Fig. 1: Schematic illustration of nonlocal transport measurements and verification of orbital response through ferromagnetic materials dependence experiments.

a Nonlocal measurement configuration to observe DOEE (direct measurement). The nonequilibrium OAM (L) is generated by the charge current I_c at the CuO_x/Cu interface through DOEE. The orbital accumulation then converts into SAM (S) via SOC in FMs and induces a nonlocal response V. b Nonlocal measurement configuration to observe IOEE (inverse measurement). A charge current brings nonequilibrium OAM from the FMs. The orbital accumulation then converts to charge current at the CuO_x/Cu interface through IOEE. In nonlocal measurement (a, b), the orbital generator and detector are sufficiently isolated in space with a separation distance d (center-to-center distance), allowing the measurement of nonlocal orbital response. c, d Typical results of direct (c, R_DOEE) and inverse (d, R_IOEE) nonlocal orbital Edelstein resistance R, which is defined as \(R\equiv V/{I}_{{{{\rm{c}}}}}\). The results are observed in sample A with separation distance d = 140 nm, Cu thickness t_Cu = 40 nm and FM = Co₂₅Fe₇₅ (Methods) while the external magnetic field B_ext swept along the hard axis of FM (Φ = 0°) from −1.25 T to 1.25 T. All the signals are globally offset to position their center at R = 0 Ω. The double-headed arrows in c, d indicate the definition of 2ΔR_DOEE and 2ΔR_IOEE, where 2ΔR_DOEE = − 2ΔR_IOEE ≈ 0.22 mΩ. e, f FM dependence results of 2ΔR_DOEE (e) and 2ΔR_IOEE (f). The solid curves represent the fitting of the data to Eq. (1), implying a long-range decay length of orbital accumulation λ_o of ~100 nm regardless of the selection of FMs. The dotted curves are guiding lines showing the value R = 0 Ω. The error bars indicate the standard deviation of R after the magnetization is saturated. All the results are measured at room temperature. All the results are in solid agreement with Onsager’s reciprocal relations.