comsol days 036:comsol光学・半導体セミナー(東京会 …...interface calculates wave...
TRANSCRIPT
Semiconductor Module
Optics SeminarJuly 18, 2018
Yosuke Mizuyama, Ph.D.
COMSOL, Inc.
The COMSOL® Product Suite
Governing Equations
Schrödinger Equation
Semiconductor
Semiconductor
Optoelectronics, FD
Semiconductor
Optoelectronics, BE
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Semiconductor
Governing Equations
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Semiconductor ModelsCarrier statistics
• Arora (LI)
• Caughey-Thomas
Heterojunction
• Quasi-Fermi continuity
• Thermionic emissions
Mobility
• Arora (LI)
• Caughey-Thomas (E)
• Fletcher (C)
• Lombardi surface (S)
• Power law (L)
Recombination
• Auger
• Direct
• Trap-assisted
Generation
• Okuto Crowell
Tunneling
• Fowler-Nordheim
Transition
• Indirect optical
• Optical
Metal contact
• Ideal ohmic
• Ideal Schottky
Band gap narrowing
• Slotboom
• Jain-Roulston
Applications
PN-Junction 1D
PN-Diode Circuit
Heterojunction 1D
Bipolar Transistor
EEPROM
MOSFET
Breakdown in a MOSFET
Small Signal Analysis of a MOSFET
Bipolar Transistor Thermal
GaAs PIN Photodiode
ISFET
GaAs PN Junction Infrared LED Diode
Lombardi Surface Mobility
Si Solar Cell 1D
Apps
Wavelength Tunable LED
Si Solar Cell with Ray Optics
Model & Features
Semiconductor ModelsCarrier statistics
• Arora (LI)
• Caughey-Thomas
Heterojunction
• Quasi-Fermi continuity
• Thermionic emissions
Mobility
• Arora (LI)
• Caughey-Thomas (E)
• Fletcher (C)
• Lombardi surface (S)
• Power law (L)
Recombination
• Auger
• Direct
• Trap-assisted
Generation
• Okuto Crowell
Tunneling
• Fowler-Nordheim
Transition
• Indirect optical
• Optical
Metal contact
• Ideal ohmic
• Ideal Schottky
Band gap narrowing
• Slotboom
• Jain-Roulston
Carrier Statistics
Maxwell-Boltzmann
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Fermi-Dirac
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Carrier Statistics
n-type
p-type
Nondegenerate Degenerate
Mobility Models
Electron mobility in a symmetric dual-gate MOSFET computed
using the Caughey-Thomas mobility model.
Electron mobility in a symmetric dual-gate MOSFET computed
using the Caughey-Thomas mobility model.
• Support arbitrary combination of multiple mobility models
• User defined
• Power-law – Effect of phonons
• Arora– Effect of phonons
– Effect of ionized impurities
• Fletcher– Effect of carrier-carrier scattering
• Lombardi Surface– Surface scattering
• Caughey-Thomas– High field velocity scattering
Generation and Recombination Models
• Recombination– User defined
– Direct
– Trap-Assisted
– Auger
• Generation– User Defined
– Impact Ionization
Summary of the implemented recombination processes for direct
(e.g. GaAs) and indirect (e.g. Si) band-gaps.
Summary of the implemented recombination processes for direct
(e.g. GaAs) and indirect (e.g. Si) band-gaps.
Tunneling
• Tunneling through
insulating boundaries
supported
– Fowler-Nordheim
tunneling model
– User defined tunnel
currents
Tunnel current into the floating contact of an EEPROM
device during program and erase events.
Tunnel current into the floating contact of an EEPROM
device during program and erase events.
