Geometrical shape design of nanophotonic surfaces for thin film solar cells

We present the effect of geometrical parameters, particularly shape, on optical absorption enhancement for thin film solar cells based on crystalline silicon (c-Si) and gallium arsenide (GaAs) using a rigorous coupled wave analysis (RCWA) method. It is discovered that the “sweet spot” that maximizes efficiency of solar cells exists for the design of nanophotonic surfaces. For the case of ultrathin, rod array is practical due to the effective optical resonances resulted from the optimum geometry whereas parabola array is viable for relatively thicker cells owing to the effective graded index profile. A specific value of thickness, which is the median value of other two devices tailored by rod and paraboloid, is optimized by truncated shape structure. It is therefore worth scanning the optimum shape of nanostructures in a given thickness in order to achieve high performance. ©2016 Optical Society of America OCIS codes: (040.5350) Photovoltaic; (080.2740) Geometric optical design; (310.6628) Subwavelength structures, nanostructures; (350.6050) Solar energy. References and links 1. A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012). 2. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). 3. S. B. Mallick, N. P. Sergeant, M. Agrawal, J.-Y. Lee, and P. Peumans, “Coherent light trapping in thin-film photovoltaics,” MRS Bull. 36(6), 453–460 (2011). 4. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 45),” Prog. Photovolt. Res. Appl. 23(1), 1–9 (2015). 5. E. Moulin, U. W. Paetzold, H. Siekmann, J. Worbs, A. Bauer, and R. 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Introduction
For decades, in order to improve solar-cell efficiency, diverse photonic designs have been proposed [1]. Thin film crystalline silicon solar cells are considered to be one of the promising techniques due to fabrication cost and the abundance. However, the relatively low light absorption in the red and near-infrared wavelength ranges limits the efficiency of thin film solar cells [2][3][4]. Therefore, effective light trapping designs are mandatory to develop the efficiency of thin film solar cells. Several light trapping schemes by using a metal back reflector [5] or managing incident light efficiently [2,6]  comparison between a regular square array of nanorod (cylindrical-shaped) [12] and a tapered submicron structure with a graded index profile [18]. This report aims to address the above question by demonstrating our recent progress on designing and analyzing different light trapping structures, i.e., rod and paraboloid arrays and a structure transformed between these two devices by depending on parabolic ratio (will be discussed in Fig. 4) based on crystalline silicon (c-Si) [26] and gallium arsenide (GaAs) [27]. We numerically investigated the effect of the optimum shape on efficiency of solar cells by varying total cell thickness. In the process, other optimum parameters of structures such as period, height and filling fraction will be also discussed. According to our results, efficiency of the structure with a thickness of 500 nm, where both rod and paraboloid cells show the similar tendency in terms of cell efficiency and reflectance, is maximized with a truncated parabola-shaped array. This shape is also known as conical frustum shape or truncated cone in other papers [15,28,29] and defined as the structure with a parabolic ratio of 50% in our paper. We then show the diverse fabrication methods to ensure reproducibility and controllability of the proposed structures.  A square symmetry has been chosen and the structures are defined by following factors: lattice period a, height of nano-patterned structure h, filling fraction FF defined by particle diameter / lattice period a and total cell thickness t which includes h as shown in Fig. 1(a). In our design, a silver (Ag) [30] metal back reflector is incorporated onto the bottom side for enhancing light trapping. Although Ag requires relatively higher cost than Al, it was selected for high absorption characteristic of solar cells. We use simulation study to examine how geometrical shape influences efficiency of thin-film c-Si solar cells using rigorous coupled wave analysis (RCWA) implemented by the commercial software package (DiffractMOD, Rsoft, USA) taking into account the AM1.5G solar spectrum. The direction of incident light is normal to the nanostructures unless otherwise specified and we considered the wavelength range varying from 300 nm to 1200 nm which covers the major solar energy spectrum. Zero to fifth diffraction orders were considered to calculate the diffraction efficiency, which is enough to numerically stabilize our results. As demonstrated in Fig. 1(b), scanning period and FF was performed for rod arrays to obtain the optimum parameters, resulting in maximizing the integrated absorption (IA) efficiency since these two parameters significantly affect the absorption characteristics of c-Si thin film solar cells [31]. These