## 4 Single Pixel Optimization
### dc Bias  
For many years, the efforts to improve the single pixels performance of TES microcalorimeters have been devoted in trying to suppress the poorly understood excess Johnson noise, typically increasing at larger \(\beta\). It was empirically observed that adding normal metal stripes on top of the TES bilayer would mitigate the overall noise [74; 109]. Detectors with very good energy resolution have been fabricated following these finding [15]. With the increasing demand of larger and larger arrays of pixels, it was soon realized that normal metal structures could strongly affect the shape of the resistive transition in a way that it was not always controllable and reproducible. Moreover, kinks in the transition  
Figure 9: Scanning electron micrographs of the (**a**): evap-Bi and (**b**): elp-Bi absorber cross-sections. The evap-Bi grain appears smaller than that of the elp-Bi, and shows a columnar structure. Reprinted from [105] by permission from AIP Publishing. Copyright ©2017, AIP Publishing. have also been regularly observed with those devices. This is partially due to the fact that the proximity effects discussed in Section 2.1 are strongly dependent on the device geometry, the transmissivity of the leads-TES interface and other parameters of the SNS structures, such as \(T_{c}\), and coherence length. In a series of very detailed experiments, it was shown, by [PERSON] et al. [34; 110] and in later works [76; 111], how the normal metal structures added to the TES, including the thermal coupling stems between TES and absorber, could dramatically change the resistive transitions shape and the current flow [112], as a complex interplay of proximity effects, non-equilibrium superconductivity, and self-induced magnetic field. An improved TES design was highly desired, in particular because the dc bias multiplexing schemes do not allow a single pixel bias optimization. Due to the common bias configuration, they require an extremely high uniformity over a large array of pixels. The traditional strategy of suppressing \(\zeta(I)=(1+2\beta)(1+M^{2})\) by adding normal stripes on top of the bilayer has the side effects of reducing \(\alpha\) as well, since \(M^{2}\),\(\alpha\) and \(\beta\) were found to be correlated [15; 34]. An alternative approach was needed. From Equation (2), it is known that, in the small signal limit, for a given detector \(T_{c}\) and heat capacity \(C\), to achieve the best energy resolution one needs to minimize the factor \(\sqrt{\zeta(I)}/\alpha\), which depends on the TES geometry and where \(\zeta(I)\) generally increases with \(\beta\). In a study on the performance of Mo(50 nm)/Au(200 nm) TESs with different stripes configurations [113], it was shown that pixels without stripes could achieve excellent resolution even with a large non-equilibrium Johnson noise in excess. An excellent energy resolution of \(1.58\pm 0.12\) eV at \(5.9\) keV was reported with a small, \(50\times 50\) um\({}^{2}\), bare TES with \(T_{c}=95.6\) mK, coupled to a Bi(\(4.04\) um)/Au(\(1.49\) um) \(240\times 240\) um\({}^{2}\) absorber with heat capacity \(C=0.69\) pJ/K. The stripe-less small device had much larger values of \(\alpha\)(\(\sim\)1600), \(\beta\)(\(\sim\)86) and \(M^{2}\)(\(\sim\)6) with respect to the larger size TESs with stripes (\(\alpha\sim 75\), \(\beta\sim 1.25\) and \(M^{2}\sim 0\)) [114], which showed energy resolution around \(2\) eV and were under consideration for X-IFU. This result was a real breakthrough in the field, because it suggested that the parameter space for the optimization of TES microcalorimeters for ultra sensitive X-ray spectroscopy, could be much larger than what it was thought before. The major advantages of removing the normal metal stripes from the top of the bilayer and of employing small TESs are the smaller sensitivity to the external and self-magnetic field, the improvement of the uniformity of the TES resistive transition itself and an increased uniformity within the array.