# Broadening of impurity bands in heavily doped semiconductors

## Abstract

The shapes of deep-lying bands of impurity states broadened by fluctuations in the local Coulomb potential are calculated for the case of randomly distributed ions in a semiconductor and are shown to determine the shapes of the spectral lines emitted by electrons or holes captured in these bands. The distribution function for the local potential is described in terms of its Laplace transform, and expressions for its first few moments, Ē, (E-Ē)2av, etc., are calculated for screened Coulomb potentials. An approximate form is given for this function and is shown to be resonably accurate for materials having a high degree of compensation or a large screening length. From the results of this analysis it is shown that the shapes of impurity emission lines can be used to aid in the identification of the nature of the states and transitions involved. In particular, the width of a line is determined principally by the product of the ion density and the screening length in the luminescent region, while the sense of its skewness depends on the signs of the domination and of the carrier which is captured. Numerical examples are given for GaAs diodes containing approximately 1018 ions/cm3. © 1965 The American Physical Society.