Abstract

Herein, the Fabry-Perot (F-P) interference of cesium silver bismuth bromide (Cs2AgBiBr6) double perovskite solar cells has been analyzed by modulating the optical path length of each layer step by step using the finite-difference time-domain (FDTD) method. The study was performed pass through three main steps. In step 1, for the fluorine-doped tin oxide (FTO)/Cs2AgBiBr6 double perovskite/gold (Au) architecture, we increased the absorption layer thickness from 150 to 600 nm at intervals of 150 nm and then predicted the optical performance including the absorption and reflection. In step 2, titanium dioxide (TiO2) layer was added between FTO electrode and Cs2AgBiBr6 double perovskite and then scaled from 20 to 200 nm at intervals of 20 nm. In the analysis process, short-circuit current density () repeatedly fell and rose as the TiO2 layer thickness increased, and when TiO2 layer thickness is 100 nm, showed the highest value of 13.91 mA/cm2. In step 3, by applying a spiro-OMeTAD layer between the Cs2AgBiBr6 double perovskite absorption layer and Au electrode, the showed a continuous increase with slight decrease as the spiro-OMeTAD layer thickness increased from 110 to 200 nm, and when the spiro-OMeTAD layer thickness was 200 nm, was calculated as 14.57 mA/cm2, which is the highest value in the range. In the case of the optimal condition of full structure device, the Fabry-Perot resonance peaks were discovered at 477, 520, 572, 659, and 778 nm five wavelength regions, and the influence of the Fabry-Perot resonance on the generation rate inside the absorption layer on the 520, 572, and 659 nm monochromatic wavelength light was analyzed according to the position in the device. Our study approaches the Cs2AgBiBr6 double perovskite solar cell from an F-P cavity perspective and shows the light trapping phenomenon of the device according to the optical path length modulation.

1. Introduction

Over the past few decades, fossil fuels are a core resource in human society, and their use is increasing dramatically in the world’s population. However, fossil fuels tend to be depleted over time, and in preparation for this, there is a need to supplement the new alternative energy. Alternative energy sources including solar, wind, hydro, and geothermal energy play an important role in supplying electricity to future generation. Among them, the solar energy is abundant, cost-free, and ecofriendly renewable energy and is considered to be a promising alternative energy currently being researched [13]. In a worldwide, the global market for solar systems is rapidly growing, with solar systems currently accounting for 26.2% of total global energy generated, with solar systems accounting for 2.4% [1]. The growth of solar power generation systems is expected to increase and develop as the years go by.

Organic–inorganic lead halide perovskite solar cells (PSCs) have attracted significant attention as new-generation photovoltaics due to their improved power conversion efficiency (PCE) comparable to photovoltaic materials such as silicon (Si) and copper indium gallium selenide [46]. Park et al. recently discovered a new method and principle for controlling the crystallinity of a perovskite photoactive layer and achieved the highest PCE of 26.08. Nevertheless, the commercialization of the typical APb2+X3 (, FA+ (CH(NH2)2+), Cs+, or mixed A-cations; , Br, or Cl) perovskite structure has a problem of the limitation due to the Pb2+ toxicity [7, 8].

Recently, research to replace Pb2+-based perovskites with Pb2+-free materials has been actively performed. One of the ways is to replace two divalent Pb2+ ions with one monovalent M+ and one trivalent M3+ ion, forming an A2M+M3+X6 double perovskite structure [8, 9]. Among this class of compounds, Cs2AgBiBr6 double perovskite has attracted much attention due to its high stability, low toxicity, and high sensitivity in the UV–blue wavelength region corresponding to the bandgap (≈1.9 eV). [1012]. Specifically, Cs2AgBiBr6 perovskite is air-stable for about 2 weeks, which is longer than the class of ASnI3 perovskite, one of the representative perovskite materials [13]. Moreover, bismuth- (Bi-) based perovskite, which is one of the attractive nontoxic perovskites, is one of the group 15 elements like Sb, with better chemical stability as compared to Pb-based perovskites. Besides, Cs2AgBiBr6 double perovskite has significant advantages of higher external quantum efficiency (≈60%), longer carrier lifetime (≈1 μs), and more moderate charge carrier mobility than other Pb2+-free perovskites [11, 14]. However, its light-harvesting ability is very poor due to its large direct bandgap, which leads to a very low short-circuit current density (), resulting in very low PCE (≈6%; Ravi et al., 2018) for Cs2AgBiBr6 double PSC commercialization. To the best of our knowledge, solar cells using Cs2AgBiBr6 double perovskite are recent research, and little information is available compared to other perovskites.

