Interedge backscattering in buried split-gate-defined graphene quantum point contacts

Quantum Hall effects offer a formidable playground for the investigation of quantum transport phenomena. Edge modes can be deflected, branched, and mixed by designing a suitable potential landscape in a two-dimensional conducting system subject to a strong magnetic field. In the present work, we demonstrate a buried split-gate architecture and use it to control electron conduction in large-scale single-crystal monolayer graphene grown by chemical vapor deposition. The control of the edge trajectories is demonstrated by the observation of various fractional quantum resistances, as a result of a controllable interedge scattering. Experimental data are successfully modeled both numerically and analytically within the Landauer-Büttiker formalism. Our architecture is particularly promising and unique in view of the investigation of quantum transport via scanning probe microscopy, since graphene constitutes the topmost layer of the device. For this reason, it can be approached and perturbed by a scanning probe down to the limit of mechanical contact.

Figure 1

(a) Optical micrograph of the device. The four split gates are indicated as SG1 to SG4. SG2 and SG4 were investigated in the present work. The contacts that were used to measure the longitudinal voltage drop $V_{xx}$ across SG2 and SG4 are indicated, as well. The transversal voltage drop $V_{xy}$ was measured in the center of the device. (b) Cross-sectional sketch of the device.

Figure 2

(a) Measured longitudinal resistance $R_{xx}$ as a function of $V_{BG}$ and $V_{SG}$. Note the logarithmic scale for $R_{xx}$. (b) $R_{xx}$ as a function of $V_{BG}$ for fixed $V_{SG}=0$ V. (c) $R_{xx}$ as a function of $V_{SG}$ for $V_{BG}=0$ V, 15 V, and 30 V. Split-gate width 400 nm, $B=0$ T, $T=270$ mK.

Figure 3

Longitudinal ($R_{xx}$) and transverse ($R_{xy}$) resistance in the bulk of the graphene flake as a function of back-gate voltage $V_{BG}$ at $B=10$ T and $V_{SG}=0$ V.

Figure 4

(a) Longitudinal resistance $R_{xx}$ (in units of $h/e^2$) measured across the 400-nm-wide QPC as a function of the back-gate ($V_{BG}$) and split-gate ($V_{SG}$) voltage biases at $B=10$ T. (b) Analytical results obtained with Eqs. (3, 4, 5, 6) reproduce well the observed resistance plateaus. The arrows in (a) indicate where the cross sections of Figs. 9 and 9 were taken.

Figure 5

Sketch of the device geometry adopted in the Landauer-Büttiker model. Four contacts and regions with three distinct filling factors ${\nu }_{BG}, {\nu }_{SG}$, and ${\nu }_{QPC}$ are assumed.

Figure 6

Edge connectivities as a function of the three filling factors ${\nu }_{BG}, {\nu }_{SG}$, and ${\nu }_{QPC}$. Each sketch implies a different edge propagation, splitting, and equilibration configuration and thus a different analytical formula for the longitudinal resistivity. Configurations with opposite chiralities are not reported explicitly. Black arrows indicate the currents for each of the spin-degenerate available transport modes and their chirality. In order to provide a specific example, the green numbers indicate one of the possible filling factor combinations giving rise to the sketched connectivity scheme. In panels (b), (c), and (d), edge equilibration in co-propagating modes plays a crucial role: the purple squares indicate the points where current branches. In the model current is assumed to partition equally on all available channels; i.e., edge modes are assumed to completely equilibrate before the branching.

Figure 7

Schematic drawing of the studied system. (a) Scheme of the nanoribbon with four horizontal zigzag leads, two vertical armchair Büttiker probes, and four Büttiker probes connected to the interior of the nanoribbon near the $n-p$ junctions, highlighted in orange. Green lines show schematic isolines of the potential. (b) The computational box for the Laplace problem. Voltage $V_{BG}$ is applied to the back gate (bottom of the computational box), and $V_{SG}$ to the split gate colored in blue. The dimensions used for the simulation are $d_{{\text{SiO}}_2}=27.6$ nm, $d_{PMMA}=9.8$ nm, $d_{\mathrm{vac}}=208.6$ nm.

Figure 8

(a) Simulated longitudinal resistance (in units of $h/e^2$) as a function of $V_{BG}$ and $V_{SG}$. (b) Resistance values calculated according to Eqs. (3, 4, 5, 6) (in units of $h/e^2$). The nonlinear shape of the boundaries of regions at fixed ${\nu }_{BG}$ is caused by the limited size of the computational box which cause the potential in the leads to slightly depend on $V_{SG}$.

Figure 9

The cross sections of the experimental [(a), (b)] and the simulated [(c), (d)] data, indicated by arrows in Figs. 4 and 8. The color of the arrow corresponds to the line color. Dashed lines show the model values of resistance given by Eqs. (3, 4, 5, 6). (a) ${\nu }_{SG}=2,V_{SG}=8$ V, (b) ${\nu }_{BG}=-6,V_{BG}=-13$ V, (c) ${\nu }_{SG}=2,V_{SG}=0.135$ eV, (d) ${\nu }_{BG}=-6,V_{BG}=-0.39$ eV.