Maximum entropy principle for hydrodynamic transport in semiconductor devices

Maximum entropy principle for hydrodynamic transport in semiconductor devicesM. Trovato and L. Reggiani Citation: Journal of Applied Physics 85, 4050 (1999); doi: 10.1063/1.370310 View online: http://dx.doi.org/10.1063/1.370310 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/85/8?ver=pdfcov Published by the AIP Publishing

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JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999

[This a

Maximum entropy principle for hydrodynamic transportin semiconductor devices

M. TrovatoDipartimento di Matematica, Universita` di Catania, Viale A. Doria 6, 95125 Catania, Italy

L. Reggiania)

Dipartimento di Scienza dei Materiali ed Istituto Nazionale di Fisica della Materia, Universita` di Lecce,Via Arnesano, 73100 Lecce, Italy

~Received 28 October 1998; accepted 14 January 1999!

A hydrodynamic ~HD! transport approach based on a closed system of balance equations isdeveloped from the maximum entropy principle. By considering a nonlinear expansion with respectto a local thermodynamic equilibrium, we determine an analytic expression for the distributionfunction as a function of macroscopic quantities such as density, velocity, energy, deviatoric stress,heat flux associated with charge carriers. From the determined distribution function and consideringthe collision interactions of carriers with phonons, all the constitutive functions appearing in thefluxes and collisional productions of the balance equations are explicitly calculated. The analyticalclosure so obtained is applied to the case of somen1nn1 submicron Si structures. Numerical HDcalculations are found to compare well with those obtained by an ensemble Monte Carlo simulatorthus validating the approach developed here. ©1999 American Institute of Physics.@S0021-8979~99!04208-5#

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I. INTRODUCTION

The development of hydrodynamic~HD! approaches forthe study of electronic devices is of great interest from ban applied and fundamental point of view.1–12 The advan-tages of this approach are the flexibility in applying to dferent operation conditions, the direct applicability to coputer aided design~CAD! simulators, and the affordablcomputational environment. The drawbacks are mainlythe less accurate microscopic description when compawith approaches at a kinetic level based on the solutionthe Boltzmann transport equation~BTE!, mostly through en-semble Monte Carlo~MC! simulators.13–15 In particular, theuse of HD approaches poses some problems, the main btheclosureof the hierarchy of equations of evolution for thmoments.8,16 This closure is usually obtained either by asuming that moments of high order~present in the constitutive functions! can be calculated by means of a drifteMaxwellian,5,6 or by introducing phenomenological constittive functions for the fluxes2,17,18 and the collisionalproductions2,7,12,19–22containing free parameters to be detemined accordingly. The presence of free parameters haways been a limit to pratical use of these models, becagenerally, these parameters are not constant but shouldetermined each time on the basis of either MC simulatior experimental data. At present, the most appropriate theto study nonequilibrium phenomena using the BTE momeis the extended thermodynamictheory.23 It provides a sys-tematic method to obtain theconstitutive equationspresentin the hierarchy of moments following two approach

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4050021-8979/99/85(8)/4050/16/$15.00

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

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~which, under suitable hypotheses, are proved toequivalent!:24 the maximum entropy principle~MEP! and theentropy principle~EP!.

The MEP allows one to derive23–33 the nonequilibriumdistribution function, associated with charge carriers, anddetermine the microstate corresponding to the given mascopic quantity. In this case the microscopic state is obtaifrom the solution of the variational problem of maximizinthe entropy of the system under the constraints correspoing to the value of some mean quantities which definemacroscopic state. Once the distribution function is knowall the unknown constitutive functions are obtained by ingrating ink space their kinetic expression.

The EP is a phenomenological method23 based on theassumption that the production of the entropy of the sysis non-negative. However, on the basis of EP it is possiblederive only the constitutive functions present in the fluxesthe considered moments, but not the terms of collisional pduction which are modeled alternatively using relaxatitimes7 derived from fitting the MC data. We note, howevethat for the constitutive functions obtained with both meods, the dependence on the moments of the distribution fution is of local-type and the evolution equations, for suitabvalues of the independent variables, determine a quasilinhyperbolic system.

The aim of this article is to present alocal and dynamicapplication of the MEP for a HD description of transpoproperties in semiconductor devices which avoids the closproblem. Previous studies on the application of MEP foone dimensional geometry have been reported in Refs.31, and 34~and references therein!. Here we extend thetheory to a full three-dimensional geometry and the theoical procedure for the local dynamical application of MEP

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4051J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 M. Trovato and L. Reggiani

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explicitely given. The novelty of the present approachtwofold.

First, use is made of a microscopic approach basedMEP which allows for a determination of an analytical epression for the distribution function. As a consequence,the unknown constitutive functions~fluxes and collisionalproductions! are calculated starting from their kinetic expresions. Accordingly, the only input data are those containethe collisional kernels~deformation potentials, phonon enegies, lattice constants, etc.! as for the kinetic approach. Second, the theoretical expressions of the constitutive functiso determined are validated for the case ofn1nn1 submi-cron structures by direct comparison with MC simulationThe theoretical approach avoids any symmetrical simplifition thus accounting for full three-dimensional real and mmentum spaces. Furthermore, as by-product of the theoryprovide a generalization of the Fourier and Navier-Stolaws of standard irreversible thermodynamics appropriathigh field transport.

The content of the article is organized as follows. In SII the problem is formulated starting from the BTE equatioIn Sec. III the application of the MEP is developed. In SeIV and V the specific case of the first 13 moments of tBTE is analyzed, the concepts of thermodynamic equirium is discussed and the explicit analytic expression fordistribution function is obtained. In Sec. VI all the unknowconstitutive functions~fluxes and collisional productions!present in the hierarchy of the evolution equations forfirst 13 moments are analytically determined. In Sec. VII,using a procedure analogous to theMaxwellian iteration, theusual HD models based on linear constitutive relationson small gradient approximations are obtained. In Sec. Vthe theory is validated by application ton1nn1 submicronSi structures. In particular, the closure condition we haformulated is verified via MC simulations of the same strutures.

Throughout this article the tensor index notation is usthus for a generic tensorA of rank n, Ai 1 . . . i n

denotes itscomponents,A( i 1 . . . i n) the symmetric part,A^ i 1 . . . i n& thetraceless symmetric part. Overall, we shall have 2n11 inde-pendent components and the constraints:

A^ i 1 . . . k . . . l . . . i n&

5A^ i 1 . . . l . . . k . . . i n& , A^ i 1 . . . k . . . k . . . i n&50.

II. FORMULATION OF THE PROBLEM

The starting point is the BTE written for the nondegeerate single particle in the conduction band as:

]F~k,r ,t !

]t1

dxi

dt

]F~k,r ,t !

]xi1

dki

dt

]F~k,r ,t !

]ki5Q~F!

~1!

with Q(F) the collision integral associated with electronphonon interaction and defined as:


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Q~F!5V

~2p!3F E dk8S~k,k8!F~k8,r ,t !

2E dk8S~k8,k!F~k,r ,t !G ~2!

V being the crystal volume andS(k,k8) the total collisionrate from statek8 to statek.

