Proceedings
"WASCOM 2003" 12th Conference on Waves and Stability in Continuous Media
Editors
Roberto Monaco Sebastiano Pennisi Salvatore Rionero Tommaso Ruggeri
World Scientific
Proceedings
"WASCOM 2003" 12th Conference on
Waves and Stability in Continuous Media
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Proceedings
"WASCOM 2003" 12th Conference on
Waves and Stability in Continuous Media Villasimius (Cagliari) Italy 1-7 June 2003
Editors
Roberto Monaco Politecnico di Torino, Italy
Sebastiano Pennisi Universitd di Cagliari, Italy
Salvatore Rionero Universitd di Napoli Federico II, Italy
Tommaso Ruggeri Universitd di Bologna, Italy
YJ? World Scientific NEW JERSEY • LONDON • SINGAPORE • SHANGHAI • H O N G K O N G • TAIPEI • CHENNAI
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PREFACE
In 1981, a group of researchers from several Italian universities had the idea of initiating an international conference on waves and stability with the specific aim to bring together Italian and foreign experts in both fields, and to discuss leading aspects of these areas of research. This meeting has taken place, since then, every two years with increasing interest and participation. The last conference, the XII edition, has been held in Sardinia and more precisely in Villasimius (Cagliari), June 1-7, 2003. Previous conferences have been held in Catania (1981), Arcavacata di Rende (Cosenza, 1983), Giovinazzo (Bari, 1985), Taormina (Messina, 1987), Sorrento (Napoli, 1989), Acireale (Catania, 1991), Bologna (1993), Altavilla Milicia (Palermo, 1995), Capitolo di Monopoli (Bari, 1997), Vulcano (Messina, 1999) and Porto Ercole (Grosseto, 2001). Each edition has published a volume of proceedings documenting the research work and progress in the area. The research groups promoting the XII edition of the conference belong to the Departments of Mathematics of the Universities of Bologna, Catania, Lecce, Messina, Napoli, Palermo, Torino-Politecnico and, of course, Cagliari which was in charge of organizing the meeting. The XII edition registered over one hundred participants coming from more than ten different countries. The topics covered were • • • • • •
Discontinuity and shock waves Stability in fluid dynamics Small parameter problems Kinetic theories towards continuum models Non equilibrium thermodynamics Numerical applications.
This volume contains 66 papers which have been presented at the XII Conference as invited lectures and short communications. The Editors of this volume would like to thank the Scientific Committee who carefully suggested the invited lectures and selected the contributed papers, as well as the members of the Organizing Committee, coming from the Departments of Mathematics of the Universities of Cagliari, Napoli
v
VI
and Bologna. A thank also to Sandra Pieraccini (Politecnico di Torino) who has carefully prepared the final editing of the manuscript. A special thank is addressed to all the participants to whom ultimately the success of the conference has been ascribed to. Finally, the Editors are especially indebted to Fondazione CRT di Torino which has partially supported the publishing expenses of this volume. January 2004
The Editors Roberto Monaco Sebastiano Pennisi Salvatore Rionero Tommaso Ruggeri
Professor Salvatore Rionero Universitd di Napoli Federico II
VIII
The International Conference WASCOM 2003 falls this year at the same time of Professor Salvatore Rionero's 70th birthday and for this reason it is dedicated to him. As already mentioned in the preface of this volume, the conference started, thanks to Professor Rionero's enterprise within some national scientific research projects, gathering different Italian researchers working in the field of waves and non linear stability in continuous media. Professor Rionero coordinated such projects for several years until today. The national research project he helped to coordinate includes more than 60 Italian researchers from seven universities in Italy. The project entitled, Non Linear Mathematical Problems of Propagation and Stability in Models of Continuous Media, over which at the moment I have the honour of coordinating, is financially supported