Violation of local reciprocity in charge-orbital interconversion

TL;DR

The paper addresses whether local reciprocity holds for charge–orbital interconversion versus global Onsager reciprocity. It uses thickness-resolved ST-FMR to quantify the direct orbital Hall effect torque and orbital pumping measurements to quantify inverse OHE in W/Ni bilayers, enabling separation of bulk and surface contributions by varying the W- and Ni-thickness. The key result is that the bulk direct OHE is positive while the bulk inverse OHE is negative, proving local nonreciprocity; moreover, two opposite sign orbital-to-charge conversions are found: a positive surface contribution at thin W and a negative bulk contribution at thicker W. The findings highlight nonconservation of orbital angular momentum due to lattice coupling and distinguish charge–orbital transport from charge–spin transport, with implications for orbitronic device design.

Abstract

We demonstrate a violation of local reciprocity in the interconversion between charge and orbital currents. By investigating orbital torque and orbital pumping in W/Ni bilayers, we show that the charge-orbital interconversion in the bulk of the W layer exhibits opposite signs in the direct and inverse processes -- the direct and inverse orbital Hall effects being positive and negative, respectively. This finding provides direct evidence of local nonreciprocity in the charge-orbital interconversion, in agreement with a theoretical prediction. These results highlight the unique characteristics of charge-orbital coupled transport and offer fundamental insights into the mechanisms underlying orbital-current-driven phenomena.

Figures (5)

• Figure 1: Schematic illustration of orbital torque induced by the direct OHE (left) and the inverse OHE induced by orbital pumping (right) in W/Ni. In the direct OHE, $j_\mathrm{C}$ represents the applied charge current, while $j_\mathrm{L}^\mathrm{bulk}$ and $j_\mathrm{L}^\mathrm{surface}$ denote the orbital currents generated by the bulk and surface contributions, respectively. The red and blue colors represent the positive and negative orbital polarizations, respectively. In the direct OHE, $j_\mathrm{L}^\mathrm{bulk}$ and $j_\mathrm{L}^\mathrm{surface}$ carry opposite orbital angular momentum because the bulk and surface orbital Hall conductivities have opposite signs in W. In the inverse OHE, $j_\mathrm{L}$ is the injected orbital current, while $j_\mathrm{C}^\mathrm{bulk}$ and $j_\mathrm{C}^\mathrm{surface}$ are the charge currents generated by the bulk and surface contributions, respectively. These charge currents, $j_\mathrm{C}^\mathrm{bulk}$ and $j_\mathrm{C}^\mathrm{surface}$, flow in opposite directions. $M$ denotes the magnetization.
• Figure 2: The damping-like torque efficiency $\xi_\mathrm{DL}^E$ determined by ST-FMR for the W/Ni bilayers. (a) Dependence of $\xi_\mathrm{DL}^E$ on the W thickness $t_\mathrm{W}$ in the W($t_\mathrm{W}$)/Ni(5 nm) bilayer. The values of $t_\mathrm{W}$, determined from the sputtering rate, are as follows: 0, 0.61, and 0.79 nm; in 0.36 nm steps from 1.15 to 2.24 nm; in 0.73 nm steps from 2.97 to 8.06 nm; and 9.15 nm. (b) Dependence of $\xi_\mathrm{DL}^E$ on the Ni thickness $t_{\rm Ni}$ in the W(3 nm)/Ni($t_\mathrm{Ni}$) (blue) and W(10 nm)/Ni($t_\mathrm{Ni}$) (red) bilayers. The values of $t_\mathrm{Ni}$, determined from the sputtering rate, range from 4.81 to 10.58 nm in steps of 1.92 nm.
• Figure 3: Charge-current signals induced by spin/orbital pumping. $H$ dependence of $I_{\rm DC}$ for the (a) W(0.4 nm)/Ni(5 nm) and (b) W(6 nm)/Ni(5 nm) bilayers at $\theta=90^\circ$, with $f=10$ GHz and $P_{\rm in}=200$ mW. The solid circles represent the experimental data, and the black solid curves show the fitting results, which are composed of the sum of symmetric (red curve) and antisymmetric (blue curve) components. In-plane magnetic field angle $\theta$ dependence of $I_{\rm{sym}}$ for the (c) W(0.4 nm)/Ni(5 nm) and (d) W(6 nm)/Ni(5 nm) bilayers. The solid circles represent the experimental data, and the black solid curves show the fitting results using Eq. (\ref{['angle dependence']}).
• Figure 4: The pumping-induced charge-current generation efficiency $\zeta$ in the W/Ni bilayers. (a) Dependence of $\zeta$ on the W thickness $t_\mathrm{W}$ in the W($t_\mathrm{W}$)/Ni(5 nm) bilayer. The values of $t_\mathrm{W}$, determined from the sputtering rate, are as follows: 0, 0.42, and 0.79 nm; in 0.36 nm steps from 1.15 to 2.24 nm; in 0.73 nm steps from 2.97 to 8.06 nm; and 9.15 and 9.88 nm. (b) Dependence of $\zeta$ on the Ni thickness $t_{\rm Ni}$ in the W(0.6 nm)/Ni($t_\mathrm{Ni}$) (blue) and W(10 nm)/Ni($t_\mathrm{Ni}$) (red) bilayers. The values of $t_\mathrm{Ni}$, determined from the sputtering rate, are as follows: for the W(0.6 nm)/Ni($t_\mathrm{Ni}$), from 5.00 to 13.30 nm in 2.08 nm steps, and 17.45 nm; for the W(10 nm)/Ni($t_\mathrm{Ni}$) structure, 4.68 and 5.72 nm, in 2.08 nm steps from 8.83 to 12.98 nm, and 17.13 nm.
• Figure 5: (a) Dependence of the W-layer resistivity $\rho_\mathrm{W}$ on $t_{\rm W}$ in the Ni(5 nm)/W($t_\mathrm{W}$) bilayer. The solid circles are the experimental data, and the solid curve is the fitting result. (b) Schematic illustration of the experimental setup used to measure the spin/orbital pumping. ${\bf H}$ and ${\bf I}_\mathrm{RF}$ denote the external magnetic field and the RF current, respectively. $\theta$ is defined as the in-plane angle between ${\bf H}$ and ${\bf I}_\mathrm{RF}$.