To pass from the kinetic level of the BTE to the Hlevel of the balance equations within a simple sphericparabolic, many-valley band model the following kinetfields should be considered:

cA~k!5H m* ,\ki 1,

\2

m*ki 1

ki 2, . . . ,

\n

~m* !n21ki 1

ki 2. . . ki n

, . . . J . ~3!

Multiplying Eq. ~1! by cA(k), integrating over thek spacewe obtain the set of balance equations formoments of thedistribution function:

]Fi 1 . . . i n

]t1

]Fi 1 . . . i nk

]xk52n

e

m*F ~ i 1 . . . i n21

Ei n

1Pi 1 . . . i n, ~4!

being

Fi 1 . . . i n5

\n

~m* !n21E ki 1ki 2

. . . ki nF~k,r ,t !dk, ~5!

Fi 1 . . . i nk5\n11

~m* !nE ki 1ki 2

. . . ki nkkF~k,r ,t !dk, ~6!

Pi 1 . . . i n5

\n

~m* !n21E ki 1ki 2

. . . ki nQ~F!dk, ~7!

respectively, the moments, the fluxes, and the collisional pductions. The set of Eq. 4 represents an infinite systemcoupled partial differential equations in the unknownFwhich should be coupled with the Poisson equation. Forealistic solution of a HD model we should assume thacertain fixed number N of moments FA ~with A51, . . . ,N) is sufficient to satisfactorily describe the themodynamical state of the physical system under considation,N being a value to be determined in such a way tothe theory appropriate for the description of the transpphenomena considered. Although we have a set of balaequations of finite order, the firstN equations contain unknown constitutive functionsHA5$FAk ,PA% represented bythe fluxesFAk and the collisional productionsPA . The sys-tem so described should be closed in a self-consistentwith the determination of theHA expressed by means of thFA . This problem of closure and solution of the set of bance equations can be tackled in general using two kindstrategies.

In a first one, use is made of some phenomenologdefinition of appropriate kinetic coefficients introduced


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4052 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 M. Trovato and L. Reggiani

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close the system of equations. For example, by introducsome average parameter obtained from MC data35,36 ~e.g.,average effective mass, momentum and energy relaxatimes, variance of velocity fluctuations, etc.!, or defining amore or less generalized Wiedmann-Franz law for the carthermal conductivity,2,8,17,18these values are introduced inthe balance equations which are numerically solved foraverage quantities.

Alternatively this problem can be solved with the helpinformation theory,10,37as introduced in the pioneer paperJaynes.25 This theory allows one to determine the closusearched for theconstitutive functionsself-consistently byusing the MEP. In this way there is no need of determinkinetic coefficients by other means like MC simulations. Tnext section will elaborate this alternative strategy in det

III. SOLUTION OF THE PROBLEM WITH THEMAXIMUM ENTROPY PRINCIPLE

For a given set ofcA(k) in Eq. ~3! and the corresponding FA we search for a method to determine an approximtion for F(r ,k,t) based on the information provided by thfirst N values ofFA . Of course, for a givenN there existdifferent distribution functions whose firstN moments coin-cide and which are thus macroequivalent.34 The MEP offersa definite procedure to provide the approximationFN(r ,k,t)for the distribution functionF(r ,k,t).23–32 From the defini-tion of entropy density,h52C*F ln(F)dk ~C being a suit-able constant!, we search the distribution function that maxmizes h under the condition that the momentsFA beexpressed by means of relation~5!. Thus, we maximize thefunctional

h85h2 (A51

N

LAF E cA~k!F~r ,k,t !dk2FA~r ,t !G ~8!

being LA5LA(r ,t) the Lagrange multipliersto be deter-mined.

By imposing that dh850 the distribution functionsearched with this method23–32 is

FN5exp~2P! ~9!

with P5(A51N cALA . For a given number of momentsFA

we have to determine the distribution function in the specformF(r ,k,t)5F@FA(r ,t),k#. For the Lagrange multipliersthis implies thatLA(r ,t)5LA@FB(r ,t)#. Including Eq.~9! inthe definition of moments of Eq.~5!, solving the integralsand inverting them, we obtain the Lagrange multipliers afunction of theFA . Consequently, both the distribution funtion and the constitutive functions can be written in the eplicit form F5F(FB ,k),HA5HA(FB) by mean of expres-sions ~9!, ~6!, and ~7!. Generally, the inversion can bperformed using either an iterative procedure34,38 which israther complicate owing to the nonlinearity ofF with respectto the Lagrange multipliers and to the many quadraturesvolved, or a more affordable series expansion around slocal equilibrium configuration.23,37 In the following weadopt this second procedure and develop the series exsion by using the first 13 moments of the distribution funtion.


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Finally, we remark that, by using MEP within a dynamcal context we differ from previous applications.39–42Indeed,besides determining the analytical expression of the distrtion function from the moments it is also necessary to clothe set of evolution equations for these moments by demining the microscopic interactions through the scatterprobabilities entering the collisional operator of the BTOnly in this way a correct evolution of the moments andturn of the distribution function can be obtained.

IV. THE FIRST 13 MOMENTS OF THE BTE AND THEIRDECOMPOSITION IN CONVECTIVE AND NONCONVECTIVE PARTS

By considering the first 13 moments of the distributiofunction, ) takes the following explicit form:

P5L1\

m*kiL i1

\2

2m*k2L l l 1

\2

m*k^ ikiL^ i j &

1\3

2~m* !2k2kiL i l l . ~10!

The set of balance equation for theFA5$n,nv i , W,S^ i j & ,Si%with the physical meaning ofn ~numerical density!, nv i ~fluxdensity!, W ~total energy density!, S^ i j & ~traceless momentumflux density!, Si ~energy flux density! is:

]n

]t1

]nvk

]xk50, ~11!

]nv i

]t1

1

m*

]S ik

]xk52

nem*

Ei1Pi~nv !1 Pi

~nv ! , ~12!

]W

]t1

]Sk

]xk52nev lEl1P~w!, ~13!

]S^ i j &

]t1

]Q^ i j &k

]xk522nev ^ iEj1P^ i j &

~S!1 P^ i j &~S! , ~14!

]Si

]t1

]Q i l lk

]xk52

3

2

e

m*S~ i l El1Pi

~S!1 Pi~S! , ~15!

with

Q^ i j &k5Q^ i jk &145 S~ id jk2 4

15 Skd i j ,

Q^ i jk &5\3

~m* !2E k^ ikjkk&Fdk,

~16!

