Nonlocal topological valley transport at large valley Hall angles

Berry curvature hot spots in two-dimensional materials with broken inversion symmetry are responsible for the existence of transverse valley currents, which give rise to giant nonlocal dc voltages. Recent experiments in high-quality gapped graphene have highlighted a saturation of the nonlocal resistance as a function of the longitudinal charge resistivity ${\rho }_{\mathrm{c},xx}$, when the system is driven deep into the insulating phase. The origin of this saturation is, to date, unclear. In this work we show that this behavior is fully compatible with bulk topological transport in the regime of large valley Hall angles (VHAs). We demonstrate that, for a fixed value of the valley diffusion length, the dependence of the nonlocal resistance on ${\rho }_{\mathrm{c},xx}$ weakens for increasing VHAs, transitioning from the standard ${\rho }_{\mathrm{c},xx}^3$ power law to a result that is independent of ${\rho }_{\mathrm{c},xx}$.

Figure 1

A sketch of the nonlocal transport setup analyzed in this work. A charge current $I$ is driven between injector and collector electrodes (green) that are in electrical contact with a 2D conductive channel (gray-shaded area). A steady-state dc voltage $\mathrm{\Delta }V\left(x\right)$ is established between two probe electrodes (red) at a lateral distance $x$ due to diffusion of valley currents (i.e., valley Hall effect) and their subsequent conversion into a regular charge voltage (inverse valley Hall effect). The nonlocal resistance is defined by $R_{\mathrm{NL}}\left(x\right)=[\varphi (x,-W/2)-\varphi (x,+W/2)]/I\equiv \mathrm{\Delta }V\left(x\right)/I$, where $\varphi (x,y)$ is the 2D electrical potential.

Figure 2

Spatial dependence of the nonlocal resistance $R_{\mathrm{NL}}\left(x\right)$ (in units of ${\rho }_{\mathrm{c},xx}$ and in logarithmic scale) for $W/{\ell }_{\mathrm{v}}=0.1$. The analytical result (dashed line) reported in Eqs. (21) and (22) is compared with the full numerical result (solid line) obtained from Eq. (19). Panel (a): ${\theta }_{\mathrm{VH}}=\pi /8$. Panel (b): ${\theta }_{\mathrm{VH}}=\pi /2-\epsilon$, with $\epsilon =\pi /30$. In panel (a) we clearly notice a biexponential behavior, where the first fast exponential decay occurs on the electrostatic length scale $W/\pi$, while the second slow decay occurs on the scale of the valley diffusion length ${\ell }_{\mathrm{v}}$.

Figure 3

Scaling between $\mathrm{\Delta }{R}_{\mathrm{NL}}\left(x\right)$ and ${\rho }_{\mathrm{c},xx}$ in bilayer graphene. The circles are experimental data extracted from Fig. 4 in Ref. [3]. These data have been taken [3] in a sample with $W=1\mu \mathrm{m}$ and at a distance $x=4.5\mu \mathrm{m}$ from the current injector. The red dashed line shows the asymptotic value (24) for ${\sigma }_{\mathrm{v},xy}=4e^2/h$ and ${\ell }_{\mathrm{v}}=4.6\mu \mathrm{m}$. The black solid line represents Eq. (21), for the same values of ${\sigma }_{\mathrm{v},xy}$ and ${\ell }_{\mathrm{v}}$.