# Fileset

[PhysRevResearch.6.023212.pdf](https://mdr.nims.go.jp/filesets/df1f4e5f-2be6-4903-93d6-792de4cc152b/download)

## Creator

Juan Salvador-Sánchez, Luis M. Canonico, Ana Pérez-Rodríguez, Tarik P. Cysne, Yuriko Baba, Vito Clericò, Marc Vila, Daniel Vaquero, Juan Antonio Delgado-Notario, José M. Caridad, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Rafael A. Molina, Francisco Domínguez-Adame, Stephan Roche, Enrique Diez, Tatiana G. Rappoport, Mario Amado

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Generation and control of nonlocal chiral currents in graphene superlattices by orbital Hall effect](https://mdr.nims.go.jp/datasets/86522511-07ee-4e8b-b6df-dfea669f05f4)

## Fulltext

Generation and control of nonlocal chiral currents in graphene superlattices by orbital Hall effectPHYSICAL REVIEW RESEARCH 6, 023212 (2024)Generation and control of nonlocal chiral currents in graphene superlattices by orbital Hall effectJuan Salvador-Sánchez ,1 Luis M. Canonico ,2 Ana Pérez-Rodríguez,1 Tarik P. Cysne ,3 Yuriko Baba,4 Vito Clericò ,1Marc Vila,2,5,6 Daniel Vaquero ,1 Juan Antonio Delgado-Notario ,1 José M. Caridad,1 Kenji Watanabe ,7Takashi Taniguchi,8 Rafael A. Molina ,9 Francisco Domínguez-Adame ,4 Stephan Roche,2,10 Enrique Diez,1Tatiana G. Rappoport,11,12,* and Mario Amado 1,†1Nanotechnology Group, USAL—Nanolab, University of Salamanca, Plaza de la Merced, Edificio Trilingüe, 37008 Salamanca, Spain2Catalan Institute of Nanoscience and Nanotechnology, CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain3Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói RJ, Brazil4GISC, Departamento de Física de Materiales, Universidad Complutense, 28040 Madrid, Spain5Department of Physics, University of California, Berkeley, California 94720, USA6Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA7Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan8International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan9Instituto de Estructura de la Materia, IEM-CSIC, E-28006 Madrid, Spain10ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain11Physics Center of Minho and Porto Universities (CF-UM-UP), Braga, Portugal12Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, 21941-972 Rio de Janeiro RJ, Brazil(Received 2 November 2023; revised 13 March 2024; accepted 3 May 2024; published 28 May 2024)Graphene-based superlattices offer a unique materials playground to exploit and control a higher numberof electronic degrees of freedom, such as charge, spin, or valley for disruptive technologies. Recently, orbitaleffects, emerging in multivalley band structures lacking inversion symmetry, have been discussed as possi-ble mechanisms for developing orbitronics. Here, we report nonlocal transport measurements in small gaphBN/graphene/hBN moiré superlattices which reveal very strong magnetic field-induced chiral response whichis stable up to sizable temperatures. The measured sign dependence of the nonlocal signal with respect tothe magnetic field orientation clearly indicates the manifestation of emerging orbital magnetic moments. Theinterpretation of experimental data is well supported by numerical simulations, and the reported phenomenonstands as a formidable way of in situ manipulation of the transverse flow of orbital information that could enablethe design of orbitronic devices.DOI: 10.1103/PhysRevResearch.6.023212I. INTRODUCTIONThe electronic properties of graphene and other two-dimensional (2D) materials with a honeycomb lattice aredictated by the low-energy physics at two inequivalent K andK ′ valleys of the reciprocal space [1]. The large momen-tum separation between these valleys allows distinguishingvalley quantum numbers that, likewise the spin degree offreedom, can be used to store and process information [2].Moreover, the valleys in graphene possess opposite orbitalmagnetic moments of topological origin [3], that at K and K ′are proportional to the inverse of the band gap. This results ingiant Zeeman splittings upon interaction with weak externalmagnetic fields, lifting valley degeneracy [4,5].*tgrappoport@gmail.com†mario.amado@usal.esPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.In gapped graphene, the application of an electric fieldhas been predicted to induce a flow of electrons, moving inopposite directions for different valleys, and thus giving riseto a valley Hall effect (VHE) [6,7] that could be detectedby nonlocal transport measurements [6–11]. However, this isalso accompanied by a transverse flow of magnetic moments.Consequently, the VHE can also be depicted as an orbital Halleffect (OHE) [12]. In this latter interpretation, the orbital mag-netic moments are physical quantities equally well-definedin the entire momentum space and in real space [13]. Theyreplace the valley quantum numbers, which depend on the ex-istence of well-defined pockets [12]. It is common knowledgethat there is a flow of magnetic moments in the VHE but theobservable is not calculated. The OHE formulation resolvesthis issue and also treats, on the same footing, situations wherek might be not a good quantum number or situations where theelectron lives in an arbitrary point of the Brillouin zone.In the case of twisted structures, for instance, one mighthave the situation where, because of the formation of mini-bands, the orbital magnetic moment can present considerablevariation around the mini Brillouin zone, which can be ac-cessible by small changes of the Fermi energy. In that case,the whole mini Brillouin zone needs to be taken into account2643-1564/2024/6(2)/023212(11) 023212-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-3043-6417https://orcid.org/0000-0001-9266-7213https://orcid.org/0000-0002-3830-3571https://orcid.org/0000-0001-6646-8309https://orcid.org/0000-0001-7025-125Xhttps://orcid.org/0000-0001-9714-8180https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0001-5728-0734https://orcid.org/0000-0002-5256-4196https://orcid.org/0000-0002-3296-5064https://ror.org/02f40zc51https://ror.org/00k1qja49https://ror.org/02rjhbb08https://ror.org/02p0gd045https://ror.org/01an7q238https://ror.org/02jbv0t02https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/05rtchs68https://ror.org/0371hy230https://ror.org/02ht4fk33https://ror.org/03490as77https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.023212&domain=pdf&date_stamp=2024-05-28https://doi.org/10.1103/PhysRevResearch.6.023212https://creativecommons.org/licenses/by/4.0/JUAN SALVADOR-SÁNCHEZ et al. PHYSICAL REVIEW RESEARCH 6, 023212 (2024)FIG. 1. (a) Optical image and schematics of a complete device, consisting of a heterostructure based on a graphite back gate (15 nm) andmonolayer graphene encapsulated between two layers of hBN with thicknesses 10 nm (top) and 50 nm (bottom). (b) Art view schematic of agraphene/hBN heterostucture with a layer alignment set to θ ≈ 30◦. A moiré wavelength of λ ∼ 0.47 nm can be extracted from the relationfound in