Optimization of Bulk Heterojunction Photovoltaic Structures with Heterocyclic Derivatives

Abstract

Photovoltaic structures of the bulk heterojunction type were fabricated, in which derivatives of N,N-diethylamine-3-Methyl-1-Phenyl-1H-pyrazolo[3,4-b]quinoxalines were used as the active layer. The compounds differed in the position of the electron-donating substituent in the carbocyclic ring. Four isomers were subjected to UV-Vis spectrophotometric measurements in solvents of different polarities. The absorption characteristics were experimentally determined for the tested derivatives. The values of HOMO-LUMO levels were determined by means of quantum chemical calculations using the HyperChem software. The current–voltage and dispersion characteristics of the produced OPV were determined. The spectral characteristics of the refractive indices and extinction coefficients of the active layers were determined using the spectroscopic ellipsometry method. These results were used in the analysis and optimization of photovoltaic structures. It was shown that the location of the N,N-diethylamine substituent affects the photophysical properties of the structure and the photovoltaic properties. The optimization of the OPV_2 photovoltaic structure using the coherent model and the 2 × 2 matrix method can be successfully used in modeling optical multilayer structures, including photovoltaic structures.

1. Introduction

In the face of the current environmental and economic situation, which aggravates energy problems, the green energy obtained from renewable energy sources (RESs) is the subject of research in many centers around the world (Figure 1) [1]. It is the best alternative to conventional sources. New technological solutions are sought, and the known ones are being modified so that the energy obtained from RESs is as profitable as possible, and in the future, it will replace conventional energy sources [2,3,4,5]. The best RES is the sun [6]. The conversion of solar energy into electricity takes place in solar cells thanks to the photovoltaic effect [7]. Despite relatively expensive installations, low efficiency and problems with energy storage, photovoltaic cells are one of the most popular and available methods of generating green energy [8]. One of the currently developed photovoltaic technologies is organic photovoltaics (OPV). This type of photovoltaic structure is characterized by advantages such as the thin-layer and flexibility of the structure, as well as the possibility of modifying compounds that absorb a wide range of solar radiation. To increase OPV performance, Wu X. et al. have proposed vertical OPV structures with a mass heterojunction (BHJ) with a field-effect transistor (VFET), improving the power conversion efficiency from 10% to 18% [9]. In turn, Aboulouard A. et al. designed new quinoxaline derivatives using the density functional theory (DFT) to evaluate, among other things, the photovoltaic properties of the proposed donor materials for potential use in OPV [10]. In the latest literature reports, you can find research works on the effect of isomers on the materials used in OPV [11,12].

Azaheterocyclic derivatives are popular compounds in medical [13] and optoelectronic applications [14]. They are characterized by many biological activities and interesting photophysical properties. Thanks to the latter, azaheterocycles have a wide range of applications, from electroluminescent materials [15,16] to fluorescent sensors [17] and sensitizers in photovoltaics [18]. These compounds are also part of the active layer in the structures of OPV cells [19]. Pyrazoloquinoxalines are an example of azaheterocyclic compounds. They are three-membered systems consisting of a quinoxaline ring and a pyrazole. The first method of synthesis of 1H-pyrazolo[3,4-b]quinoxaline derivatives was developed in 1903 by Sachs and Becherescu [20], but only recently, in 2017, Danel et al. proposed the most universal method of obtaining these derivatives [21]. Thanks to this method, these systems can be modified in a wide range. Depending on the substituents (electron-withdrawing, electron-donating), they exhibit different photophysical properties [16,22,23]. The arrangement of the substituents in the system is also important, in particular, the location of the electron-donating substituent in the carbocyclic ring [24]. The photovoltaic structures described in this article are built of 1H-pyrazolo[3,4-b]quinoxaline derivatives, which are a component of the active layer.

Optimization using the coherent model and the 2 × 2 matrix method is successfully used in modeling optical multilayer structures, including photovoltaic structures. Organic photovoltaic cells are multilayer structures in which the thickness of the layers is much smaller than the path of coherence of solar radiation, which is ~800 nm [25]. For this reason, light interference occurs in their structures, and its result may have a significant impact on the efficiency of solar energy conversion into electricity. Therefore, in the analysis of such photovoltaic cells, coherent models are used, which take into account the phenomenon of light interference. Matrices methods are very effective methods for calculating light intensity distributions in photovoltaic cell structures and the surface densities of generated excitons. These methods were used to analyze optical layered structures, including dielectric mirrors [26], anti-reflection layers [27], planar optical fibers [28], or photovoltaic cells [29]. The 2 × 2 matrix method is used here to determine the distribution of light intensity in a photovoltaic cell and to determine the influence of the thickness of the PEDOT:PSS (poly(3,4-ethylenedioxythiophene):polystyrene sulfonic acid) layer and the thickness of the active layer on the surface density of generated excitons.

