Theoretical study of pyramid sizes and scattering effects in silicon photovoltaic module stacks

Oliver Höhn; Nico Tucher; Benedikt Bläsi

doi:10.1364/OE.26.00A320

1. Introduction

Front side pyramids are the industrial standard for wafer-based monocrystalline silicon solar cells. On the one hand the reflection of light is significantly decreased by these pyramids. On the other hand they enable a strong path length enhancement inside the Silicon solar cell leading to a strong absorption enhancement of light close to the indirect bandgap of silicon. These effects are well known in literature [1] and often described via a ray tracing approach [2–5], which is a good approximation for pyramids with sizes of several micrometers [1]. In recent research and development much smaller pyramids with sizes below 1 micrometer attract more and more interest [6–8]. This has very practical reasons, as the realization of smaller pyramids could improve the process chain e.g. in terms of time or cost; furthermore, the pyramids size can also influence the cell performance, e.g. by influencing the fill factor [7,8]. Within this work, we focus on the question, if the optical performance of solar cells and modules can be improved by realizing smaller pyramids.

Recent research evaluates pyramids of different sizes mainly on cell level [9–11]. In the present work also the optical influence of the full module stack is accounted for. Additionally, the influence of the rear side reflectance and scattering, which can be introduced at the front and/or rear interface [12,13], is investigated using an extended Phong model (for details, compare reference [14]).

The simulations within this work are performed with the OPTOS formalism [15–17]. It is a matrix based formalism that allows for the efficient optical modelling of solar module stacks [15]. With the help of the OPTOS formalism interfaces that act in different optical regimes can be combined. Especially, the parameters that are crucial in this paper – the specular reflectance of the rear side mirror of the solar cell – can be varied with low computational effort. The light redistribution matrices of the pyramid interfaces that are needed as input for the OPTOS formalism are modeled with the help of the Rigorous Coupled Wave Analysis (RCWA) [18,19]. As there is no straight forward way to model small random pyramids in an efficient way, only periodically arranged pyramids are investigated. Still, from the results of this study, insights in principle differences for pyramids of different sizes can be gained and preliminary conclusions also for random pyramids can be drawn. These results shall help to understand the behavior of random pyramids in experiment, which at the moment is ongoing work. Note the outstanding importance to account for the full module stack, as the behavior of the pyramids is strongly influenced by the surrounding medium, which is in agreement with reference [15].

2. Definition of the investigated Silicon PV Systems

Four different systems were investigated that are shown in Fig. 1:

Fig. 1 Sketch of the different investigated systems. The rear side reflectance as well as the pyramid size is varied later in this work. For systems (c) and (d), EVA and glass are displayed and modeled as one medium.

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(a) The first system is a silicon solar cell facing air without an antireflective coating (ARC) that is often used for an easy experimental investigation of front side pyramids.
(b) The second system additionally has an ARC consisting of a 70nm thick Silicon nitride (SiN_x) [20] layer.
(c) As solar cells are typically embedded in EVA and glass – both having comparable refractive indices –, in the third system the same solar cell with ARC is embedded in semi-infinite non-absorbing EVA. The EVA is assumed as non-absorbing in this case as a semi-infinite absorbing material would absorb all light. The differences in reflectance compared to absorbing EVA are negligible, because the complex part of the refractive index of EVA is reasonably small.
(d) Finally, in the fourth case a solar module stack is modeled, meaning that the EVA and module glass, this time including absorption, have a finite thickness of 500µm and face an air interface. On top of the glass an ARC of 130nm with a constant refractive index of 1.27 is assumed.

The silicon solar cell thickness is set to 180 µm, as this is an industrially typical thickness [21]. Additionally, at the rear side a specular reflector with varying reflectance is implemented to also be able to investigate the influence of parasitic losses at the rear side.

Note that for the RCWA calculation the pyramids have to be divided into slices of finite thickness. As a result of a convergence analysis a thickness of 20nm was chosen for the slices in this study. That pyramid-like structures can be modeled in this way was shown e.g. in [22].

3. Influence of the pyramid size on the weighted reflectance

In a first step the AM1.5g weighted front side reflectance was calculated taking into account the spectral range from 400 nm - 1200 nm which is most relevant for silicon PV modules.

The weighted reflectance R was calculated according to

R = \frac{\int_{400 nm}^{1200 nm} AM 1.5 g_{Phot} (λ) \cdot R_{spectral} (λ) d λ}{\int_{400 nm}^{1200 nm} AM 1.5 g_{Phot} (λ) d λ},

where

AM 1.5 g_{Phot} (λ)

is the spectrally resolved number of Photons in the AM1.5g spectrum and

R_{spectral} (λ)

is the spectral reflectance of the given system.

For the investigation of the weighted front side reflectance, the silicon is assumed to be infinitely thick to avoid confusions by a possible rear side reflectance.

The spectral reflectances of the systems are calculated using a combination of rigorous coupled wave analysis (RCWA) [18,19] and the OPTOS formalism for the small pyramid sizes. For large pyramids that can be treated ray optically (RO), the reflectance was calculated using a combination of the ray tracer Raytrace3D [23] and OPTOS. In the ray optical case, the reflectance becomes independent of the actual pyramid size, meaning that all sizes, where ray optics is a good description will show the same reflectance. Typically, standard upright pyramids with sizes of several micrometers can in good approximation be treated ray optically.

