# Fileset

[s41699-024-00449-w.pdf](https://mdr.nims.go.jp/filesets/801e42df-2a5d-4b0c-90aa-6fc2ea9f9e31/download)

## Creator

Mickael L. Perrin, Anooja Jayaraj, Bhaskar Ghawri, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Daniele Passerone, Michel Calame, Jian Zhang

## Rights

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

## Other metadata

[Electric field tunable bandgap in twisted double trilayer graphene](https://mdr.nims.go.jp/datasets/5c34104f-8f80-480c-88e6-f75a24bbdbe6)

## Fulltext

Electric field tunable bandgap in twisted double trilayer graphenenpj | 2d materials and applications ArticlePublished in partnership with FCT NOVA with the support of E-MRShttps://doi.org/10.1038/s41699-024-00449-wElectric field tunable bandgap in twisteddouble trilayer grapheneCheck for updatesMickael L. Perrin 1,2,3 , Anooja Jayaraj4, Bhaskar Ghawri1, Kenji Watanabe 5, Takashi Taniguchi 6,Daniele Passerone 4, Michel Calame 1,7,8 & Jian Zhang 1Twisted van derWaals heterostructures have recently emerged as a versatile platform for engineeringinteraction-driven, topological phenomenawith a high degree of control and tunability. Since the initialdiscovery of correlated phases in twisted bilayer graphene, a wide range of moiré materials haveemerged with fascinating electronic properties. While the field of twistronics has rapidly evolved andnow includes a range of multi-layered systems, moiré systems comprised of double trilayer grapheneremain elusive. Here, we report electrical transport measurements combined with tight-bindingcalculations in twisted double trilayer graphene (TDTLG). We demonstrate that small-angle TDTLG(~1.7−2.0∘) exhibits an intrinsic bandgap at the charge neutrality point. Moreover, by tuning thedisplacement field, we observe a continuous insulator-semimetal-insulator transition at the CNP,which is also captured by tight-binding calculations. These results establish TDTLG systems as ahighly tunable platform for further exploration of magneto-transport and optoelectronic properties.The ability to tune the twist angle between different layers of two-dimensional materials has led to a paradigm shift in quantum band engi-neering. The low-energy electronic structure of twisted heterostructure isdetermined by the moiré potential originating from the rotational mis-alignment of the constituent layers. Alternatively, superlattices can also berealized based on themismatch between the lattice constants of these layers.After the initial discovery of correlated insulating states1 andsuperconductivity2 in twisted bilayer graphene, the field of twistronics hasexpanded to various moiré systems, such as ABC-stacked trilayer graphene(TLG) with hexagonal boron nitride (TLG/h-BN)3,4, twisted double bilayergraphene (TDBLG)5–7, twisted monolayer-bilayer graphene8,9, twisted tri-layer graphene10,11, twisted tetralayer and pentalayer graphene12, and tran-sition metal dichalcogenide (TMD) heterostructures13,14. A plethora ofbroken-symmetry phases, including but not limited to correlatedinsulators1,5, superconductivity2,10,15 quantum anomalous Hall effect16,ferromagnetism8,17,18, strange metal19,20, and Wigner crystals21,22 have beendiscovered in various moiré systems.Twisted double trilayer graphene (TDTLG), which has only scarcelybeen explored to date, is a natural extension of the world of moirématerials. In particular, TDTLG fabricated with two ABC-stacked TLGlayers (Fig. 1a, b) provides an appealing platform to tune the bandstructure of the hybrid system as single ABC-stacked TLG features vanHove singularities at or near the band edge where the density of states(DOS) diverges23. Moreover, the recent observation of ferromagnetism24and superconductivity25 in moiréless ABC-stacked TLGmake it a naturalchoice for exploring ABC-stacked TDTLG systems. Although theunconventional lattice relaxation effects in TDTLG have been exploredusing nano-optical and tunneling spectroscopic tools26, transportexperiments at low twist angles remain elusive. In this work, we haveperformed electric transport measurements in small-angle TDTLG(~1.7−2.0∘), where we demonstrate the exceptional tunability of the bandstructure in the presence of an external electric field. In particular, we finda continuous transition froman insulating state to a semimetallic state andagain to an insulating state with an increasing displacementfield.We referto this transition as the insulator-semimetal-insulator hereafter in the text.Results and discussionDevice fabricationThe trilayer graphene flakes were obtained via mechanical exfoliation ofgraphite crystals and theTDTLGdeviceswere fabricated using the tear-and-1Transport at Nanoscale Interfaces Laboratory, Empa, Swiss Federal Laboratories for Materials Science and Technology, 8600 Dübendorf, Switzerland.2Department of Information Technology andElectrical Engineering, ETHZurich, 8092Zurich, Switzerland. 