A General Algorithm for Calculating Irreducible Brillouin Zones

Calculations of properties of materials require performing numerical inte-grals over the Brillouin zone (BZ). Integration points in density functional theory codes are uniformly spread over the BZ (despite integration error being concentrated in small regions of the BZ) and preserve symmetry to improve computational efficiency. Inte-gration points over an irreducible Brillouin zone (IBZ), a rotationally distinct region of the BZ, do not have to preserve crystal symmetry for greater efficiency. This freedom allows the use of adaptive meshes with higher concentrations of points at locations of large error, resulting in improved algorithmic efficiency. We have created an algo-rithm for constructing an IBZ of any crystal structure in 2D and 3D. The algorithm uses convex hull and half-space representations for the BZ and IBZ to make many aspects of construction and symmetry reduction of the BZ trivial. The algorithm is simple, general, and available as open-source software.

Automated summary

Calculations of material properties require numerical integration over the Brillouin zone (BZ), with integration points uniformly spread and symmetry preserved for efficiency. Integration points over an irreducible Brillouin zone (IBZ) do not need to preserve crystal symmetry, allowing for adaptive meshes with higher point concentrations at error-prone locations. An algorithm for constructing IBZs in any crystal structure has been developed, using convex hull and half-space representations to simplify construction and symmetry reduction.