Mathematical Problems in Semiconductor Physics. Lectures C.I.M.E., Cetraro, Italy, 1998

Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris Subseries: Fondazione...

Author: Walter Allegretto | Christian Ringhofer | Angelo Marcello Anile | Angelo Marcello Anile

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Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris Subseries: Fondazione C.I.M.E., Firenze Adviser: Pietro Zecca

1823

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

A. M. Anile W. Allegretto C. Ringhofer

Mathematical Problems in Semiconductor Physics Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 15-22, 1998 With the collaboration of G. Mascali and V. Romano

Editor: A. M. Anile

13

Authors and Editors Angelo Marcello Anile Dipartimento di Matematica Università di Catania Viale A. Doria 6 95125 Catania, Italy e-mail: [email protected] Walter Allegretto Department of Mathematical and Statistical Sciences Alberta University Edmonton AB T6G 2G1 Canada e-mail: [email protected] Christian Ringhofer Department of Mathematics Arizona State University Tempe, Arizona 85287-1804, USA e-mail: [email protected]

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): 82D37, 80A17, 65Z05 ISSN 0075-8434 ISBN 3-540-40802-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10952481

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Preface

The increasing demand on ultra miniturized electronic devices for ever improving performances has led to the necessity of a deep and detailed understanding of the mathematical theory of charge transport in semiconductors. Because of their very short dimensions of charge transport, these devices must be described in terms of the semiclassical Boltzmann equation coupled with the Poisson equation (or some phenomenological consequences of these equations) because the standard approach, which is based on the celebrated driftdiffusion equations, leads to very inaccurate results whenever the dimensions of the devices approach the carrier mean free path. In some cases, such as for very abrupt heterojunctions in which tunneling occurs it is even necessary to resort to quantum transport models (e.g. the Wigner-Boltzmann-Poisson system or equivalent descriptions). These sophisticated physical models require an appropriate mathematical framework for a proper understanding of their mathematical structure as well as for the correct choice of the numerical algorithms employed for computational simulations. The resulting mathematical problems have a broad spectrum of theoretical and practical conceptually interesting aspects. From the theoretical point of view, it is of paramount interest to investigate wellposedness problems for the semiclassical Boltzmann equation (and also for the quantum transport equation, although this is a much more difficult case). Another problem of fundamental interest is that of the hydrodynamical limit which one expects to be quite different from the Navier-Stokes-Fourier one, since the collision operator is substantially different from the one in rarefied gas case. From the application viewpoint it is of great practical importance to study efficient numerical algorithms for the numerical solution of the semiclassical Boltzmann transport equation (e.g spherical harmonics expansions, Monte Carlo method, method of moments, etc.) because such investigations could have a great impact on the performance of industrial simulation codes for

VI

Preface

TCAD (Technology Computer Aided Design) in the microelectronics industry. The CIME summer course entitled MATHEMATICAL PROBLEMS IN SEMICONDUCTOR PHYSICS dealt with this and related questions. It was addressed to researchers (either PhD students, young post-docs or mature researchers from other areas of applied mathematics) with a strong interest in a deep involvement in the mathematical aspects of the theory of carrier transport in semiconductor devices. The course took place in the period 15-22 July 1998 on the premises of the Grand Hotel San Michele di Cetraro (Cosenza), located at a beach of astounding beauty in the Magna Graecia part of southern Italy. The Hotel facilities were more than adequate for an optimal functioning of the course. About 50 “students”, mainly from various parts of Europe, participated in the course. At the end of the course, in the period 23-24 July 1998, a related workshop of the European Union TMR (Training and Mobility of Researchers) on “Asymptotic Methods in Kinetic Theory” was held in the same place and several of the participants stayed for both meetings. Furthermore the CIME course was considered by the TMR as one of the regular training schools for the young researchers belonging to the network. The course developed as follows: • W. Allegretto delivered 6 lectures on analytical and numerical problems for the drift-diffusion equations and also on some recent results concerning the electrothermal model. In particular he highlighted the relationship with integrated sensor modeling and the relevant industrial applications, inducing a considerable interest in the audience. • F. Poupaud delivered 6 lectures on the rigorous derivation of the quantum transport equation in semiconductors, utilizing recent developments on Wigner measures introduced by Gérard, in order to obtain the semiclassical limit. His lectures, in the French style of pure mathematics, were very clear, comprehensive and of advanced formal rigour.The lectures were particularly helpful to the young researchers with a strong background in Analysis because they highlighted the analytical problems arising from the rigorous treatment of the semiclassical limit. • C. Ringhofer delivered 6 lectures which consisted of an overview of the state of the art on the models and methods developed in order to study the semiclassical Boltzmann equation for simulating semiconductor devices. He started his lectures by recalling the fundamentals of semiconductor physics then introduced the methods of asymptotic analysis in order to obtain a hierarchy of models, including: drift-diffusion equations, energy transport equations, hydrodynamical models (both classical and quantum), spherical harmonics and other kinds of expansions. His lectures provided comprehensive review of the modeling aspects of carrier transport in semiconductors.

Preface

VII

d) D. Levermore delivered 6 lectures on the mathematical foundations and applications of the moment methods. He presented in detail and depth the concepts of exponential closures and of the principle of maximum entropy. In his lectures he gave several physical examples of great interest arising from rarefied gas dynamics, and pointed out how the method could also be applied to the semiclassical Boltzmann equation. He highlighted the relationships between the method of moments and the mathematical theory of hyperbolic systems of conservation laws. During the course several seminars on specialized topics were given by leading specialists. Of particular interest were these of P. Markowich (co-director of the course) on the asymptotic limit for strong fieds, of P. Pietra on the numerical solution of the quantum hydrodynamical model, of A. Jungel on the entropy formulation of the energy transport model, of O. Muscato on the Monte Carlo validation of hydrodynamical models, of C. Schmeiser on extended moment methods, of A. Arnold on the Wigner-Poisson system, and of A. Marrocco on the mixed finite element discretization of the energy transport model. A. M. Anile

CIME’s activity is supported by: Ministero dell’Universit` a Ricerca Scientifica e Tecnologica; Consiglio Nazionale delle Ricerche; E.U. under the Training and Mobility of Researchers Programme.

Contents

Recent Developments in Hydrodynamical Modeling of Semiconductors A. M. Anile, G. Mascali and V. Romano . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Transport Properties in Semiconductors . . . . . . . . . . . . . . . . . . 3 H-Theorem and the Null Space of the Collision Operator . . . . . . . . . . . 4 Macroscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Maximum Entropy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application of MEP to Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Collision Term in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Balance Equations and Closure Relations . . . . . . . . . . . . . . . . . . . . . 5.3 Simulations in Bulk Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simulation of a n+ − n − n+ Silicon Diode . . . . . . . . . . . . . . . . . . . 5.5 Simulation of a Silicon MESFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Application of MEP to GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Collision Term in GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Balance Equations and Closure Relations . . . . . . . . . . . . . . . . . . . . . 6.3 Simulations in Bulk GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Simulation a GaAs n+ − n − n+ Diode . . . . . . . . . . . . . . . . . . . . . . . 6.5 Gunn Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 7 7 8 11 11 13 15 21 26 34 34 36 38 43 45 54

Drift-Diffusion Equations and Applications W. Allegretto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Classical Semiconductor Drift-Diffusion System . . . . . . . . . . . . . . . 1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Uniqueness and Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Other Drift-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Small Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 57 58 63 66 66

X

Contents

2.2 C α,α/2 Solutions and the Amorphous Silicon System . . . . . . . . . . . 2.3 Avalanche Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Degenerate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Degenerate Problems: Limit Case of the Hydrodynamic Models . 3.2 Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Degenerate Problems: Thermistor Equations and Micromachined Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Approximations, Numerical Results and Applications . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 70 70 70 73 74 80 82 89

Kinetic and Gas – Dynamic Models for Semiconductor Transport Christian Ringhofer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 1 Multi-Body Equations and Effective Single Electron Models . . . . . . . . 98 1.1 Effective Single Particle Models – The BBGKY Hierarchy . . . . . . 101 1.2 The Relation Between Classical and Quantum Mechanical Models104 2 Collisions and the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3 Diffusion Approximations to Kinetic Equations . . . . . . . . . . . . . . . . . . . . 111 3.1 Diffusion Limits: The Hilbert Expansion . . . . . . . . . . . . . . . . . . . . . 113 3.2 The Drift Diffusion Equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.3 The Energy Equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.4 The Energy Transport – or SHE Model . . . . . . . . . . . . . . . . . . . . . . 116 3.5 Parabolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4 Moment Methods and Hydrodynamic Models . . . . . . . . . . . . . . . . . . . . 120 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Recent Developments in Hydrodynamical Modeling of Semiconductors A. M. Anile, G. Mascali and V. Romano Dipartimento di Matematica e Informatica, Universit` a di Catania viale A. Doria 6 - 95125 Catania, Italy [email protected], [email protected], [email protected]

Summary. We present a review of recent developments in hydrodynamical modeling of charge transport in semiconductors. We focus our attention on the models for Si and GaAs based on the maximum entropy principle which, in the framework of extended thermodynamics, leads to the definition of closed systems of moment equations starting from the Boltzmann transport equation for semiconductors. Both the theoretical and application issues are examined.

1 Introduction Enhanced functional integration in modern electron devices requires an increasingly accurate modeling of energy transport in semiconductors in order to describe high-field phenomena such as hot electron propagation, impact ionization and heat generation. In fact the standard drift-diffusion models cannot cope with high-field phenomena since they do not comprise energy as a dynamical variable. Furthermore, for many applications in optoelectronics it is necessary to describe the transient interaction of electromagnetic radiation with carriers in complex semiconductor materials. Since the characteristic times are of order of the electron momentum or energy flux relaxation times, some higher moments of the carrier distribution function must be necessarily involved. These are the main reasons why more general models have been sought which incorporate energy as a dynamical variable and whose validity, at variance with the driftdiffusion model, is not restricted to quasi-stationary situations. These models are, loosely speaking, called hydrodynamical models and they are usually derived by suitable truncation procedures, from the infinite hierarchy of the moment equations of the Boltzmann transport equation. However, most of these suffer from serious theoretical drawbacks due to the

A.M. Anile, W. Allegretto, C. Ringhofer: LNM 1821, A.M. Anile (Ed.), pp. 1–56, 2003. c Springer-Verlag Berlin Heidelberg 2003

2

A. M. Anile, G. Mascali and V. Romano

ad hoc treatment of the closure problem [1]. Recently, in the case of silicon semiconductors, a moment approach has been introduced [2, 3] (see also [4] for a complete review) in which the closure procedure is based on the maximum entropy principle, while the conduction bands in the proximity of the local minima are described by the Kane dispersion relation. Later on, [5, 6], the same approach has been employed for GaAs. In this case both the Γ -valley and the four equivalent L-valleys have been considered. Therefore electrons in the conduction band have been treated as a mixture of two fluids, one representing electrons in the Γ -valley and the other electrons in the four equivalent L-valleys. Both in the Si and in the GaAs case, the models comprise the balance equations of electron density, energy density, velocity and energy flux. The only difference is that for GaAs both electron populations are taken into account. These equations are coupled to the Poisson equation for the electric potential. Apart from the Poisson equation, the system is hyperbolic in the physically relevant region of the field variables. In this paper we present a general overview of the theory underlying hydrodynamical models. In particular we investigate in depth the closure problem and present various applications both to bulk materials and to electron devices. The considerations and the results reported in the paper are exclusively concerned with silicon and gallium arsenide.

2 General transport properties in semiconductors Semiconductors are characterised by a sizable energy gap between the valence and the conduction bands. Upon thermal excitation, electrons from the valence band can jump to the conduction band leaving behind holes (in the language of quasi-particles). Therefore the transport of charge is achieved both through negatively charged (electrons) and positively charged (holes) carriers. The conductivity is enhanced by doping the semiconductor with donor or acceptor materials, which respectively increase the number of electrons in the conduction band or that of holes in the valence band. Therefore it is clear why the energy band structure plays a very important role in the determination of the electrical properties of the material. The energy band structure of crystals can be obtained at the cost of intensive numerical calculations (and also semiphenomenologically) by the quantum theory of solids [7]. However, for most applications, a simplified description, based on simple analytical models, is adopted to describe charge transport. In this paper we will be essentially concerned with unipolar devices in which the current is due to electrons (semiconductors doped with donor materials). Electrons which mainly contribute to the charge transport are those with energy in the neighborhoods of the lowest conduction band minima, each neighborhood being called a valley. In

Recent Developments in Hydrodynamical Modeling of Semiconductors

3

silicon, there are six equivalent ellipsoidal valleys along the main crystallographic directions ∆ at about 85 % from the center of the first Brilloiun zone, near the X points, which, for this reason, are termed the X-valleys. In GaAs there is an absolute minimum at the center of the Brillouin zone, the Γ -point, and local minima at the L-points along the Λ cristallographic orientations. As mentioned above, in the simplified description employed, the energy in each valley is represented by analytical approximations. Among these, the most common are the parabolic and the Kane dispersion relation. In the isotropic parabolic band approximation, the energy EA of the Avalley, measured from the bottom of the valley EA , has an expression similar to that of a classical free particle EA (kA ) =

2 |kA |2 . 2m∗A

(1)

In this approximation kA , the electron wave vector, is assumed to vary in all R3 , m∗A is the effective electron mass in the A-valley and the reduced Planck constant. A more appropriate analytical approximation, which takes into account the non-parabolicity at high energy, is given by the Kane dispersion relation EA (kA ) [1 + αA EA (kA )] =

2 k 2 , 2m∗A

k ∈ R,

(2)

where αA is the non parabolicity parameter. The electron velocity v(k) 1 in a generic band or valley depends on the energy E by the relation v(k) =

1 ∇k E.

Explicitly we get for parabolic band vi =

k i , m∗

(3)

while in the approximation of the Kane dispersion relation vi =

k i . m∗ [1 + 2αE(k)]

(4)

In the semiclassical kinetic approach the charge transport in semiconductors is described by the Boltzmann equation. For electrons in the conduction band it reads

1

∂f ∂f eE i ∂f + v i (k) i − = C[f ], ∂t ∂x ∂k i the valley index has been omitted for simplicity

(5)

4


where f (x, k, t) is the electron distribution function and C[f ] represents the effects due to scattering with phonons, impurities and with other electrons. In a multivalley description one has to consider a transport equation for each valley. The electric field, E, is calculated by solving the Poisson equation for the electric potential φ ∂φ , ∂xi ∇(∇φ) = −e(N+ − N− − n),

Ei = −

(6) (7)

N+ and N− denote the donor and acceptor density respectively (which depend only on the position), the dielectric constant and n the electron number density n=

f dk.

The equations (5)-(7) constitute the Boltzmann-Poisson system that is the basic semiclassical model of electron transport in semiconductors. The main scattering mechanisms in a semiconductor are the electronphonon interaction, the interaction with impurities, the electron-electron scatterings and the interaction with stationary imperfections of the crystal as vacancies, external and internal crystal boundaries. In many situations the electron-electron collision term can be neglected since the electron density is not too high. However in the case of high doping, electron-electron collisions must be taken into account because they might produce sizable effects. Retaining the electron-electron collision term greatly increases the complexity of the collision operator on the RHS of the semiclassical Boltzmann equation. In fact the collision operator for the electron-electron scattering is a highly nonlinear one, being quartic in the distribution function. After a collision the electron can remain in the same valley (intravalley scattering) or be drawn in another valley (intervalley scattering). The general form of the collision operator C[f ] for each type of scattering mechanism is C[f ] = P (k , k)f (k ) 1 − 4π 3 f (k) − P (k, k )f (k) 1 − 4π 3 f (k ) dk.(8) The first term in (8) represents the gain and the second one the loss. The terms 1 − 4π 3 f (k) account for the Pauli exclusion principle. P (k, k ) is the transition probability from the state k to the state k . Under the assumption that the electron gas is dilute, the collision operator can be linearized with respect to f and becomes C[f ] = [P (k , k)f (k ) − P (k, k )f (k)] dk. (9) As we shall see at equilibrium the electron distribution must obey the FermiDirac statistics


feq

5

−1 E −µ +1 = exp − , kB TL

kB being the Boltzmannn constant, µ the chemical potential and TL the lattice temperature which will be taken as constant. In the dilute case, one can consider the maxwellian limit of the Fermi-Dirac distribution E −µ . feq ≈ exp − kB TL In both cases from the principle of detailed balance [8], it follows that E − E P (k , k) = P (k, k ) exp − , kB TL

(10)

where E = E(k) and E = E(k ).

3 H-theorem and the null space of the collision operator In [9, 10, 11] an H-theorem has been derived for the physical electronphonon operator in the homogeneous case without electric field. The same problem has also been discussed in [12] in the parabolic case. Here we review the question in the case of an arbitrary form of the energy band and in the presence of an electric field, neglecting the electron-electron interaction and assuming the electron gas sufficiently dilute to neglect the degeneracy effects. By following [13] a physical interpretation of the results is suggested. The transition probability from the state k to the state k has the general form [14] P (k, k ) = G(k, k ) [(NB + 1)δ(E − E + ωq ) + NB δ(E − E − ωq )] (11) where δ(x) is the Dirac distribution and G(k, k ) is the so-called overlap factor which depends on the band structure and the particular type of interaction [14] and enjoys the properties G(k, k ) = G(k , k)

and G(k, k ) ≥ 0.

NB is the phonon distribution which obeys the Bose-Einstein statistics NB =

1 , exp(ωq /kB TL ) − 1

(12)

where ωq is the phonon energy. Given an arbitary function ψ(k) for which the following integrals exist, the chain of identities [9, 10, 11]

6


C[f ]ψ(k)dk =

B2

B2

[P (k , k)f (k ) − P (k, k )f (k)] ψ(k)dk dk =

P (k, k )f (k) (ψ(k ) − ψ(k)) dkd k = G(k, k ) [(NB + 1)δ(E − E + ωq ) + NB δ(E − E − ωq )] ×

f (k) (ψ(k ) − ψ(k)) dkd k = G(k, k )δ(E −E −ωq ) [(NB +1)f (k )−NB f (k)] (ψ(k)−ψ(k )) dkd k B2

holds. By following [11] if we set without loss of generality E , f (k) = h(k) exp − kB TL and in analogy with the case of a simple gas we take ψ(k) = kB log h(k), by using the definition of δ(x), one has kB C[f ] log h(k)dk = kB G(k, k )δ(E − E − ωq )NB exp −

E kB TL

(h(k ) − h(k)) (log h(k) − log h(k )) dkd k ≤ 0.

(13)

Therefore along the characteristics of eq. (5) df − log h(k) dk = − C[f ] log h(k)dk ≥ 0. dt This implies that log h(k) df dk = kB f log f − f + Ψ = kB

E f kB TL

dk. (14)

can be considered as a Liapunov function for the Boltzmann-Poisson system (5)-(7). The first two terms are equal to the opposite of the entropy arising in the classical limit of a Fermi gas, while the last term is due to the presence of the phonons. Ψ represents the nonequilibrium counterpart of the equilibrium Helmholtz free energy, divided by the lattice temperature. It is well known in thermostatics that for a body kept at constant temperature and mechanically insulated, the equilibrium states are minima for Ψ . A strictly related problem is the one of determining the null space of the collision operator. It consists in finding the solutions of the equation C(f ) = 0. The resulting distribution functions represent the equilibrium solutions. Physically one expects that, asymptotically in time, the solution to a given initial value problem will tend to such a solution.


7

The problem of determining the null space for the physical electron-phonon operator was tackled and solved in general in [11] where it was proved that the equilibrium solutions are not only the Fermi-Dirac distributions but form an infinite sequence of functions of the kind f (k) =

1 1 + h(k) exp E(k)/kB TL

(15)

where h(E) = h(E + ωq ) is a periodic function of period ωq /n, n ∈ N. This property implies a numerable set of collisional invariants and hence of conservation laws. The physical meaning is that the density of electrons whose energy E differs from a given value u by a multiple of ωq is constant. However if there are several types of phonons, as in the real physical cases, and their frequencies are not commensurable, the kernel of the collision operator reduces to the Fermi-Dirac distribution.

4 Macroscopic models

4.1 Moment equations Macroscopic models are obtained by taking the moments of the Boltzmann transport equation. In principle, all the hierarchy of the moment equations should be retained, but for practical purposes it is necessary to truncate it at a suitable order N. Such a truncation introduces two main problems due to the fact that the number of unknowns exceeds that of the equations: these are i) the closure for higher order fluxes; ii) the closure for the production terms. As in gasdynamics [15], multiplying eq. (5) by a sufficiently regular function ψ(k) and integrating over B, the first Brillouin zone, one obtains the generic moment equation ∂ ∂Mψ ∂f e + ψ(k)v i (k) i dk − E j ψ(k) j f dk = ψ(k)C[f ]dk, (16) ∂t ∂x ∂k

with Mψ =

ψ(k)f dk,

the moment relative to the weight function ψ. Since ∂f ∂ψ(k) ψ(k) j dk = ψ(k)f ndσ − f dk, ∂k ∂k j ∂B with n outward unit normal field on the boundary ∂B of the domain B and dσ surface element of ∂B, eq. (16) becomes

8


∂ ∂ψ(k) ∂Mψ e j i + E f ψ(k)v f (k)dk + dk − ψ(k)f n dσ = j ∂t ∂xi ∂k j ∂B ψ(k)C(f )dk. (17) The term

ψ(k)f ndσ ∂B

vanishes both when B is expanded to R3 , as in the parabolic and Kane approximations, ( because in order to guarantee the integrability condition f must tend to zero sufficiently fast as k → ∞ ) and when B is compact and ψ(k) is periodic and continuous on ∂B. This latter condition is a consequence of the periodicity of f on B and the symmetry of B with respect to the origin. Various models employ different expressions of ψ(k) and number of moments. 4.2 The maximum entropy principle The maximum entropy principle (hereafter MEP) leads to a systematic way of obtaining constitutive relations on the basis of information theory (see [16, 17, 18, 19] for a review). According to MEP if a given number of moments MA , A = 1, . . . , N , are known, the distribution function which can be used to evaluate the unknown moments of f , corresponds to the extremal, fM E , of the entropy functional under the constraints that it yields exactly the known moments MA ψA fM E dk = MA . (18) Since the electrons interact with the phonons describing the thermal vibrations of the ions placed at the points of the crystal lattice, in principle we should deal with a two component system (electrons and phonons). However, if one considers the phonon gas as a thermal bath at constant temperature TL , only the electron component of the entropy must be maximized. Moreover, by considering the electron gas as sufficiently dilute, one can take the expression of the entropy obtained as limiting case of that arising in the Fermi statistics (19) s = −kB (f log f − f ) dk. If we introduce the lagrangian multipliers ΛA , the problem of maximizing s under the constraints (18) is equivalent to maximizing s˜ = ΛA MA − ψA f dk − s,


9

the Legendre transform of s, without constraints, δ˜ s = 0. This gives

ΛA ψ A δf = 0. log f + kB

Since the latter relation must hold for arbitrary δf , it follows 1 A fM E = exp − ΛA ψ . kB

(20)

We stress that at variance with the monatomic gas, the integrability problem due to the fact that the sign of the argument in the exponential is not defined, does not arise here because the moments are obtained by integrating over the first Brillouin zone, which is a compact set of R3 . In order to get the dependence of the ΛA ’s on the MA ’s, one has to invert the constraints (18). Then by taking the moments of fM E and C[fM E ], one finds the closure relations for the fluxes and the production terms appearing in the balance equations. On account of the analytical difficulties this, in general, can be achieved only with a numerical procedure. However, apart from the computational problems, the balance equations are now a closed set of partial differential equations and with standard considerations in extended thermodynamics [16], it is easy to show that they form a quasilinear hyperbolic system. Let us set η(f ) = −kB (f log f − f ) . The entropy balance equation is obtained multiplying the equation (5) by η (f ) = ∂f η(f ) and afterwards integrating with respect to k, one has ∂ e i ∂ ∂ i η(f )dk + η(f )v dk − E η (f ) i f dk = η (f )C[f ]dk. ∂t ∂xi ∂k By taking into account the periodicity condition of f on the first Brillouin zone, the integral ∂ ∂η(f ) η (f ) i f dk = dk = η(f )ni dk ∂k ∂k i ∂B vanishes and the entropy balance equations assumes the usual form ∂s ∂ϕi + = g, ∂xi ∂t

with ϕi =

η(f )v i dk entropy flux

10


and g=

η (f )C[f ]dk entropy production.

The electric field contributes neither to the entropy production nor to the entropy flux. Let us now rewrite the balance equations in the form ∂MA ∂FAi = GA (MB , E), + ∂t ∂xi where

(21)

FAi

f ψA (k)v i (k)dk, ∂ψA (k) e j GA (MB , E) = − E f dk− ψA (k)f nj dσ + ψA (k)C(f )dk. ∂k j ∂B =

It is easy to prove, by multiplying (21) by ΛA and taking the sum over A, that the entropy balance equation is a consequence of the solution of (21) with the MEP closure relations. More in particular the field equations and the entropy balance equations are related by the condition that ∂s ∂ϕi ∂FAi ∂MA + + − g − ΛA − GA (MB , E) = 0, (22) ∂t ∂xi ∂xi ∂t with the lagrangian multipliers ΛA being the same as those arising by employing the maximum entropy principle. This implies [16] that MA = with If

∂s , ∂ΛA

∂ϕ , ∂ΛA i

FAi =

(23)

s = ΛA MA − s and ϕ = ΛA FAi − ϕ. i

∂ 2 s ∂ΛA ∂ΛB

is defined in sign, one can globally invert [20] and express the moments MA as function of the lagrangian multipliers ΛB . As shown in [21] the previous condition is equivalent to require that fM E =

∂ fM E (χ) ∂χ

is defined in sign, with χ = ΛA ψ A /kB . One trivally gets fM E < 0 and therefore the balance equations (21) can be rewritten in terms of the lagrangian multipliers as


∂ 2 ϕ ∂ΛB ∂ 2 s ∂ΛB + = GA ∂ΛA ΛB ∂xi ∂ΛA ∂ΛB ∂t i

∂s ,E . ∂ΛC

11

(24)

In this form it is immediate to recognize that the balance equations constitute a symmetric quasilinear hyperbolic system [22]. The main consequence of this property is that according to a theorem due to Fisher and Marsden [23] the Cauchy problem is well-posed for the system (24), at least in the simple case where the electric field is considered as an external field.

5 Application of MEP to silicon 5.1 Collision term in Silicon In Silicon the electrons which give contribution to the charge transport are those in the six equivalent valleys around the six minima of the conduction band. One assumes that the electron energy in each valley is approximated by the Kane dispersion relation. Concerning the collision term, the electronphonon scatterings which occur can be summarized as follows: • scattering with intravalley acoustic phonons (approximately elastic); • electron-phonon intervalley inelastic scatterings, for which there are six contributions: the three g1 , g2 , g3 and the three f1 , f2 , f3 optical and acoustical intervalley scatterings [24]

me electron rest mass 9.109510−10 g ∗ m effective electron mass 0.32 me TL lattice temperature 300o K ρ density 2330 g/cm3 vs longitudinal sound speed 9.18 105 cm/sec Ξd acoustic-phonon deformation potential 9 eV α non parabolicity factor 0.5 eV−1 r relative dielectric constant 11.7 0 vacuum dieletric constant 1.24 × 10−22 C/( eV cm) Table 1. Values of the physical parameters used for silicon

12


α g1 g2 g3 f1 f2 f3

Z f ω(meV ) (Dt K) (108 eV/cm) 1 12 0.5 1 18.5 0.8 1 61.2 11 4 19 0.3 4 47.4 2 4 59 2

Table 2. Coupling constants and phonon energies for the intervalley scatterings in Si.