3QuantumCenter, ETHZürich, 8093 Zürich, Switzerland.4nanotech@surfaces Laboratory, Empa, Swiss Federal Laboratories for Materials Science and Technology, 8600 Dübendorf, Switzerland. 5Research Center forFunctional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 6International Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 7Department of Physics, University of Basel, 4056 Basel, Switzerland. 8SwissNanoscience Institute, University of Basel, 4056 Basel, Switzerland. e-mail: mickael.perrin@empa.ch; michel.calame@empa.ch; jian.zhang@empa.chnpj 2D Materials and Applications |            (2024) 8:14 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41699-024-00449-w&domain=pdfhttp://orcid.org/0000-0003-3172-889Xhttp://orcid.org/0000-0003-3172-889Xhttp://orcid.org/0000-0003-3172-889Xhttp://orcid.org/0000-0003-3172-889Xhttp://orcid.org/0000-0003-3172-889Xhttp://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/0000-0001-5690-9000http://orcid.org/0000-0001-5690-9000http://orcid.org/0000-0001-5690-9000http://orcid.org/0000-0001-5690-9000http://orcid.org/0000-0001-5690-9000http://orcid.org/0000-0001-7467-9915http://orcid.org/0000-0001-7467-9915http://orcid.org/0000-0001-7467-9915http://orcid.org/0000-0001-7467-9915http://orcid.org/0000-0001-7467-9915http://orcid.org/0000-0002-2385-1882http://orcid.org/0000-0002-2385-1882http://orcid.org/0000-0002-2385-1882http://orcid.org/0000-0002-2385-1882http://orcid.org/0000-0002-2385-1882mailto:mickael.perrin@empa.chmailto:michel.calame@empa.chmailto:jian.zhang@empa.chstack dry-transfer method27,28 (see Methods for the detailed fabricationprocess). The layer thickness was identified by optical microscopy andconfirmed by Raman spectroscopy prior to the device fabrication (Supple-mentary Fig. 1). The stacking order for the TLG flakes is also determined byRaman spectroscopy, where the 2D peak of the TLG is found to be moreasymmetric, with a pronounced shoulder. This indicates an ABC-stackingorder at the measured points (Supplementary Fig. 1). A comparison ofRaman spectra between an ABC- and an ABA-stacked TLG is shown inFig. 1c. It should be noted that, although the intended stacking was ABC-ABC, the relaxation process during the transfer and device fabrication canresult in alternate stacking configurations, as discussed later. Fourdual-gatedTDTLGdeviceswithdifferent twist anglesθof 1.8 ± 0.015∘ (S1), 1.69 ± 0.003∘(S2), 2.0 ± 0.022∘ (S3), and 1.96 ± 0.02∘ (S4) were measured to reveal thetransport behavior in TDTLG. Figure 1a presents a schematic illustration ofour h-BNencapsulated dual-gatedTDTLGdevice. TheTDTLG is contactedthrough 1D edge contacts with Cr/Au electrodes. A top metal electrode isdeposited to formadual-gate device,where theTDTLGcanbe gated by boththe topmetal gate and thebottomgraphite gate.This dual-gate configurationallowed us to independently tune the carrier density (n) of the TDTLG andvertical displacement field (D) across the heterostructure. D and n aregiven by, D/ϵ0 = [CBG(VBG−VB0)−CTG(VTG−VT0)]/2 and n ¼ ½ðCBGðVBG � VB0Þ þ CTGðVTG � VT0Þ�=e, where CBG and CTG are the unit areabottom and top gate geometric capacitances, respectively, and VB0 and VT0are the gate voltages corresponding to the intrinsic doping.We note that thetop gate does not cover the entire channel (see Fig. 1a). As a result, in part ofthe channel, n and D cannot be controlled.Electric field tunable bandgapFigure 2a shows the resistance R measured in device S1 with θ = 1. 8∘ as afunction of VTG and VBG. The phase space exhibits multiple resistivefeatures in the entire range ofVTG andVBG. The central diagonal resistanceline corresponds to the charge neutrality point (CNPDG) of the dual-gatedTDTLG region, which is underneath the top gate. In contrast, the twodiagonal resistance lines at the top-right and bottom-left correspond to thefull filling of themoiré superlattice bands (±ns), which allows us to calculatethe twist angle (see Methods). We also observe three additional, horizontal,resistive features, whichwe identify as theCNP and the superlattice peaks ofthe singly-gated regions of the sample,which arenot affected by the top gate.The different peaks becomemore apparentwhen plotting a line