Abstract
Recent experiments in magicangle twisted bilayer graphene have revealed a wealth of novel electronic phases as a result of interactiondriven spinvalley flavour polarisation. In this work, we investigate correlated phases due to the combined effect of spinorbit couplingenhanced valley polarisation and the large density of states below half filling of the moiré band in twisted bilayer graphene coupled to tungsten diselenide. We observe an anomalous Hall effect, accompanied by a series of Lifshitz transitions that are highly tunable with carrier density and magnetic field. The magnetisation shows an abrupt change of sign near halffilling, confirming its orbital nature. While the Hall resistance is not quantised at zero magnetic fields—indicating a ground state with partial valley polarisation—perfect quantisation and complete valley polarisation are observed at finite fields. Our results illustrate that singularities in the flat bands in the presence of spinorbit coupling can stabilise ordered phases even at noninteger moiré band fillings.
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Introduction
Topology of the Fermi surfaces and the density of states (DOS) at the Fermi level govern various competing orders in quantum materials^{1,2}. The formation of a brokensymmetry phase, such as a magnet, is often treated as an instability in the parent electron liquid phase, driven by singularities^{3}. For example, van Hove singularities (vHSs) which are associated with saddle points of energy dispersion in momentum space, feature strongly diverging DOS and favour localisation of electronic states that stabilises phases such as density waves, ferromagnetism and superconductivity^{1,2,3,4,5}. Contrary to these ‘local’ vHSs, the whole electronic band in the magicangle twisted bilayer graphene (TBG) is nearly flat, with a large, ‘global’ DOS, that favours emergent correlated phases, including correlated insulators^{6,7,8}, orbital magnets^{8,9,10,11,12}, nonFermi liquids^{13}, and Chern insulators^{14,15,16,17,18}, typically at integer fillings of the moiré unit cell. Experiments in TBG have shown that the inversion (C_{2})^{9,10,19} or time reversal (\({{{{{{{\mathcal{T}}}}}}}}\)) symmetry breaking^{14,16} can lift the degeneracy of the flat bands and polarise spinvalley degrees of freedom leading to Chern insulators. Reports of anomalous Hall effect (AHE) at zero magnetic field at moiré filling factors ν = 3^{9,10} and most recently at ν = 1, ± 2^{20,21,22} necessitates a nonzero difference in the occupation of electronic states of the two valleys, that produces a finite Berry curvature. While AHE and Chern insulators were most significantly observed in TBG samples aligned with a hexagonal boron nitride (hBN) layer^{9,10} or with the application of a large magnetic field^{14,15,16}, spinorbit coupling (SOC) can also drive topological order and symmetrybroken phases^{20,23,24}. Proximityinduced SOC can break C\({}_{2}{{{{{{{\mathcal{T}}}}}}}}\) symmetry at zero magnetic fields and polarise the charge carriers within a single valley^{20,25}. The interplay of SOC, interactions and topology, driven by the presence of vHSs has, however, remained largely unexplored^{26}. The presence of vHSs within the nearly flat moiré bands can lead to enhanced correlations that can be manifested in the emergence of instabilities over a narrower range of tunable, noninteger moiré band fillings.
In this work, we investigate the noninteger filling regime of the moiré band in TBG proximitised by tungsten diselenide (WSe_{2}). We report signatures of valley polarisation along with Fermi surface reconstructions that suggest a Stonerlike instability favoured by vHSs in the vicinity of ν < 2. Significantly, the band reconstructions are malleable and can be tuned via a combination of carrier density and magnetic field.
