Symmetry conditions for the superconducting diode effect in chiral superconductors

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Symmetry conditions for the superconducting diode effect in chiral superconductors

Bastian Zinkl, Keita Hamamoto, and Manfred Sigrist

Phys. Rev. Research 4, 033167 – Published 29 August 2022

Abstract

We analyze the presence of nonreciprocal critical currents, the so-called superconducting diode effect, in chiral superconductors within a generalized Ginzburg-Landau framework. After deriving its key symmetry conditions we illustrate the basic mechanism for two examples, the critical current in a thin film and a Josephson junction. The appearance of spontaneous edge currents and the energy bias for the formation of Josephson vortices play an essential part in establishing a splitting of the critical currents running in opposite directions. Eventually this allows us to interpret a superconducting diode effect observed in the 3-Kelvin phase of ${Sr}_{2} {RuO}_{4}$ as evidence for spontaneously broken time-reversal symmetry in the superconducting phase.

Received 21 December 2021
Revised 17 August 2022
Accepted 18 August 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.033167

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)

Superconductivity

Josephson junctions Thin films Unconventional superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Bastian Zinkl ¹, Keita Hamamoto², and Manfred Sigrist ¹

¹Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
²Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan

Article Text

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Issue

Vol. 4, Iss. 3 — August - October 2022

Subject Areas

Superconductivity

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Images

Figure 1
Schematic picture of a thin film, which is parallel to the $x$ axis and homogeneous in the $z$ direction. The superconducting state, which is assumed to be described by the order parameter $η = η_{0} (1, \pm i)$ , breaks time-reversal symmetry. Currents running along the edges are highlighted by blue arrows and the critical current for positive and negative directions are denoted by $I_{c}^{+}$ and $I_{c}^{-}$ , respectively. The situation where the film is symmetric under both $σ_{x}$ and $σ_{y}$ [Eq. (3)] is depicted in (a). The plots at the bottom represent cases where one of these mirror symmetries is broken. In (b) the film is asymmetric under $σ_{x}$ and in (c) under $σ_{y}$ , which is marked by the $σ_{i} ✕$ and illustrated by the shaded areas.
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Figure 2
The total current in $x$ direction of the thin film as a function of the phase gradient $α$ , if the film is symmetric under $σ_{y}$ (black) or not (red and blue). In the fully symmetric system the solutions for chiralities $χ = \pm 1$ are degenerate. We used for the Ginzburg-Landau coefficients $(b_{1}, b_{2}, b_{3}) = (0.3 b, 0.4 b, 0.2 b)$ and $(K_{1}, K_{2}, K_{3}, K_{4}) = (0.6 K, 0.4 K, 0.4 K, 0.4 K)$ , which are in line with a generic 2D Fermi surface. The integrals [Eq. (14)] were taken to be $(A^{x}, A^{y}, B^{x}, B^{y}, B^{x y}, C^{x}, C^{y}, D^{x}, D^{y}) \approx (0.95, 0.6, 0.7, 0.5, 0.36, 0.01, 0.05, 0.4, 0.0)$ . Note that our parameters are connected to the order parameter magnitude through $| η_{0} |^{2} = - a / b$ and to the coherence length through $ξ^{2} = - K / a$ .
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Figure 3
Schematic picture of a Josephson junction consisting of two chiral superconductors with the same chirality (gray) and a normal state region (yellow). The Josephson current (blue) flows along the edges. The system is homogeneous in the $z$ direction.
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Figure 4
(a) The normalized magnetic flux density $B_{z} (y) \propto \partial ϕ / \partial y$ created by spontaneous edge currents for $H_{e} = 0$ and $γ ≪ 1$ . Latter is assumed to be small enough such that $ϕ ≪ 1$ . Taking $L_{y} = 10$ we consider different values of $λ_{J}$ . (b) Schematic illustration of the spontaneous magnetic field distribution inside of the Josephson junction. The field values are normalized and the location of the junction is given by the dashed line.
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Figure 5
The current density (solid) and magnetic field distribution (dashed) inside the junction for $H_{e} = 0$ and various $γ$ values. The distance is expressed in units of the Josephson penetration depth $λ_{J}$ . Note that our results converge to the ones contained in [27] for $γ \to 0$ .
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Figure 6
The schematic magnetic field pattern if a total current is running through the Josephson junction for (a) $γ = 0$ and (b) $γ \neq 0$ . Again we normalized the values by the maximum of the magnetic field. Note that the spontaneous field distribution depicted by Fig. 4 is negligibly small compared to the magnetic fields plotted here. The position of the junction is indicated by the dashed line.
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