An energy gap can be opened in the spectrum of graphene reaching values as large as 0.2 eV in the case of bilayers. However, such gaps rarely lead to the highly insulating state expected at low temperatures. This long-standing puzzle is usually explained by charge inhomogeneity. Here we revisit the issue by investigating proximity-induced superconductivity in gapped graphene and comparing normal-state measurements in the Hall bar and Corbino geometries. We find that the supercurrent at the charge neutrality point in gapped graphene propagates along narrow channels near the edges. This observation is corroborated by using the edgeless Corbino geometry in which case resistivity at the neutrality point increases exponentially with increasing the gap, as expected for an ordinary semiconductor. In contrast, resistivity in the Hall bar geometry saturates to values of about a few resistance quanta. We attribute the metallic-like edge conductance to a nontrivial topology of gapped Dirac spectra.
The gapless spectra of mono- and bilayer graphene (MLG and BLG, respectively) are protected by symmetry of their crystal lattices. If the symmetry is broken by interaction with a substrate1,2 or by applying an electric field3,4, an energy gap opens in the spectrum. In BLG, its size Egap can be controlled by the displacement field D applied between the two graphene layers. Large gaps were found using optical methods5 and extracted from temperature (T) dependences of resistivity ρ at sufficiently high T (refs 6, 7, 8, 9, 10). Their values are in good agreement with theory. On the other hand, at low T (typically, below 50 K), ρ at the charge neutrality point (CNP) in gapped graphene is often found to saturate to relatively low values that are incompatible with large Egap (refs 6, 7, 8, 9, 10, 11). This disagreement is attributed to remnant charge inhomogeneity6,8,10 that results in hopping conductivity and, therefore, weakens T dependences. Alternative models to explain the subgap conductivity were proposed, too. They rely on the nontrivial topology of Dirac bands in gapped MLG and BLG12,13,14,15, which gives rise to valley-polarized currents13,14,15. Large nonlocal resistances were reported for both graphene systems at the CNP and explained by valley currents propagating through the charge-neutral bulk16,17,18. Graphene edges12,15, p–n junctions19 and stacking boundaries20 can also support topological currents. These conductive channels were suggested to shunt the insulating bulk, leading to a finite ρ. Experimentally, the situation is even more complicated because additional conductivity may appear for trivial reasons such as charge inhomogeneity induced by chemical or electrostatic doping21,22,23. Here we show that highly conductive channels appear near edges of charge-neutral graphene if an energy gap is opened in its spectrum. We tentatively attribute the edge channels to the presence of such unavoidable defects as, for example, short zigzag-edge segments12. Their wavefunctions extend deep into the insulating bulk where they sufficiently overlap to create a quasi-one-dimensional impurity band with little intervalley scattering and high conductivity. We believe that, in certain graphene devices, the localization length can be very long, comparable to typical distances between electric contacts, which effectively results in shunting the gapped bulk.
Josephson current distributions
We start with discussing behaviour observed for superconductor-graphene-superconductor (SGS) Josephson junctions. Our devices were short and wide graphene crystals that connected superconducting Nb electrodes24 (Fig. 1). Each device contained several such SGS junctions with the length L varying from 300 to 500 nm and the width W from 3 to 5 μm. To ensure highest possible quality24, graphene was encapsulated between hexagonal boron nitride (hBN) crystals with the upper hBN serving as a top gate dielectric and the Si/SiO2 substrate as a bottom gate (Fig. 1b). For details of device fabrication and characterization, we refer to ‘Methods’ section and Supplementary Note 1 and Supplementary Fig. 1. By measuring the critical current Ic as a function of perpendicular magnetic field B, the local density Js(x) in the x direction perpendicular to the supercurrent flow can be deduced25, as illustrated in Fig. 1c,d. This technique is well established and was previously used to examine, for example, edge states in topological insulators26 and wave-guided states in graphene22. In our report, we exploit the electrostatic control of the BLG spectrum to examine how Js(x) changes with opening the gap.
