Advanced calculations for defects in materials : electronic structure methods

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Advanced calculations for defects in materials : electronic structure methods

edited by Audrius Alkauskas ... [et al.]

Wiley-VCH, c2011

  • : hbk.

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Includes bibliographical references and index

内容説明・目次

内容説明

This book investigates the possible ways of improvement by applying more sophisticated electronic structure methods as well as corrections and alternatives to the supercell model. In particular, the merits of hybrid and screened functionals, as well as of the +U methods are assessed in comparison to various perturbative and Quantum Monte Carlo many body theories. The inclusion of excitonic effects is also discussed by way of solving the Bethe-Salpeter equation or by using time-dependent DFT, based on GW or hybrid functional calculations. Particular attention is paid to overcome the side effects connected to finite size modeling. The editors are well known authorities in this field, and very knowledgeable of past developments as well as current advances. In turn, they have selected respected scientists as chapter authors to provide an expert view of the latest advances. The result is a clear overview of the connections and boundaries between these methods, as well as the broad criteria determining the choice between them for a given problem. Readers will find various correction schemes for the supercell model, a description of alternatives by applying embedding techniques, as well as algorithmic improvements allowing the treatment of an ever larger number of atoms at a high level of sophistication.

目次

List of Contributors XIII 1 Advances in Electronic Structure Methods for Defects and Impurities in Solids 1 Chris G. Van de Walle and Anderson Janotti 1.1 Introduction 1 1.2 Formalism and Computational Approach 3 1.2.1 Defect Formation Energies and Concentrations 3 1.2.2 Transition Levels or Ionization Energies 4 1.2.3 Practical Aspects 5 1.3 The DFT-LDA/GGA Band-Gap Problem and Possible Approaches to Overcome It 6 1.3.1 LDApU for Materials with Semicore States 6 1.3.2 Hybrid Functionals 9 1.3.3 Many-Body Perturbation Theory in the GW Approximation 12 1.3.4 Modified Pseudopotentials 12 1.4 Summary 13 References 14 2 Accuracy of Quantum Monte Carlo Methods for Point Defects in Solids 17 William D. Parker, John W. Wilkins, and Richard G. Hennig 2.1 Introduction 17 2.2 Quantum Monte Carlo Method 18 2.2.1 Controlled Approximations 20 2.2.1.1 Time Step 20 2.2.1.2 Configuration Population 20 2.2.1.3 Basis Set 20 2.2.1.4 Simulation Cell 21 2.2.2 Uncontrolled Approximations 22 2.2.2.1 Fixed-Node Approximation 22 2.2.2.2 Pseudopotential 22 2.2.2.3 Pseudopotential Locality 23 2.3 Review of Previous DMC Defect Calculations 23 2.3.1 Diamond Vacancy 23 2.3.2 MgO Schottky Defect 25 2.3.3 Si Interstitial Defects 25 2.4 Results 25 2.4.1 Time Step 26 2.4.2 Pseudopotential 26 2.4.3 Fixed-Node Approximation 26 2.5 Conclusion 29 References 29 3 Electronic Properties of Interfaces and Defects from Many-body Perturbation Theory: Recent Developments and Applications 33 Matteo Giantomassi, Martin Stankovski, Riad Shaltaf, Myrta Gruning, Fabien Bruneval, Patrick Rinke, and Gian-Marco Rignanese 