Interfacial orbital transmission, conversion, and mechanical torque in metals

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.

💡 Research Summary

The authors present a comprehensive theoretical study of orbital angular momentum (OAM) transport across a metallic interface, focusing on transmission, conversion, and the generation of mechanical torque. Using a minimal bilayer model, the left side is a simple free‑electron metal (HL = ℏ²k²/2m) while the right side incorporates a crystal‑field term r(L·k)² that endows the material with an intrinsic orbital texture. The Hamiltonian of the right layer (HR = ℏ²k²/2m + U + r(L·k)²) captures the coupling between the local orbital operator L (represented by the three p‑orbital matrices) and the crystal momentum k. This coupling makes the orbital helicity λ = L·k/k a good quantum number, yielding two degenerate helicity bands (λ = ±ℏ) and a non‑helical band (λ = 0).

By matching wavefunctions at the interface, the authors obtain reflection and transmission amplitudes for each helicity channel. Because the crystal‑field modifies the dispersion, the longitudinal wavevectors for transmitted states (k_tz and k_rz) differ, leading to spatial oscillations of the OAM density on the right side. The injected orbital dipole states |Lx⟩, |Ly⟩, and |Lz⟩ are treated separately. For a finite crystal‑field (r > 0) the component aligned with the injection direction (e.g., ⟨Lx⟩ for |Lx⟩ injection) exhibits damped oscillations as a function of distance from the interface, while the orthogonal components remain essentially zero. This behavior contrasts sharply with spin precession, where an injected spin component generates transverse components through the torque term. The oscillation period is set by the difference between k_tz and k_rz; larger transverse momentum (κ) makes the longitudinal wavevector imaginary, causing rapid decay.

Crucially, the dipole oscillations are accompanied by the emergence of specific torsional quadrupole moments ⟨L_iL_j⟩ (symmetrized products of orbital operators). For |Lx⟩ injection, the torsional quadrupole ⟨LxLy⟩ (and its cyclic permutations) is generated, reflecting the commutation relation