Review: Numerical simulation approaches of crystalline‐Si photovoltaics

This study reviews the current methods of numerical simulations for crystalline‐Si (c‐Si) photovoltaic (PV) cells. The increased demand for PV devices has led to significant improvements in the performance of solar cell devices. The main contribution comes from c‐Si solar cells, which constitute 90% of the industry. Numerical analysis is effective for predicting, developing, and optimizing cell performances as the cell structures become more complex and include several parameters. However, conventional methods cannot simulate improved device structures with complex configurations. Additional physics is necessary to evaluate these cell structures. This study introduces methods for evaluating these cell structures using physical modeling and highlights the latest examples of simulations of c‐Si solar cells.


| INTRODUCTION
2][3] The abundance of sunlight, easier installation, and higher light-energy output are the primary reasons for the current surge in PV device applications as sources of renewable energy. 4Currently, PV devices fabricated from crystalline-Si (c-Si) constitute approximately 90% of the market for their superior performance (over 20% efficiency), matured fabrication processes, and contamination-free c-Si structures, which are inhibitory for compound solar cells, such as galliumarsenic (GaAs) solar cells. 2 Presently (June 2022), c-Si solar cells exhibit a record cell efficiency of 26.7%, which was achieved in 2017. 5,6he cell performance has been enhanced through (a) a newly proposed advanced cell structure, (b) optimization of the cell structure, and (c) improvements in the cell fabrication processes.A novel cell structure can accentuate various unprecedented physical phenomena, such as those originating from heterojunctions 7 or carrier tunneling. 8However, the conditions that give rise to such novel phenomena may not be conducive to optimizing cell structures to achieve high-performance devices.The conditions could also affect the determination of process parameters.To overcome this shortcoming, numerical simulations are widely employed as a powerful tool to minimize the prototyping time and costs. 9This is evident from the reported studies on improving solar cell performance or investigating the physics associated with them through numerical simulations. 10However, the numerical simulation models must be validated using actual physical parameters to reproduce the realistic cell operations.Moreover, the necessity of optical and electrical self-consistent simulation models with additional parameters poses an additional challenge.2][13][14][15][16][17] The recent advancement in computing performances enabled large-scale calculation, making analyzing and optimizing solar cell devices attractive.Solar cells are among the largest semiconductor devices, with common ones typically measuring 156 mm × 156 mm.Additionally, the establishment of new simulation models enabled the evaluation of novel solar cell devices that introduced new physics, such as heterojunction or carrier tunnelings.However, reliable simulation models must be developed for evaluating the steadily growing novel c-Si solar cell structures that were recently proposed.
In this study, we present a comprehensive review of various numerical simulation approaches for c-Si solar cell devices to highlight the optimal approaches for simulating the latest cell structures.
The rest of this paper is organized as follows: Section 2 introduces the advanced cell structures, followed by the simulation methods applied to these cells in Section 3. Finally, we review the latest reported simulation-based studies (Section 4).

| NEXT-GENERATION c-SI SOLAR CELL DEVICES
The fundamental structure of a c-Si solar cell has a back surface field (BSF) layer with a heavy-doped rear-side surface diffusion to suppress carrier recombination.For a p-type cell, using aluminum as the rear contact facilitates the diffusion of Al + during the annealing process; therefore, such a cell is referred to as an "Al-BSF" cell. 18,19Figure 1 illustrates the general c-Si solar cell structures containing a BSF layer.In general, four technologies are adopted for fabricating the c-Si solar cell structures.The characteristics of the cells produced using each of these technologies are elucidated in Section 2.1, along with their differences from the BSF structure.

| Passivated emitter and rear cell (PERC)
PERCs 20 are currently mainstream in the PV industry. 21he structure of a PERC is considered the closest to the conventional BSF structure but with only a point contact.Alternatively, the PERC structure can be categorized as a point-contact BSF structure.Compared with a conventional BSF-type cell, a PERC demonstrates a cell efficiency improvement of approximately 1%.Furthermore, a PERC also involves a few additional fabrication processes that make it advantageous and a suitable replacement of the conventional BSF-type cell. 22Currently, approximately 22% cell efficiency is achieved for mass-produced PERCs. 23Furthermore, because of its nonunique rear-side structure, it is categorized as a multidimensional device, and accordingly, a minimum of 2D grid must be considered for the numerical simulations of a PERC.

