Orbital Hall effect in crystals: Interatomic versus intra-atomic contributions

The orbital Hall effect (OHE) designates the generation of a charge-neutral flow of orbital angular momentum transverse to an initial charge current. Recent theoretical investigations suggest that transition metals display sizable OHE, encouraging experimental search along this direction. Nonetheless, most of these theories assume that the orbital moment originates from the region centered on the atomic sites, adopting the so-called atom-centered approximation. In periodic crystals though, the motion of the wave packet between atoms provides a crucial contribution to the overall orbital moment, and neglecting it can lead to a severe misestimation of the OHE. By applying the ``modern theory'' of orbital magnetization to the OHE, we assess the relative importance of intra- and interatomic contributions in selected materials from first principles. We find that whereas the OHE is mostly of intra-atomic origin for wide band-gap semiconductors (e.g., ${\mathrm{MoS}}_{2}$), the interatomic contribution becomes crucial in narrow band-gap semiconductors (SnTe, PbTe) and transition metals (Pt, V etc.). These predictions invalidate the atom-centered approximation adopted in some of the previous papers and open perspectives for the realization of efficient sources of orbital currents.