Energy Bands in One-Dimensional Aperiodic Crystals

Abstract

A definition for the average momentum of an electron in a one-dimensional aperiodic crystal of finite length is presented. This momentum, when maximized with respect to the constants of integration of Schrödinger's equation, is shown to yield a useful investigative quantity, given the special name characteristic momentum, which is independent of position and from which significant information about allowed states and their transport character can be extracted. The characteristic momentum for a crystal of finite length is analogous to the momentum associated with the slope of an energy-versus-wave-number ($E$- versus-$k$) plot for a periodic crystal of infinite length. An allowed-state criterion which predicts allowed bands and forbidden gaps for finite crystals is developed and verified by computer calculations. For crystals having a fixed percentage of randomly spaced impurity cells, the characteristic momentum and bandwidth decrease with increasing length. For crystals of fixed length, the characteristic momentum and bandwidth decrease with increasing impurity-cell concentration.