The Modelling of Light Absorption and Reflection in a SiO_x/Si Structure with Al Nanoparticles for Solar Cells

Abstract

In this study, the light propagation in a structure consisting of $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ on Si substrate with Al nanoparticles regularly placed in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer is considered. Numerical modelling is performed by solving the Maxwell equations for the electromagnetic waves. In distinction from the well-known finite-difference time-domain (FDTD) simulation technique, we do not solve time-dependent wave equations here; rather, we propose a new numerical technique. This technique allows us to determine the stationary amplitudes of the electromagnetic oscillations directly from the linear algebraic equation system. The obtained results apply to silicon solar cells with an $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ + Al top layer to maximise their efficiency. We found that 26 nm and 39 nm diameters of spherical Al nanoparticles are nearly optimal for a λ = 435.8 nm wavelength of the incident light. In addition, we evaluated the (nearly) optimal parameters of their placement in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer. The results show the possibility of increasing the efficiency of solar cells by increasing the light absorption inside the active Si layer from ≈60% to ≈80%. Future perspectives on the proposed method and its possible applications are discussed.

1. Introduction

The modelling of light scattering, reflection, and absorption processes in complex structures with nanoparticles is increasingly important for various practical applications, including solar cells [1,2,3,4], surface plasmon resonance sensors [5], photodetectors [6], and optically controllable surfaces of liquid crystals [7], etc. Here, we focus on the development of a new modelling technique with an application to solar cells.

Photovoltaic solar cells play a vital role among a wide range of renewable and environmentally friendly energy resources [1]. One key problem is reducing their price. One of the possibilities is to exchange the bulk Si or GaAs wafer for a thin film deposited on an inexpensive substrate such as plastic or stainless steel [8]. It is advantageous to boost the light absorption in the active layer of such a thin-film solar cell to increase its efficiency. This can be achieved by appropriate light-trapping techniques, making it possible to reduce the reflection from the front side of the cell and lengthen the optical path inside the active layer. Based on extensive experimental and theoretical investigations, a variety of light-trapping techniques have been proposed, including antireflection coatings [9], refractive index matching [10], surface texturing [11], diffraction gratings [12], photonic crystals [13], nanostructures [14], metallic nanoparticles [2], as well as the usage of plasmonic metamaterials to increase the efficiency of solar cells due to the confinement of the optical field [15,16,17].

In this theoretical investigation, we focused on the usage of Al nanoparticles. The effect of metal nanoparticles, placed on the front surface of the solar cell, is related to the excitation of surface plasmons, making it possible to trap the incident light and scatter it into the device. The excitation of surface plasmons can increase the light trapping and absorption in thin substrates and reduce the reflection, through three primary methods: light scattering into the device from the surface of metal nanoparticles (far-field effect); near-field increment; and the direct creation of charge carriers in the semiconductor layer [18]. The spectral response of the plasmonic nanoparticles is wavelength-dependent. Hence, the optical properties of these particles can be tuned by changing some of their factors such as size, shape, and constituent material [19]. It has been argued in [20] that Al is the best candidate among the commonly used metals, such as Al, Ag, Cu, and Au, due to its potential for non-resonant light coupling in a wide range of solar spectra. Many experiments have confirmed that Al nanoparticles are good candidates for broadband plasmonic scatterers, increasing the coupling of light into silicon [21]. Our actual calculations confirm this positive effect of Al nanoparticles and provide specific information about the optimal size and distribution of these nanoparticles in the nonstoichiometric silicon oxide ($\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$) front layer of a silicon solar cell.

In addition, aluminium particles are well oxidised, and their properties change slightly with shape and size. More importantly, their dispersion properties are stronger than those of silver nanoparticles [21]. It should also be noted that aluminium and aluminium oxide particles exhibit toxic effects only in excessive concentrations and, hence, their toxicity is generally low [22,23].

