Ameliorating Properties of Perovskite and Perovskite–Silicon Tandem Solar Cells via Mesoporous Antireflection Coating Model

It is anticipated that perovskite solar cells (PSCs) will overtake other products in the market for next‐generation photovoltaics. The optical loss, however, continues to be a flaw that restricts the photocurrent (Jph) of PSCs. Mesoporous antireflection coatings (ARCs), both monolayer and multilayer, are designed using a combination of the finite element method and equivalent medium theory, and ARCs models are merged with PSCs. In the current work, mesoporous ARCs are made, the optical performance of the device is evaluated using optical modeling, and then the ARCs are integrated into solar cells. The simulation results show that the Jph of planar inverted PSCs can increase to 24.00 mA cm−2 when the front surface of PSCs adopts mesoporous ARC (via parameter optimization and sensible arrangement and combination). An increase of 0.98 mA cm−2 in Jph of PSCs is observed in comparison with flat ARC (23.02 mA cm−2). The strong light transmission and low reflection properties of the mesoporous ARCs are confirmed by the optimized solution. It is important to note that the first fusion of mesoporous and multilayer ARC offers a fresh approach to the development of perovskite and perovskite/silicon tandem solar cells with extremely high efficiency.


Introduction
Perovskite solar cells (PSCs) are an emerging photovoltaic technology that offers the potential to incorporate costeffective manufacturing with distinguished power conversion efficiency (PCE) and is at par with the silicon solar cells (SCs) market. [1][2][3][4] Owing to the unprecedented photoelectric properties of perovskite materials, such as high absorption coefficient and flexible and adjustable bandgap, the record certified PCE of PSCs has expanded by 21.7% (from 3.8% in 2009 to 25.7% in 2022) in just a decade. [5][6][7][8][9] This exceptional development in efficiency dragged PSCs to the front leads in the perovskite community. [10,11] Despite this rapid development the theoretical limit of PCE reaches 30%, which is still attractive. [6,12] A comprehensive understanding of the energy loss mechanism is mandatory to ameliorate the It is anticipated that perovskite solar cells (PSCs) will overtake other products in the market for next-generation photovoltaics. The optical loss, however, continues to be a flaw that restricts the photocurrent (J ph ) of PSCs. Mesoporous antireflection coatings (ARCs), both monolayer and multilayer, are designed using a combination of the finite element method and equivalent medium theory, and ARCs models are merged with PSCs. In the current work, mesoporous ARCs are made, the optical performance of the device is evaluated using optical modeling, and then the ARCs are integrated into solar cells. The simulation results show that the J ph of planar inverted PSCs can increase to 24.00 mA cm −2 when the front surface of PSCs adopts mesoporous ARC (via parameter optimization and sensible arrangement and combination). An increase of 0.98 mA cm −2 in J ph of PSCs is observed in comparison with flat ARC (23.02 mA cm −2 ). The strong light transmission and low reflection properties of the mesoporous ARCs are confirmed by the optimized solution. It is important to note that the first fusion of mesoporous and multilayer ARC offers a fresh approach to the development of perovskite and perovskite/silicon tandem solar cells with extremely high efficiency.
device performance. [13][14][15][16][17][18][19][20][21][22] The main energy loss in transforming solar energy into electrical energy is attributed to the Shockley-Queisser (S-Q) limit, i.e., non-absorption of low-energy photons and the thermal loss of high-energy photons. [2,23] The optical loss occurred due to the photons that cannot be gathered by the perovskite layer or converted into photo carriers. Consequently, this loss has a lasting impact on light absorption and photocurrent (J ph ) of the active layer of PSCs. [2,[13][14][15] Owing to PSCs unprecedented rapid increase in efficiency grasp the attention of the scientific community. So far, many researches have been carried out PSCs research, mainly including optical design, material selection, process optimization, energy band matching, interface regulation, and so on. [24][25][26][27][28][29][30] As a precursor and supplement to experimental studies, theoretical simulations also play an indispensable role in the study of PSCs. Numerical simulation methods of PSCs include finite time domain difference method, finite element method (FEM), etc. [24][25][26]31,32] Theoretical simulation studies of PSCs mainly focus on the optical management and factors affecting the electrical performance of the device. In terms of electricity, researchers mainly study the influence of carrier migration and recombination on the electrical properties of devices by solving the drift-diffusion equation and the carrier continuity equation. [25,26] In terms of optics, to achieve effective light management for PSCs, researchers have designed a variety of novel light-trapping structures, including photonic crystals, surface textured structures, surface plasmon enhancement effects, and other light-trapping structures to reduce the light reflection on the surface of PSCs and enhanced light absorption. [33][34][35][36][37] Using the optical transfer matrix form with knowledge of the optical constants, Liu et al. modeled the light absorption of 2D planar-structured PSCs and optimized the 2D perovskite film thickness. They established an improved drift-diffusion model to study the effects of carrier generation and recombination. [31] However, these novel light-trapping structures have complicated the fabrication processes and expensive to manufacture. [5,[38][39][40][41][42] We introduced a mesoporous antireflection coating (ARC) at the interface of air and glass. Excellent antireflection effect can be obtained by optimizing the refractive index (n), thickness and microstructure of mesoporous ARC.