cut ofR as afunction of n at D/ϵ0 = 0, as shown in Fig. 2b.Further, to understand the effect of the displacement field on the bandstructure of TDTLG, we show a high-resolution resistance map (separatemeasurement from Fig. 2a) as a function of n andD field in Fig. 2c.We findthat TDTLG shows an intrinsic bandgap in the absence of a displacementfield. The absence of a bandgap at zero field in the underlying TLG suggeststhat the gap emerges due to the intrinsic superlattice structure. However,when a displacement field is applied in either direction, the gap graduallyfades out and disappears at a finite field, indicating an insulator-to-semimetal phase transition in this region. More strikingly, when the D isincreased further, the gap opens again with increasing width.To better visualize the evolution of the bandgap as a function of thedisplacement field, we plot R as a function ofD at n = 0 in Fig. 2d. Here, aninsulator-semimetal transition is observed at around D/ϵ0 = 0 ± 0.1 V/nmand a semimetal-insulator transition at aroundD/ϵ0 = ± 0.2 ± 0.3 V/nm. Incontrast, the resistive peaks at ± ns show an insulating behavior in theinvestigated range of D. Based on these experimental observations, weschematically show the expected low-energy band structure of TDTLG atthree different values ofD (Fig. 2e). The figure shows a bandgap that is finiteatD/ϵ0 = 0, closes atD =D0, whereD0/ϵ0 ~ 0.1 V/nm, and then opens againfor D >D0. We note that all three devices show similar characteristics with2600 2700 2800Intensity (a.u.)Wavenumber (cm-1)ABA-TLGABC-TLG     (S1)TGGraphite BGhBNhBNSiO2SiS DTDTLGA BA BA BA BA BA BTopABC-TLGBottomABC-TLGacbθFig. 1 |Device structure. a Schematic cross-sectional view of the device, showing theconstituent layers, electrical contacts, and the gate assembly. The Au bar and gra-phite layer serve as top (TG) and bottom (BG) gates, respectively. The zoomed-inregion shows the TDTLG moiré superlattice. b Schematic illustration of the atomicconfiguration of TDTLG layers, with ABC-stacking order as representative.c Comparison between Raman spectra obtained from ABA and ABC TLG flakes.https://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 2intrinsic single-particle gaps at the CNP and insulating states at ±ns.Meanwhile, the insulator-semimetal-insulator transition is observed in alldevices (Supplementary Figs. 2, 4) and, thus, is intrinsic to TDTLG.The electric field tunable electronic states, especially the intrinsic gapstate in TDTLG, is interesting from a fundamental point of view. To gainfurther insight into the intrinsic gap state of TDTLG, we measured thedifferential conductance (dI/dV) at T = 255mK as a function of source-drain biasVBias andVBG at three fixedVTG (marked with three dashed linesin Fig. 2a), as shown in Fig. 3a–c. We find that the differential conductancemap for VTG = 0.16 V (Fig. 3a) reveals a large diamond-like structure cen-tered at the CNP. The line cut of dI/dV at D/ϵ0 = 0 as a function of VBias isplotted in Fig. 3d (blue color). For VBias < 3mV, dI/dV remains close to 0,while it rises sharply when VBias is increased further to 5mV, and finallyshows a linear increase with further increment of VBias. This “U” shapefeature demonstrates the presence of an intrinsic bandgap in TDTLG.Next, the stability diagram for VTG = 0.4 V (Fig. 3b) shows a muchsmaller diamond-like structure centered at the CNP. The corresponding dI/dV curve at the center of the diamond (or atD/ϵ0 = 0.05 V/nm) (yellow colora bc dθK1K’1K2K’1Δ Δθ θD = 0 D = D0 D > D0e0200400600800-10.10.050-0.05-0.10 1n (1012 cm-2)D/ε0(V nm-1)R (kΩ)CNP DGCNPSG-nsnsCNPDG4002000R (kΩ)n (1012 cm-2)-8 -4 0 4 8D/ε0 = 0insulatorinsulator insulatorsemimetal semimetal4002000R (kΩ)D/ε0 (V nm-1)-0.4 -0.2 0.0 0.2 0.4n = 0Γ0-5 50-55VTG (V)V BG (V)0100200300400R (kΩ)n sCNPSGCNPDGnD-n sFig. 2 | Electrical characterization. a Two-probe resistance R as a function of VTGandVBGmeasured at T = 300 mK for device S1 with θ = 1.8∘. b R as a function of n atD/ϵ0 = 0. The peaks corresponding to the CNP and full fillings of the bands (±ns) inthe dual-gated region aremarked for clarity. cHigh-resolutionmap showing theR asa function of D and n near the CNP. d R as a function of D at n = 0 showing theinsulator-semimetal-insulator transition. e Schematic illustration of the bandstructure of TDTLG at three different electric fields. The gap closes at a finite electricfield D0/ϵ0 ~0.1 V/nm and opens again as the D is increased further.https://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 3in Fig. 3d) presents a “V” shape feature, revealing a smaller gap at D/ϵ0 = 0.05 V/nm than the gap at D/ϵ0 = 0. Finally, no diamond-like structureis seen for the stability diagram measured at D/ϵ0 = 0.15 V/nm, and the dI/dV is predominantly independent of bias (as shown in the corresponding dI/dV curve inFig. 