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Yimeng Wang, Jihang Zhu, G. William Burg, Anand Swain, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Yuebing Zheng, Allan H. MacDonald, Emanuel Tutuc

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[Independently Tunable Flat Bands and Correlations in a Graphene Double Moiré System](https://mdr.nims.go.jp/datasets/590bd80f-92c1-4f90-923f-c5356d741b01)

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Independently Tunable Flat Bands and Correlations in a Graphene Double MoiréSystemYimeng Wang,1 Jihang Zhu,2 G. William Burg,1 Anand Swain,3 Kenji Watanabe,4Takashi Taniguchi,5 Yuebing Zheng,3 Allan H. MacDonald,2 and Emanuel Tutuc1, ∗1Microelectronics Research Center, Department of Electrical and Computer Engineering,The University of Texas at Austin, Austin, TX 78758, USA2Department of Physics, University of Texas at Austin, Austin, TX 78712, USA3Walker Department of Mechanical Engineering, Texas Materials Institute,The University of Texas at Austin, Austin, TX 78712, USA4Research Center for Functional Materials, National Institute ofMaterials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, Japan5International Center for Materials Nanoarchitectonics,National Institute of Materials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, Japan(Dated: January 10, 2025)We report on a double moiré system consisting of four graphene layers, where the top and bottompairs form small-twist-angle bilayer graphene, and the middle interface has a large rotational mis-match. This system shows clear signatures of two sets of spatially separated flat bands associatedwith the top and bottom twisted bilayer graphene (TBG) subsystems, each independently tunable.Thermodynamic analysis reveals weak correlations between bilayers that allow the chemical poten-tial to be measured as a function of carrier density for each constituent TBG. We find that correlatedinsulating states at integer number of electrons per moiré unit cell are most robust near magic angle,whereas gapped states at neutrality are more robust at larger twist angles.Materials with extremely flat energy bands have at-tracted great scientific interests due to quenched electronkinetic energy that leads to strong electron correlations.The theoretical discovery and experimental realizationof flat bands in magic-angle twisted bilayer graphene(MATBG) [1–3] have opened up paths to engineered flat-band materials. Subsequent studies have revealed theintricacy of electronic correlations in TBG flat bandswith various correlated phases, including ferromagnetism[4, 5], Chern insulators [6, 7], strange metal states [8, 9],etc., all of which reside in MATBG’s rich phase diagram.The ongoing debate on the interplay between correlatedinsulating and superconducting phases in MATBG mayhelp uncover the origin of unconventional superconduc-tivity [10–13]. Besides TBG, tunable flat bands in multi-layer graphene moiré systems have been explored, includ-ing alternating twisted multilayer graphene [14–19] andtwisted m + n layer graphene [20–24]. Here, we reportthe experimental realization of a double moiré system inwhich two independent TBG moiré bands are placed inproximity. By stacking two TBGs with a relatively largerotational mismatch we weaken hybridization betweenconstituent TBGs, leading to approximate particle num-ber conservation in each TBG. Our approach illustrates amethod to experimentally realize quantum confined flatbands in a multilayer homostructure using exclusivelytwist control. Remarkably, the constituent TBGs displaycorrelated insulators at integer number of electrons permoiré unit cell near magic angle, which shows that intra-moiré interactions remain strong despite the proximity ofthe other TBG. By utilizing the unique combination ofcompressible and incompressible states in the two TBGswe extract the chemical potential as a function of carrierdensity in the constituent TBGs. Furthermore, we findthat gaps at neutrality in individual TBGs are strongeraway from the magic angle regime, pointing to a qualita-tive difference between many-electron physics away fromvs. at magic angle.The moiré system in this study consists of fourgraphene monolayers, with two controlled twist angles(0.91◦-1.57◦) – the twist angle between the top two layers(θT), and between the bottom two layers (θB). The twistangle between the middle layers is not controlled andintentionally kept