By varying the top and bottom gate voltages (Vtg and Vbg, respectively), it is possible to keep BLG charge neutral while doping the two graphene layers with carriers of the opposite sign (see Fig. 2a). This results in the displacement field D(Vtg,Vbg) that translates directly into the spectral gap3,4,5,6. Its size Egap(D) can be deduced not only theoretically but also measured experimentally, as discussed in section 1 of Supplementary Information Note 1. To quantify proximity superconductivity in our devices, we define their critical current Ic as the current at which the differential resistance dV/dI deviates from zero above our noise level24. With reference to Fig. 2b, Ic corresponds to the edge of the dark area outlined by bright contours. At high doping (Fermi energy >Egap) and low T, Ic is found to depend weakly on D, reaching values of a few μA μm−1, in agreement with the previous reports22,24,27. The supercurrent generally decreases with increasing junction’s resistance and becomes small at the CNP. Its value depends on Egap (Fig. 2b). Accordingly, the largest Ic in the neutral state is found for zero D (no gap) reaching ≈300 nA for the junction shown in Fig. 2. The value drops to 2 nA at D=±0.07 V nm−1, which corresponds to Egap≈7 meV. For larger gaps, Ic becomes smaller than 1 nA and could no longer be resolved because of a finite temperature (down to 10 mK) and background noise24.
We analyse changes in the interference pattern, Ic(B), with increasing D (that is, increasing Egap). At zero D, we observe the standard Fraunhofer pattern at the CNP, which is basically similar to that measured at high doping (cf. two top panels of Fig. 2c). Only absolute values of Ic are different because of different ρ, as expected24. The Fraunhofer pattern corresponds to a uniform current flow (Fig. 1c,d). In contrast, the interference pattern measured at the CNP for a finite gap is qualitatively different (see Fig. 2c; D=0.055 V nm−1). The phase of the oscillations changes by 90° and the central lobe becomes twice narrower. In addition, the side lobes no longer decay with increasing B but exhibit nearly the same amplitude (additional example in Supplementary Note 2 and Fig. 2). Such a pattern resembles the one shown schematically in Fig. 1d for the case of the supercurrent flowing along edges. The only difference with Fig. 1d is that in our case the central lobe remains higher than the others. For quantitative analysis, we calculated the inverse fast Fourier transform (FFT) of Ic(B), which yielded26 the current distributions Js(x) shown in Fig. 2d. The supercurrent is progressively pushed towards device edges with increasing the gap. This is already visible for D=0.025 V nm−1 but further increase in D suppresses the bulk current to practically zero, within the experimental accuracy of our FFT analysis (Fig. 2d and Supplementary Fig. 3). The accuracy is limited by a finite range of B in which the interference pattern could be detected (Supplementary Note 3).
For completeness, we have also studied SGS junctions that were fabricated using monolayer graphene placed on top of hBN and aligned along its crystallographic axes. Such alignment (within 1–2°) results in opening of a gap of ≈30 meV at the main CNP1,2, and secondary CNPs appear for high electron and hole doping1,2,16. Unlike for the case of BLG, Egap cannot be changed in situ in MLG devices, but one can still compare interference patterns for neutral and doped states of the same SGS junction and, also, use nonaligned junctions as a reference. Figure 3a,b shows typical behaviour of Ic as a function of carrier concentration n for SGS devices made from gapped (aligned) and gapless (nonaligned) MLG. In the gapped device, the supercurrent is suppressed not only at the main CNP but also at secondary CNPs. For all electron and hole concentrations away from the CNPs, both devices exhibit the standard Fraunhofer pattern indicating a uniform supercurrent flow (cf. top panels of Fig. 3c,d). The same is valid at the CNP in gapless graphene (Fig. 3d,f). In contrast, for gapped MLG, the interference pattern at the main CNP undergoes significant changes such that the phase and period of oscillations in Ic change (Fig. 3c; bottom panel), somewhat similar to the behaviour of gapped BLG at the CNP. Quantitative analysis using FFT again shows that, in gapped MLG, the supercurrent flows predominantly along graphene edges for n<±5 × 1010 cm−2 (Fig. 3e). The figure seems to suggest a shift of conductive channels from edges into the interior. This shift originates from the increase in the Fraunhofer period at the CNP in Fig. 3c and corresponds to a decrease in the junction’s effective area. However, we believe that this shift may arise from non-uniform doping along the current direction. Our MLG devices do not have a top gate and this allows doping by metal contacts to extend significantly (tens of nm) inside the graphene channel28, which reduces the effective length of the junction.