In the elastic case

P (ac) (k, k ) = Kac δ(E − E ),

(25)

while for inelastic scatterings

(α) (α) P α (k, k ) = Kα (NB + 1)δ(E − E + ωα ) + NB δ(E − E − ωα ) , (26) (α)

where α = g1 , g2 , g3 , f1 , f2 , f3 , NB is the phonon equilibrium distribution according to the Bose-Einstein statistics (12) and ωα the phonon energy. Besides these interactions, we will also consider the scattering with impurities, which is an elastic mechanism of interaction. Its transition rate reads

P (imp) (kA , kA ) =

Kimp 2

[|kA − kA |2 + β 2 ]

δ(EA − EA ),

(27)

where β is the inverse Debye length. The parameters that appear in the scatterings rates can be expressed in terms of physical quantities characteristic of the considered material

Kac =

kB TL Ξd2 , 4π 2 ρvs2

Kimp =

NI Z 2 q 4 , 4π 2

Zf α (Dt K)2α , 8π 2 ρωα 1/2 2 q NI β= kB TL

Kα =


13

where Ξd is the deformation potential of acoustic phonons, ρ the mass density of the semiconductor, vs the sound velocity of the longitudinal acoustic mode, (Dt K)α the deformation potential realtive to the interaction with the α intervalley phonon and Zf α the number of final equivalent valleys for the considered intervalley scattering. NI and Z q are respectively the impurity concentration and charge. The deformation potentials are not known from calculations by means of first principles because the perturbation theory employed to evaluate the transition probabilities is not able to calculate them from the quantum theory of scattering. In all the simulators, even the Monte Carlo ones, these quantities are considered as fitting parameters. Their values depend on the approximation used for the energy bands, on the specific characteristics of the material and on the energy range of interest in the applications. The values of all these quantities as well as the silicon bulk constants are given in [25]. For the sake of completeness we summarize them in tables 1 and 2. 5.2 Balance equations and closure relations As already stated, the macroscopic balance equations are deduced by taking the moments of the Boltzmann transport equation for electrons in semiconductors [8]. We will consider the balance equations for density, momentum, energy and energy flux, which correspond to the weight functions 1, k, E, Ev ∂n ∂(nV i ) + = 0, ∂t ∂xi ∂(nP i ) ∂(nU ij ) + neE i = nCP i , + ∂t ∂xj ∂(nW ) ∂(nS j ) + neVk E k = nCW , + ∂t ∂xj ∂(nS i ) ∂(nF ij ) + + neEj Gij = nCS i . ∂t ∂xj

(28) (29) (30) (31)

This system is coupled to Poisson’s equation

E = −∇φ,

∇(∇φ) = −e(N+ − N− − n)

(32)

Since we consider the unipolar case, the hole concentration will be not included. The macrocopic quantities involved in the balance equations are related to the one particle distribution function of electrons f (x, k, t) as follows

14


f dk is the electron density,

n= R3

Vi = W = Si = Pi = U ij = Gij = F ij = CP i = CW = CS i =

1 f v i dk is the average electron velocity, n R3 1 E(k)f dk is the average electron energy, n R3 1 f v i E(k)dk is the energy flux, n R3 1 f k i dk = m∗ V i + 2αS i is the average crystal momentum, n R3 1 f v i k j dk is the flux of crystal momentum, n R3 1 ∂ 1 f (Evi )dk, n R3 ∂kj 1 f v i v j E(k)dk is the flux of energy flux, n R3 1 C[f ]k i dk is the production of crystal momentum, n R3 1 C[f ]E(k)dk is the energy production, n R3 1 C[f ]v i E(k)dk is the production of the energy flux, n R3

These moment equations do not constitute a set of closed relations because of the fluxes and production terms. Therefore constitutive assumptions must be prescribed. If we assume as fundamental variables n, V i , W and S i , which have a direct physical interpretation, the closure problem consists of expressing P i , U ij , F ij , Gij and the moments of the collision term CP i , CW and CS i as functions of n, V i , W and S i . If we use MEP to get the closure relations, we face the problem of inverting the constraints (18) with ψA = 1, v, E, Ev. This problem has been overcome in [2, 3] with the ansatz of small anisotropy for fM E , since Monte Carlo simulations for electron transport in Si show that the anisotropy of f is small even far from equilibrium. Formally a small anisotropy parameter δ has been introduced and explicit constitutive equations have been obtained in [2] for the higher order fluxes and in [3] for the production terms up to second order in δ. However it has been found in [26] that the first order model is sufficiently accurate for numerical applications and avoids some irregularities due to nonlinearities as occur in the parabolic band case [27]. Since the closure relations are an approximation of the exact MEP ones, the hyperbolicity is not guaranteed, but must be checked. As proved in [26],


15

the system (28)-(31) is hyperbolic in the region n > 0, W ≥ W0 , with W0 = 3 2 kB TL equilibrium energy. For the numerical integration we use the scheme developed in [28, 29, 30], see appendix, which is based on the Nessyhau and Tadmor scheme [31, 32]. 5.3 Simulations in bulk silicon The physical situation is represented by a silicon semiconductor with a uniform doping concentration, which we assume sufficiently low so that the scatterings with impurities can be neglected. On account of the symmetry with respect to translations, the solution does not depend on the spatial variables. The continuity equation gives n = constant and from the Poisson equation one finds that E is also constant. Therefore the remaining balance equations reduce to the following set of ODEs for the motion along the direction of the electric field which is chosen as x-direction

c

d eE 2αeEG c11 12 V =− ∗ + + − 2αc21 V + − 2αc22 S, ∗ ∗ ∗ dt m m m m d W = −eV E + CW , dt d S = −eEG(0) + c21 V + c22 S, dt

(33) (34) (35)

where V and S are the x-component of V and S, G = G11 and the cij , i, j = 1, 2 are production terms whose expressions can be found in [3]. As initial conditions for (33)-(35) we take

V (0) = 0, 3 W (0) = kB TL , 2 S(0) = 0.

(36) (37) (38)

The stationary regime is reached in a few picoseconds. The solutions of (33)-(35) for several values of the applied electric field are reported in figs. 1 (velocity), 2 (energy) and 3 (energy flux).

16


4.5

4

3.5

velocity (107 cm/sec)

3

2.5

2

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1 time (ps)

1.2

1.4

1.6

1.8

2

Fig. 1. velocity (cm/sec) versus time (ps) for E = 10 kV/cm, 30 kV/cm, 50 kV/cm, 70 kV/cm, 100 kV/cm, 120 kV/cm, 150 kV/cm

0.7

0.6

energy (eV)

0.5

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1 time (ps)

1.2

1.4

1.6

1.8

2

Fig. 2. energy (eV) versus time (ps) for the same values of the of the electric field as in figure 1


17

1.4

1.2

energy flux (107 eV cm/sec)

1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1 time (ps)

1.2

1.4

1.6

1.8

2

Fig. 3. energy flux (eV cm/sec) versus time (ps) for the same values of the electric field as in figure 1

The typical phenomena of overshoot and saturation velocity are both qualitatively and quantitatively well described (see [33] fig. 3.22 for a comparison with results obtained by MC simulations). Similar results were reported in [3], but there a different modeling of the collision terms has been considered and, moreover, instead of taking into account all the intervalley and intravalley scatterings, mean values of the coupling constant Ξ and Dt K have been introduced. The inclusion of all the scattering (intervalley and intravalley) mechanisms notably improves the results. For the sake of completeness, the parabolic band case has been also integrated, figs. 4, 5, 6. The differences, especially in the energy, with respect to the Kane case, confirm that the parabolic band is an oversimplification of the real band structure.

18


Fig. 4. velocity (cm/sec) versus time (ps) in the parabolic band case (dashed line) and for the Kane dispersion relation


Electric field 10 kV/cm

19


0.065

0.18 0.16

0.06 0.14 0.055 energy (eV)

energy (eV)

0.12 0.05

0.1 0.08

0.045 0.06 0.04 0.04 0.035

0

0.5

1 time (ps)

1.5

2

0.02

0

0.7

0.3

0.6

0.25

0.5

0.2

0.15

0.05

0.1

0.5

1 time (ps)

1.5

2

0.3

0.2

0

1.5

0.4

0.1

0

1 time (ps)


0.35

energy (eV)

energy (eV)


0.5

2

0

0

0.5

1 time (ps)

1.5

2

Fig. 5. energy (eV) versus electric field (kV/cm) in the parabolic band case (dashed line) and for the Kane dispersion relation

20




0.07

0.25

0.06



0.2 0.05

0.04

0.03

0.02

0.15

0.1

0.05 0.01

0

0

0.5

1 time (ps)

1.5

0

2

0


0.5

1 time (ps)

1.5

2


0.7

1 0.9

0.6


0.4

7


0.8 0.5

0.3

0.2

0.7 0.6 0.5 0.4 0.3 0.2

0.1 0.1 0

0

0.5

1 time (ps)

1.5

2

0

0

0.5

1 time (ps)

1.5

2

Fig. 6. energy flux (eV cm/sec) versus electric field (kV/cm) in the parabolic band case (dashed line) and for the Kane dispersion relation


21

5.4 Simulation of a n+ − n − n+ silicon diode Here we present the simulation of a ballistic n+ − n − n+ silicon diode as follos [26] (see also [34] for a comparison with MC data). The n+ regions are

Ò·

Ò

Ò·

Î

Fig. 7. Schematic representation of a n+ − n − n+ diode

0.1µm long, while the various lengths of the channel are taken into account. Moreover several doping profiles will be considered as reported in table 3.

+ C Channel length N+ N+ Vb 17 −3 17 Test # Lc (µm) (×10 cm ) (×10 cm−3 ) (Volt) 1 0.4 5 0.02 2 2 0.3 10 0.1 1 3 0.2 10 0.1 1

Table 3. Length of the channel, doping concentration (respectively in the n+ and n regions) and applied voltage in the test cases for the diode

Initially the electron energy is that of the lattice in thermal equilibrium at the temperature TL , the charges are averagely at rest and the density is equal to the doping concentration

n(x, 0) = N+ (x),

W (x, 0) =

3 kB TL , 2

V (x, 0) = 0,

S(x, 0) = 0,

where V and S are the only relevant component of velocity and energyflux.

22


Regarding the boundary conditions, in principle the number of independent conditions on each boundary should be equal to the number of characteristics entering the domain. However we impose, in analogy with similar cases [26, 27, 35, 36, 37, 38], a double number of boundary conditions. More precisely, we give conditions for all the variables in each boundary, located at x = 0 and x = L, the total length of the device,

n(0, t) ∂ W (0, t) ∂x ∂ V (0, t) ∂x ∂ S(0, t) ∂x φ(0) = 0

+ n(L, t) = N+ ∂ W (L, t) = 0, = ∂x ∂ = V (L, t) = 0, ∂x ∂ = S(L, t) = 0, ∂x and φ(L) = Vb ,

=

(39) (40) (41) (42) (43)

where Vb is the applied bias voltage. In all the numerical solutions there is no sign of spurious oscillations near the boundary, indicating that the conditions (39)-(42) are in fact compatible with the solution of the problem. The doping profile is regularized according to the function

x − x2 x − x1 + − tanh , − d0 tanh N+ (x) = N+ s s + + C where s = 0.01µm, d0 = N+ (1 − N+ /N+ )/2, x1 = 0.1µm, and x2 = x1 + Lc , with Lc channel length. The total length of the device is L = Lc + 0.2µm. A grid with 400 spatial nodes has been used. The stationary solution is reached within a few picoseconds (about five), after a short transient with wide oscillations. As first case we consider the test problem 1 (length of the channel 0.4 micron) with Vb = 2 Volts.

2

0.5

1.8

0.45

1.6

0.4

1.4

0.35

1.2

energy (eV)



1 0.8

0.3 0.25 0.2

0.6

0.15

0.4

0.1

0.2

0.05

0

0

0.1

0.2

0.3 0.4 micron

0

0.5

23

0

0.1

0.2

0.3 0.4 micron

0.5

0

0.1

0.2

0.3 0.4 micron

0.5

1.5 4

electric field (Volt/micron)


2

1

0.5

0 −2 −4 −6 −8 −10

0 0

0.1

0.2

0.3 0.4 micron

0.5

−12

Fig. 8. numerical results of the test case 1 after 5 picoseconds in the parabolic band case (dashed line) and for the Kane dispersion relation (continuous line)

24


1.8

0.25

1.6 0.2

energy (eV)

1.2 1

7


1.4

0.8

0.15

0.1

0.6 0.4

0.05

0.2 0

0

0.1

0.2 0.3 micron

0.4

0

0.5

0.1

0.2

0.3 micron

0.4

0.5

0.4

0.5

4 0.6

electric field (Volt/micron)

0.4

7


2 0.5

0.3

0.2

0

−2

−4

0.1 −6 0 0

0.1

0.2 0.3 micron

0.4

0.5

−8

0

0.1

0.2 0.3 micron



25

2 0.3 1.8 0.25

1.4 0.2 energy (eV)

1.2

7


1.6

1 0.8

0.15

0.1

0.6 0.4

0.05 0.2 0

0

0.1

0.2 micron

0.3

0

0.4

0.8

0.2 micron

0.3

0.4

0

0.1

0.2 micron

0.3

0.4

2

0.7 electric field (Volt/micron)

7

0.1

4

0.9


0

0.6 0.5 0.4 0.3 0.2

0 −2 −4 −6 −8

0.1

−10

0 0.1

0.2 micron

0.3

0.4

−12


The simulation for the parabolic band approximation is also shown ( fig. 8 dashed line), but it is evident, like in the bulk case, that the results are rather poor. The other test cases have been numerically integrated with Vb = 1 Volt (fig.s 9,10).

26


5.5 Simulation of a silicon MESFET

Fig. 11. Schematic representation of a bidimensional MESFET

In this section we check the validity of our hydrodynamical model and the efficiency of the above-mentioned numerical method by simulating a bidimensional Metal Semiconductor Field Effect Transistor (MESFET), see [30]. The shape of the device is taken as rectangular and it is pictured in fig. 11. The axes of the reference frame are chosen parallel to the edges of the device. We take the dimensions of the MESFET to be such that the numerical domain is Ω = [0, 0.6] × [0, 0.2] where the unit length is the micron. The regions of high doping n+ are [0, 0.1] × [0.15, 0.2] ∪ [0.5, 0.6] × [0.15, 0.2]. The contacts at the source and drain are 0.1 µm wide while the contact at the gate is 0.2 µm wide. The distance between the gate and the other two contacts is 0.1 µm. A uniform grid of 96 points in the x direction and 32 points in the y direction is used. The same doping concentration as in [39, 40, 41] is considered 3 × 1017 cm−3 in the n+ regions N+ − N − = 1017 cm−3 in the n region with abrupt junctions. We denote by ΓD that part of ∂Ω, the boundary of Ω, which represents the source, gate and drain


27

ΓD = {(x, y) : 0 ≤ x ≤ 0.1, 0.2 ≤ x ≤ 0.4, 0.5 ≤ x ≤ 0.6, y = 0.2, } . The other part of ∂Ω is labeled as ΓN . The boundary conditions are assigned as follows:

n=

n+ at source and drain ng at gate

(44)

  0 at the source φ = φg at the gate  φb at the drain

W = W0 , V · t = 0, n · ∇(V · n) = 0, S = 53 W0 V,

n · ∇n = 0, n · ∇W = 0, n · ∇φ = 0, n · ∇V i = 0, S = 53 W V

(45)

on ΓD ,

i = 1, 2

(46)

on ΓN .

(47)

Here ∇ is the bidimensional spatial gradient operator while n and t are the unit outward normal vector and the unit tangent vector to ∂Ω respectively. n+ is the doping concentration in the n+ region and ng is the density at the gate, which is considered to be a Schottky contact [42],

ng = 3.9 × 105 cm−3 . φb is the bias voltage and φg is the gate voltage. In all the simulations we set φg = −0.8V while Φb = 1V . In the standard hydrodynamical model considered in the literature (e.g. [43, 44]), the energy flux S is not a field variable and it is not necessary to prescribe boundary conditions for it. The relations (46)4 and (47)5 are not based on the microscopic boundary conditions for the distribution function, but they may be justified [30] in a heuristic way with the same approach followed in [45]. The numerical scheme can be found in [30]. We start the simulation with the following initial conditions:

28


n(x, y, 0) = N+ (x, y) − N− (x, y), V i = 0,

3 kB TL , 2 S i = 0 i = 1, 2

W = W0 =

The main numerical problems arise from the discontinuous doping and the boundary conditions at the Schottky barrier which give rise there to sharp changes in the density of several orders of magnitude. The use of a shockcapturing scheme is almost mandatory for this problem. The stationary solution is reached in a few picoseconds (less than five). The code takes about 9 minutes and 10 seconds in a PC with 1 Ghz Pentium III microprocessor. After the initial behaviour, the solution becomes smooth and no signs of spurious oscillations are present. The numerical scheme seems suitably robust and it is able to capture the main features of the solution. Only the Kane dispersion relation will be considered here because the results obtained in the parabolic band approximation are rather unsatisfactory when high electric fields are involved, as shown in the previous section for the n+ − n − n+ diode. The density is plotted in figure 12. As expected there is a depletion region beneath the gate. Moreover one can see that the drain is less populated than the source. Concerning the energy (figure 13) there are steep variations near the gate edges. The mean energy of the electrons reaches a maximum value of about 0.35 eV in the part of the gate closest to the source. The results for the velocity are shown in figure 14. The higher values of the x-component are at the edges of the gate contact. This happens also for the y-component, but with a huge peak at the gate edge closest to the source. The behaviour seems to indicate that there is a loss of regularity at the edge of the gate. The shape of the energy flux (figure 15) is qualitatively similar to that of the velocity. Very large tangential and normal components of the electric field (figure 16) are present again at the edges of the gate. For completeness the electric potential is also presented in fig. 17. The results are qualitatively similar to those presented in [40, 41] for all the variables except the y-component of the velocity in account of the huge peak at the edge of the gate.


29

30



31

32



33

34


Fig. 17. stationary solution (after 5 picoseconds) for the electric potential for Φb = 1 Volt.

6 Application of MEP to GaAs 6.1 Collision term in GaAs In GaAs as well, some interactions are intravalley transitions and others are intervalley transitions, but at variance with silicon the Γ -valley and the Lvalleys are not equivalent. The collision term may be split according to CA [fA , fB ]. CA [f ] = CA [fA , fA ] + B

The first term represents the intravalley scatterings, the second one takes into account the intervalley scatterings. As said, the main scattering mechanisms, to which electrons are subjected in semiconductors, are the ones with phonons, impurities, other electrons and stationary imperfections of the crystal such as vacancies and external and internal crystal boundaries. Since GaAs is a compound semiconductor, electrons interact with the phonons not only because of the deformation of the crystal but also through the polarization waves. The coupling through the polarization waves is due to the permanent electric dipole moment of the constituent ions in a compound material. This


35

coupling can be mediated through both the optical and the acoustic branches, but the latter contribution is marginal at room temperature in very pure semiconductors. In particular we consider the following scattering mechanisms: • • • •

the the the the

acoustic phonon scattering, non-polar optical phonon scattering, polar optical phonon scattering, impurity scattering.

All these scattering mechanisms are intravalley, except the non-polar optical phonon scattering. In the case of acoustic phonon scattering in its elastic approximation, valid when the thermal energy is much greater than that of the phonon involved in the scattering, we have P (ac) (kA , kA ) = Kac δ(EA − EA ),

(48)

where δ is the Dirac delta function and Kac a physical parameter. For the non-polar optical phonon interaction, the transition rate is given by the sum of an absorption and an emission term

− EA − ωnp ) P (np) (kA , kB ) = ZAB Knp N (np) δ(EB + (N (np) + 1)δ(EB − EA + ωnp )) , (49) where Knp is a physical parameter, ωnp is the longitudinal optical phonon energy, N (np) is the equilibrium non-polar optical phonon Bose-Einstein distribution. ZAB is the degeneracy of the final valley (Table 4), that is the valley the electron reaches after the scattering, with respect to the initial one. In GaAs, non-polar optical phonons contribute to the electron intervalley transfer between two equivalent L-valleys and between Γ -valley and L-valleys. The polar optical phonon scattering plays a very important role in compound semiconductors. It is an intravalley inelastic process whose transition rate is given by P (p) (kA , kA ) =

Kp G(kA , kA ) N (p) δ(EA − EA − ωp )+ 2 − kA | +(N (p) + 1)δ(EA − EA + ωp ) , (50)

8π 2 |kA

where Kp is a physical parameter, ωp is the polar optical phonon energy, N (p) the thermal equilibrium polar optical phonon number and the overlap factor G is given by [14] G(kA , kA ) = (aA aA + cA cA eA · eA )2 ,

(51)

36


where eA and eA are unit vectors pointing in the directions of kA and kA and 1 + αA EA 1 + αA EA aA = , aA = 1 + 2αA EA 1 + 2αA EA αA EA αA EA cA = , cA = . 1 + 2αA EA 1 + 2αA EA At last, the transition rate for the scattering with impurities, is similar to that for Si, see 27. The electron-electron scattering is not considered as well as the scattering with imperfections The values of the physical parameters appearing in the scattering rates are reported in table 4. effective electron mass in the Γ -valley 0.067 ×me m∗Γ m∗L effective electron mass in the L-valley 0.35 ×me 300 K TL lattice temperature ρ0 density 5.360 g/cm3 longitudinal sound speed 5.24 × 105 cm/sec vs non parabolicity factor in the Γ -valley 0.611 eV−1 αΓ αL non parabolicity factor in the L-valley 0.242 eV−1 r relative dielectric constant 12.90 ∞ relative dielectric constant at optical frequency range 10.92 Ξd acoustic-phonon deformation potential 7 eV Dt K non-polar optical phonon deformation potential 109 eV/cm ωnp non-polar optical phonon energy 0.03 eV ωp polar optical phonon energy 0.03536 eV EΓ (0) Γ -valley bottom energy 0 eV EL(0) L-valley bottom energy 0.32 eV ZΓ L degeneracy from Γ to L valleys 4 ZLΓ degeneracy from L to Γ valley 1 ZLL degeneracy from L to L valleys 3 Table 4. Values of the physical parameters used for GaAs

6.2 Balance equations and closure relations One assumes, as for silicon, that each valley in the conduction band [7] is described by the Kane dispersion approximation. However in this case, at a kinetic level, the system is described by two Boltzmann equations, one for the Γ -valley and the other for one L-valley. The macroscopic balance equations are deduced, as usual, by taking the moments of the Boltzmann transport equations. We consider the set of weight functions necessary to get the macroscopic balance equations for densities, momenta,


37

energies and energy-fluxes of both Γ and L-valley electrons, and the resulting equations read ∂nA ∂(nA VAi ) + = nA CnA , ∂xi ∂t ∂(nA PAi ) ∂(nA UAij ) + nA eE i = nA CPAi , + ∂t ∂xj j ) ∂(nA WA ) ∂(nA SA + nA eVAj Ej = nA CWA , + j ∂t ∂x i ) ∂(nA FAij ) ∂(nA SA i , + + nA eEj Gij A = nA CSA ∂t ∂xj where

(52) (53) (54) (55)

nA =

fA dkA is the electron density, 1 VAi = v i fA dkA the average electron velocity, nA R3 A 1 WA = EA (kA )fA dkA the average electron energy, nA R3 1 i SA = v i EA (kA )fA dkA the average energy flux, nA R3 A 1 i i PAi = the average crystal momentum, kA fA dkA = m∗A VAi +2αA SA nA R3 1 UAij = v i k j fA dkA the average crystal momentum flux, nA R3 A A Gij A

R3

1 = nA

FAij = CnA = CPAi = CWA = CSAi =

1 nA 1 nA 1 nA 1 nA 1 nA

R3

R3

R

3

R3

R3

∂ j ∂kA

1 i v EA (k) fA dkA , A

i j vA vA EA (kA )fA dkA

CA [f ]dkA

the average flux of energy flux,

the density production,

i kA CA [f ]dkA the crystal momentum production,

EA (kA )CA [f ]dkA

the energy production,

R3

i vA EA (kA )CA [f ]dkA

the energy flux production.

All these quantities refer to electrons in the A-valley, A = Γ , L. CA [f ] are the scattering operators which appear in the electron transport equations.

38


The above-written system is obviously coupled to the Poisson equation, which, in this case, becomes E = −∇φ,

∇(∇φ) = −e(N+ − N− − nΓ − 4nL )

(56)

where Φ is the electric potential, N+ and N− the donor and acceptor densities respectively, and the dielectric constant. These moment equations do not constitute a set of closed relations because of the additional fluxes and production terms. Therefore constitutive assumptions must be prescribed. i If we assume as fundamental variables nA , VAi , WA and SA , A = Γ, L, which have a direct physical interpretation, the closure problem consists of i , expressing UAij , FAij and Gij A and the moments of the collision terms CnA , CPA CWA and CSAi as functions of these variables . We again resort to the Maximum Entropy Principle in order to obtain these constitutive relations. In this case, one has to find the distribution functions which maximize the electron entropy s [fΓ , fL ] = −kB (fΓ logfΓ − fΓ ) dkΓ + 4 (fL logfL − fL ) dkL . 3

3

After that, the inversion of the constraint relations is realized assuming, as in the case of silicon, that the anisotropy of the distribution functions is small. In this way explicit constitutive equations have been obtained for fluxes and production terms up to the first order in δ. Here we do not report the constitutive relations. The interested reader is referred to [5]. One can prove that the system (52)-(55), closed with the maximum entropy principle, is hyperbolic in the physically relevant region of the dependent variables [6]. For the numerical integration we use the extension of the scheme developed in [31, 32] for homogeneous hyperbolic systems, which has been adapted in [26, 27, 28, 29, 30, 35] for balance laws with (possibly stiff) source terms, see appendix. The complete method is based on a second-order splitting technique which separately solves the system with the source put equal to zero (convection step) and the one with the flux vector put equal to zero (relaxation step). For more details see [6]. 6.3 Simulations in bulk GaAs. In this section we test the model in the case of a uniformly doped GaAs semiconductor. Two different impurity concentrations, N+ =1014 cm−3 , 1017 cm−3 are considered, and all the relevant scattering mechanisms are taken into account. Since the problem is homogeneous, in the evolution equations we can drop the spatial dependence and the balance equations reduce to the following set of ordinary differential equations


d (np)+ (np)− nΓ = nL Cn Γ L (WL ) − nΓ Cn Γ L (WΓ ), dt d m∗Γ nΓ (VΓi + 2αΓ SΓi ) = −enΓ E i + cΓ11 nΓ VΓi + cΓ12 nΓ SΓi , dt d (np)+ nΓ WΓ = −enΓ VΓk Ek + nL CW Γ L (WL ) + dt

(p) (np)− nΓ CWΓ (WΓ ) − CW Γ L (WΓ ) , d Γ i Γ i nΓ SΓi = −e nΓ Ek Gik Γ + c 21 nΓ VΓ + c22 nΓ SΓ , dt d (np)− (np)+ nL = −nL Cn LΓ (WL ) + nΓ Cn LΓ (WΓ ), dt

39

(57) (58)

(59) (60) (61)

d i L i nL (VLi + 2αL SLi ) = −enL E i + cL (62) 11 nL VL + c12 nL SL , dt

d (np)+ (p) (np) nL WL = −enL VLk Ek + nΓ CW LΓ (WΓ ) + nL CWL (WL ) + CW LL (WL ) dt (np)− −CW LΓ (WL ) , (63)

m∗L

d L i L i nL SLi = −e nL Ek Gik L + c21 nL VL + c22 nL SL , dt

(64)

where the meaning of the various productions terms which appear in the equations can be found in [5]. From (57), (61) and the expressions of the production terms, one has that n = nΓ + 4 nL = const, so that the total electron number is conserved, as it must be. In cases when a constant bias voltage is applied to the semiconductor, the Poisson equation is satisfied with n equal to the value of the doping concentration and E constant. The motion is along the direction of the electric field and, if we take this as the x-direction, the system (57)-(64) reads

40


d (np)+ (np)− nΓ = nL Cn Γ L (WL ) − nΓ Cn Γ L (WΓ ), dt Γ d 1 c11 Γ − 2 αΓ c21 nΓ VΓ + nΓ VΓ = 2αΓ GΓ − ∗ e E nΓ + dt mΓ m∗Γ Γ c12 Γ − 2 αΓ c22 nΓ SΓ , + m∗Γ d (np)+ nΓ WΓ = −enΓ VΓ E + nL CW Γ L (WL ) dt

(p)

(np)− Γ L (WΓ )

+nΓ CWΓ (WΓ ) − CW

,

(65)

(66)

(67)

d (68) nΓ SΓ = −e nΓ E GΓ + cΓ21 nΓ VΓ + cΓ22 nΓ SΓ , dt d (nΓ + 4 nL ) = 0, (69) dt L 1 c11 d nL VL = 2αL GL − ∗ e E nL + − 2 αL cL 21 nL VL + dt mL m∗L L c12 L n L SL , − 2 α c (70) + L 22 m∗L

d (np)+ (p) (np) nL WL = −enL VL E + nΓ CW LΓ (WΓ ) + nL CWL (WL ) + CW LL (WL ) dt (np)− LΓ (WL )

−CW

d L nL SL = −e nL E GL + cL 21 nL VL + c22 nL SL , dt

, (71) (72)

where VA and SA are the x-components of VA and SA and GA is the xxcomponent of Gij A , A = Γ , L. In the evolution equations the x-component E of the electric field enters as a parameter. As initial conditions for (65)-(72), we take, in suitable units, nΓ (0) + 4 nL (0) = 1, ∗ 3/2 Γ 1 (eq) d0 ( kB TL ) dkΓ nΓ (0) mΓ 3 f = R Γ(eq) = , 1 ∗ mL nL (0) dL dkL 0 ( kB TL ) 3 fL R

VA (0) = 0, WA (0) = gA SA (0) = 0,

1 kB TL

, A = Γ ,L.

The crystal temperature, TL , is assumed to be 300 K. The expressions of dΓ0 , dL 0 and gA are given in [5]. The solutions of (65)-(72) for electric fields respectively equal to 0.2, 0.5, 1, 2 and 6V /µm, are reported in Fig. 18.


41

8 6 V/mum 7

velocity (107 cm/s)

6

14 3 N+=10 /cm

2 V/mum

5 1 V/mum 4

3

0.5 V/mum

2

0.2 V/mum

1

0

0

0.5

1

1.5

2

2.5 time(ps)

3

3.5

4

4.5

5

4.5

5

8

6 V/µm

7

17 3 N+=10 /cm

6

velocity (107 cm/s)

2 V/µm 5

4 1 V/µm 3

2

0.5 V/µm

1 0.2 V/µm 0

0

0.5

1

1.5

2

2.5 time(ps)

3

3.5

4

Fig. 18. The time evolution of the electron average velocity for different values of the electric field and for N+ = 1014 /cm3 and 1017 /cm3 respectively.

The stationary regime is reached in a few picoseconds. The typical phenomena of overshoot and saturation of the velocity are both qualitatively and quantitatively well described. We also report the curves representing the electron valley occupancy, nΓ 4nL nΓ +4nL and nΓ +4nL , and average velocity as functions of the electric field (see Figs. 19 and 20) for the above impurity concentrations.