Refs. [11,26], in stark contrast with the usual λ ∼ 14 nm present in fully aligned samples where θ ≈ 0◦. (c) Local and nonlocalresistances for the terminals indicated in the schematic representation of the final device, where the solid arrow serves as the direction of thedriving (local) current as a function of the back-gate voltage in absence of external magnetic field. The two pairs of contacts are separated 2.5and 5.0 µm from the local signal.for an accurate OHE calculation. From the bulk theory of theVHE in graphene, there is not a simple route to calculate,for example, the magnetic moment accumulation at the edgesor even the possibility of having magnetic moments gener-ated by band bending that can occur in real devices. Still,dispersive edge states can also give rise to orbital magneticmoments. This phenomenon was also reported for nanorib-bons of kagome lattices, where the dispersive edge states ofthe systems are polarized in orbital momentum [14]. Thus,in sharp contrast with the VHE, the OHE refers to a trans-verse current of a physical observable that is naturally moregeneral and also physically manipulable (i.e., magnetic field).Additionally, the OHE can also occur in disordered and finitesystems, and can be used to analyze the interaction with mag-netic and electric fields in the same framework.The electronic structure of graphene-based van derWaals (vdW2Ds) heterostructures can be remarkably tai-lored by varying the twist angle between weakly interactingatomic layers to generate graphene moiré superlattices[6–8,10,11,15–17]. In this context, single or doubly alignedgraphene/hexagonal boron nitride (hBN) heterojunctions arevery interesting for the study of inversion symmetry breakingin graphene. Indeed, such systems present considerable nonlo-cality effect [8,10,18–20], whose origin, although frequentlyassociated with the VHE, is strongly questioned [21–24]. Be-sides, it has been shown that doubly aligned hBN/graphenestacks might generate a supermoiré pattern [25], leading tothe presence of small and nonuniform band gaps [23,24].vdW2Ds are, henceforth, a perfect platform for the study ofinversion symmetry breaking in graphene. Since the inducedgaps are rather small, of the order of few meV, the valleyorbital magnetic moments are large and can be manipulatedwith magnetic and electric fields.In this paper, we report the unambiguous formation of chi-ral nonlocal currents in hBN/graphene/hBN heterostructures,presenting direct evidence of their orbital magnetic origin.Our alignment design between layers minimizes the bandgap, resulting in orbital magnetic moments of the order of102μB. The interaction with weak magnetic fields lifts theirdegeneracy, generating nonlocal chiral currents. Our quantumtransport simulations in graphene nanoribbons with dispersiveedge states and linear response theory calculations supportour interpretation of the experimental findings regarding theorigin of the chiral nonlocal resistance in graphene.II. RESULTSFigure 1(a) displays an optical image of the heterostruc-ture consisting of a graphite back gate (15 nm thick) andmonolayer graphene encapsulated between a top and bottomlayer of hBN with a thickness of 10 and 50 nm, respectively.The crystals were aligned following their exfoliated straightedges using a micromechanical rotator and, for that reason,the relative twisting angles along the vertical heterostructure,if perfectly aligned, are expected to obey m × 30◦, being m =0,±1,±2 . . . . From our electrical data displayed in Fig. S5of the Supplemental Material [27], we can reasonably discardtwisting angles between graphene and hBN that are fullyaligned to 0◦ or a multiple of 60◦. On the one side, thecharge neutrality point (CNP) appears as a stand-alone mainDirac peak with no traces of secondary satellite peaks in themeasured carrier density range |n| < 3 × 1012 cm−2 that arisefrom the existence of electron-hole pockets at both sides of themain peak at such twisting angles [10,11,15,17,18,20,28,29].On the other side, CNP resistivity of heterostructures wheregraphene is aligned to 0◦ or 60◦ with the hBN exhibits a strongthermally activated behavior with values exceeding hundredsof k� at low temperature, indicating a moiré coupling-induced band gap of the order of 30 meV [11,17,18,28,30].In our case, we observe a thermally activated behavior at lowtemperature but a CNP resistivity of only ∼7 k� at room tem-perature. These characteristics are consistent with band gapssmaller than 10 meV. Charge mobility extracted from023212-2GENERATION AND CONTROL OF NONLOCAL CHIRAL … PHYSICAL REVIEW RESEARCH 6, 023212 (2024)magnetotransport Hall measurements rises to 220.000 cm2Vs, as shown in Fig. S5 of the Supplemental Material [27].From a careful Raman analysis, we underpin the relativeorientation of the flakes from the evolution of full width athalf maximum of the 2D peak (FWHM2D) as a function ofthe twisting angle. Finney et al. [29] kept the bottom hBNaligned with graphene at 0◦ and varied the relative angle ofthe top layer from 0◦ to 60◦. They showed that the verticalstructure exhibits a noticeably high value of the FWHM2D,exceeding the standard one found in isolated graphene by20 (40) cm−1 if one (both) hBN layer(s) is (are) aligned tothe graphene at the commensurate angles of 0◦ or 60◦ [29].Such broadening results from the moiré-scale relaxation of thegraphene lattice, which strongly modifies the band structure[31]. We found a value of ∼20 cm−1 for the FWHM2D (seeFig. S4 in the Supplemental Material [27]), in good agreementwith stand-alone graphene that would correspond to a twistingangle that is not aligned to neither 0◦ nor 60◦ [29]. The ratiobetween the 2D and G peaks I (2D)/I (G) > 9 also strength-ens the assumption of far-from-fully aligned layers with themost plausible scenario of twisting angles of ±30◦ betweengraphene and hBN as presented in Fig. 1(b). In both cases,the system is expected to have a very small band gap as aconsequence of the desired 30◦ desired misalignemnt betweenhBN and graphene, which will be of fundamental importancefor the present paper.Figure 1(c) shows the local (Rl) and nonlocal resistances(Rnl) for the pair of contacts indicated in the schematicview at 1.5 K. This allows us to infer the decay of thenon-local signal as a function of the distance �xi be-tween the injection (local, marked with a black arrow) andcollection (nonlocal) terminals. For �x1 = 2.5 µm (nearestnonlocal contacts), Rnl takes a maximum value of ∼1600 �which is consistent with other nonlocal measurements ingraphene/hBN heterostructures [8,11,18]. Moreover, the non-local signal gets weaker for increasing distances, reachingRnl ∼ 470 � for �x2 = 5 µm (farthest non-local contacts).In the absence of external magnetic field, the position ofthe non-local peaks are aligned around the CNP and aresymmetric with respect to electron/hole regions. The rela-tion Rnl(�x) = Vnl/I0 � πρxxe−π |�x|/W, where W = 1.5 µmis the bar