Title Light harvesting improvement of organic solar cells with self-enhanced active layer designs

We present designs of organic solar cells (OSCs) incorporating periodically arranged gradient type active layer. The designs can enhance light harvesting with patterned organic materials themselves (i.e. selfenhanced active layer design) to avoid degrading electrical performances of OSCs in contrast to introducing inorganic concentrators into OSC active layers such as silicon and metallic nanostructures. Geometry of the OSC is fully optimized by rigorously solving Maxwell’s equations with fast and efficient scattering matrix method. Optical absorption is accessed by a volume integral of the active layer excluding the metallic absorption. Our numerical results show that the OSC with a self-enhanced active layer, compared with the conventional planar active layer configuration, has broadband and wide-angle range absorption enhancement due to better geometric impedance matching and prolonged optical path. 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15. R. Biswas and C. Xu, "Nano-crystalline silicon solar cell architecture with absorption at the classical 4n 2

Introduction
Recently, thin-film organic solar cells (OSCs) have gained intensively interests for the next generation of solar cells. However, thickness of the active layer in OSCs is typically a few hundred nanometers or less due to their short diffusion length of exciton, which sets a fundamental limitation on light absorption. Consequently, improvement in light trapping is very important for their practical utilization in photovoltaics. Generally, physical guideline of light trapping involves two main approaches: geometric optics methods [1][2][3][4][5] and wave optics methods [6][7][8][9][10][11][12][13][14][15][16]. The former aims at elongating optical path inside the photoactive layer and thus enhancing photon absorption. For organic thin film solar cells with typical active layer thickness of 100nm-200nm, which is in subwavelength scale, we cannot investigate light harvesting mechanism simply by the geometric optics methods and instead wave optics properties should be considered. Photonic crystal (PC) [17], due to its strong modification of electromagnetic distribution, has attracted lots of attention and can be used to increase solar cell efficiency based on different physics mechanisms. For example, based on prolonging optical path length by diffraction, grating structure backreflector (1-D PC) [6], periodic arranged cylinders backreflector (2-D PC) [6][7][8] and periodic arranged spheres backreflector (3-D PC) [6] have been proposed. Distributed Bragg reflector (1-D PC), owing to its photonic band gaps (PBGs) where the propagating modes are completely forbidden, can perform 100% omni-directional reflection and thus strongly reduce the transmission loss [6,7]. Slow Bloch modes in flat photonic band structure have been studied in silicon solar cells [9], OSCs [10] and dye-sensitized solar cells [11] to extend dwell time of light in the active layer.
However, it is noteworthy that enhancement mechanisms based on PC properties require large refractive index contrast between the nanostructure and the background medium. This is achievable for inorganic solar cells but it is challenge for OSCs because all the organic materials have similar value of refractive index. Motivated by this issue, we explore new absorption improvement methods special for organic materials. Recently, OSCs with a PC active layer have been studied theoretically [10,12] and experimentally [13]. But we notice that square column shaped or cylinder shaped photoactive layer has relatively low absorption rate due to its abrupt change of optical impedance. If gradient structure is employed, such as pyramid type or cone type, normal incident light couples adiabatically with laterally scattered light and thus higher absorption performance can be achieved [14][15][16]. Intrigued by this problem, we present OSC designs based on periodically arranged gradient shape active layer, which can enhance light harvesting by themselves. In comparison with employing inorganic concentrators (such as silicon, silver, and gold nanostructures) into the active layer of OSC which may degrade the electrical performances of OSCs by creating recombination centers or breaking the morphology of organic materials, the self-enhanced organic active layer can both enhance the optical absorption and facilitate carrier collections simultaneously as will be described below. In modeling we employ poly-3-hexylthiophene: [17]-phenyl-C61-butyric acid methyl ester (P3HT:PCBM) blend to show the improvement because its optical and electrical properties have been well studied. As a comparison, square column type active layer and cylinder type active layer have also been discussed. Our numerical results show that both kinds of gradient-type active layers have noticeable advantages compared with traditional planar bulk active layers. Especially, only 1/3 or less amount of P3HT:PCBM will be enough to harvest same quantity of light energy if the active layer is made as pyramid shape rather than planar configuration. These designs are believed feasible experimentally because recently nanostructure patterned OSCs have been realized by the imprint methods which not only demonstrate the feasible of the making well-aligned nanoscale patterned active layer [13,18,19] but also provide new degree of freedom on introducing optical structures into the active layer directly for further improving OSCs performance for practical photovoltaics. It is therefore desirable to have theoretical studies in the well-aligned nanostructure which can show the physics and improve the device performances simultaneously.
Regarding the theoretical model, another noteworthy contribution in our work lies at the absorption power calculation based on scattering matrix method (SMM) by rigorously solving Maxwell's equations. Different from the widely-used calculation method by Absorption = 1-Transmission-Reflection, our algorism can extract the net optical absorption of the active layer rather than the whole OSCs. This improvement is very important because metal absorption, which does not contribute to exciton generation, can be excluded from the result.