(Direct)Optical Transition• Parabolic direct band gap
• Input data:– Transition strength
– Spontaneous life time
– Momentum/dipole matrix element
– Kane 4-band model
• Output:
– Optical absorption
– Spontaneous emission
– Stimulated emission
– Index changeDirect bandgap model for optical transition Direct bandgap model for optical transition
Indirect Optical Transitions
• Input:– Predefined empirical absorption data for Si, or
– Refractive index
– Electric field amplitude
• Output:– Absorption/photogeneration
Empirical absorption data for siliconEmpirical absorption data for silicon
Heterojunction ModelThermionic emissions Homojunction Heterojunction
n p
n p
Bandgap Narrowing• Slotboom: Empirical model
frequently used for Silicon
• Jain-Roulston: Physics based model, associated material properties available for most application library materials
• Arbitrary user defined models– Specify expressions the
proportion from the conduction and valence bands
Doping
• Analytic Doping Model
– Cuboidal region of uniform dopant concentration
– Decays into a background level with Gaussian, Linear, or Error Function
• Geometric Doping Model
– Define from selected boundaries
– Gaussian, Linear, or Error Function profiles
Dopant Ionization
• Complete ionization
• Incomplete ionization
– Standard/Ionization fraction
– Ramping
Dopant Ionization • Both complete and incomplete
ionization is supported– Standard model provided, or
specify user defined ionization fraction
– Continuation now supported for dopant ionization to enable easier model setup
– This enables incomplete ionization effects to be slowly ramped on automatically using the continuation machinery
Traps• Spatial distribution
• Can be added to Thin Insulator Gate, Insulator, Insulator Interface boundaries
• Discrete trap energy levels
• Multiple different discrete energy levels permitted
• Continuous energy distributions can be created
• Gaussian, rectangle, or exponential functions.
Trap Species Carriers Trapped Charge Unoccupied Charge Occupied
Donor Electrons Positive Neutral
Acceptor Holes Negative Neutral
Neutral electron Electrons Neutral Negative
Neutral hole Holes Neutral Positive
Metal-Semiconductor Contacts• Biasing options
– Voltage-driven
– Current-driven
– Power-driven
– Connect to a circuit (acting as either a current source or a voltage source)
• Ideal Schottky contact– Thermionic emission
– Ideal and non-ideal barrier height
• Ideal ohmic contactIdeal SchottkyIdeal Schottky Ideal ohmicIdeal ohmic
Assumptions
• Relaxation-time approximation
• Parabolic energy bands
• Ignore complex physics at the metal-semiconductor interface
(scattering/potential fluctuation/surface roughness/mirror
image, etc.)
• Ignore complex time-dependent conductivity
Simplifications
• Maxwell-Boltzmann (default) for nondegenerate devices
• Majority carrier devices are analyzed by one carrier (majority) only and the minority carrier concentration is estimated by mass action law
Energy Band
• Due to Bragg reflection caused by the periodic
potential of lattice
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Bragg conditionBragg condition
Schrödinger equation for the wave function
for an electron in lattice
Schrödinger equation for the wave function
for an electron in lattice
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Solution Method• Finite volume method (default)
– Gives the best accuracy for the current density
– Scharfetter-Gummel scheme
• Finite element log
• Quasi-Fermi level
Example of a finite volume discretization in 1D.Example of a finite volume discretization in 1D.
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Mesh boundaryComputational node
(0th order)
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Meshing
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Multiphysics
Optoelectronics Multiphysics Interfaces
Electromagnetic Wave
Interface calculates wave
propagation
Semiconductor Interface
calculates absorption from EM
intensity of carrier dynamics
Spontaneous & Stimulated
emission calculated, along with
change in refractive index
New refractive index fed back
into Electromagnetic Wave
Interface
Semiconductor InterfaceElectromagnetic Interface
Thermal Coupling
Semiconductor Heating Source Resulting Temperature
Schrödinger Equation
Governing Equations
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Time-dependent
Stationary
Eigenstate
Features
• Single-particle Schrödinger equation
• General quantum mechanical problems in 1D, 2D, and 3D
• Electron and hole wave functions in quantum-confined
systems
• PML for stationary problems
Applications
Quantum Wire
Harmonic Potential
Super Lattice
Double Barrier 1D
Gross-Pitaevskii Equation
Superlattice Band Gap Tool
Contact & Information
• www.comsol.com/contact
• www.comsol.com– Blog
– Reference manual
– Application Libraries
– Application Gallery