Category:Crystal momentum

A crystal is characterized by the fact that it obeys translational symmetry. In many calculations, we only consider the primitive crystal unit cell to save computational time. However in a realistic bulk material, interactions go beyond the primitive unit cell. The concept of crystal momentum is crucial in order to take into account interactions that go beyond the primitive unit cell during the simulation and it is a consequence of translational invariance.

Formally, translational invariance can be written as a translation operator that commutes with the Hamiltonian:

$${\displaystyle [T_{\mathbf R},H]=0 }$$

For the KS orbitals this implies that each translation can only add a phase factor:

$${\displaystyle T_\mathbf{R} \psi_{n\mathbf{k}}(\mathbf{r}) = \text{e}^{\text{i}\mathbf{k}\cdot\mathbf{R}}\psi_{n\mathbf{k}}(\mathbf{r}),}$$

where ${\displaystyle n}$ is the index that goes over all KS orbitals. Performing two consecutive translations should yield the sum of the individual translations:

$${\displaystyle T_{\mathbf{R}_1}T_{\mathbf{R}_2} \psi_{n\mathbf k}(\mathbf{r})=T_{\mathbf{R}_1+\mathbf{R}_2}\psi_{n\mathbf k}(\mathbf{r}).}$$

For a system with translational invariance, we obtain a periodic potential and, hence, it is most convenient to use periodic boundary conditions. The figure illustrates a periodic potential and the blue box is a possible unit cell.

This means, by virtue of the Bloch theorem, we can separate each KS orbital into

$${\displaystyle \psi_{n\mathbf{k}}(\mathbf{r})=u_{n\mathbf{k}}(\mathbf{r})\text{e}^{\text{i}\mathbf{k}\cdot\mathbf{r}} }$$

a cell periodic part ${\displaystyle u}$ and a phase factor ${\displaystyle \phi = \mathbf{k}\cdot\mathbf{r}}$. The phase ${\displaystyle \phi}$ is called Bloch factor and ${\displaystyle \mathbf{k}}$ is the crystal momentum which lives in reciprocal space. Mind that ${\displaystyle u_{n\mathbf{k}}}$ depends on ${\displaystyle \mathbf{k}}$ and recall that n is the index that goes over all KS orbitals.

The reciprocal space is spanned by reciprocal lattice vectors ${\displaystyle \mathbf{b}_i}$:

$${\displaystyle \mathbf{b}_1 = \frac{2\pi}{\Omega} \mathbf{a}_2 \times \mathbf{a}_3}$$

$${\displaystyle \mathbf{b}_2 = \frac{2\pi}{\Omega} \mathbf{a}_3 \times \mathbf{a}_1}$$

$${\displaystyle \mathbf{b}_3 = \frac{2\pi}{\Omega} \mathbf{a}_1 \times \mathbf{a}_2}$$

These are defined in terms of the real space lattice vectors ${\displaystyle a_i}$ (POSCAR). ${\displaystyle \Omega = \mathbf{a}_1 \cdot \mathbf{a}_2 \times \mathbf{a}_3}$ is the volume of the unit cell. Note that a short real space direction actually yields a long direction in reciprocal space:

$${\displaystyle \mathbf{a}_i\cdot \mathbf{b}_j = 2\pi \delta_{ij}}$$

Based on the reciprocal lattice vectors ${\displaystyle \mathbf{b}_i}$, we can identify a primitive cell in reciprocal space; This is the so-called first Brillouin zone (1. BZ). For an SCF calculation, in practice the ${\displaystyle \mathbf{k}}$ points are defined on a regular mesh with evenly spaced ${\displaystyle \mathbf{k}}$ points in the 1. BZ. These are specified in the KPOINTS file or by using KSPACING.

To get a deeper understanding, consider an integration over all of real space (as is frequently required to compute properties): The connection between real space and reciprocal space is a Fourier transformation. That is, the integral over all of space can be expressed in terms of an integral over the 1. BZ:

${\displaystyle \int_{-\infty}^{\infty} \text{d}^3 r\, f(\mathbf{r}) =\frac{1}{\Omega_{BZ}}\int_{1.\,BZ} \text{d}^3 k\, \tilde{f}(\mathbf{k}) }$

In principle one has to include an infinite number of ${\displaystyle \mathbf{k}}$ points in the 1. BZ to describe interactions with periodic replica of the unit cell. In practice, the ${\displaystyle \mathbf{k}}$ points are defined on a regular ${\displaystyle \mathbf{k}}$ mesh and beyond a certain ${\displaystyle \mathbf{k}}$-points density the result (i.e. some quantity of interest) will not change (i.e. it converges w.r.t. the number of ${\displaystyle \mathbf{k}}$ points). This is because the crystal momentum vectors that are close together are almost identical. Hence, we obtain

$${\displaystyle \int_{-\infty}^{\infty} \text{d}^3 r\, f(\mathbf{r}) =\frac{1}{\Omega_{BZ}}\int_{1.\,BZ} \text{d}^3 k\, \tilde{f}(\mathbf{k}) = \sum_{\mathbf k \in 1.\,BZ} \tilde{f}(\mathbf{k}) \Delta^3k}$$

With a closer look at the first Brillouin zone, we notice that it also has a certain symmetry. As a consequence, some ${\displaystyle \mathbf{k}}$ points on the regular mesh are equivalent. So, we only have to compute the KS orbitals on an irreducible set of ${\displaystyle \mathbf{k}}$ points. VASP automatically finds these irreducible ${\displaystyle \mathbf{k}}$ points and applies an appropriate weight ${\displaystyle w_\mathbf{k} }$ in the sum over k points.

$${\displaystyle \int_{-\infty}^{\infty} \text{d}^3 r\, f(\mathbf{r}) =\frac{1}{\Omega_{BZ}}\int_{1.\,BZ} \text{d}^3 k\, \tilde{f}(\mathbf{k}) = \sum_{\mathbf k \in 1.\,BZ} \tilde{f}(\mathbf{k}) \Delta^3k= \sum_{\mathbf{k} \in irred\, 1.\,BZ} w_\mathbf{k} \tilde{f}(\mathbf{k}) }$$

To see which symmetry VASP identified and which irreducible ${\displaystyle \mathbf{k}}$ points are used during the calculation, look at the OUTCAR and IBZKPT files.