Q i l lk 5\4

2~m* !3E k2kikkFdk,

S i j 5S^ i j &123 Wd i j ,

and$PA ,PA% denoting the collisional productions associatwith the intravalley and intervalley transitions, respective

Due to the presence of different relaxation time scalbecause of the parabolic band model it is convenient tocompose both the moments$W,S^ i j & ,Si% and the fluxes$Q^ i j &k ,Q i l lk % into their convective~i.e., terms depending ex


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plicitly on velocity! and nonconvective parts, respectiveThis decomposition has the advantage of allowing the induction of a state equation23 and as a consequence of aelectron temperature conceptT when the time scale of energy relaxationtw is much longer than that of other momerelaxationtp , ts , andtS . For the case of hot carriers, thcondition tv@tp ,tS ,ts , is verified in mostsemiconductors.7,43–46 As a consequence, during the relaation towards thermal equilibrium condition there existsintermediate state in which the electron subsystem willcharacterized by alocal equilibrium state wherev i5S^ i j &5Si50, butTÞT0 . Furthermore, with this decomposition ithe procedure of inversion for the determination of tLagrange multipliers the number of equations to be solvesignificantly reduced. For more general dispersions~e.g.,when nonparabolicity is accounted for! one needs to considemoments and fluxes without any decomposition procedTo this purpose we introduce the mean velocity

v i51

nE \

m*kiF~r ,k,t !dk

and the microscopic random velocity\/m* Ki5(\/m* ki

2v i). We thus define thenonconvective momentsMA

5$n,0,«,s^ i j & ,qi% as:

MA5E CA~k!F~r ,k,t !dk, A51,2, . . . ,13

being «5(3/2)p the specific internal energy~with p thepressure!, s^ i j & thestress deviator, qi and theheat flux. Sinceit exists a transformation23 XAB(v) such that cA(k)5XAB(v)CB(k) and consequentelyFA5XABMB , the fol-lowing inter-relations hold: Since it exists a transformation23

XAB(v) such thatcA(k)5XAB(v)CB(k) and consequentlyFA5XABMB , the following inter-relations hold:

W5 32 p1 1

2 nm* v2, ~17!

S^ i j &5s^ i j &1nm* v ^ iv j & , ~18!

Si5qi1s^ i j &v j152 pv i1

12 nm* v2v i , ~19!

Q^ i jk &5G^ i jk &13s~^ i j &vk2 25 v l~s^ l i &d jk1s^ lk&d i j

1s^ l j &dki!1nm* v ^ iv jvk& , ~20!

Q i l lk 5Gillk 1v lG^ l ik &172 pv ivk1 14

5 q~ ivk)125 ~qlv l !d ik

1v l~s^ l i &vk1s^ lk&v i !1 12 v2s^ ik&1

12 pv2d ik

1 12 m* nv2v ivk ~21!

with

G^ i jk &5\3

~m* !2E K ^ iK jKk&F~r ,k,t !dk, ~22!

Gillk 5\4

2~m* !3E K2KiKkF~r ,k,t !dk. ~23!

In this way theFA of Eq. ~5! are replaced by the new set odynamical variablesmA5$n,v i ,p,s^ i j & ,qi% and the new


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constitutive functionsGA5$G^ i jk & ,Gillk % are introduced. Interms of the new kinetic quantitiesCA(K ), it is P5(A51

13 CA(K )lA with lA the intrinsic Lagrange multipli-ers. Likewise for the momentsFA one can decompose thmultipliers LA into their convective and nonconvective~i.e.,intrinsic! parts as:

L5l2l iv i112 m* v2l l l 1m* l^ i j &v ^ iv j &

2 12 m* v2l i l l v i , ~24!

L i5l i2m* l l l v i22m* l^ i j &v j112 m* lkll~v2dki

12vkv i !, ~25!

L l l 5l l l 253 ~lkllvk!, ~26!

L^ i j &5l^ i j &2v ^ il j & l l , ~27!

L i l l 5l i l l . ~28!

We note that, in the case of semiconductorsv is the relativevelocity of carriers with respect to the lattice and thus,principle, bothGA andh could depend onv without breakingthe material objectivity principle.23 However, it has beenproven that the entropy associated with an entire systembe decomposed into the sum of the entropies associatedsingle subsystems,47,48 electrons and phonons in the presecase, and that the entropy of a single subsystem is indedent of the relative velocityv only depending upon the dynamical variables$n,p,s^ i j & ,qi%. Therefore, one directlyproves24,47,48that bothlA and$G^ i jk & ,Gillk % depend only onvariablesMA5$n,«,s^ i j & ,qi%. To determine an analyticaexpression of the distribution function, through the relatio~24!–~28!, it is sufficient to obtain an explicit representatioof the lA in terms of theMA . To this aim, we consider anexpansion of the distribution function~9!, around the localequilibrium configuration,10,23,37,48in terms of the nonequi-librium variables$v i ,s^ i j & ,qi%. The coefficients of this ex-pansion will depend on the local equilibrium variablesn(r ,t)andT(r ,t).

V. DETERMINATION OF THE NONEQUILIBRIUMDISTRIBUTION FUNCTION

In the local equilibrium configuration the distributiofunction is the Maxwellian:

F uE5FM5nS \2

2pm* KBTD 3/2

expS 2«~k!

KBT D . ~29!

By using the usual definitions for$n,p%,

n~r ,t !5E F uE dk, ~30!

p~r ,t !5\2

3m*E k2F uE dk5nKBT~r ,t !, ~31!

under conditions of local equilibrium it is:

LuE5luE52 lnFnS \2

2pm* KBTD 3/2G , ~32!


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L l l uE5l l l uE51

KBT, ~33!

L i uE5l i uE50, L^ i j &uE5l^ i j &uE50,~34!

L i l l uE5l i l l uE50.

By denoting with $LA ,lA% the nonequilibrium parts o$LA ,lA% and expanding Eq.~9! around the MaxwellianFM

up to third order, we obtain by means of the moments of~5! a set of nonlinear equations in terms of variablesLA

FA2FAuE52 (B51

13

LBE cA~k!cB~k!FMdk

11

2 (B51

13

(C51

13

LBLCE cA~k!cB~k!cC~k!FMdk

21

6 (B51

13

(C51

13

(D51

13

LBLCLD

3E cA~k!cB~k!cC~k!cD~k!FMdk. ~35!

To determine the dependence of the quantitiesLA on themomentsmA5$n,p,v i ,s^ i j & ,qi%, we use the relations~24!–~28!, ~32!–~34! and express thelA by means of the representation theorems for isotropic functions.49 In this way, weexpand to the third order and obtain:

l5d1qrqr1d2s^pq&s^qp&1d3s^pq&s^qr&s^rp&

1d4qrs^rs&qs , ~36!

l i5g1qi1g2s^ i j &qj1g3s^ ir &s^rs&qs1g4~s^rs&s^rs&!qi

1g5~qrqr !qi , ~37!

l l l 5s1qrqr1s2s^pq&s^qp&1s3s^pq&s^qr&s^rp&

1s4qrs^rs&qs , ~38!

l^ i j &5b1s^ i j &1b2q^ iqj &1b3@s^ ir &s^r j &

2 13 ~s^pq&s^qp&!d i j #1b4@ 1

2 ~qis^ j r &1qjs^ ir &!qr

2 13 ~qrs^rs&qs!d i j #1b5~qrqr !s^ i j &

1b6~s^rs&s^rs&!s^ i j & , ~39!

l i l l 5a1qi1a2s^ i j &qj1a3s^ ir &s^rs&qs

1a4~s^rs&s^rs&!qi1a5~qrqr !qi , ~40!

where all coefficients have to be determined as functionthe $n,p%.