| Interdigitated back contact (IBC)
An IBC eliminates the requirement of a front-side metal contact that prevents a large fraction of the light from reaching the device active area. 24The IBC is also considered the oldest reported advanced c-Si solar cell structure (published in 1977).Even approximately 5% front-metal contact can be a bottleneck for the optical performance of the device.Therefore, a front-shadingfree structure is essential for improving the photocurrent of the cells.Because both the emitter and BSF layer are concentrated on the rear side, this structure also requires a 2D grid.For the front side, two types of doping are performed that lead to the formation of a front-surface field (FSF) or front-floating emitter (FFE).An FSF produces a doping polarity similar to that of the substrate, whereas an FFE results in a reverse polarity doping. 25Furthermore, the FSF acts as the front-surface passivation, and the FFE enhances the lateral carrier transport.
Because IBC forms both the emitter and BSF at the same surface, the manufacturing process is more complex than that of the PERC or conventional BSFtype structures.In addition, compatibility with the old process flows is important for mass production. 26The IBC is considered the second-generation c-Si solar cell structure and is primarily studied because of its low-cost fabrication and high performance. 27

| Si heterojunction (SHJ)
In an SHJ structure, a heterojunction is juxtaposed between the substrate and emitter or the BSF regions of the c-Si/amorphous-Si (a-Si) cell with a c-Si/intrinsic a-Si/doped a-Si structure. 28The insertion of intrinsic a-Si forms the depletion region, and the removal of the pnjunction, which is the thickest layer where the hole-electron recombination events occur, effectively improves the open-circuit voltage (V OC ) of the cells. 29n SHJ features the best interface passivation quality, and consequently, the V OC reaches nearly 750 mV, which is close to the built-in potential of a Si pn-junction. 30lternatively, the heterojunction quality critically governs the performance of SHJ devices and other types of heterojunction solar cells. 31he expiry of patents and improved fabrication processes lowered the fabrication cost with several other advantages in recent years. 32Nevertheless, the fabrication of a c-Si/a-Si heterostructure is still challenging compared with the synthesis of conventional BSF, PERC, and IBC structures.Moreover, processing high-quality heterojunctions to maximize their structural characteristics is imperative for the mass production of these cells.

| Tunnel-oxide passivated contact (TOPCon)
In solar cells with TOPCons, tunnel-oxide layers, which act as carrier-selective layers, are formed at the contact regions. 33TOPCon was proposed in 2014 and is considered one of the newest solar cell structures that unraveled new physics in c-Si solar cells, that is, carrier tunneling and selective contact.In a TOPCon structure, only the majority carriers are allowed to pass, which prevents the recombination of minority carriers at the contact regions, thereby improving V OC of the device.Although SiO 2 is mostly used to form the tunnel layer, research on suitable materials, such as SiN x and AlO x , which can suppress the recombination at the Si/tunnel layer interface, is in progress. 34owever, the excessive costs of TOPCon compared with PERC is a major drawback.Nevertheless, a gradual transformation from PERC processing to TOPCon fabrication is evident TOPConCost. 35Despite the higher processing costs, TOPCon is considered more advantageous than PERC because of its higher performance (0.5% cell efficiency).

| Combined structures
The aforementioned structural fabrication approaches can be combined because they do not compete with each other.The representative structure is that with a heterojunction back contact (HBC), which is a combination of IBC and SHJ.][38] Another type of combined cell structure is polycrystalline-Si on oxide (POLO), 39 which is a combination of IBC and TOPCon.Its current efficiency of 26% is sufficiently high, despite being lower than that of the HBC structure.According to a simulation-based study reported in 2022, the efficiency of a POLO device can be increased by up to 29.1% using photonic crystals 40 and up to 27.8% without them. 41

| Bifacial cells
The bifacial method is one of the approaches for improving cell performance. 42Bifacial cells can absorb photons from both the front and rear sides of the cell surface, resulting in a remarkable increase in the photocurrent.Generally, approximately 20% of light is reflected from the ground, suggesting that the cell performance could potentially be improved by up to 20% when illumination is solely from the front side. 43ERC+, 44 which is a bifacial PERC, is considered the simplest structure.Other types of bifacial solar cells are the bifacial IBC (here, we mention it as "IBC+"), 45 bifacial SHJ (here, "SHJ+"), 46 and industrial TOPCon. 47perimentally, SHJ+ shows a 26.3% cell efficiency under a monofacial illumination. 5he bifacial scheme depends on the rear-side pointcontact design, suggesting that 2D modeling is mandatory for all types of bifacial cells.