In this article, we described photovoltaic cells built using four positional isomers of the derivative N,N-diethylamine-3-methyl-1-phenyl-1H-pyrazolo[3,4-b]quinoxaline (PQX1–PQX4), which differ in the location of the electron-donating substituent in carbocyclic ring. The aim of the work was to investigate the influence of the position of the substituent on the efficiency of the photovoltaic cell, in which the appropriate isomers constituted the material absorbing the electromagnetic radiation of the active layer. The novelty and originality of this article are the structures of the tested compounds in the produced photovoltaic cells and the assessment of the influence of the position of the electron-donating substituent in the molecule on the absorption and dispersion characteristics. The absorption characteristics were experimentally determined for the tested derivatives. The values of HOMO-LUMO (the highest occupied molecular orbital, HOMO, and the lowest unoccupied molecular orbital, LUMO) levels were determined by means of quantum chemical calculations using the HyperChem software. The current–voltage and dispersion characteristics of the produced OPV were determined. The spectral characteristics of the refractive indices and extinction coefficients of the active layers were determined using the spectroscopic ellipsometry method. These results were used in the analysis and optimization of photovoltaic structures using the coherent model and 2 × 2 matrix method.

2. Materials and Methods

2.1. Materials

The azaheterocyclic compounds used in the active layers were synthesized (Section 2.2). Other reagents for the construction of photovoltaic cells and spectroscopic measurements were commercial products and were used without prior purification. Glass substrates with an ITO (indium tin oxide) layer (sheet resistance of 8–12 Ω/sq) on which organic photovoltaic (OPV) cells were fabricated were obtained from Sigma-Aldrich (Steinheim, Germany). The solvents used at various stages of OPV preparation and for spectroscopic measurements, 2-propanol (99.5%), acetone (AR), tetrahydrofuran (THF, anhydrous, 99.9%), acetonitriles (ACN) and methylcyclohexane (MCHX), were obtained from Sigma-Aldrich (Steinheim, Germany). The polymeric materials used for the construction of the OPV: PEDOT:PSS and poly(3-octylthiophene) (P3OT) were obtained from Sigma-Aldrich (Steinheim, Germany). Demineralized water was used directly from the demineralizer (Polwater DL2-100S613 TUV, Labopol Solution&Technologies, Kraków, Poland).

2.2. Synthesis of PQX Derivatives

Figure 2 shows the individual stages of the synthesis of N,N-diethylamine-3-methyl-1-phenyl-1H-pyrazolo[3,4-b]quinoxaline (PQX) isomers, used as the active layer for the construction of photovoltaic cells.

The starting materials in the first stage were 5-amino-3-methyl-1-phenylpyrazole (2) and the corresponding chlorine isomers of 2-iodonitrobenzene (1), of which, as a result of the reaction catalyzed by the palladium:phosphine system, the so-called Buchwald–Hartwig coupling reaction yielded intermediates (3). In the second step, each of the 3 isomers was reductively cyclized using iron (II) oxalate as a reducing agent. As a result of this reaction, the corresponding isomers of n-chloro-3-methyl-1-phenyl-1H-pyrazolo[3,4-b]quinoxalines (4) were obtained. The last step was the substitution of the chlorine atom in the isomers with the N,N-diethylamino group. For this purpose, the Buchwald reaction–amination of aryl chlorides using a palladium:phosphine catalytic system was again used. The final effect of this synthesis route was four positional isomers PQX1–PQX4, as shown in Figure 3.

The general procedure for the synthesis of isomers of N,N-diethylamine-3-methyl-1-phenyl-1H-pyrazolo[3,4-b]quinoxaline (PQX1–PQX4) and the full characteristics (NMR spectra, FTIR spectra and elemental analysis) of the studied dyes are presented elsewhere [21,24].

