J t t 2,t 2,t 2,t-1'

The integral in the definition of A can be computed explicitly and we arrive after elementary computations to

JO . = 2,t

-

Vi-Vi-1 h(Vi-Vi-1)92i-92i-1 cot " 2T 2T hi

g2i+g2i-1Vi-Vi-1 +" . 2T hi

(4.163) This discretization can be seen as a non-linear Scharfetter-Gummel scheme (d. [66]). The constants Ji i are computed by using (4.160) once g2 i is given. Indeed, taking ~ = 1 in Ii ~nd ~ = 0 elsewhere in equation (4.160), i't follows (4.164)

Chapter 4. The Energy-transport Model

184

~i JIi (-JfVx + clg~)dx. Taking now (J E Xh such that where we set ri (J = Pi(x) in Ii and (J = 0 elsewhere in equation (4.159) we obtain

2 -h-A.Jl. .. 15 t 'l. 2,1. = e-VTnin/T-g2,'1,. _ e-VTnin/Tg2',Z.Tntn

(4.165)

Using (4.164) and (4.165), we can eliminate Ji,i and get

g~,i = (1' C2

+ 1)-1 (1'r i + g2,i Tni J,

(4.166)

where l' = 125hrAieVTnin. Replacing (4.166) into (4.164) we get Ji,i in terms of g2,( 1 C2 h i hi ( ) 4.167 J 2,i = - 'Y C2 + 1 g2,i m in + 'Y C2 + 1 rio Finally, equation (4.161), with qi = 1 and qk = 0 for all k -=J. i, gives i = 1, ... ,N.

(4.168)

We recall that, due to definition (4.156)-(4.158), Pi(xd = 0 (Pi+l(xd = 0, respectively) if the minimum of V on Ii is in Xi-l (Xi+ 1, respectively), otherwise it is Pi(Xi) = 1 (Pi+l(Xi) = -1, respectively). Using the expression (4.163) for and (4.167) for Ji,i' the last equation (4.168) can be written in terms of the variables g2,i only, giving rise to a tridiagonal algebraic system, with the (positive) contribution of the zero-th order term appearing only in the diagonal entry. Then the energy flux J!J: is computed locally in each interval by (4.162) and g~ is computed locally by (4.166). Discretizing the current continuity equation (4.146), (4.148) with the same scheme and applying the (simpler) static condensation procedure (Jf is piecewise constant in this case), we obtain

Jt

JP,i = JP,i+l

i

=

1, ... ,N,

(4.169)

with

JO . = - Vi - Vi-I cot h (Vi - Vi-I) gl 'i - gl ,i-I l,t 2T 2T hi

gl i + 91 i-I Vi - Vi-I . +" 2T hi

(4.170)

Moreover, the analogous of (4.166) gives the upwind expression

gl,i = gl,i m in'

i = 1, ... ,N.

(4.171)

In order to complete the scheme, we still have to specify how the piecewise constant temperature T is defined. The temperature is defined implicitly in terms of gl and g2 according to the nonlinear equation (4.41). Lemma 4.1.1

4.8. Numerical approximation

185

shows that this equation can be solved uniquely. Numerically, we define each interval Ii as approximate solution of i = 1, ... ,N,

T in

(4.172)

with T i ~ Tlli and 91,i' 92,i given by the mixed scheme (see (4.171), (4.166)). The nonlinear equation reduces to a linear one when a = 0 (parabolic band). For a > 0 a single iteration of the (scalar) Newton scheme is sufficient to obtain T with an accuracy of 10- 8 , when the initial guess is the temperature at the previous global iteration procedure step. Moreover, f' can be explicitly computed by (4.42). In contrast to the strongly coupled equations (4.8)-(4.11) in the variables MIT and -liT, the two continuity equations (4.146) and (4.148) are weakly coupled through the temperature (which varies only slowly during the iterations). Consequently, we define the global iteration procedure as follows. The temperature is frozen at the previous iteration step, and a full Newton method is used to solve the nonlinear system in gl, g2 and V. At each iteration, the temperature is updated according to equation (4.172). The associated linear system is solved by using a GMRES solver. Finally, we remark that a Gummeltype iteration procedure can be employed (instead of the Newton method) in the parabolic band case.

4.8.2 Numerical results As a numerical example we present the simulation of a one-dimensional n+nn+ ballistic silicon diode which is a simple model for the channel of a MaS transistor. The semiconductor domain is given by the interval n = (0, £*) with £* > O. In the n+-regions the maximal doping concentration is 5.10 17 cm- 3 ; in the nchannel the minimal doping profile is 2· 10 15 cm -3. The doping profile is shown in Figure 4.2. The length of the n+-regions is 0.1 Mm, whereas the length of the channel region equals 0.4 Mm. The numerical values of the physical parameters (for a silicon diode) are given in Table 4.1. On the boundary points x = 0 and x = £* we assume that the (unscaled) total space charge C - n vanishes and that the (unscaled) temperature is the ambient temperature:

n(O) = n(£*) =

C1,

T(O)

=

T(£*)

=

To,

V(O) = 0,

V(£*) = U,

where U > 0 is the applied voltage. We take the value U = 1.5 V. The unscaled relaxation time TO and the low-field mobility Mo depend on
186

Chapter 4. The Energy-transport Model

r

\

15

10

o

0.1

0.2

0.3 0.4 Position in ~m

0.5

0.6

Figure 4.2: Doping concentration in the n+nn+ diode. Parameter q Cs

j.Lo

UT £*

£0 Co Cl TO

a

Physical meaning elementary charge permittivity constant (low field) mobility constant thermal voltage at To = 300 K length of the device length of the n + region doping concentration in the n region doping concentration in the n + region energy relaxation time non-parabolicity parameter [203]

Numerical value 1.6· 101~ As 10- 12 AsV- 1 cm- 1 1.5 . 103 cm 2 V- 1 s- 1 0.026V 0.6j.Lm O.lj.Lm

2.10 15 cm- 3 5.10 17 cm- 3 0.4.10- 12 s 0.5 (eV)-l

Table 4.1: Physical parameters. We perform numerical results for a uniform mesh of 100 nodes. In Figure 4.3 we present the electron temperature for vanishing and non-vanishing nonparabolicity parameter a using Lyumkis' model. Notice that the electrons are moving leftwards. As expected the temperature in the n-channel is high, i.e. the electrons are 'hot'. The maximal temperature for a = 0 is T = 3970 K and T = 3240 K for a = 0.5 (eV)-l. The corresponding thermal energies are E th = ~kBT = 0.51 eV and Eth = 0.42 eV, respectively. Therefore, the temperature is reduced due to the non-parabolicity effects. Similar results can be observed by employing Chen's model (Figure 4.4). Here, the maximal temperature (thermal energy) values are T = 2330K (Eth = 0.30eV) for a = 0 and T = 1610K

4.8. Numerical approximation

187

(Eth = 0.21 eV) for a = 0.5 (eV)-1. The effective scaled relaxation time in the Lyumkis model is (3..[ii/4)TOVT and TO in the Chen model. Therefore, the effective relaxation time in the Chen model is smaller than that in the Lyumkis model, and we expect that the maximal temperature in the Chen model is smaller than in the Lyumkis model. This observation follows from the fact that in the vanishing relaxation-time limit, the temperature relaxes to the lattice temperature, and it is confirmed by our numerical experiments. In Figure 4.5 the electron mean velocity for the two different values of the non-parabolicity parameter a using Lyumkis' model is shown. The mean velocity u is defined by u = Jd(qn). The spurious velocity overshoot peak at the left junction becomes smaller for non-vanishing non-parabolicity parameter. The maximal mean velocity for a = 0 is u = 2.92· 107 cm/s and u = 1.51 . 107 cm/s for a = 0.5 (eV)-1. The same effect can be observed using Chen's model (see Figure 4.6), where the spurious velocity overshoot spike almost vanishes for a = 0.5 (eV)-1. The maximal velocities are u = 1.44.107 cm/s for a = 0 and u = 1.25.107 cm/s for a = 0.5 (eV)-1. The mean velocities for the non-parabolic case, using Chen's or Lyumkis' models, are compared to the mean velocity from the standard drift-diffusion model in Figure 4.7. In the latter model, no velocity saturation effects are taken into account, i.e. the drift-diffusion model equals the energy-transport equations in the case of constant temperature. The velocity overshoot from the drift-diffusion model is much larger than for the energy-transport equations (Figure 4.7). This can be explained by the fact that the total energy of the energy-transport model is composed of the thermal and the kinetic energy, whereas the total energy of the drift-diffusion model is determined only by the kinetic energy. Model Lyumkis: a = 0.0 Lyumkis: a = 0.5/eV Chen: a = 0.0 Chen: a = 0.5/eV

slope 1.00 0.88 0.90 0.90

Table 4.2: Slopes of the logarithmic current-voltage curves for U E [O.5V,1.5V]. In Figure 4.8 we present the current-voltage characteristics for the different energy-transport models. The particle current density h is always smaller in non-parabolic band situations. Its dependence on the applied voltage U seems to be sublinear. Indeed, in the voltage range U E [0.5V, 1.5V], the dependence of J1 on U is approximately J1 rv U'Y, where I is between 0.88 and 1.0 depending on the model (see Table 4.2). We remark that increasing the number of nodes does not change the values of the current.

188

Chapter 4. The Energy-transport Model 40001------,-/~--..-______;:=:::'::::====:::::::::===:::;I

I

3500

,,

~3000

,

.f:

Q)

:; 2500 ~ Q) ~2000 .$

e1500 t5 Q)

[jJ

I I

1000 I

500

/

I

0.1

0.2

\

\

\

\

\

\

\

\

,

\

\

\

\

\

,

\

~ ~ ~g.5/eV I

,,

,,

,,

0.3 0.4 Position in Jlrn

0.5

0.6

Figure 4.3. Electron temperature versus position in a ballistic diode using Lyumkis' model.

00

0.1

0.2

0.3 0.4 Position in Jlrn

0.5

0.6

Figure 4.4. Electron temperature versus position in a ballistic diode using Chen's model.

189

4.8. Numerical approximation

~

E o .£

Z:-

2.5 2

'0 o Qi

~ 1.5


Q)

,,

E c

e

t5 Q) iIi

\

\

\

\

0.5

\

\ \

0.1

0.2

\

\

\

0.5

0.4 0.3 Position in 11m

0.6

Figure 4.5. Electron mean velocity versus position in a ballistic diode using Lyumkis' model.

a=O a

,

\

= O.5/eV

\ \

\ \ \ \ \

,

0.1

0.2

0.3 0.4 Position in 11m

0.5

0.6

Figure 4.6. Electron mean velocity versus position in a ballistic diode using Chen's model.

Chapter 4. The Energy-transport Model

190

Drift-diffusion Lyumkis (0.5/aV) Chan (0.5/aV)

6

I I I

,,

I I

0.1

0.2

0.3

Position in Ilm

0.4

,, 0.5

0.6

Figure 4.7: Electron mean velocity versus position in a ballistic diode.

4.5 "'E

4

~

.£3.5

Lyumkis (0.0) Lyumkis (0.5/aV) Chan (0.0) Chan (0.5/aV)

.~ 3 c:


~2.5 c:

~

o

2

e1.5

"0 ill

1

0.5

o

0.5

1

Applied voltage in V

1.5

Figure 4.8: Current-voltage characteristics for a ballistic diode.

Chapter 5 The Quantum Hydrodynamic Model

In this chapter we are concerned with the quantum hydrodynamic model. The formal derivation of the equations is studied and an elliptic boundary value problem is derived (Section 5.1). The existence of stationary solutions is proved in Section 5.2. The main difficulty of the proof is to find a lower bound for the carrier density. The positivity of the electron density is related to the regularity of the gradient of the so-called quantum quasi-Fermi potential (Section 5.2). In Section 5.3 the uniqueness and non-uniqueness of the solutions is investigated. For special boundary conditions, a result on the non-existence of solutions to the quantum hydrodynamic model can be proved (Section 5.4). The classical limit of the thermal equilibrium state and the 'subsonic' state is performed (Section 5.5). The current-voltage characteristics of tunneling diodes to the full and two simplified quantum hydrodynamic equations are studied analytically and numerically (Section 5.6). Finally, a positivity-preserving numerical scheme for the quantum driftdiffusion model is given and numerical simulations of a tunneling diode are presented in Section 5.7.

5.1

Derivation of the model

For the modeling of semiconductor devices which are driven by quantum effects (e.g. tunneling diodes) basic quantum mechanical equations, like the Schrodinger equation, have to be employed as a starting point for the derivation of appropriate models. The single-state Schrodinger equation can be rewritten in terms of the particle density and velocity leading to a zero-temperature vortex-free quantum hydrodynamic model without relaxation. A nontrivial temperature and nontrivial vorticity fields then arise from the statistical properties of mixed quantum states.

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations © Springer-Verlag Berlin Heidelberg 2000

192

Chapter 5. The Quantum Hydrodynamic Model Consider first a single state Schrodinger equation in ~d (d 2 1):

. EN

at

ZE-

2

E --D.." 2 'f/ + V·"'f/'

=

x

E ~d, t

> 0,

subject to the initial condition

'ljJ(x,O) = 'ljJ](x), where E > 0 denotes the scaled Planck constant. The electrostatic potential V is assumed to be described self-consistently by the Poisson equation

(5.1) where A > 0 is the scaled Debye length (see Section 3.1), C = C(x) the doping profile, and the particle density n is defined by

n(x, t) = 1'ljJ(x,tWo Introducing the (scaled) phase S of the wave function and the current density J by

'ljJ(x,t) = In(x, t) exp(iS(x, t)IE),

J(x, t) = cIm(~(x, t) "V'ljJ(x, t)),

respectively (here, ~ denotes the complex conjugate of'ljJ and 1m the imaginary part), and separating real and imaginary parts in the Schrodinger equation, the following irrotational flow equations are obtained:

1 2 -as + -I"VSI -

at

2

an at + d'IV J 2

V -

E 1 --D.Vii 2 Vii

0,

(5.2)

O.

(5.3)

Taking the gradient of equation (5.3), multiplying by n, and using (5.2) and div (J ® Jln) = ~n"VI"VSI2 + "VS· div J, we can rewrite (5.3) as

oj ® J) at + . (J-ndiV

2E n"V (D.Vii) Vii = 2

n"VV -

The i-th component of div (J ® J In) is defined by d

,,~(Jdj)

LJ j=1

ax·

J

n

.

O.

(5.4)

5.1. Derivation of the model

193

The irrotational initial conditions are given by

n(x,O) = nI(x),

x E JRd ,

J(x,O) = nI(x)'VSI(X),

(5.5)

associated to the initial wave function

WI

..jni exp(iSr/c).

=

The equations (5.1)-(5.2), (5.4)-(5.5) are the zero-temperature irrotational quantum hydrodynamic equations [173, 174]. They are also known as the Madelung fluid equations [262, 263]. The connection between the Schrodinger equation and fluid mechanics was already noticed by Madelung in 1927 [265]. For c = 0, (5.2) and (5.4) are the classical zero-temperature Euler equations. For c > 0, the quantum correction term 2

2

c (I:::..vn) c . --n'V - - =--dIv(n('V0'V)lnn) 2 vn 4 can be interpreted either as internal self-potential, the so-called Bohm potential,

Q=

c2 1

-2 vnl:::....;Ti,

or as nondiagonal pressure tensor

In order to obtain a nonzero temperature we consider mixed states (cf. [173, 174]). A mixed quantum mechanical state consists of a sequence of single states with occupation probabilities Ak' kEN, for the k-th single state described by

. aWk 2c---at Wk(X,O)

c2

-2I:::..Wk+ V Wk,

(5.6)

VnI,k(X) exp(iSI,k(X)/c).

(5.7)

Writing nk = IWk 12 and Jk = nk 'V Sk, where Sk is the phase of the wave function Wk, the problem (5.6)-(5.7) is (formally) equivalent to

ank

.

7it + dIVJk

0,

c n(I:::...;nk) -aJk + d'IV (Jk 0 Jk ) -nkVnv - -nkv --

0,

2

at

nk

nk(X,O) = nI,k(X),

2.;nk

Jk(x,O) = nI,k(X)'VSI,k(X).

194

Chapter 5. The Quantum Hydrodynamic Model

The occupation probabilities clearly satisfy 2:%:0 Ak = 1. The carrier density n and the current density J of the mixed state are given by

Multiplication of (5.6), (5.7) by Ak and summation over k yields the quantum hydrodynamic equations

an . at + dlV J 2

vn) vn

( . (J 0J ) -nY'V - -nY' c !:::. -o J +dlV +n()

at

2

n

0,

(5.8)

o.

(5.9)

The total temperature () = ()e+()os is the sum of the current temperature ()e and the osmotic temperature ()os [174]. Defining the "current velocities" U e = J/n, Ue,k = Jk/nk and the "osmotic velocities" Uos = cY'n/2n, Uos,k = cY'nk/2nk (k EN), the current and osmotic temperatures are given as the tensors

In classical fluid dynamics additional assumptions (so-called closure conditions) are made to obtain a closed set of equations (cf. Section 1.1). We assume that the temperature tensor is a scalar (times identity tensor): () = TId.

For the temperature scalar T we consider two cases: (i) Isothermal case: T = To, where To

> 0 is constant.

(ii) Isentropic case: T = T(n) = Ton a where To > 0 is constant and a > 1.

1

depends on the carrier density,

In these cases the pressure r = nT is given by (identifying r and rId)

with a = 1 in the isothermal case and a > 1 in the isentropic case.

195

5.1. Derivation of the model

In analogy to the classical fluid equations, a relaxation term can be introduced such that (5.9) writes 2

. (J@ J) + V(nT) - nVV - -nV E (t1 -aJ + dlV --Vii - ) = -f3J,

at

Vii

2

n

(5.10)

where f3 > 0 is the scaled inverse relaxation time. A different way of deriving quantum hydrodynamic models is based on the so-called moment method. Taking the zeroth, first and second order velocity moments of the quantum Vlasov equation, the equations (5.8)-(5.9) can be obtained under the closure condition (i) [163]. Furthermore, the quantum hydrodynamic model (5.8)-(5.9) can be derived by taking the zero and first order moments of the Wigner equation and introducing the mixed state Wigner function [174]. In the following we consider the stationary equations (setting 82 = E 2 /2): divJ = 0, . dlV

(J@J) -n-

+ Vr(n) -

nV(V + Vext )

Vii 2 8 nV (t1Vii )

-

(5.11)

= -f3J,

(5.12)

).2t1V=n-C

(5.13)

in a bounded domain n. The external potential Vext models interior quantum barriers (see Section 2.2). The main assumption is that we consider a potential flow, i.e. we assume that the particle current can be written as J=nV8 with the quantum Fermi potential 8 (see above). This means that the velocity J In = V 8 is assumed to be irrotational. It is physically reasonable to suppose that n > 0 holds in the device. Since div (J@ Jln) = !nVIV812 + V8· div J = !n\1IV812, we can rewrite (5.12) as 1 \181 2 n\1 ( 2 1

+ Toh(n) -

V - Vext

where

82

-

t1 Vii)

Vii

= -f3nV8,

r

(5.14)

~ r'(s)ds (5.15) To Jl s is the enthalpy function. In the isothermal case, h(n) = log( n) holds; for isentropic states, we have h(n) = cx~l (n cx - 1 -1) for a> 1. Notice that the electrostatic potential and the quantum Fermi potential are fixed only up to additional constants. Since n > 0, equation (5.14) implies h(n) =

1 2 21\181 + Toh(n) -

V - Vext

-

2

8

t1Vii

Vii + f38 =

O.

196

Chapter 5. The Quantum Hydrodynamic Model

The integration constant can be assumed to be zero by choosing a reference point for the electrostatic potential. For the analysis it is convenient to use w= as variable. Then (5.11)-(5.13) can be written as

vn

w(!I\7SI 2 + Toh(w 2 ) div (w 2 \7S) = 0, A2 t::. V = w 2 - C in n.

82 t::.w

=

-

V - Vext

+ /3S),

(5.16) (5.17) (5.18)

To derive the boundary conditions we make physically relevant hypotheses. The boundary data are assumed to be the superposition of the thermal equilibrium functions (n eq , Seq, Veq ) and the applied potential U(x):

n = n eq ,

S = Seq

+ U,

V

= Veq + U

on

an.

The thermal equilibrium state is defined by J = 0 or, equivalently, S = const. (as n > 0). By fixing the reference point for S (and Seq) we can suppose that Seq = O. We assume further that the total space charge C - n eq vanishes at the boundary and that no quantum effects occur on an, i.e. t::.Vn eq / vneq = O. Finally, Vext = 0 on an, since Vext is introduced to model interior quantum wells. We get from (5.16)

or, since Seq = 0, on

an.

Therefore we get the Dirichlet boundary conditions

w = Wo, with

Wo =

v0,

S = So, So = U,

V = Vo

on

Vo = Toh(C)

an

(5.19)

+ U.

(5.20)

Clearly, mixed Dirichlet-Neumann boundary conditions are physically more realistic than pure Dirichlet conditions. However, we need to impose the boundary conditions (5.19) for technical reasons, since mixed boundary conditions exclude regularity of solutions [185, 334, 354].

5.2. Existence and positivity

5.2 5.2.1

197

Existence and positivity of steady-state solutions Existence of steady-state solutions

We prove the existence of solutions to the stationary quantum hydrodynamic equations (5.16)-(5.18) with general Dirichlet boundary data (5.19). The fluid models (5.11)-(5.13) or (5.16)-(5.18) have been studied in some special situations. For vanishing convective and quantum terms the problem (5.11)-(5.13) reduces to the isentropic drift-diffusion semiconductor model (see Chapter 3). The quantum drift-diffusion model (zero convective term, 8> 0) has been investigated in [46, 236, 305]. The classical potential flow hydrodynamic equations (8 = 0) are analyzed in, e.g., [111, 158, 161]. In the paper [375] the existence for the one-dimensional stationary quantum hydrodynamic equations (5.11)-(5.13) with non-standard boundary conditions is investigated. In the analysis of (5.16)-(5.18), two main difficulties arise. The elliptic equation (5.17) is, a priori, of degenerate type with a non-standard (since non-local) degeneracy. We will show, however, that the solution w is strictly positive and therefore, (5.17) becomes strictly elliptic. Every solution (w,5,V) of (5.16)(5.18) with positive w is a solution of the problem (5.11)-(5.13) with n = w 2 , J = n\l5. Another difficulty comes from the term t\l 51 2 on the right-hand side of (5.17), stemming from the convective term in (5.12). This difficulty also appears in the thermistor problem (see [95, 368]). However, we have to apply different techniques than used in the thermistor problem (see below). The following assumptions are needed: (HI)

n c IR d (d 2': 1) is a bounded domain with boundary an E C 1 ,1.

(H2) hE CO(O, (0) is a non-decreasing function satisfying lim h(x) = +00,

X--+OO

lim xh(x 2 ) < +00.

x--+O+

(H3) wo E W 2 ,p(n) for p > d12, infan wo > 0; 50 E c 1 ,y(O) with 'Y = 2 - dip; Vo E H 1 (n) n LOO(n); C, Vext E LOO(n). The constants (3, 8, A, and To are assumed to be positive. We call a function h E CO(O,oo) satisfying (H2) isothermal if h(O+) = -00 and isentropic if h(O+) < O. The enthalpy function h( s) = log( s) is isothermal. Furthermore, the enthalpy h(s) = a~1 (sa-1 - 1) for a > 1 is isentropic. The main results of this section are the following theorems: Theorem 5.2.1 Let (Hl)-(H3) hold and let h be isothermal. Then there exists c > 0 such that if

Chapter 5. The Quantum Hydrodynamic Model

198

then there exists a solution (w,S, V) of (5.16)-{5.19) satisfying, for some w > 0, wE W 2 ,P(0),

S E C 1,1'(0),

w(x) 2 w > 0

V E H 1 (0) n L=(O), inO.

(5.21 ) (5.22)

Theorem 5.2.2 Let (H1)-{H3) hold and let h be isentropic. Then there exists E > 0 such that if To 2 liE then there is a solution (w,S, V) of (5.16)-{5.19) satisfying {5. 21)-{5. 22). Notice that we are assuming boundary data which are independent of the parameter To. The case of the boundary functions (5.20) can also be treated, see Remark 5.2.6. We can prove existence of solutions if the electric energy which is connected with the applied potential U (and hence with So) is much smaller than the thermal energy, in some sense. The idea of the proof is to replace the equation (5.17) by div(max(m,w)2\7S) = 0 (m > 0) which is uniformly elliptic. By means of Leray-Schauder's fixed point theorem, the existence of a solution to the truncated problem will be shown. For this solution the density w turns out to be strictly positive. So we get a solution to the original problem (5.16)(5.19) by choosing the truncation parameter m > 0 smaller than the lower bound of w. We need the smallness assumption on the data in the proof of the positivity of w. We do not know if the existence of solutions can be proved without this assumption. In the stationary thermistor problem which is formally related to the quantum hydrodynamic model, it is well known that there exist solutions only if the applied potential is "small" enough (for the precise conditions see [368]). Furthermore, in the one-dimensional case it is possible to show the nonexistence of solutions for "large" applied voltages [94, 95]. We recall that the thermistor problem reads div (k(w)\7w)

div(o-(w)\7S) where wand S have here the meaning of the temperature and electrostatic potential, respectively. In the simulation of semiconductor tunneling devices where a variant of the presented quantum fluid model has been used, numerical results indicate that the density can be extremely small compared to, e.g., the boundary density, for values of the applied voltage U far from thermal equilibrium (e.g. nmin = 10- 4 , nlan = 1; see [163]). It is not clear if there is a lower bound for the density for all U and if yes, how it can be controlled. The positivity property

199

5.2. Existence and positivity

of W is connected to the regularity for 8. Indeed, we show that W is strictly positive if and only if the gradient of 8 is bounded (see Section 5.2.2). For ultra-small devices, the equations (5.16)-(5.19) can be replaced asymptotically by a simplified system (see Section 5.2.2). We show that there exists a solution of this (one-dimensional) system, where the density vanishes at some point. However, the solution is discontinuous and therefore, it is only defined in a generalized sense. Now we prove that there exists a solution of a truncated system. For this, define SK = max (0, min (s, K)) and tm(s) = max (m, s) for s E lR and 0 < m ~ K. Throughout this section (Hl)-(H3) are assumed to hold. Consider

wK(!1V7812 + Toh(wiJ- V - Vext + (38), (5.23)

82 b.w div(t m (wK)2V78) 2

oX b. V

=

0,

=

WKW -

8

W = Wo,

(5.24) =

C

80 ,

in n, V = Vo

on

(5.25) (5.26)

an.

