MATHEMATICAL AND COMPUTATIONAL PROBLEMS IN SEMICONDUCTOR DEVICE TRANSPORT

• A.M.ANILE

• DIPARTIMENTO DI MATEMATICA E INFORMATICA

• UNIVERSITA’ DI CATANIA

• PLAN OF THE TALK:

• MOTIVATIONS FOR DEVICE SIMULATIONS

• PHYSICS BASED CLOSURES

• NUMERICAL DISCRETIZATION AND SOLUTION STRATEGIES

• RESULTS AND COMPARISON WITH MONTE CARLO SIMULATIONS

• NEW MATERIALS

• FROM MICROELECTRONICS TO NANOELECTRONICS

MODELS INCORPORTATED IN COMMERCIAL SIMULATORS

• ISE or SILVACO or SYNAPSIS

• DRIFT-DIFFUSION

• ENERGY TRANSPORT

• SIMPLIFIED HYDRODYNAMICAL

• THERMAL

• PARAMETERS PHENOMENOLOGICALLY ADJUSTED--- TUNING NECESSARY- :

• a) PHYSICS BASED MODELS REQUIRE LESS TUNING

• b) EFFICIENT AND ROBUST OPTIMIZATION ALGORITHMS

THE ENERGY TRANSPORT MODELS WITH PHYSICS BASED TRANSPORT COEFFICIENTS

• IN THE ENERGY TRANSPORT MODEL IMPLEMENTED IN INDUSTRIAL SIMULATORS THE TRANSPORT COEFFICIENTS ARE OBTAINED PHENOMENOLOGICALLY FROM A SET OF MEASUREMENTS

• MODELS ARE VALID ONLY NEAR THE MEASUREMENTS POINTS. LITTLE PREDICTIVE VALUE.

• EFFECT OF THE MATERIAL PROPERTIES NOT EASILY ACCOUNTABLE (WHAT HAPPENS IF A DIFFERENT SEMICONDUCTOR IS USED ?): EX. COMPOUDS, SiC, ETC.

• NECESSITY OF MORE GENERALLY VALID MODELS WHERE THE TRANSPORT COEFFICIENTS ARE OBTAINED, GIVEN THE MATERIAL MODEL, FROM BASIC PHYSICAL PRINCIPLES

ENERGY BAND STRUCTURE IN CRYSTALS

• Crystals can be described in terms of Bravais lattices

• L=ia(1)+ja(2)+la(3) i,j,l

• with a(1), a(2) , a(3) lattice primitive vectors

EXAMPLE OF BRAVAIS LATTICE IN 2D

Primitive cellA connected subset B 3 is called aprimitive cell of the lattice

- The whole space (3) is covered by the

union of translates of B by the latticevectors.

if :

- The volume of B equals a(1).(a(2)a(3))

Diamond lattice of Silicon and Germanium

RECIPROCAL LATTICE

• The reciprocal lattice is defined by

• L^ =ia(1)+ja(2)+la(3) i,j,l

• with a(1) , a(2) , a(3) reciprocal vectors

• a(i).a(j) =2ij

Direct lattice

Reciprocal lattice

BRILLOUIN ZONE

The first Brillouin zone B is theprimitive cell of the reciprocal lattice L^

consisting of those points which arecloser to the origin than to any otherpoint of L^.

FIRST BRILLOUIN ZONE FOR SILICON

BAND STRUCTURE

Consider an electron whose motion is governed by the potential VL generatedby the ions located at the the points of the crystal lattice L. The Schrodingerequations is

H=

with H the Hamiltonian

H= -(h2/2m) -qVL

The bounded eigenstates have the form:

(x)=exp(ik.x)uk(x) , x3

uk(x+X)=uk(x) , XL

EXISTENCE OF SOLUTIONSThis is a second order self-adjoint ellipticeigenvalue problem posed on a primitive cell ofthe crystal lattice L.

