Electron Energy Bands of One-Dimensional Random Alloys

Abstract

A method for calculating the density of states for an infinite, one-dimensional random alloy is obtained by investigating the asymptotic behavior of the trace of the "transmission" matrix which relates the values taken on by the wave function and its derivative at either end of the crystal. This matrix can be calculated if the potentials of the constituent $A$ and $B$ atoms, $V_A$ and $V_B$, are given. The equations are first derived for a very general case, and then the results of a calculation for an alloy in which the $A$ and $B$ atoms have equal concentrations is shown for the case that $V_A$ and $V_B$ are $\delta$-function potentials. Certain generalizations of the method for treating other nonperiodic problems are discussed briefly.