Advances in the TCAD modelling of non-irradiated and irradiated Low-Gain Avalanche Diode sensors

The recently developed Low-Gain Avalanche Diode (LGAD) technology has gained growing interest within the high-energy physics (HEP) community, thanks to its capability of internal signal amplification that improves the particle detection. Since the next generation of HEP experiments will require tracking detectors able to efficiently operate in environments where expected fluences will exceed 1 × 1017 1 MeV n eq/cm2, the design of radiation-resistant particle detectors becomes of utmost importance. To this purpose, Technology Computer-Aided Design (TCAD) simulations are a relevant part of the current detector R&D, not only to support the sensor design and optimization, but also for a better understanding and modelling of radiation damage. In this contribution, the recent advances in the TCAD modelling of non-irradiated and irradiated LGAD sensors are presented, whose validation relies on the agreement between the simulated and experimental data — in terms of current-voltage (I-V), capacitance-voltage (C-V), and gain-voltage (G-V) characteristics, coming from devices manufactured by Hamamatsu Photonics (HPK), and accounting for different irradiation levels and temperatures.


Introduction
In recent years, the Low-Gain Avalanche Diode (LGAD) technology [1] has gained growing interest within the high-energy physics (HEP) community as detectors of choice for high precision tracking in space and time [2].For example, they have been selected to operate in the Endcap Timing Layer (ETL) of the CMS experiment [3] and in the High-Granularity Timing Detector (HGTD) of the ATLAS experiment [4] for the forthcoming High-Luminosity phase of the Large Hadron Collider (HL-LHC) at CERN.In this scenario, the maximum fluence foreseen will exceed 1 × 10 16 cm −2 1 MeV equivalent neutrons (n eq ), whereas in the next generation of high-energy and high-intensity hadron colliders (e.g., the hadronic Future Circular Collider [5]) the fluences will be even higher by more than an order of magnitude, up to about 5 × 10 17 n eq /cm 2 .
The behaviour of silicon sensors under extreme irradiation needs to be investigated and properly modelled to understand the complex physical phenomena related to radiation damage effects, and to design tracking detectors that efficiently work at that unprecedented radiation levels.To this purpose, modern Technology Computer-Aided Design (TCAD) tools allow to model and predict in advance the electrical behaviour of solid-state devices in different operating conditions, by suitably balancing physical accuracy, application versatility and computational demand.
In this work, the state-of-the-art Synopsys Sentaurus TCAD [6] suite of tools was used to validate a recently developed numerical model for the simulation and design of LGAD sensors [7].The comparison between simulation findings and experimental data, which were obtained by measuring in different operating conditions (i.e., bias, fluence and temperature) two different types of LGAD sensors coming from the second production run of Hamamatsu Photonics (HPK), as well as the detailed description of the TCAD model and the adopted experimental techniques are presented and discussed.

Experimental techniques for the characterization of LGAD sensors
This paragraph describes the experimental techniques used for the characterization of the LGAD sensors considered for this work.In the following, the tested samples and the employed measurement setups and methods are described (section 2.1), as well as the experimental data obtained from the measurements of samples before and after irradiation are presented (section 2.2).

