Simulating Monolithic Active Pixel Sensors: A Technology-Independent Approach Using Generic Doping Profiles

Simulating Monolithic Active Pixel Sensors: A Technology-Independent Approach Using Generic Doping Profiles H˚akan Wennl¨ofa,∗ , Dominik Dannheimb, Manuel Del Rio Vieraa,1, Katharina Dortb,2, Doris Ecksteina, Finn Feindta, Ingrid-Maria Gregora, Lennart Hutha, Stephan Lachnita,3, Larissa Mendesa,1, Daniil Rastorgueva,4, Sara Ruiz Dazaa,1, Paul Sch¨utzea, Adriana Simancasa,1, Walter Snoeysb, Simon Spannagela, Marcel Stanitzkia, Alessandra Tomalc, Anastasiia Velykaa, Gianpiero Vignolaa,1 aDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany bCERN, Geneva, Switzerland cUniversity of Campinas, Cidade Universitaria Zeferino Vaz, 13083-970, Campinas, Brazil Abstract The optimisation of the sensitive region of CMOS sensors with complex non-uniform electric fields requires precise simulations, and this can be achieved by a combination of electrostatic field simulations and Monte Carlo methods. This paper presents the guiding principles of such simulations, using a CMOS pixel sensor with a small collection electrode and a high-resistivity epitaxial layer as an example. The full simulation workflow is described, along with possible pitfalls and how to avoid them. For commercial CMOS processes, detailed doping profiles are confidential, but the presented method provides an optimisation tool that is sufficiently accurate to investigate sensor behaviour and trade-offs of different sensor designs without knowledge of proprietary information. The workflow starts with detailed electric field finite element method simulations in TCAD, using generic doping profiles. Examples of the effect of varying different parameters of the simulated sensor are shown, as well as the creation of weighting fields, and transient pulse simulations. The fields resulting from TCAD simulations can be imported into the Allpix2Monte Carlo simulation framework, which enables high-statistics simulations, including modelling of stochastic fluctuations from the underlying physics processes of particle interaction. Example Monte Carlo simulation setups are presented and the different parts of a simulation chain are described. Simulation studies from small collection electrode CMOS sensors are presented, and example results are shown for both single sensors and multiple sensors in a test beam telescope configuration. The studies shown are those typically performed on sensor prototypes in test beam campaigns, and a comparison is made to test beam data, showing a maximum deviation of 4% and demonstrating that the approach is viable for generating realistic results. Keywords: Shockley-Ramo, Simulation, Monte Carlo, Silicon Detectors, TCAD, Drift-diffusion, Geant4, Allpix Squared, Pixellated detectors, Charged particle tracking, Monolithic active pixel sensors, MAPS Contents 1 Introduction 2 2 General layout and assumptions 2 2.1 Doping wells ................. 2 2.2 Contacts and biasing ............. 3 2.3 Rectangular and hexagonal pixels ...... 3 3 Finite element method simulations 3 3.1 Simulation workflow ............. 4 ∗Corresponding author Email address: [email protected] (H˚akan Wennl¨of) 1Also at University of Bonn, Germany 2Also at University of Giessen, Germany 3Also at University of Hamburg, Germany 4Also at University of Wuppertal, Germany 3.2 Substrate diffusion simulation ........ 4 3.3 Impact of sensor doping concentration . . . 5 3.4 Impact of sensor geometry .......... 5 3.5 Hexagonal pixel geometry simulation . . . . 6 3.6 Transient simulations ............. 7 3.7 Generating weighting potentials ....... 7 4 Monte Carlo simulations 8 4.1 Simulation flow ................ 8 4.2 Sensor geometry and setup ......... 8 4.3 Importing results from TCAD simulations . 9 4.4 Charge carrier generation .......... 10 4.5 Charge carrier propagation ......... 11 4.6 Charge transfer ................ 12 4.7 Signal digitisation .............. 12 4.8 Transient simulations ............. 12 4.9 Effect of dopant diffusion in electric fields . 13 4.10 Simulation parameter optimisation ..... 13 Preprint submitted to NIM A August 2, 2024 arXiv:2408.00027v1 [physics.ins-det] 31 Jul 2024 5 Sensor performance studies 14 5.1 Cluster size and total charge ........ 14 5.2 In-pixel studies ................ 15 5.3 Impact of threshold value .......... 15 5.4 Transient pulse studies ............ 17 5.5 Multi-sensor studies ............. 17 6 Comparisons to data and previous simulations 19 7 Summary and outlook 19 1. Introduction Monolithic active pixel sensors (MAPS) produced using commercial CMOS imaging processes are attractive in a particle physics context, as they allow for a reduced material budget and reduction of production complexity compared to most hybrid sensors. The use of commercial processes enables relatively cheap large-scale production of sensors, but it also means that precise information of the manufacturing process may not be publicly available. Predictions of sensor behaviour are thus difficult to make, as the detailed electric field configuration in the sensitive material is highly dependent on the extent and concentration of different doping regions in the silicon. By utilising a quadruple-well technology (providing nwells, p-wells, and deep n-wells and p-wells) [1], MAPS can be constructed with a small collection electrode, which reduces sensor capacitance and improves signal-to-noise ratio while reducing power consumption compared to sensors with larger collection electrodes. However, designs with a small collection electrode lead to a highly non-linear electric field in the pixels, further complicating sensor behaviour prediction. As prototype sensor submissions and investigations are expensive and take a long time, simulations of sensor behaviour are becoming more and more important to gain insight and speed up the design process. This paper aims to demonstrate that by making simple assumptions and performing simulations based on the fundamental principles of silicon detectors and using generic doping profiles, performance parameters of MAPS can be inferred and compared for different sensor geometries. This will be done in context of simulations performed for the Tangerine project [2,3,4,5] and in collaboration with the CERN EP R&D programme on technologies for future experiments [6], but the methodology described is useful for many different silicon sensor simulations. The described method thus constitutes a toolbox for performing similar simulations, useful in extracting a realistic description of sensor behaviour without knowledge of proprietary information. The efficacy of combining detailed electric field simulations with high-statistics Monte Carlo simulations has been previously demonstrated for similar silicon sensors [7,8], and the process described in this paper is general and applicable in multiple different cases. For sensors with non-linear electric fields, simulations like those presented here are useful for gaining a deeper understanding of the sensor performance. It is important to note that the presented simulations by no means capture the intricacies of CMOS imaging processes, but merely describe the larger features of the sensor required to model an accurate signal response. Paper outline The paper aims to show guiding principles for performing detailed Monte Carlo simulations of silicon sensors, using basic assumptions and estimates. In Section 2, a general MAPS layout is described, and assumptions of the geometry and doping types used in the simulations are discussed. Then, doping concentration and electric field finite-element simulations using technology computeraided design (TCAD) are presented in Section 3, with a detailed simulation procedure using generic doping profiles and assumptions based on the physics of a semiconductor sensor. Monte Carlo simulations using the Allpix2[9] framework with electric fields and doping profiles from TCAD are described in detail in Section 4, going through the simulation setup step by step. Some example results of the high-statistics Monte Carlo simulations carried out in the Tangerine project are shown in Section 5, including in-pixel studies, transient current pulses, and simulation results from a multi-sensor setup. Finally, example comparisons of simulation results to data are shown in Section 6. 