1. Introduction

There is a global need to generate new renewable and environmentally friendly energy sources. Photovoltaic technologies are one of the main candidates are believed to solve the energy crisis and reduce the environmental problems of consuming fossil fuels. Among the different Photovoltaic technologies, organic solar cells (OSCs) also known as polymer solar cells which are based on organic polymers and low molecular weight molecules, offer several advantages compared to conventional silicon-based devices. An organic semiconductor molecule is composed of an aggregation of atoms held together by conjugated -bonds, while the molecular bonding force is weak van der Waal’s force. This bonding structure of organic semiconductors gives OSCs unique superiority such as flexibility, lightweight, easy processing, and low manufacturing cost due to low sublimation point [1]. In addition to these advantages, OSCs, because of having abundant material sources, Low environmental impact during production, and semi-transparency characteristics [2], have drawn much attention as an effective technology that can be utilized for low-cost energy harvesting devices. However, compare to their inorganic counterpart, they have serious barriers to widespread commercialization. Inferior device stability, low power conversion efficiency (PCE) (8∼19%), and fabrication techniques that must be compatible with the roll-to-roll (R2R) production are three challenges that significant work is required to improve these deficiencies. Stability studies have reported an operational lifetime of up to 30 years under certain conditions [3]. PCE is of paramount importance. OSCs have significantly low PCE due to an intrinsic limitation that restricts the thickness of the active layer to less than 100 nm [4], [5]. Exciton diffusion length is the distance that exciton travels before recombination. Organic semiconductors have short exciton diffusion lengths and low carriers’ mobility. In other words, the probability of exciton recombination before reaching the Anode/cathode increase drastically as the thickness of the absorber increase [6]. To overcome this issue, several light trapping methods have been used to increase the light-harvesting efficiency in the thin absorber layer (without increasing its thickness). These methods including antireflection coatings [7,8], photonic crystals (PCs) [9], and metallic nanostructures [10]. Utilizing metallic nanoparticles allows incident light in the active layer to have a longer optical path due to the Light-scattering characteristics of metal nanoparticles [11]. Moreover, the excitation of surface plasmons (SPs) near the metal-dielectric interface enhances the electromagnetic field near the interface. The enhanced electric field in contribution with the electric field enhanced by the effect of the photonic cavity and Bloch modes, led to increased absorption efficiency. In addition to fabrication problems, the presence of metallic nano-particles (MNPs) in the active layer should encounter the possibility of exciton quenching effect [4,12].

One of the most promising methods which is of interest here uses metal nano-gratings to manipulate light based on surface plasmon polaritons (SPPs) by which the light is being confined to subwavelength scale and leads to enhancement of the intensity of the electromagnetic field. Khai Q. Le and coworkers [13] investigated that adding square silver gratings results in broadband absorption enhancement of up to 23.4%. However, the commercialization of OSCs needs large-area printing technologies that usually lead to a significant film thickness fluctuation, and the proposed model was very susceptible. Xin Liu and coworkers [4] presented A silver grating as the back electrode, which has three different depths grooves in a period. They investigated that the multiple resonances of three groove structures results in the broadband absorption spectrum and relatively large integrated absorption efficiency under TM polarization between 350 nm to 900 nm with AM 1.5G solar spectrum. Using the two-dimensional finite-difference time-domain (2D-FDTD) method, they have reported 57.4% integrated absorption efficiency, which showed an enhancement of 13.4% with respect to the equivalent planar device (127 nm thickness of the active layer). Although we believe That the reported enhancement of absorption efficiency has lesser amount in comparison with its equivalent planar, it offers new insight into the dimensions of grooves and their influences on the different surface plasmons excitation modes. Other drawbacks of previously reported work, which decreases the quantum efficiency of the cell arises from the depth of grooves. The probability of holes created in the depth of more than tens of nanometer to travel this distance safely to the Anode decreases as the thickness of the photoactive layer increases. The conjugated polymers have a short lifetime of excitons, so as a result, the diffusion lengths are limited to less than tens of nanometers [6], [14]. If the material has low carrier mobility, electrons and holes cannot overcome the Coulomb potential and remain bounded. Finally, these bounded excitons recombine before charge collection. Although the electric field intensity causes more generation of excitons, but as long as these excitons are not able to disassociate and pass through the active material without being recombined, they won't participate in energy transfer. Free charge transportation towards electrodes occurs not only by the carrier diffusion but also by the internal electric field due to the Fermi level difference of the electrodes(drift) [6]. According to what was said, thickness of the photoactive layer is a critical factor of efficient charge generation. However, integrated absorption efficiency was extracted directly from electric field calculation and didn't consider Exciton diffusion, charge dissociation, and recombination. So, only the Absorption efficiency won't be an excellent criterion to characterize the organic solar cells. In addition to increasing the absorption efficiency, we should pay attention to the recombination of charges to avoid the generated electron-hole pairs from being wasted. The well-known figure of merit to evaluate the performance of solar cells is EQE or its equivalent, incident photon conversion efficiency (IPCE) measurement. The IPCE represents the percentage of incident photons that are converted to charge carriers and are finally transported from the interface of the donor/acceptor and collected at the electrodes under short circuit conditions [14]. IPCE is determined by the following equation:

