## Abstract

For the thin-film solar cells embedded with nanostructures at their rear dielectric layer, the shape and location of the nanostructures are crucial for higher conversion efficiency. A novel two-level hierarchical nanostructure (a sphere evenly covered with half truncated smaller spheres) can facilitate stronger intensity and wider scattering angles due to the coexistence of the merits of the nanospheres in two scales. We show in this article that the evolutionary algorithm allows for obtaining the optimal parameters of this two-scale nanostructure in terms of the maximization of the short circuit current density. In comparison with the thin-film solar cells with convex and flat metal back, whose parameters are optimized singly, the short circuit current density is improved by 7.48% and 10.23%, respectively. The exploration of such a two-level hierarchical nanostructure within an optimization framework signifies a new domain of study and allows to better identify the role of sophisticated shape in light trapping in the absorbing film, which is believed to be the main reason for the enhancement of short circuit current density.

©2013 Optical Society of America

## 1. Introduction

Even though the material consumption of conventional bulk crystalline silicon solar cells has come down considerably over many years of development, it still accounts for up to 40% of the total cost. Thus thin-film solar cells [1, 2], which reduce the thickness of active layer (the absorbent semiconductor film) from hundreds of micrometers to several micrometers or even to nanometers, become promising photovoltaic devices capable of generating electrical power at competitive cost. Moreover, thin-film solar cells have additional merits with reduced bulk recombination and photo-degradation as well as better carrier collection [3] and high open-circuit voltage [4]. As a tradeoff, however, the energy conversion efficiency becomes lower due to the light path in silicon film of absorption is shortened and the transmission losses increase. Furthermore, the absorption of thin-film solar cells seems to be weaker in the frequency range near the semiconductor band gap.

To overcome the abovementioned weaknesses, it needs to optically increase the light path within the absorbent semiconductor film so that the physically-shortened active layer could be compensated reasonably. Since conventional light trapping methods used for bulk solar cells like anti-reflection coatings [5, 6] and grated structures (e.g. pyramid-like and triangular texture) [7] are not applicable to thin-film solar cells, the localized surface plasmons (LSPs) induced by the metallic structures have been employed as they offer an alternative mechanism for trapping more light in the thin semiconductor layers [3, 8–10]. When metallic structures are scaled into sub-wavelength of incident light, their interactions with the sunlight become dramatically strong at visible range and thus produce high absorption in semiconductor even though it is as thin as hundreds of nanometers [8, 10, 11]. The presence of metallic nanoparticles can result in unwanted surface recombination and therefore lower photocurrent and open-circuit voltage. But the side-effects can be depressed as the recombination is isolated by the dielectric layer between the nanoparticles and the semiconductor.

In the original study on the role of LSPs in solar cells, irregularly-shaped particles was randomly distributed on the front surface and it is found that the absorption is significantly enhanced with respect to the flat surface [8]. Nevertheless, the abovementioned enhancement can be partially counteracted by the Fano resonance effect, i.e. the light is not absorbed but reflected if the frequency is out of the resonance range [12]. A remedy to this unwanted effect is building the nanostructures in the rear of semiconductor. It is observed that this kind of nanostructure disposition keeps the advantages of LSPs for absorption enhancement with blue and green lights while allows the long wavelength light (red) to be coupled into the Si film [12]. Thus the absorption is expected to be enhanced within the sunlight spectrum corresponding to the band gap absorption in a-Si. However, the rear-located nanostructures could induce significant Ohmic loss around them and the absorption is enhanced only within a narrow band and fluctuates near the resonant frequency. Such disadvantages can be partially overcome by optimizing the particle size and shape so that the electric intensity is relatively lower around the metallic particles and the absorption peak occurs at demanded frequency region. However, as conventional light trapping techniques, the disadvantages cannot be taken away completely in the existing structure and require further investigations [13].

The roles of LSP in the solar cells depend on the size, shape, location and coverage density of the nanostructures. Therefore these geometric parameters should be systemically investigated and whenever possible, optimally determined. For front-located irregular nanoparticles, the studies showed that small nanoparticles result in maximal overall enhancement while large ones help to improve light emission [8]. Subsequent work revealed that the shape of particle is crucial to the plasmon-enhanced solar cells and the cylindrical and hemispherical particles have better performance than spherical particles [14]. Recently, Chen et al. [15] proposed a novel two-level hierarchical nanostructure that combines the virtues of larger and smaller spheres into a single structure and therefore realizes enhancement of broadband absorption. From previous studies [12, 15], the short circuit current density (*J*_{sc}) and the conversion efficiency could be enhanced if the radius of large spheres and the coverage density of the nanostructure are properly selected.