2. The Calculation Method

We consider the light propagation in a structure consisting of $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ on a Si substrate with Al nanoparticles regularly placed in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer. The propagation of light is described by the Maxwell equations for the electromagnetic waves [24], which in SI units read

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \mathrm{E}}{\partial t}}={\displaystyle \frac{1}{{\epsilon }_0\epsilon }}\left(\nabla \times \mathrm{H}\right)\\ {\displaystyle \frac{\partial \mathrm{H}}{\partial t}}=-{\displaystyle \frac{1}{{\mu }_0\mu }}\left(\nabla \times \mathrm{E}\right)\end{array}\right.$$

(1)

Here, $\mathrm{E}$ and $\mathrm{H}$ are the electric and magnetic field vectors, ${\epsilon }_0$ and ${\mu }_0$ are the vacuum dielectric and magnetic permittivities, whereas $\epsilon$ and $\mu$ are the corresponding complex relative permittivities. These equations are based on the Mie theory of light scattering [25]. It is convenient to rewrite Equation (1) in terms of dimensionless fields, i.e., $\widehat{\mathrm{E}}=\mathrm{E}/E^{\mathrm{*}}$ and $\widehat{\mathrm{H}}=\mathrm{H}/H^{\ast }$, where $E^{\mathbf{*}}$ and $H^{\mathbf{*}}$ are the normalisation constants, which obey the relation $H^{\ast }\sqrt{{\mu }_0}=E^{\ast }\sqrt{{\epsilon }_0}$. Furthermore, we normalise all distances to ${\lambda }_0$ (a wavelength) and time to $T_0={\lambda }_0/c$, where $c=1/\sqrt{{\epsilon }_0{\mu }_0}$ is the speed of light in a vacuum. In these dimensionless variables, the Equation (1) become

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \widehat{\mathrm{E}}}{\partial \widehat{t}}}={\displaystyle \frac{1}{\epsilon }}\left(\nabla \times \widehat{\mathrm{H}}\right)\\ {\displaystyle \frac{\partial \widehat{\mathrm{H}}}{\partial \widehat{t}}}=-{\displaystyle \frac{1}{\mu }}\left(\nabla \times \widehat{\mathrm{E}}\right)\end{array} \right.$$

(2)

where $\nabla =\left({\displaystyle \frac{\partial }{\partial x}},{\displaystyle \frac{\partial }{\partial y}},{\displaystyle \frac{\partial }{\partial z}}\right)$ contains the derivatives with respect to the dimensionless coordinates x, y and z. We typically use a hat to mark the dimensionless quantities; however, we have skipped this for x, y, and z to make the notations easier.

The known FDTD (finite-difference time-domain) simulation method [24,26,27,28,29,30] is often used to solve time-dependent equations. We propose an alternative approach, looking for the solution of the form $\widehat{\mathrm{E}}={\widehat{\mathrm{E}}}_0{e}^{-2\pi i\widehat{\omega }\widehat{t}}$, $\widehat{\mathrm{H}}={\widehat{\mathrm{H}}}_0{e}^{-2\pi i\widehat{\omega }\widehat{t}}$ with stationary complex amplitudes ${\widehat{\mathrm{E}}}_0=({\widehat{E}}_{0x},{\widehat{E}}_{0y}, {\widehat{E}}_{0z})$ and ${\widehat{\mathrm{H}}}_0=({\widehat{H}}_{0x},{\widehat{H}}_{0y}, {\widehat{H}}_{0z})$, depending on the coordinates x, y, and z. Here, $\widehat{\omega }=\omega /{\omega }_0$ is the angular frequency $\omega$, which is normalised to ${\omega }_0=2\pi /T_0$. Inserting this form into Equation (1), we obtain the following equations for ${ \widehat{\mathrm{E}}}_0$ and ${\widehat{\mathrm{H}}}_0:$

$$\left\{\begin{array}{c}2\pi i\widehat{\omega }\epsilon {\widehat{\mathrm{E}}}_0+\nabla \times {\widehat{\mathrm{H}}}_0=0\\ 2\pi i\widehat{\omega }\mu { \widehat{\mathrm{H}}}_0-\nabla \times {\widehat{\mathrm{E}}}_0=0\end{array} \right.$$

