Superconducting Diode Effect in Weak Localization Regime

Abstract

We study a dirty two-dimensional superconductor with Rashba spin-orbit coupling and in-plane Zeeman fields described by the nonlinear sigma model that includes the Cooper and long-range Coulomb interactions. The renormalized Ginzburg-Landau theory, which includes the weak localization effects at the one-loop level, is constructed using the Keldysh functional formalism. It is shown that the transition temperature and magnetic field, as well as the tricritical point appearing in the phase diagram, are suppressed by the interactions. Nevertheless, we have found a universal behavior in the high-transition-temperature regime that demonstrates the robustness of the superconducting diode effect against the interactions. The conductivity of the resistive states emerging after the superconducting states are destroyed by the critical current is also calculated, and localization behaviors are demonstrated.

Figures (3)

• Figure 1: The phase diagram and diode quality factor $\eta$ in the comparable SOC case ($\alpha/p_{\rm F} = 0.01,~\tau T_{\rm c0} = 1.5\times10^{-2}$). The solid (dashed) lines show the results in the case with (without) e-e interactions, which is referred to as the interaction (free) case in the main text. (a) The color maps of $\eta$ in the vicinity of the second-order SC transition lines, which change to first-order transitions at the tricritical points represented by the black circles. (c) The transition lines from panel (a) on the normalized temperature-normalized magnetic field scale, namely the $\tau_{\rm r}$-$h_{\rm r}$ scale. (b) and (d) the diode quality factor $\eta$ as a function of $\tau_{\rm r}$ and $h_{\rm r}$, respectively. In these figures, we vary $\tau_{\rm r}$ and $h_{\rm r}$ so that the temperatures satisfy $T=0.99T_{\rm c0}(h)$ in the free case and $T=0.99T_{\rm c}(h)$ in the interaction case. Note that the same $\tau_{\rm r}$ does not imply the same $h$ or $h_{\rm r}$ between the interaction and free cases.
• Figure 2: The diode quality $\eta$ is plotted as a function of the normalized magnetic field $h_{\rm r}$ at normalized temperatures: (a) $\tau_{\rm r}=0.9$ and (b) $\tau_{\rm r}=0.25$. In panel (a), the red and blue squares denote $\eta$ in the interaction and free cases, respectively. The green line shows the fitting by Eq. (\ref{['Diode']}). In panel (b), the red solid and blue dashed lines show $\eta$ in the interaction and free cases, respectively.
• Figure 3: The WL conductivity $\sigma_{\rm WL}$ in (a) the weak SOC regime ($\alpha/p_{\rm F} = 0.002,~\tau T_{\rm c0} = 7.5\times10^{-3}$) and (b) the comparable SOC case ($\alpha/p_{\rm F} = 0.01,~\tau T_{\rm c0} = 1.5\times10^{-2}$). The solid (dashed) lines show the results in the interaction (free) case. The white circles represent the tricritical points.