Supercurrent Growth in Nonequilibrium Superconductors

Abstract

In ultrafast experiments on superconductors, a pump laser pulse often heats up the electronic system and suppresses the density of superfluid electrons. Subsequently, the electrons undergo a cooling process because of electron-phonon thermalization so that the superfluid density recovers in time. We study the nonequilibrium electromagnetic response of the system in this cooling process. We show that if a supercurrent is initiated by a probe electric field pulse, an intriguing phenomenon of `supercurrent growth' occurs, meaning that the net current grows in time with the increasing superfluid density. Using the Boltzmann kinetic equation, we uncover its microscopic origin as the momentum-relaxing scattering of Bogoliubov quasiparticles by impurities and phonons, in stark contrast to the widely accepted intuition that impurities always attenuate currents. We further show that supercurrent growth has important experimental manifestations, including the ultrafast Meissner effect and an optical reflectivity exceeding unity.

Figures (5)

• Figure 1: Illustration of the current in response to an electric field pulse (red curve) in time for different systems: normal metal (gray curve), superconductor (light blue curve), nonequilibrium superconductor (NSC) with a growing superfluid density (dark blue curve). The left inset illustrates the microscopic process of the supercurrent growth in the NSC curve: two electrons gain the velocity $\mathbf{v}$ as they form a Cooper pair. The top right inset schematically shows the time dependence of the diamagnetic current ($j_{\text{d}}$) and paramagnetic current ($j_{\text{p}}$) in the NSC current response.
• Figure 2: Microscopic mechanism for the current amplification. (a) Bogoliubov quasiparticle energy-momentum dispersion in the pump 'heated' state with a high effective temperature $T_{\text{H}}$ and a small gap. There are large amount of quasiparticles represented by the spheres whose colors represent their charges. (b) Right after the probe vector potential $\mathbf{A}(t)=-\mathbf{A}_0\Theta(t)$ is applied, there is an asymmetric quasiparticle dispersion and right-flowing diamagnetic current $\mathbf{j}_{\mathrm{d}}$. (c) Impurity scattering gives rise to the asymmetric distribution $\delta f_{\mathbf{k}}^{\mathrm{a}}$ and the left-flowing paramagnetic current $\mathbf{j}_{\mathrm{p}}$, partially canceling the diamagnetic current. (d) Pair recombination due to phonon emission reduces the number of quasiparticles. Impurities scatter the quasiparticles and relax the momentum imbalance $\delta f_{\mathbf{k}}^{\mathrm{a}}$ and the paramagnetic current, increasing the total current.
• Figure 3: (a) Illustration of the ultrafast Meissner effect during the quench to a superconducting state. The magnetic fluxes (black crosses) are expelled out of the sample as the supercurrent (blue lines) grows in time. (b) Reflectivity of a normally incident Gaussian light pulse $E(t,z)=E_0 e^{-(t-t_0+z/c)^2/(2\delta^2)-i\omega_0 (t+z/c)}$ on a nonequilibrium dirty superconductor (right inset) for three different penetration depths $d$. The time dependence of the gap is shown by the left inset, which is obtained from solving Eqs. (\ref{['eq:berta']}a) and \ref{['eq:gap_equation']} with $\gamma_{\text{E}}=1 \,\mathrm{THz}$, $T_{\mathrm{L}}=0.20\, T_{\text{c}}$ and $T_{\mathrm{H}}=0.99 \, T_{\text{c}}$. As a result, the superfluid plasma frequency grows from $0.41 \,\omega_{\mathrm{ps}}$ at $t=0 \,\mathrm{ps}$ to approach the equilibrium one $\omega_{\mathrm{ps}}=\sqrt{4\pi n_{s0} e^2 /m}=1.7 \,\mathrm{THz}$ after $t=4 \,\mathrm{ps}$. The red stripe in the left inset marks the central time and width $\delta= 0.3 \,\mathrm{ps}$ of the light pulse on the surface of the sample. The choice of parameters is motivated by those of YBCO Averitt.2001Hoegen.2022.
• Figure 4: Simulated relaxation dynamics of the energy gap from Eqs. (\ref{['eq:berta']}a) and \ref{['eq:gap_equation']} with $\gamma_{\text{E}}=1 \,\mathrm{THz}$ and different initial quasiparticle temperatures $T_{\mathrm{H}}$. The lattice temperature is fixed at (a) $T_{\mathrm{L}}=0.20 \, T_{\mathrm{c}}$ and (b) $T_{\mathrm{L}}=0.990 \, T_{\mathrm{c}}$.
• Figure 5: Schematic illustration of the reflection problem on the nonequilibrium sample.