1. Introduction

Wafer-based silicon solar cells employ surface textures for reduced reflection of the incident light as well as for the enhancement of the absorption length within the wafer [1, 2]. Cost considerations drive the production towards continuously shrinking wafer thickness which gives a strong incentive to reduce the amount of material that is lost during surface preparation. The combination of saw-damage etch and surface texturing typically removes as much as 15 to 25 µm of material [2, 3]. Saw-damage etch may be a lesser issue for kerf-free wafering, but the loss of material during texture etch should still be minimized. This may be achieved by shorter or milder etching which also results in smaller features.

Given that the surface features of current silicon solar cells are several micrometers large, ray-optical models like the one shown in the left part of Fig. 1 are traditionally used to describe the optics of solar cells with large facets at their surface. However, for smaller features in future cells this may eventually become inappropriate and scattering at tips and edges should be described adequately. Since the wafer thicknesses will likely remain in the order of hundred micrometers, the application of a full wave-optical treatment of the whole cell is not practical; instead, combined approaches are needed where propagation in the bulk may be treated with ray-optics but the surface texture is treated correctly by wave optics.

 

Fig. 1 Illustration of scattering at a surface texture; refraction at a facet is shown to the left, diffraction of partial waves by periodic repetition of a profile with length $L$ is illustrated in the middle, and definition of the phase-change introduced by the texture is shown to the right.

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Wave-optical phenomena are successfully treated by regarding unit cells and assuming a periodic continuation outside of the cell. The dashed lines in the middle part of Fig. 1 illustrate the phase change imparted to two partial waves that are emitted at a separation equal to the cell size $L$. An incident plane wave with wavelength $\lambda <L$ is thus diffracted into various orders by the periodicity $L$. For the case of a 1D-profile and refractive indices $n_1$ and $n_2$ adjacent to the interface, the angle of propagation of the $q^{\text{th}}$ order is defined by the grating equation $n_1\cdot \sin {\delta }_{\text{inc}}-n_1\cdot \sin {\delta }_{\text{q}}=q\cdot \lambda /L$ . The incident angle${\delta }_{\text{inc}}$ and the diffracted angle ${\delta }_{\text{q}}$ are measured with respect to the surface normal as illustrated in Fig. 1.

The intensity going into this angle can be described by Fourier theory [4]; it yields the radiance which is proportional to the squared modulus of the $q^{\text{th}}$Fourier coefficient $f_q$. For discrete data such as a scan of the surface with an atomic force microscope (AFM), $f_q$is given by sampling of a pupil function $U$ which contains the surface profile $\zeta (x)$, using $N$ discrete points over the length $L$. With a vacuum-wavevector given by $k_0=2\pi /\lambda$, the pupil function for transmission $U_t$ can be defined in terms of a phase modulation of $\Delta \Phi =k_0\zeta (x)\cdot (n_1-n_2)$ [5]:

Generalization of Eq. (1) to 2D is straightforward by using two indices $q$ and $r$. The result is conveniently expressed by the direction cosines $\alpha$ and $\beta$ which facilitate the treatment of incident angles by a simple shift of the diffraction pattern. Moreover, they allow a distinction between propagating and evanescent orders in terms of the unit-circle [6]. Different from the definition of $\delta$, the direction cosines are defined with respect to the interface-plane. Thus:

The angle-resolved scattering function $ARS$ is then obtained after distributing the outgoing intensity among the propagating modes (evanescent modes are obtained for orders $q^2+r^2>{(L/q)}^2$, but they do not carry energy) and multiplication with Lambert’s projection factor of $\cos {\delta }_{q,r}$.

In this contribution, we apply the Fourier model to large pyramidal textures with steep slopes. This leads to a paraxial error which can be removed by a more general definition of the pupil function [6]. We test the adapted model by comparing it to the angle-resolved scattering intensity measured for visible light from pyramidal surface textures, and we use it to predict the angular dependence for near-IR light scattered from such textures into silicon solar cells.

2. Model

Paraxial errors are expected whenever the incident beam or one of the diffracted orders propagates into a large angle. This is illustrated in Fig. 2 where Eq. (1) was applied to perpendicular incidence of light with a wavelength of 543 nm on a large period $(L\approx 20\cdot \lambda )$ that contains a single pyramid with varying facet-angle. The results are compared with the prediction of Snell’s law for refraction of light incident onto a flat interface under an angle. For a medium with refractive index of 1.5, such as glass or typical polymers, the results of Eq. (1) appear acceptable for facet angles up to about 45°.

 

Fig. 2 Variation of the refracted angle ${\delta }_{tr}$ upon perpendicular incidence on facets inclined by an angle $\phi$ w.r. to the surface plane (c.f. left panel of Fig. 1). Lines are according to Snell’s law, circles are modeled with the paraxial approach according to Eq. (1), squares after non-paraxial correction according to Eq. (3). Dashed lines and open symbols refer to incidence on a polymer, full lines and symbols refer to incidence on silicon.

