Time Resolution of the 4H-SiC PIN Detector

We address the determination of the time resolution for the $\rm 100~\mu m$ 4H-SiC PIN detectors fabricated by Nanjing University (NJU). The time response to $\rm \beta$ particles from a $\rm ^{90}$Sr source is investigated for the detection of the minimum ionizing particles (MIPs). We study the influence of different reverse voltages, which correspond to different carrier velocities and device sizes, and how this correlates with the detector capacitance. We determine a time resolution $\rm (94\pm1)~ps$ for $\rm 100~\mu m$ 4H-SiC PIN detector. A fast simulation software, termed RASER (RAdiation SEmiconductoR), is developed, and validated by comparing the waveform obtained from simulated and measured data. The simulated time resolution is $\rm (73\pm 1)~ps$ after considering the intrinsic leading contributions of the detector to time resolution.


INTRODUCTION
In the recent years, much attention has been devoted to seek the appropriate semiconductor material to be used in future particle colliders and nuclear reactors operating in harsh radiation environment (i.e., > 10 17 n eq /cm 2 ) [1]. Silicon-based detectors have the support of a sophisticated production technology and present good quality, but the leakage current sharply increases and charge collection efficiency rapidly decreases when the irradiation fluence exceeds 10 15 n eq /cm 2 [2]. To enhance performance and lifetime, most of the silicon-based detectors also need an expensive cooling system, which makes the overall detector system giant and expensive. Alternative diamond detectors have been investigated with high radiation hardness up to 3 × 10 15 particles/cm 2 [3] and have been successfully used in the ATLAS experiment at the LHC [4]. However, they are also characterized by high cost and a difficult doping process in diamond, which limit their application. On the other hand, the 4H-SiC material, owing to its potential high radiation hardness, wide bandgap energy (3.27 eV), high atomic displacement energy (25 eV), and stability at high temperature, has great potential for application in extreme radiation environments.
Owing to the currently achieved high-quality 4H-SiC epitaxy wafer, a handful of studies about charge collection, leakage current, capacitance, and deep energy levels of 4H-SiC detectors have been carried out before and after irradiation [5,6]. Due to its wide bandgap energy, which is also insensitive to visible light, SiC detectors are useful for X-ray and ultraviolet monitoring [7]. There have also been extensive studies about the SiC detector's application in neutron detection in fusion devices [6,8]. Concerning applications in high-energy physics, the detector's response to the MIPs is more relevant, but most of the previous studies focused on energy resolution and charge collection efficiency using α particles, which are distinct features from the MIPs. To the best of our knowledge, charge collection features relevant to MIPs in SiC detectors have been analyzed [9], but no investigations about time resolution have been reported, except for the 4H-SiC Schottky barrier diode (SBD), where time resolution with α particles for deuterium-tritium (D-T) applications has been analyzed [10].
In the recent years, ultrafast detectors have become a hot research topic. The main goal is to achieve a very high time and position resolution, simultaneously. A time resolution better than 20 ps has been achieved in silicon planar sensors with depletion thicknesses 133-285 μm for multiple MIP signals [11], whereas 100 μm silicon pixel detectors with 800 μm × 800 μm size have achieved a time resolution of 106 ps [12]. Currently, 50 μm silicon detectors with internal gain, usually referred to as Low Gain Avalanche Detector (LGAD), are developed by various foundries and show a time resolution of at least 50 ps [13][14][15][16][17][18]. The 4H-SiC detectors also show fast time response, coming from the highly saturated carrier velocity, but no time performance study has been reported so far.
Motivated by the abovementioned arguments, we here investigate the time resolution of the 4H-SiC PIN device using a 90 Sr source for applications in high-energy physics experiments.
A fast simulation environment to investigate time resolution is a beneficial tool to develop fast detectors and properly understand time response features. The present open-source software, for example, Weightfield2 [19] and KDetSim [20] are only available for silicon detectors. The corresponding simulation tool for silicon carbide detectors is lacking due to distinct material parameters. Therefore, we also developed a fast simulation software termed RASER [21] for applications with silicon carbide detectors, which was used in this study to reproduce measured data.

