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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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## 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.

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## 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.

## Author information

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.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## 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

• Accepted:

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• DOI: https://doi.org/10.1038/s41567-020-01142-7

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