large (> 5◦) [Fig. 1(a), (b)]. The dou-ble moiré samples are dual-gated [Fig. 1(a) inset] withhexagonal boron nitride (hBN) as gate dielectric, andthe channels are etched into a Hall bar [Fig. 1(c)]. Datafrom four samples, labeled S1-S4 are discussed in themanuscript. The large twist angle between the middlegraphene layers suppresses electron tunneling by shiftingtheir K-valleys in the momentum space. Thus, tunnelingcontributes at high order in perturbation theory and actsas a contribution to disorder.Figure 1(d) shows the longitudinal resistance (Rxx) asa function of top (VTG) and bottom (VBG) gate biasesmeasured in a double moiré sample with θT = 1.07◦and θB = 1.38◦ (sample S1). The most prominent re-sistance peaks are located at the corners of the contourplot, marked by the black dots in Fig. 1(e). At themidpoint of each two adjacent peaks, additional local re-sistance maxima are marked by yellow squares in Fig.1(e). We associate these resistance peaks with the in-teger filling states per valley and spin of the top andbottom TBG moiré bands. For example, the upper left2-8 -4 0 4 8-10-50510VTG (V)VBG (V)0612Rxx (k)θTθBKK’VTGVBG(a) (b)(c)(d) (e)FIG. 1. (a) Brillouin zones of the top (blue) and bottom (red)TBG in the double moiré system. Inset shows the biasingscheme for the dual-gated sample. (b) Schematic of the doublemoiré lattice formed by four graphene monolayers. The toptwo graphene monolayers are shifted vertically for clarity. (c)Optical micrograph of a double moiré sample. The scale baris 5µm. (d) Rxx vs. VTG and VBG measured in sample S1at T = 95mK. (e) Schematic of the resistance peaks patternseen in (d). The black dots mark the four most prominentpeaks associated with the moiré band fillings depicted in theband diagrams of the top (blue) and bottom (red) TBGs.Black dashed line represents the Fermi level. Yellow squaresmark the peaks when one of the TBG is charge neutral. Blue(red) lines mark the constant density loci for top (bottom)TBG when tracing incompressible states.peak [upper left black dot in Fig. 1(e)] corresponds tothe insulating state where the top TBG flat bands areempty and the bottom TBG fully filled. Similarly, thebottom peak around VTG = 1V [bottom yellow squarein Fig. 1(e)] corresponds to the state where the topTBG flat bands are charge neutral and the bottom TBGempty. Noticeably, in Fig. 1(d), there are local resistancemaxima connecting the resistance peaks with moderateresistance values higher than the background [blue andred lines in Fig. 1(e)], which can be associated with theinteger filling of either top (blue) or bottom (red) TBGmoiré bands. Additional local resistance maxima extend-ing along the y-axis direction around VTG = 3V are asignature of correlated insulating state in the flat bandsof the top TBG. The data in Fig. 1(d) provide evidencefor two spatially separated, independently tunable flatbands. They demonstrate that electronic confinement inmultilayer graphene stacks can be compactly realized bychanging the twist angles, in contrast to traditional bandengineering that employ dissimilar materials.We treat the double moiré as a system consisting oftwo layers with separately conserved particle numbersthat are mutually correlated. The relation between thegate biases and the chemical potentials is then [25],VBGCBG = enB + µBe (CBG + CIL)− µTe CIL ,VTGCTG = enT + µTe (CTG + CIL)− µBe CIL . (1)Here, e is the elementary charge, CTG and CBG are theareal capacitances of the top and bottom gates, respec-tively. The two TBG subsystems are assumed to be elec-trostatically coupled by an areal capacitance CIL. Thecarrier densities in the top and bottom TBG subsystemsare nT and nB, respectively, and µX = ∂ε(nT, nB)/∂nX(X = T,B) is the chemical potential in layer X. This ex-pression fully accounts for intralayer and interlayer cor-relation contributions to the energy per area ε(nT, nB).According to Eq. (1), if µT and nT (µB and nB) arekept constant, the change in the VTG (VBG) reflects thechange in µB (µT). Similar double layer systems havebeen previously utilized to probe the chemical potentialof two-dimensional electron systems, albeit with a inter-layer barrier and external interlayer bias [29–32].