Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation

The intrinsic anomalous Hall conductivity in ferromagnets depends on subtle spin-orbit-induced effects in the electronic structure, and recent ab initio studies found that it was necessary to sample the Brillouin zone at millions of $k$-points to converge the calculation. We present an efficient first-principles approach for computing this quantity. We start out by performing a conventional electronic-structure calculation including spin-orbit coupling on a uniform and relatively coarse $k$-point mesh. From the resulting Bloch states, maximally localized Wannier functions are constructed which reproduce the ab initio states up to the Fermi level. The Hamiltonian and position-operator matrix elements, needed to represent the energy bands and Berry curvatures, are then set up between the Wannier orbitals. This completes the first stage of the calculation, whereby the low-energy ab initio problem is transformed into an effective tight-binding form. The second stage only involves Fourier transforms and unitary transformations of the small matrices setup in the first stage. With these inexpensive operations, the quantities of interest are interpolated onto a dense $k$-point mesh and used to evaluate the anomalous Hall conductivity as a Brillouin zone integral. The present scheme, which also avoids the cumbersome summation over all unoccupied states in the Kubo formula, is applied to bcc Fe, giving excellent agreement with conventional, less efficient first-principles calculations. Remarkably, we find that about 99% of the effect can be recovered by keeping a set of terms depending only on the Hamiltonian matrix elements, not on matrix elements of the position operator.

Figure 1

Band structure of bcc Fe with spin-orbit coupling included. Solid lines: original band structure from a conventional first-principles calculation. Dotted lines: Wannier-interpolated band structure. The zero of energy is the Fermi level.

Figure 2

(Color online) Isosurface contours of maximally localized spin-up WF in bcc Fe (red for positive value and blue for negative value), for the $8\times 8\times 8$ $k$-point sampling. (a) $s{p}^3{d}^2$-like WF centered on a Cartesian axis and (b) $d_{xy}$-like WF centered on the atom.

Figure 3

Band structure and total Berry curvature, as calculated using Wannier interpolation, plotted along the path $\Gamma \text{−}H\text{−}P$ in the Brillouin zone. (a) Computed at the full spin-orbit coupling strength $\lambda =1$. (b) Computed at the reduced strength $\lambda =0.25$. The peak marked with a star has a height of $5\times {10}^4\mathrm{a.u.}$

Figure 4

Decomposition of the total Berry curvature into contributions coming from the three kinds of terms appearing in Eq. (32). The path in $k$-space is the same as in Fig. 3. The dotted line is the first $\left(\overline{\Omega }\right)$ term, the dashed line is the sum of second and third ($D\text{−}\overline{A}$) terms, and the solid line is the fourth $\left(D\text{−}D\right)$ term of Eq. (32). Note the log scale on the vertical axis.

Figure 5

(Color online) Lines of intersection between the Fermi surface and the plane $k_y=0$. Colors indicate the $S_z$ spin-component of the states on the Fermi surface (in units of $\hslash ∕2$).

Figure 6

(Color online) Calculated total Berry curvature $-{\Omega }_z$ in the plane $k_y=0$ (note log scale). Intersections of the Fermi surface with this plane are again shown.

Figure 7

Anomalous Hall conductivity vs spin-orbit coupling strength.