Devices under test, experimental setups and methods
The Devices Under Test (DUT) considered for this study were LGAD sensors coming from the second production run of HPK, called HPK prototype 2 (HPK2).They are characterized by a 50 μm-thick high-resistivity epitaxial layer, and by a deep and narrow Gain Layer (GL) doping profile which was designed and produced in four different splits (S1-S4) that differ in the doping concentration.The measurements reported in section 2.2 refer to the first two splits, namely S1 and S2, which were conceived for the R&D of the timing detectors in the ATLAS experiment.The samples have an active area of 1.3 × 1.3 mm 2 and a Guard Ring (GR) structure surrounding the central pad.In table 1 the main parameters of the samples before irradiation are listed.The GL depletion voltage ( GL ), the breakdown voltage ( BD ) measured at room temperature (RT), and the capacitance reached above full depletion ( FD ) are given.
The electrical characterization in absence of external stimulus was performed both at the "Laboratory for Innovative Silicon Sensors" (LISS) of the INFN Torino Unit, and at the clean room of the INFN Perugia Unit.A temperature-controlled probe station electrically connected to a device analyzer was used to extract the key physical quantities, such as the current-voltage (I-V) and capacitance-voltage (C-V) characteristics.The samples were placed and kept by a vacuum system directly on a chuck, which provided the high reverse voltage (i.e., a negative voltage for the n-in-p LGAD sensors studied in this work).During the electrical characterization the GR was connected to the ground, and a high-current resolution Medium Power Source and Monitor Unit (MP-SMU) or a Multi-Frequency Capacitance Measurement Unit (MF-CMU) were connected to the pad under test dependently on the chosen measurement (I-V or C-V, respectively).More details about the I-V and C-V setups can be found in [8].
The gain of the LGAD samples before and after irradiation was measured at LISS.It was evaluated as the ratio between the charge measured in the LGAD and the expected charge from a 50 μm-thick p-i-n diode after its full depletion (i.e., 0.5 fC).Regarding the gain measurements of non-irradiated samples, a pulsed (1 kHz) infrared (IR)-laser with a wavelength of 1060 nm in the Top Transient Current Technique (Top-TCT) setup was used.The IR-laser intensity was tuned to generate an equivalent charge of about 3 Minimum Ionizing Particles (MIP) in the detectors and the spot size was of about 10 μm.The gain of the irradiated LGAD samples, instead, was measured by using a temperature-controlled -source setup.A two-planes setup configuration (i.e., DUT and time reference, the latter used also as a trigger) was used to ensure that the -particles (i.e., the electrons emitted by a 90  source) are MIP equivalent.A dedicated read-out board with current-mode high-bandwidth amplifiers was used to readout both the DUT and the time reference signals.More details about the used TCT and -source setups can be found in [8].

Measurements before and after irradiation
The C-V measurements of the non-irradiated samples were performed at RT, and the frequency of the sinusoidal signal delivered by the device analyzer was set to 100 kHz with an amplitude of 0.5 V. Figure 1(a) and figure 1(b) show the C-V and 1/ 2 -V curves of the two different LGAD samples before irradiation.Such curves allowed for identifying the  GL and  FD values reported in table 1, as well as the extraction of the GL implant profiles (i.e., amplitude, width and depth from the junction) given in figure 1(c).In particular, the depth  and the active doping concentration   as a function of the depth  were computed by using the following equations: where  Si is the silicon dielectric constant,  is the active area of the sensor, and  is the elementary charge.The C-V measurements of the irradiated samples, instead, were performed at 298 K with a measurement frequency and amplitude of the sinusoidal signal of 2 kHz and 0.5 V, respectively.The I-V measurements of the non-irradiated and irradiated samples were performed at RT and 248 K, respectively.These are shown in figure 3(a).The curves of the non-irradiated samples allowed for identifying the  BD values reported in table 1, as well as the gain of the two splits.In particular, the different exponential growth of their leakage current ( leak ) suggests that the gain of S1 is higher than the gain of S2 when the bias voltage is fixed.This is the direct consequence of the difference between the doping concentrations in the GL region of the two splits, as given in figure 1(c).The curves related to the irradiated samples, instead, show the expected increase of the  leak and  BD when Φ increases.
The gain of the non-irradiated samples was measured at RT by using the Top-TCT setup, whereas the gain of the irradiated ones was measured at 248 K by using the -source setup.Figure 3(b) shows the gain-voltage (G-V) curves of the two different LGAD samples before and after irradiation.Before irradiation, the gain of S1 is higher than the gain of S2, as already observed in the I-V curves given in figure 3(a).The gain of the irradiated samples, instead, is almost the same when the bias voltage is fixed and, as expected, it decreases when Φ increases.-3 -  In particular, the G-V curves of the non-irradiated LGADs were obtained by using the scanning TCT-system, whereas the one of the irradiated samples were obtained by using the -source setup.

