# Fileset [2024A00124G_VOR_Insualtors at Fractional Fillings... (D.Wong et al) 2023 (submitted PDF).pdf](https://mdr.nims.go.jp/filesets/1b7528c5-46d7-4336-9746-ebf5cb04b226/download) ## Creator Dillon Wong, Kevin P. Nuckolls, Myungchul Oh, Ryan L. Lee, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Ali Yazdani ## Rights This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Dillon Wong, Kevin P. Nuckolls, Myungchul Oh, Ryan L. Lee, Kenji Watanabe, Takashi Taniguchi, Ali Yazdani; Insulators at fractional fillings in twisted bilayer graphene partially aligned to hexagonal boron nitride. Low Temp. Phys. 1 June 2023; 49 (6): 655–661 and may be found at https://doi.org/10.1063/10.0019422.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Insulators at fractional fillings in twisted bilayer graphene partially aligned to hexagonal boron nitride](https://mdr.nims.go.jp/datasets/d6066e47-a85f-470a-bf11-48dcaeb5e635) ## Fulltext Insulators at fractional fillings in twisted bilayer graphene partially aligned to hexagonal boron nitrideViewOnlineExportCitationCrossMarkRESEARCH ARTICLE | JUNE 01 2023Insulators at fractional fillings in twisted bilayer graphenepartially aligned to hexagonal boron nitride Dillon Wong ; Kevin P. Nuckolls ; Myungchul Oh ; Ryan L. Lee ; Kenji Watanabe ;Takashi Taniguchi ; Ali Yazdani Low Temperature Physics 49, 655–661 (2023)https://doi.org/10.1063/10.0019422 28 June 2023 15:06:58https://pubs.aip.org/aip/ltp/article/49/6/655/2900388/Insulators-at-fractional-fillings-in-twistedhttps://pubs.aip.org/aip/ltp/article/49/6/655/2900388/Insulators-at-fractional-fillings-in-twisted?pdfCoverIconEvent=citehttps://pubs.aip.org/aip/ltp/article/49/6/655/2900388/Insulators-at-fractional-fillings-in-twisted?pdfCoverIconEvent=crossmarkjavascript:;https://orcid.org/0000-0002-4931-4188javascript:;https://orcid.org/0000-0002-1078-7113javascript:;https://orcid.org/0000-0003-0477-1390javascript:;https://orcid.org/0009-0009-1361-9738javascript:;https://orcid.org/0000-0003-3701-8119javascript:;https://orcid.org/0000-0002-1467-3105javascript:;https://orcid.org/0000-0003-4996-8904javascript:;https://doi.org/10.1063/10.0019422https://servedbyadbutler.com/redirect.spark?MID=176720&plid=2062273&setID=592934&channelID=0&CID=749356&banID=520969615&PID=0&textadID=0&tc=1&adSize=1640x440&data_keys=%7B%22%22%3A%22%22%7D&matches=%5B%22inurl%3A%5C%2Fltp%22%5D&mt=1687964818115219&spr=1&referrer=http%3A%2F%2Fpubs.aip.org%2Faip%2Fltp%2Farticle-pdf%2F49%2F6%2F655%2F18019502%2F655_1_10.0019422.pdf&hc=1e90c8da717b1260b4c7b48f9990b4818956c793&location=Insulators at fractional fillings in twisted bilayergraphene partially aligned to hexagonalboron nitrideCite as: Fiz. Nizk. Temp. 49, 720–727 (June 2023); doi: 10.1063/10.0019422View Online Export Citation CrossMarkSubmitted: 27 April 2023Dillon Wong,1 Kevin P. Nuckolls,1 Myungchul Oh,1 Ryan L. Lee,1 Kenji Watanabe,2Takashi Taniguchi,3 and Ali Yazdani1,a)AFFILIATIONS1Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, New Jersey 08544, USA2Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan3International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japana)Author to whom correspondence should be addressed: yazdani@princeton.eduABSTRACTAt partial fillings of its flat electronic bands, magic-angle twisted bilayer graphene (MATBG) hosts a rich variety of competing correlatedphases that show sample-to-sample variations. Divergent phase diagrams in MATBG are often attributed to the sublattice polarizationenergy scale, tuned by the degree of alignment of the hexagonal boron nitride (hBN) substrates typically used in van der Waals devices.Unaligned MATBG exhibits unconventional superconductor and correlated insulator phases, while nearly perfectly aligned MATBG/hBNexhibits zero-field Chern insulating phases and lacks superconductivity. Here we use scanning tunneling microscopy and spectroscopy(STM/STS) to observe gapped phases at partial fillings of the flat bands of MATBG in a new intermediate regime of sublattice polarization,observed when MATBG is only partially aligned (θGr-hBN ≈ 1.65°) to the underlying hBN substrate. Under this condition, MATBG hostsnot only phenomena that naturally interpolate between the two sublattice potential limits, but also unexpected gapped phases absent ineither of these limits. At charge neutrality, we observe an insulating phase with a small energy gap (Δ < 5 