The preparation process of mesoporous ARC is simple and cost-effective. It can effectively reduce the refractive index of the film and greatly reduce the optical loss on the surface of PSCs. [43,44] SiO 2 is a commonly used inorganic material for preparing mesoporous thin films. SiO 2 materials have more abundant sources, more stable chemical properties and no pollution to the environment. [44][45][46] The most typical preparation method of mesoporous SiO 2 thin films is the sol-gel method. Sol-gel technology has outstanding advantages in large-area, low-cost film formation, and the technology is much developed. [45] It is currently the mainstream technology of photovoltaic glass ARC. [46] It can estimate the porosity of the thin film, so as to achieve the purpose of regulating the refractive index of the thin film. The refractive index of mesoporous SiO 2 films depends on the catalytic conditions (acid catalysis and base catalysis). [46] Acid catalysis can prepare low-refractive-index mesoporous membranes using surfactant templates, inducing self-assembly through evaporation to form ordered channels, and high-temperature annealing to remove the template agents.
The reaction temperature, pH value, and the concentration of surfactants are all effect of refractive index of the mesoporous films. [43][44][45] We used this method to prepare a single-layer mesoporous silica film. We linked the FEM and the equivalent medium theory (EMT) to design an ARC with a mesoporous structure for the first time, and combined it with PSCs. First, a physical model of inverted PSCs with a front surface covered with mesoporous ARC was established. Next, we compared the light absorption and J ph of PSCs utilizing different mesoporous materials adopted as ARC. In addition, we studied the effect of the arrangement of mesoporous materials in ARC (closely arranged and loosely arranged) on the optical properties of PSCs. Further, we researched the influence of the aperture of the porous material and the void ratio of the mesoporous ARC on the light-absorbing layer of PSCs was also studied and we also investigated the effects of the combination of different mesoporous materials and the light incident angle on the optical properties of PSCs. We used a simulation to examine the impact of light incident angle on the optical characteristics of PSCs under various mesoporous pore sizes and void ratios in order to determine if the mesoporous ARC has omnidirectional antireflection capability. Finally, we studied the effect of mesoporous ARC on the optical properties of 4T perovskite/ silicon tandem solar cells (PSK/Si TSCs) and 2T PSK/Si TSCs. In short, this work will afford an academically valuable reference for designing high-performance ARCs and promoting the response of the photosensitive layer of PSCs to photons.

Results and Discussion
This work modeled mesoporous ARC and it was applied to flat inverted heterojunction PSCs (Figure 1a). To feature the superior antireflection function of the mesoporous ARC, a comparison of the mesoporous ARC and the flat ARC is established (Figure 1b), along with a description of the light propagation path in both types of ARC. Contrastingly, mesoporous ARC has shown a significant increase in the amount of incident refracted and reflected light. Eventually, the length of the optical path increases, which gathers more photons, which are effectively absorbed by the solar cell; and ultimately exceptional improvement in the optical performance of PSCs. Figure 1c depicts the specific strategy of J ph enhancement using inverted PSCs. In contrast to flat ARC, mesoporous ARC significantly improves J ph and other optical properties of PSCs.