THE DEVICE UNDER INVESTIGATION
The 4H-SiC PIN devices under investigation are fabricated by Nanjing University and come in two different sizes: 5 mm × 5 mm and 1.5 mm × 1.5 mm. Figure 1 shows the 5 mm × 5 mm sample which has two ohmic contacts on the top and bottom. The two devices both have a 100 μm high resistive active 4H-SiC epitaxy layer and 350 μm substrate, whereas the effective doping concentration is different: we have N eff (5 mm × 5 mm) = 5.2 × 10 13 cm −3 and N eff (1.5 mm × 1.5 mm) = 2.7 × 10 13 cm −3 , respectively. These values may be extracted from the capacitance-voltage curve (see Figure 2) by where q is the electron charge, ε is the dielectric constant of 4H-SiC, and A is the area of the active region. Taking into account the doping levels and the dependence on device thickness V dep q|N eff |d 2 2ε with d = 100 μm, the full depleted voltages are given by V dep (5 mm × 5 mm) = 484 V and V dep (1.5 mm × 1.5 mm) = 248 V.
The current-voltage characteristics are shown in Figure 3. The typical unidirectional conduction characteristic of PINs is observed, and the breakdown voltage is larger than 800 V for both sizes. A higher leakage current may be collected by the 5 mm×5 mm size device due to its larger volume. The leakage current density is J < 100 nA/cm 2 for both the devices with a 500 V reverse voltage, where J (5 mm × 5 mm) = 63.2 nA/cm 2 and J (1.5 mm × 1.5 mm) = 34.6 nA/cm 2 . The lower current density of the smaller device agrees with its lower effective doping concentration, as obtained from data in Figure 2.

EXPERIMENTAL SETUP β Source Test System
A schematic diagram of the experimental setup to determine the time resolution of 4H-SiC detectors is shown in Figure 4. We choose a 33 μm silicon LGAD as the reference timing device owing to its 34 ps time resolution at U = 200 V and room temperature. The reference timing device is developed by the Institute of High Energy Physics (IHEP) of Chinese Academic Sciences and the Novel Device Laboratory (NDL) of Beijing Normal University [22][23][24]. The 90 Sr source emits β particles at 0.546 MeV from 90 Sr and at 2.280 MeV from 90 Y. Both are able to penetrate the LGAD device and deposit energy in the 4H-SiC detector. The front side readout boards are designed for LGAD devices by the University of California, Santa Cruz (UCSC). Each board has a 2 mm diameter hole in the middle. The overall front side electronics are transferred into a metal box to shield it from electromagnetic interference, with a 0.01 mm aluminum foil covering. Two 20 dB broadband amplifiers are placed before the oscilloscope to enhance the SNR. There is an additional 1 m  cable on the DUT side to delay the signal (by~5 ns) and enhance trigger efficiency. The sampling rate of the oscilloscope is 40 GSa/ s, and each channel has 20 GSa/s.

Energy Response by GEANT4 Simulation
To analyze the energy loss of MIPs in 100 μm 4H-SiC active layer tallies, we use a simulation based on GEANT4, which allows one to describe the energy deposition. Figure 5 describes the energy deposition of the β particle in the 100 μm 4H-SiC active layer in the system (see Figure 6). The energy loss in the aluminum foil may be neglected. The scattering of β particles in the aluminum foil and LGAD strongly decreases trigger efficiency for the 4H-SiC device. In turn, this explains the difference of trigger efficiency between the 5 mm × 5 mm (4.7 events/min) and the 1.5 mm × 1.5 mm (2.1 events/min) samples since a larger size corresponds to higher trigger efficiency. The most probable values (MPVs) of the energy deposition in 100 μm 4H-SiC layer are 25 and 55 MeV for the two different energy β particles from the 90 Sr source. There are little differences with the previous experimental result (~42 MeV) [25] due to scattering effects by the aluminum foil and LGAD, which make the ionization track slightly longer than 100 μm, but the average MPV of energy deposition from these two particles is close to the experimental result.

Waveform Sampling
The two channels are triggered at the same time with different trigger levels for waveform sampling. Trigger Ref = 25 mV and trigger DUT = 15 mV are determined by noise levels (see Figure 7) to suppress noise spikes. Figure 8 shows

Time Resolution
The time resolution obtained with different device sizes and reverse voltages are studied here considering the influence of capacitance and carrier velocity. For the timing method, the constant fraction discrimination (CFD) is adopted with a fraction equal to 0.5. Figure 9 show the distribution of ΔT = T DUT − T Ref for different device sizes and reverse voltages. The time resolution of DUT could be extracted by where σ(5 mm × 5 mm, U = 500 V) = 94 ± 1 ps, σ(5 mm × 5 mm, U = 300 V) = 103 ± 1 ps , and σ(1.5 × 1.5 mm, U = 300 V) = 96 ± 2 ps.
At fixed size, 4H-SiC-PIN (5 × 5 mm) shows better time resolution using higher reverse voltage due to faster carrier velocity. At fixed reverse voltage, faster rising time caused by smaller capacitance improves the time resolution if the influence of the undepleted thickness in the 5 × 5 mm size device may be neglected. Meanwhile, the mean of ΔT shifts from 5.03 to 4.81 ns for different size devices due to faster rising time of devices with  smaller capacitance. As it is apparent in Figure 10, a~400 ps difference in the rising time leads to a~200 ps time-shifting with 0.5 CFD fraction.

