First experimental evaluation of count-rate performance for micrometre resolution deep silicon detector

Objective. An ultra-fine-pitch deep silicon detector has been developed for clinical photon-counting computed tomography (CT). With a small pixel size of 14 × 650 μm2, it has shown potential to reach micrometre spatial resolution in previous simulation studies. A detector prototype with such geometry has been manufactured, and we report on the first experimental evaluation of its count-rate performance. Approach. The measurement was carried out at MAX IV synchrotron laboratory with 35 keV monochromatic x-rays. By inserting tungsten attenuators of 50, 75, 100, 150, 200, 225, 325 μ m-thicknesses into the beam, the response of the detector to fluence rates from 3.3 × 107 to 1.3 × 1011 mm−2 s−1 was characterized. Main results. The measurement result showed that the detector exhibited count rate linearity up to 6.66 × 108 mm−2 s−1 with 13% count loss and was still functional at count rate up to 2.9 × 1010 mm−2 s−1. A semi-nonparalyzable dead-time model was fitted to the count-rate behaviour of the detector, showing great agreement with the measured data, with an estimated nonparalyzable dead time of 2.9 ns. Significance. This is the first experimental evaluation of the count-rate performance for a deep silicon detector with such small pixel geometry. The results suggest that this type of detector shows the potential to be used at fluence rates encountered in clinical CT with little count loss due to pile-up.


Introduction
Photon-counting computed tomography (CT) detectors have shown significant potential in clinical use, with high spatial resolution (Kopp et al 2018), improved contrast-to-noise ratio (Gutjahr et al 2016) and radiation dose reduction (Le et al 2010).However, the high x-ray fluence rates in clinical CT systems remain a challenge for the detector design.With a reported highest fluence rate of 5.7 × 10 8 mm −2 s −1 in commercial CT system (Persson et al 2016) and a maximum of 6 × 10 8 mm −2 s −1 for routine exam types (Szczykutowicz et al 2022), this could lead to severe pulse pile-up, resulting in count efficiency degradation and deterioration of energy resolution (Kraft et al 2012, Xu et al 2013).
As an alternative to algorithmic pile-up compensation methods demonstrated in simulation studies (Taguchi et al 2010, Kappler et al 2011), minimizing the pixel area is one approach to reducing the count rate per pixel (Kraft et al 2012).Detectors with 25 m m pixel size have been reported in previous studies (Dinapoli et al 2014, Dullin et al 2018).Dividing the pixel into depth segments can also effectively decrease the count rate in each segment (Liu et al 2016).We are developing a silicon photon-counting detector for clinical CT.By having an edge-on geometry, it is also known as 'deep silicon'.To withstand the high photon fluence rate, it has a pixel area of 14 650 m 2 m ´with 14 depth segments.Such a pixel size could lead to extreme charge-sharing effects.
However, a method based on the charge cloud distribution has been proposed to benefit from such effect, potentially achieving one micrometre spatial resolution in the simulation study (Sundberg et al 2021).The proposed detector design can therefore potentially provide substantial improvements in both count-rate performance and spatial resolution.A detector prototype with the geometry mentioned above has been manufactured, and in this work we describe the evaluation of its count-rate performance performed at the MAX IV synchrotron laboratory.The Data and code is available online (Jin et al 2024).

