MATHEMATICAL AND COMPUTATIONAL PROBLEMS IN SEMICONDUCTOR DEVICE TRANSPORT

1. 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

8. 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

9. 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

10. 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

11. EXAMPLE OF BRAVAIS LATTICE IN 2D

12. Primitive cell

13. Diamond lattice of Silicon and Germanium

14. 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

15. Direct lattice

16. Reciprocal lattice

17. BRILLOUIN ZONE

18. FIRST BRILLOUIN ZONE FOR SILICON

19. BAND STRUCTURE

20. EXISTENCE OF SOLUTIONS

21. ENERGY BAND AND MEAN VELOCITY

27. PARABOLIC BAND APPROXIMATION

28. NON PARABOLIC KANE APPROXIMATION

30. DERIVATION OF THE BTE

33. THE COLLISION OPERATOR

36. 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]

37. 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

38. 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.

40. SILICON MATERIAL MODEL

41. MOMENT EQUATIONS

47. THE MOMENT METHOD APPROACHTHE LEVERMORE METHOD OF EXPONENTIAL CLOSURES

49. LEVERMORE’S CLOSURE ANSATZ: