Simulation of silicon strip detector for space-based cosmic ray experiments with Allpix2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document}

Silicon strip detectors are widely applied in space-based cosmic ray experiments and most of the silicon strip detectors deploy an analytical method for its digitization. However, the analytical method simplifies the physical process of propagation of electrons/holes generated inside the silicon detector by particles that pass through the detector. In order to simulate silicon strip detectors with different configurations comprehensively, the Allpix2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}, an open-source software, is used to study those processes. When particle passes through the silicon detector, energy is deposited based on Geant4 simulation, and electron–hole pairs are created due to the deposited energy. The Allpix2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document} simulation method and the analytical method are both used to calculate or simulate the diffusion and drift processes that electron–hole pairs propagate inside the silicon detector under internal electric field, and the number of electrons/holes accumulated at implanted strips are counted. The number of electrons/holes accumulated along the implanted strips are compared between the Allpix2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document} simulation method and analytical method for different incident angles and different incident positions, they are found to be in good agreement for proton particles, while there are discrepancies for carbon and silicon particles. The Allpix2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document} software may be an important tool for the study of response of silicon strip detectors in space. The charge resolution of single implanted strip predicted by Allpix2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document} simulation method is about 4.7% for proton, 3.8% for carbon and 1.6% for silicon particles for an incident angle of 45∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}.


Introduction
Silicon strip detectors are commonly used in all kinds of particle physics and space experiments due to its high-precision  [1][2][3][4]. For example, the Dark Matter Particle Explorer (DAMPE) silicon-tungsten tracker consists of 12 orthogonal layers of single-sided silicon micro-strip detectors and 3 layers of 1 mm thick tungsten plates [1]. The angular resolution is around 0.10 degrees for 100 GeV photons and the charge resolution is better than 0.10 charge unit for protons [2]. The Alpha Magnetic Spectrometer (AMS-02) silicon tracker has nine layers and charge resolution of the inner tracker charge is 0.1 charge units, and the position resolution is about 5 microns for carbon [3,4]. The large area telescope (Fermi-LAT) tracker module has a vertical stack of 18 (x, y) tracking planes, including two layers (x and y) of single-sided silicon strip detectors and high-Z converter material (tungsten) per tray, where the incident photons are converted into electron and positron pairs in the tungsten plate and leave a trail in the silicon strip detectors [5]. The high energy cosmic-radiation detection facility (HERD) under construction in China has five-sided silicon charge detectors that enable accurate charge measurements [6].
The silicon strip detector is usually a reverse biased PN junction with p + implanted diode strips and aluminum (Al hereafter) contacts on one side of the silicon detector. When a charged particle passes through the sensitive area of the silicon detector, electron-hole pairs are created and electrons are drifted towards the ground plane and holes are drifted towards the metal aluminum strips near the trace by an internal applied electric field, in the meanwhile, the electrons and holes are also experienced a diffusion process. Due to the electron and hole's high mobility, electron-hole pairs are quickly collected by the implanted strips on the silicon substrate surface, and are read out by electronic system. Additionally, capacitive coupling among strips redistributes the signal to neighboring strips around the particle incident position (or impact position) [7]. The angle and impact position of incident particles will affect the number distribution of accumulated electron-hole pairs at the strips, which need to be carefully corrected to get a good charge and coordinate measurement. Therefore the simulation of the propagation processes (mainly include diffusion and drift process) of the electrons and holes that created by the incident particles inside the silicon strip detector is mandatory for the charge and coordinate measurement.
Most silicon strip detectors in space experiments used an analytical method, such as AMS-02, to obtain the number of electrons/holes collected at the implanted strips. Details of this method can be found in reference [7] and a short description of this method will be given in Sect. 2.1. The analytical method simplifies the physical processes of the particles passing through the detector, and approximates the drift and diffusion of charge carriers. To obtain more detailed simulations of the detector response, the Allpix 2 software is used to study those processes [8].
Allpix 2 is a generic, open-source software framework for the simulation of silicon strip detectors [9]. It can simulate the response of the silicon strip detectors with different configuration according to the requirements of users when an incident particle passing through. In this paper, the Allpix 2 will be used to simulate the processes from generation to collection of the electron-hole pairs, the results will be compared with the analytical method's results.

