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

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]

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=-  J n =n  n  -D n  n-nD n T  T n S n = -nK n  T n -(k B  n /q)T n J n T n 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 !

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

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.

SILICON MATERIAL MODEL

FIRST BRILLOUIN ZONE FOR SILICON

MOMENT EQUATIONS

APPLICATION OF THE METHOD:

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

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

. FINAL FORM OF THE EQUATIONS

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.

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

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

A BIPOLAR JUNCTION TRANSISTOR BJT COMPARISON OF THE MEP ENERGY TRANSPORT MODEL WITH M.C. SIMULATIONS: NANOMETRIC BASE

FURTHER EXTENSIONS WIDE BANDGAP SEMICONDUCTORS WITH STRONGLY ANISOTROPIC BANDS AND SCATTERING FOR WHICH EVEN A PHENOMENOLOGICAL MODEL HAS NOT BEEN INTRODUCED : SiC. APPLICATION TO HIGH TEMPERATURE POWER DEVICES DISCRETE DOPANTS DISTRIBUTION : COUPLING THE PARTICLE KINETIC TRANSPORT SOLUTION TO THE KINETIC MONTE CARLO PROCESS DIFFUSION. APPLICATION TO NANOMETRIC DEVICES (<50 NM) WHERE THE FLUCTUATIONS OF THE DOPANT DISTRIBUTION ARE VERY IMPORTANT FOR ASSESSING THE ROBUSTNESS OF THE DEVICES.

FROM MICROTECHNOLOGY TO NANOTECHNOLOGY ACCORDING TO THE 2001 INTERNATIONAL TECHNOLOGY ROADMAP FOR SEMICONDUCTORS, MOSFETS WITH PHYSICAL CHANNEL LENGTH LESS THAN 10NM WILL BE MASS PRODUCED IN 2016 SUCH DEVICES WOULD HAVE APPROXIMATELY 10 SILICON ATOMS ALONG THE EFFECTIVE CHANNEL LENGTH RELEVANCE OF THE DISCRETE DOPANT DISTRIBUTION IN DEVICE SIMULATION FOR INTRISNSIC STOCHASTIC PARAMETER VARIATIONS RELEVANCE OF THE QUANTUM EFFECTS