| NUMERICAL SIMULATION OF c-SI SOLAR CELLS
The numerical simulation of these cells requires different physical approaches.For c-Si solar cell simulations, three fundamental semiconductor equations, namely, Poisson's, the electron current continuity, and the hole current continuity equations are solved self-consistently. 48Additionally, optical calculations are performed, and for all the calculations, the room temperature is set to ∘ 25 C.Although there are several illumination conditions, the AM1.5G spectrum (i.e., air mass 1.5 G) with an intensity of 1000 W/m 2 49 is commonly used.Both the experimental cell efficiency and theoretical limit (29.4%) 50reported to date are based on this spectrum.This global standard spectrum is available as an online source. 51

| Softwares
Table 1 shows a comparison of the merits and demerits of each software.
PC1D is available as a free software 52 and is suitable for evaluating fundamental cell characteristics, such as thickness, bulk quality, surface, and spectrum.However, modeling a multidimensional cell structure precisely using PC1D is difficult.The merit of PC1D is its free availability and quick calculations in the current computer platforms.PC1D is suitable for BSF, SHJ, and TOPCon solar cell evaluation.There is an unofficial updated version "PC1Dmod6" series, 53 with the implementation of Fermi-Dirac statistics and updated input parameters.This version is suitable for heavy-doped solar cell device simulations.Quokka series are freely available multidimensional device simulators specialized for solar cell evaluations.The first Quokka version 1 was released in 2013, 54 and then it was quickly updated to version 2. 55,56 The newest version is Quokka3, 57,58 which was released in 2018.The Quokka series is a powerful solver for evaluating solar cell operations with a full-scale device and is suitable for several advanced c-Si solar cells, including PERC, IBC, and TOPCon. 59Currently, the Quokka series is limited to only Si solar cell evaluations and cannot support the simulations for other types of solar cells, such as compound semiconductors, like, GaAs, CIGS, or perovskite, and tandem solar cells.In Quokka3, a 1D solver supports other materials in addition to Si. 60 The commercial technology computer-aided design (TCAD) software is suitable for the calculation of devices with multidimensional structures. 61The representatives are Sentaurus TCAD provided by Synopsys Inc. 62 and Victory TCAD provided by Silvaco Inc. 63 These TCAD software include a process simulator, which enables a complete evaluation starting from the processing to the device characteristics.This review article is focused on device simulations.Generally, transistor (such as Complementary Metal-Oxide-Semiconductor [CMOS]) parameters are considered the default parameters of the materials.Therefore, the calibration of material parameters or physical models is mandatory before performing simulations using TCAD.
In a previous study, MATLAB was used to simulate the diode current in a solar cell device using the fundamental pn-junction parameters as the input. 64lthough the input of the device structure is not still reported, understanding the effect of fundamental diode parameters on cell performances is crucial.
The dimension of the simulation determines the parameters that can be evaluated.Table 2 compares the dimensions of the simulations for c-Si solar cells.The dimension must be correctly selected to maximize the performance of the simulated devices; generally, most simulations require 2D grids as they include the core components of the cell designs, only missing the busbar.Most cases can be simulated using 2D grids, and the optimization of cell design is important.Conducting fullscale 3D simulations for solar cell devices, which are typically on the order of cm-squared scale, necessitates substantial computational resources.This size is notably larger than that of other semiconductor devices, such as MOSFETs, which are on the nm-scale, or even power devices, which are on the mm-scale.Threedimensional is suitable for the "loss analysis" or "characterization" of "one" cell in comparison with the experimental results (generally, it is the "champion cell").The basic properties of the cells can be confirmed via 1D simulations; however, the 1D simulation of the most advanced cell designs is challenging.

| Simulation flow
To evaluate the solar cell performance, optical and electrical simulations are required.Figure 2 illustrates the simulation flow in TCAD.First, the device structure is created either directly or through process simulations.Subsequently, the optical and electrical parts are simulated.Different solving methods may be necessary to obtain the output; for current-voltage (I-V) characteristic analysis, which is the fundamental analysis, the quasistationary DC analysis is required after the optical simulation to obtain the carrier generation profiles.However, the quantum efficiency (QE) calculation is performed in the short-circuit state, and hence, the DC bias steps are not required in this case.