Results
Figure 1a shows the schematic of our device consisting of a multilayer WSe_{2} on magicangle TBG encapsulated by two hBN layers. The fourprobe longitudinal resistance R_{xx} as a function of filling ν at a magnetic field B = 0 shows welldefined peaks at the charge neutrality point (CNP) ν = 0 and half fillings ν = ± 2 of the conduction (+) and valence (−) 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 without WSe_{2} and were found to be semimetallic rather than purely insulating^{20,24,27}. In Fig. 1c, the Hall resistance R_{xy} at low Bfields, most strikingly, shows a hysteretic behaviour over a wide range of fillings ν < 2, as the Fermi energy is swept back and forth. The hysteresis becomes narrower with increasing Bfield and vanishes at B ≈ 1.2 T. Notably, while R_{xy} shows zerocrossings at B = 50 mT, no sign change is observed for higher fields up to B = 1.2 T with R_{xy} remaining negative for B > 50 mT. This feature will be discussed in detail in Fig. 2. It is evident that R_{xy} strongly depends on the history of the sample in outofplane Bfield training, and this leads to a nonzero R_{xy} at B = 0 when the field is swept back and forth (Fig. 1d). To our surprise, we find an abrupt sign change in the hysteresis of R_{xy} at ν = 1.86 (Fig. 1e). The magnitude of the coercive field, where the hysteresis disappears, is about one order of magnitude higher than previous reports on moiré systems^{9,10,20,28,29,30,31,32}. The large coercive field suggests a more robust ferromagnetic phase than in previous experiments and may also indicate domain wall pinning due to disorder and local inhomogeneities in the twist angle. We expect that the coercive fields will couple more strongly with the orbital magnetic moments rather than the spin whose shifts in energy are typically of ~0.1 meV per Tesla. The hysteresis in R_{xy} with respect to both ν and B suggests that the sample remains magnetised without any external magnetic influence. We note that the measured R_{xy} is much lower than the quantum of resistance (h/e^{2}, where h is the Planck’s constant and e is electronic charge). The width of the hysteresis in Bfield changes as the carrier density is tuned in the vicinity of ν < 2, as evident from the colour plot in Fig. 1e, where we have plotted the difference in R_{xy} for two opposite directions of field sweep, \(\Delta {R}_{xy}={R}_{xy}(\overleftarrow{B}){R}_{xy}(\vec{B})\), as a function of B and ν. Surprisingly, we observe a significant asymmetry between positive (B^{+}) and negative coercive fields (B^{−}). The sudden switching behaviour of R_{xy} as a function of density at ν = 1.86 is accompanied by a reversal of the asymmetry between B^{+} and B^{−} coercive fields. We quantify the latter feature using the parameter α, defined as α = ∣B^{−}∣/∣B^{+}∣, obtaining α > 1 for ν ≤ 1.85 and α < 1 for ν > 1.85 (bottom panel of Fig. 1e). Additional data using various combinations of contacts can be found in the Supplementary Information (see Figs. 10–13). We also note that ferromagnetism is observed over 60% of the total area in our sample (see Supplementary Fig. 1).
Ferromagnetism at ν = 2 is unexpected in TBG since a valleypolarised ground state is energetically unfavourable due to intervalley Hund’s coupling^{19}. However, the SOC together with the gap opening terms can lead to valleypolarised isolated flat bands in TBG at ν = 2^{20,25} (Supplementary Fig. 15). Presence of proximityinduced SOC is confirmed by weak antilocalisation measurements in our device (Supplementary Fig. 8). The reversal of magnetisation is strong evidence for spontaneous switching of valley polarisation induced by tuning the carrier density. A ferromagnet can be classified as spin or orbital, depending on whether the magnetisation is due to spontaneous spin or valley polarisation. In an orbital Chern insulator, the magnetisation can jump abruptly when the chemical potential crosses the Chern gap if it can trigger reordering of the bands that are filled. The edge state contribution is sufficient to change the sign of magnetisation simply by tuning the density below the gap of an orbital Chern insulator^{30,33,34}. Therefore, the abrupt reversal of hysteresis indicates the dominance of orbital magnetism over spin magnetism, and the energetically favourable ground state is solely determined by the gate voltages in weak magnetic fields. We note that the valley polarisation is affected more strongly by subtle changes in the shape of the Fermi surface as it can abruptly modify the momentum space exchange condensation that tends to bunch together electrons that are closer to each other in kspace, whose Berry curvatures contributing to orbital magnetisation are highly variable unlike the electron spins. The observation of a nonquantised R_{xy}, however, suggests that the ground states have partially valleypolarised bands with the unequal occupation of different valleys as a function of carrier density. Partial valley polarisation is not incompatible with the intervalleycoherent phases proposed in literature^{35,36,37} that can mitigate a fully valleypolarised phase. We note, however, that the SOC terms by themselves do not mix electronic states from K and \({K}^{{\prime} }\) and are not the microscopic origin for the intervalleycoherent phases. The sign switching of valley polarisations as a function of density leads to an abrupt reversal of magnetisation (Fig. 1e) indicating a clear phase transition point between these competing phases for magnetic fields below ~0.5 T.