2. General layout and assumptions The MAPS simulated in this work consist of a highresistivity p-doped epitaxial layer grown on an electronicsgrade p-doped silicon substrate, with implanted doping wells in the epitaxial layer. The doping wells function as collection electrodes and/or shielding for the in-pixel electronics. As the epitaxial layer in the sensors is relatively thin (of the order of 10 µm), the sensor thickness is dominated by the substrate. Most of the visible signal is generated in the epitaxial layer, and the substrate is thus often thinned after sensor production, down to a total sensor thickness below 50 µm. The doping concentrations used in the simulations presented here are not values from a specific sensor or technology, but approximations derived from previous studies [10,11]. The substrate is assumed to have a doping concentration of 1 ·1019 cm−3, the epitaxial layer approximately 3 ·1013 cm−3, and doping wells ranging from 1·1015 cm−3to 1·1019 cm−3, depending on their purpose. 2.1. Doping wells In the centre of the pixels, an n-doped well is located. This well is simulated with a doping concentration of approximately 1019 cm−3, is positively biased, and serves as the collection electrode. The size of the well is of the order 2 of 1 µm across, and it has a square shape (when viewed from above). Surrounding the collection electrode is an opening without wells, and then a square deep p-well (for square pixels). This deep p-well is assumed to have a doping concentration of approximately 1015 cm−3. While none of the doping wells in the simulations contain any internal structure or electronics, the main purpose of the deep p-well in a physical sensor is to contain both NMOS transistors and internal n-wells that contain PMOS transistors. In this way, full CMOS front-end electronics are possible in the pixels. The deep p-well shields the electronics from the sensitive region, which ensures that the n-well collection electrode is the only node electrons drift to. This also allows for a higher bias voltage to be applied to the sensor bulk without damaging the electronics. The extent and shape of the wells can be used to shape the electric field, and may significantly affect the charge collection properties of a sensor. For example, the effect of changing the size of the opening between the collection electrode and the p-well is explored in the work presented here. 2.2. Contacts and biasing Ohmic contacts are essential to provide bias voltages to and extract signals from a sensor, and they are achieved by having a highly-doped region in the silicon next to the metal contact. In the sensors presented here there are contacts to the collection electrode, the p-well, and the sensor substrate. In a physical sensor, the biasing of the collection electrode and the p-well are done via metal contacts, and the substrate is biased through surface contacts outside the pixel matrix. In the simulations however, the substrate is instead biased via a contact directly on the backside as guard rings and sensor edge structures are not included. The collection electrode in the simulations presented here has a positive bias voltage of 1.2 V, whereas the p-well and substrate have bias voltages between 0 V and −6 V. The p-well and the substrate are commonly biased to the same voltage, but can also be biased separately. The bias voltage that can be applied to the p-well in a physical sensor is limited by the behaviour of the NMOS transistors, as their characteristics will change and their function may cease at high voltages. 2.3. Rectangular and hexagonal pixels Three main designs are simulated and tested in this work, labelled standard layout [12], n-blanket layout [13], and n-gap layout [14]. The standard layout is similar to what is used in the ALPIDE sensor [15], which is a MAPS used in the ALICE experiment since the ITS2 upgrade, developed in a 180 nm CMOS imaging process. This layout has a small n-type collection electrode in a p-type epitaxial layer, and depletion grows in an approximately spherical shape from this pn-junction. This depleted region tends to not extend fully below the p-well. The n-blanket layout introduces a blanket layer of n-doped silicon in the p-type epitaxial layer, which forms a deep planar pn-junction, allowing for full depletion of a pixel. This layout leaves an electric field minimum under the p-well at pixel edges and corners, however, leading to slow charge collection and possible efficiency loss in these regions. This can be amended by introducing a vertical pn-junction near the edge [14]. One way to achieve this is by leaving a gap in the blanket of n-doped silicon under the p-well, which is done in the n-gap layout. As a pn-junction is thus formed near the pixel edges, a lateral electric field is formed there, pushing charges toward the pixel centre. The n-blanket and n-gap layout modifications were originally developed for a 180 nm CMOS imaging process, but similar developments have been implemented in a 65 nm CMOS imaging process as well [16]. The three sensitive volume designs described above are applied in both rectangular and hexagonal pixel geometries in the work presented here. Using a hexagonal geometry decreases the amount of shared charge in pixel corners, as a pixel only shares a corner with two neighbours rather than three. It also reduces the maximum distance of the pixel boundary from the centre compared to rectangular geometries, while maintaining the same area. Hexagonal pixel shapes thus reduce regions with low electric fields at pixel edges. The maximum distance in a square grid between the pixel corner and the collection electrode is reduced by 12% for the same pixel area on a hexagonal grid [17]. As the pixel corner region and p-well edges have a larger opening angle, the electric fields there differ significantly compared to rectangular pixel geometries. The distance between collection electrodes is also the same for all adjacent pixels in a hexagonal configuration. 3. Finite element method simulations Technology Computer-Aided Design (TCAD) is a simulation tool that uses finite element methods to model semiconductor devices in 2D and 3D. In each node of a created mesh, calculations of the electrostatic potential and other properties are carried out by solving Poisson’s and carrier continuity equations. This work implements TCAD simulations with generic doping profiles to study effects of layout design on the electric field of CMOS sensors, and the presented simulations have been performed in 3D with Sentaurus TCAD from Synopsys [18]. The body of the sensor is created initially from simple geometrical shapes, which are then adjusted to represent the different studied layouts. To obtain insight into the effects of the adjustments, iterations of layout modifications and simulation evaluations are performed. To refine the simulations, the following principles are taken into account, to ensure a physically realistic and operational sensor: •The doping concentrations in the interfaces between 3 different doping structures (nand p-wells, epitaxial layer/substrate) should be diffused to avoid unphysical effects, such as abrupt changes in doping concentration and the corresponding electric field. •The p-well must shield its content from the electric field in the active sensor area; the doping must thus be sufficient for the depleted region to not penetrate deep into the well. •The charge carriers generated in the sensor volume have to reach the collection electrode. •There should be no conductive channel between different biased structures, i.e. punch-through in the sensor should be avoided. •The limitations on the operating voltages of the transistors in the readout electronics of a physical sensor should be respected. It should be noted that no internal structure of the doping wells is simulated in this work, so no readout electronics are included. The basic principles needed to protect them (outlined above) are included, however, in order to have a realistic sensor description. 