meV) likely related to weak sublat-tice symmetry breaking from the hBN substrate. In addition, we observe new gapped phases near fractional fillings ν = ±1/3 and ν = ±1/6,which have not been previously observed in MATBG. Importantly, energy-resolved STS unambiguously identifies these fractional fillingstates to be of single-particle origin, possibly a result of the super-superlattice formed by two moiré superlattices. Our observations empha-size the power of STS in distinguishing single-particle gapped phases from many-body gapped phases in situations that could be easily con-fused in electrical transport measurements, and demonstrate the use of substrate engineering for modifying the electronic structure of amoiré flat-band material.Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/10.0019422Van der Waals moiré materials display a variety of correlatedinsulating phases at densities corresponding to integer multiples ofthe moiré-Brillouin-zone area (integer fillings ν).1–8 Such integer-νcorrelated insulators are widely observed in a multitude of moirématerial devices, and they are often the first signatures of the strongelectronic correlations that are ubiquitous in flat-band moirésystems.9,10 Less common are correlated insulating phases at densitiescorresponding to non-integer multiples of the moiré-Brillouin-zonearea (non-integer ν). Such phases are occasionally reported intwisted graphene systems11–15 and transition metal dichalcogenide(TMD) hetero/homobilayers.16–19 Non-integer filling phases are con-jectured to involve broken space group symmetries (i.e., brokentranslational/rotational symmetry, where the total density at onemoiré site differs from the density at a neighboring moiré site) withlong-range spatial order that can only be stabilized under the highestlevels of material quality and homogeneity.20–22 However, thesenon-integer-ν phases are primarily detected through electrical resis-tivity, optical reflectance, and compressibility measurements, and thenatures of their ground states are only inferred from the simplicity oftheir rational filling factors. To date, only WSe2/WS2 and twistedWS2 have been confirmed to host generalized Wigner crystal states atν = 1/3 with direct spatial imaging.23,24Here, we use scanning tunneling microscopy and spectroscopy(STM/STS) to characterize magic-angle twisted bilayer grapheneLow TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 655Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://doi.org/10.1063/10.0019422https://doi.org/10.1063/10.0019422https://www.scitation.org/action/showCitFormats?type=show&doi=10.1063/10.0019422http://crossmark.crossref.org/dialog/?doi=10.1063/10.0019422&domain=pdfhttp://orcid.org/0000-0002-4931-4188http://orcid.org/0000-0002-1078-7113http://orcid.org/0000-0003-0477-1390http://orcid.org/0009-0009-1361-9738http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-4996-8904mailto:yazdani@princeton.eduhttps://doi.org/10.1063/10.0019422https://aip.scitation.org/journal/ltp(MATBG) that is partially aligned to its hexagonal boron nitride(hBN) substrate (θGr-hBN ≈ 1.65°). We report a sequence ofnon-integer-ν phases at simple fractions ν = ±1/3 and ν = ±1/6 in aMATBG device, seen as strong suppressions of the zero-bias con-ductance at these fillings. Although they may appear to be previ-ously unobserved correlated insulating states, tunnelingconductance dI/dV(Vs, Vg) as a function of sample bias Vs and gatevoltage Vg reveals that these non-integer-ν gapped phases are notcorrelated insulators, but rather are most likely of single-particleorigin. Scanning tunneling microscopy (STM) topographic imagesshow a significant moiré pattern between graphene and hBN,which strongly modifies the Gr-Gr moiré pattern and suggests thatthe ν = ±1/3 and ν = ±1/6 phases may be associated with a verylong wavelength super-superlattice formed from the graphene-graphene (Gr-Gr) and the graphene-hBN (Gr-hBN) moirépatterns.25–28 In addition, we observe an insulating gap at thecharge neutrality point (CNP; ν = 0) that is likely due to sublatticesymmetry breaking from the hBN substrate, as well as an extended,spatially inhomogeneous pseudogap-like feature. Our results high-light the importance of spatial imaging and spectroscopy in eluci-dating the origin of gapped states in moiré materials.Our measurements were acquired on a homebuilt dilution-refrigerator STM29 using a W tip