The proof of existence of solutions to this truncated system is based on the following a priori estimates. Lemma 5.2.3 Let (w, 8, V) E (H I (n))3 be a weak solution to (5.23)-(5.26). Then there exist positive constants w, 5-, 5, V, V, and cI(m) such that

o ~ w(x) ~ w, -5- ~ 8(x) ~ 8, IlwI12,p,0

-V

~

V(x)

~

V

~ cI(m).

in

n,

(5.27) (5.28)

Recall that 11·112,p,0 denotes the norm of the Sobolev space W 2 ,p(n). The precise dependence of the above bounds on the data is needed in the uniqueness proof in Section 5.3 and is stated here for future reference:

5-

118011 0,00,ao, 5 = 1180110,00,ao, IIVo Ilo,oo,ao + c(n, d, oX) IICllo,oo,o, max (Ilwollo,oo,ao, WI (V,5-, To, h)), IIVollo,oo,ao + c(n, d, oX) (1ICllo,oo,o + w 2 ),

V

where c(n,d,oX) > 0 and IIVext

110,00,0 + (35-)/To·

WI

(5.29) (5.30) (5.31) (5.32)

= wI(V,5-,To,h) > 0 is such that h(wr) 2: (V +

Proof. First step: L oo estimates for w, 8, and V. First observe that, using w- = min (0, w) E HJ(n) as test function in (5.23), it follows

w(x) 2: 0 a.e. in

n.

(5.33)

The maximum principle gives the bounds

-5- ~ inf 8 0 ao

~ 8(x) ~ sup 8 0 ~

ao

5

in

n.

(5.34)

200

Chapter 5. The Quantum Hydrodynamic Model

Next we show that V is uniformly bounded in Loo(O). Let Uo = 11V0110,00,an, U 2 Uo, and take (V - U)+ = max (0, V - U) as test function in (5.25). Then >.2

in

1\7(V - U)+1 2dx

in in -

<

+

WKW(V - U)+dx

in

C(V - U)+dx

C(V - U)+ dx

(5.35)

< IICllo,oo,nll(V - U)+1h.2,n(meas(V > U))1/2. Let r > 2 be such that the embedding H1(0) follows for all W > U,

'----t

U(O) is continuous. Then it

> II(V - U)+llo,r,n 2

r

>

r

J{V>W}

(V - Ufdx

(W - Ufdx

J{V>W}

(W - Ufmeas (V> W). Therefore we get from (5.35), for W > U 2 Uo, K 2 meas(V > W):S (W _ uy(meas(V > U)r/ ,

where K = co(0)>.- 2r IICllo 00 n. Since r/2 > 1, we can apply Stampacchia's Lemma (see [347, Lemma 2:9j'or [340, Ch. 4]): Lemma 5.2.4 Let ¢ : (a, b)

where a < b :S that

00.

- t JR be a nonnegative, nonincreasing function, Suppose that there exist constants K > 0, r > 0, a> 1 such

¢(w) :S K(w - u)-r¢(u)a

for a < u < W < b.

If the number u* = K 1/r2 a /(a-l)¢(a)(a-l)/r is such that a ¢(a+u*)=O.

+ u* <

Hence, we obtain for ¢(W) = meas(V > W): ¢(Uo + U*) c(0,r)>.-21ICllo,00,n. This implies

=

-

def

V(x) :S V = Uo + cdCllo,oo,n,

°

b then

with U*

=

(5.36)

where Cl = Cl (0, r, >.) > 0. Before we can find a lower bound for V, we prove that W is bounded from above (independently of K). For this set V ext = IlVext Ilo,oo,n, let W 2

5.2. Existence and positivity

201

Ilwollo,oo,an and K> wand use (w - w)+ as test function in (5.23):

f?

in

1\7(w - w)+1 2dx

-~

=

in

-in + in in

2 WK(W - w)+I\7SI dx

WK(W - w)+To(h(w 2) - h(uP))dx WK(W - w)+(V + Vext - T oh(w 2) - f3S)dx

WK(W - w)+(V + V ext - T oh(w 2) + f3SJdx,

<

using (H2), (5.34) and (5.36). Since h(s) ---t 00 as s ---t 00, there exists Ilwollo,oo,an such that h(w 2) :2 (V + V ext + f3fD/To. This implies

w(x) :S

a.e. in

W

n.

W

:2

(5.37)

Now use (-V - U)+ with U :2 Uo = 11V0 Ilo,oo,an as test function in (5.25) to get >,2

in

in

1\7(-V - U)+1 2dx:S

(WKW - C)(-V - U)+dx:S c

in

(-V - U)+dx,

where c > 0 depends on C and w. Using Stampacchia's method as above allows to conclude that

V(x) :2 - V ~f -Uo - c(n, d, >')(IICllo,oo,n

+ w 2).

Second step: HI estimate for w. Use W - Wo as test function in (5.23) to obtain

82

in

I\7W[2dx

=

in + ~ in + in in

82

\7w· \7wodx -

in

2 wKwl\7 SI dx

wK wol\7S[2dx - To

wKwVdx-

- (3

~

in

in

(5.38)

WK(W - wo)h(wkJdx

wKwoVdx+

in

wK(w-wo)Vextdx

WK(W - wo)Sdx.

With the test functions V - Vo and S - So in (5.25), (5.24) respectively, we get on the one hand

in

WKWVdx

=

_>,2

+

in

in

2 I\7V1 dx

+ >,2

C(V - Vo)dx

in

\7V . \7Vodx

+

in

VowKwdx

202

Chapter 5. The Quantum Hydrodynamic Model

using Young's and Poincare's inequalities; on the other hand, we have for K > w,

min

2 wKIV'S1 dx

<

in t in

< w2

m

2 (WK)21V'SI dx:S

in t

m

(WK)21V'SoI2dx

IV' SoI2dx.

Therefore we can estimate (5.38) as follows:

Third step: W 2 ,p estimate for w. The following elliptic estimate holds [180, Thm. 8.33 and 8.34]: for all 0 < c :S 'Y,

(5.39)

where C2 > 0 depends on n, d, m, and the Co,c (n) norm of t m ( WK)2. It can be seen from the proof of this estimate that

Furthermore, we have the elliptic estimate

021IwI12,p,n :S c5(ll woI12,p,n + Ilw(!IV'SI 2+ h(w'k) where C5 c = 'Y/2,

> 0 depends on nand

021IwI12,p,n <

V - Vext

+ /3S)llo,p,n),

d [180, 9.15 and 9.17]. Hence, using (5.39) for

+ IISIIi,2p,n) :S c(l + IISII~1''Y/2(TI») < c(l + Ilw211~o''Y/2(TI») :S c(l + w2I1wll~o''Y/2(TI») < c(l + Ilwllco''Y(TI») c(l

02

< 21IwI12,p,n+c(0,m),

5.2. Existence and positivity

203

In the last step we have used the interpolation inequality

Ilwllco,-r(Q) :S cllwI12,p,n + c(c)llwllo,oo,n, which follows from the facts that the embedding W 2,P(0) '---7 CO''Y(O) is compact (since p > d/2) and the embedding CO,'Y(O) '---7 LOO(O) is continuous [373, p. 365]. The constant c(8, m) in the above estimate depends on 0, d, 8, m, w, and V. We obtain finally

IlwI12,p,n :S 2c(8, m)/82= cI(m). o Lemma 5.2.5 There exists a solution (w, 3, V) of

82f:j.w div (t m (w)2\73) A2f:j.V

w( ~ 1\7 31 2 + Toh(w 2) - V - Vext

+ (33),

0,

(5.40) (5.41)

2

w - C in 0, (5.42) w = wo, 3 = 30, V = Vo on ao, (5.43) such that w E W 2,P(0), 3 E C1,'Y(O), V E H1(0) n LOO(O), and w(x) 2: in O.

°

Proof. We use a fixed point argument. Let u E CO''Y(O). Let V E H1(0) be the unique solution of A2f:j.V=UKU-C

in 0,

V

=

Vo on

ao,

and let 3 E H1(0) be the unique solution of

div(t m (uK)2\73)

=

°

3 = 30

in 0,

on

ao.

As in the proof of Lemma 5.2.3, we see that V E LOO(O). Since t m (UK)2 is Holder continuous of order "f, we get 3 E C1,'Y(0) [180, Thm. 8.34]. Finally, let w E HI (0) be the unique solution of

82f:j.w

= aUK(~1\7312

+ Toh(u'id -

w = awo

on

V - Vext

ao,

+ (33)

in 0,

with a E [0,1]. The right-hand side of this elliptic problem lying in LOO(O), we conclude w E W 2,P(0) and, since p > d/2, w E CO,'Y(O). Thus the fixed point operator T : CO''Y(O) x [0,1] --t CO,'Y(O), (u,a) f---+ w, is well defined. It holds T(u, 0) = for u E Co,'Y(O). Estimates similarly as in the proof of Lemma 5.2.3 give the bound

°

Il wI12,p,n :S c

204

Chapter 5. The Quantum Hydrodynamic Model

for all w E CO''''((O) satisfying T(w,(T) = w, where c > 0 is independent of w and (T. Standard arguments show that T is continuous and compact, noting the compactness of the embedding W 2 ,P(O) ~ Co''''((O). We can apply LeraySchauder's fixed point theorem to get a solution (w, S, V) of (5.23)-(5.26). Choosing K > w (see (5.31)), this tripel is also a solution of (5.40)-(5.43). 0 Proof of Theorems 5.2.1 and 5.2.2. We rewrite the elliptic estimate (5.39) for E:

= 1:

IISllc1'"YCl1) :S C3(O, d)c4(m) Iltm(w)21Ico'"YCl1) IISo Ilc1'"YCl1)'

(5.44)

It holds c4(m) ---7 00 as m ---7 0+. Now,

Iltm(W)21Ico,"Ycl1) :S c(w)llwllco,"Ycl1) :S c(w)llwI12,p,n :S Cs· From the proof of Lemma 5.2.3 it can be seen that Cs = ct;(w)c7(m) with C6 (w) ---7 00 as w ---7 00 and C7 (m) ---7 00 as m ---7 O+. The bound w depends on To such that ill ---7 00 as To ---7 0+ (see (5.31)). Thus we can write

(5.45) where f and 9 are positive non-decreasing continuous functions in [0,(0) such that f(To) ---70 as To ---70+, f(To) > 0 as To ---700, and g(m) ---70 as m ---7 0+. The constant Co > 0 does not depend on So, To, or m. Let 0 < m < infanwo and take (w-m)- = min(O,w-m) as test functions in (5.40). Then, using (H2), (5.45), and (5.27),

82ll\7(w - m)-1 2dx

=

- l w(w - m)-To(h(w 2) - h(m 2))dx - l w(w - m)- (~I\7SI2 + T oh(m2) - V - Vext

+ /3S )dx

< l w( -(w - m)-) (f(To~~(m)IISoll~l'"YCl1)

+ T oh(m2) + V -

+ V ext + /3S)dx, def

-

-

where V ext = IlVextllo,oo,n. The constant cs(To) = V + V ext + /3S depends on To through V such that cs(To) can be taken to be non-increasing as To increases (see (5.31)-(5.32)). Then

2 82ll\7(w - m)-1 dx :S

(h +

gT;)

h)

l

w( -(w - m)-)dx,

(5.46)

205

5.2. Existence and positivity where 1 2 ) "2?oh( m ) + Cs (To ,

CO 2 Tof(T ) IISollcl,'Y(O) o

1

+ "2 g (m)h(m

2

).

First case: Let h be isothermal. For arbitrary To > 0, let w E (0, infa!1 wo) be such that h(w 2) ::; -2cs(To)/To (using (H2)). This implies, for m = w, that h ::; O. Set A = -!g(w)h(w 2) > 0 and £2 = ATof(To)/CO. Then, for m = w and IISollcl''Y(n) ::; £, we obtain

Iz ::;

Co

Tof(T ) £ o

2

-

A ::; O.

Taking into account (5.46) we conclude that w 2': w in n. For arbitrary So, take m = J,Q E (0, infa!1 wo) such that h(w 2 ) ::; -2cs(1) and let A be defined as above. Choose T 1 2': 1 such that TI!(T1 ) 2: coIISoll~l''Y(n/A. Then we have for all To 2': T 1 , since h(w 2 ) < 0,

and hence h ::; O. Since the function T

I---t

T f(T) is increasing, we obtain

by definition of T 1 . This implies Iz ::; 0 and w 2': J,Q in n. Second case: Let h be isentropic. Let wE (0, infa!1 wo) be such that h(w 2) < 0, and let T 2 2': 1 be such that T z 2: -2cs(1)/h(w2) > 0 and T 2 f(T2 ) 2': coIISoll~l''Y(n/A, where A is defined as in the first case. Taking m = wand To 2': T 2, we get h ::; 0 and Iz ::; o. We conclude the proof by taking the truncation parameter m = w in (5.41).

o

Remark 5.2.6 We have assumed that the boundary functions wo, So, and Vo do not depend on the parameters, e.g. To. However, if we take Vo = Toh(C) + U(x) (see (5.20)), the above arguments also apply. Indeed, let Co > 0 be such that h(Co ) = 0 and choose a scaling of the variables and functions such that infa!1 C 2': Co (this does not affect To). Then, for isothermal or isentropic functions, h(infa!1 C) 2': O. This implies V = -To infa!1 h(C) + U ::; u, and the constant cs(To ) can be taken non-increasing as To increases. Note that now V also depends on To, but in such a way that the property w ~ 00 as To ~ 0+ remains valid.

206

Chapter 5. The Quantum Hydrodynamic Model

Remark 5.2.7 Using a relaxation scaling as in [268], Le. defining the rescaled variables n = n, S = (3B = B/7, V = V, where 7 = 1/{3 is the scaled relaxation time, we get from (5.16)-(5.17) the equations 82~W

div(w2 \7S)

7

2

'2

w( "2 1\7BI

' + Toh(w 2 )- V -

~xt

, + B),

O.

One may expect that the diffusive term Toh(w 2 ) dominates the convective term (7 2 /2) 1\7SI 2 for sufficiently small 7 > 0, which would give the existence of solutions by the presented method, for fixed To. However, we also have to transform the boundary function So = BO/7 = U/7, and it is easy to see that then the convective term is not necessarily "small" for small relaxation times. Choosing different boundary conditions, namely Bo = {3U, the above rescaling gives So = U, and the estimates of the presented proofs lead to an existence result for sufficiently small 7 > O. Remark 5.2.8 It would be very interesting to study the small dispersion (or semi-classical) limit 8 --t 0 and the relaxation time limit 7 --t O. However, the W 2 ,p(n) norm of wand therefore, the lower bound w depend on 8 such that w --t 0 as 8 --t O. Moreover, it seems difficult to identify the limits of the nonlinear functions. Concerning the relaxation time limit, it can be seen that cs(To) --t 00 as 7 --t 0 (see the proof of Theorems 5.2.1 and 5.2.2), and hence, w --t 0 as 7 --t O. Taking the boundary conditions discussed in Remark 5.2.7, we expect, however, that the limit 7 --t 0 can be performed. The relaxation time limit 7 --t 0 in the hydrodynamic equations (i.e. 8 = 0 in (5.12)) is performed in [268]. In Section 5.5 the (semi-)classicallimit 8 --t 0 is studied (also see [160]). 5.2.2

Positivity and non-positivity properties

We show that the existence of a uniform lower bound for the density w is related to the regularity of the gradient of B. Furthermore, we construct a generalized one-dimensional solution of a simplified problem, where the density w vanishes at some point. For this solution, the quantum Fermi potential B is discontinuous. Let the hypotheses (H1)-(H3) of Section 5.2.1 hold and let h be isothermal or isentropic. Proposition 5.2.9 Let (w, B, V) E (H1(n) n V Xl (n))3 be a weak solution to (5.16)-(5.19) with B E W1,OO(n). Then there exists m > 0 such that w(x)2:m>O

inn.

5.2. Existence and positivity

207

Proof. First let h be isentropic. Then the function

is bounded in O. Since w 2 0, we can apply Harnack's inequality [180, p. 199] to 82 b.w = wf to conclude that for all subsets wee 0 supw w

:s c(w) inf w. w

(5.47)

Now suppose that w vanishes in some non-empty set Wo CC O. Let Wn cc 0 be a sequence of sets with Wo C Wn and Wn -+ 0 as n -+ 00 in the set theoretic sense. Then (5.47) gives w = 0 in W n and, in the limit n -+ 00, W = 0 in O. This contradicts the positivity of Wo on ao. If h is isothermal, we proceed as in [46]. Consider Wo = {w = O} cO. Since wf E Loo(O), w is continuous, hence Wo is relatively closed in O. Suppose that Wo is nonvoid and choose Xo E Woo Then wf :S 0 in a ball B(xo) C 0 with center Xo and b.w :S 0 in B(xo). As the function w assumes its nonnegative infimum 0 in B(xo), it follows that w = 0 in B(xo). Thus Wo is relatively open in O. This implies Wo = 0 or Wo = 0. By the positivity of wo, we conclude that w > 0 in O. The existence of a uniform lower bound m > 0 for w now follows from the continuity of w in 0

n.

Corollary 5.2.10 Let (w, 8, V) be a weak solution to (5. 16}-(5.19}. Then

w(x) 2 m > 0

a.e. in 0

if and only if 8

E W1,00(0).

Now we consider the following simplified system in 0

82w xx = !W(8x )2 2

Jx = (w 8 x )x = 0

= (0,1)

C

R

in 0,

w(O) = 1, w(l) = 1,

(5.48)

in 0,

8(0) = 0, 8(1) = Uo,

(5.49)

It can be seen that the equations (5.16)-(5.17) reduce to (5.48)-(5.49) for very small domains (after an appropriate asymptotic limit; see Section 5.6.1). We only consider Uo E [0, V281l']. To solve (5.48)-(5.49) we have to distinguish the cases Uo < V281f and Uo = V281f. We say that (w,8) E H1(0) x Loo(O) is a generalized solution to (5.48)(5.49) with 8(1) = Uo if there exists a sequence of weak solutions (we, 8 10 ) E (H 1 (0))2 of (5.48)-(5.49) with 8(1) = Ue and Ue -+ Uo as c -+ 0 such that w

= 10---+0 lim We,

8

= 10---+0 lim 8 10

in the L 2 (0) sense

208

Chapter 5. The Quantum Hydrodynamic Model

and for all ¢ E HJ (0) it holds lim 82

e--->O

lim

r(wE;}x¢x dx

}n

r W;(Se)x¢x dx

o.

e--->o}n

Proposition 5.2.11 (i) Let 0::; Uo < V287r. Then there exists a smooth solution (w,S) E (C2(0))2 of (5.48)-(5.49) such that

w(x) 2: c(Uo) > 0

in O.

(ii) If Uo = V287r then there exists a generalized solution (w, S) Loo(O) of (5.48)-(5.49) such that w(!) = o.

E

Proof. Let Uo = V287r and let Ue < V287r be a sequence such that Ue c: --> O. Set a e = Ue /V28. A computation shows that

((1- 2x)2

-->

Uo as

+ 2(1 + cosae)x(1 _ X))1/2,

~f:

v2uarccos solve (5.48)-(5.49) with Se(1)

HI (0) x

1 - (1 - cosae)x

We

( ) X

,

x

E

[0,1],

= Ue (see Fig. 5.1). Furthermore,

w;(x)(Se)x(X)

=

V28 sinae

(5.50)

and in O. In the limit c:

-->

0 we get cos a e

we(x)

-->

w(x)

Se(x)

-->

=

-->

-1 and

11 - 2xl

V28 H(x)

in HI(O), in L2 (0)

(c:

-->

0),

where H(x) = 0 for x E (0,1/2) and H(x) = 7r for x E (1/2,1). Taking into account (5.50) we obtain for all ¢ E HJ(O)

(c:

-->

Therefore, (w, S) is a generalized solution to (5.48)-(5.49).

0). D

209

5.3. Uniqueness of steady-state solutions w(x) versus x

,. ,,

U = 1.5 U =2.3 U =2.9

,. ,. ,, ,. , ./0/

0.6 0.4

,

,

,,

,

,,

,

,,'

, ,, ,,

0.2 O'-------~--~~-==---~--------'------'

o

0.2

0.4

0.6

0.8

1

Figure 5.1. The function we(x) versus x for various (scaled) applied potentials U = Uo (8 = 1/V2).

5.3

Uniqueness of steady-state solutions

Uniqueness of solutions follows under the assumption that the scaled Planck constant 8 is large enough. If 8 = 0, there exists more than one solution of the thermal equilibrium state (i.e. J = 0; see Theorem 5.3.3). Theorem 5.3.1 Let (H1)-(H3) of Section 5.2.1 hold and let h be isothermal or

isentropic. Then there exists 80 > 0 such that if 8 2: 80 , there exists at most one solution (w, 8, V) to (5.16)-(5.19) satisfying {5. 21)-(5. 22).

Proof. Let (WI, 8 1 , VI) and (W2' 8 2 , V2) be two solutions to (5.16)-(5.19) satisfying (5.21)-(5.22). Take WI - W2 as test function in the difference of the equations (5.16) satisfied by WI, W2, respectively, to get

-~ 2

inr(WI 1\781

2 - w21\78 12)(Wl - w2)dx 2

1

+ in(W1V1-W2V2)(WI-W2)dX

-in - f3 in + in

TO(w1h(wi) -

w2h(w~))(Wl -

(w 18 1 - W2 8 2)(Wl - w2)dx

Vext(Wl - W2)2dx

h+···+I5 ·

(5.51) w2)dx

Chapter 5. The Quantum Hydrodynamic Model

210

The weak formulation of the difference of (5.17) for 8 1 , 8 2 , respectively, reads

LW~\7(81

- 8 2 ), \7¢dx = -

for all ¢ E HJ(O). Taking ¢ = 8 1

W2

L

1\7(81 - 82)1 2dx

<

-

L(w~

-

W~)\782' \7¢dx

8 2 we obtain

Lw~I\7(81 -l (w~ W~)\782'

- 82Wdx

-

< 2wll wl

-

\7(81 - 8 2)dx

w2110,211\782110,0011\7(81 - 82)110,2,

which implies

Now we are able to estimate h, ... , h:

using (5.52). The integral h is estimated by using (5.18):

The monotonicity of h implies

h

-To

l (wl(h(w~)

-

h(W~))(Wl -

W2)

+ (WI - W2)2h(w~))dx

< -Toh(w 2)ll w l - w2115,2' Finally, we can estimate the integral I 4 employing (5.52): I4

-

f3

l

(wl(8 1 - 8 2)(Wl - W2)

+ (WI

< f3(C(O)(wjW)2 11\782110,00 + S) Il wl

-

- W2)282)dx w2116,2'

5.3. Uniqueness of steady-state solutions

211

Let K = 11\78I11o,= + 11\782 110,= and Vm = V (5.51) and Poincare's inequality

+ V ext '

Then we get from

2- 2K (:2 + 1) - V + T oh(w - ,e(c(O) :2K + S) }llwl - w211~,2 :::; O. 2

{c(0)8

-2

m

2

)

-2

(5.53)

Only K depends on 8 (via the W 2 ,P(0) norm of w; see the third step of the proof of Lemma 5.2.3) such that K remains bounded as 8 ~ 00. Therefore there exists 80 > 0 such that if 8 ~ 80 then (5.53) implies

Ilwl Hence WI (5.18).

w211~,2

:::; O.

= W2 in O. Finally, we infer 8 1 = 8 2 from

(5.52) and VI

=

V2 from 0

Remark 5.3.2 There exists at most one weak solution (w, 8, V) in the class of functions satisfying w, V E Hl(O) n L=(O), w(x) ~ m > 0 in 0, and 8 E w1,q(0), where q = d if d ~ 3, q > 2 if d = 2 and q = 2 if d = 1, under the assumption that the scaled Planck constant 8 > 0 is large enough. The proof of this result is similar to the proof in [198]. If 8 = 0 then we can have non-uniqueness of solutions. More precisely, there are doping profiles such that the classical thermal equilibrium state admits at least two solutions. The thermal equilibrium equations are obtained by setting 8 = 0, i.e. J = 0, in (5.16)-(5.18). The equations are called classical if 8 = O.

Theorem 5.3.3 Let (H1)-(H3) of Section 5.2.1 hold and let hE C 2 ([0, (0)) be strictly increasing. Let Vo E H1(0) n L=(O). Then there exists C E L=(O) such that the classical thermal equilibrium problem >h~ V = w 2

-

C

w(h(w

2

)

in 0, V = Vo on 80, in 0, - V) = 0

(5.54) (5.55)

has at least two solutions.

Note that isentropic enthalpies (see Section 5.2.1) satisfy the assumption of the theorem but not isothermal enthalpies. Proof. Since h is strictly increasing we can define the generalized inverse of h by (t) = {h-1(t) if h(O) < t < +00 9 0 if -00 < t :::; h(O).

212

Chapter 5. The Quantum Hydrodynamic Model

Then every solution (W, V) of

g(V) - C

)h~V =

in 0,

V

w = g(V)

in 0,

2

h(w

2

) -

V {

~~

=

Vo on 00,

(5.56)

°

(5.57)

ifw > if W = 0,

(5.58)

also solves (5.54)-(5.55). The semilinear problem (5.56) with monotone righthand side has a unique solution V E H 1 (0) n LOO(O). Defining W E LOO by (5.57) we see that (w, V) is a solution of (5.56)-(5.58). Let 00, 0 1 C 0 be such that 0 1 CC 00 Cc O. Let WI E COO(O) be such that WI = in 00 and wI = g(Vo) on 00. Then define VI = h(wI) in 0\0 0 , VI = h(O) in 0 0\0 1 , and VI > h(O) in 0 1 such that VI E C 2(O). Then C 1 = wI - ~Vl E LOO(O). By construction, (WI, Vd is a solution of (5.54)(5.55) with C = C 1 . But by definition of VI, it holds h(wI) - VI < and WI = in 0 1 . Therefore, (WI, VI) is a second solution of (5.54)-(5.55). 0

°

°

°

5.4

A non-existence result for the steady-state problem

For a special set of boundary conditions we can prove the non-existence of generalized solutions to the one-dimensional stationary quantum hydrodynamic model when the data is large enough. The stationary equations read (see (5.11)-(5.13)):

(j2n + Tp(n))

x

- nVx - {Pn ((vj:x ) n

J

- T(X)'

x

n - C(x)

).2Vxx

(5.59) (5.60)

in 0 = (0,1). Notice that in one space dimension, the current density J is constant. We choose the following boundary conditions:

n(O) = no,

n(l) = nl,

V(O) = Vo,

Vx(O) = -Eo,

= ~ +Th( ) - V; +K 82 (In)xx(O) ~ 2no2 no ynO

°

.

