First-principles calculation of orbital Hall effect by Wannier interpolation: Role of orbital dependence of the anomalous position

The position operator in a Bloch representation acquires a gauge correction in the momentum space on top of the canonical position, which is called the anomalous position. We show that the anomalous position is generally orbital dependent and thus plays a crucial role in the description of the intrinsic orbital Hall effect in terms of Wannier basis. We demonstrate this from the first-principles calculation of orbital Hall conductivities of transition metals by Wannier interpolation. Our results show that consistent treatment of the velocity operator by including the correction term originating from the anomalous position predicts the orbital Hall conductivities different from those obtained by considering only the group velocity. We find the difference is crucial in several metals. For example, we predict negative signs of the orbital Hall conductivities for elements in groups X and XI such as Cu, Ag, Au, and Pd, for which the previous studies predicted positive signs. In this paper, we suggest the importance of consistently describing the spatial dependence of basis functions by first-principles methods, as it is fundamentally missing in the tight-binding approximation.

Figure 1

Schematic illustration of the matrix element for the anomalous position for $p_x$- and $p_y$-like Wannier states in a two-dimensional cubic lattice $d_{\gamma }=\langle p_x,\mathbf{0}|\widehat{y}|p_y,+\widehat{\mathbf{x}}a\rangle$ [Eq. (48)].

Figure 2

(a)–(c) Intrinsic spin Hall conductivity ${\sigma }_{\mathrm{SH}}$ and (d)–(f) orbital Hall conductivity ${\sigma }_{\mathrm{OH}}$ of body-centered cubic (bcc) W [(a) and (d); first row], face-centered cubic (fcc) Pt [(b) and (e); second row], and fcc Cu [(c) and (f); third row] as a function of the Fermi energy $E_{\mathrm{F}}$, which is varied assuming that the effective singe-particle potential is fixed to the potential at the true Fermi energy $E_{\mathrm{F}}^{\mathrm{true}}$. Blue solid lines are the results obtained by taking the full expression of the velocity operator in Eq. (20), and orange dashed lines are the results obtained by taking only the group-velocity-like term and ignoring the anomalous position contribution.

Figure 3

The intrinsic (a) spin Hall conductivity ${\sigma }_{\mathrm{SH}}$ and (b) orbital Hall conductivity ${\sigma }_{\mathrm{OH}}$ in $3d$ (red circles), $4d$ (green squares), and $5d$ (blue triangles) transition metals between groups IV and XI. For $3d$ metals, their magnetic phases (partially transparent red circles) are also considered as well as their nonmagnetic counterpart; the antiferromagnetic phase for Cr and Mn, and the ferromagnetic phase for Fe, Co, and Ni.