One can prove the existence of an infinitesequence of eigenpairs

=l(k), uk(x)=uk,l(x) , l

From(x+X)=exp(ik.X) (x), x3

it follows that the set of wavefunctions and theenergies are identical for any two wavevectors whichdiffer by a reciprocal lattice vector. Therefore one can constrain the wavevector k to theBrillouin zone B .

ENERGY BAND AND MEAN VELOCITY

The function l=l(k) on the Brillouin zone describes thel-the energy band of the crystal.One can prove that the mean electron velocity is

vl(k)=(1/h) grad l(k)

The motion of electrons in the valenceband can be described as that of quasi-particles with positive charge in theconduction band (holes)

The energy band structure of crystals can be obtained at theexpense of intensive numerical calculations (andsemiphenomenologically) by the quantum theory of solids.

For describing electron transport, for mostapplications, however one can use simpleanalytical models. The most common onesare:

PARABOLIC BAND APPROXIMATION

(k)=(h2k2)/2m*

where m* is the electron effective mass.

Notice that with this expression for theenergy, the mean electron velocity

v=hk/m*

which is the same as for a classicalparticle.

NON PARABOLIC KANE APPROXIMATION(1+)=(h2k2)/2m* =(k)where is the parabolicity parameter(0.5 for Si).

The velocity in this case is :

v=( hk/m*)/(1+4(k))

DERIVATION OF THE BTE

Under the assumption that external forces(electric field E ) are almost constant overa length comparable to the physicaldimensions of the wave packet describingthe motion of an electron for an ensembleof M electrons belonging to the sameenergy band with wavevectors ki , i=1, …M, one obtains for the joint probabilitydensity f(xi, ki ,t), with q the absolutevalue of the electron electric charge

tf + v(ki).gradi f –(1/h)qE.gradkf =0

By proceeding as in the classical theoryone obtains the hierarchy BBGKY ofequations. Then under the usualassumptions (low correlations, separationbetween long range and short range forces,etc.) one obtains formally the semiclassicalVlasov equation

tf +v(k).gradif –(1/h)qE.gradkf =0

for the one particle distribution function f(x,k,t). Here theelectric field E(x,t) is the sum of the external electric field andthe self-consistent one due to the long range electrostaticinteractions.

The above description of electron motion is valid for an idealperfectly periodic crystal. Real semiconductors cannot beconsidered as ideal periodic crystals for several reasons:

doping with impurities (in order to control the electrical conductivity); thermal vibrations of the ions off their positions in the lattice, which

destroy the periodicity of the interaction potential.

These effects are described by the collision operator C(f) and leads to theSemiclassical semiconductor Boltzmann Transport Equation:

tf +v(k).gradif –(1/h)qE.gradkf =C(f)

THE COLLISION OPERATOR

C(f)=Cld(f)+Ce(f)

where Cld(f) represents the lattice-defects

collisions (impurities and phonons)

Cld(f)=Cimp(f)+Cph(f)

and Ce(f) the electron-electron binary

collisions.

The collision operator for the

collisions with impurities is :

Cimp(f)(k)=B imp(k,k’)(‘-)(f-f’)dk’

where is the Dirac measure. Also

imp(k,k’)= imp(k’,k)

The collision operator with optical

phonons is :

Cph(f)(k)=Bph(k,k’)[(Nph+1)(-

‘+ph)+Nph(-‘-ph)]f’(1-f)-

[(Nph+1)(‘-+ph)+Nph(‘-ph)]f(1-

f’)dk’where ph is the phonon energy (acoustic

and optical branch) and Nph is the phonon

occupation number given by the Bose-

Einstein statistics

Nph = 1/(exp(ph/KBTL)-1) where TL is the lattice temperature.