by MIUR, the Italian Ministery for University and Research. Since the beginning, Professor Rionero had the intuition that a relevant research group was necessary to facilitate an exchange of ideas with foreign scientists, and WASCOM was created. The small workshop in 1981 expanded in the following years and now has become a well known and appreciated international conference that was organized and held in Italy. Professor Salvatore Rionero was born in Nola on January 1st 1933. He is full Professor of Rational Mechanics at the Faculty of Sciences of the Federico II University in Naples. He is a researcher of great international renown. His main scientific field of interest is devoted to the mathematical problems in non linear stability. He is the author of more than 120 original works that include papers and books, that were published along the lines of the above themes and related topics, with prestigious results which strongly contributed to the scientific development of the field. Professor Rionero is an active member of various academies, the Accademia Nazionale dei Lincei in particular, and has received numerous awards. The most recent one is the Honorary Degree of Doctor in Science of the National University of Ireland of Galway which was awarded to him in 2002. Besides efforts that were oriented to create a well known and remarkable school, (numbering now by many well established researchers), he also made his best to promote, encourage and stimulate the research in mathematical physics in the Italian scientific community. Professor Rionero was Vice-President of the Board of Directors of INDAM, the National Institute of High Mathematics, and also for many years, the President of the Scientific Council of the National Group of
IX
Mathematical Physics (GNFM). He also promoted the Summer School of Mathematical Physics in Ravello. The school, which Professor Rionero still directs, is today, the flagship of the National Group. Since then every year, some of the best mathematicians from all over the world would come to give courses allowing many young people, sometimes coming from peripheral areas, to keep up to date with the research progress in mathematical physics. Such an activity has produced a generation of qualified mathematical physicists, well connected to the international scientific community. Bologna, January 2004
Tommaso Ruggeri Director of GNFM
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C O N F E R E N C E DATA
W A S C O M 2003 12th International Conference on Waves and Stability in Continuous Media Villasimius (CA), Italy, June 1-7, 2003 Scientific Committee Chairmen: S. Pennisi (Cagliari), S. Rionero (Napoli) and T. Ruggeri (Bologna) C. Dafermos (Providence), D . Fusco (Messina), A. Greco (Palermo), R. Monaco (Torino), G. Mulone (Catania) I. Miiller (Berlin), B. Straughan(Glasgow), C. Tebaldi (Lecce) Organizing Committe Chairmen: F. Borghero (Cagliari) F. Brini (Bologna), G. Cantarelli (Cagliari), F. Capone (Napoli), M. Gentile(Napoli), S. Mignemi (Cagliari), S. Pennisi (Cagliari), S. Serra (Cagliari), C. Van Der Mee (Cagliari) Supported by • PRIN 2000 "Problemi Matematici non Lineari di Propagazione e Stabilita nei Modelli del Continuo" • Gruppo Nazionale per la Fisica Matematica - INDAM • CIRAM - Research Center of Applied Mathematics - University of Bologna • Universita di Cagliari, Rettorato • Facolta di Scienze MM.FF.NN. di Cagliari • Dipartimento di Matematica ed Informatica di Cagliari • Regione Sardegna - Assessorato degli Affari Generali, Personale e Riforma della Regione • Banco di Sardegna • ESIT - Ente Sardo Industrie Turistiche • SEPT ITALIA, Cagliari
XI
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TABLE OF C O N T E N T S
Preface
v
Conference Data
xi
G. Ali, A. Bartel Multi-Physics Models in Electric Network Design
1
G. Ali, I. Torcicollo Time-Periodic Solutions to Hydrodynamical Models for Semiconductors
7
A. M. Anile, G. Mascali A Two Population Model for Electron Transport in Si
13
F. Bagarello On the Time Evolution of QMX
26
Systems: An Algebraic Approach
J. Banasiak Chaos in Some Linear Kinetic Models
32
M. Basurto, C. Tebaldi Characterization of Time Periodic Behaviour by Proper Orthogonal Decomposition
