Collective modes in terahertz field response of superconductors with paramagnetic impurities

TL;DR

This work analyzes the nonlinear THz response of conventional disordered superconductors with weak paramagnetic impurities using a diffusive Usadel framework in the Keldysh formalism. By including spin-flip scattering and Coulomb corrections, it derives the dispersion relations for Anderson-Bogoliubov (phase), Carlson-Goldman (phase-plasmon coupled), and amplitude (Higgs) modes, revealing a finite momentum threshold for exciting phase-related modes and a gapless superconducting plasmon in the presence of pair breaking. The Higgs mode exhibits a red shift and broadening as the spin-flip rate increases, and the amplitude mode becomes diffusive with omega_amp ~ omega_Higgs + D q^2. These findings provide experimentally accessible signatures in THz-driven experiments and shed light on how weak magnetic impurities modify collective modes in out-of-equilibrium superconductors.

Abstract

We consider a problem of nonlinear response to an external electromagnetic radiation of conventional disordered superconductors which contain a small amount of weak magnetic impurities. We focus on the diffusive limit and use Usadel equation to analyze the collective excitations and obtain the dispersion relations for the collective modes. We determine the resonant frequency and dispersion of both amplitude and phase (Carlson-Goldman) modes for moderate strength of magnetic scattering. We find that the Carlson-Goldman and superconducting plasmon modes can only be excited at some finite value of the threshold momentum which increases with an increase in spin-flip scattering rate while the amplitude mode is diffusive and becomes strongly suppressed with the increase in spin-flip scattering. The value of the threshold momentum is determined by the distance between the two consecutive spin-flip scattering events. Furthermore, we also find that the superconducting plasmon mode becomes gapless in the presence of the pair breaking processes. Possible ways towards experimental verification of our results are also discussed.

Figures (6)

• Figure 1: Schematic diagram for the dispersion of the collective excitations produced by an external electromagnetic pulse in disordered conventional superconductor with paramagnetic impurities. The momentum is assigned to the horizontal axis and frequency is assigned to the vertical axis. In the presence of the pair breaking scattering due to weak paramagnetic impurities the excitation of the superconducting plasmon and Calrson-Goldman (phase) modes can only take place at finite momentum. In a diffusive superconductor the value of the threshold momentum $q_0$ is determined by the pair breaking rate $1/\tau_{\textrm{s}}$ and the diffusion coefficient $D$. Pair breaking processes also lead to the red shift in the frequency of the amplitude Higgs-Schmid mode and render the superconducting plasmon mode gapless.
• Figure 2: Results of the numerical solution of the equation (\ref{['Eq4ueps']}) for real and imaginary parts of $u_\epsilon$ as a function of $\epsilon/\Delta$ for various values of the dimensionless parameter $\zeta=1/\tau_{\textrm{s}}\Delta_0$. Here $\Delta_0=\Delta(\zeta=0)$ is the pairing gap in the absence of paramagnetic disorder, $\zeta_c$ denotes the critical value of $\zeta$ when the order parameter vanishes, $\Delta(\zeta_c)=0$.
• Figure 3: Dispersion of the Anderson-Bogoliubov mode at very low temperatures $T\approx 10^{-3}\Delta_0$, where $\Delta_0$ is the pairing gap in a clean superconductor. We also find that $\omega\propto q$. Here $\Delta$ is a superconducting order parameter computed for finite values of disorder $\zeta$. The critical value of $\zeta$ is determined by $2/(\tau_{\textrm{s}}^{(c)}\Delta_0)=1$ or $\zeta_{\textrm{c}}=0.855$ for our choice for the values of the coupling constant and ultraviolet cutoff.
• Figure 4: Frequency dependence of the function $\delta s(\omega)$, Eq. (\ref{['sbards']}), for various values of disorder scattering rate $\zeta=1/\tau_{\textrm{s}}\Delta_0$. This function plays an important role as it accounts for the compensation of the electronic polarizability term $2\nu_F\Phi_\omega$ by the correction due to the charge re-distribution in the Poisson equation (\ref{['Poisson']}). In the limit $\zeta\to 0$ this function vanishes identically signalling the perfect compensation in a superconductor without spin-flip scattering. However, for finite values of $\tau_{\textrm{s}}^{-1}$ this function is nonzero which means that there will be a finite contribution to the dispersion of the superconducting plasmon from the electronic polarization effects.
• Figure 5: Frequency dependence of the real and imaginary (inset) of the function $r(\omega)=a(\omega)s(\omega)-\tilde{a}^2(\omega)$, Eq. (\ref{['Plasmon']}), for various values of disorder scattering rate $\zeta=1/\tau_{\textrm{s}}\Delta_0$. Notably, the real part of the function $r(\omega)$ remains essentially constant, while the imaginary part of $r(\omega)$ shows approximately linear dependence on frequency provided it exceeds certain threshold value.