Here we increase the first term (by increasing absorption efficiency) while increasing the production of second and third terms by decreasing the thickness of photoactive layer. Thus, we ensure that the IPCE would be increased. To calculate the exact amount of IPCE, the experimental data is needed. However, this study is performed theoretically, and we are trying to find a solution to enhance the absorption efficiency while considering the thickness to keep IPCE high. As a further study, IPCE can be measured by experimental methods.

In this work, the plasmonic modes in a stopped trench metal grating in an OCS with a coating layer are studied using the 3D-FDTD methods. We believe that because of existing symmetric surface plasmon mode corresponds to the transverse component E_z which hereafter is called the tangential component of plasmon polaritons [16], two-dimensional analyses will not be sufficient. Here we investigated that the stopped-trench(groove) structure shows better absorption efficiency than the thorough-trenched (groove) structure. Another advantage of the proposed structure is that the concentration of most of the generated excitons near the cathode boundaries allows us to choose thin layers of the active layer with reasonable absorption efficiency.

2. Structure design and simulation methodology

Figure 1 demonstrates the three-dimensional schematic of the proposed stopped-trench grating with a coating layer (STGC) structure. Silver is used as back-reflecting cathode with a thickness of 500 nm. The optical constants of silver were taken from [17]. The PTB7:PC70BM with 1:1.5 weight ratio–a conjugated polymer- was used for the active absorber material and its wavelength-dependent complex optical constants are extracted from experimental previously reported data [4,18]. The extinction coefficient of PTB7:PC70BM has a maximum at the wavelength of around 670 nm, which correlates with the transition wavelength corresponding to the energy band gap of PTB7(1.84 eV) [19]. The thickness of the homogenous layer denoted by t₁ and p, w, d, and L are period, width, depth, and length of the groove, respectively. For the anode layer Indium-tin oxide (ITO) is used (denoted by t₃), and PEDOT: PSS is used for the hole transport layer. However, Molybdenum trioxide (MoO₃)- a metal oxide material – can be used due to its advantages over degradation problems [6]. Because of the similarity of results for both materials, we proceed by the conventional. The optical constants of Molybdenum trioxide, PEDOT: PSS, and Indium-tin oxide were taken from [20], [21],and [22], respectively. Figure 2(a) shows the refractive indexes of PTB7:PC70BM in green dashed line, and PEDOT: PSS in red dashed line.

 

Fig. 1. Schematic diagram of an organic solar cell with stopped-trench grating with a coating layer (STGC).

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Fig. 2. (a) refractive indexes of PTB7:PC70BM (green dashes) and PEDOT: PSS (red dashes). (b) boundary conditions for normal incidence and absorption analysis group position for (STGC) structure.

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Here we use the finite-difference time-domain (FDTD) method for numerical simulations of the absorption and EM-field calculations. The commercial software Lumerical FDTD solution is used to simulate the electromagnetic field calculations. A plane wave incident light is used with the wavelength from 350 nm to 800 nm that incidence from the top with the TM polarization (magnetic field polarized along the z-axis). The dimension of the computational domain is chosen P_x P_y 1500 nm along x, z and, y directions, respectively (Fig. 2(b)). Perfect matched layers (PMLs) boundary conditions with 12 layers were used for the top and bottom of the structure. At normal incidence, due to periodic structure along ± x and ± z periodic boundary conditions for sides at the normal incidence and the Bloch boundary conditions for oblique incidence were adopted. A mesh size of 2 nm is used for the absorption analysis area, which is composed of the homogenous photoactive layer, photoactive layer inside the grating, and the ridges of the metallic grating. The absorption analysis group area, which shows the monitors position in the simulation process is demonstrated in Fig. 2(b) with the yellow dashed line. After calculating the power absorbed at the absorption analysis area, the power absorbed by the silver ridges area can be filtered to get the power absorbed by the photoactive material.

