Computational electronics semiclassical and quantum device modeling and simulation by dragica vasileska, stephen m goodnick, gerhard klimeck

Computational Electronics Semiclassical and Quantum Device Modeling and Simulation Dragica Vasileska Stephen M Goodnick Gerhard Klimeck CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-13: 978-1-4200-6484-1 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xiii Authors xvii Introduction to Computational Electronics 1.1 Si-Based Nanoelectronics 1.1.1 Device Scaling 1.1.2 Beyond Conventional Silicon 1.1.3 Quantum Transport Effects in Nanoscale Devices 1.2 Heterostructure Devices in III–V or II–VI Technology 11 1.2.1 Modulation Doping of AlGaAs=GaAs Heterostructures with In-Plane Transport 13 1.2.2 Vertical Transport—Resonant Tunneling Devices 13 1.3 Modeling of Nanoscale Devices 15 1.4 The Content of This Book 18 References 20 Introductory Concepts 23 2.1 Crystal Structure 23 2.1.1 Classification of Crystals by Symmetry 24 2.1.2 Miller Index 26 2.1.3 Reciprocal Space 29 2.2 Semiconductors 32 2.3 Band Structure 36 2.4 Preparation of Semiconductor Materials 40 2.5 Effective Mass 45 2.6 Density of States 54 2.7 Electron Mobility 57 2.8 Semiconductor Statistics 59 2.9 Semiconductor Devices 60 2.9.1 Diode 62 2.9.2 BJT Transistor 65 2.9.3 MOSFET 71 2.9.4 SOI Devices 76 2.9.4.1 PD=FD SOI Devices 76 2.9.5 MESFET 80 2.9.6 HEMTs 81 Problems 86 References 94 Semiclassical Transport Theory 95 3.1 Approximations for the Distribution Function 96 3.1.1 Quasi-Fermi Level Concept 96 3.1.2 Displaced Maxwellian Approximation 97 v vi Contents 3.2 Boltzmann Transport Equation 98 3.2.1 Scattering Processes 100 3.3 Relaxation-Time Approximation 102 3.3.1 Solving the BTE in the Relaxation-Time Approximation 103 3.4 Rode’s Iterative Method 108 3.5 Scattering Mechanisms: Brief Description 111 3.5.1 Phonon Scattering 111 3.5.1.1 Acoustic Phonon Scattering 112 3.5.1.2 Nonpolar Optical Phonon Scattering 113 3.5.1.3 Polar Optical Phonon Scattering 119 3.5.1.4 Piezoelectric Scattering 119 3.5.2 Defect Scattering 120 3.5.2.1 Ionized Impurity Scattering 120 3.5.2.2 Neutral Impurity Scattering 121 3.5.2.3 Alloy Disorder Scattering 121 3.5.3 Electron–Electron Interactions 122 3.5.3.1 Electron–Electron Interactions: Binary Collisions 122 3.5.3.2 Electron–Plasmon Scattering 123 3.5.4 Impact Ionization 124 3.6 Implementation of the Rode Method for 6H-SiC Mobility Calculation 125 3.6.1 Relevant Scattering Mechanisms 130 3.6.2 Simulation Results 135 3.6.3 Source Code (Provided by Graduate Student Garrick Ng) 137 Problems 144 References 147 The Drift-Diffusion Equations and Their Numerical Solution 151 4.1 Drift-Diffusion Model Derivation 151 4.1.1 Physical Limitations on Numerical Drift-Diffusion Schemes 153 4.1.2 Steady-State Solution of the Bipolar Semiconductor Equations 154 4.1.3 Normalization and Scaling 155 4.1.4 Linearization of Poisson’s Equation 155 4.1.5 Scharfetter–Gummel Discretization of the Continuity Equation 157 4.1.6 Gummel’s Iteration Method 158 4.1.7 Newton’s Method 159 4.1.8 Generation and Recombination 161 4.1.8.1 Carrier Generation Due to Light Absorption 164 4.1.8.2 Band-to-Band Recombination 165 4.1.8.3 Shockley–Read–Hall Generation–Recombination Mechanism 165 4.1.8.4 Auger Recombination 166 4.1.8.5 Impact Ionization 167 4.2 Drift-Diffusion Application Examples 168 4.2.1 1D Simulation Example—Modeling of pn-Diode 168 4.2.2 3D Drift-Diffusion Example: Modeling Threshold Voltage Fluctuations Due to Discrete Impurities 182 Problems 187 References 190 Contents vii Hydrodynamic Modeling 193 5.1 Introduction 193 5.2 Extensions of the Drift-Diffusion Model 196 5.3 Stratton’s Approach 199 5.4 Hydrodynamic (Balance, Bløtekjær) Equations Model 200 5.4.1 Displaced Maxwellian Approximation 206 5.4.2 Momentum and Energy Relaxation Rates 207 5.4.2.1 Using Drifted-Maxwellian Form for the Distribution Function 208 5.4.2.2 Using Bulk Monte Carlo Simulations 209 5.4.3 Simplifications That Lead to the Drift-Diffusion Model 209 5.4.4 Discretization and Numerical Solution Schemes for the Hydrodynamic Equations 210 5.4.4.1 von Neumann Stability Analysis 212 5.4.4.2 Lax Method 212 5.4.4.3 Other Varieties of Error 214 5.4.4.4 Second-Order Accuracy in Time 215 5.4.4.5 Fluid Dynamics with Shocks 218 5.5 The Need for Commercial Semiconductor Device Modeling Tools 219 5.5.1 Key Elements of Physical Device Simulation 220 5.5.2 Historical Development of the Physical Device Modeling 220 5.6 State-of-the-Art Commercial Packages 222 5.6.1 Silvaco ATLAS 222 5.6.2 Synopsys Software 225 5.7 The Advantages and Disadvantages of Hydrodynamic Models: Simulations of Different Generation FD SOI Devices 227 Problems 234 References 239 Particle-Based Device Simulation Methods 241 6.1 Direct Solution of Boltzmann Transport Equation: Monte Carlo Method 242 6.1.1 Free-Flight Generation 243 6.1.2 Final State after Scattering 244 6.1.3 Ensemble Monte Carlo Simulation 245 6.1.4 Scattering Processes 246 6.1.5 Bulk Monte Carlo Code for GaAs 248 6.2 Multi-Carrier Effects 286 6.2.1 Pauli Exclusion Principle 288 6.2.2 Carrier–Carrier Interactions 288 6.2.3 Band-to-Band Impact Ionization 290 6.2.4 Full-Band Particle-Based Simulation 291 6.3 Device Simulations 292 6.3.1 Calculation of the Current 294 6.3.2 Ohmic Contacts 297 6.3.3 Time-Step 298 6.3.4 Particle-Mesh Coupling 299 6.3.5 Source Code for Modeling FD SOI Devices 301 viii Contents 6.4 Coulomb Force Treatment within a Particle-Based Device Simulation Scheme 306 6.4.1 Particle–Particle–Particle–Mesh Approach 311 6.4.2 Corrected Coulomb Scheme 313 6.4.3 Fast Multipole Method 315 6.4.3.1 Multipole Moment 316 6.4.3.2 Speedup of the FMM Algorithm 317 6.5 Representative Simulation Results of Multiparticle and Discrete Impurity Effects 318 6.5.1 The Role of Short-Range e–e and e–i Interactions 319 6.5.2 Fluctuations in the On-State Currents 322 6.5.3 Current Issues in Novel Devices—Unintentional Dopants 324 Problems 329 References 332 Modeling Thermal Effects in Nano-Devices 335 7.1 Some General Aspects of Heat Conduction 337 7.2 Classical Heat Conduction in Solids 342 7.3 Form of the Heat Source Term 343 7.4 Modeling Heating Effects with Commercial Simulation Packages 345 7.4.1 Thermal3D Package from Silvaco 345 7.4.2 GIGA3D—Non-Isothermal Device Simulator 347 7.5 The ASU Particle-Based Approach to Lattice Heating in Nanoscale Devices 349 7.5.1 Electrothermal Particle–Based Device Simulator Description 352 7.5.2 Thermal Degradation with Device Scaling 359 7.6 Open Problems 363 Problems 364 References 364 Quantum Corrections to Semiclassical