Transcript MODELLI MATEMATICI PER IL TRASPORTO DI CARICHE NEI

MATHEMATICAL AND COMPUTATIONAL PROBLEMS IN
SEMICONDUCTOR DEVICE TRANSPORT
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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
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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 cell
A connected subset B 3 is called a
primitive cell of the lattice
if :
- The volume of B equals a(1).(a(2)a(3)) 
vectors.
union of translates of B by the lattice
- The whole space
(3)
is covered by the
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) =2ij
Direct lattice
Reciprocal lattice
BRILLOUIN ZONE
The first Brillouin zone B is the
^
primitive cell of the reciprocal lattice L
consisting of those points which are
closer to the origin than to any other
^
point of L .
FIRST BRILLOUIN ZONE FOR SILICON
BAND STRUCTURE
Consider an electron whose motion is governed by the potential VL generated
by the ions located at the the points of the crystal lattice L. The Schrodinger
equations is
H=
with H the Hamiltonian
H= -(h2/2m) -qVL
The bounded eigenstates have the form:
(x)=exp(ik.x)uk(x)
uk(x+X)=uk(x)
, x3
, XL
EXISTENCE OF SOLUTIONS
This is a second order self-adjoint elliptic
eigenvalue problem posed on a primitive cell of
the crystal lattice L.
One can prove the existence of an infinite
sequence of eigenpairs
=l(k), uk(x)=uk,l(x) , l
From
(x+X)=exp(ik.X) (x), x3
it follows that the set of wavefunctions  and the
energies 
are identical for any two wavevectors which
differ by a reciprocal lattice vector.
Therefore one can constrain the wavevector k to the
Brillouin zone B .
ENERGY BAND AND MEAN
VELOCITY
The function l=l(k) on the Brillouin zone describes the
l-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 valence
band can be described as that of quasiparticles with positive charge in the
conduction band (holes)
The energy band structure of crystals can be obtained at the
expense of intensive numerical calculations (and
semiphenomenologically) by the quantum theory of solids.
For describing electron transport, for most
applications, however one can use simple
analytical models. The most common ones
are:
PARABOLIC BAND APPROXIMATION
(k)=(h2k2)/2m*
where m* is the electron effective mass.
Notice that with this expression for the
energy, the mean electron velocity
v=hk/m*
which is the same as for a classical
particle.
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 over
a length comparable to the physical
dimensions of the wave packet describing
the motion of an electron for an ensemble
of M electrons belonging to the same
energy band with wavevectors ki , i=1, …
M, one obtains for the joint probability
density f(xi, ki ,t), with q the absolute
value of the electron electric charge
tf + v(ki).gradi f –(1/h)qE.gradkf =0
By proceeding as in the classical theory
one obtains the hierarchy BBGKY of
equations. Then under the usual
assumptions (low correlations, separation
between long range and short range forces,
etc.) one obtains formally the semiclassical
Vlasov equation
tf +v(k).gradif –(1/h)qE.gradkf =0
for the one particle distribution function f(x,k,t). Here the
electric field E(x,t) is the sum of the external electric field and
the self-consistent one due to the long range electrostatic
interactions.
The above description of electron motion is valid for an ideal
perfectly periodic crystal.
Real semiconductors cannot be
considered 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 the
Semiclassical 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)
collisions.
the
electron-electron
binary
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)=Bph(k,k’)[(Nph+1)(‘+ph)+Nph(-‘-ph)]f’(1-f)[(Nph+1)(‘-+ph)+Nph(‘-ph)]f(1f’)dk’
where ph is the phonon energy (acoustic
and optical branch) and Nph is the phonon
occupation number given by the BoseEinstein 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
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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
FOR
THE
DISTRIBUTION
ELECTRON
TRANSPORT
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
C ld(f)
represents
the
lattice-defects
collisions (impurities and phonons)
Cld(f)=Cimp(f)+Cph(f)
and
Ce(f)
collisions.