FUNDAMENTAL DESCRIPTION:

• The semiclassical Boltzmann transport for the electron distribution function f(x,k,t)

• tf +v(k).xf-qE/h kf=C[f]

• the electron velocity

• v(k)=k(k)

• (k)=k2/2m* (parabolic band)(k)[1+(k)]= k2/2m* (Kane dispersion relation)

• The physical content is hidden in the collision operator C[f]

PHYSICS BASED ENERGY TRANSPORT MODELS

• STANDARD SIMULATORS COMPRISE ENERGY TRANSPORT MODELS WITH PHENOMENOLOGICAL CLOSURES : STRATTON.

• OTHER MODELS (LYUMKIS, CHEN, DEGOND) DO NOT START FROM THE FULL PHYSICAL COLLISION OPERATOR BUT FROM APPROXIMATIONS.

• MAXIMUM ENTROPY PRINCIPLE (MEP) CLOSURES (ANILE AND MUSCATO, 1995; ANILE AND ROMANO, 1998; 1999; ROMANO, 2001;ANILE, MASCALI AND ROMANO ,2002, ETC.) PROVIDE PHYSICS BASED COEFFICIENTS FOR THE ENERGY TRANSPORT MODEL, CHECKED ON MONTE CARLO SIMULATIONS.

• IMPLEMENTATION IN THE INRIA FRAMEWORK CODE (ANILE, MARROCCO, ROMANO AND SELLIER), SUB. J.COMP.ELECTRONICS., 2004

DERIVATION OF THE ENERGY TRANSPORT MODEL FROM THE MOMENT EQUATIONS WITH MAXIMUM ENTROPY CLOSURES

• MOMENT EQUATIONS INCORPORATE BALANCE EQUATIONS FOR MOMENTUM, ENERGY AND ENERGY FLUX

• THE PARAMETERS APPEARING IN THE MOMENT EQUATIONS ARE OBTAINED FROM THE PHYSICAL MODEL, BY ASSUMING THAT THE DISTRIBUTION FUNCTION IS THE MAXIMUM ENTROPY ONE CONSTRAINED BY THE CHOSEN MOMENTS.

STARTING POINT:

THE SEMICLASSICAL BOLTZMANN TRANSPORT

FOR THE ELECTRON DISTRIBUTION FUNCTION

f(x,k,t)

tf +v(k).xf-qE/h kf=C[f]

THE ELECTRON VELOCITY

v(k)=k(k)

(k)[1+(k)]= k2/2m* (Kane dispersion relation)

THE COLLISION OPERATOR

C(f)=Cld(f)+Ce(f)

where Cld(f) represents the lattice-defects

collisions (impurities and phonons)

Cld(f)=Cimp(f)+Cph(f)

and Ce(f) the electron-electron binary

collisions.

SILICON MATERIAL MODEL

MOMENT EQUATIONS

BY MULTIPLYING THE BTE BY A SMOOTHFUNCTION (K) AND INTEGRATING OVER THE1ST BRILLOUIN ZONE B ONE FINDS

tM +B (k)v(k).xf dk –eE. B (k)kf dk=

B (k)C[f] dk

WITH

M =B (k)f dk

IT IS CONVENIENT TO CHOOSE (k) EQUAL TO

1, k, (k), k(k). THEN ONE OBTAINS THE

FOLLOWING MOMENT EQUATIONS (ASSUMING

PARABOLIC BAND OR THE KANE DISPERSION

RELATION)

tn+i(nVi) =0

t(nPi)+j(nUij)+neEi =nCi

P

t (nW)+i(nSi) +neVr E

r=nCW

t(nNi)+j(nRij)+neEj(Uij+Wij)=nCi

N

THE DEFINITION OF THE VARIABLES IS

n =B f dk electron density

V =(1/n)B f vdk average electron velocity

P =(1/n) B f k dk average crystal momentum

W=(1/n) B (k)f dk average electron energy

U =(1/n)B fvk dk flux of crystal momentum

S= (1/n) B fv (k) dkflux of energy

N= (1/n)B fk (k) dk N-vector

R= (1/n)B f (k)vk dk R-tensor

CP =(1/n)B C[f]k dkP-production

CW=(1/n)B C[f](k) dk energy production

CN =(1/n)B C[f]k (k) dk N-production

NOW WE CAN STATE THE CLOSURE PROBLEMS:

ASSUME AS FUNDAMENTAL VARIABLES n, V, W,S, WHICH HAVE A DIRECT PHYSICAL MEANING.THEN FIND EXPRESSIONS FOR :

A) THE FLUXES U, R AND THE VECTORS P, n

AND

B) THE PRODUCTION TERMS AS FUNCTIONS OFTHE FUNDAMENTAL VARIABLES.

