## Abstract

In this paper, a graded SiN_{x} and SiO_{x}N_{y} structure is proposed as antireflection coatings deposited on top of amorphous silicon (*α*-Si) thin-film solar cell. The structural parameters are optimized by differential evolution in order to enhance the optical absorption of solar cells to the greatest degree. The optimal design result demonstrates that the nonlinear profile of dielectric constant is superior to the linear profile, and discrete multilayer graded antireflection coatings can outperform near continuously graded antireflection coatings. What’s more, the electric field intensity distributions clearly demonstrate the proposed graded SiN_{x} and SiO_{x}N_{y} structure can remarkably increase the magnitude of electric field of a-Si:H layer and hence, enhance the light trapping of a-Si:H thin-film solar cells in the whole visible and near-infrared spectrum. Finally, we have compared the optical absorption enhancements of proposed graded SiN_{x} and SiO_{x}N_{y} structure with nanoparticles structure, and demonstrated that it can result in higher enhancements compared to the dielectric SiC and TiO_{2} nanoparticles. We have shown that the optimal graded SiN_{x} and SiO_{x}N_{y} structure optimized by differential evolution can reach 33.31% enhancement which has exceeded the ideal limit of 32% of nanoparticles structure including plasmonic Ag nanoparticles, dielectric SiC and TiO_{2} nanoparticles.

©2012 Optical Society of America

## 1. Introduction

Solar cells offer a promising way to turn the solar light energy into electricity. Recent concerns about rapid depletion of crude oil and the pollution caused by fossil fuels have highlighted the need for sustainable energy generation. Nevertheless, more has to be done to further improve the efficiency and decrease the costs of solar-energy production. To address this need, amorphous silicon (*α*-Si) thin film solar cells were proposed as a low cost alternative to conventional wafer-based solar cells [1, 2]. However, the effective absorption depth of *α*-Si is only about 1 *μ*m, i.e., 2 orders of magnitude thinner than that of crystalline silicon. Furthermore, minority carriers in the *a*-Si have very short carrier diffusion length of around 300 *nm* or less [3]. Hence, it is imperative to improve the effective optical absorption of the *a*-Si:H thin-film solar cells by optimizing the nanostructure and achieve a similar absorption of crystalline silicon solar cells.

In most cases, a large portion of incident light is reflected back from the surface of *a*-Si because of the higher refractive index of *a*-Si and thus cannot be used to generate carriers in solar cells. With a thickness of a few microns or less, thin-film solar cells do not support traditional light-trapping techniques, such as the surface texturing extensively used in wafer-based silicon solar cells. Because such large scale surface texturing will cause both the scattering of light into the solar cell and the increase of minority carrier recombination in the surface and junction regions which will decrease the performance of such solar cells [4]. Thus, a novel and relatively simple method is required to enhance light trapping with minor modification and/or addition to the processing steps.

An effective solution is to use antireflection (AR) coatings deposited on top of a thin-film silicon solar cell in order to obtain an ideal broadband antireflection property in the visible and near-infrared wavelength range [5, 6]. In 1880, Lord Rayleigh analyzed reflections of waves from graded interfaces between two dissimilar media, and realized that “the transition may be so gradual that no sensible reflection would ensue” [7]. That is, for an infinitely thick, continuously graded AR coating, Fresnel reflectivity approaches zero.

Many research efforts on continuously graded refractive index structures have been carried out [8, 9]. A continuously varying porosity and effective refractive index of biomimetic nanostructures was obtained by using plasma etching or metal-assistant chemical etching, as well as nanorod arrays. An alternative approach to them is to use the multilayer graded refractive index structures [10]. Significant progresses have focused on finding refractive index profiles which minimize reflection for a given AR coating thickness. A linear profile is a reasonable starting point, but other profiles, such as the quintic profile, have been found to achieve superior performance [11, 12]. The oblique-angle depositions (OAD) of SiO_{2}/TiO_{2}, vapor deposition of materials and spin-coated organic materials with different refractive indices have been used to achieve the multilayer graded refractive index structures. However, all of the aforementioned techniques employ three-dimensional submicron structures which could cause mechanical and thermal reliability issues. Moreover, humidity can have impact upon their performance because of the presence of exposed voids.

