## 2 TES Physics and Models
### The Proximity Effects and the Resistive Transition  
A TES consists of a thin film made of a superconducting bilayer with an intrinsic critical temperature \(T_{ci}\) typically around \(100\,\mathrm{mK}\). It is directly connected to superconducting bias leads with critical temperature \(T_{cL}\gg T_{ci}\). Figure 1a shows a schematic of a TES connected to the bias superconducting leads, while more details on the TES structures and their fabrication are reviewed in Section 3. It was first reported by [PERSON] et al. [20; 21] that TES structures behave as a superconducting weak-link due to the long-range longitudinal proximity effect (LoPE) originating from the superconducting Nb leads. Their conclusion was based on three solid experimental findings, namely (i) the exponential dependence of the critical current \(I_{c}\) upon the TES length \(L\) and the square root of the temperature difference \(T-T_{ci}\):  
\[I_{c}(T,L)\propto\frac{L}{\xi(T)}e^{-\frac{L}{\xi(T)}}, \tag{3}\]  
with \(\xi(T)=\xi_{i}/(T/T_{ci}-1)^{1/2}\) for \(T>T_{ci}\), (ii) the scaling of the effective transition temperature \(T_{c}\) and the transition width \(\Delta T_{c}\) as \(1/L^{2}\), and (iii) the Fraunhofer-like oscillations of the critical current as a function of the applied perpendicular magnetic field. The longitudinal proximity effect was observed over extraordinary long (\(L>100\,\mathrm{\SIUnitSymbolMicro m}\)) distances and it is responsible for the enhancement, in the proximity of the leads, of the spatially varying superconducting order parameters \(\Psi\) of the TES bilayer, as shown in Figure 1a. It has been shown as well [21] that the order parameter is suppressed in proximity to normal metal structures deposited along or on top of the TES due to the lateral inverse proximity effect (LaiPE). Both the longitudinal and the lateral-inverse proximity effects can be used to fine tune the effective TES critical temperature \(T_{c}\). By measuring many devices with different geometries and aspect-ratios, fabricated from the same bilayer film, it is possible to separate the contribution of the longitudinal proximity effect from that of the lateral inverse proximity effect. In [22], for example, the effect of the Nb leads has been identified by measuring \(T_{c}-T_{ci}\) of TESs of equal width \(W\), but different lengths \(L\). The results are then used to correct for the longitudinal proximity effects and look at the lateral inverse proximity effects in the devices of varying widths. [PERSON] et al. [23] have recently investigated the effects of the longitudinal proximity effects in Ti/Au TES structures using Ti and Nb superconducting leads material, with low and high \(T_{CL}\) respectively. A comparison was done with the previous results obtain with Mo/Au [21] and Mo/Cu [15] TESs. The measured characteristic length scale of the proximity effect was the lowest for the Ti/Au TES with Ti leads. The reported \(T_{c}-T_{ci}\) was about 2.5 higher for the Ti/Au TES with Nb leads, and almost 8 times higher for the MoAu TES with Nb leads. Figure 1: (**a**) Schematic diagram of a TES bilayer connected to the bias leads and the absorber. The red curves represent the superconducting order parameter for an SS’S (solid) and a SNS (dashed) TES weak-link. (**b**) Thermal model of a TES-Absorber system (\(C_{1}\)) including a second thermal body \(C_{2}\). \(G_{1,b}\) is the main thermal conductance to the bath, \(G_{\mathrm{bath}}\), and \(G_{2,b}\ll G_{1,b}\). (**c,d**) The electrical circuits for the ac and dc bias read-out, respectively. The discovery of the proximity effects in TES structures naturally leads to treat these devices as superconducting SNS or SS'S weak-link, with the bilayer being N when \(T>T_{ci}\) or S' when \(T<T_{ci}\). Extensive reviews on the Josephson weak-links have been given by [PERSON] [24] and [PERSON] et al. [25]. The Josephson effect in SS'S junctions at arbitrary temperatures was analysed first by [PERSON] and [PERSON] [26; 27] in the framework of the Usadel equations.