calibrated against the surface stateof Cu(111), with scanning tunneling spectroscopy (STS) obtainedusing a standard lock-in technique. The MATBG device was pre-pared using a method similar to that described elsewhere.30MATBG rests on an hBN/SiO2/Si substrate, and Vg is applied tothe Si layer to tune the electron density in the MATBG. TheMATBG is electrically connected to prepatterned Au/Ti electrodesbiased at Vs.Figure 1(a) shows a topographic image of MATBG/hBN. Thetriangular lattice of dark sites with periodicity λGr-hBN = 7.4 nm cor-responds to the moiré pattern between the graphene and hBNatomic lattices, while the bright regions (e.g., labeled A and B) arethe AA sites of the MATBG moiré pattern (with periodλGr-Gr = 15.2 nm, θGr-Gr ≈ 0.93°). The Fourier transform of thisimage (inset) shows clear peaks for Gr-Gr and Gr-hBN reciprocallattice wave vectors, which are rotated from one another by approx-imately 30°.31,32 dI/dV(Vs, Vg) on spot A [Fig. 1(b)] shows a pair offlat bands that are pinned to the Fermi energy (EF, Vs = 0mV)between Gr-Gr filling factors ν = + 4 and ν = –4. Gap-like featuresappear at EF for 1 < ν < 3 and for –3.5 < ν < –2, possibly related tothe pseudogap observed in MATBG unaligned with hBN,33,34which we will discuss later.Focusing on the CNP, Figs. 1(c) and 1(d) show a small gap(Δ < 5meV) convolved with tip-induced charging effects thatconfirm the insulating nature of this state. This gap parallels STSon perfectly aligned MATBG, which identifies a large-energyFIG. 1. Single-particle gaps at fractional fillings in partially aligned twisted bilayer graphene. (a) STM topographic image of a magic-angle graphene moiré superlattice(bright regions) with a graphene-hBN moiré superlattice (dark regions). Inset: FFT of STM topograph, showing two moiré wavevectors, QGr-hBN and QGr-Gr. (b) dI/dV(Vs,Vg) measured at the center of the AA site labeled “A” in (a) at 200 mK and zero magnetic field. Dashed line box indicates the zoomed-in region shown in (c). (c) dI/dV(Vs,Vg) measured at the center of the AA site labeled “A” in (a). (d) Same as (c), measured at the center of the AA site labeled “B” in (a). Zero-bias conductance dI/dV(Vs = 0 V, Vg) of data in (c). (f ) Zero-bias conductance dI/dV (Vs = 0 V, Vg) of data in (d). Yellow shaded bars highlight deep conductance suppressions at ν = ±1/3. Blueshaded bars highlight weak conductance suppressions at ν = ±1/6. Tunneling parameters: (a) Vs = –80 mV, I = 10 pA, (b) Vs = –80 mV, I = 500 pA, Vac = 0.3 mV at 381.7 Hz,(d) Vs = –70 mV, I = 1 nA, Vac = 0.15 mV at 381.7 Hz.Low TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 656Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://aip.scitation.org/journal/ltpgapped spectrum (Δ > 20meV) near the CNP, again convolvedwith tip-induced charging effects.33 When the underlying hBN sub-strate is aligned to MATBG, it breaks graphene’s sublattice symme-try, gaps the Dirac nodes in the spectrum of MATBG, andproduces the insulating gap observed at this filling.35,36 A naturalinterpretation of the smaller gap observed here is that the partialalignment of the underlying hBN substrate breaks the sublatticesymmetry of MATBG, but to a lesser degree than in perfectlyaligned devices. Indeed, similar phenomena have been observed intransport experiments on graphene/hBN, where the extracted sizeof the sublattice symmetry-broken gap was observed to continu-ously increase with decreasing rotational mismatch between gra-phene and hBN.27,37,38 The gap at the CNP contrasts with STSexperiments on unaligned MATBG devices, which instead identifya small, but finite local density of states (LDOS), indicative of thecommonly observed gapless semimetallic state expected in the pres-ence of Dirac nodes.30,39In addition, zero-bias conductance dI/dV(Vs = 0 V, Vg) inFigs. 