Recently, a quasicontinuous graded index stack that uses custom tailored compositions of SiN_{x} and SiO_{x}N_{y} has been investigated [13]. The graded index SiN_{x} and SiO_{x}N_{y} thin films layers are achieved by varying the flow of reactive gases during plasma enhanced chemical vapor deposition (PECVD). The deposited thin film’s refractive index varies from high for Si-rich SiN_{x} to moderate for N-rich SiN_{x}. Therefore, a higher refractive index of SiN_{x} with *n* = 2.65 can be easily obtained from silicon rich SiN_{x} thin films by changing the gas ratio of the precursors SiH_{4} and NH_{3}. This SiN_{x} material with *n* = 2.65 has the very high dielectric constant *ε*_{high} of ~7.0, and can be used as the “bottom” layer of the graded antireflection coatings.

As N_{2}O is introduced, the material’s refractive index can be further reduced because of the oxygen component of SiO_{x}N_{y}. What’s more, the SiO_{x}N_{y} refractive index can be further controlled from *n* = 1.48 (SiO_{x}) to *n* = 1.9 (SiN_{x}) by varying the flow ratio of NH_{3} to N_{2}O. Therefore, the SiO_{x}N_{y} material with *n* = 1.48 has the lowest dielectric constant *ε*_{low} of ~2.2, which can be used as the “top” layer of the graded antireflection coatings.

Finally, by using both SiN_{x} and SiO_{x}N_{y} materials, one can conveniently obtain the thin films with a refractive index from *n* = 1.48 to *n* = 2.65 or a dielectric constant from *ε*_{low} *=* 2.2 to *ε*_{high} *=* 7.0 by varying the flow of reactive gases during plasma enhanced chemical vapor deposition.

Therefore, in this paper, a graded SiN_{x} and SiO_{x}N_{y} structure is proposed as antireflection coatings deposited on top of amorphous silicon (*α*-Si) thin-film solar cell in order to obtain an ideal broadband antireflection property in the whole visible and near-infrared wavelength range, thus enhance the light trapping and optical absorption performance of thin-film solar cells.

To improve the performance of such graded SiN_{x} and SiO_{x}N_{y} structure, a systematic study on the influence of the structural parameters on the efficiency of the enhancement is performed. The structural parameters of the proposed graded SiN_{x} and SiO_{x}N_{y} structure, including layer number *n*, graded structural coefficient *p*, thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high}, are optimized by differential evolution (DE) in order to enhance the optical absorption of *a-*Si:H thin-film solar cells to the greatest degree. Moreover, in order to verify the optical absorption enhancements of proposed graded SiN_{x} and SiO_{x}N_{y} structure, we have analyzed the electric field intensity distribution of *a-*Si:H thin film solar cell for reference cell without any antireflection coatings and cell with graded SiN_{x} and SiO_{x}N_{y} structure at the case of normal incidence at different wavelengths in the whole visible spectrum. Finally, in order to further investigate the light trapping performance of proposed graded SiN_{x} and SiO_{x}N_{y} antireflection coatings structure, we have compared its optical absorption enhancements with SiC and TiO_{2} nanoparticles structure.

## 2. Simulation model

In this paper, a graded SiN_{x} and SiO_{x}N_{y} structure is proposed as antireflection coatings deposited on top of amorphous silicon thin-film solar cell in order to obtain an ideal broadband antireflection property, thus enhance the optical absorption performance of thin-film solar cells.

To explore the effects of structural parameters for the proposed graded SiN_{x} and SiO_{x}N_{y} coatings, we perform a full-wave optical simulation of a thin-film hydrogenated amorphous silicon (a-Si:H) solar cell of the following configuration: ITO (20 *nm*)/a-Si:H (240 *nm*)/Al (80 *nm*) with graded SiN_{x} and SiO_{x}N_{y} antireflection coatings deposited on top of the ITO layer, as depicted in Fig. 1
.

