Impurity states in a disordered insulator: The Lloyd model

We have calculated the density of states for an exactly soluble model of a charged impurity in a disordered insulator. We find that the states associated with the impurity have the same mean energy as those found in the corresponding crystal, but have a spread in energies determined by the degree of the disorder. These results are compared favorably to results previously obtained for a more realistic model representing charged impurities in $a$-Si. The results also shed some light on the general nature of the localized states in the gap of an amorphous semiconductor.