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Saisab Bhowmik, Bhaskar Ghawri, Youngju Park, Dongkyu Lee, Suvronil Datta, Radhika Soni, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), Arindam Ghosh, Jeil Jung, U. Chandni

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[Spin-orbit coupling-enhanced valley ordering of malleable bands in twisted bilayer graphene on WSe2](https://mdr.nims.go.jp/datasets/37198236-ac33-4909-8854-4003faeb1228)

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Spin-orbit coupling-enhanced valley ordering of malleable bands in twisted bilayer graphene on WSe2Article https://doi.org/10.1038/s41467-023-39855-xSpin-orbit coupling-enhanced valleyordering of malleable bands in twistedbilayer graphene on WSe2Saisab Bhowmik1 , Bhaskar Ghawri 2, Youngju Park3, Dongkyu Lee3,4,Suvronil Datta1, Radhika Soni1, K. Watanabe 5, T. Taniguchi 6,Arindam Ghosh 2,7, Jeil Jung3,4 & U. Chandni 1Recent experiments in magic-angle twisted bilayer graphene have revealed awealth of novel electronic phases as a result of interaction-driven spin-valleyflavour polarisation. In this work, we investigate correlated phases due to thecombined effect of spin-orbit coupling-enhanced valley polarisation and thelarge density of states below half filling of the moiré band in twisted bilayergraphene coupled to tungsten diselenide. We observe an anomalous Halleffect, accompanied by a series of Lifshitz transitions that are highly tunablewith carrier density and magnetic field. The magnetisation shows an abruptchange of sign near half-filling, confirming its orbital nature. While the Hallresistance is not quantised at zero magnetic fields—indicating a ground statewith partial valley polarisation—perfect quantisation and complete valleypolarisation are observed at finite fields. Our results illustrate that singularitiesin the flat bands in the presence of spin-orbit coupling can stabilise orderedphases even at non-integer moiré band fillings.Topology of the Fermi surfaces and the density of states (DOS) at theFermi level govern various competing orders in quantum materials1,2.The formation of a broken-symmetry phase, such as amagnet, is oftentreated as an instability in the parent electron liquid phase, driven bysingularities3. For example, van Hove singularities (vHSs) which areassociated with saddle points of energy dispersion in momentumspace, feature strongly diverging DOS and favour localisation of elec-tronic states that stabilises phases such as density waves, ferro-magnetismand superconductivity1–5. Contrary to these ‘local’ vHSs, thewhole electronic band in the magic-angle twisted bilayer graphene(TBG) is nearly flat, with a large, ‘global’ DOS, that favours emergentcorrelated phases, including correlated insulators6–8, orbitalmagnets8–12, non-Fermi liquids13, and Chern insulators14–18, typically atinteger fillings of the moiré unit cell. Experiments in TBG have shownthat the inversion (C2)9,10,19 or time reversal (T ) symmetry breaking14,16can lift the degeneracy of the flat bands and polarise spin-valleydegrees of freedom leading to Chern insulators. Reports of anomalousHall effect (AHE) at zero magnetic field at moiré filling factors ν = 39,10and most recently at ν = 1, ± 220–22 necessitates a non-zero difference inthe occupation of electronic states of the two valleys, that produces afinite Berry curvature. While AHE and Chern insulators were mostsignificantly observed in TBG samples aligned with a hexagonal boronnitride (hBN) layer9,10 or with the application of a large magneticfield14–16, spin-orbit coupling (SOC) can also drive topological orderand symmetry-broken phases20,23,24. Proximity-induced SOC can breakC2T symmetry at zeromagnetic fields and polarise the charge carrierswithin a single valley20,25. The interplay of SOC, interactions andtopology, driven by the presence of vHSs has, however, remainedlargely unexplored26. The presence of vHSs within the nearly flatmoirébands can lead to enhanced correlations that can bemanifested in theReceived: 26 October 2022Accepted: 26 June 2023Check for updates1Department of Instrumentation andApplied Physics, Indian Institute of Science, Bangalore 560012, India. 2Department of Physics, Indian Institute of Science,Bangalore 560012, India. 3Department of Physics, University of Seoul, Seoul 02504, Korea. 4Department of Smart Cities, University of Seoul, Seoul 02504,Korea. 