42


1

0.9

0.8

L−valleys

0.7

occupancy

0.6 N =1014/cm3 +

0.5

0.4

0.3

Γ−valley

0.2

0.1

0

0

1

2

3 4 electric field (V/µm)

5

6

7

6

7

1

0.9 L−valleys

0.8

0.7

occupancy

0.6 17

3

N+=10 /cm 0.5

0.4

0.3 Γ−valley 0.2

0.1

0

0

1

2


5

Fig. 19. Electron occupancy in the Γ - and L-valleys for N+ = 1014 /cm3 and 1017 /cm3 respectively


43

1.8 14

3

N+=10 /cm 1.6 17

3 N+=10 /cm

1.4

velocity (107cm/s)

1.2

1

0.8

0.6

0.4

0.2

0

0

1

2


5

6

7

Fig. 20. Electron velocity versus electric field characteristics for N+ = 1014 /cm3 and 1017 /cm3 respectively

Fig. 19 clearly shows the electron transfer from the Γ -valley to the L-valleys. The population inversion is observed at electric fields of about 1.8V /µm. Fig. 20 shows that the low field mobility and the electron peak velocity decrease with the increase of the impurity concentration. 6.4 Simulation a GaAs n+ − n − n+ diode In this section we consider the case of a n+ − n − n+ diode which models the channel of a MOSFET. The device is made of GaAs and its temperature is 300 K. The n+ regions are 0.1 µm long, while the channel length is 0.4 µm. The doping profile is 2 × 1018 cm−3 in the n+ regions, N+ = 1016 cm−3 in the n region, An external voltage of 2 V is applied. To avoid initiating too much a complicated transient behaviour in the device, the voltage is slowly raised to this value. The following initial and boundary conditions are considered nΓ (x, 0) =

r(0) N+ (x), 4 + r(0) (0)

WA (x) = WA ,

nL (x, 0) =

N+ (x) − nΓ (x, 0) , 4

VA (x, 0) = SA (x, 0) = 0,

44


r(0) N+ (a) − nΓ (a) N+ (a), nL (a) = , (0) 4 4+r ∂SA ∂VA (0) |a = 0, A = Γ, L, |a = WA (a) = WA , ∂x ∂x a = 0, 0.6 µm.

nΓ (a, t) =

(0)

(0)

where r(0) is the equilibrium ratio between nΓ and nL , and WΓ and WL are the equilibrium energies at the lattice temperature. Boundary conditions for Poisson’s equation are imposed by specifying the electric potential at the device contacts: φ(0) = 0V φ(0.6) = 2V. Simulations are run to a final time of t = 75ps, at which the solutions are judged to reach a steady state. The results relative to the electric field, the total electron average velocity and energy are represented in Figs 21, 22, where a comparison with analogous results for a Si n+ − n − n+ diode is also shown. The great difference in the velocity, which can be seen in Fig. 21-b is essentially due to the difference between the effective masses of Silicon electrons and Γ electrons in GaAs. Indeed near the first junction the mean energy is low and electrons are mainly in the Γ -valley where they have a small effective mass and consequently high velocities. Instead close to the second junction, where the electric field reaches its maximum value, the L-valley is almost exclusively populated. Since here the effective mass is much greater than that in the Γ valley and comparable with the effective mass in silicon, the mean velocity is lower.

10

4.5

4

____ GaAs

____ GaAs

5 ****** Si

3.5 ****** Si 3

2.5

7

mean velocity (10 cm/sec)

electric field (Volt/ µm)

0

−5

−10

2

1.5

1

0.5

−15 0

−20

0

0.1

0.2

0.3

0.4 micron

0.5

0.6

0.7

−0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

micron

Fig. 21. A comparison between a Si diode and a GaAs one, electric field and velocity vs position.


45

0.7

0.6

____ GaAs

***** Si

mean energy (eV)

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

micron

Fig. 22. A comparison between a Si diode and a GaAs one, energy vs position.

6.5 Gunn oscillations In this section we consider a GaAs diode with the same doping profile as that in [46] and [47]. This is also coupled to an RLC tank circuit which stimulates Gunn oscillatory effects. The one-dimensional diode has length Ld = 2 µm and its doping profile is  17 for x < 0.125 µm , 10     for 0.125 µm < x < 0.15 µm ,  1016 N+ (x) = 0.5 × 1016 for 0.15 µm < x < 0.1875 µm , (donors/cm3 )   1016 for 0.1875 µm < x < 1.875 µm ,    17 for 1.875 µm < x . 10 (73) The transitions in the doping profile at the device junctions are discontinuous (in contrast to [46]).The same initial and boundary conditions as before have been used except for the potential. Now φ(Ld ) may be either equal to 2V or is determined by coupling the device to a system of ODE which models the circuit. These equations read dVd 1 1 dI Vd = I − Id − , = (Vb − Vd ) , (74) dt C R dt Λ where Vd is the voltage through the device, Vb = 2V the bias voltage of the circuit, and Id , the particle current in the device, is calculated as q A Ld (nΓ vΓ + 4 nL vL ) dx . (75) Id = − Ld 0

46


Simple finite difference versions of equations (74) and (75) allow the diode voltage to be updated at each simulation time step. The values used for the capacitance, C, resistance, R, and inductance, Λ, of the circuit are C = A/Ld + 0.82 × 10−12 F , R = 25 ohm , Λ = 3.5 × 10−12 henry , where the cross-sectional area, A, of the diode is equal to 1.0 × 10−3 cm2 . The oscillator equations (74) are given the initial state Vd (t0 ) = 2V ,

I(t0 ) = 0 .

(76)

The circuit is engaged at time t0 = 75ps when the Ga As diode is judged to have reached the steady state illustrated in Figs 23, 24, 25.

Fig. 23. Eletron density and velocity vs position. Fig.7a : Continuus line: Γ -electron density, dotted line: L-electron density, dotted-dashed line: total electron density


0.35

0.3

mean energy (eV)

0.25

0.2

0.15

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1 micron

1.2

1.4

1.6

1.8

2

1.4

1.6

1.8

2

0.185

0.18

current (1024 cm−2 sec−1)

0.175

0.17

0.165

0.16

0.155

0.15

0

0.2

0.4

0.6

0.8

1 micron

1.2

Fig. 24. Eletron energy and current vs position.

0.185

0.18

current (1024 cm−2 sec−1)

0.175

0.17

0.165

0.16

0.155

0.15

0

0.2

0.4

0.6

0.8

1 micron

1.2

1.4

1.6

Fig. 25. Potential vs position.

1.8

2

47

48


One observes, Fig. 26, that there are some initial oscillations that smooth out and become negligible after about 200 ps. The qualitative behaviour, at variance with other hydrodynamical models, as those presented in [47], is very similar to the MC simulation reported in [47], even though this latter has been obtained in the parabolic band approximation.

2.2

2.15 Gunn diode

potential (Volt)

2.1

2.05

2

1.95

1.9

1.85

0

50

100

150

200

250

300

350

time (ps)

Fig. 26. The potential Vd versus time for the Gunn diode.

Acknowledgments This work has been partially supported by MURST, ex fondi 60%, and by CNR grants n. 98.01041.CT01, n. 99.01714.01, n. 00.00128.ST74 and n. CNRG00DB7 (program Agenzia 2000).

Appendix: numerical method The left hand side of the moment equations represents a quasilinear hyperbolic operator, while the right hand side contains relaxation and drift terms. We make use of a splitting scheme, based on the following decomposition. Without loss of geenrality, let us consider a scalar one space dimensional system of the form ∂u ∂f (u) + = g. (77) ∂t ∂x with u : R × R → R unknown, f : R → R flux and f : R → R source term. Then, for each time step, a numerical approximation u ˜ of the solution is obtained by solving the two consecutive steps:


∂f ∂u1 + =0 ∂t ∂x

49

convection step

(78)

relaxation step

(79)

u1 (t) = u ˜(t)

∂u ˜ =g ∂t u ˜(t) = u1 (t + ∆t).

We shall call this scheme simple splitting (SP). Note that this scheme is only first order in time, independently of the accuracy of the solution of the two steps. A better scheme is obtained by a more sophisticated splitting strategy, as below. Convection step During the convection step one integrates the quasilinear hyperbolic system (78). It is well known that the solutions of such systems suffer loss of regularity and may develop discontinuities. There is a wide literature on shock capturing schemes for hyperbolic systems of conservation laws, which are second order accurate in space in the region of regularity, and give sharp shock profile. A recent account on numerical methods for conservation laws is given in[48]. Higher order methods have been developed, and used for solving problems in semiconductor device simulation, such as ENO schemes (see[36]). These schemes, however, require an exact or approximate Riemann solver, or at least the knowledge of the characteristic structure of the Jacobian matrix. For systems similar to gas dynamics, an approximate Riemann solver based on the Roe matrix is used. However, in the case of extended models, it is not possible to write down a simple analytical expression for the eigenvalues of the system. Therefore it is desirable to use a shock-capturing scheme that does not require the explicit knowledge of the characteristic structure of the system. The scheme proposed by H. Nessyahu and E. Tadmor (NT)[31] has these properties. Its building block is the Lax-Friedrichs scheme, corrected by a MUSCL type interpolation that guarantees second order accuracy in smooth regions and TVD property[49].

50


tn+1

tn xj-1

xj xj-1/2

xj+1/2

Fig. 27. NT scheme on a staggered grid

For sake of completeness, we report the derivation of NT scheme and UNO reconstrution. For more details see[31, 50]. Let us consider a system of the form ∂v ∂f (v) + = 0, ∂t ∂x

(80)

where v ∈ R and f : R → R. We introduce a uniform grid in x, x1 , x2 , · · · , xN , and in t, t1 , t2 , · · · . By integrating Eq.(80) on a cell [xj , xj+1 ] × [tn , tn+1 ], one obtains tn +∆t 1 v j+ 12 (tn + ∆t) = v j+ 12 (tn ) − f (v(xj+1 , τ ))dτ ∆x tn tn +∆t − f (v(xj , τ ))dτ , (81) tn

where v j+ 12 (tn ) =

1 ∆x

xj+1

v(y, tn )dy xj

represents the cell average of v(x, t) in [xj , xj+1 ] for t = tn . The integral of the flux f (v(x, t)) is computed by the midpoint quadrature rule:

tn +∆t

f (v(xj , τ ))dτ = ∆t f tn

∆t v(xj , tn + 2

+ O(∆t3 ).

(82)

The quantity v(xj , tn + ∆t/2), is computed according to Lax-Wendroff approach, by using Taylor’s formula:


v(xj , t +

∆t 1 ) = vj (t) − λfj + O(∆t2 ), 2 2

51

(83)

where fj /∆x is an approximation of the space derivative of the flux (yet to be specified), and λ = ∆t/∆x. In order to obtain a second order scheme we require that ∂ 1 f (v(x, t)) + O(∆x). fj = ∆x ∂x By substituting (82) into (81), one has a relation that involves both cell averages and point values of the solution. By introducing a MUSCL interpolation, we approximate v(x, t) by a piecewise linear polynomial Lj (x, t) = vj (t) + (x − xj )

1 v , ∆x j

xj− 12 ≤ x ≤ xj+ 12 .

(84)

1 (xj + xj+1 ) 2

(85)

with xj− 12 =

1 (xj−1 + xj ) 2

xj+ 12 =

and in order to ensure a second order accuracy we require that 1 ∂ v = v(xj , t) + O(∆x). ∆x j ∂x

(86)

Therefore, Eq. (81), together with (82), (83), and (84), gives 1 1 [vj (t) + vj+1 (t)] + vj − vj+1 + 2 8 1 1 ) − f (vj (tn ) − λfj ) + O(∆t3 ). −λ f (vj+1 (tn ) − λfj+1 2 2 v j+ 12 (t + ∆t) =

Because the initial state at t = tn is given by the piecewise linear function Lj (x, tn ), the fluxes remain regular functions if the solutions of the corresponding generalized Riemann problems between adjacent cells do not interact. This is obtained by imposing the following CFL condition λ · max ρ (A(v(x, t)))
0,

p = p > 0

(I.1.4)

while natural boundary conditions are to hold on ∂ΩN : ∂ϕ ∂p ∂n = =0 = ∂ν ∂ν ∂ν

(I.1.5)

where ν denotes the outward normal. With equations (I.1.2), (I.1.3), we also associate initial conditions, which for simplicity we also take to be the same as (I.1.4), i.e.: n(x, 0) = n (x),

p(x, 0) = p(x).

(I.1.6)

Equations (I.1.1) - (I.1.6) constitute the classical drift diffusion system. We observe that although equation (I.1.1) is “electrical”, equations (I.1.2), (I.1.3) reflect conservation laws. They thus apply to other problems, with possibly different functions Dn , Dp , µn , µp , R. As a final remark we observe that the functions n , p, ϕ are not chosen arbitrarily, but rather involve the doping function N and the applied bias, [109].

1.2 Existence System (I.1.1) - (I.1.6) has received considerable attention since the book by Mock, [97], first appeared, and numerous early references may be found in the books cited in Section I.1. Even a casual glance at the equations indicates that two main sources of mathematical difficulty in this prototype system are the “higher order terms”:

Drift-Diffusion Equations and Applications

59

“µn n∇ϕ” and “µp p∇ϕ” in equations (I.1.2), (I.1.3), and the mixed boundary conditions (I.1.4), (I.1.5). The latter clearly show that global high order regularity cannot be generally expected. The difficulties with the former terms are usually dealt with by employing equation (I.1.1) and using some form of the maximum principle. A variety of methods are employed, including time discretization, coefficient truncation and Liapunov functionals, monotonicity methods, [49],[50],[60],[61],[63], [76]. Some of these procedures are very lengthy and we refer the interested to the references. We illustrate the flavor of some results by considering a classical steady-state result, and a more recent one due to Beirao da Veiga, [24]. Theorem 1.1. Suppose Dn = µn , Dp = µp with: 0 < α ≤ µn = µn (x, |∇ϕ|), 1−np µp = µp (x, |∇ϕ|) ≤ β for some positive constants α, β and R = 1+n+p . Then the steady state system:  −∆ϕ = p − n + N  −∇[µn (∇n − n∇ϕ)] = R (I.2.1)  −∇[µp (∇p + p∇ϕ)] = R subject to boundary conditions (I.1.4), (I.1.5) has a (weak) solution. Proof. Since Dn = µn , and Dp = µp we can rewrite the equations in terms of new variables: Put u = ne−ϕ , v = peϕ (The Slotboom variables) then (I.2.1) becomes − ∆ϕ = N + ve−ϕ − ueϕ 1 − uv − ∇[µn eϕ ∇u] = 1 + ueϕ + ve−ϕ 1 − uv − ∇[µp e−ϕ ∇v] = 1 + ueϕ + ve−ϕ

(I.2.2) (I.2.3) (I.2.4)

with the obvious changes made to the boundary conditions (I.1.5), (I.1.6), so that , v = v. Now choose and fix constants u, u, v, v such that: on ∂ΩD we have u = u u v = u v = 1, and u ≤ u ≤ u, v ≤ v ≤ v. Observe that for any v ∈ [v, v] and any ϕ, u, u form an upper/lower solution pair for equation (I.2.3) where [v, v] denotes the obvious order interval. An identical remark applies to v, v and equation (I.2.4). If we select and fix (u0 , v0 ) ∈ I [u, u] × [v, v] and substitute this pair for (u, v) in (I.2.2) we obtain a nonlinear equation for ϕ, which can also be solved by upper/lower solutions. Furthermore since the right hand side of (I.2.2) is nonincreasing in ϕ, the solution ϕ1 is unique. We then substitute ϕ1 , u0 , v0 suitably into (I.2.3) to obtain the linear equation for u : −∇[µn eϕ1 ∇u] =

1 − uv0 . 1 + u0 eϕ1 + v0 e−ϕ1

This equation has a unique solution in [u, u], and similar remarks apply to equation (I.2.4) and v. We thus have created a map T (u0 , v0 ) : I ⊂ L2 × L2 → I and straightforward arguments show that T is both continuous and completely continuous. By Schauder’s Fixed Point Theorem, T has a fixed point and we thus have a solution to the problem. We observe that if Dn = cn µn , Dp = cp µp with cn , cp positive constants, then the proof is virtually unchanged.

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We note the significant role played in the above proof by the change of variables, the form of the righthand sides and by Maximum Principle arguments (via constant upper/lower solutions). These yield the key step: the establishment of a priori bounds, and work in more general situations. For example, cases where µn → 0 as |∇ϕ| → 0, and equations (I.2.2) - (I.2.4) hold in different domains, as discussed by Jerome, [75], or if a magnetic field is present, [117]. Often the Einstein relations are assumed to hold: Dn,p kT µn,p with k, q q constants and T denoting temperature. If T is not constant, then at least formally, the above process fails. The case where Dn,p and µn,p are not related was considered by Frehse and Naumann, [54]. They assumed that µn , µp saturate, i.e. (µn + µp )|∇ϕ| ≤ K for some constant K, and obtain existence of solutions by clever choices of test functions in the (weak) form of equations (I.2.2) - (I.2.4). In this regard, observe that if µn , µp saturate, equations (I.2.3), (I.2.4) have bounded first and second order coefficients, bounded righthand side and a positive zeroth order term. Some a priori bounds thus are to be expected via maximum principle type arguments and these are indeed obtained in [54]. Unlike the previous case, the uniform ellipticity is now essential to the arguments. If we consider the original time dependent problem (I.1.1) - (I.1.3), then the change of variable used in 1.1 does not appear to be advantageous. Better is the other possible change: ∇n − n∇ϕ = n∇( n (n) − ϕ). Thus we put u = n (n) − ϕ, v = n (p) + ϕ (quasi Fermi variables) and modify the equations and boundary conditions accordingly, leading to the procedures employed in [63]. Observe that u+ϕ now ∂n = ∂(e∂t ) . ∂t As mentioned earlier, we illustrate some of the procedures employed in a time dependent problem by presenting results of Beiro da Veiga, [24], which clearly illustrate both the uses of Maximum Principle arguments and the significance of the nature of the specific coupling between the equations. We first need the following preliminary result, which is an extension (given in [24]) of a classical result of Stampacchia, [113]. Lemma 1.1. Let ω be a nonnegative nonincreasing function such that: ω(h) ≤

ckθ [ω(k)]1+x (h − k)δ

for h > k ≥ k0 , with constants: c, θ, δ, x > 0 and θ < δ(1 + x). Then there exists a number d such that ω(d) = 0. Proof. A small variant of the proof given in [24], is as follows: Put z(h) = hµ ω 1/δ (h), with µ = θ/[δ(1 + x)] < 1, then z(h) ≤

1+χ c1/δ hµ . z(k) (h − k)

In [113], Stampacchia proved that if z is nonnegative nonincreasing and z(h) ≤

1+χ c1/δ M µ z(k) (h − k) 1+x

µ 1/δ x x for some constant M > 0,then z(k by con 0 + d) = 0 for d = M c [z(k0 )] 2 1 sidering the sequence ks = k0 + d 1 − 2s , s = 1, 2, . . . . In our case, we choose the same sequence ks for some M to be chosen, and note that

Drift-Diffusion Equations and Applications ks ≤ k0 + d = k0 + c1/δ M µ 2

1+x x

61

[z(k0 )]x < M

for M large enough, since µ < 1. We can now apply Stampacchia’s result, since our z is the product of a continuous function and a nonincreasing one. The existence results follow as a consequence of the estimate: Theorem 1.2. If Dn , Dp , µn , µp are constants then any solution (n, p) of (I.1.1) (I.1.5) is uniformly bounded. Proof. Without loss of generality, we take µn = µp . The more general case is dealt with by a simple variant of the arguments to follow. Let: 0 < D1 ≤ Dn , Dp and choose a constant k so large that n max (n − k, 0) is a suitable test function for equation (I.1.2). We multiply by n and integrate by parts to get: 1 d (n)2 + D1 |∇n|2 − µn n∇ϕ∇n = Rn 2 dt Ω Ω Ω Ω 1 d

(n)2 + D1 |∇n|2 − µn (n + k)∇ϕ∇n = Rn. 2 dt Ω Ω Ω Ω

i.e.

Equation (I.1.1) is used to estimate the third integral on the righthand side. This is a key step critical to the methods of proof, and we obtain: (n)2 1 d (n)2 + D1 |∇n|2 − µn Rn. + kn [N + p − n] = 2 2 dt Ω Ω Ω Ω We set p = max (p − k, 0), and repeat this process with equation (I.1.3). Adding, yields: 1 d (n)2 + (p)2 + D1 {|∇n|2 + |∇p|}2 2 dt Ω Ω

(p)2 (n)2 − µn R(n + p). + kn − − kp (N + p − n) = 2 2 Ω Ω

(I.2.5)

We observe:

(p)2 − (n)2 (p − n) ≥ 0

(p − n)(p − n) ≥ 0 R(n + p) ≤ (n + p) and thus:

p + n (N + p − n)[(p)2 − (n)2 ] ≥ (p − n)2 − N 2 . 2 Substituting this estimate in (I.2.5) and assuming k ≥ 1, yields: d

(n)2 + (p)2 + co (|∇p|2 + |∇n|2 ) dt Ω Ω 2 ≤ c1 N (p + n) + c2 k N (p + n) + k (p + n). Ω

Ω

Ω

Here c0 , c1 , c2 denote constants independent of n, p, N, k. Define

(I.2.6)

62

W. Allegretto Ak (t) = {x | p(x, t) > k

Then:

or

n(x, t) > k}.

1 N 2 (p + n) ≤ p + n L6 N 2 Lr1 µ Ak (t) r2 Ω

with: 16 + r11 + r12 = 1 (we recall Ω ⊂ R3 ). By the multiplicative inequality we then have for any ε > 0 :

2 (I.2.7) N 2 (p + n) ≤ ε |∇p|2 + |∇n|2 + c(ε) N 2 2r1 µ Ak (t) r2 . Ω

Ω

It is critical that r22 exceed 1, and this places limitations on r1 (and thus r2 ), see [24]. In our simplified situation, we merely take r1 = ∞, r2 = 65 and thus ( r22 ) = 1+χ with χ > 0. We estimate the remaining terms in (I.2.6) in the same way as was done for (I.2.7), choose a small ε > 0 and obtain by Poincarés Inequality:

1+x d

(n)2 + (p)2 + ν (n)2 + (p)2 ≤ ck2 µ Ak (t) (I.2.8) dt Ω Ω for some positive constants ν, c independent of n, p, k. Set: yk = Ω (n)2 + (p)2 and observe yk (0) = 0, while (I.2.8) becomes: 1+x yk + νyk ≤ ck2 µ Ak (t) or: yk (t) ≤ ck2 whence:

yk (t) ≤ ck2

t

1+x e−ν(t−s) µ Ak (s)

0

→

0≤t k, we obtain: c(h − k)2 µ Ah (t) ≤ yk (t).

Hence, setting φ(h) = → sup µ Ah (t) , yields: 0≤t k ≥ M, with M determined by the problem data. 1.1 then shows φ(d) = 0 for some finite d, i.e. p, n are bounded above. To obtain an existence result from 1.2, we proceed as follows: Let θ be a fixed constant and set  θ ω≥θ     ω 0≤ω≤θ ω =    0 ω ≤ 0. Following [24], [61] consider the auxiliary problem:

∂n ∂t ∂p ∂t

Drift-Diffusion Equations and Applications  −∆δ = N + q − m   − ∇[D ∇n − µ m∇ϕ] = R( q , m)  n

n

  − ∇[Dp ∇p + µp p∇ϕ] = R( q , m). 

63

(I.2.9)

For a given (m, q), one shows there exists a unique solution (n, p), and by Schauder’s Theorem the existence of a fixed point. The weak maximum principle yields n, p ≥ 0 , p as test functions) to show that and the arguments of 1.2can be repeated (with n the fixed points of (I.2.9) satisfy a bound independent of θ. The existence of a solution to the original problem follows. We remark that the assumed nature of µn , µp was important here. The same proof also works if (µn + µp )|∇ϕ| is bounded, or if µn , µp are the sum of constants plus saturation terms. Alternatively one may change the system of equations.

1.3 Uniqueness and asymptotics If Dn , Dp , µn , µp are, for example, constants, uniqueness for the time dependent problem is obtained by applying Gronwall’s Lemma [24], [49], [61]. The situation is different in the steady-state case. Numerical results (e.g. [118]) and physical considerations indicate that uniqueness in general does not hold. A detailed analysis of one dimensional cases where uniqueness does hold, together with more references and multiplicity results, may be found in the papers of Alabau, [1], [2], [3]. In these papers, it is also shown that uniqueness for the current driven problem does hold. In the more general three dimensional case, uniqueness is known to hold for constant Dn , Dp , µn , µp if the applied biases are small enough, and conditions ensure regularity [97]. The following is a simple example. Theorem 1.3. Under the conditions of 1.1, but with R = 0 and Dn = µn , Dp = µp functions of x, the solution to (I.2.2) - (I.2.4), (I.1.4), (I.1.5) is unique if ∇ u L2 + ∇ v L2 is small enough and ∇u, ∇v are uniformly bounded in Lτ for some τ > 3. Proof. Introduce the variables u, v as was done in 1.1. The maximum principle implies that any solution (u, v, ϕ) must be bounded in L∞ in terms of the boundary data L∞ norm. Elementary estimates then show ∇u L2 ≤ C ∇ u L2 ∇v L2 ≤ C ∇ v L2 , whence ∇u L3 + ∇v L3 ≤ C[ ∇ u α v α L2 ] L2 + ∇ for some constants C, α independent of (u, v) but dependent on the Lτ bound on ∇u, ∇v. If (u, v, ϕ) and (u , v , ϕ ) denote two solution triples, we have by subtraction u − u H 1 ≤ D ∇u L3 ϕ − ϕ L6 v − v H 1 ≤ D ∇v L3 ϕ − ϕ L6 . Finally,

−∆(ϕ − ϕ ) = (v − v )e−ϕ + v(e−ϕ − e−ϕ ) − (u − u )eϕ + u(eϕ − eϕ ).

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Using (ϕ − ϕ ) as a test function and noting (eϕ − eϕ + e−ϕ − e−ϕ )(ϕ − ϕ ) ≤ 0 gives:

ϕ − ϕ H 1 ≤ E v − v L2 + u − u L2

v α ≤ F ∇ u α L2 ϕ − ϕ H 1 L2 + ∇ and the result. Obviously a similar result holds without the Lτ assumption on ∇u, ∇v if we can conclude directly that ∇u L3 + ∇v L3 ≤ C[ ∇ u L3 + ∇ v L3 ] by regularity properties of the problem. We remark that in many practical cases ∇ u L2 + ∇ v L2 can be made small by assuming that the applied bias is small enough, [97]. Implicit function arguments can be employed to extend such a result to the case of “small” R. Uniqueness results near equilibrium which take into account the lack of regularity due to the mixed boundary conditions, may be found in [103]. It is clearly of interest to determine the asymptotic behaviour of the solutions of the time dependent problem. Under suitable regularity conditions, Fang and Ito [48], show the existence of a compact attractor (in suitable spaces), and asymptotic results have also been known for a long time, [98], and obtained by several authors, see e.g.: ([23],[63]). A special type of result is of particular significance: When does the solution of the time dependent problem converge to the (a?) solution of the steady-state problem? Mock, [98], obtained the first results in this direction. As an example of a situation when the answer to this question is known, we follow the ideas of Gajewski and Gr¨ oger [61]. Suppose that Dn = µn = Dp = µp = 1 and that the boundary data are in the “thermal equilibrium” steady state situation: Jn = Jp ≡ 0, so that the steady state problem has solution (n∗ , p∗ , ϕ∗ ) with ∇n∗ − n∗ ∇ϕ∗ ≡ 0 ∇p∗ + p∗ ∇ϕ∗ ≡ 0 and: n∗ p∗ = 1, p∗ − n∗ + N = 0 on ∂ΩD , while ϕ∗ satisfies: ∗ ∗ −∆ϕ∗ = e−ϕ − eϕ + N ϕ∗ = n (n∗ ) on ∂ΩD . Now suppose n, p, ϕ equal n∗ , p∗ , ϕ∗ respectively on ∂ΩD , and n, p = ni , pi (for some functions ni , pi ) respectively at t = 0. If this data is positive, it can be shown, [61], that n, p are bounded above and below by positive constants. In order to obtain the required estimates, we wish to obtain positive integrands. This leads us to use the function n nn∗ − (ϕ − ϕ∗ ) as a test function in (I.1.2), and the left hand side of (I.1.2) then yields:

n 2

n nt n ∗ − (ϕ − ϕ∗ ) + n ∇ n ∗ − (ϕ − ϕ∗ ) . n n Ω Ω We repeat this process with (I.1.3) using n pp∗ + (ϕ − ϕ∗ ) as a test function and add to obtain, on the left hand side:


Ω

65

n p ∗

n − n )(ϕ − ϕ ) nt n + p + (p t t t n∗ p∗

n 2 + n ∇ n ∗ − (ϕ − ϕ∗ ) n Ω 2 p + p ∇ n ∗ + (ϕ − ϕ∗ ) p Ω

To deal with the second part of the first integral, observe that: −∆(ϕ − ϕ∗ ) = (p − p∗ ) − (n − n∗ ) and thus:

∂ − ∆(ϕ − ϕ∗ ) = pt − nt . ∂t That is: ∂ 1 (pt − nt )(ϕ − ϕ∗ ) = |∇(ϕ − ϕ∗ )|2 . ∂t 2 Ω Ω On the other hand, we note that on the right hand side we have, after addition: p 1 − np n

n ∗ − (ϕ − ϕ∗ ) + n ∗ + (ϕ − ϕ∗ ) 1+n+p n p Ω 1 − np =

n (np) ≤ 0. Ω 1+n+p In summary, if we set: H

Ω

+

1 2

n

n∗

n

p y y

n dy + dy n∗ p∗ p∗

|∇(ϕ − ϕ∗ )|2

Ω

ω1 n n − ϕ

ω2 n p + ϕ

and recall that n, p are bounded above and below, we obtain: 2 dH |∇(ωi )|2 ≤ 0 +c dt i=1 Ω for some constant c. Elementary estimates based upon the equations satisfied by p∗ , n∗ , ϕ∗ , give the bound: 2 H≤c |∇(ωi )|2 i=1

Ω

and thus:

dH + mH ≤ 0 dt for some constant m. Direct calculation shows: H(t) ≥ c{ (n − n∗ )(t) 2L2 + (p − p∗ )(t) 2L2 + (ϕ − ϕ∗ )(t) 2H1 }. But since Ω ⊂ R3 , the maximum principle gives: (ϕ − ϕ∗ )(t) L∞ ≤ k (n − n∗ )(t) 2L2 + (p − p∗ )(t) 2L2 . In summary we have:

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W. Allegretto

Theorem 1.4. Let Dn = µn = Dp = µp = 1 and suppose that the boundary data equals (n∗ , p∗ , ϕ∗ ), i.e.: that of steady state thermal equilibrium. Let n, p be ni , pi > 0, respectively, at t = 0 and suppose n∗ , p∗ , ϕ∗ , ni , pi are all positive. Then as t → ∞, (n − n∗ )(t) L2 + (p − p∗ )(t) L2 + (ϕ − ϕ∗ )(t) H 1 ∩L∞ → 0 as ce−mt for some positive constants c, m.