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The second example in Fig. 2 represents perpendicular incidence on silicon, using a wavelength of 1200 nm. This wavelength was chosen because it is representative of the weakly absorbing region of silicon, but the refractive index $n_2=3.55$ can be assumed to be purely real at this wavelength [9]. Here, the model fails already for angles of 30° and diverges completely for the natural facet angle of silicon at 54.7°.

Harvey et al. pointed out that the paraxial error can be avoided by using a projection of the profile $\zeta (x,y)$ along the incoming and outgoing beams as illustrated in Fig. 1 [6]. For the 1D example, this is achieved by multiplying factors of $\cos {\delta }_{inc}$ and $\cos {\delta }_q$ in the exponent of the pupil function:

Krywonos et al. discussed the equivalent of Eq. (3) for the case of reflection [10]. For our case of large and steep facets, we noted that the model works without paraxial error for facet-angles up to 45°. Beyond that angle, scattering occurs into evanescent modes only. Based on insight from ray tracing, this case corresponds to forward reflection at the steep facets and yields a second incidence on an adjacent facet; reflection occurs only after the double rebound. The Fourier model does not account for this situation.

The corrected Fourier model is easily generalized to 2D, but it was pointed out that it becomes computationally intensive since a complete 2D Fourier-transform is required for every outgoing angle [10]. In case of isotropic surface textures, scattering can be treated with a computationally less intensive 1D Hankel-transform [10].

Nevertheless, the sheer speed of the FFT-algorithm provides a workaround for this issue without restriction to isotropic textures; it can be used to provisionally calculate a set of Fourier transforms for different outgoing angles. Subsequently, the desired radiance pattern without paraxial error can be assembled by stitching.

For example, one may calculate the FFT ten times for direction cosines equal to $\sin \delta =0$, $\sin \delta =0.1$, $\sin \delta =0.2$, etc. and stitch together an approximation of the desired radiance pattern. Thus, we can take the exact value for 0° from the first pattern and the approximately correct values of a circular domain that represents outgoing angles up to 5.7°. All other data of this pattern is discarded. Next, we take the values for all outgoing angles between 5.7° and 11.5° from the second pattern. Data inside and outside of this ring-shaped domain is again discarded. The third pattern yields values for outgoing angles between 11.5° and 17.5°, and the procedure is continued for ever greater radii until the boundary to evanescent modes is reached at 90°.

The resulting 2D radiance pattern for scattering from a square pyramidal facet with refractive index $n_2=1.5$ and an angle of 54.7° for a wavelength of 543 nm is shown in Fig. 3. Stitching was carried out from 10 pre-calculated diffraction patterns and the result is plotted over the domain of $[-1\dots 1]\times [-1\dots 1]$ of cosine-space. The pattern in Fig. 3 contains four main signatures on the principal axes at coordinates of ( ± 0.4,0) and (0, ± 0.4), corresponding to ${\delta }_{q,0}={\delta }_{0,r}=$ ± 24°. Snell’s law predicts refraction into an angle of 22.9° for this case. The main diffraction peaks are connected by weaker signatures, and there are tails that extend outwards to larger angles.

 

Fig. 3 The 2D radiance pattern of an idealized facet (dashed lines), assembled from ten diffraction patterns. Each of the patterns contributes only in an angular domain as illustrated by the by the dashed lines. The pattern extends over a domain of $[-1\dots 1]\times [-1\dots 1]$ in cosine-space, the outermost line represents the unit circle which separates propagating from evanescent modes. The data within the dashed box and its assembly is illustrated in Fig. 4.

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The stitching procedure is illustrated in more detail for the box shown along one of the principal axes in Fig. 3. Out of the ten pre-calculated radiance patterns, Fig. 4 shows data only from those five that apply for the smallest angles. Since the modeled data will later be compared to measurements of the angle-resolved scattering intensity, the data has already been multiplied with Lambert’s cosine factor. The desired $ARS$ characteristic is thus assembled by taking the first six points from the first characteristic which is valid around the origin (denoted by squares), the next six points from the second characteristic (denoted by circles) and so forth.

 

Fig. 4 Assembly of the $ARS$from the diffraction intensity of 10 provisionally calculated diffraction patterns. For the sake of clarity, only five spectra, corresponding to lowest angular domains, are shown in the graph. Depending on rounding, each one contributes 5 or 6 points to the assembled curve.

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3. Results

Before using our model to predict scattering of weakly absorbed light at the surface texture of crystalline silicon solar cells, we tested our model experimentally by using a transparent replica and illumination with 543 nm.