Approaches 367 8.1 One-Dimensional Quantum-Mechanical Space Quantization 369 8.1.1 Description of SCHRED 370 8.1.1.1 Capacitance Degradation 379 8.1.1.2 Threshold Voltage 380 8.1.2 Modification of the Effective Mass Schrödinger Equation for Heterostructures 381 8.2 Quantum Corrections to Drift-Diffusion and Hydrodynamic Simulators 383 8.2.1 Quantum Correction Approaches 383 8.2.2 Quantum Moment Methods 384 8.3 The Effective Potential Approach in Conjunction with Particle-Based Simulations 387 8.3.1 Effective Potential Approach 387 8.3.2 Effective Potential from the Wigner–Boltzmann Equation 388 8.4 Description of Gate Current Models Used in Device Simulations 394 8.4.1 Oxide Charging and Tunneling 395 8.4.2 Hot Carrier Injection 397 8.4.3 Gate Leakage Calculation in Conjunction with Particle-Based Device Simulators 399 Contents ix Monte Carlo—k Á p—1D Schrödinger Solver for Modeling Transport in p-Channel Strained SiGe Devices 405 8.5.1 Transport in SiGe p-Channel MOSFETs 406 8.5.2 The Six-Band k Á p Model Applied to Valence Band Structure of Si and Ge 411 8.5.2.1 Valence Band Structure in Si Inversion Layers—2D Dispersion Problem 413 8.5.2.2 Valence Band Structure in Strained Layer Heterostructure MOSFET Inversion Layers 416 8.5.2.3 Valence Band Structure in Inversion Layers—2D Contour Problem 420 8.5.2.4 Description of the Self-Consistent Scheme 422 8.5.2.5 Density of States 424 8.5.3 Monte Carlo Procedure and Simulation Results 425 8.5.3.1 Carrier Scattering Rates 425 8.5.3.2 2D $ 3D Transitions 429 8.5.5.3 Simulation Results 429 Problems 436 References 438 8.5 Quantum Transport in Semiconductor Systems 445 9.1 Tunneling 446 9.2 General Notation 448 9.2.1 Stationary States for a Free Particle 453 9.2.2 Potential Step 453 9.2.3 Tunneling through a Single Barrier 458 9.3 Transfer Matrix Approach 461 9.3.1 Basic Description of the Method 461 9.3.2 Piecewise Constant Potential Barrier Tool 462 9.3.2.1 Example for Quantum Mechanical Reflections 463 9.3.2.2 Is There Source-to-Drain Tunneling in Nanoscale MOSFETs? 464 9.3.2.3 Quasi-Bound States Formation in a Double-Barrier Structure 465 9.3.2.4 Formation of Energy Bands 468 9.3.2.5 More Complex PCPBT Capabilities That Utilize a Tight-Binding Approach 469 9.3.3 Limitations of Transfer Matrix Approach and Its Alternatives 470 9.4 Landauer Formula and Usuki Method 471 9.4.1 Landauer–Büttiker Formalism 472 9.4.2 Usuki Iterative Procedure 473 9.4.2.1 Spin Transport and Spin Filter 475 9.4.2.2 Theoretical Modeling of Spin Filters 477 9.4.2.3 Simulation Results for Spin Filter 479 Problems 484 References 489 x Contents 10 Far-From-Equilibrium Quantum Transport 493 10.1 Mixed States and Distribution Function 493 10.2 Irreversible Processes and MASTER Equations 495 10.3 The Wigner Distribution Function 496 10.4 Green’s Functions 498 10.4.1 Mathematical Physics Formulation of the Green’s Function Method 499 10.4.1.1 Schrodinger, Heisenberg, and Interaction Representation 499 10.4.1.2 Wick’s Theorem and Perturbation Series Generation 501 10.4.1.3 Dyson Equation for the Retarded Green’s Function 503 10.4.1.4 Equilibrium Properties of a Semiconductor—GW Approximation 505 10.5 Nonequilibrium Keldysh Green’s Functions 506 10.5.1 Need for Approximations to the NEGF Formalism—Application Targets 512 10.6 Low Field Transport in Strained-Si Inversion Layers 513 10.6.1 Theoretical Model 513 10.6.2 Calculation of the Broadening of the States and Conductivity=Mobility 517 10.6.3 Electron Mobility Results—Low Doped Samples 520 10.6.4 Electron Mobility Results—Highly Doped Samples 523 10.7 NEGF in a Quasi-1D Formulation 526 10.7.1 Tight-Binding Hamiltonian 526 10.7.2 Recursive Green’s Functions Method for the Retarded Green Function 527 10.7.3 Recursive Green’s Functions Method for the Less Than Green Function 529 10.7.4 NEGF with Incoherent Scattering 531 10.7.5 Open Boundary Condition Formulation 531 10.7.6 1D Effective Mass Hamiltonian and Boundary Conditions 533 10.8 Quantum Transport in 1D—Resonant Tunneling Diodes 534 10.8.1 RTDs with Linear Potential Drops 535 10.8.2 RTDs with Realistic Doping Profiles 542 10.8.3 Resonant Tunneling Diodes with Relaxation in the Reservoirs 547 10.8.4 RTDs with Quantum Charge Self-Consistency (Hartree Model) 551 10.8.5 Asymmetric RTDs with Charge Accumulation and Depletion 555 10.8.6 Resonant Tunneling Diode Simulations with Incoherent Scattering 561 10.8.7 RTD Simulations at Room Temperature with Full Bandstructure 565 10.9 Coherent High-Field Transport in 2D and 3D 568 10.9.1 Full Atomistic Quantum Transport in OMEN 568 10.9.2 2D Effective Mass Hamiltonian and Boundary Conditions 569 10.9.3 High-Field Transport—CBR Algorithm (Denis Mamaluy) 570 10.9.3.1 Bound States Treatment 572 10.9.3.2 CBR Energy Discretization 574 10.9.3.3 CBR Self-Consistent Solution 574 10.9.3.4 Device Hamiltonian, Algorithm, and Some Numerical Details 575 Contents xi 10.9.3.5 CBR Simulation Example—2D Results 577 10.9.3.6 CBR Simulation Example—3D Results 580 Problems 592 References 593 11 Conclusions 599 References 601 Appendix A: Electronic Band Structure Calculation 605 A.1 Spin-Orbit Coupling 606 A.2 Rashba and Dresselhaus Spin Splitting 608 A.2.1 Empirical Pseudopotential Method 608 A.2.1.1 Description of the Empirical Pseudopotential Method 611 A.2.1.2 Implementation of the Empirical Pseudopotential Method for Si and Ge 613 A.2.1.3 Empirical Pseudopotential Method for GaN 615 A.2.2 The Tight-Binding Method 616 A.2.3 The k Á p Method 619 A.2.3.1 k Á p General Description 619 A.2.3.2 k Á p Theory Near the G Point and for Bulk Materials 620 A.2.3.3 Kane’s Theory 622 A.2.3.4 Coupling with Distant Bands 626 A.2.3.5 The Luttinger–Kohn Hamiltonian 627 A.2.4 Carrier Dynamics 630 References 630 Appendix B: Poisson Equation Solvers 633 B.1 Maxwell’s Equations 633 B.1.1 Case without Magnetic or Dielectric Materials 635 B.1.2 Case of Linear Materials 635 B.1.3 General Case 635 B.2 Gauge Transformations 636 B.2.1 Lorenz Gauge 637 B.2.2 Coulomb Gauge 637 B.3 General Guidelines for Solving Partial Differential Equations 638 B.4 Finite Difference Discretization of the Poisson Equation 639 B.4.1 Finite Difference Discretization 640 B.4.2 Linearization of the Poisson Equation 642 B.4.3 Final Expressions 642 B.5 Numerical Solution Techniques for 2D=3D Poisson Equation 644 B.5.1 Direct Methods 645 B.5.1.1 Gauss Elimination Method 645 B.5.1.2 The LU Decomposition Method 646 B.5.2 Iterative Methods 648 B.5.2.1 The Gauss–Seidel Method 648 B.5.2.2 The Successive Over-Relaxation Method 649 B.5.2.3 Other Iterative Methods 650 References 670 Index Ellipse