°

(5.61) (5.62)

Here, h(s) is the enthalpy function defined in (5.15), and K > is an arbitrary constant. The condition (5.62) can be interpreted as a boundary condition for the quantum Fermi potential (or quantum velocity potential). We impose the following assumptions: (AI) h E C 1 (0,00) and p' (defined by p'(s) = h'(s)js, s > 0) are nondecreasing, and h satisfies lim h(s) 8-->00

> 0,

lim h(s) 8-->0+

< 0,

lim VSh(s) 8-->0+

> -00.

(5.63)

5.4. A non-existence result

213

(A2) C E £2(0), C ~ 0 in 0; (A3) J,wo,wl,8,)..,T

> 0; v

E V)O(O),

T

T(X)

~ TO

> 0 in O.

0; Vo,Eo E R

~

Typical examples for hare h(s) = log(s), s > 0, or h(s) = with a > 1. Our main result is the following theorem.

S",,-1 -

1, s ~ 0,

Theorem 5.4.1 Let (Al)-(A3) hold. (i) There exist J o > 0 and K E lR such that for all J :S Jo there exists a solution (n, V) E (C3 (0))2 to (5.59)-(5.62) with strictly positive n. (ii) Let in addition (5.67) hold (see below). Then there exists J 1 > 0 such that for all J ~ J 1 , the problem (5.59)-(5.62) cannot have a generalized solution.

We have to specify our notion of generalized solution to (5.59)-(5.60). Suppose that n is a positive smooth solution. Then, for a test function cP E Coo(O) in (5.59), we obtain

-l(~2 +TP(n))cPxdx-lnvxcPdx+82l(ncP)x(~XXdx=-l~cPdX. The last integral on the left-hand side equals

Thus we say that (n, V) is a generalized solution to (5.59)-(5.62), if V E W 1 ,OO(O), n ~ 0 in 0, p(n), lin E L 1 (0), Vii E H 2(O), the boundary conditions (5.61)-(5.62) and the equations [

J2

- in (~+TP(n)-8\/n(vn)xx)cPxdx =l

2

(nVx - 28 ( vn)x( vn)xx -

~)cPdX,

)..2l VxcPx dx = l (C - n)cPdx

are satisfied for all smooth test functions. It follows that every generalized solution to (5.59)-(5.62) is strictly positive. Indeed, suppose to the contrary that there exists Xo E 0 such that n(xo) = O. Then, since Vii E W 1,OO(0),

1 1

yn(x) = (x - xo)

(yn)x(sx

+ (1- s)xo)ds,

x E O.

214

Chapter 5. The Quantum Hydrodynamic Model

But

in IJ~)12 2 in (in I(

vn)xl ds ) -2 1x

-

2 xol- dx =

00,

which contradicts lin E L 1 (0). Thus n > 0 in O. We do not prove Part (i) of Theorem 5.4.1 since there is already a similar result for the multi-dimensional case (see Section 5.2.1). A detailled proof can be found in [159, Ch. 3]. Here we concentrate on the non-existence result. The idea of Part (ii) is to reformulate Eqs. (5.59)-(5.62) as an equivalent system of second-order equations, similar to Section 5.2.1, and to prove nonexistence of solutions for this problem. The second-order problem is obtained by dividing Eq. (5.59) by n > 0, integrating from 0 to x, using (5.62), and setting W = Vii:

p

-23+Twh(w2)-Vw+Kw+Jw

w

w2

lx~ -2' 0

TW

C(x),

-

(5.64) (5.65)

subject to the boundary conditions w(O)

= wo,

w(l)

= WI,

V(O)

= Yo,

Vx(O)

= -Eo,

(5.66)

vno

where Wo = and WI = .Jnl. Notice that every solution (w, V) of (5.64)(5.66) satisfying w > 0 in 0 gives a solution (n, V) to (5.59)-(5.60) subject to the boundary conditions (5.61)-(5.62). First we prove some auxiliary result for the solutions w to (5.64). Lemma 5.4.2 Let (w, V) E (Hl(O)? be a weak solution to (5.64)-(5.66) with w- 3 E L 1 (0). Then there exists m > 0 such that w(x) 2 m > 0 for all x E O.

Proof. Since 1/w 3 E L 1 (0), it holds W xx E L 1 (0), which implies w E W 2,1(0) '---t W 1,OO(0). Suppose that there exists Xo EO such that w(xo) = O. Then

1 1 Iwt:)1 3 21 (1 1

w(x) = (x - Xo)

and

1

1

wx(sx + (1 - S)Xo) ds, 1

Iwx(s)1 dS) -31x

x E 0,

- xoj-3 dx =

00 ,

contradicting the integrability of 1I w 3 . The assertion follows from the continuity of w. 0 Next, we show that w is bounded from above independently of J > O.

215

5.4. A non-existence result

Lemma 5.4.3 Let (w, V) be any weak solution to (5.64)-(5.66) with K E JR. and J> 0. Furthermore, let

h(s) ~ cO(sCt-l - 1) for s ~ 0, with a> 2, CO > 0.

(5.67)

Then there exists M o > 0, independent of J, such that w(x) :S M o for all x E O. The bound M o does not depend on K if K > 0. Proof. Take (w - J.L)+ with J.L

~

max(wO,wl) as test function in (5.64) to get

The main difficulty is to estimate the last integral. Integrating the Poisson equation gives

1

Vo -Eox+.x.- 2

V(x)

x

lY(w(z)2-C(Z))dZdY

< vo+max(-Eo,0)+.x.- 2 k w 2dx. Since

we get

and, setting V1 ~f Vo + max(-Eo,O),

k(W-J.L)+WVdX

:S

k(w-J.L)+w(V1 +J.L 2.x.- 2)dX

+ 2.x.- 2 < V1

k

(k

(w - J.L)+wdx

(w - J.L)+w dx

+.x. -2

f k

+2.x.- 2 k(w-J.L)+2W2dX

< V1

k

(w - J.L)+wdx

+ 3.x.- 2

(w - J.L)+w 3 dx

k

(w - J.L)+w 3 dx,

216

Chapter 5. The Quantum Hydrodynamic Model

since J.L ~ w on {x : w(x) 2: J.L} and (w - J.L)+ ~ w. Therefore we get from (5.68), using (5.67), 82

k

J.L)~2 dx S

(w -

k

(w - J.L)+w[VI

°

Since 2a - 2 > 2, there exists M o >

coM5 a -

2

-

+ 3>'-2W2 -

K - cO(W 2a - 2 - 1)] dx.

such that

3>' -2 M~ - VI

which implies, after taking J.L = M o, that w

+K

~

- Co 2: 0,

(5.69)

M o in O.

0

°

Proposition 5.4.4 Let (A1)-(A3) and (5.67) hold and let K E R Then there exists J1 > such that if J 2: J1 then the problem (5.64)~(5.66) cannot have a weak solution with w- 3 E L 1 (0).

Remark 5.4.5 The constant J1 >

2~3 = 882(max(wo, WI) + 1) o

° >°

and M o >

°

is defined by

Tho

+ M o(lVol + IEol + >.-2 M~ -

min(O, K)), (5.70)

is defined by (5.69).

°

Proof. Suppose that there exists a weak solution (w, V) to (5.64)-(5.66) for for some K E R Since w E H 1 (0) '----t CO(O) and w > in 0, all J by Proposition 5.4.2, we get W xx E CO(O). Thus w is a classical solution. Set ho = inf{sh(s2) : < s ~ M o} > -00 (see (5.63)). Let J 2: J1 , where J1 is defined in (5.70). Then 82w xx

°

> >

J2 M3

2

0

+ Tho -

w(sup V - min(O, K)) fl

J?3 + Tho - M o(lVol 2M 0 88 2(max(wo,wI) + 1).

Introduce p(x) = 4(max(wo, wI)

+ 1)(x -

p(O) = p(l) = max(wo, WI)'

+ IEol + >.-2 M~ -

min(O, K))

!? - 1, x E [0,1]. Then

Pxx(x) = 8(max(wo, wI)

+ 1),

for x E 0, which implies w-p~O

on

aO,

(w-p)xx2:0 in O.

°

The maximum principle gives w - p ~ in O. In particular, w(!) ~ p(!) = -1 < 0, which contradicts the positivity of w. 0 Proof of Theorem 5.4.1, Part (ii). The theorem follows immediately from Proposition 5.4.4 and the equivalence of the problems (5.59)~(5.62) and (5.64)0 (5.66).

5.5. The classical limit

5.5

217

The classical limit

We perform the classical limit in the situations: thermal equilibrium state in several space dimensions (Section 5.5.1) and 'subsonic' steady state in one space dimension (Section 5.5.2). In the 'transonic' steady state we cannot expect that the classical limit can be performed in a strong sense since fast oscillations may occur. This behavior is confirmed and illustrated by numerical experiments (Section 5.5.3).

5.5.1

The classical limit of the thermal equilibrium state

The equations of the thermal equilibrium states are obtained by setting 8 = 0 in the stationary quantum hydrodynamic equations:

82 t1w = w(h(w 2 ) - V) A2 t1 V = w 2 - C

in 0, in 0,

w =Wo on 00, V = Vo on 00.

(5.71) (5.72)

The existence of a solution (w, V) E (HI(O) n LOO(O)? follows from Theorem 5.2.1 by setting 80 = O. For 8 > 0 fixed, let (W6, V6) be a solution of (5.71)-(5.72). In this subsection we will show that, as 8 tends to zero, a subsequence of (W6, V6) converges in L 2 (0) x HI(O) to a solution of the classical thermal equilibrium equations (5.54)-(5.55). To avoid cumbersome assumptions we only consider special, physically relevant enthalpies h and physically motivated boundary conditions (see Section 5.1): (H4) hl(x)

= In(x)

(H5) Wo =

ve, Vo =

or ha(x)

=

a~1 (x a - I - 1)

(a> 1), x > O.

h(C) on 00 and C E HI(O), C(x) 2: C> 0 in O.

For simplicity, we have set To = 1. The enthalpy hI is isothermal, h a is isentropic (see Section 5.2.1 for the definition). Note that (H5) implies wolan, Volan E H I / 2(aO). Thus we can extend wo, Vo to 0 such that Wo = Vo E HI(O) (without changing the notation). For the classical limit 8 ----t 0 we need a priori estimates independent of 8.

ve,

Lemma 5.5.1 Let the assumptions (H1)-(H3) of Section 5.2.1 hold and let (w, V) E (HI(O) n L oo (0))2 be a solution of (5.71)-(5.72). Then

81Iwlll,2,n + AIIVIh,2,n + Ilw2h(w2)llo,l,n + Il wI16,2,n :s; c, where c > 0 depends on 0, h, wo, Va, C, and A such that c ----t

00

as A ----t 0+.

Chapter 5. The Quantum Hydrodynamic Model

218

Proof. Use w - Wo and V - Vo as test functions for (5.71), (5.72) respectively to obtain 8

2

L

2

IVwl dx

2

8

=

+

L

2 Vw dx

L

L

2 Vw dx -

L L

_,\2

+

L

2 2 w h(w )dx +

Vw· Vwodx -

L

L

Vwwodx,

2

IVVI dx +,\2

L

VV· VVodx +

2 wh(w )wodx

L

2 Vow dx

C(V - Vo)dx.

Therefore we have

rIVwl dx + < 8; L

2 8

2

2

in

3,\2

4

r

in

IVVj2dx

2 IVwol dx +,\2

+ Ilwollo,oo + In the following c,

L

L

L

2 IVVol dx -

L

2 Iwh(w )ldx + IlVollo,oo

C(V - Vo)dx -

L

2 2 w h(w )dx

L

2 w dx

Vwwodx.

denote positive constants independent of 8 with different values at different occurences. Observing that Ci

and setting k = max(llwollo,oo, IlVollo,oo

+ Cl)

we get

Taking into account the condition (H2) it can be shown by elementary calculations that there exists M o > 0 such that for all x E IR the inequality

holds. Thus the assertion follows.

o

219

5.5. The classical limit

The above estimate is not sufficient to perform the limit 8 ---t O. We need a gradient estimate such that compactness results can be employed. The following lemma provides this estimate which rests heavily on assumption (H5). (Indeed, this assumption avoids the formation of boundary layers as 8 ---t 0.) Lemma 5.5.2 Let the assumptions (H1)-(H3) of Section 5.2.1 and (H4)-(H5) (see above) hold, let 1 ::; d ::; 3, and let (w o , Vo) be a solution of (5.71)-(5.72). Define H a , a primitive of J2xh~(x2), by

2V2a

H ( )a S

2a -1 s

-

a-l/2

,

s 2 0,

Then the estimate

kl'V H a (wo)1 2dx + 8- 2 k wo(ha(w~) holds, where c

Vo)2dx ::;

c

> 0 is independent of 8 (but depending on >..).

Proof. Due to Hypothesis (H5) we can take ha(w~) - Vo as test function in

(5.71):

-k wo(ha(w~)

- Vo)2dx

{j2k'VW O .

'Vha(w~)dx -

{j2k 'VVo . 'V(wo - wo)dx

- {j2k 'VVo ' 'Vwodx =

{j2k l'V Ha(woWdx

+ 82 >.. -2k (w~ -

C)(w o - wo)dx

- 82k 'VVo . 'Vwodx.

Hence, by Lemma 5.5.1,

k''V Ha(woWdx + 8- 2 k wo(ha(w~) + >..-2k (wo

Vo)2dx

- WO)2(w o + wo)dx

< l/'Vwollo,2 I/'VVo110,2 < c, where c

> 0 is independent of 8.

The convergence result is stated in the following theorem.

o

Chapter 5. The Quantum Hydrodynamic Model

220

Theorem 5.5.3 Let the assumptions of Lemma 5.5.2 hold and let (W8, V8) be a solution of (5.71)-(5.72). Then there exists a subsequence, also denoted by (W8' V8), such that, as 8 ---t 0, we have W8 ---t w in L 2(0), V8 ---t V in HI(O),

where (w, V) is a solution of A2~V = w 2 - C in 0, V = Va on 0= w(h a (w 2) - V) in O.

ao,

(5.73) (5.74)

Proof. We conclude from Lemma 5.5.2 that (extracting a subsequence)

H a (W8)

---t

z in L 5 (0) as 8 ---t 0

for some z E HI(O) holds. Thus, for a subsequence,

W8

---t

W clef = H-I( a Z)

. a.e. In

n

(5.75)

H.

Since (Ha (W8)) is bounded in L (0) uniformly in 8 > 0 we have 6

in w~dx:::;

c

and therefore [259, Lemma 1.3 and p. 144]' for a subsequence,

W8 W8

---->.

w

in L 3 (0),

---t

w

in L 2 (0).

There exists c > 0 such that for w > 0

Iw In w2 1:::; c(w + w 4 / 3) holds. Hence

in IW8hl(W~)13/2dx in (w~/2 + w~)dx :::; c

:::; c,

where c> 0 is independent of 8. For ex > 1 we obtain

in IW8ha(w~Wa/(2a-l)dx

< c(in w~adx +

1)

c( in Iw~ha(w~)ldx + 1) : :; c

by Lemma 5.5.1. Thus, for ex 2: 1, extracting a subsequence, w8ha(w~)

---->.

P in LP(O)

for some p > 1. This weak convergence and the convergence a.e. (5.75) imply as above

5.5. The classical limit

221

The sequence (Vb) is bounded uniformly in H 1 (n) by Lemma 5.5.1. Hence Vb ~ V in H 1 (n) (for a subsequence) and A2~V = w 2 - C in n, V = Va on an. Standard elliptic estimates now imply Vb ----t V in H 1 (n). Let cp be a smooth test function. We can pass to the limit 8 ----t 0 in 2

8

in wb~cpdx in wbha(w~)cpdx in -

=

WbVbcpdx,

o

which shows that (w, V) solves (5.73)-(5.74).

Now we show some LOO bounds for Wb, Vb independent of 8. First we prove that Wb is positive for fixed 8 > O. Proposition 5.5.4 Let the assumptions of Lemma 5.5.2 hold and let (w, V) be a solution of (5.71)-(5.72). Then there exists W > 0 maybe depending on 8 such that w(x) ~ :!Q > 0 for x E n.

Furthermore, if h(x) ----t -00 as x ----t 0+ then we can choose W = min( y'Q, a) where a> 0 is such that h(a2) ::; -llVllo,oo,n. Proof. First let h be isentropic. Suppose that there exists wee n with meas( w) > 0 such that W = 0 in w. Let W n C c n be a sequence of sets with W C W n and W n ----t n in the set theoretic sense. From the Harnack inequality (see, e.g., [180, p. 199])

o::; sup W

::;

c(wn ) inf W = 0 Wn

Wn

we conclude W = 0 in W n , n E N. But this implies w = 0 in n, contradiction. Thus w > in n. Since w E H 2(n) '---+ CO(O) for d ::; 3 and w ~ y'Q > on an this implies the first assertion. By means of the test function (w - w) - = min(O, w - w) where w is as in the statement of the proposition, we get

°

82

in

°

1V'(w - w)-1 2dx

=

in +l -

(h(w 2) - h(w 2))w(w - w)-dx (V - h(w 2))w(w - w)-dx

< 0 since h is nondecreasing. This proves w

~

w in

n.

o

Next we state a result on Loo bounds from above and below independent of 8. As a consequence we get LOO bounds for the solution of the limiting problem (5.73)-(5.74).

222

Chapter 5. The Quantum Hydrodynamic Model

Proposition 5.5.5 Let (H1)-(H5) hold and let (W8' V8) be a solution of (5.71)(5.72). Then there exist constants w, w > 0 independent of 8 such that

(i) for ex = 1, d

~

2:

0< w ~

(ii) for ex > 1, d

~

3:

for x E

W8(X) ~ W

o ~ W8(X)

for x E

~ W

n,

n.

Proof. From Lemma 5.5.2 and Sobolev embeddings we get the estimate

Ilw81Io,6a-3,n

~

c,

where c> 0 is independent of 8. Hence, for p > d,

1!V81Io,co ~ CI!V8112,p/2 ~ c(l + Ilw811~,p) ~ Cl if either ex = 1 and d ~ 2, or ex> 1 and d ~ 3. Note that Cl > 0 does not depend on 8. Now let a > 0 be such that ha (a 2 ) :2 Cl. Set k = max (Ilwollo,co,an, a) and use (w - k) + as test function in (5.71): 82ll'V(w - k)+1 2dx

- l w(h a (w 2) - h a (k 2))(w - k)+dx

=

+ <

o.

l

Therefore W8 ~

The lower bound for

W8

k

(V - ha(e))(w - k)+dx

in n.

in case (i) follows from the proof of Proposition 5.5.4. D

5.5.2

The classical limit in the 'subsonic' steady state

We wish to perform the classical limit 8 - t 0 in Eqs. (5.16)-(5.18) in a simplified framework. We consider the case of one space dimension, such that J = wSx is a constant, and of 'small' given current densities J> O. More precisely, the stationary equations read: j2

_8 2 (n(ln n)xx)x + ( -;;;: +

Tnt -

nVx

>.2Vxx

J T

,

n-C

(5.76) (5.77)

5.5. The classical limit in

n=

223

(0,1), subject to the boundary conditions

n(O) = no,

n(l) = nl,

nx(O) = n x (1) = 0,

V(O) = Yo·

(5.78)

Eq. (5.76) follows from Eq. (5.10) after setting 82 = £2/4. In this formulation, the electron current density J is a given (positive) constant. From the equations, the applied voltage U can be computed by U = V(l) - V(O). Notice that the boundary conditions (and the definition of 8) are different compared to Section 5.2; the above boundary conditions have been used in numerical simulations (see [163]). For the limit procedure 8 ----t 0 we need appropriate uniform a priori bounds for the solution (n, V) = (no, Yo). However, since Eq. (5.76) is of third order we cannot use a symmetric variational formulation or maximum principle arguments. The idea is to transform the equation (5.76) via the exponential transformation of variables n = eU or u = In n and to differentiate the equations in order to get a (symmetric) fourth-order problem (see [64]). Let (n, V) E H 4 (n) x H 2 (n) be a strong solution to (5.76)-(5.78) such that n(x) ~!l > 0 in n. By dividing (5.76) by n, taking the derivative with respect to x, and using the Poisson equation (5.77), we get the following fourth-order equation for u:

82 ( Uxx

+ -u; ) 2

=+

J2 (e - 2u) Ux x - T Uxx

+ eU-\ 2 C A

= -J (e -U) x

(5.79)

T

with the boundary conditions

u(O) = uo,

u(l) =

UI,

ux(O) = u x (1) = 0,

(5.80)

where Uo = In no and Ul = In nl. To obtain the electrostatic potential, divide (5.76) by n, use the relation

and integrate from 0 to x: V(x) = _28 2e- u / 2 (e- U / 2)

xx

Jl

J2 2u + Tu + + _e2

T

x

0

e-u(s)ds.

(5.81)

The integration constant can be assumed to be zero by fixing the reference point for the electrostatic potential. This implies that (see (5.78))

Vo = _28 2e- uo / 2 (e- U / 2)

xx

(0)

+ J2 e- 2uo + Tuo. 2

(5.82)

Chapter 5. The Quantum Hydrodynamic Model

224

Every strong solution (n, V) to (5.76)-(5.78) satisfying 11 :S n E H 4 (n), V E H 2 (n) defines a strong solution (u, V) E H 4 (n) x H 2 (n) to (5.79)(5.81). Inversely, we show below (see Theorem 5.5.10) that every solution (u, V) E H 4 (n) x H 2 (n) to (5.79)-(5.81) defines a solution (n, V) to (5.76)(5.78) satisfying 11:S n E H 4 (n) and V E H 2 (n). The transformation of variables has two advantages. First, we can avoid the problem of not having boundary conditions for the electrostatic potential. This point is also noticed in the paper [64]. Secondly, we obtain a lower bound for the electron density by the Sobolev embedding U E HI(n) '---+ Loo(n) and the relation n = exp(u). The usual procedure is now to use Ufj - UD, where Ufj is a solution to the problem and UD is the (extended) boundary data, as a test function in (5.79) in order to get uniform estimates for 8(ufj)xx and (Ufj)x in L 2 (n). Then, by compactness, there is a subsequence of (Ufj) which converges strongly in Loo(n) and the limit 8 --t 0 in Eqs. (5.76)-(5.77) can be performed. However, due to the exponentials, we only obtain an estimate of the type 811(Ufj)xxllo,2

+ Ilufjlll,2 <

c(1 + J2 exp (21IUfjllo,00))

< c(1 + J2 exp (21Iufjlll,2)),

which does not imply the uniform boundedness of Ufj in HI, even for small J> O. Therefore we proceed in a different way. We consider a truncated version of Eqs. (5.76)-(5.77), prove uniform bounds for the solutions of the truncated problem and show that we can remove the truncation parameter. We prove the existence of solutions and the uniform a priori estimate at once. Hence we obtain the classical limit for one solution, not for every solution. First we prove the existence of solutions to the truncated problem in 0. subject to the boundary conditions (5.80), where we have set max( - K, u)) for K > O. We assume that

8, J, T,'x,

T

> 0;

UO,UI

E lR;

C E L 2 (n).

UK

(5.83) = min(K, (5.84)

For simplicity, let 8 < 1. Furthermore, for given, E (0,1), we suppose that J satisfies (5.85) where Kb) > 0 only depends on" T, T, Uo, UI, ,x, and IICllo,2,0. This constant is defined below (see (5.96)-(5.99)).

225

5.5. The classical limit

The following lemma provides a priori estimates for the solution to the truncated problem. Lemma 5.5.6 Assume (5.84) and (5.85). Let U E H 2 (O) be a solution to (5.83), (5.80) and let K = K(')'). Then

<5llu xx llo,2,n + VTllux llo,2,n :S K o,

(5.86)

where K o > 0 is defined in (5.98) and only depends on "(, T, A, IICll o,2,n. Furthermore, K(')') = Iuol + KolVT and

Ilullo,<Xl,n :S K(')').

T, UN, Ul,

and

(5.87)

Remark 5.5.7 In the case of the hydrodynamic model, i.e. <5 = 0 in (5.79), (5.81), the condition (5.85) means that the flow is subsonic. Indeed, subsonic flow is characterized by the fact that the mean velocity J In is smaller than /T. Now, the estimate (5.87) implies

!.... n

= Je- u :S J eK(-y) :S

"(VT < VT

for

<5 =

0,

that means, the flow is subsonic. Another interpretation of the condition (5.85) is the observation that the second-order term (J 2 e- 2u - T)u xx is non-positive if (5.85) holds. Thus, in general, the second-order term is of sign-changing type (cf. [70]). Remark 5.5.8 From the proof of Lemma 5.5.6 it can be seen that the condition on T can be weakened. In fact, it suffices to assume that T : n x JR - t JR is a CaratModory function satisfying T(X, u) 2: TO > 0 for all (x, u) E n x JR. In this case, the right-hand side of (5.79) has to be replaced by J(e-UIT(u))x and the integral in equation (5.81) by

(X

J

io

e-u(s)

T(S, u(s)) ds.