To overcome the low PCE of Cs2AgBiBr6 double perovskite-based solar cells, the most important research is to improve the poor light harvesting of Cs2AgBiBr6 double perovskite due to the large indirect band gap. To date, many studies have been conducted to improve the light-harvesting capability and optoelectronic properties of photovoltaics. One of the representative methods is to utilize light trapping technology [1517]. Light trapping is a strategy for controlling photon absorption by extending the optical path length (OPL) [17, 18]. One of the representative light trapping strategies is metallization for electronic applications. Embedding metallic nanoparticles such as gold (Au), silver (Ag), or copper (Cu) in the active layer plays a crucial role in reducing exciton quenching and cell resistance, significantly improving PCE [19, 20]. Localized surface plasmon resonance (LSPR), in which the surface of a metal nanostructure glows when exposed to light, utilizes the light trapping for metalized solar cell with various plasmonic structures, contributing to enhance the light absorption [2123]. By embedding these plasmonic nanostructures in solar cells and illuminating them, the light-harvesting capability can be improved using the localized surface plasmon resonance principle, which improves light scattering efficiency [24, 25].

Another strategy to improve light trapping in solar cell is to use the Fabry-Perot (F-P) effect. The F-P cavity is a structure in which two reflectors with high reflectance are placed in parallel at a certain distance, enhancing the optical absorption [2629]. The F-P cavity is used to analyze the interference phenomenon of light acting between reflecting mirrors [3032]. Light waves generated in the cavity and reflected light waves cause constructive and destructive interference; only light waves of a specific wavelength remain, and all others are canceled out, allowing only specific wavelengths of light to pass through selectively and appear as output. As a result, the optical path length (OPL) of the cell increases, and the light absorption ability is enhanced [33, 34]. Djurić and Jokić studied about F-P cavity mode PSC with p-i-n device structure, then investigated the maximum efficiency of PSC using the theoretical backgrounds, and explained the interference effects of PSC structure represented by a F-P resonator [29]. In conclusion, analysis of the F-P effect within the Cs2AgBiBr6 double perovskite with large indirect band gap-based solar cell is particularly significant for enhancing the optical absorption and light trapping.

Based on the foregoing, this study focused on the optical–electrical properties based on the F-P effect of Cs2AgBiBr6 PSCs with a fluorine-doped tin oxide (FTO) top electrode and a gold (Au) bottom electrode [35, 36]. We regarded the FTO/Cs2AgBiBr6 perovskite/Au architecture as an FP cavity and analyzed the electromagnetic wave propagation in the device via the finite-difference time-domain (FDTD) method [22, 37]. In 2023, Pritom et al. designed a Si solar cell with parabola moth-eye antireflection layer and used FDTD simulation to investigate the F-P resonance occurring in the device [38]. To construct the n-i-p structure solar cell, we formed the titanium dioxide (TiO2) and spiro-OMeTAD as an electron transport layer and a hole transport layer, respectively, and then investigated the optoelectronic properties, step by step. TiO2 is the common ETL material which is used in PSC with its suitable energy match and good transparency, while spiro-OMeTAD is the common HTL material with the matching energy level and nonreactive properties between spiro-OMeTAD and the perovskite layer [39, 40]. The OPL was modulated by scaling the layer thickness while the refractive index value at each light wavelength was fixed [41]. By scaling the thickness of Cs2AgBiBr6, TiO2, and spiro-OMeTAD layer, optimal OPL conditions of the device and resulting electrical performance have been calculated, and optimized device conditions and the resulting F-P resonance at a function of wavelength were investigated via FDTD simulation.

