Extrapolative approaches to Brillouin-zone integration

A highly efficient extrapolative Brillouin-zone integration scheme is presented that requires a very low k-point sampling density for spectral integrations. It is important to use an extrapolative approach, since at low sampling densities interpolative schemes are hindered by problems associated with band crossing, which introduce spurious singularities in the density of states (DOS). The information for the extrapolation is obtained using second-order $\mathbf{k}\mathbf{\cdot }\mathbf{p}$ perturbation theory within a set of subcells of the Brillouin zone, which can be chosen to make full use of symmetry. The resulting piecewise quadratic representation of the band structure is converted directly to a DOS using an analytic approach. It is also shown that this method can be successfully applied even in the linear extrapolative case.