Abstract
Moiré superlattices1,2 have recently emerged as a platform upon which correlated physics and superconductivity can be studied with unprecedented tunability3,4,5,6. Although correlated effects have been observed in several other moiré systems7,8,9,10,11,12,13,14,15,16,17, magic-angle twisted bilayer graphene remains the only one in which robust superconductivity has been reproducibly measured4,5,6. Here we realize a moiré superconductor in magic-angle twisted trilayer graphene (MATTG)18, which has better tunability of its electronic structure and superconducting properties than magic-angle twisted bilayer graphene. Measurements of the Hall effect and quantum oscillations as a function of density and electric field enable us to determine the tunable phase boundaries of the system in the normal metallic state. Zero-magnetic-field resistivity measurements reveal that the existence of superconductivity is intimately connected to the broken-symmetry phase that emerges from two carriers per moiré unit cell. We find that the superconducting phase is suppressed and bounded at the Van Hove singularities that partially surround the broken-symmetry phase, which is difficult to reconcile with weak-coupling Bardeen–Cooper–Schrieffer theory. Moreover, the extensive in situ tunability of our system allows us to reach the ultrastrong-coupling regime, characterized by a Ginzburg–Landau coherence length that reaches the average inter-particle distance, and very large TBKT/TF values, in excess of 0.1 (where TBKT and TF are the Berezinskii–Kosterlitz–Thouless transition and Fermi temperatures, respectively). These observations suggest that MATTG can be electrically tuned close to the crossover to a two-dimensional Bose–Einstein condensate. Our results establish a family of tunable moiré superconductors that have the potential to revolutionize our fundamental understanding of and the applications for strongly coupled superconductivity.
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The data that support the current study are available from the corresponding authors upon reasonable and well motivated request.
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Acknowledgements
We thank S. Todadri, A. Vishwanath, S. Kivelson, M. Randeria, S. Ilani, L. Fu and A. Georges for discussions. This work has been primarily supported by the US Department of Energy (DOE), Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering under award DE-SC0001819 (J.M.P.). Help with transport measurements and data analysis were supported by the National Science Foundation (DMR-1809802), and the STC Center for Integrated Quantum Materials (NSF grant number DMR-1231319; Y.C.). P.J.-H. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF9643. P.J.-H. acknowledges partial support by the Fundación Ramon Areces. The development of new nanofabrication and characterization techniques enabling this work has been supported by the US DOE Office of Science, BES, under award DE‐SC0019300. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant number JPMXP0112101001, JSPS KAKENHI grant numbers JP20H00354 and the CREST (JPMJCR15F3), JST. This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the National Science Foundation (DMR-0819762) and of Harvard’s Center for Nanoscale Systems, supported by the NSF (ECS-0335765).
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J.M.P. and Y.C. fabricated the samples, and performed transport measurements and numerical simulations. K.W. and T.T. provided hBN samples. J.M.P., Y.C., and P.J-H. performed data analysis, discussed the results, and wrote the manuscript with input from all co-authors.
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Extended data figures and tables
Extended Data Fig. 1 Stacking order in MATTG.
a, b, Illustrations of A-tw-A stacking (a) and A-tw-B stacking (b), where ‘tw’ denotes the middle twisted layer (L2, orange) and A/B represents the relative stacking order between the topmost (L3, green) and bottommost (L1, blue) layers. c–f, Continuum-model bandstructures of A-tw-A stacked (c, d) and A-tw-B stacked (e, f) MATTG at zero (c, e) and finite (d, f) displacement fields. The twist angle is θ = 1.57° for all plots. g–j, Calculated Landau level sequence corresponding to the bands in c–f. The size of the dots represents the size of the Landau level gaps in the Hofstadter spectrum. For A-tw-A stacking, the major sequence of filling factors near the charge neutrality is ±2, ±6, ±10, …, regardless of the displacement field, whereas for A-tw-B stacking the Landau levels evolve into a symmetry-broken sequence that has 0, ±8 as the dominant filling factors with largest gaps in a finite displacement field. An anisotropy term of β = −0.01 is included in all of the above calculations (see Methods). k, l, Experimentally measured Landau levels in MATTG near the charge neutrality. We find the strongest sequence of ±2, ±6, ±10, … at both D = 0 and D/ε0 = 0.77 V nm−1, consistent with the A-tw-A stacking scenario.
