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High-temperature topological superconductivity in twisted double-layer copper oxides

Abstract

Various phenomena occur when two-dimensional materials, such as graphene or transition metal dichalcogenides, are assembled into bilayers with a twist between the individual layers. As an application of this paradigm, we predict that structures composed of two-monolayer-thin d-wave superconductors with a twist angle form a robust, fully gapped topological phase with spontaneously broken time-reversal symmetry and protected chiral Majorana edge modes. These structures can be realized by mechanically exfoliating van der Waals-bonded high-critical-temperature copper oxide materials, such as Bi2Sr2CaCu2O8 + δ. Our symmetry arguments and detailed microscopic modelling suggest that this phase will form for a range of twist angles in the vicinity of 45°, and will set in at a temperature close to the bulk superconducting critical temperature of 90 K. Therefore, this platform may provide a long-sought realization of a true high-temperature topological superconductor.

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Fig. 1: Schematic view of two copper–oxygen square lattices with twist angle close to 45°.
Fig. 2: Twisted double-layer d-wave superconductor.
Fig. 3: Lattice model results.
Fig. 4: Crystal structure and results of the twisted double-layer DFT calculations.

Data availability

This manuscript contains no experimental data. All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The complete code used to obtain the results shown in Fig. 3 is available at https://github.com/ocanphys/tbcuprate/. The DFT results shown in Fig. 4 were obtained using VASP34,35. Input files are also available from GitHub.

References

  1. Yu, Y. High-temperature superconductivity in monolayer Bi2Sr2CaCu2O8 + δ. Nature 575, 156–163 (2019).

    Article  ADS  Google Scholar 

  2. Cao, Y. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  ADS  Google Scholar 

  3. Cao, Y. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  ADS  Google Scholar 

  4. Yankowitz, M. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    Article  ADS  Google Scholar 

  5. Sharpe, A. L. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    Article  ADS  Google Scholar 

  6. Lu, X. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    Article  ADS  Google Scholar 

  7. Wang, L. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    Article  Google Scholar 

  8. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    Article  ADS  Google Scholar 

  9. Kang, J. & Vafek, O. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).

    Google Scholar 

  10. Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X 8, 031089 (2018).

    Google Scholar 

  11. Xie, M. & MacDonald, A. H. Nature of the correlated insulator states in twisted bilayer graphene. Phys. Rev. Lett. 124, 097601 (2020).

    Article  ADS  Google Scholar 

  12. Koshino, M. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

    Google Scholar 

  13. Guinea, F. & Walet, N. R. Electrostatic effects, band distortions and superconductivity in twisted graphene bilayers. Proc. Natl Acad. Sci. USA 115, 13174–13179 (2018).

    Article  ADS  Google Scholar 

  14. Kang, J. & Vafek, O. Strong coupling phases of partially filled twisted bilayer graphene narrow bands. Phys. Rev. Lett. 122, 246401 (2019).

    Article  ADS  Google Scholar 

  15. Hejazi, K., Liu, C., Shapourian, H., Chen, X. & Balents, L. Multiple topological transitions in twisted bilayer graphene near the first magic angle. Phys. Rev. B 99, 035111 (2019).

    Article  ADS  Google Scholar 

  16. Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Preprint at https://arxiv.org/pdf/2008.08129.pdf (2020).

  17. Elliott, S. R. & Franz, M. Colloquium: majorana fermions in nuclear, particle and solid-state physics. Rev. Mod. Phys. 87, 137–163 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  18. Laughlin, R. B. Magnetic induction of \({d}_{{x}^{2}-{y}^{2}}+{{{id}}}_{{{xy}}}\) order in high-Tc superconductors. Phys. Rev. Lett. 80, 5188–5191 (1998).

    Article  ADS  Google Scholar 

  19. Franz, M. & Tešanović, Z. Self-consistent electronic structure of a \({d}_{{x}^{2}-{y}^{2}}\) and a \({d}_{{x}^{2}-{y}^{2}}\) vortex. Phys. Rev. Lett. 80, 4763–4766 (1998).

    Article  ADS  Google Scholar 

  20. Vishwanath, A. Quantized thermal Hall effect in the mixed state of d-wave superconductors. Phys. Rev. Lett. 87, 217004 (2001).

    Article  ADS  Google Scholar 

  21. Kuboki, K. & Sigrist, M. Proximity-induced time-reversal symmetry breaking at Josephson junctions between unconventional superconductors. J. Phys. Soc. Jpn 65, 361–364 (1996).

    Article  ADS  Google Scholar 

  22. Bille, A., Klemm, R. A. & Scharnberg, K. Models of c-axis twist Josephson tunneling. Phys. Rev. B 64, 174507 (2001).

    Article  ADS  Google Scholar 

  23. Yokoyama, T., Kawabata, S., Kato, T. & Tanaka, Y. Theory of macroscopic quantum tunneling in high-Tc c-axis Josephson junctions. Phys. Rev. B 76, 134501 (2007).

