## Abstract

This numerical study investigates the influence of rectangular backside diffraction gratings on the efficiency of silicon solar cells. Backside gratings are used to diffract incident light to large propagation angles beyond the angle of total internal reflection, which can significantly increase the interaction length of long wavelength photons inside the silicon layer and thus enhance the efficiency. We investigate the influence of the silicon thickness on the optimum grating period and modulation depth by a simulation method which combines a 2D ray tracing algorithm with rigorous coupled wave analysis (RCWA) for calculating the grating diffraction efficiencies. The optimization was performed for gratings with period lengths ranging from 0.25 µm to 1.5 µm and modulation depths ranging from 25 nm to 400 nm under the assumption of normal light incidence. This study shows that the achievable efficiency improvement of silicon solar cells by means of backside diffraction gratings strongly depends on the proper choice of the grating parameters for a given silicon thickness. The relationship between the optimized grating parameters resulting in maximum photocurrent densities and the silicon thickness is determined. Moreover, the thicknesses of silicon solar cells with and without optimized backside diffraction gratings providing the same photocurrent densities are compared.

© 2011 OSA

## 1. Introduction

In order to maximize light absorption inside silicon solar cells, on the one hand, reflection losses at the front side have to be minimized and, on the other hand, the optical interaction length of photons, in particular in the wavelength region between 900 nm and 1140 nm, which contains 25% of the photons in the useable range of the solar spectrum between 300 nm and 1140 nm but is only weakly absorbed by silicon, has to be increased. The former can be accomplished by antireflection coatings and pyramid structures fabricated by anisotropic etching [1]. The latter can to some extent be achieved by pyramid structures. Increasing the silicon thickness is a further option, however, at the cost of higher material use and increased electrical losses [2,3].

Another approach are backside gratings that diffract the incident light to large propagation angles beyond the angle of total internal reflection and, thus, significantly increase the interaction length of long wavelength photons. This light trapping scheme was first studied by Sheng for thin film solar cells with amorphous [4] and later in more detail by Morf and Heine for crystalline silicon solar cells [5–7]. During the past years this concept has attracted the interest of several research groups, which investigated different grating geometries [8–10] as well photonic crystals at the backside of silicon solar cells [11,12]. However, in all these studies the thickness of the silicon solar cell was either kept constant or the influence of the silicon thickness on the optimum grating structure was neglected. Previously published preliminary results showed that the thickness of the solar cell has to be taken into account when optimizing the grating design [13]. In this paper, we perform an optimization of the grating geometry parameters for silicon thicknesses *d _{Si}* ranging from 1 µm up to 200 µm, which provides deeper insight into the relationship of these parameters. Different to the majority of other studies, we extend our investigation to a wide range of grating periods Λ and modulation depths

*h.*Moreover, we compare the silicon thicknesses of cells with and without optimized backside gratings having the same efficiencies.

## 2. Simulation model and method

Figure 1
depicts the structures of the silicon solar cells without and with backside grating investigated in this study. The flat front surface is covered with a single layer antireflection (AR) coating made of 80 nm silicon nitride (SiN_{x}), which minimizes the reflection losses at the solar cell front side. The rear side is covered with a 100-nm thick SiO_{2} layer, which increases the reflectance of the backside aluminum electrode.

In order to accurately model the impact of a backside grating on the efficiency of a silicon solar cell, precise knowledge of the imaginary part, *i.e.*, the extinction coefficient is indispensible. Various sources provide the wavelength dependent complex refractive index of silicon determined either by ellipsometry or intensity transmission measurements [14,15]. In this study, we use Green’s data [16]. For calculating the photocurrent density generated in thesolar cell we combined a 2D ray tracing algorithm with rigorous coupled wave analysis (RCWA) for determining the diffraction efficiencies of the gratings [17]. First, a database containing the diffraction efficiencies of gratings with different period lengths and modulation depths depending on the angle of incidence, wavelength, polarization, and diffraction order *m* was prepared employing the RCWA method. In a second step, the transfer matrix method was used to calculate the reflection coefficient of the front surface antireflection coating as a function of angle of incidence, wavelength, and polarization resulting in a second database. Next, for each grating configuration the light absorbed inside the silicon was calculated by means of a ray tracing algorithm, which sums up the absorption experienced by the rays according to Beer Lambert’s law. For a given wavelength each ray was traced until its fractional intensity was smaller than a minimum limit or the number of reflections at the backside grating exceeded six. Diffraction at the backside grating was taken into account up to the fifth order (*m*=5) generating new rays with fractional intensities according to the corresponding diffraction efficiency values stored in the database. At the front side, the fractional intensity of each ray propagating inside the silicon was attenuated according to the reflection loss. This procedure was performed for 85 wavelengths in the range of 300 nm to 1140 nm in 10-nm steps for both polarizations. The photocurrent density was obtained by summing up the number of absorbed photons over the solar spectrum (AM 1.5) and averaging over both polarizations assuming that each absorbed photon generates an electron hole pair, *i.e.*, electrical loss mechanisms are omitted. The photocurrent is defined by