1(e) and 1(f ) show deep suppressions of the LDOS atν = ±1/3 (yellow-shaded bars) and weak suppressions of the LDOSat ν = ±1/6 (blue-shaded bars). In the absence of bias-resolvedspectroscopy, these zero-bias conductance suppressions would behighly suggestive of gapped insulating states derived from electron–electron interaction effects, a hypothesis that hinges on the simplic-ity of their rational filling factors and their presence at partialfillings of a flat electronic band. However, bias-resolved STS offerscrucial information that contradicts this hypothesis. dI/dV(Vs, Vg)in Figs. 1(c) and 1(d) show that the ν = ±1/3 insulating gaps areobserved away from the Fermi level as ν is tuned away from ±1/3,indicating that they are single-particle gapped phases that arerigidly tuned to the Fermi level via the gate voltage (also true forν = ±1/6). This is in strong contrast to STS measurements ofcorrelation-driven insulating phases in MATBG, which areobserved to spontaneously appear at the Fermi level, often in placeof an expected peak in the single-particle LDOS. An example ofthis is observed in the spectroscopic properties of the correlatedinsulator at ν = –2 in unaligned devices, where an insulating gap ismeasured to open and close only around EF.33 Thus, an alternatehypothesis is necessary for explaining these single-particle insula-tors at fractional fillings.A plausible explanation for the non-integer-ν gaps stems fromthe similarity of two observed moiré wavelengths, λGr-Gr and2λGr-hBN, which can produce a super-superlattice from the interfer-ence of the Gr-Gr and Gr-hBN moiré patterns. A simulated imagein Fig. 2(a) that uses the length scales measured from the Fig. 1(a)topograph shows such a super-superlattice. Here, we plot the“super-superlattice moiré function”F(r) ¼ 3�X3i¼1cos (RiGGr-hBN):r !3þX3i¼1cos (RiGGr-Gr):r !,where GGr-hBN (GGr-Gr) is a primitive reciprocal lattice vector of theGr-hBN (Gr-Gr) moiré pattern and Ri ¼ Ci3 are the 120° rotations.Visual inspection of the image reveals a large-scale approximateperiodicity of the super-superlattice [marked by the yellowrhombus in Fig. 2(a) and by blue translation vectors in Fig. 2(c)]that is roughly 7 times longer than the Gr-hBN moiré pattern (orroughly 7/2 times larger than the Gr-Gr moiré pattern), denoted Λ1.Accounting for the four-fold spin and valley degeneracies of the elec-tronic states of MATBG, a triangular super-superlattice potentialwith length scale Λ1 is expected to produce a partial gap at fillingν = ±4/(Λ1/λGr-Gr)2 ≈ ±4/(7/2)2 = ±16/49 ≈ ±1/3, consistent withour observations of deep zero-bias conductance suppressions nearFIG. 2. Schematic of two length scales in simulated moiré super-superlattice. (a) Simulated “super-superlattice moiré function” F(r) using measured lengths and anglesfrom STM topographic images. The red hexagons and the yellow rhombus depict the locally approximate super-superlattice unit cell. (b) Calculated “approximate translationsymmetry function” S(t), which shows a peak at a translation vector t when F(r) and F(r � t) are most similar. Peaks in this function are observed when t ¼ Λ1 andt ¼ Λ2. The two unlabeled vectors are equivalent to vectors Λ1 and Λ2, related to the labeled vectors by a 60° rotation about the origin. (c) Annotated version of the plotin (a), illustrating the two approximate translation symmetry vectors, Λ1 (blue vectors) and Λ2 (orange vectors), and the two-length-scale tiling that covers the completemoiré super-superlattice using these two approximate translation vectors.Low TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 657Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://aip.scitation.org/journal/ltpfillings ν = ±1/3. The “approximate translation symmetry func-tion” S(t) ¼ 1/Ð jF(r� t)� F(r)j2d2r [plotted in Fig. 2(b)], whichmeasures how well the super-superlattice moiré function F(r) ispreserved when translated by t, shows a strong local maximum att = (52.2 nm, 0), again consistent with the filling factor ν = ±4/(Λ1/λGr-Gr)2 = ±0.34 ≈ ±1/3.The origin of the ν = ±1/6 spectroscopic gaps is less clear, buthere we provide one possible hypothesis. Visual inspection of thesimulated moiré super-superlattice indicates that unit cells of sidelength Λ1 are well-approximated in many regions locally, but areinsufficient for describing the entire super-superlattice periodicity.Instead, as shown in the tiling diagram in Fig. 2(c), the completesuper-superlattice is characterized by two length scales,Λ1 = 52.2 nm and Λ2 = 77 nm, where the latter is observed to beroughly 10 times longer than the Gr-hBN moiré pattern (orroughly 5 times larger than the Gr-Gr moiré pattern). We find thatΛ2 also represents an approximate quasiperiodicity of the super-superlattice [marked by orange translation vectors in Fig. 2(c)], butto a lesser degree than the Λ1 periodicity. A triangular super-superlattice potential with length scale Λ2 is expected to produce apartial gap at filling ν = ±4/(Λ2/λGr-Gr)2≈ ±4/(5)2 = ±4/25≈ ±1/6,consistent with our observations of weaker zero-bias conductancesuppressions near fillings ν = ±1/6. In addition, the approximatetranslation symmetry function S(t) [as well as the autocorrelationof the super-superlattice moiré function F(r)] shows a localmaximum at t = (0, 77 nm), which again corresponds to fillingfactors ν = ±4/(Λ2/λGr-Gr)2 = ±0.16 ≈ ±1/6. Despite the presence ofthese suppressions at ν = ±1/6, we do not believe the suppressionsfound at ν = ±1/3 are higher order gaps associated with Λ2.Moreover, we do not observe any higher-order gaps associated withΛ1 and Λ2, which may parallel observations of satellite features ingraphene/hBN moiré superlattices.40 We note, however, that thistoy picture of the moiré super-superlattice may not be a completelyaccurate description of the system, especially since inhomogeneityin the twist angle over long length scales is reported in MATBGsamples.41Further evidence that a super-superlattice potential influencesthe tunneling conductance is shown in Figs. 3(a) and 3(c), whichshow dI/dV(Vs, Vg) for filling factors beyond full filling ν = ±4. Atthese densities, we resolve a series of LDOS peaks in the dispersiveremote bands that are unexpected from the band structure ofMATBG. These peaks are more clearly seen in fixed-gate-voltagedI/dV(Vs) spectra in Figs. 3(b) and 3(d) and are consistent with theemergence of the van Hove singularities of replica remote bands,which would result from Brillouin-zone folding in the presence ofa super-superlattice potential modulation with a wavelength longerthan the Gr-Gr moiré wavelength.Finally, we turn to the density-extended gaps at EF seenbetween gate voltages 10 V +4 (top) and ν < –4 (bottom). (c) dI/dV(Vs, Vg) measured at the center of an AA site at T = 200 mK. (d)dI/dV(Vs) line cut spectra from (c) showing replica remote band peaks at fillings ν > +4 (top) and ν < –4 (bottom). Tunneling parameters: (a) Vs = –80 mV, I = 300 pA,Vac = 0.5 mV at 381.7 Hz, (b) Vs = –100 mV, I = 1 nA, Vac = 1 mV at 381.7 Hz.Low TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 658Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://aip.scitation.org/journal/ltpposition, as seen most strikingly when comparing spectroscopyobtained in locations A and B. In some areas (e.g., location B), thegap-like feature seems not to be present at all. We speculate thatthis spatial inhomogeneity could be related to the degree of localC2-symmetry breaking, which varies among locations on theGr-hBN moiré pattern,42–44 although understanding the relation-ship between structural and electronic properties in MATBG ischallenging and requires a more careful study than was carried outhere. Particularly, further studies are required to understand thespatial dependence of these gap-like features and to correlate theirenergy widths and gate-voltage ranges to different locations on thesuperlattice.Detailed band structure calculations and further measure-ments will be required to fully understand the nuanced effects ofthe sublattice potential on MATBG in partially aligned configura-tions with the underlying hBN substrate. Such measurements willbe particularly important in understanding the relationshipbetween superconductivity and C2 rotational symmetry,45 which iscritical to several open proposals for the mechanisms of this exoticsuperconductor.46–48 Our spectroscopic measurements show thatsublattice potential asymmetry should no longer be considered amatter of presence and absence, but rather a matter of the degreeand the microscopic form of this crucial energy scale. Furthermore,our measurements highlight the importance of local spectroscopicprobes in the field of moiré materials. STM/STS offers spatially andenergy-resolved information that can vastly reduce the phase spaceof plausible explanations for new electronic phases observed inthese materials.ACKNOWLEDGMENTSWe thank B. Andrei Bernevig for helpful discussions. Thiswork was primarily supported by the Gordon and Betty MooreFoundation’s EPiQS initiative Grant Nos. GBMF9469 andARO-MURI W911NF2120147. Other support for the experimentalwork was provided by DOE-BES Grant Nos. DE-FG02-07ER46419,NSF-MRSEC through the Princeton Center for Complex MaterialsNSF-DMR-2011750, NSF-DMR-1904442. K. W. and T. T. acknowl-edge support from the Elemental Strategy Initiative conducted bythe MEXT, Japan, Grant Nos. JPMXP0112101001, JSPS KAKENHIGrant Nos. 