By introducing the relations~24!–~28!, ~32!–~34!, and~36!–~40! in the system in Eq.~35! and considering only thecubic terms in the nonequilibrium variables$v i ,s^ i j & ,qi%,we obtain a set of 24 equations48 expressed in terms of the 2unknowns $d1 ,d2 ,d3 ,d4 ,s1 ,s2 ,s3 ,s4 ,a1 ,a2 ,a3 ,a4 ,a5 ,g1 ,g2 ,g3 ,g4 ,g5 ,b1 ,b2 ,b3 ,b4 ,b5 ,b6%. The solution ofthis system is


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g15nm*

p2, g252

7

5

nm*

p3, g35

33

25

nm*

p4,

g4521

5

nm*

p4, g552

148

125

n2~m* !2

p5,

a1522

5

n2m*

p3, a25

18

25

n2m*

p4, a352

102

125

n2m*

p5,

a452

25

n2m*

p5, a55

296

625

~m* !2n3

p6,

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n

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9

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p4, b35

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p3,

b452102

125

n2m*

p5, b55

2

25

n2m*

p5, b652

1

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p4,

d1522

5

nm*

p3d252

1

4

1

p2, d35

1

6

1

p3, d45

18

25

m* n

p4,

s152

5

m* n2

p4, s25

1

3

n

p3, s352

1

3

n

p4, s452

24

25

m* n2

p5.

In this way, having determined thelA , we obtain throughEqs. ~24!–~28!, ~32!–~34!, and ~36!–~40! both theLA andthe analytical expression of the nonequilibrium distributifunction, estimated up to the third order in the variab$v i ,s^ i j & ,qi%. Similarly, we integrate the distribution function so obtained in thek space over the solid angledV anddeduce an analytic expression of the electron energy dibution ~see Appendix A! in the form f @«(k),mA(r ,t)# suchthat n5* f @«,mA(r ,t)#d«.

VI. DETERMINATION OF FLUXES AND COLLISIONALPRODUCTIONS

From the knowledge of the distribution function, thconstitutive functions $GA ,PA,PA%, can be determinedthrough their kinetic expressions. From the integrals~22! and~23!, for theGA it is:

Gillk 5H 5

2

p2

nm*d ik1

7

2

p

m* ns^ ik&J

1H 18

25

1

p~qrqr !d ik1

56

25

1

pqiqk1

1

m* ns^ ir &s^rk&J

1H 2174

125

1

p2~qrs^rs&qs!d ik1

48

25

1

p2~qrqr !s^ ik&

146

125

1

p2~qis^kr&qr1qks^ ir &qr !J , ~41!


7 Dec 2013 22:23:00

f

in

,

-ithse

-ins

ysons,

iesons

r-eydy-fal

n-

n

gen-ore

ionno-esall

la-


[This a

G^ i jk &512

5

1

pq^ is^ jk&&1H 136

125

m* n

p3q^ iqjqk&

12

5

1

p2[ ~qis^ j r &s^rk&1qks^ ir &s^r j &

1qjs^kr&s^ri &!2 35 ~s^rs&s^sr&!q~ id jk)]

216

25

1

p2@qrs^ri &s^ jk&1qrs^rk&s^ i j &1qrs^r j &s^ki&#

112

125

1

p2@qrs^rs&~s^si&d jk1s^sk&d i j 1s^s j&dki!#J .

~42!

Similarly, to evaluate the collisional production in Eq.~7!~associated with a generic bandn among the number oequivalent bands considered! we introduce the collision ratefor acoustic intravalley and intervalley scattering~withacoustic and non polar optical modes!.

For intravalley transitions with acoustic modes withthe elastic and equipartition approximations it is:13

Sac~k,k8!52pEl

2KBT0

\VrUl2

d@«~k8!2«~k!#, ~43!

whereEl is the acoustic deformation-potential parameterrthe crystal density, andUl the longitudinal sound velocity.

For intervalley transitions it is:

Sh~k,k8!5pDh

2

VrvhFNh1

1

26

1

2G3d$«~k8!2@«~k!6\vh#%, ~44!

where Dh is the intervalley deformation potential,vh thephonon angular frequency,Nh the phonon occupation number here taken as the equilibrium Planck distribution, wthe 6 signs refering to emission and absorption procesrespectively.

By inserting the Eqs.~43! and~44! and using the expansion of the distribution function previously determinedrelations ~2! and ~7!, after performing many tediouintegrals,48 we find for acoustic intravalley transitions

Pi~nv!52 4

3 jnA i , P~w!50, ~45!

P^ i j &~S!52 16

5 jpC^ i j & , Pi~S!524jpDi , ~46!

with j5 El2KBT0 /\4rUl

2 @(2m* /p)(p/n)#3/2, and for inter-valley transitions:

Pi~nv!5

4

3(h Ah(r 51

5

A i2r 11@~Nh11!H2r 11

1 1NhH2r 112 #,

~47!

P~w!5(hAhXh(

r 50

4

B2r 11@~Nh11!H2r 111

2NhH2r 112 #, ~48!


195.19.233.81 On: Sat, 0

s,

P^ i j &~S!5

4

15(h Ah(r 52

5

C^ i j &2r 11@~Nh11!H2r 11

1

1NhH2r 112 #, ~49!

Pi~S!5

4

3(h Ah(r 52

6

Di2r 11@~Nh11!H2r 11

1 1NhH2r 112 #,

~50!

with

Ah52n

\3S m* n

2ppD 3/2 Dh2

rvhZhXh , Xh5

\vh

2

n

p

whereZh is the number of possible final equivalent valleand all the coefficients, present in the previous expressi

$A i ,C^ i j & ,Di%, $A i2r 11 ,B2r 11,C^ i j &

2r 11 ,Di2r 11% are compli-

cated functions of the momentsmA5$n,p,v i ,s^ i j & ,qi% andare reported in Appendix A. The dimensionless quantitH2r 11

6 are expressed through the modified Bessel functiof the second kindK1 andK2 , and, by defining

G65Xh exp~7Xh!K2~Xh!, H165exp~7Xh!K1~Xh!

it is

H3656XhH1

61G6

and in general

H2r 116 562XhH2r 21

6 6~r 11!! F (n51

r 22 3XhH2n116

~n13!!6

G6

2 G~51!

with r>2.Collisions with intravalley acoustic phonon are impo

tant for momentum relaxation but, being nearly elastic, thlead the carriers energy to relax more slowly than othernamical variables$v i ,s^ i j & ,qi%. Indeed, from the analysis othe coefficients reported in Appendix A, we note that in locthermodynamic equilibrium all collisional productions vaish exceptP(w) which takes the form

P~w!uE58p3

n3(hAhXhNhK1

3FexpS \vh

KBT0Dexp~2Xh!2exp~Xh!G . ~52!

We can easily verify from Eq.~52! that also the productionP(w) cancels out only when thermal equilibrium conditiohas been reached~i.e., v i5s^ i j &5qi50, andT5T0).