(3)

We have set $\mu =1$ in all our further calculations, as the materials considered here do not exhibit magnetic properties. The numerical solution of Equation (2) or (3) needs to be performed on a spatial grid. A convenient choice used in the FDTD method is the Yee grid [24,26,27,28], which we also use for the solution of (3). We consider the incident light propagating along the z-axis normal to the surface of the layered $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ structure with spherical Al nanoparticles within the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer. Periodic boundary conditions are applied to model periodically placed Al nanoparticles. Namely, we consider a cell with a single Al nanoparticle and periodic boundary conditions along the x and y axes. A set of discrete coordinates $x=x_1, x_2,\dots , x_M$, $y=y_1, y_2,\dots , y_M$ and $z=z_1, z_2,\dots , z_N$ is introduced with constant grid step sizes $∆x=x_{i+1}-x_i$, $∆y=y_{i+1}-y_i$ and $∆z=z_{i+1}-z_i$. The field components ${\widehat{E}}_{0x}$, ${\widehat{E}}_{0y}$, ${\widehat{E}}_{0z}$, ${\widehat{H}}_{0x}$, ${\widehat{H}}_{0y}$ and ${\widehat{H}}_{0z}$ are calculated on the Yee grid, i.e., ${\widehat{E}}_{0x}\left(x_i,y_j+{\displaystyle \frac{∆y}{2}},z_k+{\displaystyle \frac{∆z}{2}}\right)$, ${\widehat{E}}_{0y}\left(x_i+{\displaystyle \frac{∆x}{2}},y_j,z_k+{\displaystyle \frac{∆z}{2}}\right)$, ${\widehat{E}}_{0z}\left(x_i+{\displaystyle \frac{∆x}{2}},y_j+{\displaystyle \frac{∆y}{2}},z_k\right)$, ${\widehat{H}}_{0x}\left(x_i+{\displaystyle \frac{∆x}{2}},y_j,z_k\right)$, ${\widehat{H}}_{0y}\left(x_i,y_j+{\displaystyle \frac{∆y}{2}},z_k\right)$, and ${\widehat{H}}_{0z}\left(x_i,y_j,z_k+{\displaystyle \frac{∆z}{2}}\right)$ are considered. These quantities are calculated from a system of linear algebraic equations obtained from Equation (3), representing the derivatives by finite differences. Appropriate boundary conditions at $z=z_1$ and $z=z_N$ allow us to obtain a unique solution for a given $\widehat{\omega }$. No exceptions are applied to the interfaces between different subregions of the structure when the derivatives are formally represented by finite differences. This method provides a solution, where the $\widehat{\mathrm{E}}$ and $\widehat{\mathrm{H}}$ fields are continuous at these interfaces. This is just the physical solution.

We consider an extended structure consisting of an absorbing layer having the complex refractive index $n^{\ast }=n+i\kappa$ with $n=1$ and $\kappa >0$, an air gap, an $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ structure with Al nanoparticles, an air gap, and a second absorbing layer with the same $n^{\ast }$. A schematical view of this structure is shown in Figure 1. We recall that the refractive index $n$ and the extinction coefficient $\kappa$ are related to the complex dielectric permittivity $\epsilon ={\epsilon }^{\prime }+i\epsilon ″$ via $n^2-{\kappa }^2=\epsilon \prime$ and $2n\kappa =\epsilon ″$, or $\epsilon ={n^{\ast }}^2$. For a large enough dimensionless width $l$ of the absorbing layers, i.e., for $2\pi \widehat{\omega }l\kappa \gg 1$, boundary conditions can be set in the form of a linearly polarised incident plane wave at $z=z_1=0$ and a plane wave at $z=z_N$. It is true because the light reflected from the $\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}\right)/\mathrm{S}\mathrm{i}$ structure practically does not reach the boundary $z=0$, i.e., it is absorbed within $z\in \left[0,l\right]$ at $2\pi \widehat{\omega }l\kappa \gg 1$. At the other boundary, the light intensity is relatively small, and the light reaches this boundary propagating mainly in the normal or near-to-normal direction (in the shortest ways) due to the large absorption within $z\in \left[z_N-l,z_N\right]$. Thus, it is practically a plane wave, propagating in the positive direction of the z-axis. Hence, we can use the boundary conditions:

$${\widehat{E}}_{0x}=1, {\widehat{E}}_{0y}=0\hspace{1em}\hspace{1em}\mathrm{at}\hspace{1em}z=0,$$

