# Fully Superconducting Josephson Bolometers for Gigahertz Astronomy
## 3 Theory of Nano-TES and JES Bolometers  
The behavior of superconducting bolometers strongly depends on the predominant heat exchange mechanisms occurring in their active region. The thermal model describing all the main thermal exchange mechanisms occurring in \(A\) is the same for the nano-TES and the JES, as schematically shown in Figure 3a. Here, \(P_{in}\) is the power associated with the absorption of the external radiation, while \(P_{A-B}\) represents the energy out-diffusion for the active region to the lateral leads kept at the bath temperature (\(T_{bath}\)). We notice that depending on the device operation, nano-TES or JES, the bath temperature is set to \(T_{C}(I)\) or \(T_{e}(I)\), respectively. The lateral electrodes can act as energy filters (\(P_{A-B}\to 0\)), the so-called Andreev mirrors [43], when their critical temperature (\(T_{C,B}\)) is much higher than the zero-current critical and escape temperatures of the active region (\(T_{C,B}\gg T_{C}(0),T_{e}(0)\)), that are the maximum operating temperatures. Therefore, heat exchange with lattice phonons (\(P_{e-ph}\)) is the predominant thermal relaxation channel in the active region. In the TES operation, \(A\) is always kept almost in the normal-state thus yielding an electron-phonon coupling [17,44]  
\[P_{e-ph,n}=\Sigma_{A}\mathcal{V}_{A}\left(T_{A}^{5}-T_{bath}^{5}\right), \tag{6}\]  
where \(\mathcal{V}_{A}\) is the nanowire volume, and \(\Sigma_{A}\) is its electron-phonon coupling constant. On the contrary, since the JES operates at \(T_{e}(I)\), the latter can be substantially smaller than \(T_{C}\) depending on \(I\) (see Figure 2f) with \(A\) deeply in the superconducting state. The electron-phonon heat exchange in a superconductor takes the form [45]  
\[P_{e-ph,s}\propto P_{e-ph,n}\exp{[-\Delta_{A}/(k_{B}T_{A})]}, \tag{7}\]  
where \(\Delta_{A}\) is the superconducting energy gap in \(A\), thus showing an exponential suppression with respect to the normal-state at very low temperatures. As a consequence, the performance of a JES radiation detector is expected to be considerably better than that of a nano-TES operating at the same temperature. The transition to the normal-state driven by radiation absorption generates Joule heating in \(A\), thus rising \(T_{A}\) and creating thermal instability. For TES bolometers, this issue  
Figure 3: General thermal and electrical model of the nano-transition edge sensor (TES) and Josephson escape sensor (JES). (**a**) Thermal model of the nano-sensors where the main thermal exchange channels are shown. \(P_{in}\) is the power coming from the incoming radiation, \(P_{e-ph}\) is the heat exchanged between electrons at \(T_{A}\) (violet) and lattice phonons at \(T_{bath}\) (green), and \(P_{A-B}\) is the heat current flowing towards the superconducting electrodes at \(T_{bath}\) (blue). (**b**) Schematic representation of a typical biasing circuit for the sensors. The parallel connection of the sensor (\(R\)) and the shunt resistor (\(R_{S}\)) is biased by the current \(I_{Bias}\). The role of \(R_{S}\) is to limit the Joule overheating of the active region when transitioning to the normal-state. The \(I\) variations are measured thank to the inductance \(L\), for instance by a SQUID amplifier. is solved by a biasing circuit implementing the so-called negative electrothermal feedback (NETF). Similarly, the nano-TES and the JES could be biased with the circuitry shown in Figure 3b. The shunt resistor (\(R_{S}\)) guarantees that the current (\(I\)) flowing through the sensor (\(R\)) is limited when the active region undergoes the superconducting to normal-state transition. In TES operation, the sensor is biased at \(T_{C}\) (\(R=R_{N}/2\)) so that the condition for the shunting resistor reads \(R_{S}=IR_{N}/2(1_{Bins}-I)\)], where \(I_{Bins}\) is the current provided by the generator.