Probability distribution maps of deposited energy with sub-pixel resolution for Timepix3 detectors

The Medipix3 Collaboration has developed the Timepix3 chip, which can provide information about individual particles interacting with a hybrid pixelated semiconductor detector. The Timepix3 generates a data packet including time and energy information for each pixel receiving a charge above the discriminator threshold. From a set of such packages generated by one particle the total deposited energy and the sub-pixel center of mass (c.m) position of the interaction can be calculated. However, sub-threshold losses will distort both the measured energy and c.m calculation. This work aims to generate probability maps for the most probable initial interaction position and energy for a given detected cluster. The correction maps were calculated for different energies and sub-pixel c.m positions. It is crucial to realistically simulate the front-end response of the Timepix3 detector to be able to match and understand the experimental data and generate accurate probability maps. The correction maps from the simulations using the simulation tool Allpix 2 are applied to experimental data utilizing an Am-241 source to irradiate a Minipix Timepix3, leading to sub-pixel resolution and improved energy resolution for Timepix3 detectors.


Timepix3 detectors
An important feature of the hybrid pixel detectors is the single particle processing principle.Instead of integrating the charge of many detected particles in one pixel, like in charge-integrating detectors, such as Charge-Coupled Device (CCD) or Complementary Metal-Oxide-Semiconductor (CMOS) sensors, these detectors are able to detect and separate the information of individual particles thanks to their segmentation.This is achieved because each pixel has its own signal processing chain that allows them to process discrete pulses.
The Timepix detectors directly convert photons or other ionizing radiation into charge carriers with the additional feature of assigning a time stamp on the arrival of the particles.For the application of spectroscopic X-ray imaging Medipix3 has been utilized so far due to its ability to apply many thresholds at the same time [1].Photon counting detectors like the Medipix3 offer a limited number of energy bins defined by several discrimination thresholds.These thresholds need to be set before data acquisition according to the material composition of the sample, which can be cumbersome for samples with unknown composition.The Timepix3, a hybrid pixelated semiconductor detector with 256×256 pixels of 55 μm pitch [2] can apply just one threshold, but the data driven mode provides Time-over-Threshold (ToT) data with up to 10-bits per pixel hit.This allows for post measurement re-binning of the spectral data and is a big advantage when in-homogeneous samples or samples of unknown composition are being studied.

Charge sharing
Primary ionization created by the incoming particle spreads out during the charge collection process leading to signal in several neighboring pixels forming clusters, as it can be seen in figure 1. Due to this charge sharing photons might be registered in a number of neighboring pixels, thus causing a blur of the image.Also, Contrast-to Noise Ratio (CNR) is influenced negatively, since some photons are counted in more than one pixel and add extra noise in photon counting mode.In ToT mode the spreading of the signal to neighboring pixels affects both the spatial and the energy resolution due to sub-threshold losses in pixels with low signal.

Objectives
The objective of this study is to mitigate the effect of the sub-threshold losses due to charge sharing and to improve spatial and energy resolution.For that purpose, probability maps for the initial position and initial energy of the particle need to be generated.

Methodology
Regarding the simulation part of this research, the simulation tool   2 was utilized [3].With   2 it was possible to simulate the c.m deposition maps for a sub-matrix of 7x7 silicon pixels for a wide range of monochromatic X-ray beams.
For the simulation of the diffusion and repulsion of the charge carriers inside the sensor volume the model from [4] was implemented as an   2 plugin.Following this model the diffusion and repulsion distributions are coupled to each other forming an ellipsoidal distribution.Furthermore, a new preamplifier model for the Timepix3 was used [5], which provides an improved modeling of the low input response and preamplifier undershoot.The working point parameters for the simulated Timepix, such as discriminator threshold and feedback current are configurable.The Timepix3 simulation generated the Time over Threshold (ToT) value, which was converted to energy (keV) after an energy calibration following the same procedure implemented for experimental data [6].For the center of mass (c.m.) calculation the calibrated energy of the interaction was used as weight.Center of mass deposition maps for a sub-matrix of 7x7 silicon pixels for a wide range -2 -of monochromatic X-ray beams and the corresponding detected energy spectra were calculated separately for cluster sizes from 1 to 4 pixels.By matching the cluster energy and sub-pixel position calculated from the cluster shapes obtained by the Timepix3 simulation with the respective values derived from the GEANT4 charge deposition information it was possible to generate probability maps for cluster position and energy.