calculations were conducted over a technologically reasonable range: between 300 and 700 nm for period and 0.3 to 0.7 for FF [32] with three different thicknesses (i.e., h = 100 nm for t = 200 nm, h = 200 nm for t = 500 nm and h = 400 nm for t = 800 nm). As can be seen, regions characterized by a high IA are observed around a FF of 0.7 for all cases, therefore, it is attributed to the fact that a subtle change in the parameters hardly influence the optical properties of our structures. In this regard, we fixed the value of FF for our rod structures as 0.7 which is also a practical value to gain high performance solar cells [33]. Figure 1(d) shows the change of the effective refractive index, calculated with the volume weighted average refractive index of air (n air = 1) and c-Si (n c-Si ~3.94 at a wavelength of 600 nm), of the same paraboloid structures but different FF corresponding to the height. For the cases of FF 0.5 and 0.75, the lines are steeper than the case of FF 1.0. This means that the refractive indice of these nanostructures more abruptly change at the interface between substrate and nanostructure, resulting in degrading the anti-reflection (AR) property of cells. However, the effective refractive index of the paraboloid structure with a FF of 1 more gradually increases as the height is more adjacent to the substrate. This case thus is the most viable solution among our paraboloid structures in terms of FF to reduce the Fresnel reflection [34,35], providing the improvement of solar cell efficiency. Therefore, FF is always set equal to 1 for our paraboloid structures. Figure 1(e) illustrates simulated surface reflectance of the paraboloid nanostructure as a function of wavelength with following parameters: height is varied from 100 nm to 600 nm with a period of 500 nm. The total cell thickness is negligible for this case since the back reflection which occurs between Ag reflector and c-Si substrate was not considered to analyze how height influences the reflectance of the paraboloid structure. Higher order diffraction losses are observed in the wavelength range below ~500 nm. An abrupt change of the reflectance at the wavelength of about 350 nm is caused by transition of density distribution of the diffraction orders. At the wavelength of 500 nm which is analogous to the period of the structure, the drastic decrease of reflectance occurs due to Wood-Rayleigh (WR) anomaly from the grating formed by the regular array of nanoparticles, providing the cut-off high orders [36-39]. In the wavelength range above ~500 nm, the AR performance is gradually improved as the height is increased and saturated beyond a height of around 400 nm. These phenomena are explained by the fact that if a height is not tall enough, the effective refractive index abruptly appears to the incoming light [18,34,40]. This also indicates that in order to properly implement the AR property in the visible and near-infrared region, a height should be at least around 200 nm.

Result and discussion
In order to investigate how geometrical shape influences efficiency of solar cells as the cell thickness is increased, we performed a number of simulations for the two cases of array with varying height and period since these two parameters which strongly affect the absorption characteristics should be considered to achieve high cell efficiency in c-Si solar cells [33,41].
Cell efficiency was calculated by the equation: where sc J is the short-circuit current density, oc V is the open-circuit current, f Γ is the filling factor and in P is the total incident power under the AM1.5G solar spectrum. In our calculation, the collection efficiency of 85% with a 5% shadowing effect from the electrode was used. The reverse bias saturation current was taken to be 15    Interestingly, a thickness of 500 nm shown in Fig. 2(b), the highest values of cell efficiencies of both rod and paraboloid structures are nearly the same and depicted as circle symbols, 10.64% with h = 200 nm, a = 550 nm and 10.52% with h = 200 nm, a = 400, respectively. This is expected to the fact that another optimum shape for maximizing efficiency may exist and this work will be justified in Fig. 4. Structures thicker than 800 nm such as 1100 nm in Fig. 3(d) and 2 μm, which is not shown in this paper but calculated, have optimum values with paraboloid surface. This tendency is attributed the fact that the heights are tall enough to properly implement the AR property in paraboloid structures, resulting in enhancing absorption characteristic of solar cells, which is previously mentioned in Fig. 1(e). To understand these mechanisms, as shown in Fig. 3, the reflectance spectra of nanopatterned surfaces without back reflection were performed in the visible and near-infrared region as the total cell thickness is increased with following heights and periods: (a) h = 100 nm, a = 400 nm and these are slightly adjusted from the optimum values due to difficulties in fabrication, (b) h = 200 nm, a = 500 nm and these are reasonably