coordinate system (ECS), 50–51 Empirical pseudopotential method (EPM), 409 crystal wave function, 609 description, 611–612 implementation for GaN, 615 implementation for Si and Ge, 613–615 model potentials, 610 Phillips–Kleinman cancellation theorem, 610 Ensemble Monte Carlo (EMC) simulation nanoscale dimension device, 221 particle-based device simulation methods, 245–246 EPM, see Empirical pseudopotential method Excitation source modeling hard source, 700–701 soft source, 701–702 F Face-centered cubic (FCC) system, 25, 28, 31–32 Fast multipole method (FMM) Coulombic=gravitational potential, 315 multipole expansion coefficients, 317 multipole moment, 316–317 particle-in-cell calculation, 316 quad-tree and oct-tree, 317 FBMC method, see Full-band Monte-Carlo method FCC system, see Face-centered cubic system FD SOI, see Fully depleted silicon-on-insulator FDTD method, see Finite-difference time domain method Fermi–Dirac statistics, 194–195, 379, 433, 437 Fermi Golden Rule, 495 conditions for validity matrix element for scattering, 736–737 reflection coefficient, 738 scattering potential barrier, 736 scattering probability, 737 transmission coefficient, 738 elastic scattering of electrons Bohr radius, 743 Brook’s Herring result, 745 coordinate system choice, 742–743 Debye screening length, 738 nature of Coulomb scattering, 745 perturbing potential, 738 scattering cross-section, 739–741 summary, 735–736 Feynman–Dyson perturbation theory, 503 751 FinFET double-gate vs tri-gate drain current, percentage reduction, 590–591 3D electron density, 585, 587 net gate leakage vs gate voltage, 585, 587 output characteristics, 586, 588 performance matrices, 588–589 2D potential energy profile, 590 transfer characteristics, 585–586, 590–591 unintentional dopant, 589–590 YZ plane, electron densities, 586, 588 fabrication, 579 geometry, 577–578, 581–582 2D simulation, 578 2D vs 3D simulation computational efficiency, 583 2D electron density, 584 net gate leakage, 585 output characteristics, 585–586 residuum, nonlinear Poisson equation, 582–583 transfer characteristics, 583 Finite difference discretization east-west notation, 641 final expressions, 642–644 linearization of the Poisson equation, 642 nonhomogeneous grids, 641 ‘seven-point’ stencil, 640 Finite difference method, 413 Finite-difference time domain (FDTD) method ABC CPML (see Convolutional perfectly matched layer) overview, 681–682 perfectly matched layer, 682–684 accuracy and numerical dispersion, 680 alternate-direction implicit-FDTD method background, 692–693 CPML formulation, 697–699 general formulation, 693–696 split-field PML formulation, 696–697 finite-difference update equations, 677–678 grid coordinates, 676 programming implementation, 679–680 stability, 681 temporal and spatial variations, 676 time and space derivatives, 676 Yee cell, 678 Forward time centered space (FTCS), 211–212 Fowler–Nordheim tunneling process, 395–397 Fredholm integral equations, 519 FTCS, see Forward time centered space 752 Full-band Monte-Carlo (FBMC) method, 17 Full-wave FDTD solvers 3D planar microstrip circuits analysis low-pass filter, 705–708 rectangular patch antenna, 702–705 excitation source modeling hard source, 700–701 soft source, 701–702 Fully depleted silicon-on-insulator (FD SOI) devices, 76–79 Auger generation–recombination, 227 characteristic dimension, device structure, 227–228 energy relaxation time, 230, 232–234 mesh and output characteristics, 228–229 on-state current dependence, 228, 230 particle-based simulator, 230 Silvaco input deck, 230–232 source code average carrier energy and drift velocity, 305–306 bulk Monte Carlo code, 303 conduction band variation, 303–304 cumulative charge vs time, 305, 307 current density, 305, 307 drain induced barrier lowering (DIBL), 305, 308 equilibrium electron density, 302 ID–VD characteristics, 305, 309 initialization section flowchart, 301–302 intervalley scattering, 302–303 MOSFET device structure, 301 scattering rate variation vs energy, 302–303 subroutines, 304–305 transfer characteristics, 305, 308 SRH generation–recombination, 227 G Gate current models gate leakage calculation, particle-based device simulators Airy functions, 401–403 electric field profile, 400 Monte Carlo simulation, 399 nonlinear potential barrier, 404 piecewise linear potential barrier, 400 Schottky barrier diode, 399 Schrödinger equation, 399–401 slicing, 404 SOI MESFET device structure, 404–405 transmission coefficient, 403 Index transmission probability, 404–405 wave function, 402–403 hot carrier injection barrier height, 398 erase and write mechanisms, memories, 397–398 HEI process, 397–398 oxide charging and tunneling drain current, technology generation, 395–396 Fowler–Nordheim tunneling process, 395–397 MOS device structures, 395–396 threshold voltage shifts, 395 Gauge transformations Coulomb gauge, 637–638 Lorenz gauge, 637 Gauss elimination method, 645 Gauss–Seidel method, 648–649 GCA, see Gradual channel approximation GIGA3D–non-isothermal device simulator, 347–348 Gradual channel approximation (GCA), 74–75 Green’s functions, 471 density and Wigner function, 498 Dyson equation analytic expression, 504–505 connected diagrams, 503 Feynman diagrams, structure of G, 503–504 Feynman–Dyson perturbation theory, 503 first-order proper self-energies, 504 Hartree and Fock self-energies, 503–504 proper self-energy insertion, 504 equilibrium properties, semiconductor, 505–506 formalism, 525 NEGF, 498–499 Schrodinger, Heisenberg, and interaction representation, 499–501 Wick’s theorem and perturbation series generation, 501–503 Gummel’s iteration method, 158–159, 220–221 GW approximation, 505–506 H Harmonic oscillator, linear perturbation, 727–729 HEI process, see Hot electron injection process Heisenberg representation, 499–501 Heisenberg uncertainty principle, 55, 451, 485 HEMTs, see High electron mobility transistors Index Heterostructure devices AlGaAs=GaAs heterostructures with in-plane transport, 13–14 LPE, 12 MBE, 12–13 MESFETs, 11 semiconductor lasers, 11 vertical transport-resonant tunneling devices bandgap engineering, 13 NDR, 15 quantum effects, 14 RTD, 14–15 High electron mobility transistors (HEMTs) band gap energy, 81–82 2DEG, 83–84 donor and Schottky layer, 84–85 DX centers, 85 energy band diagram, 83 epitaxial structure, 82–83 semi-insulating GaAs substrate, 82 spacer layer, 84 superlattice structure, 83 transconductance, 85 Hot electron injection (HEI) process, 397–398 6H-SiC mobility calculation anisotropy, 127 breakdown voltage, 130 Brillouin zones, 127–128 energy bands, 127–128 epi-layer, 130 mechanical