## Abstract

The optimal morphology of nanotextured interfaces, which increase the photocurrent density of thin-film solar cells, is still an open question. While random morphologies have the advantage to scatter light into a broad angular range, they are more difficult to assess with Maxwell solvers, such as the finite-element method (FEM). With this study we aim to identify necessary requirements on the unit cell design for the accurate simulation of nanotextured thin-film solar cells with FEM.

© 2015 Optical Society of America

## 1. Introduction

Thin-film solar cells have absorbers with thicknesses ranging from 0.1 to a few μm. For maximal absorption of the incident light in the absorber layer, *light trapping* techniques must be applied. Light trapping usually is done using nano-textured interfaces, which scatter the incident light. While this technology has been applied in thin-film silicon (Si) solar cells for a long time [1], it is currently also applied in other thin-film technologies such as the CIGS technology [2] and for polycrystalline silicon thin-film solar cells [3–6]. Besides randomly nanotextured interfaces, also the use of periodic nanotextures was studied theoretically [7, 8] and experimentally [9]. In recent years, structures that show both random and periodic characteristics have been investigated with great interest [10–12].

For further optimisations we need approaches that allow to estimate the optical performance of thin-film solar cells accurately. The effect of random nanotextures on these solar cells often was investigated with approximate methods such as the scalar scattering theory [13–19]. The scalar scattering theory is well suited for simulating randomly nanotextured interfaces, because it accepts data with side lengths of tens of micrometres as input. While the scalar approach works well for nanotextures with a moderate roughness, it was seen in the past that it becomes less accurate when the rms roughness significantly exceeds 100 nm [18].

Another approach is the *finite-element method* (FEM), a versatile tool that allows the simulation of various kinds of optical structures with high accuracy [20]. Because of this high accuracy, FEM and other Maxwell solvers such as the *finite difference time domain* method (FDTD) [21, 22], were often used for the study of solar cells with periodic structures: the periodicity allows the definition of unit cells with short side lengths in the range of hundreds of nm up to a few micrometres.

The simulation of randomly nanotextured solar cells with such Maxwell solvers is more involved, because these simulations are usually done with periodic boundary conditions which – naturally – do not reflect the random character of the nanotextures. Because of computational cost the unit cell should be small, hence the statistical information contained in the unit cell may not sufficient to represent the nanotexture adequately. Note that this is not a problem of the Maxwell solver itself, but purely of the limited unit cell width.

The goal of this paper is to identify guidelines for accurately and reliably simulating randomly nanotextured thin-film solar cells with FEM. To achieve this goal, we will use FEM to perform a *numerical experiment* in which we study the influence of the unit cell width and the construction of its boundaries onto the accuracy of the results. This work is very important as many groups use FEM for studying solar cells but concrete guidelines are still missing. We are not only interested in FEM because of the high accuracy of this method but also because we want to use it to assess the validity of the fast scalar approach in the future.

As a platform for this study we use the thin-film silicon technology with record efficiencies ranging from 11.4% (stabilised) [23] for single junction solar cells up to 16.3% (initial) for triple junction solar cells [24]. Despite these rather low efficiencies and the difficult market situation [25], this technology is very well suited for such studies because of its high maturity: thin-film silicon solar cells have been developed for the past 40 years [26], such that we are able to manufacture thin-film silicon solar cells in a very controlled and reproducible way.

## 2. Method

#### 2.1. Generating nanotextures

With the *Perlin noise* algorithm *random nanotextures* with well controlled lateral feature size (LFS) and root-mean-square (rms) roughness (*σ _{r}*) can be generated [27, 28]. Therefore, it was used in the past for optimisation studies based on the scalar scattering theory [29, 30].

As illustrated in Fig. 1, Perlin textures are generated by assigning uniformly distributed random numbers to a grid of points, where the mutual distance determines the LFS. After interpolating the points in-between these major grid points with a cosine interpolation and subtracting the average plane, the texture is vertically scaled to obtain the required rms roughness. The cosine interpolation ensures that the first derivatives on the major grid points vanish. Hence, also the first derivatives in normal direction to the unit cell border vanish on the borders.

In order to prevent numerical instabilities it is important to choose the LFS such that it is a multiple of the distance between two points of the grid. For example, if two grid points are 20 nm away from each other, the LFS should be a multiple of 20 nm.

