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Density functional theory
The first theory developed to deal with the electron energy and its density distribution was propounded by Thomas [32] in 1927 and Fermi [33] in 1928. Density functional theory (DFT) was invented by Kohn and Hohenberg [10, 11] in 1964 and was later made practical by Kohn and Sham [12] in 1965. DFT makes the electron density n(~r) the central quantity. Due to this reformulation, many-body problems can be simplified, because instead of dealing with the 3N spatial coordinates for the many body wavefunction, n(~r) now depends on 3 spatial coordinates only [34, 35].
The exchange-correlation energy
The exact formulation of the exchange-correlation energy is unknown and hence it needs to be approximated. Several methods have been developed to deal with this problem. The simplest and one of the most remarkable exchange-correlation approximations is the local density approximation (LDA) [12]. The generalized gradient approximation (GGA) [13, 14] is a class of widely used exchange-correlation energy functionals. The main deficiency of the LDA is it failure to accurately predict some properties of materials, especially the band gaps [40, 41, 42, 43] of most semiconductors. The GGA functionals yield more accurate atomization energies [44, 45, 46], total energies [47, 48] and barriers of chemical reactions than the LDA. Other notable exchange-correlation functionals are the LDA+U [49, 50], GGA+U [51] and hybrid functionals.
1 Introduction 1
1.1 Rationale and motivation
1.2 Research objective and goals
1.3 Overview of study
2 Density functional theory: a theoretical background
2.1 Introduction
2.2 The many-body wavefunction problem
2.3 The Hartree-Fock (HF) method
2.4 Density functional theory
2.5 The exchange-correlation energy
2.6 Periodic boundary conditions
2.7 Pseudopotentials
2.8 Brillouin zone sampling
3 Theory of defects
3.1 Introduction
3.2 Crystal structure
3.3 Electronic properties of a solid
3.4 Defects
3.5 Defects in semiconductors
3.6 Other properties of defects in semiconductors
3.7 Brief history of germanium
3.8 Defects in germanium
4 Computational background
4.1 Introduction
4.2 Computational code and techniques
4.3 Supercell and cluster method of defect modelling
4.4 Supercell correction techniques
4.5 Test of convergence
4.6 Details of the calculation
4.7 Validation of computational techniques
5 Results
5.1 Introduction
5.2 Ab initio study of the germanium di-interstitial using a hybrid functional (HSE)
5.3 A hybrid functional calculation of Tm3+ defects in germanium (Ge)
5.4 Rare earth interstitials in Ge: a hybrid density functional theory study
5.5 A first principle hybrid functional calculation of Tm3+ Ge -VGe defect complexes in germanium
5.6 Results summary
5.7 Published articles
6 Conclusion
6.1 Introduction
6.2 Future opportunities