3.1. Simulation workflow The simulation process starts with defining the sensor geometry using the Sentaurus Structure Editor tool. The materials of structures are defined together with their shapes, and the materials used in these simulations are aluminium for the electrodes, silicon oxide for the dielectric material, and silicon for the sensor bulk. In order to apply electrical boundary conditions, it is necessary to define interface regions called contacts, which correspond to physical contacts between the electrodes and the silicon bulk. For simplicity, only the top part of the sensor corresponding to the region taken up by the epitaxial layer and its interface to the substrate is used in the TCAD simulations presented here. In addition to the geometrical definition of the sensor, doping profiles and meshing parameters can be incorporated for different parts of the structure. This has been done for the epitaxial layer, the collection implant, and the p-well. Refinement/evaluation windows are defined to place the corresponding doping profiles. Analytical doping profiles are used to emulate the well structures; the wells are formed with an error function distribution in depth and a Gaussian distribution laterally. This emulates a dopant diffusion region with an extent of 0.3−0.4µm in the depth direction, and an extent of 0.4−0.5µm in the lateral direction. Flat doping concentrations are used across the full well structures, which is a simplifying assumption, but deemed sufficient for understanding the physical behaviour of the signal formation in a sensor. In the interface regions between the silicon bulk and the electrodes, it is necessary to add a highly doped region to create an Ohmic contact. Once doping regions and profiles have been defined, the refinement parameters for the mesh are established. This includes minimum and maximum mesh sizes, and the refinement function. A fine mesh provides more accurate results. However, since the number of calculation nodes can be substantial, the simulations take longer and can make subsequent calculations intractable with a finer mesh. This can be addressed by using an unstructured mesh that is refined only in certain regions. In this work, the mesh refinement is a function of the doping gradient, meaning that the mesh will be finer in places where there are significant changes in the doping concentration, e.g. at the edges of the well structures. When the geometry has been built and the mesh is defined, the device simulations can be performed with the Sentaurus Device tool. Simulations have been performed in both quasistationary mode, to obtain electric fields, and in transient mode to obtain the signal response to a charged particle traversing the sensor. The grid file created using the Structure Editor is imported into Sentaurus Device and the contacts are identified. Physical properties and solver properties are defined, as well as the boundary conditions of the simulation. The results of the quasistationary simulations are voltage-dependent curves and several electrical properties within the 3D volume. The TCAD simulations presented here were performed with a collection electrode bias voltage of 1.2 V and a p-well and substrate bias voltage of −5 V, unless otherwise stated. Properties studied in this work are the electric field magnitude, lateral electric field, charge current density, and depleted volume, shown in Sections 3.3,3.4, and 3.5. The results of transient simulations are time-dependent curves and snapshots of the perturbed electrical properties within the 3D volume, shown in Section 3.6 and compared to results of Monte Carlo simulations in Section 5.4. 3.2. Substrate diffusion simulation As the substrate is not expected to directly contribute to the electric field in the sensor, the TCAD simulations only include the epitaxial layer and the interface region between the epitaxial layer and the substrate. The only possible influence stems from the diffusion of p-dopants from the highly-doped substrate to the lower-doped epitaxial layer. During the process of semiconductor fabrication, a high difference in doping concentration and the high temperatures the device is exposed to is expected to produce a significant diffusion region at the interface of the epitaxial layer and the substrate. To simulate the diffusion of dopants from the substrate to the epitaxial layer, simulations of a sensor production process were performed using the Sentaurus Process tool of the Synopsys TCAD framework. The simulation includes 10 minutes of a chemical vapour deposition (CVD) process on the substrate with a temperature of 1050 ◦C, which results in 10 µm of epitaxial silicon [19]. The assumed doping concentration of the substrate in this simulation is 1 ·1019 cm−3, and 3·1013 cm−3for the epitaxial layer [10]. All the implanted 4 structures need to go through an annealing procedure to electronically activate the implanted ions. The simulated activation process involves heating of the structure to a temperature of 1100 ◦C for 240 min. The resulting structure is converted to a one-dimensional doping profile for the epitaxial layer, which can be seen in Figure 1. This profile is then imported for the epitaxial layer in further simulations. Figure 1: Simulated diffusion from the substrate to the epitaxial layer. Red indicates a higher doping concentration, and blue a lower one (by approximately six orders of magnitude). 3.3. Impact of sensor doping concentration Doping concentration is an important parameter in the design of silicon sensors, especially for the structures that constitute the junctions that shape the electric field inside the sensor. Studies conducted on changing the assumed doping concentrations on the pand n-wells described in Section 2for the standard layout by an order of magnitude did not significantly change the electric field forming in the sensor. Through the studies, a value of the p-well doping concentration of 5 ·1015 cm−3was selected as a baseline assumption for studies of the n-blanket layout, where the pn-junction is larger and the impact presumed greater. Studies of the impact of altering the doping concentrations of both the n-blanket and the p-well were performed using this layout, while keeping the epitaxial layer doping concentration fixed at 3 ·1013 cm−3. The limit values of the study were selected as the minimum value that would produce an effect on the depleted volume and the maximum value that would start to have an adverse effect on the surrounding doping structures (e.g. depletion deep into the p-well). The effects of altering the doping concentrations are studied by observing plots of the electric field magnitude and the depleted volume. The doping concentration of the n-blanket was studied with a fixed value of the p-well concentration of 5·1015 cm−3, varying the n-blanket concentration between 1·1014 cm−3and 4·1015 cm−3, shown in