Firstly, the electric field at a particular wavelength is calculated then the absorption efficiency (A(λ)) in the active blend is calculated according to Eq.1.

Then the integrated absorption efficiency (A_total) is calculated using Eq. (2).

The wave impedance of media is the ratio of the transverse components of the electric and magnetic fields.

To clarify how the groove enhanced the absorption efficiency, firstly we investigated the effect of surface plasmon polariton (SPP) modes in the structure. In several nano-scale waveguide architectures, part of the incident light’s energy is stored as the plasma oscillations at the metal and dielectric interfaces. SPP modes, despite intrinsic losses, are capable of manipulating light, and transporting it up to a hundred wavelengths [23]. Owing to the geometry of the structure, cavity modes, wedge plasmon polaritons (WPPs), and Gap surface plasmon (GSPs) are the main modes here that affect the absorption efficiency. Although WPPs and GSPs operate on the same plasmonic phenomenon that results in the enhancement of the electric field near the metal-photoactive material interface, they show different behavior in terms of propagation direction, propagation distance, propagation depth, and confinement.

To find an equation for the above characteristics, the dispersion equation of SSP must be solved. There are two main approaches: The first one is to use Maxwell equations and boundary conditions. The second method is based on Fresnel's equation for transmission and reflection. Hocker and coworkers represented a semi-analytical Effective Index Method (EIM) to calculate mode dispersion in channel waveguides [24]. Numerous works have been represented to apply this method to nano-grooved structures [25], [26]. These results are reached under certain assumptions for waveguides, where the medium between the grooves was a vacuum or dielectric/non-absorbent materials. So, the imaginary part for the permittivity or refractive index of the medium has not been taken to account in the calculation. Nevertheless, all of them represent that inside the groove, the refractive index of SSP mode will be affected by the geometry of grooves. On the other hand, SPP characteristics and resonant coupling on thin silver layer have been reported in special cases of metal/vacuum and metal/dielectric structures where the thickness of dielectric tends to infinity [27]. Here we have a metal grating with absorbent material inside and upward the grooves as well as several layers above, which is investigated within a range of wavelength. So, finding an analytical equation to define dispersion and other characteristics of the SPPs becomes very complicated. Here we find the intensity of the field inside the structure numerically by the FDTD method. Then we study the influence of dimensional parameters on the propagation profile, wave impedance inside the model, and absorption efficiency.

3. Results and discussion

In order To evaluate our simulation, firstly, we evaluated the three adjacent multiple depth groove structure (TAMDGS) with the results of previous work [4]. The results showed a good agreement in between. Figure 3(a) demonstrates the comparison between to results. The red line shows our simulation results and the black one is the previous work results.

 

Fig. 3. (a) evaluation of TAMDGS with the previous work results [4]. (b) convergency with different mesh size.

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Here we studied a 3-dimensional single stopped-groove grating structure depicted in Fig. 1. the structure is illuminated by a plane wave source at TM polarization (propagating along Y, electric field along the x-direction, and the magnetic field polarized along the z-axis). The thickness of the anode and hole transport layer was chosen to be t₃ = 60 nm and t₂= 10 nm. Two-layer of SiO₂ and ITO as a simple coating layer is added to the structure with thicknesses of t₄= 50 nm and t₅= 15 nm, respectively. The homogenous active layer thickness was set to 80 nm. The other parameters of the grating, such as P_x, P_y, W, L, and d, will be studied at different lengths. Simulation parameters are summarized in Table 1.

Table 1. The important parameters of simulated structure

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Figure 3(b) shows the absorption efficiency of STGC structure with the above dimensions and simulation parameters for different mesh sizes. This figure shows the simulation results for a coarse mesh size of 10 nm with a green line, red line for the medium mesh size of 6 nm and black one for the fine mesh size of 2 nm. Therefore, the convergency is ensured for the meshes below 6 nm.