Theoretical model
The studied OSC structures comprise of Indium tin oxide (ITO) anode / Zinc oxide (ZnO) injection layer / P3HT:PCBM nanostructure active layer/ P3HT:PCBM flash layer/ Lithium fluoride (LiF) electron transport layer /aluminum (Al) cathode. Our proposed structure with periodic arranged pyramids of P3HT:PCBM blend in the ZnO background are shown in Fig.  1(a). As a comparison, we also calculate an OSC with square column type active layer. Another set of OSCs with cone type active layer and cylinder type active layer are depicted in Fig. 1(b). In modeling, we employ complex refractive index of P3HT:PCBM measured by ellipsometer [20]. In some papers, ITO and ZnO are taken to be transparent over the involved wavelength range, but in our calculation we did not make this approximation, optical parameters of ITO and ZnO are taken from Ref  In view of good convergence, fast speed and low memory consumption associated with the SMM method [24,25], it is employed here to investigate the optical properties of OSCs. SMM is a convenient algorithm for modeling multilayered system with laterally patterned nano-structure. Firstly, electromagnetic field is expanded as a series of Bloch modes (plane waves). Then, the OSC is divided into sub-layers along the z-direction and the Maxwell's equations of each layer are recast into eigenvalue equations. Finally, by using the continuous tangential boundary condition of electromagnetic field, elements of the S-matrix can be obtained by recursive equations. Along the direction of propagating light, electromagnetic field of the l th layer is expressed as a superposition of forward and backward propagating waves with the complex amplitudes a l and b l . The scattering matrix related to two layers l and l' is given by: Transmission and reflection coefficients can be calculated by Poynting vectors based on the matrix S in,out , which connects the substrate region (corresponding to incoming waves) to the air region (corresponding to outgoing waves). It is common to calculate the absorption of solar cells straightforwardly by Absorption = 1-Transmission-Reflection. However, this formula essentially gives the total absorbed energy of all the layers but not the expected absorption of the active layer for photon generation. In order to handle this problem, we calculate absorption power A by taking volume integral over the active layer: where E inc refers to the total energy of incident light, L denotes the energy loss fraction including metallic and leaky losses, and E is the electric field profile at the active layer.

Optimization of parameters
where A is absorption of the OSC and I(λ) is the sun irradiation spectrum intensity of standard air mass 1.5(AM 1.5). We find that for both kinds of gradient type active layers, <A> increases monotonously as filling ratio increased (Figs. 2(a)-2(b)). Consequently in the following calculation we choose close packed structure for both pyramids type and cone type design.
It should be noted that some reports have detailed discussion about minimizing surface reflection by choosing ITO anode with proper thickness, which is also functioned as the antireflective coating (AR coating) of the solar cells. In Figs. 2(c)-2(d) we calculate average reflectivity of different ITO thickness based on a similar wavelength integral of Eq. (5). The results show that lowest reflection loss can be achieved when ITO thickness is around 240nm for both gradient type active layers. However, when we calculate the absorption power using ITO = 240nm, we found that the improvement compared with widely used 100nm is only 0.2%, consequently we still use typical 100nm thickness.