| Optical simulations
Optical simulation calculates the carrier generation by the light illumination, which is used as the input to follow electrical simulation.Several optical solvers exist to solve the optical problem, such as the transfer-matrix method (TMM), 65 raytracing, 66 beam-propagation method (BPM), 67 and finite-difference-time-domain (FDTD) 68 method.Each solver has its advantages and disadvantages, which are listed in Table 3. TMM is a 1D optical solver and is therefore difficult to apply to multidimensional cell designs for optical simulations.Therefore, it is rarely used independently.The TMM solver is combined with another optical solver mainly focused on the stacked thin-film parts.Raytracing can be the first choice because of its versatility.However, it is difficult to evaluate thin-film properties using raytracing, and thus, this solver must be used with another solver (like, TMM) to evaluate the stacked thin-film parts.BPM is also a multidimensional optical solver that performs calculations using the approximation method and thus requires suitable techniques to improve its calculation accuracy.Finally, FDTD is based on solving Maxwell's equation and is considered highly accurate.However, a tensor mesh is required for FDTD calculations, and therefore, preprocessing is necessary.

| I-V tracing
To calculate the I-V characteristics, electrical simulation with the quasistationary DC analysis is used.The electrical simulation generally solves three fundamental semiconductor equations: Poisson's equation and current continuity equations for both electrons and holes, all in a selfconsistent manner. 48Each equation is expressed as follows: with (1) of Poisson's equation, (2) of current continuity equation of electron, and (3) of current continuity equation of hole, respectively.Generally, before conducting the quasistationary DC analysis, the transient analysis is necessary to improve the convergence of the analysis by precalculating the carrier transport after the carrier generation.The bias range should be from 0 V to approximately 800 mV for a single-junction c-Si solar cell to include all the operation ranges.The performances of the four fundamental parameters, namely, the short-circuit current density (J SC ), V OC , fill factor (FF ), and cell efficiency (η) are obtained from the I-V curves.

| Quantum efficiency
QE is the wavelength response of cells under the shortcircuit state.It expresses the quality of cells at each wavelength, and its value is between 0 (the worst) and 1 (the best).There are two QEs, that is, the external QE (EQE) and internal QE (IQE).The difference between EQE and IQE is the consideration of the optical route; EQE includes all the physics from the optical illumination to internal recombination and resistive effects.However, IQE only considers the processes that occur after the optical carrier generation; therefore, it is useful to evaluate recombination or resistive-related losses.Consequently, IQE is consistently higher than EQE.EQE is calculated as Here, I is the current, q is the elementary charge, h is Planck's constant, c is the speed of light, λ is the wavelength of the illuminated light, I 0 is the intensity of the illuminated light, and S is the area of illumination.IQE is calculated in two approaches, that is, using the EQE and surface reflectance R, or the carrier generation rate G, as The former method is close to the experimental evaluation because generally only the EQE and surface reflectance R can be obtained from the experiment.The difference between the two methods appears at long wavelengths because, at long wavelengths, the carrier generation rate G becomes exceedingly small.Here, the latter method exhibits a high IQE even at 1 μm or longer wavelengths.
The related performance is obtained as the quantum yield (QY), which is the ratio of the number of generated carriers N gc to the number of absorbed photons N ac , QY N N = .
gc ac (7)   This QY is considered the pair part of IQE and indicates a pure optical performance.

| Loss analysis
Loss analysis is useful in mitigating the drawbacks in current designs and their improvement. 69The loss analysis can be performed using several methods, such as the I-V-based loss analysis, 70 free energy loss analysis (FELA), 71 and synergistic efficiency gain analysis (SEGA). 72An example of I-V-based loss analysis for IBC+ was previously reported, 73 which was useful in demonstrating the drawbacks of the design-based components.FELA uses the difference of the quasi-Fermi energy, which is called "free energy."FELA facilitates easy comparison of each component because all of the components have the common unit of mW/cm 2 .The SEGA method accounts for the shift in the maximum power point voltage, a significant factor in advanced solar cell structures, thereby providing substantial support for the analysis of recombinationrelated loss.
Currently, the loss analysis method of FELA is mainstream and widely used because it is easy to evaluate.Using SEGA might improve accuracy.However, its use is rarely reported, and further studies will be necessary.Previously, a module-level loss analysis of multidimensional c-Si solar cells has been reported. 74At a module-level loss, the light trapping by the module part can be a significant drawback and will be useful in evaluating the PV system performance.

| Mapping
Mapping enables direct visualization of internal physics from numerous perspectives and is thus advantageous for experimental evaluations.The TCAD software provides most physical data and enables integration of the volume entirely or at a certain part or cutting the device at an arbitrary plane.Mapping data is also important for loss analysis to calculate and extract the related values.

