Nonlocal Thresholds for Improving the Spatial Resolution of Pixel Detectors
NNonlocal Thresholds for Improving theSpatial Resolution of Pixel Detectors
Benjamin Nachman and Alex Spies Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94704, USA Simons Institute for the Theory of Computing, University of California, Berkeley,Berkeley, CA 94720, USA Department of Physics, University of California Berkeley, Berkeley, CA 94720, USA
March 6, 2019
Abstract
Pixel detectors only record signals above a tuned threshold in order to suppressnoise. As sensors become thinner, pitches decrease, and radiation damage reducesthe collected charge, it is increasingly desirable to lower thresholds. By making thesimple, but powerful observation that hit pixels tend to be spatially close to each other,we introduce a scheme for dynamic thresholds. This dynamic scheme can enhancethe signal efficiency without significantly increasing the occupancy. In addition topresenting a selection of empirical results, we also discuss some potential methodsfor implementing dynamic thresholds in a realistic readout chip for the Large HadronCollider or other future colliders.
Pixel detectors are designed to be thin, to be highly granular, and to have low occupancy inorder to precisely reconstruct charged-particle trajectories (tracks) from minimum ionizingparticles (MIPs). In order to achieve this goal while maintaining a high signal efficiency, onlysignals above a tuned threshold are recorded. This threshold is chosen to be small comparedto a typical signal, but large compared to the noise. For example, sensors in the currentLarge Hadron Collider (LHC) experiments ATLAS and CMS are 200-300 µ m thick, leadingto a signal at perpendicular incidence of 16k-24k electrons (e); noise levels (measured asthe equivalent noise charge or ENC) are typically 100-150e and tuned thresholds are 2k-3ke.With these settings, the noise occupancy is well below 10 − [1, 2].Given the increased instantaneous luminosity at the High Luminosity LHC (HL-LHC)and the goal of improving track reconstruction, there is a move towards thinner and narrowersensors. Such sensors will require lower thresholds to compensate for the reduced signalcharge resulting from the decreased path length of MIPs. At the same time, the high particle1 a r X i v : . [ phy s i c s . i n s - d e t ] M a r ux expected at the HL-LHC also poses challenges due to increased radiation damage; Non-ionizing energy loss results in defects in the sensor bulk that act as trapping sites and reducethe collected charge. The large fluence also increases the noise via the sensor leakage current,ENC leak ∝ √ I leak and I leak ∝ Φ (aside from annealing effects) [3]. Indeed, with the currentLHC lifetime fluence at circa 10 n eq / cm [4], the charge collection efficiency hasdropped to roughly 70% [5] and the leakage current has reached one mA or more [6–8]. TheATLAS and CMS collaborations are working together within the RD53 collaboration [9] todevelop a new readout chip for their HL-LHC pixel detectors and therefore now is a criticaltime to find solutions that address, at least in part, the challenges associated with the next-generation of pixel designs. To this end, we propose a new method for pixel thresholdingwhich stems from a simple, but significiant observation about MIP and noise hits: while theprobability for a single pixel to be hit by a MIP is 0.1% or smaller [10,11], the probability fora pixel to be hit given that one of its neighboring pixels was hit is 10% or more [12]. Whilethe neighboring pixel hits can be caused by charge sharing from diffusion and capacitivecoupling, they can also be due to an inclined primary particle traversing multiple sensors atan angle. Coupled with the fact that noise hits exhibit no spatial correlation, this suggeststhat the optimal threshold should depend on the pattern of neighboring hits. We studymultiple implementations of this idea.We are not aware of any previous efforts to utilize neighboring pixel information todynamically adjust thresholds. There have been previous proposals to implement dynamicthresholds to correct for spatial-temporal effects using information from a particular pixel [13].A related topic is dual thresholds, which have been used extensively to separate time andenergy measurements in order to make the best of both for a single detection. This tech-nique has been applied to precision timing ( O (10 ps)) applications as diverse as positronemission tomography detectors [14] and high energy physics timing detectors [15–18] as wellas ‘standard’ LHC pixel detectors with timewalk concerns at O (10 ns) timescales [19]. Dualthresholds have also been used for improving the position resolution by using one thresholdfor event triggering and one for measuring charge in regions of interest [20]. Such a scheme isnot possible