Thermal conductance and non-equilibrium superconductivity in a diffusive NSN wire probed by shot noise

We investigate diffusive nanowire-based structures with two normal terminals on the sides and a central superconducting island in the middle, which is either grounded or floating. Using a semiclassical calculation we demonstrate that both device layouts permit a quantitative measurement of the energy-dependent sub-gap thermal conductance $G_\mathrm{th}$ from the spectral density of the current noise. In the floating case this goal is achieved without the need to contact the superconductor provided the device is asymmetric, that may be attractive from the experimental point of view. In addition, we observe that the shot noise in the floating case is sensitive to a well-known effect of non-equilibrium suppression and bistability of the superconducting gap. Our calculations are directly applicable to the multi-mode case and can serve as a starting point to understand the shot noise response in open one dimensional Majorana device.