Theoretical Techniques for Impurities in Semiconductors

Theoretical techniques for studying impurities in semiconductors can be categorized according to two criteria one, the geometrical arrangement, and two, the method for treating the electron-electron interactions. 

The physics problem that needs to be addressed is that of an isolated impurity in an infinite crystal. This problem is clearly too complex to treat exactly specific geometrical arrangements have to be chosen that closely represent the physical situation while being computationally tractable. 

In a supercell geometry, which seems to have become the method of choice these days, the impurity is surrounded by a finite number of semiconductor atoms, and what whole structure is periodically repeated (e.g., Pickett et al., 1979 Van de Walle et al., 1989). This allows the use of various techniques that require translational periodicity of the system. Provided the impurities are sufficiently well separated, properties of a single isolated impurity can be derived. Supercells containing 16 or 32 atoms have typically been found to be sufficient for such purposes (Van de Walle et al., 1989). The band structure of the host crystal is well described. 

Another approach that provides a good desciption of the band structure of the host crystal is based on the Green s function determined for the perfect crystal. This function is then used to calculate changes induced by the presence of the defect (e.g., Rodriguez et al., 1979 Katayama-Yoshida and Shindo, 1983). The Green s function approach seems to be more cumbersome and less physically transparent that the supercell technique. 

Any Hamiltonian for an impurity in a semiconductor must include terms that describe the interactions between the nuclei, the interactions of electrons with the nuclei, and the electron-electron interactions. The latter are the hardest part of the problem. Typically, a one-particle description is used. 

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