Proof. We define, following [64], a function ary conditions

UD E

C2

(0')

satisfying the bound-

with piecewise linear second derivative 40<

c: 2 (1-c:) UD,xx(X) =

{

oC:2 (t:-c:)

X

for

XE[O,~)

(e - x)

for for

x x

E [~,e) E [e,~]

226

Chapter 5. The Quantum Hydrodynamic Model

and UD,xx(X) = -uD,xx(1 - x) for x E (1/2,1]' where (0,1/2). Elementary computations show that

0:

= Ul - Uo and

€ E

(5.88) (5.89) (5.90) (5.91) Use U - UD E H5(O) as test function in the weak formulation of (5.83):

8211 +T

(U xx

1

+ ~;)

(u - UD)xxdx - J2

1

ux(u - UD)xdx

+ :21\eu

-

11

e- 2UK u x (u - UD)xdx

eUD)(u - uD)dx

r (eUD _ C)(u _ uD)dx = _!.... Jor e-UK(u - UD)xdx. A Jo

+~

1

1

T

Employing Young's inequality in the first three integrals and the monotonicity of U f---+ eU yields, for any Tf > 0,

Observe that, in view of the boundary conditions for u x ,

5.5. The classical limit

227

We estimate the right-hand side of (5.92) term by term. From (5.89) we obtain 2 2 2 8211 2 48 0: 40: IUD xxi dx = ( ) <-, 2"1 0 ' 3€ 1 - E: "1 3E:"1 since 8 < 1. Using the Cauchy-Schwarz inequality and the boundary conditions for u x , we get for x E n

Iux(x)/ ~ JXlluxx llo,2,0,

lux(x)1 ~ vT=Xlluxx llo,2,O,

(5.94)

which gives, together with (5.88),

8; 1U;UD,xx dx < 0; lIu•• lli,2,n (11/2 1

XIUD,.. (x)ldx +

1,;2(1 -

X)IUD,• .(x)ldx)

82

< 211uxx 116,2,0€10:1

<

82

4""1ll uxx Il6,2,0,

choosing € = min(1/2, "1/210:/). The assumption (5.85) and the inequality (5.91) yield

and

T 11 0:2T -2 IUD xl 2dx ~ - . "1

4"1

0'

By the Poincare inequality

lIu -

uDllo,2,0 ~

1

ill(u -

uD)xII0,2,0

and the Young inequality, we conclude 1

;21 (e UD

-

C) (u - uD)dx

<

"1~ II(u -

<

"1~ Ilux 116,2,0 + "1~ IluD,x 116,2,0

uD)xI16,2,0

+ 2"1~4T Ile "1T

< 4

uD -

+ 2"1~4T Ile

2

-

CII6,2,0

CI16,2,0

r 0:2"1T Jo uxdx + -8- + 2"1)..4T 1

UD

1

II

e

U

D -

C

11 2

0,2,0·

228

Chapter 5. The Quantum Hydrodynamic Model

Finally, again using the hypothesis (5.85), we get

employing (5.90) and 8 < 1. With the above estimates, we conclude from (5.92)

where K 1 is defined by

From (5.94) follows that

Ilux 116,2,0 :::; ~ Iluxx 116,2,0 and therefore

(1 - 3417)8211 u;xdx 2:

~8211 u;xdx + 2(1 -17)8211 u;dx.

With this inequality and assumption (5.85) we obtain from (5.95)

Choosing 1-')'

17 = 2(1 + ')') E (0,1/2),

(5.96)

229

5.5. The classical limit we get

t-

r

1

~(1 -

'Y ) 82 U;xdx + 8 1 + 'Y Jo 2 Hence, setting K 2 = 8(1

+ 'Y)Kd(l -

'Y)

(T + 282 )

r u;dx ::; K Jo 1

1·

'Y), (5.97)

and the estimate (5.86) follows after defining K o = J2K 2 . The constant K o is equal to Ko = 4

+'Y 1-'Y

~

- - K1 .

(5.98)

Finally, the inequality (5.87) is obtained from

lu(x)l:S luol + Jor lux(s)lds:S luol + Ilu llo,2,n :S luol + ;K x

x

where we have set

K(-y) =

=

K(-y),

Ko

luol + ..;T'

(5.99)

o

The lemma is proved.

Remark 5.5.9 We have used in the above proof several times the assumption of one space dimension. In particular, the integral (5.93) does not vanish in the case of several space dimensions.

The existence result for the problem (5.79)-(5.80) is as follows: Theorem 5.5.10 Under the assumptions (5.84) and (5.85), there exists a weak solution u E H 2 (D.) to (5.79)-(5.80).

Proof. We show that there exists a weak solution u E H 2 (D.) to (5.83), (5.80) satisfying Ilullo,oo,n :S K(-y). Then u solves (5.79)-(5.80) after setting K = K(-y). The existence of a solution to the truncated system is shown by using the Leray-Schauder fixed point theorem. For this, let v E X = CO) ([0, 1]) and consider the linear problem 2 2 2vK v) -Tu 82 (u xx +£v 2 x ) xx +aJ (ex x xx +2.... >-2

= a 1T (e-

u(O) = auo,

VK

)X

u(l) = aUl,

V

(e V-l u

in D. ,

ux(O) = u x (1) = 0,

+1_C) (5.100) (5.101)

230

Chapter 5. The Quantum Hydrodynamic Model

where a E [0,1]. Define the bilinear form

11

2 (8 u xx¢xx

a(u, ¢) =

+ Tux¢x +

:2 ev :

1 u¢) dx

for u, ¢ E H 2 (n) and the functional F(¢) =

11(-

8:a v;¢xx

+ aJ2e-2vKvx¢x +

:2 (C - 1)¢ - a~e-VK ¢x) dx

for ¢ E H 2 (n). Since v E X '---t W 1 ,OO(n), we see that a(-,·) is continuous and coercive in H 2 (n) and that F is linear and continuous in H 2 (n). By LaxMilgram's Lemma, there exists a unique solution u E H 2 (n) to (5.100)-(5.101). This defines the fixed point operator S : X x [0, 1] ~ X, (v, a) f--+ u. It can be easily seen that S is continuous. Thanks to the compact embedding H 2 (n) '---t X, the mapping S is compact. Furthermore, S(v,O) = for all v E X, and there exists a constant c > such that for all (u, a) E X x [0, 1] satisfying S(u,a) = u it holds

°

°

Ilullx :S c.

Indeed, Lemma 5.5.6 settles the case a = 1. For a < 1, the same steps as in the proof of Lemma 5.5.6 lead to (cf. (5.95))

(1-

~ - ac~al ) 82

1 1

u;x dx + [(1- 7])T - (1

+ 7]h (T + 282))

1 1

u;dx :S K 1 .

By choosing c = min(1/2, 7]/2Ial) as in the proof of Lemma 5.5.6 and observing that a :S 1, we get

821IuxxI16,2,n + TlluxI16,2,n :S K 2 , with

Ki = K'5/2

(see (5.97)). Using Poincare's inequality, we obtain

°

where C2 > is independent of u and a. Now, the existence of a fixed point u with S(u, 1) = u follows from the Leray-Schauder fixed point theorem. 0 Corollary 5.5.11 Under the assumptions (5.84)-(5.85) there exists a solution (u, V) E H 4 (n) x H 2 (n) to (5.79)-(5.81). Proof. Let (u, V) E H 2 (n) x £2(0,) be a solution to (5.79)-(5.81) (see Theorem 5.5.10). Since u E H 2(n), it holds (u x? E H 1 (n) and thus (u;)xx E H- 1 (n). Observing that, by (5.79), 82uxxxx

= -8 2(u;)xx - J 2(e- 2u u x )x + Tu xx -

;2

(e U- C) + ~(e-U)x, (5.102)

5.5. The c1assicallimit

231

we get u xxxx E H- 1 (0), i.e. there exists W E £2(0) such that W x = U xxxx or (u xxx - w)x = 0 in O. Therefore, U xxx = w + const. E £2(0). This implies (u;)xx E £2(0) and, by (5.102), U xxxx E £2(0). We conclude that u E H 4 (0). The regularity on u and the definition (5.81) for V immediately give V E

H 2 (O).

D

The existence result is as follows: Theorem 5.5.12 Assume (5.84) and (5.85). Then there exists a solution

(n, V) E H 4 (O) x H 2 (O)

to the quantum hydrodynamic problem (5.76)-(5.78). Moreover, it holds

n(x) 2:11>0

in 0,

where 11 = exp( -Kh)) and Kh) > 0 is as in Lemma 5.5.6. Proof. Define n = exp(u) where u is as in Corollary 5.5.11. Then n E H 4 (O) and n 2: 11 > 0 in 0, where 11 = exp(-Kh)), by Corollary 5.5.11 and Lemma 5.5.6. We can rewrite (5.79) as follows:

82 [~(n(lnn)xx)x] n x

(J2n22) xx -T(lnn)xx+ n~2C = (l-) . /\ Tn x

(5.103)

We differentiate equation (5.81) for V twice with respect to x:

Vxx = -28

2

[

~ (vn)xx] xx + (2P2 ) + T(ln n)xx + (l-). n xx Tn x

yn

Since 2 [)n (vntx

Lx

=

(5.104)

[~(n(ln n)xx)xL'

we get from (5.103) and (5.104) the Poisson equation

..\2Vxx = n - C. Taking the derivative with respect to x in (5.81) and multiplying by n, we obtain

nVx =

2 -28 n()n(vn)xx)x +n(::2)x -82 (n(lnn)xx)x+

(P) n

x

+Tnx+~

J +Tn x +-, T

which is equal to (5.76). Furthermore,

V(O) = -28

2

taking into account (5.82).

1

J2

vno (vn)xx (0) + 2n5 + TIn no = Va, D

232

Chapter 5. The Quantum Hydrodynamic Model

Now we can show that there is a solution (U8' V8) to the quantum hydrodynamic model which converges, as 8 -. 0, to a solution (u, V) to the hydrodynamic model (5.105)

u(O)

(5.106)

Uo,

u(l) = U1, x J2 2u + Tu + -J l e-u(s)ds, V(x) = _e=

2

T

x E O.

0

(5.107)

Theorem 5.5.13 Let (5.84) and (5.85) hold and let (U8, V8) be the solution to (5.79)-(5.81) constructed in Theorem 5.5.10, for 8 > O. Then there exists a subsequence (U8" V8') such that

U8' ----' U U8' -. U

in H 1(0) weakly, in CO (0), in L 2(0) weakly as 8' -'0,

(5.108) (5.109) (5.110)

where (u, V) E H 1(0) x L 2(0) is a solution of (5.105)-(5.107). Proof. We conclude from Lemma 5.5.6 and Poincare's inequality that (U8) is uniformly bounded in H 1 (0), i.e. there exists a subsequence (U8') such that (5.108) and (5.109) hold. The weak formulation of (5.79) reads

r

r

1 1 (8')2 (8')2 Jo u8'
r

1 2uo - J2 Jo e- ' (U8' )x
J1

= -T

0

1

r1(U8' )x,2

+ T Jo

e- uo'
(5.111)

for

e{3u o' -. e{3u

in L 2 (0) as 8' -. 0,

for any j3 E R Furthermore, the integral view of (5.108) and (5.112),

1 1

e- 2uo ' (U8' )x
-.1

J(U8' Y;
(5.112)

remains bounded and, in

1

e-2Uux
Hence we can pass to the limit 8' -. 0 in (5.111) to get

as 8' -. O.

5.5. The classical limit

233

for all ¢ E COO(O), which gives (5.105). The boundary conditions (5.106) follow from (5.109) and (5.80). Eq. (5.81) can be rewritten as 2

2 U6 U6 (u )2 + _e8 j2 2U6 - _e0 xx 2 0 x 2 o(x) = 8 e- (u)

V;

+ Tu 0 + -JT

l

0

x

e- u6 (s)ds.

Taking into account the uniform LOO bound for u o and the uniform L 2 bound for 8(u o)xx (see (5.86)), we get a uniform L 2 bound for Vo and hence, there exists a subsequence (VOl) such that (5.110) holds. Multiply (5.81) by a test function ¢ E COO(O), integrate over (0,1) and use integration by parts in

8211 e- u6 / 2 (e- U6 / 2) xx ¢dx - 82 1

1

[(e- U6 / 2)J2 ¢dx- 82 1

1

e-U6/2(e-U6/2)x¢xdx

_8: 1e-U6(uo);¢dx+ 8; 1e-U6(uo)x¢xdx 1

1

to obtain 1 1 Vol¢dx (5.113)

we can let 8'

l

x

--t

0 in (5.113) to get

1 1

e- U61 (S)ds

V ¢dx =

--t

l

1 ~2 1

(

x

e-u(s)ds

e- 2u

in L 2(O) as 8'

+ Tu + ~

l

--t

0,

x e-U(S)dS) ¢dx

for all ¢ E COO(O). This proves the theorem.

0

Remark 5.5.14 For sufficiently small current densities J > 0, the hydrodynamic equations (5.105)-(5.107) are uniquely solvable (see [192]). Then the whole sequence (u o, Vo) converges to (u, V) in the sense of (5.108)-(5.110).

Now, we can prove the main result of this section.

Chapter 5. The Quantum Hydrodynamic Model

234

Theorem 5.5.15 Assume (5.84), (5.85). Then there is a solution (n8, V8) E H 4 (n) x H 2 (n) to (5.76)-(5.78) which possesses a subsequence (n8', V8') such that n8' ~ n

in HI(n) weakly,

(5.114)

n8'

in CO (0) ,

(5.115)

--t

n

in £2(0,) weakly as f/

V8, ~ V

--t

0,

(5.116)

where (n, V) E HI(n) x HI(n) is a solution to the hydrodynamic problem

(~ +Tn)x -nvx~, ).2Vxx

(5.117)

n- C

(5.118)

in 0, with boundary conditions n(O)

= no,

n(l)

= nl,

V(O)

= Uo.

(5.119)

Moreover, n(x) 2: 11 > 0 in 0" where 11 = exp( -K('y)) and K('y) > 0 is defined in Lemma 5.5.6. The value Uo in (5.119) is defined by Uo = j2 /2n6 + TIn no.

Notice that we only get weak £2 convergence of V8 with the limit function V being in HI(n). We cannot conclude strong convergence or convergence in HI since we have no convergence results for the boundary value Vo, defined in (5.82), which depends on 8. Proof. By Theorem 5.5.13, there exists a solution (u, V) to (5.105)-(5.107), obtained as the limit of solutions (U8"V8') to (5.79)-(5.81) as 8' --t 0, where (U8', V8') is a subsequence of (U8, V8). Setting n8' = exp(U8') and n = exp( u), the convergence results (5.114)-(5.116) follow from (5.108)-(5.110), since the embedding HI(n) '----7 CO(n) is compact in one dimension. We show that (n, V) solves (5.117)-(5.119). We rewrite the equation (5.105) in the variable n: -

(~) 2n 2

=

- T(lnn)xx

+ ~(n - C) = ).2

(~) Tn

.

(5.120)

x

Notice that n 2: 11 = exp( -K('y)) since K('y) does not depend on 8. Differentiating (5.107) twice with respect to x gives Vxx = (:22) n xx

+ T(lnn)xx +

(~) E H-I(n). Tn x

(5.121)

Thus, the relations (5.120) and (5.121) give the Poisson equation (5.118). Taking the derivative with respect to x in (5.107) gives Vx =

J2) 2 (-2 n

x

+ T(lnn)x + -J E £2(0,), Tn

(5.122)

235

5.5. The classical limit

and therefore, V E H1(fl). Multiplying (5.122) by n, the resulting equation equals (5.117). Finally, from (5.107) we get

V(O)

=

Uo

clef

J2 +Tlnno. 2n o

o

= -2

Remark 5.5.16 Under some additional assumptions it is possible to perform the zero-Debye-length limit A ----7 O. The limits 8 ----7 0 and A ----7 0 commute (if the limits exist). We refer to [192] for details. 5.5.3

Numerical examples

In this subsection, the limit 8 ----7 0 is illustrated by numerical examples. The experiments were performed by employing the general purpose boundary-valueproblem solver COLSYS, which uses piecewise polynomial collocation at Gaussian points [31]. We model an n+n diode defined by the doping profile

C(x) = 0.75 - 0.25tanh(100(x - 0.5)),

XE(O,l).

This doping concentration is a smooth approximation of the piecewise constant function C(x) = 1 for x E (0,1/2) and C(x) = 0.5 for x E (1/2,1). Note that C(O) = 1, C(l) = 0.5 and Cx(O) = Cx (l) ~ 10- 42 . We choose the scaled parameters T = 1, T = 0.1 and the boundary densities no = 1, nl = 0.5. In Figure 5.2 the electron densities for J = 0.4, A = 0.01 and different values for 8 are shown (logarithmic scaling). This situation corresponds in the classical hydrodynamic model to fully subsonic flow since Jln < 1 = vIT holds in (0,1). According to Theorem 5.5.13, the classical limit 8 ----7 0 can be performed, and the electron density converges strongly to the hydrodynamic particle density. For larger current densities, the situation becomes more involved. In Figure 5.3 the current value J = 1 is taken. This corresponds to supersonic flow in the n region since Jln > 1 = vIT. We observe high-frequency oscillations in this region. For smaller A, small-frequency oscillations appear (Figure 5.4). They are also present in the corresponding hydrodynamic problem (see [32]), whereas the large-frequency oscillations are coming from the dispersive quantum term. Figure 5.5 shows a blow-up of Figure 5.4 in the region x E (0.85,1) for two different values of 8. The frequency is of order 1/8 and the amplitude is of order 1 which shows a typical dispersive behavior. For even smaller A, the largefrequency oscillations are damped and the wavelength of the small-frequency oscillations decreases (Figure 5.6). Therefore, for the classical limit we cannot expect to get strong convergence of the particle density in the case Jln > vIT.

236

Chapter 5. The Quantum Hydrodynamic Model

J = 0.4, CJ.= 10

0.1

A=O.OI ..... 0=0.1

0 \

--0=0.02

I I I

-0.1

-0=0.005

-0.2 -0.3 :>

-0.4 I'

-0.5

I I I I I

-0.6

I

-0.7 -0.8

0

0.1

0.2

0.3

0.4

0.5

x

0.6

0.7

0.8

0.9

Figure 5.2: Logarithm of the electron density for J = 0.4. J=l, CJ.=IO 0.1 r - - - - , - - - - - , - - - , - - - - , - - - r - - - - , - - - - - , - - , - - - - - - - , - - - - , A=0.05

O f - - - - - - - - - - -________

0=0.005

-0.1 -0.2 -0.3 :>

-0.4 -0.5 -0.6 -0.7 -0.8 _0.9'------'-------L--.L..----'-------''------'------'---.L..----'-----' o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x

Figure 5.3: Logarithm of the electron density for J = 1.

5.5. The classical limit

237

J=l,

0.1

0.=10 A=0.02

0

0=0.003

-0.1 -0.2 -0.3 :::>-0.4 -0.5 -0.6 -0.7 -0.8 -0.9 0

0.1

0.2

0.3

0.4

0.5

x

0.6

0.7

0.8

0.9

Figure 5.4: Logarithm of the electron density for J J=l,

= 1.

0.=10 A=0.02 --0=0.006 -0=0.003

:::>-0.7

-0.8

0.85

0.9

x

0.95

Figure 5.5: Blow-up of Figure 5.4.

238

Chapter 5. The Quantum Hydrodynamic Model J=1.

0.1

a=lO 1,.=0.01

0

0= 0.003

-0.1 -0.2 -0.3 :J

-0.4 -0.5 -0.6 -0.7 -0.8 -0.9 0

0.1

0.2

0.3

0.4

0.5

x

0.6

0.7

0.8

0.9

Figure 5.6: Logarithm of the electron density for J = 1.

5.6

Current-voltage characteristics for tunneling diodes

5.6.1

Scaling of the equations

One main application of the quantum hydrodynamic equations is the simulation of quantum devices that depend on particle tunneling through potential barriers, like resonant tunneling diodes. One-dimensional simulations of tunneling diodes show negative differential resistance in the current-voltage characteristic (see [89, 163, 189, 365]). In this section we make evident that not only the quantum correction and the external potential but also the convection term are mathematically responsible for these effects. We consider the stationary isothermal quantum hydrodynamic equations in one space dimension including the physical parameters:

~(j2) q

n

x

+

L

qkBT x _ m * n (V m* n

2 v. ) _ qn + ext x 2( *)2 n ((vn)xx) ~

m

yn

x

--{ - T, (5.123)

Jx = 0,

E sVxx =

q(n - C)

in D.

(5.124)

The physical constants are the elementary charge q, the Boltzmann constant k B , and the reduced Planck constant n. We assume that the temperature T, the effective mass of the electrons m * , the relaxation time T, and the semiconductor

5.6. Current-voltage characteristics

239

permittivity Cs are constant. The relaxation time is given by the expression = /'jvo , where /, is the mean free path of the particles and V o = JkBTjm* is the characteristic velocity of the electrons. Interior quantum layers are modeled by the external potential Vext . Finally, the semiconductor domain is the interval n = (0, L) c JR, L > 0 being the device length. We consider different cases of the order of magnitude of the device length (for the details see below):

T

(i) If the device length is much larger than the mean free path, the convection term in (5.123) can be neglected and we get the so-called quantum driftdiffusion model. (ii) If the device length is much smaller than the mean free path and the de Broglie length Lb = fij..)2m*kBT, the convection and quantum terms dominate the remaining terms in (5.123) and we get a reduced quantum model. (iii) In the case where the device length is of the same order as the mean free path, we obtain, after an appropriate scaling, a dimensionless version of (5.123)-(5.124), the full quantum model, where no term is neglected. We choose the following boundary conditions. The electron density n, the electrostatic potential V, and the velocity potential S (defined by Sx = J j n) are prescribed on the boundary x = 0, L. In the numerical simulations, we also use Dirichlet boundary conditions for n and V and homogenous Neumann boundary conditions for n as in [163]. The parameter U = V(L) - V(O) is called the applied voltage. We are interested in the properties of the current-voltage characteristic J = J(U). We summarize the main results: (a) The current-voltage characteristic of the reduced quantum model is

J(U) =

V28

0

sin

~

y280

where 80 > 0 is some constant. Thus we get negative differential resistance, i.e. dJ j dU < 0, in some interval. (b) The current-voltage characteristic of the quantum drift-diffusion model for constant doping profile and zero external potential is given by

J(U) = U

for U 2: O.

(c) Numerical simulations of a resonant tunneling diode using the full quantum model show effects of negative differential resistance. If the convection term is neglected (quantum drift-diffusion model) and the external

240

Chapter 5. The Quantum Hydrodynamic Model potential is very small, the current J(U) increases monotonically with the applied voltage U.

However, for a different choice of parameters and choosing sufficiently large values for the external potential Vext, the quantum drift-diffusion equations are able to reproduce negative differential resistance effects (see [305] and Section 5.7). Now we scale appropriately the quantum hydrodynamic equations and we derive the quantum drift-diffusion and the reduced quantum model. Let C m be the maximal value of the doping profile and recall that ~ = TV o and L b = n/J2m*k B T are the mean free path and the de Broglie length, respectively, introduced above. Using the scaling (cf. [274]) n

V

--+

Cmn,

--+ kaTV q ,

V.ext

C

--+

CmC,

--+ kaTv. q ext,

x

--+

J --+

Lx, qkBTCrnTJ Lm*

in (5.123)-(5.124), we get 2 ~ )2(J-;;:; ) x (L Jx

+nx -

= 0,

n(V + Vext)x - (L Lb )2 n (v'rixx) v'ri x ,\2Vxx

=n

- C

in 0

= (0,1),

=

-J, (5.125)

(5.126)

where,\2 = E:skBT/q2L2Cm is the squared scaled Debye length. Notice that we have used the same notations for the scaled and unsealed variables.

1. The full quantum hydrodynamic model. Consider a device with the parameters (cf. [163]) T = lOOK, T = 1O- 12 s, L = O.lJ-lm. Then the free mean path is ~ = 150 nm, and we get ~/ L ~ 1. The parameter 8 = L b / L « 1 (here 8 ~ 0.08) is called the (scaled) Planck constant. The equations (5.125)-(5.126) can be formulated as the elliptic system (see Section 5.1):

82wxx = w(1S; + In(w 2 ) - V - Vext + S), (w 2S x )x = 0, ,\2Vxx = w2 - C.

(5.127) (5.128)

The boundary data are assumed to be the superposition of the thermal equilibrium functions and the applied potential Va(O) = 0, Va (1) = U, which implies on 80. w = JC, S = Va, V = In(C) + Va

241

5.6. Current-voltage characteristics

In the following we assume that we are modelling devices with an n+nn+ structure such that C(O) = C(I) = 1 holds. Hence we impose the boundary conditions

w(O) = w(l) = 1,

8(0)

= V(O) = 0,

8(1)

= V(I) = U.

(5.129)

In the simulation of tunneling devices, other boundary conditions for (5.125)(5.126) are also used (see, e.g., [163]):

n(O) = n(l) = 1,

nx(O) = n x (1) = 0,

V(O) = 0,

V(I) = U.

(5.130)

2. The quantum drift-diffusion model. In a quantum device with the data T

= lOOK,

T

=

1O-13

s,

L

= O.IJLm,

(T corresponds to a low-field mobility in GaAs; see [163]) the mean free path

is equal to [ = 15 nm. Thus, the parameter C = [/ L is small compared to one. Letting formally c - t 0 in (5.125) we get the equation

n x - n(V + Vext)x -

82n(~X

t=

-J,

(5.131)

or, as above,

82w xx = w(ln(w 2 ) - V - Vext + S), (w 2S x )x = 0, >.2Vxx = w 2 - C in 0,

(5.132) (5.133)

which, with the boundary conditions (5.129) or (5.130), is referred to as the quantum drift-diffusion equations. They are motivated in [10] and mathematically analyzed in [46, 237, 305] (also see Section 5.7).

3. The reduced quantum model. For ultra-small devices with data

T = 1 K,

T

= 10- 11 S,

L = 20 nm

(cf. [353]) the mean free path [ = 150 nm and the de Broglie length Lb = 80 nm are much larger than the device length. Then c = L/ [ and Cb = L/ Lb are "small" parameters. Thus, letting formally c - t 0 and Cb - t 0 such that c/cb - t 80 > 0, we obtain the reduced model equations

(5.134) with the boundary conditions (5.129) for wand 8. For such ultra-small devices we expect that quantum boundary effects may occur so that the boundary conditions (5.129) are only approximately satisfied.