width, displays an exponential decay of theohmic contribution to Rnl as a function of distance betweenthe driving current and the nonlocal pair of contacts. Thisrelation, already seen in graphene-based devices [18–20],serves to rule out the ohmic contribution to Rnl as itsprevailing mechanism. We extracted the ratio between themeasured nonlocal resistances at different distances, obtainingRnl(�x2)/Rnl(�x1)|Measured = 0.29, while the purely ohmicexpression gives us Rnl(�x2)/Rnl(�x1)|ohmic ∼ 0.005. Fromthis analysis, one can see that the nonlocal signal is orders ofmagnitude higher than the expected ohmic contribution (seeFig. S7 in the Supplemental Material [27]).III. CHIRAL NONLOCAL SIGNALTo explore the relation between the nonlocal currents andorbital magnetic moments, we apply positive and negativeperpendicular magnetic fields and analyze the nonlocal sig-nals for different injection-collection configurations. Unlessstated otherwise, all measurements were performed at a fixedelectronic temperature of 1.5 K and with an excitation currentset to 10 − 20 nA. This low current amplitude was chosen tominimize thermal contributions to the nonlocal transport dueto Joule heating and Ettingshausen effects [22] while simulta-neously maximizing the signal-to-noise ratio of the measuredvoltages. A summary of our results is shown in Fig. 2, whichcontains nine different panels divided in three columns. Eachcolumn presents a different configuration of the external mag-netic field, i.e., −0.5 T, 0 T, and 0.5 T. Each of the three rowspresents a different injection-collection setup, sketched in thediagram on the left. Each pair of contacts on the diagramhas a specific color, while the arrows show the direction ofthe current for the pair of contacts of a particular injection-collection configuration. Each panel displays the local andnonlocal resistances as a function of the voltage applied to thegraphite back gate. Their color palette matches the ones of thecorresponding contacts, irrespective of the specific injection-collection setup.Let us first comment on the effect of the magnetic fieldon the local resistances. The CNP is located at Vg ∼ 0.6 V,as extracted from the Lorentzian fit of the local resistance,and does not present a noticeable shift when the magneticfield is applied. Still, there is a sharp increase of the localresistance with the magnetic field, which is very pronouncedat the CNP. Figure 2(b1) displays the evolution of the nonlocalsignal as a function of the distance without external magneticfield. Figure 2(b2) represents a configuration of the nonlocalpairs of contacts that are placed symmetrically to the currentflow but at opposite directions. In this case, the homogeneityof the sample is demonstrated because the nonlocal signalsat both sides of the current flow have an expected matchingvalue, as the magnitude of the nonlocal signal decays withthe absolute value of the distance to the injection current.This characteristic, discussed previously, can also be seen inFigs. 2(a2) and 2(c2).A striking behavior of the nonlocal signal arises in thepresence of an external magnetic field. We first focus onthe case where the current is injected between two nonlocalcontacts. Figure 2(a2) shows the nonlocal resistances for theB = −0.5 T, where one can see a clear separation betweenthe peaks of opposite contacts. Moreover, they are mostlylocated either in the electron or hole sectors. Surprisingly,the position of the two peaks is swapped upon magnetic fieldreversal, which is a clear indication of a chiral behavior ofthe electronic response. If the two collectors are located atthe same side of the injector, as shown in the first and lastrow of Fig. 2, the situation is different. In these two cases, thetwo nonlocal resistance peaks are aligned and located either atthe electron or hole sector and switch positions with the signof the magnetic field and the relative orientation with respectto the collector. While both peaks appear at the hole sectorin Fig. 2(a1), when the sign of B is reversed, they appearat the electron sector [see Fig. 2(c1)]. If instead we switchthe position for positive B, as in Fig. 2(a3), the peaks alsoappear at the electron sector, changing to the hole sector ifthe field is reversed [see Fig. 2(c3)]. It is worth mentioningthe presence of reduced spurious peaks in Fig. 2, labeled as *,and might arise from nonperfect carrier-valley locking in ourresults. These reduced nonlocal spurious peaks appear exactly023212-3JUAN SALVADOR-SÁNCHEZ et al. PHYSICAL REVIEW RESEARCH 6, 023212 (2024)FIG. 2. Local and nonlocal resistances for three different (1–3) configurations of the injection-collection terminals in the transversaldirection as a function of the back-gate voltage. The left column includes schematic top view of the corresponding configuration for each rowof graphs where the local resistance will be measured across the pair of contacts connected by the driving current marked by the solid arrow.Each column corresponds to a different value of the magnetic field in the out-of-plane direction. The similarity of the nonlocal resistivity valuesin the absence of applied magnetic field shown in (b2) evidences the homogeneity and, in general, the good quality of the sample. Moreover,similar values arise for opposing configurations (as in rows 1 and 3) for opposite directions of the applied magnetic field, as can be observedcomparing (a1) and (c3) with (a3) and (c1), for example. Peaks labeled by * are ascribed to nonperfect carrier-valley locking within the wholesample, which can be attributed to a slight inhomogeneity in it. These peaks are nondominant and do not rule out the main findings presentedin this paper.at densities matching their dominant chiral counterparts and,while relevant, do not contradict our interpretation since theycan be ascribed to nondominant imperfections in the finaldevice arising from nonperfect valley-carrier locking withinthe whole sample. Furthermore, it is important to mentionthat in resemblance to similar experiments, the nonlocality isstrongly enhanced with the magnetic field in all configurations[15,18,20].The nine panels in Fig. 2 provide full demonstration of achiral behavior of the nonlocal signal at low magnetic fields,which has not been reported in similar graphene-basedheterostructures with twisting angles set to differentcommensurate angles [11,18,20]. In contrast to these works,the chiral response for the nonlocal signal cannot be attributedto electron-hole puddles or charge accumulation at the edges,which should be symmetrical upon the application of externalmagnetic fields. Moreover, while band-bending mechanismscould provoke measurable nonlocal signals in these structures,it has already been demonstrated to be inhomogeneous [32],i.e., measurable for one kind of carrier and depleted for theother one and, therefore, it does not align to our findingseither. To clarify the underlying mechanism, we use themodern