Distinguishing trivial and topological zero-energy states in long nanowire junctions

Letter
Open Access

Distinguishing trivial and topological zero-energy states in long nanowire junctions

Jorge Cayao and Annica M. Black-Schaffer

Phys. Rev. B 104, L020501 – Published 1 July 2021

Abstract

The emergence of zero-energy states in nontopological superconductors represents an inevitable problem that obscures the proper identification of zero-energy Majorana bound states (MBSs) and prevents their use as topologically protected qubits. In this Research Letter we investigate long superconductor-normal-superconductor junctions where trivial zero-energy states, robust over a large range of parameters, appear as a result of helicity and confinement in the normal region. We demonstrate that both equilibrium supercurrents and critical currents are sensitive to variations in the length of the superconductor regions in the topological phase hosting MBSs but, remarkably, no such length dependence exists when robust, but trivial, zero-energy states are present. This strikingly different response originates from the nonlocal nature of the MBSs, and we therefore propose it as a simple protocol for distinguishing between trivial and topological zero-energy states.

Received 24 November 2020
Revised 17 June 2021
Accepted 22 June 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L020501

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. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Andreev reflection Ballistic transport Josephson effect Majorana bound states Majorana fermions Proximity effect Quantum transport Rashba coupling Spin-singlet pairing Spin-triplet pairing Superconductivity Topological phases of matter Topological superconductors

Josephson junctions Low-temperature superconductors Nanowires

Bogoliubov-de Gennes equations Finite-element method Methods in superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jorge Cayao and Annica M. Black-Schaffer

Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden

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Issue

Vol. 104, Iss. 2 — 1 July 2021

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Images

Figure 1
(a) Semiconducting nanowire with SOC covered by $s$ -wave superconductors (S, blue) leaving an uncovered central region in the normal (N, yellow) state. S regions are connected by a grounded superconducting loop allowing control of the phase difference $ϕ$ via the flux $Φ$ . Gate voltages individually control the chemical potential in each region. (b) Energy dispersion in the normal state with an applied perpendicular magnetic field producing a gap of size $2 B$ (yellow) and centered around $μ_{N} = 0$ . Inside this gap the system exhibits only two Fermi points, representing counterpropagating states with different spins (solid and open green circles), and this is thus a helical gap. For $μ$ above or below this gap there are four Fermi points (pink circles). Visibility of the helical gap is lost when $B ≫ E_{SOC}$ , with $E_{SOC}$ being the SOC energy.
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Figure 2
(a) and (b) Low-energy spectrum in a long SNS junction as a function of Zeeman field $B$ at $ϕ = 0$ for $L_{S} = 500$ nm (a) and $L_{S} = 2000$ nm (b). Vertical green and red dashed lines mark when the N region becomes helical (yellow region), $B = μ_{N}$ , and the S regions become topological, $B = B_{c}$ , respectively. The orange curve corresponds to the minimum S bulk energy gap $Δ_{bulk}$ . (c) and (d) Phase-dependent low-energy spectrum taken at $B$ in the helical and topological regimes [vertical gray dashed lines in (a) and (b)] for both junctions with the lowest four levels color coded and the thick orange line indicating $Δ_{bulk}$ . (e) and (f) Spatial dependence of the wave-function amplitude ${| Ψ |}^{2}$ for the lowest four states [color coded in (c) and (d)] at $ϕ = π$ at $B = 0$ (black curve), in the helical phase (yellow curve), and in the topological phase (red curve). Arrows indicate the outer (white arrows) and inner (red arrows) MBSs. Here, $μ_{S} = 0.5 meV, μ_{N} = 0.25 meV$ .
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Figure 3
(a) Current-phase relationship for a long SNS junction in $B = 0$ (black curves), helical (orange curves), and topological (red curves) phases and for different S region lengths $L_{S}$ . Here, $I_{0} = (e Δ / ℏ)$ . (b) Individual contributions to $I (ϕ)$ from the first level $ɛ_{1}$ , the second level $ɛ_{2}$ , additional levels below $Δ_{bulk}$ ( $ɛ_{add}$ ), and the quasicontinuum (Quasi.; levels above $Δ_{bulk}$ ) for $L_{S} = 2000$ nm. (c) and (d) Critical current as a function of the Zeeman field $B$ for the same long junction as in (a) for different $L_{S}$ (c) and different $μ_{N}$ for $L_{S} = 500$ nm (d). Curves in (d) are offset by a constant value for visualization. Vertical gray dashed lines in (d) mark the helical phase transition. The rest of the parameters are as in Fig. 2.
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