Having established the orbital nature of the ferromagnet at ν < 2, we now turn to the zerocrossings in R_{xy} that accompany the hysteresis in ν at B = 50 mT (Fig. 1c). The νdependence of Hall density n_{H} gives insights into the Fermiology of a system. In Fig. 2a, n_{H} = − (1/e)(B/R_{xy}) is plotted as a function of ν at four different low Bfields, but at a higher temperature T = 2 K, where ferromagnetism disappears (Supplementary Fig. 6) and the Hall data is independent of the direction of density sweep. We observe a rich sequence of sign changes and resets in n_{H} around ν < 2, particularly for the lowest fields 20 mT and 50 mT. Assuming a single particle energy band diagram for TBG, the DOS is expected to show a vHS around ν = 2 (see Fig. 2b, top right panel). As the Fermi energy is swept through the vHS, a Lifshitz transition is expected that changes the topology of the Fermi surface, flipping the sign of n_{H} with a logarithmically diverging profile, as shown in the top left panel^{38}. However, when the bands are malleable, as the Fermi energy approaches the peak in the DOS, it can reset the bands and produce a split DOS profile as shown in the bottom right panel^{16,24}. This leads to a ‘reset’ of charge carriers, where n_{H} drops to a low value before rising again, without a sign change. Our experiments reveal a set of phase transitions in comparison to previous reports on magicangle TBG, where a reset is typically observed near ν = 2^{16,24,39}. A closer look at our data near ν < 2 at 50 mT shows two Lifshitz transitions that flank a reset, indicated by the colourbars in Fig. 2c. We note that the vHS within the nearly flat bands can shift the density of states weights for small changes in the twist angle^{40,41}. Surprisingly, our experiments reveal tunability of the DOS with Bfield and density, further validating the malleability of the TBG bands. The Lifshitz transitions disappear at B = 100 mT, and n_{H} shows a peaklike feature that decreases to zero and increases slowly (Fig. 2a). Such a ‘reset’ of charge carriers at a relatively higher field, with no additional Lifshitz transitions, indicates Bfielddriven changes in the DOS of the bands. Fig. 2d, e shows that these phase transitions become weaker and fade away with increasing temperature. Remarkably, these distinct features in n_{H} appear around the same density ν < 2 where we have observed ferromagnetism at lower temperatures of T ≤ 1 K. The flat band condition of onsite Coulomb interactions (U) dominating over the kinetic energy of the carriers, and the diverging DOS around ν < 2 easily satisfy the Stoner criterion of ferromagnetism UD(E_{F}) > 1, where D(E_{F}) is the DOS at the Fermi energy E_{F}^{3,11,42}. We speculate that such a strong instability in the DOS at ν < 2 favours spontaneous valley polarisation, leading to the observed AHE along with the switching of magnetisation (Fig. 1d, e). Our theoretical calculations discussed below show that spin polarisation together with SOC assists valley polarisation. Thus, a valleypolarised orbital magnet with a nonzero spin polarisation should be favoured over a valleypolarised magnetic phase without a net spin polarisation. In Fig. 2f, we illustrate the scenario where conventional Stoner spin polarising ferromagnetic phase is accompanied by valley polarisation where K and \({K}^{{\prime} }\) valleys are unevenly occupied.
