Numerical Study of Plasmonic Effects of Ag Nanoparticles Embedded in the Active Layer on Performance Polymer Organic Solar Cells

In this paper, the light absorption in the active layer of polymer solar cells (OPV) by using plasmonic nanocrystals with a hexagonal lattice structure is investigated. To study the relationship between the performance of the OPV solar cell and its active layer, a three-dimensional model of its morphology is utilized. Therefore, the three-dimensional (3D) finite-difference time-domain method and Lumerical software were used to measure the field distribution and light absorption in the active layer in terms of wavelength. OPV solar cells with bilayer and bulk heterojunction structured cells were designed using hexagonal lattice crystals with plasmonic nanoparticles, as well as core–shell geometry to govern a design to optimize light trapping in the active layer. The parameters of shape, material, periodicity, size, and the thickness of the active layer as a function of wavelength in OPV solar cells have been investigated. A very thin active layer and an ultra-thin shell were used to achieve the highest increase in optical absorption. The strong alternating electromagnetic field around the core–shell plasmonic nanoparticles resulting from the localized surface plasmon resonance (LSPR) suggested by the Ag plasmonic nanocrystals increased the intrinsic optical absorption in the active layer poly(3-hexylthiophene):phenyl-C61-butyric acid methyl ester (P3HT:PCBM). Based on the photovoltaic results, the short circuit current ranged from 19.7 to 26.7 mA/cm2.


