dc transport in superconducting point contacts:  A full-counting-statistics view

We present a comprehensive theoretical analysis of the dc transport properties of superconducting point contacts. We determine the full-counting statistics for these junctions, which allows us to calculate not only the current or the noise, but all the cumulants of the current distribution. We show how the knowledge of the statistics of charge transfer provides an unprecedented level of understanding of the different transport properties for a great variety of situations. We illustrate our results with the analysis of junctions between BCS superconductors, contacts between superconductors with pair-breaking mechanisms, and short diffusive bridges. We also discuss the temperature dependence of the different cumulants and show the differences with normal contacts.

Figure 1

Current, shot noise, and third cumulant at zero temperature as a function of the voltage for BCS superconductors of gap $\Delta$. The different curves correspond to different transmission coefficients as indicated in the panels. Here, $G_N=(2e^2∕h)T$ is the normal state conductance.

Figure 2

(a) Second cumulant and (b) third cumulant at zero temperature for BCS superconductors. Both are normalized to the first cumulant (the average current). The transmissions are indicated in the plots.

Figure 3

Average MAR probabilities ${\overline{P}}_n\left(V\right)\equiv (1∕eV){\int }_0^{eV}dE{P}_n(E,V)$ as a function of voltage for a contact between BCS superconductors at zero temperature. The two panels correspond to two different transmissions. The index of the processes is indicated in the plots. Notice the logarithmic scale in the panel (a).

Figure 4

(a) Density of states as a function of energy of a superconductor for different values of the depairing energy $\Gamma$ measured in units of gap in the absence of pair-breaking ${\Delta }_0$. (b) Order parameter $\Delta$ and spectral gap ${\Delta }_g$ in units of ${\Delta }_0$ as a function of the depairing energy $\Gamma$ normalized by ${\Delta }_0$.

Figure 5

Zero temperature current-voltage characteristics for superconductors with a depairing energy $\Gamma$ in units of ${\Delta }_0$. The current and the voltage have been normalized with the order parameter $\Delta$ at the corresponding $\Gamma$. The different panels correspond to different transmissions values.

Figure 6

Current contribution of processes $n=1,2,\dots$ for $T=1$ as a function of voltage for superconductors with a depairing energy $\Gamma$ in units of ${\Delta }_0$. The current and the voltage have been normalized with the order parameter $\Delta$ at the corresponding $\Gamma$. The order of the processes is indicated in the plots.

Figure 7

Zero temperature noise for superconductors with a depairing energy $\Gamma$ in units of ${\Delta }_0$. The current and the voltage have been normalized with the order parameter $\Delta$ at the corresponding $\Gamma$. The different panels correspond to different transmissions values.

Figure 8

Zero-temperature third cumulant for superconductors with a depairing energy $\Gamma$ in units of ${\Delta }_0$. The current and the voltage have been normalized with the order parameter $\Delta$ at the corresponding $\Gamma$. The different panels correspond to different transmissions values.

Figure 9

Zero-temperature transport properties of a short diffusive SNS junction. (a) First three cumulants: current in units of $(G_N\Delta ∕e)$, shot noise in units of $\left(2G_N\Delta \right)$, and the third cumulant in units of $(G_N\Delta t_0∕h{G}_0)$. (b) Current contribution of the different processes. (c) Ratio $C_2∕C_1$ as a function of the inverse of the voltage. (d) Ratio $C_3∕C_1$ as as a function of the inverse of the voltage.

Figure 10

Current-voltage characteristics at finite temperature for BCS superconductors. The temperature is in units of the critical temperature $T_C$. The current and the voltage are normalized with the temperature-dependent gap. The different panels correspond to different transmission values.

Figure 11

Finite temperature noise for BCS superconductors. The temperature is normalized with the critical temperature $T_C$. The different panels correspond to different transmission values. The voltage is normalized with the temperature-dependent gap, and the current with the zero-temperature gap. Note that the scaling is different from the other plots in this section.

Figure 12

Finite temperature thrid cumulant for BCS superconductors. The temperature is normalized with the critical temperature $T_C$. The different panels correspond to different transmission values. The third cumulant and the voltage are normalized with the temperature-dependent gap.

Figure 13

Average MAR probabilities ${\overline{P}}_n\left(V\right)\equiv (1∕eV){\int }_0^{eV}dE{P}_n(E,V)$ at finite temperature as a function of voltage for a contact between BCS superconductors with transmission $T=0.95$. The four panels correspond to different temperatures $T_e$ expressed in units of the critical temperature $T_C$. The index of the processes is indicated in the plots.

Figure 14

Ratio $C_3∕C_1$ for two different temperatures as a function of the voltage for a contact between BCS superconductors. The different curves correspond to different transmissions as indicated in the plots.