5Research Center for Functional Materials, National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan. 6International Centerfor Materials Nanoarchitectonics, National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan. 7Centre for Nano Science andEngineering, Indian Institute of Science, Bangalore 560012, India. e-mail: saisabb@iisc.ac.in; jeiljung@uos.ac.kr; chandniu@iisc.ac.inNature Communications |         (2023) 14:4055 11234567890():,;1234567890():,;http://orcid.org/0000-0002-1303-2437http://orcid.org/0000-0002-1303-2437http://orcid.org/0000-0002-1303-2437http://orcid.org/0000-0002-1303-2437http://orcid.org/0000-0002-1303-2437http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0001-5188-4617http://orcid.org/0000-0001-5188-4617http://orcid.org/0000-0001-5188-4617http://orcid.org/0000-0001-5188-4617http://orcid.org/0000-0001-5188-4617http://orcid.org/0000-0002-4275-2687http://orcid.org/0000-0002-4275-2687http://orcid.org/0000-0002-4275-2687http://orcid.org/0000-0002-4275-2687http://orcid.org/0000-0002-4275-2687http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39855-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39855-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39855-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39855-x&domain=pdfmailto:saisabb@iisc.ac.inmailto:jeiljung@uos.ac.krmailto:chandniu@iisc.ac.inemergence of instabilities over a narrower range of tunable, non-integer moiré band fillings.In this work, we investigate the non-integer filling regime of themoiré band in TBG proximitised by tungsten diselenide (WSe2). Wereport signatures of valley polarisation along with Fermi surfacereconstructions that suggest a Stoner-like instability favoured by vHSsin the vicinity of ν < 2. Significantly, the band reconstructions aremalleable and can be tuned via a combination of carrier density andmagnetic field.ResultsFigure 1a shows the schematic of our device consisting of amultilayerWSe2 onmagic-angle TBG encapsulated by two hBN layers. The four-probe longitudinal resistance Rxx as a function of filling ν at a mag-netic field B = 0 shows well-defined peaks at the charge neutralitypoint (CNP) ν = 0 and half fillings ν = ± 2 of the conduction (+) andvalence (−) bands (Fig. 1b). We estimate the twist angle to be θ ≈ 1.17°.The data presented in Fig. 1 were taken at a temperature T = 0.3 K.The resistive peaks at ν = ± 2 were weaker compared to TBG withoutWSe2 and were found to be semi-metallic rather than purelyinsulating20,24,27. In Fig. 1c, the Hall resistance Rxy at low B-fields, moststrikingly, shows a hysteretic behaviour over a wide range of fillingsν < 2, as the Fermi energy is swept back and forth. The hysteresisbecomes narrower with increasing B-field and vanishes at B ≈ 1.2 T.Notably, while Rxy shows zero-crossings at B = 50mT, no sign changeis observed for higher fields up to B = 1.2 T with Rxy remainingnegative for B > 50 mT. This feature will be discussed in detail inFig. 2. It is evident that Rxy strongly depends on the history of thesample in out-of-plane B-field training, and this leads to a non-zeroRxy at B = 0 when the field is swept back and forth (Fig. 1d). To oursurprise, we find an abrupt sign change in the hysteresis of Rxy atν = 1.86 (Fig. 1e). The magnitude of the coercive field, where thehysteresis disappears, is about one order of magnitude higher thanprevious reports on moiré systems9,10,20,28–32. The large coercive fieldsuggests a more robust ferromagnetic phase than in previousexperiments and may also indicate domain wall pinning due to dis-order and local inhomogeneities in the twist angle. We expect thatthe coercive fields will couple more strongly with the orbital mag-netic moments rather than the spin whose shifts in energy are typi-cally of ~0.1meVper Tesla. The hysteresis inRxywith respect to both νand B suggests that the sample remains magnetised without anyexternal magnetic influence. We note that the measured Rxy is muchlower than the quantum of resistance (h/e2, where h is the Planck’sconstant and e is electronic charge). The width of the hysteresis in B-field changes as the carrier density is tuned in the vicinity of ν < 2, asevident from the colour plot in Fig. 1e, where we have plotted thedifference in Rxy for two opposite directions of field sweep,ΔRxy =Rxyð B Þ� Rxyð~BÞ, as a function of B and ν. Surprisingly, weobserve a significant asymmetry between positive (B+) and negativecoercive fields (B−). The sudden switching behaviour of Rxy as afunction of density at ν = 1.86 is accompanied by a reversal of theasymmetry between B+ and B− coercive fields. We quantify the latterfeature using the parameter α, defined as α = ∣B−∣/∣B+∣, obtaining α > 1for ν ≤ 1.85 and α < 1 for ν > 1.85 (bottom panel of Fig. 1e). Additionaldata using various combinations of contacts can be found inthe Supplementary Information (see Figs. 10–13). We also note thatferromagnetism is observed over 60% of the total area in our sample(see Supplementary Fig. 1).-0.8 -0.4 0.0 0.4-0.30.00.3-0.30.00.3-0.30.00.3B (T)= 1.87BRxy (k) = 1.85B= 1.561.6 1.7 1.8 1.9 2.0-0.10.00.1-0.40.0-1.2-0.8-0.4B = 0.05 TRxy (k)B = 0.4 TB = 1.2 T-4 -3 -2 -1 0 1 2 3 4110Rxx (k)= 1.17T = 0.3 K, B = 0 T1.6 1.7 1.8 1.91.01.5-0.50.00.5B)T(-0.12 0.14Rxy (k )a bc deVxy VxxI VGBFig. 