Band-gap shrinkage in impure $n$-Si and Ge as a function of impurity concentration is studied theoretically at $T=0\mathrm{}$ K. Above the Mott critical density it is assumed that electrons occupy the host conduction band in the form of an electron gas. Interactions among the free carriers and with the ionized point impurities give rise to a downward shift of the conduction band. Interband matrix elements are explicitly included for the valence bands. As a consequence, the two valence bands at the center of the zone are shifted upwards by the same amount. Correlation and impurity scattering both give corrections to the band gap, which are of the same order of magnitude. In particular, one finds that the band gap depends rather sensitively on the arrangement of donor ions. The choice of dielectric screening is investigated, and comparison with previous calculations is made. A comparison between present theoretical results and experimental estimates of the band-gap narrowing is also indicated. One notes an order-of-magnitude agreement, although there is considerable scatter in the experimental data. Shortcomings of the present theory are briefly discussed.

• Received 8 January 1981

DOI:https://doi.org/10.1103/PhysRevB.24.1971