$$\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\TERA#1{\text{ T}\text{#1}}% \def\GIGA#1{\text{ G}\text{#1}}% \def\MEGA#1{\text{ M}\text{#1}}% \def\KILO#1{\text{ k}\text{#1}}% \def\UNIT#1{\,\text{#1}}% \def\CENTI#1{\text{ c}\text{#1}}% \def\MILI#1{\text{ m}\text{#1}}% \def\MICRO#1{\text{ }\mu\text{#1}}% \def\NANO#1{\text{ n}\text{#1}}% \def\PICO#1{\text{ p}\text{#1}}% \def\FEMTO#1{\text{ f}\text{#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}}}}%$$
Matías Senger
July 21th 2021
In this document I give some details on the commissioning of the new "Chubut board", which is the Santa Cruz board made smaller. This board will be used for the long term studies of irradiated LGADs.

# Introduction

To perform the long term studies of the LGAD devices sent from Torino, I developed a new board which I called the Chubut board. This is basically (almost) the same circuit as the widespread Santa Cruz board  but implemented in a smaller board, so I can fit more inside the climate chamber simultaneously. It should also make everything easier, e.g. to put them in the TCT, etc.

The files with the design of the Chubut board, and also simulations, can be found in the GitHub repository . In some pictures of the board assembled are shown.

Pictures of a Chubut board assembled, without any detector.

# Comparison with the standard Santa Cruz board

Despite it should work out of the box, a test was made to compare the new Chubut board against the standard Santa Cruz board. For this, four boards were prepared as follows:

• HPK2 PIN diode installed in Santa Cruz board.
• HPK2 PIN diode installed in Chubut board.
• HPK2 LGAD installed in Santa Cruz board.
• HPK2 LGAD installed in Chubut board.
All the devices are from wafer 31 to ensure they are similar enough (PIN with PIN and LGAD with LGAD). These boards were exposed to beta rays from the Sr-90 source. The conditions for all the boards were exactly the same. The bias voltage for all the measurements was 111 V while the bias current was between 20 and 100 nA.

## Measurements results

At a first glance it was evident that the Chubut board was performing fine, just looking at the response in the oscilloscope based in my previous experience with Santa Cruz boards. Even though, data was recorded and a more quantitative comparison between the two boards was done. In the distribution of the most relevant quantities are compared for the LGAD devices. As can be seen, the differences between each board are negligible. For the PIN devices similar results were obtained. As a conclusion, the Chubut boards are performing as expected ✅.

Comparison of the distribution of the most relevant figures between the Chubut board and the Santa Cruz board assembled each with an HPK2 LGAD device (wafer 31) and exposed to beta rays.

# Transimpedance characterization

An important characteristic of the amplifier circuit is the transimpedance. This magnitude relates the output voltage with the input current $$\text{Transimpedance} = \frac{V_\text{out}}{I_\text{in}}$$ and allows to measure the collected charge of the detector in the board in Coulomb. In order to measure the transimpedance, the board with the PIN diode was used. The reason is that because this device has unity gain, it can be used as a "standard charge producer" which depends only in the properties of the silicon. In the following paragraphs the method will be described.

If the transimpedance is a constant, which we expect it to be, then it is possible to integrate in time the voltage and the current such that the relation between them is still valid: $$\text{Transimpedance}=\frac{\intop V\DIFERENTIAL t}{\intop I\DIFERENTIAL t}=\frac{Q_{\text{a.u.}}}{Q}$$ where $Q_\text{a.u.}$ is "the charge in arbitrary units" as I am measuring it right now, and $Q$ is the charge measured in Coulomb. $Q_\text{a.u.}$ is the number I get from a measurement in the lab while $Q$ comes from the theory as explained below.

## Theory of collected charge

When a MIP particle passes through the detector, a number $N$ of electron-hole pairs is produced. $N$ is a random variable that follows a Landau distribution  . The total charge that will be measured when integrating the current coming out from the detector will be just $$Q=eN$$ where $e$ is the elementary charge.