Figure 2a and Figure 2c respectively. The figures show a cross-section of a pixel, with half a collection electrode in the upper corners and the p-well in the centre of the image. Introduction of a low-doped n-type blanket implant creates a large pnjunction in the sensor, but a bulbous shape of the depletion region below the collection electrodes is still present. The highly-doped n-type blanket implant is not fully depleted, which leads to a conductive path being present between the two collection electrodes, and thus a non-functioning sensor. This can be observed by the shape of the depletion line. The doping concentration selected for the n-blanket for use in the final simulations was 9·1014 cm−3, shown in Figure 2b, as it provides a good depletion at the bottom of the sensor, and the least depletion intrusion in the p-well. (a) 1 ·1014 cm−3(b) 9 ·1014 cm−3(c) 4 ·1015 cm−3 Figure 2: Different doping concentrations of the n-blanket for a 10 µm pitch sensor in the n-blanket layout, represented by different colours. The colour scale corresponds to the total doping concentration (with the highest p-doped regions being blue and the highest n-doped regions red), the brown line indicates the location of a pnjunction, and the white lines delimit the depleted volume. With the value of the n-blanket doping concentration fixed to 9 ·1014 cm−3, the p-well doping concentration was varied. The study started with the value employed in the n-blanket concentration study. Figure 3displays the electric field magnitude at a close-up to the edge of the p-well, with different tested doping concentrations. For a doping concentration of 5 ·1015 cm−3, there is a relatively large volume of the p-well that has a non-zero electric field, which is undesirable as that may influence the in-pixel electronics contained there in a physical sensor. A higher doping concentration than the one in the previous study should allow for a better shielding of the p-well, but a too high doping concentration produces a deeper structure than what is desired. Furthermore, Figure 3b shows a more uniform electric field outside the p-well, when compared to the simulations using the upper and lower tested limit values shown in Figures 3a and 3c. The doping concentration of 1 ·1016 cm−3was thus chosen as the value for the p-well to use in the final simulations. 3.4. Impact of sensor geometry Modifications of the sensor layout can have a significant impact on the strength and extent of the electric field inside the sensor, and on the depleted volume. To investigate this impact, studies have been performed on the size of the p-well opening, which corresponds to the distance 5 (a) 5 ·1015 cm−3(b) 1 ·1016 cm−3(c) 5 ·1016 cm−3 Figure 3: Electric field magnitude for three different doping concentrations of the p-well, for a 10 µm pitch sensor. Close-up to the corner of the p-well. The brown line indicates the location of a pn-junction and the white line delimits the depleted volume. The electric field magnitude is given by the colour scale, where a dark blue colour indicates an electric field magnitude of zero. between the edge of the collection implant and the edge of the p-well, and on the gap size in the n-gap layout. These purely geometrical features are defined by the mask design used in sensor production. The effects are studied by observing plots of the electric field magnitude, the lateral electric field strength, and the depleted volume. In the figures, the former two are represented in colour scale, while the latter one is delimited by a white line. 3.4.1. P-well opening The p-well opening extent was varied from 1 µm to 4 µm, at a pixel size of 20 ×20 µm2. It was observed that increasing the p-well opening creates a stronger lateral electric field and increases the depleted volume in the standard layout, on the order of µm in both width and depth. A larger depleted volume allows for more charge collection through drift, while a stronger lateral electric field provides a higher drift velocity for the free charge carriers produced in the edges of the pixels. However, increasing the p-well opening means decreasing the p-well size and hence the available space for front-end electronics. The study was also performed using the n-blanket layout. Here, it was observed that the increase in lateral electric field strength and depleted volume was not as significant as in the standard layout case. A large p-well opening here leads to a larger undepleted region around the collection electrode, however, which has a negative impact on the sensor capacitance and charge collection behaviour. A comparison between the lateral electric field and depletion boundary for p-well openings of 1 µm and 4 µm is shown in Figure 4for both the standard and the nblanket layouts. The region with a strong lateral electric field visibly increases in both layouts as the p-well opening increases. A larger p-well opening is expected to also directly affect the sensor capacitance, but studies of this have not been carried out. An opening size of 2 µm is selected for use in further studies, as a balance between the increased depleted region and the total p-well size. 3.4.2. Gap size in the n-gap layout A lateral electric field is observed to appear under the p-well once the vertical junctions of the n-gap layout are added to the sensor, as can be seen in Figure 5. The gap (a) Standard layout, 1 µm opening (b) Standard layout, 4 µm opening (c) N-blanket layout, 1 µm opening (d) N-blanket layout, 4 µm opening Figure 4: Lateral electric field of two p-well openings for 20 µm pitch sensors in the standard and n-blanket layouts. The brown line indicates the location of a pn-junction and the white line the boundaries of the depleted volume. size was varied from 1 µm to 4 µm, and it was found that the strength of the lateral electric field is increased with increasing gap size, as the two vertical junctions move further apart. When the junctions are close, the regions of dopant diffusion will overlap, leading to a smaller lateral field. As can be seen in Figure 5the lateral field directions of the two vertical junctions are opposite, implying that they cancel out in the centre when the distance is small, so at sufficiently large gap size the lateral field strength reaches a maximum. However, when the gap is increased, the vertical pn-junction as well as the lateral electric fields are shifted away from the pixel edges, thus reducing their usefulness in improving charge collection far from the collection electrode and leaving an electric field minimum in the gap. A gap size of 2.5 µm is sufficiently large to maximise the lateral field strength, while keeping the junction close to the pixel edge. 3.5. Hexagonal pixel geometry simulation Detailed investigations of hexagonal pixel designs require custom field maps from TCAD for hexagonal geometries. One full pixel cell with the collection electrode in the centre is used in these simulations, and the p-well and substrate are biased with a voltage of −1.2 V. In Figure 6, a pixel cell for a simulation of a sensor in the standard layout is shown, with the colour indicating the doping level. The plane indicated as C1 represents a cross-section, along which the electric field magnitude is shown in Figure 7for both the standard and n-gap layouts. The regions sticking out from the top of the sensor are the metal biasing contacts for the collection electrode and the p-well. 6 (a) 1 µm gap (b) 2.5 µm gap (c) 4 µm gap Figure 5: Lateral electric field of three n-gap sizes for a 20 µm pitch sensor in the n-gap layout. The brown line indicates the location of the pn-junction and the white line delimits the depleted volume. It can be seen that the depletion region is small for the standard layout, only extending below the opening between the collection electrode and the p-well. For the n-gap layout, however, it extends across the full pixel. 