The Monte Carlo Method for Semiconductor Device Simulation

The application of the Monte Carlo method to the simulation of semiconductor devices is presented. A review of the physics of transport in semiconductors is given, followed by an introduction to the physics of semiconductor devices. The Monte Carlo algorithm is discussed in great details, and specific applications to the modelling of semiconductor devices are given. A comparison with traditional simulators is also presented.

1 Introduction.- References.- 2 Charge Transport in Semiconductors.- 2.1 Electron Dynamics.- 2.2 Energy Bands.- 2.2.1 Relationship of Energy to Wavevector.- 2.2.2 Effective Masses.- 2.2.3 Nonparabolicity.- 2.2.4 Herring and Vogt Transformation.- 2.2.5 Actual Bands of Real Semiconductors.- 2.3 Scattering Mechanisms.- 2.3.1 Classification and Physical Discussion.- 2.3.2 Fundamentals of Scattering.- 2.4 Scattering Probabilities.- 2.4.1 Phonon Scattering, Deformation-Potential Interaction.- 2.4.2 Phonon Scattering, Electrostatic Interaction.- 2.4.3 Ionized Impurity Scattering.- 2.4.4 Carrier-Carrier Scattering.- 2.5 Transport Equation.- 2.6 Linear Response and the Relaxation Time Approximation.- 2.6.1 Relaxation Times for the Various Scattering Mechanisms.- 2.6.2 Carrier Mobilities in Various Materials.- 2.7 Diffusion, Noise, and Velocity Autocorrelation Function.- 2.7.1 Basic Macroscopic Equations of Diffusion.- 2.7.2 Diffusion, Autocorrelation Function, and Noise.- 2.7.3 Electron Lifetime and Diffusion Length.- 2.8 Hot Electrons.- 2.9 Transient Transport.- 2.10 The Two-dimensional Electron Gas.- 2.10.1 Subband Levels and Wavefunctions.- 2.10.2 Scattering Rates.- References.- 3 The Monte Carlo Simulation.- 3.1 Fundamentals.- 3.2 Definition of the Physical System.- 3.3 Initial Conditions.- 3.4 The Free Flight, Self Scattering.- 3.5 The Scattering Process.- 3.6 The Choice of the State After Scattering.- 3.6.1 Phonon Scattering, Deformation-Potential Interaction.- 3.6.2 Phonon Scattering, Electrostatic Interaction.- 3.6.3 Ionized Impurity Scattering.- 3.6.4 Carrier-Carrier Scattering.- 3.7 Collection of Results for Steady-State Phenomena.- 3.7.1 Time Averages.- 3.7.2 Synchronous Ensemble.- 3.7.3 Statistical Uncertainty.- 3.8 The Ensemble Monte Carlo (EMC).- 3.9 Many Particle Effects.- 3.9.1 Carrier-Carrier Scattering.- 3.9.2 Molecular Dynamics and Monte Carlo Method.- 3.9.3 Degeneracy in Monte Carlo Calculations.- 3.10 Monte Carlo Simulation of the 2DEG.- 3.11 Special Topics.- 3.11.1 Periodic Fields.- 3.11.2 Diffusion, Autocorrelation Function, and Noise.- 3.11.3 Ohmic Mobility.- 3.11.4 Impact Ionization.- 3.11.5 Magnetic Fields.- 3.11.6 Optical Excitation.- 3.11.7 Quantum Mechanical Corrections.- 3.12 Variance-reducing Techniques.- 3.12.1 Variance Due to Thermal Fluctuations.- 3.12.2 Variance Due to Valley Repopulation.- 3.12.3 Variance Related to Improbable Electron States.- 3.13 Comparison with Other Techniques.- 3.13.1 Analytical Techniques.- 3.13.2 The Iterative Technique.- 3.13.3 Comparison of the Different Techniques.- References.- 4 Review of Semiconductor Devices.- 4.1 Introduction.- 4.2 Historical Evolution of Semiconductor Devices.- 4.2.1 Evolution of Si Devices.- 4.2.2 Evolution of GaAs Devices.- 4.2.3 Technological Features.- 4.2.4 Scaling and Miniaturization.- 4.3 Physical Basis of Semiconductor Devices.- 4.3.1 p-n Junction.- 4.3.2 Bipolar Transistors.- 4.3.3 Heterojunction Bipolar Transistor.- 4.3.4 Metal-Semiconductor Contacts.- 4.3.5 Metal-Semiconductor Field-Effect Transistor.- 4.3.6 Metal-Oxide-Semiconductor Field-Effect Transistor.- 4.3.7 High Electron Mobility Transistor.- 4.3.8 Hot Electron Transistors.- 4.3.9 Permeable Base Transistor.- 4.4 Comparison of Semiconductor Devices.- 4.4.1 Device Parameters.- 4.4.2 Comparison of Semiconductor Devices.- References.- 5 Monte Carlo Simulation of Semiconductor Devices.- 5.1 Introduction.- 5.2 Geometry of the System.- 5.2.1 Boundary Conditions.- 5.2.2 Grid Definition.- 5.2.3 Superparticles.- 5.3 Particle-Mesh Force Calculation.- 5.3.1 Particle-Mesh Calculation in One Dimension.- 5.3.2 Charge Assignment Schemes in Two Dimensions.- 5.4 Poisson Solver and Field Distribution.- 5.4.1 Finite Difference Scheme.- 5.4.2 Matrix Methods.- 5.4.3 Rapid Elliptic Solvers (RES).- 5.4.4 Iterative Methods.- 5.4.5 Calculation of the Electric Field.- 5.4.6 The Collocation Method.- 5.5 The Monte Carlo Simulation of Semiconductor Devices.- 5.5.1 Initial Conditions.- 5.5.2 Time Cycles.- 5.5.3 Free Flight.- 5.5.4 Scattering.- 5.5.5 Carrier-Carrier Scattering.- 5.5.6 Degenerate Statistics.- 5.5.7 Statistics.- 5.5.8 Static Characteristics.- 5.5.9 A.C. Characteristics.- 5.5.10 Noise.- References.- 6 Applications.- 6.1 Introduction.- 6.2 Diodes.- 6.2.1 n+-n-n+ Diodes.- 6.2.2 Schottky Diode.- 6.3 MESFET.- 6.3.1 Short Channel Effects.- 6.3.2 Geometry Effects.- 6.3.3 Space-Charge Injection FET.- 6.3.4 Conclusions.- 6.4 HEMT and Heterojunction Real Space Transfer Devices.- 6.4.1 HEMT.- 6.4.2 Real-Space Transfer Devices.- 6.4.3 Velocity-Modulation Field Effect Transistor.- 6.5 Bipolar Transistor.- 6.6 HBT.- 6.7 MOSFET and MISFET.- 6.7.1 MOSFET.- 6.7.2 GaAs Injection-modulated MISFET.- 6.7.3 Conclusions.- 6.8 Hot Electron Transistors.- 6.8.1 The THETA Device.- 6.8.2 GaAs FET with Hot-Electron Injection Structure.- 6.8.3 Planar-doped-Barrier Transistors.- 6.9 Permeable Base Transistor.- 6.10 Comparison with Traditional Simulators.- References.- Appendix A. Numerical Evaluation of Some Integrals of Interest.- References.- Appendix B. Generation of Random Numbers.- References.