Magnetic flux induced topological superconductivity in magnetic atomic rings

Jinpeng Xiao, Qianglin Hu, and Xiaobing Luo

Phys. Rev. B 109, 205420 – Published 16 May 2024

Abstract

There have been numerous studies on topological superconductivity in magnetic atomic chains deposited on $s$ -wave superconductors. Most of these investigations have focused on spin-orbit interactions or helical spin orders. In this paper, we propose a model for achieving one-dimensional topological superconductivity in a magnetic atomic ring. This model utilizes a magnetic field and an antiferromagnetic/ferromagnetic order, under the condition that the magnetic field is perpendicular to the moments of the magnetic order. On a quasi-one-dimensional substrate surface, where the half-filled ring favors an antiferromagnetic configuration, we demonstrate that either the magnetic field itself or a Rashba spin-orbit coupling guarantees the perpendicularity. On a two-dimensional surface, where the ring favors ferromagnetic orders, the perpendicularity is achieved by introducing a minor Rashba spin-orbit coupling.

Received 22 December 2023
Revised 7 May 2024
Accepted 8 May 2024

DOI:https://doi.org/10.1103/PhysRevB.109.205420

Physics Subject Headings (PhySH)

Edge states Superconductivity Topological materials

1-dimensional spin chains

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jinpeng Xiao ¹, Qianglin Hu², and Xiaobing Luo^3,*

¹School of Mathematics and Physics, Jinggangshan University, Ji'an 343009, People's Republic of China
²Department of Physics and Electronics Engineering, Tongren University, Tong Ren 554300, People's Republic of China
³Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, People's Republic of China

^*Corresponding author: xiaobingluo2013@aliyun.com

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Vol. 109, Iss. 20 — 15 May 2024

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Images

Figure 1
(a) A schematic representation of an AFM ring with a flux threading through it. (b), (c) The normal-state bands of the AFM ring when the chemical potential $μ = 0$ . In (b), the phase $ϕ (= ϕ_{0} / L$ , where $L$ is the number of sites in the ring) is 0, while in (c), $ϕ$ is $0.06 π$ . The magnetic Zeeman field $V$ splits the bands of different spins by an energy $δ V$ . The phase $ϕ$ induces a shift of the bands along the wave vector $k$ . (d) The bands of the Hamiltonian (2) with $μ = μ_{0} = 0.1$ , $Δ = 0.3$ , and $ϕ = 0.06 π$ . The other parameters in (b)–(d) are $J S = 0.2$ and $V = 0.4$ .
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Figure 2
(a) The open boundary spectrum of the AFM ring with 200 sites as a function of the superconducting pairing strength $Δ$ . (b)–(d) The topological phase diagrams for the ring. In (a)–(c), the phase $ϕ$ is set to $0.06 π$ . (a), (b), (d) $V = 0.4$ ; (a), (c), (d) $μ = 0.1$ , $J S = 0.2$ . The blue regions represent topologically nontrivial phases. The red dashed line in (c) marks the scanning trajectory of the open boundary spectrum shown in (a).
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Figure 3
(a) An AFM ring on an $s$ -wave superconductor canted under a magnetic field $V$ applied perpendicular to the ring plane. $φ$ represents a random angle. (b) The canted AFM moments, with the AFM components along the $x$ direction and the FM components along the $z$ direction. (c) The FM moments along the $z$ direction, twisted from an initial AFM order by a large $V$ . (d) The cosine of the canted angle as a function of $V$ for different SOC strengths with $J S = 0.2$ .
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Figure 4
Self-consistently solved magnetic orders and the corresponding zero modes in real space for a system with 160 sites. (a) The FM moment components induced by a magnetic field; (b)–(d) both the induced FM and the in-plane AFM components along the $x − y$ directions under different superconducting pairing strengths. The parameters are set as $α_{R} = 0.05$ , $J S = 0.2$ , $ϕ = 0.05 π$ , and $μ = 0$ . (e), (f) The open boundary energy levels of the ring and LDOS of one zero mode, respectively. The magnetic configuration is selected from (c) and marked by open circles for $Δ = 0.35$ and $V = 0.4$ . The “N” in (e) denotes the energy-level index.
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Figure 5
(a)–(d) The topologically nontrivial phases (blue regions) in the parameter space with different patch angles $φ$ . (e) The open boundary energy spectrum with the scanning trajectory denoted by a red dashed line in (b). (f) The LDOS of the zero mode marked in (e) by the open circle. In (a)–(f), $α_{R} = 0$ , $ϕ = 0.04 π$ , and $Δ = 0.5$ . Other parameters in (a)–(d) are $φ = 0$ , $0.01 π$ , $0.1 π$ , and $0.2 π$ , respectively. In (e) and (f), $φ = 0.01 π$ , $J S = 0.2$ , and in (f), $V = 0.4$ .
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