properties, 129 physical characteristics, 129 polytypes, 125 principal axes, 125–126 relevant scattering mechanisms acoustic deformation potential scattering, 130 coupling coefficient, 131 free electron concentration, 134 intervalley phonon scattering, 132 ionized and neutral impurity scattering, 131 lattice scattering, 135 parabolic band approximation, 131 piezoelectric scattering, 131–132 POP scattering, 132–133 scattering rates vs energy, 133–134 transverse and longitudinal elastic constants, 132 wurtzite structures, 131–132 a-SiC, 126 b-SiC, 126 SiC layer structure, 125 silicon fundamental material properties, 129 753 simulation results, 135–137 source code, 137–144 stacking sequence, 126–127 Hydrodynamic modeling Bløtekjær’s approach, 194 bulk Monte Carlo simulations, 209 carrier and energy flux, 200 carrier-density balance equation, 202 commercial semiconductor device modeling tools physical device simulation, 220–222 TCAD, 219 complete hydrodynamic equations, 204–205 different generation FD SOI devices Auger generation–recombination, 227 characteristic dimension, device structure, 227–228 energy relaxation time, 230, 232–234 mesh and output characteristics, 228–229 on-state current dependence, 228, 230 particle-based simulator, 230 Silvaco input deck, 230–232 SRH generation–recombination, 227 discretization and numerical solution advective equation, 211 conserved flux, 211 fluid dynamics with shocks, 218–219 flux-conservative equation, 210 forward Euler differencing, 211 FTCS representation, 211–212 Lax method, 212–213 second-order accuracy, 215–218 varieties of error, 214–215 von Neumann stability analysis, 212 displaced Maxwellian approximation, 206–207 drift-diffusion (DD) model acoustic deformation potential scattering, 209 carrier mobility, 209 channel velocity, 197–198 deep-submicrometer devices, 197 electric field and density gradient, 196–197 electron current density, 196 kinetic energy density, 210 momentum balance equation, 209–210 nondegenerate semiconductors, 196 Silvaco simulation package, 210 drifted-Maxwellian form, distribution function, 208–209 energy balance equation, 203–204 Fermi–Dirac statistics, 194–195 generalized conservation equation, 202 754 generation–recombination term, 200–201 Kane’s dispersion relation, 195 macroscopic transport models, 194 momentum balance equation, 202–203 nondegenerate semiconductor, 201 spherical harmonic expansion method, 194 state-of-the-art commercial packages Silvaco ATLAS, 222–225 synopsys software, 225–227 Stratton’s approach, 199–200 thermal equilibrium, 193 Hydrodynamic (HD) transport equations, 17 I Interaction representation, 499–500 Irreversible processes and master equations, 495–496 J Junction field effect transistor (JFET), 71 K Kane’s dispersion relation, 195 Kane’s theory atomic Bloch states, 622–623 dispersion for the HH-band, 625 eight-band Hamiltonian, 624 light-hole and split-off bands, 626 Keldysh Green’s function, 509 k Á p method for bulk materials, 620–621 coupling with distant bands, 626–627 general description, 619–620 Kane’s theory atomic Bloch states, 622–623 dispersion for the HH-band, 625 eight-band Hamiltonian, 624 light-hole and split-off bands, 626 Luttinger–Kohn Hamiltonian, 627–629 L Landauer–Büttiker formalism, 472–473 Lightly doped drain (LDD) devices, 344 Liouville–von Neumann equation, 242 Liquid phase epitaxy (LPE), 12 Local density approximation (LDA), 478 Lorenz gauge transformations, 637 Index Low field transport, strained-Si inversion layers electron mobility results highly doped samples, 523–525 low doped samples, 520–522 states and conductivity=mobility, broadening, 517–519 theoretical model deformation potential constant, 515–516 f- and g-phonons, 517 form-factors, 513–514 graded buffer technique, 517 matrix element, 515 potential, charged center, 513 power spectrum, 514–515 scattering mechanisms, 513–514 zero-order process, 516–517 Low-pass filter, 705–708 LPE, see Liquid phase epitaxy LU decomposition method, 646–648 Luttinger–Kohn Hamiltonian, 627–629 M Maxwell–Boltzmann statistics, 273–274 Maxwell’s equations, 17, 673–674 Gauss’ law, implicit enforcement of, 674–675 Poisson equation solvers differential and integral form, 634 general case, 635–636 linear materials, 635 magnetic or dielectric materials, 635 partial differential equations, 633–634 stretched-coordinate formulation, 684–686 MBE, see Molecular beam epitaxy MESFETs, see Metal semiconductor field effect transistors Metal oxide semiconductor field effect transistors (MOSFETs) channel and depletion region formation, 73 drain electrode, 72–73 equilibrium band diagram, 72 flat-band voltage, 74 gate electrode, 72 GCA, 74–75 I–V characteristics, 73–74 junction field effect transistor (JFET), 71 n- and p-channel enhancement, 71 source and drain series resistances, 75–76 source code, 301 square-law vs bulk-charge theory, 75 threshold voltage, 74 755 Index velocity limited drain current, 76 velocity saturated device, 76–77 velocity saturation, 75 Metal semiconductor field effect transistors (MESFETs) GaAs materials technology, 11 particle distribution, 294–295 SOI devices, 80–81 Miller–Bravais index, 28 Miller index Cartesian coordinates, 28 crystallographic planes, 28–29 direct lattice vectors, 26–27 hexagonal and rhombohedral crystal systems, 28 Miller–Bravais index, 28 primitive reciprocal lattice vectors, 27 reciprocal lattice vectors, 26–27 Mixed states and distribution function coherent process, 493 incoherent effects, 493 Liouville equation, 494 Liouville operator, 494–495 operator A, 493 particle density, 493–494 single particle density matrix, 493 Molecular beam epitaxy (MBE), 12–13 Monte Carlo procedure band structure simulation results bulk strained SiGe, 429 constant energy surfaces, 429–431 3D equi-energy surfaces, 431–432 3D hole DOS, 431 triangular test potential, 2D carriers, 432–433 carrier scattering rates acoustic deformation potential, 426 Coulomb scattering matrix element, 428 effective average velocity, 425 K-vector, 426–427 longitudinal and transverse elastic constants, 425–426 optical deformation potential, 426 power spectral density, 427 surface roughness scattering, 427–428 velocity, acoustic modes, 425 drain current enhancement, 435 25 nm p-channel MOSFET, 433–434 p-channel strained SiGe MOSFET, 434–436 2D$3D transitions, 429 MOSFETs, see Metal oxide semiconductor field effect transistors Multigrid method from one-grid through two-grid, 658–661 preconditioned conjugate gradient methods CPU demand, SOI MOSFET, 669–670 error reduction, SOI MOSFET, 668 integrated electron density, 668 