WE SHALL ASSUME THE APPROACH BASED ON THE METHOD OF

EXPONENTIAL CLOSURES OR EQUIVALENTLY THE MAXIMUM ENTROPY

PRINCIPLE (MEP)

1. I.Muller & T.Ruggeri, “Extended Thermodynamics”, Springer-Verlag, 1993;

2. C.D.Levermore, J.Statistical Physics, 83, 331-407, (1996)

THE MEP IS FUNDAMENTALLY BASED ON INFORMATION THEORY AND

STATES THAT IF A SET OF MOMENTS MA IS GIVEN, FOR THE “MOST

PROBABLE “CLOSURE ONE MAY USE THE DISTRIBUTION FUNCTION FME

WHICH CORRESPONDS TO A MAXIMUM OF THE ENTROPY FUNCTIONAL

UNDER THE CONSTRAINTS THAT IT GIVES RISE TO THE GIVEN MOMENTS

MA =B A(k)f MEdk

THE MOMENT METHOD APPROACHTHE LEVERMORE METHOD OF EXPONENTIAL CLOSURES

We expound the method in the case of a simple kinetic equation .

Let F(x,v,t) be the one particle distribution function

defined on x3x

Satisfying a Boltzmann transport equation (BTE)

(L1)tf(x,v,t)+v.xf(x,v,t) =Q

where Q is the collision operator.

We assume that Q obeys the LocalDissipation Relation

(L2) Q(f)(x,v,t) log f dv 0

Let

(L3) H(f)=f log f –f

The Local Entropy is

(L4) = H(f)(x,v,t)dv

and the Local Entropy Flux

(L5) =v H(f)(x,v,t)dv

LEVERMORE’S CLOSURE ANSATZ:

substitute for f the expression

One has then = (x,t) such that exp(.m(v)) dv <

With this closure the moment equations give

(L8) t<m exp(.m)>+div(<mv exp(.m)>)=<m.Q(exp(.m)>

in the unknown .

F(,v)=exp(.m(v))

Th. The closure based on the distribution function

F(,v)=exp(.m(v)) corresponds to the formal solution

of the “entropy” constrained minimization problem

J(f) =J(exp(.m(v)))=minf {(f log f –f)dv ; fmdv

fixed}

where

={f L2 , f log f L2 , f0 }

are the Lagrange multipliers of the minimization

constraints, fixing fmdv , and -J(f) is the physical

entropy . When the solution exists it is unique (by

convexity).

HYPERBOLICITYLet us define the moments(L9) U() = <m exp(.m)> =U where

(L10) U= < exp(.m)>

is a strictly convex function of .

Also define the fluxes

(L11) A () = <mv exp(.m)> =A

where

(L12) A() = <v exp(.m)>

and the collision moments

(L13) S() = <m Q exp(.m)>

Then the moment system (L8) rewrites

(L14) t U() +div A () = S()

and for smooth solutions one has

(L15) U, t + A, x = S()

Now U, is symmetric and positive definite, A, is

symmetric and therefore the system is hyperbolic.

THEOREM1. There exists a scalar function (U) and a vector function (U)

with (U) convex function such that

(L.16) U (U).U A =U (U)

defined by the Legendre transformation of U :

(L.17) (V)=inf (.V-U())=(V).V-U((V))

(V)=(.A-A)((V))

with U((V))=V

THEOREM

2. Each smooth solution of the moment system satisfies

the entropy inequality

(L.18) t (U) +div (U) 0

3. (U) is the minimum of all entropy functions J(f)=(f

log f –f)dv subject to the constraint that the moments

<mf> are fixed.