3.6. Transient simulations Transient simulations were performed to estimate the shape, amplitude, and duration of a signal generated by a minimum ionising particle traversing the sensor. For these simulations, the p-well and substrate are biased with a voltage of −1.4 V. The particle traversing the sensor is represented by linear charge deposition along the particle track with Gaussian lateral smearing of 0.5 µm using the “Heavy Ion” charge deposition model. The mesh is adjusted for the transient simulations to have a finer cell size around the areas with a doping concentration gradient and the track of the traversing particle. For the thin sensors used in the simulations, a deposition of 63 electron-hole pairs per micrometre is assumed [20]. A matrix of 3 ×3 pixels is simulated in 3D, in order to avoid edge effects. The time step of the transient simulation is adapted to the expected shape of the signal. Two incident positions were simulated; in the centre and in the corner of a square pixel, for the three different layouts. The simulated track of the traversing particle is perpendicularly incident on the sensor for all the simulations. The absolute electron current density of the standard layout for the incidence of a MIP in the centre of the pixel is shown in Figure 8. Three adjacent pixels are shown. The depletion volume in the standard layout is limited, and the layout allows for significant charge sharing due to diffusion. The n-blanket and n-gap layouts were developed in order to improve the charge collection efficiency of the Figure 6: Simulated hexagonal pixel cell in TCAD. The colours correspond to doping level, and the plane marked C1 is a cut for display purposes. The electric field magnitude for this cut is shown in Figure 7. sensor in incident positions further from the readout implant [13,14]. In Figure 9the signals for the centre and corner incident positions for the n-gap layout are shown. The duration of the signal is dependent on the MIP incident position; the faster charge collection is observed in the centre of the pixel, due to the immediate proximity to the readout implant. The charges are deposited 0.5 ns after the start of the simulation, leading to the rising edge of the pulse. Figure 10 shows the signals for corner incident positions, for all three layouts. The n-blanket has a larger depletion region compared to the standard layout, and the n-gap has an additional area with a stronger lateral electric field component, which improves the charge collection far from the pixel centre. As the standard layout is undepleted in the pixel corners, charges formed there move slowly by diffusion, and the charge collection thus takes a comparatively long time. 3.7. Generating weighting potentials Transient simulations using TCAD are computationally intensive, and it can thus be beneficial to perform transient simulations using e.g. Allpix2instead. A weighting potential is required to be able to perform transient simulations using the Shockley-Ramo theorem [21,22]. The potential can be calculated by taking the difference of the electrostatic potentials arising from applying two slightly different bias voltages to one collection electrode in a sensor, keeping the other collection electrodes at a constant bias voltage. The two required electrostatic potentials can be simulated using TCAD, using the same sensor geometry with a difference of 0.01 V in the bias voltage of a single collection electrode. By calculating the difference between the two potentials in each mesh point, and dividing the difference by the difference in collection electrode bias voltage, the weighting potential is acquired. Before 7 (a) Standard layout (b) N-gap layout Figure 7: Electric field magnitude as output from a TCAD simulation for a hexagonal pixel in the standard and n-gap layouts. The white lines denote depletion boundaries, and the colour scale denotes the magnitude of the electric field. Figure 8: Absolute electron current density. Three adjacent pixels are shown, and the incident position is in the centre of the middle pixel readout implant. Standard layout. The dashed lines indicate pixel edges. utilising this weighting potential for simulations, the values should be constrained to be between 0 and 1, as this is the physical range of a weighting potential. Larger and smaller values may occur in the calculation due to numerical errors. 4. Monte Carlo simulations Simulation of sensor response to incident particles can be performed using TCAD, but studies with high statistical significance taking stochastic fluctuations into account are not feasible due to the long simulation time required per particle hit. By combining the doping concentrations, electric fields, and weighting potentials generated using TCAD with the Allpix2Monte Carlo simulation framework however, high-statistics simulations can be carried out [9,23]. This section demonstrates how such simulations can be performed for the monolithic sensors described earlier, using Allpix2version 3.0 [24]. 4.1. Simulation flow Allpix2is built on the concept of exchangeable modules, making it possible to flexibly change simulation aspects such as particle source and charge propagation method. 0 0.5 1 1.5 2 2.5 3 Time [ns] 0 100 200 300 400 500 600 Current [nA] TCAD simulation Pixel centre incidence Pixel corner incidence , n-gap layout 2 mµSquare pixels, 20x20 Figure 9: Signal in the n-gap layout sensor in the centre and corner incident positions. 0 5 10 15 20 25 Time [ns] 0 5 10 15 20 25 30 35 40 Current [nA] TCAD simulation Standard layout N-blanket layout N-gap layout , corner incidence 2 mµSquare pixels, 20x20 Figure 10: Signal of the sensor with the incident position in the corner of the pixel, for the standard,n-blanket,n-gap layouts. The modules also constitute different steps taken in the simulation process, and parameters of each module can be controlled by configuration files with keyword-value pairs. When providing values to keywords that represent physical quantities in module configurations, it is important to also provide a unit in order to avoid unexpected behaviour. 4.2. Sensor geometry and setup A detector model in Allpix2is defined in a configuration file, and an example can be seen in Listing 4.1. The example shows a monolithic sensor assembly with square pixels, with a pixel size of 20 ×20 µm2and a total sensor thickness of 50 µm. The sensor excess consists of sensor material without pixels, and is an important parameter to keep in mind when calculating sensor efficiency from simulation results, as particles hitting the sensor excess should not be counted as particles that should produce a signal in the sensor. 