According to the Fresnel reflection equation, the mesoporous ARC mainly slows down the sudden change of the refractive index of the light incident from the air to the PSCs surface, thereby reducing the reflection loss on the PSCs surface. On the other hand, owing to the unique mesoporous structure in the ARC, light will undergo multiple reflections and refractions in the mesoporous ARC, which is bound to increase the collection of photons by the mesoporous ARC, so that more photons will propagate into the PSCs, absorbed by the solar cell ( Figure 1a). To better reflect the path of incident light in mesoporous ARC, we ignore the influence of nanoparticles on light. On the other hand, the unique pore structure of mesoporous ARC has a good light-trapping effect. www.advelectronicmat.de

Characterization and Optical Testing of Mesoporous Silica Thin Films
To further investigate the microstructure and surface morphology of mesoporous silica, scanning electron microscopy (SEM) and atomic force microscopy (AFM) were used for observation in this work. The mesoporous silica is uniformly distributed, and the gap between the pores is 10 nm. The mesoporous silica presents the shape of a spherical particle with an average particle size of 80 nm (Figure 2a,b). The surface roughness of the ARC is critical to the optical properties of the film. The introduction of the mesoporous structure increases the surface roughness of the film and ensures that the mesoporous film has excellent optical properties. The surface roughness of the ARC deposited on ITO-coated glass was measured by AFM ( Figure 2c). The root mean square roughness (RMS) value of the mesoporous film in the scan area of 3 × 3 µm 2 is 48.3 nm. The height map of the film corresponds to the cross-sectional view, and it can be observed that the thickness of the mesoporous silica film is 150 nm (Figure 2d). Figure 2e,f depicts the measurement parameters of the optical properties of the film with different thicknesses. To examine the reflectance spectra of mesoporous silica films with various thicknesses, we used the integrating sphere technique ( Figure S1,

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Supporting Information). The test results show that when the thickness of the mesoporous ARC is 50, 100, and 150 nm ( Figure S2a-c, Supporting Information), in the 300-800 nm band, the reflection spectrum of the coated mesoporous ARC is significantly lower than that of the glass substrate, and the transmission spectrum shows the opposite trend. When the thickness of the mesoporous ARC is 200 nm ( Figure S2d, Supporting Information), the reflectance spectrum is significantly increased compared with that of the glass substrate. To validate presence of mesoporous silica, we performed EDS elemental analysis ( Figure S3, Supporting Information). The corresponding elemental mapping images of a single mesoporous silica indicate a uniform distribution of Si and O elements. This fully demonstrates that by reasonably controlling the thickness of the mesoporous ARC, more excellent antireflection properties can be obtained.

Optimization of Mesoporous ARC Thickness and Refractive Index
In this paper, a mesoporous silica film was prepared, and the film was characterized and measured. Experimental measurement results show that the mesoporous ARC has excellent antireflection properties. It is based on this excellent antireflection performance that mesoporous ARC has been widely used in the photovoltaic field. We integrated the mesoporous ARC into the SCs and simulated the overall optical performance of the device using finite element simulation and equivalent medium theory. In this paper, only pure optical simulation is considered, and the carrier transmission/recombination/collection process is not considered. The materials used to establish the model in the present work and the thickness of each layer of the material are consistent with the references. [10] To justify the rationality of the physical model and the accuracy of the simulation method, we compare the calculated absorption spectrum with literature ( Figure S4, Supporting Information). [10] We observed that the absorption spectrum of the perovskite material is lower than the absorption spectrum described in the literature (350-750 nm waveband, when HTL adopts PTAA material). To analyze the reason for the downshift of the absorption spectrum, the optical parameters of PTAA were being analyzed. PTAA has a higher extinction coefficient than NiO x . Consequently, PTAA generates more parasitic absorption in this waveband, which is detrimental to the absorption of the perovskite layer. [35][36][37]47] Therefore, from an optical point of view alone, NiO x is more suitable as a material for HTL of inverted PSCs; moreover, using this HTL with perovskite layer the absorption spectrum is consistent with the literature ( Figures S4-S6, Supporting Information).
Owing to the broad spectrum of sunlight, the ideal ARC should not only have excellent optical properties but also continue stability in complex environments, and it caters to the photovoltaic industry with low-cost extensive applications. [39][40][41][42]48] In this regard, first, we will investigate about antireflective properties of flat structure ARC using the parametric scanning function. To make our theoretical simulation work more reproducible, we give detailed boundary condition settings and physics choices in the Supporting Information. The influence of thickness and n of the ARC on the transmittance of light can also be calculated at the same time ( Figure S7, Supporting Information). The n of the ARC is between 1.2 to 1.3, and the thickness is 100-140 nm, and a transmittance of about 96.5% can be realized. Subsequently, the influence of the n and thickness of ARC on the optical performance of the PSCs can be analyzed by integrating this flat structure ARC on the PSCs. Figure 3a,b shows the J ph and J R as a function of the n and thickness of flat ARC. The J of PSCs is calculated under AM 1.5G incident light, expressed as follows. [10,[39][40][41][42] 300 800 where q is the electron charge, c is the speed of light in vacuum, h is Planck's constant, λ is the incident wavelength, φ AM1.5G (λ) is the standard solar spectral irradiance under AM1.5G. A (λ) represents the absorption efficiency of the perovskite layer.