2.3. OPV Technology

OPVs were produced according to the procedure described earlier [30]. Figure 4 shows a diagram of the photovoltaic cells produced for testing.

Photovoltaic structures with mass heterojunction were fabricated on glass substrates with dimensions of 15 × 15 mm², covered with an ITO layer. Before applying subsequent layers, these substrates were cleaned in an ultrasonic cleaner, rinsed in solvents (water, 2-propanol and acetone) and dried in the air. A layer of PEDOT:PSS was applied to the cleaned substrates with the ITO layer, which was then heated under a vacuum at 100 °C for 40 min. The PEDOT:PSS layer was covered with an active layer, which was a mixture of P3OT and the appropriate PQX1-–4 dissolved in THF. The layers were applied by the spin-coating method. An aluminum electrode was deposited on the active layer in a high vacuum (10⁻⁶ bar). Five OPV structures were made with each PQX isomer.

2.4. Characterization and Optimization

The absorption spectra of solutions and thin films of PQX dyes in the UV-Vis range were recorded on a Spectrophotometer Ocean Optics. The concentration of the tested dye was about 1.8–7.5 × 10⁻⁵ mol/dm³. Thickness and refractive index measurements of the produced OPVs were performed on a Woollam M2000 spectroscopic ellipsometer (JA Woollam Co. Inc., Lincoln, NE, USA) with CompleteEASE software.

Quantum chemical calculations for PQX1–4 structures were performed using HyperChem 7.0 software. OPV optimization was performed using a coherent model in which the 2 × 2 matrix method was used.

The current–voltage characteristics for the manufactured OPVs were tested using the Keithley 2400 SourceMeter current source. Measurements were made in a darkroom and under a tungsten lamp with a constant power density of 1.3 mW/cm² at room temperature.

3. Results and Discussion

3.1. Photophysical Properties of Isomers PQX1–4

In Figure 5, the spectrum of absorption measurement in the UV-Vis range of solutions in THF (Figure 5a) and thin films (Figure 5b) of the PQX1–4 isomers are shown. Thin films of dyes were prepared by dissolving the appropriate isomer in THF and applying the solution to a glass substrate by dripping and drying the layer in the air. In the spectra recorded in the THF solution, each isomer is characterized by three intense absorption bands. The first of them—sharp peaks in the ultraviolet range (230–340 nm)—could be found with π-π* transitions. The second band is in the violet part of the visible spectrum (395 nm for PQX1, 380 nm for PQX2, 412 nm for PQX3, and 375 nm for PQX4), and the third visible band is in the blue, green, and common yellow parts of the visible spectrum. In the spectrum, a transition band between the ground state S0 and the excited state S1 is observed, with a maximum ranging from 460 nm to 512 nm in THF solutions.

The S₀→S₁ transition band for the PQX2 dye is located at the longest wavelengths in relation to the other bands (λ_max = 512 nm) and is characterized by a high value of the molar extinction coefficient (ε = 10,199 M⁻¹·cm⁻¹). On the other hand, the S₀→S₁ transition band for the PQX3 system lies at the shortest wavelengths (λ_max = 460 nm), but this measurement showed the highest value of the molar extinction coefficient (ε = 10,911 M⁻¹·cm⁻¹). Basic parameters of absorption in solvents of different polarity, i.e., in ACN, THF, and MCHX, and for the thin films of the tested isomers, are presented in

Table 1. Basic parameters for PQX1–4 isomers measured in solvents of different polarity (ACN, THF, MCHX) and in thin films (λ_max—wavelength for maximum absorption, transition S₀→S₁, ε—molar absorption coefficient at maximum absorption).

Compound	UV-Vis λmax [nm] (ε [dm3·mol−1·cm−1])
	ACN	THF	MCHX	Thin Films
PQX1	481 (2670)	485 (2665)	479 (2730)	525
PQX2	513 (8255)	512 (10,199)	485 (8889) 504 (8770) 515 (8165)	527
PQX3	468 (10,215)	459 (10,911)	443 (10,625) 457 (8230) 471 (7320)	492
PQX4	476 (3730)	475 (3691)	474 (3675)	516

.

