## Abstract

Optimization method which is based on the ant colony algorithm (ACA) is described to optimize antireflection (AR) coating system with broadband omnidirectional characteristics for silicon solar cells incorporated with the solar spectrum (AM1.5 radiation). It’s the first time to use ACA method for optimizing the AR coating system. In this paper, for the wavelength range from 400 nm to 1100 nm, the optimized three-layer AR coating system could provide an average reflectance of 2.98% for incident angles from ${R}_{ave}^{\theta +}$ to $80\xb0$ and 6.56% for incident angles from $0\xb0$ to $90\xb0$.

© 2014 Optical Society of America

## 1. Introduction

With the increasing urgent demands in clean energy, solar cells have been gaining much attention in recent years. One of the key problems for solar cells is how to decrease the surface reflection due to the large refractive index discontinuity between the semiconductor and the air. On the basis of destructive interference between the incident and reflected light, the reflection loss of the antireflection (AR) coating system, which was applied on the crystalline silicon solar cells, can be achieved less than 5% for one specific wavelength under the normal incidence [1–3]. Broadband AR coating system over a wide incident angle range are highly desirable for solar cells, which can increase light absorption in the active region up to a factor of 4n^{2} in a relative wide wavelength range, where n is the refractive index of the material [4, 5].

By now, various approaches for broadband and omnidirectional AR coating system design have been reported. They include the use of multilayer porous films [6], the biomimetic moth’s eye structure [7, 8], subwavelength surface Mie resonators [9], and etc.. Recently, a step-graded graded-refractive-index (GRIN) AR coating system with a refractive index as low as 1.05 has been demonstrated which could eliminate Fresnel reflection [10]. However, it is difficult to optimize the GRIN profiles, because the parameter space generally includes many local minima, which makes it unsuitable to find the local minima for deterministic optimization schemes. To meet this challenge, computational genetic algorithm (GA) [2, 11–13] and simulated annealing algorithm (SA) [14] methods have been applied in order to design optimized GRIN profiles for AR coating system.

The ant colony algorithm (ACA) is a heuristic optimization method, which was developed to solve traveling salesman problem (TSP) by Dorigo [15, 16]. The searching mechanism of ACA is based on the ants’ capability of finding the shortest path from a food source to their nest. The global optimum found by ACA is insensitive to the initial values which are often critical in conventional optimization algorithms. ACA has been proved to be a useful technique to solve optimization problems in feeder bus network design [17].

In this paper, according to the demands for the broadband and omnidirectional AR coating system, the iterative method of ACA was applied to optimize the AR coating system for silicon-based solar cells, with the objective of minimizing the average reflectivity over the 400 nm to 1100 nm which can be absorbed by silicon [14] from $0\xb0$ to $90\xb0$ of all the incident angle ranges.

## 2. Optimization algorithm

Figure 1 depicts a multilayer structure used in this paper. Each layer in this structure is assumed to be homogeneous and is characterized by its thickness ${d}_{i}$ with the refractive index ${n}_{i}$, i = {1, 2, …, N}. A plane wave is incident from the semi-infinite air region with refractive index ${n}_{0}$. For simplicity, the entire absorption layer is assumed by the bottom silicon substrate with the refractive index ${n}_{Si}$.

Assuming the possible maximal thickness of each layer is ${d}_{\mathrm{max}}$, and the refractive index range is [${n}_{\mathrm{min}}$, ${n}_{\mathrm{max}}$], the refractive index of the ${i}_{th}$ layer can be calculated by [18]:

According to our ACA coding, the optimization process for the AR coating system is illustrated as Fig. 2, according to the multi city-layer TSP (MCLTSP) model [18]. In Fig. 2, the four cities of A, B, C, D in one column is called one city-layer, and the Ng city-layers form a city-matrix. In the MCLTSP model, the traveler must start the tour from the first city layer to the next city-layer one by one until to the last city-layer. In such an open loop tour, one and only one city could be visited in each city-layer. When the traveler reached the last city, the tour was completed. Each completed tour produced a solution by the MCLTSP. For example, the tour shown in Fig. 2 could be expressed by {A C … B D} to represent an AR coating system {${c}_{1}^{1}$ = 0, ${d}_{1}^{1}$ = 0, ${c}_{1}^{2}$ = 1, ${d}_{1}^{2}$ = 0, …, ${c}_{N}^{g-1}$ = 0, ${d}_{N}^{g-1}$ = 1, ${c}_{N}^{g}$ = 1, ${d}_{N}^{g}$ = 1}.

The reflectance of AR coating system could be changed to the traveled distance in the MCLTSP model. The total traveled distance of one tour was defined by the average reflectance, ${R}_{ave}^{\theta}$ [2],

The intensity of trail information, which was used to simulate the pheromone of ants, was denoted between city i in the ${d}_{th}$city-layer and city j in the ${(d+1)}_{th}$ layer as $\tau (d,i,j)$, where d = {1, 2, ..., Ng-1}, i = {1, 2, 3, 4}, j = {1, 2, 3, 4}. Since the prior trail information was not available, the intensity matrix of trail information was first initiated as a fixed number ${\tau}_{0}$. All the ants with the number of K were randomly placed in the cities in the first city-layer. For any ant in the city i of the ${d}_{th}$ city-layer, the probability to visit the city j in the ${(d+1)}_{th}$ city-layer can be written in a formula as follows [18]:

The pheromone trail now could be expressed by:

where ρ was a random number within [0,1], e was the number of elite ants, and the updated amount of pheromone $\Delta \tau $ wasFor the ${k}_{th}$ ant,

The calculation procedure using ACA can be described as following with its implementation of the AR coating system optimization, as illustrated in Fig. 3.

