Abstract
Van der Waals heterostructures display numerous unique electronic properties. Nonlocal measurements, wherein a voltage is measured at contacts placed far away from the expected classical flow of charge carriers, have been widely used in the search for novel transport mechanisms, including dissipationless spin and valley transport1,2,3,4,5,6,7,8,9, topological charge-neutral currents10,11,12, hydrodynamic flows13 and helical edge modes14,15,16. Monolayer1,2,3,4,5,10,15,16,17,18,19, bilayer9,11,14,20 and few-layer21 graphene, transition-metal dichalcogenides6,7 and moiré superlattices8,10,12 have been found to display pronounced nonlocal effects. However, the origin of these effects is hotly debated3,11,17,22,23,24. Graphene, in particular, exhibits giant nonlocality at charge neutrality1,15,16,17,18,19, a striking behaviour that has attracted competing explanations. Using a superconducting quantum interference device on a tip (SQUID-on-tip) for nanoscale thermal and scanning gate imaging25, here we demonstrate that the commonly occurring charge accumulation at graphene edges23,26,27,28,29,30,31 leads to giant nonlocality, producing narrow conductive channels that support long-range currents. Unexpectedly, although the edge conductance has little effect on the current flow in zero magnetic field, it leads to field-induced decoupling between edge and bulk transport at moderate fields. The resulting giant nonlocality at charge neutrality and away from it produces exotic flow patterns that are sensitive to edge disorder, in which charges can flow against the global electric field. The observed one-dimensional edge transport is generic and nontopological and is expected to support nonlocal transport in many electronic systems, offering insight into the numerous controversies and linking them to long-range guided electronic states at system edges.
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Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
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The COMSOL simulation codes used in this study are available from the corresponding authors on reasonable request.
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Acknowledgements
We thank M. E. Huber for the SOT readout system, S. Grover for the data acquisition setup, and G. Zhang, I. V. Gornyi, Y. Gefen and A. Uri for discussions. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation programme (grant nos 785971 and 786532), by the Israel Science Foundation (ISF) (grant nos 921/18 and 994/19), by the Sagol Weizmann–MIT Bridge Program, by the German–Israeli Foundation for Scientific Research and Development (GIF) grant no. I-1505-303.10/2019, and by Lloyd’s Register Foundation. E.Z. acknowledges the support of the Andre Deloro Prize for Scientific Research and Leona M. and Harry B. Helmsley Charitable Trust grant 2018PG-ISL006.
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Contributions
A.A.-S., A.M. and E.Z. conceived the experiments. D.J.P. and A.K.G. provided the studied devices. A.M. and A.A.-S. carried out the measurements and data analysis. K.B. and Y.M. fabricated the SOTs and the tuning fork feedback. T.H., A.A.-S. and L.S.L. developed the analytic models. A.A.-S. performed the numerical simulations. A.A.-S., A.M., T.H., E.Z., L.S.L and A.K.G. wrote the manuscript with contributions from the rest of the authors.
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Extended data figures and tables
Extended Data Fig. 1 Nonlocal transport measurements.
a, b, The two-probe resistance R2p (right axis) and the nonlocal resistances RNL and \({ {\mathcal R} }_{{\rm{NL}}}={R}_{{\rm{NL}}}/{R}_{{\rm{2p}}}\) (left axis) at B = 5 T (a) and 3 T (b) for negative Vbg at T = 4.2 K using an a.c. voltage bias V0 = 9.0 mV. The RNL is normalized by RNL(Vbg = 0) of 220 kΩ at 5 T (a) and 106 kΩ at 3 T (b). The \({ {\mathcal R} }_{{\rm{NL}}}\) emphasizes the giant nonlocality at larger |Vbg| as compared to RNL which drops much faster with |Vbg| due to the trivial drop in ρxx(|Vbg|). The R2p shows quantization at quantum Hall plateaus as indicated by the dashed lines. c, \({ {\mathcal R} }_{{\rm{NL}}}\) versus Vbg at B = 1, 1.2, and 1.4 T with dashed guide-to-the-eye envelope curves. d, \({ {\mathcal R} }_{{\rm{NL}}}\) data with envelopes of 1 T and 1.2 T data normalized to that of 1.4 T illustrating that the nonlocal mechanism is described by a smooth and continuous envelope function.