(a)(d)𝜇𝜇𝐿𝜇𝑈𝑛𝐸𝑔Incompressible𝜇𝐵𝑈𝜇𝐵𝐿𝜇𝑇𝐿𝜇𝑇𝑈(c)VTGVBG𝐸𝑔𝐵𝑒∙𝐶𝐼𝐿𝐶𝑇𝐺𝐸𝑔𝑇𝑒∙ 1 +𝐶𝐼𝐿𝐶𝑇𝐺𝐸𝑔𝑇𝑒∙𝐶𝐼𝐿𝐶𝐵𝐺𝐸𝑔𝐵𝑒∙ 1 +𝐶𝐼𝐿𝐶𝐵𝐺-8 -4 0 4 8-10-50510VTG (V)VBG (V)Rxx (k)0612(b)VTGVBG𝜈𝑇 = −10+1𝜈𝐵 = +10−1FIG. 2. (a) µ vs. n of a system displaying finite compress-ibility for µ < µL (blue shaded region), and an incompress-ible state for µL < µ < µU (pink shaded region) with a gapEg = µU − µL. (b) Schematic of the incompressible states ofthe top (orange) and bottom (green) TBG subsystems seenin the Rxx vs. VTG and VBG plot. The black dashed rectan-gle marks a diamond-shaped region where both TBGs are inincompressible states. (c) Expanded view of the dashed blackrectangle in panel (b). The diamond dimensions are deter-mined by the incompressible state gaps. (d) Rxx vs. VTG andVBG data in Fig. 1(e) showing the constant µT and µB linesat the edge of incompressible states.The TBG subsystems in the double moiré system haveboth compressible and incompressible states. A chargegap (incompressibility) in layer X results in a cusp inthe dependence of ε on nX and a corresponding jumpdiscontinuity in µX [Fig. 2(a)]. Figure 2(b) illustratesthe (VTG, VBG) gate voltage ranges over which the sub-3system is incompressible in the top (orange) and bot-tom (green) TBG; the incompressible states are labelledby the moiré band filling factors at which the chemi-cal potential jumps occur. We define the filling factorsνT,B = nT,B/nsT,sB, where nsT,sB = (8/3)(θT,B/a)2, witha = 2.46 Å the graphene lattice constant, are respectivelythe carrier densities required to fill one moiré Brillouinzone of the top or bottom TBG, including its four-foldspin and valley degeneracy. Quarter-multiple filling fac-tors correspond to integer numbers of particles per moiréunit cell. Note that a state that is incompressible inone or both TBGs corresponds to a segment, or diamondrespectively in the (VTG, VBG) plane. The black dashedrectangle in Fig. 2(b) marks a diamond where the groundstate is incompressible in both layers [Fig. 2(c)]. Alongits boundaries, the chemical potential in one layer is fixedat one edge of its chemical potential jump interval in itsincompressible state [Fig. 2(a)]. Using Eq. (1) the di-amond dimensions along the VTG and VBG axes can bereadily related to the incompressible state gaps. For ex-ample, when µB varies between the incompressible stateinterval at constant µT:∆VBG =EgBe(1 + CILCBG),∆VTG = −EgBeCILCTG. (2)Here, EgB is the size of the bottom TBG gap, ∆VBGand ∆VTG are the gate bias changes needed to movethe Fermi level across the gap. Corresponding equa-tions can be written for the top TBG gap by keepingµB constant. Since the slopes of the diamond edges aredetermined only by the capacitance values, the inter-layer capacitance CIL can be determined [25]. We extractCIL = 1.6 µF/cm2, a value corresponding to two parallelplates separated by 0.55 nm vacuum, comparable to dis-tance between the mid-planes of the TBG subsystems.When correlations between TBG subsystems are neg-ligible, the chemical potential in each TBG depends onlyon its own density and the analysis simplifies. For ex-ample at constant µT (and hence nT) Eq. (1) impliesthat∆µB = −e∆VTGCTGCIL,∆nB = 1e[∆VBGCBG +∆VTGCTG(1 + CBGCIL)]. (3)According to Eq. (3), on a constant µT trace, each pointin the (VTG, VBG) plane can be converted into a µB and acorresponding nB value, rendering the µB vs. nB relation.We exploit the property that the µT values at the edges ofgaps are easily identified experimentally by sharp changesin resistance. Because there are several incompressiblestates as the filling factor varies, multiple constant µTloci are traceable in a Rxx vs. (VTG, VBG) map. Weobserve identical µB vs. nB (µT vs. nT) dependence atall integer νT (νB) values, justifying the weak-correlationassumption which allows us to use one layer to probe theother at least when one layer is incompressible.1.0 1.2 1.4 1.60100200 () p, exp.         Ref. 31 n, exp.         Ref. 33  p, theory     Ref. 34  n, theory     Ref. 36 (meV)1.0 1.2 1.4 1.60100200 () Eg,-1, exp.  Ref. 33 Eg,0, exp.   