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Sublattice-sensitive Majorana modes

Di Zhu, Bo-Xuan Li, and Zhongbo Yan
Phys. Rev. B 106, 245418 – Published 15 December 2022
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  • INTRODUCTION
  • GENERIC THEORY FROM A BOUNDARY…
  • MAJORANA KRAMERS PAIRS AT THE SUBLATTICE…
  • DISCUSSION AND CONCLUSION
  • ACKNOWLEDGMENTS
    • Supplemental Material
    References

    Abstract

    For two- and three-dimensional topological insulators whose unit cells consist of multiple sublattices, the boundary terminating at which type of sublattice can affect the time-reversal invariant momentum at which the Dirac points of helical boundary states are located. By incorporating a generic theory and a representative model, we reveal that this interesting property allows the realization of Majorana modes at sublattice domain walls forming on the boundary when the boundary Dirac points of the topological insulator are gapped by appropriate superconductivity in proximity. Remarkably, we find that the sensitive sublattice dependence of the Majorana modes allows their positions to be precisely manipulated by locally controlling the terminating sublattices or boundary potential. Our work reveals that the sublattice degrees of freedom commonly found in materials open a different route to realize and manipulate Majorana modes.

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    • Received 8 December 2021
    • Revised 21 August 2022
    • Accepted 5 December 2022

    DOI:https://doi.org/10.1103/PhysRevB.106.245418

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Proximity effectTopological insulatorsTopological superconductors
    1. Physical Systems
    Honeycomb lattice
    1. Techniques
    Bogoliubov-de Gennes equationsTight-binding modelk dot p method
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Di Zhu1,*, Bo-Xuan Li2,3,*, and Zhongbo Yan1,†

    • 1Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
    • 2Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
    • 3University of Chinese Academy of Sciences, Beijing 100049, China

    • *These authors contributed equally to this work.
    • †yanzhb5@mail.sysu.edu.cn

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    Issue

    Vol. 106, Iss. 24 — 15 December 2022

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    Images

    • Figure 1
      Figure 1

      Sensitive dependence of boundary Dirac points on the terminating sublattice type. (a) The upper and lower zigzag edges of the lattice respectively terminate with sublattice B (red dots) and A (blue dots). (b) The lower edge keeps to be the same as in (a), but the upper edge changes to be a beard type, with the terminating sublattice type changing from B to A. Panels (c) and (d) show the corresponding normal-state energy spectra when the y-normal open boundaries follow the structures shown in (a) and (b), respectively. In (c) and (d), periodic boundary conditions are imposed in the x direction and parameters are t=1 and λso=0.1.

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    • Figure 2
      Figure 2

      Energy spectrum of the BdG Hamiltonian for a cylindrical geometry with open (periodic) boundary conditions in the y(x) direction. The upper (lower) edge in the y direction is chosen to be the beard (zigzag) type. In (a)–(d), t=1, λso=0.1, μ=0, and pairing amplitudes are as follows: (a) Δ0=0.2, Δ1=Δ2=0; (b) Δ1=0.2, Δ0=Δ2=0; (c) Δ2=0.2, Δ0=Δ1=0; (d) Δ0=0.1, Δ1=0, Δ2=0.2.

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    • Figure 3
      Figure 3

      Majorana Kramers pairs bounded at sublattice domain walls. Chosen parameters are t=1, λso=0.1, μ=0, Δ0=Δ2=0.3, and Δ1=0. With periodic boundary conditions in the x direction except for the uppermost beard-type part, the two insets in (a) and (b) show the corresponding energy spectra. The four dots highlighted by red indicate the existence of two Majorana Kramers pairs. The shade of the red color on the lattice sites reflects the weight of the probability density of Majorana Kramers pairs.

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