Nonequilibrium characteristics in all-superconducting tunnel structures

We study the nonequilibrium characteristics of superconducting tunnel structures in the case when one of the superconductors is a small island confined between large superconductors. The state of this island can be probed, for example, via the supercurrent flowing through it. We study both the far-from-equilibrium limit when the rate of injection for the electrons into the island exceeds the energy relaxation inside it, and the quasiequilibrium limit when the electrons equilibrate between themselves. We also address the crossover between these limits by employing the collision integral derived for the superconducting case. The clearest signatures of the nonequilibrium limit are the anomalous heating effects seen as a supercurrent suppression at low voltages, and the hysteresis at voltages close to the gap edge $2\Delta ∕e$, resulting from the peculiar form of the nonequilibrium distribution function.

Figure 1

Scheme of the SISIS structure studied in this work. The superconducting island (2) in the middle is connected with tunnel contacts to four large superconducting leads (1, 3, 4, and 5). The control line is biased with voltage $V$, which controls the energy distribution on the island. A supercurrent $I_S$ is driven across the island from lead 4 to lead 5, and its magnitude depends on the distribution function on the island. Each SIS junction is a tunnel contact of resistance $R_i$.

Figure 2

(Color online) Nonequilibrium distribution function for the superconducting island at $T=0.1T_C$. The cooling effect reducing the number of excited quasiparticles as the voltage is increased is evident. Here and below we denote ${\Delta }_0={\Delta }_L(T=0)$ and $T_C$ is the critical temperature of the leads.

Figure 3

(Color online) Nonequilibrium distribution function for the superconducting island at various bath temperatures for a voltage $eV∕{\Delta }_0=3$.

Figure 4

(Color online) Order parameter as a function of bias voltage at various bath temperatures (a) and ratios ${\Delta }_2∕{\Delta }_L$ (b).

Figure 5

(Color online) Supercurrent through the island in full nonequilibrium as a function of bias voltage at various bath temperatures with a ratio ${\Delta }_2∕{\Delta }_L=0.3$. The arrows indicate the direction the curve is traced when the bias voltage is varied. Thermal fluctuations may cause the discontinuous jump to occur somewhere in between the two extremes shown in the figure. The system is assumed symmetric, i.e., $R_1=R_3=R_4=R_5$ and ${\Delta }_4={\Delta }_5={\Delta }_L$.

Figure 6

(Color online) Supercurrent through the island in quasiequilibrium as a function of bias voltage with parameters identical to those of the full nonequilibrium case presented in Fig. 5.

Figure 7

(Color online) Distribution function for the superconducting island at $eV∕{\Delta }_0=1.5$ (left) and 3 (right) for various ${\mathcal{K}}_{\mathit{coll}}$ with $T=0.7T_C$. ${\mathcal{K}}_{\mathit{coll}}=\infty$ corresponds to quasiequilibrium.

Figure 8

(Color online) Supercurrent through the island as a function of voltage for various ${\mathcal{K}}_{\mathit{coll}}$ with $T=0.7T_C$. For the hysteretic curves the arrows indicate the direction the curve is traced when the bias voltage is varied.

Figure 9

(Color online) Supercurrent through the island as a function of voltage for different degrees of asymmetry in the SISIS control line. The arrows indicate the direction the curve is traced when the bias voltage is varied. The inset shows the distribution function on the island for a ratio $R_1∕R_3=2$ at $eV∕{\Delta }_0=1.5$. The distribution function exhibits small asymmetry due to finite $f_T$ as can be seen by the additional notch at negative energies.

Figure 10

(Color online) Potential difference between quasiparticles and the condensate for different degrees of asymmetry in the SISIS control line.