Abstract
Exploiting the lightlike properties of carriers in graphene could allow extreme nonclassical forms of electronic transport to be realized^{1,2,3,4,5,6,7,8}. In this vein, finding ways to confine and direct electronic waves through nanoscale streams and streamlets, unimpeded by the presence of other carriers, has remained a grand challenge^{9,10,11,12}. Inspired by guiding of light in fibre optics, here we demonstrate a route to engineer such a flow of electrons using a technique for mapping currents at submicron scales. We employ realspace imaging of current flow in graphene to provide direct evidence of the confinement of electron waves at the edges of a graphene crystal near charge neutrality. This is achieved by using superconducting interferometry in a graphene Josephson junction and reconstructing the spatial structure of conducting pathways using Fourier methods^{13}. The observed edge currents arise from coherent guidedwave states, confined to the edge by band bending and transmitted as plane waves. As an electronic analogue of photon guiding in optical fibres, the observed states afford nonclassical means for information transduction and processing at the nanoscale.
Main
Electrons in Dirac materials such as graphene can be manipulated using external fields that control electron refraction and transmission in the same way that lenses and optical elements manipulate light^{1,2,6,7}. Several of the key ingredients, including phasecoherent Klein transmission and reflection^{3,4,5}, ballistic transport^{8} and transverse focusing on micrometre scales^{14}, have already been established. One promising yet unexplored direction, which we investigate here, is the quasionedimensional confinement of electrons in direct analogy to refractionbased confinement of photons in optical fibres. Electronic guided modes formed by a line gate potential, although discussed in the literature^{9,10,11,12}, have so far evaded direct experimental realization. Extending the fibreoptic techniques to the electronic domain is key to achieving control of electron waves at a level comparable to that for light in optical communication systems.
Rather than pursuing the schemes discussed in refs 9,10,11,12, here we explore modes at the graphene edges. The atomically sharp graphene edges provide a natural vehicle for band bending near the boundary, which then confines the electronic waves in the direction transverse to the edge. The resulting guided ‘fibreoptic’ modes are situated outside the Dirac continuum (see Fig. 1a, b), propagating along the crystal edge as plane waves and decaying into the bulk as evanescent waves. This approach to carrier guiding is particularly appealing because of the ease with which band bending at the graphene edge can be realized, as well as because there is no threshold for fibreoptic states to occur: they are induced by an edge potential of either sign, positive or negative, no matter how weak (see discussion below and in the Supplementary Methods). The presence of such guided modes enhances the density of currentcarrying states at the edge. The effects of electron confinement and guiding are strongest near charge neutrality, where the edge potential is unscreened, whereas uniform behaviour is recovered away from neutrality (see Fig. 1c and Supplementary Fig. 1).
The edge currents associated with guided states, anticipated at zero magnetic field, have so far eluded experimental detection owing to the challenge of imaging current with submicron spatial resolution. In particular, scanning tunnelling spectroscopy (STS) images the density of states but not the current flow^{15,16}, whereas macroscopic conductivity cannot distinguish the edge and bulk contributions^{17,18}. With this motivation, we developed a technique to spatially image electric current pathways and applied it to highmobility graphene. We employ superconducting quantum interferometry in a graphene Josephson junction to reconstruct the spatial structure of the electronic states which transmit supercurrent, which allows edge and bulk contributions to be disentangled^{13,19}.
Our approach employs gated Josephson junctions consisting of graphene coupled to superconducting titanium/aluminium electrodes (Fig. 1d). A gate electrode is used to tune the carrier density, n, in the graphene. To access the intrinsic properties of graphene at densities near charge neutrality, flakes are isolated from substrateinduced disorder through placement on thin hexagonal boron nitride (hBN) substrates^{20}. A total of four bilayer devices and one monolayer device are investigated, BL1, BL2, BL3, BL4 and ML1, all of which exhibit similar behaviour (Supplementary Table 1). Measurements of the a.c. voltage drop dV across the junction in response to an a.c. current modulation dI were conducted using lockin techniques in a dilution refrigerator at 10 mK, well below the critical temperature of Al. Figure 1e–h shows transport data from one of the bilayer devices. On sweeping the d.c. current bias I_{d.c.}, a sharp transition in resistance between dissipationless and normal metal behaviour appears at a critical current I_{c}, a transport signature of the Josephson effect (Fig. 1e, f).
