Abstract
The robustness of the macroscopic quantum nature of a superconductor can be characterized by the superfluid stiffness, ρs, a quantity that describes the energy required to vary the phase of the macroscopic quantum wavefunction. In unconventional superconductors, such as cuprates, the low-temperature behaviour of ρs markedly differs from that of conventional superconductors owing to quasiparticle excitations from gapless points (nodes) in momentum space. Intensive research on the recently discovered magic-angle twisted graphene family has revealed, in addition to superconducting states, strongly correlated electronic states associated with spontaneously broken symmetries, inviting the study of ρs to uncover the potentially unconventional nature of its superconductivity. Here we report the measurement of ρs in magic-angle twisted trilayer graphene (TTG), revealing unconventional nodal-gap superconductivity. Utilizing radio-frequency reflectometry techniques to measure the kinetic inductive response of superconducting TTG coupled to a microwave resonator, we find a linear temperature dependence of ρs at low temperatures and nonlinear Meissner effects in the current-bias dependence, both indicating nodal structures in the superconducting order parameter. Furthermore, the doping dependence shows a linear correlation between the zero-temperature ρs and the superconducting transition temperature Tc, reminiscent of Uemura’s relation in cuprates, suggesting phase-coherence-limited superconductivity. Our results provide strong evidence for nodal superconductivity in TTG and put strong constraints on the mechanisms of these graphene-based superconductors.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
27,99 € / 30 days
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
199,00 € per year
only 3,90 € per issue
Buy this article
- Purchase on SpringerLink
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
Source data are provided with the paper. All other data that support the findings of this study are available from the corresponding authors upon reasonable request.
Change history
07 April 2025
In the version of the article initially published, the grant no. NSF DMR-2220703 appeared incorrectly and has now been amended in the HTML and PDF versions of the article.
References
Lee, P. A. & Wen, X.-G. Unusual superconducting state of underdoped cuprates. Phys. Rev. Lett. 78, 4111 (1997).
Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995).
Millis, A. J., Girvin, S. M., Ioffe, L. B. & Larkin, A. I. Anomalous charge dynamics in the superconducting state of underdoped cuprates. J. Phys. Chem. Solids 59, 1742–1744 (1998).
Yip, S. K. & Sauls, J. A. Nonlinear Meissner effect in CuO superconductors. Phys. Rev. Lett. 69, 2264 (1992).
Xu, D., Yip, S. K. & Sauls, J. A. Nonlinear Meissner effect in unconventional superconductors. Phys. Rev. B 51, 16233 (1995).
Hardy, W. N., Bonn, D. A., Morgan, D. C., Liang, R. & Zhang, K. Precision measurements of the temperature dependence of λ in YBCO: strong evidence for nodes in the gap function. Phys. Rev. Lett. 70, 3999 (1993).
Uemura, Y. J. et al. Universal correlations between Tc and ns/m* (carrier density over effective mass) in high-Tc cuprate superconductors. Phys. Rev. Lett. 62, 2317 (1989).
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Hao, Z. et al. Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).
Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021).
Park, J. M. et al. Robust superconductivity in magic-angle multilayer graphene family. Nat. Mater. 21, 877–883 (2022).
Zhang, Y. et al. Promotion of superconductivity in magic-angle graphene multilayers. Science 377, 1538–1543 (2022).
Cao, Y. et al. Nematicity and competing orders in superconducting magic-angle graphene. Science 372, 264–271 (2021).
Cao, Y., Park, J. M., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Pauli-limit violation and re-entrant superconductivity in moiré graphene. Nature 595, 526–531 (2021).
Tian, H. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023).
Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).
Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene. Nature 606, 494–500 (2022).
Schoelkopf, R. J., Wahlgren, P., Kozhevnikov, A. A., Delsing, P. & Prober, D. E. The radio-frequency single-electron transistor (RF-SET): a fast and ultrasensitive electrometer. Science 280, 1238–1242 (1998).
