Effective-mass theory in noncrystalline solids

A generalized effective-mass theory is derived that is applicable to noncrystalline as well as crystalline systems. The formalism is particularly useful for describing the electronic properties of shallow impurity states in the band gap, or pseudogap, of an amorphous semiconductor. As an example, the theory is used to study shallow acceptor states in a continuous-random-network model of $a$-Si. In the generalized theory applied to noncrystalline solids, the effective-mass parameters are functions of position. When the correlation length of the fluctuations in the values of the parameters is short compared to the size of the impurity state, the appropriate case for shallow acceptor levels in $a$-Si, the acceptor ground-state binding energy and orbital size are found to be relatively insensitive to the disorder. The theoretical predictions are compared favorably to available experimental data on doped $a$-Si.