Recursion method in the k-space representation

We show that by using a unitary transformation to k space and the special-k-point method for evaluating Brillouin-zone sums, the recursion method can be very effectively applied to translationally invariant systems. We use this approach to perform recursion calculations for realistic tight-binding Hamiltonians which describe diamond- and zinc-blende-structure semiconductors. Projected densities of states for these Hamiltonians have band gaps and internal van Hove singularities. We calculate coefficients for 63 recursion levels exactly and for about 200 recursion levels to a good approximation. Comparisons are made for materials with different magnitude band gaps (diamond, Si, α-Sn). Comparison is also made between materials with one (e.g., diamond) and two (e.g., GaAs) band gaps. The asymptotic behavior of the recursion coefficients is studied by Fourier analysis. Band gaps in the projected density of states dominate the asymptotic behavior. Perturbation analysis describes the asymptotic behavior rather well. Projected densities of states are calculated using a very simple termination scheme. These densities of states compare favorably with the results of Gilat-Raubenheimer integration.