Procedures for Comparing the Theoretical and Experimental Positions of Energy Levels of Multi-Electron Impurities in Semiconductors

Abstract

Calculations of the energy bands of semiconductors are usually based on three main approximations. The first is the independent particle approximation. The second is the reduction, by the use of local exchange, of the Hartree-Fock coupled equations to a single effective Schrödinger equation giving single-particle eigenenergies. The third is inherent in Koopmans’ theorem. Energy level diagrams of the type used in semiconductor physics also contain these approximations implicitly.

It is necessary to go a stage further when dealing with multi-electron impurities such as transition metal impurities. A method was proposed earlier which allowed one to use a few empirical parameters to give a tractable treatment of the energy levels of 3dⁿ impurities. The method when applied to GaAs, for example, gives a good account of the experimental data and allows predictions about energy levels to be made.

Recently, there have been several attempts to calculate energy levels of impurities using modifications of the methods used for energy bands, and containing the approximations listed above. The single-particle energies obtained cannot be directly compared with experiment. It is possible to resolve this difficulty either by using the calculated wave-functions as basis functions in diagonalizing a correlated Hamiltonian or, equivalently, by comparing the single-particle energies with those occurring in the semi-empirical method mentioned above.