### 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, *V*_{SG1} = *V*_{SG2}. 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, *n*_{side} ∝ (*V*_{SG1,2} − α*V*_{BG}) = 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 *V*_{SG1,2} = 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 *V*_{SG1,2} = − 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 *V*_{BG}= 4.7 V. (Taking a *V*_{SG1} = *V*_{SG2} 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*/2*e*^{2}. When either side gate is applied beyond the plateau region, the resistance drops to a different quantized value.

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 ~10^{11} cm^{−2} per volt, compared with ~7 × 10^{10} 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*/2*e*^{2} to *R* = *h*/6*e*^{2}, if a positive side-gate voltage is applied (green curve, measured along the green line in Fig. 2A). This corresponds to ν_{2}, the local filling factor on the side close to side gate 2 (SG2), reaching ν_{2} = 6 as shown schematically in Fig. 2C. The bulk filling factor remains at ν = 2, and an additional conductance of 4*e*^{2}/*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 ν_{1} = 6. Before SG2 is applied, we start with resistance of *h*/6*e*^{2}: The baseline conductance is 2*e*^{2}/*h*, and the right edge contributes additional 4*e*^{2}/*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*/10*e*^{2}, which corresponds to conductance of (2 + 4 + 4)*e*^{2}/*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*/2*e*^{2} to *R* = *h*/4*e*^{2} (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)*e*^{2}/*h*. Note that here as well, counter-propagating states are created in close proximity to each other.

### Side gates and QH supercurrent

So far, the measurements have been performed with an applied DC bias current *I* of 10 nA to suppress any supercurrent. We now switch *I* to zero and explore the emerging superconducting features, maintaining the small AC current of 50 pA used to measure the differential resistance. Figure 3A shows a map of sample resistance versus side gates similar to that in Fig. 2A. While no supercurrent is found on top of the ν = 2 plateau, once the ν = 6 state is induced by either side gate, the sample resistance develops pronounced dips that were not present at high DC current.

Figure 3B shows the sample resistance versus bias taken at the location in Fig. 3A marked by an orange asterisk, corresponding to *V*_{SG2} = 0 V and *V*_{SG1} = 2.5 V, so that ν_{2} is close to bulk filling and ν_{1} = 6. The region of suppressed resistance flanked by peaks is characteristic of a small supercurrent washed by thermal fluctuations. Notice that when the density enhancement is induced on one side only (regions in Fig. 3A corresponding to the normal resistance of *h*/6*e*^{2}), the supercurrent features appear as horizontal/vertical lines—they depend on one side gate and do not vary with the other side gate. This confirms that the supercurrent is localized at one side of the junction.

Furthermore, the supercurrent does not vary for small changes in magnetic field (Fig. 3C), indicating that the area it encompasses does not enclose additional flux quanta for a few millitesla change in field. This observation limits the distance between the counter-propagating edge channels responsible for the supercurrent to no more than ∼100 nm (see also fig. S1C). This distance is comparable to the coherence length of MoRe, which facilitates the coupling of the edge states to the superconductor and explains the appearance of a supercurrent when a side gate is turned on.

The dependence of the supercurrent on magnetic field completely changes when both side gates are applied, creating supercurrents along the two edges of the sample. Figure 3D shows a map similar to Fig. 3C, but with both side gates applied (*V*_{SG1} = 3.04 V, *V*_{SG2} = 2.11 V, marked by a white asterisk in Fig. 3A). The map demonstrates a superconducting quantum interference device (SQUID)–like interference pattern with a period of 0.6 mT, close to that of the low-field Fraunhofer pattern of this junction (0.7 mT).

We explore the device as an interferometer for QH supercurrents in Fig. 4. Here, we change the field to 1 T to observe a more robust superconducting signature. Figure 4A shows the pattern of resistance oscillations in magnetic field, measured at zero applied DC bias as a function of the back gate. The period of the oscillations is found to be the same as in Fig. 3D and independent of the gate voltage. The phase of the oscillations, however, is seen to vary with gate with an approximate slope of +150*V*_{BG}/*T*.

This gradual shift of the magnetic interference pattern with the back gate is explained by the fact that the changing electron density shifts the position of the QH edge states, thereby changing the area between the supercurrents on the two sides. The phase change from an increase in density (at more positive *V*_{BG}) is compensated for by the increase in the magnetic field, indicating that the effective area of the SQUID shrinks. This behavior can be understood from the schematic in Fig. 4B, where the blue curve (lower back-gate voltage) is compared to the green line (higher back-gate voltage). The counter-propagating edge states occur on the opposite slopes of the nonmonotonic density profile close to each edge. As the overall density increases (from the blue to the green curve), the inner states move further inward, while the outer states stay relatively stationary due to the very high density gradient close to the sample edge. As a result, on average, the location of the supercurrent moves inward with increasing density.

A similar change in the interference pattern is observed when a side gate is applied (Fig. 4C). The slope of this pattern is roughly −300*V*_{SG1}/*T*. Notably, the sign of the slope in Fig. 4C is flipped compared with the one seen in Fig. 4A. Following the discussion in the previous paragraph, this slope suggests that applying the side gate may be increasing the effective area of the SQUID. This could likely be attributed to the outward shift of the outer edge state, which is more strongly influenced by the side gate than the inner edge state. The very small size of the graphene region affected by the side gate might also result in charging effects, which are known to invert the slope of fringes in QH interferometers (*37*–*39*).

Last, an additional interference pattern is revealed in Fig. 4D, which shows the ΔR, the difference in the sample resistance between 0 and 10 nA DC bias, which highlights the superconducting regions. The map is measured as a function of both side gates at *B* = 1 T and *V*_{BG} = 3.9 V. The interference is visible at the intersections of the vertical and horizontal lines corresponding to supercurrents flowing along the SG1 and SG2 edges, respectively. The interpretation of this interference pattern is similar to the discussion above, with each gate affecting the location of the edge state on its side of the device: The gates change the phase by inducing small changes of the area of the SQUID. Lines of the constant phase correspond to the situation in which the area reduction on one side is compensated by the area increase on the opposite side, so that the total area stays constant. The contours of the constant phase at the intersections of the vertical and horizontal lines have a roughly diagonal slope, indicating that the two gates have comparable efficiency—as one gate voltage is increased; decreasing the opposite gate voltage accordingly maintains roughly the same area of the SQUID. Note that these area changes are sufficient to evolve the phase difference across the junction, but too small to create noticeable changes in the magnetic field periodicity.

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