38
A. Belleni-Morante, R. Monaco, R. Riganti, F. Salvarani Inverse Problems in Photon Transport. Part I: Determination of Physical and Geometrical Features of an Interstellar Cloud
52
A. Belleni-Morante, R. Riganti Inverse Problems in Photon Transport. Part II: Features of a Source Inside an Interstellar Cloud
60
XIII
XIV
S. Biondini, G. Borgioli Rigorous Derivation of Kane Models for Interband Tunneling
70
S. Biondini, G. Frosali, F. Mugelli Quantum Hydrodynamic Equations for a Two-Band Wigner-Kane Model
78
M. Bisi, G. Spiga Hydrodynamic Limit for a Gas with Chemical Reactions
85
F. Borghero, F. Demontis, S. Pennisi An Exact Macroscopic Extended Model with Many Moments for Ultrarelativistic Gases
94
F. Brini, T. Ruggeri The Riemann Problem for a Binary Non-Reacting Mixture of Euler Fluids
102
F. Capone, S. Rionero On the Onset of Convection for a Double Diffusive Mixture in a Porous Medium under Periodic in Space Boundary Data
109
S. Carillo, M. Chipot, G. Vergara Caffarelli A Variational Problem with Non-Local Constraints
116
F. Conforto, A. Jannelli, R. Monaco, T. Ruggeri On the Riemann Problem for a Reacting Mixture of Gases
122
H. Cornille Planar DVMS Mixtures Models with the Same Geometrical Structure
133
C. Currb, D. Fusco Shock-Like Travelling Wave Solutions for a Hyperbolic Tumour Growth Model
141
S. De Martino, G. Lauro Soliton-Like Solutions for a Capillary Fluid
148
XV
L. Desvillettes, C. Villani Rate of Convergence toward the Equilibrium in Degenerate Settings
153
S. Donati, S. Totaro, M. Lisi A Mathematical Model for a Sole Larval Population with Two Time Scales
166
M. Fabrizio Free Energies in the Materials with Fading Memory and Applications to PDEs
172
M. Favretti Computation of the Phase Fraction in a Discrete Model of a Pseudoelastic Material
185
P. Fergola, F. Aurelio, M. Cerasuolo, A. Noviello Andrews Nutrient Uptake and Linear Quorum Sensing in Allelopathic Competition Models
191
G. Ferrarese Iterative Discontinuities and Wave Propagation
204
J. N. Flavin Asymptotic and Other Properties of Positive Definite Integral Measures for Nonlinear Diffusion
221
D. Fusco, N. Manganaro Reduction Approaches for a Quasilinear Nonautonomous Diffusion Equation
230
G. Gambino, A. M. Greco, M. C. Lombardo Symmetry Reduction of a Model in Spherical Symmetry for Benign Tumor
241
M. Gentile, S. Rionero Global Stability of Flows in a Divergent Channel
247
XVI
H. Gouin Thermocapillary Fluid and Adiabatic Waves Near its Critical Point
254
G. Grioli Dynamics of the Rigid Bodies and Celestial Mechanics
269
G. Guerriero Exact Steady Solution to a Nonlinear Integral Equation of the Particle Transport Teory
275
C. O. Horgan, G. Saccomandi Constitutive Models for Atactic Elastomers
281
D. Izzo, C. Valente A Globally Stable Lyapunov Non Linear Controller for the Attitude of a Platform Equipped with a System of Actuating Gyroscopes
295
M. C. Lombardo, M. Sammartino Delayed Feedback Control for the Benard Problem
303
G. Mascali Mobility in GaAs Semiconductors
309
S. Mignemi Noncanonical Poisson Brackets in Deformed Special Relativity
316
A. Mikelic, M. Primicerio Blood Perfusion in Muscles: From Microscopic Models to Continuum Approach via Homogenization
321
R. Monaco, M. Pandolfi Bianchi, S. Pieraccini, G. Puppo Numerical Simulations of a Reacting Gas Mixture at the Hydrodynamic Scale
334
/. Mutter Considerations about the Gibbs Paradox
341
XVII
G. Mulone Nonlinear Stability in Fluid-Dynamics in the Presence of Stabilizing Effects and the Choice of a Measure of Perturbations
352
0. Muscato An Extended Hydrodynamic Model for a Bipolar Junction Transistor 366 F. Oliveri, G. Baumann A Completely Exceptional Equation Uniquely Characterized by Lie Symmetries
372
M. Pandolfi Bianchi, A. J. Soares Kinetic Approach to Transport Properties of a Reacting Gas
380
S. Pennisi, M. C. Carrisi On the Exact Macroscopic Approch to Extended Thermodynamics with 20 Moments
386
S. Pennisi, A. Scanu Judicious Interpretation of the Conditions Present in Extended Thermodynamics
393
J. Polewczak, G. Stell Transport Coefficients in Stochastic Models of the Revised Enskog and Square-Well Kinetic Theories