Verification of the model

The pyramidal texture of a alkaline-etched Si(100) surface was replicated into a UV-curable polymer which is transparent to visible light [11]. An AFM topography of the replicated surface is shown in Fig. 5 together with two scattering patterns. The one shown in panel b) is based on Ray tracing, using 1000 rays incident on randomly selected points of the surface. If reflection is in forward-direction at a facet steeper than 45°, our algorithm follows the beam towards its second incidence on the surface using reflective boundaries at the domain walls. Snell’s law of refraction is applied to calculate the directions of the directly transmitted beam and also of the secondary beam, if needed. In Fig. 5 b), the primary transmission yields four clear maxima located at ca. 19° along the principal axes. The secondary incidence yields signatures at ca 70°. Note that the number of diffracted rays is plotted. For the expected intensity shown in Fig. 6 below, we also take into account the reflection of 4% due the refractive index of 1.5 and Lambert’s projection factor of $\cos \delta$. Figure 5 c) shows results of the modified Fourier approach obtained by stitching from 30 patterns. Compared to ray tracing, the main maxima are broader and the connecting structure predicted also by Fig. 3 is clearly visible. However, the signature of the secondary beams at high angles is not predicted by the Fourier approach.

 

Fig. 5 Panel a) shows an AFM topography of a replica of the pyramids etched into a Si-100 surface (image size 15 µm x 15 µm). Panels b) and c) show corresponding 2D scattering patterns obtained with ray tracing and with the modified Fourier model, respectively. The arrows illustrate the direction of data extraction for Fig. 6.

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Fig. 6 Measurement of the angle-resolved scattering intensity (thick red line), using the geometry shown in the inset. The open squares denote results modeled with the modified Fourier method (adjacent average of the full data shown by the thin gray line), full and open circles with drop-lines represent results of ray tracing for direct incidence and the second rebound, respectively.

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In order to measure the angle-resolved scattering, we applied the replica to the flat side of a hemicylindric prism and measured the angular intensity distribution of the scattered light with the geometry shown in Fig. 6. The index-matching fluid ensures that the angular distribution is maintained during the transit from the replica into flat face of the prism. The angular distribution is also maintained on the way to the detector because the hemicylindric shape of the prism ensures perpendicular incidence without refraction upon transit into air.

The measurement was carried out with a wavelength of 543 nm along the directions of high symmetry as indicated in Fig. 5; negative angles represent a sweep of the principal axis, positive angles denote a direction along one of the diagonals. The main maxima along the principal axes are located at ca 19°, suggesting that the pyramids have shallower facet angle than the theoretically expected 54.7°. Weak local maxima are observed at 70° in agreement with the prediction of the ray tracing method. Along the diagonal represented by positive angles, a maximum is measured at an angle of 15°; this is in agreement with the Fourier model of Fig. 3 where a weak band is predicted to connect the main peaks.

The ray tracing approach reproduces the main maxima at 19° and the secondary maxima at 70°. We applied a scaling factor to reproduce the height of the main maxima. Subsequently, the expected intensities of the secondary beams are obtained by multiplying with the reflectivity of the reflecting facet and with Lambert’s cosine factor. Apart from the good correspondence of the maxima in angle and relative height, the drop-lines in Fig. 6 illustrate that the regions between the main signatures are covered only very sparsely. Likewise, ray tracing did not yield any appreciable contribution along the diagonals. We conclude that the structures that connect the main peaks are a consequence of wave-optics. Since they reach as much as 30% of the intensity scattered into the main peaks, it becomes clear that ray tracing has severe shortcomings even for structures with a size of several micrometers.

The results of the Fourier model in Fig. 6 show considerable scattering due to the randomness of the pyramidal structure. Consequently, an adjacent average is included for clarity, and the data was scaled by one single scaling factor. The agreement between measured and modeled results is excellent around the main maxima and extends over a dynamic range of more than four orders of magnitude for angles between −45° and 0°. Along the diagonals, the agreement is not as good, but it still extends over three orders between 0° and 30°. However, the weaker peaks at high angles are not reproduced because they originate from forward reflected beams as discussed above.

Besides the angular distribution upon entering a given medium, it is also interesting to test the opposite, especially for high angles of incidence where flat interfaces yield total internal reflection. This type of measurement is shown in Fig. 7; we tilted the incident beam by three different angles along one of the principal axes and measured the full range of scattered angles along the same axis.

 

Fig. 7 Measurement of the angle-resolved scattering intensity upon outcoupling into air for three different angles of incidence (thick lines), using the illustrated geometry. Small open squares denote again a 5pt average over the Fourier model, whereas full and open circles with drop-lines denote results of ray tracing.

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For normal incidence, the measurement shown in the inset of Fig. 7 yields a broad maximum at 0° and secondary maxima at angles of 83°. Ray tracing does not yield any appreciable intensity for the direct beams because they undergo total internal reflection at the facets, but secondary beams produce a weak signature between 75° and 80°. The Fourier model, again stitched from 30 patterns, does not predict the single maximum of the measurement, but yields a broad intensity distribution where secondary maxima at ± 30° appear above noise-level.