Detector geometry
Due to its low mass attentuation coefficient and low atomic number, the silicon detector is irradiated in an edgeon geometry to ensure a sufficient thickness to stop incident x-ray photons.Each pixel is divided into 14 depth segments in the x-ray incidence direction, which efficiently reduces the count rate per segment.There are four multi-channel application-specific integrated circuits(ASICs) surrounding the detector, with each channel connected to one depth segment for the processing of the generated pulses.The pulse-height discrimination is achieved using eight adjustable energy thresholds, which are used to sort the incoming pulses into eight energy bins, each of which is connected to an 11-bit counter that counts the number of registered events.As this is the first detector prototype, only the first depth segment of each pixel is wire-bonded to the ASICs for the measurement.
As shown in figure 1(a), the pixels are positioned along the front side of the wafer.The width of each electrode is 10 mm and the pixel pitch is 14 mm.The electrodes are distributed uniformly in parallel, but to make space for readout connections, the metal pads used for electrical transmission are arranged in a staggered pattern.The silicon wafer thickness is 650 mm.The backside of the wafer contains the electrode where the bias voltage 250 V is applied.Real photo of the detector as seen from the side of the wafer.The thick structure surrounding the active area is the guard ring (a series of concentric metal rings surrounding the border of the sensor, which is designed to terminate the electric field and reduce the amount of leakage current along the borders of the wafer).The small gray blocks indicate the metal pads for wire bonding, which are 50 mm wide.In the zoomed-in view, the strip on each metal pad is the oxide opening to connect to the electrode.
The detector consists of 384 pixels.However, since this is the first prototype, only the middle 36 pixels are wire-bonded to the ASICs, which are marked by the white parallelogram in figure 1(b).An enlarged view of this area is shown as well.Due to the small pixel pitch, it is not possible to fit the metal pads side by side.The solution is to stagger them vertically by a certain distance, as illustrated by the arrangement of gray blocks in the photo.The pixel pitch can be seen as the horizontal distance between the strips.In this way, every six rows refers to one depth segment, and the length of the segment is 504 mm.The total length of the detector in the depth direction is 7.6 mm including the surrounding guard ring.

Measurement setup
The measurement was carried out at beamline DanMAX at the MAX IV synchrotron facility in Lund, Sweden.The choice of using a synchrotron facility was motivated by the ease of attaining high fluence rates and producing a narrow, parallel pencil beam for alignment.The beamline could provide a maximum photon rate of 5 × 10 12 cps for unattenuated beam (Kantor et al 2017).In this measurement, slit-1 was set to 1.1 × 1.1 mm 2 , and slit-2 was set to 1.0 × 1.0 mm 2 to acheive a beam size of 1.0 × 1.0 mm 2 .This size was selected so that the beam could cover several pixels, which was essential for the pile-up measurement.A 35 keV monochromatic photon energy was applied.The beam was first carefully aligned to the detector in all directions and rotations.Photon fluence rates in the range of 3.3 × 10 7 -1.3 × 10 11 mm −2 s −1 were adjusted by placing tungsten attenuators of different thicknesses (50,75,100,150,200,225,325 m m ) into the beam.During the measurement, the lowest thresholds were swept from below zero keV to 52-56 keV (for 150-325 μm tungsten) or to 96-106 keV (for 50-150 μm tungsten) in steps of 0.8 keV, and for each step, the registered counts above the threshold was recorded during 0.5 s.As a measure of the total registered counts for each attenuator thickness, the measured value in a single pixel ('pixel 5') with the threshold just above the noise floor, approximately 7.3 keV, was used.All the higher thresholds were set to the maximum values, so that all events were collected by the first energy bin.A pulse detection time T c of 120 ns was used.

Dead-time model
To better describe the count-rate behavior of the detector under high fluence rates, a semi-nonparalyzable deadtime (SNP) model has been proposed (Xu et al 2013).In this model, the pulse detection period T c has been divided into two parts, a nonparalyzable dead time n t and a semi-nonparalyzable time T .c n t -Once a pulse detection period T c is activated, events happening within the nonparalyzable time could not lead to more counts.However, events occurring during the semi-nonparalyzable time could trigger a new count as long as the pulse tail is higher than the lowest threshold at the end of current detection period.An analytical formula of such deadtime model can be derived (refer to Xu et al 2013 for details) with n and m being the input and output count rates of one detector element.Output count rate m in equation (1) will reach a maximum value of /T 1 c for high input rates.Compared with the nonparalyzable deadtime (NP) model, SNP model allows the count coming in near the end of current dead-time period to be counted right after the period finishes.In this way, the detector could respond faster under high fluence rates.

