Electronic States in a Perturbed Periodic Lattice

Solutions of the one-dimensional Schrödinger equation for a periodic potential modified by a perturbing potential are expressed as $\psi \mathrm{}\left(x\right)=\Sigma k^{}\phi \left(x_k\right)a(x-x_k)$. The function $a(x-x_k)$ is a Wannier function localized around the $k\mathrm{th}$ atom of the crystal and associated with a particular permitted band; the coefficients $\phi$ satisfy a differential equation of infinite order. W.B.K. solutions of the equation for $\phi$ are obtained. In the permitted band, $\phi$ is oscillatory; in the neighboring forbidden bands $\phi$ decreases exponentially with distance from a band edge. Various types of perturbing potentials are considered and quantum conditions are derived for impurity states and for states in which the permitted band is crossed.