8 type = monolithic geometry = pixel sensor_material = silicon number_of_pixels = 20 20 pixel_size = 20um 20um sensor_thickness = 50um sensor_excess_right = 200um [implant] type = frontside shape = rectangle size = 2.2um 2.2um 0.8um Listing 4.1: Detector model configuration example. The geometry parameter can be used to select rectangular, radial strip, or hexagonal pixel geometries. For hexagonal geometries, the pixel size is defined from corner to corner along axes 60◦apart, i.e. the maximum distance across a hexagon. The [implant] section defines the x-, y-, and z-extent of the collection electrode of the sensor. In Allpix2, this defines the volume in which charge propagation stops, and charges are counted as “collected”. A small difference in this parameter can have a sizeable effect on the final results. A geometry configuration file defines the full simulated geometry, and can contain several sensors and passive volumes. An example configuration for a single-sensor simulation is shown in Listing 4.2. For each sensor or passive [dut] type ="detectorModel" position = 0mm 0mm 0mm orientation = 0deg 0deg 0deg Listing 4.2: Geometry configuration example, for a single sensor without random misalignment. volume, a position and orientation has to be defined, along with an alignment precision. In the given example, the name of the detector is “dut”, located at the centre of the global coordinate system. The alignment precision is more important to include when several sensors are involved. The detector type in the example is detectorModel, which is the name of a detector model configuration file such as the example shown in Listing 4.1. The global coordinate system is defined by the simulated world volume, and positions of components are defined in this system. Each detector placed in the world has a local coordinate system, with an origin defined in the centre of the lower left pixel of the sensor pixel matrix. 4.2.1. Constructing the geometry for use with Geant4 The [GeometryBuilderGeant4] module constructs the geometry for use with Geant4 [25,26,27], which allows for detailed particle interaction simulations. To visualise the geometry constructed by the module, the module [VisualizationGeant4] can be used. This opens a Geant4 graphical user interface window, with the possibility of also starting a Geant4 terminal, giving access to both a visualisation of the setup and Geant4 commands. By using the /run/beamOn Geant4 command, it is possible to see where particles from a defined source will hit the setup. This is useful for making sure that the source is aligned with the detectors in the desired way, but the command cannot be used to perform a proper simulation. 4.3. Importing results from TCAD simulations Electric fields and doping concentrations can be imported from TCAD simulations, which gives access to more detailed fields than the built-in parametric models. To be usable in Allpix2the TCAD mesh has to be adapted into a regularly-spaced grid, and this process is performed using the Mesh converter tool. The tool either performs a barycentric interpolation of values for each point in the new regularly-spaced grid or uses the value of the closest TCAD mesh point without interpolation. The second case is particularly useful for conversion of large field maps. An example of a configuration file for the mesh converter is shown in Listing 4.3. This file is used for converting the electric field of the region named “epitaxial” of a 20 ×20 µm2pixel. The model keyword defines the output format, and the units of the converted observable have to be provided; for electric fields, it is typically V/cm, and for doping concentrations cm−3(written /cm/cm/cm in the Allpix2configuration files). model ="APF" region ="epitaxial" observable = ElectricField observable_units ="V/cm" divisions = 300 300 100 xyz =xy-z Listing 4.3: Mesh converter configuration example, for the electric field in the “epitaxial” region of a sensor. The divisions parameter defines the number of points used in the regularly-spaced grid, and thus the granularity of the field map when imported into the framework. The number of grid points used can have a significant impact 9 0 5 10 15 20 25 30 35 40 45 50 m]µ x [ 0 5 10 15 20 25 30 35 40 45 50 m]µ y [ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Efficiency, 4 pixels, 200 electron threshold (a) Standard layout 0 5 10 15 20 25 30 35 40 45 50 m]µ x [ 0 5 10 15 20 25 30 35 40 45 50 m]µ y [ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Efficiency, 4 pixels, 200 electron threshold (b) N-gap layout Figure 18: In-pixel efficiency for four adjacent pixels with a pixel size of 25 ×25 µm2, at a bias voltage of −4.8 V and a threshold of 200 electrons. 0 100 200 300 400 500 600 700 Threshold [e] 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Cluster size Standard layout N-blanket layout N-gap layout Mean cluster size vs threshold Figure 19: Threshold dependency of the mean cluster size, at a bias voltage of −4.8 V for a sensor with a 25 ×25 µm2pixel size. pixels. The cluster size of the n-gap layout is significantly lower than for the other two layouts over all tested threshold values, as it has a smaller amount of charge sharing between pixels. The reduction in cluster size as the threshold increases for the standard layout is affected by the loss of efficiency in this layout at higher thresholds. The efficiency reduction is largest at pixel corners and edges, which are the main particle incidence areas that lead to a higher cluster size, as can be seen in Figure 17. The dependence of the detection efficiency on the threshold value is shown in Figure 20, where the mean efficiency value of the sensor is plotted for the three layouts. It can be observed that the n-blanket and n-gap layouts maintain efficiency over a larger threshold range than the standard layout, which is consistent with what is shown in Figure 18. The n-gap layout thus enables the largest efficient operating margin. This trend is consistent for all tested bias voltages. For a pixel size of 15 ×15 µm2 the efficient threshold range is larger for the standard and n-blanket layouts, compared to the larger pixel size of 25 ×25 µm2. This can be seen in Figure 21, where re0 100 200 300 400 500 600 700 Threshold [e] 0 20 40 60 80 100 Efficiency [%] Mean efficiency vs threshold Standard layout N-blanket layout N-gap layout Mean efficiency vs threshold Figure 20: Threshold dependency of the detection efficiency, at a bias voltage of −4.8 V for a sensor with a 25 ×25 µm2pixel size. sults are shown for a bias voltage of −1.2 V. For the n-gap layout, the efficient threshold range is slightly smaller than for the larger pixel size at this bias voltage. This is due to a reduced efficiency in the gap in the n-blanket between pixels; in a smaller pixel this region takes up a larger fractional volume, as the gap size remains the same. The trend between the different layouts remains the same, with the n-blanket layout more efficient than the standard layout, and the n-gap layout more efficient than the n-blanket layout. 0 100 200 300 400 500 600 700 Threshold [e] 0 20 40 60 80 100 Efficiency [%] Mean efficiency vs threshold mµStandard, 25x25 mµN-blanket, 25x25 mµN-gap, 25x25 mµStandard, 15x15 mµN-blanket, 15x15 mµN-gap, 15x15 Mean efficiency vs threshold Figure 21: Threshold dependency of the detection efficiency, for sensors with two different pixel sizes at a bias voltage of −1.2 V. Figure 22 shows the mean resolution of the sensor in the x-direction versus the threshold, for the same simulation setup as before. As the pixels are square and symmetric, the resolution is identical in the y-direction. The resolution is defined as the root mean square of the central 3σ(99.73%) of the residual distribution, i.e. the distribution of the difference of reconstructed particle position and Monte Carlo truth position for each event. The re16 constructed position is taken as a charge-weighted mean position of all pixel hits in a cluster. In these simulations, the full charge information is used, rather than the value from a charge-to-digital converter with limited resolution. 