## 2 TES Physics and Models
### The Proximity Effects and the Resistive Transition
They showed that for long junctions and \(T\geq T_{ci}\) the \(I_{c}(T)\) has an exponential dependence, as shown in Equation (3), with the effective coherence length \(\xi(T)\) larger than the intrinsic coherence length of the material:  
\[\xi(T)=\xi_{N}\sqrt{\frac{T_{cL}}{T}\bigg{[}1+\frac{\pi^{2}}{4}\ln^{-1}\bigg{(} \frac{T}{T_{ci}}\bigg{)}\bigg{]}}. \tag{4}\]  
Here, \(\xi_{N}=\sqrt{\hbar D_{N}/(2\pi k_{B}T)}\) is the coherence length for the normal material N in the dirty limit, with \(D_{N}=v_{F}l_{e}/3\) the electronic diffusivity for a material with Fermi velocity \(v_{F}\) and mean-free path \(l_{e}\). Equation (4) can be approximated as  
\[\xi(T)=\frac{\pi}{2}\xi_{N}\sqrt{\frac{T_{cL}}{T}}\frac{1}{(T/T_{ci}-1)^{1/2}}. \tag{5}\]  
For a 200 nm Au metal film at \(T=100\) mK, with \(v_{F}=1.39\cdot 10^{6}\) m/s and thickness limited \(l_{e}\sim 200\) nm, the diffusivity \(D_{N}\simeq 0.092\) m\({}^{2}\)s and the coherence length \(\xi_{N}\simeq\ 1\) \(\upmu\)m. When studying the physics of a TES, one has to keep in mind that its coherence length \(\xi(T)\) is enhanced by a factor \(\sqrt{T_{cL}/T}\) due to the presence of the leads with a critical temperature \(T_{cL}\) and by the fact that the detector operates close to \(T_{ci}\). For typical TESs, the coherence length can become much larger than \(10\) \(\upmu\)m.  
A microscopic model of a TES as a weak-link has been developed by [PERSON] et al. [28] using the Usadel approach. Usadel equations have been used later on by [PERSON] et al. [29; 30] to investigate the effect of lateral normal metal structures and to reproduce the TES current-to-voltage characteristics. The most successful macroscopic model developed to describe Josephson junctions and superconducting weak-links of many different kind is the resistively and capacitively shunted junction (RCSJ) model. The overdamped limit (\(C\sim 0\)) of the RSCJ model, namely the resistively shunted junction model (RSJ), was formalized by [PERSON] et al. [31] for a dc-biased TES. The RSJ model can be used to calculate analytically the TES resistive transition and to generalize the TES response to an alternating current. The most interesting prediction in this work is the existence of an intrinsic TES reactance, which can be calculated exactly. As we will see in Section 2.2, this has important implications on the TES response, in particular when operating under MHz voltage biasing. Using the Smoluchowski equation approach for quantum Brownian motion in a tilted periodic potential in the presence of thermal fluctuations [32], an analytical solution for the dependency of the TES resistance on temperature and current was found [31]. In the limit of zero-capacitance and small value of the quantum parameter, we can write:  
\[R(T,I)=R_{N}\bigg{\{}1+\frac{1}{x}\text{Im}\bigg{[}\frac{I_{1+i\gamma x}( \gamma)}{I_{i\gamma x}(\gamma)}\bigg{]}\bigg{\}}, \tag{6}\]  
with \(R_{N}\) the normal state resistance, \(\gamma=(\hbar L_{c}(T,L))/(2 ek_{B}T)\) the ratio of the Josephson coupling to thermal energy, and \(x=I/I_{c}(T,L)\). Here, \(I_{c}(T,L)\) is the critical Josephson current and \(I_{1+i\gamma x}(z)\) and \(I_{i\gamma x}(z)\) are the modified Bessel function \(I_{\mu}(z)\) of the complex order, \(\mu\), and real variable, \(z\). An example of an \(R(T,I)\) curve calculated from Equation (6) is shown in Figure 2. By differentiating Equation (6), the parameters \(\alpha\) and \(\beta\) are obtained:  
\[\alpha=-\frac{\gamma}{x}\frac{\partial\text{ln}I_{c}}{\partial\text{ln}T} \frac{R_{N}}{R}\text{Im}\Bigg{\{}\frac{I_{-1+i\gamma x}(\gamma)I_{1+i\gamma x }(\gamma)}{I_{i\gamma x}^{2}(\gamma)}\Bigg{\}}, \tag{7}\]\[\beta=-1+\frac{R_{N}}{R}\Bigg{\{}1-2\text{Re}\Bigg{[}\frac{1}{I_{t\gamma x}^{2}( \gamma)}\int_{0}^{\gamma}I_{1+i\gamma x}(z)I_{t\gamma x}(z)dz,\Bigg{]}\Bigg{\}}. \tag{8}\]  
A TES generally operates in the limit of \(\gamma\rightarrow\infty\) (Josephson coupling energy larger than the thermal energy) and \(I/I_{c}>1\).