Interfacial orbital transmission, conversion, and mechanical torque in metals

Abstract

Interfacial orbital transport remains far less understood than its bulk counterpart despite its central role in orbitronic experiments. Here, we theoretically investigate the transmission and conversion of orbital angular momentum across a metallic interface using a model Hamiltonian incorporating crystal-field effects. We show that an injected orbital dipole moment undergoes pronounced oscillations driven by the crystal field and generates characteristic quadrupole moments determined by the orbital orientation relative to the interface. Unlike spin precession, the dipole relaxes toward a finite value away from the interface. We further quantify interfacial orbital memory loss and demonstrate that orbital absorption produces a sizable mechanical torque obtained from the orbital continuity equation.

Figures (4)

• Figure 1: (a-b) Orbital dipole and (c-d) quadrupole moments as a function of $z$ in the right layer induced by incident $|L_x\rangle$ and $|L_z\rangle$ dipole moments from the left layer. The short-handed notation $\langle L_i^j\rangle$ represents $\langle L_i\rangle=\langle \Psi_\text{R}|\boldsymbol{L}_i|\Psi_\text{R}\rangle$ induced by incident $|L_j\rangle$. The parameters used are: $U/E_F =0.5$ and $r/(1/2m)=0.5$, which gives $k_t/k_F=\sqrt{1/3}\approx0.577$ and $k_r/k_F=\sqrt{1/2}\approx0.707$. Here $\alpha_R^I=0$. The inset in (a) represents a schematic diagram of the bilayer structure with energy bands.
• Figure 2: First row: (a-b) Orbital dipole moments and (c) quadrupole moments as a function of $z$ for different $r$ values. Second row: The crystal field $r$-dependence of the orbital dipole and quadrupole moments at (d) $z=0$, (e) $z=\infty$, and (f) their oscillation wavelengths. The other parameters are the same as used in Fig. \ref{['fig:L_all']}.
• Figure 3: (a-b) Interfacial orbital dipole memory loss $\delta$ as a function of the interfacial Rashba strength $\alpha_R^I$. Dipole to quadrupole interfacial conversion efficiency $\eta$ as a function of the crystal field strength $r$ and $\alpha_R^I$ in (c) and (d), respectively, in which $\alpha_R^I=0$ is used in (c) while $r/(1/2m)=0.5$ is considered in (d). The other parameters are the same as used in Fig. \ref{['fig:L_all']}.
• Figure 4: The orbital torque induced by (a) $|L_z\rangle$ and (b) $|L_x\rangle$ injections as a function of $z$ for different crystal field $r$, respectively. The other parameters are the same as used in Fig. \ref{['fig:L_all']}.