3d) andgate, suggesting the gap closes at a largerDfieldofD/ϵ0 = 0.15 V/nm.Temperature dependenceTo further quantify the magnitude of intrinsic bandgap in TDTLG, weperformed a detailed study of the temperature dependence of the resistance.Figure 3e–g show Arrhenius plots of the temperature dependence ofresistance at the CNP for the dual-gated region at three different values ofthe D/ϵ0 field (0.043, 0, and −0.041 V/nm). In all three cases, the experi-mental data above 15−20 K follow the behavior expected for thermally-activated transport, according to R ¼ R0 expðΔ=2kBTÞ, where kB is theBoltzmann constant and R0 and Δ are fitting parameters. A maximumenergy gap Δ = 37meV is extracted for D/ϵ0 = 0 V/nm, and two reducedenergy gaps Δ = 21meV for D/ϵ0 = 0.043 V/nm and Δ = 17meV forD/ϵ0 =− 0.041 V/nm. At lower values of T, the deviation from thethermally-activated behavior to a much weaker T-dependence can beattributed to a combination of theMott variable-range hopping conductionmediated by localized states and quantum tunneling through a short-0.05 0.1 0.15 0.21/T (1/K)681012ln(R) (Ω）0.05 0.1 0.15 0.21/T (1/K)4681012ln(R) (Ω)0.05 0.1 0.15 0.21/T (1/K)681012ln(R) (Ω)e f gD = 0.043 V nm-1 D = 0 V nm-1 D = -0.041 V nm-1Δ = 0.017 eVΔ = 0.037 eVΔ = 0.021 eVVTG = -0.13 V VTG = 0.16 V VTG = 0.45 V-0.2 -0.1 0 0.1-0.0200.02-6.5-6-5.5-5-0.4 -0.35 -0.3 -0.25-0.0200.02V Bias (V)-6.5-6-5.5-5VBG (V)V Bias (V)VBG (V)V Bias (V)VBG (V)-1.7 -1.6 -1.5 -1.4 -1.3-0.0200.02-6.5-6-5.5-5-4.5aVTG = 0.16 V VTG = 0.4 V VTG = 1.32 V log 10(|dI/dV|) (S)log 10(|dI/dV|) (S)log 10(|dI/dV|) (S)|dI/dV| (S)bc d0 V nm-1 0.050 V nm-1 0.15  V nm-10 V nm-10.05 V nm-10.15 V nm-1Fig. 3 | Differential conductance and temperature dependence of resistance.a–cDifferential conductance dI/dV of the device S1 as a function ofVBias andVBG ata fixed top gate voltage of 0.16, 0.4, and 1.32 V, respectively (marked in Fig. 2a). Thedevice exhibits diamond-like structure for VTG = 0.16 V, which almost vanishes atVTG = 1.32 V, suggesting the closing of the bandgap.d dI/dV as a function ofVBias forthree different values of D, marked by dashed lines in (a–c). e–g Arrhenius plotspresenting the T-dependence of resistance recorded at three different values of D.The solid lines show the Arrhenius-like fits to the data, where Δ is the extractedbandgap of the device.https://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 4channel semiconductor29. We note that the insulating state near the CNPsurvives up to temperatures as high as 40 K. (Supplementary Fig. 3). In anadditional device S4 (Supplementary Fig. 4), wemeasure the T-dependenceof resistance over a wide range of D fields. At CNP and large D fields, thedevice behaves like an insulator as the resistance decreases as T goes up,which agrees well with the literature ref. 30. At the critical fieldD =D0=0.21 V/nm, the resistance first slightly decreases and then increasesafter T drops down to a critical point (10 K), suggesting a semimetalbehavior31,32. A detailed discussion on device S4 can be found in Supple-mentary Fig. 4.Theoretical calculationFinally, we have performed tight-binding calculations to rationalize theobserved charge-transport behavior and get insights into the band structure.The band structures of TDTLG as a function of twist angles and displace-ment field were calculated using the tight-binding model implemented inWannierTools33. The tight-binding Hamiltonian was built using orbitaldistance-dependent Slater–Koster parameters and considers second-nearest-neighbor interactions. For the relaxed band structures, the atomicpositions of TDTLG were relaxed using the classical force field approx-imation implemented in LAMMPS34. The ABC-CAB domains display thelargest out-of-plane relaxation of atomic positions, while the ABC-ABCdomains have the least (Supplementary Fig. 5). This behaviorminimizes therepulsion between “C-C” regions at the twisted layer.Figure 4a shows the band structure of unrelaxed (blue) and relaxed(orange) ABC-ABC stacked TDTLG at a twist angle of θ = 1.8∘. The cal-culations show that TDTLG is intrinsically an insulator—with a bandgap of~0.01 eV, as observed experimentally in Fig. 2a. The intrinsic bandgap at theCNP may be attributed to the absence of inversion symmetry in twistedABC-ABC stacked TDTLG. This behavior is similar to that reported forTDBLG for the same twist angle35. A comparison of the unrelaxed/rigid andrelaxedbands shows that lattice relaxationhas anegligible effect on the bandstructure around the Fermi energy, again similar to that reported forTDBLG35.a b cd e0 V nm-1 0.074 V nm-1 0.156 V nm-1Fig. 4 | Theoretical calculations. a Calculated band structure of ABC-ABC stackedTDTLG with θ = 1. 