Absorption power
Obviously, besides the employed volume of P3HT:PCBM, height and lighted area also have great influence to device absorption. In order to keep a relatively fair comparison, for every pyramid or cone type OSC, we set a corresponding control planar OSC with same height and same lighted area. With this setting the planar OSC consists of three time amount of P3HT:PCBM compared to its pyramid or cone shape counterpart. Consequently, we should compare two OSCs with absorption power of per unit volume P3HT:PCBM rather than the original absorption power. Surprisingly, we found that the original absorption power of closed pack pyramid active layer is even higher than its corresponding planar control cell in 400nm-620nm wavelength range, although the later contents three times volume of P3HT:PCBM than the former (Fig. 3(a)). Unit volume absorption power of Fig. 3(a) is shown in Fig. 3(b). Average enhancement factor reaches 3.21 and maximum reaches 4.88 at 520nm. For a larger scale situation, it will reach an even higher average enhancement factor of 4.58 (Fig. 3(c)). However, as mentioned above, thick P3HT:PCBM active layer results in good optical absorption but may degrade electric performance due to low carrier mobility, and thus the power conversion efficiency will be even lower than thin OSC. Results of cone type active layer and cylinder type active layer are also shown in Fig. 3(d), indicating that cone shape performs better than cylindrical shape. It should be noted that ZnO is used as the background of all P3HT:PCBM nanostructure, while no ZnO background exists for the planar cell. Although ZnO absorption is incorporated in the results, it is too weak as compared to P3HT:PCBM absorption in λ = 400nm-800nm. Hence, we simply assume that in this wavelength range, the absorption of ZnO is zero. For our OSCs design, the transmission loss is neglectable because light transmission rate through the Al backreflector is very low, hence main energy loss of the OSC comes from reflection. That means given same quantity of incident energy, devices with lower reflectivity will result in higher absorption. As widely known, single layer quarter-wavelength AR coating is only capable to reduce reflection in limited wavelength range, while multiple layers AR coating can achieve relatively wider range reduction and the best structure is the graded index AR coating (GRIN AR coating) which has broadband effect due to better impedance matching. Our OSC design with gradient active layer can offer the improvement as from the linear GRIN system. The gradient-type active layer has a linear geometric transition of refractive index from top to bottom while GRIN structure has linear continuous changed refractive index from top to bottom. By dividing these inhomogeneous systems into N sublayers, effective refractive index of the i th sub-layer can be calculated approximately: for pyramid or cone shape situation, ff i is the filling factor of the i th sub-layer, while for GRIN system ff i is a generalized quantity denoting the weighted refractive index. Consequently, although linear GRIN system and gradient shape system are physically different, using this effective medium approximation (EMA), they can be mathematically equivalent. That is why they both have similar broadband reflection reduction compared with planar structure. If precise study is involved, GRIN system should be simulated by more complicated model such as Bruggeman EMA [30]. To verify our declaration, we calculate reflectivity of the active layer as Fig. 4 shown. Since spacer and electrode layers will affect the optical environment of the multilayered OSC structure, here we simplify the structure to be a bare patterned active layer. According to Fig.  4, since reflectivity also equals to (Z blend -Z air )/(Z blend + Z air ) (where Z blend and Z air are optical impedance of P3HT:PCBM blend and air, respectively), we can see that gradient shape active layer reflects less energy than column shape active layer due to better impedance matching. In comparison with the abrupt change in the column or planar case, better coupling can be expected between the incident light and the laterally scattered light. It should be noted that fabrication of GRIN system is a challenging work because choosing proper material and precise control of the inhomogeneous refractive index profile is rather difficult. So it is very challenging to fabricate a GRIN active layer for broadband absorption enhancement, especially for organic materials. Our structures offer possible designs which have same advantage of linear GRIN system on reducing device reflection loss and thus enhancing light absorption.
Another reason of the absorption improvement is that light in the periodic patterned nanostructure has longer optical path and longer dwell time due to multiple scatters, which can compensate for the reduced OSC thickness. According to the Beer-Lambert's law, the absorption length of photon, although depends on materials, generally has an increasing trend as wavelength increases. Firstly, this law can explain the absorption power shown in Fig. 3, which is similar with the P3HT:PCBM absorption coefficient (inset of Fig. 3(a)), but in short wavelength range the absorption is obviously enhanced. It is because photon in shorter wavelength range does not require long optical path and thus has priority of being absorbed compared with longer wavelength photon. Secondly, the law suggests that to absorb photons in longer wavelength regions, the corresponding planar active layer should be made thicker. PC structure can diffract light efficiently and thus light can traverse through the active layer for multiple times, leading to efficient absorption. Such kind of mechanism, which contributes to prolonging optical path length, has a similar effect of the situation that light propagates through a very thick planar photoactive layer. As a result, periodic nano-structures are quite suitable for photon absorption in long wavelength. Especially, pyramid has been proved to be the most efficient scatterer compared to other shapes due to its low degree of symmetry, which has similar light trapping performance as ideal Lambertian surface [31]. We also investigate the average absorption power with incident angle θ as shown in Fig.  5. The result shows that over a wide angular range, our gradient type active layer has better absorption than planar OSC with same volume of P3HT:PCBM. This is very important for OSC application because the incident angle of sunlight always changes dynamically. It is worth point out that although linear GRIN structure has similar optical property of gradient shape structure as discussed above, it cannot keep good performance at large incident angle. Quintic refractive index profile has been proposed to improve oblique incident absorption but it also faces the challenge of fabrication [32]. Electric field profiles of OSC with pyramid type active layer and cone type active layer under incident wavelength λ=500nm are shown in Fig. 6(a) and 6(b) respectively. Remarkably, there is more energy penetrating into the absorption material, which is significantly different from the cylinder case (Fig. 6(c)) and planar OSC (Fig. 6(d)) where the incident energy evanesced quickly. In other words, when same volume of P3HT:PCBM blend is employed, more light can be absorbed by gradient shape active layer.

