Parity-protected superconducting diode effect in topological Josephson junctions

Henry F. Legg, Katharina Laubscher, Daniel Loss, and Jelena Klinovaja

Phys. Rev. B 108, 214520 – Published 22 December 2023

Abstract

In bulk superconductors or Josephson junctions formed in materials with spin-orbit interaction, the critical current can depend on the direction of current flow or of applied magnetic field, an effect known as the superconducting (SC) diode effect. Here, we consider the SC diode effect in Josephson junctions in nanowire devices. We find that the $4 π$ -periodic contribution of Majorana bound states (MBSs) to the current phase relation (CPR) of single junctions results in a significant enhancement of the SC diode effect when the device enters the topological phase and can therefore be used as an indicator of the phase transition. Crucially, this enhancement of the SC diode effect is independent of the parity of the junction and therefore protected from parity-altering events, such as quasiparticle poisoning, which have hampered efforts to directly observe the $4 π$ -periodic CPR of MBSs. We show that this effect can be generalized to SQUIDs and that, in such devices, the parity-protected SC diode effect can provide a highly controllable probe of the topology in a Josephson junction.

Received 1 February 2023
Revised 20 November 2023
Accepted 21 November 2023

DOI:https://doi.org/10.1103/PhysRevB.108.214520

Physics Subject Headings (PhySH)

Josephson effect Majorana bound states Spin-orbit coupling Superconductivity Topological materials

Devices Diodes Josephson junctions Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Henry F. Legg ¹, Katharina Laubscher ^1,2, Daniel Loss¹, and Jelena Klinovaja ¹

¹Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
²Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

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Vol. 108, Iss. 21 — 1 December 2023

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Figure 1
Setups to measure the parity-protected SC diode effect in topological Josephson junctions. (a) In a single junction the SC diode effect (SDE), meaning that the critical current is directional dependent $I_{c}^{+} \neq I_{c}^{-}$ , can occur when the applied magnetic field has a component parallel to the SOI vector (here $y$ direction). If, in addition, a component of the magnetic field perpendicular to the SOI vector drives the system into the topological SC phase, then MBSs form both in the junction and at the ends of the nanowire (red dots). The $4 π$ -periodic CPR of the MBSs in the junction results in a significant enhancement of the SDE that is independent of the parity of the junction. (b) The SDE can also be probed using a SQUID that consists of a topologically trivial low-transparency junction that hosts only ABSs (blue dot) and a junction that can be driven into the topological SC phase hosting MBSs (red dots). At finite flux $Φ$ , a large parity-protected SDE occurs when the junction is in the topological phase.
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Figure 2
Parity-protected SDE in topological JJs. (a) Schematic showing the impact of parity-altering events in CPR from MBSs. In reality, parity-altering events render the theoretical $4 π$ -periodic CPR of a topological JJ effectively $2 π$ periodic. (b) The interference between the $2 π$ -periodic CPRs of ABSs and the $4 π$ -periodic CPR of MBSs, here both of amplitude $I_{0}$ , can lead to a significant SDE in both a single junction and SQUID (see Fig. 1). The full CPR remains parity dependent (red and blue lines, individual CPR contributions are indicated by dashed lines). In contrast, however, the size and sign of the SDE, governed by $Δ I = I_{c}^{+} - I_{c}^{-}$ , is independent of parity and can be used as a measure of junction topology.
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Figure 3
Parity protected SDE in a single nanowire JJ with both parallel ( $Δ_{y}$ ) and perpendicular ( $Δ_{x}$ ) Zeeman fields to the SOI vector. Top panels: energy phase relations $ɛ (φ)$ of ABSs and continuum states (light blue), interior MBSs (red/blue), and exterior MBSs (black, at zero energy and independent of $φ$ ), over a $4 π$ period as obtained from the tight-binding model in Eq. (8), for various perpendicular fields $Δ_{x}$ and fixed parallel field $Δ_{y} = 0.025 Δ_{0}$ . Bottom panels: Corresponding CPR $I (φ)$ exhibiting an SDE. (a), (b) When the junction is in the trivial phase, $Δ_{x} < Δ_{0}$ for our parameters, only a small SDE is present, even for junctions with visible deviations from a purely sinusoidal CPR, see also Figs. 4. Note that, within our Rashba nanowire model, the SDE is only present when $Δ_{x} \neq 0$ , $Δ_{y} \neq 0$ , and $α \neq 0$ are all satisfied, since these are necessary to manifest both broken inversion and time-reversal symmetry, see similar discussions in Refs. [86, 102]. (c), (d) In the topological phase $Δ_{x} > Δ_{0}$ , MBSs begin to form and a much larger SDE is present. In particular, when the exterior MBSs are well localized at the ends of the SC sections away from the junction, indicated by zero-energy states entirely independent of $φ$ (black), as in (d) but not in (c), the SDE is independent of the parity of the MBSs. The diode efficiency is given by $Δ I / {\bar{I}}_{c}$ , where $Δ I = I_{c}^{+} - I_{c}^{-}$ and ${\bar{I}}_{c} = (I_{c}^{+} + I_{c}^{-}) / 2$ . Parameters: $N_{s} = 200$ , $N_{b} = 3$ , $N_{j} = 5$ , $t = 3.4$ meV, $α = 3.33$ meV, $V = 1$ meV, $Δ_{0} = 0.25$ meV.
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Figure 4
CPRs of finite JJs and parity protected SDE in a SQUID. (a) Dependence of CPR of a single JJ of finite length on barrier height $V$ for $Δ_{x} = 0$ . Note that the CPRs are normalized by $I_{c}$ , which is also modified by $V$ . (b) Relative harmonics of the CPR in a single JJ as a function of barrier height $V$ for a finite JJ width ( $N_{j} = 15$ ) and short JJ ( $N_{j} = 1$ ) for $Δ_{x} = 0$ . In a finite JJ the relative harmonics fall off considerably faster than in a short JJ, reducing the size of the SDE in the trivial phase in both single JJs and SQUID for finite barrier heights. (c) The SC diode figure of merit $η (Φ)$ , as a function of the flux $Φ$ , through the SQUID loop when one junction is in the topological phase ( $Δ_{x} = 5 Δ_{0} / 3$ ). (d) Maximal absolute SDE $η_{\max} = max {| η (Φ) |}$ , as a function of Zeeman field $Δ_{x}$ , as one junction is tuned into the topological phase via application of a Zeeman field. In the trivial phase, $Δ_{x} < Δ_{0}$ for our parameters, the maximal $η (Φ)$ is small for all barrier heights. However, in the topological phase $Δ_{x} > Δ_{0}$ (indicated by dashed gray line), the maximal $η (Φ)$ increases substantially regardless of barrier height and is independent of the parity, therefore providing a measure of the onset of the topological phase. Parameters: $N_{s} = 480$ , $N_{b} = 6$ , $N_{j} = 15$ , $t = 30.5$ meV, $α = 5$ meV, $Δ_{y} = 0$ , $Δ_{0} = 0.15$ meV. For the SQUID (c) and (d) the trivial JJ is modeled using the same parameters but with $Δ_{x} = 0$ .
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