Interaction-driven Band Flattening and Correlated Phases in Twisted Bilayer Graphene
Youngjoon Choi, Hyunjin Kim, Cyprian Lewandowski, Yang Peng, Alex Thomson, Robert Polski, Yiran Zhang, Kenji Watanabe, Takashi Taniguchi, Jason Alicea, Stevan Nadj-Perge
IInteraction-driven Band Flattening and Correlated Phasesin Twisted Bilayer Graphene
Youngjoon Choi , , ∗ , Hyunjin Kim , , ∗ , Cyprian Lewandowski , , , Yang Peng , Alex Thomson , , ,Robert Polski , , Yiran Zhang , , , Kenji Watanabe , Takashi Taniguchi , Jason Alicea , , , StevanNadj-Perge , † T. J. Watson Laboratory of Applied Physics, California Institute of Technology, 1200 East Cali-fornia Boulevard, Pasadena, California 91125, USA Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, Cal-ifornia 91125, USA Department of Physics, California Institute of Technology, Pasadena, California 91125, USA Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Cal-ifornia 91125, USA Department of Physics and Astronomy, California State University, Northridge, California 91330,USA National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305 0044, Japan*These authors contributed equally to this work † Correspondence: [email protected]
Flat electronic bands, characteristic of magic-angle twisted bilayer graphene (TBG), host awealth of correlated phenomena. Early theoretical considerations
1, 2 suggested that, at themagic angle, the Dirac velocity vanishes and the entire width of the moir´e bands becomesextremely narrow. Yet, this scenario contradicts experimental studies that reveal a fi-nite Dirac velocity as well as bandwidths significantly larger than predicted. Although moresophisticated modeling can, in part, account for the bandwidth broadening, many essen-tial aspects of magic-angle TBG bands and emerging correlated phenomena remain elusive.Here we use spatially resolved spectroscopy in finite and zero magnetic fields to examine theelectronic structure of moir´e bands and their intricate connection to correlated phases. Byfollowing the relative shifts of Landau levels in finite fields, we detect filling-dependent bandflattening caused by strong interactions between electrons, that unexpectedly starts alreadyat ∼ . ° , well above the magic angle and hence nominally in the weakly correlated regime.We further show that, as the twist angle is reduced, the moir´e bands become maximally flatat progressively lower doping levels. Surprisingly, when the twist angles reach values forwhich the maximal flattening occurs at approximate filling of − , + , + , + electrons permoir´e unit cell, the corresponding zero-field correlated phases start to emerge. Our obser-vations are corroborated by calculations that incorporate an interplay between the Coulombcharging energy and exchange interactions; together these effects produce band flatteningand hence a significant density-of-states enhancement that facilitates the observed symmetry-breaking cascade transitions. Besides emerging phases pinned to integer fillings, we also ex-perimentally identify a series of pronounced correlation-driven band deformations and softgaps in a wider doping range around ± filling where superconductivity is expected. Our a r X i v : . [ c ond - m a t . s t r- e l ] F e b esults highlight the essential role of interaction-driven band-flattening in defining electronicproperties and forming robust correlated phases in TBG. Figure 1a sketches our scanning tunneling microscopy (STM) setup. TBG is placed on anatomically smooth dielectric consisting of monolayer WSe and a thicker ( ∼ nm) layer hexag-onal boron nitride (hBN) (see also Supplementary Information (SI), section 1). By applying agate voltage V Gate on a graphite gate underneath, we tune the TBG charge density, or equiva-lently filling factor ν corresponding to the number of electrons per moir´e unit cell. Typical TBGtopography shows a moir´e superlattice consisting of AA sites, where the local density of states(LDOS) originating from bands closest to the Fermi energy at charge neutrality is predominantlyconcentrated , and AB sites in between (Fig. 1a). Local twist angle and strain are determined bymeasuring distances between neighboring AA sites
4, 5 (see also SI, section 2). We first focus on aTBG region with local twist angle . ° to show that interactions play an important role even wellabove the magic angle of ∼ . °.To further examine the moir´e band structure, we probe Landau levels (LLs) that developwhen an out-of-plane magnetic field is applied. The tunneling conductance spectrum taken on anAB site shows two different sets of LLs observed as LDOS peaks separated by the VHSs (Fig. 1c,d). The LLs from the inner set, with energies bounded within the two VHSs