for the extreme event rate at LHC pixel detectors, but the idea is similar to ourproposal. Another body of related work is dynamic and dual thresholds for edge detection(see e.g. [21–23]). One of our proposals for implementing non-local thresholds will involvemodifying the capacitative coupling between neighboring pixels. This form of charge sharinghas been well-suited in the literature (see e.g. Ref. [24]), but is traditionally viewed as anuisance. We show that this effect may instead be an asset for improving position resolu-tion and increasing signal efficiency. In contrast to traditional dual-threshold methods, thisalgorithm requires a fast communication between neighboring pixels and therefore has morestringent timing and power constraints.This paper is organized as follows: Section 2 introduces the simulation setup and Sec-tions 3 and 4 introduce the metrics and threshold schemes, respectively; The results arepresented in Section 5 with a brief discussion on implementation in Sec. 6; the paper endswith conclusions and outlook in Section 7. Given that both ATLAS and CMS are designing their innermost layers to be replaceable, new ideasmay still see utilization in subsequent years, even if they are not fully developed in time for the upcomingproduction runs. Simulation
A standalone simulation setup using Allpix [25] built on the Geant4 package [26] is used tosimulate single particles interacting with a single planar pixel layer. The sensor specificationsare similar to those proposed for the ATLAS and CMS pixel detector upgrades for the HL-LHC [10, 11]. In particular, the sensors are 150 µ m thick with a pitch of 50 × µ m . Thesimulation of energy deposition, drift, and digitization is the same as in Ref. [27] and is brieflysummarized here for completeness. Charge deposition and straggling are provided by Geant4using the emstandard opt0 model . The ionization energy is converted into electron-holepairs assuming 3.6 eV/pair and electrons are transported to the collecting electrode, includingdrift and diffusion. Collected electrons are digitized using a the time-over-threshold (ToT)method [29], with a linear charge-to-ToT conversion. The analog threshold is varied, but thenumber of bits is fixed at 4, as suggested in Ref. [30]. Unless otherwise specified, the sensorsare modeled without radiation damage. The effects of radiation damage are approximatedby reducing the collected charge according to the n + -in- n planar sensor results based oncombining TCAD simulations from the Perugia [31] and New Delhi models [32] with drift,diffusion, and digitization presented in Ref. [10]. Three important rates that are tied to the choice of threshold are the signal efficiency, theoccupancy, and the noise rate. The signal efficiency is the fraction of collected charge froma MIP. Charge that diffuses to a neighboring pixel or is in the first or last pixel of a clustermay be below the threshold. The threshold can also be used to control the overall hit rate inorder to ensure that the occupancy is manageable. For pixel detectors at the HL-LHC, theoccupancy will be dominated by real hits and not noise. However, the total occupancy stillhas a large contribution due to non-MIP hits. Since it is difficult to accurately model thelow-energy spectrum, instead of providing the total occupancy, we report the contributionof MIPs to the occupancy. Finally, as the noise rate is well below the overall occupancy, itis important to report the error rate separately.One of the important consequences of a reduced charge collection efficiency with increasedthreshold is that the estimated position resolution degrades. Alongside the counting metricsdescribed above, we also report the position resolution as a function of the threshold setting.For a cluster of length L cluster like the one shown in the bottom of Fig. 1, all of the informationabout the position in the y (long) direction as well as the longitudinal incidence angle iscontained in y head - the location of the particle as it traversed the first pixel in the cluster.As the tail and head position resolutions are approximately the same, the resolution on theposition estimator y cluster = ( y head + y tail ) is σ y head / √
