Wave-Function Expansion in the Brillouin Zone: Silicon

Wave functions are required throughout the Brillouin zone for the treatment of many problems in solid-state physics. It therefore is of interest and considerable importance to develop analytical representations of the k dependence of wave functions throughout the Brillouin zone. In this paper the momentum wave functions $b_n(\mathbf{k},{\mathbf{K}}_s)$ in the plane-wave expansion of the Bloch function ${\psi }_n(\mathbf{k},\mathbf{r})$ are considered. An analytical representation for the $b_n(\mathbf{k},{\mathbf{K}}_s)$ based on a set of symmetrized polynomials is proposed. The specific polynomials, although they pertain to general points in the Brillouin zone, are somewhat analogous to the Kubic harmonics employed in the cellular method. A primary distinction, however, is that while formerly one was concerned with space-group operations in the group of the k vector, here we are concerned with operations in the group of the ${\mathbf{K}}_s$ vector for which ${\mathbf{K}}_s$ in $b_n(\mathbf{k},{\mathbf{K}}_s)$ is left invariant.