Nonlinear closures for hydrodynamical semiconductor transport models

23 June 1997 PHYSICS

LETTERS

A

Physics Letters A 230 ( 1997) 387-395

ELSEVIER

Nonlinear closures for hydrodynamical semiconductor transport models A.M. Anile ‘,I, a Dipartimenfu

di Matematica,

h Dipartimento

di Fisica.

Universitir Universitcl

M.

di Catania,

Trovato b*2 Viale A. Doria

6. 95125 Cufania,

Italy

di Catunia, Corso Italia 57. 9.5129 Catania. lta1.v

Received 21 February 1997; accepted for publication 2 April 1997 Communicated by L.J. Sham

Abstract Linear and nonlinear closures are explicitly determined for a model of 13 moment equations semiconductors in the case of a Fermi electron gas. @ 1997 Published by Elsevier Science B.V. PACS:

for carrier transport

in

72.10.Bg; 72.20.Ht

1. Int~duetion

hydrodynamical models for carrier transport in semiconductors have been proposed where the closure of the moment equations is achieved within the framework of extended thermodynamics (by exploiting the entropy principle in order to determine the constitutive laws providing the sought for closures) [ I-41. These closures have been checked by Monte Carlo simulations in the case of homogeneous silicon in an external electric field and the results are quite satisfactory [ 5,6]. In the hydr~yn~ical model so far considered only linear closures have been treated, i.e. the constitutive laws have been determined up to first order off a local state of thermal equilibrium (which we call partial thermodynamical equilibrium) and which will be defined more precisely later. However, in electron devices with sharp junctions the electric field can become substantially large in such a way that nonlinear effects could become very important. Therefore it is mandatory to develop the constitutive functions to higher orders in the deviations from the state of partial thermal ~uilibrium. When one does this in the usual fr~ework of extended the~~yn~ics [7] one finds that the second order terms can be determined up to an undetermined parameter. This baffling situation can be remedied by adopting an equivalent approach to the entropy principle of extended thermodynamics, the maximum entropy principle [8-lo], which, being based on a specific form for the distribution function, is capable of calculating unambiguously all the Lately

’ E-mail: ~i~e~dipmat.unict.it. 2 E-mail: trovato~dipmat.unict.it 0.77%9601/97/$17,00 @ 1997 Published by Elsevier Science B.V. All rights reserved. SO375-9601(97)00278-S

PII

388

A.M. Anile. M. TrovaiolPhysics Letters A 230 (1997) 387-395

coefficients app~ing in the expansions for the ~onstitutive laws. Such a calculation has already been done [ 11 J in the case of a M~well-Boltzmann statistics. However, in the heavily doped regions of electron devices degeneration effects can be important and then it is necessary to determine the closures also in this case.

2. Balance equations

for an electron gas

Let us consider the semiclassical conduction band [ 121, c?F:(k,r,t,v) at

+Cya.m,r,r,4

l

semiconductor

Boltzmann

+ dki d I( k, r, t, V)

d Xi

dt

a k;

transport

equation

= Qv(F),

for electrons

in the vth

(1)

where (a) P = F(k, t, t, V) is the one particle distribution function (in the pth band) in phase space (k, r), with k E B (where B corresponds to the first Brillouin zone). (b) The rate of change of the distribution function P( k, r, t, v)due to collisions is expressed as the sum of rates of changes due to particles that are scattered into (k, r, v) from any possible state (k’, r, ,u) at time t minus the sum of rates of change due to particles that are scattered out of (k, r, V) into any possible state (k’, r, p) at the same time t. Therefore one writes for the collisional pr~uction

including intraband (p = V) and interband (if p Z V) collisions for all kind of scatterings (with acoustic and optical phonons, with impurities, charged carriers,) S( k’, ,x; k, V; r) being the total (differential) scattering rate for the transition (k, V) -+ (k’, ,u). (c) The group velocity of electrons c,(k) = ( l/h)V~,( k) is expressed through the band energy eY(k) . For the sake of simplicity we shall restrict ourselves to only one conduction band, assumed to be spherical and parabolic. Therefore electrons will be described as an assembly of free charged particles with an effective

isotropic mass m* and elementary charge e. Then we have the following energy band dispersion relation and group velocity, c(k) = lik/m*,

c(k) = ~k2~2rn~

(2)

and the Brillouin zone is expanded to cover all k. It is convenient to consider the distribution function P( k, r, t) of (r, t) and the velocity c, i.e. 3( k, r, t) = 3( c, r, t) and consequently Eiq. ( 1) is rewritten in

as a function the form

(3)

(4) where 4 and j are the charge and current densities vectors.

respectively

while E, B are the electric and magnetic field

A.M. Anile,

M. Trovaio/Physics

Leiters A 230 (1997) 387-395

By multiplying Eq. (1) by $~(c) = (1,~. t, 12m *c 2, m *C(iCj), $m*c2Ci} and integrating in velocity (assuming suitable vanishing conditions at infinity for the distribution function) we obtain the following moment equations,

d

1 J Oik

?ll!i

-+-r??

in* a Xk

389

space set of

= nfi + Pi, (7)

3

@(ij) at

i3

+

a @(ij)k ~

a xk

=

2nm*U(;fj) + P(ij),

Si

t’

for the 13 fields n (numerical density), nui (carrier density flux), W (energy of density), O(ij) (traceless part of the momentum density flux), Si (energy of flux). {@(ijk) , Sik, PA} being the constitutive functions with @ij = @(ij) + i”v8ij. @i,jk= @(ijk) f :S(i8jk),

