Title:

Some problems in the theory of the solid state

During the past thirty years, a large number of calculations on the electronic structure of crystals have been made. There are several reasons why such calculations are quite attractive. From the theoretical point of view, the advantage of crystals over large molecules is that their periodicity considerably simplifies the calculations; it is much easier to calculate the oneelectron energy levels of a perfect crystal of sodium or germanium, even though it may contain some 10^{22} atoms, than to find the energy levels of a nonsymmetric molecule containing twenty or thirty atoms. Thus, it is natural to proceed from calculations on isolated atoms and small molecules to calculations on simple crystals, rather than to work on large polyatomic molecules. Since most materials used in electronics are crystalline, such calculations can also lead to important applications; the energy gap between the valence and conduction bands of a semiconductor is an extremely important parameter, and generally calculations of the band structure can enable experimental results to be explained and made use of. Typical applications include the selection of materials for use in transistors, lasers and masers. The major part of this thesis is devoted to calculations on the valence bands of semiconducting crystals having the diamond or zincblende structures, which can also be called tetrahedral semiconductors. While a lot of work had been done previously on diamond, silicon and germanium, it is only in the past few years that an appreciable number of results have been published for compounds having the zincblende structure. However, several standard methods, such as the orthogonalised plane wave and augmented plane wave methods, are now available and in use for fullscale calculations. These methods generally use at least fifty plane waves in a variational procedure to find the oneelectron eigenvalues and eigenfunctions for the valence and lowest conduction bands, and seem to be fairly accurate. We had neither the desire nor the computer facilities to perform such calculations, but were concerned with a different aspect of these crystals. Classically, the ground state of these tetrahedral semiconductors can be represented by associating all the valence electrons with covalent bonds, and it therefore seemed of interest to investigate whether an accurate quantitative treatment could employ wave functions representing such bonds. Most work done previously using this approach has been based either on the LCAO method or on a related interpolation scheme. Since ours was, to the best of our knowledge, the first attempt to find bond functions without making the rather drastic a priori assumptions of the LCAO method, we decided to attempt approximate calculations for a wide range of substances rather than an extremely accurate calculation for just one or two compounds. This decision was also prompted by the fact that only a Mercury computer was available, and unlike Edsac, Atlas, and the other newer machines this was not fast enough and did not have enough storage space to make accurate calculations really feasible. In the first part of chapter one of this thesis, we derive the HartreeFock equations in a form which is convenient for calculations on crystals. In section 1, we introduce a division of the Coulomb and exchange terms in these equations which differs from the conventional one. It is customary with a closedshell system to write the electronelectron interaction terms as the sum of (i) the Coulomb potential due to all electrons in a different orbital from the one being considered, (ii) the Coulomb potential due to an electron in the same orbital as the one being considered but having opposite spin, and (iii) the exchange interaction due to all the electrons other than the one being considered. A disadvantage of this scheme is that none of these terms individually is invariant under a unitary transformation of the oneelectron orbitals, although their sum of course is. We preferred, therefore, to divide the electronelectron interactions into just two parts, viz. the Coulomb potential due to all the electrons in the crystal, including the one being considered, and the exchange potential due to all the electrons, again including the one being considered. The total potential is then the same as previously, but now each term separately is invariant under a unitary transformation of the wave functions. The importance of this step first emerges in section 2, where we consider the effects of crystal symmetry. The section begins with a brief outline of some of the more useful aspects of group theory, and after this we introduce a oneelectron Hamiltonian H. With our division of the electronelectron interactions, this can be so defined that the electronic wave functions enter into it only through a spinindependent firstorder density matrix q(r′r). This Hamiltonian H is equivalent to the HartreeFock operator in its effect on orbitals occupied in the ground state of the crystal. If we assume that the original manyelectron ground state is nondegenerate, so that its wave function has the full crystal symmetry, then all the density matrices derived from it also have this symmetry. It therefore seems reasonable in the HartreeFock approximation to restrict our consideration to those Slater determinants for the ground state which lead to firstorder density matrices having the full crystal symmetry. If this is done, our Hamiltonian H also has the full crystal symmetry, so that all the valuable results of group theory can be applied to its eigenfunctions: this is generally assumed in calculations on crystals, but is seldom explicitly stated or proved. For orbitals not occupied in the ground state, our oneelectron Hamiltonian H is not equivalent to the HartreeFock operator; on the other hand, the approximation of a single Slater determinant is unlikely to be valid for such excited states. It is possible to give the name "generalised molecular orbital" to those eigenfunctions of H not corresponding to orbitals occupied in the ground state of the crystal, and the corresponding eigenvalues are in fact usually used to represent the conduction band. The rest of chapter one, in a slight digression from the main theme of the thesis, examines various different methods of calculating molecular orbitals in a crystal. Particular attention is paid to the orthogonalised plane wave method, and it is shown how the effect of orthogonalisation to the core states, which is often represented by a pseudopotential, can be very simply and effectively treated by a matrix routine. This approach can only be used if a finite set of basis functions is employed, but this restriction is not important in practice; moreover, our method copes with such problems as the possible overcompleteness of a basis set once orthogonalisation to the core functions has taken place. Unlike the pseudopotential method, our scheme allows exactly for the normalisation of the total wave function, and it can also be used with the modified plane wave method, L.C.A.O. methods, etc; we also used it in our bond orbital calculations. After examining the LCAO and LCBO methods, we proceed in the last section of chapter one to establish the formal relationship between these methods and the OPW one. To the extent that these methods are regarded as a means of calculating molecular orbitals (an alternative approach to the LCBO method is suggested in chapter two), we show that they differ from the OPW method only in attempting to expand the smooth parts of the molecular orbitals in terms of a few Bloch waves constructed from atomic orbitals instead of in terms of a much larger number of plane waves.