Exciton generation
Further, when energy (photons) is absorbed by P3HT:PCBM, excitons will be created. Assuming one incident photon can generate one exciton, we calculate the exciton generation profile in the photoactive layer by the time average dissipation energy <Q>: where <S> refers to the time average Poynting vector and E 0 (ω) is the energy of a single photon. Figure 7 shows exciton generation distribution at the position z=50nm of the cone shape active layer ( Fig. 7(a)) and cylinder shape active layer ( Fig. 7(b)). Due to deeper penetration of energy in cone type active layer, exciton creation is obviously more efficient and this predicts more efficient utilization of absorption material. Vertical cross view of exciton generation profile of these two structures are depicted in Figs. 7(c) and 7(d). According to the figures, in cone shape active layer most of the excitons are created at the position which is near to the interface between P3HT:PCBM and ZnO. This phenomena has important meaning for device electric property because exciton diffusion length in P3HT:PCBM is only around 10nm. Shorter exciton creation distance will result in higher carrier collection rate. Pyramid type active layer also has this advantage so here we did not show those similar results.

Conclusion
We theoretically study organic solar cells (OSCs) with periodically arranged gradient type active layer based on self-enhanced mechanism. The results show that these structures have broadband (400nm-800nm) and wide angle range absorption enhancement compared with conventional planar OSCs due to better geometric impedance matching and longer optical path. This work can contribute to improve the nano-patterned OSCs efficiency in practical photovoltaic applications.