Starting in 2025 and extending through 2028, the LHC will be upgraded to the high-luminosity LHC (HL-LHC), making the expansion of experimental particle physics research and the exploration of fundamental physics of the LHC possible. The HL-LHC will deliver an instantaneous luminosity of 7.5 × 10
34 cm
−2s
−1, a 5–7 times increase from the nominal value resulting in an average pileup of about 200 inelastic proton–proton collisions per bunch crossing. The innermost parts of the pixel detector, situated closest to the interaction point, will be exposed to unprecedented levels of radiation, reaching a non-ionizing radiation fluence of 1 − 2 × 10
16 n
eqcm
−2 during the expected lifetime of the ATLAS detector [
19,
20].
The ATLAS ITk pixels detector comprises five barrel layers (from L0 to L4) and a system of as many rings covering the forward region. The innermost pixel layer (L0) and ring (R0) will be equipped with 3D n+-on-p sensors. All the rest of the detector will be equipped with planar n+-on-p sensors, with thicknesses of either 100 μm (in L1 and R1) or 150 μm (in all other layers and rings).
The anticipated increase in particle density at HL-LHC calls for a faster algorithm to model the effects of radiation damage in the ATLAS MC events. A novel methodology centered on charge re-weighting from Look-Up Tables (LUTs) is under study and is outlined in the subsequent sections.
3.1. LUT Method
Similar to the actual radiation damage digitizer (see
Section 2.2), the Look-Up Table (LUT) method simulates radiation damage effects in ATLAS MC events after Geant4 [
12,
13,
14] simulates charge deposition and before signal digitization, where signal is discriminated and transformed into digital format by simulating the response of the readout electronics chip.
The task of the radiation damage digitizer is to estimate on which pixels the deposited charges will induce a signal and how large this signal will be. The LUT method is proposed to accomplish this task for the data taking of ATLAS during the high-luminosity phase of LHC (HL-LHC). ATLAS MC events will be simulated assuming an ideal, undamaged pixel sensor. The impact of radiation damage will then be incorporated as a correction, utilizing the LUTs.
The LUT method presented herein is inspired by the “template” method used by the CMS collaboration [
22]. The primary distinction between the CMS and LUT methods is that, while the former applies corrections to simulated clusters of pixels, the latter aims to reweigh observables of a group of drifting carriers in the silicon bulk.
The design of the LUT method to model the radiation damage effects in ATLAS MC events is based on the dynamics of charge carriers in the bulk of pixel sensors. The situation is sketched in
Figure 3 for the case of planar sensors.
Here, the sensor is assumed to be a planar pixel sensor of thickness w; y is the direction of the magnetic field and the direction of the electric field is along the bulk depth axis z; carriers are deflected in the x direction due to magnetic field.
Electrons and holes produced by a minimum ionizing particle (MIP) traversing the sensor drift toward collecting electrodes at an angle known as the Lorentz angle () with respect to the electric field.
In pristine detector—i.e., without radiation damage effects—the final position of the carriers is calculated projecting the carrier position to the respective collecting electrode. The final position is then smeared to take into account diffusion. The LUT method is intended to apply corrections to this basic method. In the following the initial position of the charge carrier will be referred to as
deposited position
, and the final one the
propagated position
; see also
Figure 3.
In
Figure 3, drifting carriers can be trapped before reaching the collecting electrode due to radiation damage in the sensor bulk. In such cases, they induce a signal that is only a fraction,
, of their charge. This fraction can be calculated using the Shockley–Ramo theorem [
23,
24], which requires the initial and final positions of the carrier as input.
The carriers that are trapped cover a distance
along the bulk depth direction, which depends on the
position of the carriers. This dependence on the carrier generation position is due to the non-uniform electric field in the sensor bulk after radiation damage [
10]. As mentioned above, the carriers will drift at an angle
relative to the direction of the electric field. This angle also depends on the
position where the carrier was created, again due to the non-uniformity of the electric field. The carrier will end up at a position in the bending direction (orthogonal to the electric and magnetic field) that is displaced—on average, (the carrier movement is affected by diffusion too which adds dispersion to the final carrier position)—by
; of course,
, if there was no diffusion. As discussed above, the carrier’s final position
will determine the amplitude of the signal induced on the electrodes. This final position depends on the carrier’s initial position
and on the combination of electric and magnetic fields. The signal will be a fraction
k of the carrier charge
q, and this fraction
k is also a function of the original carrier position
.
