Energy Bands and Projection Operators in a Crystal: Analytic and Asymptotic Properties

作者：J. des Cloizeaux · 发表于：Physical Review · 年份：1964 · DOI：10.1103/physrev.135.a685 · 被引用次数：275 · 研究领域：Photorefractive and Nonlinear Optics、Optical and Acousto-Optic Technologies、Photonic Crystals and Applications

In an $n$-dimensional crystal, an energy band is usually made of several branches which are connected with each other. Accordingly, the Bloch states of wave vector K which are eigenfunctions of a one-electron Hamiltonian $H=\ensuremath{-}\ensuremath{\Delta}+V$ and which belong to a given band $\mathcal{B}$, define a subspace $\mathcal{S}(\mathrm{K})$ of finite dimensionality. For a large class of potentials, two properties concerning the subspaces $\mathcal{S}(\mathrm{K})$ which are associated with a fixed band $\mathcal{B}$ have been proved for $n$-dimensional crystals. (1) The projection operator $P(\mathrm{K})$ on $\mathcal{S}(\mathrm{K})$ can be defined for complex values of K, and its matrix elements $〈\mathrm{r}|P(\mathrm{K})|{\mathrm{r}}^{\ensuremath{'}}〉$ are analytic in a strip of the complex K space; this strip is centered on the real K space and is independent of r and r'. (2) The projection operator $P=\ensuremath{\int}{d}^{h}\mathrm{K}P(\mathrm{K})$ (integration on the Brillouin zone) has matrix elements $〈\mathrm{r}|P|{\mathrm{r}}^{\ensuremath{'}}〉$ which decrease exponentially when the length|r-${\mathbf{r}}^{\ensuremath{'}}$| goes to infinity.