*A(λ)*is the photon absorption inside the silicon,

*S(λ)*is the incident photon flux, and e is the elementary charge. The theoretical limit of the photocurrent density is reached in the case when all photons are absorbed in the silicon layer and generate an electron-hole pair, which corresponds to a photocurrent density of about −40 mA/cm

^{2}omitting electrical losses.

The correctness of our combined RCWA/ray tracing approach we verified by comparing the absorption spectra of 3-µm and 60-µm thick silicon solar cells with results of full RCWA simulations, which are much more computationally expensive. Figure 2
plots the results fortwo structures with gratings indicating a good match between both methods. The oscillations in the full RCWA simulation results are due to interference effects not taken into account by the ray tracing method. These oscillations are averaged out when calculating the photocurrent densities. The photocurrent densities derived from the results of the two methods differ by <0.1mA/cm^{2} for both structures without grating and for the 60-µm thick structure with grating. The difference of 0.4mA/cm^{2} for the 3-µm thick structure with grating can be mainly attributed to the fact that the RCWA simulations of the grating for the ray tracing method were performed with higher accuracy (35 harmonics) than the one of the whole full structure (10 harmonics) due to limitations in computer memory and computation time.

## 3. Results

Figure 3
plots the increase of the photocurrent density Δ*J _{ph}* as a function of period and modulation depth for different silicon thicknesses

*d*ranging from 1 µm to 200 µm. The increase of the photocurrent density is defined as Δ

_{Si}*J*(

_{ph}*d*= |

_{si})*J*(

_{ph}*d*) -

_{Si}*J*(

_{ph,ref}*d*)|, where

_{Si}*J*(

_{ph,ref}*d*) is the photocurrent of a solar cell with a 100 nm thick SiO

_{Si}_{2}layer and a backside aluminum electrode but without grating structure. These simulations were first performed for grating periods ranging from 0.25 µm to 1.5 µm and modulation depths ranging from 25 nm to 400 nm in steps of 25 nm to identify the global maxima. From the gradually changing patterns in Fig. 3 it becomes obvious that with increasing silicon thickness the optimum grating parameters resulting in a maximum photocurrent density shift to different values. Figure 4 and 5 show the photocurrent densities and the increase of the photocurrent density, respectively, in the region near the global maxima in steps of 12.5 nm. The bright areas in Fig. 4 indicate geometries with relatively higher photocurrent densities while dark areas correspond to geometries with relatively lower photocurrent densities. Note that the scales are different for each plot. The corresponding minimum and maximum values for each silicon thickness are given in the table. For 1-µm thin solar cells a grating period of about 625 nm and a modulation depth of about 50 nm lead to the strongest increase of photocurrent density by 4.4 mA/cm

^{2}, which corresponds to an improvement of about 35.3%. For larger siliconthicknesses, the geometries providing maximum photocurrent density shifts to significantly larger periods while the modulation depth changes moderately by a few tens of nanometers. For a thickness of 200 µm the optimum is found at a period of 950 nm and a modulation depth of 75 nm with an increase in photocurrent density by more than 1.3 mA/cm

^{2}, which corresponds to an improvement of about 3.9%. From the practical point of view, the fabrication of symmetrical rectangular backside gratings with periods in the above stated range on 6 inch wafers appears feasible. One way to define the gratings over a large area is interference lithography, where two coherent beams produce a periodic interference pattern, which is recorded in a photoresist and subsequently transferred into silicon by means of reactive ion etching [18]. An analysis of the simulation results plotted in Fig. 5 reveals that a deviation of the actual modulation depth from the optimized value by up to ± 20% will reduce Δ

*J*by 10%, at maximum.

