Empirical laws and their theoretical justification for the crystal-field splitting and ionization energy of transition-metal ions in semiconductors

It is pointed out that the ionization energy and crystal-field splitting of substitutional transition-metal impurities (TM) in semiconductors are simply related to the interatomic distance and ionicity of the perfect crystal. A theoretical justification of these empirical laws is provided through the use of both a simple molecular model and a full tight-binding Green's-functions calculation. The results confirm the validity of Harrison's $d^{-\frac{7}{2}}$ semiempirical law for the interaction between the TM atom and its neighbors. Linear plots can then be drawn for each TM impurity giving a simple and useful method for predicting these spectroscopic quantities in unknown cases.