(4)

$${\widehat{H}}_{0y}=n^{\ast }{\widehat{E}}_{0x},\hspace{1em}{\widehat{H}}_{0x}=-n^{\ast }{\widehat{E}}_{0y}\mathrm{at} z=z_N.$$

(5)

One can set an arbitrary real value of ${\widehat{E}}_{0x}$ at $z=0$, but this is just a trivial rescaling of the solution. The boundary condition Equation (5) is used at the spatial points $\left(x_i,y_i,z_N\right)$, where the electric and magnetic field components are evaluated by the linear interpolation of the values calculated on the Yee grid. An extension of the plane-wave solution ${\widehat{E}}_{0x}=e^{2\pi i{\widehat{\omega }n}^{\ast }z}$ to $z=-{\displaystyle \frac{∆z}{2}}$ is practically used as the other boundary condition, which is a slight modification of Equation (4), convenient for the Yee grid.

The value of $\kappa$ is important in these calculations, as we are looking for the case in which $\kappa \to 0$ with no reflections from the interface between the air and the absorbing media with $n^{\ast }=1+i\kappa$. We need to take a larger $l$ (proportional to $1/\kappa$) for a smaller $\kappa$ to ensure that $2\pi \widehat{\omega }l\kappa \gg 1$. Practically, we used the value of $2\pi \widehat{\omega }l\kappa$, corresponding to the decay of the amplitude $e^{2\pi i{\widehat{\omega }n}^{\ast }z}$ of the incident light to $~0.04$ in the first absorbing layer. A linear extrapolation to $\kappa =0$ was performed, using the results obtained at $\kappa =0.2$ and $\kappa =0.1$. We modified this calculation scheme slightly by reducing the width of the second absorbing layer to $l/2$, since this had no significant effect on the results.

The reflection and absorption were evaluated by computing the averaged (over one oscillation period) energy fluxes through cross-sections near the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ and Si surfaces and the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ interface, as well as the averaged energy flux of the incident plane wave at the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ surface. The latter was calculated precisely, taking into account the decay of the amplitude of this plane wave within the absorbing layer and the reflection from the boundary of this layer. The energy fluxes were calculated using the $z$-component of the normalised Pointing vector $\mathrm{R}\mathrm{e}\widehat{\mathrm{E}}\times \mathrm{R}\mathrm{e} \widehat{\mathrm{H}}$ (the dimensionless local energy flux density), noting that the real parts used here represent the physical $\widehat{\mathrm{E}}$ and $\widehat{\mathrm{H}}$ fields. The explicit expression for the averaged energy flux reads

$$\begin{gathered} J=∆x∆y\widehat{\omega }{\sum }_{i,j=1}^M{\int }_0^{1/\widehat{\omega }}{\left(\widehat{\mathrm{E}}\prime \times {\widehat{\mathrm{H}}}^{\prime }\right)}_{z}d\widehat{t}={\displaystyle \frac{1}{2}}∆x∆y{\sum }_{i,j=1}^M\left({\widehat{E}}_{0x}^{\prime }{\widehat{H}}_{0y}^{\prime }+{\widehat{E}}_{0x}^{″}{\widehat{H}}_{0y}^{″}-{\widehat{E}}_{0y}^{\prime }{\widehat{H}}_{0x}^{\prime }-{\widehat{E}}_{0y}^{″}{\widehat{H}}_{0x}^{″}\right)= \\ {\displaystyle \frac{1}{2}}∆x∆y{\sum }_{i,j=1}^M{\left({\widehat{\mathrm{E}}}_0^{\prime }\times {\widehat{\mathrm{H}}}_0^{\prime }+{\widehat{\mathrm{E}}}_0^{″}\times {\widehat{\mathrm{H}}}_0^{″}\right)}_z \end{gathered}$$