Extended Data Fig. 2 Scanning gate microscopy and simulations away from charge neutrality.
a, b, Simultaneous scanning gate imaging of \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{L}}}=({V}_{5}-{V}_{6})/{V}_{0}\) (a) and \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{R}}}=({V}_{3}-{V}_{4})/{V}_{0}\) (b) at B = 5 T, Vbg = −1.4 V and Vtg = 8 V. The graphene device is biased with a voltage V0 = 5.5 mV and the SOT is scanned over a large fraction of the sample area in Fig. 1a. At this Vbg we estimate the decay length of the edge current to be λ ≈ 9W (based on Extended Data Fig. 3b and not taking into account the suppression of the edge currents by the edge disorder). Note the large suppression of \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{R}}}\) by the tip along the top-right edge (blue) and a hardly visible suppression at few locations along the bottom-right edge (light red), in contrast to Fig. 3a, which shows a more symmetric suppression close to the CNP. c, d, Numerical simulations of \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{L}}}\) (c) and \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{R}}}\) (d) scanning gate signal for λ ≈ 9W (η = 0.15, Vbg = −1.4 V, and vtg = 2) with disordered edge accumulation along the top-right edge (with the same disorder function f(x) as in Fig. 3l) and bottom-left edge, and uniform accumulation along the rest of the edges. The tip-induced suppression of \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{R}}}\) is visible predominantly along the top-right edge (blue) and of \({{\mathscr{V}}}_{{\rm{NL}}}^{{\rm{L}}}\) along the bottom-left edge, consistent with experimental data in a, b.
Extended Data Fig. 3 Edge accumulation ratio η and nonlocal decay length λ behaviour.
a, Plot of the calculated edge accumulation ratio η(Vbg) for hole doping with Pe = 1.8 × 108 m−1 and pdis = 2.9 × 1010 cm−2 in the limit of w = 0. b, The corresponding nonlocal decay length λ given by equation (1) versus Vbg at B = 0, 1 and 5 T. The dashed line shows λ/W = 1/π in the ohmic regime and the dotted line shows the distance to our nonlocal contacts x/W = 2.85. c, The corresponding calculated \({ {\mathcal R} }_{{\rm{NL}}}({V}_{{\rm{bg}}})\) (dashed) using λ(Vbg) from b, along with the experimental \({ {\mathcal R} }_{{\rm{NL}}}({V}_{{\rm{bg}}})\) at B = 0, 1 and 5 T from Fig. 1e. d, Calculated λ/W versus η, as in Fig. 1b but on a log–log scale using the 2D analytic expressions (blue) along with the 1D results (dotted red). The coloured region shows the span of η(Vbg) in a. e, COMSOL numerical calculation of η = |2Pe/(pbW)| versus Vbg with edge charge accumulation Pe arising due to combined effects of electrostatic gating of the graphene by the backgate potential Vbg and of the negatively charged impurities with line density Nimp = 0.18 nm−1 (blue) and 1 nm−1 (red) along the edges. A pronounced asymmetry is observed with a faster decay of η and of the corresponding nonlocality upon n bulk doping. η vanishes at a specific value of positive Vbg at which hole edge doping due to Nimp is compensated by electron edge doping due to electrostatic gating by Vbg. At larger Vbg both the edges and the bulk become n-doped (dashed curves). For the red and blue curves we have used pdis = 0, which results in diverging η at the CNP due to vanishing of pb at Vbg = 0. The green curve shows η(Vbg) for the case of pure electrostatic gating (Nimp = 0) with pdis = 2.9 × 1010 cm−2. In this case, the transport is local at the CNP with η = 0 because Pe vanishes at Vbg = 0, whereas pb = pdis is finite.