2 Other Drift-Diffusion Equations We consider now the study of some equations which involve modifications of the classical system (I.1.1) - (I.1.6). These problems have often not received the attention that has been given to the classic case.

2.1 Small devices In dealing with small devices, the following procedure was originally suggested by Thornber, [114], and modified by Blakey, Maziar and Wang. The velocity equation for electrons is replaced by v = µn ∇ϕ −

Dn ∇n dE +γ n dt

(II.1.1)

), where E denotes the electric field, and an analogous change for with γ = γ(E, dE dt holes. We note that here: ∂E dE ∂E vi . = + dt ∂t ∂x i i The modification of v has a significant effect on the mathematical analysis of the resulting equations, and these have not been studied extensively. Friedman and Liu, [58], considered the following one dimensional case: Suppose p ≡ 0, 0 < Dn = µn D, γ constants R ≡ 0 and substitute (II.1.1) for v to obtain the system: −ϕxx = N − n

(II.1.2)

γn ∂ 1 − γN ∂ D(nx − nϕx ) nt = + ϕxt ∂x 1 + γ(n − N ) ∂x 1 + γ(n − N ) 1 + γ(n − N )

(II.1.2)

for 0 < x < L. Let as before, n(x, 0) = n (x) and assume the boundary conditions: −αnx + βn = 0 ϕ(0, t) = V,

at

x=0

ϕ(L, t) = 0

αnx + βn = 0

at

x=L

(II.1.3) (II.1.4)

with V > 0, α > 0, β ≥ 0 constants. The specific nature of the boundary conditions and the assumption that γ is a constant, are important in the analysis given in [58]. To indicate how this goes, we scale the problem to the unit interval, put N ≡ 0 and once again set mathematically irrelevant constants to unity. After some elementary manipulations, equations (II.1.2) - (II.1.4) become:


ϕxx = n

67

(II.1.5)

nx nt − nxx = (−γnx − ϕx + γϕxt ) − n2 1 + γn n(x, 0) = n0 (x) − nx + n = 0 ϕ(0, t) = V,

at

x = 0,

nx + n = 0

at

(II.1.6) (II.1.7) x=1

(II.1.8)

ϕ(1, t) = 0.

(II.1.9)

An obvious difference between this system and the earlier one considered in Section I, is the presence of the term “ϕxt ” in equation (II.1.6). In the one dimensional case, this can be replaced as follows: formal direct calculations from (II.1.5) give: 1 1 ϕx = −V − n(y, t)dy + yn(y, t)dy

x 1

ϕxt = 0

ynt (y, t)dy −

0

1

nt (y, t)dy x

and equation (II.1.6) becomes:

nt − nxx = F1 (n, nx ) + g(x, t)

0

1

ynt dy −

1

nt dy

(II.1.10)

x

γnx where F1 (n, nx ) is a smooth functional of n, nx and g(x, t) = 1+γn . It is important to bound n, g(x, t). 0 Theorem 2.1. Let: 0 < n0 ≤ M0 ; dn ≤ M1 ; and, if γ < 0, 1 + γM0 > 0. If dx M0 + M1 is small enough, then any solution of (II.1.5) - (II.1.9) satisfies: 0 < n; 1 + γn ≥ 21 ; and nx |n| + ≤M 1 + γn

for some calculable constant M. If M0 + M1 is small enough, M ≤

1 2

.

Proof. The maximum principle yields: 0 ≤ n(x, t) ≤ M0 and 1 + γn ≥ 12 for all nx t < T, if γ ≥ 0. The same holds for γ < 0 if we assume 1 − γM0 > 12 . Set u = 1+γn , and note that a direct calculation gives: ut − uxx = f0 (x, t)ux − g0 (x, t)u for some f0 , g0 with g0 ≥ 0. On the boundary we have: n nx = 1 + γn 1 + γn and, at t = 0,

|n0 | nx = 1 + γn 1 + γn0 and the result is implied once again by the maximum principle.

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W. Allegretto

It follows form 2.1 that g(x, t) in (II.1.10) is bounded and we eliminate the two integrals on the righthand side of (II.1.10) by treating (II.1.10) as a first order x 1 differential equation for x nt . Setting f (x, t) = e− 0 g(z,t)dz and integrating gives: nt − nxx = F2 (n, nx ) +

g(x, t)f (1, t) f (x, t)

1

ynt (y, t)dy

(II.1.11)

0

for some smooth functional F2 . Multiplication of (II.1.11) by x and integration yields: t ynt (y, t)dy = F3 (n, nx ) (II.1.12) 1 − λ(t) 0

with F3 smooth and: λ(t) = f (1, t) 0

1

g(x, t) x dt = −f (1, t) f (x, t)

1

x 0

fx dx. f2

I.e.: 1 − λ ≥ c > 0. Equation (II.1.12) can be used to reduce (II.1.11) to: nt − nxx = F (n, nx )

(II.1.13)

with F a smooth functional. A contraction mapping argument then gives: Theorem 2.2. Under the conditions of 2.1, system (II.1.5) - (II.1.9) has a unique solution. We observe that if |nx /(1 + γn)| is not sufficiently small, the process fails. A generic counterexample is given in [58].

2.2 C α,α/2 solutions and the amorphous silicon system The mathematical model now involves drift-diffusion equations as well as equations describing the density of trapped charges, [59], [96]. The latter may be integrated in time and we obtain a drift-diffusion system with integral (i.e. “memory”) terms. Specifically we obtain: t t −∆ϕ = p − n + C1 (x) + (p − n)e− ξ (n+p+2)dη dξ (II.2.1) 0

∂n (II.2.2) − ∇[Dn ∇n − nµn ∇ϕ] ∂t t

t t t (p − n)e− ξ (n+p+2)dη dξ − n 1 + (p − n)e− ξ (n+p+2)dη dξ =1− 0

0

∂p (II.2.3) − ∇[Dp ∇p + pµp ∇ϕ] ∂t t t

t t (p − n)e− ξ (n+p+2)dη dξ − p 1 − (p − n)e− ξ (n+p+2)dη dξ =1+ 0

0


69

to be satisfied in a smooth domain Ω ⊂ R3 . We observe that the factor “2” is present in the various integrals in equations (II.2.1) – (II.2.3) to ensure charge conservation. With these equations we associate initial/mixed boundary conditions (I.1.4), (I.1.5). We follow the approach in [6], and employ this system to also show how the existence of C α,α/2 solutions can be obtained directly. For this, it is convenient to assume that if x ∈ (∂ΩN ) ∩ ∂ΩD and N is a small neighbourhood of x, we require that regularity considerations for N ∩ ∂Ω be reduced via bi-Lipschitz (resp. smooth) coordinate maps to similar problems on quarterspheres (resp. hemispheres). The reader interested in the explicit formulations of such conditions may find them for example in [89], [101], [110], [120]. We assume Dn , Dp , µn , µp are functions of (x, t, n, p, |∇ϕ|) with 0 < α ≤ Dn , Dp ≤ β for some constants α, β, and µn , µp can be expressed in the form µn,p = constant + saturation term. It is convenient to introduce the functional: t t h(p, n) = (p − n)e− ξ (n+p+2)dη dξ, 0

to choose and fix a parameter τ with 3 < τ < 4 and set QT = Ω × (0, T ). We recall Ω ⊂ R3 , and observe the results: Lemma 2.1. (a) Let −∆u(x) = f1 (x) in Ω, with f1 ∈ Lτ (Ω). If u = u (x) ∈ C 1 on ΩD , 1,τ on ΩN then u ∈ H (Ω) and

∇u Lτ (Ω) ≤ C f1 Lτ (Ω) + u C 1 (Ω)

∂u ∂n

=0

(b) Let v be a generalized solution of vt − ∇[w∇v + δv] + mv = f2

(II.2.4)

with 0 < α < w(x, t) < β (α, β constants) and |δ|2 , m, f2 in Lq,r (QT ) for some q ∈ n ( n2 , ∞], r ∈ (1, ∞], r1 + 2q < 1. Suppose v satisfies the initial/boundary conditions: ∂v 1 v = v(x) ∈ C on {∂ΩD × (0, T )} ∪ Ω × {0} , ∂ν = 0 on ∂ΩN × (0, T ) and v is bounded in L2 (QT ). Then there exists an α0 > 0 (independent of v) such that v ∈ C α0 ,α0 /2 (QT ). (c) If v solves (II.2.4) with the given initial/boundary conditions and v L2 (Ω) (t) is bounded, then v is globally bounded in L∞ . The proof of Part (a) is immediate from the results of Shamir, [110], (see also Murty and Stampacchia, [101]). Parts (b), (c) follow from the results of Ladyzhenskaya et al., [88], together with a reflection argument to establish the needed regularity on ∂ΩD ∩ ∂ΩN . From 2.1, we obtain the following a priori bound: Theorem 2.3. There exist α1 > 0 and K > 0 such that all solutions of (1I.2.1)– (II.2.4) in C α,α/2 (QT ) with 0 < α < α1 and n, p ≥ 0 actually satisfy n C α1 ,α1 /2 + p C α1 ,α1 /2 + ϕ C α1 ,α1 /2 ≤ K.

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The proof hinges on noticing that |h(p, n)| ≤ 1, and the observation that we may choose test functions in (II.2.3), (II.2.4) analogous to those selected in 1.2. From this, and the nature of the coupling, we obtain that n, p are bounded in Lξ (Lξ ) for large ξ. 2.1(a) then gives that |∇ϕ| is bounded in Lξ (Lτ ). 2.1(b), (c) then give the result. Observe that working with C α,α/2 spaces makes the process much easier. An existence theorem is immediate from 2.3, [6], by embedding the problem in a parameterized family of problems and using a Degree Theory argument in a standard way. If Dn , Dp , µn , µp depend only on (x, t), the solution is unique by Gronwall’s Lemma.

2.3 Avalanche generation If impact ionization is taken into account, [109], [92], [93], the following system is obtained: −∆ϕ = p − n + N ∂n − ∇[Jn ] = α1 (|∇ψ|)|Jn | + α2 (|∇ψ|)|Jp | ∂t ∂p + ∇[Jp ] = α1 (|∇ψ|)|Jn | + α2 (|∇ψ|)|Jp | ∂t

(II.3.1) (II.3.2) (II.3.3)

with: Jn = µ1 (∇n − n∇ϕ) Jp = −µ2 (∇p + p∇ϕ). Here µ1 , µ2 are constants and α1 , α2 are smooth bounded functions, and we keep the conditions (I.1.4)-(I.1.6). Clearly the quite significant difference between this problem and the earlier ones, lies in the form of the right hand sides of (II.3.2), (II.3.3). This problem has been studied in several papers by Naumann, (see [102]), and Frehse and Naumann, [55], [56], and the approach used is basically as follows: p n In Jn (resp. Jp ) the term n (resp. p) is first replaced by 1+εn (resp. 1+εp ) for some ε > 0. The same modification is made on the right hand side of (II.3.1) and the resulting approximate system is solved. Uniform estimates are obtained by clever choices of test functions and a limit process then gives the desired solution.

3 Degenerate Systems 3.1 Degenerate problems: limit case of the hydrodynamic models A degenerate drift-diffusion system arises as the limiting case of the hydrodynamic model, [93], [94]. Specifically we now have:

∂n − ∇ Dn ∇ r(n) − µn n∇ϕ = R ∂t

∂p − ∇ Dp ∇ r(p) + µp p∇ϕ = R ∂t −∆ϕ = p − n + N

(III.1.1) (III.1.2) (III.1.3)


71

with r(n), r(p) such that r(0) = 0, r > 0 in (0, ∞). The prototype case is r(p) = pγ with γ ≥ 1 and, of course, the case γ = 1 reduces this system to the nondegenerate case considered earlier. With these equations we associate mixed boundary conditions (I.1.4) - (I.1.6). In the steady-state case, equations (III.1.1) - (III.1.3) with the associated boundary conditions, may at first be treated by introducing “quasi-Fermi” levels and then using an upper/lower solution process formally similar to that employed for 1.1, [94]. Observe that for Dn = µn , Dp = µp constants; n, p > 0; and R = 0 equation (III.1.1), (III.1.2) can be written in the steady-state case as: n r (ξ) dξ − ϕ =0 ((III.1.1) ) ∇ n∇ ξ 1 p r (ξ) dξ + ϕ =0 ((III.1.2) ) ∇ p∇ ξ 1 p n r (ξ) dξ, 1 r ξ(ξ) dξ and the variables ψ1 = so that the enthalpy functions 1 ξ p n r (ξ) dξ − ϕ, ψ2 = 1 r ξ(ξ) dξ + ϕ play a significant role in the analysis. ξ 1 The new difficulty is now the degeneracy introduced by the terms r(n), r(p). However, if one postulates that R ≡ 0 and a suitable smallness condition on N and on the variation of the Dirichlet boundary data, then a solution may be found near the thermal equilibrium case with n, p positive, by exploiting the fact that upper/lower solutions may be chosen once again to be constants. As may be expected, if the above conditions are relaxed, then the presence of the degeneracy leads to situations where solutions of (III.1.1) - (III.1.3) exist but with n, p merely nonnegative. Such solutions are termed vacuum solutions, and the zones where n or p vanish are termed vacuum sets. Solutions of (III.1.1) - (III.1.3) in the steady state case with R = 0, are obtained under more general conditions on the data by regularizing the problem (r(n), r(p) are replaced by rε (n), rε (p) with rε (ξ) ∼ ξ near ξ = 0) and then passing to the limit as ε → 0. That the resulting solutions n, p may actually have vacuum sets is shown in [94] by example. A more detailed study of the existence of vacuum or non-vacuum (i.e. positive) solutions is given in [116], in the thermal equilibrium case. In such a case, the system under consideration can be reduced to a single (nondegenerate) elliptic semilinear equation for ϕ, for which monotonicity methods are applicable. The existence/non existence of vacuum solutions and estimates on size/location of the vacuum sets may then be obtained by the construction of suitable local upper/lower solutions to the single equation. A detailed analysis is given in [116] for the one dimensional situation. It is interesting to note that it is shown in [116] that a vacuum can appear for at most one of n, p if N is of fixed sign. The same problem, i.e. reduction to a single equation in the thermal equilibrium case followed by analysis of this equation, was also considered by J¨ ungel, [83]. In this paper the regularity of the boundary of the vacuum sets is also considered and conditions under which it is a C 1 manifold are obtained. Illustrative numerical examples, including some error estimates, are also given in [83]. The above mentioned results clearly dealt with rather specific cases of system (III.1.1) - (III.1.3): R = 0; steady state; at or near equilibrium. The case R = 0 was treated and some other results were obtained in [85], where a precise form for R and for the boundary conditions is given based on physical arguments. The existence of a solution is then shown by passing to the limit in a

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W. Allegretto

family of regularized problems. Voltage-current characteristics are also examined in [85] for one dimensional pn-diode. As mentioned above, the methods used in the above analyses often depended critically on the specific nature of the problem considered, but of course there is more scope when the approach involves the process of approximating (III.1.1) (III.1.3) with a family of regular problems and then passing to the limit. This idea was followed in [80] to obtain solutions for the time-dependent problem (for earlier one dimensional results in this direction as well as some numerical considerations, see [82]). The treatment given in this paper is actually stated for an electrophoretic system involving three charged species – see below – but will be presented here in the context of the drift diffusion situation with two charged species. Assume that by a solution n (resp. p) we mean that n ∈ L∞ , ∇r(n) ∈ L2 (resp. p ∈ L∞ , ∇r(p) ∈ L2 ). We then have: Theorem 3.1. Let r, r , be nondecreasing and r(0) = 0; µn = Dn , µp = Dp constants. Choose T > 0. Then there exists a solution (n, p, ϕ) to equations (III.1.1) (III.1.3), (I.1.4) - (I.1.6) in QT with n, p ≥ 0. Proof. The first step consists in the regularization of the problem: set rε (ξ) = r(ξ) + εξ, and consider the equations: nt − µn ∇ · ∇rε (n) − n∇ϕ = R(n, p) ((III.1.1) ) ((III.1.2) ) pt − µp ∇ · ∇rε (p) + p∇ϕ = R(n, p) −∆ϕ = p − n + N

((III.1.3) )

subject to boundary conditions (I.1.4) - (I.1.6). A solution (nε , pε , ϕε ) to this (regular) system, with nε , pε ≥ 0, can be shown by topological methods closely related to the ones we presented earlier. In particular, it can be shown that any solution must satisfy an estimate of type: nε (t) 0,∞,Ω + pε (t) 0,∞,Ω ≤ C(t) with C(t) independent of ε, since the degeneracy does not play a significant role here. The next step involves further a-priori estimates on (nε , pε , ϕε ) so that a strongly convergent in L2 (QT ) can be extracted. To do this, take: nε − n , pε − p as test functions for equations (III.1.1) , (III,1,2) , and estimate each term in turn to conclude:

(rε ) (nε )|∇nε |2 ≤ c

(nε − n )(t) 0,2,Ω + QT

2

(rε ) (nε )|∇nε |2 .

|∇rε (nε )| ≤ c QT

QT

An identical estimate holds for pε , and we have (nε )t L2 ([H 1 (Ω∪∂ΩN )] ) + (pε )t L2 ([H 1 (Ω∪∂ΩN )] ) ≤ c. 0

Finally set:

0

ρ(s) =

s

r (τ ) dτ

0

for

s ≥ 0,

1 S = {v ∈ L∞ + (Ω)|ρ(v) ∈ H (Ω)}, 1/2 M (v) = |∇ρ(v)|2 for v ∈ S. Ω


73

Note that if λ ∈ [0, 1] and v ∈ S then

2 M (λv)2 ≤ λM (v)

by the monotonicity of r , while a proof similar to [90, Chap. 1, Prop. 12.1] shows that {v|v ∈ S and M (v) ≤ 1} is relatively compact in L2 (Ω). A variant of a result of Dubinskii, [80], then shows that we may assume nε → n, pε → p in L2 (QT ). Elliptic estimates then show ∇ϕε → ∇ϕ in L2 (QT ), and we further have the convergence: rε (nε ) → r(n),

rε (pε ) → r(p)

weakly in

L2 (H 1 )

and

(nε )t → nt , (pε )t → pt weakly in L2 ([H01 (Ω ∪ ∂ΩN )] ). From these estimates, the conclusion of the theorem follows by passing to the limit as ε → 0. A variety of extensions of this result are also given in [80], as well as some uniqueness criteria. It is also shown there that if the boundary/initial data are positive then the existence of non-vacuum solutions follows. The question of the existence of uniformly bounded solutions and of an absorbing set are also considered. The temporal and spatial localization of the vacuum sets is considered in [45]. In this paper, speed of propagation, formation and lack of dilation of vacuum sets are investigated. Finally, numerical results are presented in [84].

3.2 Temperature Effects So far temperature T has been treated as a constant (or at least known). In general, however, T itself enters into the equations, [109], and we have the modifications Jn = Dn ∇n − nµn ∇ϕ + nDnT ∇T

(III.2.1)

Jp = −[Dp ∇p + pµp ∇ϕ +

(III.2.2)

pDpT ∇T }

with Dn , Dp , µn , µp , DnT , DpT also functions of T, [109]. To our fundamental system of equations we also add a heat conservation equation: ∂T − div k(T )∇T = H (III.2.3) ∂t with suitable boundary conditions, and for example, [109], 1 − np Eg H = (Jn + Jp ) · (−∇ϕ) + 1+n+p with Eg = band gap energy. We are not aware of mathematical results for this system in the given generality. Saidman and Troianiello, [108], did consider a drift-diffusion system which included temperature, but as mentioned in their paper, this was done under restrictive assumptions: µn , µp = const.; DnT = DpT = 0; Dn , Dp bounded as T → 0, ∞, etc. This enabled them to use an approach somewhat similar to what is employed for the standard drift-diffusion system. More recently, Yin, [131], also considered a related set of equations under a special form of coupling, and some other related systems are mentioned below. Note that, as given, system (III.2.1), (III.2.2) appears to preclude the possibility of introducing variables analogous to u, v so that it could be reduced as was done earlier.

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3.3 Degenerate problems: thermistor equations and micromachined structures Considerable more work has been done in recent years for a simplified version of system (III.2.1) – (III.2.3). Here we consider a unipolar situation, assume Ohm’s Law holds: J = σ(T )E and thus obtain a “drift” system, also called the “Thermistor Equations”: −∇[σ(T )∇ϕ] = 0

(III.3.1)

∂T − ∇ · [k(T )∇T ] = σ(T )|∇ϕ|2 . ∂t

(III.3.2)

For ϕ we assume that the mixed conditions (I.1.4), (I.1.5) still hold, with analogous conditions on T : T = T > 0 ∂T =0 ∂ν

on

T ∂ΩD

on

T , ∂ΩN

and

Ω × {t = 0}

T T

= ∂ΩD and ∂ΩN

= ∂ΩN . There are two problems with with, in general, ∂ΩD (III.3.1) - (III.3.2): The first is degeneracy: i.e. σ(T ), k(T ) → 0 as T → ∞ may arise in applications. The second is the quadratic term “σ(T )|∇ϕ|2 ” on the right hand side of (III.3.2). In view of (III.3.1), we note that at least formally, σ(T )|∇ϕ|2 = ∇ σ(T )ϕ∇ϕ and this substitution is commonly used to simplify the analysis. The study of these equations is much older than that of the drift-diffusion system. The first references we are aware of date from the turn of the century, e.g.: [46]. System (III.3.1) – (III.3.2) and its variations has become particular significant to microelectronics in recent years due to the importance of micromachined structures being considered for possible sensor applications, [16], [100]. Mathematical investigations have been undertaken by Cimatti and Prodi, [41], Cimatti, [37],[39],[40], Chen, [31], Chen and Friedman, [32]-[33], Fowler and Howison, [53], Howison, [72], Howison, Rodrigues and Shillor, [73], Shi, Shillor and Xu, [111], Antontsev and Chipot, [15], W. Xie, [121], Yuan and Lui, [132], Xu, [123-130], and others. Here we consider at first primarily the steady-state problem and follow the presentation of Xie and Allegretto, [122], [10], Xie, [120] , and Allegretto and Barabanova, [4]. The next result is quite similar in approach to those considered in the earlier sections.

Theorem 3.2. If T (a) ∂ΩN ⊂ ∂ΩN ;

(b)

∞ 0

k(T ) σ(T )

dT >

(max (ϕ)− min (ϕ)) 2 2

.

Then the steady state problem for (III.3.1) – (III.3.2) has a C α (Ω) solution (T, ϕ) with Tmax calculable. Proof. Write (III.3.2) as: −∇[k(T )∇T + σ(T )ϕ∇ϕ] = 0 or:

(III.3.3)


−∇ σ(T )∇

k ϕ2 dξ + = 0. σ 2

T 0

75 (III.3.4)

T 2 We conclude immediately that 0 σk dξ + ϕ2 takes on its max ./ min . on ∂Ω, and clearly the same is true for ϕ. Conditions (a), (b) then imply that T must be bounded. We can truncate T in the coefficients k, σ, and solve the regularized problem in C α for some α > 0 by means of Schauder’s Theorem. For this purpose Campanato space theory as presented in the book by Troianiello , [115], is particularly useful. Finally, we observe that – as was noticed before – the solutions of the regularized problem satisfy the same bound as before. ∞ We note that if 0 σk = ∞, the proof is easier. If condition (b) of the Theorem does not hold, then there may not be a solution, [39]. In more realistic situations, where k = (kij ), σ = (σij ) and (III.3.1) and (III.3.2) hold on different domains, equation (III.3.4) no longer applies. Nevertheless, some results can still be obtained by the Implicit Function Theorem if the applied bias is small enough, [11]. Next, we observe that the current I is often specified, not the potential. This leads to a nonlocal problem as follows; Suppose ∂ΩD = S0 ∪ S1 with T = T on S0 ∪ S1 , but ϕ = 0 on S0 , ϕ = λ on S1 . Here S0 , S1 represent disjoint “contact” regions and λ is an unknown constant. As before we assume ∂T = ∂ϕ = 0 on ∂ΩN ∂ν ∂ν and now that ∂ϕ I= σ(T ) ∂ν S1 is known. Scaling the potential by putting ϕ = λψ yields: −∇[σ(T )∇ψ] = 0 2

−∇[k(T )∇T ] = λ σ(T )|∇ψ| ∂ψ λ σ(T ) = −I ∂ν S1 T = T ψ≡0

on

S0 ,

on

(III.3.5) 2

∂T = 0 on ∂ΩN ∂ν ∂ψ S1 ; = 0 on ∂ΩN . ∂ν

S0 ∪ S1 ;

ψ≡1

on

(III.3.6) (III.3.7) (III.3.8) (III.3.9)

It is convenient to put T∗ = min T ≤ max T = T ∗ and

∞

k(ξ) dξ. σ(ξ) Similar methods to the ones employed before give the following bounds: K=

Lemma 3.1. If M is such that:

M T∗

T∗

k K λ2 ≥ ≥ σ 2 2

then 0≤ψ≤1 T∗ ≤ T ≤ M.

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W. Allegretto

This enables us to remove the degeneracy by truncating the coefficients σ, k just as before, which we now assume has been done. We then have: Let: E ∗ (v) = max (σ) Ω |∇v|2 ; E∗ (v) = min (σ) Ω |∇v|2 ; λ∗ = → min E∗ (v); v∈V

λ∗ = → min E ∗ (v) for v ∈ V = {v ∈ H 1 , v = 0 on S0 , v = 1 on S1 }. v∈V

Theorem 3.3. For any I ∈ (0, λ∗ K 1/2 ] there exists at least one solution for (III.3.5) – (III.3.9). Once again the proof follows from a priori estimates. It is interesting to observe the following nonexistence result. This should be compared with the earlier mentioned existence result for the drift-diffusion current driven system. Theorem 3.4. Assume that part of Ω away from the contacts is cylindrical. I.e.: there exists two constants a, b such that: Ω ∗ = Ω ∩{(x, y, z)| a ≤ z ≤ b} is cylindrical ∞ = 0 on ∂Ω ∩ ∂Ω ∗ . Then if 0 k < ∞, there exists an I such that if (u, ψ, λ) and ∂T ∂ν I depends only on conditions in the cylinder. solves (III.3.5) – (III.3.9) then I ≤ I.

Proof. Note I 2 ≤ λ2 A(c) σ(u) σ(u)|∇ψ|2 where A(c) = cross sectional A(c) T area of the cylinder at z = c. Put ω = T∗ k(ξ)dξ and note that: −

∂2

2 ω = λ σ(T )|∇ψ|2 ∂z 2 A(z) A(z) ≥

i.e. −

I2 σ(T ) A(z)

[ A ω] I2 . ≥ [ A ω] σ(T ) ω A(z) A(z)

∂2 ∂z 2

We then have by Barta’s Inequality: λ1 = whence:

π2 ≥ (b − a)2

→

a≤z≤b

sup

π2 I ≤ σ L∞ 2 (b − a) 2

I2 A(z)

∞

σ(u) ω A(z)

k(ξ) [µ(A)]2 I2 .

T∗

We note that if σ and u 0 are constants, then I ∗ /I = 1/π . We summarize the situation as follows: ∞ 1. If 0 k < ∞ then for large I there is nonexistence whether σ > const. > 0 or σ → 0 at ∞. ∞ 2. If 0 k = ∞ then existence for large I for σ > const. > 0. If σ → 0 at ∞ then in general the existence/nonexistence is not known, except in 1 dimension if ξ σ(ξ) k ∈ L∞ one can show nonexistence for large I.


77

3. To obtain an idea of the sharpness, one can estimate I ∗ , I for a cylinder and obtain: [ → min σ(ξ)]2 ∞ k I ∗ 2 1 T∗ ≤ξ≤M ∗ σ T∞ = 2 π σ L∞ k I T∗ M k ∞ k 1 with M defined by T ∗ σ = 2 T ∗ σ . For micromachined structures, simulations based on equations (III.3.1) – (III.3.2) or (III.3.5) – (III.3.7) can lead to significant discrepancies with experimental results, since these systems do not take into account heat losses to the surrounding gas. To compensate for this effect, it was suggested, [95], that the system should be replaced in steady state by: −∇[σ(T )∇ϕ] = 0 −∇[k(T )∇T ] + ηT = σ(T )|∇ϕ|2

(III.3.10) (III.3.11)

Ω

with η a complicated function which depends on the device geometry and on the surrounding gas type and pressure. In the simplest approach, η is a positive constant. Here Ω is viewed as a two-dimensional domain due to the thinness of the device. We observe that equation (III.3.11) is a nonlocal equation related to equations obtained earlier by DeFigueireido and Mitidieri, [43], Hernandez, [70], Freitas and Sweers [57], Lopez-Gomez, [91], Fiedler and Pol´ aˇcik, [51], Catchpole, [28], and others. It is possible to obtain some existence results for system (III.3.10) – (III.3.11) but the main feature of (III.3.11) is that if η > 0 is big enough, the maximum principle fails. We have however, Theorem 3.5. Consider the equation: −∇[aij (x)∇u] + η

u=f ≥0 Ω

u=0

on

∂Ω.