Ref. 35 Eg,1, exp. Eg,-1, theory Eg,1, theoryEg (meV)-2 0 2-100-50050100n (1012 cm-2) (meV)0.99-1 0 1-1 0 1-2000200 S4T, 1.57o S2B, 1.54o S3B, 1.47o S1B, 1.38o S1T, 1.07o S4B, 0.99o S2T, 0.95o S3T, 0.91o (meV)𝐸𝑔,−1𝐸𝑔,1(a)(c) (d)-2 0-505VTG (V)VBG (V)048Rxx (k)(b)FIG. 3. (a) µ vs. n in a 0.99◦ TBG, measured in a doublemoiré sample with θT = 1.57◦ and θB = 0.99◦ (sample S4).The shaded areas mark the incompressible states with νB =±1. Inset shows Rxx vs. VTG and VBG around νT = 0. Thepeak depicts the constant µT loci at νT = 0. (b) µ vs. nfor TBGs with varying θ, extracted from four samples. Thesuperscripts label the constituent top (T) and bottom (B)TBGs. (c) Eg,ν vs. θ. (d) ∆µp and ∆µn vs. θ. Scatteredsymbols are data from literature. Data from Ref. 34 in (d)are measured in 12T parallel magnetic field.By analyzing the Rxx vs. (VTG, VBG) data [Fig. 2(d)]in a double moiré sample, the relations µ vs. n for bothconstituent TBGs can be determined [25]. In Fig. 3(a),the chemical potential of a constituent TBG close to themagic angle is plotted as a function of carrier density andmoiré band filling factor (top axis). We observe jumpsnear half fillings ν = ±1/2 which we associate with bro-ken flavor symmetry states induced by strong electroninteractions [31, 33, 34]. The jumps occur concomitantlywith the correlated insulators in constituent TBG flatbands. The shaded areas mark the gap sizes calculatedwith Eq. (2) in the incompressible states with filling fac-tors ν = ±1. The carrier densities at these full-fillingstates are converted into the twist angle. Figures 3(b)-(d) summarize µ vs. ν with the twist angle ranging from0.91◦ to 1.57◦ measured in four double moiré samples (la-beled S1-S4), as well as the gap sizes at the integer fill-ings Eg,ν and the change of the chemical potential overthe filling factor ranges from −1 < ν < 0 (∆µp) and0 < ν < 1 (∆µn). In Fig. 3(c) and (d), we compare ourresults with previously reported chemical potential mea-4surements [31, 33–36], most of which studied near magicangle.Our study provides µ vs. n data in the TBG flat bandsover a wide θ range, but is in good agreement with pre-vious studies near magic angle [31, 33–36]. As we seein Fig. 3(b), the chemical potential dependence on fill-ing factor is consistently weaker for TBGs with θ largerthan the magic angle, suggesting flatter moiré bands, onthe hole-side compared to the electron-side. Surprisingly,we see in Fig. 3(c) that the gap sizes at ν = 0 showa minimum near the magic angle, and increase with θaway from the magic angle. Gaps at neutrality are notobserved in most MATBG experiments [31, 33–35, 37],except in rare cases where the sample has minimal strain[38]. Conversely, gaps at quarter multiples of moiré Bril-louin zone fillings are only observed near the magic an-gle, and are associated with flavor polarization in ultra-flat bands. The qualitative difference between the twistangle dependencies of the gaps at neutrality vs. thoseat quarter-multiple filling factors suggests that the for-mer originate from a different type of broken symmetry,perhaps similar to the flavor-dependent layer polariza-tion states [39–42] that are thought to produce gaps insuspended Bernal bilayer graphene between valence andconduction bands for each flavor [43]. The ∆µn (∆µp)data which measure the bandwidth of the conduction (va-lence) flat bands, show large electron-hole asymmetry inthe band flatness, especially for large twist angles. Fur-thermore, on the valence side of the flat band, the changein the chemical potential has a weaker dependence on thetwist angle compared to the conduction side.We compare our exprimental data with the predictionsof a self-consistent Hartree approximation that are sum-marized by the dashed lines in Fig. 3(c) and (d). The useof this approximation is justified by the fact that corre-lations are expected to be weak when the flat bands areempty and when they are full [44]. To qualitatively matchthe observed variations in insulating gaps across differ-ent twist angles, we set α = 0.3 for θ = 0.91◦, α = 0.6for θ = 0.99◦ to 1.57◦ and α = 1 for θ = 1.7◦, where αis the ratio between same and different sublattice inter-layer tunneling and is introduced as a phenomenologicalparameter [25]. The dependence of α on the twist anglecan be qualitatively attributed to the corrugation effect,which becomes more pronounced at smaller twist angles,leading to a smaller α for smaller twist angles. Addition-ally, we included a non-local interlayer tunneling term[44, 45] to qualitatively capture the electron-hole asym-metry in the band structure. The non-local interlayertunneling strength is defined as wNL = t′bM/Auc, wheret′ = dtk/dk at the graphene Dirac point, tk is the Fouriertransform of the two-center approximation of the inter-layer hopping amplitude t(rrr), bM is the moiré recipro-cal lattice vector length, and Auc is the moiré unit cellarea. For simplicity, we used a constant wNL = −20meVin Fig. 3(c) and (d), but note that the rigorous valueof wNL should be twist angle dependent and experimen-tally determined. The calculations