MATHEMATICAL AND COMPUTATIONAL PROBLEMS IN SEMICONDUCTOR DEVICE TRANSPORT

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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

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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

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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

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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

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EXAMPLE OF BRAVAIS LATTICE IN 2D
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Primitive cell
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Diamond lattice of Silicon and Germanium
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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

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Direct lattice
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Reciprocal lattice
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BRILLOUIN ZONE
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FIRST BRILLOUIN ZONE FOR SILICON
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BAND STRUCTURE
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EXISTENCE OF SOLUTIONS
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ENERGY BAND AND MEAN VELOCITY
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PARABOLIC BAND APPROXIMATION
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NON PARABOLIC KANE APPROXIMATION
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DERIVATION OF THE BTE
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THE COLLISION OPERATOR
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FUNDAMENTAL DESCRIPTION

• The semiclassical Boltzmann transport for the
  electron distribution function f(x,k,t)
• ?tf v(k).?xf-qE/h ?kfCf
• 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 Cf

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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, 2001ANILE, 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

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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.

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SILICON MATERIAL MODEL
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MOMENT EQUATIONS
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THE MOMENT METHOD APPROACHTHE LEVERMORE METHOD
OF EXPONENTIAL CLOSURES
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LEVERMORES CLOSURE ANSATZ
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HYPERBOLICITY
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THEOREM
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THEOREM
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APPLICATION OF THE METHOD
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TEST FOR THE EXTENDED MODEL WITH 1D
STRUCTURESMUSCATO ROMANO, 2001
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IDENTIFICATION OF THE THERMODYNAMIC VARIABLES

• ZEROTH ORDER M.E.P. DISTRIBUTION FUNCTION
• fME exp(-?/kB - ?W?)
• ENTROPY FUNCTIONAL
• s-kB?Bf 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

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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)??nL12?(1/Tn)
• TnJ sn (L21/Tn)??nL22?(1/Tn)
• WITH
• L11 -nD11/kB
• L12 -3/2 nkBTn2D12nD12Tn(log n/Nc -3/2)
• L22 -3/2 nkBTn2D22n?nD11Tn(log n/Nc
  -3/2)-L12kBTn(log n/Nc -3/2)?n
• WHERE
• ?n -?n q?
• ARE THE QUASI-FERMI POTENTIALS, ?n THE
  ELECTROCHEMICAL POTENTIALS

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. FINAL FORM OF THE EQUATIONS
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PROPERTIES OF THE MATRIX A

• A11q2L11
• A12-q2L11?-qn(3/2)D11TnkBTn2D12
• A21q2L11?nqL12
• A22 q2L11?n22qL21 ?nL22
• THE EINSTEIN RELATION D11-KBTn/Q D13 HOLDS
• BUT THE ONSAGER RELATIONS (SYMMETRY OF A) HOLD
  ONLY FOR THE PARABOLIC BAND EQUATION OF STATE.

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COMPARISON WITH STANDARD MODELS

• A11n??nqTn
• A12n??nqTn (?kBTn /q ? -?n?)
• A12 A21
• A22n??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

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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.

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THE MESFET
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MONTE CARLO SIMULATIONINITIAL PARTICLE
DISTRIBUTION
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INITIAL POTENTIAL
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INTERMEDIATE STATE PARTICLE DISTRIBUTION
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INTERMEDIATE STATE POTENTIAL
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FINAL PARTICLE DISTRIBUTION
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FINAL STATE POTENTIAL
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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

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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