In summary, for each group of carriers, the method calculates the free path , the Lorentz angle , and the fraction of the induced signal k. These three quantities will be referred to as the observables.
To determine the values of the three observables, repeated simulations of the drift of carriers deposited at precise positions
within the bulk are conducted. These simulations are performed using the Allpix
2 [
25] simulation framework, which will be presented in
Section 3.2, using precise electric field and Ramo potential maps produced using TCAD (Technology–Computer-Aided Design) simulations.
For each simulated event, the initial position
, the final position
of the carrier will be saved, together with the signal fraction
k induced on the pixel matrix. In principle, the values of three observables should be recorded as a function of the initial position
of the carrier, but due to the aforementioned limited computing resources, the LUTs will be a function of only
. For a fixed
z value, the average over all
positions will be carried out for each observable and assigned to that
z value; to make the notation lighter, “
dep” is dropped from here onward. Thus, for a single event and a single value of
z, the three LUTs will be:
Using the LUTs defined in Equation (
1), the propagated position
and the induced signal
of a charge,
q deposited at depth
z in the sensor bulk is calculated as follows:
where
is a Gaussian-distributed random number added to simulate the effect of diffusion.
The essence of the LUT method is summarized in Equation (
2): the precise dynamics of carrier drift are substituted with an “average” drift, and the same principle applies to the signal amplitude.
The procedure of charge deposition and drifting is repeated for each several times in order to assess the dispersion of the carriers dynamics.
It is worth noting that, while this study primarily focuses on planar pixel sensors, the same methodology can also be readily applied to strip sensors. Additionally, for the sake of simplicity, the method will be presented here only for planar pixel sensors, but ongoing efforts are being made to extend it to 3D sensors as well—see, for example [
10].
In the following sections, the Allpix2 simulation framework, the TCAD simulations utilized as input to Allpix2 simulations, and the process of calculating the Look-Up Tables (LUTs) will be presented.
3.2. Allpix-Squared for Radiation Damage Digitizer
Allpix
2 is a generic, open-source software framework for the simulation of silicon pixel detectors [
25]. The framework allows the user to create detailed simulations of the entire experimental chain of a testbeam, from incident radiation to digitized detector response. An extensible system of modules is responsible for executing the distinct simulation steps, such as realistic charge carrier deposition using the Geant4 toolkit and the propagation of charge carriers in silicon through a drift–diffusion model. Detailed electric field maps imported from TCAD simulations can be used to model the drift behavior of charge carriers within the silicon, introducing a higher level of realism to Monte Carlo-based simulations of particle detectors.
In this study, a planar n
+-on-p pixel sensor with a thickness of 100 μm and a pitch of 50 × 50 μm
2 is simulated. This type of sensor is representative of what will be used in the second-to-innermost pixel layer (L1) of the ITk pixel detector. The sensor is simulated after irradiation corresponding to a fluence of 4 × 10
15 n
eqcm
−2 and operated at a voltage of 600 V. These conditions are expected for the aforementioned pixel layer at the end of the HL-LHC phase of ATLAS data taking.
Figure 4 illustrates the simulation chain implemented for evaluating the LUTs, highlighting the use of electric field and Ramo potential maps from TCAD, along with the simulation of trapping effects.
3.2.1. Simulation Inputs
The electric field and Ramo potential maps calculated using TCAD tools can be inputted into Allpix
2 simulations to model the behavior of silicon radiation detectors, as outlined in
Section 3.1. The following sections provide details of the TCAD simulations used for the calculation of LUTs in this study. All simulation results presented here have been obtained using Silvaco TCAD (
https://silvaco.com/tcad/, accessed on 22 May 2024) tools.
The structure simulated in TCAD represents one-quarter of a
pixels matrix from a 100 μm n
+-on-p planar pixel sensor with a pitch of 50 × 50 μm
2; refer to
Figure 5a,c. The simulation of one pixel plus its neighbors is required to model charge sharing accurately via Ramo potential calculation. However, due to the symmetry of the design, it is sufficient to simulate just one-quarter of a
pixels matrix.
To simulate the effects of radiation damage, the LHCb TCAD radiation damage model [
26] was used. This model was developed for planar n
+-on-p pixels intended for the LHCb Velo Upgrade [
27], where irradiation fluences of several 1 × 10
15 n
eqcm
−2 are expected. Similarly, planar n
+-on-p pixel sensors in the ATLAS ITk detector will be exposed to similar fluences during their expected operational lifetime. Therefore, the LHCb TCAD radiation damage model was chosen for this study.