9, 10 , originate fromband pockets around the κ and κ (cid:48) high symmetry points of the moir´e Brillouin zone; we thereforedenote them as κ LLs (see Fig. 1e). Similarly we define LLs from the outer set as γ LLs sincethey originate from portions of the bands around the γ point. This assignment is justified bythe magnetic-field dependence of the observed LL spectrum (see also Supplementary Fig. S1). Inparticular, upon increasing the magnetic field the zeroth γ LLs ( γ LL in the valance and conductionbands) approach the VHSs, as expected from the conduction- and valence-band dispersion at the γ point; the κ LL energies, in contrast, do not change—consistent with the zeroth LLs expectedfrom the Dirac-like dispersion at the κ, κ (cid:48) points. Moreover, even though both κ LLs and γ LLs arevisible on the AB sites, only the κ LLs are resolved on AA sites (Supplementary Fig. S2a, b). Thisobservation suggests that the spectral weight of the κ, κ (cid:48) pockets is spatially located predominantlyon AA sites while the weight of the γ pocket is more distributed over AB sites and domain walls,in line with previous theoretical calculations .Importantly, the energy separation between γ LL and γ LL changes with carrier density(Fig. 1c, d), signaling significant deformation of the moir´e bands upon doping even at the large1.32° twist angle. For the conduction band, the separation is maximized at ν = − , where E ( γ LL ) − E ( γ LL ) ≈ meV, and monotonically decreases below meV near ν = +4 (Fig. 1g).For the valence band, the trend is reversed: the separation between γ LLs increases with filling fac-tor (see Supplementary Fig. S3a, b, on remote bands and extracted effective mass). Note that adisplacement field, likely present due to the single back-gate geometry of our device, might alsoslightly modify the band structure with doping. However, the displacement field would change theconduction and the valence bands symmetrically, in contrast to the observed asymmetric evolutionof γ LLs (Supplementary Fig. S3c, d and SI, sections 3, 4 and 5).2he relative γ LL shifts upon doping are well-captured within a model that includes inter-action effects deriving from the inhomogeneous charge distribution in a moir´e unit cell
18, 20–23 .Starting from charge neutrality, electrons are first added or removed from states near κ, κ (cid:48) thatlocalize primarily on AA sites—creating an associated inhomogeneous electrostatic Hartree po-tential peaked in the AA regions. States near the γ point feel this potential less strongly becausethey are relatively delocalized within the unit cell, which in turn renormalizes the energy cost forpopulating states near γ . More generally, each part of the moir´e bands experiences a differentelectrostatic potential, creating filling-dependent band deformations. Figure 1e presents the calcu-lated band structures for different integer fillings at B = 0
T (see SI, section 3 for details of thecalculation). The conduction and valence bands deform asymmetrically upon doping: the con-duction (valence) band becomes flatter and the valence (conduction) band more dispersive as thefilling factor increases (decreases). Consequently, in finite magnetic fields the energy separationbetween γ LLs also changes asymmetrically (Fig. 1g inset). The Landau level spectrum evaluatedwith only the electrostatic Hartree potential (Fig. 1f; see also SI, section 5) indeed reproduces themain features of the experimental data.Interaction-induced deformation of the moir´e bands completely flattens the γ pocket whenthe twist angle approaches the magic-angle value, as can be deduced from the measured evolu-tion of γ LLs (Fig. 2). To explore the twist-angle dependence, we focus on an area where theangle changes over a ∼ nm area and strain is relatively low ( < . ); see Fig. 2a. Spatiallyresolved measurements in this area reveal the twist angle-dependent evolution of LLs with elec-trostatic doping (Fig. 2b-e). At V Gate near the charge neutrality point (CNP) (Fig. 2b), we observethat γ LL , for both the valence and conduction moir´e bands, eventually merges with the corre-sponding VHS—signaling maximal band flattening. The onset of merging occurs at somewhatlarger angles for the valence band ( ∼ . °) compared to the conduction band ( ∼ . °). Moreover,doping strongly shifts this onset. For example, in the valence band (Fig. 2c-e) the maximal bandflattening moves considerably towards larger (smaller) twist angles for hole (electron) doping (seeSupplementary Fig. S4 for the conduction band).More