(6)

where the sum is taken over all grid points $\left(x_i,y_j,z_k\right)$ of the consider cross-section normal to the z-axis. The real and imaginary parts are marked by primes and double primes, respectively. These quantities are determined by linearly interpolating the values calculated on the Yee grid. The flux associated with the incident light at the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ surface is $J_0={\displaystyle \frac{2S\left(1+{\kappa }^2\right)}{4+{\kappa }^2}}{e}^{-4\pi \widehat{\omega }\kappa l}$, where $S=M^2∆x∆y$ is the surface area of the considered cross-section. In this calculation of $J_0$, the electromagnetic field is represented as a superposition of the incident and the reflected plane waves within $z\in \left[0,l\right]$ and as the transmitted plane wave within $z\ge l$, requiring the continuity of the resulting $\widehat{\mathrm{E}}$ and $\widehat{\mathrm{H}}$ fields at the boundary $z=l$. Let us denote by $J_1$, $J_2$, and $J_3$ the total fluxes (6) at the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ surface, the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ interface, and the rare Si surface, respectively. In these notations, we have $R=\left(J_0-J_1\right)/J_0$ for the reflection coefficient, $A_1=\left(J_1-J_2\right)/J_0$ for the absorption in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}$ layer, $A_2=\left(J_2-J_3\right)/J_0$ for the absorption in the Si layer, and $T=J_3/J_0$ for the transmission coefficient. These relations imply $R+A_1+A_2+T=1$.

An advantage of this method is that it is unconditionally stable. To the contrary, numerical instabilities can appear when solving the partial differential Equation (1). We encountered this problem when trying to solve Equation (1) straightforwardly for the structure considered here with spherical Al nanoparticles. A disadvantage is that the number of numerical operations and the required operative memory increase rapidly with the number of grid points if the linear system of algebraic equations is solved by the Gaussian exclusion method. However, thanks to the quasi-diagonal structure of the coefficient matrix, it is still possible to make calculations on a rough grid. For example, we can perform calculations with the grid step size equal to $D/4$, $D/6$, or $D/8$ in some cases, where $D$ is the diameter of a spherical Al nanoparticle. We used the step size $D/4$ in our calculations. In order to obtain more reliable results on such a rough grid, we performed an averaging over 4 possibilities on how to place the centre of the Al sphere in the Yee grid. The first possibility is to choose one of the points $\left(x_i,y_j,z_k\right)$, and the remaining 3 possibilities correspond to the shifts by $\left({\displaystyle \frac{∆x}{4}},\mathrm{0,0}\right)$, $\left(0,{\displaystyle \frac{∆y}{4}},0\right)$ and $\left({\displaystyle \frac{∆x}{4}},{\displaystyle \frac{∆y}{4}},0\right)$ from this point. Note that we set $∆x=∆y=∆z$.

3. Results

Using the method described in Section 2, we performed calculations for the $\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}\right)/\mathrm{S}\mathrm{i}$ structure with a $d=156$ nm thick $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer and a $12d=1.872 \mathsf{\mu }\mathrm{m}$ thick Si layer, with the refractive index of $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ being set as $n_{\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}}=1.56$. These values of $d$ and $n_{\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}}$ correspond to real samples investigated in our laboratory, whereas the $12d$ thickness of the Si layer was taken for convenience to simulate the situation, where the absorbance of this layer is high for the incident light with the wavelength $\lambda =435.8 \mathrm{n}\mathrm{m}$ used in our calculations.

First, we performed a test calculation of the reflection spectrum for the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ structure without Al nanoparticles, using data from the refractive index database (refractiveindex.info). The calculation results are compared with the experimentally measured spectrum in Figure 2. A reasonably good agreement is observed, with the theoretical spectrum being only slightly shifted toward shorter wavelengths. This might be related to a small uncertainty in the measured $d$ and/or $n_{\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}}$ values, as the reflection minimum is caused by the interference, which is sensitive to these values.