Extended Data Fig. 4 Analytic solution of nonlocal transport in infinite strip with edge accumulation.
a, Schematic drawing of the sample consisting of a bulk strip of width W described by conductivity tensor σ and two edge strips of width w/2 with conductivity ησW/w. All the analytic solutions are derived in the limit w → 0. b, Schematic drawing of the current flow at elevated field in presence of edge accumulation. The current flows from the source predominantly along the top-right edge in the \(\hat{x}\) direction (red) and gradually leaks into the bulk where it reverts its direction and flows in the \(-\hat{x}\) direction (green) to the left side of the sample. The current is then gradually absorbed by the bottom edge where it reverts its direction again and flows to the drain (blue). Note that the edge currents (red and blue) flow ‘downstream’ along the potential drop (red and blue arrows in c), whereas the x component of the bulk current, \({I}_{{\rm{bulk}},\hat{x}}(x)\), flows against the potential drop (green arrow in c). c, Calculated normalized potentials \({{\mathscr{V}}}_{{\rm{top}}}(x)\) (red) and \({{\mathscr{V}}}_{{\rm{bot}}}(x)\) (blue) along the top and bottom edges and \({{\mathscr{V}}}_{{\rm{bulk}}}(x,y=0)\) (green) for the case of η = 0.2 and θ = 26 (λ = 2.4W). d, The corresponding normalized currents \({ {\mathcal I} }_{{\rm{top}}}(x)\) (red), \({ {\mathcal I} }_{{\rm{bot}}}(x)\) (blue), and the x component of the bulk current integrated over the strip width, \({ {\mathcal I} }_{{\rm{bulk}},\hat{x}}(x)\), (green) showing the flow pattern described schematically in b.
Extended Data Fig. 5 Inversion of the Hall voltage owing to edge charge disorder.
Numerical simulations of the normalized potential \({\mathscr{V}}\) (left column) and of the magnitude of the normalized current density \({\mathscr{J}}\) (right column) at B = 5 T and λ = 30W (η = 0.58, Vbg = −0.2 V). a, b, Uniform edge accumulation giving rise to strong nonlocal transport and positive Hall voltage VH as defined in a (potential at the left contact is higher than at the right contact). c, d, The edge charge accumulation is suppressed to 10% of its original value (f(x) = 0.1) at two points along the edge marked by the arrows. A major part of the edge current is diverted into the bulk (d) and the potential distribution is altered markedly (c) leading to inversion of the Hall voltage (the potential at the left contact is lower than at the right contact).
Extended Data Fig. 6 Field-orientation-dependent nonlocal transport in presence of nonuniform edge accumulation.
Numerical simulations of the normalized potential \({\mathscr{V}}\) (left column) and of the magnitude of the normalized current density \({\mathscr{J}}\) (right column) at B = ±4 T and λ = 2.54W (η = 0.05, Vbg = −4 V) for the case of charge edge accumulation being present only in the top-right and bottom-left quadrants of the sample indicated by the pink outlines in b. a, b, B = 4 T and V0 applied to the top contact. Highly nonlocal transport is observed similar to the case presented in Fig. 4e, f with charge accumulation along the entire edges. c, d, Same as a, b but with V0 applied to the bottom contact. The polarities of the potentials and the currents are flipped but the spatial distributions remain the same. e, f, B = −4 T and V0 applied to the top contact. The transport becomes local resembling the ohmic case in Fig. 4c, d in absence of edge accumulation. g, h, Same as e, f but with V0 applied to the bottom contact. The transport remains local with flipped current and potential polarities. This nonuniform edge accumulation exemplifies the strong field-orientation dependence of the nonlocal transport that can arise in presence of edge accumulation disorder.
Extended Data Fig. 7 Nonlocal transport in presence of one-sided edge accumulation.