of Eg,±1 and ∆µp,nagree well with experimental measurements. Specifically,Eg,±1 increases with twist angle before an abrupt dropnear θ ∼ 1.57◦, and a noticeable dip near the magic an-gle. These features of Eg,±1 are directly influenced by thevarying α values at these twist angles. ∆µp,n show mini-mal dependence on twist angle for θ ≲ 1.38◦ followed bya visible increase as bandwidths increase for θ ≳ 1.38◦.The value we have chosen for the constant wNL resultsin a smaller electron-hole asymmetry in the calculation.From the definition of wNL above, wNL should be largerfor larger twist angles (assuming weak dependence of t′on twist angle in the small range explored in our ex-periment). We expect, therefore, larger electron-holeasymmetry at larger twist angles that is not captured byour constant-wNL approximation. The larger Eg,±1 and∆µp,n in our experimental data compared to the Hartreeapproximation results are due to contributions from ex-change interactions and correlations.0 5050100150-1 1⎯21-VBG (V) 95 mK 300 mK 530 mK 965 mK 1.4 K 2.2 KRxx (k)0 ⎯43⎯21B-10 -9 -8 -7-30369VTG (V)VBG (V)566876Rxx (k)1012-10 -5 0 5 10-8-4048VTG (V)VBG (V)07500Rxx (k)(a) (b)(c)−1234FIG. 4. (a) Rxx vs. VTG and VBG measured in sample S4 withθT = 1.57◦ and θB = 0.99◦ at T = 95mK. (b) Expanded viewof the data in the green dashed rectangle in panel (a). TheνB values corresponding to the resistance peaks are included,highlighting correlated insulators at νB = ± 12and 34. (c)Temperature dependence of Rxx vs. VBG measured along thered dashed line in (b). The top axis shows νB.Figure 4(a) shows a contour plot of Rxx as a function ofVTG and VBG in a sample with θT = 1.57◦ and θB = 0.99◦measured at T = 95mK. The most prominent resistancepeaks around VTG = ±8V signal that the top TBG isincompressible with νT = ±1. Figure 4(b) focuses onthe resistance peaks with νT = −1. Multiple insulatingstates associated with integer fillings νB = 0, 1 and cor-5related insulators at quarter-multiple fillings νB = ± 12 ,34are observed. In Fig. 4(c) the insulating behavior ofthe correlated insulators is confirmed by temperature-dependence measurements. Similar correlated insulatorswith the same νB values are present with νT = 1 aswell. We observe signatures of correlated insulators in allthe constituent TBGs close to the magic angle down to0.91◦[25]. This observation suggests that the proximityof one TBG does not suppress the correlated insulatingstates in the other TBG, even though screening suggeststhe possibility [12].We note that superconductivity is not observed inour double moiré samples down to the base tempera-ture of our dilution refrigerator (95mK), even thoughthey show clear correlated insulators. This observationis interesting in light of theoretical interest in the rela-tionship between superconducting and flavor orders inMATBG. Since superconducting phases are usually ac-companied by correlated phases with spin or valley or-der at nearby filling factors similar to high-temperaturesuperconductors [46], the two phases are sometimes as-sumed to arise from a common mechanism. However,experimental studies have also shown that superconduc-tivity can be present when the correlated insulators aresuppressed [9, 11, 12]. Furthermore, in MATBG, thesuperconducting phase is experimentally observed to bemore robust on the valence bands[2, 4, 10]. As our mea-surements show that the valence bands are flatter thanthe conduction counterpart, these findings suggest that aflatter band is favorable for superconductivity, either di-rectly through a higher density of states or indirectly bysuppressing other competing states that inhibit supercon-ductivity. Meanwhile, the substrate has also been shownto impact the critical temperature of the superconductingphase in both TBG and twisted double bilayer graphene[13, 47]. In the double moiré system studied here, eachTBG is bound by hBN on one side, and the oppositeTBG on the other, distinctly different from the case ofhBN encapsulated MATBG. This difference in environ-ment may suppress superconductivity in our samples.We demonstrated a double moiré system with spatiallyseparated, independently tunable flat moiré bands. Theunique combination of compressible and incompressiblestates in constituent TBGs that are weakly correlated en-ables the extraction of the chemical potential vs. carrierdensity relationship in each TBG subsystem. 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