| Material parameters
For c-Si solar cells, the fundamental parameter list has been published with sufficient agreements for reliabilities. 75Users should set the bulk parameters related to resistivity and recombination that express the bulk qualities.The lifetime of an n-type bulk (by a magnitude of a few milliseconds) exceeds that of a p-type bulk by a factor of almost 10.To express the bulk quality, the Shockley-Read-Hall (SRH) lifetime must be set 76 in the following equation: This equation expresses the doping-dependent SRH lifetime, and τ 0 is the bulk lifetime, which is determined by the doping concentration and type of dopant, and for p-type, it is in the range of 10 2 -10 3 μs 77 and for n-type, 10 3 -10 4 μs. 78The smaller bulk lifetime in p-type doping originates from boron-oxide (B-O) clusters in the bulk, formed during the Czochralski growth of the bulk.These B-O clusters act as recombination centers and reduce the lifetime. 79According to a prior study, the bulk lifetime of p-type Si can be improved to the magnitude of milliseconds by deactivating the B-O clusters. 80Here, impurity contamination during processes can cause severe lifetime degradation, and different impurity effects are researched via the numerical simulation. 81Cu is relatively less effective in cell performances; however, Co or Cr particularly causes substantial cell efficiency drops as the contamination level increases.
Furthermore, auger recombination must be included in the simulation to consider the heavy-doped part losses.The recombination rate of auger recombination is expressed as follows: The auger recombination rate increases as the carrier concentration is far from the intrinsic carrier concentration.The parameters C n and C p include the temperature terms, which means that the auger recombination will be affected by phonons. 82,83lthough Si is an indirect bandgap semiconductor, the radiative recombination in Si may be difficult to ignore, and including it will improve the accuracy of the model. 84The radiation recombination rate can be expressed as follows: The radiation recombination is also the function of carrier concentration.As these recombinations include n i , the smaller n i by the wide bandgap will enhance the recombination contrary to smaller carrier generation by the carrier excitations in the wide bandgap.
At the interface of materials, surface recombination should be modeled, which expresses the passivation quality and is expressed using surface recombination velocity (SRV). 85,86At a Si/passivation interface, the SRV is affected by the doping concentration and material type of the passivation, which has positive or negative fixed charges. 87,88or p-type Si, negative fixed-charge materials like AlO x , and n-type Si, positive fixed-charge materials like SiO 2 and SiN x are used for the passivation.Unsuitable materials cause approximately 10 times higher SRV.
For bandgap narrowing (BGN), generally, Schenk's BGN model provides a good agreement. 89For carrier mobilities in Si, Klaassen's mobility model is widely used. 90For contact modeling, the SRV and contact resistivity must be modeled.The SRV at a semiconductor/metal interface is extremely high, and a value of 10 6 -10 7 cm/s is used for the simulations.Contact resistivity is used to model the contact design, and 2D simulations need to incorporate the effect of busbar designs; therefore, the contact resistivity value is likely to be higher for 2D simulations than for 3D simulations or experimental values.
For solar cells, optical parameters must also be considered.The complex refractive index of Green reported in 2008 91 is useful for optical simulations.Furthermore, free carrier absorption must be included to evaluate the optical losses. 92,93Additionally, most solar cells contain surface texturing to suppress reflectance, and for c-Si, the textured reflectance has been reported. 94ather than introducing textured structures in simulations, using a combination of the light scattered at the surface and the reported reflectance will minimize the calculation time.
The models for c-Si solar cell simulations are listed in Table 4.They are composed of electrical and optical parts.This parameter list is useful to simulate the classic cells of BSF, PERC, and IBC, and for SHJ and TOPCon that involve different physics, additional models are required.

| Si heterojunction
The SHJ needs heterojunction modeling at the c-Si/a-Si interface by using the physical values of the a-Si material.
The bandgap of a-Si is approximately 1.72 eV, and a thin layer inserted at the surface interface works as a passivated region, which increases V OC to over 700 mV.The modeling of SHJ must include a transparent conductive oxide (TCO) because of the extremely small carrier mobilities in a-Si (less than 1 cm 2 /Vs). 95,96The drawback in using TCO is the low internal light reflectance of approximately 10% (see Figure 3A). 97CO materials have relatively higher resistivities-on the order of mΩ-cm-compared with metals, 98 and the TCO is used to transport carriers in the lateral direction toward the metal contacts.Therefore, an extremely high resistivity would be a bottleneck for cell performance. Th representative TCO materials are indium oxide, 99 titanium oxide (TiO 2 ), 100 and zinc oxide (ZnO) with aluminum-doping and gallium-doping.101 Recently, nickel oxide (NiO) has been reported as another suitable TCO material.However, its lower doping concentration increases the resistivity, thereby hindering its application as a TCO material.102