TCAD model for the numerical simulation of LGAD sensors
This paragraph describes the TCAD model used for the numerical simulation of the LGAD sensors considered for this work.In particular, the layout and the doping profiles, as well as the radiation damage model employed for the simulations of the irradiated devices are described in section 3.1; the simulation findings and their comparison with the experimental data are presented in section 3.2.

Layout, doping profile and radiation damage model
The layout of the LGAD device was implemented within the TCAD by using the simple quasi-1D geometry depicted in figure 4(a).The periphery structures (i.e., the JTE and GR) were neglected because the simulations showed no significant differences when considering them or not.
The sensor active thickness, the bulk doping concentration and the GL doping profiles of S1 and S2 were extracted from the C-V and 1/ 2 -V measurements presented in section 2.2 and fed into the TCAD simulations.The measured  GL and  FD values were then used to fine-tune the GL implant profile.In particular, a slight shift of its position, and small variations in the sensor thickness and -4 -bulk doping concentration resulted in the perfect agreement between the measured and simulated C-V and 1/ 2 -V curves, as displayed in figure 5(a) and figure 5(b).
The radiation damage effects were modelled by means of the "Perugia Modified Doping" (PerugiaModDoping) TCAD radiation damage model, which was already validated by comparing simulations and measurements of sensors coming from the UFSD2 and UFSD3.2 production runs of Fondazione Bruno Kessler (FBK) [7].Such model combines the traps parameterization of the "New University of Perugia" radiation damage model [9] with the fluence-dependent analytical parameterizations of the GL and bulk effective doping, called "Torino analytical parameterizations" [8]. Figure 4(b) shows the effect of such parameterizations on the effective doping, thanks to which it is possible to take into account in the TCAD simulations both the acceptor-removal mechanism in the GL [10] and the saturation of the acceptor-creation mechanism in the bulk [11].-5 -

Simulation findings and model validation
Once the model of the LGAD was implemented within the TCAD, the electrical behaviour and the response to an equivalent MIP stimulus under different operating conditions were simulated.To have a direct comparison with the experimental data, the simulations were performed in the same measurement conditions (i.e., temperature, frequency and irradiation level).Moreover, since the S1 and S2 behave at the same manner after irradiation as shown in section 2.2, all the comparisons between simulations and measurements after irradiation will refer only to one of the two splits in the following.
The C-V curves of the LGAD after irradiation were simulated at 298 K, and the frequency of the sinusoidal signal was set to 2 kHz with an amplitude of 0.5 V.In figure 6(a) and figure 6(b), the simulated C-V and 1/ 2 -V curves of S1 after irradiation are shown and compared with the measurements.The reduction of the  GL and the increase of the  FD when Φ increases were well reproduced in simulation.
The I-V curves of the non-irradiated LGAD were simulated at 290 K, instead the ones after irradiation were simulated at 248 K. Special attention were devoted to the choice of the avalanche model, by investigating among two different versions of the Van Overstraeten-De Man (vOv) model: the default TCAD-embedded one [12], and the one which uses a new set of parameters for the impact ionization coefficients [13].Figure 7(a) gives a comparison of the experimental and simulated curves of S1 before and after irradiation.A good matching was obtained in terms of  BD when using the default avalanche model before irradiation, and the new one after irradiation.No differences were observed by using the two versions of the avalanche model, instead, in terms of increase of  leak when Φ increases.
By performing transient simulations of the passage of an equivalent MIP stimulus, the gain for a given voltage was obtained as the ratio between the charge collected in the LGAD and the charge collected in the corresponding p-i-n diode under the same operating conditions.In particular, the gain of the non-irradiated LGAD was simulated at 290 K, whereas the one of the LGAD after irradiation was simulated at 248 K.As shown in figure 7(b), a good agreement between the experimental and simulated curves was obtained before and after irradiation.The curves before irradiation were simulated at 290 K by using the vOv avalanche model [12], whereas the ones after irradiation were simulated at 248 K by using a new version of such model [13].