the
electron-electron
binary
SILICON MATERIAL MODEL
MOMENT EQUATIONS
BY MULTIPLYING THE BTE BY A SMOOTH
FUNCTION (K) AND INTEGRATING OVER THE
1ST 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 =nCiP
 t (nW)+ i(nSi) +neVr Er=nCW
 t(nNi)+ j(nRij)+neEj(Uij+W ij)=nCiN
THE DEFINITION OF THE VARIABLES IS
n = B f dk
V =(1/n) B f vdk
electron density
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 fvk dk flux of crystal momentum
S= (1/n) B fv (k) dk flux of energy
N= (1/n)B fk (k) dk
N-vector
R= (1/n)B f (k)vk 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 OF
THE 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 APPROACH
THE 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 x3x 
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 Local
Dissipation 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
F(,v)=exp(.m(v))
One has then = (x,t) such that exp(.m(v)) dv <
With this closure the moment equations give
(L8)
t<m exp(.m)>+div(<mv exp(.m)>)=<m.Q(exp(.m)>
in the unknown .
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 , f0 }
 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).
HYPERBOLICITY
Let 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 () = <mv 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)
Now U,
U, t  + A, x = S()
is symmetric and positive definite, A, is
symmetric and therefore the system is hyperbolic.
THEOREM
1. 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=-kBB[f logf +(1-f) log(1-f)]dk
IF WE INTRODUCE THE LAGRANGE MULTIPLIERS
MAXIMIZING
MAXIMIZE
s
A
THE PROBLEM OF
UNDER THE MOMENT CONSTRAINTS IS EQUIVALENT TO
s’=s- AMA
THE LEGENDRE TRANSFORM OF
s’ =0
s , WITHOUT CONSTRAINTS
WHICH GIVES
fME =exp[- A /kB]
A
IF  =(1,v,,v) AND  A=(, i,  ,  i)
A
w
w
fME =exp[-(/kB+ + iv + iv )]
w
i
w
i
SMALL ANISOTROPY ANSATZ:
fME =exp[-(/kB+ w+ ivi+ w ivi)]

FORMAL
SMALL
PARAMETER.
EXPANDING:
fME =exp[-(/kB+ w)[1-X+ 2X2/2]
BY
X= ivi+ w ivi
fME positive definite and integrable in R3
CRITICISM FOR THE GAS DYNAMICAL CASE (DREYER, JUNK &
KUNIK, 2001 )AND MATHEMATICAL REMEDIES FOR THE
SEMICONDUCTOR CASE : JUNK (2003), JUNK & ROMANO (2004)
UP TO SECOND ORDER EXPANSIONS OF THE
CONSTITUTIVE FUNCTIONS FOR THE TENSORS
UijME , RijME , 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
ROMANO, 2001).
ET
AL.,
1994;
MUSCATO
&
SYSTEM
OF
CONSERVATION
LAWS
EQUIVALENT TO A SYMMETRIC HYPERBOLIC
SYSTEM, WITH A CONVEX ENTROPY.
 tn+ i(nVi) =0
 t(nPi)+ j(nUij)+neEi =nCiP
 t (nW)+ i(nSi) +neVr Er=nCW
 t(nSi)+ j(nFij)+neEjGij=nCiS
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
DISCONTINUITIES
TRANSIENT
AT
THE
DUE
TO
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
J.SCI.COMP., 2000)
and
Pidatella,
SIAM
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+1/2 j+1/2 = (1/2)(U n+1 j +Unj+1 )+(1/8)(U'j –U'j+1 )+
-(t/x)[F(Un+1/2 j+1 )–F(Un+1/2 j )]
Un+1/2 j =Unj -(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 STRUCTURES
MUSCATO & 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,
CONDITIONS:
AS
COMPATIBILITY
 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=nn-D nn-nDnT Tn
Sn = -nKn Tn -(kBn/q)Tn Jn Tn
PROPERTIES OF THE ENERGY TRANSPORT
MODEL:
- NON LINEAR PARABOLIC YSTEM WITH A
CONVEX ENTROPY SYMMETRIZABILITY IN
TERMS OF THE DUAL ENTROPY
VARIABLESEXISTENCE AND
UNIQUENESS,STABILITY OF EQUILIBRIUM
STATE
- (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 SYMMETRIC
SYSTEM; MARCHING IN TIME METHOD TO
REACH THE STATIONARY SOLUTION; IMPLICIT
EULER 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=-kBB[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+nnD11Tn(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=q2L11n+qL12
A22= q2L11n2+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=nnqTn
A12=nnqTn (kBTn /q  -n+)
A12= A21
A22=nnqTn [(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