2, while the resolution on the clusterlength y head − y tail is √ σ y head . Since the deposited charge scales with path length, one canuse the amount of deposited charge in the first pixel to estimate the location y head .The estimator for y head that minimizes the mean squared error is ˆ y head ( Q ) = (cid:104) y | Q (cid:105) , where Q is the (digitized) charge deposited in the first pixel. The top right plot in Fig. 1 shows This is not accurate for thin sensors, but 200 µ m are sufficiently thick that the total deposited charge iswell-modeled [28]. (cid:104) y | Q (cid:105) is computed assigning zero to the start of the pixel and normalizing by the pitch. In rarecircumstances, enough charge can diffuse to the pixel before the first traversed one in thepixel matrix and therefore, ˆ y can be negative. In addition, the threshold can be sufficientlyhigh that the first traversed pixel is below threshold and so ˆ y can also be bigger than 1.Each of these cases are illustrated schematically in the left diagrams of Fig. 1. The positionresolution is given by (cid:112) (cid:104) (ˆ y head − (cid:104) ˆ y head (cid:105) ) (cid:105) and is approximately [33] bounded by pitch / √ δ -rays are excluded from the analysis because they register an anomalouslyhigh charge that has little to do with the position of the original MIP. Especially for δ -raysthat travel many pixels before reaching their Bragg peak, the non-MIP signature can beidentified and removed before estimating the MIP position. The occurrence of δ -rays for thefirst pixel in a cluster is about 1%. 4 r b i t r a r y no r m a li z a t i on D i s t an c e T r a v e l ed / P i x e l P i t c h First pixel below threshold Diffusion to second pixel -ray veto), ToT bits = 4, threshold = 600e δ Geant4 (Allpix) + Digitization ( | = 1 η , radius = 39 mm, | m µ
50 X 50 x 150 M I P d e p o s i t e d cha r g e Distance < 0 M I P d e p o s i t e d cha r g e threshold M I P d e p o s i t e d cha r g e threshold diffusion threshold y en t e r y s t p i x e l y en t e r y s t p i x e l y en t e r y s t p i x e l M I P L cluster = y head y tail ! L cluster = p y head y cluster = ( y head + y tail ) ! y cluster = y head / p y cluster = ( y head + y tail ) ! y cluster = y head / p M I P d e p o s i t e d cha r g e Distance < 0 M I P d e p o s i t e d cha r g e threshold M I P d e p o s i t e d cha r g e threshold diffusion threshold y en t e r y s t p i x e l y en t e r y s t p i x e l y en t e r y s t p i x e l M I P d e p o s i t e d cha r g e Distance < 0 M I P d e p o s i t e d cha r g e threshold M I P d e p o s i t e d cha r g e threshold diffusion threshold y en t e r y s t p i x e l y en t e r y s t p i x e l y en t e r y s t p i x e l M I P M I P Figure 1: A schematic diagram to illustrate the calculation of the position resolution metric.The bottom right figure shows a pixel cluster, where the filled regions indicate the path of aMIP. All of the information about the position and length of the cluster are contained in y head and y tail ; since the resolution of these two quantities should be approximately the same, wefocus only on the former quantity. The top right plot shows the distribution of the positiontraversed by a MIP normalized per bin of measured charge. The left figures illustrate thedefinition of ˆ y , which can be negative if enough charges diffusion into the previous pixel andcan be more than 1 if the first traversed pixel is below threshold.5 Threshold Schemes
We consider three schemes for setting charge thresholds: • Nominal : If the charge is below the threshold, then the ToT is zero. This is theusual way a fixed threshold is implemented: a comparator takes the output of thepixel pre-amplifier and compares it with a fixed threshold. • f share = X %: Pixel modules already exhibit a form of dynamic thresholds due tocharge sharing (often called ‘cross-talk’) via interpixel capacitance. When a charge q isdeposited in one pixel, the neighboring pixels register f share q , where f share depends onthe capacitive coupling between pixels which in part scales with the length of the sharededge. This means that the effective threshold for the neighbor of a hit pixel is reducedby f share q [24]. The value of f share is typically specified to be as small as possible, oftenbeing on the order of a few percent. We propose to engineer f share to optimize theoccupancy and resolution. In practice, designing a pixel with a given f share while alsosimultaneously meeting other specifications may prove difficult, however, our goal is tostudy the impact of a larger f share so as to motivate future studies in a real chip. Sincewe are assuming square pixels, we add f share q to the four neighbors sharing an edgeand subtract 4 f share q from the primary pixel. For the other schemes, the cross-talk isset to zero. • f neighbor = X %: Cross-talk is an indirect method for dynamic thresholds; instead,we propose to directly set the the threshold of a given pixel based on the activity inneighboring pixels. The simplest such scheme is to have two thresholds: a nominal highthreshold and a lower threshold that is f neighbor of the high one. If any pixel is abovethe high threshold, then all of its neighbors see a lower threshold. In practice, thiswould require explicit information sharing between pixels and may require significantadded capacitance and/or power. However, Sec. 5 will show that this is a powerfulscheme for maintaining both high efficiency and good