MATHEMATICAL AND COMPUTATIONAL PROBLEMS IN SEMICONDUCTOR DEVICE TRANSPORT A.M.ANILE DIPARTIMENTO DI...
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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
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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
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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
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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
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EXAMPLE OF BRAVAIS LATTICE IN 2D
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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))
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Diamond lattice of Silicon and Germanium
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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
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Direct lattice
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Reciprocal lattice
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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^.
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FIRST BRILLOUIN ZONE FOR SILICON
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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
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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 .
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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)
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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:
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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.
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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))
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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
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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.
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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)
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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.
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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)
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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.
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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]
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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
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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.
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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.
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SILICON MATERIAL MODEL
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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
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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)
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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
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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
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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.
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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
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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.
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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
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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))
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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).
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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)>
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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.
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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
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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.
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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
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WHICH GIVES
fME =exp[-AA/kB]
IF A =(1,v,,v) AND A=(, i, w, wi)
fME =exp[-(/kB+w+ivi+w
ivi)]
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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)
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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).
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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
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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)
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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
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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.
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TEST FOR THE EXTENDED MODEL WITH 1D STRUCTURESMUSCATO & ROMANO, 2001
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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:
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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
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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
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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)
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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
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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
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.
FINAL FORM OF THE EQUATIONS
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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.
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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
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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.
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THE MESFET
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MONTE CARLO SIMULATION:INITIAL PARTICLE DISTRIBUTION
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INITIAL POTENTIAL
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INTERMEDIATE STATE PARTICLE DISTRIBUTION
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INTERMEDIATE STATE POTENTIAL
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FINAL PARTICLE DISTRIBUTION
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FINAL STATE POTENTIAL
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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
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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