    Article  ADS  Google Scholar 

  24. Pathak, S., Shenoy, V. B. & Baskaran, G. Possible high-temperature superconducting state with a d + id pairing symmetry in doped graphene. Phys. Rev. B 81, 085431 (2010).

    Article  ADS  Google Scholar 

  25. Nandkishore, R., Levitov, L. S. & Chubukov, A. V. Chiral superconductivity from repulsive interactions in doped graphene. Nat. Phys. 8, 158–163 (2012).

    Article  Google Scholar 

  26. Rice, T. M. & Sigrist, M. Sr2RuO4: an electronic analogue of 3He? J. Phys. Condens. Matter 7, L643–L648 (1995).

    Article  ADS  Google Scholar 

  27. Ishida, K. Spin–triplet superconductivity in Sr2RuO4 identified by 17O Knight shift. Nature 396, 658–660 (1998).

    Article  ADS  Google Scholar 

  28. Kallin, C. & Berlinsky, A. J. Is Sr2RuO4 a chiral p-wave superconductor? J. Phys. Condens. Matter 21, 164210 (2009).

    Article  ADS  Google Scholar 

  29. Pustogow, A. Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance. Nature 574, 72–75 (2019).

    Article  ADS  Google Scholar 

  30. Andersen, O. K., Liechtenstein, A. I., Jepsen, O. & Paulsen, F. LDA energy bands, low-energy Hamiltonians, \(t^{\prime}\), t′′, t(k), and j. Preprint at https://arxiv.org/pdf/cond-mat/9509044.pdf (1995).

  31. Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).

    Article  ADS  Google Scholar 

  32. Fischer, O., Kugler, M., Maggio-Aprile, I., Berthod, C. & Renner, C. Scanning tunneling spectroscopy of high-temperature superconductors. Rev. Mod. Phys. 79, 353–419 (2007).

    Article  ADS  Google Scholar 

  33. Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).

    Article  ADS  Google Scholar 

  34. Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Phys. Rev. B 48, 13115–13118 (1993).

    Article  ADS  Google Scholar 

  35. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    Article  ADS  Google Scholar 

  36. Furness, J. W. An accurate first-principles treatment of doping-dependent electronic structure of high-temperature cuprate superconductors. Commun. Phys. 1, 11 (2018).

    Article  Google Scholar 

  37. Zhang, Y. Competing stripe and magnetic phases in the cuprates from first principles. Proc. Natl Acad. Sci. USA 117, 68–72 (2020).

    Article  ADS  Google Scholar 

  38. Markiewicz, R. S., Sahrakorpi, S., Lindroos, M., Lin, H. & Bansil, A. One-band tight-binding model parametrization of the high-Tc cuprates including the effect of kz dispersion. Phys. Rev. B 72, 054519 (2005).

    Article  ADS  Google Scholar 

  39. Li, Q., Tsay, Y. N., Suenaga, M., Gu, G. D. & Koshizuka, N. Superconducting coupling along the c-axis of [001] twist grain-boundaries in Bi2Sr2CaCu2O8 + δ bicrystals. Phys. C Supercond. 282-287, 1495–1496 (1997).

    Article  ADS  Google Scholar 

  40. Zhu, Y. et al. Isotropic Josephson tunneling in c-axis twist bicrystals of Bi2Sr2CaCu2O8 + δ. Preprint at https://arxiv.org/pdf/1903.07965.pdf (2019).

  41. Harrison, W. A. Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover Publications, 2012).

  42. Bansil, A., Lindroos, M., Sahrakorpi, S. & Markiewicz, R. S. Influence of the third dimension of quasi-two-dimensional cuprate superconductors on angle-resolved photoemission spectra. Phys. Rev. B 71, 012503 (2005).

    Article  ADS  Google Scholar 

  43. Slezak, J. A. Imaging the impact on cuprate superconductivity of varying the interatomic distances within individual crystal unit cells. Proc. Natl Acad. Sci. USA 105, 3203–3208 (2008).

    Article  ADS  Google Scholar 

  44. Coleman, P. Introduction to Many-Body Physics (Cambridge Univ. Press, 2015).

  45. Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

  46. Feng, D. L. Bilayer splitting in the electronic structure of heavily overdoped Bi2Sr2CaCu2O8 + δ. Phys. Rev. Lett. 86, 5550–5553 (2001).

    Article  ADS  Google Scholar 

  47. Yokoyama, T., Kawabata, S., Kato, T. & Tanaka, Y. Theory of macroscopic quantum tunneling in high-Tc c-axis Josephson junctions. Phys. Rev. B 76, 134501 (2007).

