Impurity Band Conduction in Germanium and Silicon

In this paper, we attempt to establish the idea of impurity band conduction on a firmer theoretical basis for germanium and silicon. Calculations by Baltensperger of energy bands arising from the impurities are adapted for these materials by replacing wave functions of impurity states with suitable hydrogenic approximations. These are used to estimate the concentration for which the impurity band substantially merges with the remainder of the conduction band, and this estimate is found to agree reasonably well with experiment. An estimate is also made of the range of concentrations for which the usual band type of theory might be applicable to electrons in the impurity band. It is found that approximately within this range the simple band theory does well for the impurity band. This, incidentally, highlights the necessity of taking into account conduction in impurity states for degenerate and near-degenerate samples even at relatively high temperatures. For concentrations lower than these a crude treatment of conduction is described which can account for the sharp increase in impurity band resistivity with decreasing impurity concentration, the importance of compensation in these samples, and even the order of magnitude of the resistivity. It does not, however, predict the correct temperature dependence for the resistivity, and a possible origin for this is suggested. The question of the sign of carriers in the impurity band is discussed, and it is shown that the sign reversal of thermoelectric power observed for the impurity band need not imply a change in sign of the effective mass.