400
A. Pulvirenti, G. Toscani Probabilistic Treatment of Some Dissipative Kinetic Models
407
S. Rionero Asymptotic Properties of Solutions to Nonlinear Possibly Degenerated Parabolic Equations in Unbounded Domains
421
V. Romano, A. Valenti On Group Analysis of a Class of Energy-Transport Models of Semiconductors in the Two Dimensional Stationary Case
434
XVIII
T. Ruggeri Some Recent Mathematical Results in Mixtures Theory of Euler Fluids
441
T. Ruggeri, S. Simic Non Linear Wave Propagation in Binary Mixtures of Euler Fluids
455
T. Ruggeri, M. Trovato Hyperbolicity Region for Fermi and Bose Gases
463
F. Rundo A Cellular Neural Network Applied to the Equations of Mathematical Physics
470
F. Salvarani, J. L. Vazquez From Kinetic Systems to Diffusion Equations
475
M. Sammartino Asymptotic Methods in Option Pricing
482
M. Sammartino, V. Sciacca Approximate Inertial Manifolds for Thermodiffusion Equations
494
M. P. Speciale, F. Oliveri Linear Stability of Some Exact Solutions to Ideal Magneto-Gas-Dynamics Equations
500
B. Straughan Non-Boussinesq Convection in Porous Media
507
M. Sugiyama A New Continuum Model of Solids Incorporating Microscopic Thermal Vibration and its Application to Wave Propagation Phenomena
512
K. Thailert, S. V. Meleshko On Partially Invariant Solutions of the Navier-Stokes Equations
524
XIX
R. Tonelli, G. Cappellini, F. Meloni, S. Trillo Coupled-Mode versus Nonlinear Schrodinger Equations for Electromagnetic Wave Propagation in Continuous Media
535
M. Torrisi, R. Tracind, A. Volenti A New Class of Linearizable Wave Equations
541
G. Volenti, C. Currb, M. Sugiyama Wave Features for a New Continuum Model of Isotropic Solids
547
C. van der Mee Stationary Kinetic Equations with Collision Terms Relatively Bounded with Respect to the Collision Frequency
555
G. Zappald On Relative Asymptotic Stability with Two Measures via Limiting Equations
564
MULTI-PHYSICS MODELS IN ELECTRIC N E T W O R K D E S I G N
G. A L I Istituto per le Applicazioni del Calcolo "M. Picone" Consiglio Nazionale delle Ricerche, v. Pietro Castellino 111 1-80131 Naples, Italy E-mail:
[email protected] A.
BARTEL
Fachbereich Mathematik, AG Angewandte Mathematik/Numerik Bergische Universitat Wuppertal, Gausstr. 20 D-42119 Wuppertal, Germany E-mail: A ndreas.
[email protected]. uni-wuppertal. de
We consider a linear RLC network which contains a semiconductor device. In a multi-physics approach, the network is modeled by Modified Nodal Analysis, while a detailed drift-diffusion model is used for the semiconductor device. We discuss the coupling of the two models and address briefly the well-posedness of the resulting partial differential-algebraic equations, both for steady and transient drift-diffusion equations.
1. Introduction: electric network equations A linear RLC electric network is a circuit composed of linear capacitors, inductors and resistors. In classical Modified Nodal Analysis (MNA) 6 , the evolution of such a network is described by the node potentials u(t) £ Rn, and by the currents jL(t) £ RnL and jv(t) £ Rnv through inductors and voltage sources, respectively. If the network contains a semiconductor device with two Ohmic contacts, this set of unknowns is supplemented by the currents j D £ R 2 at the contacts of the device (as inflow). The topology of the network is described by means of incidence matrices: Ac £ r x " c , AL £ R n x " i and AR £ R n x n c , which describe the branch-node relationship for capacitors, inductors and resistors; Ay £ WLXnv and Ax £ R " x n i , which describe the branch-node relationship for the independent voltage and current sources, v £ M.nv and % £ Rnx; AD £ E n x 2 , which
1
2
matches the boundaries of the device with the corresponding network nodes. Using Kirkhhoff's current law and the constitutive relations for the electric components, the network equation for x = (u,jL, j v ) T can be written as /AcCAl
fARGArRALAv\
0 0\
(AD3D\
(AX%\
+
(i vM-i
+
m
::r (:) (:r
where the capacitance, inductance and conductance matrices C G K r i c X r a c , L e M.nL*nL and G e R nGXn satisfy the system 3 A
P
| + BPPy
+ BPQz
+ (
BQPy
+ BQQz
+
P
^
J
)
QTCA^I\ ^ ~ ^
+ (
P
c ^ )
= 0,
(5a)
, fQrcAn + r c Z
= 0.