The absorption efficiency spectrum TAMDGS (previous work), stopped_ Trench grating structure without coating layer (STG) and the STGC is shown in Fig. 4(a) Similarly, the absorption efficiency spectrum of STGC and planar structure with (without) a coating layer is shown in Fig. 4(b) STG and STGC refer to stopped trench grating without a coating layer and stopped trench grating with coating layer respectively. PCL and planar refer to planar structure with a coating layer and planar structure without a coating layer, respectively. In this absorption spectrum, the STGC showed 77.01% integrated absorption efficiency, which is 19.6% better than previous work (TAMDGS), while 19% less photoactive material was used. This model also shows 18% integrated absorption efficiency, more than an equivalent planar structure (with an equivalent photoactive material thickness of 107 nm) without coating layer while the hole transport layer and the anode were set arbitrarily to 50 nm and 100 nm, respectively. Adding a coating layer for planar equivalent structure enhanced the integrated absorption efficiency by 7% (shown in Fig. 4(b)) Adding a groove to the homogenized photoactive layer (80 nm) has improved the integrated absorption efficiency by 14% over the investigated wavelength range (especially in the range from 700 to 750 nm), as shown in Fig. 4(a). As a result, adding both coating layer and stopped-groove structure is a promising way to design OSCs with broad bandwidth. Figure 4(a) shows unwanted absorption efficiency in the silver material for STGC structure which is considered as thermal loss in the system.

 

Fig. 4. (a) Integrated absorption efficiency of STGC, previous work (TAMDG), and stopped trench without coating (STG) (b) comparison of integrated absorption efficiency of stopped trench groove structure with the coating (STGC), planar with(without) coating layer.

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The WPPs and GSPs modes arise from sidewall and wedge interaction. They are maximal near the tips of the structure because near the narrow regions, the coupling of the electric charges is strong. WPPs mode couples strongly to nearby atoms due to very strong local field enhancements. If the two WPPs are closed enough, then they will interact with each other with near-field and far-field coupling between surface plasmon polaritons [28], [29]. Here, our unit cell can be modeled by the combination of WPPs and GSPs modes that depends on the phase matching of plasmon polaritons propagating along the surfaces. Figure 5(a) shows the electric field vectors in the STGC model at the wavelength of 360 nm. As expected for the planar model, The intensity of the electric field depends on the superposition of incident and reflected wave, which has the same values at the same depth due to the symmetry of the cell. In the STGC model, adding a groove has caused additional asymmetry and resulted in creation of important characteristics which are not present in the planar model. The amplitude of electric field vectors has been increased above the platform and around the wedges. The field intensity is resulted from the superposition of incident electric field, cavity mode, and electric field induced by plasmon polaritons. Because of variation of the effective refractive index inside and outside of the groove, the wavenumber and wave phase have different values inside the cell. To thoroughly investigate all the modes inside the cell, the electric field vectors and field intensity at the wavelengths of 530 nm, 770 nm, and 730 nm, at Z = 0 slice as shown in Fig. 5(b-d) respectively. Figure 5(e) shows the Z = 0 slice at the wavelength of 380 nm, where the electric field vectors were sampled and scaled for clarity. It can be found that the plasmonic modes inside the grating, are comprised of coupled wedge plasmon modes as well as coupled gap surface plasmon mode. The coupling of WPPs and GSPs, causes the enhancement of electric field near the wedges up to 16-fold of incident electric field. Since the absorption efficiency has a square relationship with the field intensity, thus increasing the field intensity will result in the enhancement of the absorption efficiency to the power of two for the same wavelength and material.

 

Fig. 5. (a) Electric field distribution at λ=360 nm in the STGC structure and (b)-(e) electric field vectors at z = 0 slice of STGC, at λ=530 nm, λ=770 nm, λ=730 nm, and λ=360 nm, respectively.

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It can be found that because of the surface plasmons propagation along the surface of the metal, charge distribution can be modeled with capacitors for each frequency. J. H. Kang and coworkers [30] have introduced the concept of capacitance for certain structures of plasmonic systems. However, In the solar cell concept, we deal with a spectrum of wavelengths. Although the concept of capacitance in a plasmonic configuration is not the same as conventional static capacitance, there are common basics that justify the enhancement of the induced electric field. Since the surface current which is induced by the light and flows into the edge region, has an alternating characteristic, the current responsible for charging the metal wedge is restricted within the lambda-zone (the volume that encloses the metal edge within a distance λ from the edge) [30]. Therefore, to study the dimensional parameters which influence the enhancement of the electric field, we restricted the investigation of dimensional parameters to the lambda-zone. Two-dimensional Analysis has been shown here, which is in line with three-dimensional results for optimizing platform width, w, and d parameters.

