Matias Senger
March 9th, 2023

In this document I discuss how to fit the langauss distribution to the signal events from a silicon detector when the signals overlap with the noise. This is useful when measuring PIN detectors or heavily irradiated LGAD devices.



When an ionizing particle impinges on a silicon detector it produces ionization charge in the form of electrons and holes that can be sensed as a peaked current signal on the terminals of the device. The amount of charge that is produced determines the intensity of the signal, the more charge the more intense the signal. Normally it is desired to obtain as much charge as possible in the output, thus some technologies such as the LGAD (on which I am working during my PhD) add internal gain to multiply the charge carriers. This gain is severely degraded by radiation and so is the amount of charge produced by a particle. Since there is a lower threshold on the minimum amount of charge that the electronics can trigger on, it is one of the key factors to measure when testing different technologies for a future application.

One way to measure the charge produced by a MIP is to expose the DUT to beta particles from a 90Sr source and observe the signals. shows an example of a signal measured with an oscilloscope connected to the output of a silicon radiation detector. This signal was measured from a PIN detector from an LGAD productionTo be specific, the RD50 FBK production of TI-LGADs., i.e. a detector without gain and a thickness on the order of 40 µm. As a consequence of being so thin and without internal gain, the signals are very small (compared to the noise) as seen in . This makes it challenging to measure the charge, as will be discussed later on.

Example of a signal produced by a MIP on a silicon detector. Several features extracted from the waveform are shown such as the amplitude, the integral of the peak, etc.

Signal processing

For the signal processing I usually just feed the waveforms straight into my signal processing module This works perfectly when the SNR is high enough, e.g. with LGADs. However, when working with a PIN (as in this work) or with heavily irradiated LGADs this becomes more challenging.

Recently I developed a new board which I called Chubut 2 which has multiple readout channels thus allowing to measure multiple signals per event. In the current work I was actually characterizing this board with a 4 pixel PIN and I noticed that there is a correlated noise component on the 4 channels. Normally, with an LGAD this is not a problem since the internal gain makes the signals high enough to neglect this noise. But in the case of the PIN (and as well for heavily irradiated LGADs) this correlated noise can be removed thus making easier (and better) the later analysis. So I recently implemented a script that computes the common mode noise component as the average of the channels and then subtracts it. It allows to improve the quality of the signals, as shown in where in the left plot is hard to tell whether there is a signal at all while in the right plot (after the fix) it is evident that CH1 has a signal.

I want to emphasize that this level of processing is normally not required for regular LGADs (as they have enough gain) but it helps a lot when working with PINs (and, I predict, with heavily irradiated LGADs). Also, this is not possible with single channel readout boards.

Example of the subtraction of the common mode noise. In the left plot it is hard to tell whether there was a signal or not. In the right plot it is evident that channel CH1 has a signal while the others don't.

Charge measurement

The procedure to measure the charge produced by MIPs using a beta source is to expose the device, record a large number of events, and then fit a langauss distribution to the data. When the SNR is high enough this is easy, since a simple threshold in amplitude allows for a good separation of background and signal events. When there is no gain, as for the PIN detector, a threshold in the amplitude does not work anymore. This is evident when looking at the distribution of the amplitude, an example is shown in . Here, the first plot shows the distribution for an LGAD (labeled as DUT, ignore the MCP-PMT trace) while the second plot shows the distribution for the PIN. In the plot for the LGAD it can be seen that a threshold in the amplitud at ~20 mV provides an excellent separation between background events (less than the threshold) and signal events (higher than the threshold). In the case of the PIN not only such threshold would classify all the events as background but it is literally impossible to separate the signal from background with a threshold, as can be seen.

Amplitude distribution (ECDF), first plot is for an LGAD labeled as DUT (ignore the MCP-PMT trace), second plot is for the PIN.

Background estimation

There is another variable that can be used to which I call t_50 and measures the time at which the highest peak in the waveform was at the 50 % of its total height. This is nothing more than the Time at 50 % shown in the signal example of . Since the oscilloscope is trigged with the MCP-PMT placed underneath the DUT, then the t_50 has a very well defined value for signals that were caused by a real particle, while it has a uniform distribution for background signals. This can be well appreciated looking at the joint distribution of the amplitude and the t_50 which is shown in . In this plot each dot is an event. The vertical column of events around 14 ns corresponds to signal events, i.e. those with well defined t_50 and high amplitude. The horizontal band of events with low amplitude are background. For a regular LGAD device this plot looks similar but the vertical column is completely separated from the horizontal strip, because the gain is enough such that signal events have a very high amplitude.

Joint distribution of the amplitude and the t_50 variable, which is defined in as the Time at 50 %.

It is possible to exploit the t_50 variable to still measure the underlying Landau distribution from the signal events. This is a two step procedure. First, a representative sample of background is taken from the data, say requiring t_50 ∈ [5 ns,10 ns]. After this selection it is possible to employ KDE to obtain a functional representation of the amplitude distribution for the background, as shown in . In this figure the x axis shows the amplitude SCALED, which is just a renormalization of the amplitude so that it fluctuates around 1. The only reason for this is that the fitting algorithm from Python is more stable and has a better convergence.

Background distribution estimation using KDE. Note: The apparent discrepancy for the DUT_CH2 channel is related to an insufficient dense x axis sampling and not to a failure in the KDE.

Fitting the signal

Once the KDE model for the background is obtained, it is now possible to include it as part of the fit model for a subset of samples in a region where we expect to measure signal, say t_50 ∈ [13 ns,15 ns]. The model to fit here is simply $$\alpha ~ \text{Background KDE}(x) + \beta ~ \text{Langauss(x)}$$ where $\alpha$ and $\beta$ specify the weight of each component. When performing the fit, it is possible to constraint the values of $\alpha$ and $\beta$ using the fact that the background events (seem to) come from Poisson point process. Thus, we can obtain an estimation of the number of background events expected in the signal region by counting how many background events were found in the background-only region before. As for the number of signal events, an estimation for this would be simply the total number of events in the signal region minus the estimation of the number of background events. To allow for some flexibility I let $\alpha$ and $\beta$ to move within $\pm \sigma$, where $\sigma$ is estimated as the square root of the expected events for each, since I am assuming a Poisson point process.

shows an example of a fit performed as described above. In each of these fits, the background model component is simply the KDE estimate (from ) appropriately normalized to take into account the number of background events in the signal region (nbackground), while the signal model component is a langauss distribution fitted to the data constraining only the number of signal events in the signal region (nsignal).

Example of a fit of to data. To better appreciate the details it is suggested to zoom in horizontally, and maybe disable some traces by clicking in the legend items.


While characterizing the new Chubut 2 board using a PIN detector I faced the issue of signals overlapping with noise, and after some thinking I arrived to this method which is probably very standard but I was not using it before. This method to perform the fit allows to get an estimate of the charge even when it partially overlaps with the noise level. The procedure is not only useful for the characterization of the Chubut 2 board but also for studies of heavily irradiated LGADs.