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Intrinsic chiral topological superconductor thin films

Xi Luo, Yu-Ge Chen, Ziqiang Wang, and Yue Yu
Phys. Rev. B 108, 235147 – Published 14 December 2023
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  • INTRODUCTION
  • MODEL AND SYMMETRY ANALYSIS
  • TOPOLOGICAL PHASE DIAGRAMS
  • EFFECTS OF ZEEMAN FIELD
  • MIRROR-SYMMETRY BREAKING EFFECTS
  • 3D LATTICE MODEL AND QUASI-2D χTSC THIN…
  • DETECTING HALF-INTEGER QUANTIZED…
  • DISCUSSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Superconductors (SCs) with nontrivial topological band structures in the normal state have been discovered recently in bulk materials. When such SCs are made into thin films, quantum tunneling and Cooper pairing take place between the topological surface states (TSSs) on the opposing surfaces. Here, we find that chiral topological superconductivity with spontaneous time-reversal symmetry breaking emerges on the surface of such thin film SCs. There is a mirror symmetry that protects a novel nonunitary orbital and spin triplet pairing of the TSS. In the mirror diagonal space, the chiral topological SC manifests as two independent chiral p-wave spin-triplet pairing states, in which each is a two-dimensional superconducting analog of the Anderson-Brinkman-Morel state in superfluid He3 with in-plane exchange fields. A rich topological phase diagram governed by the nontrivial Z⊕Z topological invariant is obtained with gapless chiral Majorana edge modes and anyonic Majorana vortices. We further construct a three-dimensional lattice model with a topological band structure and SC pairings, which is motivated by Fe-based SCs such as Fe(Te,Se). We demonstrate the realization of the proposed intrinsic chiral topological superconductor in the quasi-two-dimensional thin-film limit. Our findings enable thin-film SCs with nontrivial Z2 band structures as a single-material platform for intrinsic chiral topological superconductivity with both vortex and boundary Majorana excitations for topological quantum device making.

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    • Received 21 September 2023
    • Accepted 1 December 2023

    DOI:https://doi.org/10.1103/PhysRevB.108.235147

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    SuperconductivityTopological materialsTopological superconductors
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Xi Luo1,*, Yu-Ge Chen2,*, Ziqiang Wang3,†, and Yue Yu4,5,6,‡

    • 1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
    • 2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
    • 3Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
    • 4State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, People's Republic of China
    • 5Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, People's Republic of China
    • 6Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, People's Republic of China

    • *These authors contributed equally to this work.
    • †Corresponding author: wangzi@bc.edu
    • ‡Corresponding author: yuyue@fudan.edu.cn

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    Issue

    Vol. 108, Iss. 23 — 15 December 2023

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    Images

    • Figure 1
      Figure 1

      Schematics of a χTSC thin film with coupled superconducting TSS. The mirror plane is outlined by the brown dashed line with respect to the TSS on the top and bottom surfaces labeled by 1 and 2. The red (blue) balls represent the Dirac fermions localized in the top (bottom) surface, and the arrows represent spin. The pairing terms preserving the mirror symmetry M are the spin-singlet pairing on the same surface Δ (purple dashed line) and the spin-triplet pairing across the surfaces Δt (green dashed line). The quantum tunneling t of the electrons between the surfaces also preserves the mirror symmetry and participates in realizing the χTSC thin film.

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    • Figure 2
      Figure 2

      Energy spectrum of the thin-film SC described by the BdG Hamiltonian given by Eq. (7), plotted in the qy=0 direction for Δ=1 and μ=2. The spin-triplet, orbital-singlet pairing (a) Δt=0 and (b)–(e) Δt=1.5. The interorbital tunneling evolves as (a) t=0.5, (b) t=2.06, (c) t=2.5, (d) t=3.2, and (e) t=4. In (b)–(e), the blue (red) curves correspond to the spectra of HD+ (HD−). (f) Energy spectrum of the lattice model with the same parameters as in (c) under the open boundary condition in the y direction. The red and blue curves are the gapless states of a single χMEM localized at the two edges.

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    • Figure 3
      Figure 3

      Topological phase diagrams of the thin-film SC in the Δ−μ plane for (a) t=0.5, Δt=1, (b) t=2.5, Δt=1.5, and (c) t=1.5, Δt=0. The χTSC phases are labeled by the total Chern number N≠0 originating from HD+(HD−) in regions bounded by the blue (red) curves. For comparison, the energy spectrum shown in Fig. 2 corresponds to the (Δ=1, μ=2) point in (b). The red overlapping region in (b) is a topological nontrivial phase with N=0 and mirror Chern number C=1, while the green phase region in (c) is characterized by Z2 due to the restored TRS. (d) The energy spectrum of the lattice model with open boundaries in the y direction, showing the gapless helical edge modes. The results are obtained at the Δ=1, μ=0 point in (c).