2. Simulation Procedure

We used the two-dimensional FDTD method to analyze the electromagnetic wave propagation mechanism in a user-defined medium-based architecture. The FDTD method is a methodology that solves Maxwell’s equations in the time domain, considers light as an electromagnetic wave, and analyzes it from a wave optical perspective with state-of-the-art technology. Therefore, the FDTD technique provides researchers with optimal optical modeling conditions in nanophotonics research. This method, devised by Kane Yee, simulates the interaction of electromagnetic waves in a medium by solving Maxwell’s equation [42]. Bader and Saghaei used this FDTD solution to analyze the tunability of the metamaterial absorbers under the terahertz spectrum corresponding to 0.2–20 THz [43]. In addition, Islam et al. [44] numerically studied the absorption and reflection of various optical metamaterials nanoabsorbers with the aim of improving the refractive index sensitivity sensitivity using the FDTD domain [44]. Lumerical FDTD solution software was utilized, which has a built-in technology computer-aided designing (TCAD) module for drawing the solar cell structure. Figure S1 shows the refractive index of materials which are used in the study. The refractive index of Au was based on cyclic redundancy check (CRC) data embedded in the software. We illustrate the simulation procedure step by step, as depicted in Figure 1. Referring to Islam et al., the optimal PCE of the solar cell appeared when the thickness of Cs2AgBiBr6 perovskite layer was 600 nm via simulation, and we analyzed the Cs2AgBiBr6 perovskite solar cell by scaling the absorption layer thickness from 150 to 600 nm with a 150 nm interval [44]. Next, in step 2, we designed the FTO/TiO2/optimal Cs2AgBiBr6 perovskite (t1)/Au-based architecture and investigated the electrical performance as a function of TiO2 layer thickness and optical performance. The TiO2 layer thickness was scaled from 20 to 200 nm with a 20 nm interval. We considered the range referred to other theoretical and experimental results in the literature; a lower-thickness TiO2 layer (approximately 60 nm or less) calculates a relatively lowand shows a continuous decreasing tendency, while a high-thickness TiO2 layer (approximately 100 nm or more) calculates a high [45, 46]. This result is expected to be useful data for forming compact TiO2 or mesoporous TiO2 layers when fabricating real Cs2AgBiBr6 double perovskite solar cell. Finally, in step 3, it designed the FTO/optimal TiO2 ()/optimal Cs2AgBiBr6 perovskite ()/spiro-OMeTAD/Au-based full structure device and then investigated the electrical performance as a function of TiO2 layer thickness and optical performance. The spiro-OMeTAD layer was scaled from 110 to 200 nm with a 10 nm interval, considering the less recombination or other performance such as reproducibility in real fabrication [47]. A red arrow symbols indicate the light traveling direction, which indicates that light is repeatedly reflected between two electrodes.

Two-dimensional FDTD was set to run the simulation, and the simulation time and temperature were set to 1000 fs and 300 K, respectively. In the mesh setting, the mesh type was set to auto-nonuniform, and mesh accuracy was set to 4, considering speed, memory requirements, and economy of time. The boundary condition of all simulation steps was set with a periodic boundary condition along the -axis, allowing for the calculation of the response of the entire system to the one-unit cell. The perfectly matched layer, which provides the zero-reflection coefficient for all incident angles of light, was set along the -axis to avoid the parasitic reflection. By setting the FDTD according to these conditions, we were able to perform the rigorous modeling with an appropriate trade-off between fast and accuracy using mathematical algorithms such as Maxwell’s equations even in complexly designed device structures.

3. Mathematical Algorithm

3.1. Finite-Difference Time-Domain Modeling Algorithm

can be calculated from the divergence of the Poynting vector and then converted into a more numerically stable form as where represents the angular frequency, and presents the imaginary part of permittivity. Assuming that all electron–hole pairs contribute to photocurrent, the can be calculated by where represents the electron charge, denotes quantum efficiency, and denotes a solar spectrum.

3.2. Fabry-Perot Wavelength-Dependent Power Transmission and Reflection

Before analyzing the FP effect, it is necessary to review the OPL. where is the real refractive index of the medium through which light travels and represents the thickness of the object. Based on the OPL, phase is expressed by the following equation:

From Eqs. (3) and (4), we can understand that the phase change is affected by the refractive index or thickness, or both. For the F-P interference approach to the modeled device, we discussed wavelength-dependent power transmission and reflection. where and denote the power reflectivity of the two mirrors that form the F-P cavity, denotes the roundtrip phase shift of the cavity, and denotes the power loss of each reflection upon the mirror, which can be caused by surface light scattering or absorption. Based on the FP resonance, the resonance wavelength in the cavity at normal incidence, this condition can be expressed as [25] where represents the total thickness of the cavity, represents the real refractive index in the cavity, represents the order of the resonance, and denotes the phase that occurred by reflection at both interfaces of the cavity.

