GRAPHENE

Upstream modes and antidots poison graphene quantum Hall effect


Scanning gate microscopy within the quantum Corridor regime

Right here, we use scanning gate microscopy (SGM) to construct a full microscopic image of QHECs topological safety breakdown in graphene. For this goal, we studied two units (G1 and G2), consisting in 250-nm-wide encapsulated graphene constrictions as introduced in Fig. 1a and functioning solely within the p-doped aspect at excessive magnetic area (see Supplementary Observe 1). Determine 1b shows the longitudinal resistance Rxx as a perform of again gate voltage Vbg, displaying fingerprints of the QH regime in graphene: Rxx vanishes (orange-shaded packing containers in Fig. 1b) across the filling components ν = ± 4(n + 1/2), whereas it’s maximal round ν = ± 4n (the nth LL is aligned with the Fermi vitality—see Supplementary Observe 1).

Fig. 1: Imaging the topological safety breakdown.
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a Schematic of the experimental setup. The biased tip can domestically change the cost carriers density when making use of the voltage Vtip and is moved at a distance dtip ~ 70 nm above the graphene airplane. The worldwide (bulk) cost provider density in graphene is tuned by the again gate voltage Vbg. A magnetic area B is utilized perpendicularly to the graphene airplane. b Longitudinal resistance Rxx as a perform of Vbg, at B = 10 T, measured in pattern G1. ce SGM maps of Rxx as a perform of tip place. The scanning space is sketched by the orange rectangle in a, situated ~500 nm away from the constriction. The info are recorded with Vbg = −13 V—arrow in b—and Vtip = +3 V (c), 0 V (d), and −6 V (e).

On this work, we deal with the transition between the latter two regimes, the place Rxx, whereas near zero, displays fluctuations (see Supplementary Fig. 6a), signatures of QH topological safety breakdown. Related fluctuations have been evidenced in transport via constrictions outlined in excessive mobility semiconductor-based 2DESs25,26,27,28. They’ve been ascribed to backscattering between QHECs propagating at reverse machine edges, occurring via resonant tunneling by way of an antidot localized state. This mechanism is especially efficient when the antidot is situated within the neighborhood of the constriction the place QHECs are introduced in shut proximity.

The antidots areas in actual house may be pinpointed due to SGM measurements. In SGM, native management over the potential panorama is achieved by electrically polarizing a pointy metallic tip transferring in a airplane parallel to the machine floor. Recording concurrently Rxx as a perform of tip place yields SGM maps. Within the case of resonant tunneling between QHECs, a transferring potential perturbation adjustments the resonance situations, turning on and off QHECs backscattering. This yields round options in SGM resistance maps, centered across the lively antidot28.

In distinction with observations in semiconductor-based 2DEGs, facilities of concentric SGM fringes are additionally situated away from the constriction area of our graphene machine. SGM photos displayed in Fig. 1c–e had been obtained at a distance of 500 nm from the constriction, at Vbg = −13 V, as indicated with an arrow in Fig. 1b, i.e. the place the primary deviations from Rxx = 0 emerge, comparable to the onset of the ν = −6 QH state breakdown. SGM maps enable to pinpoint the place the breakdown happens: certainly, non-zero Rxx areas draw units of concentric rings centered near the sides, whose quantity and place evolve with the tip polarization Vtip (Fig. 1c–e for pattern G1 and Supplementary Fig. 3b–f for pattern G2). Nevertheless, the remark of SGM distinction at massive distance from the constriction (about 500 nm in Fig. 1, and some μm in Supplementary Fig. 3) demonstrates that the constriction doesn’t play a big function right here, which is counter-intuitive within the textbook framework of QH impact in standard semiconductor-based 2DEGs. On this image, counterpropagating QHECs run alongside reverse machine edges, and are separated by an insulating bulk area a lot bigger than the tip-induced perturbation. Away from the constriction, the sting states can solely circumvent the perturbation and no tip-effect may be anticipated.

The important thing lacking ingredient within the image, permitting to unravel the puzzling SGM signatures alongside the units edges, is electrostatics. Certainly, as predicted by concept15, inhomogeneous screening of the again gate potential by graphene cost carriers results in non-monotonic confining potential on the edges (see Supplementary Observe 4 for additional discussions in regards to the impact that edge impurities may even have on this confining potential). Since LLs observe the identical evolution because the potential, as schematically depicted in Fig. 2a, one then expects the presence of each up- and downstream QHECs alongside the identical edge if the Fermi vitality crosses twice the identical LL. Tunneling between counterpropagating QHECs may be mediated by the presence of localized states related to antidots, which pin round QHECs “islands” in-between the QH channels (Fig. 2a). These antidots are on the origin of the attribute concentric rings of non-zero Rxx in Fig. 1c–e. Observe that these SGM signatures don’t originate from a direct coupling of the counterpropagating QHECs induced by the tip potential alone: this is able to yield iso-resistance stripes following the sting topography24,29. The absence of such stripes in SGM maps (Fig. 1) testifies that the tip perturbation is sufficiently small to keep away from inducing direct backscattering.

