$$\def\KRONEDELTA#1{\delta_{#1}} \def\MODULE#1{\left|\,#1\,\right|}% \def\PARENTHESES#1{\left(#1\right)}% \def\SQBRACKETS#1{\left[#1\right]}% \def\BRACES#1{\left\{ #1\right\} }% \def\LBRACE#1{\left\{ #1\right.}% \def\RBRACE#1{\left.#1\right\} }% \def\LSQBRACKET#1{\left[#1\right.}% \def\RSQBRACKET#1{\left.#1\right]}% \def\LPARENTHESIS#1{\left(#1\right.}% \def\RPARENTHESIS#1{\left.#1\right)}% \def\ANGLEBRACKETS#1{\left\langle #1\right\rangle }% \def\SPACELONG{\hspace{10mm}}% \def\SPACEMEDIUM{\hspace{5mm}}% \def\DEF{\overset{{\scriptscriptstyle \text{def}}}{=}}% \def\UPBRACE#1#2{\overset{{\scriptstyle #2}}{\overbrace{#1}}}% \def\UNDERBRACE#1#2{\underset{{\scriptstyle #2}}{\underbrace{#1}}}% \def\REALES{\mathbb{R}}% \def\IMAGINARIOS{\mathbb{I}}% \def\NATURALES{\mathbb{N}}% \def\ENTEROS{\mathbb{Z}}% \def\COMPLEJOS{\mathbb{C}}% \def\RACIONALES{\mathbb{Q}}% \def\DIFERENTIAL{\,\text{d}}% \def\PRIME{{\vphantom{A}}^{\prime}}% \def\ORDER#1{\mathcal{O}\PARENTHESES{#1}}% \def\DIRACDELTA#1{\delta_{D}\PARENTHESES{#1}}% \def\HEAVYSIDETHETA#1{\Theta_{H}\PARENTHESES{#1}}% \def\ATAN{\text{atan}}% \def\INDICATORFUNCTION#1{\mathbf{1}\BRACES{#1} }% \def\VECTOR#1{\boldsymbol{#1}}% \def\VERSOR#1{\hat{\VECTOR{#1}}}% \def\IDENTITY{\mathds{1}}% \def\CURL{\VECTOR{\nabla}\times}% \def\GRADIENT{\VECTOR{\nabla}}% \def\DIVERGENCE{\VECTOR{\nabla}\cdot}% \def\LAPLACIAN{\nabla^{2}}% \def\REALPART#1{\text{Re}\left(#1\right)}% \def\IMAGPART#1{\text{Im}\left(#1\right)}% \def\TENDSTO#1{\underset{{\scriptscriptstyle #1}}{\longrightarrow}}% \def\EVALUATEDAT#1#2#3{\left\lceil #1\right\rfloor _{#2}^{#3}}% \def\unit#1{\text{#1}} \def\TERA#1{\text{ T}\unit{#1}}% \def\GIGA#1{\text{ G}\unit{#1}}% \def\MEGA#1{\text{ M}\unit{#1}}% \def\KILO#1{\text{ k}\unit{#1}}% \def\UNIT#1{\,\unit{#1}}% \def\CENTI#1{\text{ c}\unit{#1}}% \def\MILI#1{\text{ m}\unit{#1}}% \def\MICRO#1{\text{ }\mu\unit{#1}}% \def\NANO#1{\text{ n}\unit{#1}}% \def\PICO#1{\text{ p}\unit{#1}}% \def\FEMTO#1{\text{ f}\unit{#1}}% \def\TIMESTENTOTHE#1{\times10^{#1}}% \def\PROB#1{\mathbb{P}\left(#1\right)}% \def\MEAN#1{\mathbb{E}\PARENTHESES{#1}}% \def\VARIANCE#1{\mathbb{V}\PARENTHESES{#1}}% \def\COLOR#1#2{{\color{#2}{\,#1\,}}}% \def\RED#1{\textcolor{red}{#1}}% \def\BLUE#1{\COLOR{#1}{blue!80!white}}% \def\GREEN#1{\textcolor{green!70!black}{#1}}% \def\GRAY#1{\COLOR{#1}{black!30}}% \def\GRAY#1{\COLOR{#1}{blue!35!white}}% \def\GUNDERBRACE#1#2{\GRAY{\UNDERBRACE{\COLOR{#1}{black}}{#2}}}% \def\GUPBRACE#1#2{\GRAY{\UPBRACE{\COLOR{#1}{black}}{#2}}}% \def\REDCANCEL#1{\RED{\cancel{{\normalcolor #1}}}}% \def\BLUECANCEL#1{{\color{blue}\cancel{{\normalcolor #1}}}}% \def\GREENCANCEL#1{\GREEN{\cancel{{\normalcolor #1}}}}% \def\BLUECANCELTO#1#2{\BLUE{\cancelto{#2}{{\normalcolor #1}}}}% \def\KET#1{\left|#1\right\rangle }% \def\BRA#1{\left\langle #1\right|}% \def\BRAKET#1#2{\left\langle \left.#1\vphantom{#2}\right|#2\right\rangle }%$$
Matías Senger
August 31, 2021
This time I studied what happens with the spacial resolution of an AC-LGAD when multiple pads are interconnected together producing one single, big and extended pad. This is potentially interesting because it may allow to reduce the density of readout channels per unit area of detector while keeping its performance, simplifying thus the electronics, reducing the power consumption and also the amount of data flowing out of the detector.

