Superconducting tunnel-junction refrigerator

The dc current through an $S_1-S_2$ tunnel junction, with ${\Delta }_2$ greater than ${\Delta }_1$, when biased with $\mathrm{eV}<\mathrm{}{\Delta }_1+{\Delta }_2$, will lower the energy in $S_1$. This energy reduction will be shared by the phonons and electrons. This device is shown to be analogous to a thermoelectric refrigerator with an effective Peltier coefficient ${\pi }^{*}\sim \frac{{\Delta }_1}{e}$. Tunneling calculations yield the cooling power $P_c$, the electrical power $P_e$ supplied by the bias supply, and the cooling efficiency $\eta =\frac{P_c}{P_e}$. The maximum cooling power is obtained for $eV=\pm ({\Delta }_2-{\Delta }_1)$ and $t_1=\frac{T_1}{T_{c1\mathrm{}}}\sim 0.9$. Estimates are made of the temperature difference $T_2-T_1$ achievable in Al-Pb and Sn-Pb junctions with an ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ tunneling barrier. The performance of this device is shown to yield a maximum cooling efficiency $\eta \simeq \frac{{\Delta }_1}{({\Delta }_2-{\Delta }_1)}$ which can be compared with that available in an ideal Carnot refrigerator of $\eta =\frac{T_1}{(T_2-T_1)}$. The development of a useful tunnel-junction refrigerator requires a tunneling barrier with an effective thermal conductance per unit area several orders of magnitude less than that provided by the ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ barrier in the Al-Pb and Sn-Pb systems.