kpoint

A k-point, or k-point in reciprocal space, refers to a point k in the first Brillouin zone that labels Bloch states in a crystalline solid. In periodic systems, electron energies form bands E_n(k) as a function of the wavevector k; many properties are obtained by integrating over the Brillouin zone rather than summing over real-space states. Since the Brillouin zone is continuous, numerical calculations sample a finite set of k-points to approximate these integrals.

Sampling strategy: A k-point grid is a mesh of k-points covering the Brillouin zone. Common schemes include Monkhorst-Pack grids (uniform grids) and Gamma-centered grids. Symmetry operations reduce the sampling to the irreducible part of the zone, the irreducible Brillouin zone, which lowers computational cost without sacrificing accuracy.

Applications and practices: For band structure plots, calculations compute E_n(k) along a path through high-symmetry points in the Brillouin zone (for example Γ–X–M in cubic lattices). For total energies, densities of states, and response functions, dense, uniform k-point grids are used to achieve convergence. In metals, convergence with respect to k-points is particularly important near the Fermi level; in insulators, larger gaps permit fewer k-points.

Terminology and related concepts: The term k-point is often used interchangeably with k-vector. In some phonon calculations, a related quantity q is used to sample the phonon Brillouin zone.