We emphasize that the observed redistribution of supercurrents towards edges is an extremely robust effect observed for all eight gapped graphene junctions we studied and in none without a gap (more than 10)24. In principle, one can imagine additional electrostatic and/or chemical doping near graphene edges21,22,23 (Supplementary Note 4), which would enhance their conductivity and, hence, favour local paths for supercurrent. This mechanism disagrees with the fact that edge supercurrents appeared independently of the CNP position as a function of gate voltage (residual doping in our devices varied from practically zero to <1011 cm−2) and were observed for devices with the top gate being only a few nm away from the graphene plane. The latter facilitates a uniform electric field distribution (Supplementary Fig. 4). Chemical doping at graphene edges was previously reported in non-encapsulated21 and, also, encapsulated but not annealed devices23. All our devices were encapsulated and thoroughly annealed, and some of them had edges that were fully covered by top hBN rather than exposed to air (Supplementary Figs 5 and 6). We also note our Josephson experiments yielded similar supercurrent densities at BLG edges, even in the case where the two edges were fabricated differently (one is etched as discussed above and the other cleaved and covered with hBN; see Fig. 1a). The latter observation in particular indicates little external doping along the edges. Importantly, we have found no evidence for enhanced transport along edges of similar but gapless graphene devices. To this end, we refer, for example, to Fig. 3e,f. In the gapped MLG device, near-edge Js reaches ≈100 nA μm−1. Such supercurrents would certainly be visible in the distribution profile of the non-gapped graphene at the CNP in Fig. 3f. All the above observations point at a critical role of the presence of the gap in creating local edge currents.
While providing important insights about the current flow, Josephson interference experiments are limited to small Egap such that junction’s resistance remains well below 1 MOhm allowing superconducting proximity. To address the situation for the larger gaps accessible in BLG devices, we compare their normal transport characteristics in the Corbino and Hall bar geometries. Because the Corbino geometry does not involve edges, such a comparison has previously been exploited to investigate the role of edge transport (for example, in the quantum Hall effect29). A number of dual-gated BLG devices such as shown in Fig. 4a were fabricated and examined over a wide range of D and T. Our experiments revealed a striking difference between ρ measured in the two geometries. In the Corbino geometry, ρ at the CNP rises exponentially with D and its value is limited only by a finite dielectric strength of ≈0.7 V nm−1 achievable for our hBN (Fig. 4b) and, at low T, by leakage currents. In contrast, in the Hall bar geometry, ρ at the CNP saturates at D as low as <0.2 V nm−1, reaching only a few tens of kOhms at all T (Fig. 4c). This disparity in the behaviour of the Hall bar and Corbino devices was observed under the same measurement conditions and despite the same or higher homogeneity in the former devices (Supplementary Note 5 and Supplementary Fig. 7). The profound difference unambiguously points at a finite conductivity caused by the presence of graphene edges, in agreement with the conclusions achieved from our Josephson experiments.
Another noteworthy distinction between the two geometries is their temperature dependences at the CNP. For T above 100 K, both Corbino and Hall bar devices exhibited the same activation behaviour ρ∝exp(Egap/2kBT) as expected for a semiconductor with the gap Egap (Fig. 4d). Our measurements over a wide range of D yielded Egap[meV]≈100 × D[V nm−1], in quantitative agreement with theory and previous reports5 (inset of Fig. 4d). At lower T, resistivity of the Corbino devices continued growing and is well described by hopping conductivity that may involve both nearest neighbour and variable range hopping6,8,9,10 (Fig. 4d). On the other hand, ρ(T) found using the Hall bars rapidly saturated below 100 K to values of a few resistance quantum RQ=h/e2 and changed little (by <30%) with decreasing T down 2 K. The saturation behaviour is similar to that observed for conductance along a p–n junction in oppositely biased BLG19, and along walls separating BLG domains with AB and BA stacking20.