#### 2.2. Calculations using the finite-element method

We performed the FEM calculations with the software Ansoft HFSS, release 13.0.2. With a custom-made algorithm, we imported AFM scans or Perlin textures to create 3D models of thin-film solar cell structures with nanotextured interfaces. HFSS determines the optimal size of the finite-element through an adaptive meshing process, which tries to keep the mesh density at a minimum to limit the simulation duration, while ensuring a high accuracy of the results.

With FEM, the electric and magnetic fields **E** and **H** inside the unit cell are calculated. The absorption of the *j*-th layer *A _{j}* is obtained by integrating across the layer volume

*V*,

_{j}*ε*

_{0}and ${\tilde{\epsilon}}_{j}$ denote the electric permittivity of vacuum and the

*j*-th layer, respectively, and

*ω*is the angular frequency.

From the absorption profile the *implied photocurrent density J*_{ph} is calculated, which we use as a figure of merit. Here, we assume that every photon absorbed in the absorber leads to the creation and subsequent collection of an electron-hole pair. Hence, all the light absorbed in layers other than the absorber is assumed to be a loss. Mathematically, *J*_{ph} is given by [8]

*A*

_{abs}denotes the absorption in the absorber layer, ΦAM1.5 is the photon flux according to the AM1.5 spectrum, and

*λ*

_{1}and

*λ*

_{2}are the lower and upper limits of the integral, respectively.

*λ*

_{2}usually corresponds to the absorber band gap. Finally,

*e*denotes the elementary charge. Using the same method, current losses in the supporting layers can be calculated, which enables us to directly assess the different optical losses in the solar cell.

By using the implied photocurrent density we neglect all the recombination losses that can lead to a reduction of the short-circuit current density. This allows us to fully concentrate on the optics of our solar cell structure.

#### 2.3. Boundary conditions

For performing the FEM simulations, we construct unit cells with periodic boundary conditions on the side and perfectly matched layers at top and bottom. In order to remove sharp edges at the corners of the simulated area, we chose two different approaches, as illustrated in Fig. 2.

### 2.3.1. Tukey window

With the *Tukey-window* approach, which is illustrated on the left hand side of Fig. 2, all values on the corners of the simulated domain are smoothly brought to zero by multiplying with a Tukey-window function, also called *tapered cosine window* [31]. This function is illustrated in Fig. 2(t1); it is given by

*r*. If

*r*= 0, the window is a rectangular window. For

*r*= 1, the window is equal to a

*von Hann*window, also called

*raised cosine window*.

For using the Tukey window in FEM simulations, *r* must not be too large as otherwise too much information is lost. On the other hand, *r* must not be too small either in order to prevent too sharp edges on the border that would lead to high-frequency resonance modes. In this work, we always use a value of *r* = 0.3. As applying the Tukey window leads to a reduction of the rms roughness, the nanotexture is rescaled in order to bring it back to the initial roughness. Figure 2(t2) shows a unit cell constructed with this approach: it was obtained by applying Eq. (3) with *r* = 0.3 to the original texture shown in Fig. 2(o).

### 2.3.2. Unit cell with *enforced periodicity*

Applying the Tukey-window approach leads to loss of information on the edges of the structure. As Perlin textures are numerically generated, we also have the possibility to generate them with periodic boundaries on purpose, as illustrated on the right hand side of Fig. 2. The construction of such textures is sketched in Figs. 2(p1)–2(p3): first a texture with dimensions slightly larger than the final texture has to be generated [Fig. 2(p1)]. Prior to the cosine interpolation, the last row and column of random points are replaced by the first row and column of random points [Fig. 2(p2)]. Then, the cosine interpolation is performed in the usual way. Finally, the last row and column are cut off, resulting in a structure that is perfectly periodic at the edges [Fig. 2(p3)]. The application of this method to the original texture [Fig. 2(o)] leads to the texture shown in Fig. 2(p4). The periodicity of this texture is confirmed in Fig. 2(p5), where 2 × 2 textures from Fig. 2(p4) are stuck together.