Firstly, the platform width (P_x - w), which is mainly responsible for the charges and the capacitance and has been investigated. The dimension of platform width is the paramount important factor that affects the intensity of charge current and consequently the electric field intensity around the wedge. Here we restricted the investigation of platform width to 900 nm. The wavenumber of plasmon polaritons is more than the wavenumber of incidence light, So the length in which the charge distribution affects the intensity of the electric field on the ridges would be even less than 900 nm. Figure 6(a) shows that extending the platform from 50 nm to 120 nm results in redshifted peak frequency with greater electric field intensity up to 20-fold in edges (d = 40 nm and w = 270 nm). Extending further shifts the peak wavelength near the cut of mode where the imaginary part of the refractive index has a negligible value and is not of interest here. It also showed that the intensity of electric field fell as we increased the platform width by more than 120 nm. Figure 6(b) demonstrates the effect of depth and width of the trench on the integrated absorption efficiency. It can be found that for depth between 40 nm to 60 nm and width of the trench between 200 to 270 nm, the ${\textrm{A}_{\textrm{total}}}$ has a maximum value of 79%. Given that we are seeking a model in which the thickness and volume of photoactive material are minimum, a structure with depth and width of 50 nm and 220 nm would be the best model to exploit the incident light. Although the ${\textrm{A}_{\textrm{total}}}$ slightly falls for the dimensions beyond the mentioned values, it still retains its high value for the tolerances of a few nanometers. We will discuss the influence of tolerances and defects of the printing process during fabrication on the absorption efficiency later on in this paper when discussing the geometry of the tip. Figure 6(c) compares the absorption efficiency of the structure with P_x= 390 nm, P_y= 390 nm, W_x= 270 nm, and d = 40 nm for different values of trench length (L). The results show that L = 270 nm would lead to better absorption efficiency in which the structure is fully symmetric to x = 0 and z = 0 planes. Figure 6(d) shows the electric field vectors in the photoactive layer (homogeneous layer plus the layer inside the trench) for the structure at the wavelength of 790 nm. The electric field at the edges of the structure reaches 16-fold of incident value due to the surface plasmon polariton's resonance and its maximum value near the tips. To understand the reason for having the best absorption efficiency at symmetric structure, we have to study deeper the surface plasmon polaritons characteristics.

 

Fig. 6. Electric field intensity at the tips of groove with different platform width (a). total absorption efficiency as a function of depth of trench (b) and width of a trench (c). Electric field vectors inside the active layer of STGC structure at the frequency of 790 nm.

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SPP modes in the Metal Insulator Metal (MIM) configuration, for sufficiently small gap widths, may couple with each other and form single gap surface plasmons, in which the normal field component, E_x, has an even symmetry while the tangential electric field component (E_z), shows odd symmetry. The y-direction component of electric fields of GSPs and CPPs is antisymmetric to the central plan. To distinguish between GSPs and Channel plasmon polaritons (CPPs), GSPs may either propagate normally to the planar surface or along the groove axis, while CPPs propagate only along the groove axis. Moreover, Plasmon's propagation direction-oriented electric field component, E_z, has the antisymmetric (or capacitor-like) charge distribution. For GSPs the difference in spacing of the gap has negligible effects on the field intensity [31]. here we have both GSPs and WPPs modes that have been hybridized in the structure. Similar to the characteristics of modes which are mentioned above, Fig. 7(a) demonstrates the structure when the light incidence and its electric field are along y and x-direction, respectively, at the wavelength of 360 nm. The E_x inside the unit cell has the even-symmetry profile to both the X = 0 and Z = 0 planes (The y-direction component of the electric field (E_y) shows an even-symmetry profile to the Z = 0 plane and an odd-symmetry profile to the X = 0 plane. E_y goes through zero along the X = 0 plane (Fig. 7(b)). The Z direction-oriented electric field, which is created due to the presence of WPPs and GSPs, shows an odd-symmetry profile to both of X = 0 and Z = 0 planes (as shown in Fig. 7(c)). E_z goes through zero along both Z = 0 and X = 0 planes.

 

Fig. 7. Electric field intensity at λ=380 nm, (a) E_x(a), (b) E_y, and (c) E_z, and (d) tangential electric field intensity (E_z) at λ=790 nm with a logarithmic scale.