In the spectra of the PQX1–4 dyes in the form of thin films (Figure 5b), single bands were observed due to the absorption of the substrate glass (λ < 400 nm). The location of the maxima of these bands is summarized in Table 1. A clear bathochromic shift of the location of the maxima of the absorption bands was observed for all PQX1–4 isomers. The largest band position shift was observed for PQX4 (41 nm) and the smallest for PQX2 (15 nm). As in the THF measurements, the absorption band for PQX2 (λ = 527 nm) lies with the longest wavelengths in the green part of the visible electromagnetic spectrum, while for PQX3—with the shortest wavelengths (λ = 492 nm) in the blue-green part of the Vis range.

Based on the information collected in Table 1, it can be concluded that the investigated PQX1–4 compounds do not show a clear solvatochromic effect with the change of solvent polarity, which is generally observed for this type of compounds with electron-donating substituents [31]. On the other hand, PQX1–4 isomers in the form of a thin layer are characterized by a shifted position of the absorption bands in relation to the measurements in THF towards longer wavelengths.

Using HyperChem software, the structures of PQX1–4 isomers were optimized, and the following levels were determined HOMO and LUMO values and dipole moments (MD). The obtained results are presented in Figure 6.

The location of the electron-donating substituent in the carbocyclic ring of the 3-methyl-1-phenyl-1H-pyrazolo[3,4-b]quinoxaline (PQX1–4) derivative has a slight influence on the value of the energy gap E_g of these structures. For each structure, the HOMO level is at the same level (−3.18 eV for PQX2–4 and −3.17 eV for PQX1), while the position of the LUMO orbital changes slightly depending on the isomer. The LUMO orbital has the highest value for PQX4 (−0.18 eV) and the lowest for PQX3 (−0.29 eV). In this way, the band gap width for PQX1–4 isomers is in the narrow range of 2.89–3.00 eV. The PQX4 derivative also has the highest dipole moment (2.54 Debyes). Low energy gap values (E_g ≤ 3 eV) provide the basis for further studies of these compounds as elements of optoelectronic devices [32].

3.2. Photovoltaic Cells

Figure 7 shows the current–voltage characteristics of the bulk heterojunction organic photovoltaic cells (OPV_1–OPV_4) for structures with the following architecture: ITO/PEDOT:PSS/active layer/Al. The active layer was a mixture of P3OT polymer and PQX1–4 derivative. The characteristics were recorded in the dark and under illumination with P_light = 1.3 mW/cm².

The basic photovoltaic parameters of OPV (Figure 7) structures were determined [33], which are summarized in

Table 2. Main photovoltaic parameters obtained for the manufactured OPV devices (d_al—active layer thickness; J_SC—current density, V_OC—open circuit voltage, FF—fill factor, η—power efficiency).

Photovoltaic Cell	dal [nm]	JSC [μA/cm2]	VOC [V]	FF	η [%]	EQE (λ = 536 nm)
OPV_1 ITO/PEDOT:PSS/P3OT + PQX1/Al	102.9 ± 1.2	16.22 ± 0.02	1.01 ± 0.01	0.21 ± 0.01	0.27 ± 0.02	2.9
OPV_2 ITO/PEDOT:PSS/P3OT + PQX2/Al	101.5 ± 1.1	22.43 ± 0.02	0.77 ± 0.01	0.21 ± 0.01	0.28 ± 0.02	4.0
OPV_3 ITO/PEDOT:PSS/P3OT + PQX3/Al	103.1 ± 1.0	19.72 ± 0.02	0.72 ± 0.01	0.23 ± 0.01	0.25 ± 0.02	3.5
OPV_4 ITO/PEDOT:PSS/P3OT + PQX4/Al	100.1 ± 1.2	28.44 ± 0.02	0.80 ± 0.01	0.21 ± 0.01	0.37 ± 0.02	5.1

. The thickness of the active layer was determined using the ellipsometric method.

One can see the influence of the position of the electron-donor substituent in the carbocyclic ring in the PQX derivative on the parameters of OPV cells produced with their participation (Figure 7, Table 2). All of the devices are characterized by photovoltaic efficiency η below 1%. The best photovoltaic properties were shown by the OPV_4 structure, for which the power efficiency η was 0.37% at a short circuit current density equal to 28.44 μA/cm², the open circuit voltage was 0.80 V, and the fill factor was equal to 0.21. The remaining OPV_1–OPV_3 structures achieved photovoltaic efficiencies in the range of η = 0.25–0.27%. Analyzing the open circuit voltage (V_OC), one can notice the preserved dependence of the V_OC value on the HOMO-LUMO levels [34]. The V_OC values increase with the bandgap width ΔE_g (Figure 6), except for PQX4. Presumably, this deviation for PQX4 (V_OC = 0.80 V, ΔE_g = 3.00 eV) is related to the comparison of ΔE_g values determined by quantum chemical calculations.