It is possible to relate the number of e-h pairs produced $N$ with the energy lost by the impinging MIP via $$N = \frac{E_\text{lost by the MIP in the detector}}{E_\text{to produce a single e-h pair}}$$ where $E_\text{lost by the MIP in the detector}$ is the total energy lost by the impinging MIP, which is a random variable following a Landau distribution, and $E_\text{to produce a single e-h pair} \approx 3.65 \text{ eV}$ is an effective energy required to produce a single e-h pair by ionization  .

Because these quantities are random variables following a Landau distribution, we can study the MPV, so each of the random variables now is a fixed number: $$Q^\text{MPV} = e\frac{E_\text{lost by the MIP in the detector}^\text{MPV}}{E_\text{to produce a single e-h pair}}$$ It turns out that $E_\text{lost by the MIP in the detector}^\text{MPV}$ has a simple expression for electrons in silicon given by   $$E_\text{lost by the MIP in the detector}^\text{MPV} = \PARENTHESES{0.031\ln\frac{d}{1\MICRO m}+0.128}\KILO{eV}\MICRO m^{-1}~d$$ where $d$ is the depleted thickness of the silicon detector. Replacing this in the expression for $Q^\text{MPV}$ we get $$Q^\text{MPV}_\text{MIP electrons in silicon} = e\frac{\left(31 \ln \left(\frac{d}{1 \MICRO{m}}\right) + 128\right) \frac{d}{1 \MICRO{m}}}{3.65}$$ As far as I understand, this expression should be quite general.

Using that $d = 45 \MICRO{m}$ for HPK2 devices, we get $$Q^\text{MPV} \approx 0.486 \FEMTO{C}$$ Just for the record, this is equivalent to $\approx 67 \UNIT{eh pairs} \MICRO{m}^{-1}$.

## Measurement of collected charge

For each signal measured by the PIN diode mounted in the Chubut board the "collected charge in arbitrary units", i.e. $Q_\text{a.u.}$, was calculated, as shown in the example in . The distribution of the collected charge for all the signals measured is shown in . Again we see a good agreement between the Chubut and the Santa Cruz boards. The collected charge follows a Landau distribution, the smaller peak at $\approx 0.6 \PICO{ a.u.}$ is the noise, so we can just ignore it.

Example of a signal from the HPK2 PIN diode mounted in the Chubut board produced by a beta particle and the different parameters extracted. Distribution of the "collected charge in arbitrary units" (i.e. not multiplied by the transconductance of the amplifier) from the measurements of the PIN diodes for both boards.

The most probable value for the collected charge in arbitrary units for the Chubut board measurement was estimated using a kernel density estimation. A value of $$Q_\text{a.u.}^\text{MPV} = 1.57\TIMESTENTOTHE{-12}\UNIT{a.u.} \equiv 1.57\TIMESTENTOTHE{-12}\UNIT V\UNIT s$$ was obtained.

## Calculation of transimpedance and comparison with simulations

Using the values from and we can calculate the transimpedance from such that $$\text{Transimpedance} = 3237 ~\Omega$$ From the simulation, however, a value of about 450 Ω was expected. The big difference between these two numbers is not yet understood.

# Conclusions

A new board with a single stage low power amplifier was designed, produced and tested. This board will be primarily used for the long term studies of irradiated LGADs, but can be used as a one-to-one replacement for the Santa Cruz board. The results from show that the performance of this board is practically identical to the Santa Cruz board.

The transimpedance of the board was measured in order to being able to convert the collected charge from "arbitrary units"Volt × second. to Coulomb. A big discrepancy between the measured value and the simulated one was found. The reason for this disagreement is still unknown.

# References

Ucsc Single Channel. https://twiki.cern.ch/twiki/bin/view/Main/UcscSingleChannel. Chubut board GitHub repository. https://github.com/SengerM/ChubutBoard. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Ferrero, Marco, Roberta Arcidiacono, Marco Mandurrino, Valentina Sola, and Nicolò Cartiglia. An Introduction to Ultra-Fast Silicon Detectors: Design, Tests, and Performances. Boca Raton: CRC Press, 2021. https://doi.org/10.1201/9781003131946. Energy loss measurement for charged particles in very thin silicon layers, S Meroli et al. 29 June 2011. http://iopscience.iop.org/1748-0221/6/06/P06013.