Numerical simulations of the normalized potential \({\mathscr{V}}\) (left column) and of the magnitude of the normalized current density \({\mathscr{J}}\) (right column) at B = ±4 T and λ = 2.54W (η = 0.05, Vbg = −4 V) for the case of charge edge accumulation being present only in the right side of the sample indicated by the pink outline in b. Highly nonlocal transport is observed solely on the right side of the sample, while the left side of the sample exhibits local ohmic transport. a, b, B = 4 T and V0 applied to the top contact. The current (b) emerges from the source (top) and flows clockwise along the top-right edge against the chiral (counterclockwise) direction. It then leaks to the bulk and flows to the left against the global potential drop (a). Since the left side of the sample has local behaviour, in contrast to Fig. 4e, f and Extended Data Fig. 6a, b, the bulk current is drained directly to the bottom contact without continuing its nonlocal flow to the left side. c, d, Same as a, b but with V0 applied to the bottom contact. The polarities of the potentials and the currents are flipped but the spatial distributions remain the same. The current emanates from the source (bottom) into the bulk, flows to the right against the potential, and is drained along the top-right edge. e, f, B = −4 T and V0 applied to the top contact. The polarity of the potential distribution (e) flips relative to a, but the transport (f) remains local in the left side and nonlocal in the right side. The current (f) flows as in d, but along inverted trajectory. g, h, Same as e, f but with V0 applied to the bottom contact. The polarities of the potentials and the currents are flipped relative to a, b, but the spatial distributions remain the same. This one-sided edge accumulation case exemplifies how the two sides of the sample behave almost independently, with nonlocal transport properties of each side being determined by its edge accumulation and disorder.
Supplementary information
Supplementary Information
This file contains supplementary text, supplementary equations s1 – s27 and supplementary references.
Supplementary Video 1 Thermal imaging of dissipation vs. Vbg at B = 0 T.
Temperature maps T2f of the central part of the sample acquired with the scanning Pb SOT at B = 0 T and Vtg = 0 V with Vbg varying from -6 V to 5 V with applied constant power V0I0 = 15 nW. At large |Vbg| most of the dissipation occurs at the top and bottom contacts outside the imaging frame, while dissipation within the sample is limited mainly to the central region where the current is expected to flow. On approaching CNP the dissipation extends from the central region into the right and left arms with enhanced thermal signal along the edges.
Supplementary Video 2 Thermal imaging of dissipation vs. Vbg at B = 5 T and Vtg = 0 V.
Temperature maps T2f of the full Hall bar structure acquired with the scanning MoRe SOT at B = 5 T and Vtg = 0 V with Vbg varying from -10 V to 10 V with applied constant power V0I0 = 15 nW. At large |Vbg| the dissipation is observed in the central region while near CNP the thermal signal extends over the entire sample. At Vbg values corresponding to QH plateaus (e.g. −8, −2, 3, and 8 V) the dissipation in the sample is reduced with most of the dissipation occurring at the top and bottom contacts outside the imaging frame.
Supplementary Video 3 Thermal imaging of dissipation vs. Vbg at B = 5 T and Vtg = 8 V.
Temperature maps T2f at B = 5 T and Vtg = 8 V with Vbg varying from -10 V to 10 V with applied constant power V0I0 = 15 nW. At large |Vbg| the dissipation is observed in the central region while near CNP the thermal signal extends over the entire sample similar to Supplementary Video 2, but with enhanced signal along the edges. The tip potential Vtg causes local depletion of the hole edge accumulation leading to diversion of the edge current into the bulk and to corresponding enhancement in dissipation as demonstrated by numerical simulations in Supplementary Video 9. The irregular pattern with enhanced thermal signal reveals the locations of suppressed edge charge accumulation along the edges.
Supplementary Video 4 Scanning gate microscopy vs. Vbg at B = 5 T.
A constant V0 = 5.5 mV is applied to contact 2 and \({\mathcal{V}}\)NL, \({\mathcal{V}}\)3 and R2p are measured as a function of the tip position in the central part of the sample (dotted area in Fig. 1a) with Vtg = 8 V at B = 5 T upon varying Vbg from −10 V to 4 V. When the SOT is located above the sample edges the local depletion of the hole edge accumulation by the tip potential Vtg causes suppression in VNL and enhancement of R2p at Vbg values corresponding to large ℛNL in Fig. 1e. \({\mathcal{V}}\)3 is enhanced along the top edge and suppressed along the bottom edge in contrast to Fig. 3b due to \({\mathcal{V}}\)0 applied to contact 2 instead of contact 1. The signals fade away at Vbg values corresponding to QH plateaus.
Supplementary Video 5 Scanning gate microscopy along the top edge of the sample vs. Vtg and Vbg at B = 5 T.
