Wide angle antireflection in metal nanoparticles embedded in a dielectric matrix for plasmonic solar cells

The photon density in solar cells is usually optimized through tailored antireflection coatings (ARCs). We develop an analytical model to describe metal hybrid nanoparticles (NPs)‐based ARC, where metal NPs are embedded in a standard ARC on a Si‐substrate. A point dipole approach is implemented to calculate diffuse reflectance by NPs, while transfer matrix method is used for specular reflectance from front surface. We found that embedding metal NPs in SiN ARC enhances the antireflection property of the former at non‐normal angles of incidence (AOI) of light. Electric field distribution patterns of radiation in the substrate by NPs are calculated for various AOI, which support the improvements in the antireflection property. Weighted solar power transmittances from ARCs are calculated, which show that Ag‐NPs (radius = 35 nm) embedded in SiN (thickness = 70 nm) performs better than SiN for AOI over 74°, whereas Al‐NPs (radius = 35 nm) embedded in SiN (thickness = 70 nm) performs better for AOI over 78°.


| INTRODUCTION
Solar cells suffer loss from front surface reflection of incident sunlight. To reduce this loss, an antireflection coating (ARC) is typically used. [1][2][3][4][5][6][7][8][9] As an alternative, metal nanoparticles (NPs), which show plasmonic effect and scatter the sunlight efficiently at resonance wavelength, have been suggested. [10][11][12][13][14][15] The scattering and absorption properties of such NPs have been studied and modeled extensively in previous years. [16][17][18][19][20][21][22][23] Gold NPs (Au-NPs) at front side of Si solar cells 10 and silver NPs (Ag-NPs) at front side of GaAs solar cells 11 have been demonstrated with improved performance in efficiency of the solar cell device. In this configuration, metal NP layers act as an ARC. Lesina et al have modeled and characterized ARCs based on Ag-NPs embedded in a SiO 2 dielectric matrix on silicon solar cells. 24 They used finite-difference time-domain (FDTD) simulation and have stated that Ag-NPs within SiO 2 gives promising results, although the broadband performance of SiO 2 -based ARC remains unbeaten at normal incidence. They also suggested that to complete the analysis, other metal NPs such as Al, Cu, and another ARC material such as SiN should be explored with thinfilm solar cells. Thus far, most studies have been performed only for normal incidence, since commercially available solar cells are usually coated with an ARC, which is such that the reflectance is minimal for a wavelength near the maximum of the solar spectrum under normal incidence. But in reality, a fixed solar cell on a house roof receives the sunlight throughout the day at various angles of incidence (AOI) of a wide range of wavelengths. Thus, it is important to compare the performance of standard (pure dielectric-based ARC) and hybrid ARC (NPs embedded in the dielectric-based ARC) at various AOI.
In this paper, we present an analytical model that combines transfer matrix method (TMM) 25,26 with Mie theory 27,28 to model the optical properties of NPs-based ARCs on a Si-substrate. When the sunlight incident on a rough interface (the hybrid ARC layer in our case), reflectance from interface divides into two parts-specular reflectance (R s ) and diffuse reflectance (R d ), similarly for transmittance as shown in Figure 1. 29 The specularly reflected light is obtained by efficient TMM method of multilayer structures, whereas for the diffusely reflected light, angular power distribution of radiation by an NP in the substrate is calculated. 30 It is assumed in the current study that the specular and diffuse reflectance occurrences are independent of each other and do not interfere.

