Robust Majorana bound states in magnetic topological insulator nanoribbons with fragile chiral edge channels

Open Access

Robust Majorana bound states in magnetic topological insulator nanoribbons with fragile chiral edge channels

Declan Burke, Dennis Heffels, Kristof Moors, Peter Schüffelgen, Detlev Grützmacher, and Malcolm R. Connolly

Phys. Rev. B 109, 045138 – Published 22 January 2024

Abstract

Magnetic topological insulators in the quantum anomalous Hall regime host ballistic chiral edge channels. When proximitized by an $s$ -wave superconductor, these edge states offer the potential for realizing topological superconductivity and Majorana bound states without the detrimental effect of large externally applied magnetic fields on superconductivity. Realizing well-separated unpaired Majorana bound states requires magnetic topological insulator ribbons with a width of the order of the transverse extent of the edge state, however, which is expected to bring the required ribbon width down to around $100 nm$ . In this regime, it is known to be extremely difficult to retain the ballistic nature of chiral edge channels and realize a quantized Hall conductance. In this paper, we study the impact of disorder in such magnetic topological insulator nanoribbons and compare the fragility of ballistic chiral edge channels with the stability of Majorana bound states when the ribbon is covered by a superconducting film. We find that the Majorana bound states exhibit greater robustness against disorder than the underlying chiral edge channels.

Received 4 April 2023
Revised 24 October 2023
Accepted 27 November 2023

DOI:https://doi.org/10.1103/PhysRevB.109.045138

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Majorana bound states Quantum anomalous Hall effect Topological materials

Nanostructures Topological superconductors

S-matrix method in transport Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Declan Burke¹, Dennis Heffels^2,3, Kristof Moors^2,3,*, Peter Schüffelgen^2,3, Detlev Grützmacher^2,3,4, and Malcolm R. Connolly^1,†

¹Blackett Laboratory, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
²Peter Grünberg Institute 9, Forschungszentrum Jülich, 52425 Jülich, Germany
³JARA-Fundamentals of Future Information Technology, Jülich-Aachen Research Alliance, Forschungszentrum Jülich and RWTH Aachen University, 52425 Jülich, Germany
⁴JARA-FIT Institute: Green IT, Jülich-Aachen Research Alliance, Forschungszentrum Jülich and RWTH Aachen University, 52425 Jülich, Germany

^*k.moors@fz-juelich.de
^†m.connolly@imperial.ac.uk

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Issue

Vol. 109, Iss. 4 — 15 January 2024

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Images

Figure 1
(a) Schematic of the MTINR setup: (top) proximitized MTINR with MBSs at opposite ends (MBS wave function density shown above); (middle) a spatially correlated electrostatic disorder profile; (bottom) interedge scattering of CECs. (b) The spectral gap in a proximitized $40 − n m$ -wide MTINR as a function of magnetization $M_{z}$ and chemical potential $μ$ , with trivial (nontrivial) regions indicated by a gray (red) color scale, delineated with green boundaries. (c) The spectral gap and chemical potential window of the proximitized CEC phase as a function of magnetization $M_{z}$ for different widths of the MTINR. (d) A Gaussian distribution for fluctuations of the MTINR spectrum due to electrostatic disorder, with relevant energy scales (proximity-induced spectral gap $E_{gap}$ and extent of the CEC phase $Δ μ_{CEC}$ ) indicated. (e) Distributions of normal-state and zero-bias tunneling conductance over an ensemble of disordered MTINRs.
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Figure 2
(a), (b) The normal-state two-terminal conductance (top) mean and (bottom) standard deviation of 200 MTINRs with $M_{z} = 15 meV$ in the single-channel regime as a function of (a) width and (b) length, considering disorder strength $S_{dis} = 3 m e V$ and different spatial correlation lengths. (c), (d) The local current density along the transport direction of a forward-propagating CEC in a $40 − n m$ -wide and $1 − µ m$ -long MTINR with the same disorder statistics and averaging as in (a) and (b) is shown as a function of (c) the coordinate along the MTINR (integrated over the transverse coordinate $y$ [see Fig. 1] and of (d) the transverse coordinate (integrated over the transport direction).
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Figure 3
(a), (b) The tunneling conductance (top) mean and (bottom) standard deviation as a function of (a) width and (b) length of 200 proximitized MTINRs with identical magnetization, dimensions, and disorder statistics as considered in Fig. 2. (c) The tunneling conductance $G_{T}$ of a pristine MTINR as a function of ribbon length $L$ and width $W$ . (d) The MBS wave function density of $1 − µ m$ -long proximitized MTINRs with different widths [indicated in (c)] in the pristine ( $S_{dis} = 0 m e V$ ) and disordered ( $S_{dis} = 3 m e V$ ) case, with different spatial correlation lengths [legends in (a) and (b)].
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Figure 4
(a) The normal-state and tunneling conductance (top) mean and (bottom) standard deviation of 100 MTINRs with $M_{z} = 30 m e V, W = 30 n m$ , (A) $L = 4.5 µ m$ and $| Δ | = 5 meV$ , or (B) $L = 15 µ m$ and $| Δ | = 1.5 m e V$ , as a function of disorder strength with $λ = 35 n m$ . (b) The (top) local extent of the CEC phase around the chemical potential (horizontal dashed line) and (bottom) wave-function density of the two lowest-energy solutions along the length of a MTINR with a chemical potential in the single-channel regime [similar to the horizontal linecut in Fig. 1], considering different disorder strengths [indicated in (a)]. The wave-function density is presented relative to the maximum density of the solution of a pristine MTINR (not shown, but nearly indistinguishable from the density with $S_{dis} = 5 meV$ ).
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Figure 5
(a) The tunneling conductance shown in Fig. 3, with a contour line of constant tunneling conductance $G_{T} = 1.95 e^{2} / h$ . (b) The contour line (shown as dots) and the exponential fitting function, which is extrapolated to larger ribbon widths and lengths.
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