theory of magnetism to address the effect of thefields in bulk calculations. Although the orbital magneticmoments of electrons in moiré heterostructures remain to beunderstood, their phenomenology can be tentativelly relatedto the effective g factor of semiconductors, where the orbitalmotion of electrons can lead to enhanced magnetic responses[33]. The theoretical description [3,34,35] derives fromtreating Bloch electrons as self-rotating wave packets and023212-4GENERATION AND CONTROL OF NONLOCAL CHIRAL … PHYSICAL REVIEW RESEARCH 6, 023212 (2024)(d) (e) (f)(a)(g)(b) (c)FIG. 3. (a) Comparison between the energy bands of gaped graphene systems with and without out-of-plane magnetic fields with B = 0.2 Tand a gap � = 14 meV. Comparison between the energy bands of zigzag graphene nanoribbons for B = 0 and B = −30 T (b) and B = +30 T(c) with a sublattice staggered potential Vab = 5 meV. (d)–(f) Nonlocal resistance signals for lateral (d) and central (e), (f) current injectioncomputed using 60 Anderson disorder realizations with U = Vppπ/128 and Vppπ = −3.26 eV, and sublattice staggered potential Vab = 5 meV,as a function of the energy for B = 0 T, B = −30 T and B = 30 T, respectively. (g) Schematic representation of the device used in thesimulations, with Lch = 512 nm, W = 13 nm, �x1 = Lch/3 − W and �x2 = 2Lch/3 − W .through the incorporation of the Berry phase theory to expressthe magnetic moment purely in terms of bulk quantities.Following Ref. [36], for gapped Dirac materials the orbitalmagnetic moment m(k) = (τeh̄/2m∗)(1 + h̄2v2F k2/�2)−1 ẑ,where τ = ±1 is the valley quantum number, m∗ = �/v2F isthe effective mass at the Dirac point (DP), � is the system gapand vF is the Fermi velocity. At the Dirac points, the orbitalmagnetic moment is inversely proportional to the gap width,while it can generally be seen as inversely proportional to theeffective mass as well. Similarly to spins, this orbital momentcan couple directly with weak magnetic fields. This gives riseto a k-dependent Zeeman splitting that, in first-order pertur-bation theory, renormalizes the energy spectrum close to theDirac points [36,37], as depicted in Fig. 3(a). As the valleyshave opposite magnetic moments, the Zeeman effect producesa relative shift between the valleys. For small gaps, it can evenlead to situations where the Fermi energy lies inside the gapfor one valley while it is located in the electron (hole) sectorfor the other, as illustrated in Fig. 3(a). Given the couplingwith the external fields, one can argue that the chiral behavior023212-5JUAN SALVADOR-SÁNCHEZ et al. PHYSICAL REVIEW RESEARCH 6, 023212 (2024)observed in the nonlocal resistance appears as a manifestationof the Zeeman effect but with enhanced g factor of the orderof 100, allowed by the formation of a small gap in the doublyencapsulated hBN/graphene/hBN heterostructure. The OHEinterpretation is in fully agreement with the chiral behaviorobserved experimentally, shown in Fig. 2 (see details in theSupplemental Material [27]). The magnetic moments of eachvalley flow in opposite directions. Because of the sign of theBerry curvature, the flows invert directions when switchingfrom hole to electron sectors. At the same time, the Zeemanshift between the valleys leads to a relative shift betweenthe nonlocal peaks that should invert with the change in themagnetic field orientation and collector’s locations. Using thisreasoning, we can estimate the band gap from nonlocal peaksof Fig. 2(a2), obtaining � ∼ 5 − 8 meV. One should mentionthat for this formulation, in general, the orbital angularmomentum current is not conserved. Other approaches,useful for for spin currents, allow alternative formulation ofconserved currents of nonconserved quantities [38,39].To validate our scenario, we further performed quantumtransport simulations using the Landauer-Büttiker formalismimplemented in KWANT [40]. In our calculations, we consid-ered a multiterminal device containing a graphene nanoribbongeometry with contacts [Fig. 3(g)] in the presence of an ex-ternal magnetic field. The advantage of this approach is thatit does not rely on any assumption about the orbital Zeemaneffect, as the magnetic field is included through the Peierls’substitution. Moreover, this geometry can connect directlywith the experiments, as it considers contributions from thebulk and edge states and allows the calculation of nonlocalresistances. Still, here we are not particularly interested in thespecific channels carrying the orbital currents. As will becomeclear below, the orbital Zeeman effect is present in bulk andedge states and either of them can, in principle, convey chiralnonlocal currents.Previous numerical simulations showed the need for dis-persive edge states near the Dirac point for nonlocal transportin gapped graphene [21]. They are absent in theories basedon the simplistic Hamiltonian considering a single pz orbital.Here, with full generality, we used a six-band tight-bindingmodel that also takes into account the d orbitals [41] and astaggered sublattice potential to break the inversion symmetryof the system. It is important to note that other effects suchas nonuniform potential and coupling to hBN layers can alsolead to dispersive edge states and a similar nonlocal response(see details in the Supplemental Material [27]).Figures 3(b) and 3(c) depicts the comparison betweenthe energy bands of the nanoribbon with negative (positive)magnetic field and the energy states of the nanoribbon inthe absence of magnetic field. The energy bands from thismultiorbital model exhibit a dispersive behavior similar to theones observed from ab initio calculations of Ref. [21]. It doesnot present a band gap, although there is a clear valence-conduction band separation due to the inversion symmetrybreaking and very well-defined electron pockets at oppositesides of the nanoribbon Brillouin zone. The most relevantfeature displayed by the two panels is the stark k-dependentcoupling with the magnetic field, which is similar to the oneobserved in the bulk bands. Although the modern theory ofmagnetism is well developed only for bulk systems, it is clear,from the results in panels 3(b) and 3(c), that the behavior dis-played by the energy bands of the nanoribbon is in qualitativeagreement with this theory.Aiming to reproduce results from the middle columns inFig. 2, in Figs. 3(d)–3(f) we used the same injection-collectionscheme for our nonlocal resistance simulations. Figure 3(d)portrays the case in which the injection occurs at one side ofthe device in the absence of magnetic field. As in the mea-surements, the simulation results show considerable decay ofthe nonocal signal with the channel length. However, the moststriking behavior appears when we inject current in the middlecontact of the device and calculate the nonlocal resistance atthe two lateral contact pairs in the presence of an externalmagnetic field. Comparing these figures with the measure-ments shown in Fig. 2, we find convincing agreement betweenthe chiral behavior displayed by the nonlocal resistance sim-ulations in Figs. 3(e) and 3(f) with the measurements fromFigs. 