Por José Luís Martins (Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Portugal).

The Brillouin zone and vectors in reciprocal space (k-vectors) are key concepts in the calculation of the properties of crystals. The calculation of electron states from first principles of quantum mechanics at a given point in the Brillouin zone is computationaly expensive, and therefore it is highly desirable to have efficient interpolation methods for the Brillouin zone. Two interpolation methods developed in the past few years, the modified tight binding (MTB) and the generalized Luttinger-Kohn (GLK) methods will be presented. They are related to two simple solid state models, the tight-binding and the k.p approximations. The calculation of some crystal properties, for example optical response, require an integration over the whole three-dimensional Brillouin zone, but others can be calculated on lower dimensionality regions, for example transport properties have only relevant contributions from the two-dimensional Fermi surface or from band edges, while phonon life-times have relevant contributions from one-dimensional lines. Recent developments using concepts from graphics computing to integrate efficiently on those low-dimensional domains will be presented. Topological properties are often associated with zero-dimensional small regions of the Brillouin zone and will be briefly discussed.