Title:

Problems concerning the electonic structure of solids

The major emphasis of this thesis relates to the calculation of the electronic structure of the isolated vacancy in the diamond and silicon crystal using the defect molecule approach. The electronic energy levels from an undistorted vacancy calculation are corrected, to a first approximation, for symmetric relaxation and JahnTeller distortion effects. The electronic properties of the undistorted neutral divacancy in diamond are also determined. A description of a dynamic relaxation program developed to estimate the displacement of atoms surrounding a point defect in the diamondtype crystal as the result of distortion effects is also reported. A discussion of the theoretical methods available for the calculation of the electronic structure of point defects in diamondtype crystals and the reasons for choosing the defect molecule approach is presented in section 1.2 of Chapter 1. In the remainder of the chapter the experimental evidence available for the existence of single vacancies, divacancies and the interstitial atom in irradiated diamond and silicon crystals is briefly reviewed. Chapter 2 contains a detailed account of the fundamental aspects of the defect molecule method first proposed by Coulson and Kearsley. The limitations as well as the advantages of this method are discussed and the modifications made to the model in the present study are outlined. The basic formal relationships required for the calculation of the electronic structure of the undistorted vacancy in its various charge states are developed within this framework. The results of applying this method to a calculation for the single isolated vacancy in the diamond crystal are presented and discussed in Chapter 3. Four possible charge states of the vacancy; namely, the single positive, the neutral, the single negative and the double negatively charged state are considered. An investigation of the sensitivity of the relative ordering of the energy levels for the various centres to two different choices of basis functions to represent the 2s and 2p orbitals on the tons in the crystal is undertaken. For the neutral vacancy the ^{1}E electronic level is predicted to be the ground state of this centre with the ^{3}T_{1} level the first excited state when simple Slatertype functions are chosen for the 2s and 2p orbitals. However, when the atomic HartreeFock functions proposed by Clementi are employed the order of these two levels is inverted. A simple model to examine the influence of delocalizing the electronic wavefunction of the defect electrons on the relative ordering of the lowest levels is presented in section 3.5. For the neutral vacancy no change in the ordering of the lowest levels results when a reasonable amount of delocalisation is incorporated. For the negatively charged centre the ^{4}A_{2} level is reinforced as the lowest electronic level. The calculation for the electronic structure of the uto distorted neutral divacancy in the diamond crystal using a modified antisymmetrized molecular orbital method, based upon the bond method, is reported in Chapter 4. Using simple Slatertype functions for the 2s and 2p orbitals the ^{3}A_{2g} level is predicted to be the ground state of this centre, with spin and orbitallyallowed transitions to the ^{3}A_{1u} level corresponding to optical absorption in the uv region of the spectrum. In Chapter 5 the development of a dynamic relaxation method which enables the displacement of the atoms surrounding a point defect in the diamondtype lattice to be determined free from the constraints of previous models is described. The method relies upon having a valenceforce potential function which includes noncentral as well as central force terms to describe the interactions between the atoms of the perfect crystal. The interaction between the defect electrons is simulated by applying external forces to the atoms nearest the defect. The method is applicable to a wide range of systems; however, here it has been applied only to the neutral single vacancy in the diamond and silicon crystal which undergoes a tetragonal distortion as a result of the nature of the rebonding forces. Displacements of the atoms in the vicinity of the defect are much greater for the silicon crystal than for the diamond crystal. It has been demonstrated that the displacement of atoms which are fifth neighbour to the defect is greater than that for third or fourth neighbours. As a byproduct of the calculation the formation energy of the single vacancy has been evaluated. The theory of the JahnTeller effect, to first order, for the degenerate levels of the vacancy system is developed in Chapter 6 using the rigid atom approximation proposed by Lidiard and Stoneham. The differences between the method used in the present calculation and that developed by Lannoo and coworkers is discussed. The extent of JahnTeller splitting for all the lowest levels of the various centres is determined. Contrary to previous suggestions the values obtained for the splittings were found to be insensitive to the choice of basis functions for the valence orbitals. A correction to the lowest energy levels for the symmetry relaxation of the atoms nearest the defect is also made. This latter modification was different for various electronic levels belonging to a particular charge state of the vacancy and consequently is very important in determining the relative ordering of the lowest levels. This is again contrary to what has previously been tacitly assumed. The corrected electronic structure of the neutral vacancy suggests that this centre is not responsible for the GR1 band observed in irradiated diamonds as the ^{3}T_{1} level is predicted to be the ground state of this centre from all calculations. On the basis of this work it is also unlikely that this band is associated with the negatively charged vacancy; however, this possibility cannot be definitely eliminated. The electronic structure for the four possible charged states of the single vacancy in the silicon crystal is determined in Chapter 7. The atomic HartreeFock functions proposed by Clementi are used to approximate the 3s and 3p orbitals on the silicon atoms in the crystal. Using both an unmodified and a modified antisymmetrized molecular orbital theory it has been demonstrated that electronelectron interaction terms and configuration mixing effects are important in determining the lowest levels of each centre. Again the energy levels resulting from the undistorted calculation are corrected for symmetric relaxation and JahnTeller distortion effects. The modified theory predicts ground state levels for the various centres which have the same total spin as those suggested by Watkins from his electron spin resonance experiments. The roost important findings and their implications on further work in this area are reported in a final general discussion.
