Quasiparticle relaxation in superconducting nanostructures

We examine energy relaxation of nonequilibrium quasiparticles in “dirty” superconductors with the electron mean free path much shorter than the superconducting coherence length. Relaxation of low-energy nonequilibrium quasiparticles is dominated by phonon emission. We derive the corresponding collision integral and find the quasiparticle relaxation rate. The latter is sensitive to the breaking of time reversal symmetry (TRS) by a magnetic field (or magnetic impurities). As a concrete application of the developed theory, we address quasiparticle trapping by a vortex and a current-biased constriction. We show that trapping of hot quasiparticles may predominantly occur at distances from the vortex core, or the constriction, significantly exceeding the superconducting coherence length.

Figure 1

Complex plane of $\vartheta \left(\epsilon \right)$ for $\gamma /{\mathrm{\Delta }}_0=0.01;0.05;0.2;0.456;0.49$ from right to left. In the gapped case (i.e., $\gamma /{\mathrm{\Delta }}_0<0.456)$ $\vartheta \left(0\right)=-i\pi /2$ and there is a cusp at $\epsilon ={\epsilon }_g$; eventually $\vartheta \left(\infty \right)\to 0$. Full dots indicate $\epsilon =\mathrm{\Delta }$, notice $\mathrm{Im}\vartheta \left(\mathrm{\Delta }\right)\to -i\pi /6$ as $\gamma \to 0$.

Figure 2

DOS as a function of energy for the same values of TRS breaking parameter $\gamma$ as in Fig 1.

Figure 3

Order parameter $\mathrm{\Delta }/{\mathrm{\Delta }}_0$ (green) and gap energy ${\epsilon }_g/{\mathrm{\Delta }}_0$ (red) as functions of TRS breaking parameter $\gamma /{\mathrm{\Delta }}_0$. The blue line is the supercurrent density $j_s\left(\gamma \right)$ (in arbitrary units), given by Eq. (63).

Figure 4

$2u_{\epsilon }{v}_{\epsilon }$ as a function of energy for the same values of TRS breaking parameter $\gamma$ as in Fig 1.