Lattice Boltzmann Scheme Semiconductor Dynamicsdownloads.hindawi.com/archive/1998/054940.pdf · Wediscuss an extension ofthe Lattice Boltzmann method which mayprove useful for the

VLSI DESIGN1998, Vol. 6, Nos. (1-4), pp. 137-140

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A Lattice Boltzmann Scheme for Semiconductor DynamicsS. SUCCI and P. VERGARI

IBM European Centerjbr Scientific and Engineering Computing, v. Oceano Pacifico 171, Rome, 00144, ItalyMathematics Department, University ofCatania, via Andrea Doria, Catania, Italy

We discuss an extension of the Lattice Boltzmann method which may prove useful for thenumerical study of electron transport in semiconductors.

Keywords: semiconductor, discrete kinetic theory

1. INTRODUCTION

Lattice-gas (LG) models obeying cellular automata

rules and the related Lattice-Boltzmann (LB) tech-nique have known a rapid expansion in the recent

years mostly in connection with the simulation ofcomplex hydrodynamic phenomena [1, 2, 3, 4] Thedriving force behind these methods is their outstand-ing amenability to parallel processing which stems

from the extreme space-time locality of cellularautomata rules. So far, LG and LB methods have beendirected chiefly to the investigation of hydrodynamicproblems. However, in the recent years, extensions ofthe LB method have been developed which are capa-ble of handling several generalizations of the Navier-Stokes equations, including transport of passive sca-

lars, and flows with thermal and chemical effects.Precisely in this spirit of "generalized" hydrodynam-ics, we present here an extension of the LB modelwhich accounts for the generalized hydrodynamicequations describing the motion of hot electrons insemiconductors.

137

2. THE LATTICE BOLTZMANN EQUATION

The Lattice Boltzmann (LB) method is based on thefollowing finite-difference kinetic equation [5]:

b

Ni(Xk -k- Cik,t -t- 1) Ni(x,t) 2ij(Nj Nq) (1)j=l

where N represent particle populations moving alongthe direction identified by the discrete speed cik,

denoting the propagation direction and k runs over thespatial dimension, f2ij is the scattering matrix mediat-

ing two-body collisions between state and j and Nq

are local equilibrium populations. The large-scalelimit of the equation (1) is shown to converge to theNavier-Stokes equations of fluid dynamics, providedthe following mass and momentum conservation lawsare fulfilled:

b b

E "2iJ O, E ’)iJCik 0 (2)i= i=

In addition, in order to recover isotropy at the macro-scopic level, the set of discrete speeds must ensure

isotropy of the fourth order tensor Tijkl cicf?kCl.

This latter condition places rather restrictive

requirements on the class of lattices suitable to hydro-dynamic purposes.

138 S. SUCCI and E VERGARI

The derivation of the Navier-Stokes equations for-mally proceeds by contracting over the discrete

speeds degrees of freedom so as to generate evolutionequation for the moments of the discrete distributionfunction Ni. These equations are then closed via theusual assumption of weak departures from smoothlyvarying thermodynamic equilibria. A sensible choiceof the local equilibrium distribution function is alsoinstrumental to achieve the correct hydrodynamicequations. A suitable choice is a local maxwellianexpanded to 2nd order in the Mach number:

9Nq (1 + 2CikMk + "QiklMkMl) (3)

where M =- u#/cs, cs being the sound speed and Oiklikil- Cs2kl is the projector along the i-th propaga-tion direction.

On the analytical side one can prove a global H-theorem which guarantees numerical stability in thelinear regime (flow speed u << Cs) provided the spec-trum of f2ij be confined within the strip -2 < L < 0.

3. HYDRODYNAMIC SEMICONDUCTOREQUATIONS (HSE)

In the recent years, hydrodynamical models [6, 7, 8]describing carrier flow in semiconductor devices haveattracted considerable attention as an alternative toolto kinetic theory to describe carrier flow in semicon-ductor devices. In this paper, we shall restrict our

attention to the hydrodynamic model proposed byAnile and coworkers [9].