APPLICATION OF THE METHOD:

THE EXPRESSION FOR THE ENTROPY DENSITY

s=-kBB[f logf +(1-f) log(1-f)]dk

IF WE INTRODUCE THE LAGRANGE MULTIPLIERS A THE PROBLEM OF

MAXIMIZING s UNDER THE MOMENT CONSTRAINTS IS EQUIVALENT TOMAXIMIZE

s’=s-AMA

THE LEGENDRE TRANSFORM OF s , WITHOUT CONSTRAINTS

s’ =0

WHICH GIVES

fME =exp[-AA/kB]

IF A =(1,v,,v) AND A=(, i, w, wi)

fME =exp[-(/kB+w+ivi+w

ivi)]

SMALL ANISOTROPY ANSATZ:

fME =exp[-(/kB+w+ivi+w

ivi)]

FORMAL SMALL PARAMETER. BYEXPANDING:

fME =exp[-(/kB+w)[1-X+2X2/2]

X= ivi+w

ivi

fME positive definite and integrable in R3

CRITICISM FOR THE GAS DYNAMICAL CASE (DREYER, JUNK &KUNIK, 2001 )AND MATHEMATICAL REMEDIES FOR THESEMICONDUCTOR CASE : JUNK (2003), JUNK & ROMANO (2004)

UP TO SECOND ORDER EXPANSIONS OF THE

CONSTITUTIVE FUNCTIONS FOR THE TENSORS

UijME , Rij

ME , IN TERMS OF THE ANISOTROPY

PARAMETER . COMPARISON OF THE 0-TH

ORDER TERM WITH THE RESULTS OF MONTE

CARLO SIMULATIONS FOR THREE BENCHMARK

DIODES (TANG ET AL., 1994; MUSCATO &

ROMANO, 2001).

SYSTEM OF CONSERVATION LAWS

EQUIVALENT TO A SYMMETRIC HYPERBOLIC

SYSTEM, WITH A CONVEX ENTROPY.

tn+i(nVi) =0

t(nPi)+j(nUij)+neEi =nCi

P

t (nW)+i(nSi) +neVr E

r=nCW

t(nSi)+j(nFij)+neEjGij=nCi

S

SCALING (V.ROMANO, M2AS,

2000) :

t=O(1/2), x=O(1/), V=O(), S=O()

W =O(1/2) where

W is defined from the energy

production rate

Cw =-(W-W0)/W

NUMERICAL TECHNIQUES

The aim is to solve the full non stationary equations.

REQUIREMENTS:

ACCURATE NUMERICAL SOLUTION OF THE

TRANSIENT ; SHOCK WAVES MIGHT ARISE

DURING THE TRANSIENT DUE TO

DISCONTINUITIES AT THE JUNCTIONS

NECESSITY OF HIGH ORDER TVD SCHEMES.

ANALYTICAL SOLUTION OF THE RIEMANN

PROBLEM FOR THE SYSTEM OF HYPERBOLIC

CONSERVATION LAWS NOT AVAILABLE

SCHEMES WHICH DO NOT USE THE RIEMANN

SOLVERS.

(Anile, Romano and Russo, SIAM J.APPL.MATH.

2000; Anile, Nikiforakis and Pidatella, SIAM

J.SCI.COMP., 2000)

NESSAYAHU-TADMOR SCHEME:

GENERAL NON LINEAR HYPERBOLIC SYSTEM

OF CONSERVATION LAWS

Ut +F(U)x =G(U,x,t)

U(x,0)=U(0)(x)

U(0,t)=Ul(t) , U(L,t)=Ur(t)

SPLITTING STRATEGY (STRANG). FOR THE

CONVECTIVE STEP

(staggered grid)

Un+1/2j+1/2 = (1/2)(Un+1

j +Unj+1 )+(1/8)(U'

j –U'j+1 )+

-(t/x)[F(Un+1/2j+1 )–F(Un+1/2

j )]

Un+1/2j =Un

j -(t/x)F'j

The values U'j and F'j are computed from cell averages

using UNO reconstruction.