SOI device structure, 667 simulated structure and practical implementation charge densities, 664 conduction band profile, 664 electric field profiles, 665 intergrid transfer operator, 663 uniform grid organization, 662 smoothing, restriction, and prolongation operators, 661–662 N Nano-devices, thermal effects applications, 363–364 Boltzmann transport equations, 349 classical heat conduction, solids, 342–343 collision terms, relaxation time approximation (RTA), 350–351 commercial simulation packages GIGA3D–non-isothermal device simulator, 347–348 Thermal3D package, Silvaco, 345–347 conservation equations, optical and acoustic phonons, 350–351 electrothermal particle-based device simulator description acoustic phonon temperature, 358–359 ASU electrothermal simulator, 352–353 average velocity, channel, 358 current vs thermal iterations, 357 device cross section and simulation domain, 355–356 Dirichlet=Neumann conditions, 354–355 electron kinetic energy, 354–355 ensemble Monte Carlo (EMC) simulation, 352 heterogeneous structures, 353–354 isothermal and non-isothermal current, 357–358 lattice heating effect, 358 nanoscale SOI MOSFETs, 355 phonon system, grid, 352 sample electron density, 354 scattering events count, 356 scattering mechanism table, 353, 356 756 steady-state behavior, 354 steady-state solutions, 352 energy transfer, 349 field effect transistor technology evolution, 335–336 heat conduction boundary scattering, 339–340 BTE, 339 continuum theory, 340 depth-dependence, thermal conductivity, 340–341 electrothermal design, 342 experimental data, 338 Fourier law, 337–338 heat flow, 337 heat transfer, 337 lattice vibrational waves, dielectric materials, 338 mean free path (MFP), 338–339 phonons and electrons, 342 phonon transport, ultrathin silicon layers, 339 silicon film thickness dependence, 341 temperature distribution, 337 thermal conductivity, 337–340 heat removal, 335–336 heat source term electron and lattice heating, 343–344 electron-lattice scattering model, 344 Joule heating model, 344 phonon-model, 344–345 microprocessor power density vs year, 335–336 primary path, energy transport, 349–350 self-heating effects, 335, 337 SOI technology, 335, 337 thermal degradation, device scaling channel direction, silicon layer, 360, 362 lattice temperature profiles, silicon layer, 360–361 Neumann thermal boundary conditions, 360, 362 simulated fully depleted SOI MOSFETs, 359 vs isothermal value, 359–360 Nanoelectronic device simulation, 599 Nanoelectronics modeling (NEMO), 535, 600 Nanoscale devices device simulation sequence, 16–17 product cycles, 16 quantum transport effects Born approximation, 11 classical vs quantum charge, 9–10 Index de Broglie wavelength, dynamical quantum effects, 9, 11 Fermi Golden Rule, potential and electric fields, 9–10 Schrödinger equation, spatial=size-quantization, semi-classical Boltzmann transport, 17 TCAD, 15 transport model hierarchy, 17–18 Negative differential resistance (NDR), 15 NEGF, see Non-equilibrium Green’s function NEMO, see Nano electronics modeling Newton’s law, 45 Newton’s method block Gauss–Seidel iteration, 160 convergence, 160 Gummel’s method, 161 Jacobian matrix, 159–160 Newton–Richardson approach, 160 successive over-relaxation (SOR) method, 160–161 Nonatomistic theory, 600 Non-equilibrium Green’s function (NEGF), 221–222 1D effective mass Hamiltonian and boundary conditions, 533–534 incoherent scattering, 531 open boundary condition formulation contacts and Hamiltonian matrix, 531 coupled matrix equations, 533 1D mesh, A- or B-type atoms, 532 recursive Green’s functions (RGFs) method less-than Green function, 529–531 retarded Green function, 527–529 tight-binding Hamiltonian, 526–527 Non-equilibrium Keldysh Green’s functions bare particle, 509 contour-ordered Greens function, 508 Dyson equation, 510 integration path, 506–507 NEGF formalism, approximations, 512–513 one-point functions, 509–510 perturbation expansion, 508 rotation, Keldysh space, 509 self-energy functions, 509–511 time arguments, 507–508 Nonpolar optical phonon scattering average electron drift velocity, 118 central valleys, 117 first-order intervalley scattering rate, 116 first-order process, 115 g- and f-processes, 114–115 intervalley transition, 113 Index nonparabolic band structure, 115–116 phonon occupancy factor, 114 satellite valleys, 117–118 zeroth-order intervalley scattering, 115–116 O OMEN, 568–569, 600 Optical absorption, 164–165 P Particle-based device simulation methods BTE bulk Monte Carlo code, GaAs (see Bulk Monte Carlo code, GaAs) definition, 241 electron and hole distribution, 242 ensemble Monte Carlo (EMC) simulation, 245–246 final energy and momentum, 244–245 free-flight generation, 243–244 random sampling, 242 scattering processes, 246–248 type of scattering, 244–245 Coulomb force treatment corrected Coulomb scheme, 313–315 device on-state current, 311 3DMCDS code, 310–311 FMM, 315–317 nanoscale devices, 306 P3M algorithms, 311–313 Poisson’s equation, 306 threshold voltage fluctuation, 309–310 device simulations current calculation, 294–297 flowchart, 294 Monte Carlo phase, 292–293 ohmic contacts, 297–298 particle distribution, MESFET structure, 294–295 particle-mesh coupling, 299–301 Poisson’s equation, 292–293 source code, FD SOI devices (see Fully depleted silicon-on-insulator devices, source code) steady-state solution, 293 three valley model, 293 time-step, 298–299 transport kernel and field solver, 292 757 hydrodynamic model, 242 multi-carrier effects band-to-band impact ionization, 290–291 carrier–carrier interactions, 288–290 full-band particle-based simulation, 291–292 particle distribution function, 286 Pauli exclusion principle, 288 multiparticle and discrete impurity effects doping concentration, 318 high-field characteristics, 319 on-state current fluctuation, 322–323 short-range e–e and e–i interactions, 319–322 threshold voltage standard deviation, 318–319 unintentional dopants, 324–329 probability distribution function, 241 quantum mechanical solution, 241 Particle–particle–particle–mesh (P3M) algorithms Coulomb’s law, 312 radial approximation, 313 short-range Coulomb force, 312 SRD, 312–313 total force, 311–312 Particle’s effective mass carrier group velocity, 49 CCS, 51 conductivity calculation, 48 constant energy ellipsoid, 50 constant energy surfaces, 46–47 cyclotron resonance, 49 DCS, 51 DOS calculations, 48 ECS, 50–51 E-k diagram, 46 energy band minima and maxima, 46–47 energy–wave vector, 48 external electric field, 45 Fortran code, 51–54 Newton’s law, 45 parabolic and nonparabolic energy band structure, 49 