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Our main observation here is that the adjacent walls, which are created by stopping the groove (trench), contribute to a strong resonance of the z-direction oriented electric field (E_z) and cause the enhancement of absorption efficiency in the unit cell (as shown in Fig. 6(d)). Figure 7(d) shows the y = -75 nm plane (depth of 75 nm from the top of the unit cell), which demonstrates the intensive amount of E_z created at λ=790 nm in the logarithmic scale. Regions near the sidewalls show E_z up to 16-fold. In addition to the mentioned characteristic of the stopped groove, it has another important characteristic that plays an important role in the absorption efficiency of OCSs. To clarify the role of the stopping groove (trench) on the symmetry of structure and its useful characteristic, let's compare the stopped trench with a trenched structure concerning the different polarization of incident light.

In order to compare the Thorough-trenched grating with the Stopped-trenched grating, the behavior of both structures with different polarization of incident light has been studied. Actually, when we use gratings, the polarization of incident light matters. We denote the angle of an incident electric field concerning the normal axis of the surface plane of walls by polarization angle (α) and the angle between the propagation light and the line normal to the cell's surface by incidence angle (θ)). Figure 8(a) shows the absorption efficiency of trench grating structures at different polarization angles, where the incident light is normal to the surface (θ = 0). At θ = 0° (the light is normal to surface). when the Electric field is perpendicular to the groove axis (α= 0°), ${\textrm{A}_{\textrm{total}}}$ has its maximum value due to the plasmon polaritons excitation. At this incidence angle, the incident light forms TE polarization concerning the bottom and top surfaces and TM polarization with the sidewalls. Where the electric field is no more perpendicular to the groove axis, ${\textrm{A}_{\textrm{total}}}$ starts to decrease. More α leads to more decrease in the absorption efficiency. α = 90° (where the electric field of incident light is parallel to the groove axis) causes the least amount of absorption efficiency, and the absorption efficiency drops at once, at the wavelength of λ = 700 nm (as shown in Fig. 8(a)), because the TE mode doesn’t support plasmonic modes and the absorption coefficient of the material drops near the cut of mode. In other words, there is no surface on which the polarization of incident light can be formed as a TM polarization. Both the walls and top and bottom surfaces are parallel to the electric field. But in the case of Stopped-trenched grating (STGC), when the incident light is normal to the surface of the cell (θ = 0°), all polarization angles between α = 0° to α = 90° has almost the same absorption efficiency profile (as shown in Fig. 8(b)). Actually, in stopped trench grating, due to the symmetry of the structure, for each value of α, both sidewall pairs will support the plasmon polaritons excitation due to the TM polarization of normal component of the electric field to the surfaces, and the results will be very similar for both the TM and TE polarized incident light. Both structures exhibit the best value of absorption efficiency when θ = 0°. As θ increases, the absorption efficiency starts to fall at both structures. Figure 8(b) shows that the absorption efficiency has slightly less amount when θ = 30°. Since the sunlight isn't polarized so, the average value of different polarization has to be considered to compare the efficiency of two structures. It is shown that comparing the average value of integrated absorption efficiency for both structures, the stopped trench grating (STG) has a better profile than Thorough-trenched grating (TG) for the different polarization of incident light.

 

Fig. 8. (a) Integrate absorption efficiency of TGC structure and STGC comparison and (b) integrate absorption efficiency of STGC structure at different incident light polarization angles and comparison with blunt structure.

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To investigate the influence of the tip’s geometry on the absorption efficiency, the tips and corners of the structure have been rounded with a curvature radius of 30 nm. Figure 8(b) shows the comparison of the blunted model with the previous sharp tips model. Although confinement properties of WPPs strongly depend on the sharpness of tips and making the tips blunt reduces the intensity of resonance at the resonance frequency, the blunt model exhibits a well-integrated absorption efficiency. These results may lead us to conclude that the capacitor model of charge distribution for plasmon polaritons which dominantly connect the electric field intensity to the dimension of the surfaces, matches the results better than the theory of geometry of tips (for WPPs). Since the fabrication of the SCs at the nanometer scale suffer certain limitations such as printing tolerances and difficulties in making sharp tips, the proposed model has to be less prone to the mentioned limitations. As discussed above, our STGC model shows strong stability in the imperfection of fabrication of sharp tips. Moreover, considering Fig. 6(b) and (c) shows that tolerances in printing for the width or depth of the trench will not affect the results drastically for integrated absorption efficiency. Thus, taking to account the fabrication limitations of sharp tips, the stopped trench gratings with a coating layer (STGC) is a promising way to produce OCSs with an extremely thin active layers and high absorption efficiency.