Reilly, D. J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Fast single-charge sensing with a RF quantum point contact. Appl. Phys. Lett. 91, 162101 (2007).
Vigneau, F. et al. Probing quantum devices with radio-frequency reflectometry. Appl. Phys. Rev. 10, 021305 (2023).
Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann–Franz law in graphene. Science 351, 1058–1061 (2016).
Annunziata, A. J. et al. Tunable superconducting nanoinductors. Nanotechnology 21, 445202 (2010).
Singh, G. et al. Competition between electron pairing and phase coherence in superconducting interfaces. Nat. Commun. 9, 407 (2018).
Weitzel, A. et al. Sharpness of the Berezinskii–Kosterlitz–Thouless transition in disordered NbN films. Phys. Rev. Lett. 131, 186002 (2023).
Haller, R. et al. Phase-dependent microwave response of a graphene Josephson junction. Phys. Rev. Res. 4, 013198 (2022).
Phan, D. et al. Detecting induced p ± ip pairing at the Al–InAs interface with a quantum microwave circuit. Phys. Rev. Lett. 128, 107701 (2022).
Bøttcher, C. G. L. et al. Circuit quantum electrodynamics detection of induced two-fold anisotropic pairing in a hybrid superconductor-ferromagnet bilayer. Nat. Phys. 20, 1609–1615 (2024).
Tanaka, M. et al. Superfluid stiffness of magic-angle twisted bilayer graphene. Nature https://doi.org/10.1038/s41586-024-08494-7 (2025).
Kreidel, M. et al. Measuring kinetic inductance and superfluid stiffness of two-dimensional superconductors using high-quality transmission-line resonators. Preprint at https://arxiv.org/abs/2407.09916 (2024).
Tinkham, M. Introduction to Superconductivity 2nd edn (Dover Publications, 2004).
Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019).
Probst, S., Song, F. B., Bushev, P. A., Ustinov, A. V. & Weides, M. Efficient and robust analysis of complex scattering data under noise in microwave resonators. Rev. Sci. Instrum. 86, 024706 (2015).
Nelson, D. R. & Kosterlitz, J. M. Universal jump in the superfluid density of two-dimensional superfluids. Phys. Rev. Lett. 39, 1201 (1977).
Turkel, S. et al. Orderly disorder in magic-angle twisted trilayer graphene. Science 376, 193–199 (2022).
Uri, A. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581, 47–52 (2020).
Homes, C. C. et al. A universal scaling relation in high-temperature superconductors. Nature 430, 539–541 (2004).
Dordevic, S. V., Basov, D. N. & Homes, C. C. Do organic and other exotic superconductors fail universal scaling relations? Sci. Rep. 3, 1713 (2013).
Božović, I., He, X., Wu, J. & Bollinger, A. T. Dependence of the critical temperature in overdoped copper oxides on superfluid density. Nature 536, 309–311 (2016).
Bidinosti, C. P., Hardy, W. N., Bonn, D. A. & Liang, R. Magnetic field dependence of λ in YBa2Cu3O6.95: results as a function of temperature and field orientation. Phys. Rev. Lett. 83, 3277–3280 (1999).
Oates, D. E., Park, S.-H. & Koren, G. Observation of the nonlinear Meissner effect in YBCO thin films: evidence for a d-wave order parameter in the bulk of the cuprate superconductors. Phys. Rev. Lett. 93, 197001 (2004).
Wilcox, J. A. et al. Observation of the non-linear Meissner effect. Nat. Commun. 13, 1201 (2022).
Dahm, T. & Scalapino, D. J. Theory of microwave intermodulation in a high-Tc superconducting microstrip resonator. Appl. Phys. Lett. 69, 4248–4250 (1996).
Bae, S. et al. Dielectric resonator method for determining gap symmetry of superconductors through anisotropic nonlinear Meissner effect. Rev. Sci. Instrum. 90, 043901 (2019).
de Visser, P. J. et al. Evidence of a nonequilibrium distribution of quasiparticles in the microwave response of a superconducting aluminum resonator. Phys. Rev. Lett. 112, 047004 (2014).