To gain further insights into the possible ground state at halffilling, we have measured R_{xx} and R_{xy} simultaneously in a Bfield up to B = 10 T, at T = 0.3 K. A series of symmetrybroken Chern insulators in the form of minima in R_{xx} and wedgelike features in R_{xy} emerge from different fillings (Fig. 3a, b). The Chern insulators can be characterised by fitting the Diophantine equation, n/n_{0} = Cϕ/ϕ_{0} + s, where n_{0} is the density corresponding to one carrier per moiré unit cell, C is the Chern number, ϕ is the magnetic flux per moiré unit cell, ϕ_{0} = h/e is the fluxquantum, and s is the band filling index or the number of carriers per unit cell at B = 0 T. For sufficiently strong magnetic fields the four fold spinvalley degeneracy is completely lifted near the CNP: (C, ν) = (±1, 0), (±2, 0), (±3, 0), (±4, 0). In addition, we observe states emanating from 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/2e^{2} at a high Bfield (Fig. 3d). In TBG devices, such topological incompressible insulators have been described within the picture of isolated eight fold bands with broken \({{{{{{{\mathcal{T}}}}}}}}\) symmetry, where the Chern numbers of the bands are the same in the two valleys^{14,15,16}, but opposite for valence and conduction bands (Fig. 3e). The valley imbalanced filling of the bands results in the net Chern number observed, consistent with previous studies. While a large Bfield is usually used to break \({{{{{{{\mathcal{T}}}}}}}}\)symmetry, proximityinduced SOC in graphene due to WSe_{2} breaks \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) symmetry inherently at B = 0^{25} and together with an exchange field it can generate spinvalley degeneracylifted bands (Fig. 3e). Moreover, the spinvalley flavour resolved Chern numbers can be tuned by varying the sublattice splitting energy and Ising SOC in TBGWSe_{2} systems. The values of exchange fields are expected to change with the degree of spin polarisation at the onset of magnetism that depends on specific system parameters and Coulomb interactions, see Supplementary Information and Fig. 16 that shows how different initial conditions for a meanfield selfconsistent Hubbard model result in metastable spinpolarised states at different carrier densities. We illustrate in Fig. 3f, the spinvalley resolved band degeneracy lifting introduced by a finite exchange field of λ_{ex} = 0.5 meV in the Hamiltonian that models the spin polarisation of a ferromagnetic phase, together with a proximitised SOC term discussed in the methods section. In the SOC model in Eqs. ((1)–(2)) we have used the Rashba coupling term λ_{R} = 0.56 meV following ref. ^{43}, while the proximity induced λ_{R} in Bernal bilayers is expected to be almost an order of magnitude larger. A range of λ_{R}, including a larger value comparable to those of Bernal bilayers, were also considered in models of twisted bilayer graphene in contact with WSe_{2} when calculating the valley Chern numbers phase diagram of the low energy bands^{25} as a function of other SOC and sublattice potential parameters. The associated spinvalley resolved bands develop a well defined Chern number that will lead to a finite orbital moment when they are filled. We illustrate by using frozen bands how the total orbital moment evolves with filling density giving rise to a magnetisation of the order of ~10^{−2}μ_{B}/A_{M} that can change its sign depending on the specific carrier density value. Since the orbital magnetisation depends sensitively on the actual exchange field for the specific spin configuration, see Supplementary Figs. 17, 18, it is expected that its integrated magnitude, as well as the local values in experiments, vary considerably with density, especially near the phase transition points. Experimentally, the orbital magnetisation maps can reach local values as large as a few μ_{B}/A_{M}^{44,45}. In our experiments, an abrupt change in orbital moment as a function of filling indicates that reordering of levels at ν = 1.86 takes place due to a close competition between the magnetic phases of opposite signs.