| DEVICE STRUCTURE AND THEORY
A hybrid ARC made of nanocomposite (NPs and host medium), in which NPs are in a two-dimensional (2-D) array of equal period in a dielectric SiN matrix bounded with air and the substrate, is investigated. A schematic of the simulated device structure is shown in  absorption efficiency (Q abs ). The sum of both efficiencies is called extinction efficiency (Q ext ) of NPs. Radiative efficiency (Q rad ) of NPs equals to Q sca /Q ext , which tells how much light get scattered from and absorbed in the NP.
The TMM, also called 2 × 2 matrix method, is generally applied to determine reflection, transmission, and absorption in a multilayer thin film. 25,26 A plane wave incident on the proposed device is assumed. The SiN layer has a thickness of d 1 with refractive index n 1 . Therefore, a transmission matrix for the SiN layer can be written as where t jk and r jk are the Fresnel transmission and reflection coefficients at an interface jk. A propagation matrix for the wave propagating through the SiN layer is given by where λ and θ 1 are wavelength of the incident wave and angle of refraction in the SiN layer, respectively. By using the transmission matrix and the propagation matrix, the total transfer matrix, M, for the device is given by The transmission and reflection coefficients can be expressed from F I G U R E 4 Reflectance from Ag-NPs/SiN layer versus the light operating wavelength. Bottom layer is Si-substrate as shown in Figure 1. AOI = 00 corresponds to normal incidence. Ag(40%)/SiN refers that 40% surface cross section is covered by Ag-NPs and the rest is SiN. AOI, angle of incidence; NPs, nanoparticles [Colour figure can be viewed at wileyonlinelibrary.com] From above results, the specular reflectance is expressed as R s = | r| 2 and the specular transmittance is expressed as T s = nscos θs where s and 0 are corresponding to the substrate and incident medium (air).
Mie 27 , 28 obtained a solution for the interaction of a plane electromagnetic wave by homogeneous spheres of arbitrary index of refraction embedded in a homogeneous dielectric medium. The solutions are expressed in infinite series as extinction (Q ext ) and scattering efficiency (Q sca ) of a homogeneous sphere: where a n = ð Þ , and size parameter x is 2πn 1 r/λ. m is ratio of refractive index of NP (n r ) to that of surrounding medium (n 1 ); ψ n and ξ n are Riccati-Bessel functions; and r is radius of the sphere. In a dipole model, the NP is assumed as a dipole oscillating at its resonant frequency. The dipole generally excites in the direction of polarization of the incident light (regardless of front and back illumination). Therefore, the excitation angle changes with the AOI of light.
Mertz 30 has formulated a dipole radiating at a distance near a substrate and found an expression for the angular power distribution of scattered light from the NP in air and in the neighboring substrate, which is as follows: where θ is the observation angle and ϕ is the inclination of the dipole from vertical. The derivation of Equation (5) and explanations of L s,p , L p ⊥ , and L p X can be found in Mertz. 30 F I G U R E 5 Reflectance from Al-NPs/SiN layer versus the light operating wavelength. Bottom layer is Si-substrate as shown in Figure 1. AOI = 00 corresponds to normal incidence. Al(40%)/SiN refers that 40% surface cross section is covered by Al-NPs and the rest is SiN. AOI, angle of incidence; NPs, nanoparticles [Colour figure can be viewed at wileyonlinelibrary.com] The diffused reflectance and transmittance can be given from above equations as The sum of both reflectance (specular and diffuse) is the total reflectance from the front surface of device such as where f is fraction of cross section area covered by NPs at the front surface. The dipole approximation is valid when distance between the NPs are of the order of or more than their diameters such that Optical constants of metal NPs and refractive indices of SiN for the calculations are taken from Palik. 31 Linear interpolation is used to fit the optical constants to desired step size. The solar spectrum used in calculation is taken from Buie et al 7 and is shown in Figure 2. An angular dependent solar spectrum is ignored as it has no effect on   (Figure 3c). Al has a weak interband region near 875 nm. 16 The most useful quantity of NP for the ARC application is Q rad . The Q rad graph shows that Ag-NP has high radiation at 575 nm where the solar spectrum is at maximum. Al-NP also follows the solar spectrum graph. This makes them promising to incorporate in conventional

