Vertex corrections to nonlinear photoinduced currents in 2D superconductors

TL;DR

This work addresses the nonlinear, transverse photoresponse (SC photodiode) in a 2D $s$-wave superconductor with a built-in DC supercurrent under circularly polarized light, showing that impurity scattering and gauge invariance are essential for a finite response. The authors develop a microscopic theory within the Keldysh formalism that includes a BCS interaction–induced vertex correction, giving an explicit expression for the rectified current $j_p$ and the accompanying vertex $\oldsymbol{\hat{\Lambda}}(\omega)$. The main contributions come from type-a and type-b diagrams, with the vertex correction introducing a $g$-dependent modification to the spectrum and a cubic low-frequency behavior, while enabling a spectroscopic handle on relaxation times via zero-current and extremum conditions. Overall, the work restores gauge invariance, refines the nonlinear photoconductivity description in 2D superconductors, and suggests a practical route to measure $\tau_E$ and $\tau_i$ using the photoinduced current spectrum, with implications for light-controlled superconducting devices.

Abstract

The emergence of a rectified steady-state supercurrent as a response to the photoexcited current of the quasiparticles constitutes the concept of a superconducting photodiode. This phenomenon occurs in a two-dimensional thin superconducting film with a built-in DC supercurrent that is exposed to a circularly polarized external electromagnetic field. The flow of a Cooper-pair condensate, resulting as a second-order photo-response in a direction transverse to the initially built-in supercurrent, represents a superconducting counterpart to the photogalvanic effect. In this paper, we examine the photodiode supercurrent by restoring gauge invariance within the mean-field BCS framework. To achieve this, we derive an impurity-sensitive BCS-interaction-induced correction to the vertex function by performing self-consistent calculations within the Keldysh Green's function technique. The resulting photodiode current can be utilized for spectroscopic analysis of typical relaxation times in superconducting films.

Figures (3)

• Figure 1: (a) Feynman diagrams providing the highest-order contribution to the photoinduced quasiparticle electric current. Blue lines are the Green's functions of quasiparticles, red wavy lines describe the external EM field $\mathcal{A}$; green dots stand for the quasiparticle velocity vertices $\textbf{v}$, and dashed lines show the supercurrent momentum ${\bf p}_s$. Orange triangles indicate the vertex renormalization $\hat{\Lambda}$ due to the BCS interaction. (b) Graphical illustration of the self-consistent equation for the vertex correction, see Eq. (\ref{['vertex']}). Here, the red dash-dotted line indicates the BCS two-electron interaction of strength $\lambda$.
• Figure 2: The contributions to the electric current density Eqs. (\ref{['expr1']})--(\ref{['expr3']}) as functions of normalized frequency $\omega\tau_E$. Black, green and gray solid curves show the contribution of $(j_y^{(a)}+j_y^{(b)})/j_0$, $j_I/j_0$, $j_{II}/j_0$ currents without the vertex correction ($g=0$), while red, blue and orange dashed curves corresponds to the same currents with $g=0.1$. Here, we used $T=9$ K, $T_c=10$ K ($\Delta_0\equiv \Delta(T=0)=1.5$ meV) and $\tau_i=10$ ps, $\tau_E=100$ ps. Inset: Maximum current densities $(j_y^{(a)}+j_y^{(b)})$ and $j_{II}/j_0$ as functions of impurity scattering time. Maximum for $j_{\rm y}$ current occurs at $\omega\tau_E\approx 1/\sqrt{3}$ (thus, $j_{m}=(3\sqrt{3}/4) j_0$), while for $j_{II}$ current, the maximum is at $\omega\tau_E=1$ (with $j_m=j_0/8$). The color scheme is the same as in the main plot: solid curves indicate $g=0$ case and dashed curves correspond to $g=0.1$.
• Figure 3: Spectrum of the total photodiode current density $j_p=j_y^{(a)}+j_y^{(b)}+j_y^{(c)}$ normalzied by $j_0$ for different values of the BCS-induced vertex correction $g$. All parameters are the same as for Fig. \ref{['Fig2']}. Note, current $j_I$ gives negligible contribution for $g\geq0.2$.