# Fileset

[s41467-022-35421-z.pdf](https://mdr.nims.go.jp/filesets/b0530625-e3f2-4c46-95df-77fae2a3b044/download)

## Creator

Pratap Chandra Adak, Subhajit Sinha, Debasmita Giri, Dibya Kanti Mukherjee, Chandan, L. D. Varma Sangani, Surat Layek, Ayshi Mukherjee, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), H. A. Fertig, Arijit Kundu, Mandar M. Deshmukh

## Rights

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

## Other metadata

[Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands](https://mdr.nims.go.jp/datasets/87e08a94-f12e-4c9b-abed-114ce014501e)

## Fulltext

Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bandsArticle https://doi.org/10.1038/s41467-022-35421-zPerpendicular electric field drives Cherntransitions and layer polarization changes inHofstadter bandsPratap Chandra Adak 1 , Subhajit Sinha 1, Debasmita Giri2,Dibya Kanti Mukherjee3,4,5, Chandan1, L. D. Varma Sangani1, Surat Layek 1,Ayshi Mukherjee1, Kenji Watanabe 6, Takashi Taniguchi 7, H. A. Fertig3,4,Arijit Kundu 2 & Mandar M. Deshmukh 1Moiré superlattices engineer bandproperties and enable observationof fractalenergy spectra of Hofstadter butterfly. Recently, correlated-electron physicshosted by flat bands in small-angle moiré systems has been at the foreground.However, the implications of moiré band topology within the single-particleframework are little explored experimentally. An outstanding problem isunderstanding the effect of band topology on Hofstadter physics, which doesnot require electron correlations. Ourwork experimentally studies Chern stateswitching in the Hofstadter regime using twisted double bilayer graphene(TDBG), which offers electric field tunable topological bands, unlike twistedbilayer graphene. Here we show that the nontrivial topology reflects in theHofstadter spectra, in particular, by displaying a cascade of Hofstadter gapsthat switch their Chern numbers sequentially while varying the perpendicularelectric field. Our experiments together with theoretical calculations suggest acrucial role of charge polarization changing concomitantly with topologicaltransitions in this system. Layer polarization is likely to play an important rolein the topological states in few-layer twisted systems. Moreover, our workestablishes TDBG as a novel Hofstadter platform with nontrivial magneto-electric coupling.The 2D moiré lattice, when subjected to a magnetic field, loses itsperiodicity due to spatial dependence of the gauge potential. How-ever, when the applied magnetic field is such that the magnetic fluxquantum per unit cell of the moiré lattice is a rational number, thediscrete translational symmetry of the lattice is restored with a largermagnetic unit cell. The energy spectrumof sucha system, as a functionof the magnetic field, has a self-similar fractal structure known asHofstadter’s butterfly1. The observation of this quantum fractal islimited by the requirement of high magnetic flux Φ through the unitcell, such that Φ/Φ0 ~ 1. Here, Φ0 = h/e is the magnetic flux quantum,with h being Planck’s constant and e being the electron charge. Hof-stadter’s butterfly was first observed in graphene aligned to hexagonalboron nitride (hBN)2,3. The large unit cell in such moiré superlatticesrealizes Φ/Φ0 ~ 1 with available lab magnets.Received: 25 November 2021Accepted: 2 December 2022Check for updates1Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India.2Department of Physics, Indian Institute of Technology, Kanpur 208016, India. 3Department of Physics, Indiana University, Bloomington, IN 47405, USA.4QuantumScience andEngineeringCenter, IndianaUniversity, Bloomington, IN47408,USA. 5Laboratoire dePhysiquedes Solides, Univ. Paris-Sud,UniversitéParis Saclay, CNRS, UMR 8502, F-91405 Orsay Cedex, France. 6Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan. 7International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044,Japan. e-mail: pratapchandraadak@gmail.com; kundua@iitk.ac.in; deshmukh@tifr.res.inNature Communications |         (2022) 13:7781 11234567890():,;1234567890():,;http://orcid.org/0000-0003-4719-161Xhttp://orcid.org/0000-0003-4719-161Xhttp://orcid.org/0000-0003-4719-161Xhttp://orcid.org/0000-0003-4719-161Xhttp://orcid.org/0000-0003-4719-161Xhttp://orcid.org/0000-0002-8219-0287http://orcid.org/0000-0002-8219-0287http://orcid.org/0000-0002-8219-0287http://orcid.org/0000-0002-8219-0287http://orcid.org/0000-0002-8219-0287http://orcid.org/0000-0002-1719-2542http://orcid.org/0000-0002-1719-2542http://orcid.org/0000-0002-1719-2542http://orcid.org/0000-0002-1719-2542http://orcid.org/0000-0002-1719-2542http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0001-8911-4829http://orcid.org/0000-0001-8911-4829http://orcid.org/0000-0001-8911-4829http://orcid.org/0000-0001-8911-4829http://orcid.org/0000-0001-8911-4829http://orcid.org/0000-0002-1401-1080http://orcid.org/0000-0002-1401-1080http://orcid.org/0000-0002-1401-1080http://orcid.org/0000-0002-1401-1080http://orcid.org/0000-0002-1401-1080http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-35421-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-35421-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-35421-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-35421-z&domain=pdfmailto:pratapchandraadak@gmail.commailto:kundua@iitk.ac.inmailto:deshmukh@tifr.res.inRecently, the