# Beta source vs TCT: Real life

In a previous post I presented a brief bibliographical research on what is supposed to happen in a silicon detector when 1) a MIP particle passes through it and 2) when an infrared laser pulse is shined onto it. Today I want to bring this to real life and see what I get. I am also interested in performing some kind of calibration of the TCT setup such that we know “how many MIPs we are shining into the detector”, this however will probably go into a different post.

## Introduction

All the results I am presenting here were obtained by using the following two detectors:

1. PIN diode:
• AIDA V2 RUN 12916 IP47 W3-DB45.
• Thickness = 50 µm.
• Vbias = 99 V.
• AIDA-2020 V1 CNM RUN 11478 W5-DA11 (alias Speedy Gonzalez).
• Thickness = 35 µm.
• Vbias = 100 V.

Ideally, the PIN diode is better for this comparison/calibration because it is a simpler device and it has no gain or strange things that can affect the results. I proceeded with both devices, however, because I am learning new stuff with every single thing I do with these things. The objective of this post is to see if the theory I wrote in this post is valid. I want to use this understanding to later perform a MIP calibration for the intensity of the laser in the TCT to do a timing characterization of the TCT (a future post, not today). For this purpose I want to use our “best LGADs” which are the Speedy Gonzalez brothers[1]The “Speedy Gonzalez brothers” are two LGAD diodes which had a temporal resolution of around 27 ps. See this post for more detailed information.. Unfortunately we have no PIN diodes with the same thickness to compare, so the plan is to do the MIP calibration specifically with the LGAD detector.

A block diagram of the setup for today would be like this:

The “amplifier → oscilloscope” part was left unchanged during the measurements. The silicon detector was installed in the TCT, even for the beta source measurements, in order to have as most as possible the same conditions for both cases. Here some photos of the setup inside the TCT with the radioactive source on top:

To switch to the TCT, I just removed the radioactive source and the metallic cap, and moved the stages such that the optic window of the PIN/LGAD was in the focus of the laser. The PIN diode (and also the LGAD) looks like this:

The material that is surrounding the optical window is the metallic pad on top of the device. In this post there are more details on the structure of these devices.

## Beta source in real life

### Beta source with PIN diode

Let’s start with the beta source, which is a Sr-90 source of beta particles. A typical signal coming out of the PIN diode due to a beta particle looks like this:

Some quantities that are calculated from this signal are indicated in the plot, you can zoom in for more detail. Today we are mainly interested in the distribution of the collected charge. This distribution (in arbitrary units because I have no knowledge of the exact amplifier’s gain) looks like this:

As can be seen, it resembles a Landau distribution. I have fitted a Moyal distribution, wich apparently is a good approximation to the Landau distribution[2]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.3.2., in order to qualitatively check the “Landauness of the collected charge”. Though the fit is not perfect, it looks reasonable to me.

In the LGAD diode the number of electron-hole pairs should follow exactly the same distribution as for the PIN. The collected charge, however, will be different due to the presence of the gain layer in the LGAD. In this layer a gain of around 10-20 is achieved by an avalanche process. Since this process has a random component too, the expected Landau distribution for the collected charge may be distorted. I did not stop to think on how should it be distorted, I just measured it and compared it to the Moyal distribution. Here are the results:

In this case we clearly see that the fit is not as good as for the PIN diode.

Though the LGAD and the PIN have a different thickness, we can still roughly estimate the gain of the LGAD diode as

\begin{aligned}\text{LGAD gain} & =\left.\frac{\text{Collected charge by LGAD}}{\text{Collected charge by PIN}}\right|_{\text{both have the same thickness}}\\ & \sim\frac{\text{Collected charge by LGAD}}{\text{Collected charge by PIN}}\times\frac{\text{PIN thickness}}{\text{LGAD thickness}}\\ & =\frac{397\text{ pico a.u.}}{60\text{ pico a.u.}}\times\frac{35\text{ µm}}{50\text{ µm}}\\ & \approx9.5. \end{aligned}

This should be approximately true since the number of electron-hole pairs produced by a MIP is “approximately proportional” to the thickness of the device assuming that the fraction of energy lost by the MIP in the device is negligible[3]Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Chapter 3., which for thin detectors should be the case.

## TCT in real life

### TCT with PIN diode

Now let’s move to the TCT. According to my (not necessarily correct) theoretical analysis in this post, the collected charge should follow a Gaussian distribution

$Q\sim\text{Gaussian}\left(\mu=gnp,\sigma=g\sqrt{np\left(1-p\right)}\right)$

where $$n$$ is the number of photons in each pulse, $$p\sim1-\exp\left(\frac{\text{detector thickness}}{1\text{ mm}}\right)\approx5\text{ %}$$, and $$g$$ is a factor that converts the number of electron-hole pairs produced to the measured collected charge, i.e. $$g \approx e \times \text{amplifier gain}$$ with $$e$$ the electron charge. To verify if $$Q$$ has such distribution it is possible to proceed like this: Measure the distribution of $$Q$$ for different values of $$n$$, estimate $$\mu_i$$ and $$\sigma_i$$ for each $$n_i$$ and then observe the ratio

$\frac{\mu_{i}}{\sigma_{i}^{2}}=\frac{1}{g(1-p)}$

which turns out to be a constant. To vary $$n$$ I basically used the “LASER pulse width” in the Particulars LASER control software, for which a screenshot is shown below. I don’t exactly understand how this parameter works or should be interpreted, in my experience low percentage values yield a large number of photons while high percentage values yield a low number of photons. I have also observed that for $$\text{LASER pulse width} \gtrsim 66 \text{ %}$$ the signal coming out of the detector completely disappears.