There exists η0 > 0, independent of f, u, such that if 0 < η < η0 then u ≥ 0. The constant η0 depends on the ellipticity bounds of (aij ) and can be characterized as: −1 L−1 (1) η0 = K − Ω

where K=

→

sup

L−1 (1)(x)L−1 (1)(y)

(x,y)∈Ω×Ω G(x, y) G = Green’s Function for − ∇ aij (x)∇ .

If aij = δij and Ω = unit sphere ⊂ Rθ η0 =

θ2 (θ + 2) . σθ 2θ−1 (θ + 2) − 1

where σθ is the surface area of the unit ball.

78

W. Allegretto The same methods as used before then give, as an example:

Corollary 3.1. If T = 0 and ∂ΩD = ∂Ω, then the steady state version of (III.3.10) – (III.3.11) has a solution with T > 0 for η small enough. We conclude with a discussion of the time-dependent problem (III.3.1) – (III.3.2). If we also assume that α ≤ σ(u), k(u) ≤ β, then existence follows by methods similar to those employed for the drift-diffusion equations. In particular, Antontsev and Chipot, [15], show: Theorem 3.6. If α ≤ σ(T ), k(T ) ≤ β then (III.3.1) – (III.3.2) with the associated boundary/initial conditions has a (weak) solution. Proof. In [15], only the case of Dirichlet or Neumann conditions is discussed in detail, but for this part the result still holds for mixed conditions. Suppose for convenience that T = 0, and note first of all that from (III.2.1) we have: ϕ L∞ ≤ K and ∇ϕ L2 ≤ K for almost all t. Next, we write σ|∇ϕ|2 as ∇(σϕ∇ϕ), whence 1 T2 + α |∇T |2 ≤ − σ(T )ϕ∇ϕ∇T 2 Ω Ω Ω α ≤K+ |∇T |2 . 2 Ω We then that Tt is bounded as a map into the dual space and T is bounded obtain in L2 L2 (Ω) . If ω is any given function in L2 L2 (Ω) , we replace σ(T ) and k(T ) by σ(ω) and k(ω) in (III.2.1) and (III.2.2) and solve the resulting linear equation. This set up a map Q, from a ball B in L2 L2 (Ω) to B, which is compact and continuous. Again the fixed point of Q is the desired T. Next, suppose we allow σ(T ) → 0 as T → ∞, but keep k(T ) bounded above and away from zero, so that (III.3.1) is degenerate, but (III.3.2) is not. It seems intuitively clear that (III.3.1) will give a bound for |∇ϕ| when T is finite, but difficulties will arise when (and if) T is infinite (and thus σ(T ) = 0). One may thus expect that σ(T )∇ϕ will have better properties than ∇ϕ and thus ought to be treated as a single function g, which one can then hope to show is actually σ(T )∇ϕ almost everywhere. This leads to the concept of a capacity solution introduced by X. Xu, [124]. Specifically: a triplet (σ, ϕ, g) is a capacity solution of (III.3.1) - (III.3.2) if (III.3.1) - (III.3.2) hold with σ(T )∇ϕ replaced by g and the equation: ρ(u)g = σ(u) ∇(ρ(u)ϕ) − ϕ∇(ρ(u)) (III.3.12) added to the system where ρ denotes any function in C01 (R). The connection between this system and the earlier one, in particular the role of g, can be seen from equation (III.3.12): choose ρ such that ρ ≡ 1 on [−k, k]. Then (III.3.12) means g = σ(u)∇ϕ a.e. on the set {(x, t) |u| ≤ k}, and, letting k → ∞, we obtain g = σ(u)∇ϕ a.e. We have: Theorem 3.7. System (III.3.1), (III.3.2), (III.3.12) has a capacity solution. The proof involves replacing σ by σk = σ + k1 and then showing that the regularized problem’s solution (Tk , ϕk ) converges to the required solution as k → ∞.


79

If we suppose both k and σ are allowed to degenerate, then X. Xu [128] has shown that for any given time interval there is a solution if the initial data is small enough (this is the extension of the steady state results of [11]). Here by a solution we mean an actual weak solution, not a capacity solution. Finally, X. Xu, [129], has also shown that if only σ degenerates at ∞ and it does so according to the rules introduced by Chen, [31]; → lim

τ →0

σ(s + τ ) =1 σ(s)

uniformly on R, then there exists a relatively open subset Q0 of QT , with the interior of QT − Q0 empty, such that u is locally H¨ older continuous in Q0 and essentially bounded in QT − Q0 . As may be expected, the difficulty with allowing degeneracy in both k, σ is that solutions may blow up in finite time. As an example, we give the following result of Barabanova, [22], (see also [15], [87]) which is the analogue of the steady-state result shown earlier. Theorem 3.8. Consider the time-dependent current driven problem, i.e. equations (III.3.5) – (III.3.9) with (III.3.6) replaced by ∂T − ∇[k(T )∇T ] = λ2 σ(u)|∇ψ|2 . ∂t

(III.3.6)

Assume Ω has a cylindrical part, just as in 3.4, and that I is constant. ∞ (a) If 0 σ(s) < ∞, σ (s) ≤ 0 then there exists a value I such that if I > I then all solutions ∞ (classical except at ∂ΩN ∩ ∂ΩD ) have T blow up in finite time. (b) If 0 k(s) < ∞ then there exists I ∗ such that if I > I ∗ then such solutions have T unbounded in Ω × {t > 0}. I ∗ depend only on the cylindrical part of Ω. This theorem can be used Again I, to also obtain nonexistence of solutions to voltage driven problems with “reasonable” T since it also gives an estimate on the largest temperature in the cylinder. Much as for the drift-diffusion equations, uniqueness can be shown for the timedependent problem, see e.g.: [125], but the same cannot be said for the steady-state case. In some special cases, Cimatti, [39], Howison, Rodrigues, and Shillor, [73], have shown the existence of only one solution. Continuity arguments, [11], also give some modest uniqueness results. We note that it is possible to give mathematical examples of situations with multiple steady state solutions. In [39], Cimatti constructed such an example in which there are three solutions. Finally, we remark that Cimatti and Xu have also obtained results for systems which take into account the Thomson effect, [125], [38], i.e. J = σ(E − α∇T ) with α = α(T ). We conclude this section with a reference to the recent formally related papers [81], [68]. In [81] the hydrodynamic equations governing the evolution of a fluid are considered in steady state and a system of three equations obtained whose existence and uniqueness is shown under suitable conditions. In [68] the system:

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W. Allegretto −∇(µn∇u) = R

(III.3.7)

−∇[νp∇v) = R

(III.3.8)

−∆ψ = p − n + D 2

(III.3.9) 2

−∇(χ∇θ) = µn|∇u| + νp|∇v| − (u + v)R

(III.3.10)

is examined, subject to mixed boundary/initial conditions. Here µ, ν denote mobilities and u, v the respective quasi-Fermi levels. While (III.3.7)-(III.3.9) can be recognized as a drift-diffusion model, equation (III.3.10) is the “temperature” equation, so that heuristically this system includes both the classic drift diffusion model and the thermistor equations. Campanato Space theory and the Implicit Function Theorem are employed to show the existence and uniqueness of H¨ older continuous weak solutions near thermodynamic equilibria.

4 Related Problems There is a variety of other problems clearly related – either physically or mathematically or both – to the ones we have already mentioned. Here we mentioned four: electrochemistry; chemotaxis; Stefan problems; in situ vitrification. We also allude briefly to other related problems. Electrochemistry problems can be viewed as extensions of the semiconductor equations obtained by replacing n, p by m interacting species, [105], [35], [36], [52], [17], [12], [62], [14], [80], [29], [44], [67]. A typical system is: ∂ui − ∇[di ∇ui + Ωi zi ui ∇ϕ] = 0, i = 1, . . . , m ∂t m −ε∆ϕ = e zk u k .

(IV.1.1) (IV.1.2)

k=1

Here zi denotes the electric charge of the particle ui , must be integer value and may be positive, negative, or zero. The parameters di , Ωi denote the diffusion and mobility constants of the particle ui and it is commonly assumed that the Einstein relation holds: Ωi = µdi with µ > 0. Finally, ε, e denote positive physical constants. With equations (IV.1.1) - (IV.1.2) we associate mixed boundary conditions and initial conditions identical to the earlier ones. We mention that it is also of interest to consider purely Neumann conditions (i.e. ∂ΩD = φ) in which case the steady-state problem is augmented by the nonlocal condition: ui dx = ci > 0, i = 1, . . . , m Ω

with prescribed constants ci . Various existence/uniqueness results for these (and related) problems may be found in the cited references. There are also several chemotaxis systems. For example the system of equations proposed by Chaplain and Stuart, [30], to describe the chemotactic response of endothelial cells under the angiogenesis stimulus, is:


81

∂u −auv − ∆u = = λu ∂t γ+u

(IV.2.1)

∂v − D∆v + k∇ [v∇u] − bv(1 − v)G(u) + βv = 0. ∂t

(IV.2.2)

With a, b, λ, γ, D, β, k relevant physical constants, and  0 if s ≤ c∗ G(s) = s − c∗ if s > c∗ .  Physically, c∗ is a threshold concentration below which proliferation does not occur. Some mathematical results for this system (subject to suitable boundary/initial conditions) are given in [13]. A different chemotaxis system was considered in [64]. Namely the system: ∂u − ∆u = −χ ∇ · (u ∇ v) ∂t ∂v − α∆v + βv = δu ∂t

(IV.3.1) (IV.3.2)

was examined by means of suitable Lyapunov functionals. Convergence and conditions excluding blow-up were obtained in terms of the data and the geometry of Ω. Stefan problems with Joule Heating are clearly related to the thermistor equations. As an example, we give the system: ∂h − ∆K(u) + λ v · ∇u = σ(u)|∇ϕ|2 ∂t −∇ σ(u)∇ϕ = 0 ∂v − ν∆ c + v · ∇ v + ∇p = f (u) in {u > 0} ∂t v=0 in {u < 0}

(IV.4.1) (IV.4.2) (IV.4.3) (IV.4.4)

div ( v ) = 0

(IV.4.5)

h ∈ α(u)

(IV.4.6)

subject to boundary/initial conditions, and in which α in a maximal monotone graph. This system was considered in [130], where the existence and regularity of a capacity solution were obtained. Finally, a related model has been proposed to describe in situ vitrification, i.e. the melting of soil by electrical currents, followed by solidification into a rock. This is a free boundary problem given by: Find (u, ϕ) and domain Ωm ⊂ Ω with free boundary Γ such that −∇ k(u)∇u = σ(u)|∇φ|2 in Ωm (IV.5.1) (IV.5.2) −∇ σ(u)∇φ = 0 in Ωm (IV.5.3) −∇ k(u)∇u = 0 in Ωs = Ω − Ω m subject to suitable boundary/free boundary conditions. This system of equations was treated in [65], where existence of a solution and of a “molten region” (u > 0) were obtained.

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We conclude by mentioning briefly other related problems. In [26], a mathematical model is developed for the laser beam induced current method, employed to detect active regions and defects in semiconductors. Here the generation-recombination term R is modified to account for the generation due to the laser beam. In [66], a system describing the transport of foreign atoms in semiconductors is considered. In [119], a system with both Auger and avalanche effects is examined, while in [79], Poisson’s equation is replaced by Maxwell’s equations in the drift-diffusion system. Finally, in [60] the modifications needed to deal with a small magnetic field are presented.

5 Approximations, Numerical Results and Applications By far the largest number of articles to be found in the literature on these topics, deals with the classical drift-diffusion system. The other problems which we have mentioned, have usually received much less attention, if any at all. For example, for the thermistor problem, we mentioned [5], [47], [133], besides the related papers already cited. We recall that two of the issues which distinguish the classical drift-diffusion system (and which are also present in many of the related problems) from standard numerical procedures for partial differential equations are as follows: The first has to do with the actual equation coefficients which so far we have reduced to unity for convenience, but in reality may be far from unity. In practical problems this necessitates the introduction of exponential fitting procedures, the most common known as the Scharfetter-Gummel algorithm, mentioned below. There are numerous description of these methods given in the above references, and we also mention [107], [106], [25], [134], [34] on these topics and related matters. The second issue – which is also present in theoretical considerations – has to do with the lack of regularity due to the mixed boundary conditions. A perusal of some of the above references, eg. [74], indicates that it is desirable to be in situations where estimates of type: u 1+α ≤ C f 0 for some α with α > 1/2 hold for solutions of the elliptic problem: −∆u = f 0 denote the in Ω, subject to mixed boundary conditions, where 1+α , norm in H 1+α,2 , L2 respectively. This requires some form of restriction on the specific problem considered. While many approaches have been used, possibly the most common discretization procedure employed in practice is the box (or “control volume”) method, which can be briefly described as follows: the domain Ω is decomposed into elements, usually: triangles (in two dimensions) or tetrahedrons (in three dimensions). To be definite and for simplicity of presentation, we consider here the two dimensional situation. Each triangle vertex determines a node, and with each node Pi we construct an associated “box”, consisting of those points which are nearer to Pi than to any other node Pj . In practice, this means that the boundary of each cell is determined by the perpendicular bisectors of the sides of the triangles which share Pi as a vertex, see Fig. 1.


83

Figure 1

Let Ωi denote the cell associated with Pi and suppose Ω is the union of the Ωi . The basic idea behind the classic “flux conservation” box method is to discretize an equation such as: −∇[a(x)∇u] = f

(V.1.1)

by integrating over each interior cell, assuming ∇u to be constant in each element and applying Gauss’s Theorem so that the left hand side becomes: ∂u −∇[a(x)∇u]dx = − a(x) ds (V.1.2) ∂ν Ωi ∂Ωi ∂u indicates the outward normal to the cell Ωi , and dS the surface measure. where ∂ν The same principle is applied to boundary cells. In the standard box method, the ∂u term ∂ν is easily approximated in terms of the nodal values of u by assuming ∇u to be constant and a(x), Ω f (x)dx are dealt with by “lumping”, and a suitable i matrix is thus obtained. It is already clear from this brief description that obtuse angle triangles should be avoided in the triangularization. A description of this discretization as a piecewise-linear Petrov-Galerkin method may be found in [21], [27], [69], [86]. In these papers, precise formulations of the geometric requirements on the triangularization as well as error analysis are presented. As mentioned earlier, the approach employed for the current equations involves, besides the box method, a type of exponential fitting, which is briefly described in the classical case as follows: We observe that the elliptic part of the left hand side of the “n equation” (I.1.2) may be expressed in divergence form as Jn = eϕ ∇(e−ϕ n) if µn , Dn are set to unity. The main idea involves treating Jn as a constant and ϕ as piecewise linear, so that integrating from node i to node j gives:

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W. Allegretto Jn · ν ij =

nj ni B(ϕj − ϕi ) − B(ϕi − ϕj )

ij

ij

where B denotes the Bernoulli function, B(ξ) = ξ/[eξ − 1], and the subscripts i, j mean that the quantity is to be evaluated at node i, j respectively, see Fig. 1. If we now apply the box procedure, then we find that the contribution to the “n matrix” from this term and this element is (Jn · ν ij )dij , with Jn · ν ij as given in the previous formula. A similar procedure is employed for p. One of the advantages of the box method is its current conservation properties. These can also be obtained by the employment of mixed methods, [25], briefly described as follows: Consider once again equation (V.1.1), with – for simplicity – ∂u u = 0 on ∂ΩD , ∂ν = 0 on ∂ΩN . If we set σ = a∇u then we may rewrite (V.1.1) as ∇·σ = f (V.1.3) σ = a∇u We assume Ω has been decomposed into a family of triangles Th and introduce the spaces: RT (T ) = {δ = (δ1 , δ2 )| δ1 = a + bx, δ2 = c + by; a, b, c ∈ R} Σ = {δ ∈ L2 (Ω), div (δ) ∈ L2 , δ·ν = 0 on ∂ΩN , δ|T ∈ RT (T ); for all T ∈ Th} Φ = {φ ∈ L2 (Ω), φ|T ∈ P0 (T ) for all T ∈ Th } Then the discretized version of (V.1.1) via (V.1.3) becomes: Find σ ∈ Σ, u ∈ Φ such that φ div (σ) = fφ for φ ∈ Φ Ω Ω a−1 σ · τ = − u div (τ ) for τ ∈ Σ. Ω

Ω

Due to the assumptions on Σ, this scheme preserves (strong) current continuity, but the associated matrix will not be definite. To deal with this situation, enlarge Σ = { δ ∈ L2 (Ω), δ|T ∈ RT (T )}. Then if u, σ are as given in (V.1.3) and τ ∈ Σ to Σ we have: −1 a σ·τ =−→ u div (τ ) − λτ · ν (V.1.4) → T

T

T

T

∂T

where λ = u|∂T , and → T

φ div (σ) =→

T

Ω

f φ.

(V.1.5)

Ω

In order to ensure current continuity, we adjoint the equation µ σ · ν = 0 for µ ∈ Λ → T

(V.1.6)

∂T

where Λ = {µ|µ ∈ L2 (Eh ); µ| ∈ P0

for all

∈ Eh ; µ| = 0

and Eh denotes the set of edges associated with Th .

if

∈ ∂ΩD }


85

u ∈ Φ, An approximate solution to (V.1.4) - (V.1.6) is then sought with σ ∈ Σ, λ ∈ Λ. It can be shown, [25], that if the triangularization of Ω is of acute type, then static condensation procedures yield an M -matrix equation for the unknowns, so that the resulting discrete system can be solved. For the classical drift-diffusion case, in equation (V.1.1) we have for example u = e−ϕ n, a = eϕ and in these cases e±ϕ is replaced in the system by e± ϕ , its average over the edges, [25], i.e.: e±ϕ = e±ϕ i

i

where i ∈ Eh . Alternatively, (see [84], where this process is applied to the degenerate case) it is possible – at least if R is zero, or small – to also treat Jn as a constant in each triangle (much as was done in the classical case) to simplify the discretized system. In the classical approach the currents densities are assumed to be constant in each element, but in the recent work, [106], a different view is taken. Here it is assumed that the current densities are linear with divergence equal to zero, and a nonconforming exponentially fitted finite element approximation obtained as a consequence. In [106], some convergence results are also given for this discretization. We remark that at an abstract level, it is possible to consider the mathematical approximation of the various systems we have mentioned without dealing in detail with the practical effectiveness of the suggested schemes. As an example, we consider the approximation of the classical drift-diffusion system, as given in [134], by means of a mixed finite element method for the potential equation combined with a finite element method for the charge density equations. The approximation considered is given by the equations: Find (uh , ψh , nh , ph ) ∈ Σ × Φ × Sh × Sh such that for J = (0, T ) there holds: (uh , v) − div (v), ψh = 0

div (uh ), η = (ph − nh + N, η)

∂n

h

∂t ∂p

h

∂t

, z +(Dn ∇nh , ∇z) + (µn nh uh , ∇z) = R(nh , ph ), z , z + Dp ∇ph , ∇z)−(µp ph uh , ∇z) = R(nh , ph ), z

v ∈ Σ,

t∈J (V.1.7)

η ∈ Φ,

t∈J (V.1.8)

z ∈ Sn ,

t∈J (V.1.9)

z ∈ S,

t∈J (V.1.10)

with the obvious treatment of the initial conditions, and with the boundary conditions chosen to vanish on ∂ΩD . Here Σ, Φ and (V.1.7), (V.1.8) are as given above for the mixed method, while Sh is a finite dimensional subspace of S = {z ∈ H 1 (Ω), z|∂ΩD = 0}. It is also assumed that the solution of the original system enjoys certain regularity conditions, namely:

86

W. Allegretto ϕ L∞ (H 1+α ) + ∇ϕ L∞ (Lµ ) ≤ K n L∞ (H 1+β ) + n L∞ (L∞ ) ≤ K p L∞ (H 1+β ) + p L∞ (L∞ ) ≤ K

∂p ∂n 2 2 + 2 2 ≤K ∂t L (L ) ∂t L (L )

with 0 < α, β < 1 and µ > 2. Under boundedness conditions on Dn , Dp , µn , µp , and suitable approximation properties of Σ, Φ, Sh , it is then obtained that system (V.1.7) - (V.1.10) has a unique solution and that furthermore for σ = min (α, β) and h small, there holds: n − nh L∞ (L2 ) + p − ph L∞ (L2 ) + ∇(n − nh ) L2 (L2 ) + ∇(p − ph ) L2 (L2 ) + ϕ − ϕh L∞ (L2 ) + ∇ϕ + uh L∞ (L2 ) ≤ Khσ . We pass to the question of actual numerical simulations in applications. Obviously, numerical results are of paramount importance in practical problems. The actual implementation of various numerical schemes and the comparison of the results with experimental measurements are given in the Engineering Literature. We refer the interested reader in particular to the issues of the Institute of Electrical and Electronic Engineers Transactions on Computer Aided Design, where numerous such articles may be found. Some of the papers and books we have cited earlier contain numerical analyses and simulations, and we also refer to the books in the [NASECODE] series and [112], [20], [19], [99], [18], [42], [71]. In these sources detailed descriptions of such critical practical topics as: grid generation, nonlinear discretization schemes, matrix solvers, etc. may be found. We mention two of these topics briefly here, drawing upon the descriptions given in [104], where many other references are given. The first step – and arguably the most important – involves the grid generation. The mesh obtained should be able to deal with “arbitrary” geometries and with variations of the material parameters. The classic finite difference approach, [109], decomposes the domain into rectangles or cubes and is less flexible in dealing with complicated regions and/or mesh refinement. However, it is relatively simple to implement since adjacency information is obvious. The more common approaches involve triangles or tetrahedrons, constructed by a variety of means: curvilinear transformations, local grid refinement, etc. As mentioned earlier, it is also important to be able to avoid obtuse angles and to be able to interactively refine part of the mesh. Such refinements are often present a priori, based on the specific problem and on past experience, and may be combined with a posteriori steps based on error estimates. In practice, after suitable scaling, the discretized equations are often solved by using a nonlinear Newton procedure which is often reduced to a decoupled scheme (Gummel’s Algorithm), which works well for small currents and R, and is described as follows: Starting with a guess n0 , p0 , we calculate ψ0 from the potential equation using n0 , p0 for n, p on the right hand side. After that, given nk , pk , ϕk we calculate ϕk+1 from the linearization of the potential equation employing the quasi-Fermi potentials (calculated using nk , pk ) on the right hand side. The current densities are then updated (as linear equations) in turn to yield nk+1 , pk+1 by using the most up to date values of the other two variables, and the process is repeated. An


87

extension of this classical procedure to deal with degenerate problems is given in detail in [84]. The iterations usually begin at thermal equilibrium and then the potential is incremented in suitable steps until the desired applied bias is reached. Often acceleration procedures are also employed. The recent book by Jerome, [74], (see also [77], [78]) deals in detail with some of these topics for the steady state case, and has many references on the subject. The significant role of analysis and numerical simulations in practical applications is obvious: by these means the performance of contemplated devices may be quickly and cheaply assessed as may the effect of geometric and other variations on the response of the structures presently being constructed. In this way, the number of design and fabrication cycles needed to meet the desired specifications can be reduced. Besides discretization errors, and in view of the assumptions involved in obtaining the mathematical models and of the tolerances present in the fabrication processes – see below – the obvious question arises as to whether the simulations are accurate enough to satisfactorily describe device performance. It appears essential to determine the ultimate validity of the results obtained by direct comparison with experimental measurements on prototype structures. We conclude with a description of the simulation of an actual microsensor application taken from [7], [8], [9]. Volume micromachining of silicon by anisotropic etching has produced several interesting and possibly truly useful microelectromechanical systems. These have appeared in the popular scientifical literature as tiny cogs, wheels, comb drives, etc. The more limited but less expensive micromachining compatible with standard complimentary metal-oxide-semiconductor (CMOS) technology has significant potential for the production of microsensors and microactuators. We present two such possibilities. After etching, the two structures considered consisted of a thin platform ( 2.5µm) suspended over a deep trench ( 50µm), and shown in Fig. 2. They were fabricated by the Northern Telecom 3µm 13-layer process.

Figure 2a

The first device – shown in Fig. 2(a) – consists of a polysilicon resistor of width 10 µm and length 1000µ m supported by an oxide platform of width 20 µm, with 16 oxide supporting arms. The etched region was 90 µm wide. The serpentine layer shown in Fig. 2(a) is a polysilicon resistor. It has a width of 6µm and from this the other dimensions may be estimated. Polysilicon has a moderate temperature coefficient of resistance, and thus measuring the resistance change of the polysilicon allows any physical process affecting this resistance to be monitored. This has been demonstrated for temperature sensors, thermal radiation emitters, gas flow sensors and gas pressure sensors, etc. We focus on the last application: A current is passed through the resistor and in short order (a few milliseconds) the temperature reaches

88

W. Allegretto

Figure 2b

steady state. If the surrounding gas pressure is reduced, then the temperature of the resistor will increase, leading to a change in resistance. Monitoring resistance changes can thus be used to estimate the surrounding gas pressure, at least for low pressures (around 1 Torr.). The equations employed to simulate this situation are basically the time dependent version of those given by the system (III.3.10) - (III.3.11). Observe however that while equation (III.3.10) holds for the resistor only, equation (III.3.11) applies to the much broader region where thermal effects are significant, and as a consequence the mathematical analyses earlier presented no longer seem applicable. Nevertheless the simulation gives good results in the most important test: comparison with experiment. We employed an abstract Gauss-Seidel procedure, an adaptive grid generator and the box method as described earlier, with ∂T ∂t discretized by a simple backward Euler, and σ(T )|∇ϕ|2 replaced by ∇(σ(T )ϕ∇ϕ). The resulting matrix equations were solved by a family of direct solvers, since the matrices involved were fairly small. Fig. 3 and Fig. 4 give the comparison between simulation and experiment for resistance changes as a function of current at various pressures.

We note that due to manufacturing and etching tolerances, it has been our experience that devices which ought to be in theory exact copies actually are somewhat different. This can lead to noticeable discrepancies (possibly 10%) in their performance, and thus renders the need for very high precision simulations somewhat questionable, at least for these problems, unless uniformity in construction can be achieved. The simulation results we presented here were not obtained from the design parameters but from actual measurement of the constructed device parameters.