Fig. 2: Artist’ view of QHECs at graphene edge.
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a The 2 lowest LLs arising as a result of perpendicular magnetic area (inexperienced arrow) are represented as blue semi-transparent surfaces. Due to electrostatics on the graphene edge (on the proper aspect), they’re bent and the n = 1 LL crosses twice the Fermi vitality EF (crimson airplane) yielding two downstream QHECs (in crimson) and to an upstream QHEC (in blue). An antidot is situated between the counterpropagating QHECs and pins a QHEC island. b, c Line profile throughout the QHEC island (blue dotted line in a) for the n = 1 LL (electron cost carriers). Discrete vitality ranges are represented in black. The tip-induced potential φtip(xtip) (grey line) tunes discrete vitality ranges positions with respect to EF when various the tip place xtip. When EF lies between two discrete vitality ranges, transport will not be allowed by way of the QHEC island (b) whereas when a discrete vitality stage is aligned with EF, cost carriers can tunnel between the counterpropagating QHECs (crimson and blue dots) via the QHEC island (c).

Transport via antidots

Subsequent, we element how the tip influences tunneling via such an antidot, whose digital construction has been extensively studied in graphene with scanning tunneling microscopy30,31,32. Antidots host discrete vitality ranges within the QH regime, whose positions are decided by measurement confinement within the resultant QHEC island on one hand (quantum contribution) and by Coulomb charging vitality however (electrostatic contribution). A extra in-depth dialogue on the totally different contributions is given in Supplementary Observe 2.2. Discrete vitality ranges are shifted underneath the tip-induced native modification of potential panorama, as sketched in Fig. 2b, c. The excessive Rxx rings in Fig. 1c–e are the loci of tip positions resulting in an alignment between one of many antidot’s discrete vitality ranges and the QHECs potential (Fig. 2c), whereas low Rxx between the rings corresponds to Coulomb blockade33,34 (Fig. 2b). This image is confirmed by the emergence of Coulomb diamonds in scanning gate spectroscopy35: making use of a DC bias between supply and drain permits to beat Coulomb blockade as quickly because the source-drain vitality home windows overlaps a localized state vitality (see Supplementary Observe 3). On this framework, the place of the antidot corresponds to the middle of the Coulomb rings (at low Vtip, screening results can nevertheless distort and shift Coulomb rings, as mentioned in Supplementary Observe 2). Primarily based on Fig. 1c–e, we pinpoint antidots positions at a distance between 50 and 150 nm from pattern G1 boundaries. That is in settlement with the estimated upstream QHEC place extracted from latest native probe outcomes23,24.

A basic query rising at this level considerations the origin of the noticed antidots. Atomic defects on the edges of graphene have typically been invoked as supply of perturbation for cost transport. Nevertheless, in the event that they had been concerned within the current case, it will stay to clarify how they might yield potential landscapes just like the one introduced in Fig. 2a, with a possible extremum situated 50–150 nm from the sting. Extra realistically, such potential panorama may originate from two identified attainable sources: (1) nanoscale random pressure fluctuations, identified to induce cost density inhomogeneities in graphene36 (2) distant charged impurities within the dielectric hBN layer37. Each sources result in native variations of Dirac level energies (sometimes ~50–100 meV at B = 0 T, over typical distances ~50–100 nm38), most likely ubiquitous in all hBN/graphene/hBN heterostructures. It’s noteworthy {that a} potential fluctuation giving rise to an antidot on the p-doped aspect would yield a dot on the n-doped aspect. Whereas our experiment doesn’t enable to discriminate between strain- or impurity-induced potential fluctuations, it supplies knowledge on antidots distance from machine borders, in addition to on their spatial distribution alongside the borders of graphene units : the everyday distance between neighboring antidots is within the vary 100–500 nm, from knowledge in Fig. 1c–e and Supplementary Fig. 3, i.e. appropriate with knowledge from ref. 38. For the reason that tip-induced potential perturbation extends past 500 nm, Coulomb rings originating from distant antidots can superimpose, as proven on Fig. 1c–e and Supplementary Figs. 3.