Chapter 5. The Quantum Hydrodynamic Model

242

5.6.2

Analytical and numerical current-voltage characteristics

We are able to compute explicit solutions for the reduced quantum model and the quantum drift-diffusion model, from which the current-voltage characteristics can be derived. The current J = J(U) is defined by J = w 2S x E IR (see (5.128)). Let Vext = a in O.

Proposition 5.6.1 For the reduced quantum model (5.129), (5.134) it holds

Proof. Let a = U/ V28o. A computation shows that the functions w(x) S(x)

((1-2x)2+2(I+cosa)x(l-x))1/2, ~ 1-(I-cosa)x v28oarccos w(x) , x E (0,1),

solve (5.129), (5.134), and that J(U) = w(x)2Sx(x) = V280 sina for a E [0,11") (also see Section 5.2.2). 0 For applied voltages near the limit value U = V28 0 1l" , the so-called valley current can be very small (compared to the peak current V28o). We expect that the diffusion term in (5.123) (and hence a non-vanishing temperature) leads to a positive current for the full model. Physical experiments show that the valley current can be very small (compared to the peak current) and decreases as the temperature decreases [353].

Proposition 5.6.2 Let C(x) = 1 for x E (0,1). Then, for the quantum driftdiffusion model, it holds J(U) = U for all U ~ o.

Proof. The functions w(x) = 1, S(x) = V(x) = Ux, x (5.133), (5.129). Thus J(U) = w2S x = U.

E (0,1), solve (5.132)0

Clearly, the equations with constant doping profile do not model a diode. However, we present now a numerical example for a tunneling diode, where a similar behavior of the current-voltage curve as in Proposition 5.6.2 can be observed. The experiments were performed employing the general purpose two-point-boundary-value-problem solver COLSYS, which uses piecewise polynomial collocation at Gaussian points [31]. The scaled parameters and functions for the diode are as follows. The doping profile is given by C(x) = 1 for x < 0.3 and x > 0.7, and C(x) = 0.1 for 0.3 < x < 0.7. The external potential is Vext(x) = 1 for 0.4 < x < 0.45 and 0.55 < x < 0.6, Vext(x) = a

5.6. Current-voltage characteristics

243

for x < 0.4 and x> 0.6, and Vext(x) is a quadratic polynomial in (0.45,0.55) defined by Vext (0.45) = Vext (0.55) = 1 and Vext (0.5) = O. Furthermore, b = 0.5 and>. = 0.1 (see [330]). These values correspond to the unsealed parameters L = 0.11 p,m, T = 4K, T = 10- 12 s, and em = 1.8.1015 cm- 3 . In Figure 5.7 the current-voltage curves for the full quantum hydrodynamic model (QHD) and the quantum drift-diffusion model (QDD) for the above parameters and boundary conditions (5.130) are shown. The characteristic for the quantum hydrodynamic equations show negative differential resistance in some region, whereas the curve for the quantum drift-diffusion model is nearly linear. This means that if the convection term in (5.123) is neglected, the negative differential resistance disappears. These results do not contradict the numerical results of [305] (see Section 5.7). Indeed, here we have used an unsealed external potential with maximal value of 3.5mV which is very small compared to the external potential used in quantum semiconductor simulations (usually, Vext = 0.2 ... O.4V). Therefore, we conclude that the external potential is necessary (in the quantum driftdiffusion equations) to get negative differential resistance in the current-voltage curves (see Section 5.7). This fact is very reasonable from a physical point of view. However, negative differential resistance effects can also be obtained, without external potential, from the convective term as shown in Proposition 5.6.1 for the reduced quantum model.

J

.

5.00

(jHo

(job

4.50 4.00 3.50 3.00

.

2.50 2.00 1.50 1.00 0.50 0.00 0.00

20.00

40.00

V

Figure 5.7. Current-voltage characteristics of a tunneling diode (J in 250Acm- 2 , U in O.4mV).

244

5.7

Chapter 5. The Quantum Hydrodynamic Model

A positivity-preserving numerical scheme for the quantum drift-diffusion model

In this section we discretize the (scaled) quantum drift-diffusion equations

atn - div I n In =

-E

2

n\l

=

0,

(fl.:) +

A2 fl.V = n - C(x)

in

O\ln - n\lV,

n x (0,00),

with the bounded domain n C jRd, d 2: 1, 0 > 0 beeing the temperature constant. Inserting the second equation in the first, we obtain after elementary computations (5.135) (5.136)

where ai = a/aXi, yielding a fourth order nonlinear parabolic equation for the electron density n, which is self-consistently coupled to Poisson's equation for the potential V. To get a well posed problem, the system (5.135)-(5.136) has to be supplemented with appropriate boundary conditions. We assume that the boundary an of the domain n splits into two disjoint parts f D and f N, where f D models the Ohmic contacts of the device and f N represents the insulating parts of the boundary. Let /.I denote the unit outward normal vector along an. The electron density is assumed to fulfill local charge neutrality at the Ohmic contacts: n

=C

on fD.

(5.137)

Concerning the potential we assume that it is a superposition of its equilibrium value and an applied biasing voltage U at the Ohmic contacts, and that the electric field vanishes along the Neumann part of the boundary:

V=Veq+U

onfD,

\lV·/.I=O

onfN.

(5.138)

Further, it is natural to assume that the normal component of the current along the insulating part of the boundary and additionally the normal component of the quantum current have to vanish:

I n · /.I = 0,

\l

(fl.:) .

/.I = 0

on f

N.

(5.139)

5.7. A positivity-preserving numerical scheme

245

Lastly, we require that no quantum effects occur at the contacts

b.y/ri = 0

on

rD.

(5.140)

These boundary conditons are physically motivated and commonly employed in quantum semiconductor modelling. The numerical investigations in [304] underline the reasonability of this choice. Finally, the System (5.135)-(5.136) is supplemented by an initial condition

n(x,O) = no(x)

in

n.

(5.141)

This model was first investigated in [303] with a slightly different set of boundary conditions. There the dynamic stability of stationary states was established, at least for small scaled Planck constants and small applied biasing voltages. So far, there are only a few results available concerning the solvability of the system (5.135)-(5.136) due to the lack of an appropriate maximum principle ensuring the positivity of the electron density n. Nevertheless, for zero temperature and vanishing electric field the system (5.135)-(5.136) simplifies to _ 10

2

2 2

- 10

2

b.2n +

~ f).f). (f)inf)jn) 2L...-t J n

10

2

(5.142)

i,j=I

L f)if)j(nf)if)jlog(n)). d

i,j=I

This equation, in the case of one space dimension, also arises as a scaling limit in the study of interface fluctuations in a certain spin system. BIeher et al. [57] showed that there exists a unique positive classical solution locally in time, assuming strictly positive HI (Q) data and periodic boundary conditions. Under much weaker assumptions the existence of a nonnegative global (in time) solution n could be deduced in [236] for the one-dimensional problem. The preservation of nonnegativity or positivity is not only challenging from an analytical point of view, also the derivation of sign-preserving numerical schemes for fourth-order equations is a field of intensive research. Even for strictly positive analytical solutions, the solution of a naive discretization scheme may become negative, causing unwanted numerical instabilities [51]. In the last years this question was thoroughly investigated in the context of lubrication-type equations, which read [50, 52, 99] f)th + div (J(h)"Vb.h) = O.

(5.143)

They arise in the study of thin liquid films and spreading droplets (for an overview see [51] and the references therein). Here, the main ingredient for

246

Chapter 5. The Quantum Hydrodynamic Model

the proof of the nonnegativity or positivity property is to exploit the special nonlinear structure of Eq. (5.143), especially the degeneracy of the mobility f(h), i.e. f(h) = hQ as h ----t 0 for some a > O. Numerically, there are two ways of dealing with Eq. (5.143): Bertozzi et at. [53] designed a space discretization using finite differences, which exhibits the same properties as the continuous equation, while Barrett et at. [40] proposed a nonnegativity preserving finite element method, where the nonnegativity property is imposed as a constraint such that at each time level a variational inequality has to be solved. Concerning Eq. (5.142) a different approach was used in the existence proof of [236]. After an exponential transformation of variables, n = e2u , a semidiscretization in time was performed for J:l

Ute

2u = -10 2( e 2u U ) xx xx·

As the resulting sequence of elliptic problems is uniquely solvable in each time step, this yields intrinsically a global nonnegative solution. However, due to the introduced additional nonlinearity in the time derivative this scheme meets some difficulties in numerical simulations. We introduce a new approach to the numerical solution of the fully coupled system (5.135)-(5.136), which consists of two main ideas: Firstly, we write Eq. (5.135) in conservation form

Otn = div (n\l (-c

f:

2t1

+ Blog(n) - V))

and introduce the quantum quasi-Fermi level F=-c

2

t1 y'Ti y'Ti +Blog(n)-V.

Here, -c 2 t1y'Tily'Tiis the so-called quantum Bohm potential (see Section 5.1). Employing the boundary conditions (5.137)-(5.140) we learn that the equilibrium value of the potential is given by Veq = log( C) and that F fulfills

F= U

on

r D,

\lF . v = 0

on

r N.

Secondly, motivated by the results for the stationary problem, where the positivity of solutions follows immediately from Harnack's inequality [46], we employ an implicit time discretization by a backward Euler scheme on the system

Otn = div (n \lF), 2 t1 y'Ti -10 y'Ti + Blog(n) - V >.2t1V=n-C

(5.144) =

F,

innx(O,oo).

(5.145) (5.146)

5.7. A positivity-preserving numerical scheme

247

In the following we prove that the discretized version of (5.144)-(5.146) admits at each time level a strictly positive solution n(x, tk)' Unfortunately, we cannot derive a uniform lower bound on the electron density such that this property does not hold in the limit and weakens to nonnegativity. Further, it is worth noting that the entropy (or free energy) S(t) =

C;2

rIV' Vn(tWdx + inrH(n(t))dx + >.22 inrIV'V(tWdx

in

(5.147)

is (formally) non-increasing in time, as long as the boundary data F D for the quantum quasi Fermi level is nonpositive. Here,

H(s) ~f Bs(log(s) - 1) + B denotes a primitive of the logarithm. This observation allows us to derive a stability bound for the numerical scheme in any space dimension. Therefore, without additional assumptions, this is only sufficient to prove convergence of the scheme in one space dimension, since in the proof the Sobolev embedding H 1 (0,) ~ £"0(0,) plays a crucial role. As a by-product we establish the existence of a global nonnegative solution to the system (5.144)-(5.146) in one space dimension. Imposing stronger assumptions on the regularity of the continuous solutions it is possible to give an estimate on the order of convergence, which proves to be optimal (see [237]). Finally, let us give some comments on the numerical advantages of (5.144)(5.146) compared with (5.135)-(5.136), which are twofold: On the one hand we do not have to discretize a higher order differential operator and on the other hand it is now possible to introduce an external potential, modeling discontinuities in the conduction band, which occur for example in resonant tunneling structures [163, 305]. It is common to replace in (5.145) the potential V f--+ V + B, where B is a step function. Clearly, such a replacement in (5.135) causes extreme numerical problems due to the second derivative of B. 5.7.1

Semidiscretization in time: existence of the discrete system

In this subsection we derive the implicit semidiscretization of (5.144)-(5.146) and prove the existence of solutions to the resulting system on each time level. In particular, we show that the approximation of the electron density is strictly positve. For the following investigations we introduce the new variable p = Vii. Then (5.144)-(5.146) reads: (5.148)

248

Chapter 5. The Quantum Hydrodynamic Model 2/:1P p

-£0 -

+ () log(p2 ) -

>.2/:1V = p2 -

C

V = F, in

n x (0,00).

(5.149) (5.150)

For the numerical treatment of (5.148)-(5.150) we employ a vertical line method and replace the transient problem by a sequence of elliptic problems. Let T > 0 be given. We divide the time interval [0, T] into N subintervals by introducing the temporal mesh {tk : k = 0, ... , N}, where 0 = to < t1 <

... < tN = T. We set Tk ~f tk T

d~f

tk-1 and define the maximal subinterval length

maxk=l,... ,N Tk. We assume that the partition fulfills T ---7

0,

as N

---7

00.

(5.151)

For any Banach space B we define CN(O, T; B)

d~f {vT : (0, T]

---7

B : vTI(tk_l,tk] = const. for k

= 1, ... ,N}

and introduce the abbreviation Vk = vT(t) for t E (tk-1, tk] and k = 1, ... , N (see Section 4.3). Now we discretize (5.148)-(5.150) in the following way: Set Po = jn(O). For k = 1, ... , N solve recursively the elliptic systems (5.152) (5.153) (5.154)

subject to the boundary conditions

= PD, Fk = FD, Vk = VD \lPk . v = \lF k . V = \lVk . v = 0 Pk

where PD

=

JC,

FD

= U,

VD

(5.155) (5.156)

= ()log(C) + U.

Then the approximate solution to (5.148)-(5.150) is given by (pT , FT, V T ). For the subsequent considerations we impose the following assumptions. (HI) Let n c ]Rd, d = 1,2 or 3 be a bounded domain with boundary an E C 1 ,1. The boundary an is piecewise regular and splits into two disjoint parts rN and rD. The set rD has nonvanishing (d - I)-dimensional Lebesgue measure. r N is closed.

249

5.7. A positivity-preserving numerical scheme

(H2) The boundary data fulfills

PD E H 2 (D),

infpD>O,

an

FD E C 2 ,,·(0)

'lPD'V=O onf N ,

for 'Y E (0,1 - d/2),

FD:s -F D < 0,

VD E C ,,.(0), 2

and for the initial condition holds Po E H 2 (D). Further, C E Co,"(O). (H3) Let 'Y E (0,1) and a E CO,,.(O) with a 2 Q > O. Then there exists a constant K = K (D, r D, r N, a, d, 'Y) > 0 such that for f E Co,,. (0) and UD E C 2 ,,. (0) there exists a solution u E C 2 ,,. (0) of

div(a'lu)

=

f,

U-UD

E H6(DUrN),

which fulfills

Remark 5. 7.1 (i) Assumption (H3) is essentially a restriction on the geometry of D. It is fulfilled in the case where the Dirichlet and Neumann boundary do not meet, i.e. I'D n r N = 0 [347]. (ii) The restriction FD :S FD on the quantum quasi-Fermi level is purely technical. From the physical point of view the device behavior is independent of a shift F I---t F + a, V I---t V + a, a E R (iii) For a smoother presentation we assume that the boundary conditions are independent of time. We can prove the existence of solutions to the discrete system (5.152)(5.154) in two special situations: First, we employ different boundary conditions than (5.155)-(5.156). Secondly, we assume that the temperature constant is large enough. Theorem 5.7.2 Assume (H2) and d = 1. Furthermore, let k E {I, ... ,N} and Pk-l E CO'''(D). Then there exist constants Ck > 0 such that there is a solution (pk' Fk' Vk) of the system (5.152)-(5.154), with boundary conditions Pk = 1, Pk,x = 0, V = VD on aD, fulfilling Pk, Fk' Vk E c 2 ,,.(0), and

Pk 2 Ck > 0

in D.

Proof. The idea of the proof is the following: We eliminate F k from (5.153), introduce an exponential transformation of variables and employ LeraySchauder's fixed point theorem on the resulting system.

Chapter 5. The Quantum Hydrodynamic Model

250

Elimination of Fk and some calculus yields (for positive p) 2) £2 2 2) -1 (2 P - Pk-l = --2 (p (log P xx)xx Tk ),2Vxx = p2 - C in 0, p = 1, Px = 0, V = VD on ao,

+ O(p2 (log P2 )x)x -

(p 2 Vx)x, (5.157) (5.158)

(5.159)

which has to be solved for (p, V). Since there is no maximum principle available we employ the exponential transformation of variables p = eU as in [192] and get the system 1 _(e 2u _ e2Z ) = -£2(e 2u u xx )xx + 20(e 2u u x )x - (e 2u Vx )x, (5.160) Tk ),2Vxx = e2u - C in 0, (5.161) (5.162) u=O, ux=O, V=VD onaO,

where z = logPk-l' We show that there exists a solution u E H 2 (0) to (5.160)(5.162). Since H 2 (0) <......t V'O(O), we can set P = eU and P solves the system (5.157)-(5.159). Moreover, P is strictly positive in O. To establish the existence of a solution to (5.160)-(5.162) we use LeraySchauder's fixed-point theorem. For this purpose we define a fixed point mapping G : H1(0) x [0,1] ----t H1(0) as follows. Let W E H1(0) and (J E [0,1] be given. Let V E H1(0) n £00(0) be the unique solution to

V = VD

on

ao.

in H5(0). Notice that H1(0) <......t COCO) such that the above equation is strictly elliptic. In order to see that this problem has a unique solution u E H5(0), we introduce the bilinear form 2w 2 2w II 2w 1 e - 1 ) a u,

( ) 1(

for u,

F(
=

in

(J

(e 2w Vx
+ T;1(e 2z -

1)
for

0 such that a(u, u) 2: clluIIH2(O)' i.e. the form a(-, .) is coercive. (The above construction avoids the use of the

251

5.7. A positivity-preserving numerical scheme

Poincare inequality.) Moreover, a(·,·) is continuous in H 2 (n) and F is linear and continuous in H 2 (n). By Lax-Milgram's lemma, there exists a unique solution u E HS(n) to a(u, ¢) = F(¢) for ¢ E H5(n), i.e. u solves (5.163). Thus the mapping G, given by G(w,O') = u, is well defined. Moreover, G(w,O) = 0 for any w E H 1 (n). It can be easily seen that G is continuous. Thus, G is compact, noting the compact embedding H2(n) '--t H 1 (n). We show that there is a constant c > 0 such that for all (u, 0') E H 1 (n) x [0,1] satisfying G(u,O') = u it holds IluIIH2(!1) ~ c. In order to prove the uniform bound, let (u, 0') be such that G( u, 0') = u and use 1 - e- 2u E H 2 (n) as a test function in (5.163) to obtain

7k11o (e

2u

-2c

=

2

-

1 - 0'(e 2z -1)) (1 - e- 2U ) dx

(5.164)

10 u;x dx + 4c 10 uxxu;dx - 40 10 u;dx + 20' 10 2

U xVxdx.

The inequality eX 2: 1 + x for all x E IR implies (e 2U

1 - 0'(1- e 2z )) (1- e- 2u )

_

(e 2u

-

+ (1 > (e

2u

-

2u) - 0'(e 2z 0')(e-

2u

-

2z)

+ 2u -

2u) - 0'(e

2z

-

+ 0'(e2 (z-u)

-

2(z - u) - 1)

2)

2z) - (1 - 0').

From the Poisson equation we get, since u = 0 on an,

Furthermore, the boundary conditions give

10 uxxu;dx = ~ 10 (u~)xdx = O. This step is crucial; it does not hold if we impose the boundary conditions (e 2u )xx = 0 on an. Therefore we obtain from (5.164) 7

1

10 (e

~

7

k

k

2u

1

0'

-

2u)dx + 2c

k

(e

2z

-

2

10 u;x dx + 40 k u;dx

2z)dx

+ 20'A- 2

k

(C - e 2U )udx + 7 k 1 (1- O')meas(n).

252

Chapter 5. The Quantum Hydrodynamic Model

The elementary inequalities e2u

-

2u ~ 1 + 2u 2 ~ 2u

for u

~

> -2u

for u

<0

e

2u

-

2u

0,

imply for u E JR. Then, using

r (C - e2U )udx r in =

for some constant

i{u>o}

(C - e2U )udx +

r

i{u
C> 0, we get from (5.165)

2Tk 1 1lull£lcn) + 2E21Iuxxll£2cn) + 40lluxll£2cn)

~ Tk 1

(C - e2U )udx

L (e

2z

-

~ c,

2z)dx + C(Tk)'

Therefore, employing the generalized Poincare inequality we conclude

Il u llH

2

cn) ~

Ck,

where Ck > 0 only depends on Pk-l, >., and IICIILOOcn). Now the Leray-Schauder fixed-point theorem ensures the existence of a fixed point to (5.163) with a = 1, i.e. to (5.152)-(5.154). Since P = eU and P > 0 in 0, the quantum quasi-Fermi level F in (5.153) is well defined and as F fulfills div(p 2 VF)

= ~(p2 - pLl) Tk

E Co,'"!(O)

we get from (H3) the desired regularity F E C 2 ,,"!(0). An analoguous argument gives V E C 2 ,,"!(0). This implies P E C 2 ,,"!(0). 0 Remark 5.7.3 Let (H1)-(H3) hold. Working directly with the system (5.152)(5.154), we can prove the existence of solutions if 0 is sufficiently large. To see this, we define the fixed point operator as follows. Let w E H 3 / 2 +a (0) (0 < a < 1/2) be given, with w ~ m > 0 in 0 for some m > 0 to be determined, and let V E H1(0) be the unique solution to >.2~V=w2_C

V

=

VD

on fD,

in 0,

VV· v

= 0

on fN.

Since u E H 3 / 2 +a (0) '--t Co,'"!(0) with 'Y < a, we have V E C 2 ,'"!(0), in view of Assumption (H3). Further, the unique solution F to div (w 2 V F)

= T k 1 (w 2 - pLl) in 0,

F=FD onfD,

VF·v=O

onf N ,

(5.165)

5.7. A positivity-preserving numerical scheme

253

satisfies the regularity property F E CZ,'Y(O). Finally, let p E HI(O) be the unique solution to

-eZb.p = p(()logpz - V - F) p

"V p . v =

= PD on r D,

°

in 0, on

(5.166)

r N·

°

The main difficulty now is to prove that p 2: m > holds in O. For this, we can use (p - m)- = min(O, p - m) as a test function in (5.166) for 0< m ::; infrD PD. Then we can achieve p 2: m if info F > -00 is independent of m, by choosing m > small enough. For an estimate for info F take (F - f)for appropriate f as test function in (5.165). Using Stampacchia's truncation method (see Section 5.2.1), however, it is only possible to show that F 2: f - c(llwIILq)/m z (for some q > 1). We need to choose () > large enough in order to get the bound p 2: m. We proceed similar as in [228]. Indeed, let m = min(1/2, inf rD PD) > 0, PD E WZ,P(O) with p > d, and assume (for regularity) I'D n I'N = 0. It is easy to check that sUPo F ::; CI (m) and sUPo V ::; Cz with Cz > only depending on 0, C, and VD. This gives sUPo p ::; M ~f c3(m), by the truncation method, and thus info V 2: -c4(M) and info F 2: -c5(m, M). Hence, employing the test function (p - m)- in (5.166), we obtain

°

°

°

eZkl"V(p -

m)-Izdx

< k p(O log m Z - V - F)( -(p - m)-)dx <

°

k

p(Ologm Z + c4(m)

+ cs(m, M))( -(p -

m)-)dx

< 0,

if we choose 0 > large enough. Therefore, p 2: m in n. This provides L= bounds for p, F and V from which we conclude uniform HI bounds for p, F and V. From elliptic regularity, we obtain uniform HZ bounds for p, F and V. This shows that the fixed-point operator

G :X

-+

X,

G(w) = p,

with X = {w E H3/2+ a (0) : m ::; w ::; M in O}, is well defined. Moreover, G is continuous and the image G(X) is compact, in view of the compact embedding HZ(O) ~ H 3 / Z+a(o), (J < 1/2. Hence, Schauder's fixed-point theorem gives the existence of a fixed point, i.e. a solution (p, F, V) to (5.152)-(5.154), satisfying p, F, V in CZ,'Y(O). 5.7.2 Stability bounds and convergence results First we prove a stability estimate on the solution which is valid in all space dimensions. In the following, we set () = 1.

Chapter 5. The Quantum Hydrodynamic Model

254

Lemma 5.7.4 Assume (H1)-{H2). For fixed k E {I, ... , N} let (pk' Fk , Vk) E H 2(D) x C 2 ,"Y(O) x C 2 ,"Y(O) be a solution of (5.152)-{5.156). Then the following discrete entropy estimate holds 2

c LIV Pkl 2dx + L H(p%)dx

:S

c2

+ ~2LIVVkI2dX - L FDP%dx

rIVPk_11 2dx + inrH(pLl)dx + inrIVVk_ 112dx - inrFDPLI dx. ,\2

in

2

Proof. We use ¢ = Fk - FD = -c 2 !j.Pk/Pk + log(p%) - Vk - F D as test function in (5.152). Note that ¢ satisfies homogeneous boundary conditions. This yields

~ Tk

r(p% -

in

pLl)¢dx = -

r p%VFk' V(Fk -

in

FD)dx.

First, we estimate the left-hand side

~ Tk

r(p% - pLl)¢dx

in

~ Tk

(_c inr 2

-L

(p% - pLl) !j.Pk dx Pk

(p% - pLl)Vk dx

-L

+

r(p% - pLl) log(p%)dx

in

(p% - pLl)FD dX)

1

-(h+12 +h+14 ). Tk

We estimate termwise. Integration by parts yields

h

=

c2LIVPkl2dX - c

2

L

VPk' v(P;:l )dx 2

2

2 2 c LIV Pkl 2dx - c LIV Pk_11 2dx + c LIv Pk-l - P;t V Pk 1 dx

> c2 LIV Pk 12dx - c 2 LIV Pk_11 2dx . Employing some straight-forward calculus we get 12

=

L (p%(log(p%) - 1) + l)dx - L (pLl (log(pLl) - 1) + l)dx

+ L~pLl (log(pLl) - 1),,- pL1log(p%) + p%), dx >

L

H(p%)dx

-L

2:0

H(pLddx.

5.7. A positivity-preserving numerical scheme

255

Integration by parts yields

and using the identity 2r(r -

8) =

r2

-

8

2

+ (r -

8)2,

Now we estimate the right-hand side by Young's inequality:

-l/~'VFk''V(Fk-FD)dx

-

inP~I'VFkI2dX+ inP~'VFk''VFDdX

-~

<

in p~I'VFkI2dx + ~ in p~I'VFDI2dx.

Define the discrete entropy

Combining the above estimates we get

S(Pk) -

in FDp~dx + ~ in p~I'V in ~k inP~I'VFDI2dX in in FDp~dx Fkl 2dx

< S(Pk-l) -

FDPL1dx+

< S(Pk-l) -

FDPL1dx - cl(FD)Tk

(recall that F D < 0), where

_ IIFDIIi,oo,n . 2F D

Cl (FD ) -

Thus consecutively we get

(5.167)

Chapter 5. The Quantum Hydrodynamic Model

256

Note that S 2': 0. Hence, it holds

Now the discrete Gronwall Lemma implies

from which we immediately deduce the uniform boundedness of the entropy

This finishes the proof.