VII. TRANSITION TO LINEAR IRREVERSIBLETHERMODYNAMICS

The closed HD system in Eqs.~11!–~15!, obtained in theprevious sections, can be used to derive a rigorous anderal closure in a reduced system of balance equations makin to standard HD models. In this way, using an iterattechnique, we provide a generalization of the phenomelogical constitutive equations of Fourier and Navier-Stokfor the heat flow and viscous stress, respectively, in whichtransport coefficients are explicitly calculated. Using re


7 Dec 2013 22:23:00

e

th

s

lio

thr

ob-

eassti-ardhed ofn-


[This a

tions ~17!–~21! it is possible to rewrite the set of balancequations ~11!–~15! in terms of the new variablesmA

5$n,p,v i ,s^ i j & ,qi% as:

dn

dt1n

]vk

]xk50, ~53!

dv i

dt1

1

nm*

]s^ ik&

]xk1

1

nm*

]p

]xi52

e

m*Ei1Ri

~v!1Ri~v! , ~54!

dp

dt1

2

3

]qk

]xk1

5

3p

]vk

]xk1

2

3s^ lk&

]v l

]xk5R~p!1R~p!, ~55!

ds^ i j &

dt1

]G^ i jk &

]xk1

4

5

]q^ i

]xj12sk^ i

]v j &

]xk5R^ i j &

~s!1R^ i j &~s! , ~56!

dqi

dt1

]Gillk

]xk1

7

5S qi

]vk

]xk1qk

]v i

]xkD1

2

5qk

]vk

]xi1G^ ikl &

]v l

]xk

23

2

1

nm*s~ i l

]s l )k

]xk5Ri

~q!1Ri~q! ~57!

with the constitutive functionsG^ i jk & , andG^ i l lk & , expressedby Eqs.~41! and~42! and the new collisional productionsRA

and RA expressed through the relations

Ri~v!5

1

nPi

~nv! , Ri~v!5

1

nPi

~nv! ,

R~p!5 23 ~P~w!2m* Pl

~nv!v l !, R~p!52 23 m* Pl

~nv!v l ,

R^ i j &~s!5P^ i j &

~S!22m* P^ i~nv!v j & ,

R^ i j &~s!5 P^ i j &

~S!22m* P^ i~nv!v j & ,

Ri~q!5 Pi

~S!25

2

p

nPi

~nv!21

ns^ i l &Pl

~nv!2 P^ i l &~S!v l

1 12 m* ~v lv l !Pi

~nv!1m* ~ Pl~nv!v l !v i ,

Ri~q!5Pi

~S!25

2

p

nPi

~nv!25

3P~w!v i2

1

ns^ i l &Pl

~nv!2P^ i l &~S!v l

1 12 m* ~v lv l !Pi

~nv!1m* ~Pl~nv!v l !v i .

In order to obtain a simplified model which resemblesusual HD equations for the quantities$n,v i ,p%, it is possibleto employ an approximated procedure analogous to theMax-wellian iteration.23,50 Through this procedure the quantitie$s^ i j & ,qi%, present in the balance Eqs.~53!–~55! can be ex-pressed in terms of the variables$n,v i ,T%. The first iterationis obtained by inserting into Eqs.~56! and~57!, the values ofs^ i j & , qi , v i , G^ i jk & , andGillk evaluated in the local thermaequilibrium state. In this way, from the linearized expressfor the production termsRA5$R^ i j &

(s) ,R^ i j &(s) ,Ri

(q) ,Ri(q)% one ob-

tains the first iterated constitutive functions fors^ i j & andqi .Analogously, in the second iteration we substitute intoleft-hand side of Eqs.~56! and~57! the linear expressions fo


195.19.233.81 On: Sat, 0

e

n

e

the $G^ i jk & ,Gill j % describing the quantitiess^ i j & and qi bymeans of the first iteration. In this way, from theRA , drop-ping all terms of higher order than the second one, onetains the second iteration fors^ i j & andqi . The results of thefirst iteration givess^ i j &50 and:

qi52k]T

]xi2

a

bpv i ~58!

with

k55

2

KB2nT

m* b~59!

a524

3

p2

n3(h Ah$210Xh@~Nh11!H112NhH1

2#

25@~Nh11!H311NhH3

2#12@~Nh11!H511NhH5

2#%

12

3j

n

p, ~60!

b524

3

p2

n3(h Ah$5@~Nh11!H311NhH3

2#

24@~Nh11!H511NhH5

2#1 45 @~Nh11!H7

11NhH72#%

126

15j

n

p, ~61!

where k is a thermal conductivity of hot carriers and thcoefficientsa andb are average collision rates expresseda function of electron temperature. We note that the contutive relation~58! represents a generalization of the standFourier law for heat conduction. In fact, the heat flux is tsum of a term proportional to the temperature gradient ana convective term which is usually the most important cotribution in practical applications.

In the second iteration fors^ i j & it is:

FIG. 1. Spatial profile of the electric field for devicesA, B, and C at T0

5300 K obtained from HD calculations.


7 Dec 2013 22:23:00

:

e

s-in

-

ro

csio

la

er

cle.m,

-onsed

e

m-tiblen-nsnsd of

ps

nel

de-


[This a

s^ i j &522m]v ^ i

]xj &2

4

5

1

g

]q^ i

]xj &2

16

15

j

gL ^ i j &

116

15

1

g(hAh(

r 51

4

L ^ i j &2r 11@~Nh11!H2r 11

1

1NhH2r 112 #, ~62!

wherem5p/g is the shear viscosity of hot carriers withg,in analogy witha andb, an average collision rate given by

g5216

15

p2

n3(hAh@~Nh11!H5

11NhH52#1

8

5j

n

p,

~63!

and

L ^ i j &52n2m*

pv ^ iv j &1

21

50

n2m*

p3q^ iqj &

11

10

n2m*

p2v ^ iqj & ,

L ^ i j &~3! 525

p2m*

n2v ^ iv j &15

pm*

n2v ^ iqj & ,

L ^ i j &~5! 5

p2m*

n2v ^ iv j &1

7

25

m*

n2q^ iqj &2

24

5

pm*

n2v ^ iqj & ,

L ^ i j &~7! 5

4

5

pm*

n2v ^ iqj &2

4

5

m*

n2q^ iqj & ,

L ^ i j &~9! 5

4

25

m*

n2q^ iqj & .

We remark that Eq.~62! represents a generalization of thusual Navier-Stokes law for shear diffusion.23

Figure 1 shows the average collision ratesa,b and g,given in Eqs.~60!, ~61!, and ~63!, as a function of electrontemperature for the case ofn-Si when the conduction band icharacterized by six equivalentX valleys at the lattice temperature of 300 and 77 K, respectively. The pronouncedcrease ofb andg is similar to the behavior of a momentumrelaxation rate, whilea behaves similarly to an energy relaxation rate.7,16,43,45,46

VIII. APPLICATION TO n 1nn 1 SI STRUCTURES

For the purposes of validating the approach here pposed, we consider one-dimensionaln1nn1 Si structures,and compare analytical results with MC simulations. Acordingly, the conduction band is characterized byequivalentX valleys and the intervalley scatterings by twdifferent groups of phonons~f andg type! with six differenttransitions (h5g1 , g2 , g3 , f 1 , f 2 , and f 3). All the valuesof the physical parameters used for Si, both in MC simutions and in the evaluation of constitutive functionsHA , arethose already reported in the literature13 and are summarizedfor completeness in Table I. The MC simulations are pformed with the Damocles code, using 1.53104 particles and


195.19.233.81 On: Sat, 0

-

-

-x

-

-

the same physical approximations described in this artiFor the HD calculations, due to the symmetry of the systeit is:

v5$v,0,0%, s^ i j &5diag$s,2 12 s,2 1

2 s%q5$q,0,0%,

~64!E5$E,0,0%

~analogously it is S^ i j &5diag$S,21/2S,21/2S% and S5$S,0,0%). The system in Eqs.~11!–~15! reduces to fiveequations9,10 whose independent quantities are$n,W,v,S,S%and is coupled with Poisson’s equation

«Df5e~ND2n!

with f the electrical potential andND the donor concentration. For a one-dimensional geometry, both the equatiand the closure for the constitutive functions are obtainfrom the general three-dimensional relations~11!–~15!, ~20!and~21!, ~41! and~42!, ~45!–~50!, and~A2! and~A3! usingthe expression~64! for the momentsmA 5 $n, p, v i , s^ i j & ,qi%. Accordingly, the only components of the constitutivfunctions to be considered are: for the fluxes$Q^11&1 ,Q1l l 1%,and for the collisional productions

P1~nv!1P1

~nv!5Pnv, P~w!5Pw,

P^11&~S! 1P^11&

~S! 5PS, P1~S!1P1

~S!5PS.