detailed examples of the evolution of LLs, along with the development of correlatedgaps in finite fields, can be seen in the doping vs. bias maps of Fig. 2f-h (See also SupplementaryFig. S5 for more data). At . ° (Fig. 2f), the γ LL energies are well-resolved in both the valenceand conduction bands between ν ≈ − . and ν ≈ +3 . , beyond which one of the two mergeswith the corresponding VHS (and also Fermi energy V Bias = 0 mV; see discussion of Fig. 3below). The difference in filling factors of the merging points for the conduction and valence bandsreflects an appreciable electron-hole asymmetry. At this large twist angle, aside from quantumHall ferromagnetism in the zeroth κ LL (responsible for the structure between ν = − and +1 ),no pronounced correlated gaps are observed at the Fermi energy. As the twist angle is decreasedto . °, γ LL in the valence band merges at a lower | ν | , and an additional correlated gap appearswhen the merged γ LL crosses the Fermi energy (black arrow in Fig. 2g). For an even lower angleof . °, correlated gaps also begin to emerge on the electron-doped side after γ LL is mergedwith conductance band VHS (black arrows in Fig. 2h). These correlated gaps correspond to Chern3nsulators that emanate from integer filling factors when electron-electron interactions are strongcompared to the width of the Chern bands in finite magnetic fields
10, 24 . Our observations suggestthat such correlated phases emerge only once portions of the bands around γ become maximallyflat and join with the VHS.To explore the development of zero-field correlated phases as a function of twist angle, weperform angle-dependent gate spectroscopy to trace out the evolution of the LDOS at the Fermilevel ( V Bias ≈ ) as a function of charge density ( V Gate ); see Fig. 3a-b. Pronounced LDOS sup-pression near the Fermi energy occurs at certain integer filling factors ν . For all angles, prominentsuppression is observed at ν = ± , , reflecting the small LDOS around the CNP ( ν = 0 ) and bandgaps at full fillings ( ν = ± ). At ν = ± , +3 , +1 we additionally observe sharp LDOS drops thatcan be attributed to emerging correlated gaps similar to those resolved in transport
3, 25–27 . Impor-tantly, the observed LDOS suppressions at integer fillings begin to emerge within the same rangeof angles that displayed considerable band flattening in finite fields (Fig. 2, Fig. 3a and Supple-mentary Fig. S6). The angle onsets of the insulating regions for the conduction and valence bands(marked by dashed lines and arrows) have an electron-hole asymmetry that is also consistent withthe band flattening. Spectra at ν = ± taken at various twist angles (Fig. 3c,d) indeed showthat LDOS suppressions originate from the development of a gap at the Fermi energy, and thatthe ν = − gap emerges at slightly higher angles—corroborating electron-hole asymmetry. Themaximal size of this half-filling gap is ∼ . meV, lower than the initial reports from spectroscopicmeasurements , but slightly larger than the activation gap extracted from transport
3, 25–27 . We notethat the LDOS suppression from the observed gaps in Fig. 3c,d may also be, in part, related to theFermi surface reconstruction due to flavour symmetry-breaking cascade
28, 29 (see also discussionof Fig. 4). As we show in the following, the band flattening also creates conditions that favor thecascade.A theory analysis of the continuum-model band structure
1, 30 with interactions treated at amean-field level in part accounts for the observed band flattening and related symmetry-breakingcascade instabilities near the magic angle (see SI, section 6 for details). While the doping depen-dence of the moir´e band deformation at larger angles is well-modeled by assuming only a Hartreecorrection (Fig. 1), near the magic angle a more complete Hartree-Fock treatment is necessary. Forexample, including only the Hartree term
18, 20, 22 predicts a dramatic band inversion at the γ pocketthat is not observed experimentally (Supplementary Fig. S10a-d). The Fock term in part coun-teracts this band inversion—thereby stabilizing the band flattening; see Fig. 3e-g, SupplementaryFig. S10e-h and SI, section 3. Importantly, our calculations predict that near the magic angle, thedensity of states (DOS) at the Fermi level ( E F ) is significantly amplified (by up to a factor of ∼ for ν = 2 and ∼ for ν = 3 ) relative to expectations from non-interacting models (Fig. 3h). Thisconsiderable interaction-driven DOS increase magnifies the already high DOS around the VHSand accordingly promotes the symmetry-breaking cascade of electronic transitions . For a