1 | Ferromagnetism and valley polarisation at ν < 2. a Schematic of hBN-encapsulatedTBG-WSe2 heterostructureon SiO2/Si substrate. The carrier density inthe system is tuned by applying top gate voltage VG. The longitudinal Vxx andtransverseVxy voltages aremeasuredbydriving current I through the channel of theHall bar device in the presence of an out-of-planemagnetic field B. The black arrowindicates the direction of the magnetic field. b Four-probe longitudinal resistanceRxx as a function of filling ν measured at T =0.3 K and B =0 T. c Hall resistance Rxyfor three perpendicular magnetic fields B =0.05, 0.4 and 1.2 T for density beingswept back and forth. The red and blue arrows indicate directions of density sweep.The horizontal black dashed lines at Rxy =0 are drawn for a better illustration of thezero crossings in Rxy that disappear with increasing B-field. d Rxy at three differentfillings for B swept back and forth, as indicated by the arrows. A reversal ofhysteresis is seen at ν = 1.86. The coercive fields are indicated as B+ and B−. e Colourplot of ΔRxy, defined as the difference between the values of Rxy for the oppositefield sweeps, as a functionof ν andB. The change in colour along the vertical dashedline shows the reversal of magnetisation that accompanies the occupation ofelectrons in different K and K 0 valleys (see the inset schematics). The red and bluecircles around two valleys represent the electronic orbital, and the difference inthickness of these circles indicates a valley imbalance. In the bottompanel, the ratioof the magnitude of negative and positive coercive fields α is plotted as a functionof ν. Coercive fields are asymmetric in positive and negative B. The asymmetry flipsexactly when the magnetisation is reversed as evident from the data points above(α > 1) and below (α < 1) the horizontal black dashed line at α = 1. The error bars indetermining the coercive fields are negligible.Article https://doi.org/10.1038/s41467-023-39855-xNature Communications |         (2023) 14:4055 2Ferromagnetism at ν = 2 is unexpected in TBG since a valley-polarised ground state is energetically unfavourable due to inter-valleyHund’s coupling19. However, the SOC together with the gap openingterms can lead to valley-polarised isolated flat bands in TBG at ν = 220,25(Supplementary Fig. 15). Presence of proximity-induced SOC is con-firmed by weak antilocalisation measurements in our device (Supple-mentary Fig. 8). The reversal of magnetisation is strong evidence forspontaneous switching of valley polarisation induced by tuning thecarrier density. A ferromagnet can be classified as spin or orbital,depending on whether the magnetisation is due to spontaneous spinor valley polarisation. In an orbital Chern insulator, the magnetisationcan jump abruptly when the chemical potential crosses the Chern gapif it can trigger reordering of the bands that are filled. The edge statecontribution is sufficient to change the sign ofmagnetisation simplybytuning the density below the gap of an orbital Chern insulator30,33,34.Therefore, the abrupt reversal of hysteresis indicates the dominanceoforbital magnetism over spin magnetism, and the energeticallyfavourable ground state is solely determined by the gate voltages inweak magnetic fields. We note that the valley polarisation is affectedmore strongly by subtle changes in the shape of the Fermi surface as itcan abruptly modify the momentum space exchange condensationthat tends to bunch together electrons that are closer to each other ink-space, whose Berry curvatures contributing to orbital magnetisationare highly variable unlike the electron spins. The observation of a non-quantised Rxy, however, suggests that the ground states have partiallyvalley-polarised bandswith the unequal occupation of different valleysas a function of carrier density. Partial valley polarisation is notincompatible with the intervalley-coherent phases proposed inliterature35–37 that can mitigate a fully valley-polarised phase. We note,however, that the SOC terms by themselves do not mix electronicstates from K and K 0 and are not the microscopic origin for theintervalley-coherent phases. The sign switching of valley polarisationsas a function of density leads to an abrupt reversal of magnetisation(Fig. 1e) indicating a clear phase transition point between these com-peting phases for magnetic fields below ~0.5 T.Having established the orbital nature of the ferromagnet at ν < 2,we now turn to the zero-crossings in Rxy that accompany the hysteresisin ν at B = 50 mT (Fig. 1c). The ν-dependence of Hall density nH givesinsights into the Fermiology of a system. In Fig. 2a, nH = − (1/e)(B/Rxy) isplotted as a function of ν at four different low B-fields, but at a highertemperature T = 2 K, where ferromagnetism disappears (Supplemen-tary Fig. 6) and the Hall data is independent of the direction of densitysweep. We observe a rich sequence of sign changes and resets in nHaround ν < 2, particularly for the lowest fields 20 mT and 50 mT.Assuming a single particle energy band diagram for TBG, the DOS isexpected to show a vHS around ν = 2 (see Fig. 2b, top right panel). Asthe Fermi energy is swept through the vHS, a Lifshitz transition isexpected that changes the topology of the Fermi surface, flipping thesign of nH with a logarithmically diverging profile, as shown in the topleft panel38. However, when the bands are malleable, as the FermiFig. 