The n and thickness of flat ARC were varied from 1.10 to 2.00 nm and 40 to 200 nm, respectively. The effective J ph of PSCs is increased by 4.56 mA cm −2 , and the reflection loss (J R ) is also reduced by 4.58 mA cm −2 . The ideal ARC with the highest J ph (23.56 mA cm −2 ) and the lowest J R (3.12 mA cm −2 ) should have a thickness of 120 nm and n of about 1.20. This value is extremely consistent with the thickness and n values that ARC satisfies when having the maximum average transmittance. To understand the effect of n on the optical properties of PSCs, when the refractive index of the ARC is changed from 1.2 to 1.6, the thickness of the ARC is also reduced from 100 to 60 nm. As the refractive index of the ARC continues to increase (from 1.6 to 1.8), the thickness of the ARC doubles. This shows that only by matching different ARC thicknesses, ARCs with different refractive indices can achieve lower reflectivity ( Figure S8, Supporting Information). The absorption and reflectance were calculated for PSCs and FEM is used to obtain the J ph and J R values ( Figure 3c). As the n value of ARC increases, the absorption spectrum gradually shifts down, and the effective J ph value also declines. Meanwhile, the reflectance spectrum and J R show the opposite trend. To effectively reduce the reflection on the glass surface, researchers usually design a multilayer ARC, utilizing materials with lower n from the bottom up to reduce the abrupt change of the n at the interface of the two materials. This strategy will help in mitigating the reflection of the interface. [48] ARC antireflection mechanisms are exhibited in Figure S8 (Supporting Information). In this work, we analyzed the absorption and reflection spectra of the PSCs when the PSCs use flat and mesoporous ARCs, and the front surface of the PSCs without ARC (Figure 3d). To highlight the superior optical properties of mesoporous ARCs, we quantified the absorption and reflectance spectra using the average absorptivity and reflectivity ( Figure 3e). It is not difficult to insight that the average absorptivity of mesoporous ARC is improved by 2.8% and 1.2% compared to without ARC and flat ARC, respectively. This fully demonstrates that mesoporous ARC can effectively reduce the refractive index of the film and improve the light absorption of PSCs. Harnessing SiO 2 as antireflection material (ARM) for www.advelectronicmat.de the mesoporous (diameter of the mesopore is 30 nm) structure and the flat structure; blue-shifted the absorption spectra and descent the reflection spectra. When the mesoporous structure of the ARC is used on the front surface of the PSCs an increase in the J ph by 0.39 mA cm −2 (from 23.02 to 23.41 mA cm −2 ) is recorded contrasting to the flat structure of the ARC (Figure 3f). Mesoporous structure tends to reduce the reflection losses of PSCs as in mesoporous structure n of the individual coatings are likely to be commutative. This near equivalency of n of two materials (from air to glass) halted abrupt change in n that eventually reduce reflection at the interface of the two materials. This outcome is consistent with Fresnel's reflection theory. [49] The equivalent n of the mesoporous ARC is calculated using the following equation. [17][18][19][20]40] where n m represents the equivalent n of the mesoporous ARC, n o is the n of the material used in the dense ARC, and V represents the void ratio of the mesoporous ARC. Moreover, when the incident light passes through the mesoporous ARC, owing to its special hole structure and void ratio, the light will be refracted and reflected multiple times in the mesoporous ARC. For instance, when the mesoporous material is SiO 2 (n = 1.46) and the porosity is 52%, the equivalent refractive index of the mesoporous ARC is 1.24, which is relatively closer to the ideal refractive index value (1.23). This increases the propagation path of the light inside the PSCs resultantly the absorption of photons is strengthened by the perovskite layer. [4,[50][51][52][53]

Optimization of the Material and Arrangement of Mesoporous ARC
Compared with the flat structure of the ARC, the mesoporous structure of the ARC has incomparable advantages in reducing light reflection and increasing the photon utilization rate of PSCs. After discussing the action mechanism of flat structure ARC and mesoporous structure ARC on light, we concentrated on the materials used in mesoporous ARCs (with different n); the arrangement of mesopores; the aperture; void ratio; and the influence of the combination of different mesoporous materials on the optical performance of PSCs. At first, the response of PSCs to incident light when different ARMs are used to fill the ARC is discussed. ARMs are arranged in two ways i) tightly arranged, i.e., there is no gap between each hole, and each hole is in a tangent relationship with the surrounding holes (Figure 4a); ii) loosely arrangement of ARMs (Figure 4d). Figure 4b compares the absorption and reflection spectra of the perovskite layer in the 300-800 nm band when the ARM is closely arranged and the materials successively adopted ARCs; Al 2 O 3 , SiN x , SiO 2 , PMMA, and LiF. It is not difficult to find that when the ARM is different, the changing trend of the Abs of the perovskite layer is consistent in the whole band. It is worth noticing that as the n of ARM gradually increases, the absorption spectrum shifts downward and the reflection spectrum shifts upward. Correspondingly, the effective J ph value declines, and the J R increases (Figure 4b,c). For the same ARM, the reflection of PSCs is approximately close to zero at a wavelength of 660 nm, which proves that the incident wavelength also has a certain influence on the reflection  77). [17,[20][21][22] When ARM is closely arranged in the ARC, no matter how the aperture changes, the value of the void ratio is fixed (22%), so the equivalent n of the ARM is only related to the nature of the material. When the n value of the