Local gating of a graphene Hall bar by graphene side gates

We have investigated the magnetotransport properties of a single-layer graphene Hall bar with additional graphene side gates. The side gating in the absence of a magnetic field can be modeled by considering two parallel conducting channels within the Hall bar. This results in an average penetration depth of the side gate created field of approximately $90\mathrm{nm}$. The side gates are also effective in the quantum Hall regime and allow to modify the longitudinal and Hall resistances.

Figure 1

Scanning force micrograph of the Hall bar with graphene side gates. Numbers 1–6 indicate contact pads for transport measurement. Electrodes SG 1–SG 4 label the side gates.

Figure 2

(a) Conductivity versus backgate voltage $V_{\mathrm{BG}}$ for various settings of the side gate voltage $V_{\mathrm{SG}}$ as indicated. The same voltage is applied to all four side gates, and the measurement is performed with an ac of $20\mathrm{nA}$ rms. The black curves represent the measured traces, while the gray lines are calculated with the model explained in Sec. 3B taking into account two parallel conducting channels with different influences from the side gates. (b) Value of the minimum in conductivity as a function of applied side gate voltage, determined from the smoothed measured curves (circles) and calculated traces (line). [(c) and (d)] Grayscale plot of the conductivity as a function of applied side gate and backgate voltage, (c) representing the measured date and (d) the calculations. (e) True to scale schematic of the Hall bar, with illustration of the two channels considered in the model. (f) Schematic section of the induced carrier density over the width of the Hall bar, with indication of the two regions (gray areas), and the simplified steplike density distribution used in the model and a realistic distribution (solid line).

Figure 3

(a) Schematic of the Hall bar with indication of the voltages applied to the side gates. An ac of $100\mathrm{nA}$ rms is used for these measurements. [(b) and (c)] Conductivity as a function of side gate and backgate voltages, (b) measured and (c) calculated. (d) Conductivity as a function of backgate voltage for different voltages applied to the side gates. The black curves represent the measured traces and the gray lines the calculated ones. Individual curves are vertically offset by $4\times 4e^2∕h$ for clarity and the horizontal dashed lines indicate $4e^2∕h$.

Figure 4

[(a) and (b)] Magnetoresistance and Hall resistance as a function of backgate voltage, recorded at $B=8\mathrm{T}$ and $T=2\mathrm{K}$ with an ac of $100\mathrm{nA}$ rms. (a) All side gates are at $0\mathrm{V}$. The dashed lines indicate the expected positions for plateaus in the Hall resistance for single-layer graphene. (b) Different curves correspond to different voltages applied to all four side gates and are offset by $7\frac{h}{4e^2}$ for clarity. The vertical dashed lines indicate the center of the middle peak in the longitudinal resistance. The short horizontal dashed lines mark a resistance value of $\pm \frac{h}{2e^2}$ where the first plateau in the Hall resistance is expected. (c) Backgate voltage position of the charge neutrality point measured in longitudinal and Hall resistance. The solid line indicates the charge neutrality point shift as calculated by the model described in Sec. 3B (offset arbitrarily chosen).