Majorana bound states in a $d$-wave superconductor planar Josephson junction

Letter

Majorana bound states in a $d$ -wave superconductor planar Josephson junction

Hamed Vakili, Moaz Ali, Mohamed Elekhtiar, and Alexey A. Kovalev

Phys. Rev. B 108, L140506 – Published 24 October 2023

Abstract

We study phase-controlled planar Josephson junctions comprising a two-dimensional electron gas with strong spin-orbit coupling and $d$ -wave superconductors, which have an advantage of a high critical temperature. We show that a region between the two superconductors can be tuned into a topological state by the in-plane Zeeman field, and can host Majorana bound states. The phase diagram as a function of the Zeeman field, chemical potential, and the phase difference between superconductors exhibits the appearance of Majorana bound states for a wide range of parameters. We further investigate the behavior of the topological gap and its dependence on the type of $d$ -wave pairing, i.e., $d, d + i s$ , or $d + i d^{'}$ , and note the difficulties that can arise due to the presence of gapless excitations in pure $d$ -wave superconductors. On the other hand, the planar Josephson junctions based on superconductors with $d + i s$ and $d + i d^{'}$ pairings can potentially lead to realizations of Majorana bound states. Our proposal can be realized in cuprate superconductors, e.g., in a twisted bilayer, combined with the layered semiconductor ${Bi}_{2} O_{2} Se$ .

Received 4 August 2023
Accepted 10 October 2023

DOI:https://doi.org/10.1103/PhysRevB.108.L140506

Physics Subject Headings (PhySH)

Majorana bound states Topological superconductors d-wave

Josephson junctions

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Hamed Vakili , Moaz Ali , Mohamed Elekhtiar, and Alexey A. Kovalev

Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA

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Issue

Vol. 108, Iss. 14 — 1 October 2023

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Images

Figure 1
(a) Schematic diagram of the planar Josephson junction heterostructure. The 2DEG is covered by $d$ -wave superconductors, except for a junction of width $W_{J}$ between the two superconductors. In (b) the probability function ( ${| ψ |}^{2}$ ) of MBS at $μ = 1.32 t$ and $h_{J} = 0.6 t$ is plotted. (c) The energy gap as a function of the Zeeman field $h_{J}$ in the junction and $μ$ . (d) The topological invariants $Q$ ( $Z_{2}$ ) and $W$ ( $Z$ ) as a function of of the Zeeman field $h_{J}$ in the junction and $μ$ . The colors show the value of the $W$ invariant. The black lines separate the regions with $Q = - 1$ and 1. The $W$ invariant only changes between odd (even) values for $Q = - 1$ (1) regions. We used the parameters $L_{y} = 400 a_{c}, W_{J} = 5 a_{c}, W_{L} = W_{R} = 20 a_{c}, Δ_{0} = 0.3 t, Δ_{0}^{'} = 0.0, h_{SC} = 0$ , and the phase difference $Δ ϕ = π$ .
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Figure 2
Phase diagrams for $d + i s$ pairing ( $β = π / 4$ ). (a) The $Q$ topological number ( $Z_{2}$ ) as a function of $h_{J}$ and $Δ ϕ$ for $μ = 1 t$ . (b) The energy gap $E_{gap}$ as a function of $h_{J}$ and $Δ ϕ$ for $μ = 1 t$ . [The dark blue line extending from left to right has a small gap, which is why no corresponding topological phase change exists in (a).] (c) Phase diagram of the $Q$ topological number as a function of $h_{J}$ and $μ$ for the phase difference $Δ ϕ = π$ . (d) The energy gap $E_{gap}$ as a function of $h_{J}$ and $μ$ for the phase difference $Δ ϕ = π$ . We used the parameters $h_{SC} = 0, Δ_{0} = 0.3 t, Δ_{0}^{'} = 0.0, W_{J} = 5 a_{c}, W_{L} = W_{R} = 40 a_{c}$ .
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Figure 3
The Zeeman field is applied to the whole system. (a) The $Q$ topological invariant as a function of $h$ and $μ$ where $h_{J} = h_{SC} = h$ for the phase difference $Δ ϕ = π$ . The blue color shows the topological region ( $Q = - 1$ ). (b) The energy gap $E_{gap}$ as a function of $h$ and $μ$ where $h_{J} = h_{SC} = h$ for the phase difference $Δ ϕ = π$ . (c), (d) The $Q$ topological number and the energy gap as a function of the ratio $h_{SC} / h_{J}$ and $ϕ$ for $μ = 1 t$ . We used the parameters $Δ_{0} = 0.3 t, Δ_{0}^{'} = 0.0, β = π / 4, h_{J} = 0.2 t, W_{J} = 5 a_{c}$ , and $W_{L} = W_{R} = 40 a_{c}$ .
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Figure 4
(a) The topological invariant $W$ as a function of $h_{J}$ and $μ$ for $Δ ϕ = π$ . The black line shows the outlines of $Q = - 1$ and $Q = 1$ regions. (b) The gap $E_{gap}$ of a JJ with periodic boundaries as a function of $h_{J}$ and $μ$ for $Δ ϕ = π$ . (c) The topological number $W$ as a function of $h_{J}$ and $Δ ϕ$ for $μ = 1 t$ . (d) The gap $E_{gap}$ of a JJ with periodic boundaries as a function of $h_{J}$ and $Δ ϕ$ for $μ = 1 t$ . We used the parameters $W_{J} = 5 a_{c}, W_{L} = W_{R} = 20 a_{c}$ . (e), (f) Modified JJ structure and ${| ψ |}^{2}$ of the lowest eigenvalue corresponding to MBS. The phases of four superconducting regions are set to $ϕ_{1} = ϕ_{2} = π / 2, ϕ_{3} = ϕ_{4} = - π / 2$ . Dimensions used: $W_{U} = W_{D} = W_{C} = 50 a_{c}, W_{S} = 600 a_{c}, W_{J} = 5 a_{c}$ . Other parameters used: $h_{J} = 0.4 t, h_{SC} = 0, μ = 2 t, Δ_{0} = 0.3 t, Δ_{0}^{'} = 0.06 t, θ = arctan (1 / 9) \approx π / 9$ .
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