2 | Fermi surface reconstructions and malleability of bands at ν < 2. a Halldensity plotted as a function of ν for B =0.02, 0.05, 0.1 and 0.2 T at T = 2 K. Lifshitztransitions and reset of charge carriers are seen for B =0.02 and 0.05 T, whereasonly reset of charge carriers occurs forB =0.1 and0.2T.b Expected behaviour ofnHas a function of ν and the corresponding density of states D(E) vs energy E profile.The purple line indicates a Lifshitz transition with a sign reversal in Hall density,while the green line shows a reset where the Hall density reaches a minimum valuewithout a sign-change consistent with the splitting of D(E) shown by the greenarrow. The vertical orange dashed lines indicate the position of Fermi energy EF.c nH vs ν atB =0.05T for the shaded region in Fig. 2a. Different colour bars are usedto indicate Lifshitz transitions (purple) flanking the reset of carriers (green).d, e Temperature dependence of Lifshitz transitions at B =0.05 T and reset atB =0.2 T. f D(E) (black parabola) as a function of E showing a finite imbalance inoccupation of states between K and K 0 valleys leading to orbital magnetism. Thehorizontal orange dashed line indicates the position of Fermi energy EF.Article https://doi.org/10.1038/s41467-023-39855-xNature Communications |         (2023) 14:4055 3energy approaches the peak in the DOS, it can reset the bands andproduce a split DOSprofile as shown in the bottom right panel16,24. Thisleads to a ‘reset’ of charge carriers, where nH drops to a low valuebefore rising again,without a sign change.Our experiments reveal a setof phase transitions in comparison to previous reports onmagic-angleTBG,where a reset is typically observed near ν = 216,24,39. A closer look atour data near ν < 2 at 50mT shows two Lifshitz transitions that flank areset, indicated by the colourbars in Fig. 2c. We note that the vHSwithin the nearly flat bands can shift the density of states weights forsmall changes in the twist angle40,41. Surprisingly, our experimentsreveal tunability of the DOS with B-field and density, further validatingthemalleability of the TBG bands. The Lifshitz transitions disappear atB = 100mT, and nH shows a peak-like feature that decreases to zeroand increases slowly (Fig. 2a). Such a ‘reset’ of charge carriers at arelatively higher field, with no additional Lifshitz transitions, indicatesB-field-driven changes in the DOS of the bands. Fig. 2d, e shows thatthese phase transitions become weaker and fade away with increasingtemperature. Remarkably, these distinct features in nH appear aroundthe same density ν < 2 where we have observed ferromagnetism atlower temperatures of T ≤ 1 K. The flat band condition of on-siteCoulomb interactions (U) dominating over the kinetic energy of thecarriers, and the diverging DOS around ν < 2 easily satisfy the Stonercriterion of ferromagnetism UD(EF) > 1, where D(EF) is the DOS at theFermi energy EF3,11,42. We speculate that such a strong instability intheDOS at ν < 2 favours spontaneous valley polarisation, leading to theobserved AHE along with the switching of magnetisation (Fig. 1d, e).Our theoretical calculations discussed below show that spin polarisa-tion together with SOC assists valley polarisation. Thus, a valley-polarised orbital magnet with a non-zero spin polarisation should befavoured over a valley-polarised magnetic phase without a net spinpolarisation. In Fig. 2f, we illustrate the scenario where conventionalStoner spin polarising ferromagnetic phase is accompanied by valleypolarisation where K and K 0 valleys are unevenly occupied.To gain further insights into the possible ground state at half-filling, we have measured Rxx and Rxy simultaneously in a B-field up toB = 10 T, at T =0.3 K. A series of symmetry-broken Chern insulators inthe form of minima in Rxx and wedge-like features in Rxy emerge fromdifferent fillings (Fig. 3a, b). The Chern insulators can be characterisedby fitting the Diophantine equation, n/n0 =Cϕ/ϕ0 + s, where n0 is thedensity corresponding to one carrier permoiré unit cell,C is the Chernnumber, ϕ is the magnetic flux per moiré unit cell, ϕ0 = h/e is the flux-quantum, and s is the band filling index or the number of carriers perunit cell at B =0 T. For sufficiently strong magnetic fields the four foldspin-valley degeneracy is completely lifted near the CNP: (C, ν) = (±1,0), (±2, 0), (±3, 0), (±4, 0). In addition, we observe states emanatingfrom different integers ν as (C, ν) = (+3, +1), (±2, ±2), (+4, +2), (±1, ±3)(Fig. 3c). The Chern insulator C = 2 at ν = 2 is perfectly quantised to h/2e2 at a high B-field (Fig. 3d). In TBG devices, such topological incom-pressible insulators have been described within the picture of isolatedeight fold bands with broken T symmetry, where the Chern numbersof the bands are the same in the two valleys14–16, but opposite forvalence and conduction bands (Fig. 3e). The valley imbalanced fillingFig. 