Introduction
In the last decade, organic photovoltaic cells (OPVs) have attracted great attention due to their promising potential for next-generation solar cells because they are lightweight, flexible, and low-cost, and for their semi-transparency, solution process ability, and large surface area [1]. Therefore, OPV solar cells are capable of alternating with conventional photovoltaic devices, since they reduce the cost of energy consumption. Photovoltaic materials are fabricated in thin-film form to decrease the production cost. Various approaches have been employed to enhance light harvesting in films. Nevertheless, the absorption of light, especially at a long wavelength, is rather low in the OPV solar cells, since they have a thin active layer thickness and indirect bandgap of polymers. Consequently, this prevents the cell from absorbing low-energy photons in the solar energy spectrum [1,2]. Several light trapping and photon management methods have been suggested to improve the performance of OPV solar cells [3,4], to improve the collection efficiency of incident light inside thin films such as antireflection coating [5,6], photonic crystals [7,8], plasmonic excitation using metallic nanoparticles [3,4,9], optimization of the electromagnetic field distribution and periodic metallic grating [1,10,11], which are traditionally based on geometrical optics. The efficiency of these polymer cells is low compared to other cells [1], and this is a significant barrier to widespread commercialization of them.
The efficiency of OPV solar cells has been increased by more than 13-14% [6,12]. In OPV materials, owing to their short diffusion length in organic semiconductor materials [13], while fairly low carrier mobility (10 4 cm 2 /V s ), as well as the photoactive layer is generally rather thin, less than 100 nm to facilitate charge carrier diffusion and light extraction [12] is achieved. Even though much photon energy is lost because of the poor absorption of the incident light [13], numerous research has been done to further light absorption in the OPVs.
Among these approaches, the ideas of plasmonic metallic nanoparticles and gratings play an important role in absorption enhancement [14][15][16]. Different kinds of light scattering methods, from random to periodic plasmonic nanocrystals, at the front and/or backside of the active layer, have been utilized in which light is diffracted, refracted, or reflected back to increase the total path length of incident light inside the active layer. However, placing metallic plasmonic nanocrystals on top of the active layer would block a huge amount of total incident solar power. Currently, using multiple plasmonic nanocrystals to enhance absorption, on top and/or at the bottom of the active layer, has attracted much attention because of the broadband absorption enhancement, as well as less sensitivity to the angle of incident light [17,18]. Metallic nanoparticles have been exploited to increase the photocurrent as well as efficiency by 33% of thin-film silicon solar cells [19]. The advantages of metallic nanoparticles in OPVs have been investigated since they can be used in the device architecture without any significant effort.
Localized surface plasmons (LSP), which are collective oscillations of free electrons in metallic nanoparticles in response to an incident field [3,5,[20][21][22], are a type of plasmon, have extensively attracted researchers' attention for different field-enhanced spectroscopy, photovoltaic, and sensing applications [9,12]. In regulation, light excitation, a resonance, which is named Plasmonic, happens as soon as the frequency of the incident photons reaches the frequency of collective oscillation of conduction electrons of metallic particles, and can interact strongly with external electromagnetic waves [23]. Generation rate to recombination rate ratio for PSC with and without nanoparticles in their active layer is shown the improvement of the cell absorption with the incorporation of nanoparticles [3]. The influence of different nanoparticles incorporated in active layer of PSC yield the highest PCE and the current density-voltage measurements as well as high efficiency [4]. The semiconductor nanoparticles around the stacked plasmonic Ag nanowires enhanced the photocatalytic activities over the visible range through synergistic effects between the strong localized surface plasmon resonance excitation and the efficient hot electron transfer at the interface of them [14,15]. Therefore, the temperature polarization at this interface is lower than the bulk temperature and, as a result, the net driving force to the mass transport decreases [15,16]. The electron inelastic scattering is generally a utilized tool (HREELS) to determine surface electronic excitations. The measured data cover a wide energy range from the infrared (phonons; vibrations of atoms or molecules) to the visible-ultraviolet (plasmonic excitations; inter-band transitions) [20][21][22].
Currently, surface plasmon has become a significant light trapping technique in solar cells [24]. Owing to the near-field nature of surface plasmons, they are essentially appropriate for application in thin-film devices. Recently, photonic crystals have also been applied to OPV devices, the same as inorganic solar cells [25,26]. However, this method will be restricted by the relatively low refractive indices of the organic materials, since the proper guided modes in photonic crystal devices can be achieved by high index contrast as well as a reasonably thick layer [23][24][25]. Hence, plasmonic metallic nanoparticles are more appropriate for application in OPV devices [27]. Metallic nanoparticles could enhance scattering and local field from the excitation of localized surface plasmon resonances to improve the OPV performance. Patterned electrodes could enhance local fields as well as optical path length by coupling into the surface plasmon to improve the OPV performance. Recently, many papers on plasmonic nanoparticles have been published with applications in OPV technology. The objective of our paper is to use localized plasmon excitation of metallic nanoparticles to trap or confine light inside the photoactive material layer, or to acquire the resonance of the internal light scattering on these plasmonic metallic nanoparticles. This excitation of localized plasmons depends on several parameters, including size, material, the surface density of the metallic nanoparticles, as well as wavelength, polarization, and incident angle of light. Finally, light absorption could be enhanced, especially in the desired spectral ranges, where the photovoltaic material absorbs light weakly. For small periods less than 300 nm, the reason for the improvement in light is mostly LSP [23,26]. The large grating periods could scatter light into surface plasmons [26,28,29].
The report on performance enhancement through the combination of gold plasmonic nanoparticles in the poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT-PSS) active layer has been presented by Chen et al. [30] and a 20% improvement has been observed. Ion et al. achieved an enhancement in photocurrent due to LSPR generation by embedding the layer of Ag nanosphere between PEDOT-PSS and the active layer P3HT-PCBM of the OPV solar cell [31]. Kulkarni et al. obtained three times more generation of charge carriers by inserting a thin layer of Ag material at the bottom of the active layer [32]. Zhu et al. reported on the 1.40 times absorption enhancement by plasmonic nanostructures in the hybrid active layer and optimizing the fabricated structure [28]. Bai et al. published the first reports on a combination of 2D nanotubes on OPV solar cells, resulting in effective polarization-independent absorption [33]. Diukman et al. showed an improvement in the external quantum efficiency of OPV solar cells after embedding Au nanoparticle arrays in the active layer [34]. Kekeli et al. reported an enhancement in intrinsic optical absorption in the active layer with and without the square lattice of the core-shell (Ag@SiO 2 ) by 24.7% [29].
The Au nanoparticles, which are on the order of 5-50 nm, act as antennas, accumulating incident light energy in localized surface plasmon modes with up to a 100-fold increase in electric field. The generation of a strong near-field distribution around the gold nanoparticles embedded in the active layer was simulated and gained an enhancement in light absorption [24]. These plasmonic nanostructures immersed in the active layer can improve the plasmon and exciton coupling, as well as the generation of hot excitons. This generation can enhance the free charges [35][36][37].
In this paper, to increase the efficiency of polymer solar cells using surface plasmon, factors such as the size and material of plasmonic nanocrystals and the effect of these factors on parameters such as absorption and intensity distribution of electromagnetic fields in the active layer of polymer solar cells are investigated.
In "Theoretical and Simulation Methods," the theoretical and modeling methods, as well as simulation layout, are studied in detail. In the "Results and Discussions," the optical effects of plasmonic nanoparticles in OPV solar cells are investigated, including various morphology sizes and materials of this plasmonic nanocrystal in the active layer of a polymer solar cell with bulk heterojunction. The effects of plasmonic nanocrystals on the efficiency of the OPV solar cell by trapping the incident light into the active layer are investigated. Tuning the morphology of a plasmonic nanostructure to achieve higher efficiency is one of the key issues in our research.