4. Results and Discussion

4.1. Analysis of Cs2AgBiBr6 Double Perovskite-Based Architecture

Figures 2(a)2(d) illustrate the optical performance spectra of Cs2AgBiBr6 double perovskite-based architecture under plane wave source illumination with 400–900 nm wavelength of light. Each top data indicates the optical performance of the Cs2AgBiBr6 double perovskite-based architecture without an Au electrode (W/O Au electrode), and each bottom data indicates the optical performance of the Cs2AgBiBr6 double perovskite-based architecture with an Au electrode (W/ electrode) during the step 1 design process. Focusing on the W/O Au electrode conditions, light absorption can be calculated using the following equation: , where and represent the reflection and transmission of the architecture, respectively. As the thickness of Cs2AgBiBr6 double perovskite layer increases, the absorption spectrum showing a narrow absorption peak in the wavelength region of approximately 430 nm broadens to the wavelength region of 660 nm. Substituting this result into the transmission calculation, almost zero transmission was exhibited in the shorter-wavelength regime light; however, the transmission increased as the longer-wavelength regime light propagated into the absorption layer. As the thickness of the absorption layer increases, the transmitted light also decreased in the shorter-wavelength light regime. In the case of reflection, the spectrum is nearly identical to the FTO reflection spectrum; however, it exhibited a value close to zero in the wavelength regime of 400–500 nm, because light absorption is high in this regime, and light is barely reflected due to the optical properties of the Cs2AgBiBr6 double perovskite. As shown in Figure 2, the reflection differs depending on the absorption layer thickness, because of the phase difference depending on the layer thickness [48]. As light absorption increased, light transmission decreased correspondingly. After forming the Au electrode on the absorption layer, the absorption spectrum became broader and enhanced, with several peaks affected by the F-P effect. The absorption enhancement over the wavelength regime can be described as the Beer–Lambert effect and cavity effect [49, 50]. For the wavelength regime of less than 500 nm, due to the Beer–Lambert effect, light absorption is dominated by thickness-dependent properties. Above the 780 nm regime, the light absorption of the architecture becomes dominated by the cavity effect. From 500 to 780 nm regimes, the architecture is affected by both effects. For the W/ Au electrode architecture, light transmission calculations were not performed; therefore, light absorption can be calculated using the following equation: , unlike W/O electrode architecture. We quantitatively summarize the average optical performance of the W/O Au electrode-based architecture until 660 nm continuous wavelength light in Table 1. Compared with the W/O Au electrode condition, light absorption increased with the formation of the Au electrode, and light reflection by the Au electrode was calculated higher than that of the W/O Au electrode condition.

4.2. Analysis of TiO2 Electron Transport Layer Formed in Cs2AgBiBr6 Double Perovskite-Based Architecture

In the previous simulation, we confirmed that the 600 nm absorption layer thickness exhibited the best absorption properties. Based on this result, we formed a TiO2 layer between the FTO electrode and Cs2AgBiBr6 double perovskite absorption layer, scaling the thickness of the TiO2 layer, and examined the optical and electrical properties. Figure 3 illustrates the of the designed architecture with different TiO2 layer thicknesses performed in step 2. For a TiO2 layer with a thickness of 20 nm, was calculated to be 13.76 mA/cm2. However, when the TiO2 layer thickness was increased to 40 nm, dropped sharply. The architecture exhibited the lowest of 13.54 mA/cm2 when the TiO2 layer thickness was 40 nm. As the TiO2 layer thickness became 60 nm, increased again, and at 100 nm, the maximum was calculated as 13.91 mA/cm2. Based on this result, we examined the optical performance of the absorption layer in the designed architecture. As shown in Figure 4, the optical performance showed resonance peaks, and peak appearance is different depending on the TiO2 layer thickness. For an in-depth study of the degree of absorption for the thickness of the TiO2 layer, we analyzed the based on Figure 4. Figure 5 shows the according to wavelength until 900 nm within the absorption layer of the architecture. The wavelength interval was divided into 100 nm, and Figures 5(a)5(e) indicate 400–500, 500–600, 600–700, 700–800, and 800–900 nm wavelength of light, respectively. It showed the distribution in the Cs2AgBiBr6 perovskite layer with different three thicknesses of TiO2 layers. Black, red, and blue spectra indicate the distribution of 40 nm, 100 nm, and 160 nm thickness of TiO2 layer, respectively. When lights of each wavelength regime were illuminated to the architecture, the waveforms were different according to the thickness, which corresponded to the absorption spectra in Figure 4. The optical performance change according to the electron transport layer (ETL) thickness can be attributed to the light transmission change between the FTO/TiO2 layer and the Cs2AgBiBr6 perovskite layer interface [51, 52]. The average of each architecture were quantitatively summarized in Table 2.

