Abstract
Electronic states in quasicrystals generally preclude a Bloch description1, rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals2,3, the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This ‘moiré quasicrystal’ allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition4,5, the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals6,7,8,9,10,11,12. We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals.
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Data availability
Source data for all figures in the main text are hosted by the repository given in ref. 47. All other data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
We thank B. E. Feldman, M. Koshino, Z. Zhu, D. K. Bediako, A. H. MacDonald, F. Guinea, L. Levitov, L. Glazman and E. Berg for illuminating discussions. A.U. acknowledges support from the MIT Pappalardo Fellowships and from the VATAT Outstanding Postdoctoral Fellowship in Quantum Science and Technology. M.T.R. acknowledges support from the MIT Pappalardo Fellowships. This work was supported by the Army Research Office MURI W911NF2120147 (A.U.), the National Science Foundation (DMR-1809802; M.T.R. and D.R.-L.), the STC Center for Integrated Quantum Materials (NSF grant no. DMR-1231319; S.C.d.l.B., T.D., P.J.D.C. and N.P.) and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF9463 to P.J.-H. This work was performed in part at the Harvard University Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the NSF under NSF ECCS award no. 1541959. K.W. and T.T. acknowledge support from JSPS KAKENHI (grant nos. 19H05790, 20H00354 and 21H05233). R.L. is supported by the Israel Science Foundation (ISF) through grant no. 1259/22.
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M.T.R., S.C.d.l.B. and D.R.-L. conceived the project. M.T.R. and D.R.-L. fabricated the sample. S.C.d.l.B., A.U., D.R.-L. and M.T.R. performed the measurements. A.U., S.C.d.l.B. and T.D. analysed the data and wrote the manuscript, with input from all co-authors. T.D., P.J.D.C. and N.P. performed the numerical calculations. R.L., L.F., R.C.A. and P.J.-H. discussed and analysed the results, together with A.U., S.C.d.l.B. and T.D. K.W. and T.T. provided hBN crystals. R.C.A. and P.J.-H. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Extraction of twist angle, θ12.
a, Measured Rxx versus ntot and perpendicular field B⊥ at temperature T = 4 K, with fixed D = 0. LLs beyond full filling of the moiré unit cell formed by layers 1 and 2 (ns,12 = 4.7 × 1012 cm−2) emerge from ns,12. b, Extracted LLs used to calibrate the geometric capacitances (together with ntot–D behaviour) and used to determine ns,12 (ν12 = ±4).
Extended Data Fig. 2 Layer-resolved constant density traces.
Rxx versus ntot and D at B⊥ = 1 T with important estimated traces as guides to the eye, indicated by labels. The dashed red line traces approximate charge-neutrality of layer 3, n3 = 0, indicated by the N = 0 LL of layer 3. A resistance peak appears at the crossing point of n3 = 0 and full filling of the pairwise moiré of layers 1 and 2, ν12 = 4. At this point, ntot ≈ n1 + n2 = ns,12 = 4.7 × 1012 cm−2, confirming our value of θ12 = 1.42° (see b). Semi-transparent traces indicate less well-defined layer character in regions in which the hybridization between the layers is more pronounced. Superconducting pockets (see Fig. 4) are outlined by dotted black lines.
Extended Data Fig. 3 Phenomenological model.
a, Detail of the dashed blue box region in Fig. 2e showing three sets of LLs with different slopes. b, Simulation of the DOS for three Dirac cones with layer hybridization giving rise to renormalized Fermi velocities (v1, v2, v3) = (0.51, 0.2, 1)v3 at B⊥ = 1 T. Displacement field is accounted for by adding to each LL spectrum a linear energy shift Δεi = αiD for cones i = {1, 2, 3}. The three colour ranges indicate the partial DOS on the three effective layers, L1, L2, L3. c, Schematic Dirac cones with Fermi velocities vi and rates of potential shift, αi = ∂εi/∂D. d, Example DOS from each effective layer for D = 0, with energy spacings differing between layers owing to different Fermi velocities vi. e, Applying non-zero D shuffles the sequence of LLs because of the different potential shifts, Δεi.
Extended Data Fig. 4 Further characterization of the superconductivity.
a, Temperature dependence of Rxx with D = 0, showing the two superconducting domes for ntot < 0 (measured between contacts 1 and 2 shown in Fig. 2). b, Critical current of the superconductivity versus B⊥ measured at ntot = 3.5 × 1012 cm−2 and D = 0.08 V nm−1 (measured between contacts 4 and 5).