dense material is larger, the difference between the n m value and 1.23 is also larger, so the reflection loss on the surface of PSCs is also greater. Figure 4d is a schematic diagram of the loose arrangement of the ARMs. The aperture is the same as the tightened arrangement, but there is a certain gap between each small hole, which raises the void ratio of the mesoporous ARC. With the change in n of ARM (Material with lower n), J ph and J R trend does not vary much and is the same as in the tight arrangement illustrated in Figure 4e,f. However, the equivalent n of ARM is affected by the real n and void ratio of the filling material when it is loosely arranged. When the material with a higher refractive index is used as a mesoporous ARC, the J ph gain of PSCs is obvious. For instance, when the ARM is SiN x , the effective J ph value of PSCs is increased by 9.6% (from 20.63 to 22.61 mA cm −2 ), while for LiF as the ARM, the effective J ph is only increased by 0.7% (from 23.23 to 23.41 mA cm −2 ). We designed the mesoporous spheres as densely arranged, loosely oriented, cone-shaped, and disordered to be more similar to the experimental genuine mesoporous ARC. To investigate the Abs spectra, R spectra, and optoelectronic characteristics of PSCs under these four configurations, we carried out numerical simulations on these four arrangements ( Figure S9, Supporting Information). It is important to note that when the mesoporous materials are disordered, the optical characteristics of PSCs demonstrate extraordinary advantages. Disordering the mesoporous material brings it closer to the actual experimental setting. Therefore, in the future, we will experimentally create disordered mesoporous ARCs and incorporate them onto actual photovoltaic devices.

Optimization of Pore Size and Porosity of Mesoporous ARC
In this part, we conducted a systematic and comprehensive study on the aperture and void ratio of ARC and analyzed the influence of different apertures and void ratios of different materials on the optical performance of PSCs. Figure 5 depicts the optical properties of PSCs under different apertures and void ratio when ARM is SiO 2 , LiF, Al 2 O 3 , and ZrO 2 . These optical performance metrics include effective J ph , reflection loss (J R ), absorption spectrum, reflection spectrum, average absorption, and reflectance ( Figure 5; Figures S10 and S11, Supporting Information). For materials with low refractive index in LiF and SiO 2 , the pore size has a weak effect on the J ph and J R of PSCs; for materials with high refractive index in ZrO 2 , the pore size has a significant effect on the absorption and reflection of PSCs (Figure 5a,b). To deeply understand the effect of mesoporous materials with different refractive indices on the optical properties of PSCs, we calculated the absorption and reflection spectra of PSCs with different pore sizes when the mesoporous materials were SiO 2 and ZrO 2 , respectively (Figure S10, Supporting Information). We quantified absorption and reflection spectra using average absorptivity and reflectivity. For mesoporous SiO 2 materials, when the pore size increases from 30 to 90 nm, the average absorptivity and reflectance change with pore size is negligible, with only a 1% change. The corresponding absorption spectrum and reflection spectrum are almost coincident in the whole waveband (300-800 nm). For the mesoporous ZrO 2 material, with the increase of pore size (from 30 to 90 nm), the average absorption decreases from 72.4% to 67.6%, while the average reflectance increases from 16.2% to 20.9%. The www.advelectronicmat.de simulation data show that for mesoporous materials with low refractive index, the pore size has a marginal effect on the optical properties of PSCs, and for mesoporous materials with high refractive index, the pore size has an important effect on the light absorption of PSCs. Owing to the effective proportion of high-refractive-index materials in the mesoporous ARC increases as the aperture increases, resulting in an abrupt increase in the refractive index at the air/ARC interface and an increase in reflection loss. [21,22,54] When the ARM is loosely arranged, the optical influence of the void ratio on PSCs is studied (Figure 5c-f). The aperture of the mesopores is fixed at 30 nm, and SiO 2 and LiF represent mesoporous materials with high and low n, respectively. For mesoporous materials with low n, when the void ratio is around 53%, J ph is maximized, and at the same time, J R is reduced to a minimum. For mesoporous materials with high n, the void ratio must be greater to ensure that the equivalent n of the ARC is constant. [19] Therefore, when SiO 2 is utilized as the filling material of the ARC, and the J ph of PSCs is in the region of peak value, and the J R is in the region of minimal value when the void ratio of ARC maintains at 60%-70%. For mesoporous materials with low refractive index (LiF and SiO 2 ), the porosity has a weak effect on the optical properties of PSCs, and the difference between the maximum J ph and the minimum J ph is only 0.4 mA cm −2 ; For high refractive index mesoporous materials (Al 2 O 3 and ZrO 2 ), the difference between the maximum J ph and the minimum J ph reaches 2.0 mA cm −2 , and high refractive index mesoporous materials need to match higher porosity to achieve the best optical performance of PSCs. This phenomenon can also be mapped from the absorption and reflection spectra of PSCs with different void ratio. For SiO 2 , optimizing the porosity of the mesopores only increased the average absorptivity by 1.7%, while for ZrO 2 , the average absorptivity increased by 7.1% ( Figure S11, Supporting Information). According to the calculation formula of the equivalent refractive index of mesoporous ARC, the mesoporous material with high refractive index needs to match a higher void ratio, so that the equivalent refractive index of mesoporous ARC can be closer to the ideal refractive index (n = 1.23).