Again gate and tip management of antidots

The spatial areas of the antidots being unveiled, we now study how their signatures emerge and evolve as a perform of Vbg. For this goal, we scan the tip throughout one of many antidots as indicated in Fig. 3a (the scan space in Fig. 3a corresponds to the crimson rectangle in Fig. 3b) and plot in Fig. 3c the SGM line profile as a perform of Vbg within the neighborhood of ν = −6 for a relentless Vtip (see Supplementary Observe 2.3). It’s well-known from earlier SGM experiments on Coulomb blockaded islands that such a plot permits to deduce the tip potential perturbation from the Vbg-shift of Coulomb blockade resonances33,39. Coulomb resonances endure a Lorentzian evolution, as proven by the matches in Fig. 3c, as anticipated for a tip-induced potential perturbation (see Supplementary Observe 2.3). Analyzing Fig. 3a, c, d collectively, one can get the total image of the destiny of Coulomb resonances related to antidots: Fig. 3c, d proof that peaks recognized by the crimson and blue dashed strains endure a parallel evolution with the approaching tip perturbation, and are subsequently related to the identical antidot, whose location is clearly recognized within the SGM map in Fig. 3a. Importantly, the Coulomb resonances are additionally noticed when the tip is much away from the machine edges which signifies that the tunneling via the antidot will not be essentially triggered by the tip potential. Certainly, the Coulomb resonance signatures may be tuned by Vbg as proven in Fig. 3d.

Fig. 3: Coupling counterpropagating QHECs by way of an antidot.
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The info are obtained in pattern G1 for Vtip = 0 V and B = 14 T. a SGM map obtained at Vbg = −20.85 V, by scanning the tip contained in the crimson rectangle indicated within the schematic image of the machine sketched in b. b The QHECs are represented by crimson (downstream) and blue (upstream) steady strains, and dashed line delineate the constriction. c Rxx recorded as a perform of Vbg and the tip place xtip, alongside the sunshine blue dotted line in a. The resonances signatures (highlighted with the crimson and blue dashed strains) enable to measure the tip-induced potential variation on the QHEC island location as a perform of xtip. Blue and crimson dashed strains are matches obtained with two merged half-Lorentzian capabilities. Above graphene, the half width at half most is 140 nm whereas it’s 280 nm when the tip is above the etched space. The black dashed line signifies the Vbg restrict past which one of many resonances disappears. d Longitudinal resistance Rxx as a perform of Vbg round ν = −6—zoom on the inexperienced rectangle of the inset. Schematics of the three lowest LLs, following the potential profile (thicker line) alongside xtip-axis in map c for Vbg > −21.5 V (e) and Vbg < −21.5 V (f). g Schematic of the QHECs in actual house, on the Fermi vitality indicated by the crimson dash-dotted line in e (downstream in crimson and upstream in blue). The round QHEC is pinned on the location of the antidot. h Actual house schematics of QHECs comparable to Fermi vitality indicated by the crimson dash-dotted line in g, the place the upstream channel vanishes.

A extra intriguing conduct can be revealed for the resonance highlighted by the crimson dashed strains in Fig. 3c : under Vbg < −21.5 V (black dashed line), signatures of this resonance vanish. This Vbg threshold is impartial of Vtip as demonstrated in Supplementary Fig. 4. We suggest the next image to know this phenomenon. Resonances are solely seen offered that (1) a discrete state related to an antidot is tunnel-coupled to up- and downstream QHECs as depicted in Fig. 3e, g and (2) the upstream QHEC permits cost carriers to be despatched again to the injection contact. Various Vbg has a powerful affect on the place of the upstream QHEC (blue in Fig. 3g, h). As quickly because the tunnel coupling turns into too small as illustrated in Fig. 3f, h or the upstream QHEC is not any extra related to the injection contact, backscattering via the antidot is now not efficient and the resonance signature disappears. These knowledge are essential as they verify the presence and the contribution of forward- and backward-propagating QH states on the machine border.

The coupling between the upstream QHEC and the injection contact is important to know the hyperlink between the QHECs construction and the filling issue deduced from transport measurements. Contemplating that this coupling will not be good, the obvious filling issue will not be outlined by the majority (darkish purple in Fig. 3b) however somewhat by the incompressible area between the up- and downstream QHECs (mild purple in Fig. 3b). In Fig. 3e, f, the filling issue subsequently takes a price ν ~ −6 even when the majority filling issue is −2. Now we have additional mentioned the coupling between QHECs and the contacts in graphene samples in40.

