Fig. 5: Analysis on temperature-dependent orbital transport.

a Fitting of the temperature-dependent 2ΔR_DOEE data with Eq. (1). Here, t_Cu = 40 nm and FM = Co₂₅Fe₇₅. The solid curves represent the fitting results. The error bars indicate the standard deviation of R after the magnetization is saturated. b Fitting result of λ_o for 2ΔR_DOEE (blue squares) and 2ΔR_IOEE (red squares) as functions of temperature. The error bars indicate the 95% confidence intervals from fitting results. All the results obey Onsager’s reciprocal relations. c Schematic of the proposed mechanism for the temperature-dependent orbital propagation. At low temperatures (upper panel), hopping between the localized states in the oxidized Cu (CuO_x layer) is limited, so the itinerant electrons (green arrow) are confined to the unoxidized Cu layer without carrying OAM (L = 0). Thus, continuous orbital Rashba bands are not forming. At high temperature (lower panel), the localized states are thermally broadened, establishing the conductive channel in the CuO_x layer. The electron wavefunctions are extended into the CuO_x layer, creating hybridized states with OAM (L ≠ 0), which form the continuous orbital Rashba bands and allow the long-range orbital propagation.