| Tunnel-oxide passivated contact
The carrier tunnel transport model at the Si/tunnel-oxide interface is required modeling TOPCons (see Figure 3B).To express the tunneling current, the direct tunneling model 103 is used, 104 with dielectric constant ε ox , carrier effective masses m e h , , and barrier heights at the interface ΔΦ C V , as the input parameters.Here, the use of the direct tunneling model assumes the uniform formation of the tunneling layer as the trapezoidal barrier is estimated, which the TOPCon will be applicable.Mostly, the tunnel layer material SiO 2 is widely used in TOPCon, and the fundamental parameters, that is, the effective masses and barrier heights, have been reported. 105
The most important difference from the monofacial structures is the internal light reflectance (or light trapping) for both sides.Unlike monofacial illumination, the bifacial configuration may experience a light escape from the side opposite to the illuminated one, as the bifacial module allows light passage from both sides (see T A B L E 4 Fundamental physics and corresponding models for c-Si solar cell simulations.

Electrical
Carrier recombination SRH, auger, radiative, surface 75 Carrier mobility Klaassen's model 90 Figure 4). 44This becomes more severe when TCO is used because its internal reflectance is significantly low (approximately 10% as aforementioned).A low internal reflectance degrades the long-wavelength response, resulting in a low EQE.
To simulate bifacial solar cells, both the front-side and rear-side illuminated carrier generation rates are required.The common simulation condition entails 100% front-side and 20% rear-side illumination intensities as the standard albedo situation.As the indicator of cell performance, the power density rather than the cell efficiency is used, with mW/cm 2 as the unit.Most cell structures demonstrate an increased power density of approximately 15% compared with the monofacial illumination condition.This increase in power density is substantial considering the challenges of improving cell performances under monofacial structures.Therefore, bifacial solar cells are expected to become standard structures in the future.Furthermore, bifaciality is expressed as the ratio of rear-side to front-side cell efficiency (or short-circuit current density) under 100% illumination intensity on both sides.
Bifaciality is strongly affected by the rear-side design, as it is almost proportional to the rear-side contact area.Generally, the bifaciality becomes smaller than one.However, depending on the designs and structures, values over one can also be obtained.

| MODELING EXAMPLES
In this section, we introduce some examples of c-Si solar cell models from our prior works.The models and their associated fundamental physics are summarized in Table 4.

| PERC
The PERC model must consider the SRV and internal light reflectance of oxide and contact regions.When evaluating PERC+, internal light reflectance is affected as most rear-side metals are removed.In our previous work, 106 the model was based on the experimenting on PERC+ under three illumination conditions. 44Table 5 lists the models used in this study, and Figure 5 provides the 2D schematic of PERC+ incorporating these models.Table 6 presents comparisons with experimental results under three different conditions, demonstrating a high degree of accuracy in the reproduction of these results.The most significant difference in the 2D-modeling contact structure from the realistic cell is the large distributed contact resistance (22.5 mΩcm 2 in this study).

| IBC
The IBC model is similar to that of PERC.However, as both p +and n + -regions are formed at the rear side and as the two regions are close, specific models should be established.In our previous study, 73 the model was based on the experiment of IBC+ under three illumination conditions. 107Table 7 lists the simulation model of IBC+, 73 and Figure 6 provides the 2D schematic of IBC+ with this model.The specific model parameter is extremely high SRV at the pitch regions between p +emitter and n + -BSF region, by reflecting the closed p +and n + -regions 108 to fit V OC parameter.Additionally, the area of rear-side metals is enlarged to reflect the difference between the 2D-modeling and 3D realistic cells.Table 8 presents the comparisons with the experiment under three conditions, demonstrating wellreproduced results from the experiment.Here, the comparison is mainly focused on the monofacial condition and J SC parameter under three conditions, therefore modification of SRV specifically for rear-side illumination may enhance the reliability of the model.Abbreviations: BSF, back surface field; PERC, passivated emitter and rear cell; SRV, surface recombination velocity.
F I G U R E 5 Two-dimensional schematic of PERC+ with simulation modeling in Table 5. BSF, back surface field; PERC, passivated emitter and rear cell.
T A B L E 6 Comparison between the experiment 44 and the simulation 106 with the modeling in Table 5 (the values are of simulation with the differences between the experiment).
Illumination η (%) F I G U R E 6 Two-dimensional schematic of IBC+ with simulation model in Table 7. IBC, interdigitated back contact.