In Fig. 1, the proposed graded SiN_{x} and SiO_{x}N_{y} antireflection coatings have five structural parameters, including layer number *n*, graded structural coefficient *p*, antireflection coatings thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high}. The whole graded SiN_{x}/SiO_{x}N_{y} structure is divided into *n* layers with equal thickness and the thickness of each layer is *d*/*n* (*d* is the total thickness of graded SiN_{x}/SiO_{x}N_{y} structure). The top and the bottom layers stand for the outer and inner layer of the graded SiN_{x} /SiO_{x}N_{y} structure respectively. As a result, from the top to the bottom, the dielectric constant increases from the lowest value *ε*_{low} to the highest value *ε*_{high}. Thus, the dielectric constant of each layer can be expressed by Eq. (1) defined as follows:

In the Eq. (1), the *x* is the relative thickness, and the *p* is the structural coefficient of the graded SiN_{x} and SiO_{x}N_{y} material. The dielectric constant (*ε*) of each layer as a function of the relative thickness (*x*) and structural coefficient (*p*) is calculated and shown in Fig. 2
. In Fig. 2, when structural coefficient *p* is equal to 1, the graded dielectric constant of each layer will submit to a linear profile. However, when structural coefficient *p* is bigger or smaller than 1, the graded dielectric constant of each layer will vary with a nonlinear profile.

Furthermore, the solar cell is simulated using the finite-difference time-domain (FDTD) method [14, 15], by solving 3D Maxwell’s equations with perfectly matched layer boundary conditions for monochromatic, normally incident plane waves. The boundary conditions of the solar cell enhanced with graded SiN_{x} and SiO_{x}N_{y} structure are presented in Fig. 3
.

In order to analyze the optical performance of the solar cell enhanced with graded SiN_{x} and SiO_{x}N_{y} coatings, we assume that frequencies *ω* of the incident waves cover the whole solar spectrum with irradiance *F*(*ω*) corresponding to the AM1.5G standard. The optical data for all the materials used in the simulated calculations are taken from the SOPRA database [16], and the complex refractive index data for amorphous silicon, indium tin oxide and aluminum are taken from SIAM1.mat, ITO2.mat and AL.mat respectively.

In our optical simulations, we calculate the spectral power absorbed by the amorphous silicon layer, defined as follows:

In the Eq. (2), the *V* represents the element volume, characterized by complex and frequency-dependent relative dielectric permittivity *ε*(*ω*), *E* is the electric field, whose distribution can be obtained in the simulation, and *ε*_{0} is the permittivity of free space. Moreover, to quantify performance of the optical absorption, it is more convenient to use the spectral absorption rate which is defined as following:

In the Eq. (3), the *Q*_{inc} represents the incident spectral power coming from the Sun over the solar cell surface *S*, and ${Q}_{inc}(\omega )=S\cdot F(\omega )$. Therefore, the *A*(*ω*) describes optical response of the element to the incoming waves of frequency *ω* regardless of the irradiance *F*(*ω*).

Finally, in order to investigate the broadband enhancement of the light trapping, we calculate the overall power absorbed inside the amorphous silicon layer by integrating the *a*-Si:H absorption rate over the AM1.5G solar spectrum, defined as follows:

The overall enhancement *G* then can be calculated using the following equation:

In the Eq. (5), ${Q}_{abs}^{total}(\text{Ref})$is denoted for the reference cell without any graded SiN_{x} and SiO_{x}N_{y} structure.

## 3. Single factor analysis

To improve the performance of proposed graded SiN_{x} and SiO_{x}N_{y} structure, a systematic study on the influence of the structural parameters on the efficiency of the enhancement is performed. In this section, we have analyzed single factor influence of structural parameters respectively, including layer number *n*, graded structural coefficient *p* and antireflection coatings thickness *d*.

#### 3.1 Effect of layer number (n) on the optical absorption enhancement efficiency of the graded SiN_{x} and SiO_{x}N_{y} structure

The initial conditions are: *p* = 4, *d* = 50 *nm*, *ε*_{low} = 2.2 and *ε*_{high} = 7.0, and the optimization range of layer number *n* is set to$2\le n\le 15$.