As boundary conditions null gradient to all moments are iposed at boundary points. These conditions seem compawith the effective configuration of moments, both in the trasient and in the final stationary state. For the HD calculatiowe use 250 cells for a total time of 6 ps. In these conditiothe HD code requires about three seconds for a picoseconsimulation on an AlphaStation 600, 333 MHz. Two grouof submicron structures are analyzed atT05300 K. A firstgroup, labeled as A, B, and C, is biased at 1 V and is char-acterized by different doping concentrations and chanlengths as given in Table II~Fig. 1!. This group is used tovalidate the constitutive functionsHA of the HD model givenby expressions~20! and~21!, ~41! and~42!, and~45!–~50!. Asecond group, labeled as D, E, F, and G, is used for a

TABLE II. Device parameters with sharp doping profile.

N1 N Channel biasDevice ~cm23) (cm23) (mm! ~V!

A 1018 1016 0.2 1B 1018 1016 0.3 1C 531017 231015 0.4 1

TABLE I. Physical parameters for electrons in silicon.

Sym. Val. Unit Sym. Val. Unit Sym. Val. Unit

D f 10.33108 @eV/cm# Tf 1

220 @K# El 9 @eV#D f 2

23108 @eV/cm# Tf 2550 @K# Ul 9.03105 @cm/s#

D f 323108 @eV/cm# Tf 3

685 @K# r 2.33 @gr/cm3#Dg1

0.53108 @eV/cm# Tg1140 @K# ml* /me 0.91 ¯

Dg20.83108 @eV/cm# Tg2

215 @K# mt* /me 0.19 ¯

Dg3113108 @eV/cm# Tg3

720 @K# «/«0 11.7 ¯


7 Dec 2013 22:23:00

th

oa-leesl-

-

heoanvs

an

sti-

ja-ndan-bitMCthelize

s

es


[This a

tailed investigation of the dependence of moments fromdoping profile at the interfaces between then and then1

regions as given in Table III.Figures 2 to 5 report the results of the first group

structures~for a detailed description of the results of simultions see Ref. 10. Figure 2 shows the electric-field profiobtained from HD calculations. By using the valu$n,p,v,s,q%, obtained from MC simulations, we have caculated the constitutive functionsHA by means of relations~16!, ~20! and ~21!, ~41! and ~42!, ~45!–~50! and ~A2! and~A3!. The same functionsHA have been also directly evaluated by MC simulations, and the comparison is reportedFigs. 3–4. All functions mimic the triangular shape of tfield profile. As we can see from Figs. 3–4 there is a goagreement between the results of direct MC simulationsthose obtained by calculating the analytic expressions ein proximity of the critical regions adiacent to the junctionIn the case of structuresB and C we have also verified theconstitutive relations for the heat flux and stress deviator

TABLE III. Device parameters.

N1 N Channel xs biasDevice (cm23) (cm23) (mm! (mm! ~V!

D 1019 1017 '0.3 0.06 1–2E 1019 1017 '0.3 0.05 1–2F 1019 1017 '0.3 0.03 1–2G 1019 1017 '0.3 0.01 1–2


195.19.233.81 On: Sat, 0

e

f

s

in

dd

en.

d

results are reported in Fig. 5. Here we note that the contutive relation for the heat flux in Eq.~58! is well verified inall points of the structure except for the region strictly adcent to the second junction where both the electric field aits gradient exhibit very high values at decreasing the chnel length. In this region, the constitutive relations exhipeak values greater than those directly calculated bysimulations for up to one order of magnitude. Indeed,application of the present iterative procedure to genera

FIG. 2. Spatial profile of the collisional production$Pnv/n,Pw/n,PS/n,PS/n% for the devices reported in Table II atT05300 K.Symbols refer to results directly obtained from MC simulations. Curvrefer to results obtained from the MEP by substituting in Eqs.~45!–~50! thevalues of the moments$n,p,v,s,q% obtained by MC simulations.

C
FIG. 3. Spatial profile of the fluxes$Q^11&1 /n,Q1l l 1 /n% for the devices reported in Table II atT05300 K. Symbols refer to results directly obtained from Msimulations. Curves refer to results obtained from the MEP by substituting in Eqs.~16!, ~20!, and ~21!, and ~41! and ~42! the values of the moments$n,p,v,s,q% obtained by MC simulations.

7 Dec 2013 22:23:00

ng in Eqs.


[This a

FIG. 4. Spatial profile of the heat flux and stress deviator for the devicesB andC reported in Table II atT05300 K. Symbols refer to an explicit evaluatioof heat flux and stress deviator directly obtained from MC simulations. Curves refer to results obtained from the constitutive functions by substitutin~58!–~63! the values of the moments obtained from MC data.

FIG. 5. Spatial profile of electric fields and doping for devicesD, E, F, andG reported in Table III atT05300 K with a bias of 1~left! and 2 V ~right!,respectively. Curves refer to HD simulation, points to MC simulations~deviceD).

FIG. 6. Spatial profile of velocity, energy, traceless momentum flux, and heat flux for the devicesD, E, F, andG reported in Table III atT05300 K with abias of 1~left! and 2 V ~right!, respectively. Curves refer to HD simulation, points~for velocity and energy! to MC simulations~deviceD).

rticle is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

195.19.233.81 On: Sat, 07 Dec 2013 22:23:00

el used


[This a

FIG. 7. Average collision ratesa, b, andg for electrons in Si as a function of electron temperature at lattice temperatures of 77 and 300 K. The modrefers to six spherical and parabolic equivalentX valleys with all values of the physical parameters entering Eqs.~60!, ~61!, and~63! taken from Table I.

o-la-

tlyesomediss

ne-veing

nd

d

the Fourier’s law involves a linearization of collisional prductions which leads to a modellization of constitutive retions strictly valid only within small gradient approximations. From HD calculations we have already observed9,10

that with a nonlinear description~up to the third order! of thecollisional productions it is possible to improve significanthe agreement with MC simulations. Concerning the strdeviator, whose magnitude is detectable only starting frthe second iteration, the constitutive relation is verifiwithin 25% by MC simulations. The overall agreementconsidered to be satisfactory and thus validating the con


195.19.233.81 On: Sat, 0

-

s

ti-

tutive relations presented here. We remark that the odimensional HD simulations performed with this model habeen extended to different lattice temperatures, other dopprofiles, and applied biases.9–11

Figures 6 and 7 report the HD calculations of the secogroup of submicron structures~labeledD, E, F, andG! withdifferent doping profiles and applied biases~see Table III!.The structures have highly doped~but always within nonde-generate conditions! n1 regions of length 0.2mm and achannel of about 0.3mm. The doping profiles are smootheat the junction points by anerfc function10 scaled by the


