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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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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Schrödinger equation for the wave function

for an electron in lattice

Schrödinger equation for the wave function

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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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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