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1 Semiconductor Module Optics Seminar July 18, 2018 Yosuke Mizuyama, Ph.D. COMSOL, Inc.
2 The COMSOL Product Suite
3 Governing Equations Semiconductor Schrödinger Equation Semiconductor Optoelectronics, FD Semiconductor Optoelectronics, BE 1 1 & Semiconductor+! 0 " Semiconductor+ ' 2) ' $&' ð $% $%! " 0
4 Semiconductor
5 Governing Equations 1 1
6 Semiconductor Models Carrier statistics Arora (LI) Caughey-Thomas 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 Heterojunction Quasi-Fermi continuity Thermionic emissions Band gap narrowing Slotboom Jain-Roulston Metal contact Ideal ohmic Ideal Schottky
7 Applications
8 PN-Junction 1D
9 PN-Diode Circuit
10 Heterojunction 1D
11 Bipolar Transistor
12 EEPROM
13 MOSFET
14 Breakdown in a MOSFET
15 Small Signal Analysis of a MOSFET
16 Bipolar Transistor Thermal
17 GaAs PIN Photodiode
18 ISFET
19 GaAs PN Junction Infrared LED Diode
20 Lombardi Surface Mobility
21 Si Solar Cell 1D
22 Apps
23 Wavelength Tunable LED
24 Si Solar Cell with Ray Optics
25 Model & Features
26 Semiconductor Models Carrier statistics Arora (LI) Caughey-Thomas 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 Heterojunction Quasi-Fermi continuity Thermionic emissions Band gap narrowing Slotboom Jain-Roulston Metal contact Ideal ohmic Ideal Schottky
27 Maxwell-Boltzmann Carrier Statistics +,-. exp2/ -. Fermi-Dirac exp / -. 1
28 Carrier Statistics n-type p-type Nondegenerate Degenerate
29 Mobility Models 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 Electron mobility in a symmetric dual-gate MOSFET computed using the Caughey-Thomas mobility model.
30 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.
31 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.
32 (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 change Directbandgap model for optical transition
33 Indirect Optical Transitions Input: Predefined empirical absorption data for Si, or Refractive index Electric field amplitude Output: Absorption/photogeneration Empirical absorption data for silicon
34 Heterojunction Model Thermionic emissions Homojunction Heterojunction n p n p
35 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
36 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
37 Complete ionization Incomplete ionization DopantIonization Standard/Ionization fraction Ramping
38 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 Dopant Ionization
39 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
40 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 contact Ideal Schottky Ideal ohmic
41 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
42 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
43 Energy Band Due to Bragg reflection caused by the periodic potential of lattice < & 2) & 2) Vcos29: ; ' <' Schrödinger equation for the wave function for an electron in lattice 9 ; 29 ; 39 ; k % )9 ; Bragg condition
44 Solution Method Finite volume method (default) Gives the best accuracy for the current density Scharfetter-Gummel scheme Finite element log Quasi-Fermi level : : >, >?/ : >? Mesh boundary Computational node (0 th order) :, >?/ > B C E >? 1 B C E A basic Scharfetter-Gummel scheme Example of a finite volume discretization in 1D.
45 Meshing Mesh needs to resolve the Debye length -.F F
46 Multiphysics
47 Optoelectronics Multiphysics Interfaces Electromagnetic Interface Electromagnetic Wave Interface calculates wave propagation Semiconductor Interface Semiconductor Interface calculates absorption from EM intensity of carrier dynamics New refractive index fed back into Electromagnetic Wave Interface Spontaneous & Stimulated emission calculated, along with change in refractive index
48 Thermal Coupling Semiconductor Heating Source Resulting Temperature
49 Schrödinger Equation
50 Governing Equations Time-dependent & ' 2) ' $&' ð Stationary Eigenstate & ' 2) ' 2'
51 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
52 Applications
53 Quantum Wire
54 Harmonic Potential
55 Super Lattice
56 Double Barrier 1D
57 Gross-PitaevskiiEquation
58 Superlattice Band Gap Tool
59 Contact & Information Blog Reference manual Application Libraries Application Gallery
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