D

Hence, the approximate solution is stable in the following sense.

Corollary 5.7.5 Assume (H1)-(H2). For k = 1, ... , N let (pk' Fk' Vk) be the recursively defined solution of (5.152)-(5.156) and (pT,FT, V T) E CN(O,T; H 2(n) x C 2 ,1'(n) x C 2 ,1'(n)). Then pT E Loo(0,T;H 1(n)) and pTT\lF T E L 2 (0, T; L 2 (n)). Further, there exists a positive constant c, independent of T, such that

Proof. The bounds on pT and VT are immediate consequences of Lemma 5.7.4, D while the one on F T follows from (5.167). Now we are able to prove the convergence of the scheme in one space dimension. Our argument depends crucially on a uniform Loo(n) bound on pT, which follows from Corollary 5.7.5 only in one space dimension due to the embedding H 1 (n) '-+ Loo(n). First, we derive the following energy estimate.

an

rD.

Lemma 5.7.6 Assume (H1)-(H2) and let d = 1, = For k = 1, ... , N let (pk' Fk' Vk) be the recursively defined solution of (5.152)(5.156) and (pT,F T, V T) E C N (0,T;H 2 (n) x C 2 ,1'(0) x C 2 ,1'(0)). Then pT E L 2 (0, T; H 2 (0,)) and there exists a positive constant c, independent of T, such that (5.168) IlpT II £2(H2) ::; c.

Proof. We start with (5.152), which can be equivalently written as

257

5.7. A positivity-preserving numerical scheme

Since Pk > 0, we can divide by Pk which yields 2 1 (Pk - Pk_l)2 1 . (2 ) - (Pk - Pk-l) - = -dlV Pk V' Fk Tk Tk Pk Pk

and after elimination of Fk

_c 2 t:~.z Pk

+ c 2 (L~Pk)2 + 2~Pk

Pk 2 + 2IV'PkI - - - 2V'Pk' V'Vk - Pk~Vk' Pk

Now we use 1> = Pk - PD as test function, observing that, in view of (H2), V'PD'// = 0 on r N :

We estimate termwise. The left-hand side can be written as

~ Tk

Inr (Pk ~ Tk

~

Pk-l) (Pk - PD) dx

Inr (Pk -

r (Pk -

~Jn

PD - (Pk-l - PD)) (Pk - PD) dx PD)2 dx -

Define 'fl

def

=

~

r (Pk-l -

~k

PD)2 dx

+~

r (Pk -

~Jn

Pk_I)2 dx.

minn PD >0 maXk=l,... ,N IlpkIIL=(fl) ,

which is independent of N due to Corollary 5.7.5 and the embedding H1(n) LOO(n) in one space dimension. Note, that for k = 1, ... , N it holds Pk - PD :::; 1- 'fl. Pk

'--+

Chapter 5. The Quantum Hydrodynamic Model

258

Then we have the following estimates: Young's inequality yields

h +h <

_c

2

_c

2

in in

< _ TJ2c2

fi.pkfi. (Pk - PD) dx + c (fi.Pk)2 dx + c

2

in

2

in

(fi.Pk)2 (Pk ;k

fi.Pkfi.PDdx

+ (1 -

TJ)c 2

r (fi.Pk)2 dx + 2TJc inr (fi.PD)2 dx. in

PD

in

) dx (fi.Pk)2 dx

2

By integration by parts and use of Young's inequality we get h

-2

in in

\7 Pk . \7 (Pk - PD) dx

< -2TJ < -TJ

r

in

1\7 Pkl 2dx + 2

in

+2

=

in -in in -in -

2 1\7 Pkl (Pk ;/D) dx

\7 Pk . \7 PDdx

r

1\7 Pkl2dx + ~ 1\7 PD1 2dx. TJ in

From Holder's inequality we derive

14

in

\7 (Pk - PD)2 . \7Vk dx - 2

in

\7 PD' \7VdPk - PD) dx

pdPk - PD) fi.Vk dx

(Pk - PD)2 fi.Vk dx - 2 (Pk - PD)2 fi.Vk dx -

in in

\7PD' \7Vk (Pk - PD)dx PD (Pk - PD) fi.Vk dx

< 21I\7PDIILOO(n) II\7Vkll£2(n) Ilpk - PDII£2(n)

- .x- 2

in

PD (Pk - PD) (Pk -

for some positive constant Cl = dimension the embedding H 2(D.)

Cl '-7

C) dx

(.x, D., PD, Po, C). Note that in one space W1,OO(D.) holds. Finally, we get directly

259

5.7. A positivity-preserving numerical scheme

Combining these estimates we arrive at

~ Tk

r(Pk - PD)2 dx +!LTk inr(Pk - Pk_l)2 dx

in

+ TJ E 2

<

r (tlPk)2 dx + TJ inr IV' Pkl 2dx

2

in

r(Pk-l - PD)2 dx + 2TJ inr(tlPD)2 dx + ~TJ inrIV'PDI 2dx + Tk in 2

~

E

Cl,

from which we immediately deduce

E2T 2 2 2 TJ k I tl (Pk - PD ) 1£2(0.) Ilpk - PDIIP(n) + TJllpk - Pk-Iil£2(n) + -2+ TJTkllV' (Pk - PD) Ili2(n) ::; Ilpk-l - PDlli2(n) + Tk C2· 1

Now (5.168) follows from Gronwall's Lemma.

o

For the convergence result we also need some bound on the time derivative. To this purpose we introduce the linear interpolant of (p T )2 E CN(O, T; L 2(D)), defined by t- tk ( Pk2 (X) n t, X ) clef = -

-T (

Tk

-

2 ()) Pk-l X

2 () + Pk-l X ,

Lemma 5.7.7 Let the assumptions of Lemma 5.7.6 hold. Then

and there exists a positive constant c, independent of T, such that

lin; 11£2(H-l) ::; c. with the norm IIV'tl- l . 11£2(0.),

Proof. We supply H-l(D) where tl- l : H-l(D) --+ HJ(D) is the inverse Laplacian [344]. Using ¢ = -tl-ln[ as test function in (5.152) yields after integration by parts

in

lV'tl- l n;1 2dx =

in p~V'

F k . V' tl-ln;dx.

Employing Holder's inequality combined with Holder's inequality we get

which is uniformly bounded according to Corollary 5.7.5. We state the desired convergence result.

o

260

Chapter 5. The Quantum Hydrodynamic Model

Theorem 5.7.8 Assume (H1)-(H2) and let d = 1, an = rD. For k = 1, ... ,N let (pk' Fk' Vk) be the recursively defined solution of (5.152)-(5.156) and (pT,F T, VT) E CN(0,T;H 2(n) x C2 ,'"Y(O) x C 2 ,'"Y(O)). Then, there exists a subsequence, again denoted by (pT, FT, VT), such that

pT

~

weakly in £2(0, T; H 2(n)), strongly in CO(O, T; CO,'"Y(O)) , weakly in £2(0, T; £2(0,)), strongly in CO(O, T; C 2 ,'"Y(O)) ,

P

pT ~ P (pT)2F; ~ J VT~V

as

T

~

r

JQT

0, where (p, J, V) is a solution of

p2at
r

JQT

J
JrQT [c pxx(2Px
_,\2

r J

Vx
QT

for all
E

r J

=

0,

p2
(5.169)

(p2
JrQT J
C)
(5.170) (5.171)

QT

COO(QT), where QT

= 0,

X

(0, T).

Remark 5.7.9 The above result shows that (p, V) is a weak solution of the problem

at (p2) =

_

(c p2 (Pxx) 2

_ (p2)x

P x in QT,

,\2Vxx = p2 - C p = PD, Pxx = 0, V pC 0) = Po in 0"

=

VD

in the sense of Eqs. (5.169)-(5.171), satisfying p 2:

+ p2 Vx ) on

, x

an x (0, T),

°

in 0, (see [237] for details).

Proof. We choose a sequence of partitions of [0, T] satisfying (5.151). According to Lemma 5.7.6 we have the boundedness of (pT) in £2(0, T; H 2 (n)). We may choose a subsequence, again denoted by (pT), such that pT ~ P

weakly in £2(0, T; H 2 (n)).

Further, we have due to Lemma 5.7.7 and Corollary 5.7.5 that fiT E £00(0, T; H 1 (n)) n H 1 (0, T; H- 1 (n)). Since the embedding H 1 (n) '---7 CO,'"Y(O) is compact in one space dimension for 'Y E (0, 1/2) we deduce from Aubin's Lemma [337] that

£00(0, T; H 1 (n))

n H 1 (0, T; H- 1 (n)) '---7 CO(O, T; CO,'"Y(O))

compactly.

261

5.7. A positivity-preserving numerical scheme Hence, there exists a subsequence, not relabeled, such that

TiT

---t

n

strongly in CO(O, T; CO,'1' (0)).

It is not difficult to verify that n = p2. Moreover, the compact embedding

implies that (up to a subsequence)

Standard results from elliptic theory and (H2) imply now

Defining F = (pT)2\l FT we deduce from Corollary 5.7.5 that (F) is bounded in L 2 (0, T; L 2 (0)), such that

F -' J

weakly in L 2 (0, T; L 2 (0)).

Note that the discrete solutions satisfy

r ((pT)28

JQT

r

JQT

t

[c2p~x(2p~

(pT)2
+ (pT)2Vx
r F.2 r V;
JQT

=

QT

QT

for all test functions

262

Chapter 5. The Quantum Hydrodynamic Model 9

X 10 2.5,---------,----,------r------,----.------,-----y---,-------,

2

0.5

O'-==--...J------l.------'-----'----'----'----'------.L.-----'

o

0.05

0.1

0.15

0.2

0.25

Voltage U [V]

0.3

0.35

0.4

0.45

Figure 5.8: Stationary current-voltage characteristic of a resonant tunneling diode.

5.7.3 Numerical examples In this section we employ the transient quantum drift-diffusion model for the simulation of the switching behavior of a resonant tunneling diode. Such devices proved to be well suited for the validation of quantum models, since their performance is completely determined by quantum mechanical phenomena [248]. Their main characteristic is the appearance of negative differential resistance in the stationary current-voltage characteristic (see Section 2.2). During the last years most simulations focused on the stationary models and the computation of current-voltage characteristics. There, negative differential resistance was recovered by many authors in a varying set of models, such as the quantum drift-diffusion [305], the quantum hydrodynamic model [89, 163] and recently the "smoothed" quantum hydrodynamic model [166] (see Section 1.1). A typical stationary current-voltage characteristic, which was computed using the quantum drift-diffusion model, is depicted in Fig. 5.8. From experiments it is well-known that the switching time of the device is correlated with the peak-tovalley ratio of the current-voltage characteristic, where a large ratio corresponds to a small switching time [286].

5.7. A positivity-preserving numerical scheme

263

Electron Density n (m1

70

60

0.8 0.6 0.4 10

0.2

oL...--------~---------------' o 0.2 0.4 0.6 0.8 Timet [s1

Figure 5.9: Transient electron density. In the following we present the first simulations of the switching behavior of a resonant tunneling diode computed by a macroscopic quantum model. The GaAs-AIGaAs double barrier structure consists of a quantum well GaAs layer sandwiched between two AlxGa1_xAs layers, each 5 nm thick (see Section 2.2). This resonant structure is itself sandwiched between two spacer layers of 5 nm thickness and supplemented with two contact GaAs layers, each 25 nm thick. The contact region ist highly doped with C = 1018 cm- 3 , while the channel is moderately doped with C = 1015 cm- 3 . The barrier height is assumed to be B = O.4eV and the relaxation time is fixed at Trel ax ~ 1O- 12 S. These device parameters yield for the scaled Planck constant and the scaled Debye length, respectively,

In the simulation we switched the resonant tunneling diode from the equilibrium

state (U = OV) to the valley state (U = O.3V), see Fig. 5.8. For the computations we used the one-dimensional version of (5.144)-(5.146) and replaced in (5.145) the potential V by V + B, where B is a step function modeling the

264

Chapter 5. The Quantum Hydrodynamic Model

108 '-~~u..uJ.~~~u.ll-~~~"'-~~~-'--~~u..uJ.~~~u.ll-~~~ 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 Time I [s]

Figure 5.10: Transient current density.

barriers. Then we employed the vertical line method given by (5.152)-(5.154) as time discretization. The discretization in space was done by finite differences on a uniform grid with 300 points. To solve the resulting nonlinear systems we used a Newton iteration, where the solution on the previous time level was used as initial guess. Due to this fact only two to three Newton iterations were needed on each time level. The initial time step was set to TO = 10- 4 and afterwards a heuristic adjustment of the time step was used, which significantly reduced the total number of time steps required to reach the stationary state. In Fig. 5.9 we present the computed transient electron density over a period corresponding to the relaxation time. Note that the electrons move top-down. One clearly identifies an initial time layer, where the electrons accumulate in front of the first barrier. After this short delay they start to tunnel through this barrier and accumulate dramatically in the quantum well. This charge build-up is more than three orders of magnitude larger than the background doping and was also reported by other authors [163, 305, 306]. Lastly we discuss the transient current density at the left contact (x = 0), which is depicted in Fig. 5.10. As we switch at time t = 0 instantaneously out

5.7. A positivity-preserving numerical scheme

265

of the equlibrium state (J = 0) there is a jump in the current density. During the evolution to the stationary valley state the current density does not change monotonically, apparently an oscillation occurs. This oscillatory behavior was also reported in [248], where the resonant tunneling diode was simulated by the (microscopic) Wigner-Poisson model. There the transient current density proved to be even highly oscillatory on account of ballistic effects. We cannot expect this in our case, since we are working in a diffusive regime [73], where the small relaxation time prevents ballistic phenomena. Note that the stationary state is reached after 10- 11 seconds, which is ten times the relaxation time.

References

[1] R. Adams. Sobolev spaces. Academic Press, New York, 1975.

[2] F. Alabau. New uniqueness theorems for the one-dimensional drift-diffusion semiconductor device equations. SIAM J. Math. Anal., 26:715-737, 1995. [3] G. Albinus. A thermodynamically motivated formulation of the energy model of semiconductor devices. Preprint No. 210, WeierstraB-Institut, Berlin, Germany, 1995. [4] G. Albinus. Convex analysis of the energy model of semiconductor devices. Preprint 285, WeierstraB-Institut, Berlin, Germany, 1996. [5] W. Allegretto and H. Xie. Nonisothermal semiconductor systems. In X. Liu and D. Siegel, editors, Comparison methods and stability theory, volume 162 of Lecture Notes in Pure and Applied Mathematics, New York, 1994. Marcel Dekker. [6] H. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Math. Z., 183:311-341, 1983. [7] N. Aluru, K. Law, P. Pinsky, and R. Dutton. An analysis of the hydrodynamic semiconductor device model - boundary conditions and simulations. Compel, 14:157-185, 1995. [8] P. Amster, M. P. Beccar Varela, A. Jiingel, and M. C. Mariani. Subsonic solutions to a one-dimensional non-isentropic hydrodynamic model for semiconductors. Submitted for publication, 1999. [9] M. Ancona. Diffusion-drift models of strong inversion layers. Compel,6:11-18, 1987. [10] M. Ancona and G. Iafrate. Quantum correction to the equation of state of an electron gas in a semiconductor. Phys. Rev. B, 39:9536-9540, 1989. [11] J. Anderson. Local existence and uniqueness of solutions of degenerate parabolic equations. Comm. P. D. E., 16:105-143, 1991. [12] A. Anile and O. Muscato. Improved hydrodynamical model for carrier transport in semiconductors. Phys. Rev. B, 51:16728-16740, 1995.

268

References

[13] A. Anile and O. Muscato. Extended thermodynamics tested beyond the linear regime: the case of electron transport in semiconductors. Cont. Mech. Thermodyn., 8:131-142, 1996. [14] M. Anile and V. Romano. Non parabolic band transport in semiconductors: closure of the moment equations. Contino Mech. Thermodyn., 11:307-325, 1999. [15] S. Antontsev. On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. Sov. Math. Dokl., 24:420-424, 1981. [16] S. Antontsev and J. I. Diaz. New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by the energy method. Sov. Math. Dokl., 38:535-539, 1989. [17] S. Antontsev and J. I. Diaz. On space an time localization of solutions of nonlinear elliptic and parabolic equations via energy methods. In P. Benilan et al., editor, Recent advances in nonlinear elliptic and parabolic problems, pages 3-14. Longman, 1989. [18] S. Antontsev and J. I. Diaz. Space and time localization in the flow of two immiscible fluids through a porous medium: energy methods applied to systems. Nonlin. Anal., 16:299-313, 1991. [19] S. Antontsev and J. I. Diaz. Energy Methods for Free Boundary Problems in Continuum Mechanics. In preparation, 2000. [20] S. Antontsev, J. I. Diaz, and S. Shmarev. The support shrinking in solutions of parabolic equations with nonhomogeneous absorption terms. Ann. Fac. Sci. Toulouse, VI. Ser., Math. 4:5-30, 1995. [21] S. Antontsev, J. I. Diaz, and S. Shmarev. Energy Methods in Continuum Mechanics. Kluwer Academic Publishers, Dordrecht, 1996. [22] S. Antontsev, A. Domansky, and J. I. Diaz. Continuous dependence and stabilization of solutions of the degenerate system in two-phase filtration. Dinamika sploshnoi sredy, 107:11-25, 1993. [23] Y. Apanovich, P. Blakey, R. Cottle, E. Lyumkis, B. Polsky, A. Shur, and A. Tcherniaev. Numerical simulations of submicrometer devices including coupled nonlocal transport and nonisothermal effects. IEEE Trans. El. Dev., 42:890-897, 1995. [24] F. Arimburgo, C. Baiocchi, and L. Marini. Numerical approximation of the I-D nonlinear drift-diffusion model in semiconductors. In V. Boffi, F. Bampi, and G. Toscani, editors, Nonlinear Kinetic Theory and Mathematical Aspects of Hyperbolic Systems. World Scientific, Singapore, 1992. [25] L. Arkeryd. On the Boltzmann equation. Arch. Rat. Mech. Anal., 45:1-34, 1972. [26] L. Arkeryd, M. Esposito, and M. Pulvirenti. The Boltzmann equation for weakly inhomogeneous data. Comm. Math. Phys., 111:393-407, 1987.

References

269

[27J 1. Arlotti and N. Bellomo. On the Cauchy problem for the Boltzmann equation. In N. Bellomo, editor, Lecture Notes on the Mathematical Theory of the Boltzmann Equation, pages 1-64. World Scientific, Singapore, 1995. [28] A. Arnold, J. Lopez, and P.A. Markowich. An analysis of quantum FokkerPlanck models: a Wigner function approach. Preprint, Technische Universitat Berlin, Germany, 1999. [29] D. Arnold and F. Brezzi. Mixed and non-conforming finite element methods: implementation, post-processing and error estimates. Model. Math. Anal. Num., 19:7-32, 1985. [30] D. Aronson, M. Crandell, and L. Peletier. Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlin. AnaL, 6:1001-1022, 1982. [31J U. Ascher, J. Christiansen, and R. D. Russel. Collocation software for boundary value ODEs. Trans. Math. Soft., 7:209-222, 1981. [32J U. Ascher, P. Markowich, P. Pietra, and C. Schmeiser. A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model. Math. Models Meth. Appl. Sci., 1:347-376, 1991. [33] N. Ashcroft and N. Mermin. Solid State Physics. Sauners College, Philadelphia, 1976. [34J V. Babskii, Z. Zhukov, and V. Yudovich. Mathematical Theory of Electrophoresis - Applications to Methods of Fractionation of Biopolymers. Naukova Dumka, Kiev, 1983. [35J C. Baiocchi and A. Capelo. Variational and Quasivariational Inequalities. John Wiley and Sons, Chichester, 1984. [36] N. Balkan, editor. Hot Electrons in Semiconductors. Clarendon, Oxford, 1998. [37] V. Barcilon, D. Chen, R. Eisenberg, and J. Jerome. Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study. SIAM J. Appl. Math., 57:631-648, 1997. [38J C. Bardos, F. Golse, and C.D. Levermore. Fluid dynamical limits of kinetic equations. I. Formal derivations. J. Stat. Phys., 63:323-344, 1991. [39J C. Bardos, F. Golse, and C.D. Levermore. Fluid dynamical limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math., 46:667-753, 1993. [40J J. Barrett, J. Blowey, and H. Garcke. Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Num. Math., 80:525-556, 1998. [41] N. Ben Abdallah and P. Degond. On a hierarchy of macroscopic models for semiconductors. J. Math. Phys., 37:3308-3333, 1996. [42] N. Ben Abdallah, P. Degond, and S. Genieys. An energy-transport model for semiconductors derived from the Boltzmann equation. J. Stat. Phys., 84:205231, 1996.

270

References

[43] N. Ben Abdallah, P. Degond, and P. Markowich. On a one-dimensional Schrodinger-Poisson scattering model. Z. Angew. Math. Phys., 48:135-155, 1997. [44] N. Ben Abdallah, P. Degond, P. Markowich, and C. Schmeiser. High field approximations of the spherical harmonics expansion model for semiconductors. To appear in Z. Angew. Math. Phys., 2000. [45] N. Ben Abdallah, L. Desvillettes, and S. Genieys. On the convergence of the Boltzmann equation for semiconductors towards an energy transport model. Submitted for publication, 1999. [46] N. Ben Abdallah and A. Unterreiter. On the stationary quantum drift-diffusion model. Z. Angew. Math. Physik, 49:251-275, 1998. [47] A. Bermudez and C. Saguez. Mathematical formulation and numerical solution of an alloy solidification problem. In A. Fasano, editor, Free Boundary Problems: Theory and Applications, volume 1, pages 237-247. Pitman, 1983. [48] F. Bemis. Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension n. Comm. P. D. E., 9:271-312, 1984. [49] F. Bemis. Finite speed of propagation and continuity of the interface of thin viscous films. Adv. Diff. Eqs., 1:337-368, 1996. [50] F. Bemis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Diff. Eqs., 83:179-206, 1990. [51] A. Bertozzi. The mathematics of moving contact lines in thin liquid films. Notices of the AMS, 45:689-697, 1998. [52] A. Bertozzi and M. Pugh. Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math., 51:625-661, 1998. [53] A. Bertozzi and L. Zhomitskaya. Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Num. Anal., 37:523-555, 2000. [54] M. Bertsch and D. Hilhorst. A density dependent diffusion equation in population dynamics: stabilization to equilibrium. SIAM J. Math. Anal., 17:863-883, 1986. [55J P. Biler, A. Krzywicki, and T. Nadzieja. Self-interaction of Brownian particles coupled with thermodynamic processes. Preprint, 1999. [56J J. Blakemore. Semiconductor Statistics. Pergamon Press, Oxford, 1962. [57J P. Bleher, J. Lebowitz, and E. Speer. Existence and positivity of solutions of a forth-order nonlinear PDE describing interface fluctuations. Comm. Pure Appl. Math., 47:923-942, 1994. [58J K. Blllltekjrer. Transport equations for electrons in two-valley semiconductors. IEEE Trans. El. Dev., 17:38-47, 1970. [59J A. Bobylev and J. Struckmeier. Implicit and iterative methods for the Boltzmann equation. Transp. Theory Stat. Phys., 25:175-195, 1996.

References

271

[60] A. Bobylev and J. Struckmeier. Numerical simulation of the one-dimensional Boltzmann equation by particle methods. Europ. J. Mech., B/Fluids, 15:103118, 1996. [61] K. Boer. Survey of Semiconductor Physics. Van Nostrand Reinhold, New York, 1990. [62] V. Bonc-Bruevic and S. Kalashnikov. Halbleiterphysik. VEB Verlag der Wissenschaften, Berlin, 1982. [63] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, New York, 1991. [64] F. Brezzi, I. Gasser, P. Markowich, and C. Schmeiser. Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension. Appl. Math. Lett., 8:47-52, 1995. [65] F. Brezzi and G. Gilardi. Fundamentals of P. D. E. for Numerical Analysis. Technical report, lstituto di Analisi Numerica, Pavia, Italy, 1984. [66] F. Brezzi, L. Marini, and P. Pietra. Methodes d'elements finis mixtes et schema de Scharfetter-Gummel. C. R. Acad. Sci. Paris, 305:599-604, 1987. [67] F. Brezzi, L. Marini, and P. Pietra. Numerical simulation of semiconductor devices. Compo Meth. Appl. Mech. Engrg., 75:493-514, 1989. [68] F. Brezzi, L. Marini, and P. Pietra. Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Num. Anal., 26:1342-1355, 1989. [69] W. Bullis, D. Seiler, and A. Diebold. Semiconductor Characterization. Present Status and Future Needs. AlP Press, Woodbury, New York, 1996. [70] L. Caffarelli and N. Muler. An L oo bound for solutions of the Cahn-Hilliard equations. Arch. Rat. Mech. Anal., 133:129-144, 1995. [71] R. Caflisch. The fluid dynamical limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math., 33:651-666, 1980. [72] T. Carleman. Sur la theorie de l'equation integro-differentielle de Boltzmann. Acta Math., 60:91-146, 1933. [73] J.A. Carrillo, I. Gamba, and C.-W. Shu. Computational macroscopic approximations to the 1-D relaxation-time kinetic system for semiconductors. Submitted for publication, 1999. [74] J.A. Carrillo, A. Jiingel, P. Markowich, G. Toscani, and A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Submitted for publication, 2000. [75] J.A. Carrillo and G. Toscani. Asymptotic £l-decay of solutions of the porous media equation to self-similarity. To appear in Indiana Math. Univ. J., 2000. [76] F. Castella. L 2 solutions to the Schrodinger-Poisson system: Existence, uniqueness, time behaviour, and smoothing effect. Math. Models Meth. Appl. Sci., 7:1051-1083, 1997.

