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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 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 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) =2ij 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) , x3 , XL 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), x3 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)=Bph(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 • • • • 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 fvk 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)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 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 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 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(<mv 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 , 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). 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 () = <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) 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=-kBB[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 A =(1,v,,v) AND A=(, i, w, wi) fME =exp[-(/kB+ + iv + iv )] w i w i SMALL ANISOTROPY ANSATZ: fME =exp[-(/kB+ w+ ivi+ w ivi)] FORMAL SMALL PARAMETER. EXPANDING: fME =exp[-(/kB+ w)[1-X+ 2X2/2] BY 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 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)(Un+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=nn-D nn-nDnT Tn Sn = -nKn Tn -(kBn/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 VARIABLESEXISTENCE 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=-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