Extended Data Fig. 2 Device schematics and device optical picture.
a, Our device consists of hBN-encapsulated MATTG etched into a Hall bar, Cr/Au contacts on the edge, and top/bottom metallic gates. For transport measurements, we measure current I, longitudinal voltage Vxx, and transverse voltage Vxy, while tuning the density ν and displacement field D by applying top gate voltage Vtg and bottom gate voltage Vbg. b, Optical picture of devices A and B. Device C is lithographically similar.
Extended Data Fig. 3 Robust superconductivity in other MATTG devices (devices B and C).
a, Rxx–T curve. b, Vxx–I and dVxx/dI–I curves. c, I–B map in device B with a smaller-than-magic-angle θ ≈ 1.44°. In this device, maximum TBKT ≈ 0.73 K. The choice of ν is to display the Fraunhofer-like Josephson interference, which demonstrates the superconducting phase coherence. d–f, As in a–c, for device C, with a twist angle θ ≈ 1.4°. Device C has a maximum TBKT of ~0.68 K. f shows a regular B-suppression of Ic with B. Both devices show sharp peaks in dVxx/dI at their critical currents.
Extended Data Fig. 4 Vxx–I curves and critical current Ic in MATTG.
a, Forward (red) and backward (blue) sweeps of Vxx–I curves for the optimal point ν = −2.4 and D/ε0 = −0.44 V nm−1. Inset, A clear hysteresis loop exists in the curve at I ≈ 550–600 nA. b, Map of Ic versus ν and D in the major superconducting regions. c, Evolution of Ic over D at ν = −2.4, showing that Ic initially increases as finite D is applied, and quickly decreases beyond local maxima near |D|/ε0 ≈ 0.48 V nm−1. d, Ic versus D at ν = +2.26 shows that the maximum Ic occurs near |D|/ε0 ≈ 0.71 V nm−1, after which Ic quickly decreases. The modulation of superconducting strength in D may be due to change in the band flatness, as well as the interactions with the electrons in the Dirac bands. e–g, Vxx–I and dVxx/dI–I curves for certain points in superconducting domes near ν = −2 + δ (e), ν = +2 – δ (f), and ν = +2 + δ (g), all showing sharp peaks in dVxx/dI at the critical current.
Extended Data Fig. 5 Rxx versus ν at T = 70 mK, 5 K and 10 K.
a–d, Measured at D/ε0 = 0.77 V nm−1 (a), D/ε0 = 0.52 V nm−1 (b), D/ε0 = 0.26 V nm−1 (c) and D/ε0 = 0 V nm−1 (d).
Extended Data Fig. 6 Hall density analysis.
a–c, Linecuts of Rxx, Rxy and νH (right axis) versus ν at representative D from high to zero, showing the bounding of major superconducting phases within the Hall density features. Vertical red, yellow, and dark blue bars denote ‘gap/Dirac’, ‘reset’ and ‘VHS’ features, respectively, and the light-blue regions denote superconductivity. Purple dashed lines show the expected Hall density. We note that there are some small regions right before ν = +1 and ν = +2 where for certain D values there are signatures of a more complex behaviour in νH, with VHSs possibly very close to the ‘resets’, as shown in Fig. 2b. d, The Hall density νH extracted from smaller magnetic fields of B ≈ 0.1–0.3 T reveals a VHS boundary close to the weak superconducting phase boundary near ν = −2 + δ, which is absent in the Hall density shown in a–c and Fig. 2b extracted from a higher magnetic field of B ≈ −1.5 T to 1.5 T. e, Rxx in the same region as shown in d, where the superconducting boundary is close to the VHSs. All measurements are performed at the base temperature T ≈ 70 mK. SC, superconducting.