ability to stack multiple layers of 2D materialsrotated with sub-degree precision has opened up a new frontier. Inaddition to tuning the moiré length scale, the twist angle between twoadjacent layers tunes the symmetry and the topology of the emergentmoiré bands, providing new experimental knobs. Furthermore, magic-angle twisted bilayer graphene (TBG) hosts low-energy flat bands4–6that support correlated-electron phenomena such as correlated insu-lator states7–9, ferromagnetism10,11, and superconductivity9,12. Topolo-gical properties of the twisted systems are of particular interest, asseveral recent studies have explored correlated Chern insulator statesin the Hofstadter regime in TBG13–18. These states are interpreted asarising due to the occupation of subsets of underlying Chern bands13–15or Hofstadter subbands16–18, a mechanism similar to Quantum Hallferromagnetism. In the physics of Chern insulator states, as also in thequantumHall physics due to the formation of Landau levels, gaps withdifferent Chern numbers can be accessed by changing the Fermienergy by varying the charge density. Recently a pure electrical con-trol, such as the perpendicular electric field, to open up a Cherninsulating state from a bulk gapless state has been demonstrated usinga correlated system19. Similar electrical control over Chern stateswithout requiring electron correlation will be novel.Twisted double bilayer graphene (TDBG), made by twisting twocopies of Bernal stacked bilayer graphene (BLG), provides suchopportunities as the electric field can tune the band structure and itstopological properties20–26. Notably, the flat bands in TDBG possess anonzero valley Chern number that changes with the electric field,offering a unique correlated Hofstadter platform27. While earlierexperiments in TDBG have a major focus on electron correlationsphysics28–34, the tunability of the topological flat bands in the Hof-stadter regime is little explored35. An ability to tune theChern numbersof these bandswould provide further insight on the role of topology inthese correlated states. Thus TDBG is a rich platform as the electricfield plays an important role, unlike in TBG.In this work, we study electron transport in TDBGwith small twistangles around 1.1∘–1.5∘ under a highmagnetic field uptoΦ/Φ0 ~ 1/3. Weobserve a cascade of gaps that change their Chern numbers sequen-tially as the perpendicular electric field is varied. This contrasts withTBG for which the band structure is unaffected by such an electricfield, and so cannot induce Chern transition. The Hofstadter fan dia-gramswemeasure show additional features that reveal the topologicalnature of the underlying band structure. Our calculation of the Hof-stadter energy spectrum in TDBG confirms the key experimentalobservations. Interestingly, we find that a small exchange enhancedspin Zeeman term plays a role in determining the sequence of Cherngaps. Furthermore, our analysis shows that the electric field varies thelayer polarization and provides the underlying mechanism of theChern transition.ResultsLow-temperature transportWe now present our magneto-transport measurements in TDBGdevices. To fabricate TDBG devices, we cut two pieces of BLG from asingle exfoliatedflake and sandwich thembetween twohBNflakeswitha relative rotation36. A schematic of our device structure is shown inFig. 1a. Using the metal top gate and Si++ bottom gate we can inde-pendently control the charge density, n, and the perpendicular electricdisplacement field, D. Using the multiple electrodes in the devices wemeasure low-temperature electron transport in a Hall bar geometryunder a perpendicular magnetic field, B. See Methods section fordetails about fabrication and measurement.We first discuss electron transport in zero magnetic field. InFig. 1b, we calculate the zero magnetic field band structures of TDBGwith a twist angle of 1.10∘ for two different electric fields (see Supple-mentary Note 1 for calculation details). In addition to the tunability ofthe band gaps andwidth, the valley Chern numbers of themoiré bandschange with the electric field. In Fig. 1c, we show a color-scale plot ofthe longitudinal conductivity, σxx, as a functionof ν andD atB = 0Tanda temperature of 300mK for aTDBGdevicewith twist angle 1.10∘. Here,ν = 4n/nS is the moiré filling factor, where nS is the number of chargecarriers required to fill an isolated moiré band and the factor 4 incor-porates the spin and valley degeneracy. In the color-scale plot of σxx,conductivity dips are observed corresponding to two moiré gaps atν = ±4 and the CNP gap at ν = 0. The electric field tunability of theunderlying band structure is evident as two moiré gaps close at a highelectric field, and the CNP gap opens up only above a finite electricfield value. We also see additional regions with low conductance – across-like feature around D/ϵ0 = 0 and ν ~ −2 on the hole side and twohalo regions around D/ϵ0 = ±0.3 V/nm and ν ~ 2 on the electron side.These are characteristic features of small-angle TDBG37. Here we note,while the correlated gaps at partial filling develop in TDBG with twistangle ~ 1.2∘ − 1.4∘, for smaller twist angle ~ 1. 