Methods
The calculation or simulation in this paper is based on the silicon detector shown in Fig. 1. It is a N-type silicon substrate with a thickness of 0.15 mm. One surface of the detector is doped with n + layer and aluminum layer. On the other surface the sensitive area of the detector is covered with p + implanted strips and aluminium contacts (180 strips of 100 Fig. 1 Schematic diagram of the silicon strip detector. On the upper surface, aluminum strips and silica insulation strips are contacted alternately, below the aluminum strip, there is heavy doping p + strip. On the lower surface, there are heavy doping n + layer and aluminum layer. Between them are the detector sensitive area. The purple line in the plot indicates the incident particle track, black line represents the drifting direction of the holes and white line denotes the drifting direction of electrons µm pitch and 50 µm width). Silica insulation strips are contacted between the adjacent strips. The depletion voltage is 35 V, which makes the entire detector full depletion. An reverse bias voltage 80 V is applied, which allows it to operate in a reverse bias state.
There is a two-dimensional coordinate system was created for the detector, the origin of this coordinate system is the most left strip of the detector, the index of this strip is 0, the x axis is along the implanted strips, and go for the direction with increasing strip index. One unit of x coordinate is the strip pitch. All positions along the strips can either be stated in absolute coordinate number or the strip number. The z axis is perpendicular to the x axis, and goes from the bottom surface to the upper surface, as shown in Fig. 1.
When a charged particle passes through the silicon detector, energy is deposited along the track of the incident particle, which creates the electrons and holes inside the detector. The amount of the deposited energy can be well described by the Bethe-Bloch formula [10], as shown in formula 1. where, W max is the maximum energy transferred in one single collision, z is the charge number of the incident particle, Z is atomic number of the absorber, A is atomic mass of the absorber, N A is the Avogadro's number, r e is the classical electron radius, m e is the electron mass, M is the incident particle mass, −dE/dx is the average ionization energy loss of the charged particle in absorber, which is proportional to the charge number of the incident particle z 2 . However, this formula only describes the average deposited energy, in reality the deposited energy fluctuates for each event, to simulate the fluctuation of the deposited energy, a Monte Carlo simulation software named Geant4(version 10.5.1) is used to simulate the deposited energy for each event [11]. The simulation process of the silicon strip detector is shown in Fig. 2. Firstly, the deposition energy of the particle that incidents into the detector is obtained from the Geant4 simulation software. The electrons and holes are created inside the silicon detector from a simple calculation that every 3.64 eV deposited energy creates one electron-hole pair [12]. Secondly, the propagating process of electron/hole pairs inside the detector is resolved with the analytical method and Allpix 2 simulation method, respectively. Both of them are described in detail below and their results are compared with each other in section 3.

Analytical method
For the analytical method, the steps of the calculation are shown in Fig. 3. The Poisson equation was solved to obtain the electric field distribution inside the detector, and the time spent from the electron-hole pair creation to reaching the implanted strips is calculated to obtain the width (σ in formula 5) of the space distribution of electrons/holes according to formula 5, details of the calculation can be found in reference [7]: where k is the Boltzmann constant, T is the effective temperature, q is the meta charge, U depl is the silicon detector depletion voltage, U bias is the internal applied bias voltage to the detector, d is the detector thickness and z 0 is z coordinate when electron-hole pair is created. The track of the incident particle is evenly divided into n parts, the number of electrons/holes created of each part is N e /n, where N e is the total number of electrons/holes created inside the detector. The space distribution of electrons/holes accumulated on the surface after propagation for each part is a Gaussian function as shown in the middle panel of Fig. 3. The number of electrons/holes collected by the ith strip from the jth part (named as I i j ) is calculated according to for- x j is the x coordinate of jth part's midpoint, σ is the sigma value of the Gaussian function calculated according to formula 5. The accumulated number of electrons/holes at the ith strip is the sum of all the n parts, then the ratio of the number of electrons/holes at ith strip over the total number of electrons/holes (named charge ratio or R i hereafter) can be calculated according to formula 7, the space distribution of charge ratio is shown in the lower plot of Fig. 3.