Metallic Nanoparticles
Metallic nanoparticles are investigated as plasmonic structures since the LSPR could be tuned by their physical properties such as size, material, and shape. These nanostructures can increase light absorption by cumulative forward scattering cross-section, as well as near-field light enhancement. When the size of the metallic nanoparticles is equal to or smaller than the incident light wavelength, a strong interaction happens between the electromagnetic field and the free conduction electrons in the metallic nanoparticles [38].
Surface plasmon resonance is the collective oscillations in the electron density at the metal surface due to their interaction with the incident light. The resonance happens when the light coupled to the electrons oscillates at the same wavelength, referred to as LSPR in nanostructures, and results in a strong enhancement of the localized electromagnetic field. Surface plasmons, which are coupled with photons, can excite a collective of conduction electrons to propagate along an interface between a metallic nanostructure and a dielectric d > 0 [38]. The LSP at the vicinity of the interface can be propagated along the metal surface until the energy dissipates. Metallic nanoparticles with a small amount of inter-band absorption. The dielectric function is defined by the Drude and Lorentz model, which elucidates the response of damped and free electrons to an applied electromagnetic field [39]: The coupling strength between photons and the LSPR is determined by the absorption and scattering cross-sections of the nanoparticles [40]. In Eq. 2, the Drude model for metal is used to describe the inter-band transition [41]. p is called the plasma frequency and is defined by where N e is the density of free electrons, e is the charge, m e is the mass of electrons, 0 is the dielectric constant in free space, γ is the attenuation constant. The second term in Eq. 2, the Lorentz, describes the inter-band transition for particles, such as holes in occupied or unoccupied states [38]. Electronic inter-band transitions significantly influence the surface plasmon resonance in metallic nanoparticles. As soon as the nanoparticle size is much smaller than the incident wavelength, the scattering ( C scat ) and the absorption ( C abs ) cross-sections of spherical nanoparticles can be defined as follows: where α is the polarizability of the spherical nanoparticle with radius a, m is the dielectric constant of the metal, d is the dielectric constant of the environment. If resonance occurs in the system when the following condition is met, the denominator of Eq. 5 tends to zero [41,42]. If m = −2 d , the polarization of the particle turns out to be considerable, and the frequency is the same as the SPR sp permitting light to interact over a bigger region than the cross-section of the nanoparticle. The scattering efficiency can be calculated using the following formula: Silver nanoparticles in the air have a scattering crosssection approximately ten times larger than larger particles, allowing them to fully absorb and scatter incident light with just 10% of metallic nanoparticles covering the substrate [41]. Therefore, the bigger the scattering cross-section, the more light trapping happens by a scattering effect, and this is more important than light absorption. This effect can be achieved and is suitable for solar cell devices. In addition, the shape, material of the particles, and surrounding medium contributed to enhancing the scattering effect [12,[43][44][45].
Metal nanoparticles directly increase the absorption in solar cells, which occurs through coupling to near fields [41]. In addition, the absorption in the near field of metal nanoparticles is about hundreds or thousands of times higher than in the surrounding field. The near field strength around nanoparticles depends on the following factors [46]: the stored energy in plasmonic modes, the sharpness and edge of the nanostructure; hence, the sharper the edges in the nanostructure, the higher the near field strength around it. The stored energy also depends approximately on the cross-sectional area of the absorption on quasi-static particles (where the size of the particle is much smaller than the incident photon wavelength) [47]. Generally, metallic nanoparticles are made of silver, aluminum, gold, or copper due to their strong light. SPR of aluminum and silver occurs in the ultraviolet region, while SPR of gold and copper happens in the visible region [42]. Mostly, silver nanoparticles for their low cost as well as low absorption are employed in OPV, although gold nanoparticles have recently been largely used due to their lack of oxidation effects.
Although the use of nanoparticles in solar cells increases the absorption of photons in the cell, mostly, a semiconductor is coated on the nanoparticles to protect them from corrosion and recombination. Semiconductors such as TiO 2 , SiO 2 , and Al 2 O 3 coated with gold and silver nanoparticles as a protective layer are employed in solar cells [48]. Altering the thickness of the coating layer and the dielectric of the surrounding environment ( d ) and the dielectric constant of the coating layer ( c ) are influenced by resonance. Therefore, as the thickness of the coating layer increases, the resonance shifts toward higher wavelengths. Consequently, the longer the wavelength, the greater the number of photons and the higher the intensity can be achieved. The wavelength intensity and position of the SPR depend on the dielectric function of the metal and is expressed by Eq. 2 [38,41].