For an incident angle of 22°, the measurement shows a maximum at −5°, a shoulder at −30°, and another local maximum at −75°. Ray tracing clearly reproduces the maximum by direct beams, but the signature due to secondary beams appears between −55° and −60° which accounts neither for the shoulder, nor for the local maximum at −75°. The Fourier model reproduces the main peak and predicts wings to both sides which agree with the measurement between −20° and + 20°. Towards higher positive angles, the prediction remains too high by about one order of magnitude, towards lower negative angles, the shoulder is not reproduced.

The incident angle of 45° is a decisive test because one would expect total internal reflection from a flat surface. The measurement shows a similar shape as in the case of 22°, but all features are compressed into a narrower angular range with the maximum appearing at −37° and the shoulder at −54°, whereas the local maximum is again recorded at −75°. The direct beams of ray tracing describe the global maximum, the secondary beams predict signatures at −30° and at + 30°. The Fourier model provides a better description of this case except for an exaggerated signal around 0°.

Overall, we are led to conclude that ray tracing can predict the location of maxima that are due to primary and secondary beams, but it cannot give a valid description over the full angular range. The modified Fourier approach also reproduces the main maxima and additionally provides a good description of their decay over large angular domains. However, it does not account for high-angle signatures that are explained by secondary beams of the ray tracing approach.

4. Scattering of weakly absorbed light in silicon solar cells

We applied the non-paraxial model to the case of surface textures in silicon in order to assess differences in the predictions of ray-optics and wave-optics. Unfortunately, an experimental verification would require to etch the structure directly into prisms made of silicon since there are no index-matching materials for $n=3.55$.

The left column of Fig. 8 shows measured AFM data of two etched Si(100) surfaces; the upper one was measured on a conventionally textured surface with average pyramid size of ca. 5 µm. This surface texture is nominally the same as the master surface of the replica shown in Fig. 5. The second texture was obtained after an etch that yields smaller pyramids with average size of 1 µm [12].

 

Fig. 8 Panels a) and d) show AFM topographies of two differently etched Si-100 wafers on an area of 20 µm x 20 µm. Panels b) and e) denote corresponding 2D refraction patterns obtained by ray tracing, and panels c) and f) show corresponding data obtained with the modified Fourier model. The AFM image d) with the small pyramids was taken from ref [12], the arrow in panel f) illustrates the extraction of azimuthal data through the maxima which is shown in Fig. 9.

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The middle column of Fig. 8 illustrates results of ray tracing using 1000 rays. Overall, the ray tracing results are very similar for the two surfaces, expressing the fact that both cases are dominated by incidence on 111-oriented facets. Similar to the results of the previous section, there are only very few rays along the diagonals and at small angles around the origin.

Finally, the right column in Fig. 8 shows the corresponding results of the Fourier model. Again, perpendicular incidence of a plane wave with a wavelength of 1200 nm was used. For the large pyramids, the result of the ray tracing is essentially reproduced. However, for the case of the small pyramids, the four main spots are no longer resolved as clearly, suggesting that scattering events at the tips and at the edges of the texture start having a significant influence on the angular distribution of the transmitted light intensity.

This is more clearly illustrated in Fig. 9 by plotting the intensity at a polar angle of 40° for the full range of azimuthal angles between 0 and 360°. Ray tracing predicts four clear peaks for the large as well as for the small facets. The shown contrast ratio is about 20 and almost exactly the same for both textures. The Fourier approach yields a contrast ratio of more than 100 for the large pyramids, but only little above 10 for the small ones.

 

Fig. 9 Cuts through the 2D data of Fig. 8 for a polar angle of 40°. Full and dashed lines correspond to Fourier modelling of large and small facets, respectively, Full and open symbols with drop-lines denote corresponding results of ray tracing (shifted vertically for clarity).

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5. Conclusion

We implemented a modified Fourier-scattering model that is capable of treating large textures and scattering into high angles without paraxial error. Since the model is no longer compatible with the FFT routine, we propose a workaround that applies to full 2D geometries without undue computation cost. A test with visible light applied to a replica of the pyramidal structure etched into Si-100 wafers showed excellent agreement between measurements of the angle resolved scattering intensity and the predictions of the modified Fourier model. A comparison with ray tracing revealed that some measured features can only be reproduced by a wave-optical treatment; geometric ray-optics cannot explain their existence. Similar issues are expected for the more realistic situation of scattering into silicon even though this case could not be verified experimentally. Despite the drawback that the Fourier model cannot yet describe absolute values of total reflection or transmission, we are nevertheless led to conclude that ray tracing is not fully adequate to treat light scattering of this surface structure which is technologically relevant for solar cell fabrication and many other fields.