Calculations for the $\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}\right)/\mathrm{S}\mathrm{i}$ structure were performed at the wavelength $\lambda =435.8 \mathrm{n}\mathrm{m}$, indicated in Figure 2 by the vertical dashed line. This corresponds approximately to the maximum of the solar irradiance in space. The refractive index of Si is $n_{\mathrm{S}\mathrm{i}}=4.8482$ and the extinction coefficient of Si is ${\kappa }_{\mathrm{S}\mathrm{i}}=0.1813$ in this case [31]. We used $n_{\mathrm{A}\mathrm{l}}=0.58738$ and ${\kappa }_{\mathrm{A}\mathrm{l}}=5.2806$ for Al at $\lambda =435.8 \mathrm{n}\mathrm{m}$ [32]. Al nanoparticles of different diameters, i.e., $D=19.5 \mathrm{n}\mathrm{m}$, $D=26 \mathrm{n}\mathrm{m}$, $D=39 \mathrm{n}\mathrm{m}$ and $D=52 \mathrm{n}\mathrm{m}$, were considered.

Our method allowed us to calculate the absorption in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}$ layer and the Si layer separately. Since $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ is transparent, the absorption in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}$ layer is simply the absorption on Al nanoparticles. The absorption in the Si layer has a practical importance for applications in solar cells. The larger the fraction of the incident light absorbed in Si, the higher the efficiency of a silicon solar cell covered by the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}$ layer. Hence, we are looking for optimal parameters, which maximise the absorption in the Si layer. The relevant parameters considered here are the nanoparticle diameter $D$, the distance $L$ between the centres of the neighbouring nanoparticles, and the distance $h$ between the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ interface and the bottom of a nanoparticle. Moreover, the results essentially depend on the ratio $L/D$. The results depending on $L/D$ and $D$ are presented in Figure 3, where the absorption both in the Si layer and on Al nanoparticles (left), as well as the reflection $R$ from the whole $\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}\right)/\mathrm{S}\mathrm{i}$ structure (right), are shown, comparing these absorption and reflection values with those for a pure $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ structure without the Al nanoparticles (dotted lines). We set $h={\displaystyle \frac{D}{2}}$ in this calculation series. According to these results, the diameters $D=26 \mathrm{n}\mathrm{m}$ and $D=39 \mathrm{n}\mathrm{m}$ with $L/D$ around 1.5 (for $D=26 \mathrm{n}\mathrm{m}$) or 1.7 (for $D=39 \mathrm{n}\mathrm{m}$) appear to be nearly optimal for the wavelength $\lambda =435.8 \mathrm{n}\mathrm{m}$, making it possible to increase the absorption in the Si layer from ≈0.6 to ≈0.8 by adding the Al nanoparticles in the top $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer. Some absorption on the Al nanoparticles also takes place (dashed lines). However, this is accompanied by a decrease in the total reflection from the structure (Figure 3, right), resulting in a larger absorption in the underlying Si layer at a good choice of $D$ and $L/D$. This is known as the plasmon resonance effect [2,5,20,33]. We compared the R values with that of a structure with a continuous $15.7 \mathsf{\mu }\mathrm{m}$ thick Al layer (as in our real samples) between the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ and Si layers instead of the Al nanoparticles (the dot-dashed line in Figure 3, right). This comparison shows an undesirably large reflection (about $80\%$) from the structure with a continuous Al layer.

The same quantities depending on the gap $h$ between Al nanoparticles and the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}/\mathrm{S}\mathrm{i}$ interface are shown in Figure 4 for $D=26 \mathrm{n}\mathrm{m}$ and $D=39 \mathrm{n}\mathrm{m}$ at $L/D=1.5$ and $L/D=2$. As we can see, the optimal value of $h$, maximising the absorption in the Si layer is about 10 or 20 nm. In fact, the choice $h=D/2$, used in the calculation series presented in Figure 3, appears to be good or nearly optimal. In any case, a small gap appears to be useful.

