Side gate influence in the QH regime
As a magnetic field B is applied perpendicular to the sample, the junction enters the QH regime. By 1.8 T, the QH effect is very well developed, and we stay at that field in Figs. 1 to 3. The influence of the side gates is substantial in this regime, since the edges of the device dominate the transport properties. Figure 1C maps the influence of the back gate and the two side gates, applied symmetrically, VSG1 = VSG2. This and subsequent measurements in this section are performed with a DC (direct current) bias of 10 nA, enough to suppress any supercurrent that may be flowing between the contacts in the QH regime. An additional, negligibly small alternating current (AC) of 50 pA is applied to measure the differential resistance with a lock-in amplifier. The large central red (high resistance) features in Fig. 1C mark the ν = ± 2 QH plateaus. Above and below these are the standard ν = ± 6 states. Only the
sequence of filling factors is visible at this field.
The regions of quantized conductance have a diamond shape, whose boundaries in the back gate direction are flat (horizontal), which means that they are not affected by the side gates. The inclined side boundaries of the red diamonds indicate that they depend both on the side gates and the back gate. These boundaries are interpreted as a line of constant carrier density along the edges of the device, nside ∝ (VSG1,2 − αVBG) = const, where α ∼ 2 is a constant determined by the relative gate efficiencies. The overall shape of the map in Fig. 1C is well reproduced by a simple electrostatic simulation, as shown in Fig. 1D.
Last, the centers of the diamond-shaped plateaus in Fig. 1C are shifted from VSG1,2 = 0 V, indicating that the “neutral” side-gate voltage is close to −1 V. This differs from the back-gate position of the charge neutrality point (3.5 V) not only in magnitude but also in polarity, indicating a carrier buildup along the edges of the junction distinct from the doping of the bulk. The side gate influence is illustrated in Fig. 1E, which demonstrates that the resistance plateaus of the device, as a function of the back gate, are better formed at VSG1,2 = − 1 V than at −3 or +1 V.
More insight into the device’s phenomenology is gained by applying the side gates independently. Figure 2A shows a resistance map of the device as a function of both side gates at VBG= 4.7 V. (Taking a VSG1 = VSG2 diagonal line in Fig. 2A would correspond to a horizontal line going through the middle of the ν = 2 diamond in Fig. 1C.) The prominent feature of Fig. 2A is a square central region with resistance quantized at R = h/2e2. When either side gate is applied beyond the plateau region, the resistance drops to a different quantized value.
(A) dV/dI map plotted versus side-gate voltages VSG1 and VSG2 at B = 1.8 T. The back-gate voltage is fixed at VBG = 4.7 V, corresponding to the bulk ν = 2 state. The numbers mark the sample conductance in units of e2/h. (B) Sample resistance measured as a function of a single side gate. Green and red curves correspond to the vertical lines in (A) at VSG1 = 0 and 3 V, respectively (with VBG = 4.7 V). The blue curve shows a similar trace with a bulk filling factor ν = − 2 (VBG = 1.5 V), sweeping VSG1 with VSG2 = 0 V. (C and D) Schematics corresponding to the green and blue curves in (B) for VSG greater than ∼2 V. Additional edge channels are created near the gate, with local filling factor ν2 = 6 (C, green region) and ν1 = 2 (D, blue region). Additional conductance is equal to 4e2/h and 2e2/h in (C) and (D), respectively, on top of the base conductance of 2e2/h, as is observed for the blue and green curves in (B). (E) Schematic of the carrier density within the graphene junction as a function of position when SG2 (1) is active (passive), akin to (C).
The observed influence of the side gates on the QH conductances is similar to the impact of local out-of-plane gates (33, 34). The fact that the features in Fig. 2A are purely horizontal or vertical shows that the influence of the two side gates is highly local: The left gate has a negligible effect on the right edge and vice versa. This negligible cross-talk is different from that typically found in samples with out-of-plane gates. Furthermore, the side gates are efficient and tune the local density by ~1011 cm−2 per volt, compared with ~7 × 1010 cm−2 per volt for the back gate. In particular, we are able to change the filling factor along either edge.
Figure 2B shows that the measured resistance drops from R = h/2e2 to R = h/6e2, if a positive side-gate voltage is applied (green curve, measured along the green line in Fig. 2A). This corresponds to ν2, the local filling factor on the side close to side gate 2 (SG2), reaching ν2 = 6 as shown schematically in Fig. 2C. The bulk filling factor remains at ν = 2, and an additional conductance of 4e2/h is contributed by the additional fourfold degenerate edge states induced near SG2. Note that in this case, the spatial separation between counter-propagating QH states in the side-gated region is less than 100 nm, as detailed further in the text. The observation of quantized resistance plateaus suggests that backscattering between these counter-propagating states is suppressed, despite their close proximity. Indeed, robust QH plateaus were previously observed in graphene nanoribbons of comparable width (35, 36).
Next, the red line of Fig. 2B demonstrates that each side gate can induce an independent ν = 6 state along its edge. Here, SG1 is fixed at 3 V; this corresponds to a local filling factor near SG1 of ν1 = 6. Before SG2 is applied, we start with resistance of h/6e2: The baseline conductance is 2e2/h, and the right edge contributes additional 4e2/h, much like at the end point of the green curve in Fig. 2B. Applying SG2 then adds an additional fourfold degenerate channel on the other edge of the sample, resulting in the drop of resistance to h/10e2, which corresponds to conductance of (2 + 4 + 4)e2/h.
Last, we tune the back gate to 1.5 V (instead of 4.7 V), resulting in a bulk filling of ν = − 2. Applying SG1 now yields a transition from R = h/2e2 to R = h/4e2 (blue curve in Fig. 2B.) The schematics in Fig. 2D shows that in this case, the side gate locally induces a QH state with an opposite filling factor of ν = 2, and the resulting plateau has a conductance of (2 + 2)e2/h. Note that here as well, counter-propagating states are created in close proximity to each other.