Gapped Dirac materials and quantum valley currents in dual-gated hBN/bilayer-graphene heterostructures

In gapped Dirac materials, the topological current associated with each valley can flow in opposite directions creating long-range charge-neutral valley currents. We report valley currents in hexagonal boron nitride (hBN)/bilayer-graphene heterostructures with an energy gap, which is tunable by a perpendicular electric (displacement) field in a dual-gated structure. We observed significant nonlocal resistance, consistent with the scaling theory of the valley Hall effect. In the low-temperature limit, the nonlocal resistance approaches a saturated value near the “quantum limit,” indicating the emergence of quantum valley currents.

Figure 1

(a) Schematic cross section of the device. (b) Landau fan diagram: the intensity map of ${\rho }_{xx}$ as a function of $V_{\mathrm{bg}}$ and $B$ at $T=1.7$ K and $V_{\mathrm{tg}}=0$ V. (c) ${\rho }_{xx}$ (black), $R_{\mathrm{nl}}$ (red), and $R_{\mathrm{Ohm}}$ (blue) as a function of $V_{\mathrm{bg}}$ at $T=1.7$ K, $B=0$ T, and $V_{\mathrm{tg}}=-0.32$ V. The inset shows the optical image of the device. The scale bar corresponds to 5 µm. (d) ${\rho }_{xx}$ as a function of $V_{\mathrm{bg}}$ for $V_{\mathrm{tg}}$ from −0.6 to 1 V. The inset shows the schematic of the local measurement configuration. (e) Same plot as (d) for the nonlocal configuration.

Figure 2

(a),(b) Temperature dependence of (a) ${\rho }_{xx}$ and (b) $R_{\mathrm{nl}}$ for various $D$. The insets show the Arrhenius plots for (a) ${\rho }_{xx,max}$ and (b) $R_{\mathrm{nl},\max }$, respectively. The solid lines show the fitting to $\sim exp\left(\text{–}{E}_{\mathrm{g}}/2k_{\mathrm{B}}T\right)$. (c) Energy gap extracted from the Arrhenius fit for ${\rho }_{xx,max}$ (local, black) and $R_{\mathrm{nl},\max }$ (nonlocal, red) as a function of $D$. The error bars correspond to the ambiguity in the fitting procedure.

Figure 3

Scaling analysis in the log-log plot between ${\rho }_{xx,max}$ and $R_{\mathrm{nl},\max }$. The data are obtained from each $T$ ranging 1.7–40 K. (a) At $D=27.3$ mV/nm. The dashed line represents a fitting to the cubic relation ($\sim {\rho }_{xx,max}^3$). (b) The same as (a) with various $D$. The dashed, dotted, and dot-dashed lines correspond to the relation $\sim {\rho }_{xx,max}^3$, $\sim {\rho }_{xx,max}^2$, and $\sim {\rho }_{xx,max}^1$, respectively. (c) At $D=27.3$ mV/nm. The solid line illustrates the fitting result with $p=3.09$, $L=2.5\mu \mathrm{m}$, $W=1.6\mu \mathrm{m}$, and $l_{\mathrm{v}}=2.5\mu \mathrm{m}$.