2(a2) and 2(c2), respectively. Moreover, the comparisonbetween numerical results and the renormalized energy bandsin Fig. 3 indicates that the mechanism at play in the genera-tion of these nonlocal signals observed in the experiment isthe orbital valley Hall effect [12]. The chirality and energyselectivity are directly related to the coupling between theorbital magnetic moment and the external magnetic field. Still,our analysis suggests that Fermi surface edge currents carrythe nonlocal signal once the absence of dispersive edge statesdestroys the nonlocal signal.Figures 4(b) and 4(c) display the contour plot of Rnl as afunction of both B and Vg − VDP for two symmetrical con-figurations for the local and nonlocal contacts as sketchedin Fig. 4(a). The electronic temperature at which the curveswere recorded was T = 250 mK and the voltage has beencentered at the Dirac peak for B = 0. Panels 4(b) and 4(c)show a clear chiral behavior and an apparent valley-carrierlocking in the nonlocal signal for low magnetic fields rangingfrom −0.5 to 0.5 T. In Fig. 4(b), we can observe a distincttransition from an electron-mediated nonlocal transport fornegative magnetic fields towards a holelike one when themagnetic field is reversed. Moreover, a strong asymmetry inthe nonlocal curves is clearly visible, with a sudden decaywhile approaching the DP from the dominant carrier speciestowards the prohibited one. Figure 4(c) displays the mirroredconfiguration for the pairs of contacts, compared to Fig. 4(b)as sketched in Fig. 4(a). Consequently, we can argue thatthe valley-carrier locking, visible in the nonlocal signal, isa quite robust phenomenon and implies a flow of carrierscharacterized by different orbital magnetic moments. As thenonlocal signal should originate from the flow of orbital mag-netic moments from a single valley, we compare Rnl with thevalley-filtered OHE, calculated according to Ref. [12], shownin the Figs. 4(d) and 4(e). To confirm this hypothesis, we alsocompare the measured shift in the maximum of the nonlocalresistance signals in neighboring contacts for various mag-netic fields and opposite injection-detection configurations forsample 1. The agreement between the OHE for gaps of 8 meVand the measured shift up to ±0.5 T, where the low-energytheory remains valid, is very convincing (see details in theSupplemental Material [27]). Consequently, the interpretationof the experimental results is fully consistent with the changesin the OHE resulting from the orbital Zeeman effect.023212-6GENERATION AND CONTROL OF NONLOCAL CHIRAL … PHYSICAL REVIEW RESEARCH 6, 023212 (2024)FIG. 4. (a) Sketch for the two different injection and collection configurations for the experimental and theoretical results shown in (b) and(c). (b), (c) Heat maps of the sample nonlocal resistance for two symmetrical configurations scanned in the space of back-gate voltage,corrected by the position of the Dirac peak at B = 0 (VG − VDP), and magnetic field measured at 250 mK. (d), (e) Numerical results for theOHE conductivity using Eq. (S8) in the Supplemental Material [27] for an energy gap � = 8 meV; the red markers correspond to the measuredpeak in the nonlocal signal as a function of the parameters, showing extraordinary agreement with the experimental results.Importantly, the origin of the nonlocal resistance signalscould also be related to some spin-dependent effect, which canbe typically probed by applying in-plane magnetic fields andproducing field-driven oscillations of non-local resistance. Todiscard such effect, we performed local and nonlocal mea-surements for a configuration where the excitation currentlies symmetrically between two different pairs of contactsfor Rnl at a fixed perpendicular magnetic field of B⊥ = 0.5 Tand for varying in-plane component (B‖) measured at 1.5 K(see Fig. 5). B‖ ranges from 0 to 12 T and its evolution havebeen marked with an arrow as a guide to the eye in the threedifferent panels (a)–(c). For B‖ = 0, the charge carrier type iseffectively coupled to one of the valleys and the asymmetricchiral behavior is clearly seen (see Fig. 5). In contrast, theincrease of B‖ reduces the intensity of the nonlocal and localsignals, which additionally becomes symmetric. From thesedata, it becomes clear that spin-dependent effect cannot ex-plain the measured chiral nonlocal currents.On the other hand, in-plane magnetic fields can actuallybe used as a tuning parameter to manipulate the orbital char-acteristics of 2D multilayers and twisted bilayers and moirésuperlattices [42,43]. B‖ affects the quasimomentum of eachlayer differently [44–47], modifying the effective coupling be-tween the layers and thus the resulting band structure. Indeed,B‖ introduces a layer-dependent gauge field Al = B‖ × zl thatmodifies the electron momentum p → p + (e/c)Al , where lindexes the layer. For graphene encapsulated by two hBN lay-ers and located at z = 0, the magnetic field shifts the momentaof electrons in each hBN layer along opposite directions. Inaddition, since time-reversal symmetry is broken, the mo-menta of the electrons of different valleys are shifted in thesame direction, altering the band structure and the resulting023212-7JUAN SALVADOR-SÁNCHEZ et al. PHYSICAL REVIEW RESEARCH 6, 023212 (2024)FIG. 5. Evolution of the nonlocal [(a), (c)] and local [(b)] resistances as a function of the external in-plane magnetic field for a fixedout-of-plane field B⊥ = 0.5 T. Panel (d) sketches the sample, showing the direction of the driving current, the two pairs of contacts fornonlocal measurements, and the two components for the external magnetic field. Solid arrows show the evolution of the in-plane componentfor B ranging from 0 to 12 T.orbital magnetic moments with a direct impact on local andnonlocal signals.In conclusion, we have presented nonlocal transport mea-surements on hBN/graphene/hBN narrow gap heterostruc-tures at low magnetic fields, which clearly indicate thepresence of chiral effects. Such chiral response is inferredfrom the nonlocal resistance when reversing both the magneticfield and the injection-collection configurations. The interac-tion between large orbital magnetic moments arising in smallgap graphene-based superlattices and external magnetic fieldproduces a relative Zeeman shift between the two valleys inboth bulk and edge electron states. Furthermore, based onour experimental and theoretical analysis, regardless of thedetails about the location of the current flow, the manifes-tation of a strong chiral effect originates from the interplaybetween the Zeeman shift and the transverse flow of orbitalmagnetic moments. Importantly, the analysis of the nonlocaltransport as a function of the magnetic field direction rules outspin effects, whereas its dependence on the distance