3 | Quantised orbital Chern insulator at ν = 2 and possible Chern bands.a, b Landau fan diagram in longitudinal resistance Rxx and transverse resistance Rxyplotted as a function of ν for different B up to 10 T. c Fitting of the Diophantineequation along the minima in Rxx and wedge-like states in Rxy. The slopes of thestraight lines give the Chern numbers C. d Rxy data at B = 10 T plotted for a smalldensity range near ν = 2 marked by the blue colourbar on the top axis in panel 3c.Rxy is quantised to h/2e2 accompanied by aminima inRxx at B = 10 T. e Schematic offlat bands with different Chern numbers via valley polarising T -symmetry breaking(TRS) and proximity-induced SOC. The dark and light colours represent each spin-valley flavour’s lower and upper bands, respectively. In the presence of SOC, thespin-valley flavour degeneracy can be completely lifted, leading to 8 isolated bandsfor appropriate system parameters. Blue(green) and red(yellow) indicate the upanddown spincomponents for valleyK(K 0). ThenetChernnumberCnet obtainedbyadding the Chern numbers of the filled bands at ν = 2 (horizontal black dashed line)is +2 in both cases. f Schematic of valley Chern bands giving rise to Cnet = + 2 atν = + 2. The band degeneracy can be lifted completely in the presence of SOC and aspin splitting exchange field that naturally accompanies a ferromagnetic spinpolarised phase.Wehaveused the SOCparameters forgrapheneonWSe2 followingref. 43 togetherwith the exchangefield λex = 0.5meV as summarised in themethodssection. The degeneracy split Chern bands lead to finite orbital magnetism M(orb)that changes with the filling density ν, shown here from −4 to 4 in the right-mostsub-panel. We note the changing signs in the total orbital magnetisation due to therelative filling of K↓(K 0 ") bands near ν = + 2, suggesting that delicate changes inlevel orderingwith carrier density due toCoulomb interactions can strongly impactthe net magnetisation.Article https://doi.org/10.1038/s41467-023-39855-xNature Communications |         (2023) 14:4055 4of the bands results in the netChernnumber observed, consistentwithprevious studies. While a large B-field is usually used to break T -sym-metry, proximity-induced SOC in graphene due to WSe2 breaks C2Tsymmetry inherently at B =025 and together with an exchange field itcan generate spin-valley degeneracy-lifted bands (Fig. 3e). Moreover,the spin-valley flavour resolved Chern numbers can be tuned byvarying the sublattice splitting energy and Ising SOC in TBG-WSe2systems. The values of exchangefields are expected to changewith thedegree of spin polarisation at the onset ofmagnetism that depends onspecific system parameters and Coulomb interactions, see Supple-mentary Information and Fig. 16 that shows how different initial con-ditions for a mean-field self-consistent Hubbard model result in meta-stable spin-polarised states at different carrier densities. We illustratein Fig. 3f, the spin-valley resolved band degeneracy lifting introducedby a finite exchange field of λex = 0.5 meV in the Hamiltonian thatmodels the spin polarisation of a ferromagnetic phase, together with aproximitised SOC term discussed in the methods section. In the SOCmodel in Eqs. ((1)–(2)) we have used the Rashba coupling termλR =0.56meV following ref. 43, while the proximity induced λR inBernal bilayers is expected to be almost an order of magnitude larger.A range of λR, including a larger value comparable to those of Bernalbilayers, were also considered inmodels of twisted bilayer graphene incontact with WSe2 when calculating the valley Chern numbers phasediagram of the low energy bands25 as a function of other SOC andsublattice potential parameters. The associated spin-valley resolvedbands develop a well defined Chern number that will lead to a finiteorbital moment when they are filled. We illustrate by using frozenbands how the total orbital moment evolves with filling density givingrise to a magnetisation of the order of ~10−2μB/AM that can change itssign depending on the specific carrier density value. Since the orbitalmagnetisation depends sensitively on the actual exchange field for thespecific spin configuration, see Supplementary Figs. 17, 18, it isexpected that its integrated magnitude, as well as the local values inexperiments, vary considerably with density, especially near the phasetransition points. Experimentally, the orbital magnetisation maps canreach local values as large as a few μB/AM44,45. In our experiments, anabrupt change in orbital moment as a function of filling indicates thatreordering of levels at ν = 1.86 takes place due to a close competitionbetween the magnetic phases of opposite signs.In Fig. 4a, we have presented a diagram with the summary ofvarious phases discussed throughout this report. The Lifshitz transi-tions and reset of carriers at B = 20 and 50mT occur at the densitiesnear the extreme left boundary of the ferromagnetic domain as well aswithin the domain. At B ≥ 100mT the Lifshitz transitions disappear,and we find a second reset near the extreme right boundary of theferromagnetic domain (inset in Fig. 4). As evidenced from the diagram,ferromagnetism is accompanied by a series of Fermi surface recon-structions at the vHSs.We also highlight that our