3.1 Introduction 33 3.2 Many-Body Perturbation Theory 34 3.2.1 Hedin.s Equations 34 3.2.2 GW Approximation 36 3.2.3 Beyond the GW Approximation 37 3.3 Practical Implementation of GW and Recent Developments Beyond 38 3.3.1 Perturbative Approach 38 3.3.2 QP Self-Consistent GW 40 3.3.3 Plasmon Pole Models Versus Direct Calculation of the Frequency Integral 41 3.3.4 The Extrapolar Method 44 3.3.4.1 Polarizability with a Limited Number of Empty States 45 3.3.4.2 Self-Energy with a Limited Number of Empty States 46 3.3.5 MBPT in the PAW Framework 46 3.4 QP Corrections to the BOs at Interfaces 48 3.5 QP Corrections for Defects 54 3.6 Conclusions and Prospects 57 References 58 4 Accelerating GW Calculations with Optimal Polarizability Basis 61 Paolo Umari, Xiaofeng Qian, Nicola Marzari, Geoffrey Stenuit, Luigi Giacomazzi, and Stefano Baroni 4.1 Introduction 61 4.2 The GW Approximation 62 4.3 The Method: Optimal Polarizability Basis 64 4.4 Implementation and Validation 68 4.4.1 Benzene 69 4.4.2 Bulk Si 70 4.4.3 Vitreous Silica 70 4.5 Example: Point Defects in a-Si3N4 72 4.5.1 Model Generation 72 4.5.2 Model Structure 73 4.5.3 Electronic Structure 74 4.6 Conclusions 77 References 77 5 Calculation of Semiconductor Band Structures and Defects by the Screened Exchange Density Functional 79 S. J. Clark and John Robertson 5.1 Introduction 79 5.2 Screened Exchange Functional 80 5.3 Bulk Band Structures and Defects 82 5.3.1 Band Structure of ZnO 83 5.3.2 Defects of ZnO 85 5.3.3 Band Structure of MgO 89 5.3.4 Band Structures of SnO2 and CdO 90 5.3.5 Band Structure and Defects of HfO2 91 5.3.6 BiFeO3 92 5.4 Summary 93 References 94 6 Accurate Treatment of Solids with the HSE Screened Hybrid 97 Thomas M. Henderson, Joachim Paier, and Gustavo E. Scuseria 6.1 Introduction and Basics of Density Functional Theory 97 6.2 Band Gaps 100 6.3 Screened Exchange 103 6.4 Applications 104 6.5 Conclusions 107 References 108 7 Defect Levels Through Hybrid Density Functionals: Insights and Applications 111 Audrius Alkauskas, Peter Broqvist, and Alfredo Pasquarello 7.1 Introduction 111 7.2 Computational Toolbox 112 7.2.1 Defect Formation Energies and Charge Transition Levels 113 7.2.2 Hybrid Density Functionals 114 7.2.2.1 Integrable Divergence 115 7.3 General Results from Hybrid Functional Calculations 117 7.3.1 Alignment of Bulk Band Structures 118 7.3.2 Alignment of Defect Levels 120 7.3.3 Effect of Alignment on Defect Formation Energies 122 7.3.4 The Band-Edge Problem 124 7.4 Hybrid Functionals with Empirically Adjusted Parameters 125 7.5 Representative Case Studies 129 7.5.1 Si Dangling Bond 129 7.5.2 Charge State of O2 During Silicon Oxidation 131 7.6 Conclusion 132 References 134 8 Accurate Gap Levels and Their Role in the Reliability of Other Calculated Defect Properties 139 Peter Deak, Adam Gali, Balint Aradi, and Thomas Frauenheim 8.1 Introduction 139 8.2 Empirical Correction Schemes for the KS Levels 141 8.3 The Role of the Gap Level Positions in the Relative Energies of Various Defect Configurations 143 8.4 Correction of the Total Energy Based on the Corrected Gap Level Positions 146 8.5 Accurate Gap Levels and Total Energy Differences by Screened Hybrid Functionals 148 8.6 Summary 151 References 152 9 LDA p U and Hybrid Functional Calculations for Defects in ZnO, SnO2, and TiO2 