The first Brillouin zone can be mapped out by a continuous set of points, , throughout that region of reciprocal space (k-space). The occupied states at each k-point contribute to the electronic potential of the bulk solid. Since the set is dense, there are an infinite number of k-points in the Brillouin zone at which the wavefunctions must be calculated. Therefore if a continuum of plane wave basis sets were required, the basis set for any calculation would still be infinite, no matter how small the plane wave energy cut-off was chosen.
For this reason electronic states are only calculated at a set of k-points determined by the shape of the Brillouin zone compared to that of its irreducible part. The reason that this can be done is that the electronic wavefunctions at k-points that are very close together will almost be identical. It is therefore possible to represent the electronic wavefunctions over a region of reciprocal space at a single k-point. This approximation allows the electronic potential to be calculated at a finite number of k-points and hence determine the total energy of the solid.
The error incurred by this approximation can be made arbitrarily small by choosing a sufficiently dense set of k-points. This is easily made clear by the use of an example. Each k-point has energy eigenvalues associated with it which is usually presented in the form of a band structure - the band structure of GaAs in the zincblende structure along several lines through the Brillouin zone is given in Figure 3.2.
Figure 3.2: The band structure of GaAs (zincblende) along several lines of high symmetry.
At each k-point the energy eigenvalues are averaged, which gives a measure of the energy associated with the electrons which occupy that region of k-space. These average energies are given in Figure 3.3.
Figure 3.3: Averaged energy eigenvalues of GaAs corresponding to the above band structure.
It can be seen that there is a large variation in energy over the Brillouin zone. In this case it is found that the average band energy is -4.8eV with a standard deviation of 1.7eV. This shows that there is a large dispersion in the band energies of GaAs in the zincblende structure. Therefore, in order to converge the total energies of the structure to within 0.01eV it seems that k-points randomly spread throughout the first Brillouin zone is necessary. Fortunately, there are more efficient methods of choosing the k-point set in which to sample the Brillouin zone.
Methods have been devised for obtaining very accurate approximations to the electronic potential from a filled electronic band by calculating the electronic wavefunctions at special sets of k-points. The two most common methods are those of Chadi and Cohen and Monkhorst and Pack which will be described below.