Semiconductor Module User’s Guide Semiconductor Module User’s Guide © 1998–2018 COMSOL Protected by patents listed on www.comsol.com/patents, and U.S. Patents 7,519,518; 7,596,474; 7,623,991; 8,457,932; 8,954,302; 9,098,106; 9,146,652; 9,323,503; 9,372,673; and 9,454,625. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/comsol-license-agreement) and may be used or copied only under the terms of the license agreement. COMSOL, the COMSOL logo, COMSOL Multiphysics, COMSOL Desktop, COMSOL Server, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/trademarks. Version: COMSOL 5.4 Contact Information Visit the Contact COMSOL page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: • Support Center: www.comsol.com/support • Product Download: www.comsol.com/product-download • Product Updates: www.comsol.com/support/updates • COMSOL Blog: www.comsol.com/blogs • Discussion Forum: www.comsol.com/community • Events: www.comsol.com/events • COMSOL Video Gallery: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase Part number: CM024101 Contents Chapter 1: Introduction About the Semiconductor Module 12 Modeling Semiconductor Devices . 12 What Can the Semiconductor Module Do? . 13 The Semiconductor Module Physics Interface Guide . 13 Common Physics Interface and Feature Settings and Nodes . 15 The Semiconductor Module Study Capabilities by Physics Interface . 15 The Semiconductor Materials Database . 17 Where Do I Access the Documentation and Application Libraries? . 17 Overview of the User’s Guide 21 Chapter 2: Modeling Guidelines Physics for Semiconductor Modeling 24 Units of Band Gap and Electron Affinity . 25 Finite Element and Finite Volume Discretization 26 Defining the Carrier Mobility 29 Doping 31 Using the Analytic Doping Model. 31 Using the Geometric Doping Model . 33 Visualizing the Dopant Distribution . 36 Example of Multiple Superimposed Doping Profiles . 36 Connecting to Electrical Circuits 40 About Connecting Electrical Circuits to Physics Interfaces . 40 Connecting Electrical Circuits Using Predefined Couplings . 41 Connecting Electrical Circuits by User-Defined Couplings . 41 CONTENTS | 3 Solving . 42 Postprocessing. 42 Nonisothermal Models 44 SPICE Import and Export 46 SPICE Import . 46 SPICE Export . 47 Reference . 47 Small-Signal Analysis 48 Meshing 50 Solving 52 Quantum-Confined Systems 55 Chapter 3: Semiconductor Branch Interface The Semiconductor Interface 58 Domain, Edge, Boundary, Pair, and Point Nodes for the Semiconductor Interface . 61 Semiconductor Material Model . 63 Optical Transitions . 68 Indirect Optical Transitions . 72 Insulation. 74 Continuity/Heterojunction . 74 WKB Tunneling Model, Electrons . 75 WKB Tunneling Model, Holes . 76 Initial Values . 77 User-Defined Mobility Model . 77 Power Law Mobility Model (L). 78 Fletcher Mobility Model (C). 79 Caughey-Thomas Mobility Model (E) . 81 Lombardi Surface Mobility Model (S) . 83 4 | CONTENTS Arora Mobility Model (LI) . 85 Analytic Doping Model . 86 Geometric Doping Model . 90 Boundary Selection for Doping Profile . 91 Analytic Trap Density . 91 Geometric Trap Density . 93 Boundary Selection for Trap Density Profile . 94 Trap-Assisted Recombination . 94 Discrete Energy Level (Domains). 97 Continuous Energy Levels (Domains) . 98 Auger Recombination . 100 Direct Recombination . 101 Impact Ionization Generation . 102 User-Defined Recombination . 103 User-Defined Generation . 103 Metal Contact . 104 Thin Insulator Gate . 106 Trap-Assisted Surface Recombination . 108 Discrete Energy Level (Boundaries) . 108 Continuous Energy Levels (Boundaries) . 108 Floating Gate . 109 Insulator Interface . 109 Theory for the Semiconductor Interface 111 The Sommerfeld Model and the Density of States . 112 The Effect of Periodicity . 115 Electrons in a Perturbed Periodic Potential . 130 The Semiclassical Model . 135 The Boltzmann Equation . 139 Doped Semiconductors . 141 Specifying the Dopant Distribution . 143 Equilibrium Carrier Concentrations. 148 The Semiconductor Equations . 151 Mobility Models . 164 Traps . 169 Recombination and Generation . 174 Incomplete Ionization . 180 Band Gap Narrowing . 181 CONTENTS | 5 Optical Transitions . 183 Boundary Conditions . 201 Metal Contacts . 201 Thin Insulating Gates . 208 Continuity/Heterojunction . 211 Boundary Conditions for Charge Conservation . 214 Tunneling. 214 References for the Semiconductor Interface . 217 The Semiconductor Optoelectronics, Beam Envelopes Interface 221 Semiconductor-Electromagnetic Waves Coupling . 223 The Semiconductor Optoelectronics, Frequency Domain Interface 224 Chapter 4: The AC/DC Branch The Electrostatics Interface 228 Domain, Boundary, Edge, Point, and Pair Nodes for the Electrostatics Interface . 229 Charge Conservation . 232 Initial Values . 233 Space Charge Density . 233 Zero Charge . 233 Ground . 233 Electric Potential . 234 Surface Charge Density . 235 External Surface Charge Accumulation . 235 Electric Displacement Field . 236 Periodic Condition . 236 Thin Low Permittivity Gap . 237 Dielectric Shielding . 238 Terminal . 238 Floating Potential . 240 Distributed Capacitance . 242 6 | CONTENTS Line Charge . 242 Line Charge (on Axis) . 243 Line Charge (Out-of-Plane) . 243 Point Charge . 244 Point Charge (on Axis) . 245 Change Cross Section . 245 Change Thickness (Out-of-Plane). 246 Charge Conservation, Piezoelectric . 246 Electrostatic Point Dipole . 247 The Electrical Circuit Interface 248 Ground Node . 250 Voltmeter . 250 Ampère Meter. 251 Resistor . 251 Capacitor. 251 Inductor . 252 Voltage Source . 252 Current Source . 253 Voltage-Controlled Voltage Source . 254 Voltage-Controlled Current Source. 255 Current-Controlled Voltage Source. 255 Current-Controlled Current Source . 256 Subcircuit Definition . 256 Subcircuit Instance . 257 NPN BJT and PNP BJT. 257 n-Channel MOSFET and p-Channel MOSFET . 258 Mutual Inductance . 259 Transformer . 259 Diode . 260 External I vs. U . 260 External U vs. I . 261 External I-Terminal . 262 SPICE Circuit Import . 263 SPICE Circuit Export . 263 Theory for the Electrostatics Interface 265 Charge Relaxation Theory . 265 CONTENTS | 7 Electrostatics Equations . 268 The Electrostatics Interface in Time Dependent or Frequency Domain Studies . 269 Theory for the Electrical Circuit Interface 270 Electric Circuit Modeling and the Semiconductor Device Models. 270 Bipolar Transistors . 271 MOSFET Transistors . 274 Diode . 277 Reference for the Electrical Circuit Interface . 279 Chapter 5: Schrödinger Equation Interfaces The Schrödinger Equation Interface 282 Domain and Boundary Nodes for the Schrödinger Equation Interface . 283 Effective Mass . 284 Electron Potential Energy. 284 Initial Values . 285 Zero Flux . 285 Zero Probability . 286 Periodic Condition . 286 Open Boundary . ..