The Finite Difference Time Domain Method
In this method, the understudied structure and interacting fields are defined by discretizing the simulation space and allocating of optical properties to cells of the mesh. The technique was introduced in 1966 by Yee [49] but did not receive much attention until the late 1980s owing to a lack of access to computers with adequate speed and memory.
With advances in the production of personal computers, this technique has come to prominence for studying the propagation and interaction of electromagnetic waves and matter in many branches of science, including physics, electronics, communications, and photonics [50].
In the standard FDTD method, simulation space is divided into identical cubic cells as shown in Fig. 1. The positions of components of the electric and magnetic fields on each cell are alternately determined in a way that each component of the electric field is enclosed by components of the magnetic field and vice versa. By discretizing Maxwell's equations through applying finite-difference to spatial and time components of the field, six updating equations are achieved for six components of electric and magnetic fields.
The structure is discretized with a non-uniform grid mesh, which is subdivided into rectangular unit cells. The H-and E-field fields are shown in time steps, where during each time step they are iteratively updated. To avoid reflections at the domain boundary, perfectly matched layers (PML) are applied. A three-dimensional non-uniform lattice is introduced as follows. The vertices of the lattice are defined by one-dimensional coordinates: The cell and edge centers in the non-uniform space are introduced as: The E-and H-fields in the non-uniform grid are denoted as follows: This algorithm is based on a discretization of Maxwell's equations in their integral form [23]. The surface integrals are performed over a lattice cell face, and the contour integral is performed over the edges of the bounding face. The equation over cell faces using the discrete field approximations in Eq. (9) and evaluating the time derivatives using central difference approximations leads to: where h x i = Δx i + Δx i−1 ∕2;i = 2, N x , and the other component is introduced in the same manner. i,j,k+1∕2 , i,j,k+1∕2 and i+1∕2,j+1∕2 ,k are the averaged permittivity, conductivity, and permeability about grid edges, respectively. Similar updates for other components are easily derived. In particular, for three-dimensional simulation, this process is very time-consuming and requires a lot of computer memory resources. To acquire good stability between accuracy and computational resources, 10 grid points per lattice constant "a" are utilized. Several detectors are used to record the electric and magnetic fields of light. A commercial 3D-FDTD code is used (Lumerical) to enable us to free position, move, and tilt the structure. The power flow is calculated as follows:

OPV Solar Cells
Organic solar cells, which are called organic photovoltaics (OPV), have been considerably used by industry and researchers in the recent past [51][52][53][54]. The five critical procedures for an effective OPV are summarized as follows: (1) Light absorption goes together with exciton creation, (2) diffusion of the excitons through the active layer interface, (3) the dissociation and separation of charge, (4) transporting the separated charges, (5) charge gathering (Fig. 2) [55].
Carbon-based organic compounds or polymers are commonly used as active semiconducting materials in organic or polymer solar cells, which can convert solar energy into electric energy. Three types of OPV are categorized as single-layered, bilayer, and bulk heterojunction structured cells. The first group of OPVs is mainly made of single organic layers inserted between two metal electrodes [56]. Since the tremendously low charge separation is caused by the presence of tightly bound excitons in organic semiconductors, these layers, such as p-type or n-type sandwiches between electrodes, are non-capable of appropriately creating charges in the single-layer structure [57]. Kippelen B et al. reported single-layer OPVs with efficiencies as high as 5.9% recently [58].
In bilayer/multilayer OPVs, the active layers containing donor and acceptor organic semiconductors absorb light and, as a result, generate photocurrents. The donor material is capable of donating electrons and resultant holes, while the acceptor materials can accept electrons. These active layers collect photons from light to form excitons, which produce a concentration gradient, which diffuses them towards the donor or acceptor interface and separates them into free holes and electrons. Therefore, free electrons and holes transfer to the electrodes, resulting in photovoltaic. In a bilayer heterojunction device, p and n-type materials are consecutively placed next to each other [57].
The third one, bulk heterojunction, is arranged by bulk volume mixing of donor and acceptor components, which results in very slight exciton diffusion length at their interface [56]. Excitons are formed as soon as photons are absorbed by organic materials. Finally, separation of excitons bounces charge gathering at the heterojunction interface at the electrodes, resulting in photovoltaic [57].
Although inorganic semiconductors have a suitable bandgap, the main problem with them is that they possess less light absorptivity than organic materials [58]. Therefore, inorganic semiconductors require a bigger absorbing layer, producing thicker devices. Moreover, one of the vital features of inorganic semiconductors is purity in attaining higher performance, which increases the cost of production. The inorganic semiconductors, alternatively, have a lower energy bandgap for the creation of excitons to produce charge carriers. As a result, for fabricating OPVs, we should attempt better electron and hole mobility, which will result in higher performance. Figure 3 depicts the schematic structure of an OPV solar cell, where a hexagonal periodic array of nanoparticles is embedded inside the P3HT:PCBM active layer on the top of the aluminum. The PML boundary conditions on the bottom of the aluminum layer were used to avoid reflections in the stack direction. To create multiple scattering and cross-linking between adjacent nanoparticles, the boundary conditions along the X and Y directions were considered periodically. The simulation mesh size was fixed at 0.5 nm in the region encircling the nanoparticles. The plane wave was illuminated at normal incidence along the z-direction to propagate through the structure, with a wavelength range between 300 and 900 nm. The OPV solar cell is designed with three thin layers, which include an aluminum substrate as a cathode with a thickness of 100 nm, the active layer of polytrihexyl thiophene (P3HT:PCBM) with a thickness of 40 nm, a highly conductive polymer layer, and the polyethylene dioxide thiophene:polysulfonate (PEDOT:PSS) as the anode with a thickness of 30 nm, and plasmonic metal nanospheres, which were used as a hexagonal lattice. The refractive index (n for the real part and refractive index k for the imaginary part) PEDOT:PCBM and P3HT:PCBM are taken from Abass [59]. To enhance the performance of OPV solar cells by improving light absorption, plasmonic nanocrystals were employed as a hexagonal lattice inside only the active P3HT:PCBM layer (as shown in Fig. 3). Intrinsic adsorption inside the active layer is calculated without silver material absorption on plasmonic nanocrystals. Furthermore, the increase in optical absorption in multilayer cells can be evaluated at high efficiencies.

