## 4 Single Pixel Optimization
### ac Bias
The geometry of these devices is rectangular with dimensions (length\(\times\) width) \(100\times 20\), \(120\times 20\), \(140\times 30\), \(80\times 20\), and \(80\times 40\,\upmu\)m\({}^{2}\), which correspond to aspect ratios ranging from 2:1 up to 6:1. The results on the characterization and uniformity of a kilo-pixels array of \(140\times 30\,\upmu\)m\({}^{2}\) is reported in [121; 122]. More details on the TES design and fabrication are given in [89] and in Section 3. The TES bilayer had a \(T_{c}\sim 90\,\)mK and a squared normal resistance of \(26\,\upmu\)m\({}^{2}\). The scaling of the devices \(T_{c}\) and the thermal conductance \(G_{bath}\), with the TES geometry has been studied carefully. It is shown that the \(T_{c}\) is decreasing with increasing TES length and decreasing width, in agreement with the LaiPE and LoPE models described in [21]. The thermal conductance, evaluated at the critical temperature, was shown to scale with the perimeter of the device, as expected when the dominant process for the thermal conductance is two-dimensional radiative transport in the silicon nitride membrane [123; 124]. The thermal characterization of the devices is essential for the development of large uniform arrays tailored for the multiplexing read-out and the instrument requirements. The devices showed excellent energy resolution at 5.9 keV, with a mean of \(2.03\pm 0.17\,\)eV and median of 2.07 eV over all the tested geometries and bias frequencies from 1 to 4 MHz [22]. The best results were obtained with the \(120\times 20\,\upmu\)m\({}^{2}\), with AR = 6:1, with energy resolution consistently below 2 eV. These results show that, even with high aspect ratio and very high normal resistance devices, there appears to be no degradation in the performance and in the signal-to-noise ratio. This seems to indicate that the internal thermal fluctuation noise, for example, do not simply scale with the TES normal resistance. The good results have motivated the team to explore the performance of extreme aspect ration TESs and more exotic geometries, given the assumption that the Josephson effects should be minimized in high normal resistance devices. Devices based on TiAu TESs have been fabricated at SRON with aspect ratio as high as AR = 15:1, including TES with small width \(W=10\,\upmu\)m and meandering geometries. The characterization of these new devices is currently on going, but the first results are very promising. In Figure 12, we show the energy spectrum at 5.9 keV of \(80\times 20\,\upmu\)m\({}^{2}\) and \(80\times 10\,\upmu\)m\({}^{2}\) Ti(33)Au(210) TESs with, respectively, \(G_{bath}\sim 65\) and \(50\,\)pW/K, optimized for the MHz frequency-division multiplexing readout of X-IFU-like instruments. The \(T_{c}\) of these TESs is \(\sim\)86.5 mK, and they are both coupled to a \(240\times 240\,\upmu\)m Au (\(2.3\,\upmu\)m) absorber with a total heat capacity of \(C\sim 0.75\,\)pJ/K at \(T_{c}\). Excellent energy resolution at 5.9 keV has been achieved. An example is given with the two spectra in Figure 12 taken at about \(R/R_{N}\sim 0.1\). An energy resolution of \(dE=1.83\pm 0.07\,\)eV and \(dE=1.69\pm 0.07\,\)eV, has been achieved, respectively, with the \(80\times 20\,\upmu\)m\({}^{2}\) TES biased at 2.5 MHz and the \(80\times 10\,\upmu\)m\({}^{2}\) TES biased at 3.5 MHz. The spectra are the result of many hours of acquisition at a count-rate of about 0.8-1 pulse/s. They have been taken with the set-up described in [22] hosted in a shared facility where two other experiments were running simultaneously. Multiplexing demonstrations for X-IFU-like X-ray instruments are currently being done with SRON \(8\times 8\) and \(32\times 32\) uniform array of the high aspect ratio devices discussed above. As it will be shown in Section 5.2, the first results are very encouraging. It is likely that a further improvement in the energy resolution and multiplexing performance will be shown in the near future with the new generation of TESs.