Optimization of the Combination Mode of Mesoporous ARC Materials
In this way, the light will be refracted and reflected numerous times in the mesoporous ARC, which enlarges the path of light propagation and develops the optical characteristics of the ARC. When the porosity of the mesoporous ARC is excessively extensive, the gap between each hole and the adjoining hole is too large. As discussed above, in addition to the refraction of the light in the ARC, the reflection is still present. [8,20,22,[54][55][56] The FEM simulation results exhibited that the design of mesoporous ARC with reasonable porosity is of great significance for enhancing the light-trapping ability of ARC and promoting the light absorption of PSCs.
Investigating the aperture and void ratio helps in understanding the effect of the arrangement and combination of different mesoporous materials in ARC and their optical behavior www.advelectronicmat.de in perovskite photovoltaic devices. Employing different materials in mesoporous ARC, ARC not only maintains the excellent optical properties of mesopores, but also incorporates the multilayer film structure, thus forming a multilayer mesoporous film structure with graded refractive index. The effects of singlelayer ARC, multilayer ARC and mesoporous ARC on light have been vividly demonstrated ( Figure S11, Supporting Information). The mesoporous materials, which are generally used (SiO 2 , Al 2 O 3 , PMMA) are selected as the ARC material. A schematized diagram is drawn that depicts how light is propagated in ARC (i.e., composed of different mesoporous materials). Owing to the difference in n of these three materials, according to the law of refraction, the angle of refraction between light propagating from voids to different mesoporous materials is also different (Figure 6a). Figure 6b,c demonstrated the absorption spectrum, reflection spectrum, effective J ph , and J R of PSCs when three different mesoporous materials are closely arranged in ARC. These objective functions are compared with a single mesoporous material and are used as ARC. It can be seen that when the refractive indices of the three materials are arranged closely from top to bottom in the ARC (aperture is 30 nm) in the order of small to large, the performance indicators of PSCs are all lower than that of a single mesoporous material (SiO 2 , PMMA) as ARC. This is because Al 2 O 3 has a larger n compared to the other two materials. Its addition is equivalent to enlarging the equivalent n of the mesoporous ARC, which affects the excellent optical properties of ARC, resulting in the weak response of PSCs to incident light. Figure 6d,e depicts when the n value of the three materials is arranged loosely from top to bottom in the ARC (aperture is equal to 30 nm, and the void ratio is 53%) and exhibits the absorption spectra, reflection spectra, J ph, and J R of PSCs are compared with a single material. It is not difficult to the insight that when the mixed material is used as a mesoporous ARC, the optical performance of PSCs is slightly improved. This is mainly attributed to the fact that when the mixed mesoporous materials are loosely arranged in the ARC, there is a gap between the two adjacent layers of materials, which causes the n of the different materials in the ARC to jump along the direction of light propagation. Therefore, a small amount of reflection occurs at the gap between the two layers of mesoporous materials. The loose arrangement increases the void ratio of the mesoporous ARC ultimately, reduces the equivalent n of the ARC, and improves the light transmission with a reduction in J R . The simulation results reveal that when mesoporous materials with different n are arranged and combined in ARC and to ensure ARC has excellent optical properties. It is necessary to avoid large abrupt changes in the n of every two adjacent layers of materials. Materials having lower n should be considered such as MgF, SiO 2 , PMMA. This simulation provides an effective theoretical solution for the experimental design of multilayer mesoporous ARC. Although this research has proposed an exciting theoretical design method for ARC, yet many hidden gaps still need to be deeply investigated. Such as, the combination of different mesoporous materials and the selection of mesoporous materials, and what kind of combination can perfectly solve the problem of optical loss in the solar cell and the interface. This will be our tireless pursuit. Based on the optimized mesoporous ARC parameters (SiO 2 is selected as the material of mesoporous ARC, with loose arrangement, aperture of 30 nm, and void ratio of 53%) influence of the thickness of the perovskite film, light incident angle, and the band gaps of the perovskite material on the optical properties of PSCs are studied (Figures S12-S14, Supporting Information). [17,53,57,58] We analyze the effects of ITO thickness and back electrode on the optical properties of PSCs and elucidate the light-harvesting mechanism of the device through the electromagnetic field distribution (Figures S15-S17, Supporting Information). These parameters are then compared with the front surface of the PSCs covered in the flat structure ARC (Figures S18-S22, Supporting Information).