1∘ correlated gaps developunder a parallel magnetic field23,28,30. See Supplementary Fig. 8 for datafrom a 1.46∘ device, where we observe correlated gap at a zeromagnetic field.Observation of electric field tunable Chern gaps in high mag-netic fieldTo study the effect of a perpendicular magnetic field B, we now mea-sure σxx as a function of ν and D for different values of B as shown inFig. 1d–f. In contrast to the case of B =0, the CNP gap emerges even atD = 0 as we apply a finite magnetic field (see Fig. 1d). Furthermore, asevident from the σxx plot at 5 T in Fig. 1e, the CNP gap undergoesmultiple closing and reopening as the electric field is varied. At highermagnetic fields, such as at 9 T in Fig. 1f, the formation of Landau Levels(LL) leads to multiple lines of conductivity peaks and dips parallel totheD-axis. Ourmost interesting observation is that the positions of theσxx dips shift discretely on the ν-axis as the electric field is varied. Thiseffect of the electric field is most prominent in the region ∣D∣/ϵ0≲0.25V/nm (indicatedby thedashed rectangle in Fig. 1f). Interestingly,withinthis D-range, marked by the closing of the hole-side moiré gap at ∣D∣/ϵ0 ~ 0.25 V/nm at zero magnetic field, both the flat bands are isolatedfrom the remotemoirébands. At higher electricfields, as the flat bandsmerge with the remote moiré bands, the electric field tunabilitybecomes weaker.We now study the step-like evolution of the σxx dips with D indetail by focusing on σxx as a function of ν andD at B = 9T in Fig. 2a. Toelucidate the nature of these dips as they evolve in the three-dimensional parameter space of ν, D, and B, we perform a systematicanalysis. We first identify two groups of most prominent σxx dipscorresponding to the larger gaps (see the Supplementary Note 3 andSupplementary Fig. 2 for estimation of the gaps), which shift further onthe hole side as D is increased, by line segments of different colors.Then to track the evolution of these dips with B, we measure σxx as afunction of ν andB at different values of constantD. In Fig. 2b, we showthree such plots, namely fan diagrams, for three different D. We focuson the pair of prominent σxx dips in each fan diagram. The position ofthe dips on the ν-axis evolve with B along linear trajectories markedwith lines of the same colors as in Fig. 2a. Besides the marked pair ofdips, there are other dips which are also tuned by the electric field(Supplementary Fig. 4). See SupplementaryNote 4 and SupplementaryFigs. 7–9 for similar data from devices with different twist angles.The linear trajectories of the conductivity dips can be understoodin terms of Hofstadter physics. Within the Hofstadter picture, the gapswith integer Chern number C follow linear trajectories in n-B diagram,i.e., the Wannier diagram, given by the Diophantine equation,ν =CΦΦ0+ s: ð1ÞArticle https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 2Here, s is an integer denoting the moiré filling factor corresponding tothe number of carriers per moiré unit cell in zero magnetic field, andΦ =BA, with A being the area of the moiré unit cell. We extract C and sfrom the slope and ν-axis intercept in the fan diagrams at different Dfor all the marked lines. The extracted values of (C, s) are marked inFig. 2b. Then we assign these (C, s) values inferred from Fig. 2b to thecorresponding σxx minima in Fig. 2a. Here we note that from the evo-lution of σxx dips with temperature we extract the Hofstadter gaps tobe ~0.1−0.5 meV (see Supplementary Note 3 and SupplementaryFig. 2). Our experimental data of σxy (Supplementary Fig. 5) showweakquantization possibly due to the smallness of these gaps together withangle inhomogeneity disorder38. See Supplementary Note 6 and Sup-plementary Fig. 11 for the role of flat band energy scale in Hofstadterspectra. Also, moiré commensurability aspects can lead to absence ofquantization39.From Fig. 2a, we find two interesting trends in the transition of(C, s) as a function of D. Firstly, for both the groups corresponding tos =0 and s = −2, the Chern number decreases sequentially by 1 as themagnitude of D is increased; crucially, we observe both even and oddChern numbers. Secondly, the difference in C for s =0 and s = −2remains 2. The sequential change of Chern number by D can beempirically understood by considering a simple schematic in Fig. 2c,where apartof theChernbandpeels off due to varying electricfield. AsD is varied, branches with Chern number 1 are separated from a groupof Hofstadter subbands and merge with another group, resulting in asequence of Chern gaps with different C. A dip in σxx, observedexperimentally, corresponds to a gap between two groups of Hof-stadter subbands. This simple picture of the sequential evolution ofthe Chern gaps with electric field as Hofstadter subbands peel off isconfirmed by