Collapse of the superconducting order parameter in Ising superconductors with Rashba spin-orbit coupling

TL;DR

The paper studies Ising superconductors with strong spin-orbit coupling subjected to Rashba SOC in a two-valley $s$-wave framework. It shows that Rashba SOC introduces a finite-field collapse of the superconducting order parameter at low temperatures when $\alpha p_F H \sim \beta|\Delta|$, with a quantitative low-$T$ scaling and a Lambert $W$-based expression for the critical field. In contrast, the high-temperature Ginzburg-Landau limit reveals that this collapse vanishes near $T_c$, yielding a GL-type equation that governs the field dependence without a sharp gap closure. The results highlight a pronounced difference between the low- and high-temperature regimes in Ising superconductors and propose practical means to extract Rashba SOC strength from magnetic-field-driven gap behavior, with implications for tunneling spectroscopy measurements.

Abstract

Ising superconductors have attracted quite some attention recently, due to their resilience against magnetic fields way beyond the Pauli-paramagnetic limit. Their protection against external magnetic field relies on strong Ising spin-orbit coupling, which originates from in-plane inversion symmetry breaking. Due to the heavy atom nature of Ising SCs, a smaller but sizable Rashba SOC could be present through gating or interfacial effects. Here, we consider the effect of Rashba SOC in a two valley model of Ising superconductors with an attractive $s$-wave interaction. We show that Rashba SOC gives a critical magnetic field, above which the superconducting order parameter collapses at low temperatures. This effect, however, disappears at high temperatures. Our findings demonstrate that the low- and high temperature physics of Ising SCs is quantitatively and qualitatively different in our two-valley model, and may lead to new ways to determine the strength of the Rashba SOC in Ising SCs.

Figures (4)

• Figure 1: Graphical depiction of the in-plane component of the spin-splitting at the Fermi surface due to Rashba SOC and an external magnetic field in the $x$-direction. The out of plane spin splitting is given by the Ising SOC.
• Figure 2: The self-consistent superconducting order parameter $\bar{\beta}|\Delta|$ as a function of the external magnetic field $H$ for different values of Rashba SOC $\alpha p_F$ at the temperature $T/T_c=10^{-2}$ and Ising SOC strength $\beta=7\Delta_0$. The effect of Rashba SOC is clear, as the gap reaches $\alpha p_FH=\beta|\Delta|$ it collapses.
• Figure 3: Dependence of the critical magnetic field on the strength of Rashba for $\beta=7\Delta_0$. We notice that Rashba SOC is still significant if it is one order smaller than the superconducting gap $\alpha p_F/\Delta_0\gtrsim\mathcal{O}(10^{-1})$.
• Figure 4: Density of states of the Ising SC for different values of $\alpha p_F H/\beta|\Delta|$. This describes BCS density of states for $\alpha p_H =0$ with a gap given by $\bar{\beta}|\Delta|$ and becomes cone-shaped within $|\epsilon-\bar{\beta}|\Delta||<\alpha p_FH$ for increasing values of $\alpha p_FH/\beta|\Delta|$. For $\alpha p_FH=\beta|\Delta|$ the SC OP collapses for low temperatures, which gives a constant density of states. For high temperatures, however, the SC OP doesn't collapse beyond the critical magnetic field, rendering the density of states gapless and cone-shaped for higher magnetic fields.