Allpix 2 simulation method
For the simulation with Allpix 2 software, the detector was constructed also according to the detector shown in Fig. 1.
The internal electric field applied on the detector is shown in formula 8, where z is the z coordinate of the position required; U bias is the bias voltage applied; U depl is the depletion voltage, d is the thickness of the detector. Allpix 2 software simulates the transport of electrons and holes that pass through the sensitive area of the detector. The propagation consists of a combination of drift and diffusion simulation. The drift process is calculated by using the charge carrier velocity derived from the charge carrier mobility parameterization from Jacoboni [14]. The correct mobility for either electrons or holes is automatically chosen based on the type of the charge carrier under consideration. A fourth-order Runge-Kutta-Fehlberg method is used to integrate the particle motion through the electric field. After each Runge-Kutta step, the diffusion is accounted for by applying an offset drawn from a Gaussian distribution calculated from the Einstein relation [15]. The main parameters of electrons and holes used in the simulation are listed in Table 1, where Temperature represents the temperature of the detector, the default value is room temperature 20 • C; Charge perstep represents the maximum number of charge carriers propagated together. Defaults to 10 charge carriers. It can be set manually and thus the accuracy of the propagation can be controlled, which is set 5 charge carriers to improve the accuracy of the simulation. Time step min and Time step max are the minimum and maximum step respectively in time to use for the Runge-Kutta integration regardless of the spatial precision. Integration time is the time within which charge carriers are propagated.
The propagation stops when the set of charges reaches any surface of the sensor. Figure 4 shows the concentration distribution of the charge carriers, where electrons are marked in blue colors and propagating downward, while holes are pre-

Comparison between two methods
The space distribution along x-axis of electrons accumulated at each implanted strip are calculated or simulated by the analytic method and Allpix 2 simulation method, respectively, and they are compared with each other. To remove the influence of the difference of total number of electrons/holes created, only the charge ratio (the number of electrons/holes at single strip over the total electrons/holes) is compared. The sample for the comparison is proton with an energy equal to 10 GeV. There are totally 10,000 events simulated with Allpix 2 simulation method, while for analytic method, the distribution is only calculated once, since there is no fluctuation. Figure 5 shows the distribution of charge ratio for the The blue line indicates the distribution of the charge ratio at the 90th implanted strip, the red line represents a Gaussian fit to the distribution and the blue vertical line denotes the same charge ratio calculated by the analytical method. As can be seen, the ratio between the two methods agrees with each other within one sigma error for this incident angle and position. The charge ratio distribution comparison for different incident angles but the same incident position (fixed at x = 90) are shown in Fig. 6. The blue line is the result of analytical method, only the 90th strip has signal when θ = 0 • (Fig. 6a), and the charge ratio is 1, it is ∼99.6% when θ = 30 • (Fig. 6b), and ∼0.04% of the total electrons/holes is collected by the neighboring strips. It is about 66.3% and 38.3% for incident angle θ = 45 • (Fig. 6c) and θ = 60 • (Fig. 6d) respectively.
The red line is the result from Allpix 2 simulation method, and the red area is the one sigma error calculated similar to Fig. 5. The ratios at different implanted strips between the two methods are approximately equal when the incident angle θ = 0 • and θ = 30 • . The ratio at the 90th implanted strip for Allpix 2 simulation method are slightly smaller than that of the analytical method, but they are still within one sigma error giving by Allpix 2 software.
The charge ratio distribution comparison for different incident positions but the same incidence angle (fixed at θ = 30 • ) are shown in Fig. 7. For analytical method, the charge ratio is To understand the behavior of charge ratios for heavier nuclei and higher energy, Carbon and silicon particles with 10,000 events are also simulated with an energy of 60 GeV, the charge ratios are shown in Fig. 8 with incident angle θ = 45 • and incident position x = 90. As seen, the charge ratio obtained by the analytical method is different from the Allpix 2 simulation method, for the 90th strip, the charge ratio is 66.3% and 66.3% for carbon and silicon particle respectively for analytical method, while they are 57.2% and 57% for Allpix 2 simulation method. The error of the charge ratio at the 90th strip from Allpix 2 simulation method decreases steadily as the charge Z increases. It is 8.5% for proton, 3.3% for carbon and 1.6% for silicon particles. The discrepancy between the two methods for heavy nuclei may be compared and verified with the test beam data or experimental data in the future.
In terms of computing efficiency, for a sample of 10,000 vertically incident protons, the analytical method takes less than 20 s and the Allpix 2 simulation method takes 4 min 35 s with one CPU core of Intel i5-1135G7@ 2.40 GHz. The analytical method is more efficient, due to that it simplified the diffusion and drift process with a Gaussian function. The Allpix 2 simulation method takes more time, due to that it solved the diffusion and drift process numerically, which may be more accurate.
For systematic errors, both analytical method and Allpix 2 simulation adopts the linear electric field inside the detector, it is an approximation of the true electric field, which may cause systematic error. The analytical method simplifies the diffusion and drift process with a Gaussian function, which may also cause systematic error and assumes that the self-repulsion of the charge cloud is negligible, which is correct for minimum ionization particles, but is no proper for heavy ions [7]. The Allpix 2 simulation describes the physical process of charge carriers inside the detector more comprehensively, which in principle should have less systematic error. However, to evaluate the systematic errors of the two methods, the test beam data or experimental data is needed to compare with the result from those two methods, which will be studied in the near future.