19H05790 and JP20H00354.D. W., K. P. N., and M. O. contributed equally to this work.REFERENCES1Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo,J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori,and P. Jarillo-Herrero, “Correlated insulator behavior at half-filling inmagic-angle graphene superlattices,” Nature 556, 80 (2018).FIG. 4. Signatures of local pseudogap-like features. (a) STM topograph of MATBG partially aligned to hBN. (b) dI/dV(Vs, Vg) measured at the center of AA sites at T = 200mK, at locations A (left), B (middle), and C (right) as labeled in (a). Tunneling parameters: Vs = –80 mV, I = 500 pA, Vac = 0.3 mV at 381.7 Hz for spot A; Vs = –80 mV,I = 1 nA, Vac = 0.4 mV at 381.7 Hz for spot B; Vs = –80 mV, I = 1 nA, Vac = 0.4 mV at 381.7 Hz for spot C.Low TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 659Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://doi.org/10.1038/nature26154https://aip.scitation.org/journal/ltp2M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi,D. Graf, A. F. Young, and C. R. Dean, “Tuning superconductivity in twistedbilayer graphene,” Science 363, 1059 (2019).3X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, Y. Ronen, H. Yoo, D. H. Najafabadi,K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, “Tunable spin-polarizedcorrelated states in twisted double bilayer graphene,” Nature 583, 221 (2020).4G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, B. L. Chittari, K. Watanabe,T. Taniguchi, Z. Shi, J. Jung, Y. Zhang, and F. Wang, “Evidence of a gate-tunableMott insulator in a trilayer graphene moiré superlattice,” Nat. Phys. 15, 237(2019).5H. Polshyn, “Electrical switching of magnetic order in an orbital Chern insula-tor,” Nature 588, 66 (2020).6L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen,D. M. Kennes, Y. Bai, B. Kim, K. Watanabe, T. Taniguchi, X. Zhu, J. Hone,A. Rubio, A. N. Pasupathy, and C. R. Dean, “Correlated electronic phases intwisted bilayer transition metal dichalcogenides,” Nat. Mater. 19, 861 (2020).7Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K. Watanabe, T. Taniguchi,A. H. MacDonald, J. Shan, and K. F. Mak, “Simulation of Hubbard modelphysics in WSe2/WS2 moiré superlattices,” Nature 579, 353 (2020).8T. Li, S. Jiang, L. Li, Y. Zhang, K. Kang, J. Zhu, K. Watanabe, T. Taniguchi,D. Chowdhury, L. Fu, J. Shan, and K. F. Mak, “Continuous Mott transition insemiconductor moiré superlattices,” Nature 597, 350 (2021).9R. Bistritzer, and A. H. MacDonald, “Moiré bands in twisted double-layer gra-phene,” Proc. Natl. Acad. Sci. USA 108, 12233 (2011).10E. Suárez Morell, J. D. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, “Flatbands in slightly twisted bilayer graphene: Tight-binding calculations,” Phys.Rev. B 82, 121407 (2010).11H. Polshyn, Y. Zhang, M. A. Kumar, T. Soejima, P. Ledwith, K. Watanabe,T. Taniguchi, A. Vishwanath, M. P. Zaletel, and A. F. Young, “Topologicalcharge density waves at half-integer filling of a moiré superlattice,” Nat. Phys.18, 42 (2021).12S. Wu, Z. Zhang, K. Watanabe, T. Taniguchi, and E. Y. Andrei, “Chern insula-tors, van Hove singularities and topological flat bands in magic-angle twistedbilayer graphene,” Nat. Mater. 20, 488 (2021).13H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, E. M. Spanton,T. Taniguchi, K. Watanabe, E. Berg, M. Serbyn, and A. F. Young, “Half- andquarter-metals in rhombohedral trilayer graphene,” Nature 598, 429 (2021).14P. Siriviboon, J.-X. Lin, X. Liu, H. D. Scammell, S. Liu, D. Rhodes,K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. I. A. Li, “A newflavor of correlation and superconductivity in small twist-angle trilayer gra-phene,” arXiv:2112.07127 (2021).15G. Chen, A. L. Sharpe, E. J. Fox, S. Wang, B. Lyu, L. Jiang, H. Li, K. Watanabe,T. Taniguchi, M. F. Crommie, M. A. Kastner, Z. Shi, D. Goldhaber-Gordon,Y. Zhang, and F. Wang, “Tunable orbital ferromagnetism at noninteger filling ofa moiré superlattice,” Nano Lett. 22, 238 (2022).16E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B. Gao, X. Wei, S. Zhao,W. Zhao, Z. Zhang, K. Yumigeta, M. Blei, J. D. Carlström, K. Watanabe,T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, and F. Wang, “Mott and general-ized Wigner crystal states in WSe2/WS2 moiré superlattices,” Nature 579, 359(2020).17Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser,K. F. Mak, and J. Shan, “Correlated insulating states at fractional fillings of moirésuperlattices,” Nature 587, 214 (2020).18X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T. Taniguchi,K. Watanabe, S. Okamoto, D. Xiao, S.