sampleexhibits valleypolarisation in two different scenarios: First, anomalous Hall sig-natures at ν < 2. Second, time reversal symmetry-broken Chern insu-lator at exactly ν = 2.While bothmechanisms are expected to give a netChern number of C = 2 (Fig. 3e), the experimental signatures are radi-cally different in the sense that Rxy is hysteretic and non-quantised atB =0, whereas it becomes fully quantised to h/2e2 at high B-field (seeSupplementary Fig. 14). These observations indicate that the nature ofthe valley-polarised ferromagnetic ground state is distinct from that oftheChern insulators at highB-field. Finally, in the context of a veryhighcoercive field in our data, we have plotted coercive field reported inseveralmoiré graphene systems at different ν (Fig. 4b). The plot clearlyindicates the coercive field observed in our work is the highest incomparison to other reports to date.DiscussionOur AHE results differ from those reported recently at ν = ± 2 in hBN-aligned TBG without WSe2, having twist angles slightly away from themagic angle21. It was speculated that the combined effect of increasingbandwidth away from the magic angle and staggered sublatticepotential arising from the hBN alignment stabilises the magnetic phaseat half-filling. In our experiments, while hBN was not aligned with gra-phene layers as is evident from the low resistance at the CNP, we cannotcompletely rule out the effects of the increased band dispersion, whichalong with SOC, may act as an alternative mechanism that polarises thevalleys. Our data, alongwith previous reports, also indicate that deviceswith identical twist angles may show widely varying properties,1.6 1.8 2.00.10.21.6 1.8 2.0 2.20123B (T)ab-2 -1 0 1 2 3 40.00.20.40.60.8TBG [10] [9] [21] [22]TMBG [4, 30] [31]TDBG [28]ABC-TLG/hBN [29]TBG/WSe2 [20] [32] This workCoercive field (T)FerromagnetismLifshitz transitionReset of carriersFig. 4 | Summary of the various phases observed. a Four different colours areused to indicate the extent of the phases in the density and magnetic field sub-space; Ferromagnetism (orange), Chern insulator (sky blue), Lifshitz transitions(red) and reset (blue). The inset shows low field phase diagram up to B =0.2 T for adensity range of ν = 1.5−2 in the shaded region of the main panel. The densitiescorresponding to Lifshitz transitions and reset are marked by red and blue circles,respectively.While ferromagnetism, Lifshitz transitions and reset occur below ν = 2,the Chern insulator emanates from exactly ν = 2 outside the ferromagnetic region.b Coercive field as a function of ν for various reports on graphene-based moirésystems including TBG, twisted monolayer bilayer graphene (TMBG), twisteddouble bilayer graphene (TDBG), ABC-trilayer graphene (TLG) aligned with hBN,and TBG/WSe2 clearly indicates the large coercive field observed in this work.Article https://doi.org/10.1038/s41467-023-39855-xNature Communications |         (2023) 14:4055 5suggesting the presence of an unknown set of parameters that governsthe band structures. Although the discrepancies among samples areoften attributed to twist angle disorder, strain, and dielectric screening,it is not clear how these factors affect the driving mechanisms for thevarious correlated phases observed. Notably, AHE has been reported byonly a few groups, mainly in single devices, with only a fraction of thetotal area of the samples exhibiting ferromagnetism (see Supplemen-tary Table 1). The absence of hysteresis in electrical transport mea-surements does not guarantee that a magnetic phase is truly absent.Disorder, substrate potentials, twist angle inhomogeneity, and straincan interrupt the propagation of edge modes between the transverseprobes in the device and obstruct the measurement of actual magne-tisation. The entire sample can still retain local magnetic moments,although it becomes difficult to measure the magnetisation globallyusing electrical transport. Recent experiments using a scanning super-conducting quantum interference device on the tip have demonstratedimaging of such local orbital magnetic domains44,45. Surprisingly, asubstantial part of the sample was found to acquire local Berry curva-tures andCherngaps, even in the absenceof local hysteresis45. Althoughthe lack of reproducibility across devices and experimental groups is asignificant issue, our observation of AHE near ν = 2 over a wide region inour sample clearly indicates the robustness of our results. Futureexperiments with well-controlled external perturbations and betterfabrication protocols will be essential to gain insights into the origin ofthe observed variability among the samples.To summarise, our experiments have revealed a phase diagramofcompeting phases in TBG near its first magic angle, which indicatesvHSs within the quasi-flat bands. This diverging DOS within the nearlyflat bands of TBG reveals a finer internal structure of the bands that ismanifested through multiple phase transitions as a function of mag-netic fields and carrier densities in the vicinity of ν ~ 1.8. Uncovering