Calculating the Berry curvature of Bloch electrons using the KKR method

We propose and implement a particularly effective method for calculating the Berry curvature arising from adiabatic evolution of Bloch states in $\mathbf{k}$ space. The method exploits a unique feature of the Korringa-Kohn-Rostoker (KKR) approach to solve the Schrödinger or Dirac equations. Namely, it is based on the observation that in the KKR theory the wave vector $\mathbf{k}$ enters the calculation only via the structure constants which reflect the geometry of the lattice but not the crystal potential. For both the Abelian and non-Abelian Berry curvature we derive an analytic formula whose evaluation does not require any numerical differentiation with respect to $\mathbf{k}$. We present explicit calculations for Al, Cu, Au, and Pt bulk crystals.

Figure 1

The absolute value of the Fermi velocity of Cu (in a.u.) obtained using three different methods: (a) the numerical derivative of the dispersion relation ${\nabla }_{\mathbf{k}}{\mathcal{E}}_n\left(\mathbf{k}\right)$, (b) the expectation value of the relativistic velocity operator given by Eq. (2.6), and (c) implementation of Eq. (2.8).

Figure 2

The length of the diagonal components of the Berry curvature on the Fermi surface of Au in a.u.: (a) the KKR part ${\Omega }_{ii}^{\mathit{KKR}}\left(\mathbf{k}\right)$ according to Eq. (3.25), (b) the contribution from ${\Omega }_{ii}^r\left(\mathbf{k}\right)$ given by Eq. (3.28), and (c) the part ${\Omega }_{ii}^v\left(\mathbf{k}\right)$ introduced by the energy dependence of the basis functions according to Eq. (3.26).

Figure 3

The absolute value (in a.u.) of the diagonal component of the Berry curvature for the Fermi surface of several metals. From left to right: Al (3rd and 5th bands), Cu (11th band), and Pt (7th, 9th, and 11th bands). For Al we used a logarithmic scale to visualize the important regions.

Figure 4

The energy-resolved spin Hall conductivity for Au (top) and Pt (bottom) according to Eq. (5.3).

Figure 5

The energy-resolved Berry curvature for Au. The red (solid) and blue (dashed) curves show the separate contributions from the KKR part ${\Omega }_{11}^{\mathit{KKR},z}\left(\mathcal{E}\right)$ and the dipole part ${\Omega }_{11}^{r,z}\left(\mathcal{E}\right)$ of the Berry curvature, respectively.