Helical phases and Bogoliubov Fermi surfaces probed by superconducting diode effects

TL;DR

The paper analyzes SDE and JDE in clean, Rashba‑spin–orbit–coupled noncentrosymmetric superconductors under in‑plane fields using a quasiclassical Eilenberger approach. It finds that the bulk diode efficiency can peak near the critical endpoint of a Lifshitz transition between weak and strong helical phases, where finite‑momentum pairing and BFS emerge. In SNS junctions, JDE arises from finite‑momentum pairing in short devices and from the Zeeman field in long devices, with η oscillating in field and BFSs strongly suppressing current when aligned with BFS momenta, yielding pronounced anisotropy. This BFS‑induced anisotropy offers a practical route to detect BFSs and characterize helical superconductivity, even in systems where a JDE is absent or masked by disorder.

Abstract

Noncentrosymmetric superconductors (NCSs) with Rashba spin-orbit coupling (SOC) and in-plane magnetic fields have emerged as natural platforms for realizing both the bulk superconducting diode effect (SDE) and the Josephson diode effect (JDE) - phenomena characterized by unequal critical currents in opposite directions due to the simultaneous breaking of time-reversal and inversion symmetries. Using the quasiclassical Eilenberger formalism, we systematically investigate both the bulk SDE and the JDE in a clean NCS with Rashba SOC and in-plane magnetic fields. For the bulk system, we find that the diode efficiency can nominally approach its maximal value at the critical endpoint of the first-order Lifshitz transition between weak and strong helical phases featuring finite-momentum Cooper pairs, the latter marked by the emergence of Bogolyubov Fermi surfaces (BFSs). In a Josephson junction, we show that finite-momentum pairing in the superconducting leads is the dominant mechanism behind the JDE in short junctions, whereas in long junctions it is primarily governed by the Zeeman field in the normal region. In the long-junction regime, the diode efficiency additionally oscillates between positive and negative values as a function of magnetic field at low fields, providing a route toward a highly tunable Josephson diode. At higher fields, the onset of BFSs in the strong helical phase leads to a sharp suppression of both the JDE and the Josephson current when the current direction is aligned with momenta along the BFS, resulting in strong anisotropy. We propose that this anisotropy in the Josephson current offers an alternative method for detecting BFSs, applicable to systems with or without a JDE.

Figures (10)

• Figure 1: (a) The schematic setup of the S-N-S junction, where a helical metal is sandwiched between two helical superconductors; a homogeneous in-plane magnetic field $h$ is applied throughout the junction. (b) The Fermi surface of the normal metal when the magnetic field is absent (left) and present (right).
• Figure 2: (a) The supercurrent $j(q)$ vs Cooper pair momentum $q$ for different $h/T_c$ at $T/T_c=0.01$. (b-c) Zoomed-in plot of $j(q)$ near the first-order transition line between weak and strong helical phase at different temperatures, with different colored curves corresponding to different values of $h/T_c$ (indicated by the numbers in the plots).
• Figure 3: The Cooper pair momentum $q$ and gap $\Delta_q$ as a function of magnetic field $h$ for a homogeneous bulk superconductor. The dashed and dotted line denotes the approximate expression $\tilde{q}=2\tilde{h}$ and $\tilde{q}=2\tilde{\alpha} \tilde{h}$ respectively.
• Figure 4: The phase diagram of bulk helical superconductor where the color denotes the magnitude of $\Delta_q/T_c$, for $\tilde{\alpha}=0.25$. The dashed line denotes the superconducting phase tranition, and the solid (dotted) line denotes the first-order (crossover) Lifshitz transition between the weak and strong helical phases. The crossover line is determined by $\tilde{q}+2\tilde{h}=2$ at which the superconducting gap of the $\lambda=-$ band closes.
• Figure 5: The single-band energy spectrum in the normal region as a function of $\phi_\lambda$ for different $\tilde{L}$ and $\tilde{q}_\lambda$. The darker (lighter) color denotes higher (lower) local density of states.