(5b)
3
These equations can be obtained by left-multiplying the first line of (1) by Pc and Qc, and combining the resulting equations with the second and the third line of (1), respectively. By definition, the linear map defined by the matrix Ap is invertible when restricted to ( P c K n ) x R™L- Then, system (5) is a DAE of differential index-1 if (5b) can be uniquely solved for z. The simplest index-1 conditions require that I(Au) does not depend on z, and that the matrix BQQ is invertible in (QcRn) x E™L. Recalling (4), the first condition is equivalent to 3 AlQc The invertibility of BQQ
=
(6a)
(QIARGARQC
\
the conditions 4,7
= 0. QlAv\ ^ ig 0 J
AVQC
kGi{Ac,AR,Av)T
equivalent
= {0},
to
(6b)
kerQ"£.Av = {0}.
(6c)
The aim of this note is to show how the above theory is modified when a detailed modeling of the semiconductor device is taken into account. 2. Coupling of network and semiconductor equations To begin the study of the new nonlinear effects arising from the coupling of electric circuits and semiconductor devices, we consider a simple onedimensional device of length /. Using non-dimensional variables, this device is modeled by the scaled, transient, drift-diffusion system 5 : 9n
-, and a characteristic time associated to the current through the circuit, i = CR. Typically, we have ip i = UT logn^). The potential urj in (9a) is the external electric potential applied to the device, which is given by
C$3)-^
0, l e (0,1).
(16)
We approximate (15) by means of an suitable discretization procedure. For any integer M, we introduce the space step AX — 1/M, and the variables vi{T) = MT)
v{(i-\)AX,T)1
=
) is a solution of (15), the functions (i>i:Ui>,fa)are approximated by the solution of the following system of ordinary differential equations: — Vi
Ui-Ui-i
dT duj/
dT
AX
+
UT(e
q
^
AX
\K
= -E,
~
l
Jv
*e
e
AX
where Uo(T) = UM(T) = 0, and Ei = - e " 1 ^ 1
= 0, i-e—'i')
AX AX
.
"i' +
'
(17) AX
J
12 To prove existence of periodic solutions of this discretized system, we rewrite it as dT AX ~~ "*' fry _ CM-V-n-^,') dT AX q
1
AX
—-
JV
y_ /ly+i-iy _ »i'-"i'-i\ _ AX \ AX AX J ~ »* ' *
e
n a^ i0
l
J
'
7
It is possible to prove t h a t , for assigned periodic functions hi, g^, the above linear system admits a unique periodic solution with same period if a n d only if Yli=i Jo hi(s)ds = 0, and t h a t this condition is satisfied. Then, we can build an iterative m a p based on t h e linearized system (18). T h e existence of a fixed point of this m a p , which can be achieved by a Leray-Schauder theorem, implies t h e existence of a periodic solution for t h e discretized system (17). Finally, passing t o t h e limit as M —> oo, t h r o u g h uniform a priori estimates, we can ascertain t h e existence of periodic solutions for t h e continuous system (15).
Acknowledgments T h e Authors t h a n k gratefully Prof. Salvatore Rionero for his helpful suggestions and for his continuous encouragement and support.
References 1. G. All, D. Bini and S. Rionero, SIAM J. Math. Anal. 32, 572 (2000). 2. G.-Q. Chen, J. Jerome, J. W. Shu and D. Wang, in: J. Jerome (ed.), Modelling and Computation for Application in Mathematics, Science and Engineering, Clarendon Press, Oxford, 103 (1998). 3. A. M. Anile, S. D. Hern, VLSI Design 15 (2002). 4. J. B. Gunn, Solid State Commun. 1, 88 (1963). 5. J. B. Gunn, Procs. Symposium on Plasma Effects in Solids, Dunod, Paris, 199-207 (1965). 6. J. B. Gunn, J. Phys. Soc. Japan (Supp.) 2 1 , 505 (1966). 7. A. Matsumura and T. Nishida, Lecture Notes in Num. Appl. Anal. 10, 49 (1989).
A T W O P O P U L A T I O N MODEL FOR ELECTRON T R A N S P O R T I N SI
A. M. ANILE Dipartimento
di Matematica ed Informatica, Universita di Catania, Viale A. Doria 6- 95125 Catania, Italy E-mail:
[email protected] G. M A S C A L I Dipartimento di Matematica, Universita della Calabria, Via Ponte Bucci, cubo SOB, 87036-Arcavacata di Rende(Cs), Italy E-mail:
[email protected] In this work we present a fluid dynamical model for electron transport in silicon which takes direct account of highly energetic electrons by introducing macroscopic quantities averaged over the tail electron population. The model is based on the maximum entropy principle and is free of any fitting parameter.