8∘ in the absence of an external electric field. The blue (orange)lines denote the bands of an unrelaxed (relaxed) TDTLG. b, c Calculated bandstructure in the presence of an external field. The displacement field D has beencalculated using the dielectric constant of h-BN and screening effects have beenignored. d Bandgap as a function ofD for both the relaxed (red curve) and unrelaxed(green curve) structure of ABC-ABC stacked TDTLG. e Bandgap as a function oftwist angle for unrelaxed ABC-ABC stacked TDTLG in the absence of an externaldisplacement field.https://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 5Figure 4a–c show the closing and reopening of the bandgap withincreasing displacement field, indicating an insulator-semimetal-insulatortransition, consistent with our experimental observation. This is in con-trast with the behavior of twistedABA-ABA stacked graphene, which is anintrinsic insulator in the absence of an external field, becomes metallic inthe presence of low external fields, but does not revert to insulatingbehavior upon a further increase of the external field (Supplementary Fig.6). However, as illustrated in ref. 26, the device can exhibit several stackingconfigurations. To correlate the charge-transport behavior with stackingconfiguration, we have performed the calculations on Thirty-two slidingconfigurations, as discussed in ref. 26 (Supplementary Fig. 7).We find thatthere are eight configurations that exhibits a variation in the bandgapsimilar toABC-ABCstacking. In all other configurations, the systemeitherremains semimetallic or shows a very small bandgap upon crossing acritical displacement field. Based on the symmetry arguments, we cannarrow down the possible configurations in our devices to ABC-ABC,CAB-ABC, CAB-ACB, and CBA-BCA (see Supplementary Informationfor discussion). Further, as demonstrated in ref. 26, CAB-ABC/BCA-BCAare energetically the least favorable configurations with respect to theconfiguration with no sliding and are therefore unlikely to exist in ourdevices. Additionally, theoretical calculations show that CAB-ACB, andCBA-BCA have large stacking energy26, but it is impossible to rule outthese possibilities as stacking energy data are not available for all config-urations. In conclusion, while the ABC-ABC stacking configuration ismore likely to exist in our devices, CAB-ACB, CAB-ABC, or CBA-BCAconfigurations, or a combination thereof, may also exist.Further, a comparison of Fig. 4b, c to Fig. 4a shows that the presence ofa displacement field leads to a splitting in the valence bands, as well as theconduction bands. The position and dispersion of valence bands below−0.02 eVremain largelyunaffected by the externalfield,while the increaseddispersion and splitting of both the degenerate conduction and valencebands around 0 eV lead to the closing of the bandgap at displacement fieldsaround 0.074 V/nm, as observed in panel Fig. 4b. A similar behavior hasbeen reported in TDBLG35 and was attributed to layer polarization andcorrelation effects. Itwas demonstrated that thedisplacementfield in the+zdirection leads to an asymmetry in the layer contribution to the bandsaround the Fermi energy in TDBLG. The valence bands become confinedwithin the top layer, while the conduction bands become confined to thebottom layer. In addition to layer polarization, there is an enhancement inthe density of states in the conduction bands around the Fermi energy. Thismakes the correlation effects stronger in the conduction bands than in thevalence bands, leading to an electron-hole asymmetry in the external fields.As shown in Fig. 4d, lattice relaxation has a negligible effect on theoverall trend of the bandgap as a function of the displacement field. Both theunrelaxed and relaxed ABC-ABC stacked TDTLG are intrinsic insulators inthe absence of an external displacement field. An insulator-to-semimetaltransition is observed for a displacement field slightly below 0.1 V/nm, and asemimetal-to-insulator transition isobserved for adisplacementfield slightlyabove 0.1 V/nm.The bandgap increasesmonotonically forD/ϵ0 > 0.1 