overlapping values, almost the mean value of the optimum parameters for both structures in Fig. 2(b), 2(c) h = 400 nm, a = 600 nm. We also calculated the reflectance of a flat surface for comparison. In Fig. 3(a), despite the fact that the dips of reflection resulting from the WR anomaly occur in both structures at a wavelength of 400 nm which is equivalent to the period, the reflectance of the rod-shaped one is still far lower than that of paraboloid one. This is because in the wavelength range above ~400 nm, dips are clearly observed due to optical resonances, which are specific to the square rod arrays and depend on aspect ratio and period of structures. These mutually adjacent dips and the WR anomaly consequently broaden the AR property. Mie resonance which redshifts for increasing particle diameter is also observed at a wavelength of 1100 nm since the diameter of the particle is 280 nm resulted from a FF of 0.7 and a period 400 nm) [43,44]. However, due to the fact that this property weakly appears and c-Si has low light absorption in the region (~1100 nm), this result hardly affects the efficiency of solar cells. For the case of paraboloid, although the parabola-shaped one is a sub-wavelength structure (SWS) in the wavelength range above ~400 nm, which requires dimension smaller than the optical wavelength, and exhibits a linearly graded index profile, the AR property is weakly observed since the height is not tall enough to proceed the AR property in the broadband wavelength range as previously mentioned in Fig. 1(e). The electric field distributions in Figs. 3(d) and 3(e) provide useful information for understanding the reflectance on the surfaces. In Fig. 3(b), the reflectance of both structures are remarkably similar in the range of below ~500 nm and the sharp dips at a wavelength of about 500 nm are consistent with the WR anomaly. Above ~500 nm, although the rod one is slightly more practical than the paraboloid one in terms of the AR property, it is hard to assert that the former is more viable than the latter as there is only a subtle improvement.
As depicted in Fig. 3(c), the paraboloid structure with a cell thickness of 800 nm exhibits a high performance of the AR property in the broadband spectrum. The light with the wavelengths greater than 600 nm, the structure is a SWS which suppresses the Fresnel reflection at the interface between air and surface due to the gradually increasing refractive index. Although the rod structure shows dips of reflectance, the paraboloid one shows the far more effective AR property. These results therefore indicate that, in effective medium theory, the paraboloid structure can considerably reduce the surface reflection and additionally, be efficient to achieve a high broadband absorption. The electric field distributions are also simulated in Figs. 3(f) and 3(g). The result displays that the AR property of the paraboloid structure is visible compared to the rod one. Parabolic ratio (%) (a) In order to provide further insight into the case of 500 nm thickness where both rod and parabola arrays show the almost equal values of cell efficiencies and the similar reflectance characteristics, we conducted more simulations by transforming shape of the nanostructures. A simple quadratic function is used to describe the variations in parabolic ratio of the structures. The function was designed on the basis of the tapered function of paraboloid. To obtain gradual change of geometrical shape from rod to paraboloid, the tapered order (TO) was defined by the parabolic ratio raised to the power of 1.5.
The equation of describing paraboloid raised to the power of the tapered order describes parabolic ratio of structures. The final equation is defined as: In this equation, the range of 'z' value is limited to from 0 to 1, normalized values of heights. Figure 4(a) illustrates the geometrical transformation process of nanostructure shape. For instance, if the ratio is 100%, a surface is composed of parabola-shaped nanostructure whereas 0% of parabolic ratio means that the structure is perfect rod-shaped one. Figure 4(b) demonstrates simulated absorption efficiency as a function of the wavelength and parabolic ratio with following factors (a = 500 nm, h = 200 nm and t = 500 nm). The result shows that, in the spectral range 800-900 nm in which c-Si has low light absorption depicted as a white rectangular in Fig. 4(b), the absorption efficiencies of the structures with parabolic ratio range 40-80% are higher than both pure rod and paraboloid ones whose parabolic ratio are 0% and 100%, respectively. The averaged values of this result is also shown as a line graph (green) in Fig. 4(c). In this regard, this tendency can be worth being studied since the efficiency of thin-film c-Si solar cells strongly depends on optical absorption. Only absorption efficiency above 40% are plotted to highlight the differences of absorption efficiencies among the structures. Further simulation was carried out to analyze the effect of the parabolic ratio.
