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  i j

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)  t f +v(k).  x f-qE/h  k f=C[f] the electron velocity v(k)=  k  (k)  (k)=k 2 /2m* (parabolic band)  (k)[1+  (k)]= k 2 /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 APPROACH THE LEVERMORE METHOD OF EXPONENTIAL CLOSURES

49. LEVERMORE’S CLOSURE ANSATZ:

51. HYPERBOLICITY

53. THEOREM

55. APPLICATION OF THE METHOD:

64. TEST FOR THE EXTENDED MODEL WITH 1D STRUCTURES MUSCATO & ROMANO, 2001

76. IDENTIFICATION OF THE THERMODYNAMIC VARIABLES ZEROTH ORDER M.E.P. DISTRIBUTION FUNCTION: f ME =exp(- /k B - W  ) ENTROPY FUNCTIONAL: s=-k B  B [f logf +(1-f) log(1-f)]dk WHENCE ds= dn+ k B W du COMPARING WITH THE FIRST LAW OF THERMODYNAMICS 1/T n =k B W ;  n =- T n

77. FORMULATION OF THE EQUATIONS WITH THERMODYNAMIC VARIABLES THEOREM : THE CONSTITUTIVE EQUATIONS OBTAINED FROM THE M.E.P. CAN BE PUT IN THE FORM J n =(L 11 /T n )  n +L 12  (1/T n ) T n J s n =(L 21 /T n )  n +L 22  (1/T n ) WITH L 11 = -nD 11 /k B ; L 12 = -3/2 nk B T n 2 D 12 +nD 12 T n (log n/N c -3/2); L 22 = -3/2 nk B T n 2 D 22 +n n D 11 T n (log n/N c -3/2)-L 12 [k B T n (log n/N c -3/2)+ n ] WHERE  n =- n +q  ARE THE QUASI-FERMI POTENTIALS, n THE ELECTROCHEMICAL POTENTIALS

78. . FINAL FORM OF THE EQUATIONS

79. PROPERTIES OF THE MATRIX A A 11 =q 2 L 11 A 12 =-q 2 L 11  -qn(3/2)[D 11 T n +k B T n 2 D 12 ] A 21 =q 2 L 11  n +qL 12 A 22 = q 2 L 11  n 2 +2qL 21  n +L 22 THE EINSTEIN RELATION D 11 =-K B T n /Q D 13 HOLDS BUT THE ONSAGER RELATIONS (SYMMETRY OF A) HOLD ONLY FOR THE PARABOLIC BAND EQUATION OF STATE.

80. COMPARISON WITH STANDARD MODELS A 11 =n  n qT n A 12 =n  n qT n (  k B T n /q  -  n +  ) A 12 = A 21 A 22 =n  n qT n [(  k B T n /q  -  n +  ) 2 +(  -c)(k B T n /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

81. 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.

85. THE MESFET

88. MONTE CARLO SIMULATION: INITIAL PARTICLE DISTRIBUTION

89. INITIAL POTENTIAL

90. INTERMEDIATE STATE PARTICLE DISTRIBUTION

91. INTERMEDIATE STATE POTENTIAL

92. FINAL PARTICLE DISTRIBUTION

93. FINAL STATE POTENTIAL

102. 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

126. 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