Turneaure, S. J., Lemberger, T. R. & Graybeal, J. M. Effect of thermal phase fluctuations on the superfluid density of two-dimensional superconducting films. Phys. Rev. Lett. 84, 987–990 (2000).
Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).
Peotta, S., Huhtinen, K.-E. & Törmä, P. Quantum geometry in superfluidity and superconductivity. Preprint at https://arxiv.org/abs/2308.08248 (2023).
Verma, N., Guerci, D. & Queiroz, R. Geometric stiffness in interlayer exciton condensates. Phys. Rev. Lett. 132, 236001 (2023).
Mendez-Valderrama, J. F., Mao, D. & Chowdhury, D. Low-energy optical sum rule in moiré graphene. Phys. Rev. Lett. 133, 196501 (2023).
Mao, D. & Chowdhury, D. Diamagnetic response and phase stiffness for interacting isolated narrow bands. Proc. Natl Acad. Sci. USA 120, e2217816120 (2023).
Mao, D. & Chowdhury, D. Upper bounds on superconducting and excitonic phase stiffness for interacting isolated narrow bands. Phys. Rev. B 109, 024507 (2024).
Ioffe, L. B. & Millis, A. J. d-wave superconductivity in doped Mott insulators. J. Phys. Chem. Solids 63, 2259–2268 (2002).
Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Benfatto, L., Castellani, C. & Giamarchi, T. Kosterlitz–Thouless behavior in layered superconductors: the role of the vortex core energy. Phys. Rev. Lett. 98, 117008 (2007).
Hetel, I., Lemberger, T. R. & Randeria, M. Quantum critical behaviour in the superfluid density of strongly underdoped ultrathin copper oxide films. Nat. Phys. 3, 700–702 (2007).
Hazra, T., Verma, N. & Randeria, M. Upper bounds on the superfluid stiffness and superconducting Tc: applications to twisted-bilayer graphene and ultra-cold Fermi gases. Phys. Rev. X 9, 031049 (2019).
Mahmood, F., He, X., Bozovic, I. & Armitage, N. P. Locating the missing superconducting electrons in the overdoped cuprates La2−xSrxCuO4. Phys. Rev. Lett. 122, 027003 (2019).
Hu, X., Hyart, T., Pikulin, D. I. & Rossi, E. Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 123, 237002 (2019).
Törmä, P., Peotta, S. & Bernevig, B. A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat. Rev. Phys. 4, 528–542 (2022).
Ganiev, O. K. & Yavidov, B. Superfluid density and critical current density in superconducting cuprates with an extended d-wave pairing symmetry. Eur. Phys. J. B 94, 116 (2021).
Acknowledgements
We thank S. Kivelson and E. Berg for discussions. The major experimental work is supported by ARO MURI (W911NF-21-2-0147). P.K. and A.B. acknowledge support from the DOE (DE-SC0012260). M.K. is supported by the STC Center for Integrated Quantum Materials, NSF Grant number DMR-1231319. A.V., P.L. and P.A.V. are supported by a Simons Investigator grant (A.V.) and by NSF DMR-2220703. P.A.V. acknowledges support by a Quantum-CT Quantum Regional Partnership Investments (QRPI) Award. K.C.F. thanks M. Randeria for her musings inspiring this research direction. A.Y. and M.E.W. are supported by the Quantum Science Center (QSC), a National Quantum Information Science Research Center of the US Department of Energy (DOE). A.Y. is also partly supported by the Gordon and Betty Moore Foundation through grant GBMF ID #12762. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant numbers 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. Nanofabrication was performed at the Center for Nanoscale Systems at Harvard, supported in part by an NSF NNIN award ECS- 00335765.