0 100 200 300 400 500 600 700 Threshold [e] 1 2 3 4 5 6 7 m]µResolution in x [ Residual RMS in x vs threshold Standard layout N-blanket layout N-gap layout Residual RMS in x vs threshold Figure 22: Threshold dependency of the spatial resolution, at a bias voltage of −4.8 V for a sensor with a 25 ×25 µm2pixel size. The resolution deteriorates as the threshold increases, and for a large range of threshold values the standard layout has the best (lowest) resolution. At low thresholds, this is due to the larger amount of charge sharing compared to the other two layouts. The reconstructed position is more accurate when the cluster size is larger, due to the charge-weighted position reconstruction occurring between more pixels. At high thresholds, the resolution for the standard layout decreases as threshold increases. This is an effect of the reduction of the efficiency, as can be seen in Figures 18 and 21. As efficiency is reduced at pixel edges when the threshold increases, only particle hits close to the pixel centre can be reconstructed, and thus the effective pixel size is reduced. The smaller cluster sizes of the n-blanket and n-gap layouts deteriorate their resolutions, but as can be seen in Figures 20 and 18, their efficiency is improved. The n-gap layout has the highest efficiency, but the largest intrinsic resolution. 5.4. Transient pulse studies Performing transient simulations as described in Section 4.8 can considerably reduce simulation time in comparison to transient simulations using TCAD [8]. Using Monte Carlo simulations also allows inclusion of stochastic effects, such as Landau fluctuations and secondary particles. A validation between both approaches has been performed, utilising the same parameters and setup, to ensure that using Allpix2mimics the results of TCAD transient simulations as described in Section 3.6. In these validation studies, only an epitaxial layer with a thickness of 10 µm was simulated. The mobility model parameter values in Allpix2were changed to match the extended Canali mobility model used in TCAD [37]. The simulated geometry consisted of a matrix of 3×3 pixels, with a pixel size of 20 ×20 µm2. Charges were injected along a straight line in the corner between four pixels using the [DepositionPointCharge] module, with 63 electron-hole pairs deposited per µm. Electric fields, doping concentrations, and weighting potentials from TCAD were imported into Allpix2using their respective module readers. Using the [TransientPropagation] module as described in Section 4.8, an integration time of 40 ns was used for all simulations. A coarse value of the timestep parameter may lead to smaller pulses than expected, so a timestep of 15 ps was used in the presented results. As the charge is injected in the corner, pulses are expected to be induced in all four pixels sharing the corner. The total pulses were calculated as the average of the induced pulses in the four collection electrodes for each event. Figure 23 shows the resulting pulses for both TCAD and Allpix2simulations, for the standard and n-blanket layouts. The Allpix2pulses are the average of 10 000 events, whereas the TCAD pulses come from single events. The plots show that the two methods agree in terms of pulse height and peaking time, which indicates that the Allpix2 method largely yields compatible results with the TCAD method. At the falling edge of the pulses, there is a small difference between the approaches, however. The peak structure in the TCAD pulse for the n-blanket layout between 0 and 1 ns is an artefact of the TCAD simulations from the initial charge deposition, and its integral is zero and does not affect the rest of the pulse. A noticeable difference in the pulse rise time and duration is present between the shown standard and n-blanket layouts, with the n-blanket pulse being faster, which is expected due to the larger depleted region and thus more charge collection by drift. This also increases the charge collection efficiency per pixel and results in a higher peak and higher integrated charge for the n-blanket layout. 5.5. Multi-sensor studies Several sensors can be simulated simultaneously in Allpix2, in for example a beam telescope setup. By using the [CorryvreckanWriter] module, the results of the Allpix2simulation can be exported in a format suitable for the Corryvreckan test beam reconstruction framework [38]. This framework can then be used to extract parameters such as telescope resolution at different positions for the setup. The simulation of multi-sensor setups enables studies of the tracking performance of different setups and sensor designs, and construction of a beam telescope represents a possible use case of the sensors described in this work. Simulations were carried out with a six-plane beam telescope surrounded by air, using sensors in the three different layouts with a pixel size of 20 ×20 µm2. A beam of 17 0 5 10 15 20 25 30 35 40 Time [ns] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Current [nA] TCAD simulation simulation 2 Allpix (a) Standard layout 0 5 10 15 20 25 30 35 40 Time [ns] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Current [nA] TCAD simulation simulation 2 Allpix (b) N-blanket layout Figure 23: Comparison between pulses obtained with TCAD (blue) and Allpix2using TCAD fields (red). A TCAD pulse corresponds to a single event, while the Allpix2pulse is the average of 10 000 events. electrons was fired at the setup, with a single electron per event. The distance between telescope planes was varied, and the spatial resolution at the device-under-test (DUT) position (in the middle of the setup) extracted. The resolution was determined from the distribution of the difference of track intercept locations and the true particle positions at the DUT, where the tracks were reconstructed using the telescope plane hits and the general broken lines method [39]. Figure 24a shows the resulting telescope tracking resolution at the DUT position for different distances between telescope planes, for the three sensor layouts. The distance is the same between any two adjacent planes, and the presented results are at a threshold of 200 electrons for each of the six sensors. An estimate of the telescope tracking efficiency is shown in Figure 24b. The calculation is performed by dividing the number of reconstructed tracks by the total number 0 20 40 60 80 100 120 140 160 dz [mm] 1 2 3 4 5 6 m]µ Telescope resolution in x [ N-gap layout N-blanket layout Standard layout (a) Resolution at the DUT position 0 20 40 60 80 100 120 140 160 dz [mm] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Tracking efficiency N-gap layout N-blanket layout Standard layout (b) Tracking efficiency estimate Figure 24: Telescope resolution at the DUT position, and tracking efficiency estimate, as a function of the distance between telescope planes for the three different layouts at a threshold of 200 electrons. of simulated events for each data point. At least five of the six telescope planes have to register hits for an event to be considered for track reconstruction, and not all such events will have a track successfully reconstructed due to scattering leading to hits outside of the spatial cut. A spatial cut equal to the pixel size (20 µm) is used in the final track reconstruction. From the telescope tracking resolution, it can be seen that the standard layout provides the smallest resolution, while the resolution for the n-blanket and n-gap layouts is slightly larger. This agrees qualitatively with the results shown for single-sensors in Figure 22, but the difference is smaller than indicated there. As the distance between telescope planes increases, the tracking resolution deteriorates. Both of these effects agree well with expectations; as multiple sensors are used, the resolution of the full system is better than that of an individual sensor, and when the distance between planes increases the scattering in air increases along with the uncertainty in deflection angle. The telescope tracking efficiency estimation qualitatively 18 agrees with the single-sensor results shown in Figure 20; at a threshold of 200 electrons, the tracking efficiency is low for the standard layout due to the low efficiency of each individual sensor. The n-blanket layout shows a significantly higher tracking efficiency, and the n-gap layout is the most efficient. The tracking efficiency estimate has a weak dependence on the distance between telescope planes, with a decrease due to the increased scattering in air as the distance between planes increases. In conclusion, the spatial resolution of a beam telescope consisting of the investigated sensors is comparable to that of the EUDET-type beam telescopes [40]. While the tracking efficiency is low for the standard layout, it can be improved without significant loss of tracking resolution by utilising one of the other layouts. 