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 one-electron energy levels of a perfect crystal of sodium or germanium, even though it may contain some 1022 atoms, than to find the energy levels of a non-symmetric 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 poly-atomic 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 semi-conductors. 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 full-scale calculations. These methods generally use at least fifty plane waves in a variational procedure to find the one-electron 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 Hartree-Fock 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 closed-shell system to write the electron-electron 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 one-electron orbitals, although their sum of course is. We preferred, therefore, to divide the electron-electron 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 one-electron Hamiltonian H. With our division of the electron-electron interactions, this can be so defined that the electronic wave functions enter into it only through a spin-independent first-order density matrix q(r′|r). This Hamiltonian H is equivalent to the Hartree-Fock operator in its effect on orbitals occupied in the ground state of the crystal. If we assume that the original many-electron ground state is non-degenerate, 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 Hartree-Fock approximation to restrict our consideration to those Slater determinants for the ground state which lead to first-order 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 one-electron Hamiltonian H is not equivalent to the Hartree-Fock 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 over-completeness of a basis set once orthogonalisation to the core functions has taken place. Unlike the pseudo-potential 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. This procedure is not inherently unreasonable, but requires justification by comparing the resulting wave functions with those obtained by a more accurate method such as the OPW one, just as the use of Slater orbitals for atoms can be justified by comparing them with Hartree-Fock orbitals. A formalism for carrying out such a comparison is given, and is applied to silicon; incidentally, it might be possible to develop this method into an interpolation scheme for wave functions, which could be considerably more accurate than the more conventional method of interpolating by use of energy parameters in order to find the shape of the energy bands away from points of symmetry in reciprocal space. We also show that the LCAO method, as usually employed, cannot be expected to give good values for the effective masses of the electrons, so that it is not justifiable to attack the method as a whole because of this failure.
Chapter 2 of the thesis is devoted to a derivation of the equations which equivalent orbitals must satisfy. We first show how these equations can be derived from those for molecular orbitals, and how a self-consistent one-electron Hamiltonian can, in principle, be found much more easily for equivalent orbitals than for molecular orbitals. One problem with equivalent orbitals such as Wannier functions is that they are not very well localised, and since they do not in general correspond to any intuitive ideas of chemical bonds it is difficult to know what sort of basis functions should be used for calculating them. Fortunately for tetrahedral semiconductors the position is rather simpler, since a set of equivalent orbitals exists having the same symmetry as the classical chemical bonds. These equivalent orbitals are henceforth called bond orbitals; the L.C.A.O.-type bond orbitals used by Stocker and by Cohan can be regarded as approximations to them, even though these approximations (especially Stocker's) do not satisfy the required orthonormality conditions. At the end of section 1, we introduce the model substance springium in an attempt to show how the relationship between equivalent and molecular orbitals resembles that between localised and normal coordinates in the classical theory of mechanical vibrations. In section 2 of this chapter we present an alternative derivation of the equivalent orbital equations, based on that given by Koster but derived somewhat more rigorously for the special case of bond orbitals. This variational treatment of equivalent orbitals leads to a much clearer picture of what they should look like, and the molecular orbitals and their associated energies can easily be derived. Finally, we set up in section 3 an exact formalism for finding the bond orbitals and deriving the one-electron energies for the valence bands of tetrahedral semiconductors.
So far our treatment has been completely rigorous, and it is satisfying to know that such a sound basis exists for the calculation of bond orbitals. In chapter three, we describe the various approximations which we found it necessary or advisable to make so as to be able to carry out actual calculations on the Mercury computer. We first treat the problem of adapting the variational procedure so as to be able to obtain self-consistent solutions of the equivalent orbital equation: the problem of obtaining convergence was far more tricky than than been foreseen initially. We also introduce the localisation potential D, which is a pseudopotential representing the effect of requiring the bond orbitals to be orthogonal to each other. This localisation potential is the extra term in the equation for equivalent orbitals as compared with that for molecular orbitals, and while its expectation value for an exact bond orbital must be zero it has a profound effect on the shape of the self-consistent solutions of the equivalent orbital equation.
The next problem is the choice of basis functions in terms of which to expand the bond orbitals. We decided not to use Slater orbitals, such as Stocker and Cohan used, because of the difficulty of evaluating exactly many of the integrals that would be needed. The idea of using Gaussian orbitals centred on the atoms at each end of the bond was also rejected, because it seemed likely to require too large a set of basis functions. Instead, we chose to use a set of Gaussian functions centred about the mid-point of the bond, and to orthogonalise them to the core orbitals by the matrix method derived in chapter one. We used a total of twelve basis functions, six of them symmetric about the mid-point of the bond and the remainder anti-symmetric about it. Because our basis set was so small, and could not conveniently be enlarged because of the limited storage facilities on the Mercury computer, we restricted ourselves to axially symmetric functions instead of only requiring them to have the C3v symmetry required by group theory. If we denote distance along the bond by ζ, our functions were e-Cr2 × 1, r2, ζ2, r4, r2ζ2, ζ4; ζ, r2ζ, ζ3, r4ζ, r2ζ3, and ζ5. The choice of the parameter C proved very difficult, as is not unusual in calculations involving Gaussian functions. It has the dimensions of (length)-2, and so we decided to make it inversely proportional to the square of the bond length. It was not obvious a priori how the constant of proportionality should vary with the atomic numbers and electronegativities of the atoms composing the crystal, and so we took it to be independent of these quantities; we chose it by examining the results of calculations on silicon, germanium, and tin using various different values of C. Our results suggest that this was not a terribly good choice for C, but this knowledge only became available after the calculations had been carried out.