V/nm.The displacement fields required to observe insulator-metal-insulatortransitions are slightly lower than that those observed experimentally. Weattribute this to the absence of screening effects in the tight-bindingmodel36.Figure 4e shows the bandgap of ABC-ABC stacked TDTLG as afunction of twist angle in the absence of an external displacement field.Without a twist, the system has no bandgap and behaves like an ABC-stacked trilayer graphene37. The bandgap increases with twist angle to amaximum around θ ≈ 2.5∘ and then decreases for larger twist angles. Wespeculate that this non-monotonic behavior originates from an interplaybetween the reduced coupling in some regions of the trilayer-trilayerinterface and the periodicity of those regions due to angle-induced latticemismatch.ConclusionsIn summary,wehaveperformed thefirst electricalmeasurementson twisteddouble trilayer graphene for small twist angle devices (~1.7−2.0∘). Ourmeasurements show the presence of an intrinsic bandgap in this hetero-structure, which is supported by tight-binding calculations. In addition, wealso observe an insulator-semimetal-insulator transition,which results fromthe closing and reopening of the bandgap upon an increase of the dis-placement field. The presence of an intrinsic bandgap in TDTLG opens upthe possibility of exploiting such devices for optoelectronic applications38, inparticular, because its size is highly tunable by the applied displacementfield.Moreover, we anticipate that further tuning of devices with twist anglecloser to ~1.2−1.5∘ will lead to stronger correlation effects and, in turn,symmetry-broken many-body phases.MethodsDevice fabricationThe TDTLG devices were fabricated following a typical “tear and stack”technique27,28. Trilayer graphene, h-BN (20−35 nm thick) and graphiteflakes were mechanically exfoliated on SiO2/Si (100 nm oxide thickness).The heterostructure was assembled using standard dry-transfertechniques39. We used poly bisphenol A carbonate (PC) supported bypolydimethylsiloxane (PDMS)onaglass slide tofirst pickuph-BNand thentear andpickup the trilayer graphene.A commercialmicromanipulatorwasused to control the rotation angle. The graphiteflakewasfinally pickedup asabottomgate.The fabricationof themetal topgate and electrodes followedastandard electron-beam lithography process and electron-beam metalevaporation. The devices were designed into a four-terminal structureshaped by traditional reactive ion etching with a CHF3 and O2 gas mixture.Finally, all the contacts were shaped as 1D edge contacts with Cr/Auelectrodes39.Electronic measurementsTransport measurements were performed in a He3 cryostat with a basetemperature of 255mK.Weapplied standard lock-in techniques tomeasurethe resistance, with 17.777Hz excitation frequency and 1 nA excitationcurrent.Although thedeviceshad four contacts, theywerenot patterned in afour-terminalHall-bar orvanderPauwgeometry for accurate four-terminalmeasurements. Hence, we conducted measurements in a two-terminalconfiguration to avoid artifacts that can arise froma “pseudo” four-terminalgeometry.Twist angle extractionTo estimate the twist angle, we used the relation ns ¼ 8θ2=ffiffiffi3pa2 (includingvalley and spin degeneracies), where a = 0.246 nm is the lattice constant ofgraphene and ns is the charge carrier density corresponding to a fully filledsuperlattice unit cell1.Band structure calculationsTight-BindingModel Calculations—The tight-bindingmodel Hamiltonianfor carbon atoms in WannierTools considers only the pz orbitals.H ¼Xijεiayi aj þXi≠jVijayi aj ð1Þwith εi ¼ Ezi, where E is the strength of the electric field, zi is the atomicz-axis coordinate, and ayi and aj are the creation and annihilation operators.Vij ¼ Vppπsin2θ þ Vppσcos2θ ð2ÞwhereVppσ andVppπ are σ-type andπ-type Slater–Koster parameters, θ is theangle between the orbital axes and Rij = Ri− Rj connecting the two orbitalcenters.TheSlater–Koster parameters dependon thedistance r between twoorbitals asVppπðrÞ ¼ V0ppπeqπð1� raπÞFcðrÞ ð3ÞVppσðrÞ ¼ V0ppσeqσ ð1� raσÞFcðrÞ ð4Þhttps://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 6whereaπ is thefirst nearest neighbordistance in theplane,aσ is the interlayerdistance, and V0ppπ and V0ppσ are the corresponding coupling values. Thesmooth cutoff function that takes into account the distance between orbitalsis given byFcðrÞ ¼ 1þ er�rclc� ��1¼ 1 r≪ rc0 r≫ rc� �ð5ÞThe on-site energy of pz orbitals is set to ϵi =− 0.78 eV to adjust thereference. The following constants have been used in the aboveequations;V0ppπ ¼ �2:81 eV,V0ppσ ¼ 0:48 eV, qσ = 7.428, aπ = 1.418Å,qπ = 3.1451, lc = 0.265 Å, aσ = 3.5 Å, and rc = 6.165 Å36,40.Classical Force—Field Simulations—The second generation REBOpotential41 and the Kolmogorov– Crespi potential42 were used to describethe intralayer and interlayer interactions respectively in the lattice relaxationof TDTLG in LAMMPS. The relaxation was performed until the total forceacting on each atomwas less than 10−6 eV/atomusing the FIRE algorithm43.Data availabilityThe datasets generated during and/or analyzed during the current study areavailable from the authors on reasonable request.Received: 28 June 2023; Accepted: 8 February 2024;References1. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80 (2018).2. Cao, Y. et al. Unconventional superconductivity in magic-anglegraphene superlattices. Nature 556, 43 (2018).3. Chen, G. et al. Signatures of tunable superconductivity in a trilayergraphene moiré superlattice. Nature 572, 215–219 (2019).4. Chen, G. et al. Evidence of a gate-tunable mott insulator in a trilayergraphene moiré superlattice. Nat. Phys. 15, 237–241 (2019).5. Shen, C. et al. Correlated states in twisted double bilayer graphene.Nat. Phys. 16, 520–525 (2020).6. Liu, X. et al. Tunable spin-polarized correlated states in twisteddoublebilayer graphene. Nature 583, 221–225 (2020).7. Cao, Y. et al. Tunable correlated states and spin-polarized phases intwisted bilayer–bilayer graphene. Nature 583, 215–220 (2020).8. Chen, S. et al. Electrically tunable correlated and topological statesin twisted monolayer-bilayer graphene. Nat. Phys. 17,374–380 (2021).9. Xu, S. et al. Tunable van hove singularities and correlated states intwisted monolayer–bilayer graphene. Nat. Phys. 17, 619–626 (2021).10. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P.Tunable strongly coupled superconductivity in magic-angle twistedtrilayer graphene. Nature 590, 249–255 (2021).11. Hao, Z. et al. Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).12. Park, J. M. et al. Robust superconductivity in magic-angle multilayergraphene family. Nat. Mater. 21, 877–883 (2022).13. Wang, L. et al. Correlated electronic phases in twisted bilayertransition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).14. Zhang, Z. et al. Flat bands in twisted bilayer transition metaldichalcogenides. Nat. Phys. 16, 1093–1096 (2020).15. Lu, X. et al. Superconductors, orbitalmagnets and correlated states inmagic-angle bilayer graphene. Nature 574, 653–657 (2019).16. Serlin, M. et al. Intrinsic quantized anomalous hall effect in a moiréheterostructure. Science 367, 900–903 (2020).17. Sharpe, A. L. et al. Emergent ferromagnetism near three-quartersfilling in twisted bilayer graphene. Science 365, 605–608 (2019).18. Bhowmik, S. et al. Spin-orbit coupling-enhanced valley ordering ofmalleable bands in twisted bilayer graphene on wse2. Nat. Commun.14, 4055 (2023).19. Ghiotto, A. et al. Quantum criticality in twisted transition metaldichalcogenides. Nature 597, 345–349 (2021).20. Ghawri, B. et al. Breakdown of semiclassical description ofthermoelectricity in near-magic angle twisted bilayer graphene. Nat.Commun. 13, 1522 (2022).21. Regan, E. C. et al. Mott and generalized wigner crystal states in wse2/ws2 moiré superlattices. Nature 579, 359–363 (2020).22. Li, H. et al. Imaging two-dimensional generalized wigner crystals.Nature 597, 650–654 (2021).23. Zhang, F., Sahu, B., Min, H. & MacDonald, A. H. Band structure of a bc-stacked graphene trilayers. Phys. Rev. B 82, 035409 (2010).24. Zhou, H. et al. Half-and quarter-metals in rhombohedral trilayergraphene. Nature 598, 429–433 (2021).25. Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F.Superconductivity in rhombohedral trilayer graphene. Nature 598,434–438 (2021).26. Halbertal, D. et al. Unconventional non-local relaxation dynamics in atwisted trilayer graphene moiré superlattice. Nat. Commun. 13,7587 (2022).27. Kim, K. et al. van der waals heterostructures with high accuracyrotational alignment. Nano Lett. 16, 1989–1995 (2016).28. Cao, Y. et al. Superlattice-induced insulating states and valley-protectedorbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016).29. Khodkov, T., Khrapach, I., Craciun, M. F. & Russo, S. Directobservation of a gate tunable band gap in electrical transport in abc-trilayer graphene. Nano Lett. 15, 4429–4433 (2015).30. Dean, C. R. et al. Boron nitride substrates for high-quality grapheneelectronics. Nat. Nanotechnol. 