# Introduction

Continuing in my work with AC-LGAD devices     now I wanted to test if there could be any improvement by changing the shape of the pads. It turns out that in AC-LGADs there is almost total freedom on what the shape of the pads is, it is not restricted to squares but instead they can take any 2D shape.
Microscope picture of the AC-LGAD used for this test, also showing the way the pads were interconnected resulting in a device with less pads but covering the whole surface.
For the tests presented in this document the used device was the one labeled RSD1 W15-A 5,3 3× 3 200.

# Results

In it is shown the collected charge in each of the four pads, each connected to one channel in the oscilloscope, as a function of position. It can be seen each of the individual "dual-pads" as was described before, specifically the CH1 in is the pad 1 from , the CH2 is pad 2, and so on.
Collected charge, in arbitrary units, measured as a function of position along the AC-LGAD using the TCT.
Unfortunately the measurement was affected by an intermittent source of noise. This can be seen in the presence of vertical noisy and no-noisy strips in the plots of . This noise, of course, worsened the results that will be shown later on. The process of taking this measurement takes several hours/days, thus each vertical strip represents a different moment in time. It was not possible to locate the origin of this noise yet, and it is still affecting new measurements in the lab.
The procedure I followed was the same as for my other measurements (see e.g.  ). Namely, I produced one training and one testing dataset which I then used with the MLE algorithm. Results can be seen in . Here we see the average reconstruction error as a function of position. In each measured $xy$ point $4$ events were recorded for the testing dataset, so at each point in the plot in the average is of $4$ events.
Average reconstruction error for the MLE algorithm as a function of $xy$ position.
In order to study the performance of the detector, I defined a total of 5 regions numbered from 1 to 4 and one "main region", as shown in . The regions 1 to 4 are just squares while the region named "main region" is the difference between the outer square and the inner square, i.e. the area between the two squares. For each of these regions the reconstruction error distribution in $x$, in $y$ and in absolute value $\sqrt{x^{2}+y^{2}}$ is shown in , and respectively. To ease visualization and compare different regions it is possible to enable/disable traces by clicking in the legend of each plot. We see that events from regions 1 and 2 have better (smaller) $y$ error, given by the width of each distribution, than events from regions 3 and 4, and the inverse is true in the $x$ direction. This can be understood by the position of each region with respect to the pads: regions 1 and 2 are closer to pads in the $y$ direction and far from pads in the $x$ direction, creating this a steeper gradient of collected charge in the $y$ direction and an almost flat dependency in the $x$ direction, as can be seen in the charge color maps of . The same applies to regions 3 and 4 inverting $x$ and $y$.
Events in the region called "main region" show, obviously, a combined effect from the other regions. Both the $x$ and $y$ components of the reconstruction error follow a strange distribution (see and ) which seems to have two components: 1) a Gaussian like main peak in the middle and 2) tails that extend further from this Gaussian main peak. A Gaussian distribution was fitted to the main peak for each coordinate (see the figures) obtaining from each \LBRACE{\begin{aligned} & \sigma_{x\text{ Gaussian fit}}^{\text{main region}}=36.5\MICRO m\\ & \sigma_{y\text{ Gaussian fit}}^{\text{main region}}=35.2\MICRO m \end{aligned} } and thus $\sigma_{\sqrt{x^{2}+y^{2}}\text{ Gaussian fit}}^{\text{main region}}\approx50.7\MICRO m\text{.}$ If, instead, we look at all the data and calculate its standard deviation we get \LBRACE{\begin{aligned} & \sigma_{x\text{ all data}}^{\text{main region}}=78.8\MICRO m\\ & \sigma_{y\text{ all data}}^{\text{main region}}=76.8\MICRO m \end{aligned} } and $\sigma_{\sqrt{x^{2}+y^{2}}\text{ all data}}^{\text{main region}}=110\MICRO m\text{.}$
Distribution of the $x$ component of the reconstruction error. The different regions are defined in . Traces can be enabled-disabled by clicking in the legend.
Distribution of the $y$ component of the reconstruction error. The different regions are defined in . Traces can be enabled-disabled by clicking in the legend.
Distribution of the combined $\sqrt{x^{2}+y^{2}}$ reconstruction error. The different regions are defined in . Traces can be enabled-disabled by clicking in the legend.

