Figure 1

(Color online) Theoretical (Ref. 12) two-terminal QH conductance $g$ as a function of filling factor $\nu$ for (a) single-layer graphene, (b) bilayer graphene, and (c) gapped bilayer graphene, for effective aspect ratios $\xi =L/W=2$ (lower curve) and 0.5 (upper curve). Finite longitudinal conductivity due to the states in the middle of each Landau level distorts the plateaus into N-shaped structures, which are of opposite sign for $\xi <1$ and $\xi >1$. Local extrema of $g$ at filling factors $\nu =\pm 2,\pm 6,\pm 10,\dots$ for single layers and at $\nu =\pm 4,\pm 8,\pm 12,\dots$ for bilayers are either all maxima $\left(\xi >1\right)$ or all minima $\left(\xi <1\right)$. (a) and (b) For gapless monolayer and bilayer samples, $g\left(\nu =0\right)$ is a maximum for $\xi <1$ and minimum for $\xi >1$ and (c) for the gapped bilayer $g$ vanishes at $\nu =0$ for all $\xi$.Reuse & Permissions

Figure 2

(Color online) (a) Inset: conductance $g$ in the quantum Hall regime as a function of $B$ and $V_{\text{bg}}$ at $T=250\text{mK}$ for sample A1. Dashed lines correspond to filling factors $\nu =-6,-10,-14,-18$ and align with the local maxima of conductance. Main: horizontal cut of inset giving $g\left(V_{\text{bg}}\right)$ at $B=8\text{T}$ and calculated $g$ for the best-fit equivalent aspect ratio ${\xi }_{\text{fit}}=1.7$ (solid curve) and for the measured sample aspect ratio ${\xi }_s=0.7$ (dashed curve) using Landau-level broadening parameter $\lambda =1.2$. (b) Inset: conductance $g$ in the quantum Hall regime as a function of $B$ and $V_{\text{bg}}$ at $T=250\text{mK}$ for sample A2. Dashed lines correspond to $\nu =-6,-10,-14,-18$ and align with the local minima of conductance. Main: horizontal cut of inset giving $g\left(V_{\text{bg}}\right)$ at $B=8\text{T}$ and calculated $g$ for ${\xi }_{\text{fit}}=0.2$ (solid curve) and ${\xi }_s=0.2$ (dashed curve) ($\lambda =1.2$, the same as sample A1). The dashed curve was vertically displaced for clarity.Reuse & Permissions

Figure 3

(Color online) Inset: measured $g$ of sample B1 as a function of $B$ and $V_{\text{bg}}$ at $T=4\text{K}$. Dashed lines, corresponding to $\nu =-12,-16,-20$, align with local minima of $g$. No minima are observed at $\nu =8$ for $B=5\text{T}<B<B=8\text{T}$. Main: horizontal cut of inset at $B=8\text{T}$ and calculated $g$ using $\lambda =0.7$ for ${\xi }_s=2.5$ (dashed curve) and ${\xi }_{\text{fit}}=0.8$ (solid curve).Reuse & Permissions

Figure 4

(Color online) Measured $g\left(V_{\text{bg}}\right)$ for sample B2 and the calculated $g$ using $\lambda =0.25$ for ${\xi }_s=0.2$ (dashed trace) and ${\xi }_{\text{fit}}=0.3$ (solid trace). Two key features in the curve suggest this sample is a gapless bilayer, namely, a pronounced peak in $g$ near the CNP and the larger spacing between the two minima straddling the CNP compared to the spacing $\delta V_{\text{bg}}\sim 9.5\text{V}$ between other consecutive minima.Reuse & Permissions

Figure 5

(Color online) Measured $g\left(V_{\text{bg}}\right)$ for sample C and calculated conductance (solid curve) for ${\xi }_{\text{fit}}=0.9$ $\left(\lambda =0.7\right)$. The asymmetric contacts of this sample can be conformally mapped onto a rectangle, producing a device aspect ratio of ${\xi }_s=0.9$ (dashed curve). The dashed curve was vertically displaced for clarity.Reuse & Permissions

Figure 6

A polygon representing sample C (see Fig. 5). Black regions correspond to contacts (length scale $a=200\text{nm}$).Reuse & Permissions

Figure 7

(Color online) Three steps used to map the polygon in Fig. 6 (sample C) onto the upper half plane (schematic). (a) First, the rectangle in Fig. 6 is replaced by a half-infinite strip, extending indefinitely to the right. (b) Next, we map the domain shown in (a) onto a rectangle with contact 3-5-6-4 straightened out. Under this mapping, the sample is slightly distorted, as indicated by the gray polygon in (b). (c) Because the deviation of the gray polygon boundary from the original sample boundary [serrated rectangle in (b), red online] is fairly small, it can be neglected, giving a half-infinite strip. (d) Finally, the domain (c) is mapped onto the upper half plane, which allows to find the cross ratio ${\Delta }_{1234}$, Eq. (9), and evaluate the effective aspect ratio, Eq. (10).Reuse & Permissions