We obtain realspace information by applying a magnetic flux ϕ through the junction area, which induces a positiondependent superconducting phase difference parallel to the graphene/contact interface^{21}. As a result, the critical current I_{c} exhibits interference fringes in the magnetic field B (Fig. 1e). The measured interference patterns feature welldefined nodes, which indicates the absence of field inhomogeneity such as that due to vortices^{22}. The critical current I_{c} can be expressed as the magnitude of the complex Fourier transform of the current density distribution J(x), providing a simple and concise description of our system. That is, , where where ϕ_{0} = h/2e is the flux quantum, h is Planck’s constant, e is the elementary charge, W is the width of the flake (Fig. 1d). Here, following the conventional treatment for wide junctions such that L ≪ W, where L is the distance between contacts, we ignore the y dependence. The spatial distribution of supercurrent thus dictates the shape of the interference pattern^{13,21,23}.
The results obtained with this technique show strikingly different behaviour at high and low carrier densities. We observe the conventional uniformcurrent behaviour at high density, I_{c}(B)/I_{c}(0) ∼ sin(πϕ/ϕ_{0})/(πϕ/ϕ_{0}), which mimics singleslit Fraunhofer diffraction (Fig. 1e). Defining features of such interference include a central lobe of width 2ϕ_{0} and side lobes with period ϕ_{0} and amplitude decaying as 1/B. However, near the Dirac point, our results exhibit a striking departure from this picture and show a twoslit ‘SQUIDlike’ interference (Fig. 1f)^{24}. Such behaviour arises when supercurrent is confined to edge channels and is characterized by slowly decaying sinusoidal oscillations of period ϕ_{0}. Importantly, these two regimes are easily distinguishable without much analysis by the width of the central lobe, which is twice as wide for the uniform case as for the case of edge flow.
The realspace current distribution can be obtained by inverting the relation in equation (1) with the help of the Fourier techniques of Dynes and Fulton^{13} (see Supplementary Methods). The resulting current density map reveals strong confinement of supercurrent to the edges of the crystal near the Dirac point (Fig. 1h), a robust experimental feature seen in all five devices. The width of the edge channel, extracted quantitatively from Gaussian fits, is of the order of the electron wavelength (∼200 nm) and consistent across multiple samples. This value is probably an upper bound because the peak width is manifested in the decay envelope of the interference pattern; external factors that suppress the critical current amplitude at high B, such as thermal activation of quasiparticles and suppression of the Al superconducting gap, may also contribute to peak broadening. At high electron density, a conventional Fraunhoferlike behaviour is recovered (Fig. 1e), suggesting a uniform distribution of supercurrent (Fig. 1g). A more numerically expensive Bayesian estimation of the current distribution produces current maps and standard error estimates that agree with the Fourier techniques (see Supplementary Methods and Supplementary Fig. 2).
By tuning carrier concentration with a gate electrode, our measurements reveal coexistence of edge and bulk modes at intermediate densities. This is illustrated in Fig. 2, for a monolayer graphene device. The SQUIDlike quantum interference at charge neutrality, which is similar to that for bilayer graphene in Fig. 1, gives the spatial image of supercurrent that confirms edgedominated transport (Fig. 2a–c, f). As density is increased, bulk current flow increases monotonically and crosses over to mostly uniform flow across the sample (Fig. 2d), signified by conventional Fraunhoferlike interference at high electron density (Fig. 2e). To track the evolution of edge and bulk currents with density, line cuts of the corresponding contributions are provided in Fig. 2f. Notably, the gate voltage corresponding to the charge neutrality point, identified as a dip in the current amplitude for edge and bulk, appear at similar carrier density values, which indicates the absence of edge doping in our system.