betweencontacts clarifies that ohmic and thermal contributions aremarginal. Finally, our computational results show that theorbital valley Hall effect displays fingerprints in both bulk andedge transport, being of topological nature or not. All thesefacts convincingly support the interpretation that the originof the giant nonlocality in the studied graphene superlatticesis linked to the OHE resulting from the valley magnetic mo-ments. Beyond shining light on a fierce debate concerning theformation of topological versus nontopological valley-drivenphenomena to explain previously reported nonlocal signals[24,48], our findings pave the way towards future develop-ments in graphene orbitronics.IV. METHODSThe device fabrication of the superlattices follows thestandard dry transfer technique with a polycarbonate filmfabricated and deposited onto a polydimethylsiloxane stamp.The relative rotation between the different layers, followingtheir natural edges, was controlled using a heated stage witha micromenchanical rotator with an accuracy better than 0.5◦.The heterostructure rested atop a commercial Si/SiO2 sub-strate. The fabricated stack was patterned using electron beamlithography followed by a dry-etching process in an ICP-RIEto define the sample geometry. The sample was patternedinto a form of a multiterminal Hall bar (optical image inFig. 1) with the central horizontal bar of width W = 1.5 µm,total length of ∼11 µm, and a distance between the centersof the contacts of 2.5 µm. Electrical contact to all deviceswas made by Cr/Au (10 nm/50 nm) deposited by electronbeam evaporation (see Fig. S1 and S2 in the SupplementalMaterial [27]). We extracted the all Raman spectra and theirassociated FWHM2D from a Lorentzian fit (see details inthe Supplemental Material [27]). Transport measurements inthe multiterminal device were conducted in two- and four-terminal geometries with ac current excitation of 10-20 nAusing the standard lock-in technique at 17.7Hz. The graphitelayer was gated by applying a direct bias to it in the range of±12 V.023212-8GENERATION AND CONTROL OF NONLOCAL CHIRAL … PHYSICAL REVIEW RESEARCH 6, 023212 (2024)All data needed to evaluate the findings of this study arepresent in the paper and/or the Supplemental Material. Dataare available from the corresponding authors on reasonablerequest.ACKNOWLEDGMENTSM.A. and E.D. acknowledge financial support fromthe Ministerio de Ciencia e Innovación of Spain (Span-ish Ministry of Science, Innovation, and Universities) andFEDER (ERDF: European Regional Development Fund) un-der Research Grants No. PID2019-106820RB-C21/22, No.PID2022-136285NB-C32, and FEDER/Junta de Castilla yLeón Research Grant No. SA121P20. A.P.-R. acknowl-edges financial support received from the Marie SkłodowskaCurie-COFUND program under the Horizon 2020 researchand innovation initiative of the European Union, withinthe framework of the USAL4Excellence program (GrantAgreement No. 101034371). D.V. acknowledges finan-cial support from the Ministerio de Universidades (Spain)(Ph.D. Contract No. FPU19/04224), including funding fromERDF/FEDER. J.S.-S. acknowledges financial support fromthe Consejería de Educación, Junta de Castilla y León,and ERDF/FEDER. L.M.C. acknowledges funding fromMCIU/AEI/10.13039/501100011033 and European UnionNextGenerationEU/PRTR under Grant No. FJC2021-047300-I. S.R. acknowledges funding from the European UnionSeventh Framework Programme under Grant No. 881603(Graphene Flagship). The Catalan Institute of Nanoscienceand Nanotechnology (ICN2) is funded by the CERCAProgramme/Generalitat de Catalunya and supported by theSevero Ochoa programme (MINECO Grant No. SEV-2017-0706. S.R is also supported by MCIU with European funds-NextGenerationEU 324 (PRTR-C17.I1) and by Generalitat deCatalunya. T.G.R. acknowledges funding from FCT-Portugalthrough Grant No. 2022.07471.CEECIND/CP1718/CT0001[63] and in the framework of the Strategic FundingUIDB/04650/2020. T.P.C. acknowledges financial supportfrom Brazilian agency CAPES. J.M.C. acknowledges sup-port from the MICINN Ramón y Cajal program (ProjectNo. RYC2019-028443-I). M.V. was supported as part of theCenter for Novel Pathways to Quantum Coherence in Ma-terials, an Energy Frontier Research Center funded by theU.S. Department of Energy, Office of Science, Basic EnergySciences. K.W. and T.T. acknowledge support from JSPSKAKENHI (Grants No. 19H05790, No. 20H00354, and No.21H05233). J.A.D.-N. acknowledges support from Junta deCastilla y León cofunded by FEDER under Research GrantNo. SA103P23 and the support from the Universidad de Sala-manca for the María Zambrano postdoctoral grant fundedby the Next Generation EU Funding for the Requalificationof the Spanish University System 2021–23, Spanish Min-istry of Universities. F.D.-A. and Y.B. were supported bythe Recovery, Transformation and Resilience Plan, funded bythe European Union NextGenerationEU (Grant No. MAD2D-CM-UCM5) and Ministerio de Ciencia e Innovación of Spain(Grant No. PID2022-136285NB-C31).M.A., E.D. and F.D.-A. developed the concept of the exper-iment. T.T. and K.W. provided hBN crystals. J.S.-S., A.P.-R.,V.C., D.V., and J.A.D.-N. performed device fabrication andcarried out Raman spectroscopy. J S.-S., A.P.-R.E.D., andM.A. performed transport measurements. J.S.-S. and A.P.-R.performed experimental analysis. J.S.-S., A.P.-R., E.D., andM.A. interpreted results with help from L.M.C., T.G.R., S.R.,J.M.C., Y.B., R.A.M, and F.D.-A. Theoretical calculationswere performed by L.M.C., T.G.R., T.P.C., M.V., and S.R.The paper and the Supplemental Material were written byT.G.R., M.A., S.R., L.M.C., J.S.-S., A.P.-R., and F. D.-A.,with additional contributions from all authors.The authors declare that they have no competing interests.[1] A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, andA. K. Geim, The electronic properties of graphene, Rev. Mod.Phys. 81, 109 (2009).[2] D. Xiao, W. Yao, and Q. Niu, Valley-contrasting physics ingraphene: Magnetic moment and topological transport, Phys.Rev. Lett. 99, 236809 (2007).[3] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects onelectronic properties, Rev. Mod. Phys. 82, 1959 (2010).[4] Z. Ge, S. Slizovskiy, F. Joucken, E. A. Quezada, T. Taniguchi,K. Watanabe, V. I. Fal’ko, and J. Velasco, Control of gianttopological magnetic moment and valley splitting in trilayergraphene, Phys. Rev. Lett. 127, 136402 (2021).[5] J. Yin, C. Tan, D. Barcons-Ruiz, I. Torre, K. Watanabe, T.Taniguchi, J. C. W. Song, J. Hone, and F. H. L. Koppens,Tunable and giant valley-selective Hall effect in gapped bilayergraphene, Science 375, 1398 (2022).[6] M. Sui, G. Chen, L. Ma, W.-Y. Shan, D. Tian, K. Watanabe, T.Taniguchi, X. Jin, W. Yao, D. Xiao, and Y. Zhang, Gate-tunabletopological valley transport in bilayer graphene, Nat. Phys. 11,1027 (2015).[7] M. J. Zhu, A. V. Kretinin, M. D. Thompson, D. A. Bandurin, S.Hu, G. L. Yu, J. Birkbeck, A. Mishchenko, I. J. Vera-Marun,K. Watanabe, T. Taniguchi, M. Polini, J. R. Prance, K. S.Novoselov, A. K. Geim, and M. B. Shalom, Edge currents shuntthe insulating bulk in gapped graphene, Nat. Commun. 