1. Introduction Describing the functioning of modern electron devices requires increasingly accurate physical models of carrier transport in semiconductors in order to deal with high-field phenomena such as impact ionization, thermal selfheating, etc. Hot electron phenomena are of particular interest for the accurate evaluation of the degradation and breakdown of devices. Current calculations using Monte Carlo methods (MC) are extremely CPU intensive and therefore not practical for routine design applications. On the other hand traditional hydrodynamical models for carrier transport in semiconductors cannot describe hot electrons since they deal only with average values over the whole carrier population. Several considerations, x , support the existence of two thermal distributions at different temperatures for electrons having energies respectively lower and higher than a suitable threshold energy, the so-called cold and
13
14
hot electrons. This is the reason why several authors, 2 ' 3 have introduced new fluid dynamical models in which two well-defined subpopulations of electrons are considered, each subpopulation being described by the respective macroscopic quantities. Here we present a two population model which is obtained by utilizing the moment method and a closure technique by which one can obtain both the constitutive fluxes and the production terms, appearing in the moment equations, as functions of the fundamental hydrodynamical variables, without resorting to MC. All the closure relations are shown, the model is tested by applications to bulk Si and the results are compared with those obtained by an usual hydrodynamical model. 2. The Boltzmann equation for semiconductors We treat the case of silicon unipolar devices for which the electrons contributing to charge transport are those in the six equivalent valleys around the six minima of the conduction band. We assume that, for those electrons, the relation between the energy, £, and the quasi-wave vector, k, both measured from the bottom of the conduction band, is given by the Kane dispersion relation ?72lkl2
£(k)[l + a£(k)} = -±±-,
keS3,
(1)
which involves a parameter a, called the non-parabolicity factor, while m* is the electron effective mass. At a kinetic level, electrons in a semiconductor are described by a oneparticle distribution function, / ( x , t, k), whose evolution is governed by the semiclassical Boltzmann equation coupled to the Poisson equation for the electric field E
V x • (eE) = q \N+(x) - AL(x) - n(x)] , here, q represents the absolute value of the electron charge, h the reduced Planck constant, e the dielectric constant, N+ and AL the donor and acceptor concentrations respectively, and n the total electron number density, v, the electron group velocity, depends on the energy £ through the relation v(k) = ^ V k 5 ( k ) .
15 C[f] is the collision term, which reflects the various scattering mechanisms the electrons undergo in a semiconductor. In the non-degenerate case, its form is
c[f]= f
Hk',k) /(k') - «,(k, k')/(k)] dk',
where w(k, k') represents the sum of the various electron scattering rates from a state with wave vector k to one with wave vector k'. We will take into account the following scattering mechanisms for silicon • electron - acoustical phonon intravalley scattering, for which the transition rate, in its elastic approximation, reads wac(k,k')=!Cac5(£"-£), with /Cac acoustical intravalley scattering kernel coefficient, • electron - phonon intervalley scattering, for which there are six contributions tu a (k,k') = Ka \na S(£' -£-£a)
+ (na + 1)S{£' -£ + Sa)} ,
where a runs over the three gi,g2,#3 and the three / i , fi, f-z intervalley scatterings, K.a are the corrispondent optical or acoustical intervalley scattering kernel coefficients and na is the occupation number of phonons with energy £a, • electron-impurity scattering, which is an elastic mechanism of interaction whose transition rate reads w p(k k,)=
-
'
[[k-kT + / 3 2 ] 2 ^ / - g ) '
where K.imp is a physical parameter, and /3 is the inverse Debye length.