In Fig. 4a, we have presented a diagram with the summary of various phases discussed throughout this report. The Lifshitz transitions and reset of carriers at B = 20 and 50 mT occur at the densities near the extreme left boundary of the ferromagnetic domain as well as within the domain. At B ≥ 100 mT the Lifshitz transitions disappear, and we find a second reset near the extreme right boundary of the ferromagnetic domain (inset in Fig. 4). As evidenced from the diagram, ferromagnetism is accompanied by a series of Fermi surface reconstructions at the vHSs. We also highlight that our sample exhibits valley polarisation in two different scenarios: First, anomalous Hall signatures at ν < 2. Second, time reversal symmetrybroken Chern insulator at exactly ν = 2. While both mechanisms are expected to give a net Chern number of C = 2 (Fig. 3e), the experimental signatures are radically different in the sense that R_{xy} is hysteretic and nonquantised at B = 0, whereas it becomes fully quantised to h/2e^{2} at high Bfield (see Supplementary Fig. 14). These observations indicate that the nature of the valleypolarised ferromagnetic ground state is distinct from that of the Chern insulators at high Bfield. Finally, in the context of a very high coercive field in our data, we have plotted coercive field reported in several moiré graphene systems at different ν (Fig. 4b). The plot clearly indicates the coercive field observed in our work is the highest in comparison to other reports to date.
Discussion
Our AHE results differ from those reported recently at ν = ± 2 in hBNaligned TBG without WSe_{2}, having twist angles slightly away from the magic angle^{21}. It was speculated that the combined effect of increasing bandwidth away from the magic angle and staggered sublattice potential arising from the hBN alignment stabilises the magnetic phase at halffilling. In our experiments, while hBN was not aligned with graphene layers as is evident from the low resistance at the CNP, we cannot completely rule out the effects of the increased band dispersion, which along with SOC, may act as an alternative mechanism that polarises the valleys. Our data, along with previous reports, also indicate that devices with identical twist angles may show widely varying properties, suggesting the presence of an unknown set of parameters that governs the band structures. Although the discrepancies among samples are often attributed to twist angle disorder, strain, and dielectric screening, it is not clear how these factors affect the driving mechanisms for the various correlated phases observed. Notably, AHE has been reported by only a few groups, mainly in single devices, with only a fraction of the total area of the samples exhibiting ferromagnetism (see Supplementary Table 1). The absence of hysteresis in electrical transport measurements does not guarantee that a magnetic phase is truly absent. Disorder, substrate potentials, twist angle inhomogeneity, and strain can interrupt the propagation of edge modes between the transverse probes in the device and obstruct the measurement of actual magnetisation. The entire sample can still retain local magnetic moments, although it becomes difficult to measure the magnetisation globally using electrical transport. Recent experiments using a scanning superconducting quantum interference device on the tip have demonstrated imaging of such local orbital magnetic domains^{44,45}. Surprisingly, a substantial part of the sample was found to acquire local Berry curvatures and Chern gaps, even in the absence of local hysteresis^{45}. Although the lack of reproducibility across devices and experimental groups is a significant issue, our observation of AHE near ν = 2 over a wide region in our sample clearly indicates the robustness of our results. Future experiments with wellcontrolled external perturbations and better fabrication protocols will be essential to gain insights into the origin of the observed variability among the samples.