our theoretical calculation which we discuss next.Calculation of Hofstadter spectra in TDBGTo calculate the Hofstadter energy spectrum in TDBG, we construct aHamiltonian for each valley in the basis of bare Landau levels of gra-phene, indexed by the Landau level index, guiding center, and layerindex. Inter-bilayer tunneling then couples states with various guidingcenters and Landau levels of each layer and one can diagonalize theHamiltonian to find the spectra. The prominent Chern gaps are char-acterized as a functionof thefilling factor ν, calculated from the chargeneutrality point and flux quanta per moiré unit cell,Φ/Φ0, obtained bysolving the Diophantine equation (Eq. (1)). In Fig. 3a, we plot the Hof-stadter spectra for both K and K 0 valleys in two different colors for aninterlayer potential of V = 20meV. Here we note that at nonzero V thevalley degeneracy is already lifted due to the nontrivial topology of theunderlying band structure, as we discuss later. The observation of oddChern numbers further implies that the spin degeneracy is also lifted.To incorporate this in our calculation we include an exchangeenhanced spin Zeeman term ΔS = 2.5meV; such term can arise due toFig. 1 | Magneto-transport in twisted double bilayer graphene (TDBG).a Schematic of the dual gated TDBG device. b Calculated band structure for TDBGwith twist angle 1.10∘ in absence of themagnetic field for two different values of theinterlayer potential V. Interlayer potential simulates the application of the electricfield. The numbers in brackets adjacent to each band indicate the correspondingvalley Chern number,which changeswithV. c σxx as a function ofmoiré filling ν andperpendicular electric displacement field D at zero magnetic field at 300mK. d–fVariation of σxx vs. ν andD at three different values of perpendicularmagnetic field,B. The region inside the dashed rectangle in f is discussed in details in Fig. 2.Article https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 3electron correlations of the flat bands. See Supplementary Note 2 forthe details of the calculation. In Fig. 3b, we plot the Wannier diagramshowing the evolution of the Chern gaps as a function of ν and Φ/Φ0for V = 20meV. Here the width of the line segments indicates thestrength of the corresponding gaps considering both K and K 0 valleys.See Supplementary Fig. 1 for calculation at other electric field values.Similar to our experimental observation, we find that two prominentChern gaps originate from s =0 and s = −2 and their Chern numberschange with the electric field.To further elucidate the electric field-induced quantum phasetransition of Chern numbers we plot the evolution of the Hofstadterenergy spectrum as a function of the inter-layer potential V at a con-stant Φ/Φ0 = 0.32 ~B = 9T in Fig. 3c. We find that the energy levelsdisperse nonmonotonically with V. As a result, branches of a Chernband peel off and merge with another band, giving rise to gaps withdifferent Chern numbers at different electricfields. The correspondingplot of extracted Chern gaps as a function ν and V is shown in Fig. 3d.Interestingly, the sequence of changes in the prominent Chern gaps inFig. 3d match quite well with our experimental data in Fig. 2a. More-over, the Chern gaps are more prominent on the hole side, again asseen in experiment.Role of electric field-tunable layer polarizationTo understand the underlying mechanism further, we now discuss therole of an induced layer polarization in tuning the Hofstadter spectra.Under an applied electric field, electrons occupying the four differentlayers of TDBG are at different values of on-site potential, Vi,i = 1, 2, 3, 4. This contributes a mean-field energy of the form E ~∑ρiVi,where ρi is the electron density of i-th layer. As Vi changesproportionally with the electric field, the nonmonotonic change in Ewith the electric field in Fig. 3c suggests an asymmetric change in thedistribution of ρi. To measure the asymmetric change in the distribu-tion, we define an energy band specific charge polarization acrosslayers, P = ρ1 + ρ2 − ρ3 − ρ4. The role of this layer polarization in deter-mining the energy vs. electricfield dispersion is depicted schematicallyin Fig. 3e. For an electron distribution polarized toward the top layer,the energy increases with an increasing electric field, and hence theenergy vs. electric field dispersion has a positive slope. The slope isnegative for the opposite polarization. As the layer polarization isvaried by the electric field, the energy state evolves nonmonotonicallywith the electric field. When two energy bands with different Chernnumbers cross each other, the Chern number of the gap between thetwo bands changes. The electric field tunable layer polarization isindeed confirmed in Fig. 3f whereweplot the polarization vs. V for twodifferent energy levels across the CNP. Figure 3e further suggests howthe tunable layer polarization can manifest itself into a complex evo-lution of a gap, with the possibility of multiple closings andreopenings40. Indeed, we verify this interesting implication as wemeasure σxx at the CNP gap as a function of B andD. We find a complexevolution of the CNP gap with multiple closings and reopenings, afeature distinct from other materials like BLG, suggesting importantrole of field tunable layer polarization in twisted systems (details in theSupplementary Note 5 and Supplementary Fig. 10).Role of topology on Hofstadter spectraFinally, we discuss the important role of the nontrivial topology of theTDBG band structure in determining the Hofstadter spectra. At finitevalues of the electricfield, the valley Chernnumbers ofK andK 0 valleysFig. 