The determination of optimal parameters in the conventional work is largely based on a trial-and-error exercise, in which interactions of different parameters are barely considered. To a certain extent, such an approach could help accelerate the design process for a range of solar cells, but attaining the optimized parameters using the trial-and-error approach could be difficult as the number of permutations can be astronomic even for a small number of parameters. For this reason, a more methodical optimization technique is needed for the efficient design of solar cells. Nevertheless, little systematic study has been conducted in this regard. Recently, great efforts have been directed towards the optimization of physical properties by rearranging its microstructures using topology optimization techniques [16–21]. Thin-film solar cells could be considered as a class of new composites comprised by periodically repeated Represented Volume Elements (RVE). For the devised nanostructures in the present study, only the location and a few geometric parameters of a given shape are considered as the design variables.

Differential evolutionary algorithm [22] offers a viable approach to size optimization for a parameter-based and sophisticated cost function, in which the gradient information may be difficult to be obtained. Unlike traditional gradient-based algorithms that can be susceptible to being trapped at local minima, malfunction in cases of insensitive gradient, and slack-off if the initial guess is far from the optimum [23], the evolutionary algorithm proves to be versatile, powerful and efficient for optical design problems [23]. Compared with other gradient-free algorithms like genetic algorithm [24] that is based upon natural selection and survival-of-the-fittest, the evolutionary algorithm usually exhibits faster convergence as shown by extensive numerical tests [23]. It is noted that the prevalence of using genetic algorithms in the design of optical devices has been seen in the metamaterial design for negative refractive index [25], the photonic crystal with large absolute band gaps [26], and the superstructure with different electronic and optical gaps [27]. However, the application of more powerful algorithms, such as the differential evolutionary algorithm, to improve the solar cells has not been reported even though its recent applications in solar devices, such as nonimaging Fresnel lens [28], seem promising.

## 2. Statement of the problem

As shown in the schematic (Fig. 1(a)
), a typical thin-film solar cell consists of four layers: a 80 nm SnO_{2}:F coated glass which is used as a transparent conductive oxide on the front; followed by a 350 nm hydrogenated amorphous silicon (a-Si:H) layer; subsequently a layer of Al-doped ZnO (ZnO:Al) thin-film embedded with periodically-repeated Ag particles. Noticeably, the LSP resonant frequency red-shifts when the surrounding medium of metallic particle yields large refractive index. But it cannot be the design variable as we are concerned with the size and shape of the nanostructures in this study. Finally, a 120 nm Ag layer is used as the back reflector. It is noted that the back surface is not flat but distributed with bulging dots (Figs. 1(b) and (c)), where the distance between the bottommost points of ZnO:Al and Ag back layers maintains the same value (120 nm) as the planar part, otherwise it could lead to electricity leakage. Such a design also allows to consistently compare the roles of nanospheres in different sizes [15]. For the same reason, the thickness of ZnO:Al layer is not given in Fig. 1(a) and it should be determined in terms of the size of nanostructure. The location of larger and small spheres is illustrated by the cross section of this solar sell in Fig. 1(b) with an inset (red dashed region) to zoom in inner geometrical characterizations. Figure 1(c) gives a 3D view of this thin-film solar cell.

Since the nanostructures are periodically-repeated and the incident light is normal to the front surface, it is sufficient to have the computational region limited to the RVE with periodic boundary conditions on its bilateral surfaces (Fig. 2(a)
). The open boundaries on the input and output sides are truncated by the perfectly matched boundary layers (PMLs), whose distances to the top and bottom of solar cell are 200 nm and 100 nm, respectively. The electromagnetic field within the RVE is governed by the well-known Maxwell’s equations and solved by the finite difference time domain (FDTD) algorithm [29]. In the present study, we utilize the same FDTD software from Lumerical [30] as used by Chen *et al*. [15]. By taking the advantage of the symmetry and anti-symmetry at *x* and *y* directions, the computational region can be reduced to a quarter of the base cell without destroying its periodicity. As a result, the computing time is reduced significantly. Once the Maxwell’s system is solved, the short circuit current density, electric intensity, absorption as well as other relevant factors can be computed to any pre-defined level of accuracy.

