Abstract
The physics of superconductivity in magic-angle twisted bilayer graphene (MATBG) is a topic of keen interest in moiré systems research, and it may provide an insight into the pairing mechanism of other strongly correlated materials such as high-critical-temperature superconductors. Here we use d.c. transport and microwave circuit quantum electrodynamics to directly measure the superfluid stiffness of superconducting MATBG through its kinetic inductance. We find the superfluid stiffness to be much larger than expected from conventional Fermi liquid theory. Rather, it is comparable to theoretical predictions1 and recent experimental indications2 of quantum geometric effects that are dominant at the magic angle. The temperature dependence of the superfluid stiffness follows a power law, which contraindicates an isotropic Bardeen–Cooper–Schrieffer (BCS) model. Instead, the extracted power-law exponents indicate an anisotropic superconducting gap, whether interpreted in the Fermi liquid framework or by considering the quantum geometry of flat-band superconductivity. Moreover, a quadratic dependence of the superfluid stiffness on both d.c. and microwave current is observed, which is consistent with the Ginzburg–Landau theory. Taken together, our findings show that MATBG is an unconventional superconductor with an anisotropic gap and strongly suggest a connection between quantum geometry, superfluid stiffness and unconventional superconductivity in MATBG. The combined d.c.–microwave measurement platform used here is applicable to the investigation of other atomically thin superconductors.
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Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request and with the cognizance of our US government sponsors who funded the work.
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Acknowledgements
We acknowledge helpful discussions with P. Törmä, L. Levitov, S. Todadri, M. Roppongi, K. Ishihara, T. Kitamura and Y. Yanase. We thank L. Ateshian, D. Rower, P. Harrington and the device packaging team at the MIT Lincoln Laboratory for technical assistance. This research was funded in part by the US Army Research Office grant no. W911NF-22-1-0023, by the National Science Foundation QII-TAQS grant no. OMA-1936263, by the Air Force Office of Scientific Research grant no. FA2386-21-1-4058 and by the Under Secretary of Defense for Research and Engineering under Air Force contract no. FA8702-15-D-0001. M.T. acknowledges support from the JSPS KAKENHI (grant no. 23K19026), Murata Science and Education Foundation, and JST, PRESTO (grant no. JPMJPR24H7). P.J.-H. acknowledges support by the National Science Foundation (DMR-1809802), the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF9463, the Fundacion Ramon Areces and the CIFAR Quantum Materials program. D.R.-L. acknowledges support from the Rafael del Pino Foundation. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant nos. 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Under Secretary of Defense for Research and Engineering or the US Government.
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J.Î-j.W. conceived and designed the experiment. M.T., J.Î-j.W., T.H.D. and M.H. performed the microwave simulation. M.T., J.Î-j.W., T.H.D., S.Z., D.R.-L., D.K.K., B.M.N., K.S. and M.E.S. contributed to the device fabrication. M.T., J.Î-j.W., T.H.D., S.Z., D.R.-L., A.A. and B.K. participated in the measurements. M.T., J.Î-j.W. and M.H. analysed the data. K.W. and T.T. grew the hBN crystal. J.Î-j.W., M.T. and W.D.O. led the paper writing, and all other authors contributed to the text. J.A.G., T.P.O., S.G., P.J.-H., J.Î-j.W. and W.D.O. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Microwave and DC characterization setup.
Wiring diagram in the dilution refrigerator, which includes attenuators, amplifiers, isolators, and filters.
Extended Data Fig. 2 Model and simulation of the resonator with termination.
a, Model used for the microwave simulation, including the through line, resonator, termination, and backgate. The termination includes both the MATBG (resistance or inductance, depending on bias point) and the proximitized edge inductance (independent of bias point). See main text. b, Out-of-plane magnetic field dependence of the resonance of Bernal-stacked bilayer graphene biased near its charge neutrality point. c, Simulated |S21| as a function of ∆fr = fr − 3.8 GHz for different values of CBG of the model. d, Simulated resonance frequency as a function of 1/LTBG for different values of Lprx. CBG is fixed at 3 fF. e, Data in panel d, with the baseline resonance frequency subtracted. Dashed lines are the linear fit to the simulation data. Inset depicts the slope of the linear fit as a function of offset frequency fr0, which depends on Lprx. f,g, Simulations of possible series (f) and parallel (g) contact resistances between the MATBG and the resonator, ground, and leads. Our results are most consistent with superconducting contacts, i.e., zero series resistance and large parallel resistance.
Extended Data Fig. 3 Temperature and power dependence of the resonant frequency for the aluminum-only λ/4 resonator (control resonator).
a, Microscope image of the aluminum-only λ/4 resonator, terminated directly to ground with Al. b, Temperature dependence of the resonance for the aluminum-only λ/4 resonator. c, Temperature dependence of fr(Tbase)/fr(T)-1 for the aluminum-only λ/4 resonator as fit with the isotropic BCS model (exponential fit, purple dashed line) and a power-law fit (black dashed line). d, |S21| for the aluminum-only λ/4 resonator at Tbase = 20 mK for different microwave powers. The power independence indicates the negligible kinetic inductance for the thick (250 nm) aluminum resonator and termination.