The spectral absorption rate *A(ω)* of the *a*-Si:H active region as functions of incident wavelength for different layer number *n* is shown in Fig. 4
. It is seen that the greater the *n* is, the higher the spectral absorption rate is, indicating that the near continuous gradient structure maybe has better spectral absorption rate than the multilayer gradient structure. When *n* is less than 5, the spectral absorption rate increases greatly with the increase of *n*, but it fluctuates little when *n* is greater than 5. Thus, the layer number of *n* = 5 will be more feasible for the practical preparation of the graded SiN_{x} and SiO_{x}N_{y} structure.

In order to further investigate the broadband enhancement of the light trapping, we analyze the dependence of the broadband light-trapping enhancement *G* on layer number *n* of the graded SiN_{x} and SiO_{x}N_{y} structure, which is depicted in Fig. 5
. According to Fig. 5, it is seen that there exist optimum value of layer number *n* which maximizes the overall broadband enhancement *G*. This figure reveals that with the increasing of layer number *n* from 2 to 15, the broadband light-trapping enhancement *G* firstly increases dramatically, and reaches the optimal values of *G* = 31.95% when layer number *n* is 5, then it begins to decrease slowly. When the layer number *n* is 10 or higher, its enhancement is worse than that of *n* = 5.

Therefore, the optimization results of Fig. 5 clearly demonstrate that the discrete multilayer graded antireflection coatings with the layer number of *n* = 5 can outperform near continuously graded antireflection coatings with the layer number of *n* = 10 or higher, by taking advantage of interference effects arising from reflections within the graded SiN_{x} and SiO_{x}N_{y} coating. According to this conclusion, the layer number *n* is set to 5 in the subsequent optimization calculation.

#### 3.2. Effect of graded structural coefficient (p) on the optical absorption enhancement efficiency of the graded SiN_{x} and SiO_{x}N_{y} structure

The initial conditions are: *n* = 5, *d* = 50 *nm*, *ε*_{low} = 2.2 and *ε*_{high} = 7.0, and the optimization range of graded structural coefficient *p* is set to$\text{1/16}\le p\le \text{16}$.

The spectral absorption rate *A(ω)* of the *a*-Si:H active region as functions of incident wavelength for different graded structural coefficient *p* is shown in Fig. 6
. It is seen that the greater the *p* is, the higher the spectral absorption rate is.

When *p* is less than 1, for example, *p* = 1/4 or 1/16, a major portion of SiN_{x}/SiO_{x}N_{y} layers have higher dielectric constant *ε* which are close to 7.0 according to the dielectric constant distribution description in Fig. 2. Therefore, a large portion of incident light is reflected back from the surface of SiN_{x}/SiO_{x}N_{y} layers which will lead to a very low spectral absorption rate in the *a*-Si:H active region.

On the contrary, when *p* is bigger than 1, for example, *p* = 8 or 16, a major portion of SiN_{x}/SiO_{x}N_{y} layers have lower dielectric constant *ε* which are close to 2.2, except the bottom SiN_{x}/SiO_{x}N_{y} layer has a high dielectric constant of *ε* = 7.0 which will raise a better antireflection performance, thus can achieve a very high spectral absorption rate in the *a*-Si:H active region.

In order to further explore the broadband enhancement of the light trapping, we analyze the dependence of the broadband light-trapping enhancement *G* on structural coefficient *p* of the graded SiN_{x} and SiO_{x}N_{y} structure, which is shown in Fig. 7
. From the optimization results of Fig. 7, it is seen that there exist optimum value of structural coefficient *p* which maximizes the overall broadband enhancement *G*. Figure 7 reveals that with the increasing of structural coefficient *p* from 1/16 to 16, the broadband light-trapping enhancement *G* firstly increases greatly, and reaches the optimal values of *G* = 31.95% when structural coefficient *p* is equal to 4, then it begins to decrease slightly. When the structural coefficient *p* is 16 or higher, its enhancement is worse than that of *p* = 4.

Finally, we compare the broadband enhancement for linear dielectric constant profile with structural coefficient of *p* = 1 and nonlinear dielectric constant profile with structural coefficient of *p* = 4. When *p* = 1, its broadband light-trapping enhancement *G* is only 29.05%. However, when *p* = 4, its enhancement *G* is 31.95% which obtains obvious improvement.