Simulated deposition maps for center of mass of different cluster sizes
In figures 2, 3, 4, 5 the c.m values calculated from the simulation are compared for one-pixel, two-pixel, three-pixel and four-pixel clusters respectively for the case of 59.54 keV photons.The plotted c.m values were normalized to the center of the pixel.The borders of the pixel are depicted with red dashed lines.
Sub-figures 2a, 2b, 2c demonstrate the c.m calculation utilizing the simulation outputs for three separate stages in the simulation.The first output takes into account the charge that is deposited in the sensor, the second output the information for the total charge collected by the sensor pixels and finally the third output the pixel charge after the processing with the electronic chain of the simulated Timepix3.The c.m deposition calculations from these stages in the signal formation are compared.The reason for this comparison lies on the fact that during a measurement the user receives only information for the charge after its processing in the electronic chain of the Timepix3 readout.This means that the output is the result after the sub-threshold losses due to the charge sharing.
For example, for the case of the one-pixel clusters, in sub-figure 2a the c.m deposition map calculated from the simulated output that provided information on the charge that was deposited in the sensor of Timepix3 is presented.Sub-figure 2b shows the c.m deposition calculated from the charge collected in the sensor pixels, which already shows a significant loss of position accuracy due to Gaussian weighting of the individual pixel contribution.Finally, the sub-figure 2c depicts the c.m deposition map calculated from the charge after the process from the electronic chain of the Timepix3 readout.Similarly, the deposition maps for the different simulation outputs is shown in figures 3a, 3b, 3c for two-pixel clusters, in figures 4a, 4b, 4c for three-pixel clusters and in figures 5a, 5b, 5c for four-pixel clusters respectively.
It is interesting to notice that for the case of two-pixel clusters the c.m deposition map calculated with the charge after the simulated Timepix3 readout (figure 3c) and the c.m deposition map calculated with the charge collected by pixels (figure 3b) have similar shapes, but the c.m deposition map from the simulated charge deposited in the sensor (figure 3a) has a different distribution.This is explained by the fact that for the two-pixel clusters it is possible to calculate the c.m with information only from two pixels for the case of the simulation output that contains information of the charge after its process from the electronic chain of the Timepix3 readout.This results in a calculation that is limited in one direction at the time.However, for the c.m deposition map calculated by the charge that was deposited initially in the sensor there is information from the pixels that were previously ignored in the readout analysis, due to the sub-threshold losses.That means that the c.m from the charge deposited in the sensor can be calculated in both directions (lateral and vertical) simultaneously.For the three-and four-pixel clusters, as expected, the difference between the c.m deposition maps is less noticeable.

Probability maps for sub-pixel regions
From inversion of the c.m deposition maps one can derive the probability map for the initial position for a given detected pixel cluster c.m.The process of the inversion corresponds to matching events from the c.m deposition map, which was calculated by the simulated output from the charge after the process of the electronic chain, to the events of the c.m deposition map, which was calculated by the simulated output of the charge that was deposited in the sensor.
In figure 6, an example of the sub-pixel analysis for a two-pixel cluster for 59.54 keV is presented.Figure 6 depicts the absolute values of hit c.m normalized to the pixel pitch, taking advantage of the symmetry of the pixel and folding the cluster c.m values into the upper right quadrant of the pixel.The sub-pixel events for the output of the Timepix3 readout were matched with the output of the simulated charge that was deposited in the sensor.For the analysis the normalized pixel quadrant was subdivided into a 5x5 sub-pixel matrix for the Timepix3 cluster position, whereas the normalized quadrant for the charge deposition used a 200x200 binning.
As example, the c.m events calculated from the simulated output of charge after the Timepix3 readout for a small sub-pixel area between the normalized vertical position of the pixel 0.3-0.4 and from the normalized horizontal position 0 to 0.1 were matched to the events for the simulated output of the charge that was deposited in the sensor.Then the probability of these events being located in the particular sub-pixel region was calculated, as it can be seen in the plot on the right of figure 6.The same procedure was followed for all sub-pixel regions with the same step covering the whole pixel area.Figure 6.Sub-pixel analysis of two-pixel cluster for 59.54 keV on a quadrant of the pixel.On the left plot the c.m deposition calculated with the simulated events from the charge after the Timepix3 readout is shown for two-pixel clusters.The blue square depicts the sub-pixel area c.m events that are matched to the events from the simulated c.m deposition for the charge deposited in the sensor, as shown in the middle plot.The plot on the right shows the c.m probability map of the events of this specific sub-pixel region.

Probability map for initial energy
The energy spectra are distorted due to the effect of charge sharing, leading to sub-threshold losses in the final detected pixel cluster, which causes asymmetric distortion of the detected energy spectra towards the lower energies.For example, for detected energy of 59 keV there are contributions from the tails of other energies, as well as contributions from electronic noise of lower energy photons, as it can be seen in the left plot of figure 7.In figure 7 the calculated probabilities for one-pixel hits that the detected 59 keV event is generated by other energies is presented on the plot on the right.The same procedure is repeated for a wide range of monochromatic energies.