To summarise, our experiments have revealed a phase diagram of competing phases in TBG near its first magic angle, which indicates vHSs within the quasiflat bands. This diverging DOS within the nearly flat bands of TBG reveals a finer internal structure of the bands that is manifested through multiple phase transitions as a function of magnetic fields and carrier densities in the vicinity of ν ~ 1.8. Uncovering the underlying physics of the various quasidegenerate, competing ground states will require an overarching theoretical analysis of strongly interacting manybody physics. Our primary findings of AHE and Fermi surface reconstructions are reported away from the usual commensurate filling of ν = 2. The various features in our data, including abrupt reversal of magnetisation, and nonquantised R_{xy} are clear signs of orbital magnetism, and partially valleypolarised ground state, where both bulk and edge modes are expected to contribute to the transport. The bulk transport may be affected by percolating conduction channels between topological domains of closely competing phases where external electric or magnetic fields can be used as control knobs to favour certain phases over the other. Varying the twist angle between the graphene sheet and the WSe_{2} can modify the proximity SOC strength that would, in turn, modify the phase diagram of the expected ground states. The high sensitivity of the electronic structure to experimental conditions makes it a challenge to perform experiments reproducibly and simultaneously provides an opportunity to explore the physics near multiphase transition points where the electronic response functions will be particularly sensitive to external perturbations. An interesting future research direction would be to identify the connection, if any, between the valleypolarised Stoner magnet found in our work and the superconducting phase in the vicinity of ν = ± 2^{7,8,27,39,46}.
Methods
Device fabrication
The wellknown ‘tear and stack’ method was used to assemble the heterostructure in this work. Polypropylene carbonate (PPC) film coated on a polydimethylsiloxane (PDMS) stamp was used for picking up individual layers of hBN of thickness 25–30 nm, WSe_{2}, and graphene. The sharp edge of the top hBN was used to tear the graphene, following which one half of the torn graphene was picked up, leaving the other half on the substrate. The sample stage was then rotated by 1. 2^{°} (marginally higher than the magic angle), the second half of the graphene layer was picked up. Next, the bottom hBN was picked up, and the heterostructure was released on a 285 nm SiO_{2}/Si substrate at 90 ^{∘}C. The final device was etched into a multiterminal Hall bar by reactive ion etching using CHF_{3}/O_{2} followed by electronbeam lithography, and thermal evaporation of Ohmic edge contacts and top gate using Cr/Au (5 nm/60 nm). WSe_{2} layer with a thickness of ~3 nm was exfoliated from bulk crystals procured from 2D Semiconductors.
Transport measurements
Electrical transport measurements were performed in a He^{3} cryostat with a 10 T magnetic field and a cryogenfree, pumped He^{4} cryostat with a 9 T magnetic field. Magnetotransport measurements were carried out with a bias current of 10 nA, using an SR830 lowfrequency lockin amplifier at 17.81 Hz. The carrier density in the system was tuned by the top gate. The twist angle was estimated using the relation, \({n}_{s}=8{\theta }^{2}/\sqrt{3}{a}^{2}\) where a = 0.246 nm is the lattice constant of graphene and n_{s} (ν = 4) is the charge carrier density corresponding to a fully filled superlattice unit cell. The twist angle determined from R_{xx} data at B = 0 T and Landau fan diagram are in good agreement. For the measurements of hysteresis in R_{xy}, Onsager reciprocity theorem^{9,47} was used, details of which are given in the Supplementary Information.
Proximity SOC in Graphene/WSe_{2}
The interlayer coupling, in particular, the proximity SOC induced in the graphene sheet on top of a WSe_{2} can be modelled by combining sublattice dependent site potential differences together with Rashba and intrinsic SOC terms^{48}. In the following, we briefly outline how the bands of graphene can be altered under the proximity SOC effects of WSe_{2}.
Continuum model bands
We model the singlelayer graphene Hamiltonian contacting a TMD layer through the staggered potential (U), exchange field (ex), intrinsic (I) and Rashba (R) SOC, and pseudospin inversion asymmetry (PIA)^{43}.
