Spectral and Fermi surface properties from Wannier interpolation

We present an efficient first-principles approach for calculating Fermi surface averages and spectral properties of solids, and use it to compute the low-field Hall coefficient of several cubic metals and the magnetic circular dichroism of iron. The first step is to perform a conventional first-principles calculation and store the low-lying Bloch functions evaluated on a uniform grid of $k$ points in the Brillouin zone. We then map those states onto a set of maximally localized Wannier functions, and evaluate the matrix elements of the Hamiltonian and the other needed operators between the Wannier orbitals, thus setting up an “exact tight-binding model.” In this compact representation the $k$-space quantities are evaluated inexpensively using a generalized Slater-Koster interpolation. Owing to the strong localization of the Wannier orbitals in real space, the smoothness and accuracy of the $k$-space interpolation increases rapidly with the number of grid points originally used to construct the Wannier functions. This allows $k$-space integrals to be performed with ab initio accuracy at low cost. In the Wannier representation, band gradients, effective masses, and other $k$ derivatives needed for transport and optical coefficients can be evaluated analytically, producing numerically stable results even at band crossings and near weak avoided crossings.

Figure 1

(Color online) Wannier-interpolated bands of bcc Fe along $\Gamma \text{−}H$. The bands are colorcoded according to the value of the spin projection $⟨S_z⟩$: red for spin-up and blue for spin-down. The energies are given in eV and the Fermi level is at $0\mathrm{eV}$. The vertical dashed lines indicate $k$ points on the ab initio mesh used for constructing the WFs. For comparison, points from a full ab initio calculation are shown as open circles.

Figure 2

Upper panel: Dispersion of the three $p$-like energy bands of Pb along the $\Gamma \text{−}K$ direction, obtained from a nonrelativistic ab initio calculation. Lower panel: Inverse effective masses of those bands along the same direction, calculated in two ways: from the ab initio eigenenergies on a regular mesh of points using a spline fit (circles), and from perturbation theory in the Wannier representation (solid lines).

Figure 3

(Color online) Density of states of bulk diamond calculated in the range $8–18\mathrm{eV}$ using the conventional Gaussian broadening approach (light thin lines) with a fixed width of $0.4\mathrm{eV}$, versus the adaptive broadening approach (dark thick lines). The inset shows the density of states in the full valence band range computed with the adaptive broadening approach.

Figure 4

Magnetic circular dichroism spectrum of bcc iron. The calculated spectrum (solid lines) is compared with the experimental spectrum from Ref. 33 as reproduced in Ref. 34 (open circles).

Figure 5

(Color online) Upper panel: decomposition of the magnetic circular dichroism spectrum into the three terms defined by the Wannier-interpolation procedure. Lower panel: cumulative anomalous Hall conductivity (AHC) versus the energy $\hslash \omega$, defined as the contribution to the AHC from frequencies higher than $\omega$ in Eq. (43).

Figure 6

Convergence of the Wannier-interpolated band energies as a function of the linear dimensions $N_0^{1∕3}$ of the ab initio $\mathbf{q}$-point grid. We plot the maximum error (squares) and mean absolute error (circles), where the error is the difference between the Wannier-interpolated band energy and the value obtained from a full non-self-consistent diagonalization of the plane-wave Hamiltonian. The lines are linear fits to the points with $N_0^{1∕3}>8$.

Figure 7

(Color online) Comparison of interpolated band energies for Pb obtained using sets of lattice vectors defined within Wigner-Seitz (WS) and parallelepipedal $\left(P\right)$ supercells. Upper figure: Energy bands interpolated using a $4\times 4\times 4$ $\mathbf{q}$-point grid (WS cell, solid lines; $P$ cell, dashed lines). Full ab initio results are shown as open circles. Lower figure: Convergence of the Wannier-interpolated band energies for the two supercells as a function of the linear dimensions $N_0^{1∕3}$ of the $\mathbf{q}$-point grid. Details as in Fig. 6.