k sampling

For a periodic system, the k points appearing in the wave function belong to the first Brillouin zone, by virtue of the Bloch's theorem. In the case of samples with defects, which are by definition aperiodic, the cell that contains the defecs is periodically repeated, using periodic boundaries. The Bloch theorem can be therefore applied to this supercell, the dimension of the Brillouin zone being determined by the dimension of the supercell itself, i. e. the larger the supercell, the smaller the Brillouin zone should be.

To calculate the energy band, or the charge density for example, the sum over these k points has to be done. Therefore, choosing a sufficiently dense mesh of summation is crucial for the convergence of the results. Monkhorst and Pack [104] proposed a scheme where the k are distributed homogeneously in the Brillouin zone according to

$${\bf k} = x_1 {\bf b}_1 + x_2 {\bf b}_2 + x_3 {\bf b}_3$$ (7.28)

where ${\bf b}_1, {\bf b}_2, {\bf b}_3$ are the reciprocal lattice vectors, and

$$x_i = \frac{l}{n_i}, \ \ \ \ \ \ \ \ l = 1,...,n_i$$ (7.29)

where n_i are the folding parameters.

Usually, total energies of different stuctures are compared. Therefore, if the two structures have the same unit cell, the same set of k-points should be used. Since only the difference in the energies of the two structures is required, possible errors from a non-converged k-point sampling may cancel out. The computational effort could therefore be reduced by using a carefully chosen and small k-point set.

An alternative method for choosing k-point mesh has been proposed by Chadi and Cohen [105], on the basis of ``shells'' analysis. This concept can be explained by considering the Bloch funtion for a specific band (see equation 7.7)

$$\Psi_{\bf k}({\bf r}) = \frac{1}{\sqrt{N}} \sum_{{\bf R}_i}{\rm exp}(i{\bf k \cdot R}_i) \psi({\bf r}-{\bf R}_i).$$ (7.30)

The charge density for this specific state is therefore

$$\rho_{\bf k}({\bf r}) = \frac{1}{N} \sum_{{\bf R}_i,{\bf R}_j... ... R}_j)] \psi^\ast({\bf r}-{\bf R}_j) \psi({\bf r}-{\bf R}_i),$$ (7.31)

which can be rewritten as

$$\rho_{\bf k}({\bf r}) = \frac{1}{N} \sum_{{\bf R}_i} \vert\p... ...si^\ast({\bf r}+{\bf R}_l-{\bf R}_i) \psi({\bf r}-{\bf R}_i),$$ (7.32)

where the prime in the second sum is over all l, except the ${\bf R}_l = 0$ term. Equation (7.32) should be compared to the total charge density, given by

$$\rho({\bf r}) = \sum_{\bf k} \rho_{\bf k}({\bf r}) = \sum_{{\bf R}_i} \vert\psi({\bf r}-{\bf R}_i)\vert^2.$$ (7.33)

One can see that the first term in equation (7.32) is the average charge density for the band considered. It could therefore be a reliable approximation to the total charge density, providing a good choice of k-points are chosen that minimize the second term in equation (7.32).

To find these special k-points, we first consider the 48 possible symmetry operations T on one k-point. Each new k-point generated leads to a new charge density $\rho_{T{\bf k}}({\bf r})$. Therefore, we obtain from equation (7.32)

 

$$\sum_{T} \rho_{T{\bf k}}({\bf r}) = \frac{1}{N} \sum_{T} \sum_{{\bf R}_i} \vert\psi({\bf r}-{\bf R}_i)\vert^2$$

$$+ \frac{1}{N} \sum_{{\bf R}_l} \ '\sum_{{\bf R}_i} \sum_{T} {\rm ... ... {\bf\cdot R}_l] \psi^\ast({\bf r}+{\bf R}_l-{\bf R}_i) \psi({\bf r}-{\bf R}_i)$$

$$= \frac{48}{N} \sum_{{\bf R}_i} \vert\psi({\bf r}-{\bf R}_i)\vert^2$$

$$+ \frac{1}{N} \sum_{{\bf R}_l} {\rm exp}[i(T{\bf k}) {\bf\cdot R}... ...R}_i} \sum_{T} \psi^\ast({\bf r}+T{\bf R}_l-{\bf R}_i) \psi({\bf r}-{\bf R}_i).$$ (7.34)

The T operations on ${\bf R}_l$ do not change the norm of the vectors. Therefore, the first term in equation (7.34) would be a good approximation to the charge density if the k vectors satisfy the set of equations:

$$\sum_{\vert{\bf R}_l\vert = c_m} {\rm exp}(i {\bf k \cdot R}_l) = 0, \ \ \ \ \ m = 1,2,3,...,$$ (7.35)

where c_m is the nearest neighbor distance corresponding to index m. That is, for a good k-point set, the contribution from the leading shells should vanish.