3. Moment equations Introducing a threshold energy , £, for the electrons, which we take to be equal the energy gap in Si, one can consider electrons as consisting of two subpopulations: electrons having energy less than and greater than £. Considering the kinetic quantities: l,v,£, £} and A c = 5ft3 — A # in the k-space. Henceforth the subscripts H and C will indicate quantities referring to hot and cold electrons respectively. Multiplying the Boltzmann equation by the kinetic quantities 1, hk, £, £ v and integrating over AH, one can obtain the moment equations describing the behaviour of hot electrons. Subtracting these from the usual moment equations, one finds the corresponding equations for cold electrons i
qEiN at """ dxi 9n„P'H , dn„ U% -qEjT'i dxi at at . "+ +
at
dxx
= CnH, +qEinH=C•Ph
H '
- qSE'ATi + qEinHVH
= CWH,
(5)
d
-^+qEjnHGiH-q£E]ri=CSh,
i dn. no-qEiX , dt "•" *«L%--C dx dn dngPh , c Ug i ++ "-"^^ +qE nc dt dxi
^ % P + ^§^
+qEincV£
= C* -qEj T « , = CWc -
qE&Mi,
(6)
where summation over repeated lowercase letters is understood and P\ = — / hk% JAdk = m* (VX + 2aS^) is the average crystal momentum, n A JAA i l J the average crystal momentum flux, ™AJAA v hk /yidk
n
ij = ±_ f A
r
A
UAJA
d (\:vl£(k) dki \ h
1
i i£(k)fAdk
=
nA
f
/ VV JAA
fAdk, the average flux of energy flux,
17
CnA = /
C[f}dk
the density production,
(7)
JAA
CPi = I
hklC[f\dk
CWA = /
£(k)C[f]dk
the crystal momentum production, the energy production,
JAA
CSi = /
vl£(k)C[f]dk
the energy flux production,
all these quantities referring to electrons in zone A, with A = H,C. The surface terms Mi = \ J
fH Sdcr
Tij = J klfH vjda,
(8)
with £ = {k : £(k) = £}, and vl inner normal to E, represent the increasing rate of the corresponding macroscopic quantities due to the net migration of carriers from one energy zone to the other owing to the driving electric field. In equations (5), (6), in addition to the fundamental variables (3), (4), the extra-unknowns (7), (8) appear, so that, in order to have a closed system of equations, it is necessary to express the latter variables in terms of the former ones. A way to get constitutive relations, which lies on sound physical bases, is to use the maximum entropy principle 4 ' 5 . This principle furnishes the form of the distribution functions that make the best use of the knowledge of a finite number of moments. They, as well known, are given by 6
/JT=exp
±\A
+ Aj £A + AX* V\ + Xs; vA £;
H,C,
where the A's are Lagrange multipliers that take care of the constraints (3), (4) and KB is the Boltzmann constant. To determine the Lagrange multipliers in terms of nAl V ^ , WA, SA, A = H, C, one has to insert the expressions of the maximum-entropy-distribution functions into (3)-(4) and solve the resulting system. Then the closure relations can be obtained by evaluating (7), (8), with fA, A = H, L, replaced by the corresponding maximum entropy functions. However, on account of the algebraic difficulties, we can get only approximate explicit expressions for the Lagrange multipliers under reasonable physical assumptions on the distribution functions. On the basis of Monte Carlo results, we assume that the anisotropy of the
18
/ A , A = H ,C remains small even out of equilibrium. We formally introduce a small anisotropy parameter 5, assume that the Lagrange multipliers are analytic in 5 and expand the maximum entropy distributions, obtaining tfE
= exp(-^-\%S>)[l-6(\Kvi
+ \Stvi£)],
A =
H,C.(9)
4. Inversion of the constraint relations In order to express the Lagrange multipliers in terms of the fundamental moments, we have to invert the system of equations (3), (4) where JH and fc are substituted by the maximum entropy distributions. By retaining only the terms up to the first order in S a , we get \Av=g-A1(WA),
A^ = - f c B l o g ( f % — A \ 4 7rm* \Z2m*a,Q J *VA = Ki VX + b?2 SA,
(10)
XSA = bf2 V\ + # 2 SA,
where d£(\Av)=[
£k^£(l
+ a£)(l
+ 2a £) e x p ( - A ^
with A£H = (£, +oo) and A£c = (0,£). gA ^(Ajf) = | ^ ! j ,
£)d£,
are the inverse functions of A = H,C,
(11)
and bfj, A = H ,C, are the coefficients of the matrices
-
HP?)2-P$PU
^"3
mAd-
P$ -P?
l-pf^J'
(12)
with pi = JAA £k \%+2aa££r2 e x p ( - A y £)d£, A = H,C. For the inversion of the functions (11) we have resorted to a numerical approach. The results are shown in Fig. l tt . 5. Fluxes and surface terms. Once the Lagrangian multipliers are expressed as functions of the fundamental variables, the constitutive equations for the fluxes can be obtained. a
VA and SA, A = H , C, are consistently considered as terms of order 5.
19 Up to the first order terms, one has 6ij
A
&{
{£(l+a£)]3/2exp{-X^£)d£,
f
V 2
JASA JAeA
P?
3 m*
d^
5