Therefore, the optimization results of Fig. 7 clearly demonstrate that the nonlinear profile of dielectric constant with the graded structural coefficient of *p* = 4 or higher is superior to the linear profile with the graded structural coefficient of *p* = 1.

#### 3.3. Effect of antireflection coatings thickness (d) on the optical absorption enhancement efficiency of the graded SiN_{x} and SiO_{x}N_{y} structure

The initial conditions are: *n* = 5, *p* = 4, *ε*_{low} = 2.2 and *ε*_{high} = 7.0, and the optimization range of antireflection coatings thickness *d* is set to$\text{10}nm\le d\le \text{100}nm$.

The spectral absorption rate *A(ω)* of the *a*-Si:H active region as functions of incident wavelength for different antireflection coatings thickness *d* is shown in Fig. 8
. According to the optimization results of Fig. 8, it is seen that with the increasing of antireflection coatings thickness *d* from 30 *nm* to 80 *nm*, the spectral absorption rate decreases gradually at the short wavelength of ~400 *nm*, but it increases gradually at the long wavelength of ~700 *nm*. Therefore, there should exist an optimum value of antireflection coatings thickness *d* which maximizes the overall broadband enhancement *G*.

Figure 9
analyzes the dependence of the broadband light-trapping enhancement *G* on antireflection coatings thickness *d* of the graded SiN_{x} and SiO_{x}N_{y} structure. It is seen that with the increasing of thickness *d* from 10 *nm* to 100 *nm*, the broadband light-trapping enhancement *G* firstly increases remarkably, then it begins to decrease slowly. When *d* = 60 *nm*, it reaches its optimal values of *G* = 32.30%.

Therefore, the optimization results of Fig. 8 and Fig. 9 reveal that the control of the antireflection coatings thickness *d* as a design parameter can significantly enhance the performance of light trapping for *α*-Si thin film solar cells.

## 4. DE-based design and optimization

#### 4.1 Differential evolution algorithm

In section 3, we have analyzed the single factor influence of the layer number *n*, graded structural coefficient *p* and antireflection coatings thickness *d* on the efficiency of light trapping enhancement respectively. When the single factor influence of one variable is analyzed and optimized, we assume that the other four variables are fixed constants. For example, when the influence of layer number *n* is analyzed, the other four variables such as graded structural coefficient *p*, thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high} are all set to the fixed values. In fact, these five variables are highly correlated with each other, hence they should not be analyzed separately but be considered all together in order to obtain the global optimal solution.

Heuristic search techniques have been successfully applied to many electromagnetic problems to pursue novel solutions which are difficult to obtain using the conventional design approaches. For instance, genetic algorithms (GAs) have been used to optimize wide frequency band of negative index metamaterials [17], fractal antenna-array [18], and so on.

Differential Evolution (DE) algorithm [19] is a new heuristic approach which mainly has three advantages: finding the true global minimum regardless of the initial parameter values, fast convergence, and using few control parameters.

Therefore, in this paper, differential evolution algorithm is employed to design and optimize the structural parameters of the proposed graded SiN_{x} and SiO_{x}N_{y} structure, including layer number *n*, structural coefficient *p*, thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high} in order to enhance the optical absorption of *a-*Si:H thin-film solar cells to the greatest degree. However, in order to simplify the optimization model and consider the feasibility of practical preparation for the graded SiN_{x} and SiO_{x}N_{y} structure, the layer number *n* is set to 5 in the following parameters optimization.

The block diagram of the DE design for parameters optimization of electromagnetic problems has been illustrated in detail in our previous literature [20, 21]. Firstly, an initial population is generated randomly. Then the four parameters including structural coefficient *p*, thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high} of each individual are used to generate the geometrical structure of graded SiN_{x} and SiO_{x}N_{y} for the finite-difference time-domain (FDTD) method to call. The fitness function is then calculated based on the broadband light-trapping enhancement of the graded SiN_{x} and SiO_{x}N_{y} structure. The DE judges if the termination criteria is met. If not, the population is regenerated through a DE process including selection, crossover and mutation. In this paper, the exponential crossover strategy of *DE/rand-to-best/1/exp* is applied in the DE. The DE process is repeated until the maximum number of the generation is reached.