2 | Electric field tunable Chern gaps. a Color-scale plot of σxx as a function νand D at B = 9T. The overlayed lines of different colors indicate the (C, s) values ofthe corresponding σxx dips as in the legend. (C, s) values are inferred from b with Cbeing the Chern number and s being themoiré filling factor corresponding to B =0T (see Supplementary Fig. 6 for fitting details to extract (C, s) values). b Color-scaleplots of σxx as a function of ν and B, i.e., fan diagrams, at three different values ofD.The overlayed lines trace the trajectory of the σxx minima. The slope and the ν-axisintercept give C and s, respectively. (C, s) values are indicated adjacent to the linesand assigned in a accordingly. c Schematic representation of the origin of electricfield tunable Chern gaps. Chern number of the gaps changes when Hofstadtersubbands with Chern number 1 peel off from a group of Hofstadter subbands andmerge with another group as the electric field is varied.Article https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 4are nonzero and opposite in sign. As seen form Fig. 3a, the Hofstadterenergy spectra from two valleys disperse differently with themagneticfield due to the different topologies of the two valleys27. This has twoimportant implications. Firstly, the two-fold valley degeneracy is lifted,as we noted earlier. Secondly, only themost prominent gaps survive asgaps in the Hofstadter spectrum of one valley can be filled by theenergy states of the other. This is evidenced in our experimental dataas well.We find additional signatures of the nontrivial band topology inthe Hofstadter energy spectra in TDBG. The Hofstadter spectrum of atopologically trivial band is confined within the band, i.e., dis-connected from the spectra of the neighboring bands. Conversely, theHofstadter spectrumof a topological band connects to thatof a nearbyband, such that the total Chern number of the bands with connectedHofstadter spectra is zero41,42. Consequently, in a Hofstadter energyspectrum for a topological band, the gap with a nonzero Chern num-ber C closes at Φ/Φ0 ≤ 1/∣C∣. Indeed, our calculation in Fig. 3a, b con-firms this as the CNP gap (0, 0) closes at Φ/Φ0 < 0.5, consistent withvalleyChern number 2 at theCNPgapof AB-ABTDBGat afinite electricfield. This contrasts with the Hofstadter spectrum for AB-BA TDBG at afinite electric field, for which the CNP gap has a zero valley Chernnumber, and find the CNP gap open throughout 0 <Φ/Φ0 < 1(Fig. 4a, b).To see experimental signatures of nontrivial band topology, weplot Rxx as a function of ν and B for D/ϵ0 = −0.02 V/nm in Fig. 4c fromthe 1.10∘ AB-AB TDBG device. We find that the moiré gap of ν = 4 isFig. 3 | Calculation of Hofstadter spectra and layer polarization in TDBG.a Calculated Hofstadter energy levels for both K and K 0 valleys in TDBG with twistangle 1.10∘ for an interlayer potential of V = 20 meV. Energy levels disperse differ-ently for two valleys due to different topology resulting in fewer number of gaps--e.g., energy gap for K valley at location ⋆ is filled with energy levels from K 0 valley(vice versa for location #). b Corresponding Wannier diagram showing the evolu-tion of Hofstadter gaps considering both the valleys as a function ν and Φ/Φ0.c Evolution of energy spectra as a function of interlayer potential V at a constantmagnetic field corresponding to Φ/Φ0 =0.32 ~ 9T with an exchange enhancedZeeman splitting of 2.5meV. This confirms the change in Chern numbers as sche-matically depicted in Fig. 2c. d Evolution of gaps as a function of ν and V extractedfrom c. The width of the line segments in b and d are proportional to the values ofthe gaps in the Hofstadter spectrum. The color of the line segments denotes thecorresponding Chern numberC, as indicated in the color-scale legend. e Schematicdepicting the role of energy band-specific charge polarization across layers inenergy vs. electric field dispersion. When electrons are polarized more toward top(bottom) layers, as for points A and D (B and C), the energy of the band increases(decreases) with increasing electric field. As the layer polarization is varied by theelectric field, energy disperses nonmonotonically with the electric field inducingmultiple gap closing andopening -- this can lead to changes in the Chern number ofthe gap. f Calculated layer polarization (bottom panel) corresponding to twoHofstadter bands (top panel) across the CNP gap (0,0) (indicated by two arrows inc). Panels a and b are shaded in the same color to indicate that they are extractedfrom the same