theunderlying physics of the various quasi-degenerate, competingground states will require an overarching theoretical analysis ofstrongly interacting many-body physics. Our primary findings of AHEand Fermi surface reconstructions are reported away from the usualcommensurate filling of ν = 2. The various features in our data,including abrupt reversal of magnetisation, and non-quantised Rxy areclear signs of orbital magnetism, and partially valley-polarised groundstate, where both bulk and edge modes are expected to contribute tothe transport. The bulk transport may be affected by percolatingconduction channels between topological domains of closely com-peting phases where external electric ormagnetic fields can be used ascontrol knobs to favour certainphases over theother.Varying the twistangle between the graphene sheet and the WSe2 can modify theproximity SOC strength that would, in turn, modify the phase diagramof the expected ground states. The high sensitivity of the electronicstructure to experimental conditions makes it a challenge to performexperiments reproducibly and simultaneously provides an opportu-nity to explore the physics near multi-phase transition points wherethe electronic response functions will be particularly sensitive toexternal perturbations. An interesting future research direction wouldbe to identify the connection, if any, between the valley-polarisedStoner magnet found in our work and the superconducting phase inthe vicinity of ν = ± 27,8,27,39,46.MethodsDevice fabricationThe well-known ‘tear and stack’ method was used to assemble theheterostructure in this work. Polypropylene carbonate (PPC) filmcoated on a polydimethylsiloxane (PDMS) stamp was used for pickingup individual layers of hBN of thickness 25–30nm, WSe2, and gra-phene. The sharp edge of the top hBN was used to tear the graphene,following which one half of the torn graphene was picked up, leavingthe other half on the substrate. The sample stage was then rotated by1. 2° (marginally higher than the magic angle), the second half of thegraphene layer was picked up. Next, the bottom hBN was picked up,and the heterostructure was released on a 285 nm SiO2/Si substrate at90 ∘C. The final device was etched into a multi-terminal Hall bar byreactive ion etching using CHF3/O2 followed by electron-beam litho-graphy, and thermal evaporation of Ohmic edge contacts and top gateusing Cr/Au (5 nm/60 nm). WSe2 layer with a thickness of ~3 nm wasexfoliated from bulk crystals procured from 2D Semiconductors.Transport measurementsElectrical transport measurements were performed in a He3 cryostatwith a 10 T magnetic field and a cryogen-free, pumped He4 cryostatwith a 9 T magnetic field. Magnetotransport measurements were car-ried out with a bias current of 10 nA, using an SR830 low-frequencylock-in amplifier at 17.81 Hz. The carrier density in the system wastunedby the top gate. The twist anglewas estimatedusing the relation,ns =8θ2=ffiffiffi3pa2 where a =0.246nm is the lattice constant of grapheneandns (ν = 4) is the charge carrier density corresponding to a fullyfilledsuperlattice unit cell. The twist angle determined from Rxxdata at B = 0T and Landau fan diagram are in good agreement. For the measure-ments of hysteresis in Rxy, Onsager reciprocity theorem9,47 was used,details of which are given in the Supplementary Information.Proximity SOC in Graphene/WSe2The interlayer coupling, in particular, the proximity SOC induced in thegraphene sheet on top of a WSe2 can be modelled by combining sub-lattice dependent site potential differences together with Rashba andintrinsic SOC terms48. In the following,we briefly outline how the bandsof graphene can be altered under the proximity SOC effects of WSe2.Continuum model bandsWe model the single-layer graphene Hamiltonian contacting a TMDlayer through the staggered potential (U), exchange field (ex), intrinsic(I) and Rashba (R) SOC, and pseudospin inversion asymmetry (PIA)43.hi ! hi +hU +hI +hex +hR +hPIA ð1ÞwithhU =Δσz1s1τhI =12λAI ðσz + σ0Þ+ λBI ðσz � σ0Þh iηszhex = λex1σsz1τhR = λRðησxsy � σysxÞ1τhPIA =a2λAPIAðσz + σ0Þ+ λBPIAðσz � σ0Þh ikxsy � kysx� �1τð2Þwhere σ and s are Pauli matrices that represent the A/B sublattice and↑/↓ spin, and 1 is a 2 × 2 identity matrix. The simultaneous presence ofa spin-splitting exchange field and a SOC term is known to introducevalley polarisation. For instance, an intrinsic SOC captured through theKane-Mele model together with sublattice staggering potentialintroduces unequal gaps at the K and K 0 valleys that leads to ananomalous Hall effect by populating a given spin-valley bandwhen thesystem is carrier doped. Similarly, a bilayer graphene system subject toa Rashba SOC and spin-splitting exchange field is known to give rise toan anomalous Hall effect for appropriate system parameters49. Bothexamples illustrate different mechanisms for the onset of ananomalous Hall effect when SOC is accompanied by an exchange fieldthat separates the spin-up and down bands in a ferromagnetic phase.The Rashba SOC mixes spins but does not mix valleys. The