Orbital Rashba effect in a surface-oxidized Cu film

Recent experimental observation of an unexpectedly large current-induced spin-orbit torque in surface oxidized Cu on top of a ferromagnet pointed to a possibly prominent role of the orbital Rashba effect (ORE) in this system. Here, we use first principles methods to investigate the ORE in a system of oxygen monolayer deposited on top of a Cu(111) film. We show that surface oxidization of the Cu film leads to a gigantic enhancement of the ORE near the Fermi energy. The resulting chiral orbital texture in the momentum space is exceptionally strong, reaching as much as $\sim 0.5\hslash$ in magnitude. We find that resonant hybridization between O $p$ states and Cu $d$ states is responsible for the emergence of the ORE at the interface. We also present a minimal model that captures the emergence of the ORE through the $pd$ hybridization mechanism. We demonstrate that an application of an external electric field to the system generates colossal orbital Hall currents which are an order of magnitude larger than the spin Hall currents found in heavy metals. This implies that the “orbital torque” mechanism may be significant in surface oxidized Cu/ferromagnet structures. Our results encourage an experimental verification of the rich orbital physics in surface oxidized Cu films through optical measurements such as angle-resolved photoemission spectroscopy and momentum microscopy.

Figure 1

(a) Crystal structure of surface oxidized Cu in fcc stacking geometry, where only a few top surface layers are shown. Red and blue spheres represent O and Cu atoms, respectively. (b) Hybridization between an O $p$ orbital and Cu $d$ orbitals mediates the formation of a superposition of the type $d_{zx}\pm i{d}_{xy}$, which carries a finite OAM. (c) As a result, electrons moving in opposite directions have an opposite sign of OAM expectation value, which constitutes the ORE.

Figure 2

Orbital character analysis of the electronic states of surface oxidized Cu. On top of the energy bands (grey lines), weights of (a) O $p$ and (b) Cu1 $d$ orbitals are shown in the color map.

Figure 3

(a) Electronic band structure of the surface oxidized Cu (grey lines), where the color indicates the OAM expectation value of each state projected onto the Cu1 layer. (b) OAM texture at the Fermi surface. The arrows indicate the direction of the OAM expectation value, while the color represents its absolute value.

Figure 4

Electric field response of (a) OAM accumulation $〈L_y〉$ and (b) orbital Hall current $\langle J_z^{L_y}\rangle$ (blue circles). Spin responses, which are much smaller than the orbital counterparts, are also shown for comparison (orange squares). An external electric field ${\mathcal{E}}_x$ is applied along the $x$ direction, and the $z$ direction is defined such that it points toward the top surface (O/Cu1). The observed orbital responses are gigantic, comparable or larger than the spin responses in heavy elements.