| SIMULATION RESULTS
SiN ARC. Figure 4 shows the reflectance spectra of the simulated device structure with Ag-NPs in SiN at various AOI. At normal incidence (AOI = 00 ), SiN shows the best performance. The reflectance minimum in this case is obtained at 536 nm. Ag(70%)/SiN shows the worst performance at normal incidence and reflects 10% to 20% of the incident light on average from the entire solar spectrum. As AOI of the light changes from normal to non-normal angle, we obtained increase in the reflection of SiN ARC. At 60 AOI, the reflectance curves of SiN ARC and NPs-based ARC become almost equal. And after this, NPs-based ARC outperform SiN ARC. Ag(70%)/SiN shows better performance among all the ARCs at higher AOI. SiN ARC shows around 40% reflection at 75 AOI. When SiN ARC reaches near to its critical angle at higher AOI, the reflectance becomes higher, whereas NPs do not show this behavior and performs better at higher AOI. Al-NPs also exhibits improvement in ARC performance at higher AOI. Figure 5 shows the reflectance spectra of the simulated device structure with Al-NPs at various AOI. The difference in reflectance between Ag-NPs and Al-NPs can be seen only at higher AOI, where the reflectance curves of NPs-based ARC shows the same behavior as the radiative efficiency curves of NP.
The electric field distribution profile of the light radiated from Ag-NP in the substrate is shown in Figure 6. At higher AOI, the Ag-NP radiates with the same order of intensity as at normal incidence, which is the reason that the NPs-based ARC performs F I G U R E 7 Transmittance from Ag-NPs/SiN layer into Si-substrate versus the light operating wavelength. AOI = 00 corresponds to normal incidence. Ag(40%)/SiN refers that 40% surface cross section is covered by Ag-NPs and the rest is SiN. AOI, angle of incidence; NPs, nanoparticles [Colour figure can be viewed at wileyonlinelibrary.com] relatively better than SiN ARC at higher AOI. The figure at normal incidence looks same as reported in Schmid et al. 22 The angular distribution of radiation in the substrate also benefits in waveguidebased trapping of the incident light. Soller and Hall reported that more than 80% of light radiated from dipole directs into the waveguide mode. 32 And Catchpole and Pillai measured experimentally enhancement in absorption by a factor of 7.5 due to the waveguide mode coupling. 33 The ARCs in solar cells are designed to minimize the reflectance and maximize the transmittance across the wavelength range of interest. Since the transmittance needs to be maximized where the solar spectrum has maximum intensity, we have calculated weighted solar power transmittance (T w ): where T(λ) and S(λ) are the transmittance and intensity of the AM1.5D solar spectrum at wavelength λ. λ 1 and λ 2 are the minimum and maximum allowed wavelengths. Figure 9 shows the T w curves of SiN ARC and metal NPs-based ARC. SiN ARC shows more than 90% transmittance at normal incidence as expected (Figure 9a

| DISCUSSION
The scattering and absorption in the NPs are highly influenced by the size and host medium of the NPs. The geometric cross section of a metal NP is generally smaller than the optical cross section, which means that smaller NPs absorb more than bigger ones. 37  NPs. 16 In our study, we focused to SiN as a host medium to present our idea, albeit it is not limited.
In this study, we have restricted ourselves to spherical NPs for less complexity in NP structure and because these are easy to model analytically. However, the shape of NPs plays a major role in the scattering of sunlight to the neighboring substrate. Atwater and Polman have published a review article and have shown that the cylindrical shape of Ag-NP scatters more fraction of the incident sunlight in the substrate than the spherical one. 20 The fraction of radiation by a dipole also varies with the position of NPs. It radiates more fraction of light in the substrate when in contact with the substrate than when in air. 19 Therefore, NPs embedded in an ARC must be in contact with the substrate.
Our study can be used to guide future experimental design. The optimizations of experimental work such as the optimization of nanoparticles size, the optimization of nanostructures etc. might lead to even better performance than that predicted by the simulations. 20 In this study, we performed the analysis to calculate the antireflection property of metal NPs embedded in a SiN layer and weighted solar power transmittance from the ARC layers into the substrate under incidence of the AM1.5D solar spectrum. However, this model can be extended to the analysis of net power gain in a device from 1 day to a whole year for a specific location and orientation.

| CONCLUSIONS
We performed a theoretical study on metal NPs with a Si-substrate over a wide AOI of the AM1.5D solar spectrum. An analytical model was developed for the analysis of ARCs based on metal NPs embedded in a SiN dielectric matrix in a 2-D array on the surface of substrate. A point dipole approach with TMM method was implemented to calculate the total reflectance by NPs-based ARC. We found that metal NPs enhance the antireflection property of conventional SiN ARC at non-normal AOI. At normal incidence, SiN ARC still performs the best. Electric field distribution patterns of radiation by NPs in the substrate support the improvements in antireflection performance.
We also obtained weighted solar power transmittance curves, which shows that Ag-NPs in SiN performs better than SiN over 74 AOI,