with
where 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 of a spinsplitting exchange field and a SOC term is known to introduce valley polarisation. For instance, an intrinsic SOC captured through the KaneMele model together with sublattice staggering potential introduces unequal gaps at the K and \({K}^{{\prime} }\) valleys that leads to an anomalous Hall effect by populating a given spinvalley band when the system is carrier doped. Similarly, a bilayer graphene system subject to a Rashba SOC and spinsplitting exchange field is known to give rise to an anomalous Hall effect for appropriate system parameters^{49}. Both examples illustrate different mechanisms for the onset of an anomalous Hall effect when SOC is accompanied by an exchange field that separates the spinup and down bands in a ferromagnetic phase. The Rashba SOC mixes spins but does not mix valleys. The band structure of Graphene (G)/WSe_{2} in Supplementary Fig. 17 illustrates how the twofold low energy nearly flat bands are split into eight different bands due to spinvalley degeneracy lifting using a model system, whereas explicit meanfield calculations for the Hubbard model are presented in the Supplementary Information to illustrate the sensitivity of the calculated results to different initial conditions and carrier densities. The high sensitivity of the orbital level ordering to spin configuration also leads to sensitive changes in the orbital moments with exchange field parameters. We illustrate in Supplementary Fig. 18 the small changes in λ_{ex} up to ~4 meV in magnitude, which is sufficient to change the number of nodes crossing zero and therefore resulting in sign flips of the orbital magnetisation slopes as a function of filling, in turn, related with the spinvalley Chern numbers. Below, we write the pristine TBG continuum model Hamiltonian for one spinvalley flavour as 8 × 8 matrix to emphasize the three dominant interlayer tunneling^{50}
with
where k and k_{j} are the wave vectors measured from the K^{(η)} of the layer 1 and 2 with the valley index η = ± 1, and k_{j} = k + G_{j} with j = 0, + , − are connected with the three moiré reciprocal lattice vectors, which adjusts the momentum difference between two different Dirac points from layer 1 and 2. The h_{1} and h_{2} describe the Dirac bands of each layer where we use the Fermi velocity υ_{F} = 1 × 10^{6} m/s and the twist angle θ. The t_{12} term is the tunneling matrix between the two graphene layers with tunneling constants ω = 0.12 eV and \({\omega }^{{\prime} }=0.0939\) eV. For the actual calculation, we use 392 × 392 Hamiltonian matrices for each valley and each k point, which includes the hopping terms between the two spins, and use four sublattices, and 49 reciprocal lattice points^{51}. The valley mixing terms are not included in our spinorbit coupling models.
To understand the topological phase transition along with the band filling ν, we obtain the orbital magnetisation M_{(orb)}(μ) as a function of chemical potential μ^{33} using
where f(E) is the FermiDirac distribution, ε_{n} and \(\leftn\right\rangle\) are the eigen energy and vector of the nth band. The Chern number C = (2πℏ/e)dM_{(orb)}/dμ can be estimated by the slope of the M_{(orb)} v.s. μ graphs (See Supplementary Fig. 18).
Data availability
Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon request. Source data are provided with this paper.
Code availability
The code that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We gratefully acknowledge the usage of the MNCF and NNFC facilities at CeNSE, IISc. U.C. acknowledges funding from SERB via SPG/2020/000164 and WEA/2021/000005. Y.J.P. was supported by the Korean National Research Foundation grant NRF2020R1A2C3009142, and D.L. was supported by grant NRF2020R1A5A1016518, as well as the Korean Ministry of Land, Infrastructure and Transport (MOLIT) from the Innovative Talent Education Programme for Smart Cities. J.J. was supported by the Samsung Science and Technology Foundation under project SSTFBAA180206. We acknowledge computational support from KISTI through the grant KSC2021CRE0389 and the resources of Urban Big Data and AI Institute (UBAI) at the University of Seoul and the network support from KREONET. K.W. and T.T. acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790, 20H00354, and 21H05233).
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S.B. fabricated the device, performed the measurements, and analysed the 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.
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Bhowmik, S., Ghawri, B., Park, Y. et al. Spinorbit couplingenhanced valley ordering of malleable bands in twisted bilayer graphene on WSe_{2}. Nat Commun 14, 4055 (2023). https://doi.org/10.1038/s4146702339855x
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DOI: https://doi.org/10.1038/s4146702339855x
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