Numerical experiments illustrate some parameters are unlikely to be influenced by others (e.g. the thickness of Si Film invariantly converges to its upper bound), therefore we herein only consider more dependent factors such as the location and geometrical parameters of the abovementioned two-scale nanostructure (Fig. 2(b)). Because of the central points of small spheres are evenly distributed on the large sphere, this novel structure exhibits the combined virtues of a large sphere, stronger scattering intensity, and small spheres, wider scattering angles. It is believed that the stronger intensity induced by large sphere contributes to the improvement of the scattering/absorption cross-section ratio and wider scattering angles owing to small spheres help the light take a prolonged path when it is scattered into the absorbing Si layer. As a result, the conversion efficiency of the solar cell is improved considerably [15]. Moreover, the combination of large and small spheres effectively prevents significant Ohmic loss as well as generate a larger scattering rate because the Ohmic loss is proportional to the volume of metal object (υ) whilst scattering rate is scaled with υ [2, 9].

The radius of the large sphere and its periodical distribution length (denoted as *p* in Fig. 1(b)) were studied singly by Chen *et al*. [15], and their optimal values were determined to be 100 nm and 792.67 nm (corresponding 10% optimal nanostructure coverage) as a result of investigating their roles. In their pioneering study, the radius of small sphere was assumed to be 1/5 of the large one and the location of the nanostructure in the ZnO:Al layer is fixed with *d*_{1} = 20 nm and *d*_{2} = 80 nm (Fig. 1(b)). It should be pointed out that, however, these factors interact with each other and create different effects on solar cells. Thus they should be designed simultaneously so that their roles can be balanced. In addition to the size and shape of nanoparticles, the incident angle of light is another important factor affecting the performance of thin-film solar cell as the light polarization changes the LSP characterizations (e.g. the shift of resonance frequency). Recent studies in [31] suggested that the optimal incident angle should be confined to a well-defined range of 0° to 35° for a thin-film solar cell whose Ag nanospheres are located on the top of a-Si substrate directly. However, for the thin-film solar cell considered herein that consists of nanostructures in the rear of semiconductor, relevant research is deficient. Thus we assume the solar cell is placed exactly perpendicular to the solar rays in this paper.

In the spherical coordinate, the azimuthal angle *θ _{k}* and polar angle

*φ*of the k

_{k}^{th}small sphere on the large sphere can be determined using the following simple formulae [32].

*N*is the number of evenly distributed small spheres. The radius of small sphere is calculated as ${R}_{2}=1.44{R}_{1}/\sqrt{N}$. The periodic length

*p*, radius

*R*

_{1}, location parameters

*d*

_{1}and

*d*

_{2}, and number of small spheres form a hyperspace for the optimization problem.

The optimization of the nanostructures is aimed at searching for the stationary point in that hyperspace spanned by the design variables so that a figure of merit is achieved. Either a larger short circuit current density or a higher open circuit voltage can enhance the conversion efficiency. A larger *J*_{sc} commonly means a higher absorption and better light trap. In such a configuration, we define *J*_{sc} as the cost function (figure of merit) to be maximized. If all light absorbed into the a-Si:H layer is collected, *J*_{sc} is formulated as

*e*denotes the charge on an electron,

*h*Plank’s constant,

*λ*wavelength and

*c*the speed of light in vacuum. The quantum efficiency $QE\left(\lambda \right)={P}_{abs}\left(\lambda \right)/{P}_{in}\left(\lambda \right)$ is the ratio of the power of the absorbed light ${P}_{abs}\left(\lambda \right)$ to that of the incident light ${P}_{in}\left(\lambda \right)$ within the Si film.

*I*stands for the relevant part of the solar spectral irradiance. In the simulation, 300 equally spaced frequency points are considered in the solar spectrum between

_{AM1.5}*λ*= 300nm and

*λ*= 840nm.

The constraints on the design variables should be considered in the optimization to reflect the real physical environment. Table 1 gives the lower and upper bounds for these parameters. It is emphasized that it generally takes longer time for the evolution algorithm to find the optimal values if the parameters are located on the bounds. Thus it is necessary to set wide ranges for these parameters, otherwise feasible solutions would be hard to achieve.

