Mesoscopic effects in the heat conductance of superconducting-normal-superconducting and normal-superconducting junctions

We study the heat conductance of hybrid superconducting junctions. Our analysis involves single-channel junctions with arbitrary transmission as well as diffusive connectors and shows the influence of the superconducting gaps and phases of the contacts on the heat conductance. If the junction is diffusive, these effects are completely quenched on average, however, we find that their influence persists in weak-localization corrections and conductance fluctuations. While these statistical properties strongly deviate from the well-known analogs for the charge conductance, we demonstrate that the heat conductance fluctuations maintain a close to universal behavior. We find a generalized Wiedemann-Franz law for Josephson junctions with equal gaps and vanishing phase difference.

Figure 1

Sketch of an SNS junction across which a heat current $J$ flows. The contacts are characterized by different temperatures, $T_{\text{L}}$ and $T_{\text{R}}$, superconducting gaps, ${\mathrm{\Delta }}_{\text{L}}$ and ${\mathrm{\Delta }}_{\text{R}}$ (one of which can even be suppressed to 0), and phases, ${\phi }_{\text{L}}$ and ${\phi }_{\text{R}}$. The normal part of the junction has length $\mathcal{L}$ and supports $\mathcal{N}$ scattering channels. In this paper, we treat the case of $\mathcal{N}=1$ and the case of a diffusive region with a nonspecified number of channels.

Figure 2

Heat conductance as function of $T/T_{\text{crit}}$, with the critical temperature of the larger gap $T_{\text{crit}}\equiv T_{\text{R,crit}}$. Panels (a)–(c) show the single-channel case with $D_n=D=0.1$ for different values of $\phi$ and ${\mathrm{\Delta }}_{0,\text{L}}/{\mathrm{\Delta }}_{0,\text{R}}$. The grey dashed line at ${\kappa }_{\text{SNS}}/{\kappa }_{\text{N}}=1$ serves as a guide for the eye.

Figure 3

The average heat conductance $\langle {\kappa }_{\text{SNS}}\rangle /\langle {\kappa }_{\text{N}}\rangle$ is gap-ratio and phase independent and exactly coincides with the transmission-independent single-channel result ${\kappa }_{\text{SNS}}/{\kappa }_{\text{N}}$ at $\phi =0$ and equal gaps; see Fig. 2.

Figure 4

Weak localization correction to the heat conductance $\delta {\kappa }_{\text{SNS}}$ normalized with respect to the (length-independent) average heat conductance ${\kappa }_{\text{ref}}$ per channel, as function of the temperature. (a) Equal gaps ${\mathrm{\Delta }}_{0,\text{L}}={\mathrm{\Delta }}_{0,\text{R}}$ at different phase differences $\phi$ and (b) different gap ratios ${\mathrm{\Delta }}_{0,\text{L}}/{\mathrm{\Delta }}_{0,\text{R}}$ at vanishing phase difference, $\phi =0$.

Figure 5

Heat conductance fluctuations normalized with respect to the (length-independent) average heat conductance per channel $\text{Var}\left[{\kappa }_{\text{SNS}}/{\kappa }_{\text{ref}}\right]$, as function of the temperature. (a) Equal gaps ${\mathrm{\Delta }}_{0,\text{L}}={\mathrm{\Delta }}_{0,\text{R}}$ at different phase differences $\phi$ and (b) different gap ratios ${\mathrm{\Delta }}_{0,\text{L}}/{\mathrm{\Delta }}_{0,\text{R}}$ at vanishing phase difference, $\phi =0$.