Count multiplicity caused by charge sharing
With the proposed detector geometry, the size of generated charge clouds could cover several pixels, up to a maximum of 5.In this case, one incident photon would generate signals on multiple pixels, leading to the difference between the count rate and fluence rate.This count multiplicity caused by charge sharing is estimated from another measurement during the beamtime, which used a beam collimated to approximately 10 × 10 mm 2 and 35 keV photon energy, with the beam center calibrated at different positions shown in figure 2. By applying the small beam size and attenuating the beam by a factor of 8.89 using an attenuator, the pile-up effect could be ignored.Same threshold levels as in the pile-up measurement were applied.Pixel 5 was considered the center pixel.
The count multiplicity due to charge sharing is calculated as the ratio of registered counts to the number of real incident photons.Based on the detector geometry, in total 9 pixels (including the center pixel) are considered as contributors to the number of registered counts.Since pixel 1 was not working properly, the counts registered on pixel 9 was used to represent in the calculation, as these two pixels are symmetrical to the middle pixel.The multiplicity estimation is based on two assumptions.
(1) When the beam is in the 'middle' position (positions 2, 5 and 8), all photons will be registered at least once in the center pixel.The center of the charge cloud is expected to be inside the center pixel, since the beam is calibrated in the 'middle' position (positions 2, 5 and 8) and the beam size is comparable to pixel size.In this way, the generated pulse height will be highest (or with little possibility equally highest) in the center pixel compared to neighboring pixels, which means that all photons will be registered at least once in the center pixel.
(2) The beam profile is constant despite of the positions.The incident fluence rate remains the same no matter whether the beam is collimated at the 'middle' position (positions 2, 5 and 8) or the 'edge' positions (either positions 1, 4 and 7 or 3, 6 and 9).This assumption is based on the fact that the beam was collimated to approximately 10 × 10 mm 2 , which is comparable to the size of the pixel.
When the beam is collimated at the 'edge' positions, it will generate more charge sharing compared to when it is collimated at the 'middle' positions due to the separation of charge clouds into neighboring pixels.To estimate the count multiplicity on a unit pixel area more precisely, we proposed a calculation method.
First, when the beam is collimated at the 'middle' position of the center pixel, record the number of registered counts on the center pixel.According to assumption (1), this value equals the number of interacting photons, N .photon Then, with the same beam position, record the registered counts on all pixels, N .
middle Next, according to assumption (2), when the beam is collimated at the 'edge' positions, the number of interacting photons is also N .photon This time, record the registered counts on all pixels, N .edge Finally, sum up the registered counts and the total number of incident photons, respectively, for these two beam position cases.An equivalent count multiplicity on a unit pixel area can be calculated as  From the experiment, the result showed that one incident photon could generate an average of 3.15 counts.This factor is used to correct the incident fluence rate from the count rate in the analysis.

Silicon transmission correction
As mentioned in section 2.1, only the first depth segment of each pixel is wire-bonded to the ASICs for read out.According to figure 3, only 10.07% of the total incident photons will be collected in this segment, as demonstrated by the red curve between the guard ring and the segment 1 data points.Under the circumstances, a translation factor of / 100% 10.07 % 9.93 = is applied to estimate the total fluence rate of incident photons from the measurement result carried out on the first depth segment.
It is also worth mentioning that, with the applied 35 keV photon energy incident on silicon detector, only photoelectric interactions were registered in the measurement.Compton interactions deposited energies lower than the threshold levels, while Rayleigh scattering did not contribute to the measured signals.In this case, the ratio of photoelectric interactions to total interactions becomes another factor contributing to the estimation of the total fluence rate.This ratio is calculated as the photoelectric absorption coefficient divided by the total attenuation coefficient, which is 0.7148 cm 2 g −1 /0.9652 cm 2 g −1 = 0.74, with the data acquired from the NIST database (Berger et al 2010).The true fluence rate could be estimated from the measured fluence rate by dividing this photoelectric absorption factor.noting that the nonparalyzable deadtime is the minimum time difference that allows the tail of the second pulse to trigger a second count after the end of current pulse detection period, and this time can be much shorter than T .c Previous works (Xu et al 2013, Liu et al 2016) have also demonstrated that it is realistic to encounter a nonparalyzable deadtime that is significantly shorter than the total pulse detection time.In addition, a possible explanation for such a short n t is that, due to the pile-up, the pulses overlap with each other, causing the spectrum to shift to higher energy.More counts are registered as they end up above the threshold.Such effect somewhat counteracts the count loss caused by pile-up, increasing the output count rate so that it becomes nearly linear in the input count rate and thus yielding a short effective .