## 3. Experimental and simulation details

The experiment and the simulations were performed with the layer structure shown in Fig. 3(a) and detailed in Table 1. Light enters the solar cell via a 1.8 mm thick glass layer, which is covered with nanotextured fluorine-doped tin oxide (SnO_{2}:F) that acts as electrical front contact. The SnO_{2}:F layer is followed by the p-i-n layer stack. The p- and n-doped layers are made from hydrogenated silicon oxide (SiO* _{x}*:H) [32–35], while the i-layer, which is the

*absorber layer*, is made from hydrogenated amorphous silicon with a bandgap of

*E*≈ 1.7 eV (a-Si:H). Last comes the metal back layer, which acts as a back reflector and electric back contact. Experimentally, it consists of a Ag-Cr-Al stack. As the Cr and Al layers do not affect the optical performance of the solar cell, they are omitted in the simulations.

_{g}The optical properties of the different layers are implemented in the simulations using the complex refractive index (*n* + *ik*). The *n, k*-data for SnO_{2}:F of Asahi U-type [36] was determined by Sap *et al.* [37]; the *n, k*-data for glass (*n* = 1.43) was obtained with the same method as average of experimental data from the Asahi U-type glass substrate. For the p-, i-, and n-materials, the *n, k*-data was determined using ellipsometry. For the back silver layer, data from the book of Palik was used [38].

For all the simulations the unit cells were constructed with conformal layer growths, *i. e.* the same interface morphology was used for all nanotextured interfaces. In the FEM simulations, light was incident directly into a 300-nm-thick glass layer, which is capped by a perfectly-matched layer made from glass. Reflection losses due to the air glass interface were then taken into account using a post process.

The FEM simulations were performed on a workstation with a dual six-core Intel^{®} Xeon^{®} X5670 Processor and 24 GB DDR3-1333 RAM. With this hardware a complete wavelength spectrum is simulated in between approx. three hours and several days.

## 4. Results and Discussion

#### 4.1. Results on a real solar cell

Before we perform simulations of solar cells with Perlin textures, we compare simulated values to experimentally obtained results. For determining *J*_{ph} with the simulations, the absorption profile between wavelengths of 300 and 900 nm is obtained in steps of 10 nm.

Note that for a real solar cell it is not possible to determine the absorption of the absorber layer experimentally. Instead, we use the external quantum efficiency (EQE) of the solar cell from which the short-circuit current density can be determined. The EQE cannot be compared directly to the *A*_{abs} as a fraction of the absorbed photons may not contribute to the EQE because of recombination effects. On the other hand, photons absorbed in the doped p- and n-layers, may contribute to the EQE but are not present in *A*_{abs}.

However, for a good thin-film silicon device the fraction of electron-hole pairs lost due to recombination is very low as recombination mainly affects the open-circuit voltage, which is not considered in this work. Therefore it is justified to compare the *A*_{abs} spectrum obtained from the simulations to the measured EQE spectrum of fabricated devices.

The validation is based on a solar cell deposited onto pyramid-like structured SnO_{2}:F of VU type from Asahi Glass Company, which is shown in Fig. 3(b). This nanotexture has an rms roughness of *σ _{r}* ≈ 40 nm and a correlation length of

*ℓ*≈175 nm.

_{c}We performed FEM simulations with unit cells of three different side lengths, namely 400 nm, 600 nm, and 1000 nm, *i. e.* about 2, 3 and 5 times the *lateral* correlation length. Using a Tukey window function, the heights at the unit cell borders were set to 0. The thickness of the p-layer was set to 20 nm (instead of 12 nm) as for a too thin layer the mesh density could become very high resulting in a very long calculation time. The p-layer is made from SiO* _{x}*:H which only is absorbing up to 400 nm, where the AM1.5 photon flux is very small. Hence, the effect of the thicker p-layer on the implied photocurrent density can be neglected. Further, the SnO

_{2}:F layer was constructed such that its minimal thickness is the layer thickness mentioned in Table 1. This was done to prevent “negative thicknesses” that otherwise could appear at some points for structures with a very high rms roughness.

Figure 4(a) shows the measured EQE (symbols), and the simulated *A*_{abs} spectra (lines). All four data sets are very close to each other. In contrast to the measured data, the FEM curves show some oscillations, which are likely caused by lateral periodicity due to the small unit cell dimensions used for FEM. In addition, in FEM simulations all the light is treated as coherent, while light will propagate incoherently in thick layers of real solar cell structures. The short-circuit current density of the measured solar cell is *J*_{sc} = 17.2 mA/cm^{2}. For the calculations performed with FEM method, the implied photocurrent densities *J*_{ph}are 17.4 mA/cm^{2}, 17.4 mA/cm^{2} and 17.8 mA/cm^{2}, for the 400 nm, 600 nm and 1000 nm unit cells, respectively.

