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Advanced Calculations for Defects in Materials: Electronic Structure Methods

ISBN: 978-3-527-41024-8
402 pages
June 2011
Advanced Calculations for Defects in Materials: Electronic Structure Methods (3527410244) cover image

Description

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.
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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 DFT-LDA/GGA Band-Gap 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 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 Grüning, 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 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 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, Jörg 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 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 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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Author Information

Chris G. Van de Walle is Professor at the Materials Department of the University of California in Santa Barbara. Before that he worked at IBM Yorktown Heights, at the Philips Laboratories in New York, as Adjunct Professor at Columbia University, and at the Xerox Palo Alto Research Center. Dr. Van de Walle has published over 200 articles and holds 18 U.S. patents. In 2002, he was awarded the David Adler Award by the APS. Dr. Van de Walle's research focuses on computational physics, defects and impurities in solids, novel electronic materials and device simulations.

Jörg Neugebauer is the Director of the Computational Materials Design Department at the Max-Planck-Institute 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 scale-bridging 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 atomic-scale 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 oxide-semiconductor 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.
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