Energy States of Overlapping Impurity Carriers in Semiconductors

An analytical solution of the Schr\"odinger equation for the spherically symmetrical wave function of an impurity carrier in a zero wave number state is given on the basis of the Wigner-Seitz cellular approximation. An approximate method is described for calculating the small deviations from the hydrogenic values. The variation of the lower band-edge energies with impurity concentration shows that the excited impurity bands may overlap each other strongly, but a continuous decrease of the activation energy from its hydrogenic value to zero cannot be accounted for by interaction alone.