The strongest deviations between the three FEM curves are observed above 700 nm: while the 600 nm curve is rather smooth, the 400 nm curves shows a sharp peak and a sharp valley, which cancel each other out such that the total *J*_{ph} is similar for both. The 1000 nm curve exhibits two peaks in this region that lead to the additional 0.4 mA/cm^{2} in *J*_{ph}. All the three *J*_{ph} values are higher than the measured one, which likely is due to the difference between *J*_{sc} and *J*_{ph}, as discussed above. A closer look on the absorption profile for the FEM simulation with 600 nm unit cell width is taken in Fig. 4(b).

The good agreement of experimental and numerical results – at least for the rather shallow Asahi VU-type nanotexture – is a good starting point for simulations using Perlin textures. However, these results do not yet allow to give guidelines on the required unit cell size. Therefore, simulations with Perlin textures are required that we present below.

#### 4.2. Results on solar cells with Perlin textures

Because calculations with FEM are very time-consuming, we have to limit ourselves to a small number of simulations. We perform these simulations for cells with a 300 nm thick a-Si:H absorber and for Perlin textures with a constant aspect ratio of LFS : *σ _{r}* = 3 : 1. This ratio is chosen, because we expect that on these rather shallow features cells with similar good electrical properties can be deposited, regardless of the absolute values of the LFS and

*σ*. All the

_{r}*J*

_{ph}values obtained from FEM are the average values of at least three simulations of Perlin textures with the same

*σ*and LFS values.

_{r}We aim to keep the simulation time short by keeping the side length of the simulated domain low. On the other hand, the unit cells must be large enough to allow a sufficient statistical representation of the structure. Note that the correlation length of Perlin textures is related to the lateral feature size by a factor of approximately $\sqrt{2}$.

Figure 5 shows the FEM results obtained on solar cells with Perlin textures with unit cells constructed with the Tukey window approach [Fig. 5(a)] and enforced periodicity [Fig. 5(b)]. The results are presented for different unit cell widths. In principle. we wanted to show results for a unit cell with of of 3×LFS and 5×LFS. However, with 5×LFS side length we could not go further than LFS = 420 nm because of limited computer power. To obtain results for larger LFS values, we reduced the unit cell size to 4×LFS.

The curves for 3 × LFS give significantly lower values for short LFS. For longer LFS the values agree well with those obtained with 4/5×LFS. We expect that the values obtained with larger unit cells are closer to reality as they contain a better statistical sample of the Perlin textures. However, also for these values we see different trends: While for the Tukey window approach the *J*_{ph} reaches a maximum at LFS = 240 nm, and decreases for longer LFS, for enforced periodicity the current seems to be rather independent of the LFS. Further, in Fig. 5(b) the 4 × LFS and the 5 × LFS lines are closer to each other than in Fig. 5(a), which might be an indication that textures with enforced periodicity allow for a better sampling than textures with a Tukey window, even if a smaller unit cell is used.

#### 4.3. Discussion

As seen in Fig. 5, the *J*_{ph} trends observed with FEM differ considerably depending on whether the unit cells are constructed with enforced periodicity or with a Tukey window. To better understand the underlying reasons for these differences, we perform 2D Fourier transforms in order to analyse the spatial frequency spectrum.

Figure 6 shows two-dimensional fast Fourier transform (2D-FFT) images for three different cases. In Fig. 6(a), the 2D-FFT of a large Perlin texture is shown, which would be used as input for the scalar approach. We see that the 2D-FFT modes are very dense and concentrated in a circle that is indicated in white. Further, weak modes extending on the north-south and east-west axes are observed; their extension is indicated with white lines.

For enforced periodicity, as shown in Fig. 6(b), the number of modes is much smaller, because the distance between two adjacent modes is inversely proportional to the unit cell size. The shape of these modes is similar to what we have seen in Fig. 6(a): The strong (white colored) modes are within a central circle and some weaker modes are observed along the north-south and east-west axes.