3. Thermal

equilibrium

and moment decomposition

A complete description of transport in semiconductors requires considering the mutually interacting subsystems of electrons, holes and phonons under the action of an external and a self-consistent electromagnetic field. The mutual interactions among the subsystems will drive the carriers towards the state of global thermal equilibrium. In this state all the productions PA vanish and ~alogously, for each carrier fluid, one has vi = W = O~ij) = Si = T - TL = 0. To each collisional production PA, in the standard way, one can associate a characteristic relaxation time which can be estimated by Monte Carlo simulations (both for silicon [ 2} and for the other semiconductors [ 131) . From the results one recognizes two distinct time scales: the momentum relaxation time rP (and likewise the energy flux relaxation time ry, and that for anisotropic stresses 7, ), and the energy relaxation time r,. One notices the following ordering among the relaxation times: 7, >> TV, ry, 7,, which is a consequence of the fact that most collisions, with acoustic phonons and impurities, are elastic so that it is necessary to have many collisions in order to relax the carrier energy to the global thermal equilibrium value W, = ;nkeTi (corresponding to the lattice temperature). In the process driving the whole system to global thermal ~uilibrium each carrier subsystem will reach an intermediate state, which can be caBed partial thermal equilibrium, characterized by the following values for the macroscopic fields: vi = W = @(ii) = Si = 0, T # TL. After this state has been reached the carrier temperature will further relax (mainly through optical phonon scattering) until it will be equal to the equilibrium lattice one. Because of the sizable difference between the energy relaxation time and the others it is convenient to separate (as is usual in gas dynamics, although for different reasons related to the objectivity principle [ 141) both the moments {W O(ij) , Si} and the constitutive fUnCtiOnS {o~ijk), Sik, PA} in a convective and in a nonconvective Component, @[, = W = ip + 4nm*v2, O(i,q) = &(ijq) + 3h(ijVy)

O(ij)

= d(ij)

+ nm*U(iUjV~),

+ U.m*U(iUj), Sic = $St,

Si = qi t + 4q,u,

+

$&)(i[V[) + $nM*V2Vi,

3~~~,~~~~~ + $m*nv4,

(10)

390

A.M. Anile. M. Trovuto/Physics

s(i.i)= i&l'lr(ij)+ Likewise

2q(iuj)

+

for the productions

P = R,

d(ij)lUl

+

3&(uU(iUj))

+

Letters A 230 (1997) 387-395

im*?W2U(iuj).

we can write

Pi = R, -t RVi,

Pw = ;Ru + m*RpI + im*Ru2, fi = ;Rill + iRcilu[) + im*Rciuioj,

P(i,i) = R(ij) + 2m* R(iuj) + m* Ru(iuj),

+ irn*RU2Ui,

where p = f&

=

im*C23dC

(pressure),

J n

&(i,i)=

(traceless

m*C(iCj)3dC

deviatory

part of the stress tensor),

.I 4i

jm*C2Ci3dC

=

(heat flux),

(13)

J

,.

0 llpp =

m*C43dC,

&ll(ij) =

&tijk)

m*C2C(iCj)3dC, J

J

m*C(iCjCk)3dC,

=

RA

J

=

J

$A

(c)Q(F)

(14)

dc,

where Ci = ci - Ui is the microscopic peculiar velocity. Therefore it is convenient to consider the new set of fields {n, Ui, p, &(ij), qi} and constitutive

functions

GA = {8(,,) , &jjij, RA} which will be expressed as functions of the fields. We notice that, at variance with gas dynamics, Ui in the case of semiconductors will describe the carrier velocity relative to the lattice and therefore, in principle, the constitutive functions GA could depend also on ui without violating the material objectivity principle. However, it is possible to show (both in the case of a mixture of interacting charged fluids [ 151, as in the specific case of semiconductors [4] ) that the entropy associated to each carrier fluid will depend only on the dynamical variables {n, p, 8(,) , qi} but not on the relative velocity ui. Consequently also the constitutive functions &(ijk), &nij will be independent general this will not be the case for the collisional productions RA.

4. Entropy

maximization

Let us consider h=-kB

the entropy density

for the carrier gas,

GA

(15)

J~ln(~3)+y(l-~3)ln(l-~3)]dC,

where kB is Boltzmann’s constant and y = 2(m*/h)3. In order to determine the (unknown) constitutive functions, {mA}:

of Di while in

=

GAh?),

RA

=

we assume that these are functions

of the fields

hhd.

The quantities GA and RA are defined by the integrals ( 14) and therefore they can be determined once the distribution 3( r, C, t) is known. Now we shall specify the distribution function according to the maximum entropy principle [ 8,161. Accordingly we shall assume that 3 has the functional dependence 3[ r, C, t] = 3[ mA (C, t) , C] , and is such that it maximizes the entropy density (15) subjected to the constraints mA(r,

f> =

+A(C)3’(r, J

c, l> K.

t

16)

A.M. Anile, M. Trovato/Physic-s Letters A 230 (1997) 387-395

391

Let us define the quantity 13

h’ = h -

x

iA

@A(C)FC(r,C,t)dC--mA

, >

A=1

with iA = iA( mB) Lagrange

multipliers.

Then the maximum

entropy principle

implies

Sh’ = 0,

(17)

where Sh’ is the first variation

F=

of ( 17) is

of h’. The solution

y

exp(2)

+

f 18) I ’

A=1

One of the aims of this article is to obtain an explicit representation of the Lagrange multipliers iA on the fields mA = {n, p, q;, &,}. As a consequence of such a representation we shall: (i) guarantee the hyperbolicity of the closed system of balance equations and (ii) obtain an explicit representation of the distribution function on the fields {mA}. In this way it is possible, in principle, to determine the COnStitUtiVe fUnCtiOnS (&