Impurity Band in Semiconductors with Small Effective Mass

The energy levels of ordered impurities in semiconductors are formally equivalent to the energy levels of metallic hydrogen if a number of simplifying approximations are made. An approximate calculation of the energy states for the $1s$ band of metallic hydrogen is carried out for smaller lattice constants than those considered by Wigner and Huntington, or by Baltensperger. A simple transformation of the distance and energy scales converts the calculation for metallic hydrogen to one applying to impurities in a semiconductor, if values for the effective mass and the dielectric constant are given. Experimental results for the optical energy gap in InAs are reported as a function of impurity concentration. The effective mass required to fit the optical data for InSb published by other workers is about $0.03m$, as compared to the value $0.013m$ found by cyclotron resonance measurements.