Results and discussion
n t From figure 4(b), a near-linear relationship between the input and output count rates is observed up to 1.42 10 cps 6 ´(with 13% count loss), and 3.02 10 cps 6 ´(with 23% count loss).Taking the pixel size, the charge sharing factor, the silicon transmission factor and the photoelectric absorption factor into account, the corresponding fluence rate is 1.42× ( / 10 14 6

×
)/ 650 10 3.15 6 ´-× / 9.93 0.74 mm s ´-in agreement with the semi- nonparalyzable model.Since similar experimental data from clinical CT systems reported highest fluence rate with centered patients to be 3.4 10 mm s 8 2 1 ´--in Persson et al (2016) and 3.5 10 mm s 8 2 1 ´-in Szczykutowicz et al (2022), the evaluated detector could work linearly in these regions.
To gain further insights about how the detector responds to different fluence rates, figure 6 shows the registered count rate as a function of threshold for three pixels (pixels 4-6).Since these plots show the total number of counts above each threshold levels, they can be viewed as integrals of the deposited energy spectra.The large peak below the threshold is the noise floor.Since the incident spectrum is monochromatic, the count rate should ideally be constant between the noise floor and 35 keV, where it should drop to zero.The fact that the count rate decreases gradually with threshold energy indicates severe distortion of the energy spectra.At lower count rate (figures 6(a)-(b), this distortion is primarily due to charge sharing, but as the count rate increases (figures 6(c)-(e), the spectrum is increasingly shifted towards higher energies due to pile-up, which can cause the energies of two or more photons arriving simultaneously to be added together and misregistered as a higherenergy event.

Conclusion
In this work, the count-rate performance of a novel deep silicon detector prototype for photon-counting CT is evaluated at MAX IV synchrotron facility.By having small pixel size and segmented structure, the detector demonstrates linearity up to an incident photon fluence rate of 6.7 10 mm s 8 2 1 ´-with 13% count loss, and can withstand a maximum fluence rate of 2.9 10 mm s 10 2 1 ´-before saturation.The fluence rates that can be measured without major performance loss due to pile-up correspond to the highest rates occurring in typical clinical CT examinations, although it should be noted that the photon energy used here is lower than that in CT.
A semi-nonparalyzable dead-time model is applied to describe the count-rate behavior of the detector and is shown to fit well with the measured data.Further investigation includes the evaluation of count-rate performance of different depth segments.In this work, all the recorded counts were collected by the first depth segment, then the total counts on all 14 segments were predicted based on the x-ray attenuation of silicon at different depths.The real performance of the depth segments needs to be evaluated.Since the 14 depth segments are connected to different ASICs channels and processed separately, we do not expect any degradation on the resolution when using all depth segments.On the other hand, better detection efficiency is expected as more segments are used for counting.One possible effect is the alteration of the electric field in a particular segment when activating neighboring depth segments, but we do not expect this to be a major influence.More experiments would be needed to quantify such effect.
Furthermore, how the detector performs at the higher incident photon energies encountered during clinical use needs to be studied.Although synchrotron radiation was selected for this experiment due to the ease of performing the experiment in its setting, the synchrotron x-ray beam differs from the spectra used in x-ray CT in that it is monochromatic and has lower photon energy.In this experiment, the photon energy was set to 35 keV which is the maximum available energy at this beamline.For this reason, with the applied threshold levels, we could only detect photoelectric events.With higher energies and future lower-noise electronics, it would be possible to register Compton events as well.
Finally, investigation of the spatial and spectral resolution is necessary.Previous simulation studies (Sundberg et al 2021) have demonstrated the potential to achieve a detector with 1 μm spatial resolution based on Compton interactions using a realistic CT spectrum with clinical photon energies.This is achieved using a charge-cloud fitting procedure, a method that could also mitigate the severe spectral distortion due to charge sharing observed in our measurements, and in future work we plan to investigate this techniqe experimentally and assess how sensitive it is to the overlapping of charge clouds that can occur at high fluence rates.It should be noted, however, that the system resolution in x-ray CT is also influenced by other factors, such as the focal spot size and the quantum noise, so that the resolution that can be achieved in practice remains to be investigated.Even though the photon statistics in such small pixels will be limited at low count rates, it is always possible to bin together pixels to attain good statistics at a lower resolution if desired.Determining the optimal count rate of the detector without decreased spectral and spatial performance is crucial in the future work.
Since this is the first prototype, some possible modifications may be applied to future detectors as well.For example, the current absorption length of x-ray is not enough to stop all incident photons, even if all depth segments are used.This can be solved by using a longer silicon wafer, leading to improved total detection efficiency and better count-rate performance.Several technical challenges remain to be addressed before a fullfield of view detector with high absorption efficiency could be achieved, including processing the signals from a large number of channels which requires more ASICs along with the associated heat generation, as well as the need to read out and process the generated data.We hypothesize that integration of CMOS electronics into the silicon and online data processing before readout from the front-end electronics may be part of the solution to some of these challenges.