position resolution.Before presenting results, we note that the latter two schemes affect only a fractionallysmall number of pixels for any given event, and therefore have a negligible impact on theoverall noise rate. Figure 2 shows the MIP efficiency and charge efficiency for first traversed pixel in a cluster,as well as the MIP effeciency measured over all pixels, as a function of the threshold forthe three schemes introduced in Sec. 4. In the case where the efficiencies are only given forthe first traversed pixel in the cluster, the MIP charge efficiency is much higher than theefficiency to register any hit; this is a consequence of the fact that the charge in the firstpixel is small when the path length is short. As expected, increasing the threshold degradesboth the (charge) efficiency. For the chosen values of f share = 5% and f neighbor = 50%, thehit efficiency is improved for every threshold. The f neighbor scheme also has a higher MIPcharge efficiency than the nominal approach. 6dditionally, the f share approach appears to have a lower MIP charge efficiency than thenominal approach, but this is an artifact caused by the increased charge from the neighboras after digitization, it cannot be distinguished from the primary charge. Notably, the MIPefficiency is 5-10% higher with the new threshold schemes. The plot on the right of Fig. 2essentially shows the average fractional amount of pixels which go over threshold in a givenscheme. As expected, this shows the same trend, but with the cross-talk scheme causingan increased rate of hit-losses relative to the other schemes (an effect which manifests mostsignificantly on the edges of clusters, and is thus supressed in the plot on the left).For reference, the left plot of Fig. 2 also shows the noise rate, assuming ideal Gaussiannoise, in which the rate decreases exponentially with increasing threshold. In practice, thenoise is not exactly Gaussian, and the suppression with increased threshold is not as strongas indicated. However, the fact that the noise rate is still significantly suppressed withincreasing threshold, coupled with the trends shown in Fig. 2, indicate that it is possible tohave a higher threshold without compromising the MIP (charge) efficiency.
200 400 600 800 1000 1200Threshold [e]00.511.5 M I P ( C ha r ge ) E ff i c i en cy Nominal = 5% share f = 50% neighbor f MIP EfficiencyMIP Charge EfficiencyNoise (assuming Gaussian)
Geant4 (Allpix) | = 1.5, 4 bits @ 32 ToT / MIP h , | m m
50 X 50 x 150 - - - - - N o i s e R a t e
200 400 600 800 1000 1200Threshold [e]0.20.40.60.811.2 M I P E ff i c i en cy ( a ll p i x e l s ) Nominal = 5% share f = 50% neighbor fGeant4 (Allpix) | = CHECK, 4 bits @ 32 ToT / MIP η , | m µ
50 X 50 x 150 R e s o l u t i on / p i t c h Geant4 (Allpix) | = 1.5, thresh. = 600 e, 4 bits @ 32 ToT / MIP η , | m µ
50 X 50 x 150
Figure 2: Left: The MIP (charge) efficiency as a function of the threshold for the threethreshold schemes; for reference, the noise rate is also given assuming an ideal 150e Gaussiannoise profile. Right: The MIP Effeciency measured over all pixels in which charge wasdeposited or shared. In both cases, MIPs are incident a slight angle (corresponding to η = 1 .
5) in order to increase the pixel multiplicity in clusters along the longitudinal direction.The ToT is tuned so that a MIP at perpendicular incidence would correspond to a ToT of32 if 15 were not the maximum value.Fig. 3 contains two plots: the first illustrating the variation of position resolution withthreshold in the three schemes under investigation, and the second showing the resolutionas a function of f share .Focussing first on the left plot, we see that the two new schemes improve the resolutionfor all values of threshold, and that the resolution worsens with increasing threshold, akinto the MIP efficiency. Furthermore, with a value of f neighbor = 50%, the triangle points in7he left plot are the same as the nominal points with a threshold reduced by 50%. Theimprovement from the f share = 5% scheme is more modest, but is still a few percent for allthresholds. Additionally, the shallow trend of the direct-talk scheme implies that increasedthresholds could be applied with relatively less detriment to the resolution than in the othertwo schemes..The right plot highlights the sensitivity of the resolution to the exact amount of cross-talk. Interestingly, there is an optimal amount of charge sharing at 5% for the given incidenceangle, pitch, threshold, and charge tuning. This is to be expected, as increasing f share fromzero improves the resolution until information about the charge from the first pixel is washedout by the contribution from the neighbor that went over the threshold. The absolute changein the resolution is about 2%, but subtracting in quadrature, the additional resolution isabout 20%.