| SHJ
As SHJ forms the heterojunction of c-Si/a-Si, the additional model for the a-Si layer is necessary.The important parameters in a-Si are the optical bandgap energy and mobility as mentioned in Section 3.4.1.In our previous studies, 109 the proposal of a new cell structure was based on HBC with the highest cell efficiency. 6able 9 lists the simulation model of HBC, 109 and Figure 7 depicts the 2D schematic of HBC with this model.Table 10 presents the comparisons with the experiment under monofacial illumination, demonstrating well-reproduced results from the experiment.The calculation of J 0 of recombination current is extracted by kT q = T (25.7 mV at ∘ 25 C).

| Tunnel-oxide passivated contact
The modeling of TOPCon requires the tunnel current parameters as introduced in Section 3.4.2.These tunnel parameters include the effective masses of electrons and holes, and barrier heights at conductive and valence bands, and dielectric constant.Additionally, the oxide possesses a unique fixed charge, both positive and negative, which determines the effective SRV at the interface. 110The proposed partial carrier-selective contact structure named advanced industrial TOPCon (Ai-TOPCon) 111 was based on the conventional i-TOPCon structure. 47The model details are summarized in Table 11, and Figure 8 depicts the 2D schematic of the cell structure.Table 12 shows the comparison between the experiment and simulation under two illumination conditions.Here, the result may slightly differ from the experiment, indicating that precise modeling of a TOPCon cell is challenging.There are several anticipated reasons for this observation: (1) Given that TOPCon forms a thin rear film, rigorous optical modeling may be necessary.(2) The rear doping overlaps with the silicon bulk, as explained in Sugiura et al., 111 which could influence the SRV parameters.

| Summary
This section presents examples of models with wellreproduced experimental results from our previous works.These simulation models are useful in establishing a reliable simulation setup to design c-Si solar cell devices.
T A B L E 8 Comparison between the experiment 107 and the simulation 73 with the model in Table 7 (the values are of simulation with the differences between the experiment).
Illumination η (%) J SC (mA/cm F I G U R E 7 Two-dimensional schematic of HBC with simulation model in Table 9. a-Si, amorphous-Si; BSF, back surface field; HBC, heterojunction back contact. T A B L E 10 Comparison between the experiment 6 and the simulation 109 with the models in Table 9 (the values are of simulation with the differences between the experiments).
η (%) J SC (mA/cm 2 ) V OC (mV) FF (%) J 0 (A/cm This section introduces the latest reported numerical simulation studies of c-Si solar cells.

| Design and physics
Design optimization is crucial in manufacturing.However, simulation is also useful in revealing the internal device physics and is particularly useful for PV devices, because they include both electrical and optical physics and are more complex than the general electron devices.Design optimization mainly focuses on new cell structures, which still require further research.For a monofacial-type structure, TOPCon is considered a viable candidate.The main topic of interest is the physics of carrier tunneling, which includes the tunnel layer material and its thickness.Our study revealed that the choice of the tunnel layer material and limitation of thickness is determined by the effective mass of the tunnel layer. 110Currently, under monofacial conditions, the simulation-based efficiency of over 29% with photonic crystal and 27.8% without it on POLO structure 41 is considered to be one of the highest values; the data herein are therefore useful for creating a roadmap in the PV field.Mass-production oriented efficiency of the POLO structure is reported at nearly 26% by the POLO + IBC structure. 112The roadmap of the TOPCon-series structure is based on the numerical simulation, 113 and research on these new structures will improve reliability and importance.Another target is the heterojunction structure, and the modeling of heterojunction c-Si solar cell with nanocrystalline 3C-Si carbide emitter is reported. 114This structure aims to reduce absorption loss, 115 and the potential efficiency of 23% is predicted by the simulation.Another topic regarding temperature evaluations has been researched.Generally, the evaluations are conducted under room temperature; however, the light illumination would increase the device temperature and the actual temperature can be significantly higher than the room temperature.The proposal and introduction of temperature modeling specifically targeting silicon solar cells have been made; 116 however, there are still unresolved issues in this field.Temperature coefficients are used to model these effects, and studies are being conducted on evaluating them 117 and the comparison of temperature effect between the p-type and the n-type cells, 118 using PC1D software.A prior study proposed the improved input parameters of a temperature-dependent bulk lifetime that affects V OC and FF with validation by Quokka3. 119The SHJ is known for its high robustness to F I G U R E 8 Two-dimensional schematic of TOPCon with simulation model in Table 11.BSF, back surface field; TOPCon, tunnel-oxide passivated contact.
T A B L E 12 Comparison between the experiment 47 and simulation 111 model in Table 11 (the values are of simulation with the differences between the experiment).
Illumination η (%) J SC (mA/cm increased temperature; therefore, further research on heterojunction structures is desirable.Bifacial-type structures are still under investigation, and our results regarding the optimizations of PERC+ 106 and IBC+ 73 have been reported.There are examples of simulating bifacial solar cell structures as well. 120ompared with the monofacial structure, there are limited reports regarding the bifacial structure; in other words, it is predicted that further improvement in the cell performances is expected in this field, and this may lead to the development of novel structures.