Charge resolution
The charge resolution of incident particles is the key parameter of silicon detector performance. The number of electrons accumulated at the implanted strips, which is simulated by the Allpix 2 simulation method, is proportional to the energy deposition inside the detector, which is further proportional to the square of the nuclei charge Z of the incident particle. Therefore, the width of the electron/hole number distribution (named charge distribution hereafter) could be used to characterize the charge resolution of the silicon detector (ignore the nonlinear response of the electronics and other detector effects) according to formula 9, where Z is the nuclei charge of the incident particle, Z is charge resolution, N e is the width of the charge distribution and N e is the number of electrons/holes accumulated.
The charge distribution at the implanted strip with maximum signal for incident angle θ = 0 • and θ = 45 • of proton, carbon and silicon particles are shown in Fig. 9. A Landau convoluted Gaussian function is used to fit the charge distribution [16], N e in formula 9 is the sigma parameter of the function, N e in formula 9 is the MPV value of the function. The derived charge resolution Z /Z according to formula 9 are 4.5% and 4.7% for θ = 0 • and θ = 45 • respectively when primary particle is proton. Those two numbers are 3.3% and 3.8% for carbon, 1.3% and 1.6% for silicon particle. As seen, the resulting charge resolution increases steadily as functions of charge Z. And they are similar to the charge resolution derived by other experiments [2,4].

Conclusion and outlook
The analytic method and Allpix 2 simulation method were used to study the response of silicon strip detector when particle passes through the detector with different impact positions and incident angles, respectively. The analytical method uses an analytic formula to obtain the charge ratio (electron/hole number at single strip over the sum of all strips), which assume a Gaussian space distribution of the electrons/holes on the surface of the detector after diffusion and drift processes. Allpix 2 simulation method simulates the detailed reaction process of drift and diffusion when particle incidence into the detector, and accumulate the number of electrons/holes at the implanted strips. There is no sysmetric errors in the analytical method and for Allpix 2 simulation method's sysmetric errors need test beam data or experimental data to obtain.
For proton particle, when the incident position at x = 90, the charge ratio of the 90th strip is 1 for 0 • incident angle,  which means that one strip collects all the electrons/holes. However, as the incident angle increases, there are more and more strips that collect signals, it is 38.3% when incident angle θ = 60 • . For events with the same incident angle θ = 30 • , the charge ratio is ∼99.6% at the 90th strip when the incident position x = 90. When the incident position is in the middle of neighboring strips, i.e. x = 89.5, the charge ratio of the adjacent strips is 50%. When the incident position is x = 89.25 and x = 89.75, the charge ratio of the two adjacent strips is about 78% and 22%. The charge ratios between the two methods are agreement with each other within one sigma error for different incident angles and incident positions when the primary particle is proton, however, the discrepancy begin to exist for heavier nuclei, e.g. carbon and silicon. At the 90th strip with incident angle θ = 45 • , the charge ratio is 66.3% and 66.3% for carbon and silicon particle respectively for analytical method, while they are 57.2% ± 3.3% and 57% ± 1.6% for Allpix 2 simulation method, which may be compared and verified with the test beam data or experimental data in the future.
The charge resolution simulated by Allpix 2 simulation method is around 4.7% for proton, 3.8% for carbon and 1.6% for silicon when incident position x = 90 and incident angle θ = 45 • . As seen, the charge resolution increases steadily with the nuclei charge number Z, and the charge resolutions are similar to the charge resolution derived by other experiments.
In summary, the analytical method process is simple, the computational efficiency is high but less comprehensive, and the Allpix 2 process is more comprehensive, but the computational efficiency is lower. The comparison of the results between the analytical method and the Allpix 2 simulation method shows that Allpix 2 may be an important tool for the simulation of silicon strip detectors for future space experiments.