Calculation of cell efficiency was conducted as the parabolic ratio is increased demonstrated as a line graph (red) in Fig. 4(c). The peak value of cell efficiency is observed at a parabolic ratio of 50%, therefore, this result is expected to the fact that the structure whose parabolic ratio is 50% may possess combined benefits from both rod and paraboloid structures. It also means that in order to maximize the performance of solar cells, a geometrical shape should be considered in a given thickness.  To ensure the fact that the nanostructure patterned with PR 50% is more adequate for maximizing cell efficiency than both conventional rod and paraboloid structures for the case of a cell thickness of 500 nm, diverse simulations were conducted with a period of 500 nm. Figure 5(a) shows the simulated absorption efficiency for the three different light trapping structures with heights varying from 50 nm to 350 nm. The noticeable increase of absorption efficiency for all structures at the wavelength of 500 nm, which is equal to lattice period a, is observed due to the WR anomaly. The paraboloid one not only has WR anomaly, but also shows that the higher absorption efficiencies can be obtained at the taller heights. The reason is already mentioned in Fig. 1(e). However, this structure exhibits relatively low absorption efficiency in the red and near-infrared region compared with other two devices. As can be seen, as a whole, PR 50% is expected to partly have the advantages of both rod and paraboloid structures. To further understand this, Fig. 5(b) demonstrates the calculated cell efficiency of the three light trapping structures as a function of the height. The optimum heights of rod and palaboloid in order to obtain the maximum cell efficiency are 150 nm and 250 nm, respectively, corresponding to a period of 500 nm. As we expected, PR 50% has the maximum cell efficiency with a height of 200 nm which is the middle point of optimum heights of the other two structures. In order to consider periods of the structures, additional simulations were carried out. In Fig. 5(c), PR 50% shows the highest performance with a cell thickness of 500 nm which is the exactly median value of 200 nm optimized by rod-shaped and 800 nm optimized by parabola-shaped shown in Figs. 2(a) and 2(c). 10.94% of cell efficiency is achieved, 46% boosted comparing to the structure with flat surface.  To study further the effect of geometrical shape on solar cells, we conducted simulations using GaAs which is widely used in high-efficiency photovoltaics and other devices. The result shown in Fig. 6 exhibits the fact that the highest cell efficiencies of solar cells based on GaAs also depend on geometrical shape as the cell thickness is increased. The PR 50% structure has the maximum cell efficiency with a cell thickness of 350 nm, which is the exactly median value of 200 nm and 500 nm. However, the tendency is more sensitive than the cases of solar cells based on c-Si. This is explained by the fact that although the cell thicknesses are thinner than c-Si based solar cells, which reduce optical path length in substrate, high extinction coefficient of GaAs [4] dominantly affects the tendency. This is far from ideal for achieving the maximum cell efficiency of solar cells owing to the sensitivity of the optimum shape to slight changes in cell thicknesses. Also, the gaps of the maximum values among the structures with different thicknesses are small-scale compared to the structures based on c-Si. It is therefore inadequate to consider the optimum geometrical shape for maximizing the performance of GaAs solar cells. In this regard, studying with this concept for other material used for solar cells is worth taking into account to decide whether scanning the optimum shape for maximizing efficiency of photovoltaics is meaningful or not.

Conclusions
In summary, c-Si nanostructures with various shapes were investigated by simulations. Our studies indicate that the maximum cell efficiency of structures depends on their geometrical shapes as the total cell thickness is increased. The calculated reflectances reveal that the AR property should be considered to design high performance solar cells. The results demonstrate that for the case of ultrathin solar cell with a thickness of 200 nm, a rod-shaped structure can be an appropriate design due to optical resonances from the optimum parameters. The PR 50% thin film c-Si solar cell can be used to maximize the cell efficiency with a cell thickness of 500 nm. With increasing thickness more than 800 nm, a parabolashaped structure can be more applicable than other structures owing to the effective graded index profile, resulting in the entire AR performance. In this regard, in order to obtain the highest performance for solar cells, it is crucial to determine the optimum geometrical shape of structures in a given thickness. The dependence of geometrical shape on cell thickness can provide useful controlling parameters in designing photovoltaic and optical devices.