morequantitative treatment of the cascade instabilities, we compare the energies of unpolarized andflavor-polarized states at integer fillings for different twist angles. In a non-self-consistent treat-ment that neglects band renormalization (see Supplementary Fig. S10), depending on the model,4ither no cascade is expected, or flavor polarization is preferred only in a narrow window aroundthe magic angle. A self-consistent treatment that incorporates band flattening (Fig. 3i), by contrast,predicts cascading over broader twist-angle windows whose widths depend strongly on filling—inagreement with experimental data on the electron-doped side of Fig. 3a (see also SupplementaryFig. S9).Our observations reconcile the apparent discrepancy between the emergence of correlatedphases around the magic angle and experimental observations at odds with seemingly crucial as-pects of the band theory predictions: (i) large Dirac velocity around the CNP
3, 10, 11 , (ii) total band-width exceeding 40 meV , and (iii) large separation of the VHSs . These quantities, Diracvelocity, total bandwidth and the VHS separation, appear not to be essential; for example Supple-mentary Fig. S7 shows that measured Dirac velocity and VHS separation are even smaller at 1.18°than 1.04°, while signatures of strongly correlated phases (e.g. symmetry-breaking cascade andcorrelated gaps) are only present in 1.04° comparing these two angles. Instead, we identify theinteraction-driven flattening of the moir´e bands around the γ pockets, with the consequent increasein the density of states, as the decisive feature needed for the formation of correlated phases.In addition to correlated gaps pinned to integer fillings ν = ± , +1 , +3 , we also observeseveral other interaction-driven features near the magic angle that have not been discussed in pre-vious STM measurements (Fig. 4). For example, prominent LDOS suppressions at the Fermi levelare visible in both the < ν < and − < ν < − doping regions (Fig. 4b-d). Of the tworegions, the feature between ν = +2 and +3 has a ’tilted V’ shape at small bias voltages, whichcan be largely understood as a consequence of the relative prominence of the flavour polarizationin the conduction band compared to the valence band (see also Fig. 3a-d). There, two of the fourflavors are pushed away from the Fermi energy by strong interactions , consistent with the LDOSbeing predominately shifted to higher energy (Fig. 4a,c) and resulting in a highly asymmetricspectrum, as seen in compressibility measurements . In contrast, the spectrum between ν = − and − exhibits a slightly different shape that can not be fully explained by a simple flavour-symmetry-breaking cascade that produces large overall asymmetry around the Fermi energy. Addi-tionally, this doping range also shows a clear, more symmetric gapped feature at small bias voltages(Fig. 4e). The corresponding gap, spanning almost the entire filling range − < ν < − (Fig. 4d),reaches its maximal size of ≈ . meV. Also, it becomes prominent only below θ = 1 . °(see Supplementary Fig. S6), and gradually recedes with temperature and disappears above − K (Fig. 4e). The gap size as well the temperature dependence of this feature is similar to the gapat ν = − ; however, the filling range observed here is unusually large and cannot be explainedsimply by non-interacting effects (see Supplementary Fig. S8). Furthermore, the gap extends overa filling range comparable to that at which superconductivity has been observed in transport atsimilar angles for TBG on WSe as well as in many hBN-only encapsulated devices
25, 26, 31 . Thiscorrespondence indicates that the observed feature may be related to superconductivity itself orsignals the possible existence of a pseudo-gap phase that precedes superconductivity . Regardlessof its exact origin, the pronounced suppression (instead of increase) of the LDOS near the Fermienergy, together with symmetry-breaking cascade features, for fillings where superconductivity is5nticipated may suggest either an electronic pairing origin or a regime of strong-coupling su-perconductivity as recently pointed out for magic-angle trilayer graphene
34, 35 . Another interestingobservation is that, while the gap-like features at the Fermi energy become weaker with increasingtemperature, features at higher energies—previously identified to be related to symmetry-breakingcascade transitions
9, 10, 29 —are enhanced, see the temperature evolution in Fig. 4f-i. The relationof these features and various recently reported phases that emerge at elevated temperatures
36, 37 remains a subject for future investigations.