-F. Shi, and Y.-T. Cui, “Correlated insulat-ing states at fractional fillings of the WS2/WSe2 moiré lattice,” Nat. Phys. 17, 715(2021).19Y. Xu, K. Kang, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, “A tunablebilayer Hubbard model in twisted WSe2,” Nat. Nanotechnol. 17, 934 (2022).20E. Wigner, “On the interaction of electrons in metals,” Phys. Rev. 46, 1002(1934).21C. C. Grimes, and G. Adams, “Evidence for a liquid-to-crystal phase transition ina classical, two-dimensional sheet of electrons,” Phys. Rev. Lett. 42, 795 (1979).22E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, and E. Paris, andB. Etienne “Observation of a magnetically induced Wigner solid,” Phys. Rev.Lett. 60, 2765 (1988).23H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei,T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang,“Imaging two-dimensional generalized Wigner crystals,” Nature 597, 650(2021).24H. Li, Z. Xiang, E. Regan, W. Zhao, R. Sailus, R. Banerjee, T. Taniguchi,K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang,“Mapping charge excitations in generalized Wigner crystals,” arXiv:2209.12830(2022).25D. Wong, Y. Wang, J. Jung, S. Pezzini, A. M. DaSilva, H.-Z. Tsai, H. S. Jung,R. Khajeh, Y. Kim, J. Lee, S. Kahn, S. Tollabimazraehno, H. Rasool,K. Watanabe, T. Taniguchi, A. Zettl, S. Adam, A. H. MacDonald, andM. F. Crommie, “Local spectroscopy of moiré-induced electronic structure ingate-tunable twisted bilayer graphene,” Phys. Rev. B 92, 155409 (2015).26M. Yankowitz, J. Xue, D. Cormode, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, P. Jarillo-Herrero, P. Jacquod, and B. J. LeRoy, “Emergence ofsuperlattice Dirac points in graphene on hexagonal boron nitride,” Nat. Phys. 8,382 (2012).27B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. Leroy,K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, andR. C. Ashoori, “Massive Dirac fermions and Hofstadter butterfly in a van derWaals heterostructure,” Science 340, 1427 (2013).28X. Zhang, K.-T. Tsai, Z. Zhu, W. Ren, Y. Luo, S. Carr, M. Luskin, E. Kaxiras,and K. Wang, “Correlated insulating states and transport signature of supercon-ductivity in twisted trilayer graphene superlattices,” Phys. Rev. Lett. 127, 166802(2021).29D. Wong, S. Jeon, K. P. Nuckolls, M. Oh, S. C. J. Kingsley, and A. Yazdani, “Amodular ultra-high vacuum millikelvin scanning tunneling microscope,” Rev.Sci. Instrum. 91, 023703 (2020).30D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe,T. Taniguchi, B. A. Bernevig, and A. Yazdani, “Cascade of electronic transitionsin magic-angle twisted bilayer graphene,” Nature 582, 198 (2020).31S. J. Ahn, P. Moon, T.-H. Kim, H.-W. Kim, H.-C. Shin, E. H. Kim, H. W. Cha,S.-J. Kahng, P. Kim, M. Koshino, Y.-W. Son, C.-W. Yang, and J. R. Ahn, “Diracelectrons in a dodecagonal graphene quasicrystal,” Science 361, 782 (2018).32T. Yu and L. Liao, “Comment on “Dirac electrons in a dodecagonal graphenequasicrystal”,” arXiv:1808.08730 (2018).33M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, X. Liu, K. Watanabe, T. Taniguchi,and A. Yazdani, “Evidence for unconventional superconductivity in twistedbilayer graphene,” Nature 600, 240 (2021).34H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski,K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, “Evidence for uncon-ventional superconductivity in twisted trilayer graphene,” Nature 606, 494(2022).35A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi,and M. A. Kastner, and D. Goldhaber-Gordon, “Emergent ferromagnetism nearthree-quarters filling in twisted bilayer graphene,” Science 365, 605 (2019).36M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe,T. Taniguchi, L. Balents, and A. F. Young, “Intrinsic quantized anomalous Halleffect in a moiré heterostructure,” Science 367, 900 (2020).37A. A. Zibrov, E. M. Spanton, H. Zhou, C. Kometter, T. Taniguchi, K. Watanabe,and A. F. Young, “Even-denominator fractional quantum Hall states at an isospintransition in monolayer graphene,” Nat. Phys. 14, 930 (2018).38T. Han, J. Yang, Q. Zhang, L. Wang, K. Watanabe, T. Taniguchi,P. L. McEuen, and L. Ju, “Accurate measurement of the gap of graphene/h-BNmoiré superlattice through photocurrent spectroscopy,” Phys. Rev. Lett. 126,146402 (2021).39U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi,K. Watanabe, Y. Oreg, F. von Oppen, A. Stern, E. Berg, P. Jarillo-Herrero, andS. Ilani, “Cascade of phase transitions and Dirac revivals in magic-angle gra-phene,” Nature 582, 203 (2020).Low TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 660Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://doi.org/10.1126/science.aav1910https://doi.org/10.1038/s41586-020-2458-7https://doi.org/10.1038/s41567-018-0387-2https://doi.org/10.1038/s41586-020-2963-8https://doi.org/10.1038/s41563-020-0708-6https://doi.org/10.1038/s41586-020-2085-3https://doi.org/10.1038/s41586-021-03853-0https://doi.org/10.1073/pnas.1108174108https://doi.org/10.1103/PhysRevB.82.121407https://doi.org/10.1103/PhysRevB.82.121407https://doi.org/10.1038/s41567-021-01418-6https://doi.org/10.1038/s41563-020-00911-2https://doi.org/10.1038/s41586-021-03938-whttp://arxiv.org/abs/arXiv.2112.07127https://doi.org/10.1021/acs.nanolett.1c03699https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/s41567-021-01171-whttps://doi.org/10.1038/s41565-022-01180-7https://doi.org/10.1103/PhysRev.46.1002https://doi.org/10.1103/PhysRevLett.42.795https://doi.org/10.1103/PhysRevLett.60.2765https://doi.org/10.1103/PhysRevLett.60.2765https://doi.org/10.1038/s41586-021-03874-9http://arxiv.org/abs/arXiv.2209.12830https://doi.org/10.1103/PhysRevB.92.155409https://doi.org/10.1038/nphys2272https://doi.org/10.1126/science.1237240https://doi.org/10.1103/PhysRevLett.127.166802https://doi.org/10.1063/1.5132872https://doi.org/10.1063/1.5132872https://doi.org/10.1038/s41586-020-2339-0https://doi.org/10.1126/science.aar8412http://arxiv.org/abs/arXiv:1808.08730https://doi.org/10.1038/s41586-021-04121-xhttps://doi.org/10.1038/s41586-022-04715-zhttps://doi.org/10.1126/science.aaw3780https://doi.org/10.1126/science.aay5533https://doi.org/10.1038/s41567-018-0190-0https://doi.org/10.1103/PhysRevLett.126.146402https://doi.org/10.1038/s41586-020-2373-yhttps://aip.scitation.org/journal/ltp40G. L. Yu, R. V. Gorbachev, J. S. Tu, A. V. Kretinin, Y. Cao, R. Jalil, F. Withers,L. A. Ponomarenko, B. A. Piot, M. Potemski, D. C. Elias, X. Chen, K. Watanabe,T. Taniguchi, I. V. Grigorieva, K. S. Novoselov, V. I. Fal’ko, A. K. Geim, andA. Mishchenko, “Hierarchy of Hofstadter states and replica quantum Hall ferro-magnetism in graphene superlattices,” Nat. Phys. 10, 525 (2014).41A. Uri, S. Grover, Y. Cao, J. A. Crosse, K. Bagani, D. Rodan-Legrain,Y. Myasoedov, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino,P. Jarillo-Herrero, and E. Zeldov, “Mapping the twist-angle disorder and Landaulevels in magic-angle graphene,” Nature 581, 47 (2020).42J. Jung, A. M. Dasilva, A. H. Macdonald, and S. Adam, “Origin of band gapsin graphene on hexagonal boron nitride,” Nat. Commun. 6, 1 (2015).43J. C. W. Song, P. Samutpraphoot, and L. S. Levitov, “Topological Bloch bandsin graphene superlattices,” Proc. Natl. Acad. Sci. USA 112, 10879 (2015).44B. Uchoa, V. N. Kotov, and M. Kindermann, “Valley order and loop currentsin graphene on hexagonal boron nitride,” Phys. Rev. B 91, 121412 (2015).45Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, andP. Jarillo-Herrero, “Unconventional superconductivity in magic-angle graphenesuperlattices,” Nature 556, 43 (2018).46E. Khalaf, S. Chatterjee, N. Bultinck, M. P. Zaletel, and A. Vishwanath,“Charged skyrmions and topological origin of superconductivity in magic-anglegraphene,” Sci Adv. 7, eabf5299 (2021).47M. Christos, S. Sachdev, and M. S. Scheurer, “Superconductivity, correlatedinsulators, and Wess-Zumino-Witten terms in twisted bilayer graphene,” Proc.Natl. Acad. Sci. USA 117, 29543 (2020).48E. Lake, A. S. Patri, and T. Senthil, “Pairing symmetry of twisted bilayer gra-phene: A phenomenological synthesis,” Phys. Rev. B 106, 104506 (2022).Low TemperaturePhysics ARTICLE scitation.org/journal/ltpLow Temp. Phys. 49, 655 (2023); doi: 10.1063/10.0019422 49, 661Published under an exclusive license by AIP Publishing 28 June 2023 15:06:58https://doi.org/10.1038/nphys2979https://doi.org/10.1038/s41586-020-2255-3https://doi.org/10.1038/ncomms7308https://doi.org/10.1073/pnas.1424760112https://doi.org/10.1103/PhysRevB.91.121412https://doi.org/10.1038/nature26160https://doi.org/10.1126/sciadv.abf5299https://doi.org/10.1073/pnas.2014691117https://doi.org/10.1073/pnas.2014691117https://doi.org/10.1103/PhysRevB.106.104506https://aip.scitation.org/journal/ltp