In treating the crystal potential, our method of dividing up the electron-electron interaction was extremely valuable. Since our Coulomb potential term has the full crystal symmetry, we were able to represent it by its Fourier transform; a similar procedure was adopted with the exchange potential for which we used the Slater approximation. It was thus extremely simple to find the matrix elements of the one-electron Hamiltonian between our basis functions. The Coulomb potential could easily be made self consistent since it is additive for the core and valence electrons, but since the Slater exchange potential is proportional to the cube root of the charge density we had to use an approximation of dubious validity in order to make it nominally self-consistent. In fact an exact exchange potential can be used fairly easily with Gaussian functions – this is one of the chief advantages of such functions for the calculation of wave functions – but we felt that to do so would take more time on the computer than would be justified for our approximate calculations. Considerations of computing time also led us to make several other approximations, including a rather drastic one in connection with the localisation potential D. None of these approximations is essential to the method, and our computer programmes can very easily be adapted to avoid all of them except that to the exchange potential, for which a new programme would have to be written. For our core orbitals, we used the Hartree-Fock wave functions and energies for the free atoms which Dr Mayers has calculated,, and introduced only a slight shift in the energies to allow for the crystal environment.
In chapter four, we present the results of our calculations; while we did not expect to obtain definitive results, the energy bands that emerged from our work were in fact a lot poorer than we had hoped. It was found that the shape of the energy bands depended critically on the choice of the parameter C and on the contribution of the valence electrons to the crystal potential. The effect was not so noticeable for the homonuclear compounds silicon, germanium, and gray tin, for which our results agree; qualitatively with those of other authors, but it was of crucial importance in the heteronuclear III-V and II-VI compounds. We have drawn the energy bands for a number of compounds in which the results did not seem too unreasonable, but they must be regarded more as possible band shapes for heteronuclear compounds than as definitive energy bands; incidentally, we found that the anti-symmetric part of the bond orbital had a far greater effect on the energies than would have been predicted from k.p perturbation theory. Our bond orbitals may well be quite close to the correct shapes, but it would be dangerous to draw any specific conclusions from them without first performing some calculations on a much larger scale, perhaps using one of the larger and faster computers now coming into service.
However, there are one or two general conclusions that can be drawn from our results. Our calculations on Si, Ge, and Sn support the view of Mooser and Pearson that the bonds tend to become less localised as the principal quantum number n of the valence electrons of the constituent atoms increases, but their suggestion about the effect of the ionicity of the bond on its degree of localisation cannot be properly judged from our results for the heteronuclear compounds. It seems that it is certainly justifiable to talk of covalent chemical bonds in III-V compounds, and that such bonds can be represented by accurate analytical wave functions and are not just loose concepts. It is not really practicable to talk about bond ionicities or effective charges in our method, and the only meaningful concept of this nature in our scheme would seem to be the charge distribution in the bond orbital. This quantity, unlike the others, can be defined without the use of the LCAO approximation and without attempting to allocate among the atoms electrons which essentially belong to bonds and are shared by pairs of atoms. Our results for the II-VI compounds are less reliable, and more unexpected, than those for the III-V crystals. It may be that our choice of C and approximate exchange potential introduced a greater error for these compounds. It is also possible that the Gaussian functions should not be centred about the mid-point of the bond in view of the difference in the number of valence electrons which the neutral atoms would possess: we note that our whole theory applies with only a few minor alterations if the Gaussian functions are centred about some other point on the bond. However, it is quite possible that the bond orbital picture is not very suitable for II-VI compounds; there may be a tendency for the group II atom to surrender some of its electrons to the group VI atom, thereby producing ionic bonding, rather than for charge to be transferred in the opposite direction to produce covalent bonding. One could perhaps distinguish between the two types of bonding by examining the bond charge density: for covalent bonding one would expect it to show no minima and just one maximum, while for ionic binding one might find maxima in the valence charge density at the two ends of the bond and a minimum in between them. On this basis, our results suggest that in all the II-VI compounds we considered, with the exception of CdTe, there is ionic binding. It should be remembered that for certain ratios of the atomic radii the zincblende structure is the most favourable energetically for a classical ionic compound, so that this structure does not necessarily imply the existence of covalent bonds.
The last chapter of our thesis is concerned with an entirely separate problem, namely the calculation of the overlap between the modulating part u, (r) of the wave function ψk(r) = eik.ruk(r) for electrons in the valence and conduction bands of planar crystals. These overlap functions are of some interest in connection with the Auger effect in semiconductors, as well as in other contexts; they had previously been calculated for a one-dimensional Kronig-Penney model, but not for any real crystal using wave functions obtained from a variational calculation of the energy bands. The crystals we considered were graphite and planar boron nitride, for which wave functions had been found by Coulson and Taylor. The calculation of these integrals involved the use of bipolar coordinates and the summation of a double series, and this work was done on a Mercury computer. The results confirm that for bands which arc not degenerate it is a reasonable approximation to set the overlap integral for electrons in the same band equal to one provided that the k-vectors of the electrons do not differ by too much; however, where (as in graphite) the band is degenerate, this is not a valid approximation if k is anywhere near the point of degeneracy. An account of this work has been published in the Journal of the Physics and Chemistry of Solids.