5, 722–726 (2010).31. Scherwitzl, R. et al. Metal-insulator transition in ultrathin lanio 3 films.Phys. Rev. Lett. 106, 246403 (2011).32. Yang,W. et al. Hofstadter butterfly andmany-body effects in epitaxialgraphene superlattice. Nano Lett. 16, 2387–2392 (2016).33. Wu, Q., Zhang, S., Song, H.-F., Troyer, M. & Soluyanov, A. A.Wanniertools: anopen-sourcesoftwarepackage for novel topologicalmaterials. Comput. Phys. Commun. 224, 405–416 (2018).34. Thompson, A. P. et al. Lammps-a flexible simulation tool for particle-basedmaterialsmodeling at the atomic,meso, andcontinuumscales.Comput. Phys. Commun. 271, 108171 (2022).35. Choi, Y.W. &Choi, H. J. Intrinsic band gap and electrically tunable flatbands in twisted double bilayer graphene. Phys. Rev. B 100,201402 (2019).36. Haddadi, F.,Wu,Q., Kruchkov,A. J. &Yazyev,O. V.Moiré flat bands intwisted double bilayer graphene. Nano Lett. 20, 2410–2415 (2020).37. Lui, C. H., Li, Z., Mak, K. F., Cappelluti, E. & Heinz, T. F. Observation ofan electrically tunable band gap in trilayer graphene. Nat. Phys. 7,944–947 (2011).38. Deng, B. et al. Strong mid-infrared photoresponse in small-twist-angle bilayer graphene. Nat. Photon. 14, 549–553 (2020).39. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).40. De Laissardiere, G. T., Mayou, D. & Magaud, L. Numerical studies ofconfined states in rotated bilayers of graphene. Phys. Rev. B 86,125413 (2012).41. Brenner, D. W. et al. A second-generation reactive empirical bondorder (rebo) potential energy expression for hydrocarbons. J. Phys.Condens. Matter 14, 783 (2002).42. Kolmogorov, A. N. & Crespi, V. H. Registry-dependent interlayerpotential for graphitic systems. Phys. Rev. B 71, 235415 (2005).43. Bitzek, E., Koskinen, P., Gähler, F., Moseler, M. & Gumbsch, P.Structural relaxationmade simple.Phys. Rev. Lett. 97, 170201 (2006).AcknowledgementsJ.Z. acknowledges support from the Empa Directorate under the EmpaInternal Research Call 2021. M.C. acknowledges support from the SwissNational Science Foundation under the Sinergia grant no. 189924https://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 7(Hydronics) and the ECH2020 FETOpenproject no. 767187 (QuIET).M.L.P.acknowledges funding from the Eccellenza Professorial Fellowship no.PCEFP2_203663, as well as supported by the Swiss State Secretariat forEducation, Research and Innovation (SERI) under contract numberMB22.00076.A.J. acknowledges support by theNCCRMARVEL, aNationalCenter of Competence in Research, funded by the Swiss National ScienceFoundation (grant number 205602). K.W. and T.T. acknowledge supportfrom the JSPS KAKENHI (Grant Numbers 19H05790 and 20H00354). Theauthors thank the Cleanroom Operations Team of the Binnig and RohrerNanotechnology Center (BRNC) for their help and support.Author contributionsJ.Z. conceived and designed the experiments. J.Z. fabricated the devicesand performed the Raman measurements. J.Z. and M.L.P. performed theelectrical measurements. K.W. and T.T. provided h-BN crystals. J.Z. andM.L.P. analyzed the data. A.J., under the supervision of D.P., did the theo-retical calculations. B.G., J.Z., M.L.P., and M.C. discussed the figures andwrote the manuscript. All authors discussed the results and their implica-tions and commented on themanuscript. M.L.P., A.J., and B.G. contributedequally to this work.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41699-024-00449-w.Correspondence and requests for materials should be addressed toMickael L. Perrin, Michel Calame or Jian Zhang.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional 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 anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this article areincluded in the article’s Creative Commons licence, unless indicatedotherwise in a credit line to the material. If material is not included in thearticle’sCreativeCommons licence and your intended use is not permittedby statutory regulation or exceeds the permitted use, you will need toobtain permission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024https://doi.org/10.1038/s41699-024-00449-w Articlenpj 2D Materials and Applications |            (2024) 8:14 8https://doi.org/10.1038/s41699-024-00449-whttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/ Electric field tunable bandgap in twisted double trilayer graphene Results and discussion Device fabrication Electric field tunable bandgap Temperature dependence Theoretical calculation Conclusions Methods Device fabrication Electronic measurements Twist angle extraction Band structure calculations Data availability References Acknowledgements Author contributions Competing interests Additional information