Similarly, we systematically explore the interplay between edge and bulk flow in bilayer graphene (Fig. 3). As the Fermi energy approaches the Dirac point from the hole side, the bulk contribution is suppressed faster than the edge contribution, leading to emergence of robust edge currents near zero carrier density (Fig. 3a, b). In this device, current distributions are not plotted at the immediate Dirac point owing to suppression of proximityinduced superconductivity at high normal state resistances. We note that the range in hole density over which the bulk contribution is recovered varies in different devices. Further, application of an interlayer electric field E breaks crystal inversion symmetry and induces a bandgap^{25,26}, manifested as a gatetunable insulating state at the Dirac point (Fig. 3c, d). In this regime, conductance is mediated by edge currents that enclose the bulk, even in the presence of a fieldinduced gap (Fig. 3e, f).
In both the monolayer and the bilayer cases, raw interference near the Dirac point and at high electron concentration exhibit the salient features that distinguish edgedominated from bulkdominated transport, including a width of φ_{0} versus 2φ_{0} of the central lobe, as well as Gaussian versus 1/B decay of the lobe amplitudes for low and high densities, respectively.
As a simple model of electronic fibreoptic states we consider massless Dirac particles in graphene monolayer in the presence of a line potential: where p is momentum, σ_{i} are pseudospin Pauli matrices and v ≈ 10^{6} m s^{−1}. We seek planewave solutions of the Schrödinger equation, ψ(x, y) = e^{iky}φ(x), where k is the wavevector component along the line and φ(x) is a twocomponent spinor wavefunction depending on the transverse coordinate. This problem can be tackled by a matrix gauge transformation , which eliminates the potential V (x) and generates a mass term in the Dirac equation. Namely, U(x) = ${\text{e}}^{i\theta (x){\sigma}_{x}}$, with yields As a simple example, we consider the case of an armchair edge, for which the problem on a half plane for carriers in valleys K and K′ is equivalent to the problem on a full plane for a single valley. Applying the above method to a potential localized in an interval −d < x < d and focusing on the longwavelength modes such that kd ≪ 1, we can use a step approximation θ(x) ≈ (u/2v) sgn (x) with the parameter We arrive at the seminal Jackiw–Rebbi problem for a Dirac equation with a mass kink where , m(x) = −ksin(u/v) sgn (x). This problem can be solved directly and explicitly^{27}, yielding guidedwave states as products of the zeromode state found from for and the planewave factors e^{iky}. The energies of these states are simply with the sign η = sgn (m(0^{+}) − m(0^{−})). This gives a linear dispersion , with the velocity As , for each k the energies of these states lie outside the bulk continuum ε ≥ ℏv k (see Fig. 1a). Decoupling from the bulk states ensures confinement to the region near the x = 0 line. The connection with the theory of zero modes renders robustness to our confinement mechanism. Similar guidedwave states are obtained for an edge potential in graphene bilayer (see Fig. 1b).
In Fig. 4, we compare supercurrent density measurements with a theoretical prediction for density of states. Current density traces J(x) measured in the bilayer device BL3 at different densities have a strong edge component near neutrality, gradually evolving to the bulk flow away from neutrality. Traces of the density of states, obtained from the above model, exhibit qualitatively similar behaviour (Fig. 4b). For the simulation, a delta function potential approximation was used with the bestfit value ℏu = 0.7 eV nm (see Supplementary Methods).
Another key feature borne out by the above model is the robustness of the guided states in the presence of edge disorder. Indeed, as the length scales defined by the evanescent waves are of the order of electron wavelength λ, the resulting modes are weakly confined to the edge at low carrier density. Such modes tend to decouple from the shortrange edge disorder by diffracting around it. In particular, our analysis of monolayer graphene yields a mode damping that quickly vanishes at long electron wavelengths near charge neutrality, scaling as γ(k) ∼ k^{2} (see Supplementary Methods). This resembles the behaviour of optical guided states in socalled ‘weakly guiding’ optical fibres, where a similar suppression of disorder scattering occurs due to evanescent waves diffracting around edge disorder.