89

Figure 5

References 1. F. Alabau On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case; Part II: a constructive method for the electroneutral case, Trans. Amer. Math. Soc. , 350, 1998, 47094756 2. F. Alabau Structural properties of the one-dimensional drift-diffusion models for semiconductors, Trans. Amer. Math. Soc. , 348, 1996, 823-871 3. F. Alabau Uniqueness results for the steady-state electrodiffusion equations in the case of monotonic potentials and multiple junctions, Nonlinear Analysis , TMA29, 1997, 849-887

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4. W. Allegretto and A. Barabanova Positivity of solutions of elliptic equations with nonlocal terms, Proc. Roy. Soc. Edinb. , 126, 1996, 643-663 5. W. Allegretto, Y. Lin and A. Zhou A box method for coupled systems resulting from thermistor problems to appear 6. W. Allegretto, Y. Lin and A. Zhou C α,α/2 solutions for the amorphous silicon system and related problems , Electronic Journal Diff. Eq. , to appear 7. W. Allegretto, B. Shen, P. Haswell, Z. Lai and A. Robinson Numerical modeling of a micromachined thermal conductivity gas pressure sensor, IEEE Trans. Computer-Aided Design , 13, 1994, 1247-1256 8. W. Allegretto, B. Shen, T. Kleckner and A. Robinson Micromachined polysilicon power dissipation: simulation and experiment, IEEE Trans. ComputerAided Design , 16, 1997, 627-637 9. W. Allegretto, B. Shen, Z. Lai and A. Robinson Numerical modelling of the time response of CMOS micromachined thermistor sensor, Sensors and Materials , 6, 1994, 71-83 10. W. Allegretto and H. Xie A non-local thermistor problem, Euro. Jnl. Appl. Math. , 6, 1995, 83-94 11. W. Allegretto and H. Xie Solutions for the microsensor thermistor equations in the small bias case, Proc. Roy. Soc. Edinburgh , 123A , 1993, 987-999 12. W. Allegretto, H. Xie and S. Yang Existence and uniqueness to an electrochemistry system, Appl. Anal. , 59, 1995, 27-48 13. W. Allegretto, H. Xie and S. Yang Properties of solutions for a chemotaxis system, J. Math. Biol. , 35, 1997, 949-966 14. H. Amann and M. Renardy Reaction-diffusion problems in electrolysis, Nonlinear Differential Equations and Applications , 1, 1994, 91-117 15. S. Antontsev and M. Chipot The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal. , 25, 1994, 1128-1156 16. S. Audet and S. Middelhoek Silicon Sensors, Academic Press, New York, 1989 17. J.D. Avrin Global existence for a model of electrophoretic separation, SIAM J. Math. Anal. , 19, 1988, 520-527 18. G. Baccarani and M. Rudan Simulation of Semiconductor Devices and Processes, Tecnoprint, 1988 19. R. Bank , R. Bulirsch, H. Gajewski and K. Merten (Eds.) Mathematical Modelling and Simulation of Electrical Circuits and Semiconductor Devices, Birkh¨ auser, Boston, 1994 20. R. Bank , R. Bulirsch and K. Merten (Eds.) Mathematical Modelling and Simulation of Electrical Circuits and Semiconductor Devices, Birkh¨ auser, Boston, 1990 21. R. Bank and D. Rose Some error estimates for the box method, SIAM J. Numer. Anal. , 24, 1987, 777-787 22. A. Barabanova The blow-up of solutions of a nonlocal thermistor problem, Appl. Math. Lett. , 9, 1996, 59-63 23. H. Beirao da Veiga On the semiconductor drift diffusion equations, Differential and Integral Equations , 9, 1996, 729-744 24. H. Beirao da Veiga Remarks on the flow of holes and electrons in crystalline semiconductors in Navier-Stokes Equations and Related Non-Linear Problems ed. A. Sequeira, Plenum Press, New York, 1995 25. F. Brezzi and M. Fortin Mixed and Hybrid Finite Element Methods, Springer, New York, 1991


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26. S. Busenberg, W. Fang and K. Ito Modeling and analysis of laser-beam-induced current images in semiconductors, SIAM J. Appl. Math. , 53, 1993, 187-204 27. Z. Cai, J. Mandel and S. McCormick The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal. , 28, 1991, 392-402 28. E. Catchpole A Cauchy problem for an ordinary integro-differential equation, Proc. Roy. Soc. Edinb. , 72, 1972, 39-55 29. D. Chae and O. Imanuvilov Global unique existence of a positive solution for a system of equations in electrochemistry, Proc. Roy. Soc. Edinb. , 128, 1998, 507-524 30. M. Chaplain and A. Stuart A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med. and Bio. , 10, 1993, 149-168 31. X. Chen Existence and regularity of solutions of a nonlinear degenerate elliptic system arising from a thermistor problem , J. Partial Differential Equation , 7, 1994, 19-34 32. X. Chen and A. Friedman The thermistor problem for conductivity which vanishes at large temperature, Quart. Appl. Math. , 51, 1993, 101-115 33. X. Chen and A. Friedman The thermistor problem with one-zero conductivity, IMA Preprint Series , 793 34. Z. Chen and B. Cockburn Analysis of a finite element method for the driftdiffusion semiconductor device equations: the multidimensional cases, Numer. Math. , 71, 1995, 1-28 35. Y. Choi and R. Lui Multidimensional electrochemistry model, Arch. Rational Mech. Anal. , 130, 1995, 315-342 36. Y. Choi and R. Lui Uniqueness of steady state solutions for an electrochemistry model with multiple species, J. Differential Equations , 108, 1994, 424-437 37. G. Cimatti A bound for the temperature in the thermistor problem, IMA J. Appl. Math. , 40, 1988, 15-22 38. G. Cimatti On the problem of the electrical heating of a conductor, Riv. Mat. Univ. Parma , 14, 1988, 53-59 39. G. Cimatti Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math. , XLVII, 1989, 117-121 40. G. Cimatti The stationary thermistor problem with a current limiting device, Proc. Roy. Soc. Edinb. Sect. , A116, 1990, 79-84 41. G. Cimatti and G. Prodi Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor, Ann. Math. , 151-152, 1988, 227-236 42. W. Coughran, J. Cole, P. Lloyd and J. White (eds.) Semiconductors Part I and Part II, Springer, New York, 1994 43. D. DeFigueiredo and E. Mitidieri A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. , 17, 1986, 836849 44. P. Degond, S. Genieys, A. J¨ ungel A steady-state system in non-equilibrium thermodynamics including thermal and electrical effects, Math. Meth. Appl. Sci. , 21, 1998, 1399-1413 45. J. Diaz, G. Galiano and A. J¨ ungel On a quasilinear degenerate system arising in semiconductor theory. Part II: localization of vacuum solutions, Nonlin. Anal. TMA , to appear

92

W. Allegretto

46. H. Diesselhorst Uber das probleme eines elektrisch erwarmter leiters, Ann. Phy. , 1, 1900, 312-325 47. C. Elliott and S. Larsson A finite element model for the time-dependent Joule heating problem, Math. Comp. , 212, 1995, 1433-1453 48. W. Fang and K. Ito Asymptotic behaviour of the drift-diffusion equations, J. Differential Equations , 123, 1995, 567-587 49. W. Fang and K. Ito Global solutions of the time-dependent drift-diffusion semiconductor equations , J. Differential Equations , 123 , 1995 , 523–566 50. W. Fang and K. Ito On the time-dependent drift-diffusion model for semiconductors, J. Differential Equations , 117, 1995, 245-280 51. B. Fiedler and P. Pol´ aˇcik Complicated dynamics of a scalar reaction-diffusion equation with a nonlocal term, Proc. Roy. Soc. Edinb. , 115, 1990, 167-192 52. P. Fife, O. Palusinksi and Y. Su Electrophoretic traveling waves, Trans. Amer. Math. Soc. , 310, 1988, 759-780 53. A. Fowler and S. Howison Temperature surges in thermistors in Proceedings of the 3rd European Conference on Mathematics in Industry, eds J. Manley et al., Kluwer Academic Publishers and B.G. Teubner, Stuttgart, 1990, 197-204 54. J. Frehse and J. Naumann An existence theorem for weak solutions of the basic stationary semiconductor equations, Appl. Anal. , 48, 1993, 157-172 55. J. Frehse and J. Naumann On the existence of weak solutions to a system of stationary semiconductor equations with avalanche generation, Math. Models Meth. Appl. Sci. , 4, 1994 , 273-289 56. J. Frehse and J. Naumann Stationary semiconductor equations modeling avalanche generation, J. Math. Anal. Appl. , 198, 1996, 685-702 57. P. Freitas and G. Sweers Positivity results for a nonlocal elliptic equation, Proc. Roy. Soc. Edinburgh , 128, 1998, 697-715 58. A. Friedman and W. Liu An augmented drift-diffusion model in a semiconductor device, J. Math. Anal. Appl. , 168, 1992, 401-412 59. J. Furlan Charge carrier dynamic nonequilibrium in amorphous semiconductors, IEEE Transactions on Electron Devices , 39, 1992, 448-450 60. H. Gajewski and K. Garner On the discretization of Van Roosbroeck’s equation with magnetic field, Z. Angew. Math Mech. , 76, 1996, 247-264 61. H. Gajewski and K. Gr¨ oger On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. , 113, 1986, 12-35 62. H. Gajewski and K. Gr¨ oger Reaction-diffusion processes of electrically charged species, Math. Nachr. , 177, 1996, 109-130 63. H. Gajewski and K. Gr¨ oger Semiconductor equations for variable mobilities based on Boltzmann Statistics or Fermi-Dirac Statistics , Math. Nachr. , 140 , 1989 , 7–36 64. H. Gajewski and K. Zacharias Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr. , 195, 1998, 77-114 65. R. Gariepy, M. Shillor and X. Xu Existence of generalized weak solutions to a model for in situ vitrification, Euro. Jnl. Appl. Math. , 9, 1998, 543-559 66. A. Glitzky, K. Gr¨ oger and R. Hunlich Discrete-time methods for equations modelling transport of foreign atoms in semiconductors, Nonl. Anal. TMA , 28, 1997, 463-487 67. A. Glitzky and R. Hunlich Energetic estimates and asymptotics for electroreaction-diffusion systems, Z. Angew. Math. Mech. , 77, 1997, 823-832 68. J. Griepentrog An application of the implicit function theorem to an energy model of the semiconductor theory, Z. Angew. Math. Mech. , 79, 1999, 43-51


93

69. W. Hackbush On first and second order box schemes, Computing , 41, 1989, 277-296 70. J. Hernandez Continuation and comparison methods for some nonlinear elliptic systems in Contributions to Nonlinear Partial Differential Equations, Vol. II, Longman, Essez, 1986, 137-147 71. K. Hess, J. Leburton and U. Ravaioli (eds.) Computational Electronics, Kluwer Academic Publishers, 1991 72. S. Howison A note on the thermistor problem in two space dimensions, Quart. Appl. Math. , XLVII, 1989, 509-512 73. S.D. Howison, J.F. Rodrigues and M. Shillor Stationary solutions to the thermistor problem, J. Math. Anal. Appl. , 174, 1993, 573-588 74. J.W. Jerome Analysis of Charge Transport, Springer-Verlag, New York, 1996 75. J.W. Jerome Consistency of semiconductor modeling: an existence/stability analysis for the stationary Van Roosbroeck system, SIAM J. Appl. Math. , 45, 1985, 565-590 76. J. Jerome Evolution systems in semiconductor modeling: a cyclic uncoupled line analysis for Gummel map , Math. Meth. Appl. Sci. , 9 , 1987 , 455–492 77. J. Jerome The approximation problem for drift-diffusion systems, SIAM Review , 37, 1995, 552-572 78. J. Jerome and T. Kerkhoven A finite element approximation theory for the drift-diffusion semiconductor model, SIAM J. Numer. Anal. , 28, 1991, 403-422 79. F. Jochmann Uniqueness and regularity for the two dimensional drift-diffusion model for semiconductors coupled with Maxwell’s equations, J. Differential Equations , 147, 1998, 242-270 80. A. J¨ ungel A nonlinear drift-diffusion system with electric convection arising in electrophonetic and semiconductor modeling, Math. Nach. , 185, 1997, 85-110 81. A. J¨ ungel A steady-state quantum Euler-Poisson system for potential flows, Commun. Math. Phys. , 194, 1998, 463-469 82. A. J¨ ungel Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion, Z. Angew. Math. Mech. , 95, 1995, 783-799 83. A. J¨ ungel The free boundary problem of a semiconductor in thermal equilibrium, Math. Meth. Appl. Sci. , 18, 1995, 387-412 84. A. J¨ ungel and P. Pietra A discretization scheme for a quasi-hydrodynamic semiconductor model, Math. Mod. Meth. Ap. Sci. , 7, 1997, 935-955 85. A. J¨ ungel and C. Schmeiser Voltage-current characteristics of a pndiode from a drift diffusion model with nonlinear diffusion, Quart. Appl. Math. , 55, 1997, 707-721 86. T. Kerkhoven Piecewise linear Petrov-Galerkin error estimates for the box method, SIAM J. Numer. Anal. , 33, 1996, 1864-1884 87. A. Lacey Thermal runaway in a nonlocal problem modelling Ohmic heating: Part I: Model derivation and some special cases, Euro. Jnl. Appl. Math. , 6, 1995, 127-144 88. O. Ladyzenskaja, V. Solonnikov and N. Uralceva Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, Volume 23 , American Mathematical Society, Providence , 1968 89. G. Lieberman Intermediate Schauder theory for second order parabolic equations IV: time irregularity and regularity, Diff. Int. Eq. , 5, 1992, 1219-1236 90. J. Lions Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires, Dunod, Paris, 1969

94

W. Allegretto

91. J. Lopez-Gomez On the structure and stability of the set of solutions of a nonlocal problem modeling ohmic heating, Journal of Dynamics and Differential Equations , 10, 1998, 537-566 92. P. Markowich The Stationary Semiconductor Equations, Springer-Verlag, New York , 1986 93. P. Markowich, C. Ringhofer and C. Schmeiser Semiconductor Equations , Springer-Verlag, New York , 1990 94. P. Markowich and A. Unterreiter Vacuum solutions of the stationary driftdiffusion model, Ann. Scuola Norm. Sup. Pisa , 20, 1993, 371-386 95. C. Mastrangelo Thermal applications of microbridges in Ph.D. Dissertation, Univ. California, Berkeley, 1991 96. J. McMacken and S. Chamberlain A numerical model for two dimensional transient simulation of amorphous silicon thin-film transistors, IEEE Transactions on Computer Aided Design , 11, 1992, 629-637 97. M.S. Mock Analysis of Mathematical Models of Semiconductor Devices, Boole Press, Dublin, 1983 98. M. Mock Asymptotic behaviour of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl. , 49, 1975, 215-225 99. H. Molenaar Multigrid Methods for Semiconductor Device Simulation, Centrum voor Wiskunde en Informatica, Amsterdam, 1992 100. R. Muller, R. Howe, S. Senturia, R. Smith and R. White Microsensors, IEEE Press, New York, 1991 101. M. Murthy and G. Stampacchia A variational inequality with mixed boundary conditions, Israel J. Math. , 13, 1972, 188-224 102. J. Naumann Semiconductor equations modelling non-stationary avalanche generation in 3D, Diff. Int. Eq. , 7, 1994, 1511-1526 103. J. Naumann and M. Wolff A uniqueness theorem for weak solutions of the stationary semiconductor equations, Appl. Math. Optim. , 24 , 1991, 223-232 104. Lj Ristic (ed.) Sensor Technology and Devices, Artech. House, Boston, 1994 105. I. Rubinstein Electro-Diffusion of Ions, SIAM Studies in Applied Mathematics, New York, 1990 106. R. Sacco, E. Gatti and L. Gotusso A nonconforming exponentially fitted finite element method for two dimensional drift-diffusion models in semiconductors, Numer. Methods Partial Differential Equations , 15, 1999, 133-150 107. R. Sacco and F. Saleri Mixed finite volume methods for semiconductor device simulations, Numer. Methods Partial Differential Equations , 13, 1997, 215-236 108. T. Seidman and G. Troianiello Time-dependent solutions of a nonlinear system arising in semiconductor theory, Nonl. Anal. TMA , 9, 1985, 1137-1157 109. S. Selberherr Analysis and Simulation of Semiconductor Devices, SpringerVerlag, New York , 1984 110. E. Shamir Regularization of mixed second order boundary value problems, Israel J. Math. , 8, 1968 , 151-168 111. P. Shi, M. Shillor and X. Xu Existence of a solution to the Stefan problem with Joule’s heating , J. Differential Equations , 105, 1993, 239-263 112. C. Snowden (ed.) Semiconductor Device Modelling, Springer-Verlag, New York, 1989 113. G. Stampacchia Equations Elliptiques du Second Ordre a Coefficients Discontinus, Les Presses de l’Universite de Montreal, Montreal, 1965 114. K. Thornber Current equations for velocity overshoot, IEEE Electron Device Letters , 3, 1982, 69-71


95

115. G.M. Troianiello Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987 116. A. Unterreiter Vacuum and non-vacuum solutions of the quasi-hydrodynamic semiconductor model in thermal equilibrium, Math. Meth. Appl. Sci. , 18, 1995, 225-254 117. Y. Wang and G. Ping Some problems on semiconductor equations, AMS/IP Studies in Advanced Mathematics , 3, 1997, 583-587 118. M. Ward, L. Reyna and F. Odeh Multiple steady-state solutions in a multijunction semiconductor device, SIAM J. Appl. Math. , 51, 1991, 90-123 119. D. Wrzosek Existence and uniqueness of solutions for a semiconductor device model with current dependent generation-recombination term, Math. Meth. App. Sci. , 19, 1996, 1199-1216 120. H. Xie L2,µ -estimate to the mixed boundary value problem for second order elliptic equations and applications in thermistor problems , Nonl. Anal. TMA , 24 , 1995 , 9-28 121. W. Xie On the existence and uniqueness for the thermistor problem, Adv. Math. Sci. Appl. , 2, 1993, 63-73 122. H. Xie and W. Allegretto C α (Ω) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SIAM J. Math. Anal. , 22, 1991, 1491-1499 123. X. Xu A compactness theorem and its application to a system of partial differential equations, Diff. Int. Eq. , 9, 1996, 119-136 124. X. Xu A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type, Comm. Part. Diff. Eq. , 18, 1993, 199-213 125. X. Xu Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the Joule-Thomson effect, Internat. J. Math. and Math. Sci. , 16, 1993, 125-138 126. X. Xu Local and global existence of continuous temperature in the electrical heating of conductors, Houston J. Math. , 22, 1996, 435-455 127. X. Xu Local partial regularity theorems for suitable weak solutions of a class of degenerate systems, Appl. Math. and Optim. , 34, 1996, 299-324 128. X. Xu On the existence of bounded temperature in the thermistor problem with degeneracy, Nonlinear Analysis TMA, to appear 129. X. Xu Partial regularity of solutions to a class of degenerate systems, Trans. Amer. Math. Soc. , 349, 1997, 1973-1992 130. X. Xu and M. Shillor The Stefan problem with convection and Joule’s heating, Advance in Diff. Eq. , 2, 1997, 667-691 131. H.-M. Yin The semiconductor system with temperature effect, J. Math. Anal. Appl. , 196, 1995, 135-152 132. G. Yuan and Z. Lui Existence and uniqueness of the C α solution for the thermistor problem with mixed boundary values, SIAM J. Math. Anal. , 25, 1994, 1157-1166 133. X.Y. Yue Numerical analysis of nonstationary thermistor problem, J. Compu. Math. , 12, 1994, 213-223 134. J. Zhu, H. Wu and Y. Wang A mixed method for the mixed initial boundary value problems of equations of semiconductor devices, SIAM J. Numer. Anal. , 31, 1994, 731-744

Kinetic and Gas - Dynamic Models for Semiconductor Transport Christian Ringhofer

Department of Mathematics, Arizona State University Tempe, Arizona 85287-1804, USA, [email protected]

Summary. Transport of electrons in semiconductors is described by a wide range of transport models ranging from the time dependent Schr¨ odinger equation to diffusion equations for an ’electron - gas’. This paper gives an overview over existing transport models and their connection and derivation.

Introduction The transport picture of electrons in semiconductors is described by a variety of model equations. The use of this wide variety is necessary since, other than ten years ago, today’s devices operate under conditions where they are influenced by effects which range from high - energy - tail effects of kinetic distribution functions, such as ’hot electrons’ to quantum mechanical phenomena, such as resonant tunneling. This paper tries to give an overview over the most common models and a unified explanation how they are connected and derived from each other. We start the discussion in Section 1 with the most basic model of classical transport of an ensemble of charged particles (’F = ma’) and derive from there the effective single body Liouville and Wigner equations. In Section 2 we introduce the concept of scattering and derive the semiclassical Boltzmann and Wigner Boltzmann equation, which are the basic kinetic equations used to describe electron transport. While kinetic equations generally provide a good description of semiconductor transport, the complexity involved in their numerical solution [23][24] requires the use

Supported by NSF Grants DMS 970-6792 and INT 960-3253

A.M. Anile, W. Allegretto, C. Ringhofer: LNM 1821, A.M. Anile (Ed.), pp. 97–131, 2003. c Springer-Verlag Berlin Heidelberg 2003

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of macroscopic approximations whenever possible. (We refer to kinetic equations as transport equations whose solutions depend on the wave vector or momentum of the electron as well as on its position and time, and to macroscopic equations as those where the solution depends only on position and time.) Sections 3 and 4 are devoted to a more or less unified presentation of of the derivation of such macroscopic equations. They can be grouped into two general types. The first type consists of diffusion approximations which result from asymptotic analysis (the Hilbert expansion) for situations where collisions of electrons with the crystal lattice dominate the transport picture. The general Hilbert expansion in the linear and nonlinear case, its properties and results are therefore the topic of Section 3. The second type of macroscopic approximations results from moment closures which are, more or less, based on ad hoc assumptions on the kinetic density functions. These moment closure equations are of hyperbolic or dispersive type. Section 4 is devoted to moment closures and the derivation of the so called hydrodynamic and various quantum hydrodynamic models, as well as to the connection of hydrodynamic models to diffusion approximations.

1 Multi - Body Equations and Effective Single Electron Models In this section we will give an overview over the most basic laws describing the motion of an ensemble of particles under the influence of internal and external forces from the viewpoint of classical trajectories and quantum mechanical wave functions. This leads to the multi- particle Liouville and Schr¨ odinger equations. Since these equations are too high - dimensional to be amenable to any computational approach, they are reduced to effective single body equations via the BBGKY hierarchy. The underlying assumption of this reduction is the assumption of a large sample of identical and statistically essentially independent particles. The resulting models are the classical Vlasov - Poisson equation and the quantum mechanical Schr¨ odinger Poisson equation. These will then describe the motion of an ensemble of electrons in a vacuum. In order to describe the transport picture in a crystal, where electrons interact with each other and the vibrations of the crystal lattice, two more model extensions will be needed. These are the introduction of energy bands, leading to the semi - classical Liouville equation, and the modeling of collision events, leading to the semi - classical Boltzmann equation, which will be the topic of the next section. Since transport processes in semiconductors often are studied in regimes on the border between the classical and quantum mechanical view of the world, it is important to relate these two descriptions. Probably the best tool for this is the introduction of Wigner functions. Therefore we will at the end of this section briefly, and on a purely formal level, describe the classical limit → 0 which reduces the quantum mechanical picture to the classical one. We start with the study of the evolution of M particles under the influence of an external force. This evolution is basically governed by Newton’s second law F = ma, where F is the force between the particles, m is the mass of the particle, ∂ and a = dt v is the acceleration (v is therfore the velocity). Denoting with xj (t), vj (t) position and velocity of particle j, j = 1, .., M we obtain

Kinetic and Gas - Dynamic Models for Semiconductor Transport

d xj = vj , dt

(1.1)

99

d 1 vj = a = Fj (x), dt m

Since our particles are electrons, it is beneficial to change notation and to define pj = mvj as the momentum of the electron and to set Fj = qEj , where Ej is the electric field acting on electron j and q denotes the unit charge (the charge of an electron). Assuming that the field E = (E1 , .., EM ) is a gradient field, so E = ∇x V, x = (x1 , .., xm ), p = (p1 , .., pm ) holds, this gives the Hamiltonian system

(1.2)

(a)

d x = ∇p H(x, p), dt

d p = −∇x H(x, p), dt

1 |p|2 + qV (x, t), E(x, t) = −∇x V (x, t). 2m To obtain the classical multi - particle Liouville equation we introduce the probability density f (x, p, t) for the probability that electron j is at position xj with momentum pj . In the coordinate system given by the particle trajectories this density does not change in time and we obtain (b)

H(x, p) =

d f (x(t), p(t), t) = 0 dt

(1.3)

which gives the Liouville equation (1.4)

∂t f +

1 p · ∇x f + qE · ∇p f = 0 m

In the quantum mechanical description of the ensemble electrons cannot be described by position and momentum simultaneously because the two cannot be pinpointed at the same time due to the Heisenberg principle. Instead they are described by a wave function ψ(x, t)which obeys the multi - particle Schr¨ odinger equation

(1.5)

∂t ψ =

i iq ∆x ψ − V ψ 2m

The probability density for the position x in the classical regime is given by the integral (1.6)

f (x, p, t)dp

n(x, t) =

and the flux (the current) is given by

(1.7)

J(x, t) =

q m

pf (x, p, t)dp

and we obtain the continuity equation, describing the conservation of charge qn by integrating (1.4) with respect to p

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q∂t n + ∇x · J = 0

(1.8)

In the quantum mechanical regime the electron density n and the current density J are given by

(1.9)

(a)

n(x, t) = |ψ(x, t)|2 ,

(b)

q Im[ψ ∗ ∇x ψ(x, t)] m

J(x, t) =

also yielding the continuity equation (1.8). To illustrate the difference between the classical trajectories and the quantum mechanical picture, it is instructive to consider a simple example which can be found in any introductory exposition of quantum mechanics, namely th reflection / transmission of an electron at a potential barrier: We consider only one particle (M = 1) and one spatial dimension. The potential barrier is given by a ramp of the form 0 (1.10)

V (x) =

xε

bx ε

where b denotes the height of the barrier and ε parameterizes the steepness of the ramp. In the classical regime a particle trajectory is given by the solution of

(1.11)

p d x= , dt m

(a) (b)

d d p = −q V (x) dt dx

x(0) = −x0 < 0,

p(0) = p0 > 0

where we start the particle at −x0 with a positive momentum, sending it towards the barrier. Before reaching the barrier at x = 0 the solution of (1.11) is given by p = p0 ,

(1.12)

x=

p0 t − x0 , m

x < 0,

t < t1 =

mx0 p0

d At t = t1 the electron reaches the barrier, and then the force is given by − dx V (x) = b − ε which gives

(1.13)

p=−

qb (t − t1 ) + p0 , ε

x=−

qb p0 (t − t1 )2 + (t − t1 ), 2εm m

0<x 2qbm (1.14) has an intersection with x = ε and the electron can overcome √ the barrier and continue its journey with a reduced momentum. For p0 < 2qbm the parabola (1.14) has no intersection with x = ε and and only intersects x = 0 again. So, in this case the electron is reflected at the potential ramp. Notice that the


101

result of this consideration does not depend on the slope e of the ramp and therefore the outcome remains the same in the limit ε → 0 giving a discontinuous potential step. In contrast, in the the quantum mechanical picture, the electron traveling to the right is given by a plane wave solution of the Schr¨ odinger equation of the form (1.15)

ψ(x, t) =

A+ eiωt+iξL x + A− eiωt−iξL x x < 0 B+ eiωt−iξR x x>0

where ωξ gives the speed of the wave. A+ is the amplitude of the wave traveling to the right (the incoming wave), and A− is the amplitude of the reflected wave travelling to the left. Since there after the barrier, the electron travels to the right without encountering any obstacle, there is only a right - traveling wave with an amplitude B+ to the right of x = 0. Since the potential V is discontinuous we seek a solution ψ of the Schr¨ odinger equation which has a continuous first derivative and a jump discontinuity in the second derivative. This gives the matching conditions (1.16)

A+ + A− = B+ ,

ξL (A+ − A− ) = ξR B+

which implies for the wave speeds and frequencies (1.17)

ω=−

2 1 2 ξL = − ξR − b, 2m 2m

ξR =

2 ξL −

2mb 2

Computing the reflection coefficient, the quotient of the amplitudes of incoming and reflected wave, we obtain (1.18)

A− = A+

ξL − ξR , ξL + ξR

which shows that there is always a certain amount of reflection and transmission in the quantum mechanical case. In the classical case (for → 0 ) the reflection coefficient becomes either 0 (transmission) or 1 (reflection). The quantum mechanical phenomenon that the electron can with a certain probability surmount the barrier, regardless of the height, is referred to as tunneling and plays an important role in the simulation of semiconductor devices.

1.1 Effective Single Particle Models -the BBGKY Hierarchy The density function f in the Liouville equation (1.4) is a function of 6M +1 variables and therefore not amenable to computations. In order to arrive at a more feasible model one has to make certain assumptions. These assumptions lead to the so called BBGKY hierarchy ( named after Bogoliubov, Born, Green, Kirkwood and Yvon) [16] which replaces the Liouville equation (1.4) by an effective one particle equation via a mean field approximation. We write (1.4) as

(1.19)

∂t f +

M M 1 ∇xj · (pj f ) + q ∇pj · (Ej f ) = 0 m j=1 j=1

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and assume that the field E = (E1 , .., EM ) is given by a sum of two particle interactions, so Ej (x1 , ..xM ) =

(1.20)

1 E(xj , xl ) M l =j

holds , where the E(xj , El ) denotes a two particle interaction (Coulomb )force. Note that the scaling factor M in (1.20) implies that the total force on one electron remains finite for M → ∞. The first main assumption is that the particles are indistinguishable to the observer. This means that any permutation of the arguments in the density function f does not change its value. So (1.21)

f (x, p, t) = f (π(x), π(p), t)

holds for any permutation π. We define the effective one - electron density f 1 by integrating over all but one electron coordinate. f 1 (x1 , p1 , t) =

(1.22)

dx2 ..dxM dp2 , .., dpM f (x, p, t)

Integrating the Liouville equation (1.4) gives (a)

(1.23)

∂t f 1 +

1 ∇x · (p1 f 1 )+ m 1

dx2 ..dxM dp2 , .., dpM [q∇p1 ·(E1 f 1 )+

M 1 [∇xj ·(pj f )+q∇pj ·(Ej f )] = 0 m j=2

Now using the fact (1.21) that the particles are indistinguishable we obtain for the influence of the other particles on electron number 1: E1 (x) =

(1.24)

M 1 E(x1 , xj ), M j=2

Using the fact that all the particles are indistinguishable again, we can write (1.24) in terms of a two particle density f 12 : (1.25) (a) dx2 ..dxM dp2 , .., dpM ssM j=2 E1 (x1 , xj )f (x, p, t) = (M − 1) (b)

f 12 =

dx2 dp2 E(x1 , x2 )f 12 (x1 , x2 , p1 , p2 , t)

dx3 ..dxM dp3 , .., dpM f (x, p, t)

So altogether: (1.26)

∂t f 1 +

1 M −1 ∇x · (p1 f 1 ) + q∇p1 · m 1 M

dx2 dp2 (E( x1 , x2 )f 12 ) = 0


103

holds. Now we assume that the particles are independent. This is reasonable, since in a large ensemble the dependence of two individual particles on each other will be small. This means for the probability density f 12 that (1.27)

f 12 (x1 , x2 , p1 , p2 , t) = f 1 (x1 , p1 , t)f 1 (x2 , p2 , t)

holds. Now letting M → ∞ gives the effective one particle Liouville equation (1.28)

∂t f 1 +

1 ∇x · (p1 f 1 ) + q∇p1 · (E ef f (x1 )f 1 ) = 0, m 1

where the effective field E ef f is given by (1.29) E ef f (x1 ) = dx2 dp2 (E(x1 , x2 )f 1 (x2 , p2 , t)) We now assume that the influence of two of the electrons on each other is given by the Coulomb force. So (1.30)

q (x1 − x2 ) 4πε|x1 − x2 |3

E(x1 , x2 ) =

holds with εd the dielectric constant of the medium. The function in (1.30) is the Green’s function of the Laplace operator in three dimensions. Therefore the effective field E ef f in (1.28) can be computed from solving a Poisson equation (1.31) E ef f = −∇x V, ε∆x1 V (x1 ) = q dp1 f 1 (x1 , p1 ). Equations (1.28) and (1.31) now constitute the classical Vlasov - Poisson system (1.32)

(a)

∂t f +

1 ∇x · (pf ) + q∇p · (E ef f (x)f ) = 0 m (b)

ε∆x V ef f = qn,

(c)

E ef f = −∇x V ef f ,

n(x, t) =

f (x, p, t)dp

in which the force of electrons on each other is calculated through the mean field E ef f . The same sort of BBGKY Hierarchy can be carried out in the quantum mechanical case. There the Schr¨ odinger equation for the multi - particle wave function ψ(x, t)

(1.33)

∂t ψ =

i i ∆x ψ − V ψ 2m

is replaced by an effective one - particle Schr¨ odinger equation for the wave function

(1.34)

ψ 1 (x1 , t) =

dx2 , .., dxM

ψ(x, t)

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Christian Ringhofer

and the Poisson equation. However, in this case, we cannot assume that the situation is described by a single wave function ψ 1 (x1 , t), but we have to introduce a mixed state as a superposition of effective one electron single state wave functions. The resulting system is then of the form (1.35)

i i ∆x ψj − V ef f (x1 , t)ψj 2m 1

(a)

∂t ψj (x1 , t) =

(b)