155 Anderson Janotti and Chris G. Van de Walle 9.1 Introduction 155 9.2 Methods 156 9.2.1 ZnO 158 9.2.2 SnO2 160 9.2.3 TiO2 161 9.3 Summary 163 References 163 10 Critical Evaluation of the LDA p U Approach for Band Gap Corrections in Point Defect Calculations: The Oxygen Vacancy in ZnO Case Study 165 Adisak Boonchun and Walter R. L. Lambrecht 10.1 Introduction 165 10.2 LDA p U Basics 166 10.3 LDA p U Band Structures Compared to GW 168 10.4 Improved LDA p U Model 170 10.5 Finite Size Corrections 172 10.6 The Alignment Issue 173 10.7 Results for New LDA p U 174 10.8 Comparison with Other Results 176 10.9 Discussion of Experimental Results 178 10.10 Conclusions 179 References 180 11 Predicting Polaronic Defect States by Means of Generalized Koopmans Density Functional Calculations 183 Stephan Lany 11.1 Introduction 183 11.2 The Generalized Koopmans Condition 185 11.3 Adjusting the Koopmans Condition using Parameterized On-Site Functionals 187 11.4 Koopmans Behavior in Hybrid-functionals: The Nitrogen Acceptor in ZnO 189 11.5 The Balance Between Localization and Delocalization 193 11.6 Conclusions 196 References 197 12 SiO2 in Density Functional Theory and Beyond 201 L. Martin-Samos, G. Bussi, A. Ruini, E. Molinari, and M.J. Caldas 12.1 Introduction 201 12.2 The Band Gap Problem 202 12.3 Which Gap? 204 12.4 Deep Defect States 207 12.5 Conclusions 209 References 210 13 Overcoming Bipolar Doping Difficulty in Wide Gap Semiconductors 213 Su-Huai Wei and Yanfa Yan 13.1 Introduction 213 13.2 Method of Calculation 214 13.3 Symmetry and Occupation of Defect Levels 217 13.4 Origins of Doping Difficulty and the Doping Limit Rule 218 13.5 Approaches to Overcome the Doping Limit 220 13.5.1 Optimization of Chemical Potentials 220 13.5.1.1 Chemical Potential of Host Elements 220 13.5.1.2 Chemical Potential of Dopant Sources 222 13.5.2 H-Assisted Doping 223 13.5.3 Surfactant Enhanced Doping 224 13.5.4 Appropriate Selection of Dopants 226 13.5.5 Reduction of Transition Energy Levels 229 13.5.6 Universal Approaches Through Impurity-Band Doping 232 13.6 Summary 237 References 238 14 Electrostatic Interactions between Charged Defects in Supercells 241 Christoph Freysoldt, Jorg Neugebauer, and Chris G. Van de Walle 14.1 Introduction 241 14.2 Electrostatics in Real Materials 243 14.2.1 Potential-based Formulation of Electrostatics 245 14.2.2 Derivation of the Correction Scheme 246 14.2.3 Dielectric Constants 249 14.3 Practical Examples 250 14.3.1 Ga Vacancy in GaAs 250 14.3.2 Vacancy in Diamond 252 14.4 Conclusions 254 References 257 15 Formation Energies of Point Defects at Finite Temperatures 259 Blazej Grabowski, Tilmann Hickel, and Jorg Neugebauer 15.1 Introduction 259 15.2 Methodology 261 15.2.1 Analysis of Approaches to Correct for the Spurious Elastic Interaction in a Supercell Approach 261 15.2.1.1 The Volume Optimized Aapproach to Point Defect Properties 262 15.2.1.2 Derivation of the Constant Pressure and Rescaled Volume Approach 264 15.2.2 Electronic, Quasiharmonic, and Anharmonic Contributions to the Formation Free Energy 266 15.2.2.1 Free Energy Born Oppenheimer Approximation 266 15.2.2.2 Electronic Excitations 269 15.2.2.3 Quasiharmonic Atomic Excitations 271 15.2.2.4 Anharmonic Atomic Excitations: Thermodynamic Integration 272 15.2.2.5 Anharmonic Atomic Excitations: Beyond the Thermodynamic Integration 274 15.3 Results: Electronic, Quasiharmonic, and Anharmonic Excitations in Vacancy Properties 278 15.4 Conclusions 282 References 282 16 Accurate Kohn Sham DFT With the Speed of Tight Binding: Current Techniques and Future Directions in Materials Modelling 285 Patrick R. Briddon and Mark J. Rayson 16.1 Introduction 285 16.2 The AIMPRO Kohn Sham Kernel: Methods and Implementation 286 16.2.1 Gaussian-Type Orbitals 286 16.2.2 The Matrix Build 288 16.2.3 The Energy Kernel: Parallel Diagonalisation and Iterative Methods 288 16.2.4 Forces and Structural Relaxation 289 16.2.5 Parallelism 289 16.3 Functionality 290 16.3.1 Energetics: Equilibrium and Kinetics 290 16.3.2 Hyperfine Couplings and Dynamic Reorientation 291 16.3.3 D-Tensors 291 16.3.4 Vibrational Modes and Infrared Absorption 291 16.3.5 Piezospectroscopic and Uniaxial Stress Experiments 291 16.3.6 Electron Energy Loss Spectroscopy (EELS) 292 16.4 Filter Diagonalisation with Localisation Constraints 292 16.4.1 Performance 294 16.4.2 Accuracy 296 16.5 Future Research Directions and Perspectives 298 16.5.1 Types of Calculations 299 16.5.1.1 Thousands of Atoms on a Desktop PC 299 16.5.1.2 One Atom Per Processor 299 16.5.2 Prevailing Application Trends 299 16.5.3 Methodological Developments 300 16.6 Conclusions 302 References 302 17 Ab Initio Green.s Function Calculation of Hyperfine Interactions for Shallow Defects in Semiconductors 305 Uwe Gerstmann 17.1 Introduction 305 17.2 From DFT to Hyperfine Interactions 306 17.2.1 DFT and Local Spin Density Approximation 306 17.2.2 Scalar Relativistic Hyperfine Interactions 308 17.3 Modeling Defect Structures 311 17.3.1 The Green.s Function Method and Dyson.s Equation 311 17.3.2 The Linear Muffin-Tin Orbital (LMTO) Method 313 17.3.3 The Size of The Perturbed Region 315 17.3.4 Lattice Relaxation: The AsGa-Family 317 17.4 Shallow Defects: Effective Mass Approximation (EMA) and Beyond 319 17.4.1 The EMA Formalism 320 17.4.2 Conduction Bands with Several Equivalent Minima 322 17.4.3 Empirical Pseudopotential Extensions to the EMA 322 17.4.4 Ab Initio Green.s Function Approach to Shallow Donors 324 17.5 Phosphorus Donors in Highly Strained Silicon 328 17.5.1 Predictions of EMA 329 17.5.2 Ab Initio Treatment via Green.s Functions 330 17.6 n-Type Doping of SiC with Phosphorus 332 17.7 Conclusions 334 References 336 18 Time-Dependent Density Functional Study on the Excitation Spectrum of Point Defects in Semiconductors 341 Adam Gali 18.1 Introduction 341 18.1.1 Nitrogen-Vacancy Center in Diamond 342 18.1.2 Divacancy in Silicon Carbide 344 18.2 Method 345 18.2.1 Model, Geometry, and Electronic Structure 345 18.2.2 Time-Dependent Density Functional Theory with Practical Approximations 346 18.3 Results and Discussion 351 18.3.1 Nitrogen-Vacancy Center in Diamond 351 18.3.2 Divacancy in Silicon Carbide 353 18.4 Summary 356 References 356 19 Which Electronic Structure Method for The Study of Defects: A Commentary 359 Walter R. L. Lambrecht 19.1 Introduction: A Historic Perspective 359 19.2 Themes of the Workshop 362 19.2.1 Periodic Boundary Artifacts 362 19.2.2 Band Gap Corrections 367 19.2.3 Self-Interaction Errors 370 19.2.4 Beyond DFT 372 19.3 Conclusions 373 References 375 Index 381

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