Conclusions
In this work, the validation of a TCAD model for the numerical simulation of LGAD sensors conceived for particle detection in the next generation of HEP experiments is presented.The model robustness and generality (e.g., in terms of operating conditions of the sensor, vendors and/or technology options) were evaluated with the aid of TCAD simulations based on a recently proposed model for the numerical simulation of radiation damage effects on LGAD sensors, called "PerugiaModDoping" [7].To this end, extensive measurement campaigns were performed on two different splits of LGAD sensors coming from the second production run of Hamamatsu Photonics (HPK), in order to extrapolate the electrical characteristics (i.e., I-V, C-V and G-V curves), and the responses of the sensors to different stimuli (laser and -particles source) under different operating conditions (i.e., temperature, frequency and irradiation level).The model validation relies on the comparison between simulation results and experimental data which were found to be in good agreement for all the tested devices and for all the considered irradiation fluences.This not only qualifies the suitability of the adopted model to accurately reproduce the electrical characteristics of LGAD sensors, but also enables the prediction of their performance and the advancement of new sensor technologies to be used in the future HL-LHC scenario.

Figure 2 (
a) and figure 2(b) show the C-V and 1/ 2 -V curves of the two different LGAD samples after irradiation.It is evident the reduction of the  GL and the increase of the voltage of full depletion ( FD ) when the fluence (Φ) increases.Since the HPK2 production run was conceived for the R&D of the timing detectors of the ATLAS and CMS experiments, the sensors were irradiated up to the fluence of 2.5 × 10 15 n eq /cm 2 , which is of interest for these two experiments.Therefore, all measurements shown in this work are performed up to this fluence value.

Figure 1 .
Figure 1.(a) C-V and (b) 1/ 2 -V characteristic curves of the two LGAD types used for this work, before irradiation.They were obtained at RT with a measurement frequency of 100 kHz.(c) GL doping profiles extracted from the C-V and 1/ 2 -V measurements by employing (2.1).

Figure 2 .
Figure 2. (a) C-V and (b) 1/ 2 -V characteristic curves of the two LGAD types used for this work, after irradiation.They were obtained at 298 K with a measurement frequency of 2 kHz.

Figure 3 .
Figure 3. (a) I-V and (b) G-V characteristic curves of the two LGAD types used for this work, before and after irradiation.The characteristics before irradiation were obtained at RT, whereas the one after irradiation at 248 K.In particular, the G-V curves of the non-irradiated LGADs were obtained by using the scanning TCT-system, whereas the one of the irradiated samples were obtained by using the -source setup.

Figure 4 .
Figure 4. (a) Layout of the LGAD device implemented within the TCAD and (b) fluence-dependent evolution of the GL and bulk doping profiles according to the analytical parameterizations of the "PerugiaModDoping" radiation damage model.Gaussian profiles were used to model the highly doped n-type an p-type implants.

Figure 5 .
Figure 5. (a) C-V and (b) 1/ 2 -V simulated (solid lines) and experimental (markers) curves of the two LGAD types, carried out at 290 K with a frequency of 100 kHz to have a direct comparison with experimental data.

Figure 6 .
Figure 6.(a) C-V and (b) 1/ 2 -V simulated (solid lines) and experimental (markers) curves related to S1, carried out at 298 K with a frequency of 2 kHz, after irradiation.

Figure 7 .
Figure 7. (a) I-V and (b) G-V simulated (solid lines) and experimental (markers) curves related to S1 and S2, respectively.The curves before irradiation were simulated at 290 K by using the vOv avalanche model[12], whereas the ones after irradiation were simulated at 248 K by using a new version of such model[13].

Table 1 .
Measured parameters of the LGAD samples used in this work before irradiation.The GL depletion voltage ( GL ), the breakdown voltage ( BD ) at room temperature (RT), and the capacitance reached above full depletion ( FD ) are given.