CFL CONDITION 0.5

INITIAL CONDITIONS: n(x,0)=C(x) , T(x,0)=TL ,

V(x,0)=0, S(x,0)=0 .

TRANSMISSIVE BOUNDARY CONDITIONS FOR

THE HYDRODYNAMICAL VARIABLES

q(0)=TLln(C(x)/ni) , q(L)=TLln(C(x)/ni)+qVbias

THE DOPING PROFILE C(x) IS REGULARIZED

ACCORDING TO

C(x)=C(0)-d0(tanh(x-x1)/s - tanh(x-x2)/s)

x1=0.1 micron, x2=0.5 micron, s=0.01 micron

d0=C(0)(1-ND/N+D) , L=0.6 micron

or with a gaussian convolution integral.

TEST FOR THE EXTENDED MODEL WITH 1D STRUCTURESMUSCATO & ROMANO, 2001

SCALING (V.ROMANO, M2AS, 2000) :

t=O(1/2), x=O(1/), V=O(), S=O()

W =O(1/2) where

W is defined from the energy production rate

Cw =-(W-W0)/W

ONE OBTAINS, AS COMPATIBILITY

CONDITIONS:

tn+i(nVi) =0

t (nW)+i(nSi) +neVr Er=nCW

WITH THE CONSTITUTIVE EQUATIONS IN THE

FORM OF THE ENERGY TRANSPORT MODEL.

V=D11(W)log(n)+D12(W)W+D13(W)

S=D21(W)log(n)+D22(W)W+D23(W)

with Dij calculated with the MEP

and the submatrix Dij ,i,j=1,2 negative definite.

NO FREE PARAMETERS !!

TO BE COMPARED WITH THE

CONSTITUTIVE EQUATIONS IN

THE STANDARD FORM OF THE

ENERGY TRANSPORT MODEL

Jn=nn-Dn

n-nDnT Tn

Sn = -nKn Tn -(kB

n/q)Tn Jn Tn

To these equations one must add the thePoisson equation

NO FREE PARAMETERS !!

TO BE COMPARED WITH THE CONSTITUTIVE

EQUATIONS IN THE STANDARD FORM OF THE

ENERGY TRANSPORT MODEL

Jn=nn-Dn

n-nDnT Tn

Sn = -nKn Tn -(kB

n/q)Tn Jn Tn

PROPERTIES OF THE ENERGY TRANSPORTMODEL:- NON LINEAR PARABOLIC YSTEM WITH A

CONVEX ENTROPY SYMMETRIZABILITY INTERMS OF THE DUAL ENTROPYVARIABLES EXISTENCE ANDUNIQUENESS,STABILITY OF EQUILIBRIUMSTATE

- (ALBINUS, 1995; DEGOND, GENIEYS, &JUNGEL,1997; 1998)

- NUMERICAL SOLUTION: MARROCCO&MONTARNAL, 1996, 1998; MARROCCO,MONTARNAL &PERTHAME, 1996; DEGOND,PIETRA & JUNGEL, 2001;

USE ENTROPY VARIABLES FOR THE SYMMETRICSYSTEM; MARCHING IN TIME METHOD TOREACH THE STATIONARY SOLUTION; IMPLICITEULER WITH VARIOUS COUPLING SCHEMES;MIXED FINITE ELEMENTS DISCRETIZATION(RT0)

IDENTIFICATION OF THE THERMODYNAMIC VARIABLES

• ZEROTH ORDER M.E.P. DISTRIBUTION FUNCTION:

• fME =exp(-/kB - W)• ENTROPY FUNCTIONAL:

