# Fully Superconducting Josephson Bolometers for Gigahertz Astronomy
## 4 Experimental Realization of JES and Nano-TES
### Density of States and Thermal Properties of the Active Region
The measurement at \(T_{bath}=250\) mK (purple) shows a value of the zero-temperature energy gap of the Al probe \(\Delta_{0,P}\simeq 200\)\(\mu\)eV, indicating a critical temperature \(T_{C,P}=\Delta_{P,0}/(1.764k_{B})\simeq 1.3\) K. Furthermore, the difference between the results obtained at 20 mK and 250 mK leads to \(\Delta_{A,0}\simeq 23\)\(\mu\)eV pointing towards a critical temperature \(T_{C,A}\simeq 150\) mK. We now focus on the heat exchange properties of the active region. The schematic representation of the experimental setup employed is shown in Figure 4d. The left Josephson junction operates as a thermometer: it was current-biased at \(I_{t}\) and the resulting voltage drop (\(V_{t}\)) reflects the variations of the electronic temperature in \(A\)[44]. Instead, the right JJ was voltage-biased at \(V_{h}>\Delta_{A,0}+\Delta_{P,0})/e\) to work as a heater [44]. The geometry of the device guarantees that the electronic temperature of the lateral superconducting electrodes and the tunnel probes are equal to the phonon bath temperature (\(T_{bath}\)). By contrast, the quasiparticles temperature of the active region is the fundamental thermal variable in the experiment. Since \(T_{A}\ll T_{C,B}\) for all the measurements, the lateral banks serve as perfect Andreev mirrors (\(P_{A-B}=0\)). Therefore, the power injected by the heater (\(P_{h}\)) relaxes only via electron-phonon interaction \(P_{e-ph}\) and out-diffuses through the tunnel junction acting as thermometer (\(P_{t}\)). The resulting quasi-equilibrium thermal exchange model is  
\[P_{h}=P_{e-ph}+P_{t}, \tag{22}\]  
as schematically represented in Figure 4e. Since the active region is in the normal-state, the power exchanged via electron-phonon interaction is described by Equation (6), while the power flowing through the thermometer reads [44]  
\[P_{t}=\frac{1}{e^{2}R_{t}}\int_{-\infty}^{+\infty}\mathrm{d}EE\ DOS_{P}(E,T_{ bath})\big{[}f_{0}(E-eV_{t},T_{A})-f_{0}(E,T_{bath})\big{]}, \tag{23}\]  
where \(R_{t}=11.6\) k\(\Omega\) is the normal-state tunnel resistance of the thermometer, and \(f_{0}(E,T_{A,bath})=[1+\exp{(E/k_{B}T_{A,bath})}]^{-1}\) is the Fermi-Dirac distribution of the active region and the superconducting probe, respectively. Above, the normalized density of states of the superconducting probe takes the form [44]:  
\[DOS_{P}(E,T_{bath})=\frac{|E|}{\sqrt{E^{2}-\Delta_{P}^{2}(T_{bath})}}\Theta( E^{2}-\Delta_{P}^{2}(T_{bath})). \tag{24}\]  
Figure 4f shows the dependence of electronic temperature in \(A\) on the injected power (\(P_{h}\)) acquired at \(T_{bath}=150\) mK for five different sets of measurements. The value of \(T_{A}\) rises monotonically from \(T_{bath}=150\) mK to about 270 mK by increasing \(P_{h}\) up to \(\sim\)7 fW. By solving Equation (22), it is possible to fit the experimental values of \(T_{A}\) as a function of \(P_{h}\). This simple model is in good agreement with the data, as shown by the black line in Figure 4f. The extracted value of the electron-phonon coupling constant of the Al/Cu bilayer is \(\Sigma_{A}\simeq 1.3\times 10^{9}\) W/m\({}^{3}\)K\({}^{5}\) in good agreement with the average of \(\Sigma_{Cu}=2.0\times 10^{9}\) W/m\({}^{3}\)K\({}^{5}\) and \(\Sigma_{Al}=0.2\times 10^{9}\) W/m\({}^{3}\) K\({}^{5}\)[44], weighted with the volumes of the copper and the aluminum layer forming the active region: \(\Sigma_{A,theo}=(\Sigma_{Cu}\mathcal{V}_{Cu}+\Sigma_{Al}\mathcal{V}_{Al})/ \mathcal{V}_{A}=1.38\times 10^{9}\) W/m\({}^{3}\) K\({}^{5}\), with \(\mathcal{V}_{Al}\simeq 1.58\times 10^{-21}\) m\({}^{-3}\), \(\mathcal{V}_{Cu}\simeq 2.25\times 10^{-21}\) m\({}^{-3}\) and \(\mathcal{V}_{A}=\mathcal{V}_{Al}+\mathcal{V}_{Cu}\simeq 3.83\times 10^{-21}\) m\({}^{-3}\) the resulting total volume of \(A\). Since the fit provides \(P_{e-ph}\gg P_{t}\), the presence of the thermometer tunnel barrier has a negligible impact on the determination of \(\Sigma_{A}\).