Experimental verification 2.4.1 Application of correction maps to the c.m of experimental data
For the experimental verification a 7×7 pixel matrix of Minipix Timepix3 [7] was irradiated with photons of an Am-241 source.The configuration of the simulation which was utilized to generate the probability maps matched the one used in the measurement (bias = 200 V for 500 μm Silicon sensor, DAC value for Ikrum = 5).The simulation also followed the geometrical configuration of irradiating a 7×7 pixel matrix.The choice of unmasking only a sub-matrix of the Timepix3 in the measurement instead of the whole matrix was made in order to reduce the amount of data needed for the analysis of the simulation results.
First, by placing the Am-241 source in front of the Timepix3 detector it is possible to measure the cluster size counts from each sub-pixel region for the energy of the gamma peak 59.54 keV.The total counts of each cluster size from each sub-pixel region were multiplied to the correction maps of the respective sub-pixel and respective cluster size.This resulted in the application of the correction map to each sub-pixel region.The sum of the corrected c.m deposition maps for all pixel cluster sizes is presented in the plot on the left of figure 8, for the quadrant of the pixel.
The plot on the left of figure 8 shows the result of the sum of the c.m deposition maps from all of the cluster sizes after the irradiation with Am-241, without the application of correction maps.Figure 8 represents the quadrant of a pixel with normalized values to the center of the pixel with borders from 0 to 0.5.From the mean value profile of the plot on the left of figure 8 it is clear that the majority of the clusters are located in the center and borders of the sub-pixel due mostly to the one-and two-pixel clusters.Without including the pixel borders the zoomed version of the mean value profile of the quadrant of the pixel matrix shows that the rest of the c.m distribution of the clusters is also not uniform over the pixel area.However, with the application of the correction maps obtained from the simulations a uniform irradiation pattern could be restored, as it can be seen on the plot on the right of figure 8. Indeed, the projection of the mean values for each side of the pixel matrix demonstrates a nonfluctuating behavior.

Application of correction maps to energy spectrum
On the left of figure 9 the uncorrected measured spectrum for one-pixel clusters of a single pixel of Minipix-Timepix3, which was irradiated with Am-241, is fitted with a Gaussian combined with -6 - an error function.The sigma of the Gaussian fit is equal to 1.2 keV.The initial energy probability maps (like the one presented in figure 7) were applied to their respective energy bin of the initial spectrum using spectrum stripping [8].This resulted in a corrected energy spectrum improved sigma equal to 0.63 keV, as it can be seen in the right of figure 9.

Discussion and conclusion
In conclusion, the exact pattern of the charge sharing split determines (together with the energy response due the pixel electronics) the precision of the calculation for the position and the energy of each photon.The depth at which the photon was converted modifies the pattern of the activated pixels.Therefore, the corrections for the position and the energy for a given center of mass of a pixel cluster are not unequivocally defined.Thus, each photon hit should be registered along with the energy and position probability distributions.In order to find the optimal distributions simulations and measurements have to be matched.
The c.m and initial energy probability maps from the simulation tool of   2 with the implementation of new plugin for the simulation of the pre-amplifier, noise and charge collection in the sensor were applied to the experimental results utilizing the Minipix Timepix3 detector.
-7 -The c.m correction maps lead to a more uniformly irradiated final image.Therefore, it is possible to get a "flat" response on sub-pixel resolution level.The energy correction maps were applied to experimental energy spectra and proved that it is possible to improve spectral fidelity.
The next step of the research will include the calculation of the probability maps for different detector materials (CdTe, GaAs, etc), which are of interest for medical applications due to their high atomic number.Finally, the material reconstruction with the application of correction maps to radiography images as well as tomographic X-ray images is planned.Using these probability maps to correct measured X-ray data-sets should lead to higher energy and spatial resolution spectroscopic X-ray imaging.

Figure 1 .
Figure 1.The effect of charge sharing in a pixelated semiconductor.The effect of diffusion and repulsion inside the sensor is depicted with the red cone.

Figure 7 .
Figure 7. Figure on the left shows the histogram of normalized gamma energy spectra for the energies of 58 keV, 59 keV and 60 keV.The overlap of the spectra as well as the asymmetric distortion in the lower energy tail is evident.Figure on the right shows an example of the initial energy probability map for the case of 59 keV.

Figure 8 .
Figure 8.Comparison between the uncorrected deposition map of experimental data and the application of correction maps to the experimental data.The mean value profiles of each matrix side are plotted for both cases.It is evident from the plot on the left that the distribution of the clusters is not uniform.On the contrary, the mean values of the sides of the corrected plot on the right demonstrate a steady behavior showcasing a uniform distribution of the clusters.

Figure 9 .
Figure 9.Comparison between initial Am-241 energy spectrum and the corrected energy spectrum.