Tetrahedron methods for Brillouin zone integration Original Research Article

Calculation of the Greenʹs function in perfect crystals and crystals with impurities requires the integration inside the Brillouin zone of a complex matrix. The irreducible part of the zone is divided in non-overlapping tetrahedra and the integration is done numerically inside each tetrahedron. The integral over the Brillouin zone is obtained by the rotation of the one in the irreducible part with the appropriate rotational matrices. In the present work the most common methods of integration inside a tetrahedron are reviewed and compared for their convergence. New formulas of the weighting factors for limiting cases are presented that agree with the results of analytically calculated integrals.