References:
1. Bistritzer, R. & MacDonald, A. H. Moir´e bands in twisted double-layer graphene.
Proceedingsof the National Academy of Sciences , 12233–12237 (2011).2. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene Bilayer with aTwist: Electronic Structure.
Physical Review Letters , 256802 (2007).3. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlat-tices.
Nature , 80–84 (2018).4. Kerelsky, A. et al.
Maximized electron interactions at the magic angle in twisted bilayergraphene.
Nature , 95–100 (2019).5. Choi, Y. et al.
Electronic correlations in twisted bilayer graphene near the magic angle.
NaturePhysics , 1174–1180 (2019).6. Xie, Y. et al. Spectroscopic signatures of many-body correlations in magic-angle twistedbilayer graphene.
Nature , 101–105 (2019).7. Jiang, Y. et al.
Charge order and broken rotational symmetry in magic-angle twisted bilayergraphene.
Nature , 91–95 (2019).8. Tomarken, S. L. et al.
Electronic Compressibility of Magic-Angle Graphene Superlattices.
Physical Review Letters , 046601 (2019).9. Nuckolls, K. P. et al.
Strongly correlated Chern insulators in magic-angle twisted bilayergraphene.
Nature , 610–615 (2020).10. Choi, Y. et al.
Correlation-driven topological phases in magic-angle twisted bilayer graphene.
Nature \ ’e Flat Bands. arXiv:2008.12296[cond-mat] (2020). .12. Uchida, K., Furuya, S., Iwata, J.-I. & Oshiyama, A. Atomic corrugation and electron local-ization due to Moir \ ’e patterns in twisted bilayer graphenes. Physical Review B , 155451(2014). 63. Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory of moir \ ’e superlatticebands in layered two-dimensional materials. Physical Review B , 205414 (2014).14. Nam, N. N. T. & Koshino, M. Lattice relaxation and energy band modulation in twisted bilayergraphene. Physical Review B , 075311 (2017).15. Carr, S., Fang, S., Zhu, Z. & Kaxiras, E. Exact continuum model for low-energy electronicstates of twisted bilayer graphene. Physical Review Research , 013001 (2019).16. Guinea, F. & Walet, N. R. Continuum models for twisted bilayer graphene: Effect of latticedeformation and hopping parameters. Physical Review B , 205134 (2019).17. Rademaker, L. & Mellado, P. Charge-transfer insulation in twisted bilayer graphene. PhysicalReview B , 235158 (2018).18. Guinea, F. & Walet, N. R. Electrostatic effects, band distortions, and superconductivity intwisted graphene bilayers. Proceedings of the National Academy of Sciences , 13174–13179 (2018).19. Carr, S., Fang, S., Po, H. C., Vishwanath, A. & Kaxiras, E. Derivation of Wannier orbitalsand minimal-basis tight-binding Hamiltonians for twisted bilayer graphene: First-principlesapproach.
Physical Review Research , 033072 (2019).20. Rademaker, L., Abanin, D. A. & Mellado, P. Charge smoothening and band flattening due toHartree corrections in twisted bilayer graphene. Physical Review B , 205114 (2019).21. Cea, T., Walet, N. R. & Guinea, F. Electronic band structure and pinning of Fermi energyto Van Hove singularities in twisted bilayer graphene: A self-consistent approach.
PhysicalReview B , 205113 (2019).22. Goodwin, Z. A. H., Vitale, V., Liang, X., Mostofi, A. A. & Lischner, J. Hartree theory calcu-lations of quasiparticle properties in twisted bilayer graphene. arXiv:2004.14784 [cond-mat] (2020). .23. Calder´on, M. J. & Bascones, E. Interactions in the 8-orbital model for twisted bilayergraphene. arXiv:2007.16051 [cond-mat] (2020). .24. Saito, Y. et al.