In this paper, the goal of DE is to pursue the optimal structural parameters, including structural coefficient *p*, thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high} in order to enhance the optical absorption of *a-*Si:H thin-film solar cells to the greatest degree. Therefore, the fitness function of DE can be defined as follows:

#### 4.2 Parameters for the DE optimization

In this paper, the graded SiN_{x} and SiO_{x}N_{y} structure design is optimized to operate at the wavelength of 300 *nm* to 1000 *nm*, and the parameters for the DE optimization are listed in Table 1
.

The DE optimization experiments are executed under Microsoft Windows Server 2003 with 3.00 GHz of Intel(R) Pentium(R) 4 CPU and 2 GB of RAM. In our optimization experiments, the CPU calculation time for fitness evaluation of one individual is about 7 minutes or so. Therefore, the whole CPU calculation time for 200 generations with 30 individuals is about four weeks.

The convergence curves of DE for graded SiN_{x} and SiO_{x}N_{y} structure optimization are presented in Fig. 10
. From Fig. 10 it can be seen that the differential evolution algorithm almost converges before the maximum number of the generation is reached.

#### 4.3 Optimization results and discussion

The optimal solutions of graded SiN_{x} and SiO_{x}N_{y} structure design at 50, 100, 150 and 200 generations are listed in Table 2
. From the optimization results of Table 2, it can be seen that the objective function *G* of DE is getting higher and higher which increases from 32.19% to 33.31%, that means the optical absorption enhancement of a-Si:H thin-film solar cells can be improved to the greatest extent by the structural parameters optimal design using differential evolution method.

For example, when the generations of DE is 50, the optimal structural parameters including graded structural coefficient *p*, thickness *d*, dielectric constant of top layer *ε*_{low}, and dielectric constant of bottom layer *ε*_{high} for graded SiN_{x} and SiO_{x}N_{y} structure design are 3.34, 55.66 *nm*, 2.28, and 6.75 respectively, and the calculated dielectric constants for five layers are *ε*_{1} = 2.28, *ε*_{2} = 2.65, *ε*_{3} = 3.12, *ε*_{4} = 3.80 and *ε*_{5} = 6.75. The optical absorption enhancement of a-Si:H thin-film solar cells can achieve the value of *G* = 32.19%. However, when the generations of DE is 200, the optimal structural parameters for graded SiN_{x} and SiO_{x}N_{y} structure design are 20.00, 73.68 *nm*, 2.20, 7.00 respectively, and the calculated dielectric constants for five layers are *ε*_{1} = 2.20, *ε*_{2} = 2.27, *ε*_{3} = 2.36, *ε*_{4} = 2.52 and *ε*_{5} = 7.00. The optical absorption enhancement of a-Si:H thin-film solar cells has increased to *G* = 33.31% which has obtained obvious improvement.

Therefore, the design methodology presented in this paper that optimizes the structural parameters of graded SiN_{x} and SiO_{x}N_{y} antireflection coatings using a differential evolution technique is a more convenient means to pursue the optimal structural parameters in order to enhance the optical absorption of a-Si:H thin-film solar cells to the greatest degree.

Finally, in order to verify the optical absorption enhancements of proposed graded SiN_{x} and SiO_{x}N_{y} structure, we have compared the electric field intensity distributions of *a-*Si:H thin film solar cell for reference cell without any antireflection coatings and cell with optimal graded SiN_{x} and SiO_{x}N_{y} structure at the case of normal incidence at the red light wavelength of 700 *nm*, orange light of 625 *nm*, yellow light of 600 *nm*, green light of 525 *nm*, blue light of 500 *nm* and violet light of 400 *nm*, respectively, which are depicted in Fig. 11
.

