Quasi-hydrodynamic Semiconductor Equations

Progress in Nonlinear Differential Equations and Their Applications Volume 41 Editor Haim Brezis Universite Pierre et M...

Author: Ansgar Jüngel

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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


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


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)


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


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)


~~ 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


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.


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 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 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


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


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


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 '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)


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


Analogous results hold for pC. We get for

E L 2 (V)

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

JQT

(n Jr QT

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


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
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


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

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)


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


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


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 O}
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


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 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


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


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),


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,


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

- 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


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(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+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


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 1. ps

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


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

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).


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


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


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


75

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

r

J&lP

. 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


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)


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

-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)


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)

0). From (3.103) we conclude

h

< M
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


°

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


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

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


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 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)


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)


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


(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


3.5.3 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


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


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


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 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


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.


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


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.


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).


110

Emitter

Base

p

N

p

Collector Figure 3.19: Geometry of the pnp transistor.

Figure 3.20: Mesh for the transistor.


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


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)


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.


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).


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


(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


~

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:


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


(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


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


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


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


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)


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


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)


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)


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

30(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


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


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


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 =


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.


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'


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


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)
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

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

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.


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.) 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


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)

-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

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


>

>

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:


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£

>.

~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


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) 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", (

../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,


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)
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

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


Remark 4.5.3 (i) It is not difficult to see that we even get the regularity \lu E £ 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


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


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

(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
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)


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


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 .


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

> 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

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


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


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


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


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 -

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*
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


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,

,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


using Young's and Poincare's inequalities; on the other hand, we have for K > w,

min

2 wKIV'S1 dx

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
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


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.


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


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 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


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

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


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


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


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+.


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


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


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


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


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. 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)


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
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


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

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. -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


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


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' 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


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.


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


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


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.


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


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


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


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)


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
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.


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) 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)


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
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.


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.


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

-~

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

'--+


258

Then we have the following estimates: Young's inequality yields

h +h

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