The one-dimensional field equations read as fol-lows

tP+x(pU) --0 (4)

),pu + )x(pu2) -epE (oT) J/Zp + xVOxpU (5)

tW + )x(Wu) -epuE )(puT) +3

T,3xK3xT + (p o W)/’rw (6)

where 9 is the carrier mass density, u the mean flowspeed and W is the total energy per unit volume.

In the following, we shall deal with the problem offinding an appropriate LB scheme yielding the eqs.(4-6) in the large-scale limit.

4. GENERALIZED HYDRODYNAMICEQUATIONS (GHE)

Since the GHE’s do involve energy as a dynamic vari-

able, the corresponding LB scheme must necessarily

cater for at least two energy levels Ei mic. The

minimal choice is the following 4-state (2-energy)discrete-speed scheme:

Ci--4-1, mi--1, i: 1,2 (Ei-- 1) (7)

Ci-- 4-2, mi-- m, i= 3,4 (Ei-- 4m) (8)This discrete set gives rise to the following definitionof temperature

T-- 91c + 929 ---c22 (9)

where 91 (N! + N2), 92 (N3 + N4)m. This results in

a dynamical range of temperatures between T andT 4 corresponding to full occupation of thelower(higher) energy levels respectively. The equilib-rium temperature lies in between and depends on the

l+4mmass ratio m according to T

l+m

This set generates a 4x4 collision matrix by six dis-tinct matrix elements a 1, a2, b l, b2, c, d

al bl c d

2ij bl al d c(10)

c d a2 b2d c b2 a2

Clearly ai, b describe "elastic" collisions among par-ticles with the same energy, while c,d are associatedwith "inelastic" cross-energy collisions. The latter are

responsible for temperature evolution since theychange 91, 92 separately, while leaving the total den-sity 9 91 + 92 unchanged. The next step is to iden-

tify the eigenvectors values of the collision matrixwhich generate the hydrodynamic fields 9, J, W incompliance with the GHE’s. Using the mass ratio rn

as a free parameter, some algebra yields the following

A’LATTICE BOLTZMANN SCHEME FOR SEMICONDUCTOR DYNAMICS 139

orthogonal set of eigenvectors (scalar products beingweighted with the array m (1, 1, m, m).

V/(1)- 1i--(1,1,1,1)" V/(2) -ci- (1,-1,2,-2)" (11)

V/(3) c/2 2 (- 1,- 1,2,2)" 1//(4)Ci(C2i 3) (--2,2,2,--2) (12)

corresponding to the specific choice m . yielding

Te= c. =2.

Straightforward projection upon the four eigenvec-tors yields:

ap + axpU o ( 3)

)tpu + axS -2ax9 (14)

)tS + axh -2axJ + )3 (S Se) (5)

a,h+2axS=M(h-h9 (16)where:

p imiNiVi(1) J- imiNiVi(2)

S ZimiNiVi(3) h ZimiNiVi(4) (1)

According to standard practice, we now make an adi-abatic assumption on both the deviatoric stress tensorS and the heat-like flux h

S Se + f )xJ + )Q axh (17)

h he + 2,-1{9xS (18)Since fluctuations do not contribute to conservedhydrodynamic fields, one has p foe, J je, qe O,which yields Se pu2 + 9(T- Te), Te 2.By plugging eq.(26) into (23), the momentum

equation reads

atpu+axpu2- -axpr )v-’ax2(pu) (19)

This is exactly the Navier-Stokes equation for a one-dimensional fluid with viscosity x) -9vAs to the energy equation, by using (27) and (25)

we obtain

atpw +a.pw -2a.puT + 2)v2’ax2(W- 2@)3 (W pu2 pTe) (20)+5-

where T’= T- Te is the temperature fluctuation

This is "almost" the correct energy equation, with a

Baccarani-Wordeman type of collisional relaxation, adetailed definition of "almost" to be clarified in the

sequel.Let us now discuss how to model the effects spe-

cific to the physics of semiconductors: the presence ofan external electric field and collisional relaxation.