200 400 600 800 1000 1200Threshold [e]0.150.20.25 R e s o l u t i on / p i t c h Nominal = 5% share f = 50% neighbor fGeant4 (Allpix) | = 1.5, 4 bits @ 32 ToT / MIP h , | m m
50 X 50 x 150 R e s o l u t i on / p i t c h Geant4 (Allpix) | = 1.5, thresh. = 600 e, 4 bits @ 32 ToT / MIP h , | m m
50 X 50 x 150
Figure 3: Left: the y head position resolution as a function of the threshold for the threeschemes. Right: The y head position resolution as a function of the charge sharing ( f share ). Inboth cases MIPs are incident at η = 1 .
5, with the same charge tuning as in Fig. 2.The intense radiation environment of current and future hadron colliders is one of thegreatest challenges for silicon-based pixel detectors. Figure 4 shows the position resolutionas a function of the non-ionizing energy loss for a fixed threshold. Since charge is lost fromcharge trapping, the resolution degrades with fluence. The innermost layers of the HL-LHCdetectors will need to cope with about 10 n eq / cm . Given the assumptions goinginto Fig. 4, the f neighbor = 50% scheme has the same position resolution after the full HL-LHC fluence as the nominal scheme does with an unirradiated sensor. The clear superiorityof the direct scheme with respect to radiation hardness is unsurprising as it is the leastseverely affected by changes in signal-size which are not significant enough to drive pixelsunder threshold.The results presented thus far were based on the simulation of pixels with symmetric sidelengths (i.e. square). However, there has been considerable investigation of the potential8 /cm eq Fluence [100.150.20.25 R e s o l u t i on / p i t c h share f = 50% neighbor fGeant4 (Allpix) | = 1.5, Thresh. = 600e, 4 bits @ 32 ToT / MIP h , | m m
50 X 50 x 150
Figure 4: The position resolution as a function of the silicon 1 MeV n eq / cm fluence for thethree threshold schemes at a fixed threshold of 600e and with the same tuning as Fig. 2.The average charge loss from Ref. [10] is given as a second axis.use of asymmetric pixels at the LHC, such as 25 × µ m , which can trade off the positionresolution in the longitudinal direction ( z ) for increased resolution in the transverse direction( d ), which is more important for flavor tagging. Importantly, when pixels are not square,the charge sharing will not be the same for the long and the short sides. While the completecalculation of charge sharing is complicated and sensor-specific, the capacitance (and thus thecharge sharing) is approximately proportional to the side length of the pixel. For example,in the 25 × µ m case, the short sides will exhibit 4 times less charge sharing than thelong sides.Figure 5 illustrates how the position resolution changes with asymmetric pixels. In orderto control for effects related to the actual amount of charge deposited due to the pixel size, allresults are actually simulated with the 50 × µ m setup from earlier. However, the amountof sharing in the x and y directions is now different and is set proportional to the side length.If the sharing before was f share , then the new sharing is f (cid:48) share = 2 f share / (1 + pixel asym.),which is chosen so that the total charge lost by the primary pixel is still 4 f share . The pixelasymmetry is the ratio of the transverse to longitudinal pixel pitch.The left plot of Fig. 5 shows how the relative resolution changes for different configurationsas a function of the amount of charge sharing. A charge sharing value of 5% means thatthe primary pixel loses 4 ×
5% of its charge to its neighbors, divided up in a way that isproportional to the shared side length. When the pitch is smaller, the optimal amount ofcharge sharing increases. In the 25 × µ m configuration, the optimal sharing for the longside is about 3% while there is no optimal value for the short side (larger value than the 10%cutoff is desired). The right plot of Fig. 5 fixes the total charge sharing and varies the pixelasymmetry. For a total charge sharing of 5%, the down-triangles and circles from the leftplot of Fig. 5 are nearly the same, which is consistent with the broad minimum in the right9lot for the up-triangles. In contrast, there is a strong dependence on the pixel asymmetryin the sub-optimal case of 10% charge sharing. R e s o l u t i on / p i t c h Geant4 (Allpix)m thick, thresh. = 600 e, 4 bits m m m · m m · m m · R e s o l u t i on / p i t c h · = 4 share,tot f 10% · = 4 share,tot f Geant4 (Allpix)m thick, thresh. = 600 e, 4 bits m Figure 5: Left: the relative position resolution as a function of the amount of charge sharingfor three different pixel configurations (the resolution always corresponds to the first dimen-sion given in the legend). Right: for a fixed amount of total charge sharing (20% charge lossfrom the primary pixel), the position resolution is shown as a function of the asymmetry inthe pixel pitches. A value of 0.25 corresponds to 25 × µ m . Using capacitive coupling to implement the dynamic threshold has the advantage that theinformation from the primary hit is transferred nearly instantly to the neighboring pixels.The disadvantage is that designing a specific amount of capacitive coupling is challenging,especially given the tight constraints from other design requirements (including noise andpower consumption).In the alternative scheme where active logic is used to reduce the threshold on the neigh-bors, information must be quickly sent to the neighboring pixels. Figure 6 illustrates thistime constraint when one hit passes the initial high threshold and a neighboring hit wouldonly pass a reduced threshold. The first clock cycle where this hit is recorded to be abovethe high threshold is t and the first clock cycle for which the smaller hit would be abovethe reduced threshold is t , while it goes below this threshold at t . This second hit willbe recorded as long as the second threshold can be reduced in a time t − t . The chargeresolution of the second hit will be optimal when the communication time is only t − t ;any will result in a degraded resolution. 10 t t p r ea m p li fi e r ou t p u t time
40 MHz clock
ToT = 11ToT = 6 high thresholdreduced threshold
Figure 6: A schematic illustration of the charge digitization for two particles going throughneighboring pixels in the same bunch crossing: one with a large charge (blue) and one witha small charge (red). The small charge particle does not pass the initial high threshold, butwould pass the reduced threshold if the new threshold could be set before t . The ToT forthe small charge in this scheme would be at most 6, when the threshold is reduced by t .11 Conclusions
The HL-LHC presents significant challenges for pixel module design and now is the timeto consider new possibilities for optimizing the information saved for offline analysis. Wehave presented two schemes for dynamic thresholds which use information from neighboringpixels in order to increase the MIP efficiency, with little or no increase in the noise rate. Onescheme exploits the natural interpixel capacitance cross-talk to lower the effective thresholdof pixels neighboring those with a large charge deposition. Furthermore, from the perspectiveof position resolution, there is an optimal amount of charge sharing. While in practice itmay be difficult to engineer a particular level of charge sharing, given other specifications,these results suggest that design studies are worthwhile, especially in light of the challengesposed by the LH-LHC conditions. One drawback of the capacitive coupling scheme is thatthe effective decrease in the threshold is random and varies significantly with the stragglingof MIP charge depositions.Secondly, we propose a scheme which instead utilizes two fixed thresholds, thus circum-venting the aforementioned challenge. This algorithmically simple scheme presents a lowerthreshold to all pixels next to a pixel that passes a high, nominal threshold. This proce-dure significantly improves the resolution and MIP efficiency, but practical implementationswould depend on a mechanism for rapid communication between neighboring pixels. Indeed,adding circuitry for this purpose would certainly increase the capacitance and/or the powerconsumption, so such tradeoffs require a thorough investigation.Signal efficiency and position resolution are crucial for both track reconstruction andflavor tagging at the LHC, and thus it is conceivable that the trade-offs of the proposeddyamic threshold schemes may be outweighed by the gains. Certainly, considerable amountsof potentially useful information are present in the neighborhood around pixels which arenot being explicitly used, and which could significantly improve detector performance forthe HL-LHC and beyond.
We would like to thank Maurice Garcia-Sciveres and Timon Heim for many useful discussionsas well as the RD53 collaboration for encouragement and feedback. This work was supportedby the U.S. Department of Energy, Office of Science under contract DE-AC02-05CH11231.