The electric field intensity distributions comparison results of Fig. 11 clearly demonstrate the proposed optimal graded SiN_{x} and SiO_{x}N_{y} structure in this paper can remarkably increase the electric field intensity of *a-*Si:H layer and hence enhance the optical absorption of *a-*Si:H thin-film solar cells in the whole visible and near-infrared wavelength range.

## 5. Performance comparison with nanoparticles structure

In order to further investigate the light trapping enhancement performance of proposed graded SiN_{x} and SiO_{x}N_{y} antireflection coatings structure, we have compared its optical absorption enhancements with nanoparticles structure.

Recent research into the field of plasmonics has found an alternative way to improve light-trapping of thin film solar cells by using plasmon-enhanced light scattering from Ag metal nanoparticles deposited on the cell surface [22]. However, the Ag metal nanoparticles also exhibit very high parasitic absorption of incident light at plasmon-resonance frequency which limits the effectiveness of Ag nanoparticles to improve light absorption in the silicon layer.

Yu. A. Akimov, et al, have proven, with the aid of numerical modeling, that the dielectric particles such as SiC and TiO_{2} nanoparticles can also result in similar and even several times higher enhancements compared to the commonly used Ag plasmonic metal [23]. In their optimization results, the light absorption enhancements obtained for Ag nanoparticles, silicon carbide (SiC) nanoparticles, and titanium dioxide (TiO_{2}) nanoparticles are 10%, 29%, and 23%, respectively. Moreover, in their literature [23], the overall influence of dispersion and dissipation of the nanoparticle material have been analyzed by comparing the enhancements caused by nanoparticles of existing materials with complex permittivity *ε*(*ω*) and those of ideal materials, characterized by constant permittivity *ε*_{0} = Re[*ε*(*ω*_{0})] and zero dissipation. It reveals that the enhancements by ideal nanoparticles grow with the absolute value of the permittivity and saturate for highly negative and positive *ε*_{0} at a limit of 32%.

In this paper, we have also optimized the structural parameters of SiC and TiO_{2} nanoparticles using the differential evolution technique mentioned in previous section in order to enhance the optical absorption of *a-*Si:H thin-film solar cells to the greatest degree.

Figure 12
depicts the sketch of the thin-film *a*-Si:H solar cell and geometric structure of nanoparticles deposited on top of the indium tin oxide (ITO) layer. In Fig. 12, the deposited nanoparticles are assumed to be spherical and distributed in a square arrangement on top of the ITO layer. The nanoparticles structure has two geometrical parameters including the nanoparticle radius *r* and the lattice constant *a* which need to be optimized by differential evolution. The goal of DE is to pursue the optimal structural parameters *r* and *a* of nanoparticles which maximize the optical absorption enhancement *G* of *a*-Si:H thin-film solar cells. The optimization ranges for the nanoparticle radius *r* and the lattice constant *a* are $0nm<r\le a/2nm$ and $0nm<a\le 200nm$ respectively.

In this paper, the thicknesses of ITO, *a*-Si:H and Al for the *a*-Si:H thin-film solar cells are set to 20 *nm*, 240 *nm* and 80 *nm* respectively which refer to the literature [23]. The optical data for all the materials used in the simulated optimization are taken from the SOPRA database, and the complex refractive index data for silicon carbide (SiC) and titanium dioxide (TiO_{2}) are taken from SIC.mat and TIO2.mat respectively.

The optimal solutions of SiC nanoparticles, TiO_{2} nanoparticles and graded SiN_{x} and SiO_{x}N_{y} structure optimized by DE are listed in Table 3
. From the optimization results of Table 3, it is seen that the SiC nanoparticles and TiO_{2} nanoparticles can obtain the optimal light trapping enhancement of *G* = 28.68% and *G* = 24.18% repectively which are similar to the optimal solutions of *G* = 29% and *G* = 23% in the literature [23]. The optimal parameters for SiC nanoparticles and TiO_{2} nanoparticles are both *a* = 50 *nm* and *r* = 25 *nm*.

