Electronic States in Perturbed Periodic Systems

The occurrence in perturbed periodic systems, such as impure crystals, of localized states with discrete energies is discussed from both qualitative and quantitative points of view. Semiclassical considerations, modified in the usual way by wave concepts, make it clear that impurities will give rise to impurity states above or below corresponding permitted bands of energy, according as the ionic charge of the impurity is less than or greater than that of the ion it replaces. Localized states at crystal interfaces and free surfaces can be discussed in the same way. Consideration of the behavior of wave packets leads to formulation of the effective-mass wave equation of Peckar. Complete solutions of the perturbed-periodic wave equation are then constructed by joining together solutions valid for a single period of the unperturbed potential. When the perturbation is slowly varying (though not necessarily small in its total effect) this approach leads to an analytic solution of the problem involving errors of the order of the ratio of the change in the perturbation potential across a single cell to the total kinetic energy of the particle. The effective-mass equation appears in connection with an approximate form of this solution, but the relation of its solution $\varphi \mathrm{}\left(x\right)\mathrm{}$ to the correct wave function $\psi \mathrm{}\left(x\right)\mathrm{}$ is more complex than has previously been realized. To construct $\psi \mathrm{}\left(x\right)\mathrm{}$ in any small region one should resolve $\varphi \mathrm{}\left(x\right)\mathrm{}$ locally into the sum of two exponentials $C\exp \{\pm (\frac{i}{\hslash }\left)p_{d}x\right\}$, multiply each by the appropriate periodic function, and add the results. A quadradically integrable $\varphi \mathrm{}\left(x\right)\mathrm{}$ corresponds to a quadratically integrable $\psi \mathrm{}\left(x\right)\mathrm{}$ with the same energy; thus stationary-state energies determined by solving the effective-mass wave equation are found to be surprisingly reliable.