Dimensional tuning of Majorana fermions and real space counting of the Chern number

Eric Mascot, Sagen Cocklin, Stephan Rachel, and Dirk K. Morr

Phys. Rev. B 100, 184510 – Published 21 November 2019

Abstract

We demonstrate that it is possible to use atomic manipulation techniques to adiabatically tune between one-dimensional and two-dimensional topological superconductors in magnet-superconductor hybrid (MSH) structures. This allows one to change the nature of the associated Majorana fermions between localized bound states and chiral Majorana edge modes, and provides a new approach for counting the topological invariant of the system – the Chern number – in real space. Moreover, we show that the topological nature of finite-size MSH structures can be characterized using the Chern number density which adiabatically connects MSH structures to macroscopic, translationally invariant topological superconductors characterized by an integer Chern number. These results open new possibilities for studying the nature of topological superconductivity in MSH structures using atomic manipulation techniques.

Received 13 February 2019
Revised 5 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.184510

Physics Subject Headings (PhySH)

Majorana fermions

Interfaces Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Eric Mascot¹, Sagen Cocklin¹, Stephan Rachel², and Dirk K. Morr ¹

¹Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA
²School of Physics, University of Melbourne, Parkville, VIC 3010, Australia

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Vol. 100, Iss. 18 — 1 November 2019

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Images

Figure 1
MSH structure consisting of a Shiba island of radius $R = 15 a_{0}$ and chain with parameters $(μ, Δ, α, J) = (- 4 t, 1.2 t, 0.45 t, 2.6 t)$ , yielding a topological phase with $C = - 1$ for the island (black circles denote sites with magnetic adatoms; $a_{0}$ is the lattice constant). (a)–(c) LDOS of the lowest energy Majorana mode with increasing chain length, corresponding to arrows (1)–(3) in (e). (d) LDOS of the second lowest energy Majorana mode, corresponding to arrow (4) in (e). (e) Evolution of the four lowest energy levels with increasing chain length. Inset: log-plot of the lowest energy level with increasing chain length, red line represents a linear fit. (f) Evolution of the four lowest energy levels with decreasing island radius. (g) and (h) LDOS of the lowest energy Majorana mode with decreasing island radius, corresponding to arrows (5) and (6) in (f). The distance between the end of the chain and the center of the island is kept constant.
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Figure 2
(a) Line cut of the Chern density $C (r)$ through Shiba islands with different radii $R$ . Spatial plot of $C (r)$ for the MSH structures in Fig. 1 with chain length (b) $L = 0$ , (c) $L = 10 a_{0}$ , and (d) $L = 20 a_{0}$ . Parameters are the same as in Fig. 1 with $C = - 1$ .
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Figure 3
(a) LDOS of the lowest energy Majorana mode for a Shiba island with $C = - 1$ and two chains of length $L = 30 a_{0}$ attached (same parameters as in Fig. 1). (b)–(e) MSH structure with $C = 2$ and parameters $(μ, Δ, α, J) = (- 0.5 t, 0.7 t, 0.45 t, 2 t)$ . Lowest energy LDOS for a Shiba island with radius $R = 15 a_{0}$ and with (b) two, (c) three, and (d) four chains of length $L = 31 a_{0}$ attached. (e) The points where the chains are attached are rotated by $45^{\circ}$ from (d).
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Figure 4
Shiba island with radius $R = 20 a_{0}$ on a triangular lattice with parameters $(μ, Δ, α, J) = (0.4, 1.2, 0.45, 2.6) t$ yielding a topological phase with $C = 3$ . Zero-energy LDOS for the island with (a) one, (b) five, and (c) six chains of length $L = 20 a_{0}$ attached. Zero-energy LDOS for the island with (d) the ends of two neighboring chains connected, giving rise to a windmill-like structure and junctions with an even number of chains (even junction), (e) the ends of neighboring chains connected, giving rise to a spider weblike structure and junctions with an odd number of chains (odd junction), and (f) the ends of two opposite chains connected with the ends of their two neighboring chains giving rise to a TIE fighterlike structure with both even and odd junctions.
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Figure 5
Experimentally realized $Fe / Re − O (2 \times 1)$ MSH structure (see Ref. [7]) with $C = 4$ . LDOS of zero-energy Majorana modes with (a) no attached chain, (b) one, (c) seven, and (d) eight attached chains. Hamiltonian and parameters for this ten-band model are given in the supplemental material of Ref. [7] and the Appendix pp1.
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