rotation matrix, 51 three-dimensional equi-energy surfaces, 46–47 valence band, 46 wurtzite material systems, 51 zinc-blende material systems, 51 PCPBT, see Piece-wise constant potential barrier tool Perfectly matched layer (PML), 682–684 758 Perturbation theory stationary degenerate states, 722–725 first-order approximation, 719–720 harmonic oscillator with linear perturbation, 727–729 second-order approximation, 720–722 smallness parameter, 717 stark effect in a potential well, 725–727 zero-order approximation, 718 time-dependent amplitude calculation, 733–734 Fermi Golden Rule (see Fermi Golden Rule) matrix methods, 732 simple harmonic perturbation, 731 time-dependent SWE, 729 unperturbed system, 730 Phillips–Kleinman cancellation theorem, 610–611 Photonic crystals 3D photonic crystal waveguides, 709 excitation and propagation bipolar pulse continuous-train, 713 continuous-wave sinusoid, 712 2-cycle bipolar pulse, 711 stair-casing effect, 710 Piece-wise constant potential barrier tool (PCPBT) energy band formation, 468–469 quantum mechanical reflections gate electrode role, 463 potential barrier shape, 464 transmission coefficient vs energy, 464 quasi-bound states formation nonsymmetric barriers case, 467–468 symmetric barriers case, 465–467 source-to-drain tunneling, nanoscale MOSFET, 464–465 tight-binding approach, 469–470 PML, see Perfectly matched layer Poisson equation solvers finite difference discretization east-west notation, 641 final expressions, 642–644 linearization of the Poisson equation, 642 nonhomogeneous grids, 641 ‘seven-point’ stencil, 640 gauge transformations Coulomb gauge, 637–638 Lorenz gauge, 637 Maxwell’s equations differential and integral form, 634 general case, 635–636 Index linear materials, 635 magnetic or dielectric materials, 635 partial differential equations, 633–634 numerical solution techniques conjugate gradient method, 650–657 Gauss elimination method, 645 Gauss–Seidel method, 648–649 incomplete LU decomposition, 650 LU decomposition method, 646–648 multigrid method, 657–670 successive over-relaxation method, 649–650 solving Laplace equations, 639 Poisson’s equation, 17 corrected Coulomb scheme, 313 Coulomb force treatment, 306 device simulations, 292–293 linearization convergence, 157 nonlinear Poisson equation, 156 stable convergence, 155 Polar optical phonon (POP), 108 Predictor carrier density, 575 Predictor–corrector approach, 574 Pseudomorphic HEMTs (pHEMTs), 81–85 Q QBSs, see Quasi-bound states QPCs, see Quantum point contacts Quantum corrections, semiclassical approaches drift-diffusion and hydrodynamic simulators effective bandgap, 383 function, F(y), 384 intrinsic carrier concentration, 384 quantum moment methods, 384–387 surface potential, 383 effective potential approach, 387–388 gate current models gate leakage calculation, particle-based device simulators, 399–405 hot carrier injection, 397–399 oxide charging and tunneling, 395–397 inversion layer capacitance, 367–368 Monte Carlo procedure and simulation results band structure simulation results, 429–433 carrier scattering rates, 425–428 device simulation results, 433–436 2D$3D transitions, 429 MOS capacitor, 367–368 one-dimensional quantum-mechanical space quantization Index bulk Hamiltonian, electron, 370 capacitance degradation, 379–380 conduction-band, silicon, 369–370 correlation energy correction, 375 Coulomb hole, 372 density-functional theory, 372, 376 effective external potential, Veff (r), 373 effective mass Schrödinger equation, 381–383 electron density, 374 electronic structure theory, 372 Euler condition, 373 exchange–correlation energy, 374 exchange–correlation potential, 374 exchange energy and hole, 371 exciton and depolarization shift, 376 experimental data, 375–376 ground state energy, 372 Hartree–Fock theory, 372 Hohenberg–Kohn–Sham (HKS) equation, 374 local-density approximation (LDA), 374 potential diagram, p-type semiconductor, 369 p-type semiconductor, potential diagram, 369–370 p-type silicon inversion layer, 375 SCHRED tool, 377–378 Schrödinger equation, 371 sheet electron concentration, ith subband, 371 stationary functional, 372–373 subband energy difference, 376–377 threshold voltage, 380–381 total energy, electrons, 371 transverse part, 370 two-dimensional electron gas (2DEG), 369 SiGe p-channel MOSFETs band structure, 408–409 bulk (3D) transport, 408 confined (2D) carrier transport, 408 energy band diagram, 407–408 enhanced performance, 406 full band Monte Carlo approach, 410 geometro-analytical model, 409 hole mobility, 407 hole transport calculations, 409–410 pseudopotentials, 408–409 self-consistency, 410 source=drain (S=D) region, 406–407 subband structure, numerical techniques, 410 759 subsurface leakage current suppression, 406–407 thin-body transistor, heterostructure channel, 407–408 six-band k Á p model, valence band structure biaxial compression and tension, 412 constant of proportionality, 412 density of states (DOS), 424 Hamiltonian, 411 inversion layers–2D contour problem, 420–421 self-consistent scheme, 422–424 Si inversion layers–2D dispersion problem, 413–416 strained layer heterostructure MOSFET inversion layers, 416–420 strain, epi-layer, 412 Wigner–Boltzmann equation, effective potential Bloch equation, 389 DG SOI device output characteristics, 393–394 doping scheme, 392–393 double gate (DG) SOI device structure, 392–393 effective quantum potential, 389, 391 Green’s function, 389 Hartree potential, 389–391 intrinsic device, 392–393 Liouville equation, 388–389 size-quantization effects, 391 threshold voltage vs SOI film thickness, 391–392 Weyl quantization, 388 Quantum dots, 471–472 Quantum moment methods coupled equations, motion, 384 degeneracy factor, 386 device structure, 386–387 electron density, 387 Madelung transformation, 385 quantum=Bohm potential, 385 quantum-hydrodynamic (QHD) equations, 385 simulated I–V characteristics, 386–387 Wigner–Boltzmann equation, 385 Wigner potential, 385 Quantum point contacts (QPCs), 472, 476 Quasi-bound states (QBSs), 545, 572–573 Quasi-Fermi level formulation, 154–155, 220 Quasi-particle self-consistent GW (QSGW) approximation, 505–506 760 R Rashba and Dresselhaus spin splitting empirical pseudopotential method crystal wave function, 609 description, 611–612 implementation for GaN, 615 implementation for Si and Ge, 613–615 model potentials, 610 Phillips–Kleinman cancellation theorem, 610 k Á p method for bulk materials, 620–621 coupling with distant bands, 626–627 general description, 619–620 Kane’s theory, 622–626 Luttinger–Kohn Hamiltonian, 627–629 tight-binding method Chadi and