6. Comparisons to data and previous simulations Comparisons of the outlined simulation procedure to previously published data are performed, using results from a test beam carried out in the framework of the CLICdp Collaboration on the TowerJazz Investigator 1 sensor, with a pixel size of 28 ×28 µm2[7]. This sensor is designed in the standard layout in a 180 nm CMOS imaging process with an epitaxial layer thickness of approximately 25 µm, and the studies are made at a bias voltage of −6 V. The sensor investigated here is thus different from what was previously used as an example in developing the simulation procedure, demonstrating the versatility of the approach. Figure 25 shows comparisons between data taken with the sensor and results using the simulation procedure outlined in this paper. Figure 25a shows the cluster charge at a threshold of 120 electrons, and Figure 25b shows the cluster size versus threshold. In the figures, data are shown in blue and the results of simulations using the generalised procedure outlined in this paper are shown in red. Figure 25a indicates that the simulation result cluster charge is shifted slightly higher compared to the data, while the rising and falling slopes of the distributions match in shape. A fit is performed using a convolution of a Gaussian and Landau function, which gives a most probable cluster charge value of 1.47 kiloelectrons for the simulation results. For the data, the value is 1.42 kiloelectrons [7]. The width of the Gaussian part is 0.22 kiloelectrons in the simulation results, and 0.21 kiloelectrons in the data. In Figure 25b the simulations and data match across the full investigated threshold range, with a slight deviation at thresholds smaller than 300 electrons. The errors shown are purely statistical for both data and simulations, and the maximum deviation between the data and simulations is 4%, at a threshold of 120 electrons. Comparative studies have also been carried out in the frame of the Tangerine project, using test beam data for sensors in a 65 nm CMOS imaging process [5]. These 0 500 1000 1500 2000 2500 3000 3500 4000 Cluster charge [e] 0 0.005 0.01 0.015 0.02 0.025 0.03 Events (norm.) Cluster charge Data Simulations (a) Cluster charge distribution at a threshold of 120 electrons 0 100 200 300 400 500 600 700 Threshold [e] 1 1.5 2 2.5 3 3.5 Mean cluster size Mean cluster size Data Simulations (b) Threshold dependency of the cluster size Figure 25: Comparison between simulation results obtained using the method described in this paper and test beam data [7]. studies show an agreement between data and simulations within 1% for the n-gap layout. In conclusion, the simulations using the method presented in this paper match data well. There is a maximum deviation of approximately 4% in both the charge distribution and the cluster size. The qualitative trends agree and to a level sufficient to draw conclusions concerning sensor performance and its origins without use of any proprietary information. The results are also compatible with simulations carried out at CERN using more realistic fields from TCAD, which have been compared to the same data [7]. 7. Summary and outlook In this paper, a simulation procedure for silicon sensors with complex non-uniform electric fields has been described, starting from first principles of a simple pnjunction, and going to high-statistics simulations of a multi-sensor beam telescope. Three-dimensional electrostatic TCAD simulations were produced, based on generic doping profiles and first principles of sensor operation, 19 without knowledge of proprietary information. Studies of the impact of varying different sensor parameters have been carried out, observing their impact on the electric fields. Three different sensor layouts have been tested and compared, in several different pixel sizes for both rectangular and hexagonal pixel geometries. The geometries used only describe the large-feature geometry of the sensors, and do not attempt to mimic the intricacies of a CMOS imaging process, but they are sufficient for modelling a signal response describing observed sensor behaviour to an accuracy within a few percent for key observables. By importing the TCAD fields and doping profiles into Allpix2, fast and complete simulations of particle interactions and charge transport can be performed, taking stochastic fluctuations stemming from the underlying physics processes into account. Through this process, sensor performance observables such as efficiency, cluster size, and resolution can be extracted. Example results of such simulations have been presented, and agree well with expectations and studies of similar sensors. Transient simulations have been carried out in both TCAD and Allpix2, and the results match well. Using the induced charge given by transient simulations is more accurate than using the notion of “collected charge”, and when charge pulses are available more sophisticated digitisation simulation can be performed to extract accurate values for time-of-arrival and time-over-threshold. This work is foreseen to continue in the near future, also including more accurate simulation of the sensor front-end response. The described simulation procedure is applicable in multiple different cases, and constitutes a generic toolbox for performing similar studies without using proprietary information. These simulations are able to provide accurate predictions of sensor behaviour and trade-offs with different designs, and can thus be used to inform decisions taken for future sensor designs. Acknowledgements The presented simulation studies have been performed in the frame of the Tangerine project, and in collaboration with the CERN EP R&D programme. CRediT authorship statement Dominik Dannheim: Conceptualisation, Resources, Writing - Review & Editing. Manuel Del Rio Viera: Formal analysis, Investigation, Writing - Original Draft, Visualisation. Katharina Dort: Methodology, Resources, Writing - Review & Editing. Doris Eckstein: Resources, Funding acquisition. Finn Feindt: Writing - Review & Editing. Ingrid-Maria Gregor: Resources, Funding acquisition. Lennart Huth: Methodology. Stephan Lachnit: Software. Larissa Mendes: Formal analysis, Investigation, Writing - Original Draft, Visualisation. Daniil Rastorguev: Writing - Review & Editing. Sara Ruiz Daza: Formal analysis, Investigation, Writing - Original Draft, Visualisation. Paul Sch¨utze: Methodology, Software. Adriana Simancas: Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualisation. Walter Snoeys: Conceptualisation, Writing - Review & Editing. Simon Spannagel: Conceptualisation, Methodology, Software, Writing - Review & Editing, Project administration. Marcel Stanitzki: Funding acquisition. Alessandra Tomal: Supervision. Anastasiia Velyka: Methodology, Software, Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualisation. Gianpiero Vignola: Writing - Review & Editing. H˚akan Wennl¨of: Methodology, Software, Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualisation. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Funding This work has been carried out within Tangerine, a Helmholtz Innovation Pool Project. 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