• s=-kBB[f logf +(1-f) log(1-f)]dk• WHENCE

• ds= dn+ kB Wdu• COMPARING WITH THE FIRST LAW OF THERMODYNAMICS

• 1/Tn =kB W ;n =- Tn

FORMULATION OF THE EQUATIONS WITH THERMODYNAMIC VARIABLES

• THEOREM : THE CONSTITUTIVE EQUATIONS OBTAINED FROM THE M.E.P. CAN BE PUT IN THE FORM

• Jn =(L11/Tn)n+L12(1/Tn)

• TnJ sn =(L21/Tn)n+L22(1/Tn)

• WITH

• L11= -nD11/kB ;

• L12= -3/2 nkBTn2D12+nD12Tn(log n/Nc -3/2);

• L22= -3/2 nkBTn2D22+nnD11Tn(log n/Nc -3/2)-L12[kBTn(log n/Nc -3/2)+n]

• WHERE n =-n +q

• ARE THE QUASI-FERMI POTENTIALS, n THE ELECTROCHEMICAL POTENTIALS

.

FINAL FORM OF THE EQUATIONS

PROPERTIES OF THE MATRIX A

• A11=q2L11

• A12=-q2L11-qn(3/2)[D11Tn+kBTn2D12]

• A21=q2L11n+qL12

• A22= q2L11n2+2qL21 n+L22

• THE EINSTEIN RELATION D11=-KBTn/Q D13 HOLDS

• BUT THE ONSAGER RELATIONS (SYMMETRY OF A) HOLD ONLY FOR THE PARABOLIC BAND EQUATION OF STATE.

COMPARISON WITH STANDARD MODELS

• A11=nnqTn

• A12=nnqTn (kBTn /q -n+)

• A12= A21

• A22=nnqTn [(kBTn /q -n+)2+(-c)(kBTn /q)2]• THE CONSTANTS , , c, CHARACTERIZE THE MODELS OF

STRATTON, LYUMKIS, DEGOND, ETC. n IS THE MOBILITY AS FUNCTION OF TEMPERATURE. IN

THE APPLICATIONS THE CONSTANTS ARE TAKEN AS PHENOMENOLOGICAL PARAMETERS FITTED TO THE DATA

NUMERICAL STRATEGY

•Mixed finite element approximation (the classical Raviart-Thomas RT0 is used for space discretization ).•Operator-splitting techniques for solving saddle point problems arising from mixed finite elements formulation .•Implicit scheme (backward Euler) for time discretization of the artificial transient problems generated by operator splitting techniques.•A block-relaxation technique, at each time step, is implemented in order to reduce as much as possible the size of the successive problems we have to solve, by keeping at the same time a large amount of the implicit character of the scheme.•Each non-linear problem coming from relaxation technique is solved via the Newton-Raphson method.

THE MESFET

MONTE CARLO SIMULATION:INITIAL PARTICLE DISTRIBUTION

INITIAL POTENTIAL

INTERMEDIATE STATE PARTICLE DISTRIBUTION

INTERMEDIATE STATE POTENTIAL

FINAL PARTICLE DISTRIBUTION

FINAL STATE POTENTIAL

COMPARISON

• THE CPU TIME IS VERY DIFFERENT (MINUTES FOR OUR ET-MODEL; DAYS FOR MC) ON SIMILAR COMPUTERS.

• THE I-V CHARACTERISTIC IS WELL REPRODUCED

• NEXT:

• COMPARISON OF THE FIELDS WITHIN THE DEVICE

PERSPECTIVES

• DEVELOP MODELS FOR COMPOUND MATERIALS USED IN RF AND OPTOELECTRONICS DEVICES

• INTERACTIONS BETWEEN DEVICES AND ELECTROMAGNETIC FIELDS (CROSS-TALK, DELAY TIMES, ETC.)

• DEVELOP MODELS FOR NEW MATERIALS FOR POWER ELECTRONICS APPLICATIONS : Sic

• EFFICIENT OPTIMIZATION ALGORITHMS