Advanced Calculations for Defects in Materials: Electronic Structure MethodsISBN: 9783527410248
402 pages
June 2011

Description
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.
Table of Contents
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 DFTLDA/GGA BandGap Problem and Possible Approaches to Overcome It 6
1.3.1 LDAþU for Materials with Semicore States 6
1.3.2 Hybrid Functionals 9
1.3.3 ManyBody 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 FixedNode 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 FixedNode Approximation 26
2.5 Conclusion 29
References 29
3 Electronic Properties of Interfaces and Defects from Manybody Perturbation Theory: Recent Developments and Applications 33
Matteo Giantomassi, Martin Stankovski, Riad Shaltaf, Myrta Grüning, Fabien Bruneval, Patrick Rinke, and GianMarco Rignanese
3.1 Introduction 33
3.2 ManyBody 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 SelfConsistent 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 SelfEnergy 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 aSi3N4 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 BandEdge 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 Deák, Adam Gali, Bálint 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 þ 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 þ 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 þ U Basics 166
10.3 LDA þ U Band Structures Compared to GW 168
10.4 Improved LDA þ U Model 170
10.5 Finite Size Corrections 172
10.6 The Alignment Issue 173
10.7 Results for New LDA þ 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 OnSite Functionals 187
11.4 Koopmans Behavior in Hybridfunctionals: 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. MartinSamos, 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
SuHuai 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 HAssisted 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 ImpurityBand Doping 232
13.6 Summary 237
References 238
14 Electrostatic Interactions between Charged Defects in Supercells 241
Christoph Freysoldt, Jörg Neugebauer, and Chris G. Van de Walle
14.1 Introduction 241
14.2 Electrostatics in Real Materials 243
14.2.1 Potentialbased 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 Jörg 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 GaussianType 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 DTensors 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 MuffinTin Orbital (LMTO) Method 313
17.3.3 The Size of The Perturbed Region 315
17.3.4 Lattice Relaxation: The AsGaFamily 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 nType Doping of SiC with Phosphorus 332
17.7 Conclusions 334
References 336
18 TimeDependent Density Functional Study on the Excitation Spectrum of Point Defects in Semiconductors 341
Adam Gali
18.1 Introduction 341
18.1.1 NitrogenVacancy 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 TimeDependent Density Functional Theory with Practical Approximations 346
18.3 Results and Discussion 351
18.3.1 NitrogenVacancy 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 SelfInteraction Errors 370
19.2.4 Beyond DFT 372
19.3 Conclusions 373
References 375
Index 381
Author Information
Jörg Neugebauer is the Director of the Computational Materials Design Department at the MaxPlanckInstitute for Iron Research in Düsseldorf, Germany. Since 2003 he has been the Chair of Theoretical Physics at the University of Paderborn.Before that, he held positions as Honorary Professor and Director of the advanced study group 'Modeling' at the Interdisciplinary Center for Advanced Materials Simulation (ICAMS) at the Ruhr University in Bochum, Germany. His research interests cover surface and defect physics, ab initio scalebridging computer simulations, ab initio based thermodynamics and kinetics, and the theoretical study of epitaxy, solidification, and microstructures.
Alfredo Pasquarello is Professor of Theoretical Condensed Matter Physics and Chair of Atomic Scale Simulation at EPFL, Switzerland. His research activities focus on the study of atomicscale phenomena with the aim to provide a realistic description of the mechanisms occurring on the atomic and nanometer scale. Specific research projects concern the study of disordered materials and oxidesemiconductor interfaces, which currently find applications in glass manufacturing and in the microelectronic technology, respectively.
Peter Deák was Professor and Head of the Surface Physics Laboratory at the Budapest University of Technology & Economics and is currently Group Leader at the Center for Computational Materials Science in Bremen, Germany. His research interests cover materials science and the technology of electronic and electric devices, functional coatings and plasma discharges, and atomic scale simulation of electronic materials. Peter Deák has published over 150 papers, eight book chapters, and six textbooks.
Audrius Alkauskas holds a position at the Electron Spectrometry and Microscopy Laboratory of the EPFL, Switzerland. His scientific interests cover computational material science, theoretical solid state spectroscopy and surface and interface science with respect to applications in renewable energy, photovoltaics, energy conversion, and molecular nanotechnology.