The main six steps were:

- Setting up the parameters and initializing the pheromone trails,
- Putting the ants to the first city-layer,
- Each ant must go to the next city through a chosen path in available paths depending on the probability given in Eq. (7),
- Calculating the thickness and refractive index of the n-layer AR coating system by the traveled distance of all ant paths, and getting the value of ${R}_{ave}^{\theta}$,
- If the iteration cyclecounter reached the maximum value, stop the process, otherwise repeat steps 2-5. ${R}_{ave}^{\theta +}$ was obtained when the iteration process finished.

## 3. Numerical results

Before the extensive optimizations of AR coating system for typical crystalline silicon solar cells, comparisons were made between the GA, SA and ACA in AR coating applications with the published theoretical and experimental results, by using the same AR coating system and corresponding refractive indices optimized in Ref [14]. and [20] under the same range of wavelength and incident angle. The refractive index of a bulk crystalline silicon in [21]. was taken into account here. There is no a rigorous theory about how to select the parameters used in ACA-based method. Considering the calculate speed and stability, the number K of ants was set by 50 [22]. The ${R}_{ave}^{\theta +}$ was close to the published results in [14]. and [20] by roughly adjusting the parameters, and then the ${R}_{ave}^{\theta +}$ was minimized by fine adjusting the parameters several times. The parameters used in this paper for ACA-based method were shown in Table 1.The optimized average reflectance ${R}_{ave}^{\theta +}$ using ACA, as shown in Fig. 4(c), was 1.89% for λ = [400, 750] nm and θ = [$40\xb0$, $80\xb0$], as opposed to 4.90% for the same wavelength and incident angle ranges reported in [20]. and 3.54% in [14], as shown in Fig. 4(a) and 4(b), respectively. The detailed layer thickness and performance comparisons were given in Table 2.The thickness of each layer was changed a lot compared with other two calculation results by GA and SA methods. Further, both of the thickness and refractive index of three-layer AR coating system were optimized by ACA method at the same time. The index domain was defined by some practically realizable refractive indices ranging from 1.05 to 2.66 [2, 10, 14]. The optimized average reflectance ${R}_{ave}^{\theta +}$ was further decreased to 1.68% for λ = [400,750] nm and θ = [$40\xb0$, $80\xb0$], with the detailed structure parameters as shown in Table 3.The calculated reflectance performance was shown in Fig. 4(d). It can be seen from Fig. 4(a) and 4(b) that the AR coating system designed by GA has a high reflectance if the incident angle was larger than $75\xb0$, and SA optimized AR coating system has a high reflectance for λ<435 nm over the span of θ. Meanwhile, it can be observed that the ACA method could minimize the reflectance for most of the wavelengths and incident angles, which is a valuable property in the practical solar cell systems.

Using the ACA method incorporated with the solar spectrum (AM1.5G radiation), AR coating system for bulk crystalline silicon solar cells with three layers was optimized while the spectral range was from 400 nm to 1100 nm and incident angle range was $0\xb0$ to $90\xb0$. The detailed structure was shown in Table 4.The calculated reflectance performance was shown in Fig. 5. It can be seen that the overall reflectance was less than 10% over the incident angle less than $80\xb0$. The reflectance increased when the incident angle was larger than $85\xb0$. In Fig. 5, there were three areas with the reflectance less than 1%, one of which located the peak of solar spectrum at 495 nm. The optimized average reflectance ${R}_{ave}^{\theta +}$ was 6.56% for λ = [400, 1100] nm and θ = [$0\xb0$, $90\xb0$], and 2.98% for λ = [400, 1100] nm and θ = [$0\xb0$, $80\xb0$], as opposed to 3.40% for the same wavelength and incident angle ranges reported in [14] by using SA method.

## 4. Conclusion

In this paper, the ACA-based design method for broadband and omnidirectional AR coating system by optimizing the thickness and refractive index of each layer was theoretically demonstrated. The calculated reflectance performance showed that the ACA optimized AR coating system for silicon solar cells could in general minimize and flatten the angle-averaged reflectance over the spectral range from 400nm to 1100nm, which dominated the whole solar spectrum. The optimized three-layer AR coating system was shown to reduce the average reflectance to 6.56% over λ = [400, 1100] nm and θ = [$0\xb0$, $90\xb0$], and 2.98% over λ = [400, 1100] nm and θ = [$0\xb0$, $80\xb0$], respectively. The results obtained for this study showed that ACA-based optimization method was a very efficient design tool for the AR coating system design, which was applicable to other wavebands and material systems for solar cells or photodetectors.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61222501 and 61335004).

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