272

References

[77J T. Cazenave. An Introduction to Nonlinear Schrodinger Equations. Textos de Metodos Matematicos 26. Universidade Federal do Rio de Janeiro, Rio de Janeiro, 3rd edition, 1996. [78J C. Cercignani. The Boltzmann Equation and Its Applications. Springer, Berlin, 1988. [79J C. Cercignani and R. Illner. Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions. Arch. Rat. Mech. Anal., 134:1-16, 1996. [80] C. Cercignani, R. Illner, and M. Pulvirenti. The Mathematical Theory of Dilute Gases. Springer, New York, 1994. [81] D. Chen, E. Kan, U. Ravaioli, C. Shu, and R. Dutton. An improved energy transport model including nonparabolicity and non-Maxwellian distribution effects. IEEE Electr. Dev. Letters, 13:26-28, 1992. [82] D. Chen, E. Sangiorgi, M. Pinto, E. Kan, U. Ravaioli, and R. Dutton. Analysis of spurious velocity overshoot in hydrodynamic simulations. NUPAD IV, pages 109-114, 1992. [83] G.-Q. Chen. Convergence of Lax-Friedrichs scheme for isentropic gas dynamics III. Acta Math. Sci., 6:75-120, 1986. [84] G.-Q. Chen. The theory of compensated compactness and the system of isentropic gas dynamics. Preprint MCS-PI54-1990, University of Chicago, 1990. [85J G.-Q. Chen, J. Jerome, C.-W. Shu, and D. Wang. Two carrier semiconductor device models with geometric structure and symmetry properties. In Modeling and Computation for Applications in Mathematics, Science and Engineering, pages 103-140. Oxford University Press, 1998. [86J G.-Q. Chen, J. Jerome, and B. Zhang. Particle hydrodynamic models in biology and microelectronics: singular relaxation limits. Nonlin. Anal., 30:233-244, 1997. [87] G.-Q. Chen and D. Wang. Convergence of shock capturing schemes for the compressible Euler-Poisson equations. Comm. Math. Phys., 179:333-364, 1996. [88J G.-Q. Chen and D. Wang. Shock-capturing approximations to the compressible Euler equations with geometric structure and related equations. Z. Angew. Math. Phys., 49:341-362, 1998. [89] Z. Chen, B. Cockburn, C. Gardner, and J. Jerome. Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode. J. Compo Phys., 117:274-280, 1995. [90] Z. Chen, B. Cockburn, J. Jerome, and C. Shu. Finite element computation of the hydrodynamic model of semiconductor devices. Preprint, University of Minnesota, Minneapolis, USA, 1993. [91] Y. Choi and R. Lui. Multi-dimensional electrochemistry model. Arch. Rat. Mech. Anal., 130:315-342, 1995.

References

273

[92J P. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. [93] P. Ciarlet and J.L. Lions, editors. Handbook of Numerical Analysis, Vol. 2. Finite Element Methods, Part 1. North-Holland, Amsterdam, 1991. [94J G. Cimatti. Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. Appl. Math., 47:117-121,1989. [95] G. Cimatti and G. Prodi. Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor. Ann. Mat. Pum Appl., 152:227-237, 1988. [96] S. Cordier. Global solutions to the isothermal Euler-Poisson plasma model. Appl. Math. Lett., 8:19-24, 1995. [97] S. Cordier. Hyperbolicity of the hydrodynamic model of plasmas under the quasi-neutrality hypothesis. Math. Meth. Appl. Sci., 18:627-647, 1995. [98J R. Courant and K. O. Friedrichs. Supersonic Flow and Shock Waves. Interscience, 1967. [99J R. Dal Passo, H. Garcke, and G. Grun. On a fourth-order degenerate parabolic equation: Global entropy estimates, existence and quantitative behavior of solutions. SIAM J. Math. AnaL, 29:321-342, 1998. [100] R. Dautray and J. L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology, volume 5. Springer, Berlin, 1992. [101] P. Degond, S. Genieys, and A. Jungel. An existence and uniqueness result for the stationary energy-transport model in semiconductor theory. C. R. Acad. Sci. Paris, 324:29-34, 1997. [102] P. Degond, S. Genieys, and A. Jungel. An existence result for a strongly coupled parabolic system arising in nonequilibrium thermodynamics. C. R. A cad. Sci. Paris, 325:227-232, 1997. [103J P. Degond, S. Genieys, and A. Jungel. Symmetrization and entropy inequality for general diffusion equations. C. R. Acad. Sci. Paris, 325:963-968, 1997. [104] P. Degond, S. Genieys, and A. Jungel. A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects. J. Math. Pures Appl., 76:991-1015, 1997. [105J P. Degond, S. Genieys, and A. Jungel. A steady-state system in nonequilibrium thermodynamics including thermal and electrical effects. Math. Meth. Appl. Sci., 21:1399-1413, 1998. [106] P. Degond and A. Jungel: High-field approximation of energy-transport models for semiconductors with non-parabolic band structure. Submitted for publication, 2000. [107J P. Degond, A. Jungel, and P. Pietra. Numerical discretization of energytransport model for semiconductors with non-parabolic band structure. SIAM J. Sci. Comp., 22:986-1007, 2000.

274

References

[108] P. Degond and B. Lucquin-Desreux. The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Meth. Appl. Sci., 6:405-436, 1996. [109] P. Degond and B. Lucquin-Desreux. Transport coefficients of plasmas and disparate mass binary gases. Transp. Theory Stat. Phys., 25:595-633, 1996. [110] P. Degond and P. Markowich. On a one-dimensional steady-state hydrodynamic model for semiconductors. Appl. Math. Lett., 3:25-29, 1990. [111] P. Degond and P. Markowich. A steady state potential flow model for semiconductors. Ann. Mat. Pum Appl., 165:87-98, 1993. [112] P. Degond, F. Poupaud, and A. Yamnahakki. Particle simulation and asymptotic analysis of kinetic equations for modeling a Schottky diode. Model. Math. Anal. Num. (RAIRO),30:763-795, 1996. [113] P. Degond and C. Schmeiser. Macroscopic models for semiconductor heterostructures. J. Math. Phys., 39:1-30, 1998. [114] M. Del Pino and J. Dolbeault. Generalized Sobolev inequalities and asymptotic behaviour in fast diffusion and porous medium problems. Submitted for publication, 1999. [115] A. DeMasi, R.Esposito, and J. Lebowitz. Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math., 42:11891214, 1989. [116] Z. Deyl, editor. Electrophoresis: A Survey of Techniques and Applications, Amsterdam, 1979. Elsevier. [117] J. 1. Diaz. Qualitative study of nonlinear parabolic equations: an introduction. Preprint No. 16, Universidad Complutense de Madrid, Spain, 1999. [118] J. 1. Diaz, G. Galiano, and A. Jungel. Space localization and uniqueness of vacuum solutions to a degenerate parabolic problem in semiconductor theory. C. R. Acad. Sci. Paris, 325:267-272, 1997. [119] J. 1. Diaz, G. Galiano, and A. Jungel. On a quasilinear degenerate system arising in semiconductor theory. Part II: localization of vacuum solutions. Nonlin. Anal., 36B:569-594, 1999. [120] J. 1. Diaz, G. Galiano, and A. Jungel. On a quasilinear degenerate system arising in semiconductor theory. Part I: existence and uniqueness of solutions. To appear in Nonlin. Anal., 2000. [121] J. 1. Diaz and R. Kersner. On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. J. Diff. Eqs., 69:368-403, 1987. [122] J. 1. Diaz and L. Veron. Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc., 290:787-814, 1985.

References

275

[123J X. Ding, G.-Q. Chen, and P. Lou. Convergence of the fractional step LaxFriedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Comm. Math. Phys., 121:63-84, 1989. [124J R. DiPerna. Convergence of approximate solutions of conservation laws. Arch. Rat. Mech. Anal., 82:27-70, 1983. [125J R. DiPerna. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys., 91:1-30, 1983. [126J R. DiPerna and P.L. Lions. On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. Math., 130:321-366, 1989. [127J G. Dong. Nonlinear Partial Differential Equations of Second Order. Amer. Math. Soc. Providence, Rhode Island, 1991. [128] J. A. Dubinskii. Weak convergence in nonlinear elliptic and parabolic equations. Amer. Math. Soc. Transl., 67:226-258, 1968. [129J R. Dutton and Z. Yu. Technology CAD: computer simulation of IC processes and devices. Kluwer Academic Publishers, Dordrecht, 1993. [130J R. Eisenberg and D. Gillespie. A singular perturbation of the Poisson-NernstPlanck system: application to ion channels. Preprint, 2000. [131J J. Fan. Uniqueness for the three-dimensional time dependent drift diffusion semiconductor equations with £2 initial data. J. Math. Anal. Appl., 233:437444, 1999. [132] W. Fang and K. Ito. Asymptotic behavior of the drift-diffusion semiconductor equations. J. Diff. Eqs., 123:567-587, 1995. [133J W. Fang and K. Ito. Global solutions of the time-dependent drift-diffusion semiconductor equations. J. Diff. Eqs., 123:523-566, 1995. [134] W. Fang and K. Ito. On the time-dependent drift-diffusion model for semiconductors. J. Diff. Eqs., 117:245-280, 1995. [135] W. Fang and K. Ito. Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors. J. Diff. Eqs., 133:224-244, 1997. [136] W. Fang and K. Ito. Weak solutions to a one-dimensional hydrodynamic model for semiconductors. Nonlin. Anal., 28:947-963, 1997. [137] E. Fatemi, J. Jerome, and S. Osher. Solution of hydrodynamic device model using high-order nonoscillatory shock capturing algorithms. IEEE Trans. CAD Integr. Circuits Systems, 10:232-244, 1991. [138J E. Fatemi and F. Odeh: Upwind finite difference solution of Boltzmann equation applied to electron transport in semiconductor devices. J. Compo Phys., 108:209-217, 1993. [139J D. Ferry and C. Jacobini, editors. Plenum Press, New York, 1992.

Quantum Transport in Semiconductors.

276

References

[140] R. Feynman. Statistical Mechanics, A Set of Lectures. Frontiers in Physics. W.A. Benjamin, 1972. [141] J. Filo and J. Kaeur. Local existence of general nonlinear parabolic systems. Nonlin. Anal., 24:1597-1618, 1995. [142] M. Fournie. Construction et analyse de schemas compacts d'ordre eleve pour des problemes fortement convectifs. Application a la simulation de semiconducteurs. PhD thesis, Universite Paul Sabatier, Toulouse, France, 1999. [143] B. Fraeijs de Veubeke. Displacement and equilibrium models in the finite element method. In O. Zienkiewicz and G. Hollister, editors, Stress Analysis. Wiley, 1965. [144] D. Fraser. The Physics of Semiconductor Devices. Clarendon Press, Oxford, 3rd edition, 1983. [145] J. Frehse and J. Naumann. On weak solutions to the non-stationary van Roosbroeck's system with discontinuous permittivities. Math. Meth. Appl. Sci., 20:689-706, 1997. [146] W. Frensley. Simulation of resonant-tunneling heterostructure devices. J. Vacuum Sci. Tech. B, 3:1261-1266, 1985. [147] G. Gagneux and M. Madaune-Tort. Sur la question de l'unicite pour les inequations des milieux poreux. C. R. Acad. Sci. Paris, 314:605-608, 1992. [148] G. Gagneux and M. Madaune-Tort. Unicite des solutions faibles d'equations de diffusion-convection. C. R. Acad. Sci. Paris, 318:919-924, 1994. [149] G. Gagneux and M. Madaune-Tort. Analyse mathematique de modeles non lineaires de l'ingenierie petroliere. Springer, Paris, 1996. [150] H. Gajewski. On a variant of monotonicity and its application to differential equations. Nonlin. Anal., 22:73-80, 1994. [151] H. Gajewski. On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. Math. Models Meth. Appl. Sci., 4:121-133, 1994. [152] H. Gajewski and K. Gartner. On the discretization of Van Roosbroeck's equations with magnetic field. Z. Angew. Math. Mech., 76:247-264, 1996. [153] H. Gajewski and K. Groger. On the basic equations for carrier transport in semiconductors. J. Math. Anal. Appl., 113:12-35, 1986. [154] H. Gajewski and K. Groger. Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics. Math. Nachr., 140:736, 1989. [155] G. Galiano. Sobre algunos problemas de la Mecanica de Medios Continuos en los que se originan Pronteras Libres. PhD thesis, Universidad Complutense de Madrid, Departamento de Matematica Aplicada, Spain, 1997. [156] G. Galiano and M. A. Peletier. Spatial localization for a general reactiondiffusion system. Ann. Fac. Sci. Toulouse, 7:419-441, 1997.

References

277

[157] 1. Gamba. Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors. Comm. P. D. E., 17:553-577, 1992. [158] 1. Gamba. Sharp uniform bounds for steady potential fluid-Poisson systems. Pmc. Roy. Soc. Edinb., Sect. A, 127:479-516, 1997. [159J 1. Gamba and A. Jiingel. Positive solutions to singular second and third order differential equations for quantum fluids. To appear in Arch. Rat. Mech. Anal., 2000. [160] 1. Gamba and A. JUngel: Asymptotic limits in quantum trajectory models. In preparation, 2000. [161] 1. Gamba and C. Morawetz. A viscous approximation for a 2D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow. Comm. Pure Appl. Math., 49:999-1049, 1996. [162] C. Gardner. Numerical simulation of a steady-state electron shock wave in a submicron semiconductor device. IEEE Trans. El. Dev., 38:392-398, 1991. [163J C. Gardner. The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math., 54:409-427, 1994. [164J C. Gardner, J. Jerome, and D. Rose. Numerical methods for the hydrodynamic device model: subsonic flow. IEEE Trans. CAD, 8:501-507, 1989. [165] C. Gardner and C. Ringhofer. The smooth quantum potential for the hydrodynamic model. Phys. Rew. E, 53:157-167, 1996. [166J C. Gardner and C. Ringhofer. Approximation of thermal equilibrium for quantum gases with discontinuous potentials and applications to semiconductor devices. SIAM J. Appl. Math., 58:780-805, 1998. [167] 1. Gasser. The initial time layer problem and the quasi-neutral limit in a degenerate drift-diffusion model for semiconductors. To appear in Nonlin. Diff. Eqs. Appl. (NoDEA), 2000. [168] 1. Gasser. A note on traveling wave solutions for a quantum hydrodynamic model. To appear in Appl. Math. Lett., 2000. [169J 1. Gasser and A. Jiingel. The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys., 48:45-59, 1997. [170J I. Gasser, A. Jiingel, M. C. Mariani, and D. Rial. hydrodynamic equations. In preparation, 2000.

The transient quantum

[171] I. Gasser and P. A. Markowich. Quantum hydrodynamics, Wigner transforms and the classical limit. Asympt. Anal., 14:97-116, 1997. [172J I. Gasser, P. A. Markowich, and C. Pohl. A dispersive numerical scheme for quantum hydrodynamics. Proceedings of ISSDT 96, pages 178-181, 1995. [173J I. Gasser, P. A. Markowich, D. Schmidt, and A. Unterreiter. Macroscopic theory of charged quantum fluids. In P. Marcati, editor, Mathematical Problems in Semiconductor Physics, volume 340, pages 42-75, Rome, Italy, 1995. Pitman.

278

References

[174] I. Gasser, P. A. Markowich, and A. Unterreiter. Quantum hydrodynamics. In Proceedings of the SPARCH CdR Conference, St. Malo, 1995. [175] I. Gasser and R. Natalini. The energy-transport and the drift-diffusion equations as relaxation limits of the hydrodynamic model for semiconductors. Quart. Appl. Math., 57:269-282, 1999. [176] E. Gatti, S. Micheletti, and R. Sacco. A new Galerkin framework for the driftdiffusion equation in semiconductors. East- West J. Num. Math., 6:101-135, 1998. [177] S. Genieys. Energy-transport model for a non degenerate semiconductor. Convergence of the Hilbert expansion in the linearized case. Asympt. Anal., 17:279308, 1998. [178] P. Gerard. Solutions globales du probleme de Cauchy pour l'equation de Boltzmann. Seminaire de Bourbaki, Asterisque, 161-162, 257-280, 1988. [179] M. Giaquinta. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, New Jersey, 1983. [180] D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 2nd edition, 1983. [181] A. Gnudi, F. Odeh, and M. Rudan. Investigation of non-local transport phenomena in small semiconductor devices. Europ. Trans. Telecomm., 1:307-312, 1990. [182] F. Golse and F. Poupaud. Limite fluide des equations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac. Asympt. Anal., 6:135-160, 1992. [183] H. Grad. Asymptotic equivalence of the Navier-Stokes and non-linear Boltzmann equation. In Proceedings of the AMS, volume 17, pages 154-183,1965. [184] J. Griepentrog. An application of the implicit function theorem to an energy model of the semiconductor theory. Z. Angew. Math. Mech., 79:43-51, 1999. [185] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. [186] K. Gr6ger. A W1'P-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann., 283:679-687, 1989. [187] K. Gr6ger and J. Rehberg. Uniqueness for the two-dimensional semiconductor equations in case of high carrier densities. Math. Z., 213:523-530, 1993. [188] S. De Groot. Thermodynamik irreversibler Prozesse. Bibliographisches Institut, Mannheim, 1960. [189] H. Grubin and J. Kreskovsky. Quantum moment balance equations and resonant tunneling structures. Solid-State Electr., 32:1071-1075, 1989. [190] G. Griin and M. Rumpf: Nonnegativity preserving convergent schemes for the thin film equation. To appear in Num. Math., 2000.

References

279

[191] H. Gummel. A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. El. Dev., ED-11:455-465, 1964. [192] M. T. Gyi and A. Jungel. A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Diff. Eqs., 5:773-800, 2000. [193] H. Hang and A.P. Jauko. Quantum Kinetics in Transport and Optics of Semiconductors. Springer, Berlin, 1996. [194] W. Hansch. The Drift-Diffusion Equation and Its Applications in MOSFET Modeling. Springer, Wien, 1991. [195] F. Hecht and A. Marrocco. Mixed finite element simulation of heterojunction structures including a boundary layer model for the quasi-Fermi levels. In J. Miller, editor, Proc. of the Nasecode X Conf. Boole Press, Dublin, 1994. [196] R. Hills, D. Loper, and P. Roberts. A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Math., 36:505-539, 1983. [197] R. Hiptmair and R. Hoppe. Mixed finite element discretization of continuity equations arising in semiconductor device simulation. In R. Bank, editor, Mathematical modelling and simulation of electrical circuits and semiconductor devices. Birkhauser, 1994. [198] S. Howison, J. Rodrigues, and M. Shillor. Stationary solutions to the thermistor problem. J. Math. Anal. Appl., 174:573-588, 1993. [199] L. Hsiao and K.J. Zhang. The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations. J. Diff. Eqs., 165:315-354, 2000. [200] G. Iafrate. Quantum transport and the Wigner function. In H. Grubin, D. Ferry, and C. Jacobini, editors, The Physics of Submicron Semiconductor Devices. Plenum Press, New York, 1988. [201] R. Illner, O. Kavian, and H. Lange. Stationary solutions of quasi-linear Schrodinger-Poisson system. J. Diff. Eqs., 145:1-16, 1998. [202] C. Jacobini and P. Lugli: The Monte Carlo Method for Semiconductor Device Simulation. Springer, Wien, 1989. [203] C. Jacoboni, R. Minder, and G. Majni: Effects of band non-parabolicity on electron drift-velocity in silicon above room temperature. J. Phys. Chem. Solids, 36:1129-1133, 1975. [204] J. Jerome. Approximation of Nonlinear Evolution Systems. Academic Press, New York, 1983. [205] J. Jerome. The approximation problem for drift-diffusion systems. SIAM Review, 37:552-572, 1995. [206] J. Jerome. Analysis of charge transport. A mathematical study of semiconductor devices. Springer, Berlin, 1996. [207] J. Jerome and T. Kerkhoven. A finite-element approximation theory for the drift-diffusion semiconductor model. SIAM J. Num. Anal., 28:403-422, 1991.

280

References

[208J J. Jerome and C.-W. Shu. Energy models for one-carrier transport in semiconductor devices. In W. Coughran, J. Colde, P. Lloyd, and J. White, editors, Semiconductors, Part II, volume 59 of IMA Volumes in Mathematics and its Applications, pages 185-207, New York, 1994. Springer. [209] J. Jerome and C.-W. Shu. Energy transport systems for semiconductors: Analysis and simulation. In Proceedings of the First World Congress of Nonlinear Analysts, Berlin, 1995. Walter de Gruyter.

[210] J. Jerome and C.-W. Shu. Transport effects and characteristic models in the modeling and simulation of submicron devices. IEEE Trans. CAD Integr. Circuits Systems, 14:917-923, 1995. [211] X. Jiang. Numerical Simulations of the Semiconductor Devices by StreamlineDiffusion Methods. PhD thesis, University of British Columbia, Vancouver, Canada, 1995. [212] X. Jiang. A streamline-upwindingjPetrov-Galerkin method for the hydrodynamic semiconductor device model. Math. Models Meth. Appl. Sci., 5:659-681, 1995. [213J F. Jochmann. On the Time-Dependent Hydrodynamic Model for Semiconductors. PhD thesis, TU Berlin, Germany, 1992. [214] F. Jochmann. Global weak solutions of the one-dimensional hydrodynamic model for semiconductors. Math. Models Meth. Appl. Sci., 3:759-788, 1993. [215J O. John and J. Stara. On the regularity and non-regularity of elliptic and parabolic systems. In J. Kurzweil, editor, Equadiff 7, Teubner-Texte Math. 118, pages 28-36, Leipzig, 1990. Teubner. [216J S. Junca and M. Rascle. Relaxation of the isothermal Euler-Poisson system to the drift-diffusion equations. To appear in Quart. Appl. Math., 2000. [217] S. Junca and M. Rascle. Strong relaxation of the isothermal Euler-Poisson system to the heat equation. Preprint, Universite de Nice, France, 1999. [218] A. Jungel. On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Models Meth. Appl. Sci., 4:677-703, 1994. [219J A. JUngel. The free boundary problem of a semiconductor in thermal equilibrium. Math. Meth. Appl. Sci., 18:387-412, 1995. [220] A. Jungel. Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. Z. Angew. Math. Mech., 75:783-799, 1995. [221] A. JUngel. Qualitative behavior of solutions of a degenerate nonlinear driftdiffusion model for semiconductors. Math. Models Meth. Appl. Sci., 5:497-518, 1995. [222J A. Jungel. Asymptotic analysis of a semiconductor model based on Fermi-Dirac statistics. Math. Meth. Appl. Sci., 19:401-424, 1996.

References

281

[223] A. Jiinge!. A degenerate quasi-hydrodynamic model for semiconductor devices. Proceedings of ICIAM 95, Z. Angew. Math. Mech., 76:563-564, 1996. [224] A. Jiinge!. Stationary transport equations for charge carriers in semiconductors including electron-hole scattering. Appl. AnaL, 62:53-69, 1996. [225] A. Jiinge!. The energy-transport model for semiconductors: some analytical and numerical results. In Proceedings of the International Workshop on "Recent Progress in the Mathematical Theory on Vlasov-Maxwell Equations" in Paris, pages 140-158, 1997. [226] A. Jiinge!. A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr., 185:85-110, 1997. [227] A. Jiinge!. A note on current-voltage characteristics from the quantum hydrodynamic equations for semiconductors. Appl. Math. Lett., 10:29-34, 1997. [228] A. Jiinge!. A steady-state quantum Euler-Poisson system for potential flows. Comm. Math. Phys., 194:463-479, 1998. [229] A. JUnge!. Regularity and uniqueness of solutions to a system of parabolic equations in nonequilibrium thermodynamics. Nonlin. Anal., 41:669-688, 2000. [230] A. JUngel and Y.-J. Pengo A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations. To appear in Annales I.H.P., Anal. non lin., 17:83-118,2000. [231] A. Jiingel and Y.J. Pengo A hierarchy of hydrodynamic models for plasmas: quasi-neutral limits in the drift-diffusion equations. Submitted for publication, 1999. [232] A. Jiingel and Y.J. Pengo A hierarchy of hydrodynamic models for plasmas: zero-relaxation-time limits. Comm. P.D.E., 24:1007-1033, 1999. [233] A. JUngel and Y.J. Pengo Rigorous derivation of a hierarchy of macroscopic models for semiconductors and plasmas. To appear in Equadiff' 99, Berlin, 2000. [234] A. JUngel and Y.J. Pengo Zero-relaxation-time limits in hydrodynamic models for plasmas revisited. Z. Angew. Math. Phys., 51:385-396, 2000. [235] A. Jiingel and P. Pietra. A discretization scheme of a quasi-hydrodynamic semiconductor mode!. Math. Models Meth. Appl. Sci., 7:935-955, 1997. [236] A. Jiingel and R. Pinnau. Global non-negative solutions of a fourth-order nonlinear parabolic equation for quantum systems. To appear in SIAM J. Math. Anal.,2000. [237] A. Jiingel and R. Pinnau. A positivity-preserving numerical scheme for a fourthorder parabolic equation. Submitted for publication, 2000. [238] A. Jiingel and R. Pinnau. Optimal convergence rate of a positivity-preserving numerical scheme for a fourth-order parabolic equation. In preparation, 2000.

282

References

[239] A. Jungel and C. Schmeiser. Voltage-current characteristics of a pn-diode from a drift-diffusion model with nonlinear diffusion. Quart. Appl. Math., 55:707-721, 1997. [240] N.-C. Kaiser and J. Rehberg. On stationary Schrodinger-Poisson equations modelling an electron gas with reduced dimension. Math. Meth. Appl. Sci., 20:1283-1312, 1997. [241] A. S. Kalashnikov. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russ. Math. Surveys, 42:169-222, 1987. [242] E. Kane. Band structure of Indium-Antimonide. J. Phys. Chem. Solids, 1:249, 1957. [243] J. Kocur. Method of Rothe in Evolution Equations. Teubner, Leipzig, 1985. [244] S. Kawashima and Y. Shizuta. On the normal form of the symmetric hyperbolicparabolic systems associated with the conservation laws. Tohoku Math. J., II. Ser. 40:449-464, 1988. [245] T. Kerkhoven and Y. Saad. On acceleration methods for coupled nonlinear elliptic systems. Num. Math., 60:525-548, 1992. [246] M. Kern. Nonzero space charge approximation of thyristors. Math. Nachr., 157: 169-183, 1992. [247] R. Kersner and A. Shishkov. Instantaneous shrinking of the support of energy solutions. J. Math. Anal. Appl., 198:729-750, 1996. [248] N. Kluksdahl, A. Kriman, D. Ferry, and C. Ringhofer. Self-consistent study of the resonant tunneling diode. Phys. Rev. B, 39:7720-7735, 1989. [249] H. Kreuzer. Nonequilibrium Thermodynamics and Its Statistical Foundation. Clarendon Press, Oxford, 1981. [250] U. Krumbein, P. Yoder, A. Benvenuti, A. Schenk, and W. Fichtner. Full-band Monte-Carlo transport calculation in an integrated simulation platform. In H. Ryssel and P. Pichler, editors, Simulation of Semiconductor Devices and Processes, volume 6, pages 400-403, 1995. [251] S. Kruzhkov. First order quasilinear equations in several independent variables. Math. USSR Sbornik, 1O:217~243, 1970. [252] M. Lachowicz. On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Meth. Appl. Sci., 9:342-366, 1987. [253] O. A. Ladyzenskaya, V. A. Solonnikov, and N. N. Ural'ceva. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, 1968.

