PERSPECTIVES FOR SEMICONDUCTOR DEVICE SIMULATION: A KINETIC APPROACH

1. PERSPECTIVES FOR SEMICONDUCTOR DEVICE SIMULATION:A KINETIC APPROACH • A.M.ANILE • DIPARTIMENTO DI MATEMATICA E INFORMATICA • UNIVERSITA’ DI CATANIA • PLAN OF THE TALK: • MODELS IN COMMERCIAL DEVICE SIMULATORS • PHYSICS BASED MODELS • NEW NUMERICAL APPROACHES • EXTENSION TO NEW MATERIALS

2. 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]

3. SKETCH OF THE DERIVATION OF THE ENERGY TRANSPORT STRATTON MODEL • DECOMPOSE THE D.F. IN ISOTROPIC AND ANISOTROPIC PARTS: • f= F0+F1 • ASSUME RELAXATION TIME APPROXIMATION FOR C[f]=-F1/ • F1=-q E/h. k F0 -v .r F0; T ELECTRON TEMPERATURE, E=-  • Jn=nn-Dnn-nDnTTn • Sn = -nKnTn -(kBn/q)Tn Jn Tn • THE COEFFICIENTS  ,D, K,  ARE OBTAINED FROM THE EXPRESSION OF  AS A POWER LAW FUNCTION OF MICROSCOPIC ENERGY (PHENOMENOLOGICAL FROM BULK M.C. SIMULATIONS) • FURTHER PHENOMENOLOGICAL PARAMETERS ,  FOR NON PARABOLICITY AND NON MAXWELLIAN EFFECTS (Chen, Lyumkis, Ravaioli et al.) FINE TUNING OF THE PARAMETERS REQUIRED FOR DATA FITTING !

5. PHYSICS BASED ENERGY TRANSPORT MODELS • STANDARD SIMULATORS COMPRISE ENERGY TRANSPORT MODELS WITH PHENOMENOLOGICAL CLOSURES : STRATTON. FINE TUNING REQUIRED !! • OTHER MODELS ( DEGOND et al.) 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 • CHECK WITH M.C. SIMULATIONS FOR MESFET AND BJT

6. 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.

9. SILICON MATERIAL MODEL

10. FIRST BRILLOUIN ZONE FOR SILICON

11. MOMENT EQUATIONS

17. APPLICATION OF THE METHOD:

25. 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

26. 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

27. . FINAL FORM OF THE EQUATIONS

28. 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.

29. 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

30. 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.

34. THE MESFET

36. MONTE CARLO SIMULATION:INITIAL PARTICLE DISTRIBUTION

37. INITIAL POTENTIAL

38. INTERMEDIATE STATE PARTICLE DISTRIBUTION

39. INTERMEDIATE STATE POTENTIAL

40. FINAL PARTICLE DISTRIBUTION

41. FINAL STATE POTENTIAL

50. 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