Extended Data Fig. 4 Temperature and power dependence of the resonant frequency for the resonator terminated by the Al-proximitized graphene edge.
a, Gate dependence of the resonance in the MATBG-terminated resonator. The insulating region is indicated by a vertical white dashed line VBG = 2.44 V. The frequency at this point is used as the reference point when determining a frequency shift. b, Temperature dependence of the resonance for the MATBG-terminated resonator in the insulating region VBG = 2.44 V. c, Temperature dependence of fr(Tbase)/fr(T) − 1 for the MATBG-terminated resonator in the insulating region VBG = 2.44 V and fit with the isotropic BCS model (purple dashed line) and the power-law fit (black dashed line). The resonance no longer shifts below about 0.5 K. d, Microwave power dependence of fr at VBG = −6.8, −6.9, −7.0 V (where the MATBG is superconducting) and VBG = −6.5 V (where the MATBG is insulating). In contrast to the MATBG traces, the proximitized edge does not vary substantially with power, indicating that its kinetic inductance is adopted from the aluminum leads and is relatively small compared with the bulk MATBG.
Extended Data Fig. 5 Estimation of Fermi velocity.
a, Differential resistance dV/dIDC as a function of DC bias current IDC and gate voltage VBG. The blue dashed line indicates Icn. b, Differential resistance dV/dIDC as a function of DC bias current IDC at VBG = − 6.82 V. c, Gate dependence of the Fermi velocity, extracted from vn = Jcn/\(\tilde{n}\) e.
Extended Data Fig. 6 Determination of TC.
a, The temperature dependence of the MATBG resistance at VBG = −6.7 V. Gray dashed lines represent linear fits. Red arrows and a red dashed line indicate the half value of line 1 at zero temperature. Critical temperatures are marked by arrows of different colors. b, c, Backgate dependence of critical temperatures T C(onset), TC (0.5), and TC (zero) as obtained from DC resistance measurements; and TC (MW) as obtained from microwave measurements in the hole-doped (b) and the electron-doped (c) regimes.
Extended Data Fig. 7 Gate dependence of DC and microwave response in the second device.
a, DC resistance R as a function of the backgate voltage VBG. Top axis represents the filling factor ν. Inset shows the optical microscope image of the second device. b, Microwave transmission coefficient |S21| versus VBG. The resonant frequency (bright line) shifts within the zero-resistance region in panel (a), near filling factors ν = ±2. The resonance remains essentially constant within the high resistance region. c, Differential resistance dV/dIDC as a function of VBG and DC bias current IDC.
Extended Data Fig. 8 Superfluid stiffness, carrier density, and critical temperature in the second device.
a,b, Frequency shift and inverse of the kinetic inductance as a function of VBG in hole-doped (a) and electron-doped (b) regimes. c, Superfluid stiffness Ds at base temperature Tbase as a function of effective carrier density \(\tilde{n}\), measured with respect to |ν| = 2. The black dashed curve estimates the conventional contribution to the superfluid stiffness from single-band Fermi liquid theory: Ds(conv) = e2\(\tilde{n}\)vF/ħkF. d, Critical temperature TC and corresponding superfluid stiffness Ds at base temperature Tbase as tuned by VBG. The black dashed line represents the BKT limit TC = πħ2Ds(Tbase)/8e2kB.
Extended Data Fig. 9 Temperature dependence in the second device.
a, Resonance frequency fr as a function of temperature and VBG. b, Temperature dependence of the measured Ds at VBG = −6.6, −7.2, and −7.8 V and 8e2kBT/πħ2 (black dashed line), where TBKT is determined by their intersection. c, VBG dependence of the exponent determined from the power-law fitting at T < 0.3 TC. d, e, f, Power-law fitting of (∆fr(Tbase) − ∆fr(T))/∆fr(Tbase) at VBG = −6.6, −7.2, and −7.8 V.
Extended Data Fig. 10 DC bias current dependence in the second device.
a, IDC dependence of the DC resistance dV/dIDC at VBG = −7.3 V. Inset depicts the DC resistance as a function of IDC and VBG. b, IDC dependence of fr at VBG = −7.3 V. The blue dashed curve is the quadratic fit. Inset depicts fr as a function of IDC and VBG. c, IDC dependence of the DC resistance over the hole-doped SC region. d, IDC dependence of fr(I) − fr(I = 0) over the hole-doped SC region using a logarithmic scale. The black dashed line indicates the quadratic dependence.
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Supplementary Notes 1–6, Figs. 1–8 and references.
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Tanaka, M., Wang, J.Îj., Dinh, T.H. et al. Superfluid stiffness of magic-angle twisted bilayer graphene. Nature 638, 99–105 (2025). https://doi.org/10.1038/s41586-024-08494-7
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DOI: https://doi.org/10.1038/s41586-024-08494-7
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