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 R_xx as a perform of again gate voltage V_bg, displaying fingerprints of the QH regime in graphene: R_xx 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).

On this work, we deal with the transition between the latter two regimes, the place R_xx, 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 2DESs^25,26,27,28. 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 R_xx 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 V_bg = −13 V, as indicated with an arrow in Fig. 1b, i.e. the place the primary deviations from R_xx = 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 R_xx areas draw units of concentric rings centered near the sides, whose quantity and place evolve with the tip polarization V_tip (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 R_xx 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 topography^24,29. 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.

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 microscopy^30,31,32. 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 R_xx 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 R_xx between the rings corresponds to Coulomb blockade^33,34 (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 V_tip, 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 outcomes^23,24.

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 V_bg. 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 V_bg within the neighborhood of ν = −6 for a relentless V_tip (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 V_bg-shift of Coulomb blockade resonances^33,39. 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 V_bg as proven in Fig. 3d.

A extra intriguing conduct can be revealed for the resonance highlighted by the crimson dashed strains in Fig. 3c : under V_bg < −21.5 V (black dashed line), signatures of this resonance vanish. This V_bg threshold is impartial of V_tip 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 V_bg 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 R_xx when scanning the tip alongside the dashed line in Fig. 1d and ranging V_tip. The totally different seen resonances comparable to the identical antidot endure parabolic evolution with V_tip as anticipated for localized states³⁹. At low V_tip, these resonances are separated by R_xx ~ 0 areas (comparable to Coulomb blockade), whereas a finite R_xx area (in darkish in Fig. 4a) is reached at bigger optimistic V_tip. This evolution can be clearly seen in Fig. 4b displaying R_xx versus the utmost tip-induced lower in gap density ∣Δn_tip∣ deduced from V_tip (see Supplementary Observe 5), for a set x_tip (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.

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 ∣Δn_tip∣ : at low ∣Δn_tip∣, finite R_xx peaks are separated by R_xx ~ 0 states and at bigger ∣Δn_tip∣, R_xx stays finite. The ∣Δn_tip∣ 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 ∣Δn_tip∣ 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 R_xx (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 R_xx. At a lot larger ∣Δn_tip∣ (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.