Hofstadter subband ferromagnetism and symmetry broken Chern insulators intwisted bilayer graphene. arXiv:2007.06115 [cond-mat] (2020). .25. Yankowitz, M. et al.
Tuning superconductivity in twisted bilayer graphene.
Science ,1059–1064 (2019).26. Lu, X. et al.
Superconductors, orbital magnets and correlated states in magic-angle bilayergraphene.
Nature , 653–657 (2019).27. Arora, H. S. et al.
Superconductivity in metallic twisted bilayer graphene stabilized by WSe2.
Nature , 379–384 (2020). 78. Zondiner, U. et al.
Cascade of phase transitions and Dirac revivals in magic-angle graphene.
Nature , 203–208 (2020). .29. Wong, D. et al.
Cascade of transitions between the correlated electronic states of magic-angletwisted bilayer graphene. arXiv:1912.06145 [cond-mat] (2019). .30. Koshino, M. et al.
Maximally Localized Wannier Orbitals and the Extended Hubbard Modelfor Twisted Bilayer Graphene.
Physical Review X , 031087 (2018).31. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices.
Nature , 43–50 (2018).32. Hofmann, J. S., Berg, E. & Chowdhury, D. Superconductivity, pseudogap, and phase separa-tion in topological flat bands.
Physical Review B , 201112 (2020).33. Bernevig, B. A. et al.
TBG V: Exact Analytic Many-Body Excitations In Twisted BilayerGraphene Coulomb Hamiltonians: Charge Gap, Goldstone Modes and Absence of CooperPairing. arXiv:2009.14200 [cond-mat] (2020). .34. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable Phase Bound-aries and Ultra-Strong Coupling Superconductivity in Mirror Symmetric Magic-Angle Tri-layer Graphene. arXiv:2012.01434 [cond-mat] (2020). .35. Hao, Z. et al.
Electric field tunable unconventional superconductivity in alternating twistmagic-angle trilayer graphene. arXiv:2012.02773 [cond-mat] (2020). .36. Saito, Y. et al.
Isospin Pomeranchuk effect and the entropy of collective excitations in twistedbilayer graphene. arXiv:2008.10830 [cond-mat] (2020). .37. Rozen, A. et al.
Entropic evidence for a Pomeranchuk effect in magic angle graphene. arXiv:2009.01836 [cond-mat] (2020). .38. Walkup, D. et al.
Tuning single-electron charging and interactions between compressibleLandau level islands in graphene.
Physical Review B , 035428 (2020).39. Cea, T. & Guinea, F. Band structure and insulating states driven by Coulomb interaction intwisted bilayer graphene.
Physical Review B , 045107 (2020).40. Xie, M. & MacDonald, A. H. Weak-field Hall Resistivity and Spin/Valley Flavor SymmetryBreaking in MAtBG. arXiv:2010.07928 [cond-mat] (2020). .41. Bultinck, N. et al.
Ground State and Hidden Symmetry of Magic-Angle Graphene at EvenInteger Filling.
Physical Review X , 031034 (2020).42. Khalaf, E., Chatterjee, S., Bultinck, N., Zaletel, M. P. & Vishwanath, A. Charged Skyrmionsand Topological Origin of Superconductivity in Magic Angle Graphene. arXiv:2004.00638[cond-mat] (2020). . 83. Xie, M. & MacDonald, A. H. Nature of the Correlated Insulator States in Twisted BilayerGraphene. Physical Review Letters , 097601 (2020).44. Fang, S., Carr, S., Zhu, Z., Massatt, D. & Kaxiras, E. Angle-Dependent {\ it Abinitio } Low-Energy Hamiltonians for a Relaxed Twisted Bilayer Graphene Heterostructure. (2019).45. Goodwin, Z. A. H., Corsetti, F., Mostofi, A. A. & Lischner, J. Attractive electron-electroninteractions from internal screening in magic-angle twisted bilayer graphene.