7 Dec 2013 22:23:00

e

arIn

t

satsina-

ecreeev

iotth

as

nsMCveuc-henrtiesantre-

r athed

s-in

orII

u-


[This a

parameterxs as reported in Table III. Figure 6 reports thelectric fields and the doping profiles for devicesD, E, F,and G. We note that in these devices conditions whichvery far from thermodynamic equilibrium are reached.deed, especially near the second junction, the values ofelectric field~see Figs. 2, 6! range between 104 and 23105

V/cm with gradients such thatuEu/u¹Eu'100 Å. Figure 7reports the HD values of velocityv, energyW, traceless partof the momentum flux densityS, and heat fluxq, as a func-tion of position for devicesD, E, F, andG with a bias of 1and 2 V, respectively~Figs. 2 and 3!. For validation pur-poses, MC values for velocity and energy for deviceD arealso reported. The agreement between HD and MC valuesatisfactory, showing only a minor discrepancy (10%worst! for the energy peak in the second homojunction aV. The expected smoothing of the moments at decreavalues ofxs is quantitatively evaluated by the HD calcultions ~Figs. 4–7!.

IX. CONCLUSIONS

Within the MEP we have formulated a closed HD modadequate to describe transport phenomena in submisemiconductor devices under conditions very far from thmodynamic equilibrium. We stress that the MEP has bhere used in a local dynamic context. Accordingly, we hadetermined both: an explicit espression of the distributfunction in terms of the relevant macrovariables, and a seevolution equations for these macrovariables containing


195.19.233.81 On: Sat, 0

e-he

ist2g

lonr-nenofe

underlying physical processes in an explicit way~i.e., thecollision rates of scattering mechanisms are the samethose of the BTE!. The analyticalconstitutive relationsob-tained for both the fluxes and the collisional productiohave been validated for a one-dimensional structure bysimulations. Mostly important, transport properties habeen calculated at a HD level without the need of introding external parameters. The HD model so developed is tproposed as valuable method to describe transport propeof hot carriers in submicrometric devices, having the relevadvantage of a reduced computational environment withspect to competitive simulative methods. Of course, fowider application of this model, one needs to better refinephysical description~e.g., by introducing nonparabolic banstructure and further mechanisms of scattering!, and eventu-ally including the contribution of higher moments of the ditribution function. Some of these issues will be the matopics of further research.

ACKNOWLEDGMENTS

Dr. M. V. Fischetti and Dr. O. Muscato are thanked fproviding the MC data. Partial support from MADESSproject of the Italian National Research Council~CNR! isgratefully acknowledged.

APPENDIX A

The analytic expression of the electron energy distribtion as a function of momentsmA5$n,p,v i ,s^ i j & ,qi% is

f @«,mA~r ,t !#52

ApnS n

pD 3/2

«1/2expS 2n

p« D H 11F2

1

2

m* n

pv rv r1

2

5

m* n

p3qrqr1

m* n

p2qrv r1

1

4

1

p2 s^rs&s^sr&

1S 1

3

m* n2

p2v rv r2

1

15

m* n2

p4qrqr2

4

3

m* n2

p3qrv r2

1

3

n

p3s^rs&s^sr&D «

1S 24

15

m* n3

p5qrqr1

4

15

m* n3

p4qrv r1

1

15

n2

p4s^rs&s^sr&D «21S 4

75

m* n4

p6qrqr D «3G

1F1

2

m* n

p2v rs^rs&vs2

18

25

m* n

p4qrs^rs&qs2

7

5

m* n

p3v rs^rs&qs2

1

6

1

p3 s^rs&s^sn&s^nr&

1S 22

3

m* n2

p3v rs^rs&vs1

2

75

m* n2

p5qrs^rs&qs1

14

5

m* n2

p4v rs^rs&qs1

1

3

n

p4s^rs&s^sn&s^nr&D «

1S 2

15

m* n3

p4v rs^rs&vs1

334

375

m* n3

p6qrs^rs&qs2

28

25

m* n3

p5v rs^rs&qs2

2

15

n2

p5s^rs&s^sn&s^nr&D «2

1S 2112

375

m* n4

p7qrs^rs&qs1

8

75

m* n4

p6v rs^rs&qs1

4

315

n3

p6s^rs&s^sn&s^nr&D «3

1S 8

375

m* n5

p8qrs^rs&qsD «4G J . ~A1!


7 Dec 2013 22:23:00

s

n


[This a ] IP:

The coefficients$A i ,C^ i j & ,Di% present in the production terms of Eqs.~45! and ~46!, associated with the intravalleytransitions, obtained through an expansion~around the local Maxwellian! and evaluated at the third order in the variable$v i ,s^ i j & ,qi%, are

A i5H n

pv i1

1

5

n

p2qiJ 1H 2

1

25

n

p3s^ ir &qr1

1

5

n

p2s^ ir &v rJ 1H 1

10

n2m*

p2~v rv r !v i2

59

625

n2m*

p5~qrqr !qi

21

25

n2m*

p4~qrqr !v i2

1

25

n2m*

p3~v rqr !v i2

1

50

n2m*

p3~v rv r !qi2

1

25

n2m*

p4~v rqr !qi1

3

875

n

p4s^ ir &s^rs&qs

21

35

n

p3s^ ir &s^rs&vs2

1

140

n

p4~s^rs&s^sr&!qi2

1

140

n

p3~s^rs&s^sr&!v iJ ,

C^ i j &51

2

n

p2s^ i j &1H 1

2

n2m*

p2v ^ iv j &1

7

50

n2m*

p4q^ iqj &1

1

5

n2m*

p3v ^ iqj &1

1

14

n

p3Fs^ ir &s^r j &21

3~s^rs&s^sr&!d i j G J

1H 1

28

n2m*

p3~v rv r !s^ i j &1

19

175

n2m*

p5~qrqr !s^ i j &1

3

70

n2m*

p4~v rqr !s^ i j &2

1

168

n

p4~s^rs&s^sr&!s^ i j &

19

875

n2m*

p5 F1

2~qis^ js&qs1qjs^ is&qs!2

1

3~qrs^rs&qs!d i j G2

13

175

n2m*

p4 F1

2~v is^ js&qs1v js^ is&qs!

21

3~v rs^rs&qs!d i j G1

1

7

n2m*

p3 F1

2~v is^ js&vs1v js^ is&vs!2

1

3~v rs^rs&vs!d i j G

13

35

n2m*

p4 F1

2~qis^ js&vs1qjs^ is&vs!2

1

3~qrs^rs&vs!d i j G J , ~A2!

