Application of the Quantum-Defect Method to Optical Transitions Involving Deep Effective-Mass-Like Impurities in Semiconductors

Approximate ground-state wave functions for effective-mass impurity centers of arbitrary binding energy are derived by the quantum-defect method and applied to calculate optical absorption and emission processes involving impurities in semiconductors. The dependences of band-impurity and impurity-ionization cross sections on impurity binding energy are calculated and shown to limit to those of the hydrogenic model and Lucovsky's $\delta$-function model for shallow and deep impurity centers, respectively. Formulas relating the cross sections to the absorption coefficients and radiative recombination rates are also presented.