Electric field effects are readily incorporated byadding a suitable forcing term F to the rhs of eq.(1).

This term must deposit )e -egE units of momen-

tum and uJe units of energy per unit volume of the

flow. Straightforward decomposition of the forcingterm F onto the four eigenvectors defined in yields

the desired expression:

JF. c UJF C2i 2)+ (21)/=ml M2

M1- micZi- 6; M2 ZmiQ2i- 2 (22)

A similar trick can also be applied to mimic theeffects of the collisional term with a relaxation term

of the form

Gi -(Ui Ni /’r, (23)

where N/ is a zero-flow (global) equilibrium distri-bution function.

This yields a momentum relaxation -J/z, and an

energy relaxation (W- Wo)/’c where W0 9Te/2. Thefinal energy equation reads now as follows

atW -I- axuW -axpuT -Jr" - lax2 (W pTe-7-)+)-23 (W 9u

2 9Te) 9eEu (24)2

This is exactly the GHE energy equation with

K2

provided the following conditions are

met:

1. )x2p << ax2W2. u2 <<Te=2

3. ,3X -1

140 S. SUCCI and E VERGARI

The first two conditions both subsume the sameassumption of weak-compressibility which lies at

very heart of the LBE theory. In this respect, theydon’t set any further limitation beyond those currentlyaccepted for usual fluid dynamics purposes.The third condition restricts the applicability of the

model to situations where energy and momentumrelax on the same time scale, this time scale beingalso the one controlling the thermal conductivity.

5. FINAL CONSIDERATIONS

We have presented a minimal extended Lattice-Boltz-mann scheme catering for the inclusion of the physi-cal effects required to model the behaviour of hotelectrons in semiconductor devices: the presence ofan external electric field and local relaxation due to

electron collisions with the phononic background.Owing to its minimality, the scheme doesn’t allow formore than two independent time scales. Once theseare chosen so as to control the momentum and energydiffusivity, no additional degrees of freedom areavailable to describe collisional relaxation of momen-tum and energy independently. This limitationshouldn’t bear any fundamental character, and can

possibly be lifted by enlarging the set of discrete

speeds.

A more fundamental limitation, which is sharedwith "plain" LBE schemes, is that the model onlyoperates for weakly co.mpressible flows characterized

by Mach numbers well below one.

In view of such a limitation the scheme mightprove viable only for the description of the bulk flowdynamics, while regions of strong heterogeneity are

likely to call for more sophisticated schemes involv-

ing more speeds and/or a non-uniform grid discretiza-

tion. A careful investigation of this issue is currentlyunderway.

AcknowledgementsThe work was

n 92.00.53801partially supported by CNR

References[1] U. Frisch, D. d’Humires, B. Hasslacher, P. Lallemand,

Y. Pomeau, J. P. Rivet, Complex Systems (1987) 649[2] Physica D Vol. 47 (1991), n. 1-2 special issue[3] J. Stat. Phys., 68, 3/4 (Aug. 1992), special issue[4] U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56

(1986) 15.[5] R. Benzi, S. Succi, M. Vergassola, Physics Reports 222 (3)

1992, and references therein[6] A. Forghieri, R. Guerrieri, E Ciompoli, A.Gnudi, M.Rudan,

G.Baccarani, IEEE Trans.on Comp.Aided Desig. vol.7, 1988[7] C. Gardner, J. Jerome, D. Rose, IEEE Trans. on Comp. Aided

Desig. vol.8, 1989[8] C. Gardner. IEEE Trans. on Comp.Aided Desig. vol.38, 1991[9] A.M. Anile, S. Pennisi, Phys. Rev. B vol. 46, n.20, 13 186,

1992.

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Lattice Boltzmann Scheme Semiconductor Dynamicsdownloads.hindawi.com/archive/1998/054940.pdf · Wediscuss an extension ofthe Lattice Boltzmann method which mayprove useful for the