3.3. Optimization of Photovoltaic Cells

A theoretical analysis of photovoltaic cells was carried out using a coherent model. The calculations were carried out using the transfer 2 × 2 matrix method [29]. Due to the fact that for all PQX, the determined extinction coefficients have similar values, we present below the results only for the PQX2 molecule. In the calculations made using the 2 × 2 matrix method, we took into account the complex refractive indices of the component layers of the OPV_2 photovoltaic cell, whose dispersion characteristics are shown in Figure 8.

All of the calculations were made assuming that the photovoltaic cell is illuminated from the ground side. The maximum value of the extinction coefficient of the PQX2 material is 0.657 at a wavelength of 547 nm. For this reason, calculations of the dependence of the density of generated excitons on the thickness of the PEDOT:PSS and PQX2 layers, respectively, were performed for wavelengths of 500, 550, 600, and 650 nm [29]. In each case, it was assumed that radiation with a unit intensity density of 1 Wm⁻²·nm⁻¹ falls perpendicularly on the photovoltaic cell. Moreover, it was assumed that the thickness of the ITO layer is 120 nm and the thickness of the Al layer is 100 nm.

Figure 9 shows the effect of the thickness of the PEDOT:PSS and PQX2 layers on the surface density of excited excitons G_surf for the selected wavelengths of 500 nm (a), 550 nm (b), 600 nm (c), and 650 nm (d), respectively. The surface density of excited excitons is defined as follows:

$$G_{surf}={\displaystyle \underset{0}{\overset{d}{\int }}G(x)dx}$$

(1)

Taking into account all calculation results, it can be seen that the maximum densities of excited excitons, when illuminated with radiation with a continuous spectrum, will be achieved for the thickness d_PQX2 of the PEDOT:PSS layer below 100 nm and the thickness of the active d_PQX2 layer of ~100 nm. For these reasons, we decided to make photovoltaic cells with a layer thickness of d_PEDOT:PSS = 60 nm and d_PQX2 = 100 nm.

The light intensity in the j^th layer can be written as [29]:

$$I_j(x\prime )=\frac{n_j}{n_c}\frac{{\left|E_j(x\prime )\right|}^2}{{\left|E_0\right|}^2}{I}_0,$$

(2)

where n_j is the refractive index of the j^th layer, n_c is the refractive index of the environment, and I₀ is the intensity of the light falling on the photovoltaic cell.

Figure 10a,b show the calculated distributions of the normalized squares of the electric field and the density of excited excitons G(x) as a function of the distance x from the glass substrate, calculated for perpenicolar and oblique illumination of the structure, respectively. The distributions of the refractive index (green line) and extinction coefficient (magenta line) were also plotted. In both figures, solid lines show properly normalized squares of the electric field (solid red lines) and densities of excited excitons G(x) (solid blue lines) for perpenicolar illumination of the structure. As can be seen, the maximum density of excited excitons in the active layer occurs near the PEDOT:PSS layer. Considering the short diffusion path of excitons excited in the active layer, this phenomenon is beneficial, as it means that more excitons can reach the PQX2/PEDOT:PSS interface and dissociate, generating electric charges.

The calculated characteristics for oblique illumination of the structure were plotted with the same colors, but with discontinuous lines, for light with s polarization (Figure 10a) and p polarization (Figure 10b), respectively. It can be seen that with oblique illumination of the solar cell, for both polarizations of the incident light, the densities of excited excitons in the area of the PQX2 active layer decrease. However, these changes are stronger for the s polarization. At the PEDOT:PSS/PQX2 interface, there is a clear decrease in the density G of excited excitons, while for the p polarization, a slight increase in G can be seen with the increase in the illumination angle. As a result, based on the results presented here, it can be concluded that in real conditions, when a solar cell is illuminated with non-polarized light, its efficiency will decrease with the increase in the illumination angle.