bandstructure of Graphene (G)/WSe2 in Supplementary Fig. 17 illustrateshow the two-fold low energy nearly flat bands are split into eightdifferent bands due to spin-valley degeneracy lifting using a modelsystem, whereas explicit mean-field calculations for the Hubbardmodel are presented in the Supplementary Information to illustrateArticle https://doi.org/10.1038/s41467-023-39855-xNature Communications |         (2023) 14:4055 6the sensitivity of the calculated results to different initial conditionsand carrier densities. The high sensitivity of the orbital level orderingto spin configuration also leads to sensitive changes in the orbitalmoments with exchange field parameters. We illustrate in Supplemen-tary Fig. 18 the small changes in λex up to ~4meV inmagnitude, which issufficient to change the number of nodes crossing zero and thereforeresulting in sign flips of the orbital magnetisation slopes as a functionoffilling, in turn, relatedwith the spin-valleyChernnumbers. Below,wewrite the pristine TBG continuum model Hamiltonian for one spin-valley flavour as 8 × 8 matrix to emphasize the three dominantinterlayer tunneling50Hη0ðkÞ=h1ðkÞ t12ðk0Þ t12ðk + Þ t12ðk�Þty12ðk0Þ h2ðk0Þ 0 0ty12ðk+ Þ 0 h2ðk + Þ 0ty12ðk�Þ 0 0 h2ðk�Þ0BBBB@1CCCCA ð3Þwithh1ðkÞ= _υF0 ðηkx � ikyÞeið+θ2Þðηkx + ikyÞeið�θ2Þ 0 !,h2ðkÞ= _υF0 ðηkx � ikyÞeið�θ2Þðηkx + ikyÞeið+θ2Þ 0 !,t12ðkjÞ=ω0 ωeið2π=3Þjωeið2π=3Þj ω0 !ð4Þwherek and kj are the wave vectorsmeasured from the K(η) of the layer1 and 2 with the valley index η = ± 1, and kj =k +Gj with j =0, + , − areconnected with the three moiré reciprocal lattice vectors, whichadjusts the momentum difference between two different Dirac pointsfrom layer 1 and 2. The h1 and h2 describe the Dirac bands of each layerwhere we use the Fermi velocity υF = 1 × 106 m/s and the twist angle θ.The t12 term is the tunneling matrix between the two graphene layerswith tunneling constantsω = 0.12 eV andω0 =0:0939 eV. For the actualcalculation, we use 392 × 392 Hamiltonianmatrices for each valley andeachkpoint, which includes the hopping termsbetween the two spins,and use four sublattices, and 49 reciprocal lattice points51. The valleymixing terms are not included in our spin-orbit coupling models.To understand the topological phase transition along with theband filling ν, we obtain the orbital magnetisation M(orb)(μ) as a func-tion of chemical potential μ33 usingMðorbÞðμÞ=XnZd2kð2πÞ2f ðμ� εnðkÞÞ ðεn + εn0 � 2μÞ×e_ImXn0≠nhn∣∂kxH∣n0ihn0∣∂kyH∣niðεn � εn0 Þ2" #,ð5Þwhere f(E) is the Fermi-Dirac distribution, εn and ∣ni are the eigenenergy and vector of the n-th band. The Chern number C = (2πℏ/e)dM(orb)/dμ canbe estimated by the slope of theM(orb) v.s. μ graphs (SeeSupplementary Fig. 18).Data availabilitySource data are provided with this paper. All other data that supportthe plots within this paper and other findings of this study are availablefrom the corresponding author upon request. 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B 89, 205414 (2014).AcknowledgementsWe gratefully acknowledge the usage of the MNCF and NNFC facilitiesat CeNSE, IISc. U.C. acknowledges funding from SERB via SPG/2020/000164 and WEA/2021/000005. Y.J.P. was supported by the KoreanNational Research Foundation grant NRF-2020R1A2C3009142, andD.L. was supported by grant NRF-2020R1A5A1016518, as well as theKoreanMinistry of Land, Infrastructure and Transport (MOLIT) from theInnovative Talent Education Programme for Smart Cities. J.J. wassupported by the SamsungScience and Technology Foundation underproject SSTF-BAA1802-06. We acknowledge computational supportfromKISTI through the grant KSC-2021-CRE-0389 and the resources ofUrban Big Data and AI Institute (UBAI) at the University of Seoul and thenetwork support from KREONET. K.W. and T.T. acknowledge supportfrom JSPS KAKENHI (Grant Numbers 19H05790, 20H00354, and21H05233).Author contributionsS.B. fabricated the device, performed the measurements, and analysedthe data. B.G. contributed to measurements and analysis of data. Y.J.P.,D.L., and J.J. performed the theoretical calculations. S.D. and R.S.assisted in measurements. K.W. and T.T. grew the hBN crystals. A.G.advised on experiments. U.C. supervised the project. S.B., J.J., and U.C.wrote the manuscript, with inputs from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-39855-x.Correspondence and requests for materials should be addressed toSaisab Bhowmik, Jeil Jung or U. Chandni.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to thepeer reviewof thiswork. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-39855-xNature Communications |         (2023) 14:4055 8https://arxiv.org/abs/2205.05225https://arxiv.org/abs/2205.05225https://doi.org/10.1038/s41467-023-39855-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Spin-orbit coupling-enhanced valley ordering�of malleable bands in twisted bilayer graphene on WSe2 Results Discussion Methods Device fabrication Transport measurements Proximity SOC in Graphene/WSe2 Continuum model bands Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information