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    • Figure 4
      Figure 4

      Topological phase diagrams in the Δt−λ plane for Δ=1, μ=0, and (a) t=0.5 and (b) t=1.5. The χTSC phases are labeled by the total Chern number N≠0 originating from HD+(HD−) in the regions bounded by the blue (red) curves. The rectangle enclosing the origin in (b) is a mirror TSC with N=0 and C=1.

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    • Figure 5
      Figure 5

      Topological phase diagrams in the presence of (a),(b) mirror-symmetry breaking pairing interaction Δs and (c),(d) substrate potential V. The parameters are (a),(c) Δ=1, μ=0, t=0.5 and (b),(d) t=1.5. The χTSC phases bounded by the blue (red) curves are adiabatically connected to those of HD+(HD−) at V=0 and Δs=0. The pink lines in (b) at V=0 and (d) at Δs=0 correspond to the mirror topological SC with N=0 and C=1 shown in Fig. 3.

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    • Figure 6
      Figure 6

      The band structure of the lattice model for the 3D TI with open boundaries along the z direction plotted along the qx direction at qy=0. The parameters are (v,m0,1,2)=(1.0,−8.5,−3.0,3.0), and V=0.5 at the top and bottom surfaces. (a) The band dispersion in the bulk limit obtained with nz=50 layers, showing the TSS split by the SIA potential V into two Rashba Dirac cones (red and blue lines). (b) The spatial distribution of the in-gap states at the 2D Γ point along the z direction, which are independently localized on the top and bottom surfaces. (c) The thin-film limit band dispersion obtained with nz=4 layers. An energy gap opens due to the coupling between the top and bottom weakly split Rashba Dirac cones with the corresponding spatial distribution of the states at the 2D Γ point shown in (d).

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    • Figure 7
      Figure 7

      Energy spectra of BdG quasiparticles in the SC state of the 3D lattice model with the same band parameters as in Fig. (6). Open boundary conditions are applied in the z and y directions for nz number of layers of width Ly. The x direction is periodic with continuous momentum qx. The pairing parameters are Δl=1.3, Δsl=1.5, and Δtl=1.5. (a) The spectrum for nz=20 and Ly=20 is fully gapped, consistent with a topologically trivial bulk superconductor. (b) The spectrum for nz=4 layers and Ly=40 shows a SC gap with in-gap states, realizing a thin-film χTSC with χMEMs. (c) Zoom-in of (b). The blue and red curves represent the χMEMs. (d) The spatial distribution of the layer-averaged wave functions of the χMEMs at an excitation energy E=0.05 in (c), showing localization at the two edges in the y direction.

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    • Figure 8
      Figure 8

      (a) Schematics of the device for detecting the half-quantized conductance of a single χMEM at the boundaries of the χTSC. The red, blue, brown, and orange layers are the antidot, χTSC, dielectric substrate, and back-gate antidot, respectively. The arrows indicate the chiral χMEM. (b) The top view of the device in (a). An external magnetic field B is applied. V1 and V2 are the voltages applied to leads 1 and 2. (c) The spatial distributions of |u| and |v| for the zero-energy bound state. The red curves are for |u| and |v| when B=0; both are localized at the edges. For B/μ̃=0.5 and in the Landau gauge, |u| (blue) is localized and |v| (green) merges into the bulk at one edge, while |v| is localized and |u| merges into the bulk at the opposite edge.

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    • Figure 9
      Figure 9

      (a),(c) Energy spectrum of the lattice model under open boundary conditions in the y direction for Δ=1. (a) μ=2, t0=2.5, t1=1.0, Δt=1.5; (c) μ=0, t0=1.5, t1=−0.5; and Δt=0. (b),(d) Spatial distributions of the gapless edge states in (a) and (c) along the y direction (50 sites) at a fixed energy Ea=0.37 and Ec=0.48, respectively. The red and blue curves in (b) are the gapless chiral edge states of a single χMEM. In (d), there are two counterpropagating χMEMs, namely, a helical Majorana edge mode at each boundary. They are represented by the solid and dotted lines.

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