The collected charge by the PIN diode for different values of the parameter $$\text{LASER pulse width}$$ looks like this (please click/double click on the legend to hide/show traces):

I have plotted, along with the measured distributions, fits of Gaussian functions too. From this fits we see that the collected charge is effectively very Gaussian, as opposed to the “Landau-ian” shape with the beta source. Can we say that these Gaussian distributions come from a binomial distribution with very high $$n$$, as “the theory says”? Let’s have a look at how the quantity $$\mu/\sigma^2$$ varies as the $$\text{LASER pulse width}$$ is changed. The two plots below show the parameters $$\mu$$ and $$\sigma$$ for the fitted Gaussians as a function of $$\text{LASER pulse width}$$, and also the ratio $$\mu/\sigma^2$$:

As said before, if these Gaussian distributions are due to a binomial with big $$n$$, the quantity $$\mu/\sigma^2$$ should be a constant, independent of $$n$$, i.e. independent of $$\text{LASER pulse width}$$. We observe here that this is not the case. I still don’t have an explanation for this result. Possible reasons that come to my mind:

• The number of photons delivered by the laser has non-negligible fluctuations.
• The number of photons in each pulse is very high such that the electron-hole pairs produced in the detector is high enough to saturate it, making the absorption of each photon no longer independent from the others.
• Other reasons?

Let’s now have a look at the exact same process but replacing the PIN diode by the LGAD. The collected charge for different values of $$\text{LASER pulse width}$$ is:

At a first glance this behavior looks different from the one with the PIN. If we look at the dependence of $$\mu/\sigma^2$$ with $$\text{LASER pulse width}$$

we now see that there are two regions; specifically for $$\text{LASER pulse width} > 65.8\text{ %}$$ this ratio seems to be constant, pointing towards the binomial distribution. As I said before, higher values of $$\text{LASER pulse width}$$ imply less number of photons (via shorter pulses). The binomial characteristic, thus, seems to be present when the number of photons in the pulse is smaller, and something opaques this behavior for pulses with higher number of photons. In fact, for the PIN diode the binomial characteristic was not observed and now I realize that $$\text{LASER pulse width}$$ was measured far outside the range where it shows the binomial characteristic with the LGAD. When setting up the measurement these ranges were chosen based on signal to noise considerations. It seems that the internal gain of the LGAD allows for bigger values of $$\text{LASER pulse width}$$ to be used (i.e. less photons).

Is it possible that the laser is sending such a big number of photons when $$\text{LASER pulse width} \lesssim 65.8 \text{ %}$$ that the silicon gets saturated and can no longer absorb photons in an independent way? Still don’t know, this will require an estimation of the number of electron-hole pairs that silicon can produce per unit volume per unit time, an estimation of the number of photons delivered by the laser and an estimation of the pulse width in time. The pulse size in space has already been estimated, see this post.

## TCT vs beta source (MIP calibration)

Now I want to compare the collected charge in each setup, and find a value of $$\text{LASER pulse width}$$ such that the TCT becomes “equivalent” to the beta source. As I analyzed in my previous post, this is possible because the absorption length for $$1064 \text{ nm}$$ photons in silicon is $$\sim 1 \text{ mm}$$ and our devices are $$\sim 50 \text{ µm}$$. The laser beam, thus, makes its way through the whole thickness of the device and the charge carriers are produced approximately uniformly along all the height of the device[4]This “infrared laser scenario” is the opposite case to a red laser which has an absorption length $$\sim 5 \text{ µm}$$ and so the carriers are produced very close to the surface of the … Continue reading, in a similar way as it happens for a MIP. There are some important differences, however, that make the signals have different distributions.

The plot below compares the distribution of the collected charge with the PIN diode both with the beta source and with the TCT, for some different values of $$\text{LASER pulse width}$$.

We see that no matter which $$\text{LASER pulse width}$$ the signals will always have a different distribution. The most appropriate value for $$\text{LASER pulse width}$$ I think is the one that makes the most probable value for both distributions to coincide.

If we do this comparison for the LGAD detector:

we see that the distribution with the beta source is less “Landau-ian”, as pointed out before, making smaller the difference between the radioactive source and the TCT.

## Conclusions

• The distribution of the collected charge in a PIN diode and in an LGAD detector was studied both using a radioactive Sr-90 beta source and an infrared pulsed laser in a TCT setup.
• The collected charge by the PIN diode with the beta source follows clearly a Landau distribution.
• The collected charge by the LGAD detector with the beta source also shows a “Landau-ian” distribution but deformed with more weight for small values. Is this related with the gain process?
• In the TCT both devices exhibit a Gaussian distribution, as expected. However, this distribution should be the Gaussian limit of a binomial and this was not observed for the PIN, and was partially observed for the LGAD. The reason for this is yet unknown.
• A comparison between the TCT and the radioactive source was done for both PIN and LGAD. Though the TCT is capable of producing signals with the same “most probable value”, the long tail of the Landau distribution produced by the beta source is not possible to obtain (trivially) with the TCT. How will this impact results (timing/space) is unknown.

References

↑1 The “Speedy Gonzalez brothers” are two LGAD diodes which had a temporal resolution of around 27 ps. See this post for more detailed information. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Sec. 3.2.3.2. Kolanoski, Hermann, and Norbert Wermes. Particle Detectors: Fundamentals and Applications. Particle Detectors. Oxford University Press, 2020. https://cds.cern.ch/record/2721300. Chapter 3. This “infrared laser scenario” is the opposite case to a red laser which has an absorption length $$\sim 5 \text{ µm}$$ and so the carriers are produced very close to the surface of the silicon.