Fig. 1: Schematic Illustration of the OHE and orbital torque.

a When an electric field is applied along the horizontal direction, a transverse orbital current is generated owing to the OHE. This orbital current is converted to a spin current through the spin-orbit coupling. Depending on the sign of spin-orbit correlation \(R=\left\langle {{{{{\bf{L}}}}}}\cdot {{{{{\bf{S}}}}}}\right\rangle\), the spin polarization of the resulting spin current is either parallel (left panel) or antiparallel (right panel) to the orbital polarization of the orbital current. b Two channels for generating the torque in FM/NM bilayers. The first channel: An orbital current \({J}_{L}\) created through the OHE is converted to a spin current \({J}_{S}\) within the NM. For a negative spin-orbit correlation of NM (\({R}_{{{{{\rm{NM}}}}}}\) < 0), the direction of spin polarization carried by \({J}_{S}\) is the opposite to that of the orbital polarization carried by \({J}_{L}\), which is the case of Ta. This spin current is injected into a FM and exerts a torque on the magnetization \(\hat{{{{{{\bf{m}}}}}}}\). The second channel: \({J}_{L}\) created through the OHE in the NM is injected into a FM in which \({J}_{L}\) is converted to \({J}_{S}\). This \({J}_{S}\) exerts a torque on \(\hat{{{{{{\bf{m}}}}}}}\), which we call the orbital torque. For FMs with a positive orbital-to-spin conversion efficiency of FM \({C}_{{{{{\rm{FM}}}}}}\) (such as Fe, Co, CoFe, and Ni), the direction of the spin polarization carried by \({J}_{S}\) is the same with that of the orbital polarization carried by \({J}_{L}\). When the second channel supplies a stronger torque than the first channel and the contributions from the two channels have the opposite signs, the sign of the net torque is the opposite to that expected for the spin Hall effect of NM.