ε∆x V ef f = qn,

n(x, t) =

∞

αj |ψ(x, t)|2

j=0

where the αj denote the probability of the state j being occupied. We refer the reader to [17] for details. One necessary extension of the model above is the inclusion of a background density of ions, which is always present in a semiconductor due to the doping concentration. Similarly to the BBGKY hierarchy above this background density is described by an additional Coulomb force between electrons and background ions. This results in an additional field term E ext and an additional density N in the Vlasov - Poisson system giving (1.36)

E tot (x, t) = E ef f (x, t) + E ext (x, t),

(a)

(b)

E ext = −∆x V ext

εd ∆x V ext = qN (x)

Adding the two Poisson equations the gives (1.37)

(a) (b)

∂t f +

1 ∇x · (pf ) + q∇p · (E tot (x)f ) = 0 m

E tot = −∇x V tot ,

ε∆x V tot (x, t) = q(n(x, t) + N (x)), n(x, t) = f (x, p, t)dp

In the quantum mechanical case this becomes

(1.38)

(a) (b)

∂t ψ 1 (x1 , t) =

i i ∆x ψ 1 − V tot (x1 , t)ψ 1 2m 1

ε∆x V tot (x, t) = q(n(x, t) + N (x)),

n(x, t) =

αj |ψ(x, t)|2

j

1.2 The Relation Between Classical andquantum mechanical Models One of the basic principles of quantum mechanics is that the quantum picture reduces to the classical description in the limit → 0. Of course is a constant, and this limit has to be understood in the sense that, in some suitable scaling, a


105

scaled and dimensionless version of the Planck constant becomes small. However, this limit is highly oscillatory, and therefore not easily carried out directly on the level of the Schr¨ odinger equation. It is therefore useful to introduce two concepts, namely density matrices [26] and Wigner functions [28] . Given an ensemble of wave functions ψj , we define the density matrix ρ by (1.39)

ρ(r, s, t) =

αj ψj (r, t)ψj∗ (s, t)

j

where ’*’ denotes the complex conjugate, and the aj are again the occupation probability for the state j. Direct calculation yields that, if the wave function ψj satisfy the Schr¨ oedinger equation (1.35) the density matrix satisfies the so called Heisenberg equation

(1.40)

∂t ρ =

i iq [∆r − ∆s ]ρ − [V (r) − V (s)]ρ. 2m

(For notational simplicity we simply set V tot = V from now on.) The Wigner function is now introduced as the Fourier transform of the rotated density matrix. (1.41)

w(x, k, t) =

ρ(x −

1 1 η, x + η, t)eiη·k dη. 2 2

The variable k is called the wave vector of the electron. Carrying out the same transformation in the Heisenberg equation (1.40) yields the Wigner or quantum Liouville equation

(1.42)

∂t w(x, k, t) +

(a)

k · ∇x w − θw = 0, m

with the operator θ given by 1 1 iq [V (x + η) − V (x − η)]w(x, k , t)eiη·(k−k ) dk dη. (b) θw = 2 2 The operator θ can be written in pseudo differential operator notation as 1 1 qi [V (x + ∇k ) − V (x + ∇k )]. 2i 2i In order to carry out the classical limit → 0 we first choose typical scales k0 and x0 for the wave vector k and the spatial variable x. ( They are given by the shape of the potential V and the initial conditions for the Wigner function w .) The average 0 0 velocity is then given by k and we choose the time scale t0 = mx such that on m k0 this time scale the the average velocity of a particle equals one. The free energy of an electron in a vacuum (computed from the eigenvalues of the Schr¨ odinger equation (c)

θ=

without a potential is given by qV accordingly and set V (x, t) (1.43)

(a)

2 2 k0 , and we choose the scale of the potential energy 2m 2 2 k0 = qm V0 ( xx0 , tt0 ). The so scaled equation reads

∂t w(x, k, t) + k · ∇x w − θw = 0,

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Christian Ringhofer

h h i [V0 (x + ∇k ) − V0 (x + ∇k )] 2i h 2i where the scaled Planck constant h is given by h = x01k0 . And the operator θ is a (b)

θ=

pseudo differential operator (1.44)

θw(x, k, t) =

[27]

, which in its integral representation is given by

h h i [V0 (x + η) − V0 (x + η)]w(x, k , t)eiη·(k−k ) dk dη h 2 2

If the scaled Planck constant h now tends to zero, the Wigner equation reduces, at least formally, to (1.45)

∂t w(x, k, t) + k · ∇x w − ∇x V0 ∇k w = 0,

which after reversal of the scaling and setting p = k is the same as the Liouville equation (1.32). So, the classical limit means that we consider the Wigner equation on a length and time and wave vector scale where where x0 k0 → ∞. Of course, to make this formal limit precise, and to show that the solution of the Wigner equation (1.42) converges to the solution of the Liouville equation (1.32) in all possible instances is by no means trivial. We refer the reader to [19] and [18] for details. Another extension to the effective one body Liouville equation (1.32) is the modeling of the crystal lattice. Equation (1.32) only describes the evolution of an ensemble of electrons in a vacuum. Since, in the modeling of semiconductor transport the electron ensemble evolves in a crystalline material (usually silicon), the forces exerted by the crystal lattice have to be included in the transport picture. This is involved a technically quite complicated homogenization procedure leading to the semiclassical Liouville equation. We present here only the result and refer the reader to [2],[21] and [22] for details. One starts with the case of a single electron where E ef f = 0 holds, by considering the single body Schr¨ odinger equation

(1.46)

∂t ψ =

i i ∆x ψ − VL ψ, 2m

where VL denotes a rapidly varying periodic potential modeling the forces exerted by the periodic lattice L. So (1.47)

V (x) = V (x + γLz),

z ∈ Z,

holds, where L is a 3 × 3 matrix with determinant 1 modeling the lattice structure. γ is the size of the lattice. Decomposing the solution of the Schr¨ odinger equation into so called Bloch waves [21] yields a Wigner equation of the form

(1.48)

∂t w(x, k, t) =

i γ γ [ε(k − ∇x ) − ε(k + ∇x )]w, γ 2i 2i


107

where ε(k) is an eigenvalue of the Schr¨ odinger equation (1.46) for the interaction with the lattice, parametrized by the wave vector k [2] . ε is called the energy band. Letting γ → 0 gives the Wigner equation in a crystal of the form

∂t w(x, k, t) +

(1.48)

1 ∇k ε(k) · ∇x w = 0.

In a vacuum, without the presence of the lattice L, the eigenvalues ε(k) are given in terms of the wave vector by

(1.49)

ε(k) =

2 |k|2 2m

and (1.48) reduces to (1.42) in the absence of the external potential V . The external potential and the lattice potential can be combined [22] to give the Wigner equation

∂t w(x, k, t) +

(1.50)

1 ∇k ε(k) · ∇x w − θw = 0.

Again, as above, the classical limit → 0 can be carried out, at least formally, in the sense x0 k0 → ∞, where x0 and k0 denote the scales for space and wave vector. This yields the semi - classical Liouville equation (1.50)

(a)

∂t w(x, k, t) + v(k) · ∇x w −

q ∇x V · ∇k w = 0, hp

1 ∇k ε(k). Here we have again set the momentum p to p = k. The macroscopic effect of the lattice is now that the velocity of an electron now points not in the same direction as its wave vector but is given by (1.50)(b). This results in the particle trajectories being bent by the presence of the lattice. (b)

v(k) :=

2 Collisions and the Boltzmann equation To this point we have only considered the evolution of electrons under the influence of a force which is generated either externally or due to the Coulomb interaction of electrons. In a semiconductor crystal there is another, even more important mechanism to consider, namely the interaction of electrons with vibration of the crystal lattice. Due to the collision with the atoms in the lattice, the electron will abruptly change its momentum- or wave vector. Since this is a frequent event, it is modeled statistically. Let S(x, k, k ) be the probability that the electron changes its wave vector from k to k. Then the collision events are modeled by adding an integral term of the form (2.1)

Q(f ) = [QG − QL ](f )

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Christian Ringhofer

to the right hand side of the Liouville equation. The gain term QG consists of the summation of all electrons scattered into the state k under the condition that the state k is not yet occupied. (This is the Pauli principle that no two electrons can occupy the same state k at the same position.) Therefore QG is of the form (2.2) QG (f )(x, k, t) = S(x, k, k )f (x, k , t)[1 − f (x, k, t)]dk . Similarly, the loss term QL is the summation over all k of electrons scattered from k into k (2.3)

QL (f )(x, k, t) =

S(x, k , k)f (x, k, t)[1 − f (x, k , t)]dk .

Adding the collision term, we obtain the semiclassical Boltzmann equation

(2.4)

∂t f (x, k, t) +

1 q ∇k ε0 · ∇x f − ∇x V · ∇k f = QG − QL .

So far we have only considered th interaction of electrons with the crystal lattice vibrations , the so called phonons, it is actually possible to model the collision of electrons with each other. To this end one has to carry the BBGKY hierarchy of the previous section not to the very end but model collisions on the level of a two - electron density function (k, k1 ). (We suppress the dependence of on x and t for simplicity.) The gain term for the two electron density is then of the form Qee G (f )(k, k1 ) =

(2.5)

S ee (k, k1 , k , k1 )f [1 − f ]dk dk1

where we use the notation f := f (k , k1 ). Again, we make the assumption of statistical independence and write as f (k, k1 ) = f (k)f (k1 ). Integrating the collision operator with respect to k1 integrate w.r.t. k1 gives (2.6)

(a)

QG (f )(k) =

S ee (k, k1 , k , k1 )f f1 [1 − f f1 ]dk1 dk dk1

with f1 = f (k1 ), f1 = f (k1 ) etc. Similarly we obtain for the loss term ee S ee (k , k1 , k, k1 )f f1 [1 − f f1 ]dk1 dk dk1 , (b) QL (f )(k) = and the collisions of electrons with each other is now modeled by the threefold integral in (2.6). Other than in the theory of rarefied gases, electron - electron collisions is usually a very rare event and the collision term Qee is hardly ever used. One property the kernel S of the collision operator Q has to have is the existence of an equilibrium. This equilibrium is usually given by the Fermi - Dirac distribution

(2.7)

fe (x, k, t) = FD (

ε0 (k) − φ ), kB T

FD (z) =

1 1 + ez


109

where φ(x, t) : is the local Fermi - energy, kB is the Boltzmann constant and T is the temperature of the lattice. That the Fermi - Dirac distribution is an equilibrium (meaning it is in the nullspace of the operator Q) is guaranteed by the principle of detailed balance S(x, k, k )fe (1 − fe ) = S(x, k , k)fe (1 − fe )

(2.8)

Using the form (2.7) of the Fermi - Dirac distribution function this is equivalent to S(x, k, k )M (k ) = S(x, k , k)M (k)

(2.9)

with M being the Maxwell distribution given by ε0 (k) 1 exp[− ] M0 kB T where M0 is a scaling factor chosen such that M (k)dk = 1 holds. The symmetry property (2.9) will be very important for the further analysis of the Boltzmann equation.

(2.10)

M (k) =

It is useful to consider some simplifications of the Boltzmann equation (2.4). The first consists of neglecting the Pauli principle, which makes (2.4) linear. For low densities the case that the state k is already occupied will be quite improbable. Therefore for f c1 , c−1 in (2.17) holds. This procedure is quite involved and we will replace it for the purpose of brevity by assuming that the collision operator is a nonlinear relaxation time operator conserving charge and energy. The resulting macroscopic equations will be the same. So we assume the collision operator in (3.9) to be of the form

(3.24)

Q(f ) =

1 [W (uf , k) − f ] τ

where W is some given equilibrium density dependent on a two dimensional parameter uf , and uf is chosen dependent on f such that Q preserves charge and energy. So, uf satisfies the equations

(3.25)

1 W (uf , k) 1 f dk = dk τ ε ε τ

The (nonlinear) Fredholm alternative (3.12) is then of the form that the equation dQ(f0 )f1 = g has a solution if and only if

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Christian Ringhofer

(3.26)

(a)

1 g ε

dk = 0

holds. In this case the solution f1 is given by f1 = −τ g + ∇u W (u, k)T u1

(3.27)

for some arbitrary parameter vector u1 , where we write u in short for uf0 , and the pseudo - inverse Q+ is just given by multiplication with τ . The diffusion equation (3.17) obtained from the Hilbert expansion is then of the form

(3.28)

1 { ∂t W (u, k)] − (∇x · v + ∇k · E)Q+ (∇x · v + ∇k · E)W (u, k)}dk = 0. ε

There are two forms of equilibrium density W which are used. The first is the one 2 based on Boltzmann statistics and is in the case of parabolic bands ε = |k|2 of the form

(3.29)

W (u, k) =

|k|2 1 nT −3/2 exp(− ), M0 2T

u = (n, T ),

where n is the density and T is the temperature. The second form of W is based on Fermi - Dirac statistics (which includes the Pauli principle in the Boltzmann equation), and is of the form

(3.30)

W (u, k) =

1 , 1 + ez

z = exp(

2φ − |k|2 ), 2T

u = (φ, T ),

where φ is the Fermi level. In the case of Boltzmann statistics (3.29) the diffusion equations (3.28) are of the form (3.31) ∂t

n 3 nT 2

−

∇Tx τ [∇x

nT 5 nT 2 2

−E

n 0 0 ]+ − =0 5 nT τ E T ∇x (nT ) |E|2 τ n 2

3.4 The Energy Transport - or SHE Model The energy transport model (or SHE- model for spherical harmonics expansion) lies somewhere in between the above energy model and the full Boltzmann equation in the sense that the resulting equation retains the energy as an independent variable. The derivation is most easily presented by assuming a parabolic band structure and writing the Boltzmann equation in polar coordinates in the wave vector dimension. So we set

Kinetic and Gas - Dynamic Models for Semiconductor Transport (3.32)

k = (ry, r 1 − y 2 sin φ, r 1 − y 2 cos φ),

117

y ∈ [−1, 1], φ ∈ [0, 2π]

where we write y instead of the usual cos(θ). Integrals with respect to dk are appropriately transformed by (3.33)

∞ 0

1

dr

f dk =

dy −1

2π

dφ(r2 f ).

0

The Boltzmann equation in polar coordinates is of the form

(3.34)

(a)

 T  a E λ 2 T λ ∂t f + λ∇x · (vf ) + 2 ∇ryφ [r b E  f ] = Q(f ) r cT E 2

(b)

(a, b, c) = (a, b, c)(r, y, φ),

with the three - vectors a, b and c arising from the transformation of ∇k . We now assume that the dominant collision mechanism is given by the energy conserving part of the collision operator. So Q is of the form

(3.35)

1 τ (x, ε)

Q(f ) =

δ(ε − ε )[f − f ]dk,

which in polar coordinates reads 4πr [F (x, r, t) − f (x, r, y, φ, t)], τ 1 f (x, r, y, φ, t)dydφ. F (x, r, t) = 4π In order to carry out the Hilbert expansion we re- group the variables in (3.36)

Q(f ) =

(3.37)

(a)

X = (x, r),

Y = (y, φ),

and change to F1 f1 , F = 2. 2 r r The variable X now plays the role of the spatial variable x, and Y plays the role of k in the Hilbert expansion. This gives the Boltzmann equation of the form (b)

(3.38)

(a)

f=

∂t f 1 + λ∇X · (Af 1 ) + λBf 1 = Q(f 1 ) v , aT E

(b)

A=

B = ∇Y ·

bT E . cT E

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Christian Ringhofer

As in the drift diffusion model, the collision operator Q is linear, and the Fredholm alternative is now of the form that the equation Q(f 1 ) = g has a solution if and only if gdY = 0 holds. In this case the solution is given by f 1 (X, Y, t) = −

(3.39)

τ g(X, Y, t) + u(X, t). 4πr

So, the integral invariant κ = 1, the nullspace parameter W (u) satisfies W = u and τ the pseudo- inverse operator Q+ is given by multiplication with 4πr . The resulting diffusion equation is of the form (3.40)

dy

dφ{∂t u(x, r, t)−[∇x ·v +∂r aT E]

τ [∇x ·v +∂r aT EB]u(x, r, t)} = 0. 4πr

The dependent variable u now depends on x, t and the energy variable r. (ε = holds for parabolic bands. )

r2 2

3.5 Parabolicity The Hilbert expansion produces a system of differential equations which are first order in time and second order in space. The question is now whether this system is well posed, i.e. parabolic. This question is answered by Theorem 1 below which requires a certain relationship between the integral invariants κ and the parameter function W , and is essentially a linearized version of Boltzmann’s famous HTheorem. In leading order the system is given by (3.41)

κ(k){∂t W (u, k) − ∇x · vQ+ ∇x · vW (u, k)}dk + ...

In the case that W depends nonlinearly on the parameters u we have to consider the linearization of the system (3.17) around some solution u, whose leading order term is given by (3.42)

κ(k){∇u W T ∂t δu− ∇x · vQ+ ∇x · v∇u W T δu}dk + ...

This can be written in matrix form as H∂t δu − ∂xµ D(µ, ν)∂xν δu = 0

(3.43)

where the matrices H and D are given by

(3.44)

H=

κ(k)∇u W T dk,

D(µ, ν) =

κ(k)vµ Q+ vν ∇u W T dk

The resulting problem is well posed and parabolic if the generalized eigenvalue problem

Kinetic and Gas - Dynamic Models for Semiconductor Transport (3.45)

3 3

eHz =

119

ξµ ξν D(µ, ν)z

µ=1 ν=1

only has positive eigenvalues for all numbers ξm , m = 1, 2, 3. In order to guarantee this, we need a certain relation between the integral invariants κ and the parametrization W of the nullspace of Q. We have the following Theorem 1 : If the integral invariants κ are expressible in terms of W in the sense that there exists a nonsingular matrix C only dependent on u and a non - negative function ω such that (3.46)

κ(k) = C(u)∇u W (u, k)ω,

holds, and if the pseudo inverse operator Q+ is positive semi- definite with respect to the L2 scalar product induced by ω, then all eigenvalues e in (3.45) are nonnegative and the linearized equation (3.43) is parabolic. Proof: Setting H = CH 1 and D = CD1 we obtain

(3.47)

(a)

(b)

H = C(x)H 1 ,

D(µ, ν) = CD1 (µ, ν),

H1 =

D1 (µ, ν) =

ω∇u W ∇u W T dk

ω∇u W vµ Q+ vν ∇u W T dk

and the eigenvalue problem becomes eH 1 z =

(3.48)

ξµ ξν D1 (µ, ν)z

µν

multiplying by z T and dividing through gives " (3.49)

µν

e=

ξµ ξν z T D1 (µ, ν)z zT H 1z

Using the form of H 1 and D1 gives zT H 1z =

(3.50)

(3.51)

µν

ξµ ξν z T D1 (µ, ν)z =

ω[

ω(z T ∇u W )2 dk > 0

z T ∇u W ξµ vµ ]Q+

µ

[ξν z T ∇u W ]dk > 0, ν

+

where the last inequality holds because the operator Q is positive definite. Thus the eigenvalues are nonnegative, and the problem is parabolic. QED

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Christian Ringhofer

For the drift - diffusion equations (3.23) the terms in the assumption (3.46) of Theorem 1 are of the form (3.52)

κ = 1,

W (u, k) = ue−ε ,

∇u W = e−ε

Therefore setting ω(k) = eε and C = 1 does the job, which implies that the diffusion matrix D in (3.23) is always positive semi - definite. Similarly , for the SHE model (3.51) we have (3.53)

κ = 1,

W (u, k) = u,

∇u W = 1,

and the assumption of Theorem 1 can be satisfied setting ω = 1 and C = 1. More interesting are the energy models arising from tha assumptions of Boltzmann and Fermi - Dirac statistics. In the case of Boltzmann statistics we obtain

(3.54)

(a)

1 κ= , ε

W =

|k|2 1 nT −3/2 exp(− ), M0 2T

(b)

n ∇u W = M0

T −3/2 3 −5/2 −2T + 12 T −7/2 |k|2

exp(−

|k|2 ), 2T

which gives for ω and C

(3.55)

ω = exp(

|k|2 ), 2T

C=

# −3/2 $−1 M0 T 0 n − 32 T −5/2 T −7/2

A similar expression holds for the energy model based on Fermi - Dirac statistics and all of these models are parabolic.

4 Moment Methods and Hydrodynamic Models A different but related approach to obtaining macroscopic equations is given by the method of moments [14][15] . In this approach one simply takes moments of the Boltzmann equation (3.9) obtaining conservation laws for charge, momentum etc. We introduce the notation (4.1)

< g > (x, t) :=

gf (x, k, t)dk

for the expectation of a quantity g under the probability distribution f . Taking the moments corresponding to charg, momentum and energy yields (4.2)

(a)

λ2 ∂t < 1 > +λ∇x · < v >=< Q >= 0

Kinetic and Gas - Dynamic Models for Semiconductor Transport (b)

λ2 ∂t < k > +λ∇x · < vk T > −λE < 1 >=< kQ >

(c)

λ2 ∂t < ε > +λ∇x · < εv > −λ < v > ·E =< εQ >

121

Of course, this process never terminates since each moment equation involves another moment of higher order. Therefore one makes a closure assumption. The simplest assumption is the assumption of a displaced a Maxwellian of the form

f (x, k, t) ≈ a(x, t) exp(−

(4.3)

ε(k − p(x, t)) ) T (x, t)

Under the assumption of parabolic bands, this gives for the unknown moments in (4.2). (4.4)

(a) (b) (c)

< v >=< k >= p < 1 >, < vk T >=< 1 > (T I + ppT ), < ε >=

(3T + |p|2 ), 2

(|p|2 + 5T )p. 2 Changing from the unknown variables < 1 >, < k > and < ε > to < 1 >, p and T and renaming < 1 >= n gives the hydrodynamic equations (d)

(4.5)

(a) (b)

(c)

< εv >=

λ2 ∂t n + λ∇x · (np) =< Q >= 0

λ2 ∂t (np) + λ∇x · [n(T I + ppT )] − λEn =< kQ >

n n λ2 ∂t [ (3T + |p|2 )] + λ∇x · [ (|p|2 + 5T )p] − λnp · E =< εQ > 2 2

which give the conservation of charge, momentum and energy. These are the compressible Euler equations for a Plasma under the influence of an external field. An important extension of the hydrodynamic model should be mentioned here which is the work of Anile et. al. (see [1]), which justifies the addition of a heat conduction term in (4.5)(c). With this heat conduction term (4.5)(c) becomes (4.5) (c )

n n λ2 ∂t [ (3T + |p|2 )] + λ∇x · [ (|p|2 + 5T )p − κ∇x T ] − λnp · E =< εQ > 2 2

where κ denotes the thermal conductivity. Other than in the field of gas dynamics, the hydrodynamic equations do not really have a good justification in the semiconductor case, since the dominant collision event in semiconductors is not particle -

122

Christian Ringhofer

particle scattering, as is the case in gases. There is some work by Poupaud [20] justifying the hydrodynamic model in the case of extremely high external fields, which only justifies them in the case of purely ballistic transport. However, the hydrodynamic model is closely related to the energy model for small velocities (or momenta) p. If we make the same assumptions on the collision operator , which lead to the energy model (l.42) for Boltzmann statistics, the moments of the collision operator in (4.2) become 1 < kQ >= − np, τ

(4.6)

< εQ >= 0.

Re- scaling the momentum p → λp gives (4.7)

(a)

(b)

(c)

λ2 ∂t n + λ2 ∇x · (np) = 0

1 λ3 ∂t (np) + λ∇x · [n(T I + λ2 ppT )] − λEn = − λnp τ

n n λ2 ∂t [ (3T + λ2 |p|2 )] + λ2 ∇x · [ (λ2 |p|2 + 5T )p] − λ2 np · E = 0 2 2

dividing the first and third equation by λ2 and the second by λ, and letting λ → 0 gives (4.8)

(a)

(b)

∂t n + ∇x · (np) =< Q >= 0 1 ∇x · [nT I] − En = − np τ

3n 5nT T ] + ∇x · [ p] − np · E = 0 2 2 which, after eliminating the momentum p by inserting (4.8)(b) into (a) and (c) , give precisely the energy equations (3.31). It should be mentioned here, that there is some confusion between the mathematical and the engineering literature. Engineering codes claiming to solve the ’hydrodynamic model’ usually neglect precisely the terms left off in (4.8), and thus solve the energy equations. This seems to be justified since velocities are usually small in devices for which the ’hydrodynamic’ model is used. Mathematically, of course, this makes a huge difference since the energy model is strictly parabolic, while the compressible Euler equations are hyperbolic[6][7] , allow for shock solutions [12] and are generally much richer in structure, and therefore also much more difficult to solve numerically. (c)

∂t [

Quantum Moment Equations As in the classical case, one can derive macroscopic approximations to the quantum Boltzmann equation (2.19). One is, however here more or less restricted to the


123

moment equation approach, since the Hilbert expansion depends crucially on the structure of the collision operator, which is not known very well in the quantum mechanical regime. Also, one can expect quantum effects to play a role mainly in situation where the length scales are so small that an approximation based on small Knudsen numbers is not valid. We start by bringing the quantum Boltzmann equation in an appropriately scaled form. Carrying out the same sort of scaling for the quantum Boltzmann equation as for the classical one using the scaling parameters

(4.9)

(a)

x→

x , x0

k→

k , k0 = k0

√

mT0

t→

t , t0

t0 =

τ0 , λ2

λ=

k0 τ 0 x0

2 k02 1 x t V ( , ), Q → Q qm t0 t0 τ0 and obtain the scaled quantum Boltzmann equation (b)

(4.10)

(a)

V (x, t) →

λ2 ∂t w(x, k, t) + λk · ∇x w − λθw = Q(w)

h h i [V (x + ∇k ) − V (x − ∇k )] h 2i 2i with the dimensionless Planck constant h given by (b)

θ=

(4.11)

h=

1 = √ . x0 k0 x0 mT0

The Wigner function w is constructed from a mixed state of solutions of the Schr¨ odinger equation, which in scaled form reads

(4.12)

(a)

(b)

w(x, k, t) =

λ∂t ψj =

j

ih i ∆x ψj − V ψj 2 h

αj ψj (x −

h h η, t)ψj∗ (x + η, t)eiη·k dη. 2 2

The following will not rely on any asymptotics for small Knudsen numbers λ. Therefore we might as well chose a length and time scale which results in a Knudsen number λ = 1. As in the classical regime we take moments of the quantum Boltzmann equation (4.10) with respect to the wave vector k to obtain macroscopic conservation laws. As it turns out, the first three moments of the pseudo differential operator θ coincide with the moments of its classical equivalent. So (4.13)

j

[k θw]dk =

[kj ∇k · (∇x V w)]dk,

j = 0, 1, 2

holds. Therefore the quantum moment equations are the same as the classical ones and are of the form (with λ = 1)

124

Christian Ringhofer

(4.14)

∂t < 1 > +∇x · < k >= 0

(a) (b)

∂t < k > +∇x · < kkT > −E < 1 >=< kQ >

(c)

∂t < ε > +∇x · < kε > −E· < k >=< εQ >

and the difference to the classical hydrodynamic equations can arise only from the way we close the system. There are essentially two ways to close the system. One is the ’single state closure’, which corresponds to zero temperature or a perturbation thereof. The other one is the closure with a displaced quantum equilibrium density, which corresponds in some sense to asymptotics for high temperatures. The puzzling fact is that they give essentially the same result. We first change variables from < 1 >, < k >, < ε >→ n, u, < ε > by the formula (4.15)

< 1 >= n,

< k >= nu,

yielding the moment equations (4.16)

(a)

∂t (nu) + ∇x · < kkT > −En =< kQ >

(b) (c)

∂t n + ∇x (nu) = 0

∂t < ε > +∇x · < kε > −nE · u =< εQ >

In the single state closure one assumes that the Wigner function w is made up of a single state. So (4.17)

w(x, k, t) =

α0 ψ0 (x −

h h η, t)ψ0∗ (x + η, t)eiη·k dη 2 2

holds. This gives for the unknown moments < kk T >, < kε > in (4.16) the relations

(4.18)

< kkT >= n[uuT −

(a)

(b)

< kε >=

h2 2 (∂x ln n)], 4

n h2 [u|u|2 − (u∆x ln n + 2(∂x2 ln n)u + ∆x u)] = 2 4

u−

h2 n [2(∂x2 ln n)u + ∆x u)], 8

h2 n [|u|2 − (∆x ln n)], 2 4 and the resulting moment system reads (c)

(4.19)

< ε >=

(a)

∂t n + ∇x (nu) = 0


(b)

(c)

∂t (nu) + ∇x · {n[uuT −

∂t < ε > +∇x · {u < ε > −

125

h2 2 ∂x ln n]} − En =< kQ > 4

h2 n [2(∂x2 ln n)u + ∆x u)]} − nE · u =< εQ > . 8

One immediately notices that the third equation is actually not needed (assuming for instance a relaxation time approximation for the moment of the collision operator < kQ >= − τ1 nu), and one could only solve (4.19) (ab). These two equations can also be derived directly from the Schr¨ odinger equation for a single state, without the detour over the Wigner function and the moments by expressing the single wave function ψ0 in complex polar coordinates as ψ0 (x, t) =

(4.20)

√

z n exp(i ), h

u = ∇x z.

which is the classical WKB expansion, and yields an irrotational velocity field u. The reason for this is that a single wave function (a single particle) cannot produce a temperature, and therefore the energy < ε > just equals the kinetic energy nuuT plus a quantum perturbation term. Temperature is generally defined as the variation of the velocities of single particles around the group velocity. One ’engineering type’ remedy is to introduce the temperature by brute force. If we define the pressure tensor P and the temperature T by

< kk T >= nuuT + P,

(4.21)

P = n(T I −

h2 2 ∂x ln n) 4

and write the system (4.19) in terms of the variables n, u and T we obtain the moment equations (4.22)

(a)

(b)

(c)

∂t n + ∇x (nu) = 0

∂t (nu) + ∇x · {n[uuT + T I −

∂t < ε > +∇x · {u < ε > −

h2 2 ∂x ln n]} − En =< kQ > 4

h2 n [2(∂x2 ln n)u + ∆x u)]} − nE · u =< εQ > . 8

n h2 (∆x ln n)], [|u|2 + 3T − 4 2 where the last equation is now just an expression for the energy < ε > and the variables are n, u and T . Of course, since the system can be closed after only two moments the temperature T will remain zero for all time if we really start with a single state initial condition. However, the system (4.22) is widely used with a finite initial temperature (see [8] for an overview). Since this is somewhat unsatisfying , the next possible avenue would be to derive some form of perturbation theory around (d)

< ε >:=

126

Christian Ringhofer

the single state for small temperatures. For a mixed state the Wigner function w is made up of a superposition of single state Wigner functions.