Extended Data Fig. 5 Calculated DOS maps.
a, Inverse DOS as a function of density and layer potential, calculated for (θ12, θ23) = (1.4°, −1.9°). Shown are the pairwise moiré angles inferred from the inverse DOS peaks, in excellent agreement with the angles input to the calculation, validating the procedure of extracting the twist angles from the magnetotransport data. b, DOS as a function of density and layer potential. The high DOS region in ntot < 0 generally overlaps with the right superconducting pocket.
Extended Data Fig. 6 Comparison of spectral function approximations.
The spectral function calculated for the tight-binding intralayer dispersion (a), the effective k ⋅ σ Dirac cone dispersion (b) and momentum-independent interlayer tunnelling (keeping only t0 in the expansion of t(k)) for the k ⋅ σ dispersion (c). The three methods show excellent agreement.
Extended Data Fig. 7 Matching function for twist angles and model parameters.
a, Matching function d from equation (3) plotted for the target velocities (0.2, 0.51, 1) with t0/ħvF = 0.02 Å−1. b, Matching function versus t0/ħvF assuming the same target velocities, averaged over 100 points in the range 1.35° < θ12 < 1.45°, −1.95° < θ23 < −1.85°. The dominant peak is at 0.016 Å−1.
Extended Data Fig. 8 Electrostatic layer potential calculation.
a, Calculated electric potential energies ϕi (for layer index i) as a function of externally applied electrical displacement field D/ε0 for ntot = 0 using Fermi velocities extracted from the spectral function calculation; used to estimate correspondence between layer potential imbalance Δ and D/ε0 in the main text. Solid lines are shown for εint = 2.5, whereas the shaded regions indicate the extreme cases described in the ‘Estimating layer potentials’ section in Methods. b, Potential energy difference ϕi − ϕ2 between layers i and 2. The relative energy shifts between the layers is the relevant quantity, to first order. Linear fits (dashed lines) illustrate the slight nonlinear deviations.
Extended Data Fig. 9 Flavour-symmetry-breaking phase transition.
a, Rxx versus ntot and B⊥ taken at T = 500 mK, showing a Landau fan emerging from moiré filling ν12 ≈ −2, indicating a flavour-symmetry-breaking phase transition near that carrier density. b, Illustration of the layer 3 LLs and their contributions to the Chern number, \({\nu }_{\,{\rm{LL}}}^{3}\), in the gaps between the LLs. c, The situation at ν12 ≈ −2 (a, dashed cyan line). The Fermi energy is in the N = 0 LLs of layers 1 and 2 and in the gap between LL N3 = 0 and N3 = −1 of layer 3. Semi-transparent LLs indicate inaccessible levels before the phase transition (ν12 > −2). d, In the first LL gap beyond the phase transition (a, dashed white line), the Fermi level is in the gap between LLs 0 and −1 in all three layers. Layer 3 contributes C = −2, whereas layers 1 and 2 each contribute C = −1 owing to their reduced degeneracy, accounting for the observed slope C = −4. e, In the C = −12 gap (a, dotted green lines), the Fermi energy is between LLs −1 and −2 in all three layers. The Chern contributions are −1, −2 = −3 for layers 1 and 2 and −2 − 4 = −6 for layer 3, in total −3 − 3 − 6 = −12.
Extended Data Fig. 10 DOS under finite magnetic field.
a, DOS calculation for a 3–4 approximant (see ‘DOS calculation under finite magnetic field’ section in Methods) under B⊥ = 1 T. b, Same as a, zooming in on moderate Δ. Dashed black lines approximately enclose the electronic quasiperiodic regimes as a guide to the eye. c, Same as a, zooming in on Δ < 0 and low energies.
Supplementary information
Supplementary Information
The Supplementary Information file includes six sections, Supplementary Figs. 1–4 and further references.
Supplementary Video 1
Calculated 1D spectral function as a function of layer potential.
Supplementary Video 2
Constant energy cuts of the computed spectral function with varying energy.
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Uri, A., de la Barrera, S.C., Randeria, M.T. et al. Superconductivity and strong interactions in a tunable moiré quasicrystal. Nature 620, 762–767 (2023). https://doi.org/10.1038/s41586-023-06294-z
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DOI: https://doi.org/10.1038/s41586-023-06294-z
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