Fabrication of nano-gratings include (laser, electron-beam and focused ion beam) and pattern-transfer lithography. direct-write methods suffer some drawbacks like being expensive, time consuming and require specialized equipment. Nanoimprint lithography (NIL) was reported as an alternative to direct-write methods with low-cost, high throughput advantages. However, in order to transfer the pattern, the high-resolution mold is required. Moreover, The residual imprint layer can be left behind on the gratings. Inkjet printing can overcome some of these limitations. F. McGrath and his coworkers has reported a solution-based process, using NIL and metallic ink for fabrication of silver gratings. This approach, is energy efficient, cheaper, less time consuming, and does not require specialized equipment [32]. In addition to the mentioned method, there is other methods to fabricate metallic Nano-gratings. Wang, D., et al reported a study in which the metal grating trenches, claddings and upper deposit layers can be prepared by inductively coupled plasma (ICP) etching and plasma-enhanced chemical vapor deposition (PECVD) systems.

To further investigate, the overall generation rate as a function of dimension at the spectrum of 350 to 900 is depicted in Fig. 9(a) To have a better image of the generation rate inside the cell, the figure has been cut from the z = 0 plane. The generation rate was calculated by assuming that each absorbed photon generates one electron-hole pair. More than 95% of the active layer has the generation rate in the order of 1∼3e28 charge pairs per cubic meter per second. Interestingly less than 2% of the volume is responsible for up to 14.8% of the overall generated hole pares. (As shown in Fig. 9(b)). Our main idea in this article is that confining the light and specifying the hotspots on the structure helps us to manage and reduce the thickness and volume of the active layer by which both the recombination losses and cost would be controllable. The results of this work can be used to introduce organic solar cells with extremely reduced thickness and make full absorption efficiency in OCSs. Another advantage is the reduction of recombination losses related to the thickness of the active layer.

 

Fig. 9. (a) Generation rate of STGC (pair/cubic meter/second) (b). hotspots with a generation rate above 3 × 10²⁸.

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4. Physical description

Because of the far-field and near-field coupling of WPPs and GSPs, here we calculated the wave impedance along the electric field of incident light (x-direction), (Z_x). The surface plasmon polariton waves propagate with a low phase velocity and a high wave impedance between dielectrics and metal [25]. Figure 10(a) shows the real part of wave impedance (Z_x) for a planar structure at λ = 700 nm, which has an amplitude up to 5000 (Ω). STGC structure at λ=700 nm to 900 nm shows a layer of extremely high wave impedance (Z_x) up to 40,000(Ω) due to the nearfield coupling of wedges (Shown in Fig. 10(b)). This extremely high impedance layer affects the effective index of the material. It causes to great mismatch of refractive index (or impedance mismatch) between the layers at the top of the grating. The mismatch between the layers helps to the trapping the incident light ray at the top of the grating and slowing the incident and reflected wave.

 

Fig. 10. (a) The real part of wave impedance for a planar structure at λ=700 nm, and (b) areas with extremely high wave impedance (Zx).

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5. Conclusion

The influence of dimensional parameters of the trench grating with a coating layer on the absorption efficiency of metallic grating-based organic solar cells (OSCs) was evaluated. The numerical study was performed using the 3D-FDTD method under different polarization angles. We found that the platform width of grating has a great influence on the intensity of wedge plasmon polaritons (WPPs) and Gap surface plasmon (GSPs) due to the capacitance-like charge distribution in a plasmonic configuration. We have also investigated the effects of the variation of depth and length of the trench on the absorption efficiency. We have discussed the plasmonic modes and found that stopped trench grating would lead to better absorption efficiency for two main reasons. Results show that less than 2% of the volume is responsible for up to 14.8% of the overall generated hole pares. We believe by specifying the hotspots on the structure, we would be able to manage and reduce the thickness and volume of the active layer. The results show that although for the WPPs, the confinement and propagation length is strongly dependent on the geometry of the tip. It concluded that the stopped-trench gratings with a coating layer (STGC) is a promising way to produce OCSs with an extremely thin active layers and high absorption efficiency. Based on our observations, some OCSs with less active layers and almost full absorption efficiency can be designed.