Michael Christopher Gibson

A thesis submitted for the degree of

Doctor of Philosophy

Department of Physics

University of Durham

University of Durham

2006

Density functional theory (DFT) is a method of effectively solving the many-electron Schrödinger equation, enabling the properties of condensed matter systems to be calculated from first principles. With the commonly used local density approximation (LDA), and generalised gradient approximations (GGAs), to the exchange correlation functional, it is currently possible to perform calculations on systems containing several hundred atoms. The accuracy of such calculations depends on the system under study and on which particular properties one wishes to calculate. The use of more advanced functionals has the potential to improve accuracy, at the expense of greater computational demand. In this work we use the LDA to calculate certain properties of GaN, such as geometry, band structure, and surface properties, including the reconstruction of GaN surfaces under the presence of hydrogen. We then describe our computational implementation of advanced density functionals, including screened exchange (sX-LDA), Hartree-Fock (HF), and exact exchange (EXX), within an efficient, fully parallel, plane wave code. The implementation of sX-LDA and HF is used to calculate band structure properties of Si, GaN, and other simple semiconductors, and it is found that sX-LDA can improve results significantly beyond the LDA. We also derive and implement the theory that allows one to calculate directly the contribution to the stress tensor from exchange and correlation when using these functionals, and demonstrate this with some simple test cases. Finally, we introduce some new theoretical ideas that may pave the way for yet more accurate density functionals in the future.

The work presented here has contributed to the following publications:

*Screened Exchange Stress Tensor in Density Functional Theory*, M. C. Gibson, S. Brand, S. J. Clark, Phys. Rev. B.**73**, 125120 (2006).*Screened Exchange Calculations of Semiconductor Band Structures*,

[4] M. C. Gibson, S. J. Clark, S. Brand, R. A. Abram, AIP Conf. Proc.**772**, 1125 (2005).*First-principles calculations of reconstructions of GaN*(0001)*surfaces involving N, Al, Ga, In, and As atoms*, V. Timon, S. Brand, S. J. Clark,

[4]M. C. Gibson, R. A. Abram, Phys. Rev. B**72**, 035327 (2005).*Defect Energy Levels in Hf, High-Dielectric-Constant Gate Oxide*,

[4]A. K. Xiong, J. Robertson, M. C. Gibson, S. J. Clark, App. Phys. Lett.**87**, 183505 (2005).

The work presented here was undertaken within the Department of Physics at the University of Durham between August 2002 and August 2005. I confirm that no part of this work has previously been submitted for a degree at this or any other institution and, unless otherwise stated, it is the original work of the author.

Michael C. Gibson

May 2006

The copyright of this thesis rests with the author. No quotation, figure,
or any other part of it should be published in any format, including
electronic and the Internet, without his prior written consent. All
information derived from this thesis must be acknowledged appropriately.

First and foremost I would like to thank my supervisor Stewart Clark for his
help, advice, curry, and malt whisky over the past few years. Thanks also to my co-supervisor
Stuart Brand, head of research group Richard Abram, and everyone else who has
been involved with the Durham condensed matter theory group during my time here.
In particular I would like to thank Ian Bolland for helping learn the basics
of Linux when I arrived, Paul Tulip for his entertaining office rants, and Dom Jochym for
some Linux-related tips that have proved very useful in writing this thesis.
Outside of work I would like to thank my girlfriend and my family for their constant
support and encouragement, without which the completion of this work would not have
been possible.

- Contents
- List of Figures
- List of Tables
- Introduction
- Physics from First Principles
- Basics of Density Functional Theory
- The Plane Wave Pseudopotential Approach
- Periodic Boundary Conditions
- Removing Infinities with Periodic Boundary Conditions
- Kohn-Sham Orbitals with Periodic Boundary Conditions
- Plane Waves and Reciprocal Space
- Evaluating Quantities in Reciprocal Space
- Operators in Reciprocal Space
- Convergence of the Plane Wave Basis Set
- Monkhorst-Pack Grids
- Introduction to Pseudopotentials
- Basic Pseudopotential Theory
- Disadvantages of Pseudopotentials

- Solving the Kohn-Sham Equations
- The Kohn-Sham Band Structure
- Extensions of Kohn-Sham Theory
- The CASTEP Code
- Summary and Outline of Chapters

- GaN Calculations with the LDA
- About GaN
- Calculations on Bulk GaN
- Theory of Surface Energetics
- GaN Surface Calculations
- Summary and Conclusions

- Theory of Non-Local Functionals
- sX-LDA and HF
- Exact Exchange
- Improving Brillouin Zone Integration
- Other Non-Local Functionals
- Summary

- Computational Implementation
- Introduction
- Preparation of Basis Set Data
- Elements of Non-Local Functional Calculations
- Additional Considerations
- Parallelisation, Symmetry, and Other Issues
- Performance Tests
- EXX and the OEP Method
- Summary and Conclusions

- Band Structure Calculations
- Full Band Structures for Silicon and GaN
- Other Group IV and III-V Semiconductors
- Discussion of Results

- Calculating Stress

- Conclusions and Further Work

- Symbols and Abbreviations

- Units and Physical Constants

- Implicit Mathematical Elements

- Derivations
- The Particle Density
- The Pair-Density and Related Quantities
- The Non-Interacting Kinetic Energy
- The Non-Interacting Internal Potential Energy

- Bibliography
- About this document ...