(4.23)

∞

w=

αj wj

j=0

holds, where each of the wj satisfy the moment equations with the single state closure conditions. The total charge and velocity are then defined as (4.24)

n=

αj nj ,

nu =

j

αj nj uj .

j

If we introduce the variations around the mean charge and velocity by nj = naj ,

(4.25)

u j = u + bj

the variational variables aj , bj satisfy

(4.26)

αj aj = 1,

j

αj aj bj = 0.

j

In order to close the system we define the temperature tensor T and the vector q by

(4.27)

< kk T >= n[uuT + T −

(a)

h2 2 (∂x ln n)] 4

nh2 [2(∂x2 ln n)u + ∆u] + nq. 8 For T = 0 and q = 0 we obtain the single state closure. The temperature tensor T and the vector q are given in terms of the variational variables aj and bj by (b)

< kε >= u < ε > +nTu −

(4.28)

(a)

T=

αj aj (bj bTj −

j

(b)

q=

j

αj aj [bj (|bj |2 −

h2 2 ∂x ln aj ) 4

h2 h2 2 h2 ∆aj ) − ∂x ln aj bj − ∆bj ]. 4 4 4

The formulas (4.28) show that temperature is really produced by the variation bj of velocities around the group velocity u. An extension to the single state closure is to assume that the occupation probabilities αj decay very rapidly and therefore the mixed state is almost a single state giving a closure for low temperatures. This has been done in [9]. Assuming Boltzmann statistics the occupation probabilities αj are of the form (4.29)

αj = exp(−

ej ), T0


127

where T0 is the ambient temperature and ej are the energy levels, the eigenvalues of the Schr¨ odinger operator Therefore the αj decay rapidly for small ambient tem0 0 perature T0 . An asymptotic analysis in the parameter γ = α = exp( e1T−e ) gives α1 0 2 that T = O(γ) and q = O(γ ) holds. Moreover T is a rank one matrix produced by a temperature potential. Altogether this gives the closure

(4.30)

(a)

< kkT >= n[uuT + T − (b) u < ε > +nTu − (c)

h2 2 (∂x ln n)], 4

< ε >=

1 tr < kkT > 2

< kε >=

nh2 [2(∂x2 ln n)u + ∆u] + O(γ 2 ) 4 T = γ∇x z∇x z T .

Unfortunately, this is not any agreement with the classical picture for h → 0, since in the classical case the temperature tensor should be a scalar temperature times the identity matrix and not a rank one matrix of the form (4.30)(c). The reason for this discrepancy lies in the fact that the approximate single state expansion really is an approximation for extremely small temperature and if one computes the eigenvalues ej for simple structures γ stops to be small somewhere in the neighborhood of 10K. An alternative to the single state closure approach is to proceed in the same way as for the classical hydrodynamic equations and to close the moment hierarchy (4.14) via a momentum displaced equilibrium density, which is now defined as the quantum mechanical thermal equilibrium, by assuming that w(x, k, t) ≈ wE (x, k − u(x, t), T (x, t))

(4.31) [10][11]

. The quantum mechanical thermal equilibrium is given by the integral holds 2 kernel of the operator F (H) , where H = − h2 ∆x + V is the Hamilton operator and , in the case of Boltzmann statistics, F is the distribution function

(4.32)

F (z) = exp(

φ−z ) T

with φ the Fermi level and T the temperature. The definition of F (H) is the classical one : Given an orthonormal L2 eigen- system {ψj }of the self adjoint operator H ,F (H) applied to a function g is defined by

(4.33)

(a)

[F (H)g](r) =

F (ej )ˆ gj ψj (r),

gˆj =

[g(s)ψj (s)]ds

j

(b)

Hψj = ej ψj ,

j = 0, 1..

The equilibrium density matrix ρE is the given by the integral kernel of F (H) as

128

Christian Ringhofer ρE (r, s) =

(4.34)

F (ej )ψj (r)ψj (s)

j

The way to find an asymptotic approximation to the equilibrium density matrix is via the so called Bloch equation. The concept of the Bloch equation is to interpret e−H/T as a semigroup. The operator e−H/T applied to a function g can also be computed by solving the evolution problem [e−H/T g](r) = z(r, γ =

(a)

(4.35)

∂γ z = −Hz,

(b)

1 ), T

z(r, γ = 0) = g(r).

The integral kernel of the semigroup operator ρE is now computed by solving the evolution equation

(4.36)

(a)

ρE (r, s) = ρ(r, s, γ =

∂γ ρ = −Hr ρ,

(b)

(c)

1 ), T

ρ(r, s, γ = 0) = δ(r − s)

[e−H/T g](r) =

ρE (r, s)g(s)ds.

Since the equilibrium density matrix ρE is symmetric, ρE (r, s) = ρE (s, r) holds, (4.36) (b) is conveniently replaced by

(4.37)

1 ∂γ ρ = − [Hr + Hs ]ρ, 2

ρ(r, s, γ = 0) = δ(r − s),

where Hr , Hs denote the Hamilton operator applied in the r and s directions respectively. The equilibrium Wigner function is obtained by the transformation (4.38)

wE (x, k) =

ρE (x −

h h η, x + η)eiη·k dη, 2 2

and is computed from the Bloch equation as

(4.39)

(a)

wE (x, k) = w(x, k, γ =

∂γ w = −Hw w,

(b)

1 ) T

w(x, k, γ = 0) = 1,

where Hw denotes the Hamiltonian after the Wigner transform

(4.40)

(a)

(b)

Hw = [−

Ω=

|k|2 h2 w + Ωw], ∆x w + 2 8

1 h h [V (x + ∇k ) + V (x − ∇k )]. 2 2i 2i


129

Closure in the moment hierarchy is now achieved by making suitable approximations to the solution of the Bloch equation (4.39). The simplest one is due to Wigner and consists of an expansion around the classical limit h → 0. If we formally expand Ω in powers of h we obtain

Ωw = V w −

(4.41)

h2 tr[(∂x2 V ) · (∂k2 w)] + O(h4 ), 8

where ∂x2 V and ∂k2 w are the 3 × 3 matrices of second order partial derivatives and tr is the trace operator. So we rewrite (4.39) in

(4.42) ∂γ w + (

|k|2 ˜ h2 + V )w = {∆x w + tr[∂x2 V ∂k2 w]} + O(h4 ), 2 8

w(x, k, γ = 0) = 1,

and expand w = w0 + h2 w1 + .. This gives for the first two terms w0 and w1 :

(4.43)

(a)

(b)

w1 (x, k, γ) =

w0 (x, k, γ) = exp[−γ(

|k|2 + V )] 2

γ3 1 0 γ3 tr[∂x2 V kkT ]} w (x, k, γ){ |∇x V |2 − γ 2 ∆x V + 3 3 8

In equilibrium the pressure tensor P in (4.21) is defined by P =< kk T >, since u = 0 holds. Using the asymptotic expression (4.43) we obtain that the equilibrium pressure tensor P and the closure vector q in (4.27)(b) are given by

(4.44)

(a)

P = n[T I − (b)

h2 2 (∂x ln n)] 12

q = 0,

These relations are then used to close the moment system (4.45)

(a) (b)

(c)

∂t n + ∇x (nu) = 0

∂t (nu) + ∇x · {nuuT + P} − En =< kQ >

h2 n ∂t < ε > +∇x · {u < ε > − [2(∂x2 ln n)u+∆x u)]}−nE · u+nq =< εQ > . 8

n [|u|2 + tr(P)], 2 in the nonequilibrium case. After inserting (4.44) in (4.45) and assuming the appropriate terms for the moments of the collision operator, a closed system for the variables n, u and T is obtained. Notice the difference of a factor 3 between (4.44) and (4.21). This is not due to a misprint or an algebra error. The puzzling fact is (d)

< ε >:=

130

Christian Ringhofer

that the quantum hydrodynamic models (4.22) and (4.45) agree up too this factor, although one is derived by single state closure, corresponding to zero temperature, and the other by the classical limit approximation for small h, which corresponds, if anything, to high ambient temperature because of (4.11). This fact is of yet unexplained. We close the section about quantum hydrodynamic models by mentioning a recent extension of the above approach which uses a different sort of asymptotics for the Bloch equation. This work, given in [13], is preferable for the case of discontinuous potentials. In this case the validity of the asymptotic solution (4.43) of the Bloch equation is no longer given, since it involves second derivatives of potentials, which exist only in a distributional sense. This approach leads to an expression for the pressure tensor of the form

(4.46)

P = n[T I −

h2 2 ˜ (∂x V ], 12

where V˜ is a smoothed potential, depending non - locally on the potential V .

References 1. M Anile, M. Trovato: Extended thermdynamics of charge carrier transport in semiconductors, Phys. Rev.B 51,pp 16728 (1995). 2. A. Arnold, P.Degond, P.Markowich, H. Steinrueck: The Wigner Poisson equation in a crystal, Appl. Math. Let. 2, pp. 187-191 (1989). 3. N. Ashcroft, M. Mermin: Solid State Physics, Holt - Saunders, New York (1976). 4. G. Baccarani, M. Wordeman: An investigation of steady state velocity overshoot effects in Si and GaAs devices, Solid State Electr. 28, pp. 407-416 (1985). 5. N. Ben - Abdallah, P. Degond, F. Genieys: The derivation of energy models from the Boltzmann transport equation, preprint, Universite Paul Sabatier, Toulouse , France. 6. S. Cordier: Hyperbolicity of Grad’s extension of hydrodynamic models for ionospheric plasmas I: The single species case, Math. Mod. Meth. Appl. Sci. 4, pp.625-645 (1994). 7. S. Cordier: Hyperbolicity of Grad’s extension of hydrodynamic models for ionospheric plasmas II: The two species case, Math. Mod. Meth. Appl. Sci. 4, pp.647667 (1994). 8. D. Ferry, H. Grubin: Modelling of quantum transport in semiconductor devices, Solid State Phys. 49, pp.283-448 (1995). 9. I. Gasser ,P. Markowich),C.Ringhofer: Closure conditions for classical and quantum moment hierarchies in the small temperature limit, Transport Theory and Statistical Physics 25, pp.409-423 (1996). 10. C. Gardner: The classical and the quantum hydrodynamic models Proc. Int. Workshop on Computational Electronics, Leeds 1993, J. Snowden ed. pp. 25-36 (1993). 11. C. Gardner: The quantum hydrodynamic model for semiconductor devices SIAM J. Appl. Math. 54, pp. 409-427 (1994).


131

12. C. Gardner: Shock waves in the hydrodynamic model for semiconductor devices, in IMA volumes in Mathematics and its Applications 59 pp. 123-134 (1991). 13. C. Gardner, C. Ringhofer: The smooth quantum potential for the hydrodynamic model , Phys. Rev. E 53, pp.157-167 (1996). 14. H. Grad: On the kinetic theory of rarefied gases , Comm. Pure Appl. Math. 2, pp.331-407 (1949). 15. H. Grad: Principles of the kinetic theory of gases , Handbooks. Phys. 12, pp.205294 (1958). 16. A. Kersch, W. Morokoff: Transport simulation in microelectronics, Birkhaeuser, Basel (1995). 17. P. Markowich, C. Ringhofer, C. Schmeiser: Semiconductor equations, Springer (1990). 18. P. Markowich, C.Ringhofer: An analysis of the quantum Liouville equation, ZAMM 69, pp. 121-127 (1989). 19. P. Markowich, N. Mauser, F. Poupaud: A Wigner function approach to semiclassical limits, J. Math. Phys. 35,pp. 1066-1094 (1994). 20. F. Poupaud: Diffusion approximation of the linear Boltzmann equation: Analyisis of boundary layers, Asympt. Anal. 4, pp. 293-317 (1991). 21. F.Poupaud, C. Ringhofer: Quantum hydrodynamic models in semiconductor crystals,Appl. Math. Lett. 8, pp.55-59 (1995). 22. F.Poupaud, C. Ringhofer: Semi - classical limits in a crystal with exterior potentials and effective mass theorems, Communications in Partial Differential Equations 21, pp.1897-1918 (1996). 23. C. Ringhofer: A spectral method for the numerical solution of quantum tunneling phenomena, SIAM J. Num. Anal. 27, pp.32-50 (1990). 24. C. Ringhofer: On the convergence of spectral methods for the Wigner - Poisson problem, Math. Models and Meth. in Appl. Sci. 2, pp.91-111 (1992). 25. S. Selberherr: Analysis of semiconductor devices 2nd ed., Wiley, New York (1981). 26. V. Tatarski : The Wigner representation of quantum mechanics, Soviet. Phys. Uspekhi 26, pp. 311-372 (1983). 27. M. Taylor: Pseudodifferential operators, Princeton University Press, Princeton (1981). 28. E. Wigner: On the quantum correction for thermodynamic equilibrium , Phys. Rev. 40, 749-759 (1932).

List of Participants

1. Fredrik Abrahamsson, Chalmes University of Technology, Goteborg, Sweden [email protected] 2. Walter Allegretto, (lecturer) University of Alberta, Canada [email protected] 3. Marcello Anile, (editor, lecturer) Universit` a di Catania, Italy [email protected] 4. Anton Arnold, Technical University Berlin, Germany [email protected] 5. Luigi Barletti, Universit` a di Firenze, Italy [email protected]fi.it 6. Philippe Bechouche, Lab. Mathématiques Univ. de Nice, France [email protected] 7. Stefano Bianchini, SISSA, Trieste, Italy [email protected] 8. Chiara Eva Catalano, Parma, Italy [email protected] 9. Stephane Codier, Lab. Analyse Numerique Paris 6, Paris, France [email protected] 10. Andrea Corli, Universit` a di Ferrara [email protected] 11. Concettina Rita Drago, Universit` a di Catania, Italy [email protected]

134

List of Participants

12. Matthias Ehrhardt, Technical University Berlin, Germany [email protected] 13. Asma El Ayyadi, Univ. Paul Sabatier Labo. MIP, Toulouse, France [email protected] 14. Roberta Fabbri, Univerist` a di Firenze, Italy [email protected]fi.it 15. Klemens Fellner, Technical University Wien, Austria klemens@[email protected] 16. Florian Frommlet, Technical University Berlin, Germany [email protected] 17. Stephane Genieys, Univ. Paul Sabatier, Toulouse, France [email protected] 18. Thirry Goudon, Univ. Nice-Sophia Antipolis, Nice, France [email protected] 19. Arthur Iordanidi, Univ. of Twente, Enschede, The Netherlands [email protected] 20. Ansgar Jungel, Technical University Berlin, Germany [email protected] 21. David Levermore, Univerisity of Arizona, Tucson, USA [email protected] 22. Fabio Salvatore Liotta, Universit` a di Catania, Italy [email protected] 23. Simona Mancini, Universit` a di Firenze, Italy [email protected]fi.it 24. Peter A. Markowich, Technical University Berlin, Germany markowich@@math.TU-Berlin.DE 25. Americo Marrocco, INRIA, Le Chesnay, France [email protected] 26. Giovanni Mascali, Universit` a di Catania, Italy [email protected] 27. Serena Matucci, Universit` a di Firenze, Italy [email protected]fi.it

List of Participants 28. Orazio Muscato, Universit` a di Catania, Italy [email protected] 29. Valerii Obukhovskii, Voronezh StateUniversity, Russia [email protected] 30. Vladislav Panferov, Chalmers University of Technology, Goteborg, Sweden [email protected] 31. Rosa Maria Pidatella, Universit` a di Catania, Italy [email protected] 32. Paola Pietra, Ist. Anal. Num. CNR, Pavia, Italy [email protected] 33. Carsten Pohl, Technical University Berlin, Germany [email protected] 34. Eric Polizzi, Centre de Mathématique INSA, Toulouse, France [email protected] 35. Frédéric Poupaud, Laboratoire J.A. Dieudonné, Nice, France [email protected] 36. Kamel Rachedi, INRIA, Le Chesnay, France [email protected] 37. Christian Ringhofer, (lecturer) Arizona state University, Tempe, USA [email protected] 38. Vittorio Romano, Universit` a di Catania, Italy [email protected] 39. Christian Schmeiser, Technical University Wien, Austria [email protected] 40. Henning Struchtrup, Technical University Berlin, Germany henning@@math.TU-Berlin.DE 41. Bernt Wennberg, Chalmers University of Technology, Goteborg, Sweden [email protected] 42. Pietro Zecca, Universit` a di Firenze, Italy pzecca@ingfi1.ing.unifi.it

135

LIST OF C.I.M.E. SEMINARS

1954

1. Analisi funzionale 2. Quadratura delle superficie e questioni connesse 3. Equazioni differenziali non lineari

1955

4. 5. 6. 7.

1956

8. 9. 10. 11.

Teorema di Riemann-Roch e questioni connesse Teoria dei numeri Topologia Teorie non linearizzate in elasticit` a, idrodinamica, aerodinamic Geometria proiettivo-differenziale Equazioni alle derivate parziali a caratteristiche reali Propagazione delle onde elettromagnetiche Teoria della funzioni di pi` u variabili complesse e delle funzioni automorfe Geometria aritmetica e algebrica (2 vol.) Integrali singolari e questioni connesse Teoria della turbolenza (2 vol.)

C.I.M.E " " " " " " " " " " " " "

1957

12. 13. 14.

1958

15. Vedute e problemi attuali in relativit` a generale 16. Problemi di geometria differenziale in grande 17. Il principio di minimo e le sue applicazioni alle equazioni funzionali 18. Induzione e statistica 19. Teoria algebrica dei meccanismi automatici (2 vol.) 20. Gruppi, anelli di Lie e teoria della coomologia

" " "

1960

21. Sistemi dinamici e teoremi ergodici 22. Forme differenziali e loro integrali

" "

1961

23. Geometria del calcolo delle variazioni (2 vol.) 24. Teoria delle distribuzioni 25. Onde superficiali

" " "

1962

26. Topologia differenziale 27. Autovalori e autosoluzioni 28. Magnetofluidodinamica

" " "

1963

29. Equazioni differenziali astratte 30. Funzioni e variet` a complesse 31. Propriet` a di media e teoremi di confronto in Fisica Matematica

" " "

1959

" " "

138


1964

32. 33. 34. 35. 36. 37. 38.

Relativit` a generale Dinamica dei gas rarefatti Alcune questioni di analisi numerica Equazioni differenziali non lineari Non-linear continuum theories Some aspects of ring theory Mathematical optimization in economics

39. 40. 41. 42. 43. 44.

Calculus of variations Economia matematica Classi caratteristiche e questioni connesse Some aspects of diffusion theory Modern questions of celestial mechanics Numerical analysis of partial differential equations Geometry of homogeneous bounded domains Controllability and observability Pseudo-differential operators Aspects of mathematical logic

1965

1966

1967

1968

1969

1970

1971

45. 46. 47. 48.

49. Potential theory 50. Non-linear continuum theories in mechanics and physics and their applications 51. Questions of algebraic varieties 52. Relativistic fluid dynamics 53. Theory of group representations and Fourier analysis 54. Functional equations and inequalities 55. Problems in non-linear analysis 56. Stereodynamics 57. Constructive aspects of functional analysis (2 vol.) 58. Categories and commutative algebra

C.I.M.E " " " " " " Ed. Cremonese, Firenze " " " " " " " " " " " " " " " " " " "

1972

59. Non-linear mechanics 60. Finite geometric structures and their applications 61. Geometric measure theory and minimal surfaces

" " "

1973

62. Complex analysis 63. New variational techniques in mathematical physics 64. Spectral analysis 65. Stability problems 66. Singularities of analytic spaces 67. Eigenvalues of non linear problems

" "

1975

68. Theoretical computer sciences 69. Model theory and applications 70. Differential operators and manifolds

" " "

1976

71. Statistical Mechanics 72. Hyperbolicity 73. Differential topology

1977

74. Materials with memory 75. Pseudodifferential operators with applications 76. Algebraic surfaces

1974

" " " "

Ed. Liguori, Napoli " " " " "


139

1978

77. Stochastic differential equations 78. Dynamical systems

Ed. Liguori, Napoli & Birkh¨ auser "

1979

79. Recursion theory and computational complexity 80. Mathematics of biology

" "

1980

81. Wave propagation 82. Harmonic analysis and group representations 83. Matroid theory and its applications

" " "

1981

84. Kinetic Theories and the Boltzmann Equation 85. Algebraic Threefolds 86. Nonlinear Filtering and Stochastic Control

(LNM 1048) Springer-Verlag (LNM 947) " (LNM 972) "

1982

87. Invariant Theory 88. Thermodynamics and Constitutive Equations 89. Fluid Dynamics

(LNM 996) (LN Physics 228) (LNM 1047)

" " "

1983

90. Complete Intersections 91. Bifurcation Theory and Applications 92. Numerical Methods in Fluid Dynamics

(LNM 1092) (LNM 1057) (LNM 1127)

" " "

1984

93. Harmonic Mappings and Minimal Immersions 94. Schr¨ odinger Operators 95. Buildings and the Geometry of Diagrams

(LNM 1161) (LNM 1159) (LNM 1181)

" " "

1985

96. Probability and Analysis 97. Some Problems in Nonlinear Diffusion 98. Theory of Moduli

(LNM 1206) (LNM 1224) (LNM 1337)

" " "

1986

99. Inverse Problems 100. Mathematical Economics 101. Combinatorial Optimization

(LNM 1225) (LNM 1330) (LNM 1403)

" " "

1987

102. Relativistic Fluid Dynamics 103. Topics in Calculus of Variations

(LNM 1385) (LNM 1365)

" "

1988

104. Logic and Computer Science 105. Global Geometry and Mathematical Physics

(LNM 1429) (LNM 1451)

" "

1989

106. Methods of nonconvex analysis 107. Microlocal Analysis and Applications

(LNM 1446) (LNM 1495)

" "

1990

108. Geometric Topology: Recent Developments 109. H∞ Control Theory 110. Mathematical Modelling of Industrial Processes

(LNM 1504) (LNM 1496) (LNM 1521)

" " "

1991

111. Topological Methods for Ordinary Differential Equations 112. Arithmetic Algebraic Geometry 113. Transition to Chaos in Classical and Quantum Mechanics 114. Dirichlet Forms 115. D-Modules, Representation Theory, and Quantum Groups 116. Nonequilibrium Problems in Many-Particle Systems

(LNM 1537)

"

(LNM 1553) (LNM 1589)

" "

(LNM 1563) (LNM 1565)

" "

(LNM 1551)

"

1992

140


1993

117. Integrable Systems and Quantum Groups 118. Algebraic Cycles and Hodge Theory 119. Phase Transitions and Hysteresis

(LNM 1620) Springer-Verlag (LNM 1594) " (LNM 1584) "

1994

120. Recent Mathematical Methods in Nonlinear Wave Propagation 121. Dynamical Systems 122. Transcendental Methods in Algebraic Geometry 123. Probabilistic Models for Nonlinear PDE’s 124. Viscosity Solutions and Applications 125. Vector Bundles on Curves. New Directions

(LNM 1640)

"

(LNM (LNM (LNM (LNM (LNM

1609) 1646) 1627) 1660) 1649)

" " " " "

(LNM 1684)

"

(LNM 1713)

"

(LNM 1656) (LNM 1714) (LNM 1697)

" " "

(LNM (LNM (LNM (LNM (LNM

1716) 1776) 1740) 1784) 1823)

" " " " "

(LNM 1715)

"

(LNM 1734)

"

(LNM 1739)

"

(LNM 1804)

"

to appear

"

(LNM 1775) to appear

" "

(LNM 1822) (LNM 1819) (LNM 1812) to appear to appear to appear (LNM 1802) (LNM 1813) (LNM 1825)

" " " " " " " " "

to appear to appear to appear

" " "

1995

1996

1997

1998

1999

2000

2001

2002

126. Integral Geometry, Radon Transforms and Complex Analysis 127. Calculus of Variations and Geometric Evolution Problems 128. Financial Mathematics 129. Mathematics Inspired by Biology 130. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations 131. Arithmetic Theory of Elliptic Curves 132. Quantum Cohomology 133. Optimal Shape Design 134. Dynamical Systems and Small Divisors 135. Mathematical Problems in Semiconductor Physics 136. Stochastic PDE’s and Kolmogorov Equations in Infinite Dimension 137. Filtration in Porous Media and Industrial Applications 138. Computational Mathematics driven by Industrial Applications 139. Iwahori-Hecke Algebras and Representation Theory 140. Theory and Applications of Hamiltonian Dynamics 141. Global Theory of Minimal Surfaces in Flat Spaces 142. Direct and Inverse Methods in Solving Nonlinear Evolution Equations 143. Dynamical Systems 144. Diophantine Approximation 145. Mathematical Aspects of Evolving Interfaces 146. Mathematical Methods for Protein Structure 147. Noncommutative Geometry 148. Topological Fluid Mechanics 149. Spatial Stochastic Processes 150. Optimal Transportation and Applications 151. Multiscale Problems and Methods in Numerical Simulations 152. Real methods in Complex and CR Geometry 153. Analytic Number Theory 154. Imaging

LIST OF C.I.M.E. SEMINARS 2003

2004

155. 156. 157. 158.

Stochastik Methods in Finance Hyperbolik Systems of Balance Laws Sympletic 4-Manifolds and Algebraic Surfaces Mathematical Foundation of Turbulent Viscous Flows 159. Representation Theory and Complex Analysis 160. Nonlinear and Optimal Control Theory 161. Stochastic Geometry

announced announced announced announced announced announced announced

141

Fondazione C.I.M.E. Centro Internazionale Matematico Estivo International Mathematical Summer Center http://www.math.unifi.it/∼cime [email protected]fi.it

2004 COURSES LIST Representation Theory and Complex Analysis June 10–17, Venezia Course Directors: Prof. Enrico Casadio Tarabusi (Universit` a di Roma “La Sapienza”) Prof. Andrea D’Agnolo (Universit` a di Padova) Prof. Massimo A. Picardello (Universit` a di Roma “Tor Vergata”)

Nonlinear and Optimal Control Theory June 21–29, Cetraro (Cosenza) Course Directors: Prof. Paolo Nistri (Universit` a di Siena) Prof. Gianna Stefani (Universit` a di Firenze)

Stochastic Geometry September 13–18, Martina Franca (Taranto) Course Director: Prof. W. Weil (Univ. of Karlsruhe, Karlsruhe, Germany)

Mathematical Problems in Semiconductor Physics. Lectures C.I.M.E., Cetraro, Italy, 1998

Mathematical problems in semiconductor physics: lectures given at the C.I.M.E. summer school held in Cetraro, Italy, July 15-22, 1998

Mathematical problems in semiconductor physics: lectures given at the C.I.M.E. summer school held in Cetraro, Italy, July 15-22, 1998

Obstacle problems in mathematical physics

Contemporary Problems in Mathematical Physics

Lectures on Geometric Methods in Mathematical Physics

Lectures on Geometric Methods in Mathematical Physics

Lectures on Geometric Methods in Mathematical Physics

Teoria dei numeri: CIME; Varenna , Italy, 1955

Geometric topology: recent developments. Lectures CIME

Holomorphic dynamical systems: Cetraro, Italy, 2008

Partial Differential Equations in Classical Mathematical Physics (CUP 1998)

Problems in mathematical statistics

Problems in mathematical analysis

Mathematical Problems in Elasticity

Problems in mathematical statistics

Mathematical Problems in Plasticity

Problems in Mathematical Analysis

Problems in Mathematical Analysis

Problems in mathematical analysis

Problems in mathematical statistics

Problems in mathematical analysis

Semiconductor Physics And Devices

Physics of Semiconductor Devices

Problems in Elementary Physics

Problems in general physics

Problems in Laser Physics

Problems in Physics

Problems in Elementary Physics

Problems in general physics

Problems in General Physics

Mathematical Problems in Semiconductor Physics. Lectures C.I.M.E., Cetraro, Italy, 1998

Mathematical problems in semiconductor physics: lectures given at the C.I.M.E. summer school held in Cetraro, Italy, July 15-22, 1998

Mathematical problems in semiconductor physics: lectures given at the C.I.M.E. summer school held in Cetraro, Italy, July 15-22, 1998

Obstacle problems in mathematical physics

Contemporary Problems in Mathematical Physics

Lectures on Geometric Methods in Mathematical Physics

Lectures on Geometric Methods in Mathematical Physics

Lectures on Geometric Methods in Mathematical Physics

Teoria dei numeri: CIME; Varenna , Italy, 1955

Geometric topology: recent developments. Lectures CIME

Holomorphic dynamical systems: Cetraro, Italy, 2008

Partial Differential Equations in Classical Mathematical Physics (CUP 1998)

Problems in mathematical statistics

Problems in mathematical analysis

Mathematical Problems in Elasticity

Problems in mathematical statistics

Mathematical Problems in Plasticity

Problems in Mathematical Analysis

Problems in Mathematical Analysis

Problems in mathematical analysis

Problems in mathematical statistics

Problems in mathematical analysis

Semiconductor Physics And Devices

Physics of Semiconductor Devices

Problems in Elementary Physics

Problems in general physics

Problems in Laser Physics

Problems in Physics

Problems in Elementary Physics

Problems in general physics

Problems in General Physics

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