_{ph}Finally, we performed a 2D interpolation for locating the global maxima of the photocurrent densities. Figure 6 plots the resulting optimized period lengths and modulation depths as a function of the silicon thickness. The fact that the optimized period lengths and modulation depths shift to higher values with increasing silicon thickness directly relates to the diffraction characteristics of the gratings. Since the absorption in silicon decreases with increasing wavelength the shorter infrared wavelengths (λ~900 nm) will not penetrate as deeply into silicon as the longer infrared wavelengths (λ~1100 nm). Thus, for thicker silicon solar cells the wavelength region that is relevant for light trapping by diffraction shifts to longer wavelengths. Consequently, also the optimum grating period and modulation depth will increase.

Figure 7(a)
plots the absolute and relative increase in photocurrent density Δ*J _{max}* with optimized gratings as function of the silicon thickness. For thin silicon solar cells (

*d*< 10µm) with backside diffraction gratings the increase of the photocurrent density is higher than in the case of thicker cells. Figure 7(b) plots the maximum photocurrent density of solar cells with backside gratings as a function of the silicon thickness in comparison with solar cells without backside gratings. The two dotted curves show the calculated photocurrent density for grating geometries optimized for a silicon thickness of 1 µm (Λ=625 nm,

_{Si}*h*=50 nm) and 200 µm (Λ=950 nm,

*h*=75 nm), respectively. The photon absorption inside the silicon layer of solar cells without backside grating but comprising a SiO

_{2}layer above the aluminum can be calculated by

*R*and

_{f}*R*are the wavelength dependent reflectance at the front side of the silicon and at the backside, and α is the absorption coefficient of silicon [19]. The grating optimized for asilicon thickness of 1 µm strongly improves the performance for solar cells with a thickness of up to 10 µm. Above this thickness the improvement starts to drop notably. On the other hand, the grating optimized for a silicon thickness of 200 µm enhances the efficiency of solar cells with thicknesses larger than 20 µm.

_{b}By comparing the simulation results of silicon solar cells with optimized backside gratings with cells without grating one finds that the silicon thickness can be strongly reduced without sacrificing efficiency. For example, a 75 µm thick silicon solar cell with optimized backside grating provides the same efficiency as a 200 µm thick cell without backside grating, which corresponds to a thickness scaling factor of about 2.7. Apart from reduced silicon raw material costs thinner solar cells also suffer lower electrical losses, which further will improve the overall efficiency. For thinner solar cells, the distance that the electrons and holes generated by the light have to travel to the electrodes is shorter. This reduces the probability of intrinsic loss processes in silicon such as Auger recombination and radiative recombination resulting in a better carrier collection with a higher open circuit voltage [2,3]. Previous studies have indicated that silicon solar cells with thicknesses of 55 µm to 90 µm lead to maximum cell efficiencies [2]. Figure 8
displays the silicon thickness of a solar cell with an optimized grating as a function of the silicon thickness of cells without gratings which lead to the same efficiencies neglecting electrical losses. A linear regression in the form of *d _{Si,grat}(d_{Si,plan})* =

*a d*+

_{Si,plan}*b*gives a regression coefficient

*a*=0.35 and an offset

*b*=1.97 leading to an average scaling factor of about

*s*= 1/

*a*~2.83. The table in Fig. 8 summarizes the data of the silicon thicknesses from cells with optimized gratings as well as the corresponding silicon thickness of cells without gratings resulting in the same efficiencies and the scaling factors, respectively.

## 4. Conclusion

In this numerical study we showed that the optimum period length and modulation depth of backside diffraction gratings used as light trapping structures in silicon solar cells strongly depend on the silicon thickness. The optimized grating parameters leading to maximum photocurrent densities were determined for silicon thicknesses ranging from 1 µm to 200 µm. For thin silicon solar cells (*d _{Si}* < 10 µm) the increase of the photocurrent density is higher than 10%. For 200-µm thick solar cells, the photocurrent density still can be increased by more than 1.3 mA/cm

^{2}, which corresponds to an improvement of about 3.9%. Compared to silicon solar cells without backside gratings the silicon thickness can be strongly reduced by up to a factor of 2.8 without sacrificing efficiency, which is beneficial both in terms of silicon material costs as well as reduction of electrical losses.

## Acknowledgments

This work was supported by the Austrian NANO Initiative under the grant PLATON Si solar (project no. 819660).

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