4.3. Analysis of Spiro-OMeTAD Hole Transport Layer Formed in TiO2/Cs2AgBiBr6 Double Perovskite-Based Architecture

Previously, we completed the device optimization of the TiO2 layer and Cs2AgBiBr6 perovskite-based architecture via FDTD simulation. Based on this, we investigated the electrical and optical properties of the spiro-OMeTAD hole transport layer (HTL). Figure 6 shows the variation of the device according to spiro-OMeTAD layer thickness. At a thickness of 200 nm, the of the device was calculated to be a maximum value of 14.57 mA/cm2. As the spiro-OMeTAD layer thickness increased from 20 to 200 nm, unlike the TiO2 layer modulation process, continuously increased with slightly decreased characteristics. Next, we confirmed how the spiro-OMeTAD layer thickness affects the optical–electrical characteristics through absorption and average distribution. Wavelength regime conditions of step 3 are the same as step 2, and the thicknesses of the spiro-OMeTAD layer were analyzed with 110, 150, and 200 nm, representatively. Figure 7 illustrates the absorption under each wavelength light in the designed device according to different spiro-OMeTAD layer thicknesses. These values appeared in black, red, and blue lines, respectively. As the thickness of the spiro-OMeTAD layer is different, the wavelength regime where the absorption peaks appear is different. Then, the average distribution depending on the position in the absorption layer is calculated by using the transfer matrix method. Figures 8(a)8(e) show the result of distribution within the absorption layer according to specific spiro-OMeTAD layer thicknesses under the 400–500, 500–600, 600–700, 700–800, and 800–900 nm wavelengths of light, respectively. The black, red, and blue spectra indicated the spiro-OMeTAD layer thickness with 110, 150, and 200 nm, respectively. Just as average distribution changes depending on the thickness of the TiO2 layer, it can be confirmed that the distribution also varies depending on the thickness of spiro-OMeTAD. However, the change in the spiro-OMeTAD layer thickness led to a larger change in average distribution than the change in the TiO2 layer thickness variation. This result indicates that the spiro-OMeTAD layer between the absorption layer and Au played a more crucial role as an optical space layer than the TiO2 layer between the FTO electrode and absorption layer [5355]. The average of each architecture according to spiro-OMeTAD layer thickness were quantitatively summarized in Table 3.

Thus far, we have investigated the performance change of Cs2AgBiBr6 PSCs through layer thickness modulation. Figures 9 and 10 depict the additive optical–electrical performances of the device under optimal conditions. As shown in Figure 9, light absorption improved accordingly as additional layers are formed. Note that numerous peaks appear in the absorption spectrum when the Au reflector is formed at the full structure. By forming the optical spacer layer, multiple reflections were involved, resulting in resonances within the device [56]. As shown in Figure 9, the resonance peaks appeared under the monochromatic lights of 520, 572, 659, and 778 nm. Based on this result, we examined the generation rate for each absorption peak. The generation rate of the architecture was calculated with and without Au electrode. Focusing on the monochromatic light of 520, 572, and 659 nm, the generation rates were distributed within the absorption layer with different positions. As shown in Figure 10, the generation rate in the absorption layer showed a higher value when forming the Au electrode. Back Au electrode contact provides much higher generation rate within the absorption layer associated with the FP effect. The wavelength of the illuminating light is longer, and the lower generation rate was calculated. At monochromatic light of 778 nm, it was confirmed that no charge generation occurred in the absorption layer. We quantitatively summarize the maximum generation rate for each resonance peak in the absorption layer of the device in Table 4.

5. Conclusion

In summary, in-depth analysis of optical–electrical properties influenced by the F-P effect of Cs2AgBiBr6 PSCs was performed using the FDTD method. In the FTO/Cs2AgBiBr6 perovskite/Au architecture, the OPL of the Cs2AgBiBr6 perovskite was modulated to determine the wavelength regime where resonance occurred, and the intensity or the number of resonances increased as the OPL increased. Subsequently, going through steps 2 and 3, the change in optical–electrical characteristics according to the modulation of the OPL in the TiO2 layer or spiro-OMeTAD layer was analyzed by calculating the , absorption, and average . Finally, we investigated the resonance peaks according to the optimization of the device and analyzed the distribution of the generation rate according to the peaks in each wavelength region. This insight is meaningful in predicting the F-P effect that occurs theoretically by scaling the layer thickness of conventional n-i-p perovskite solar cells, which are commonly addressed, and provides the guideline for optical modeling in the fabrication of real nanosized standard photovoltaics.

Data Availability

The data used to support the findings of this study are included within the supplementary information file(s).

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Acknowledgments

This work was partly supported by the innovative Human Resource Development for Local Intellectualization program through the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (IITP-2024-RS-2022-00156389, 50%) and the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (2021R1A2C1011429, 50%).

Supplementary Materials

Figure S1: complex refractive index of each material. Black spectra indicate the real part, and blue spectra indicate the extinction coefficient part. The complex refractive index of FTO, Cs2AgBiBr6, and spiro-OMeTAD referred to literature, and TiO2 and Au referred to Siefke and CRC which are embedded in the Lumerical FDTD software, respectively. (Supplementary Materials)