In the ATLAS ITk detector, planar pixels will be utilized everywhere except in the innermost section of the pixel detector, where 3D sensors will be employed [
19].
The largest fluence expected for planar pixels is Φ
max~4 × 10
15 n
eqcm
−2 [
28]. Electric field maps were obtained for a sensor irradiated at the fluence Φ
max once polarized at the maximal expected voltage (600 V). The extracted electric field profile along the bulk depth is shown in
Figure 5a,b. In
Figure 5a,c 2D projection of the weighting field is depicted; here, the result is presented for the full 3 × 3 pixels matrix as it is already in the format for Allpix
2.
The electric field and weighting potential maps from TCAD are integrated into the Allpix
2 simulation chain, as illustrated in
Figure 4, to generate the three LUTs introduced in
Section 3.1 for correcting pixels response in ATLAS simulations. In the following section, the evaluation of LUTs using Allpix
2 will be presented.
3.2.2. Look-Up Tables Generation
Charged particles produced in LHC collisions will ionize the pixels sensors volume homogeneously. So in order to obtain a realistic representation of the pixel response, it is important to study it for all possible charge deposition locations
. The LUTs are generated using the “scan” mode of the “DepositionPointCharge” module in Allpix
2 [
25]. This module places a specified quantity of charge carriers at a precise location within the detector’s active volume. In the studies reported here, the scan mode was configured to deposit
1000 electron–hole pairs every 2 μm along the
z direction and every 1 μm for the other two directions for the simulated device.
Once carriers have drifted to their final position
, the induced signal is evaluated for the pixel cell where they ended up and for the first eight neighbors, as illustrated in
Figure 6. It is important to note that, while the propagated charge may land in the same pixel cell as the deposited charge, this is not always the case.
In the following, the propagated position will always correspond to that of the electrons. This assumption is made in calculating the LUTs due to the negligible impact of holes on signal “position” compared to electrons. This simplification is based on the principle known as the “small-pixel effect” [
29]. In heavily segmented sensors such as pixel sensors, the weighting potential profile along the bulk has a steep gradient near the readout side. Consequently, only carriers in close proximity to the pixel electrode can induce a significant signal. Given that holes move away from the readout electrode, their influence on signal position can be safely neglected.
The pixel with the largest induced signal is identified, its position and signal
are retained for the subsequent steps. The CCE per deposition position
is defined as
divided by the deposited charge
:
The CCE(
z) is then evaluated by extracting the most probable CCE values (MPV) for each
z. An example of CCE distribution for charges deposited at a
z position close to the midplane is reported in
Figure 7a.
The
observable is evaluated by taking the average of the distribution of the free path traveled by the carriers as a function of the generation depth. This process is illustrated in
Figure 7b.
While for the distribution, the average over repeated simulations is a good representative of the distribution itself, it is not the case for charge collection efficiency; for that, the most probable value is a better choice.
Finally, the Lorentz angle
value is calculated by fitting a straight line to the electron drift distribution
vs.
for each
position for various
and
positions, as shown in
Figure 7c. The slope of the fit at each
is then extracted, and the LUT is constructed by plotting the slope (
) as a function of the generation depth, as illustrated in
Figure 8c.
The large spread of values observed when electrons reach the electrode is an artifact of the simulation. At this position, electrons are already within the pixel implant and cease to diffuse further. To prevent biasing the results, the upper limit of the fit range for extracting is constrained to a few micrometers away from the pixel implant.
Finally,
Figure 8 displays the LUTs for charge collection efficiency (CCE(z)), average free path (
), and tangent of Lorentz angle (
) for the case presented in this paper.
The CCE peaks at over 85% at approximately 20 μm below the pixel implant. This phenomenon occurs because holes become trapped near the pixel, thereby partially screening a portion of the signal. However, the influence of holes becomes negligible when charge is deposited farther away from the pixel electrode.
The average distance covered by electrons becomes significantly less than the distance to the electrode rather quickly. For instance, at 20 μm, the average distance is approximately 15 μm, which is 25 % less.
The Lorentz angle deflection is predicted to be very small everywhere in the bulk compared to the typical value in the ATLAS sensors before irradiation—about 0.22 radians. The main reason for this is the large electric field inside the sensor, especially at the pixel side. This reduction in the Lorentz angle after irradiation has already been observed with collision data in ATLAS pixels [
30]. The largest deflection in
x will be for charges deposited at the backside, and it will be just 2 μm.
The following section will present the first validation of the LUT algorithm.