## 2 TES Physics and Models
### The Proximity Effects and the Resistive Transition
[35] have shown, with pixels developed for solar astronomy application, that TESs as large as \(140\,\upmu\)m\({}^{2}\) and with normal metal noise mitigation structure do follow the weak-link prediction as in Equations (3) and (12). It is clear that it is not always easy to discriminate between the regime of operation of a TES-based device, in particular in the presence of complicated normal metal structures affecting the current distribution in the TES bilayer. As it will be discussed in Section 4.2, there is a wish to fabricate detectors that operate away from the weak-link regime to minimize non-ideal effects that can deteriorate the detector performance. This is particularly true for ac biased devices. It is then very important to understand the underlying physical mechanisms that shape the \(I_{c}(T,B)\) curve and the TES resistive transition. To predict the exact \(I_{c}(T,B)\) behaviour in the cross-over region between the RSJ and the phase-slip models, it might be necessary to solve the Usadel equations in 2D for a realistic TES design (with normal metal stems, bars, and stripes), as a follow up of the work done in one dimensional structures [29; 30; 31]. From the experimental point of view, the full characterization of the new generation of devices, discussed in Section 4, might help in the understanding of the resistive transition. The theoretical framework reviewed by [PERSON] [24] and [PERSON] et al. [25] could guide the interpretation of the experimental data. As described in their work, the possible physical states in which a TES of different size likely operate can be illustrated in the diagram of Figure 3, simplified and reproduced after [24; 43]. The TES length \(L\) and width \(W\) are normalized with respect to the coherence length \(\xi(T)\). The solid blue line shows the boundary between the regions of large size superconducting structures, where PSL/Abrikosov vortices could be formed, and the small weak-link where the pure Josephson effect is observed. The dashed red line corresponds to a TES critical length \(L_{c}\), estimated by [PERSON] to be \(\sim\)3.49\(\xi\), above which, for small TES width \(W\), a transition from a Josephson effect, with single valued current-phase relation \(I(\phi)\) to a one-dimensional depairing state (phase slip centers), with multivalued current-phase relation, might occur. The exact shape and position of these boundaries are not trivial to calculate for a TES, since they depend on many parameters, such as the self magnetic field, the bilayer coherence length and the difference between the bilayer \(T_{c}\) and the \(T_{cL}\) of the leads. The threshold is temperature dependent as well, so the TES might switch from a weak to a strong superconducting state depending on the actual bias conditions. We have seen before that for a TES with an Au thickness of 200 nm, the coherence length close to \(T_{ci}\sim\,90\) mK could reach values, enhanced by the electrodes with \(T_{cL}/T_{ci}\simeq 70\), of \(\xi(T=1.1T_{ci})\sim 50\) um. This leads to values of \(L_{c}=3.49\xi>150\) um. This estimation, however, is rather approximated, since the cross-over boundary strongly depends on the current distributions, which in turn depend on the width and the thickness of the bilayer, as well as on the presence or not of normal metal structure in a real TES. The common practice to discriminate whether a TES behaves as a weakly-linked or strong superconductor has been so far to look for the exponential behaviour of the \(L_{c}(T)\), its Fraunhofer-like behaviour as a function of the perpendicular magnetic field, the oscillatory features in the TES reactance as a function of the MHz voltage bias TES [36], or the presence of Shapiro-steps in dc bias TES under MHz current or magnetic field excitation [44; 45; 46]. The latter effect, for example, manifest itself with a specific current step pattern in the TES \(IV\) curves at \(V_{m,n}=m/n\hbar\omega/2e\), according to the Josephson law.