Optical conductivity of a dirty current-carrying superconductor

TL;DR

This work delivers a complete microscopic theory for the optical conductivity $\sigma(\omega)$ of a dirty superconductor carrying a dc current, using the Keldysh nonlinear sigma model to obtain a general expression that includes MB-like dissipation and two current-induced channels: $\sigma_2^{\text{qp}}$ and $\sigma_2^{\text{SH}}$ from quasiparticle redistribution and the Schmid–Higgs amplitude mode, respectively. It reveals a current-driven peak in $\mathrm{Re}\,\sigma(\omega)$ above the optical gap and a dip in $\mathrm{Im}\,\sigma(\omega)$ near $2E_g$, with the behavior controlled by the inelastic relaxation rate $\gamma$ and a smaller scale $\gamma_Q$; at low frequencies, giant absorption can occur due to these relaxation processes, and the SH mode can be directly probed through transport measurements. The authors provide general expressions near the transition temperature and in the dynamic and quasistatic regimes, connecting to and clarifying prior results by Ovchinnikov and others, while resolving discrepancies with previous phenomenological approaches. The study highlights the practical observability of the Schmid–Higgs mode via conductivity measurements in current-biased superconductors and establishes a detailed framework for interpreting microwave responses in disordered superconducting devices.

Abstract

We develop a full microscopic theory for the optical conductivity, $σ(ω)$, of a dirty current-carrying superconductor. Within the Keldysh sigma model formalism, we obtain the general analytical expression for $σ(ω)$, applicable for arbitrary frequency $ω$, temperature $T$, and dc supercurrent $I$. In addition to altering the usual Mattis-Bardeen conductivity, $σ_1(ω)$, a finite supercurrent introduces two new contributions: $σ_2^\text{qp}(ω)$ from quasiparticle redistribution and $σ_2^\text{SH}(ω)$ from the amplitude (Schmid-Higgs) mode excitation by the ac field. We investigate, both analytically and numerically, the main features of the optical conductivity in the presence of a dc supercurrent. They include a peak in $\text{Re}\,σ(ω)$ above the optical gap and a sign change of $\text{Im}\,σ(ω)$, with both effects becoming more pronounced at higher $I$ and lower $T$. We also elucidate the role of inelastic relaxation, which governs the low-frequency response, leading to a giant microwave absorption and a suppression of the apparent superfluid density at the critical current. The optical conductivity measurement of a superconductor biased by a finite dc supercurrent enables the direct observation of the Schmid-Higgs mode via transport measurements.

Figures (7)

• Figure 1: A superconducting film carrying a dc supercurrent (with momentum $2e\mathbf{A}_0/\hbar c$) under an applied microwave field.
• Figure 2: Frequency dependence of the real (a, b, c) and imaginary (d, e, f) parts of the conductivity $\sigma(\omega)=\sigma_1(\omega)+\sigma_2(\omega)$ in the collinear geometry $\mathbf{A}_1{\parallel}\mathbf{A}_0$ at temperatures $T=0$ (a, d), $T=0.7\,T_c$ (b, e), and $T=0.99\,T_c$ (c, f). Three curves at each panel correspond to different dc supercurrents: $I/I_c(T)=0$, 0.5, and 1. Dashed lines mark the positions of $2E_g$ for given $T$ and $I$.
• Figure 3: Frequency dependence of $\mathop{\mathrm{Re}}\nolimits\sigma(\omega)$ for $\mathbf{A}_1{\parallel}\mathbf{A}_0$ at $T=0$, as shown in Fig. \ref{['F:ReIm-plots']}(a), with emphasized contributions of $\sigma_1(\omega)$, $\sigma_2^\text{qp}(\omega)$, and $\sigma_2^\text{SH}(\omega)$. Note that the absorption peak observed above the optical gap entirely originates from the excitation of the Schmid-Higgs mode.
• Figure 4: Logarithmic plot of the low-frequency dissipative conductivity at $T=0.99\,T_c$ [Fig. \ref{['F:ReIm-plots']}(c)], computed for the inelastic rate $\gamma=10^{-6} \Delta_0(0)$ [$\Delta_0(0)$ is the gap at $T=0$ and $I=0$].
• Figure 5: Temperature dependence of the dissipative conductivity in the quasistatic limit, $\mathop{\mathrm{Re}}\nolimits\sigma(0)\propto1/\gamma$, computed for $I/I_c(T)=0.1$, 0.2, $\dots$, 1, and the temperature-independent inelastic relaxation rate $\gamma = 10^{-6}\Delta_0(0)$. In this case, the giant absorption is exponentially suppressed at low temperatures due to the freezing of quasiparticles.