However, the proposed graded SiN_{x} and SiO_{x}N_{y} structure can achieve the optimal light trapping enhancement of *G* = 33.31% which is superior to the light trapping enhancement of *G* = 28.68% and *G* = 24.18% for SiC nanoparticles and TiO_{2} nanoparticles. Furthermore, the optimal graded SiN_{x} and SiO_{x}N_{y} structure optimized by differential evolution have already reached 33.31% enhancement which has exceeded the ideal limit of 32% of nanoparticles structure including plasmonic Ag nanoparticles, dielectric SiC and TiO_{2} nanoparticles reported in the literature [23].

Finally, in order to further investigate the broadband enhancement performance of the light trapping, we compare the spectral absorption rate *A(ω)* of the *a*-Si:H active region for optimal SiC nanoparticles, TiO_{2} nanoparticles and graded SiN_{x} and SiO_{x}N_{y} structure, which is depicted in Fig. 13
.

Figure 13 clearly illustrates that three structures improve light trapping in different spectral regions. Optimal SiC and TiO_{2} nanoparticles can enhance the optical absorption of silicon for wavelengths above 415 *nm* and 440 *nm*, which only cover some violet light spectrum and a little violet light spectrum, respectively. However, the proposed optimal graded SiN_{x} and SiO_{x}N_{y} structure can improve the optical absorption of silicon for wavelengths above 370 *nm* which cover the whole violet light spectrum. Therefore, the graded SiN_{x} and SiO_{x}N_{y} structure almost can enhance the optical absorption of *α*-Si thin film solar cells in the whole visible and near-infrared wavelength range.

## 6. Conclusion

In this paper, a graded SiN_{x} and SiO_{x}N_{y} structure is proposed as antireflection coatings deposited on top of amorphous silicon (*α*-Si) thin-film solar cell in order to obtain an ideal broadband antireflection property in the whole visible spectrum. The structural parameters of proposed graded SiN_{x} and SiO_{x}N_{y} structure are optimized by differential evolution (DE) in order to enhance the optical absorption of *a-*Si:H thin-film solar cells to the greatest degree. The optimal design of the structural parameters theoretically demonstrates that the nonlinear profile of dielectric constant with the graded structural coefficient of *p =* 4 or higher is superior to the linear profile with *p =* 1. What’s more, discrete multilayer graded antireflection coatings with the layer number of *n* = 5 can outperform near continuously graded antireflection coatings with *n* = 10 or higher, by taking advantage of interference effects arising from reflections within the graded SiN_{x} and SiO_{x}N_{y} coating.

Furthermore, in order to verify the optical absorption enhancements of proposed graded SiN_{x} and SiO_{x}N_{y} structure, we have compared the electric field intensity distributions of *a-*Si:H thin film solar cell for reference cell without any antireflection coatings and cell with graded SiN_{x} and SiO_{x}N_{y} structure at the case of normal incidence at different wavelengths in the whole visible spectrum. The electric field intensity distributions clearly demonstrate the proposed graded SiN_{x} and SiO_{x}N_{y} structure in this paper can remarkably increase the magnitude of electric field of *a-*Si:H layer and hence enhance the light trapping of *a-*Si:H thin-film solar cells in the whole visible and near-infrared wavelength range. Finally, we have compared the optical absorption enhancements of proposed graded SiN_{x} and SiO_{x}N_{y} structure with SiC and TiO_{2} nanoparticles structure, and demonstrated that graded SiN_{x} and SiO_{x}N_{y} structure can result in higher enhancements compared to the dielectric SiC and TiO_{2} nanoparticles. We have shown that the optimal graded SiN_{x} and SiO_{x}N_{y} structure optimized by differential evolution can reach 33.31% enhancement which has exceeded the ideal limit of 32% of nanoparticles structure including plasmonic Ag nanoparticles, dielectric SiC and TiO_{2} nanoparticles.

In conclusion, we have proven, with the aid of numerical modeling and differential evolution algorithm optimization, that the deposition of graded SiN_{x} and SiO_{x}N_{y} antireflection coatings on top of a thin-film silicon solar cell can significantly enhance photoelectron generation and hence, result in superior performance of thin film solar cells.

## Acknowledgments

This work is supported by the Research Fund of the International Science & Technology Cooperation Program of China (Grant No. 2011DFA52650) and the National Natural Science Foundation (Grant No. 50972111).

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