Cohen TB parameters, 618 implementation, 617 quantitative description, 616 Rectangular patch antenna, 702–705 Recursive-convolution technique, 688–691 Reflection probability R(E), 457 Resonant-tunneling diode (RTD), 14–15 charge accumulation and depletion 0.6 V bias, 560 current–voltage characteristics, 556–557 forward bias direction, 560–561 Hartree vs Thomas–Fermi model, 557–559 I–V curve, 555–556 resonance energy traces, 556–557 reverse bias direction, 559–561 tunneling rate, 556 III–V compound material systems, 534–535 incoherent scattering current–voltage characteristic, GaAs=AlAs RTD, 562–563 forward and reverse bias current, 563–564 interface roughness, 563–564 NEMO 1D tool, 562 scattering process, 561–562 valley current, 562–563, 565 wave functions, emitter bound state, 563–564 linear potential drops 0.08 and 0.38 V biases, 540 central device region, 535 current density and normalized integrated current, 540 function of bias, 539, 541 low temperature, 536–538 Index resonance energy, Fermi level, 538–539 resonance widths, 541 operation, schematic description, 534 quantum charge self-consistency vs current voltage, 553–554 doping and charge profile, 551–552 emitter resonance alignment, 554–555 Hartree (HA) model, 551 physical reasoning, 553–554 Poisson solution, 553 quantum-charge transport model, 553 resonance width, function bias., 554–556 semiclassical charge, 551–552 realistic doping profiles 0.42 V bias, 545–546 central resonance approach, 545–546 conduction band edge profile, 542–543 emitter resonance, 547 perfect charge balance, 542 potential profiles and charge distributions, 543–544 QBS, 545 quasi-Fermi levels, 543 resonance energies, 546 resonance width, 546–547 RTDnegf calculation, 544 triangular emitter quantum well, 544–545 undoped spacer layer, 542 relaxation, reservoirs charge accumulation, 547–548 charge densities, 548 conduction band edge, 549–550 vs current–voltage characteristics, 548–549 NEMO, 548 relaxation rate, h, 550–551 resonance broadening, 548 resonance line width, 547–549 total resonance width, 548 room temperature, full bandstructure, 565–568 Si=SiGe materials system, 535 Wigner distribution function, 497–498 Rode’s iterative method anisotropic scattering process, 109 arbitrary magnetic field, 110 drift and Hall mobility, 111 elastic scattering process, 108 Hall scattering factor, 111 inelastic scattering process, 109 Index iterative sequences, 110–111 POP scattering, 108 rate of convergence, 110 RTD, see Resonant-tunneling diode S Scanning tunneling microscope (STM), 447–448 SCCM, see Space-charge control model Scharfetter–Gummel discretization, 157–158, 221 Schockley–Read–Hall (SRH) generation– recombination, 227 Schrödinger equation, 446, 473, 477, 578, 581, 619 Airy function approach, 400 band structure, 38, 40 density of states (DOS), 55 quantum transport effects, semiclassical transport theory, 95 semiconductor application, 381 tunneling coefficient calculation, 400 Semiclassical transport theory BTE Fermi–Dirac function, 100 momentum space, 98 scattering processes, 100–102 six-dimensional phase space, 99 thermal equilibrium, 100 total rate of change, 99 defect scattering alloy disorder scattering, 121–122 ionized impurity scattering, 120–121 neutral impurity scattering, 121 distribution function displaced Maxwellian approximation, 97–98 quasi-Fermi level concept, 96–97 electron–electron interactions binary collisions, 122–123 Coulomb forces, 122 electron–plasmon scattering, 123 6H-SiC mobility calculation (see 6H-SiC mobility calculation) impact ionization, 124 Newton’s equations, 95 phonon scattering acoustic phonon scattering, 112–114 approximations, 111–112 electron–phonon scattering, 111 lattice displacement, 112 nonpolar optical phonon scattering, 113–118 piezoelectric scattering, 119–120 polar optical phonon scattering, 119 761 relaxation-time approximation collision integral equation, 102 coordinate system, 104–105 current density, 106 electron mobility, 107 first-order differential equation, 103 Legendre polynomials, 104 Mathiessenn’s rule, 107 parabolic dispersion, 104 scattering process, 105–106 spheroidal harmonics expansion, 102 Taylor series expansion, 106 uniformly doped semiconductor, 103 Rode’s iterative method (see Rode’s iterative method) Schrödinger equation, 95 Semiconductor Industry Association (SIA), 2–3 Semiconductors arsenic and boron impurity, 32–33 band diagram, 35 carrier concentration, 34 covalent bond, 32–33 definition, 32 digital circuits, 61–62 diode current–voltage characteristics, 65 depletion charge approximation, 63 drift and diffusion components, 64–65 energy-band diagram, 63 forward and reverse component, 65–66 minority carrier density, 64 minority carrier diffusion equation, 64 photodiodes, 62 pn junction diode, 62 dopant-site bonding energy, 34 doping, 32 elemental materials, 36 four-terminal devices, 61 majority and minority carrier, 33 material preparation crystalline inorganic solids, 41 device-type fabrications, 42–44 electronic properties, 40 GaAs, 44 gallium nitride (GaN), 45 LED, 44–45 material properties, 40–41 selenium sulfide, 45 silicon carbide (SiC), 44 zone refining, 41 MESFET, 80–81 multi-terminal devices, 61 n- and p-type designations, 33 762 partially depleted=fully depleted silicon-oninsulator (PD=FD SOI) devices depletion regions, 78 device structures, 76–77 n-channel and p-channel SOI device, 78 output characteristics, 78–79 subthreshold slope vs Si film thickness, 79 photoemission process, 35 statistics, 59–60 thermionic devices, 60 two-and three-terminal devices, 61 Semiconductor systems, quantum transport amplitude function, 450 Born interpretation, 449 expectation value, 449 first postulate, quantum mechanics, 448 Fourier transform, 450 Heisenberg uncertainty principle, 451 Landauer–Büttiker formalism, 472–473 mathematical operators, 450–451 microscopic system, 450 normalization condition, wave function, 449 potential step E > V0, 455–458 E < V0, 454–455 piecewise-constant potentials, 453–454 Schrödinger wave equation (SWE), 451 second postulate, quantum mechanics, 448 separable solutions, 452 separation of variables, 452 stationary states, free particle, 453 stationary-state wave functions, 452 TDSWE, 451–452 third postulate, quantum mechanics, 450 TISE, 453 transfer matrix approach double-barrier structure, 465–468 energy band formation, 468–469 limitations, 470–471 nanoscale MOSFETs, 464–465 PCPBT, tight-binding approach, 469–470 propagation matrix, 462 quantum mechanical reflections, 463–464 tunneling quantum-mechanical tunneling, 446–448 resonant tunneling diodes, 