Di5H n

pv i1

3

5

n

p2qiJ 1H 3

25

n

p3s^ ir &qr1

3

5

n

p2s^ ir &v rJ 1H 3

25

n2m*

p3~v rqr !v i1

3

50

n2m*

p3~v rv r !qi1

21

125

n2m*

p4~v rqr !qi

118

125

n2m*

p4~qrqr !v i1

246

625

n2m*

p5~qrqr !qi1

3

10

n2m*

p2~v rv r !v i1

3

35

n

p3s^ ir &s^rs&vs1

3

140

n

p3~s^rs&s^sr&!v i

13

125

n

p4s^ ir &s^rs&qs1

3

100

n

p4~s^rs&s^sr&!qiJ ,

The coefficients$A i2r 11 ,B2r 11,C^ i j &

2r 11% present in the production terms of Eqs.~47!–~50! and obtained through an expansio~around the local Maxwellian! analyzed at the third order in the variables$v i ,s^ i j & ,qi%, are

A i35H 2

p2

n2v i22

p

n2qiJ 1H 14

5

1

n2s^ ir &qr22

p

n2s^ ir &v rJ 1H 28

25

m*

pn~qrqr !v i2

14

25

m*

pn~v rqr !qi12

1

n2 s^ ir &s^rs&vs

11

2

1

n2~s^rs&s^sr&!v i1

7

5

m*

n~v rv r !qi1

14

5

m*

n~v rqr !v i2

66

25

1

pn2s^ ir &s^rs&qs2

1

10

1

pn2~s^rs&s^sr&!qi

1196

125

m*

p2n~qrqr !qi2

pm*

n~v rv r !v iJ ,

A i55

4

5

p

n2qi1H 4

5

p

n2s^ ir &v r2

56

25

1

n2 s^ ir &qrJ 1H 444

125

1

pn2s^ ir &s^rs&qs2

2

5

1

n2 ~s^rs&s^sr&!v i198

625

m*

p2n~qrqr !qi

111

25

1

pn2~s^rs&s^sr&!qi1

14

125

m*

pn~qrqr !v i2

56

25

m*

n~v rqr !v i1

308

125

m*

pn~v rqr !qi2

28

25

m*

n~v rv r !qi

28

5

1

n2 s^ ir &s^rs&vs12

5

pm*

n~v rv r !v iJ ,

rticle is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

195.19.233.81 On: Sat, 07 Dec 2013 22:23:00


[This a

A i75

8

25

1

n2s^ ir &qr1H 4

625

m*

p2n~qrqr !qi2

38

175

1

pn2~s^rs&s^sr&!qi2

152

125

m*

pn~v rqr !qi2

984

875

1

pn2s^ ir &s^rs&qs

256

125

m*

pn~qrqr !v i1

4

25

m*

n~v rv r !qi1

8

25

m*

n~v rqr !v i1

8

35

1

n2 s^ ir &s^rs&vs12

35

1

n2 ~s^rs&s^sr&!v iJ ,

A i952

24

125

m*

p2n~qrqr !qi1

8

125

m*

pn~qrqr !v i1

16

125

m*

pn~v rqr !qi1

16

175

1

pn2s^ ir &s^rs&qs1

4

175

1

pn2~s^rs&s^sr&!qi ,

A i115

16

625

m*

p2n~qrqr !qi ,

B158p3

n31H 8

pm*

n2qrv r24

p2m*

n2v rv r12

p

n3s^rs&s^sr&1

16

5

m*

n2qrqrJ

1H 4pm*

n2v rs^rs&vs2

144

25

m*

pn2qrs^rs&qs2

56

5

m*

n2v rs^rs&qs2

4

3

1

n3 s^rs&s^sn&s^nr&J ,

B35H 28

15

m*

n2qrqr2

8

3

p

n3s^rs&s^sr&2

32

3

pm*

n2qrv r1

8

3

p2m*

n2v rv rJ

1H 112

5

m*

n2v rs^rs&qs1

16

75

m*

pn2qrs^rs&qs1

8

3

1

n3 s^rs&s^sn&s^nr&216

3

pm*

n2v rs^rs&vsJ ,

B55H 32

15

pm*

n2qrv r2

32

15

m*

n2qrqr1

8

15

p

n3s^rs&s^sr&J

1H 2672

375

m*

pn2qrs^rs&qs2

224

25

m*

n2v rs^rs&qs2

16

15

1

n3 s^rs&s^sn&s^nr&116

15

pm*

n2v rs^rs&vsJ ,

B75H 32

75

m*

n2qrqrJ 1H 64

75

m*

n2v rs^rs&qs2

896

375

m*

pn2qrs^rs&qs1

32

315

1

n3 s^rs&s^sn&s^nr&J ,

B9564

375

m*

pn2qrs^rs&qs , ~A3!

C^ i j &5 54

p2

n3s^ i j &1H 4

p2m*

n2v ^ iv j &1

28

25

m*

n2q^ iqj &2

56

5

pm*

n2v ^ iqj &24

p

n3Fs^ ir &s^r j &21

3~s^rs&s^sr&!d i j G J

1H 4m*

n2~qrv r !s^ i j &22

pm*

n2~v rv r !s^ i j &1

24

25

m*

pn2~qrqr !s^ i j &13

1

n3 ~s^rs&s^sr&!s^ i j &2584

125

m*

pn2F1

2~qis^ js&qs

1qjs^ is&qs!21


424

25

m*

n2 F1

2~v is^ js&qs1v js^ is&qs!2

1

3~v rs^rs&qs!d i j G

28pm*

n2 F1


1

3~v rs^rs&vs!d i j G18

m*

n2 F1


1

3~qrs^rs&vs!d i j G J ,

rticle is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

195.19.233.81 On: Sat, 07 Dec 2013 22:23:00


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C^ i j &7 5

16

5

pm*

n2v ^ iqj &2

16

5

m*

n2q^ iqj &1

8

7

p

n3Fs^ ir &s^r j &21

3~s^rs&s^sr&!d i j G1H 4

7

pm*

n2~v rv r !s^ i j &2

12

7

1

n3 ~s^rs&s^sr&!s^ i j &

216

5

m*

n2~qrv r !s^ i j &2

12

25

m*

pn2~qrqr !s^ i j &2

224

25

m*

n2 F1


1


1272

25

m*

pn2F1


1


32

5

m*

n2 F1


1

3~qrs^rs&vs!d i j G

116

7

pm*

n2 F1


1

3~v rs^rs&vs!d i j G J ,

C^ i j &9 5

16

25

m*

n2q^ iqj &1H 16

35

m*

n2~qrv r !s^ i j &2

16

35

m*

pn2~qrqr !s^ i j &2

3616

875

m*

pn2F1


1

3~qrs^rs&qs!d i j G

132

35

m*

n2 F1


1

3~qrs^rs&vs!d i j G1

32

35

m*

n2 F1


1


14

21

1

n3 ~s^rs&s^sr&!s^ i j &J ,

C^ i j &11 5

64

175

m*

pn2F1


1


16

175

m*

pn2~qrqr !s^ i j & .

The remaining coefficientsDi2r 11 can be expressed by means of the coefficientsA i

2r 21 through the relationsDi2r 11

5(p/n)A i2r 21 ~with r 52, . . . ,6).

G.

.

ng

t inLS

-

r

-

n

es

T

rd

s

-

1K. Blotekjaer, IEEE Trans. Electron Devices17, 38 ~1970!.2G. Baccarani and M. R. Wordeman, Solid-State Electron.28, 407 ~1985!.3M. Rudan and F. Odeh, COMPEL5, 149 ~1986!.4A. Forghieri, R. Guerrieri, P. Ciampolini, A. Gnudi, M. Rudan, andBaccarani, IEEE Trans. Comput.-Aided Des.7, 231 ~1988!.

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