The evolution algorithm starts from a set of parent vectors, which are randomly selected in the given parameter space. For each parent vector, a fitness (figure of merit) is obtained by solving Maxwell’s equations. Following this, the offspring are produced by a certain mutation rule that is defined as the summation of the weighted difference between a pair of parent vectors and the third one. It is required that the offspring should be mended by a cross-over process to improve their diversity. In this step the weighing factor and cross-over probability are set to be 0.85 and 1, respectively, for following examples. The mutation and cross-over processes are repeated until the best performance is obtained.

## 3. Results and discussion

Two examples are aimed at demonstrating the capabilities of the proposed evolution algorithm for the two-level hierarchical nanosphere design. We abbreviate Example 1 with a bulging back (Fig. 1(b)) to ‘convex’, as opposed to ‘flat’ with planar back Example 2. In terms of the maximization of *J*_{sc}, the best parameters (Table 2
) are found within 55 generations. In comparison with the data obtained from the parameters reported in [15], it is remarkable that *J*_{sc} has increased from 17.78 (mA/cm^{2}) to 19.11 (mA/cm^{2}), around 7.48% enhancement. The equivalent coverage of the nanostructure in terms of *r*_{1} and *p* approximates to 3.77%, much less than the previously assumed optimum 10% [9, 15]. Interestingly, the distance between the top of nanostructure and Si film (*d*_{1} = 48.81 nm) is of a moderate value, indicating a consequence of the compromise between the destructive interference effect and the near-field coupling effect [14]. The former prefers a larger *d*_{1} to avoid the incidence being offset by the reflected fields, whilst the later wants a smaller *d*_{1} to radiate a strong near-field into the Si film.

A near-field optical picture is required to explain the enhancement of short circuit currency density. Figure 3
shows a side-by-side comparison of the near-field electric intensity (|**E**(*x*,y)|^{2} in the *x*-*y* plane and |**E**(*x*,z)|^{2} in the *x*-*z* plane) distribution of the separately-optimized and optimized nanostructures at a time step where the maximal |**E**(*x*,y)|^{2} is found in the *x*-*y* plane. It is emphasized that both planes should cover the center of the large sphere (if there is one) for a realistic comparison. We can observe that the field intensity is evidently boosted after the optimization both in vertical (Fig. 3(b)) and horizontal (Fig. 3(e)) directions. It is noted the electric intensity in the absorbing Si film also increases considerably, but it is not reflected in Fig. 3 unless plotted in logarithmic scale as its magnitude is relatively smaller than that in the near-field around nanoparticles. The gradient direction (the white arrows in Fig. 3(b), only the left part is arrowed because of the symmetry) of the electric field intensity in vertical symmetric plane illustrates that the nanoparticles help scatter and diffract light at wider angles before impinging upon the absorbing Si film.

Apart from easy fabrication, it is noted that changing the convex back to flat back for solar cells can generate a larger short circuit current through the proposed optimization process. Table 3
summarizes the parameters for four types of flat solar cells and their optical performance. The first has no rear-located nanostructure and can be used as a reference for quantifying the influence of the nanostructure on *J*_{sc}. The second consists of only large sphere and its *J*_{sc} (18.38 mA/cm^{2}) is a small fraction less than the third one (18.45 mA/cm^{2}) embedded with separately-optimized two-level spheres. These two cases indicate the significance of optimization without which the two-level hierarchical nanostructure might have little effect (or even negative influence). The last highlights the role of the optimization as *J*_{sc} is increased dramatically to 20.15 (mA/cm^{2}), indicating a 10.23% enhancement with respect to the first generation. If consider the worst case (*J*_{sc} = 17.19 mA/cm^{2}) in the first generation that has tested 100 configurations, the *J*_{sc} is increased by 17.22%. Note that it takes 65 generations for *J*_{sc} to reach convergence (Fig. 4(a)
), which is slightly longer than that in Example 1. The nanostructure coverage for this example is 11.09%, which is close to the 10% optimal value assumed in the previous study [15]. It is interesting to note that *d*_{1} = 1.2 nm is very small, indicating that the flat solar cell more relies on the near-field to improve *J*_{sc} than the convex solar cell. However, *d*_{1} should not be equal to zero, as a dielectric gap between the nanoparticles and the Si film is essential to maintain electrical isolation so that additional surface recombination owing to the presence of the metal is prohibited [14].