Model Configuration
For this calculation, the dissipated power L (λ) into the volume (x, y, z) of material from the Poynting vector П (x, y, z, λ) is expressed as the following [23]: The intrinsic absorption A (λ) in each layer is governed by the ratio of the L (λ) in each volume to the total power of the incident light source P Source (λ) as follows (14) [60]: In the wavelength range of 300 to 900 nm, the optical absorption enhancement A E is calculated as follows (15) [61]:  where, Planck's constant and c is the speed of light in a vacuum, while I(λ) denotes the AM 1.5 G solar spectral irradiance. A (λ) and A without NSs (λ) are the intrinsic optical absorption (inside the active layer) with and without nanospheres, respectively. The dissipated power (D·p) via the active layer of the plasmonic nanocrystal of intrinsic absorption is defined by the AM 1.5 G illumination of light in the wavelength range of 300 to 900 nm [62].
The square of the electric field integration over the volume is employed for the calculation of the absorption of the total energy of light by the substrate.
This denotes the imaginary part of the dielectric function of the substrate with and without metallic nanoparticles. The ratio of light absorbed with and without metallic nanoparticles is defined as light absorption (efficiency).
Finally, there is the absorption enhancement. For the measurement of the induced short circuit current I SC in the cell (the presence of metallic nanoparticles), an integration over the solar spectrum is used: where F( ) represents the AM1.5 G solar light [63].

Results and Discussions
The metallic nanoparticles embedded in the OPV solar cell in three dimensions are simulated using Lumerical FDTD, and their plasmonic effects are analyzed as the model configuration is shown in Fig. 3 a and  Several parameters, such as material, alternating period, and size of Ag nanoparticles, the thickness of the polymer active layer, as well as the dielectric shell with different thicknesses around Ag nanoparticles were investigated in detail. The wavelength range from 300 to 900 nm is taken into account here.

Ag Nanoparticles Embedded Inside the P3HT:PCBM layer
The comparison of light absorption of OPV solar cells with and without Ag nanoparticles revealed that the Ag nanoparticles in a hexagonal array embedded inside the P3HT:PCBM layer increased the plasmonic effect on the light absorption of the OPV solar cell. Recent papers have shown that the LSP mode, which traps induced light, enhances the light absorption of plasmonic metallic nanoparticles, and the electric field inside the metallic nanoparticles is proportional to that outside [24]. Hence, the light absorbed inside the Ag nanoparticles traps most of the incident light and is absorbed by the active layer of the OPV solar cell. The peak of the increase in absorption due to plasmonic nanosphere crystals is observed at approximately 680 nm. Moreover, an additional spectral peak is observed at 620 nm. These peaks result from the LSPR between neighboring Ag nanoparticles in a hexagonal array, as is shown in Fig. 3. Figure 7 demonstrates the thickness dependence of the absorptance spectra. The increase in light absorption is due to the increase in the near field in the active P3HT:PCBM layer. The magnetic field distribution is shown in Fig. 4. The enhanced field between the plasmonic nanospheres is polarized in the direction of light.