Physical ReviewB , 235424 (2019).46. Bistritzer, R. & MacDonald, A. H. Moir \ ’e butterflies in twisted bilayer graphene. PhysicalReview B , 035440 (2011).47. Hejazi, K., Liu, C. & Balents, L. Landau levels in twisted bilayer graphene and semiclassicalorbits. Physical Review B , 035115 (2019).48. Klebl, L., Goodwin, Z. A. H., Mostofi, A. A., Kennes, D. M. & Lischner, J. Importance oflong-ranged electron-electron interactions for the magnetic phase diagram of twisted bilayergraphene.
ArXiv:2012.14499[cond.mat] (2020).49. Efros, A. L. Coulomb gap in disordered systems.
Journal of Physics C: Solid State Physics ,2021–2030 (1976).50. Jung, S. et al. Evolution of microscopic localization in graphene in a magnetic field fromscattering resonances to quantum dots.
Nature Physics , 245–251 (2011). Acknowledgments:
We acknowledge discussions with Francisco Guinea, Felix von Oppen, andGil Refael.
Funding:
This work has been primarily supported by NSF grants DMR-2005129 andDMR-172336; and Army Research Office under Grant Award W911NF17-1-0323. Part of theSTM characterization has been supported by NSF CAREER program (DMR-1753306). Nanofab-rication efforts have been in part supported by DOE-QIS program (DE-SC0019166). S.N-P. ac-knowledges support from the Sloan Foundation. J.A. and S.N.-P. also acknowledge support of theInstitute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of theGordon and Betty Moore Foundation through Grant GBMF1250; C.L. acknowledges support fromthe Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF8682. A.T. and J.A. aregrateful for the support of the Walter Burke Institute for Theoretical Physics at Caltech. Y.P. ac-knowledges support from the startup fund from California State University, Northridge. Y.C. andH.K. acknowledge support from the Kwanjeong fellowship.
Author Contribution:
Y.C. and H.K. fabricated samples with the help of R.P. and Y.Z., and per-formed STM measurements. Y.C., H.K., and S.N.-P. analyzed the data. C.L. and Y.P. implementedTBG models. C.L., Y.P., and A.T. provided theoretical analysis of the model results supervised9y J.A. S.N.-P. supervised the project. Y.C., H.K., C.L., Y.P., A.T., J.A., and S.N.-P. wrote themanuscript with input from other authors.
Data availability:
The data that support the findings of this study are available from the corre-sponding authors on reasonable request. 10 igure 1 | Filling-dependent band structure deformation of TBG at twist angle θ = . ° .a , Device schematics and a TBG surface topography. TBG is placed on a monolayer WSe , thinhBN layer and graphite back gate. A bias voltage V Bias is applied through a graphite contactplaced on top. Blue and yellow circles respectively indicate AA- and AB/BA-stacked regions inthe TBG moir´e pattern (tunneling set point parameters: V Bias = 100 mV,
I = 20 pA). b , Pointspectroscopy at B = 0
T near the CNP taken at an AA and an AB site; AA sites show large LDOSpeaks corresponding to VHSs. c , Tunneling conductance ( dI / dV ) spectroscopy on an AB site asa function of V Gate at a magnetic field of
B = 7
T (
T = 2
K) showing the evolution of LLs withelectrostatic doping. The LLs originating from γ and κ pockets ( γ LLs and κ LLs) of the flat bandsas well LLs from remote bands ( r LLs) are identified. The energy separation between different LLs,as marked by black lines, changes with V Gate . See SI, section 2, for conversion between V Gate and ν . d , Linecuts of data in ( c ) at V Gate = 4
V, 1 V, -5 V further illustrate the LL spectrum and itschange with electrostatic doping. e , Calculated TBG band structure with Hartree corrections for θ = 1 . ° and B = 0
T. Electron doping flattens the conduction band while hole doping flattensthe valence band. f , Calculated density of states with Hartree corrections as a function of filling for B = 7
T (see SI, sections 4 and 5). g , Measured energy separation between γ LL and γ LL as afunction of filling factor showing conductance (valence) band flattening for electron (hole) doping.11 igure 2 | Evolution of LLs with twist angle and correlated gaps at B = T. a , A 590 nm × . ° to . ° (scale bar corresponds to 20 nm, setpoint parameters: V Bias = 100 mV,