data (similarly for panels c and d).Article https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 5weakened atΦ/Φ0 = 1/3, as the Hofstadter gap of (12,0) crosses it (alsosee Fig. 4e for a line-plot of Rxx vs. B at ν = 4). The nonzero Rxy at thecrossing point of the moiré gap, where Rxy is otherwise zero, furthercorroborates the dominance of the Hofstadter gap of (12,0) over themoiré gap of (0,4) (Fig. 4f). This suggests that the Hofstadter spectraacross the moiré gap are connected due to the nontrivial topology.DiscussionIn conclusion, we have presented a comprehensive study of magneto-transport in TDBG in Hofstadter regime, complemented by theoreticalcalculations of Hofstadter spectra. We identified the manifestation ofunderlying nontrivial topology of the TDBG flat bands on these spec-tra. The tunable layer polarization plays a key role in determining theHofstadter spectra and the quantum phase transition betweenChern gaps.Our central result, that the Chern gap can be controlled by varyingthe electric field, rather than the charge density, has important impli-cations for magnetoelectric coupling: a Chern gap Δg with Chern num-ber C gives rise to a change in magnetization, δM∝CΔg43. Thus thephysics we have identified allows electrical control of the system mag-netization. Here we note that the control over Chern states has beenrecently demonstrated in twistedmonolayer-bilayer graphene by tuningthe charge density44 and in hBN-aligned ABC trilayer graphene using theelectric field19. Our work demonstrates a novel pathway to control Chernstates using the electric field without requiring electron correlation as aprerequisite. Furthermore, the Hofstadter platform of TDBG offers aplethora of Chern transitions over a broad region of electric field. It isinteresting to speculate that ferroelectric correlations, as seen in recentexperiments45, could stabilize Chern bands and the physics we discuss inthis study even at zero magnetic field.MethodsDevice fabricationTo fabricate TDBG devices, we first exfoliated graphene flakes and cutthe selected bilayer graphene flake into two halves by using a scalpelmade from an optical fiber36. The bilayer graphene flakes were chosenbased onoptical contrast and later confirmed by Raman spectroscopy.We chose exfoliated hBN flakes of 20 nm to 40nm in thickness, firstbased on optical inspection of the color and later measured by AFM.Then we made the hBN-BLG-BLG-hBN stack using the standard poly-carbonate (PC) based dry transfer method46 and dropped on SiO2/Si++substrate. The twist anglewas introduced by rotating the bottom stagewhile picking up the second half of the bilayer graphene flake. After-ward, wemade the top gate by e-beam lithography and depositing Cr/Au by e-beam evaporation. Subsequently, we defined the geometry ofthe devices by e-beam lithography followed by etching in CHF3+O2plasma. Finally, we made 1D edge contact by etching in CHF3+O2plasma and then depositing Cr/Pd/Au.Transport measurementWe carried out the low-temperature transport measurements at300mK in a He-3 insert inside a liquid He flow cryostat under a per-pendicular magnetic field from 0T to 13.6 T. A current of ~10 nA wassent, and the four-probe voltage was measured using lock-in amplifierusing low frequency (~13–17 Hz) after amplifying with a preamplifier.Themeasurement of themagneto-resistance at the CNP gappresentedFig. 4 | Role of topology on Hofstadter spectra. a Calculated Hofstadter energylevels for both K (red) and K 0 (blue) valleys in AB-BA TDBGwith twist angle 1.10∘ foran interlayer potential of V = 20 meV. b Evolution of gaps corresponding to thespectrum in a. Unlike AB-AB case (see Fig. 3a, b), the CNP gap (0, 0) is open for anyflux0 <Φ/Φ0 < 1. cColor-scaleplot ofRxx as a function ν andB atD/ϵ0 = −0.02V/nmfrom the 1.10∘ AB-AB TDBG device. d Schematic of c, showing the weakening of themoiré gap (0, 4) atΦ/Φ0 = 1/3 when a Hofstadter gap (12, 0) crosses. e Line slices ofRxx vs. B at ν = 4 showing a clear dip at the crossing. f Color-scale plot of Rxycorresponding to c. The Hofstadter gap (12,0) (with finite Rxy) dominates over themoiré gap (0,4) (zero Rxy), indicating that Hofstadter spectra across the moiré gapare connected to each other.Article https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 6in Supplementary Fig. 10 was carried out at 20mK in a dilution fridgeupto B = 12 T. The charge density n and the perpendicular electricdisplacement field D were calculated using the formula,n = (CBGVBG+CTGVTG)/e − n0 and D = (CBGVBG−CTGVTG)/2 −D0. Here CTGand CBG are the capacitance per unit area of the top and the back gate,respectively, e being the charge of an electron. n0 and D0 are the smalloffsets in the charge density and the electric displacement field. Thecapacitance values were calculated at first by noting the dielectricthickness and later estimated more precisely using the magneto-transport features such as the positions of the Brown-Zak oscillationson B-axis. To avoid artifacts associated with lead asymmetry we sym-metrize the longitudinal resistance as �RxxðBÞ= ðRxxðBÞ+Rxxð�BÞÞ=2,and antisymmetrize