Recursion method for the extended impurity problem

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Abstract
The application of the recursion method of Haydock et al. (1975) to the problem of potential scattering on a lattice by an extended impurity is discussed. The approach is an alternative to the Koster-Slater method and yields bound state energies and lattice phaseshifts corresponding to the irreducible representations of the impurity point group symmetry, as well as the total density of states. The method avoids huge determinants by taking advantage of a fixed point in the recurrence relations and by using the asymptotic behaviour of the recurrence coefficients. The extrapolation of continued fraction coefficients and the construction of 'seed states' for the recursion method are discussed. Illustrative applications are made to the case of an extended impurity in a triangular lattice and to a shallow Coulomb impurity in an FCC lattice, the latter of which is compared with effective mass theory.