Topological superconductivity in planar Josephson junctions: Narrowing down to the nanowire limit

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Topological superconductivity in planar Josephson junctions: Narrowing down to the nanowire limit

F. Setiawan, Ady Stern, and Erez Berg

Phys. Rev. B 99, 220506(R) – Published 25 June 2019

Abstract

We theoretically study topological planar Josephson junctions (JJs) formed from spin-orbit-coupled two-dimensional electron gases (2DEGs) proximitized by two superconductors and subjected to an in-plane magnetic field $B_{∥}$ . Compared to previous studies of topological superconductivity in these junctions, here we consider the case where the superconducting leads are narrower than the superconducting coherence length. In this limit the system may be viewed as a proximitized multiband wire, with an additional knob being the phase difference $ϕ$ between the superconducting leads. A combination of mirror and time-reversal symmetry may put the system into the class BDI. Breaking this symmetry changes the symmetry class to class D. The class D phase diagram depends strongly on $B_{∥}$ and chemical potential, with a weaker dependence on $ϕ$ for JJs with narrower superconducting leads. In contrast, the BDI phase diagram depends strongly on both $B_{∥}$ and $ϕ$ . Interestingly, the BDI phase diagram has a “fan”-shaped region with phase boundaries which move away from $ϕ = π$ linearly with $B_{∥}$ . The number of distinct phases in the fan increases with increasing chemical potential. We study the dependence of the JJ's critical current on $B_{∥}$ , and find that minima in the critical current indicate first-order phase transitions in the junction only when the spin-orbit coupling strength is small. In contrast to the case of a JJ with wide leads, in the narrow case these transitions are not accompanied by a change in the JJ's topological index. Our results, calculated using realistic experimental parameters, provide guidelines for present and future searches for topological superconductivity in JJs with narrow leads, and are particularly relevant to recent experiments on InAs 2DEGs proximitized by narrow Al superconducting leads [A. Fornieri et al., Nature (London) 569, 89 (2019)].

Received 3 March 2019

DOI:https://doi.org/10.1103/PhysRevB.99.220506

Physics Subject Headings (PhySH)

Majorana bound states Proximity effect Topological materials Topological superconductors

Josephson junctions Nanowires

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

F. Setiawan ^1,*, Ady Stern², and Erez Berg^1,2

¹The James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
²Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

^*setiawan@uchicago.edu

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Vol. 99, Iss. 22 — 1 June 2019

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Images

Figure 1
(a) A JJ made of two narrow SC leads in contact with a 2DEG. By applying an in-plane Zeeman field $E_{Z, | |} = g μ_{B} B_{| |} / 2$ parallel to the JJ and a superconducting phase difference $ϕ$ , the system can be tuned into a TSC supporting MZMs $γ$ . (b) Class D and (c) class BDI phase diagrams as functions of $E_{Z, | |}$ and $ϕ$ . Regions with odd and even $Q_{Z}$ topological index in the class BDI [(c)] correspond respectively to the topological $(Q_{Z_{2}} = - 1)$ and trivial $(Q_{Z_{2}} = 1)$ regions of the class D [(b)]. The phase diagrams are obtained from numerical simulations performed using the kwant package [60] of a tight-binding version of Eq. (1) (see Sec. I of Ref. [61]). The parameters used correspond to the experimental parameters of recent experiments on InAs 2DEGs [50], i.e., $m^{*} = 0.026 m_{e}, α = 0.1$ eV Å, $μ = 0.6$ meV, $Δ = 0.15$ meV $[ξ = ℏ v_{F} / (π Δ) = 126$ nm], $W = 80$ nm, and $W_{SC} = 160$ nm.
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Figure 2
BDI phase diagrams as functions of Zeeman field $E_{Z, | |}$ and superconducting phase difference $ϕ$ for different chemical potentials: (a) $μ = 0.6$ meV, (b) $μ = 4.4$ meV, and (c) $μ = 8.4$ meV. The phase boundary lines inside the fan-shaped region emanate from $ϕ = π$ and $E_{Z, | |} = 0$ with slopes which are linearly proportional to $E_{Z, | |}$ and decrease with increasing $μ$ . The parameters used are the same as in Fig. 1.
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Figure 3
(a) Fermi surfaces of the 2DEG. An applied magnetic field along $x$ shifts the two spin-orbit split Fermi surfaces (labeled by $η_{1} = \pm$ for the outer and inner Fermi surface) oppositely along $k_{y}$ . The arrows show the spin orientation on the Fermi surfaces. The Zeeman field tilts the spin-orientation angle towards its direction. (b) Energy spectrum of an infinitely long 2DEG with a finite width. Each $n th$ band consists of two subbands labeled by $m = \pm$ , denoting the eigenvalues of the mirror operator $M_{y}$ . We label the gap $Δ_{n}^{m}$ by the band index $n$ and the mirror eigenvalue $m = \pm 1$ of the right-moving state $k_{F_{n}} > 0$ .
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Figure 4
Upper panel: Critical current $I_{c}$ as a function of Zeeman energy $E_{Z, | |}$ for different SOC strengths: (a) $α = 0.1$ eV Å and (b) $α = 1$ eV Å. Lower panel: Josephson current as a function of $ϕ$ and $E_{Z, | |}$ for (c) $α = 0.1$ eV Å and (d) $α = 1$ eV Å. Here, $μ$ is the same as in Fig. 2, and the temperature is $T = 0.3 Δ / k_{B}$ . For $α = 0.1$ eV Å, $I_{c}$ exhibits a minimum at $E_{Z, | |}^{c} \approx 0.55$ meV. The minimum of the critical current does not coincide with the class D TPT, which occurs at $E_{Z, ∥} \approx 0.27$ meV [see Fig. 2]. As $α$ increases, the minimum becomes more shallow [(b)]. For small $α$ , there is an abrupt shift by nearly $π$ in the current-phase relation $I (ϕ)$ at $E_{Z, | |}^{c}$ , while for large $α, I (ϕ)$ has a gradual phase shift with $E_{Z, | |}$ [see (b) and (d), respectively].
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