8, 14552(2017).[8] R. V. Gorbachev, J. C. W. Song, G. L. Yu, A. V. Kretinin,F. Withers, Y. Cao, A. Mishchenko, I. V. Grigorieva, K. S.Novoselov, L. S. Levitov, and A. K. Geim, Detecting topologi-cal currents in graphene superlattices, Science 346, 448 (2014).[9] Y. Shimazaki, M. Yamamoto, I. V. Borzenets, K. Watanabe, T.Taniguchi, and S. Tarucha, Generation and detection of purevalley current by electrically induced Berry curvature in bilayergraphene, Nat. Phys. 11, 1032 (2015).[10] K. Komatsu, Y. Morita, E. Watanabe, D. Tsuya, K. Watanabe,T. Taniguchi, and S. Moriyama, Observation of the quantumvalley Hall state in ballistic graphene superlattices, Sci. Adv. 4,eaaq0194 (2018).[11] Y. Li, M. Amado, T. Hyart, G. P. Mazur, and J. W. A. Robinson,Topological valley currents via ballistic edge modes in graphene023212-9https://doi.org/10.1103/RevModPhys.81.109https://doi.org/10.1103/PhysRevLett.99.236809https://doi.org/10.1103/RevModPhys.82.1959https://doi.org/10.1103/PhysRevLett.127.136402https://doi.org/10.1126/science.abl4266https://doi.org/10.1038/nphys3485https://doi.org/10.1038/ncomms14552https://doi.org/10.1126/science.1254966https://doi.org/10.1038/nphys3551https://doi.org/10.1126/sciadv.aaq0194JUAN SALVADOR-SÁNCHEZ et al. PHYSICAL REVIEW RESEARCH 6, 023212 (2024)superlattices near the primary Dirac point, Commun. Phys. 3,224 (2020).[12] S. Bhowal and G. Vignale, Orbital Hall effect as an alternativeto valley Hall effect in gapped graphene, Phys. Rev. B 103,195309 (2021).[13] R. Bianco and R. Resta, Orbital magnetization in insulators:Bulk versus surface, Phys. Rev. B 93, 174417 (2016).[14] O. Busch, I. Mertig, and B. Gobel, Orbital Hall effect andorbital edge states caused by s electrons, Phys. Rev. Res. 5,043052 (2023).[15] L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias,R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R.Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M.Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I.Fal’ko, and A. K. Geim, Cloning of Dirac fermions in graphenesuperlattices, Nature (London) 497, 594 (2013).[16] C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao,J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K.Watanabe, K. L. Shepard, J. Hone, and P. Kim, Hofstadters but-terfly and the fractal quantum Hall effect in moiré superlattices,Nature (London) 497, 598 (2013).[17] R. Ribeiro-Palau, C. Zhang, K. Watanabe, T. Taniguchi, J.Hone, and C. R. Dean, Twistable electronics with dynamicallyrotatable heterostructures, Science 361, 690 (2018).[18] D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V.Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T.Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim,Giant nonlocality near the Dirac point in graphene, Science 332,328 (2011).[19] M. Ribeiro, S. R. Power, S. Roche, L. E. Hueso, and F.Casanova, Scale-invariant large nonlocality in polycrystallinegraphene, Nat. Commun. 8, 2198 (2017).[20] A. Aharon-Steinberg, A. Marguerite, D. J. Perello, K. Bagani,T. Holder, Y. Myasoedov, L. S. Levitov, A. K. Geim, and E.Zeldov, Long-range nontopological edge currents in charge-neutral graphene, Nature (London) 593, 528 (2021).[21] J. M. Marmolejo-Tejada, J. H. Garcia, M. D. Petrović, P. H.Chang, X. L. Sheng, A. Cresti, P. Plecháč, S. Roche, andB. K. Nikolić, Deciphering the origin of nonlocal resistancein multiterminal graphene on hexagonal-boron-nitride withab initio quantum transport: Fermi surface edge currents ratherthan Fermi sea topological valley currents, J. Phys. Mater. 1,015006 (2018).[22] J. Renard, M. Studer, and J. A. Folk, Origins of nonlocality nearthe neutrality point in graphene, Phys. Rev. Lett. 112, 116601(2014).[23] T. Aktor, J. H. Garcia, S. Roche, A.-P. Jauho, and S. R. Power,Valley Hall effect and nonlocal resistance in locally gappedgraphene, Phys. Rev. B 103, 115406 (2021).[24] S. Roche, S. R. Power, B. K. Nikolić, J. H. García, and A.-P.Jauho, Have mysterious topological valley currents been ob-served in graphene superlattices? J. Physics: Mater. 5, 021001(2022).[25] Z. Wang, Y. B. Wang, J. Yin, E. Tóvári, Y. Yang, L. Lin, M.Holwill, J. Birkbeck, D. J. Perello, S. Xu, J. Zultak, R. V.Gorbachev, A. V. Kretinin, T. Taniguchi, K. Watanabe, S. V.Morozov, M. Anđelković, S. P. Milovanović, L. Covaci, F. M.Peeters et al., Composite super-moiré lattices in double-alignedgraphene heterostructures, Sci. Adv. 5, eaay8897 (2019).[26] M. Yankowitz, J. Xue, D. Cormode, J. D. Sanchez-Yamagishi,K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, P. Jacquod,and B. J. LeRoy, Emergence of superlattice Dirac pointsin graphene on hexagonal boron nitride, Nat. Phys. 8, 382(2012).[27] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.023212 for a step-by-step expla-nation of the sample fabrication (as shown in Ref. [49]), adetailed Raman characterization, quantum Hall characterizationat low temperatures, and the estimation for the residual carrierdensity (see Refs. [50–52]), Ohmic and thermal contribution toRnl , the detailed perturbation theory and linear response for-malism for the orbital Hall effect (see Refs. [36,53–59]), andin-depth technical details of the nonlocal resistance simulations(see Refs. [60–62]).[28] L. Wang, Y. Gao, B. Wen, Z. Han, T. Taniguchi, K. Watanabe,M. Koshino, J. Hone, and C. R. Dean, Evidence for a fractionalfractal quantum Hall effect in graphene superlattices, Science350, 1231 (2015).[29] N. R. Finney, M. Yankowitz, L. Muraleetharan, K. Watanabe,T. Taniguchi, C. R. Dean, and J. Hone, Tunable crystal symme-try in graphene-boron nitride heterostructures with coexistingmoiré superlattices, Nat. Nanotechnol. 14, 1029 (2019).[30] B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz,B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino,P. Jarillo-Herrero, and R. C. Ashoori, Massive Dirac fermionsand Hofstadter butterfly in a van der Waals heterostructure,Science 340, 1427 (2013).[31] A. Eckmann, J. Park, H. Yang, D. Elias, A. S. Mayorov,G. Yu, R. Jalil, K. S. Novoselov, R. V. Gorbachev, M.Lazzeri, A. K. Geim, and C. Casiraghi, Raman fingerprintof aligned graphene/h-BN superlattices, Nano Lett. 13, 5242(2013).[32] A. Marguerite, J. Birkbeck, A. Aharon-Steinberg, D. Halbertal,K. Bagani, I. Marcus, Y. Myasoedov, A. K. Geim, D. J. Perello,and E. Zeldov, Imaging work and dissipation in the quantumHall state in graphene, Nature (London) 575, 628 (2019).[33] Y. Yafet, g Factors and spin-lattice relaxation of conductionelectrons, Solid State Phys. 14, 1 (1963).[34] T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Or-bital magnetization in periodic insulators, Phys. Rev. Lett. 95,137205 (2005).[35] R. Bianco and R. Resta, Mapping topological order in coordi-nate space, Phys. Rev. B 84, 241106(R) (2011).