    Article  ADS  Google Scholar 

  48. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  ADS  Google Scholar 

  49. Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 27, 1787–1799 (2006).

    Article  Google Scholar 

Download references

Acknowledgements

We are indebted to D. A. Bonn, D. M. Broun, J. A. Folk, C. Kallin, C. Li, É. Lantagne-Hurtubise, S. Plugge, S. Sahoo, O. Vafek and Z. Ye for valuable discussions and correspondence. The work described here was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Research Chairs Program (A.D.) and the CIFAR Quantum Materials Program (A.D.). This research was undertaken thanks in part to funding from the Max Planck-UBC-UTokyo Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. O.C. is supported by an International Doctoral Fellowship from UBC.

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Authors and Affiliations

Authors

Contributions

O.C. and T.T. carried out the tight-binding model calculations and the corresponding data analysis. R.P.D. and I.E. performed the density functional theory computations and the stability analysis. M.F. designed the study and, together with A.D., was responsible for the supervision of the project and writing of the manuscript, with suggestions from all authors.

Corresponding author

Correspondence to Marcel Franz.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Ashvin Vishwanath and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Analysis of the energy integrands.

(a) Integrand fB(ξ) entering the expression Eq. (15) for the GL theory coefficient B. (b) Integrand fC(ξ) entering the expression for the GL coefficient C compared to fB(ξ)2 Note that the amplitude of the latter is scaled by a constant factor of 1/30 for easier comparison. The temperature is chosen such that kBT=0.2g but the qualitative properties of the integrands remain the same in a wide range of temperatures.

Extended Data Fig. 2 Geometry of the twisted square lattice bilayers.

Commensurate unit cells for bilayers with relative twists of (a) θ1,2 = 53.13°, (b) θ2,5 =  43.60° and (c) θ5,12 = 45.24°. Twisting procedure is illustrated in panel (a) where orange and blue arrows indicate integer valued vectors v=(m,n) and v’=(-m,n) introduced in the text. Dashed and solid arrows represent vectors v and v’ before and after the twist, respectively. Moire unit cell is shown by the shaded area.

Extended Data Fig. 3 Physics of the C=2 phase.

Left: Schematic picture for the normal state band structure of the continuum model defined in Eq. (10). Both electron and hole bands that enter the BdG Hamiltonian are shown while the Fermi level is indicated by a dashed red line. Panel (a) shows decoupled layers with doubly degenerate bands yielding a circular Fermi surface. In the presence of a d-wave order parameter we have eight Dirac cones. (b) Interlayer coupling g splits the bands and Fermi surface now consists of a pair of concentric circles. (c) The inner Fermi circle shrinks to a point and half of the Dirac cones disappear. (d) Four Dirac cones survive in the strong interlayer coupling (g > μ) case. e) Evolution of the density of states with temperature in the C=2 phase with parameters g0=40 meV and μ=-1.35t. Temperature T/Tc shown as color scale. The superconducting gap persists up until T=Tc.

Extended Data Fig. 4 The topological phase transition.

The topological phase transition between C=0 (μ < -1.34t) and C=4 (μ > -1.34t) phases is marked by a closing of the gap as the chemical potential is tuned while keeping the interlayer coupling fixed at g0=20 meV and θ2,5 = 43. 60.

Extended Data Fig. 5 Finite-temperature effects.

The phase diagram of the lattice model for θ2,5 = 43. 60 at T=0 and T=Tc/2 is shown in panels (a) and (b) respectively. Panel (c) shows the minimum gap as a function of temperature for g0=32 meV and μ=-1.35t, parameters indicated in panels (a) and (b) by a dashed circle.

Extended Data Fig. 6 Physics of systems with multiple CuO2 planes per monolayer.

a) The structure and notation used for the case with N=2 CuO2 planes. Panels b) and c) show phase diagrams of the lattice BdG model for bilayers with N=2, relevant to the Bi2212 crystal structure with the intra-bilayer coupling set to tz=40 meV.

Extended Data Fig. 7 Bilayer in density functional theory.

In panel (a), comparison of cohesive energy of twisted bilayer Bi2201, as computed in both the GGA and SCAN meta-GGA. Both exchange correlation potentials recover the same equilibrium spacing. (b) shows the X-point splitting, computed as in LDA described in the main text.

Extended Data Fig. 8 The Josephson effect.

(a) The current-phase relation in units of the critical current at twist θ = 41. 40 when K=0.125. (b) Twist angle dependence of the critical Josephson current normalized by the current in the untwisted case. While the two curves show similar behaviour close to θ = 00 and 900, a non-negative K results in a finite value of the current at θ = 450.

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Can, O., Tummuru, T., Day, R.P. et al. High-temperature topological superconductivity in twisted double-layer copper oxides. Nat. Phys. 17, 519–524 (2021). https://doi.org/10.1038/s41567-020-01142-7

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