| Novel cells
We propose three novel cell structures based on numerical simulations: bifacial-type HBC, referred to as HBC+; 109 the improved i-TOPCon structure, referred to as advanced i-TOPCon; 111 and back-contact-type Ai-TOPCon, referred to as back-contact interdigitated carrier-selective (BICS). 121HBC+ is produced by introducing rear-surface TCO layers in a bifacial-type HBC.Currently, approximately 29.5 mW/cm 2 power density under bifacial illumination is predicted using TiO 2 TCO layers. 98Ai-TOPCon is an upgraded version of the i-TOPCon structure, produced by applying the local-TOPCon technology, and shows a low absorption loss and a small auger recombination volume.Finally, the BICS combines the advantages of PERC, IBC, and TOPCon, and shows over 30 mW/cm 2 power density at room temperature under bifacial illumination with a single-junction c-Si solar cell.The cell structures are illustrated in Figure 9 with the small instructions.
Recently, a novel structure, called "HAC," was proposed using numerical simulations.This HAC structure was simulated by removing the intrinsic layer of a-Si from one side. 122This structure exhibited a reduced absorption loss and an improved J SC of approximately 2 mA/cm 2 , compared with the SHJ structure, under bifacial illumination.
The examples discussed herein are focused on SHJ-or TOPCon-included structures, which exhibit improved cell performances.The proposal of novel structures based on numerical simulations is still limited.However, the improved accuracy of the models as well as the enhanced computational performance are expected to drive future studies in this field.
Another type of c-Si solar cells is known as "largescale integration (LSI) chip integrated devices" for power sources of integrated circuits.As they are designed on standard CMOS process and not specialized to design PV cells, there are a few restrictions for enhancing cell performances. 123The simplest type is the single pnjunction formation on the front surface of LSI chips. 124here is a report of on-chip back-contact PV cells that recorded 4.6% efficiency at infrared illumination. 125The proposed on-chip PV cells are "on-chip TOPCon" that utilized gate active region to form the tunnel-oxide layer, 126 and double-ring contact structure that realized shading-free to obtain the highest cell performances. 127

| Tandem cells
The tandem technique enhances cell performance by stacking different types of cells to absorb light effectively.
F I G U R E 9 Two-dimensional schematic of novel cell structures; Ai-TOPCon, HBC+, and BICS.Ai-TOPCon, advanced industrial TOPCon; a-Si, amorphous-Si; BICS, back-contact interdigitated carrier-selective; c-Si, crystalline-Si; HBC, heterojunction back contact; i-TOPCon, industrial TOPCon; TOPCon, tunnel-oxide passivated contact.As c-Si has a narrow bandgap of 1.12 eV (at room temperature), cell(s) of wider bandgap energy semiconductor is stacked on a c-Si solar cell.Conventionally, Ga-compound solar cell tandem solar cell that sought to achieve 30% total cell efficiency has been examined. 128urrently, perovskite solar cells have attracted considerable attention as the next-generation high-efficiency solar cells, given their low cost and easy availability and installation owing to fabrication by spin-coating. 129The tandem cell of perovskite/c-Si solar cells has been studied, 130 and an example of tandem cell featured 27.6% total cell efficiency. 131As the perovskite film is thin and included several TCO films, precise optical simulation is necessary.The cell efficiency of c-Si solar cells has approached the theoretical limitation of 29.4%; the tandem technique will be crucial in expanding its limitation to further higher regions, such as 30%-40%.

| CONCLUSION
In this study, we reviewed the methods of simulating c-Si solar cell devices.Emphasizing device simulations, we suggest comparisons between various software and optimized simulation plans.These include the structures currently in use and will prove beneficial in establishing simulation models.The latest reports on numerical simulations have been introduced, including both design improvements and characterizations of novel structures.These examples will be used to establish a roadmap of performances by predicting or revealing new insights for the devices.

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I G U R E 4 Effect of modules of monofacial and bifacial illuminations.

Comparison of software for c-Si solar cell evaluation.
Dimension of simulations for c-Si solar cell evaluations.