[254] H. Lange and P. Zweifel. Mixed boundary-value problems for the twodimensional Schrodinger-Poisson system. Rep. Math. Phys., 39:283-297, 1997. [255] C. Lattenzio: On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit. Math. Models Meth. Appl. Sci., 10:351-360, 2000.

References

283

[256] C. Lattenzio and P. Marcati: The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors. Discrete Contino Dynam. Systems, 5:449-455, 1999. [257] P. Lax. Shock waves and entropy. In E. Zarantello, editor, Contributions to Nonlinear Functional Analysis, pages 603-634, New York, 1971. Academic Press. [258] C.D. Levermore. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83:1021-1065, 1996. [259] J. L. Lions. Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Paris, 1969. [260] P.L. Lions, B. Perthame, and E. Souganidis. Existence of entropy solutions for the hyperbolic system of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math., 44:599-638, 1996. [261] P.L. Lions, B. Perthame, and E. Tadmor. Kinetic formulation for the isentropic gas dynamics and p-system. Comm. Math. Phys., 163:415-431, 1994. [262] M. Loffredo and L. Morato. Lagrangian variational principle in stochastic mechanics: Gauge structure and stability. J. Math. Phys., 30:354-360, 1989. [263] M. Loffredo and L. Morato. On the creation of quantum vortex lines in rotating Hell. Il nouvo cimento, 108B:205-215, 1993. [264] E. Lyumkis, B. Polsky, A. Shur, and P. Visocky. Transient semiconductor device simulation including energy balance equation. Compel, 11:311-325, 1992. [265] E. Madelung. Quantentheorie in hydrodynamischer Form. Z. Physik, 40:322, 1927. [266] A. Majorana. Space homogeneous solutions of the Boltzmann equation describing electron-phonon interactions in semiconductors. Transp. Theory Stat. Phys., 20:261-279, 1991. [267] A. Majorana and G. Russo. Stationary solutions of hydrodynamic models for semiconductor device simulation. Compel, 12:81-93, 1993. [268] P. Marcati and R. Natalini. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift diffusion equations. Arch. Rat. Mech. Anal., 129:129-145, 1995. [269] P. Marcati and R. Natalini. Weak solutions to a hydrodynamic model for semiconductors: The Cauchy problem. Proc. Roy. Soc. Edinb., Sect. A, 125:115131, 1995. [270] L.D. Marini and P. Pietra. New mixed finite element schemes for current continuity equations. Compel, 9:257-268, 1990. [271] P. A. Markowich. A singular perturbation analysis of the fundamental semiconductor device equations. SIAM J. Appl. Math., 44:898-928, 1984. [272] P. A. Markowich. The Stationary Semiconductor Device Equations. Springer, Wien, 1986.

284

References

[273J P. A. Markowich. On steady state Euler-Poisson models for semiconductors. Z. Angew. Math. Phys., 42:385-407, 1991. [274J P. A. Markowich and C. A. Ringhofer. A singularly perturbed boundary value problem modelling a semiconductor device. SIAM J. Appl. Math., 44:231-256, 1984. [275J P. A. Markowich, C. A. Ringhofer, and C. Schmeiser. Semiconductor Equations. Springer, 1990. [276J P. A. Markowich and A. Unterreiter. Vacuum solutions of the stationary driftdiffusion model. Ann. Sc. Norm. Sup. Pisa, 20:371-386, 1993. [277J A. Marrocco, P. Montarnal, and B. Perthame. Simulation of the energytransport and simplified hydrodynamic models for semiconductor devices using mixed finite elements. In Proceedings ECCOMAS 96, London, 1996. John Wiley. [278] N. Maslova. Nonlinear Evolution Equations. Kinetic Approach. World Scientific, Singapore, 1993. [279J A. Matsumura and T. Nishida. Initial boundary value problems for the equations of motion of compressible viscons and heat-conductive fluids. Comm. Math. Phys., 89:445-464, 1983. [280J M. Mock. Transport equations in heavily doped silicon, and the current gain of a bipolar transistor. Solid-State Electr., 16:1251-1259, 1973. [281J M. Mock. An initial value problem from semiconductor device theory. SIAM J. Math. AnaL, 5:597---612, 1974. [282J M. Mock. Analysis of Mathematical Models of Semiconductor Devices. Book Press, Dublin, 1983. [283J J. Molenaar. Multigrid Methods for Semiconductor Device Simulation. PhD thesis, Universiteit van Amsterdam, The Netherlands, 1992. [284J T. Moss, editor. Handbook on Semiconductors, volume 4. North-Holland, Amsterdam, 1981. [285] 1. Muller and T. Ruggeri. Rational Extended Thermodynamics. Springer, New York,1998. [286] S. Muto, T. Inata, H. Ohnishi, N. Yokoyama, and S. Hiyamizu. Effect of silicon doping profile on I-V characteristics of an AlGaAsjGaAs resonant tunneling barrier structure grown by MBE. Japanese J. Appl. Phys., 25:L557-L579, 1986. [287] A. Nakagawa. One-dimensional device model of the npn bipolar transistor including heavy doping effects under Fermi statistics. Solid State Electr., 22:943949, 1979. [288] R. Natalini. The bipolar hydrodynamic model for semiconductors and the driftdiffusion equations. J. Math. Anal. Appl., 198:262-281, 1996. [289J J. Naumann. On the existence, uniqueness and interior regularity of weak solutions to the stationary drift-diffusion equations in semiconductor theory. Preprint, Fachbereich Mathematik der Universitat Bonn, Germany, 1993.

References

285

[290] J. Naumann. Stationary semiconductor equations modelling avalanche generation. J. Math. Anal. Appl., 198:685-702, 1996. [291] J. Naumann and M. Wolff. On a global Lq estimate on weak solutions of nonlinear elliptic systems. Preprint, Dipartimento di Matematica, Universita di Catania, Italy, 1991. [292] B. Niclot, P. Degond, and F. Poupaud. Deterministic particle simulations of the Boltzmann transport equation of semiconductors. J. Compo Phys., 78:313-350, 1988. [293] T. Nishida. Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Comm. Math. Phys., 61:119-148,1978. [294] R. Nochetto. A note on the approximation of the free boundaries by finite element methods. Model. Math. Anal. Num. (RA IR 0), 20:355-368, 1986. [295] F. Otto. L1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. DiJJ. Eqs., 131:20-38, 1996. [296] F. Otto. The geometry of dissipative evolution equations: the porous medium equation. To appear in Comm. P.D.E., 2000. [297] R. Van Overstraeten, H. Deman, and R. Mertens. Transport equations in heavy doped silicon. IEEE Trans. El. Dev., ED-20:290-298, 1973. [298] F. Pacard and A. Unterreiter. A variational analysis of the thermal equilibrium state of charged quantum fluids. Comm. P. D. E., 20:885-900, 1995. [299] J. Park and J. Jerome. Qualitative properties of steady-state Poisson-NernstPlanck systems: mathematical study. SIAM J. Appl. Math., 57:609-630, 1997. [300] R. Paul. Halbleiterphysik. Hiithig-Verlag, Heidelberg, 1975. [301] J. Peng and J. Frey. Necessity of inclusion of electron-electron scattering in Monte-Carlo and energy-transport models. In J. Miller, editor, Proc. of the Nasecode X ConJ., Dublin, 1994. Boole Press. [302] P. Pietra and C. Pohl. Weak limits of the quantum hydrodynamic model. VLSI Design, Proceedings of the "Workshop on Quantum Kinetic Theory", 1999. [303J R. Pinnau. The linearized transient quantum drift diffusion model - stability of stationary states. Z. Angew. Math. Mech., 80:327-344, 1999. [304] R. Pinnau. A note on boundary conditions for quantum hydrodynamic models. Appl. Math. Lett., 12:77-82, 1999. [305] R. Pinnau and A. Unterreiter. The stationary current-voltage characteristics of the quantum drift-diffusion model. SIAM J. Num. Anal., 37:211-245, 1999. [306] C. Pohl. On the numerical treatment of dispersive equations. PhD thesis, Technical University of Berlin, Germany, 1999. [307] B. Polsky and J. Rimshans. Two-dimensional numerical simulation of bipolar semiconductor devices taking into account heavy doping effects and Fermi statistics. Solid-State Electr., 26:275-279, 1983.

286

References

Etude matMmatique et simulations numeNques de quelques equations de Boltzmann. PhD thesis, Universite de Nice, France, 1988.

[308] F. Poupaud.

[309] F. Poupaud. Diffusion approximation of the linear semiconductor Boltzmann equations: Analysis of boundary layers. Asympt. Anal., 4:293-317, 1991.

[310] F. Poupaud. Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory. Z. Angew. Math. Mech., 72:359-372, 1992. [311] F. Poupaud. Mathematical theory of kinetic equations for transport modeling in semiconductors. In B. Perthame, editor, Advances in Kinetic Theory and Computing. World Scientific, Singapore, 1994. [312J F. Poupaud, M. Rascle, and J. Vila. Global solutions to the isothermal EulerPoisson system with arbitrarily large data. J. Diff. Eqs., 123:93-121, 1995. [313] F. Poupaud and C. Schmeiser. Charge transport in semiconductors with degeneracyeffects. Math. Meth. Appl. Sci., 14:301-318, 1991. [314J F. Poupaud and A. Yamnahakki. Conditions aux limites du second ordre pour les equations de derive-diffusion des semiconducteurs. Technical report, E. N. S. Cachan, Paris, France, 1992. [315J Y. Qiu and KJ. Zhang. On the relaxation limits of the hydrodynamic model for semiconductor devices. Preprint, Universite Paul Sabatier, Toulouse, France, 1999. [316] C. Rafferty, B. Biegel, Z. Yu, M. Ancona, J. Bude, and R. Dutton. Multidimensional quantum effect simulation using a density-gradient model and script-level programming techniques. In Proceedings ISISPAD98, page 137, Lueven, Belgium, 1998. [317J P. Raviart and J. Thomas. A mixed finite element method for second order elliptic equations. In Mathematical Aspects of the Finite Element Method, volume 606 of Lecture Notes in Math., pages 292-315. Springer, 1977. [318J V. Romano and G. Russo. Numerical solution for hydrodynamic models of semiconductors. Preprint, Universita dell'Aquila, Italy, 1997. [319J W. van Roosbroeck. Theory of flow of electrons and holes in germanium and other semiconductors. Bell Syst. Techn. J., 29:560-607, 1950. [320] M. Rudan, A. Gnudi, and W. Quade. A generalized approach to the hydrodynamic model of semiconductor equations. In G. Baccarani, editor, Process and Device Modeling for Microelectronics, Amsterdam, 1993. Elsevier. [321J M. Rudan and F. Odeh. Multi-dimensional discretization scheme for a hydrodynamic model of semiconductor devices. Compel, 5:149-183, 1986. [322] M. Rudan, F. Odeh, and J. White. Numerical solution of the hydrodynamic model for a one-dimensional semiconductor device. Compel, 6:151-170, 1987. [323J J. Rulla. Weak solutions to Stefan problems with prescribed convection. SIAM J. Math. Anal., 18:1784-1800, 1987.

References

287

[324] R. Sacco and F. Saleri. Mixed finite-volume methods for semiconductor device simulation. Num. Meth. P.D.E., 13:215-236, 1997. [325] C. Sah, R. Noyce, and W. Shockley. Carrier generation and recombination in p-n junctions and p-n junction characteristics. Proc. of the IRE, 45:1228-1242, 1957. [326] D. Scharfetter and H. Gummel. Large signal analysis of a Silicon Read diode oscillator. IEEE Trans. El. Dev., ED-16:64-77, 1969. [327] C. Schmeiser. A singular perturbation analysis of reverse biased pn-junctions. SIAM J. Math. Anal., 21:313-326, 1990. [328] C. Schmeiser. A model for the transient behaviour of long-channel MOSFETs. SIAM J. Appl. Math., 54:175-194, 1994. [329] C. Schmeiser and A. Zwirchmayr. Elastic and drift-diffusion limits of electronphonon interaction in semiconductors. Math. Models Meth. Appl. Sci., 8:37-53, 1998. [330] D. Schmidt. Der dispersive Grenzwert der quantenhydrodynamischen Gleichungen. Diploma thesis, Technical University Berlin, Germany, 1995. [331] T. I. Seidman. The transient semiconductor problem with generation terms. In R. Bank, editor, Computational aspects of VLSI design with an emphasis on semiconductor device simulation, Proc. SIAM/AMS, volume 25 of Lect. Appl. Math., pages 75~87, 1990. [332] T. I. Seidman and G. M. Troianiello. Time-dependent solutions of a nonlinear system arising in semiconductor theory. Nonlin. Anal., 9:1137-1157, 1985. [333] S. Selberherr. Analysis and Simulation of Semiconductor Devices. Springer, Wien,1984. [334] E. Shamir. Regularization of mixed second-order elliptic problems. Israel J. Math., 6:150-168, 1968. [335] N. Shigesada, K. Kawasaki, and E. Teramoto. Spatial segregation of interacting species. J. Theor. Biol., 79:83-99, 1979. [336] C.-W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Report No. 97-65, NASA Langley Research Center, Hampton, USA, 1997. [337] J. Simon. Compact sets in the space U(O, T; B). Ann. Math. Pura Appl., 146:65-96, 1987. [338] K. Souissi, F. Odeh, and A. Gnudi. A note on current discretization in the hydromodel. Compel, 10:475-485, 1991. [339] K. Souissi, F. Odeh, H. Tang, and A. Gnudi. Comparative studies of hydrodynamic and energy transport models. Compel, 13:439-453, 1994. [340] G. Stampacchia. Equations elliptiques du second ordre Ii coefficients discontinus. Les Presses de l'Universite de Montreal, Canada, 1966.

288

References

[341J J. Stara, O. John, and J. MalY. Counterexample to the regularity of weak solutions of the quasilinear parabolic system. Comment. Math. Univ. Carolin., 27:123-136, 1986. [342J S. Sze. Physics of Semiconductor Devices. John Wiley, New York, 1981. [343] M. Taylor. Pseudodifferential Operators. Princeton University Press, Princeton, 1981. [344J R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988. [345] E. Thomann and F. Odeh. On the well-posedness of the two-dimensional hydrodynamic model for semiconductor devices. Compel, 9:45-57, 1990. [346] F. Treves. Introduction to Pseudodifferential Operators and Fourier Integral Operators, Vol. 1 and 2. Plenum Press, New York, 1980. [347J G. M. Troianiello. Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York, 1987. [348J H. Tsuchiya and T. Miyoshi. Quantum transport modeling of ultrasmall semiconductor devices. IEICE Trans. Electron., E82-C:88Q-888, 1999. [349] S. Ukai and K. Asano. The Euler limit and the initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J., 12:311-332, 1983. [350J G. Ulich. Existenz einer Losung fur ein hydrodynamisches Modell aus der Halbleiterphysik. Diploma thesis, Technical University Berlin, Germany, 1991. [351J A. Unterreiter. Vacuum and non-vacuum solutions of the quasi-hydrodynamic semiconductor model in thermal equilibrium. Math. Models Appl. Sci., 18:225254, 1995. [352J A. Unterreiter. The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model. Comm. Math. Phys., 188:69-88, 1997. [353] O. Vanbesien. Simulation et caracterisation electrique des diodes double barriere d effet tunnel resonnant. PhD thesis, Lille, France, 1991. [354J H. B. da Veiga. On the W 2 ,p -regularity for solutions of mixed problems. J. Math. Pures Appl., 53:279-290, 1974. [355J H. B. da Veiga. On the semiconductor drift-diffusion equations. Differ. Int. Eqs., 9:729-744, 1996. [356] J. Viallet and S. Mottet. Transient simulation of heterostructure. In J. Miller, editor, Proc. of the Nasecode IV Conf., Dublin, 1985. Boole Press. [357J P. Visocky. A method for transient semiconductor device simulation using hotelectron transport equations. In J. Miller, editor, Proc. of the Nasecode X Conf., Dublin, 1994. Boole Press. [358] P. Visocky. On decoupled algorithm for steady state semiconductor device simulation based on Stratton's energy balance model. Preprint, University Rostock, Germany, 1996.

References

289

[359] W. Walus. Computational methods for the Boltzmann equation. In N. Bellomo, editor, Lecture Notes on the Mathematical Theory of the Boltzmann Equation, pages 179-223. World Scientific, Singapore, 1995. [360] D. Wang. Global solutions to the Euler-Poisson equations of two-carrier types in one dimension. Z. Angew. Math. Phys., 48:680-693, 1997. [361] D. Wang and G.-Q. Chen. Formation of singularities in compressible EulerPoisson fluids with heat diffusion and damping relaxation. J. Diff. Eqs., 144:4465, 1998. [362] S. Wang. A new exponentially fitted triangular finite element method for the continuity equations in the drift-diffusion model of semiconductor devices. Model. Math. Anal. Num., 33:99-112, 1999. [363] M. Ward, L. Reyna, and F. Odeh. Multiple steady-state solutions in a multijunction semiconductor device. SIAM J. Appl. Math., 51:90-123, 1991. [364] A. Wettstein: Quantum Effects in MOS Devices. Series in Microelectronics, volume 94, Hartung-Gorre, Konstanz, 2000. [365] A. Wettstein, A. Schenk, A. Scholze, and W. Fichtner. The influence oflocalized states on gate tunnel currents - modeling and simulation. In SISPAD97, International Conference on Simulation of Semiconductor Processes and Devices, pages 101-104. IEEE, New York, 1997. [366] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749-759, 1932. [367] D. Woolard, M. Stroscio, M. Littlejohn, R. Trew, and H. Grubin. A new nonparabolic hydrodynamic model with quantum corrections. In K. Hess, editor, Computational Electronics: Semiconductor Transport and Device Simulation, Boston, 1991. Kluwer Academic Publishers. [368] H. Xie and W. Allegretto. COCO) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. AnaL, 22:1491-1499, 1990. [369] C. Yazhe. Existence and uniqueness of weak solutions of uniformly degenerate quasilinear parabolic equations. Chin. Ann. of Math., 6B:131-145, 1985. [370] L. Yeh. Well-posedness of the hydrodynamic model for semiconductors. Math. Meth. Appl. Sci., 19:1489-1507, 1996. [371] L. Yeh. Subsonic solutions of the hydrodynamic model for semiconductors. Math. Meth. Appl. Sci., 20:1389-1410, 1997. [372] Z. Yu, R. Dutton, and R. Kiehl. Circuit/Device modeling at the quantum level. In Proceedings of the 6th International Workshop on Computational Electronics, pages 222-229, Osaka, Japan, 1998. [373] E. Zeidler. Nonlinear Functional Analysis and its Applications, Vol. II. Springer, New York, 1990.

290

References

[374J B. Zhang. Convergence of the Gudonov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices. Comm. Math. Phys., 157:1-22, 1993. [375J B. Zhang and J. Jerome. On a steady-state quantum hydrodynamic model for semiconductors. Nonlin. Anal., 26:845-856, 1996. [376] K.J. Zhang. Global weak solutions of the Cauchy problem to a hydrodynamic model for semiconductors. To appear in Journal of P.D.E., 1999. [377J J.R. Zhou and D. Ferry. Modeling of quantum effects in ultrasmall HEMT devices. IEEE Trans. Electron. Dev., 40:421-427, 1993. [378J C. Zhu and H. Hattori. Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors. J. Diff. Eqs., 144:353-389, 1998.

Index

acceptor, 22 adiabatic, 30 alloy solidification process, 15 avalanche, 24 ballistic diode, 25, 185 band structure, 21 biophysics, 15 bipolar model, 9 bipolar transistor, 24 Bohm potential, 193 Boltzmann equation, 1, 120 Boltzmann statistics, 29 boundary conditions, 31, 127, 196, 223, 244 Brillouin zone, 119 built-in potential, 23, 96, 105 bulk, 25 Caratheodory function, 37 channel, 25, 185 Chavent's theorem, 47 chemical potential, 30 Chen's model, 132, 186 classical limit, 18, 217 closure condition, 5, 194 COLSYS, 235 conduction band, 21 convergence rate, 35 current conservation, 17 current-voltage characteristic, 17, 112, 187, 238, 242, 262 de Broglie length, 239 Debye length, 33, 180 degenerate equation, 8, 16, 18, 37

degenerate semiconductor, 32 density of states, 28, 124 depletion region, 106 diffusion potential, 96 diffusivity, 28 diode, 94, 235 donor, 22 doping profile, 22 drain, 25 drift-diffusion model, 8 dual entropy variable, 135 dual method, 16, 18, 36, 48, 166 Dubinskii's theorem, 46 Einstein relations, 29, 32, 95 electro-chemical potential, 135 electro-chemistry, 15 electron-electron collisions, 120 energy gap, 21 energy method, 16,63 energy-transport model, 6, 15, 119 enthalpy, 195 entropy, 133, 136, 141, 247 entropy inequality, 136 entropy production, 137 entropy variable, 134 Euler equation, 193 Euler method, 84, 91 existence of solutions, 36, 137, 170, 197, 224 exponential fitting, 16, 90, 99, 181 exponential transformation, 223, 250 extensive variable, 137 extrinsic semiconductor, 22

Index

292 Fermi energy, 32 Fermi integral, 28 Fermi-Dirac statistics, 16, 28 finite speed of propagation, 16, 62 Fokker-Planck approximation, 126 formation of vacuum, 16, 62 forward biased, 23, 106 fourth-order equation, 223, 244 free boundary, 16, 104 Gagliardo-Nirenberg inequality, 163 Gamma function, 131 gate, 25 generalized solution, 38, 207, 213 generation, 23 Gummel method, 102 Harnack's inequality, 207 high injection, 17, 24, 32, 106, 113 Hilbert expansion, 120, 123 hole, 22 hot electrons, 18 hydrodynamic model, 3 ideal gas, 30 impact ionization, 24 initial time layer, 264 insulating boundary, 29 insulator, 22 intensive variable, 137 interface fluctuations, 245 internal energy, 134 interpolation-trace lemma, 69 intrinsic density, 22 intrinsic semiconductor, 22 inversion layer, 25 irrotational, 195 isentropic, 30, 194, 197 isentropic drift-diffusion model, 8, 15, 27 isentropic hydrodynamic model, 5 isothermal, 30, 194, 197 isothermal hydrodynamic model, 6

long-time behavior, 153 low injection, 24 lubrication-type equation, 245 Lyumkis' model, 132, 186 Madelung's fluid equations, 193 mean free path, 239 mixed boundary conditions, 28, 157 mixed finite element method, 16, 90, 99 mixed state, 193 mobility, 28, 128 moment method, 3, 195 MOSFET,25 Mott's law, 114 n-type semiconductor, 22 negative differential resistance, 19, 26, 239, 243, 262 Nernst-Planck model, 15 Newton method, 185, 264 non-existence of solutions, 212 non-parabolic band, 17 non-parabolicity parameter, 130, 187 non-uniqueness of solutions, 211 nonequilibrium thermodynamics, 15 Ohm's law, 114 Ohmic contact, 23, 29 Onsager's principle, 122, 134 outer sphere condition, 56 oxide layer, 25

kinetic model, 1

p-region, 23 p-type semiconductor, 22 parabolic band, 3, 17, 21, 131 phonon collisions, 120 plasma, 15 pn-junction diode, 23, 83 Poisson equation, 3 polygon, 99 porous media, 15, 63 positivity, 53, 166, 206, 245 positivity-preserving numerical scheme, 244

Legendre transform, 137 Leray-Schauder's theorem, 147, 174 localization, 62

quantum Boltzmann equation, 10 quantum drift-diffusion model, 14, 197, 239, 241, 244

Index

293

quantum energy-transport model, 14 quantum hydrodynamic model, 11, 15, 18, 191 quantum quasi-Fermi level, 246 quantum Vlasov equation, 11 quasi Fermi potential, 28, 136 quasi-hydrodynamic model, 1

thin liquid films, 245 threshold voltage, 17, 112 transistor, 25, 106 tunneling diode, 25

Raviart-Thomas scheme, 93 Raviart-Thomas space, 100 recombination, 23 recombination-generation rate, 31 reduced quantum model, 239 regularity of solutions, 156 resonant tunneling diode, 15, 25, 262 reverse biased, 24, 105

vacuum set, 16, 62 vacuum solution, 27, 62 valence band, 21

Scharfetter-Gummel scheme, 93 Schauder's theorem, 38 Schrodinger equation, 11, 18, 191 semiconductor, 22 semidiscretization in time, 139, 247 Shockley equation, 17, 24, 112 Shockley-Read-Hall model, 23, 32 single state, 193 Slotboom variable, 92, 99 smooth quantum hydrodynamic model, 13 source, 25 spherical harmonic expansion, 123 Stampacchia truncation method, 16, 37, 253 Stampacchia's lemma, 200 standard drift-diffusion model, 10 static condensation, 101, 183 subsonic, 225 superconductivity, 15 superfluidity, 15 symmetrization, 133 temperature tensor, 194 thermal equilibrium, 30, 153, 196, 209, 217 thermal voltage, 24, 105 thermistor, 15, 198 thermodynamic flux, 137 thermodynamic force, 137

unipolar, 25 uniqueness of solutions, 48, 166, 177, 209

waiting time, 16, 62 Wigner function, 10 Wigner-Poisson model, 265