One other strategy to tune the place and configuration of QHECs, however on the native scale, consists in various each tip voltage and place. That is realized in Fig. 4a displaying the evolution of Rxx when scanning the tip alongside the dashed line in Fig. 1d and ranging Vtip. The totally different seen resonances comparable to the identical antidot endure parabolic evolution with Vtip as anticipated for localized states39. At low Vtip, these resonances are separated by Rxx ~ 0 areas (comparable to Coulomb blockade), whereas a finite Rxx area (in darkish in Fig. 4a) is reached at bigger optimistic Vtip. This evolution can be clearly seen in Fig. 4b displaying Rxx versus the utmost tip-induced lower in gap density Δntip deduced from Vtip (see Supplementary Observe 5), for a set xtip (with the tip on high of the antidot—black dotted line in Fig. 4a). At decrease tip perturbation, transport is decided by tunneling via the antidot as mentioned above (left inset of Fig. 4b). Because the tip-induced perturbation will increase, the antidot grows and merges with up- or downstream QHECs. The confinement of cost carriers within the antidot is then suppressed and the backscattering is barely induced by the coupling between the counterpropagating QHECs, as depicted in the proper inset of Fig. 4b and additional detailed in Supplementary Observe 6.

Fig. 4: Tip-controlled tuning of transport via a QHEC island.
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a Evolution of Rxx as a perform of Vtip (measured alongside the black dotted line in Fig. 1c at Vbg = −13 V and B = 12 T). b Rxx profile for xtip ~ −60 nm (tip on high of the antidot, i.e. alongside the black dotted line in a). Vtip has been transformed within the most tip-induced gap density lower Δntip. c Simulations of Rxx as a perform of Δntip on the decrease fringe of ν = −6 plateau, at B = 12 T. d Scheme of the simulated system, with colours comparable to the onsite potential panorama. The antidot corresponds to the round area the place the potential is decrease, centered at 45 nm from the graphene edge. The 4 leads required to compute Rxx are represented in yellow. e Profile of the three lowest LLs (n = 0, −1, −2) alongside the black dashed line in d. This graph is just like Fig. 3f, g, aside from the infinitely sharp confinement potential on the edge (proper aspect of the determine) within the simulation, which yields two downstream QHECs (crimson straight arrows). fh Simulated maps of present density (JDOS) obtained for the three Δntip values indicated with arrows in c. On high of the Rxx peak (f), the JDOS across the antidot is maximal in comparison with the scenario of zero Rxx (g). The excessive JDOS within the antidot highlights that the resonance situation is reached. h The area of finite Rxx in c corresponds to direct backscattering of QHECs. The coloured arrows point out the route of the native present density.

Simulations

Tight-binding simulations reproduce the noticed phenomenology and supply additional insights within the underlying physics via actual house photos of the native present density (JDOS) within the totally different backscattering regimes. Utilizing the KWANT package deal41 (see Supplementary Observe 7), we mannequin one fringe of the machine as a 150-nm-wide graphene ribbon represented in Fig. 4d the place the colours correspond to the onsite potential panorama. In our simulations, we deal with a single aspect of the machine, and neglect the majority area contribution. The antidot potential is positioned near the middle of Fig. 4d. On this geometrical configuration, counterpropagating QHECs (straight arrows in Fig. 4e) embody the QHEC island (curved arrows in Fig. 4e) for the Fermi vitality comparable to the crimson dashed line of Fig. 4e. The tip potential shifts the relative place of the LLs with respect to the Fermi vitality, thereby tuning the space and coupling between the QHECs and the antidot.

Noteworthy, we observe a hanging qualitative correspondence between the measured (Fig. 4b) and simulated (Fig. 4c) longitudinal resistance as a perform of Δntip : at low Δntip, finite Rxx peaks are separated by Rxx ~ 0 states and at bigger Δntip, Rxx stays finite. The Δntip scale (distance between the peaks) relies upon primarily on the scale of the thought of antidot in addition to on Coulomb interactions, not captured in our simulations. Since all of the parameters fluctuate among the many antidots, the comparability between experimental and simulated typical Δntip scales will stay qualitative. The sequence of JDOS maps proven in Fig. 4f, g supplies an actual house illustration of the peaks’ origin. Evaluating Fig. 4f, g, corresponding respectively to finite and 0 Rxx (see Fig. 4c), we observe that, whereas in each instances the antidot is coupled to downstream QH channel (proper of the figures), present via the antidot is considerably bigger within the case of Fig. 4f (as indicated by the brighter distinction in log scale on the antidot place). Coupling between up- and downstream QH channels is subsequently rather more environment friendly, yielding finite Rxx. At a lot larger Δntip (Fig. 4h), the JDOS map reveals that the raised antidot potential ends in the merging of the antidot with the upstream QHEC, confirming the schematic image sketched in the proper inset of Fig. 4b.



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