References [1] ATLAS IBL Collaboration, Production and Integration of the ATLAS Insertable B-Layer, JINST 13 (05) (2018) T05008. arXiv:1803.00844 .[2] CMS Collaboration, Commissioning and Performance of the CMS Pixel Tracker withCosmic Ray Muons, JINST 5 (2010) T03007. arXiv:0911.5434 .[3] L. Rossi, P. Fischer, T. Rohe, N. Wermes, Pixel detectors: from fundamentals to appli-cations, Particle Acceleration and Detection, Springer, Berlin, 2006.124] ATLAS Collaboration, Fluence versus time in ATLAS pixel detector, ATL-INDET-2018-015.URL https://cds.cern.ch/record/2649731 [5] ATLAS Collaboration, Modeling Radiation Damage to Pixel Sensors in the ATLASDetector, ATL-INDET-2018-020.URL https://cds.cern.ch/record/2650494 [6] ATLAS Collaboration, IBL LV currents and sensor leakage currents, PIX-2016-006.URL https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/PIX-2016-006/ [7] ATLAS Collaboration, Measurements and Predictions of Pixel Detector Leakage Cur-rent, PIX-2018-008.URL https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/PIX-2018-008/ [8] CMS Collaboration, Pixel module leakage current evolution, Pixel2018.URL https://indico.cern.ch/event/669866/contributions/3234992/attachments/1767902/2871309/Talk-pixel2018.pdf [9] RD53 Collaboration, RD53A Integrated Circuit Specifications, CERN-RD53-PUB-15-001.URL https://cds.cern.ch/record/2113263 [10] ATLAS Collaboration, Technical Design Report for the ATLAS Inner Tracker PixelDetector, CERN-LHCC-2017-021.URL https://cds.cern.ch/record/2285585 [11] CMS Collaboration, The Phase-2 Upgrade of the CMS Tracker, CERN-LHCC-2017-009.URL https://cds.cern.ch/record/2272264 [12] ATLAS Collaboration, Pixel neighbour occupancy plots, ITK-2016-003.URL https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/ITK-2016-003/ [13] M. Garcia-Sciveres, T. Heim, Self-Adjusting Threshold Mechanism for Pixel Detectors,Nucl. Instrum. Meth. A867 (2017) 209. arXiv:1701.01459 .[14] M. D. Rolo, R. Bugalho, F. Gon¸calves, G. Mazza, A. Rivetti, J. C. Silva, R. Silva,J. Varela, TOFPET ASIC for PET applications, Journal of Instrumentation 8 (2013)C02050.[15] CMS Collaboration, Technical proposal for a MIP timing detector in the CMS experi-ment phase 2 upgrade, CERN-LHCC-2017-027.URL https://cds.cern.ch/record/2296612 [16] CMS-TOTEM, Collaboration, CMS-TOTEM Precision Proton Spectrometer, CERN-LHCC-2014-021.URL https://cds.cern.ch/record/1753795 β imaging, in: 2008 IEEE Nuclear Science Symposium ConferenceRecord, 2008, p. 1.[21] P. He, et al., A novel dynamic threshold method for unsupervised change detection fromremotely sensed images, Remote Sensing Letters 5 (2014) 396.[22] N. Nain, G. Jindal, A. Garg, A. Jain, Dynamic thresholding based edge detection (2008).[23] N. Otsu, A threshold selection method from gray-level histograms, IEEE Transactionson Systems, Man, and Cybernetics 9 (1979) 62.[24] L. Rossi, P. Fischer, T. Rohe, N. Wermes, Pixel Detectors: From Fundamentals toApplications, Springer-Verlag Berlin Heidelberg, 2006.[25] J. Idarraga, M. Benoit, Generic Geant4 implementation for pixel detectors, The AllPixSimulation Framework (2006) [twiki.cern.ch:AllPix].[26] GEANT4 Collaboration, GEANT4: A Simulation toolkit, Nucl. Instrum. Meth. A506(2003) 250–303.[27] Y. Chen, et al., Optimal use of Charge Information for the HL-LHC Pixel DetectorReadout, Nucl. Instrum. Meth. A (2018), arXiv:1710.02582 .[28] F. Wang, et al., The Impact of Incorporating Shell-corrections to Energy Loss in Silicon,Nucl. Instrum. Meth. A899 (2018) 1–5. arXiv:1711.05465 .[29] I. Kipnis, et al., A time-over-threshold machine: the readout integrated circuit for thebabar silicon vertex tracker, IEEE Transactions on Nuclear Science 44 (1997) 289–297.[30] Y. Chen, E. Frangipane, M. Garcia-Sciveres, L. Jeanty, B. Nachman, S. P. Griso,F. Wang, Optimal use of charge information for the hl-lhc pixel detector readout, Nu-clear Instruments and Methods in Physics Research Section A: Accelerators, Spectrom-eters, Detectors and Associated Equipment 902 (2018) 197?210. doi:10.1016/j.nima.2018.01.091 .[31] F. Moscatelli, et al., Effects of interface donor trap states on isolation properties ofdetectors operating at high-luminosity lhc, IEEE Transactions on Nuclear Science 64(2017) 2259.[32] R. Dalal, Simulation of Irradiated Detectors, PoS Vertex2014 (2015) 030.1433] F. Wang, B. Nachman, M. Garcia-Sciveres, Ultimate position resolution of pixel clusterswith binary readout for particle tracking, Nucl. Instrum. Meth. A899 (2018) 10. arXiv:1711.00590arXiv:1711.00590