To compare the convergence of different parameters, they are normalized with respect to their maximal values in the optimization. As shown in Fig. 4(b), the radius of the large sphere *r*_{1} converges almost straightaway, which indicates that it is the most significant factor to the nanostructure according to the evolutionary algorithm. Another key factor to the solar cell is the thickness of Si film which converges to its upper bound within a few generations in the other examples that are not shown in the paper. The parameters that have less impact on the enhancement of performance, such as *d*_{1}, take more generations to converge. It is not surprising to observe that the cost function does not fluctuate significantly even though *d*_{1} drops at the late stage. The number of small spheres is another important factor as they exhibit the strongest electric intensity as shown in Figs. 3(c) and 3(f). These two figures also illustrate that the flat solar cell has stronger electric intensity than the convex one. Thus it can be regarded as one of the explanations for obtaining larger *J*_{sc} because the light path in Si film is improved accordingly by a stronger resonance.

The visualization of the relative absorption per unit volume (A(*x*,*z*) = log10[Im(*ε*)|**E**(*x*,*z*)|^{2})] helps reveal the mechanism of the enhancement of *J*_{sc}. The snapshots in Fig. 5
present the profiles of the relative absorption for three types of solar cells with the same arrangement as in Fig. 2. For the flat solar cell without nanostructure, Fig. 5(a) exhibits parallel absorption pattern in the Si film as light is reflected from the back metal in an inverse direction of the incident wave, indicating that the light travels in the Si film via the shortest path. The reflection from the convex back Ag layer, together with the scattering and diffraction caused by the nanostructure in Fig. 5(b), results in a ripple-like absorption in the Si film. Thus there exists a region (on the upper middle region of the Si layer) where absorption is rather weak (blue color) for the convex solar cell. For the flat solar cell with the rear-located optimized nanostructure, since the gap between the nanostructure to Si layer is relative small, the near-field of the surface plasmon resonance is capable of coupling into the lower part of Si film (red region in Fig. 5(c)) and therefore the effective absorption is improved. The absorption distribution in Fig. 5(c) clearly exhibits better absorption than the one in Fig. 5(b).

Figure 6
depicts *J*_{sc} in the sunlight spectrum for the abovementioned three solar cells. For the flat solar cell embedded with optimized nanostructure, the *J*_{sc} (red curve) almost matches that of the flat solar cell without nanostructure (black curve) at short wavelength. This is because the light is sufficiently absorbed within the front Si film before it reaches the nanostructure at this wavelength range. While in a longer wavelength region, the flat solar cell without nanostructures exhibits small *J*_{sc} owing to a lower absorption coefficient of Si in this spectrum [33]. However, when the light at a longer wavelength encounters nanostructure, plasmonic resonance is induced on the metal surface and thus the light is scattered and diffracted into the Si film at wider angles and a longer light path is achieved. This is the reason why both solar cells embedded with nanostructures show large *J*_{sc} than the flat solar cell between 650 and 840 nm over spectrum. An explanation for the difference between blue and red curves in Fig. 6 is that the propagating mode, one of the modes of the surface plasmon polaritons at back metal, is influenced by the inner concave metal surface. Therefore more energy is absorbed in the metal film due to the Ohmic loss. However, further investigation into the difference is needed in the future studies.

## 4. Conclusion

This work has investigated the benefits of large sphere and small spheres in thin-film solar cells, which yield a stronger intensity and wider scattering angles, can coexist in a single two-level hierarchical nanostructure consisting of evenly distributed small spheres on the surface of a large sphere. The short circuit current density of the thin-film solar cells can be improved considerably if the location and key parameters of those nanostructures embedded in the real dielectric layer are optimized by the evolutionary algorithm. We have found the coverage of the nanostructure is only 3.77% which is much smaller than the optimum 10% suggested by other researchers. Of five parameters in the design, the size of the large sphere is most significant, whilst the distance between the nanostructure and absorbing Si film is less important. The findings from the present study can be indicative to further enhancement of the conversion efficiency of thin-film solar cells by seeking the optimal values of their key factors.

## Acknowledgment

This work was supported by an Australian Research Council Discovery Early Career Researcher Award (project number DE120102906) and an Australian Research Council Discovery Project grant (DP110104698). The authors gratefully acknowledge Dr. Baohua Jia and Prof. Min Gu, from Centre for Micro-Photonics, Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, for helpful discussions.

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