[36] T. Cai, S. A. Yang, X. Li, F. Zhang, J. Shi, W. Yao, and Q. Niu,Magnetic control of the valley degree of freedom of massiveDirac fermions with application to transition metal dichalco-genides, Phys. Rev. B 88, 115140 (2013).[37] H. Zhou, C. Xiao, and Q. Niu, Valley-contrasting orbital mag-netic moment induced negative magnetoresistance, Phys. Rev.B 100, 041406(R) (2019).[38] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper definition ofspin current in spin-orbit coupled systems, Phys. Rev. Lett. 96,076604 (2006).[39] C. Xiao and Q. Niu, Conserved current of nonconserved quan-tities, Phys. Rev. B 104, L241411 (2021).[40] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal,Kwant: A software package for quantum transport, New J. Phys.16, 063065 (2014).023212-10https://doi.org/10.1038/s42005-020-00495-yhttps://doi.org/10.1103/PhysRevB.103.195309https://doi.org/10.1103/PhysRevB.93.174417https://doi.org/10.1103/PhysRevResearch.5.043052https://doi.org/10.1038/nature12187https://doi.org/10.1038/nature12186https://doi.org/10.1126/science.aat6981https://doi.org/10.1126/science.1199595https://doi.org/10.1038/s41467-017-02346-xhttps://doi.org/10.1038/s41586-021-03501-7https://doi.org/10.1088/2515-7639/aad585https://doi.org/10.1103/PhysRevLett.112.116601https://doi.org/10.1103/PhysRevB.103.115406https://doi.org/10.1088/2515-7639/ac452ahttps://doi.org/10.1126/sciadv.aay8897https://doi.org/10.1038/nphys2272http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.023212https://doi.org/10.1126/science.aad2102https://doi.org/10.1038/s41565-019-0547-2https://doi.org/10.1126/science.1237240https://doi.org/10.1021/nl402679bhttps://doi.org/10.1038/s41586-019-1704-3https://doi.org/10.1016/S0081-1947(08)60259-3https://doi.org/10.1103/PhysRevLett.95.137205https://doi.org/10.1103/PhysRevB.84.241106https://doi.org/10.1103/PhysRevB.88.115140https://doi.org/10.1103/PhysRevB.100.041406https://doi.org/10.1103/PhysRevLett.96.076604https://doi.org/10.1103/PhysRevB.104.L241411https://doi.org/10.1088/1367-2630/16/6/063065GENERATION AND CONTROL OF NONLOCAL CHIRAL … PHYSICAL REVIEW RESEARCH 6, 023212 (2024)[41] T. B. Boykin, M. Luisier, G. Klimeck, X. Jiang, N. Kharche,Y. Zhou, and S. K. Nayak, Accurate six-band nearest-neighbortight-binding model for the π -bands of bulk graphene andgraphene nanoribbons, J. Appl. Phys. 109, 104304 (2011).[42] B. Roy and K. Yang, Bilayer graphene with parallel magneticfield and twisting: Phases and phase transitions in a highlytunable Dirac system, Phys. Rev. B 88, 241107(R) (2013).[43] Y. H. Kwan, S. A. Parameswaran, and S. L. Sondhi, Twistedbilayer graphene in a parallel magnetic field, Phys. Rev. B 101,205116 (2020).[44] S. S. Pershoguba and V. M. Yakovenko, Energy spectrum ofgraphene multilayers in a parallel magnetic field, Phys. Rev. B82, 205408 (2010).[45] Y. Asakawa, S. Masubuchi, N. Inoue, S. Morikawa, K.Watanabe, T. Taniguchi, and T. Machida, Intersubband Landaulevel couplings induced by in-plane magnetic fields in trilayergraphene, Phys. Rev. Lett. 119, 186802 (2017).[46] M. M. Denner, J. L. Lado, and O. Zilberberg, Antichiral statesin twisted graphene multilayers, Phys. Rev. Res. 2, 043190(2020).[47] W. Qin and A. H. MacDonald, In-plane critical magnetic fieldsin magic-angle twisted trilayer graphene, Phys. Rev. Lett. 127,097001 (2021).[48] L. F. Torres and S. O. Valenzuela, A valley of opportunities,Phys. World 34, 43 (2021).[49] J. A. Delgado-Notario, W. Knap, V. Clericò, J. Salvador-Sánchez, J. Calvo-Gallego, T. Taniguchi, K. Watanabe, T.Otsuji, V. V. Popov, D. V. Fateev, E. Diez, J. E. Velázquez-Pérez, and Y. M. Meziani, Enhanced terahertz detection ofmultigate graphene nanostructures, Nanophotonics 11, 519(2022).[50] N. J. G. Couto, D. Costanzo, S. Engels, D.-K. Ki, K. Watanabe,T. Taniguchi, C. Stampfer, F. Guinea, and A. F. Morpurgo,Random strain fluctuations as dominant disorder source forhigh-quality on-substrate graphene devices, Phys. Rev. X 4,041019 (2014).[51] L. Wang, P. Makk, S. Zihlmann, A. Baumgartner, D. I. Indolese,K. Watanabe, T. Taniguchi, and C. Schönenberger, Mobilityenhancement ingraphene by in situ reduction of random strainfluctuations, Phys. Rev. Lett. 124, 157701 (2020).[52] D. Vaquero, V. Clericò, M. Schmitz, J. A. Delgado-Notario, A.Martín-Ramos, J. Salvador-Sánchez, C. S. A. Müller, K. Rubi,K. Watanabe, T. Taniguchi, B. Beschoten, C. Stampfer, E. Diez,M. I. Katsnelson, U. Zeitler, S. Wiedmann, and S. Pezzini,Phonon-mediated room-temperature quantum Hall transport ingraphene, Nat. Commun. 14, 318 (2023).[53] W. Kohn, Theory of Bloch electrons in a magnetic field: Theeffective Hamiltonian, Phys. Rev. 115, 1460 (1959).[54] D. Faílde and D. Baldomir, Orbital dynamics in 2D topologicaland Chern insulators, New J. Phys. 23, 113002 (2021).[55] M. Kindermann, B. Uchoa, and D. L. Miller, Zero-energymodes and gate-tunable gap in graphene on hexagonal boronnitride, Phys. Rev. B 86, 115415 (2012).[56] Y. D. Lensky, J. C. W. Song, P. Samutpraphoot, and L. S.Levitov, Topological valley currents in gapped Dirac materials,Phys. Rev. Lett. 114, 256601 (2015).[57] J. C. W. Song, P. Samutpraphoot, and L. S. Levitov, TopologicalBloch bands in graphene superlattices, Proc. Natl. Acad. Sci.USA 112, 10879 (2015).[58] T. P. Cysne, S. Bhowal, G. Vignale, and T. G. Rappoport,Orbital Hall effect in bilayer transition metal dichalcogenides:From the intra-atomic approximation to the Bloch states or-bital magnetic moment approach, Phys. Rev. B 105, 195421(2022).[59] N. M. R. Peres, Colloquium: The transport properties ofgraphene: An introduction, Rev. Mod. Phys. 82, 2673(2010).[60] S. Konschuh, M. Gmitra, and J. Fabian, Tight-binding theory ofthe spin-orbit coupling in graphene, Phys. Rev. B 82, 245412(2010).[61] J. C. Slater and G. F. Koster, Simplified LCAO method for theperiodic potential problem, Phys. Rev. 94, 1498 (1954).[62] Y.-W. Son, M. L. Cohen, and S. G. Louie, Energy gaps ingraphene nanoribbons, Phys. Rev. Lett. 97, 216803 (2006).[63] https://doi.org/10.54499/2022.07471.CEECIND/CP1718/CT0001.023212-11https://doi.org/10.1063/1.3582136https://doi.org/10.1103/PhysRevB.88.241107https://doi.org/10.1103/PhysRevB.101.205116https://doi.org/10.1103/PhysRevB.82.205408https://doi.org/10.1103/PhysRevLett.119.186802https://doi.org/10.1103/PhysRevResearch.2.043190https://doi.org/10.1103/PhysRevLett.127.097001https://doi.org/10.1088/2058-7058/34/11/40https://doi.org/10.1515/nanoph-2021-0573https://doi.org/10.1103/PhysRevX.4.041019https://doi.org/10.1103/PhysRevLett.124.157701https://doi.org/10.1038/s41467-023-35986-3https://doi.org/10.1103/PhysRev.115.1460https://doi.org/10.1088/1367-2630/ac29fchttps://doi.org/10.1103/PhysRevB.86.115415https://doi.org/10.1103/PhysRevLett.114.256601https://doi.org/10.1073/pnas.1424760112https://doi.org/10.1103/PhysRevB.105.195421https://doi.org/10.1103/RevModPhys.82.2673https://doi.org/10.1103/PhysRevB.82.245412https://doi.org/10.1103/PhysRev.94.1498https://doi.org/10.1103/PhysRevLett.97.216803https://doi.org/10.54499/2022.07471.CEECIND/CP1718/CT0001