A constant V0 = 3.8 mV is applied to contact 1 and \({\mathcal{V}}\)NL is measured as a function of the tip position along the top-right edge of the sample marked by the yellow line in Fig. 3a vs. Vtg from −2 V to 8 V. Each frame corresponds to a different Vbg that varies from −10 V to 4 V. A positive Vtg depletes the hole edge accumulation causing suppression of \({\mathcal{V}}\)NL. This suppression is strongly position dependent due to edge accumulation disorder. Locations with weaker edge accumulation show a lower Vtg threshold for \({\mathcal{V}}\)NL suppression. Negative Vtg causes additional hole accumulation along the edges which has no significant effect except at the weakest point of edge accumulation, which acts as a bottleneck for the edge current flow. At this point, a negative Vtg can “repair” the suppressed edge accumulation, thus enhancing VNL as observed at few values of Vbg. The overall \({\mathcal{V}}\)NL signal fades away at Vbg values corresponding to QH plateaus.
Supplementary Video 6 Simulations of nonlocal potential and current distributions vs. B with and without edge accumulation.
Numerical simulations of the normalized potential \({\mathcal{V}}\) (top) and of the magnitude of the normalized current density \({\mathcal{J}}\) (bottom) upon increasing B for η = 0 (left column) and η = 0.1 (right column). At low fields up to B≅ 0.2 T the normalized potential and the current density distributions for η = 0 and η = 0.1 evolve very similarly. At higher fields in absence of edge accumulation (left) the transport remains ohmic and essentially field independent. A small edge accumulation (η = 0.1), which has little effect at low fields, causes to a dramatic change in the potential and current distributions with increasing field (right). The potential becomes highly nonlocal and the spatial extent of the edge currents grows rapidly with field. At B = 5 T the nontopological edge currents extend all the way from the source (top contact) to the drain (bottom contact). In regions of \({\mathcal{V}}\) > 1.2 the red color is saturated for clarity.
Supplementary Video 7 Simulations of nonlocal potential and current distributions vs. Vbg at B = 2 T.
Numerical simulations of the normalized potential \({\mathcal{V}}\) (top) and of the magnitude of the normalized current density \({\mathcal{J}}\) (bottom) at B = 2 T upon varying Vbg from −10 V (λ = 0.57W) to 0 V (λ = 13.72W). At small λ the transport is similar to the ohmic regime. Upon increasing λ the current along the edges expands, and the bulk current rotates it direction from vertical to horizontal, while the electric field rotates from horizontal to vertical. When the enhanced edge current extends all the way from source (top) to drain (bottom) the magnitude of the bulk current drops sharply. In regions of \({\mathcal{J}}\) > 1.2 the red color is saturated for clarity.
Supplementary Video 8 Numerical simulation of scanning gate microscopy at B = 5 T.
Numerical simulations of the normalized potential \({\mathcal{V}}\) (top) and the normalized current density \({\mathcal{V}}\) (bottom) at B = 5 T and Vbg=−1 V (λ = 11.9W) upon scanning a positively biased tip (marked by a black circle) that causes local depletion of the hole carriers. When the tip resides in the bulk, the potential and the current are deformed only locally. When the tip is above the edge in the right-hand-side of the sample, the enhanced current flows along the edge from the source up to the tip position only, where it is diverted into the bulk with almost no current flowing along the rest of the edge. In the left-hand-side of the sample, the situation is opposite where the tip blocks the current from the source up to the tip position, whereas along the rest of the edge the current is gathered from the bulk into the edge and channeled towards the drain.
Supplementary Video 9 Simulations of scanning gate microscopy in presence of edge disorder.
Numerical simulations of the normalized potential \({\mathcal{V}}\) (top) and the normalized current density \({\mathcal{V}}\) (bottom) where the edge accumulation is position dependent along the top-right edge giving rise to a non-uniform potential. When the tip resides in the bulk, the potential and the current are deformed only locally. When the tip scans above the disordered top edge the suppression of the edge current is strongly position dependent with largest suppression occurring at the points of weakest edge charge accumulation. These types of simulations were used to produce Figs. 3j-l.
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Aharon-Steinberg, A., Marguerite, A., Perello, D.J. et al. Long-range nontopological edge currents in charge-neutral graphene. Nature 593, 528–534 (2021). https://doi.org/10.1038/s41586-021-03501-7
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DOI: https://doi.org/10.1038/s41586-021-03501-7
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