Omnidirectional Antireflection Performance of Mesoporous ARC
The light incident angle also has a non-negligible effect on the light absorption and light reflection properties of PSCs. [59,60] In the above chapters, we analyzed and discussed the case of direct light irradiation on PSCs (Figures S23-S26, Supporting Information). In order to study the optical properties of PSCs under real conditions, we calculated the Abs, R, and J ph of PSCs under different light incident angles. In addition, to further analyze the excellent optical properties of mesoporous ARC and whether it has omnidirectional antireflection performance, we carefully studied the response of PSCs to the incident angle of light under different mesoporous pore sizes and void ratio. Figure S24 (Supporting Information) makes it abundantly evident that the mesoporous aperture only little affects the J ph and J R when the mesoporous material is SiO 2 . It is possible to draw out the coincident incidence angle change curves. This is because changing the mesopore pore size does not affect the porosity and arrangement of the mesoporous ARC when the mesoporous material is densely stacked. Consequently, the change in aperture has no impact on the light absorption and reflection by PSCs under direct or oblique sunlight. The equivalent refractive index of mesoporous ARC is calculated using the equivalent refractive index calculation formula, and it does not change as pore size changes. We also looked into how incidence angle and mesoporous porosity affected the optical characteristics of PSCs. Here, we examine the impact of porosity on the optical performance of the device using the high-refractive-index material SiN as an example. Because the impact of the porosity on the equivalent refractive index of the mesoporous ARC will be more noticeable when the high refractive index material fills the mesoporous ARC. The equivalent refractive index of mesoporous ARCs is impacted by variations in the void ratio. Light absorption and light reflection at the air/ glass interface are impacted by changes in the equivalent refractive index of mesoporous ARCs. Figures S25 and S26 (Supporting Information) demonstrate that when the mesoporous material is SiN, the optical performance of the perovskite solar system improves with increasing mesoporous ARC porosity. This is so that, when high-refractive-index mesoporous materials are used, the equivalent refractive index of the mesoporous ARC approaches that of an ideal ARC (n = 1.23) the higher the voids of the mesoporous ARC. Less reflection loss occurs at the air/glass interface. Compared with flat ARC, when mesoporous ARC is integrated on PSCs, the photovoltaic device highlights excellent omnidirectional performance. Especially in the large incident angle range, the mesoporous ARC exhibits unique light-harvesting performance. Figure S27 (Supporting Information) depicts the effects of different structures of ARC on the photovoltaic performance of PSCs and four important electrical parameters. It can be clearly seen that mesoporous ARC can effectively promote the effective absorption of light by PSCs, improve the photocurrent value of PSCs, and then increase the power conversion efficiency of PSCs. Mesoporous ARC can not only promote the response of PSCs to light but also enhance the light absorption of perovskite/silicon tandem solar cells (TSCs) and improve the effective utilization of light in photovoltaic devices.

Mesoporous ARC Improves the Optical Properties of Perovskite/Silicon Tandem Solar Cells
Application of mesoporous ARC other than single-junction PSCs is that they can also be applied to perovskite-based tandem solar cells (TSCs); further breaking the theoretical limit of single-junction solar cells. [59] In the present work, 4T and 2T PSK/Si TSCs are taken as a model to study the influence of mesoporous ARC on the optical properties of these two types of TSCs (Figure 7). The mesoporous ARC has particularly improved the optical performance of 4T PSK/Si TSCs. Since the top cell and the bottom cell are independent of each other, the mesoporous antireflection layer can act on the two sub-cells separately to improve the light absorption of the top cell in the short wavelength range and strengthen the light utilization of the bottom cell in the long-wavelength range. In contrast to flat ARC, mesoporous ARC acts on two sub-cells of 4T PSK/Si TSCs. The J ph of the top cell is increased by 0.45 mA cm −2 , and the J ph of the bottom cell is increased by 1.58 mA cm −2 (Figure 7b,c). In 300-1200 nm band, the increase of the absorption spectrum and the changing trend of the reflection spectrum are consistent (Figure 7d). For 2T PSK/Si TSCs top cell and bottom cell are connected in series, the J ph value of TSCs depends on the smaller cell. Mesoporous ARC applied to 2T PSK/Si TSCs can promote the light absorption of the top cell in the 550-700 nm band, and the J ph has is improved by 3.40% (from 18.35 to 18.97 mA cm −2 ), but for 2T devices the overall optical performance of the light is not significantly improved (Figure 7f-h). This shows that the mesoporous ARC enhances the photon absorption of PSCs at 550-700 nm, which will inevitably weaken the absorption of crystalline silicon SCs in this waveband, resulting in a slight change in the overall J ph of the device.