Introduction of Fast Simulation Tool-RASER
We have developed a fast simulation tool, termed RASER, to study the time resolution performance of SiC detectors [21]. We use FEniCS [26], an open-source computing platform for solving partial differential equations (PDEs) to calculate the electric field and weighting potential of SiC detectors. MIPs with nonuniform charge deposition and amplitude variability are considered. The induced current is calculated by Shockley-Ramo's theorem [27] where the carrier drift is simulated using 0.1 μm steps and taking into account both magnetic field and thermal diffusion. We also assume the use of a simplified charge-sensitive amplifier (CSA) for the read-out [28].

Weighting Field Potentials and Electric Field Calculation by FEniCS
The electric potential and weighting potential can be computed by solving Poisson's and Laplace's equations: where U(r) is the electric potential, U w (r) is the weighting potential, ϵ is the electric permittivity of SiC, and ρ is the charge density. The electric and weighting field are then denoted as E(r) −∇ U(r) and E w (r) −∇ U w (r), respectively.

Current Calculation
The induced current in the SiC detector is produced by the motion of the electron-hole pairs. The current appears when electron-hole pairs begin to move and disappears when all the pairs reach the electrode or the boundary of the detector. The instantaneous current induced by an electron or a hole can be calculated with Shockley-Ramo's theorem: where r is the position of the electron or hole, v(r) is the drift velocity, and E w (r) is the weighting potential. The drift velocity v(r) is given by μ SiC · E(r), where μ SiC is the mobility in SiC. The mobility model of SiC in RASER is based on [29], and the sum of the currents induced by all electrons and holes is the total current. In the simulation, the influence of nonuniform charge deposition and impact position on the time resolution is simulated by GEANT4. The simulation contained the divergence angle from the scattering of ( Figure 4) the upper PCB board, aluminum foil, and LGAD detector.

Comparison Between the Simulated and Measured Time Resolution
The time resolution of the SiC detector can be expressed as follows [13]: where the correlations among the different items are ignored. In simulations, the time walk σ Time Walk is dominated by Landau variation in signal amplitude and is eliminated by the CFD method. The nonuniform charge deposition and scattering effects cause Landau noise σ Landau Noise . The nonuniform weighting potential and the electric field cause signal distortion σ Distortion . The electronics noise, which leads to σ Jitter and σ TDC , is dominated by the binning of signal waveforms. All these time resolution contributions, except the distortion term, are considered in the simulation process. We simulated the time resolution of the NJU detector with 5 mm × 5 mm sizes with a 500 V bias voltage at room temperature T = 300 K. In these conditions, the average carrier velocities are V electron = 150 μm/ns and V hole = 50 μm/ ns. We use RASER to model an ideal planar detector and calculate the electric field so that σ Distortion is not considered in the simulation. Based on GEANT4 simulation, the nonuniform charge deposition and scattering effects are applied in RASER to reproduce the contribution of σ Landau Noise . The random noise is added in each signal waveform to estimate σ Jitter . The contribution of σ TDC has been considered in binning the signal (the same bin interval 50 ps/bin with sampling time step). The program has been validated by comparing the induced current for MIPs of RASER simulations and measured signals. Figure 11 shows the comparison between waveforms obtained from RASER and TCAD simulation and measured data, where the mobility model applied in TCAD is the Masetti model with parameters from [30,31]. Good consistency is found in all three cases.
The results of simulations for the time resolution of the NJU detector are shown in Figure 12. We use 20, 000 events, and the same CFD fraction 0.5 as in the measurements is used to obtain   the time of arrival (ToA). The time resolution obtained by Gaussian fitting is (73 ± 1 ps), where the simulated σ Jitter is 66 ps and leftover σ Landau Noise and σ TDC is 30 ps in total due to σ Time Walk being absent. The simulated time resolution is (24 ± 1) ps less than the measurement result. The origins of this difference are likely because there is no shielding box of the beta source measurement, which leads to an increase in time resolution. RASER software can effectively simulate the time resolution of SiC to some extent, but measurements and simulations both need further optimization.

CONCLUSION
The best time resolution of the NJU 5 mm×5 mm 4H-SiC-PIN detector with the 90 Sr source is 94 ± 1 ps. On using higher reverse voltage and smaller capacitance, better time resolution is obtained. The waveform simulated by RASER has been validated against measurements. The simulated time resolution indicates that all the leading contributions of the test system should be considered to obtain reliable results. Our study about measured and simulated time resolutions is useful to develop ultrafast 4H-SiC LGAD and 3D 4H-SiC detectors [32] which are expected to achieve improved time resolution in the near future.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.