Thermal Conductance by the Inverse Proximity Effect in a Superconductor

We study heat transport in hybrid lateral normal-metal–superconductor–normal-metal structures. We find the thermal conductance of a short superconducting wire to be strongly enhanced beyond the BCS value due to the inverse proximity effect, resulting from contributions of elastic cotunneling and crossed Andreev reflection of quasiparticles. Our measurements agree with a model based on the quasiclassical theory of inhomogeneous superconductivity in the diffusive limit.

Figure 1

(a) Scanning electron micrograph of a typical sample, together with the configuration for thermal conductance measurements. Two Cu islands are connected via a short superconducting Al wire with transparent NS interfaces. Four S electrodes (top of the image) are connected to each of the two N islands through tunnel barriers for electronic thermometry and temperature control. Inset: Sketch of the side profile of the NSN structure, consisting of an S wire connected via overlap junctions to two N reservoirs. (b) Heat transparencies and (c) thermal conductances from a numerical calculation (solid line) and an analytical approximation (dotted line) for an NSN structure with an S wire of the indicated length $l_{\mathrm{S}}=L_{\mathrm{S}}/{\xi }_0$. (d) Thermal model for the experimental setup as detailed in the text. Arrows indicate direction of heat flows at temperatures $T_1<T_2<T_0$.

Figure 2

Measured electronic temperatures $T_1$ (blue solid lines) and $T_2$ (red dotted lines) as a function of the voltage $V$ across the SINIS refrigerator on island 1. Each panel shows data acquired at three different bath temperatures $T_0$. Sample I was refrigerated with only a single NIS junction; hence the voltage axis was scaled up by a factor of 2.

Figure 3

(a) Temperature dependence of the relative temperature drop $\Delta T_2/\Delta T_1$. The symbols show the measured data, whereas the solid, dashed, and dash-dotted lines correspond to the thermal model with $G_{\mathrm{th}}$ based on Eq. (3), the $l_{\mathrm{S}}\gg 1$ limit of Eq. (2), and a numerical solution of the Usadel equation, respectively. The error bars are based on the uncertainty in the temperature calibration of the NIS thermometers. (b),(c) $T$ dependence of the $G_{\mathrm{th}}$ employed to produce the solid lines in (a), normalized to $G_{\mathrm{th}}^{\mathrm{N}}$ in (b) and to $G_{\mathrm{th}}^{\mathrm{BCS}}$ in (c).