Single-particle energy levels in doped semiconductors at densities below the metal-nonmetal transition

The single-particle density of states is calculated for a random distribution of shallow donors in semiconductors at densities below the Mott density. The donors are described within the effective-mass approximation. Calculations of the energy bands to add an electron ($D^{-}$ band) and to remove an electron ($D^{+}$ band) are presented in three parts: (i) At low densities, where the broadening of the $D^{+}$ and $D^{-}$ levels arises primarily from donor pairs which are closer together than the average, we have employed the donor-pair approximation to calculate the energy bands. This leads only to a small broadening in single-valley semiconductors but in a many-valley semiconductor a $D_2$ complex can have an electron affinity as large as 0.4 Ry. (ii) Next, the band edges are presented for a simple cubic lattice of donors, which would be relevant in estimating mean band positions, mobility edge, etc., in the disordered case in the intermediate doping region. The $D^{-}$ band is calculated using a potential derived by the method of polarized orbitals. The energy gap does not shrink appreciably until a factor of 4 below the Mott density. The energy gap is found to go to zero at a density very close to the Mott criterion ($n_D^{\frac{1}{3}}{a}_B=0.25\mathrm{}$) for single-valley semiconductors and at a lower density in many-valley semiconductors. (iii) Finally, to estimate the localized tail states in the many-valley case, the energies of small dense donor clusters are calculated using a local-density approximation. Because many electrons can be placed in the bonding orbitals of these clusters without violating the Pauli principle, they are found to have very large electron affinities. We find that clusters of four donors or more can attract an electron from an isolated donor. As a result there is no Mott-Hubbard gap due to correlation in many-valley semiconductors and their insulating property is due to Anderson localization. The very large fluctuations in the one-electron potential imply an Anderson transition to the metallic state.