Effect of Period on Absorption Enhancement
The OPV solar cells were simulated with periodic arrays of Ag nanoparticles (with a radius of 15 nm and a 25-55 nm period) embedded inside the P3HT:PCBM layer. The effect of plasmonic enhancement by altering the period of absorption is displayed in Fig. 6. By tuning the period and size of the Ag plasmonic nanocrystals, the SPR peak can be designed to optimize the absorption of the whole incident light in the active polymer layers. Figure 6 shows the absorption on the polymer active layer as a function of the Ag plasmonic nanocrystal period. This parameter displays the most effects on near-field enhancement in Ag plasmonic nanocrystals. When the period of the nanoparticles array is much larger than the size of the nanoparticles, the absorption factor decreases, since the field scattering of the adjacent nanoparticles at a further distance is weaker. As the period of the nanoparticle array reduced, the absorption factor enhanced, as long as it reached a maximum value in a suitable period. This maximum value occurs where the field scattering is very large, where the nanoparticles are coupled to the electromagnetic field. [29,64]. As it is clear in Fig. 6, the optimal period of Ag plasmonic nanocrystals was observed at around 25 to 30 nm.

Effect of Plasmonic Nanocrystal Size
The radius of the plasmonic nanocrystal is another significant parameter. As the radius of the Ag plasmonic nanocrystal increased, the absorption peak enhanced and shifted to higher wavelengths, as is shown in Fig. 7. It can be concluded that with growing the size of the plasmonic nanocrystal, the absorption as well as the scattering cross-section increased.

Effect Material Plasmonic Nanocrystals
The absorption factor in terms of wavelength for different plasmonic nanocrystals is shown in Fig. 8. A hexagonal array of plasmonic nanocrystals with different materials such as silver, gold, aluminum, and copper embedded inside the P3HT:PCBM active layer investigated. The results revealed that silver is the best material with higher absorption [65]. An interface Schottky barrier is formed between Ag, Au, and others, since the nanoparticles are adjacent to each other. Its depth depends on the metal sizes, work functions, and, accordingly, the induced electron transfer, as well as the efficiency, would be higher. The separation of excited charges in the presence of an electric field improved the device's efficiency for solar energy applications. The work function 5.1, 4.26, 4.67, and 4.18 eV for Au, Ag, Cu, and Al, respectively. As a result, Au with higher work function is more active, while their morphology, the incident light intensity, as well as excitation wavelength influence electron transfer. As long as the excitation is shifted to a higher wavelength, the steady-state polarization is increased, while the Fermi levels are shifted to reach the conduction band of their embedded medium to enhance charge transport. The size of the metallic nanoparticles shifted the Fermi level upward or downward. Since nanoparticles are well attached to the surface of the embedded medium in comparison to large particles, the Fermi level is shifted from a more negative level, which could be tuned to an appropriate location by choosing and controlling the actual material, shape, as well as size of the metallic nanoparticles.

Effect of the Polymer Active Layer Thickness on Absorption Enhancement
The absorption of plasmonic nanocrystals in the P3HT:PCBM active layer is simulated as a function of the thickness of the active layer, as shown in Fig. 9. The P3HT:PCBM layer usually provides better hole transport from the active layer to the anode. This active layer is presented in the OPV solar cell with a thickness of 40 nm. The thickness variation of the P3HT:PCBM active layer from 20 to 70 nm was chosen, which has not observed substantial improvement from 40 to 70 nm in the absorption in the P3HT:PCBM active layer, as shown in Fig. 9. However, as the thickness of the P3HT:PCBM active layer increases from 20 to 70 nm, the amount of absorption increases. Because of the growing thickness of the active layer, the multiple peaks correspond to the LSPR inside or between neighboring holes. In addition, the resonances get weaker as thickness grows due to less light-matter interaction. Therefore, we expect that the sensitivity to the thickness variation in the optimum active layer is small.