the transverse resistance as �RxyðBÞ= ðRxyðBÞ�Rxyð�BÞÞ=2. The longitudinal conductivity σxx and the transverse con-ductivity σxy were calculated using the formula σxx = ðw=lÞ�Rxx=ð�R2xy +ðw=lÞ2�R2xxÞ and σxy = �Rxy=ð�R2xy + ðw=lÞ2�R2xxÞ, respectively. Here, w is thewidth and l is the length of the Hall bar geometry. In the text, thesymmetrized longitudinal resistance and the antisymmetrized trans-verse resistance are denoted by Rxx and Rxy for brevity.Twist angle determinationWedetermine the twist angle θbased on our low-temperature electrontransport measurement, using the relation nS =8θ2=ffiffiffi3pa2. Here nS isthe charge carrier density corresponding to the full filling of themoiréband (ν = ±4), a = 0.246 is the lattice constant of graphene. To deter-mine nS, we locate ν = ±4 by tracing the sequence of Landau levels tothe n-axis at B =0.Data availabilityThe experimental data used in the figures of themain text are availablein Zenodo with the identifier https://doi.org/10.5281/zenodo.565368847. Additional data related to this study are available fromthe corresponding authors upon reasonable request.References1. Hofstadter, D. R. Energy levels and wave functions of Bloch elec-trons in rational and irrational magnetic fields. Phys. Rev. B 14,2239–2249 (1976).2. Hunt, B. et al. Massive Dirac Fermions and Hofstadter Butterfly in avan der Waals Heterostructure. Science 340, 1427–1430 (2013).3. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantumHalleffect in moiré superlattices. Nature 497, 598–602 (2013).4. Suárez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic,Z. Flat bands in slightly twisted bilayer graphene: Tight-bindingcalculations. Phys. Rev. B 82, 121407 (2010).5. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H.Continuummodel of the twisted graphene bilayer. Phys. Rev. B 86,155449 (2012).6. Bistritzer, R.&MacDonald, A.H.Moirébands in twisteddouble-layergraphene. Proc. Natl Acad. Sci. 108, 12233–12237 (2011).7. Kim, K. et al. Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene. Proc. Natl Acad. Sci. 114,3364–3369 (2017).8. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).9. Cao, Y. et al. Unconventional superconductivity in magic-anglegraphene superlattices. Nature 556, 43–50 (2018).10. Sharpe, A. L. et al. Emergent ferromagnetism near three-quartersfilling in twisted bilayer graphene. Science 365, 605–608 (2019).11. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiréheterostructure. Science 367, 900–903 (2020).12. Lu, X. et al. Superconductors, orbitalmagnets andcorrelated statesin magic-angle bilayer graphene. Nature 574, 653–657 (2019).13. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).14. Das, I. et al. Symmetry-broken Chern insulators and Rashba-likeLandau-level crossings in magic-angle bilayer graphene. Nat. Phys.17, 710–714 (2021).15. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Cherninsulators, van Hove singularities and topological flat bands inmagic-angle twisted bilayer graphene. Nat. Mater. 20,488–494 (2021).16. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17,478–481 (2021).17. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P.Flavour Hund’s coupling, Chern gaps and charge diffusivity inmoiré graphene. Nature 592, 43–48 (2021).18. Choi, Y. et al. Correlation-driven topological phases in magic-angletwisted bilayer graphene. Nature 589, 536–541 (2021).19. Chen, G. et al. Tunable correlated Chern insulator and ferro-magnetism in a moiré superlattice. Nature 579, 56–61 (2020).20. Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted doublebilayer graphene. Phys. Rev. B 99, 235417 (2019).21. Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearlyflat Chern bands in moiré superlattices. Phys. Rev. B 99,075127 (2019).22. Koshino, M. Band structure and topological properties of twisteddouble bilayer graphene. Phys. Rev. B 99, 235406 (2019).23. Lee, J. Y. et al. Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene. Nat.Commun. 10, 5333 (2019).24. Choi, Y. W. & Choi, H. J. Intrinsic band gap and electrically tunableflat bands in twisted double bilayer graphene. Phys. Rev. B 100,201402(R) (2019).25. Liu, J., Ma, Z., Gao, J. & Dai, X. Quantum Valley Hall effect, orbitalmagnetism, and anomalous hall effect in twisted multilayer gra-phene systems. Phys. Rev. X 9, 031021 (2019).26. Wang, Y.-X., Li, F. & Zhang, Z.-Y. Phase diagram and orbital Cherninsulator in twisted double bilayer graphene. Phys. Rev. B 103,115201 (2021).27. Crosse, J. A., Nakatsuji, N., Koshino, M. & Moon, P. Hofstadter but-terfly and the quantum Hall effect in twisted double bilayer gra-phene. Phys. Rev. B 102, 035421 (2020).28. Burg, G. W. et al. Correlated insulating states in twisted doublebilayer graphene. Phys. Rev. Lett. 123, 197702 (2019).29. Shen, C. et al. Correlated states in twisted double bilayer graphene.Nat. Phys. 16, 520–525 (2020).30. Adak, P. C. et al. Tunable bandwidths and gaps in twisted doublebilayer graphene on the verge of correlations. Phys. Rev. B 101,125428 (2020).31. Cao, Y. et al. Tunable correlated states and spin-polarized phases intwisted bilayer–bilayer graphene. Nature 583, 215–220 (2020).32. Liu, X. et al. Tunable spin-polarized correlated states in twisteddouble bilayer graphene. Nature 583, 221–225 (2020).33. Sinha, S. et al. Bulk valley transport and Berry curvature spreadingat the edge of flat bands. Nat. Commun. 