# Discussion

How can we tell if these results are good or not? In reference   I measured the spacial resolution of the exact same device when it was connected "normally". In that opportunity I obtained a spacial resolution of \LBRACE{\begin{aligned} & \sigma_{x\text{ interpad region}}^{\text{regular connection}}=6.24\MICRO m\\ & \sigma_{y\text{ interpad region}}^{\text{regular connection}}=5.96\MICRO m \end{aligned} } and so $\sigma_{\sqrt{x^{2}+y^{2}}\text{ interpad region}}^{\text{regular connection}}=8.63\MICRO m$ where the "interpad region" is the region between the four pads. Of course in this case we expect a better resolution because the pitch is $200\MICRO m$ against about twice in the current work. The question now is how to compare these two results. One possibility is to define a "spacial resolution efficiency" $\eta\DEF\frac{\sqrt{A}}{\sigma N}$ where $A$ is the area of some region $\mathcal{R}$(e.g. any of the regions in ), $\sigma$ is the spacial resolution obtained for region $\mathcal{R}$ and $N$ the number of channels. With this definition $\eta$ increases with the covered area, it is also bigger for smaller $\sigma$ which is what we want, and it becomes worse as we increase the number of channels. Using this spacial resolution efficiency we can compare the two measurements: \eta=\LBRACE{\begin{aligned} & 0.624 & & \text{for the current work}\\ & 2.39 & & \text{for the normal configuration} \end{aligned} }\text{.} Based in this quantity, the "normal configuration" from reference   seems to be better than the current configuration mixing pads.

# Conclusion

A crazy interconnection of the pads in an AC-LGAD was measured with the objective of studying how the spacial resolution degrades while keeping the same number of channels but covering a bigger area. The results showed that the spacial resolution was severely affected, becoming a factor of about $\gtrsim10$ bigger, while the area covered with this approach increased in a factor of about $8$. The results, however, may not be accurate due to the presence of an intermittent noise in the measuring setup, as was mentioned in the text. A further study on this subject may be worth.

# References

Spacial resolution of $200\MICRO m$ pitch AC-LGAD, https://msenger.web.cern.ch/spacial-resolution-of-200-µ m-pitch-ac-lgad/. First time-space characterization of an AC-LGAD, https://msenger.web.cern.ch/first-time-space-characterization-of-an-ac-lgad/. First application of the empirical likelihood function to position reconstruction in AC-LGAD detectors, https://msenger.web.cern.ch/first-application-of-the-empirical-likelihood-function-to-position-reconstruction-in-ac-lgad-detectors/.