Extended Data Fig. 7 Quantum oscillations and effective-mass analysis.
All data shown here are measured at D/ε0 = −0.44 V nm−1. a, b, Quantum oscillations at ν = −2.86 (a) and ν = −2.5 (b) at different T. Grey dashed lines show the peaks used for analysis. Inset, Fit to the Lifshitz–Kosevich formula for the extraction of the effective mass, yielding m*/me = 1.25 ± 0.13 (a) and m*/me = 0.95 ± 0.03 (b). c, d, Quantum oscillations sampled at coarser points in T for the same ν as in a, b. Extracted effective-mass values with these coarser data are m*/me = 1.2 ± 0.2 (c) and m*/me = 0.96 ± 0.09 (d), matching the values from a, b within the uncertainty. e, Quantum oscillations at ν = −2.4 (optimal doping). f, Lifshitz–Kosevich fits for the data shown in c–e, showing δR normalized with its value at the lowest temperature. The peaks chosen for extraction are marked with triangles in c–e. Amp., amplitude; a.u., arbitrary units.
Extended Data Fig. 8 Analysis of the Ginzburg-Landau coherence length.
a, b, Superconducting transitions at perpendicular magnetic fields from B = 0 T to B = 0.2 T (40 mT between curves) for ν = −2 − δ (ν = −2.4; a) and ν = −2 + δ (ν = −1.84; b), from which the Ginzburg–Landau coherence length ξGL is extracted. D/ε0 = −0.44 V nm−1 for both plots. Inset shows \({T}_{{\rm{c}}}^{50 \% }\), \({T}_{{\rm{c}}}^{40 \% }\) and \({T}_{{\rm{c}}}^{30 \% }\) as a function of B, from which we extracted the coherence length ξGL as 9.4 nm, 12.4 nm and 16.1 nm, respectively, for ν = −2 − δ. For ν = −2 + δ, we obtained 38.0 nm, 39.1 nm and 37.1 nm, respectively. We note that for ν = −2 − δ, the Rxx–T curves develop an extra transition (‘knee’) below Tc at finite B, which is possibly related to the melting transition between a vortex solid and a vortex liquid48.
Extended Data Fig. 9 Landau fans for intermediate D.
a, b, Landau fans on the hole-doped (a) and electron-doped sides (b). They show the evolution between small D and large D, which exhibits a hybridization of the features. In a, the Landau fan diagram at D/ε0 = −0.34 V nm−1 for the hole-doped side shows the fans emanating from all integer fillings. An inward-facing fan from ν = −4 starts developing, which meets the outward-facing fan from ν = −3. Note also the appearance of an inward-facing fan from ν = −2, which meets the outward-facing fan from ν = −1. These observations agree with the formation of VHSs around these two regions in the intermediate |D|, where the electron-like carriers become hole-like, as illustrated in Fig. 4d, and identified in Fig. 2b. A small region of superconductivity starts appearing at ν = −2 + δ while the carriers from ν = −2 are present, as shown in Fig. 2a. In b, the Landau fan diagram at D/ε0 = −0.52 V nm−1 on the electron-doped side shows similar VHSs between ν ≈ +1–2 and ν ≈ +3–4. Similar to the hole-doped side, an inward-facing fan from ν = +2 develops and meets with the outward-facing fan from ν = +1. The density range of the inward-facing fan encompasses the appearance of a superconducting region at ν = −2 + δ at this D.
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Park, J.M., Cao, Y., Watanabe, K. et al. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021). https://doi.org/10.1038/s41586-021-03192-0
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DOI: https://doi.org/10.1038/s41586-021-03192-0
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