Simple geometric generation of special points in brillouin‐zone integrations. Two‐dimensional bravais lattices

Abstract A clear tabulation of sets of special points in numerical integrations over Brillouin zones ( BZ ) is given for the five two‐dimensional Bravais lattices. The special points derived on the basis of plain geometrical criteria coincide with a special case of the large unit cell ( LUC ) method. This similarity is also found for other generation procedures of k‐grids reported in the literature; that is, the corresponding techniques have their common origin in the LUC approach. The underlying integration procedure of the described scheme is based on the discrete summation, which implies that the weighting factors at the k ‐points are only determined by the point symmetry in reciprocal space not by details of the adopted numerical integration formalism. The method can be applied to semiconducting and insulating solids—that is, to materials with finite band gaps. Grids in reciprocal space are generated on the basis of simple geometric criteria. The numerical effort of the present geometrical approach is negligible in comparison to the computational steps necessary in generation procedures of optimized k ‐grids that have been published in the literature. The application of the suggested integration procedure is extremely simple; the generation of the k ‐grids is only based on a few definitions and plain geometrical formulas. Modifications of the grid size are possible without difficulties. A collection of weighting factors ω i and special and general positions in the BZ is given; different grids are tabulated together with their geometrical efficiencies. The generated integration domains are compared with the results of elaborate schemes of finding grids with maximal efficiency.