I = 20 pA). b , spectroscopic map near CNP ( V Gate = 0 . V)taken over the same area, averaged along the horizontal axis while the vertical axis is convertedinto the local twist angle. Evolution of LLs from the flat ( | V Bias | < mV) and remote bands( | V Bias | > mV) is clearly resolved. c-e , The same plot as ( b ) focusing on the evolution of thevalence-band γ LL for: ( V Gate = 1 . V ( c ), electron doping; V Gate = 0 . V ( d ), near the CNP; V Gate = − . V ( e ), hole doping). Merging between γ LL and VHS occurs at higher twist angleas V Gate is reduced. The insets in ( b-e ) sketch the band structure and Fermi level near θ = 1 . °and θ = 1 . °. A smooth signal background is subtracted to enhance LL visibility. f-h , Pointspectroscopy for θ = 1 . ° ( f ), . ° ( g ), and . ° ( h ). Black arrows in ( g ) and ( h ) indicateemerging correlated Chern phases
9, 10 after the γ LL merges with the VHS. Color coded linesshow V Gate values used in ( c-e ). 12 igure 3 | Emergence of zero-field correlated gaps and symmetry-breaking cascade. a , Angleand filling-factor dependence of dI / dV near E F ( V Bias = 0 . mV) at B = 0
T taken on the samearea as in Fig. 2a,b. Correlated gaps at ν = − , +1 , +2 , +3 , observed as an abrupt drop in dI / dV (i.e., LDOS), develop only below certain twist angle—in contrast to regions between the flat andremote bands ( ν = ± ) where LDOS is small for any angle. Black lines near ν = ± mark theupper bound where gaps begins to emerge, while colored arrows indicate corresponding LDOSsuppression regions at ν = +1 , +2 , +3 . b , Schematic showing how data in ( a ) is taken: V Bias ≈ is fixed so that the STM tip probes LDOS near the Fermi energy while V Gate is swept. c, d , Spec-troscopy for ν = − ( c ) and ν = +2 ( d ) at different local twist angles ranging θ = 1 . ° − . °,taken at AB sites. Color coding of the lines correspond to the angles marked by horizontal barsin ( a ). Clear correlated gaps that open only at the Fermi energy are observed only below a certainangle (small wiggles above this angle originate from trivial origins, see Supplementary Fig. S8).Each spectrum is normalized by an average dI / dV value and offset for clarity. e-g , Interaction-renormalized band structures at different integer fillings, calculated assuming unpolarized groundstates (see main text and SI, section 6). Horizontal lines indicate the relevant chemical poten-tials. In cases where polarized states are favorable, dotted lines are used. The non-interactingband structure is shown in black. h , Twist-angle dependence of the DOS at E F obtained from theinteraction-corrected unpolarized band structure, normalized by the non-interacting DOS. Peakssignal maximal band flattening as seen in e-g . i , Relative energy change for polarized (cascaded)states relative to unpolarized states. Interaction-driven band flattening significantly extends therange of angles, marked by arrows in ( j ) and ( i ), where this relative energy change is negative andpolarization becomes energetically favourable (see SI, section 6).13 igure 4 | Temperature dependence of correlated gaps around ν = ± and the symmetry-breaking cascade . a , Point spectra as a function of V Gate for θ = 1 . ° at B = 0 T. b,c , dI / dV spectra for filling factors ranging from ν = − . to ν = − . ( b ) and from ν = 2 . to ν = 3 . ( c ). d , High-resolution point spectra of ( a ) focusing on the soft gap between ν = − and -3. e , dI / dV spectra at V Gate = − . V (as indicated by black lines in ( a,d )) for temperatures rangingfrom mK to K at the same tip location as ( a ). f-i , Point spectra as a function of V Gate for θ = 1 . ° at temperatures T = 2
K ( f ), T = 7
K ( g ), T = 9 . K ( h ), and T = 20
K ( i ). Astemperature increases, the cascade features become more pronounced, and their onset more closelyfollows integer filling factors, hinting at a characteristic cascade temperature scale of T ≈ K aspreviously noted28