Author information
Authors and Affiliations
Contributions
A.B., Z.H., M.K., K.C.F, and P.K. conceived of the experiment. Z.H., I.P., J.M.P. and A.Z. fabricated the devices, A.B., Z.H., M.E.W. and M.K. conducted the measurements and analysed the data. P.L., P.A.V. and A.V. conducted the theoretical analysis. R.M.W., A.Y., P.J.-H, P.A.V., A.V., K.C.F. and P.K. supervised the project. All authors discussed and wrote the paper. K.W. and T.T. supplied hBN crystals.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature thanks Fan Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Device DC transport characterization.
a, TTG 4-terminal R as a function of ν and magnetic field B at zero displacement field and a temperature of 2 K. The Landau fans show two set of structures: one set with large slopes and the other that appear at low B with shallow slopes. b, TTG 4-terminal R as a function of ν and displacement field D at zero B and a temperature of 30 mK.
Extended Data Fig. 2 Measurement setup.
Circuit model of measurement setup for superfluid stiffness measurements. The moiré graphene sample is indicated by dashed pink lines while the circuit board–containing sample, LC matching network, and bias tees to allow impedance matching for microwave measurement–is indicated by dashed purple lines. DC and RF filtering is used to reduce noise throughout the measurement chain. The RF signal is sent through an attenuated input line before entering a directional coupler. Sample contact resistance of ≃ 3.2 kΩ is given by Rc. Graphite top and bottom gates are represented by “gate”. The final signal is amplified both at 4 K and room temperature before being measured by a vector network analyzer (VNA).
Extended Data Fig. 3 Circle fitting.
(a) Circle fitting of S21 in the complex plane. (b) Fitting of the reflection amplitude ∣S21(f)∣. (c) Fitting of the phase expressed in degrees \(\arg {S}_{21}(f)\).
Extended Data Fig. 4 Uemura’s law.
Scaling of ρs0 versus Tc (obtained from resistance measurements) for both electron and hole side superconductors in all measured devices. A roughly linear trend is generally observed, even for devices with twist-angle disorder.
Extended Data Fig. 5 Superfluid stiffness in NbN thin films.
(a) Optical micrograph of a NbN thin film device with a meandering Au strip. The Au strip provides an additional resistance of 2.9 kΩ to mimic the contact resistance of TTG devices. We measured NbN samples thickness of 5 nm (NbN-5nm) and 8 nm (NbN-8nm) (b) Normalized superfluid stiffness ρs(T)/ρs0 as a function of temperature T normalized with respect to the superconducting transition temperature Tc for TTG and NbN devices. TTG shows monotonically rising stiffness as the temperature is lowered, with a linear behavior for T/Tc≤0.3. On the other hand, the NbN devices show a robust saturation of the stiffness for T/Tc≤0.3. The NbN stiffness data obeys the BCS expectation. The BCS fit is given by \({\rho }_{s}(T)/{\rho }_{s0}=\Delta (T)/{\Delta }_{0}\tanh (\Delta (T)/T)\) where \(\Delta (T)={\Delta }_{0}\tanh ({\rm{\pi }}{T}_{c}/{\Delta }_{0}\sqrt{{T}_{c}/T-1}),{\Delta }_{0}=1.76{k}_{B}{T}_{c}\).
Supplementary information
Supplementary Information
The Supplementary Information file contains Supplementary Notes 1–6 and Figs. 1–9.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Banerjee, A., Hao, Z., Kreidel, M. et al. Superfluid stiffness of twisted trilayer graphene superconductors. Nature 638, 93–98 (2025). https://doi.org/10.1038/s41586-024-08444-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41586-024-08444-3
This article is cited by
-
Superfluid stiffness of magic-angle twisted bilayer graphene
Nature (2025)
-
Quasiparticle and superfluid dynamics in Magic-Angle Graphene
Nature Communications (2025)
-
Superconducting magic-angle twisted trilayer graphene with competing magnetic order and moiré inhomogeneities
Nature Materials (2025)
-
High-quality-factor viscoelastic nanomechanical resonators from moiré superlattices
Nature Communications (2025)
-
Novel insights into the field-flow fractionation characterization and their electrical properties of graphene oxide and reduced graphene oxide materials
Emergent Materials (2025)