The aim of the optical optimization of the photovoltaic cell is to achieve the highest possible efficiency in the conversion of optical radiation into electric current. The aim is to ensure that the largest possible part of the incident radiation on the photovoltaic cell is absorbed in it. The absorption of radiation in a photovoltaic cell is given by (1-−R). This dependence for the photovoltaic cell with the PQX2 active layer is plotted with the green line in Figure 11.

The visible maxima and minima are the result of light interference in the photovoltaic cell. Strong absorption of radiation occurs for ~350 nm and in the spectral range from ~400 nm to ~650 nm. The Al electrode is responsible for the absorption of radiation above this range.

The efficiency of the photovoltaic cell is determined by the generation of excitons in the active layer, and the quantity characterizing the efficiency of this phenomenon is the External Quantum Efficiency (EQE), which is drawn with a blue line in Figure 11. In the calculations, it was assumed that each of the photons absorbed in the layer excited an exciton. This characteristic defines the upper limit of the conversion efficiency of optical radiation into electric current in a photovoltaic cell with an active PQX2 layer. The maximum value of the calculated EQE is 0.900, which is achieved at a wavelength of ~536 nm, while for the same wavelength, the experimental EQE is 0.035. The spectral characteristics of the short circuit current density J_sc(λ) are plotted with a red line. For the calculations of this characteristic, the calculated characteristics of the EQE and the characteristics of the AM1.5 solar radiation photon density flux were taken. As can be seen, the current J_sc(λ) reaches a maximum value of ~0.055 mA·cm⁻²·nm⁻¹ for a wavelength of ~600 nm. This characteristic sets the upper theoretical limit of short circuit current density J_sc(λ). By integrating J_sc(λ) with respect to the wavelength λ, the maximum short circuit current density J_{sc_max} = 11.53 mA·cm⁻² was calculated. The calculated maximum theoretical efficiency of the analyzed solar cell is η = 20.6%. The experimental value of the short circuit current density J_sc for the PQX2 photovoltaic cell is 19.72 μA·cm⁻² when illuminated with light with a power density of 1.3 mW·cm⁻². It is easy to calculate that for the power density of the incident radiation of 100 mW·cm⁻² (AM1.5), the short circuit current density J_sc will be 1.52 mA·cm⁻², i.e., 7.6 times less than the maximum theoretical value. This is because not all charges resulting from exciton dissociation are collected by the electrodes.

High theoretical values of short circuit current density J_sc and efficiency η indicate the attractiveness of the PQX active layers presented in the paper. On the other hand, the lower value of the short circuit current density than the theoretical values, as well as the low efficiency of the produced solar cells, indicate the need for further optimization, including both their production technology and structure.

4. Conclusions

Four photovoltaic structures of the bulk heterojunction type were fabricated, in which positional isomers of N,N-diethylamine-3-Methyl-1-Phenyl-1H-pyrazolo[3,4-b]quinoxalines were used in the active layer. The compounds differed in the position of the electron-donating substituent in the carbocyclic ring. The UV-Vis absorption of four isomers in solvents of different polarities and in the form of thin films was measured. In the measurements in solution, the PQX2 isomer has the most shifted absorption band (λmax = 515 nm in MCHX), while none of the tested compounds showed the solvatochromic effect. All compounds showed a shift of the absorption bands in the measurements in the form of thin films towards longer wavelengths (the largest shift for PQX4 by 41 nm). The values of HOMO-LUMO levels were determined by means of quantum chemical calculations using the HyperChem software. The highest value of the band gap width was obtained for the PQX4 isomer (E_g = 3.00 eV). The produced OPV with the architecture: glass/ITO/PEDOT:PSS/P3OT + PQX/Al, current–voltage characteristics were determined. The best photovoltaic parameters were obtained for OPV_4 (η = 0.37 %, J_SC = 28.44 μA/cm², V_OC = 0.80 V, FF = 0.21). The spectral characteristics of the refractive indices and extinction coefficients of the active layers were determined using the spectroscopic ellipsometry method. The determined extinction coefficients have similar values for all isomers (for PQX2 κ_max = 0.657 (λ = 547 nm)). These results were used in the analysis and optimization of photovoltaic structures and determining the maximal possible efficiency in the conversion of optical radiation. It was shown that the location of the N,N-diethylamine substituent affects the photophysical properties of the structure and optimization of the OPV_2 photovoltaic structure with the use of the coherent model using the 2 × 2 matrix method can be a method successfully used in modeling optical multilayer structures, including photovoltaic structures.