## 2 TES Physics and Models
### The Proximity Effects and the Resistive Transition
The device is superconducting (\(R(T,I)=0\)) for \(I/I_{c}<1\) and normal (\(R(T,I)=R_{N}\)) for \(I/I_{c}\gg 1\). Under these conditions, the above equations and the relation between \(\alpha\) and \(\beta\) can be greatly simplified following the work of [PERSON] and [PERSON] [33]. The resistive transition \(R(T,I)\) can then be written as [34]  
\[\frac{R(T,I)}{R_{N}}=\Bigg{(}1-\bigg{(}\frac{I_{c}(T)}{I}\bigg{)}^{2}\Bigg{)}^{ 1/2}, \tag{9}\]  
and \(\alpha\) and \(\beta\) reduce to  
\[\alpha=-\Bigg{(}\Big{(}\frac{R_{N}}{R}\Big{)}^{2}-1\Bigg{)}\frac{\partial\ln I _{c}(T)}{\partial\ln T}, \tag{10}\]  
Figure 1:0  
A simple relation for the \(\alpha/\beta\) ratio holds:  
Figure 1:1  
which could turn out to be useful for the pixel design optimization [35]. The \(I_{c}(T,B)\) curve can be measured rather accurately, very close to \(T_{c}\), for many orders of magnitude of the current flowing in the TES [20]. By studying the shape of \(I_{c}(T,B)\) for different TES geometry and as a function of the perpendicular magnetic field \(B_{\perp}\), in relation with the TES bias current \(I\), one can derive detailed information on the physical mechanism involved in a TES [20; 21; 34; 35; 36]. A careful TES current and voltage calibration is however required to accurately calculate the ratio \(x=I/I_{c}(T)\) at each bias point, which is needed to validate experimentally the resistive transition models. In [35], for example, the \(I_{c}(T)\) curve measured for small pitch TES X-ray calorimeters optimized for solar astronomy and with two different design, has shown a very good agreement with the Equation (12) derived from the RS] model. Figure 1:2 Example of an R(T,I) curve calculated for an X-ray TES calorimeter developed for X-IFU. The red curve shows the TES bias trajectory. There exists a class of devices, where the weak-link effects described above are less dominant. TESs could for example operate in a stronger superconducting regime, depending on the exact shape of the \(I_{c}(T)\) and the bias current, which in turn depends on the device saturation power required by the applications. It has been reported [15], for example, that large Mo/Cu TESs, \(400\times 400\,\upmu\)m\({}^{2}\) in size, with several normal metal features on top, operating at high bias current \(I\sim 200\,\upmu\)A, and with the lead \(T_{CL}\) closer to the bilayer \(T_{ci}\), typically show a \(I_{c}(T)\) curve which is well described by the Ginzburg-Landau (G-L) critical current. The latter takes the form:  
Figure 1:3  
where \(I_{c0}\) is the value of \(I_{c}\) at zero temperature. These devices have typically lower \(\alpha\) and \(\beta\) values than the ones predicted by the RSJ model (Equations (7) and (8)) [37]. The physical mechanism of phase slip lines (PSL) [38] has been proposed, in the simplified form of a two-fluid model [39; 40], as a way to explain the resistive transition in devices where the ideal weak-link effects were not observed. The PSL model reproduces the general behaviour of the IV characteristic of these devices as a function of temperature and provides a better prediction of \(\alpha\) and \(\beta\) in the transition. The PSL model has also been considered to explain kinks and steps observed along the transition in dc bias TESs [38] and is used to guide the optimization of X-ray detectors by studying the dependence of the transition width on the TES \(I/I_{c}(T)\)[41; 42]. [PERSON] et al. [37] made a comparison, updated and expanded later on [15], between the RSJ and the PSL models for many TESs with different geometries. The predictions are not always consistent with all the data presented, and the influence of the TES geometry and the effect of the connected normal metal structures is still not clear. The main result of their comparison is that, over the range of operation of a TES, for TESs larger than \(80\,\upmu\)m, the measured \(I_{c}(T)\) curves are more consistent with Equation (13). On the contrary, [PERSON] et al.