11, 5548 (2020).34. Liu, L. et al. Isospin competitions and valley polarized correlatedinsulators in twisted double bilayer graphene. Nat. Commun. 13,3292 (2022).35. Burg, G. W. et al. Evidence of emergent symmetry and valley chernnumber in twisted double-bilayer graphene. Preprint at https://arxiv.org/abs/2006.14000 (2020).36. Sangani, L. D. V. et al. Facile deterministic cutting of 2Dmaterials fortwistronics using a tapered fibre scalpel. Nanotechnology 31,32LT02 (2020).37. He,M. et al. Symmetry breaking in twisteddouble bilayer graphene.Nat. Phys. 17, 26–30 (2021).38. Tschirhart, C. L. et al. Imaging orbital ferromagnetism in a moiréChern insulator. Science 372, 1323–1327 (2021).Article https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 7https://doi.org/10.5281/zenodo.5653688https://doi.org/10.5281/zenodo.5653688https://arxiv.org/abs/2006.14000https://arxiv.org/abs/2006.1400039. Shi, J., Zhu, J. & MacDonald, A. H. Moiré commensurability and thequantum anomalous Hall effect in twisted bilayer graphene onhexagonal boron nitride. Phys. Rev. B 103, 075122 (2021).40. Sanchez-Yamagishi, J. D. et al. QuantumHall effect, screening, andlayer-polarized insulating states in twisted bilayer graphene. Phys.Rev. Lett. 108, 076601 (2012).41. Lian, B., Xie, F. & Bernevig, B. A. Landau level of fragile topology.Phys. Rev. B 102, 041402 (2020).42. Herzog-Arbeitman, J., Song, Z.-D., Regnault, N. & Bernevig, B. A.Hofstadter topology: noncrystalline topological materials at highflux. Phys. Rev. Lett. 125, 236804 (2020).43. Zhu, J., Su, J.-J. & MacDonald, A. Voltage-controlled magneticreversal in orbital chern insulators. Phys. Rev. Lett. 125,227702 (2020).44. Polshyn, H. et al. Electrical switching ofmagnetic order in an orbitalChern insulator. Nature 588, 66–70 (2020).45. Zheng, Z. et al. Unconventional ferroelectricity in moiré hetero-structures. Nature 588, 71–76 (2020).46. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).47. Adak, P. C. et al. Experimental data for “Electric field drives Cherntransition in Hofstadter bands of twisted double bilayer graphene".Zenodo https://doi.org/10.5281/zenodo.5653688 (2021).AcknowledgementsWe thank Justin C W Song, Allan H MacDonald, Ajit C Balram, and G JSreejith for helpful discussions. We acknowledge Nanomission grantSR/NM/NS-45/2016 and DST SUPRA SPR/2019/001247 grant along withDepartment of Atomic Energy of Government of India 12-R&D-TFR-5.10-0100 for support. K.W. and T.T. acknowledge support from the Ele-mental Strategy Initiative conducted by theMEXT, Japan (Grant NumberJPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790 andJP20H00354). D.K.M. would like to acknowledge financial support fromAgence Nationale de la Recherche (ANR project “Dirac3D”) under GrantNo. ANR-17-CE30-0023. D.K.M. and H.A.F. acknowledge support fromNSF Grant No. DMR-1914451 and the Research Corporation for ScienceAdvancement through a Cottrell SEED award. H.A.F. further acknowl-edges the support of NSFGrant No. ECCS-1936406, and of the US-IsraelBinational Science Foundation (Grant Nos. 2016130 and 2018726). A.K.acknowledges support from the SERB (Govt. of India) via sanction no.ECR/2018/001443, DAE (Govt. of India) via sanction no. 58/20/15/2019-BRNS, as well as MHRD (Govt. of India) via sanction no. SPARC/2018-2019/P538/SL. D.G. acknowledges the use of HPC facility at IIT Kanpur.D.G. acknowledges the CSIR (Govt. of India) for financial support.Author contributionsP.C.A., S.S., C., and L.D.V.S. fabricated the devices. P.C.A. and S.S. didthe measurements and analyzed the data. S.L. and A.M. helped in fab-rication. D.G., D.K.M, H.A.F., and A.K. did the theoretical calculations.K.W. and T.T. grew the hBN crystals. P.C.A., S.S., A.K., and M.M.D. wrotethe manuscript with inputs from everyone. M.M.D. supervised theproject.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-022-35421-z.Correspondence and requests for materials should be addressed toPratap Chandra Adak, Arijit Kundu or Mandar M. Deshmukh.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of thiswork. Peer reviewer reports are available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons license and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-35421-zNature Communications |         (2022) 13:7781 8https://doi.org/10.5281/zenodo.5653688https://doi.org/10.1038/s41467-022-35421-zhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands Results Low-temperature transport Observation of electric field tunable Chern gaps in high magnetic field Calculation of Hofstadter spectra in TDBG Role of electric field-tunable layer polarization Role of topology on Hofstadter spectra Discussion Methods Device fabrication Transport measurement Twist angle determination Data availability References Acknowledgements Author contributions Competing interests Additional information