Conclusions
The implementation of mesoporous ARC on the photovoltaic performance of planar inverted PSCs is studied through FEM and EMT. It can investigate the relationship between the material properties, structural properties of mesoporous ARC and the arrangement and combination of mesoporous materials, and the optical properties of PSCs. First, an optical model of inverted PSCs with a front surface coated with mesoporous ARC was established. The effects of mesoporous ARC materials, aperture, void ratio, arrangement, and combination on the light absorption and light reflection of PSCs have been investigated. The simulation results indicate that, compared with the planar structure of the ARC, the mesoporous ARC has the optical characteristics of higher light transmittance and lower reflection, and the J ph of PSCs is enhanced by 0.98 mA cm −2 (from 23.02 to 24.00 mA cm −2 ). In addition, the outstanding optical properties of mesoporous ARC directly boost the absorption of photons by PSCs and heighten the response of PSCs to photons. From a long-term perspective, mesoporous ARC can be applied not only to single junction PSCs but also to perovskite-based tandem SCs that enhance the optical performance of tandem SCs. In addition, for many applications that demand high light transmittance, or involve relatively uniform www.advelectronicmat.de light intensity distribution, or remained in harsh environments, mesoporous ARC is more excellent choice than mostly planar ARCs. Our simulation results can pave the way for the design of ARCs with excellent optical characteristics, enhancing light stability and uniformity, and then optically pointing out the direction for promoting the light absorption and efficiency of PSCs.

Experimental Section
Finite Element Method Simulation: The optical performance of PSCs was calculated based on reliable experimental thickness and previously reported materials. The values of the n and extinction coefficient of each layer of material were also reflected in the supporting information. PSCs adopt p-i-n structure. PSCs consisted of a glass substrate with front indium tin oxide (ITO) (80 nm), PTAA/NiO x (10 nm), perovskite (500 nm), PCBM (60 nm), C 60 (20 nm), and Cu (100 nm). An optical simulation was implemented based on the finite element method (FEM). By solving Maxwell's equations, the spatial electromagnetic field information coupled with the frequency domain response could be obtained. Perform a 2D simulation using a pair of Floquet periodic indices to enforce periodic boundary conditions, and use a perfectly matched layer (PML) as the absorbing boundary condition. The standard AM1.5G spectrum with a wavelength range of 300 to 800 nm was used as the solar incidence, which was consistent with the perovskite bandgap of 1.55 eV. When solving the reflection of the system, the contact surface of the two media was taken as the integration object and calculated the reflected energy flow divided by the incident energy flow through integration.
Mesoporous ARC Model and Equivalent Medium Theory (EMT): To depict the ARC of the mesoporous structure more vividly in the simulation, a rectangle with a length of 300 nm and a width of 120 nm was first drawn, and then a circle with a radius of 10 nm, and moved the circle to the inside of the rectangle and made it tangent to the edge of the rectangle. The array command was used to copy multiple circles so that these circles fill the entire rectangle, which was also a tight arrangement. By adjusting the distance between adjacent circles and the number of circles, a loose arrangement was formed.
EMT was a concise and intuitive approximation theory in the study of sub-wavelength structure theory. EMT was applicable when the period of the microstructure grating was smaller than the incident wavelength. The smaller the ratio of the period length of the grating to the wavelength of the incident light, the more accurate the result of the theoretical analysis and calculation using the equivalent medium. When the period of the microstructure was much smaller than the incident wavelength, only zero-order diffraction existed, which would produce zero-order reflected waves and zero-order transmitted waves. The highorder diffraction orders of light could not propagate to the free space and were bound inside the structure.
According to the EMT, the equivalent n of the mesoporous ARC was deduced. The specific derivation process has been given in the supporting information.
where n eff represents the equivalent refractive index of the mesoporous ARC, n 1 is the refractive index of the dense ARC, and V is the void ratio of the mesoporous ARC.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.