Figure 1 .
Figure1.(a) Schematic plot of the detector geometry as seen from the perspective of the x-ray source.The image is not to scale, and the number of pixels is just for indication.(b) Real photo of the detector as seen from the side of the wafer.The thick structure surrounding the active area is the guard ring (a series of concentric metal rings surrounding the border of the sensor, which is designed to terminate the electric field and reduce the amount of leakage current along the borders of the wafer).The small gray blocks indicate the metal pads for wire bonding, which are 50 mm wide.In the zoomed-in view, the strip on each metal pad is the oxide opening to connect to the electrode.

3 .
Calculated transmission of incident photons through the silicon detector at different depths.The mass attenuation coefficient of silicon at 35 keV is acquired from the NIST database(Hubbell and Seltzer 2004).

Figure 2 .
Figure 2. Illustration of beam positions for the charge sharing measurement.

Figure 4 .
Figure 4. Measurement result of the count-rate performance, compared with ideal linarity in (a) an overall view.(b) Enlarged view of the low count-rate region.

Figure 5 .
Figure 5. Measurement result of the fluence rate, translated from the count rate, and compared with ideal linearity in (a) an overall view.(b) Enlarged view of the low fluence-rate region.

Figure 4
Figure 4 demonstrates the measured output count rates as a function of input count rates.For measurements with input fluence rates attenuated by 50 m m and 75 m m tungsten attenuators, the saturation effect occurred, thus the corresponding data points are not included in figure 4. The count-rate based data points are fitted to SNP dead-time model using nonlinear least squares regression model.The input count rates are obtained by extrapolating the linear relationship between input and output count rates at low photon fluence rates, where pulse pile-up is unlikely to happen.To find the relative attenuation between the data points, the x-ray attenuation data of tungsten is acquired from (Jørgensen et al 2019).In figure 4(a), it is clear that the SNP dead-time model accurately describes the count-rate behavior of the detector.The maximum output count rate derived from the model is /T 1 8.33 10 cps, c 6 = ´which differs only 2.4% from the measurement result of 8.13 10 cps.6´The fitting implies a nonparalyzable dead time n t of 2.93 ns.The remarkably short n t compared to the full pulse detection period T c = 120 ns can be understood by

Figure 6 .
Figure 6.Measured count rate as a function of threshold energy for different thickness of the tungsten attenuators.The center pixel used for the count rate measurements and its two nearest neighbors are plotted.The vertical dashed line is the energy threshold of 7.3 keV (in pixel 5) used for the count rate measurements of figure 4.
the latter case.As shown in figure 5(a), the detector is still functional at a photon fluence rate of 2.87 10 mm s ,