A Soluble Problem in Energy Bands

Abstract

The problem of an electron moving in a periodic simple cubic potential of the form $cosx+cosy+cosz$ is investigated, with particular attention to the nature of the wave functions, Wannier functions, degeneracy of overlapping bands, etc. It is well known that this problem is separable, leading to Mathieu's equation for the functions of $x$, $y$, and $z$. This equation can be solved, making use of the Fourier expansion of the wave function (the momentum eigenfunction). Solutions are set up, for a number of interatomic distances. The momentum eigenfunctions are similar to Hermite functions for the tightly-bound levels but very different for free-electron-like levels. Those for the levels near the top of the potential barrier, corresponding to the valence and conduction bands of a real crystal, are very complicated; we illustrate their properties by means of a test of the completeness relation for orthogonal functions. It is proved that the Wannier function is the Fourier transform of the momentum eigenfunction, and special cases are discussed. The effect of various perturbations on the one-dimensional problem is considered. It is shown that the usual assumption made in band theory, connecting the width of the gap introduced by a perturbation with the matrix component of the perturbative potential, is not generally justified. The case of a perturbation of double periodicity is taken up, and it is shown that this results in a spreading out of the Wannier function, which sometimes has large contributions on many atoms. In the two- and three-dimensional cases, the interesting feature is the degeneracy between overlapping bands, for example, between the different components of a $p$ or a $d$ band. These are discussed in detail. For the unperturbed Mathieu case, the energy surfaces cut without interaction, but almost any reasonable perturbation will cause them to interact, resulting in perturbed bands of great complication, which will not cut each other, though they may adhere at certain points of crystal symmetry, as can also be proved by group theory. The Wannier functions in these cases become combinations of the Wannier functions of the unperturbed problem, located on many different atoms, and possessing the rotational symmetry of the crystal, which the unperturbed Wannier functions do not have. The type of perturbation is also taken up which would convert the simple cubic potential of the unperturbed problem into a face-centered or body-centered structure.