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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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
Yu, Y. High-temperature superconductivity in monolayer Bi2Sr2CaCu2O8 + δ. Nature 575, 156–163 (2019).
Cao, Y. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
Cao, Y. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Yankowitz, M. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).
Sharpe, A. L. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).
Lu, X. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).
Wang, L. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).
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).
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).
Xie, M. & MacDonald, A. H. Nature of the correlated insulator states in twisted bilayer graphene. Phys. Rev. Lett. 124, 097601 (2020).
Koshino, M. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).
Guinea, F. & Walet, N. R. Electrostatic effects, band distortions and superconductivity in twisted graphene bilayers. Proc. Natl Acad. Sci. USA 115, 13174–13179 (2018).
Kang, J. & Vafek, O. Strong coupling phases of partially filled twisted bilayer graphene narrow bands. Phys. Rev. Lett. 122, 246401 (2019).
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).
Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Preprint at https://arxiv.org/pdf/2008.08129.pdf (2020).
Elliott, S. R. & Franz, M. Colloquium: majorana fermions in nuclear, particle and solid-state physics. Rev. Mod. Phys. 87, 137–163 (2015).
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).
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).
Vishwanath, A. Quantized thermal Hall effect in the mixed state of d-wave superconductors. Phys. Rev. Lett. 87, 217004 (2001).
Kuboki, K. & Sigrist, M. Proximity-induced time-reversal symmetry breaking at Josephson junctions between unconventional superconductors. J. Phys. Soc. Jpn 65, 361–364 (1996).
Bille, A., Klemm, R. A. & Scharnberg, K. Models of c-axis twist Josephson tunneling. Phys. Rev. B 64, 174507 (2001).
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).
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).
Nandkishore, R., Levitov, L. S. & Chubukov, A. V. Chiral superconductivity from repulsive interactions in doped graphene. Nat. Phys. 8, 158–163 (2012).
Rice, T. M. & Sigrist, M. Sr2RuO4: an electronic analogue of 3He? J. Phys. Condens. Matter 7, L643–L648 (1995).
Ishida, K. Spin–triplet superconductivity in Sr2RuO4 identified by 17O Knight shift. Nature 396, 658–660 (1998).
Kallin, C. & Berlinsky, A. J. Is Sr2RuO4 a chiral p-wave superconductor? J. Phys. Condens. Matter 21, 164210 (2009).
Pustogow, A. Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance. Nature 574, 72–75 (2019).
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).
Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).
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).
Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Phys. Rev. B 48, 13115–13118 (1993).
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).
Furness, J. W. An accurate first-principles treatment of doping-dependent electronic structure of high-temperature cuprate superconductors. Commun. Phys. 1, 11 (2018).
Zhang, Y. Competing stripe and magnetic phases in the cuprates from first principles. Proc. Natl Acad. Sci. USA 117, 68–72 (2020).
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).
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).
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).
Harrison, W. A. Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover Publications, 2012).
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).
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).
Coleman, P. Introduction to Many-Body Physics (Cambridge Univ. Press, 2015).
Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).
Feng, D. L. Bilayer splitting in the electronic structure of heavily overdoped Bi2Sr2CaCu2O8 + δ. Phys. Rev. Lett. 86, 5550–5553 (2001).
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).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 27, 1787–1799 (2006).
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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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.
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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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DOI: https://doi.org/10.1038/s41567-020-01142-7
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