446–447 single potential barrier, 458–459 STM, 447–448 total transmission, 460–461 transmission coefficient, 460–461 Index uncertainty, 449–450 Usuki iterative procedure, spin filters simulation results, 479–484 spin transport, 475–477 theoretical modeling, 477–479 Sentaurus device device structures, 226 electrical characteristics, 227 features, 225–226 tool flow, 227 types of simulation, 226 virtual device structure, 225 Shockley–Read–Hall (SRH) generation– recombination mechanism, 165–166 Short-range domain (SRD), 312–313 SIA, see Semiconductor Industry Association Si-based nanoelectronics conventional silicon band-engineered transistor, CV=I metric, new device structures, 6–7 new materials, 5–6 nonclassical CMOS devices, 6, PMOS and NMOS transistors, device scaling dimensional integrity, lithography improvement, 2–3 Moore’s law, nonbulk MOSFET structures, silicon FETs and integrated circuits, quantum transport effects Born approximation, 11 classical vs quantum charge, 9–10 de Broglie wavelength, dynamical quantum effects, 9, 11 Fermi Golden Rule, potential and electric fields, 9–10 Schrödinger equation, spatial=size-quantization, semiconductors history, 1–2 single-crystal materials, transistor scaling, SiGe p-channel MOSFETs band structure, 408–409 bulk (3D) transport, 408 confined (2D) carrier transport, 408 energy band diagram, 407–408 enhanced performance, 406 full band Monte Carlo approach, 410 geometro-analytical model, 409 hole mobility, 407 hole transport calculations, 409–410 pseudopotentials, 408–409 763 Index self-consistency, 410 source=drain (S=D) region, 406–407 subband structure, numerical techniques, 410 subsurface leakage current suppression, 406–407 thin-body transistor, heterostructure channel, 407–408 Silicon-on-insulator (SOI) transistor, 348 Silvaco ATLAS ATLAS command file, 223 DECKBUILD tools, 222–223 definition, 222 device structure, 224 Gummel method, 224–225 input and output information, 222–223 Newton method, 224–225 sequence of statements, 223 VWF interactive tools, 222 Single channel Landauer formula, 473 Slotboom variables, 154–155 Space-charge control model (SCCM), 75 Spintronics, 475–476 Staggered leapfrog method, 215–216 Stair-casing effect, 710 Standard deviation, 449 Stark effect diagonal matrix elements, 726 infinite square well, 725 off-diagonal matrix elements, 726 results, 727 Stationary perturbation theory degenerate states, 722–725 first-order approximation, 719–720 harmonic oscillator with linear perturbation, 727–729 second-order approximation, 720–722 smallness parameter, 717 stark effect in a potential well diagonal matrix elements, 726 infinite square well, 725 off-diagonal matrix elements, 726 results, 727 zero-order approximation, 718 STM, see Scanning tunneling microscope Strained InGaAs=AlAs RTD system, 566–567 Stratton’s approach, 199–200 0.7 Structure, 476 Successive over-relaxation method, 649–650 Surface-roughness scattering, 427–428, 433, 514, 522 Synopsys software, see Sentaurus device T TCAD, see Technology computer aided design TDSWE, see Time-dependent Schrödinger wave equation Technology computer aided design (TCAD), 15, 219 Thermal3D package, Silvaco GaN HEMT device, 346 key features, 345 optimal space determination, 346–347 thermal conductivity, GaAs and Si, 346 Tight-binding (TB) method Chadi and Cohen TB parameters, 618 implementation, 617 quantitative description, 616 Time-dependent perturbation theory amplitude calculation, 733–734 Fermi Golden Rule conditions for validity, 736–738 elastic scattering of electrons, 738–746 summary, 735–736 matrix methods, 732 simple harmonic perturbation, 731 time-dependent SWE, 729 unperturbed system, 730 Time-dependent Schrödinger wave equation (TDSWE), 451–452, 729–730 Time-independent Schrödinger equation (TISE), 453–454, 458 Transfer matrix approach, 413 double-barrier structure, 465–468 energy band formation, 468–469 limitations, 470–471 nanoscale MOSFETs, 464–465 PCPBT, tight-binding approach, 469–470 propagation matrix, 462 quantum mechanical reflections, 463–464 Transmission probability T(E), 457 Tridiagonal block matrix, 414 Two-step Lax–Wendroff scheme, 217 U Ultrasmall electronics, 19 Ultrathin body (UTB) devices, 568–569 Umklapp (U) process, 345, 517 Unintentional dopants conduction band profile, 325 drain current fluctuation, 325–326 electrostatics, 327 764 gate-oxide layer, 324 impurity position dependence, 327–328 localized barrier, 325 nonuniform carrier quantization, 327 screening impact, 326–327 size-quantization effect, 326 source-drain doping, 324 ultra-narrow channel FD-SOI device structure, 324 uniform and discrete impurity model, 329 velocity and energy plots, 325 Usuki iterative procedure, spin filters 2D lattice model, quantum wire, 473–474 linear operator, 474–475 simulation results calibration, parameters, 479 conductance–split gate voltage characteristics, 482–483 GaAs=AlGaAs heterostructure, 480–481 Hartree potential, 481 relevant scattering mechanisms, 484 sheet density vs gate voltage, 479–480 simulated structure, 479–480 spin polarization, 482–483 split-gate structures, 480 spin transport, 475–477 theoretical modeling correlation energy, 478–479 2DEG, 477–478 electron density, 477–478 exchange energy, 478 QPC, 477 Index V Van Dort model, 383–384 Virtual Wafer Fab (VWF) interactive tools, 222 von Neumann stability analysis, 212, 216 W Wick’s theorem, 501–503, 508 Wigner–Boltzmann equation, effective potential Bloch equation, 389 DG SOI device output characteristics, 393–394 doping scheme, 392–393 double gate (DG) SOI device structure, 392–393 effective quantum potential, 389, 391 Green’s function, 389 Hartree potential, 389–391 intrinsic device, 392–393 Liouville equation, 388–389 size-quantization effects, 391 threshold voltage vs SOI film thickness, 391–392 Weyl quantization, 388 Wigner distribution function potential operator, 497 RTDs, 497–498 tunneling and quantization, 496 unitary evolution, 497 Wigner–Seitz cell, 24, 30 Wigner–Weyl transformation, 496 Y Yee cell, 678 .. .Computational Electronics Semiclassical and Quantum Device Modeling and Simulation Dragica Vasileska Stephen M Goodnick Gerhard Klimeck CRC Press Taylor & Francis... Hiroki, and K Matsuzawa, Device modeling and simulation toward sub-10 nm semiconductor devices, IEEE Trans Nanotechnol., 1, 63 (2002) 26 I Knezevic, Memory Effects and Mesoscopic Quantum Transport,... computer aided design (TCAD) involves (1) process simulation followed by (2) device simulation finalized with a (3) circuit simulation Device simulation itself is the process 16 Computational Electronics