Figure 6(c) shows the 2D-FFT for a texture constructed with the Tukey-window approach. The distance of two adjacent modes is the same as in Fig. 6(b), because the unit cell size is the same. But here we also see a number of strong (white colored) modes outside the central circle. Further, in contrast to Figs. 6(a) and 6(b), more modes are observed outside the white lines, *i. e.* in the top-left, top-right, bottom-left, and bottom-right corners. As the Tukey-window function attenuates information on the borders of the unit cell, a smaller area within the unit cell contains all the information. Hence, the spatial frequency modes are smeared out across a larger frequency range.

As the 2D-FFT is strongly connected to the scattering profile of the structure [13, 18], we may conclude that the scattering profile of the unit cell with enforced periodicity resembles the scattering profile of the original structure better than a unit cell with a Tukey window. Following this assumption, FEM results shown in Fig. 5(b) are more reliable than those shown in Fig. 5(a).

One reason for the difference in results between the two approaches might be the choice of *r* = 0.3 [see Eq. 3] when building the Tukey window. Hence, only (1−*r*)^{2} = 0.49 are completely unaffected by the Tukey window. When assuming that about 50% of the Tukey window belt contribute to the nanotexture, we find that about 70% of the unit cell constructed with the Tukey window contain valuable information, no matter how large the unit cell is. In contrast, when using enforced periodicity, only a belt with a width of LFS is altered on two sides of the structure. In Fig. 2(p1), this belt would be enclosed by the points 13, 14, 44, 41, 31 and 33. The fraction covered by this belt decreases when increasing the number of LFS per unit cell widths. Further, using enforced periodicity gives less distortion to the texture than the Tukey window approach. One method to increase the accuracy of the Tukey window would be to keep the absolute width of the belt constant, hence decreasing *r* with increasing unit cell width. Investigating this approach could be a topic of future research.

Now as we have an indication that enforced periodicity leads to more reliable results than the Tukey window approach, we perform a convergence study with unit cells constructed with enforced periodicity. Fig. 7 shows the average *J*_{ph} and the standard deviation for unit cell widths ranging between 3 × LFS and 10 × LFS. In order to keep the simulation time short, we performed this study with LFS = 120 nm and *σ _{r}* = 40 nm. We see that the current stabilises for 5 × LFS, which corresponds to about three times the correlation length. Hence, for unit cells constructed with enforced periodicity a unit cell width of three times the correlation length is sufficient for good sampling. The simulation time for one spectrum with 5×LFS is about 5 hours, while it is about 83 hours for 10 × LFS. Hence, choosing 5×LFS is much better when looking at the computational cost.

## 5. Conclusions

For optimising the morphology of nanotextures that are utilised in thin-film solar cells, modelling is essential. Rigorous methods such as the finite-element method (FEM) are well suited for simulating periodic structures with small periods in in the range of up to about 1 μm. Modelling of random nanotextures with FEM is more involved, as larger unit cells must be used to obtain statistically representative samples of the nanotexture.

In this study, we performed FEM simulations of two types of thin-film silicon solar cells: first, real solar cells with a Asahi VU-type nanotexture and secondly, solar cells with artificially generated nanotextures based on Perlin noise.

The finite-element method was well suited for simulating the optical properties of the experimental reference [see Fig. 4(a)]. For FEM simulations of Perlin nanotextures, we used two methods for constructing the borders: a Tukey window approach and unit cells constructed with enforced periodicity. The trends observed with these two approaches differ considerably. However, an evaluation of the FFT spectra of the differently constructed unit cells revealed that unit cells with enforced periodicity resemble large Perlin textures better.

Finally, a convergence study on unit cells with enforced periodicity showed that a unit cell width of 5×LFS, which is comparable to 3 times the lateral correlation length, is sufficient for obtaining stable results.

## Acknowledgments

We thank Mostafa El-Shinawy for the automated import of arbitrary morphologies in HFSS via his Matlab-based custom script. We also thank Hairen Tan from Delft University of Technology for providing us with the EQE data of the experimental reference solar cell and with the *n, k* data of the p-, i- and n-layers. Further, we want to acknowledge Carlo Barth and Daniel Lockau from the Helmholtz-Zentrum Berlin and Martin Hammerschmidt from the Zuse Institute Berlin for the helpful discussions.

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