Onset of superconductivity in a voltage-biased normal-superconducting-normal microbridge

We study the stability of the normal state in a mesoscopic NSN junction biased by a constant voltage $V$ with respect to the formation of the superconducting order. Using the linearized time-dependent Ginzburg-Landau equation, we obtain the temperature dependence of the instability line, $V_{\mathrm{inst}}\left(T\right)$, where nucleation of superconductivity takes place. For sufficiently low biases, a stationary symmetric superconducting state emerges below the instability line. For higher biases, the normal phase is destroyed by the formation of a nonstationary bimodal state with two superconducting nuclei localized near the opposite terminals. The low-temperature and large-voltage behavior of the instability line is highly sensitive to the details of the inelastic relaxation mechanism in the wire. Therefore, experimental studies of $V_{\mathrm{inst}}\left(T\right)$ in NSN junctions may be used as an effective tool to access the parameters of the inelastic relaxation in the normal state.

Figure 1

(a) Real (solid blue line) and imaginary (dashed red line) parts of the lowest eigenvalues ${\lambda }_{1,2}\left(v\right)$ of the Hamiltonian (5). The spectrum is entirely real until $v=v_c\approx 49.25$. (b)–(e) Spatial dependence of the absolute values of the eigenfunctions, $|{\psi }_1\left(\tilde{x}\right)|$ (blue) and $|{\psi }_2\left(\tilde{x}\right)|$ (red), for $v=0.8v_c$, $1.2v_c$, $5v_c$, and $10v_c$, respectively.

Figure 2

Instability voltage as a function of temperature, $V_{\mathrm{inst}}\left(T\right)$, obtained numerically for a wire of length $L=15{\xi }_0$ for three limiting types of the distribution function: without inelastic relaxation (solid blue line) and with dominant $e$-$e$ (dot-dashed line) or $e$-ph (dashed line) relaxation. The dotted curve illustrates the suppression of $V_{\mathrm{inst}}^{\left(\mathrm{free}\right)}\left(T\right)$ by a finite terminal resistance, $\beta =0.1$ (see text). The inset shows the behavior in the vicinity of the bifurcation point.