4. Discussion

In this paper, the light reflection and absorption in a structure consisting of an $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer with spherical Al nanoparticles on the Si substrate are calculated. A new method is proposed for a numerical solution to this problem. Contrary to the well-known FDTD method, which solves the time-dependent Maxwell equations for electromagnetic waves, here we find the stationary oscillation amplitudes from a system of linear equations. These equations are obtained from Equation (3), representing the derivatives by finite differences and applying appropriate boundary conditions to a certain extended structure, as described in Section 2. An advantage of this method is its numerical stability. One problem, however, is that the required computational resources (the number of operations and the operative memory) increase very rapidly with the number of grid points if the system of linear equations is solved by the Gaussian exclusion method. Finding a more efficient method could lead to a breakthrough in solving such physical problems. For example, if the system of linear equations could be solved by applying a limited number of iterations of the kind $X=F\left(X\right)$, where $X$ is the solution vector, then the required computational resources would increase linearly with the number of grid points. Such a method would be relatively fast since it does not require integration over time.

The calculation results obtained here are applicable to the optimisation of solar cells. We have found that the light absorption in the Si layer can be increased from ≈60% to ≈80% at the wavelength $\lambda =435.8 \mathrm{n}\mathrm{m}$ by an appropriate choice of the parameters for the system of Al nanoparticles placed in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ top layer of a silicon solar cell. The diameters of these spherical Al nanoparticles $D=26 \mathrm{n}\mathrm{m}$ and $D=39 \mathrm{n}\mathrm{m}$ appear to be nearly optimal, placing them at the distance $L$ (measured between the centres of spheres) about $1.5D$ or $1.7D$ from each other. Moreover, according to our results, placing these Al nanoparticles not just on the Si surface but at a small distance from it is optimal. The optimal gap between the Si surface and the bottom of Al nanoparticles appears to be around 10 or 20 nm.

Our results are complementary to the existing theoretical results about the effect of Al nanoparticles. Routinely, this effect is judged from estimates of the extinction cross-section of a nanoparticle depending on its size and the wavelength of light (see, e.g., [25,33]). However, this does not give a precise and unambiguous answer to the relevant question: how useful are these nanoparticles in a specific application for a specific structure? In our particular application to solar cells, a relevant parameter is the absorption in the Si layer, since the rate of generation of the electron-hole pairs in the solar cell is proportional to this quantity. Knowing only the extinction cross-section, we cannot say much about the absorption in the Si layer, because a part of light energy is transformed into heat due to the absorption on Al nanoparticles. Our method allowed us to calculate the absorption in the Si layer separately from the total absorption, thus providing useful new information about the positive effect of Al nanoparticles on the efficiency of solar cells. In a future perspective, our calculations could be extended to a wide range of solar spectra. Our method could also be applied to various metal nanoparticles of different shapes.

5. Conclusions

In summary of this study, several conclusions can be made. Firstly, the light reflection and absorption in the $\left(\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}\right)/\mathrm{S}\mathrm{i}$ structure were studied using a new numerical technique, allowing us to find the stationary amplitudes of the electromagnetic oscillations by solving linear algebraic equations. In distinction from the well-known FDTD method, this new approach does not require the numerical integration of the Maxwell equations over time. An advantage of this method is its numerical stability, while improving its efficiency will require further investigation. Secondly, the new method of numerical modelling allowed us to calculate the absorption on Al nanoparticles and the absorption in the Si layer separately. The latter quantity is directly related (practically, is proportional) to the efficiency of a Si solar cell with an $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}+\mathrm{A}\mathrm{l}$ top layer, since the rate of generation of the electron-hole pairs in the solar cell is proportional to this quantity. Finally, nearly optimal sizes and placement of spherical Al nanoparticles in the $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{x}}$ layer were found (see Section 3 for details) for the wavelength $\lambda =435.8 \mathrm{n}\mathrm{m}$, making it possible to increase the absorption in the Si layer from ≈60% to ≈80% in this example. A more detailed investigation for various $\lambda$ and various shapes of metal nanoparticles is a prospective future application of our method.