Effect of The Hexagonal Plasmonic Nanocrystals Shell Thickness on Absorption Enhancement
Finally, the increase in light absorption is investigated by using plasmonic nanocrystals of silver nanospheres with dielectric shells. In some electron-hole recombination materials from plasmonic nanocrystals in application systems, they may significantly reduce conversion efficiency by LSPR, another reason for the use of nanoshells in applications to prevent plasmonic nanoparticle corrosion. To solve this problem, nanospheres are separated from the active polymer layer by dielectric shells. Dielectric shells with different refractive indices and thicknesses also change the location of the SPR peak [66]. Figure 8 shows the absorption factor in the P3HT:PCBM active layer as a function of shell thickness using two dielectric shells made of SiO 2 and Al 2 O 3 . The absorption factor decreases with increasing thickness of both shells. This decrease is more severe for SiO 2 than for Al 2 O 3 . This is because the change in shorter wavelengths for silicon dioxide is larger than for Al 2 O 3 , and this is because SiO 2 has a smaller refractive index.
As a result, the surface plasmon resonance improves the enhancement of the scattering and absorption cross-sections of the metallic nanoparticles. Meanwhile, scattering and absorption procedures are happening, through radiative and non-radiative plasmon decay processes, respectively. Their decay rate ratio depends on the nanoparticle morphology of nanoparticles. The radiative plasmon decay originates the scattering, and it is more common in Ag nanoparticles with larger sizes. The non-radiative plasmon decay originates from the absorption, which leads to the robust absorption of light and happens via intra-band excitations within the conduction band or inter-band excitations below the conduction band. It is more common in smaller sizes of Ag nanoparticles. The robust absorption of light leads to more electron-hole pair production, which enhances the light absorption as well as the efficiency of the OPV solar cell.

Photovoltaic Performance
As discussed later, absorption enhancement is mostly attained in the short wavelength range using hexagonal Ag plasmonic nanocrystals embedded inside the P3HT:PCBM active layer in Fig. 2. Although Ag plasmonic nanocrystals influence the short wavelength region, the main enhancement occurs at long wavelengths. OPV solar cell fabrication using Ag plasmonic nanocrystals as an absorber in the active layer is analyzed by using the advantage of the density of state of Ag plasmonic nanocrystals by arranging them in the non-absorbing region at the weakly absorbing region. Table 1 summarizes the characteristics of the I-V curve of the OPV solar cell structure with and without Ag plasmonic

Conclusion
Light absorption enhancement by an Ag plasmonic nanocrystal hexagonal array embedded inside the P3HT:PCBM active layer at the bottom of the OPV solar cell was analyzed. FDTD simulations were performed to further investigate the influence of different geometry parameters on the absorption enhancement. Simulation results revealed that Ag plasmonic nanocrystals can significantly improve total absorption by exciting numerous optical resonances. The outcome of each active layer and nanoparticle morphology design on the absorption enhancement was debated. Light scattering by Ag plasmonic nanocrystals enhances light absorption in the long wavelength range and also generates a strong electromagnetic field. As proved in the simulation, plasmonic nanostructures with various morphology parameters, such as size and materials, display different behaviors in absorption enhancement. For instance, a different range of Ag nanoparticles embedded in the P3HT:PCBM active layer, using Lumerical FDTD, makes it possible to estimate the amount of light scattered into the P3HT:PCBM active layer for various materials and sizes of the Ag nanoparticles. It was seen that the scattered light scattered inside the active layer was influenced by the size, the material, and the embedded environment of the Ag nanoparticles. Finally, the resonant plasmon excitation in an OPV solar cell is to exploit the strong local field enhancement around the Ag nanoparticles, which improves the light absorption in the P3HT:PCBM active layer. In conclusion, we have revealed that the I SC , consequently, efficiency, was enhanced due to light scattering, strong absorption, and the resonant plasmon excitation of Ag plasmonic nanocrystals. The clear enhancement in the I-V curve is attained with the Ag plasmonic nanocrystals embedded in the active layer of the OPV solar cells.