Anomalous Nonlocal Resistance and Spin-Charge Conversion Mechanisms in Two-Dimensional Metals

We uncover two anomalous features in the nonlocal transport behavior of two-dimensional metallic materials with spin-orbit coupling. First, the nonlocal resistance can have negative values and oscillate with distance, even in the absence of a magnetic field. Second, the oscillations of the nonlocal resistance under an applied in-plane magnetic field (the Hanle effect) can be asymmetric under field reversal. Both features are produced by direct magnetoelectric coupling, which is possible in materials with broken inversion symmetry but was not included in previous spin-diffusion theories of nonlocal transport. These effects can be used to identify the relative contributions of different spin-charge conversion mechanisms. They should be observable in adatom-functionalized graphene, and they may provide the reason for discrepancies in recent nonlocal transport experiments on graphene.

Figure 1

Nonlocal transport on a Hall bar device. A current $I$ is injected at $x=0$ between the electrodes $V_2$ and $V_1$. This gives rise to both a local resistance $R_{xx}$ and a nonlocal resistance $R_{\mathrm{nl}}$. The latter is enhanced by a process involving (i) a spin-charge conversion (current-induced spin polarization or spin Hall effect) at $x=0$, (ii) diffusion from $x=0$ to $x=L$, and (iii) the inverse spin-charge conversion at $x=L$. The green-dotted line indicates the direction of electron flow, following the convention of Ref. [40].

Figure 2

Nonlocal resistance of graphene decorated with adatoms that induce SOC [33]. (a) Nonlocal resistance $R_{\mathrm{nl}}$ versus distance, in the absence of a magnetic field. When the SOC magnetic length ${\ell }_R$ (i.e., the length scale associated with Rashba SOC) becomes comparable to the spin relaxation length ${\ell }_s$, $R_{\mathrm{nl}}$ can become negative due to spin precession, even in the absence of a magnetic field. (b) $R_{\mathrm{nl}}$ versus the chemical potential $\mu$ in the absence of a magnetic field, showing that $R_{\mathrm{nl}}>0$ when the SHE dominates over DMC effects, whereas $R_{\mathrm{nl}}<0$ when DMC dominates. (c) Hanle precession at different values of $\mu$, in the regime where the SHE dominates over DMC [8]. The oscillations are suppressed for values of $\mu$ away from the Dirac point due to increases in the elastic scattering time. In all plots, $R_{\mathrm{nl}}$ and ${\mathcal{R}}_{\mathrm{nl}}$ are normalized to the local resistivity $R_{xx}$. See the Supplemental Material for details of the microscopic model, such as the SOC potential strengths which determine the conversion factors ${\theta }_{\text{sH}}$, ${\ell }_{\mathrm{DMC}}$ [37].

Figure 3

Anomalous Hanle precession. The two main spin-charge conversion mechanisms are the spin Hall effect (SHE) and current-induced spin polarization. The latter is characterized by the direct magnetoelectric coupling (DMC) length ${\ell }_{\mathrm{DMC}}$ [33]. The Hanle precession behaves very differently in different limits and can serve as a guide to identify the dominant spin-charge conversion mechanism. The parameters used in the plots are ${\ell }_s={10}^{-6}\mathrm{m}$ [8] and ${\ell }_R={10}^{-5}\mathrm{m}$. For $\mathrm{SHE}\gg \mathrm{DMC}$ scattering, ${\theta }_{\text{sH}}=0.2$ and ${\ell }_s/{\ell }_{\mathrm{DMC}}={10}^{-3}$; for $\mathrm{DMC}\gg \mathrm{SHE}$, ${\theta }_{\text{sH}}={10}^{-3}$; and when ${\ell }_s/{\ell }_{\mathrm{DMC}}=0.1$ and for $\mathrm{SHE}\sim \mathrm{DMC}$ scattering, ${\theta }_{\text{sH}}={\ell }_s/{\ell }_{\mathrm{DMC}}=0.1$.