Two possible scenarios for shunting the insulating state of gapped graphene have previously been put forward. Both rely on nontrivial topology of the gapped Dirac spectrum. One of them considers electronic states due to short zigzag segments15 that may be present even at relatively random edges12. Although these states decay exponentially into the gapped bulk, their penetration length ξ is very long with respect to the lattice constant a. For MLG and BLG, ξ can be estimated as ≈ħv/Egap and ħ/, respectively, where ħ is the reduced Planck constant, v the Fermi velocity in MLG and m the effective mass in BLG. For our typical gaps, ξ is about 10–20 nm, much larger than a. This suggests that wavefunctions of isolated zigzag states should strongly overlap inside the bulk creating a quasi-one-dimensional band. Moreover, because ξ/a≫1, the wavefunctions mostly reside in the bulk where there are little defects, which ensures that impurity bands are effectively protected against backscattering. The situation resembles the modulation doping used to achieve high-carrier mobilities in semiconductor quantum wells. The observed saturation of ρ to ∼RQ and the long-range nonlocal resistance reported previously16,17,18 imply that the mean free path along the quasi--one-dimensional channels can reach a micrometre scale for high-mobility graphene. Although numerical simulations12 yielded zero-T localization lengths at least an order of magnitude shorter than this scale, localization in the edge channels may be suppressed by a finite T and electron–electron interactions that are prominent especially in low-dimensional conductors. Such delocalization effects have so far not been investigated theoretically. The invoked edge channels would be consistent with our experimental observations. Obviously, the mean free path can vary from sample to sample and strongly depend on fabrication procedures, which may explain only weakly saturating behaviour that was reported in some gapped graphene devices16,17,19. In addition, there is a complementary scenario that also relies on the nontrivial topology of the gapped Dirac spectra but may not require zigzag segments. The valley Hall effect is inherent to gapped graphene and generates valley currents that flow perpendicular to applied electric field13,16. If injected from electric contacts into the gapped bulk, they are expected to become squeezed towards weakly conductive edges, similar to what is known for the case of the quantum Hall effect30 and in agreement with recent simulations for gapped MLG14. Lastly, let us mention another relevant suggestion that a weak confining potential at graphene edges may guide electronic states over large distances, independently of its strength22,31. In the latter scenario, an enhanced edge conductance is expected irrespectively of the gap size, which seems to contradict our experimental observation that there is little enhancement of near-edge supercurrent in the absence of the gap.
Our results can explain low apparent resistivity often observed for charge-neural gapped graphene at low temperatures, especially in devices made from high-quality graphene in which the bulk is expected to contribute little to either hopping conductivity or backscattering of edge modes5,6,7,8,9,10,11,19. Further experiments and theory are needed to distinguish between the possible scenarios described above and elucidate the nature of the reported edge conductance.
Mono- or bilayer graphene crystals were encapsulated between hBN crystals (typically, ≈30 nm thick) using the dry transfer technique as detailed previously32. The hBN-graphene-hBN stack was assembled on top of an oxidized Si wafer (300 or 90 nm of SiO2) and then annealed at 300 °C in a forming gas (Ar–H2 mixture) for 3 h. As the next step, we used the standard electron-beam lithography to create a poly (methyl methacrylate) (PMMA) mask that defined contact regions. Reactive ion etching (Oxford Plasma Lab 100) was employed to make trenches in the heterostructure through the mask. We used a mixture of CHF3 and O2, which provided easy lift-off of PMMA, so that metal contacts could be deposited directly after plasma etching. This also allowed us to minimize contamination of the exposed graphene edges24. After this, for BLG devices, another metal film (typically, Au/Cr) was deposited on top of the heterostructure to serve as the top gate. To avoid the edges of graphene extending out of the metal gate, the latter is used as a part of the final etch-mask; the uncovered graphene between the contacts and the gate is protected by a second PMMA mask, allowing the metal gate to extend slightly at the crucial edge locations. For the Hall bar geometry, we often used an additional hBN crystal to cover the hBN-graphene-hBN stack after plasma etching, which allowed the metal film for the top gate to go over exposed graphene edges without touching them. To provide the central contact in Corbino devices, we used air bridges33. In some of our Josephson devices, graphene was not etched but made directly from cleaved crystals selected to have a strip-like shape. In this case, graphene edges were not exposed but fully encapsulated in hBN. Similar transport and Josephson behaviour was found in all cases, independent of the variations in fabrication procedures.
All electrical measurements were carried out in a He3 cryostat (Oxford Instruments) for T down to 0.3 K and, for lower T, in a dilution refrigerator with the base temperature of 10 mK (BlueFors Cryogenics). The differential resistance was measured in a quasi-four-terminal configuration (two superconducting leads for driving the current and the other two for measuring voltage) using a low-frequency lock-in technique. All electrical connections to our devices passed through a cold RC filter (Aivon Therma) placed close to the sample and additional AC filters were used outside the cryostats. At large displacement fields, our Corbino devices exhibited high resistivity such that the lock-in technique became inappropriate. In this case, we used dc measurements.
The data that support the findings of this study are available from the corresponding author upon request.
How to cite this article: Zhu, M. J. et al. Edge currents shunt the insulating bulk in gapped graphene. Nat. Commun. 8, 14552 doi: 10.1038/ncomms14552 (2017).
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We acknowledge support from EPSRC - EP/N010345/1, and from the European Research Council ARTIMATTER project - ERC-2012-ADG.
Supplementary Figures, Supplementary Notes and Supplementary References