One appealing aspect of the fibreoptic model is that it can naturally accommodate a wide range of different microscopic physical mechanisms discussed theoretically in the literature^{27,28,29,30,31} that may produce an edge potential. Examples include pinning of the Fermi energy to the lowenergy states due to broken A/B sublattice symmetry^{29,30,31}, density accumulation caused by dangling bonds or trapped charges at the boundaries, or electrostatics^{27,28}. The competition of these effects can produce a complex dependence of the edge potential V (x) on carrier density. Pinpointing the precise microscopic origins of the edge potential requires further study.
To the best of our knowledge, the fibreoptic modes is perhaps the simplest model fully consistent with the observations. In particular, we eliminate edge density accumulation, which can influence the edge potential and in principle also support guided edge currents. The fact that the charge neutrality points for both edge and bulk roughly coincide in density n suggests an absence of a positional charge imbalance on a large scale (Fig. 2f). In addition, edgedominated current flow is observed near the Dirac point, but not at higher densities, behaviour not expected for strong edge doping. Explanations involving electron–hole puddles are excluded by the reproducibility of edge currents with widths of the order of the electron wavelength across many samples, as well as the observation that edge currents tend to be stronger in clean samples with ballistic Fabry–Pérot interference^{32} (see Supplementary Fig. 6 for normal state characterization of the graphene). Large charge inhomogeneities across the sample would suppress Fabry–Pérot interference and are thus unlikely.
Lastly, it is widely known that the A/B sublattice imbalance for broken bonds at the edge can lead to edge modes in pristine graphene at neutrality. Such dispersing zeromode states can exist even in the absence of a line potential, forming edge modes for an atomically perfect zigzag edge^{29,30,31}. However, our simulations for a disordered edge show that these states are highly localized on the disorder length scale, and also that edge roughness induces strong scattering between the states at the boundary and in the bulk, which hinders ballistic propagation. Similarly, valley Hall currents predicted at the boundaries of a gapped bilayer due to momentumspace Berry curvature of the bands^{33,34} are eliminated because they are highly sensitive to disorder scattering at the boundaries.
Our measurements establish that edge currents are present in graphene even at zero magnetic field, near the Dirac point. The observed edge currents are linked to electronic guidedwave states formed owing to band bending at the edge. This demonstrates confinement of electron waves at a level comparable to that for light in photonic systems and defines a new mode for the transmission of electronic signals at the nanoscale. We anticipate this work will inspire more detailed investigations of boundary states in graphene and other materials. Such capabilities are also of great interest owing to the predicted topological nature of edge states along stacking domain boundaries in bilayer graphene^{35,36}.
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Acknowledgements
The authors thank O. Dial, B. Halperin, V. Manucharyan and J. Sau for helpful discussions. This work is supported by the Center for Integrated Quantum Materials (CIQM) under NSF award 1231319 (L.S.L. and O.S.) and the US DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DESC0001819 (P.J.H., M.T.A., A.Y.). Nanofabrication was performed at the Harvard Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN) supported by NSF award ECS0335765. A.R.A. was supported by the Foundation for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO/OCW). I.C.F. was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Project MUNATOP, the US–Israel Binational Science Foundation, and the Minerva Foundation.
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Affiliations
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
 M. T. Allen
 & A. Yacoby
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
 O. Shtanko
 , P. JarilloHerrero
 & L. S. Levitov
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
 I. C. Fulga
Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1 2628 CJ Delft, The Netherlands
 A. R. Akhmerov
Environment and Energy Materials Division, National Institute for Materials Science, 11 Namiki, Tsukuba Ibaraki 3050044, Japan
 K. Watanabe
 & T. Taniguchi
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Contributions
M.T.A. and A.Y. designed and fabricated the devices, performed the experiments, analysed the data, and wrote the paper. O.S. and L.S.L. developed the theoretical model of guided edge modes and wrote the paper. A.R.A. and I.C.F. performed the Bayesian analysis of the Fraunhofer patterns. P.J.H. contributed to discussions of the results and wrote the paper. K.W. and T.T. provided the hexagonal boron nitride crystals used in device fabrication.
Competing interests
The authors declare no competing financial interests.
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Correspondence to A. Yacoby.
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