In Fig. 2 the weighted reflectance is shown for varying pyramid sizes. The ray optical limit is shown as “RO”. It becomes obvious that for pyramids with 1µm base length the ray optical description is not applicable. All four investigated systems behave differently. In the end, the performance of a Silicon solar cell in a solar module is the relevant case, so further investigations focus on case d (full solar module stack). In this system almost no difference due to different pyramid sizes can be observed, meaning that the remaining relevant parameter is the light-trapping efficiency that is ideally described via the useful absorption (generated photocurrent) in the system.

Fig. 2 Weighted Reflectance of the different investigated systems shown in Fig. 1 depending on the pyramid size. The ray optically treated pyramids are shown as “RO”

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4. Extension of the OPTOS formalism to account for additional scattering

In the following the absorption within an 180µm thick Silicon solar cell including a rear side mirror with varying rear side reflectance ( $R_{rear}$ ) is modeled. Therefore, a full solar module stack (case (d)) is modeled with the help of the OPTOS formalism. As the absorption of experimental solar cells with pyramidal front and planar rear side is well described by assuming a reasonable amount of light that is scattered in the system [12,13], also the influence of scattering is investigated.

Brendel et al [12]. and the Module Ray Tracer by PV-Lighthouse [3] describe this scattering via a Lambertian fraction, meaning that a certain amount of the reflected or transmitted light is scattered in a Lambertian way and the other fraction is not scattered at all. Brendel et al. use a fraction of 20% of Lambertian scattering at the rear side. A value of 50% was reported by Keith McIntosh in [24]. For the described systems, 50% Lambertian scattering leads to the same absorption results as assuming a Phong-scatterer with a scatter angle of 11.4°. This angle corresponds to a W-value of 50 (see Eq. (2) that is also used to model solar cells with scattering in [25]. Both ways of describing partial scattering lead to comparable results. Nevertheless, we are convinced that Phong-scattering is closer to the real behavior, since an angular distribution centered around the main propagation directions can be observed at rough rear surfaces [25] and pyramidal interfaces [26]. Therefore we use the Phong-scattering approach. Note that for the absorption results of the investigated systems it makes hardly any difference, if the scattering is implemented at the front or the rear side. A discussion whether the scattering occurs mainly at the front interface, at the rear interface or as a combination of both is beyond the scope of this work. We assume the scattering to occur at the pyramid interface as described in the following:

In order to account for scattering, a Phong-scatterer was implemented in the OPTOS formalism. For this purpose OPTOS matrices for a Phong-scatterer were realized using the phong formula [27,28]

f_{R, P h o n g} = \cos^{W} (θ_{o u t} - θ_{i n}) .

The resulting value

f_{R, P h o n g}

defines the relative amount of reflected light into the angle

θ_{o u t}

depending on the angle of incidence (

θ_{i n}

). The W-value defines the width of the scatter distribution around the specular reflectance or transmittance angle (

θ_{o u t}

). Increases in W lead to a decrease in the width of the scatter distribution. More intuitive is the description of this width via a scatter angle

γ_{s c a t t e r}

relative to

θ_{o u t} = θ_{i n}

that describes the decay of the scattered intensity to 1/e as compared to the intensity at the angle

θ_{o u t} = θ_{i n}

:

γ_{s c a t t e r} = arc \cos [{(\frac{1}{e})}^{1 / W}] .

A depiction of the scatter distribution

f_{R, P h o n g}

and the scatter angle

γ_{s c a t t e r}

can be found in Fig. 3.

Fig. 3 Depiction of the scatter distribution $f_{R, P h o n g}$ and the scatter angle $γ_{s c a t t e r}$ . For an angle on incidence of 20° and a W-value of 50 that corresponds to a scatter angle of 11.4°, the scatter distribution is shown in the right graph.

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Light redistribution matrices of phong scattering as input to the OPTOS formalism are calculated by integrating $f_{R, P h o n g}$ for each angle channel corresponding to a set of channels that cover the angular half space (see [17] for a detailed description of the angle channels). Subsequently, each column of outgoing angles corresponding to one angle of incidence is normalized to one in order to maintain energy conservation.

One exemplary matrix of a phong-scatterer with scatter angle 11.4° is shown in Fig. 4. Note that all entries corresponding to one polar angle but different azimuth angles are summed to simplify the interpretation.

Fig. 4 Graphical representation of a scattering redistribution matrix as used in the OPTOS formalism. Each column corresponds to a constant angle of incidence, while each row corresponds to an outgoing angle. Light that hits the virtual phong interface (see description below) is scattered and thereby distributed around the original angle. The scatter angle was chosen to be $θ_{s c a t t e r} = 11.4 °$ for the depicted case.

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The scattering occurs in transmission as well as in reflection and for light incident from the top as well as from the bottom (when light is reflected from the rear side of the solar cell). This implies that also the front side reflectance of the full system can be influenced by this scattering.

For the sake of comparison all pyramids themselves are assumed to be non-absorbing, so the absorber thickness of each of the systems is always exactly the same. The system with included scatterer is depicted in Fig. 5. The scattering angle is assumed to be the same in EVA as well as in Silicon.

Fig. 5 Sketch of the implementation of scattering. After each interaction with the pyramidal interface the light is scattered.

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To include scattering at the pyramids, the light – after each interaction with the pyramids – is additionally scattered with a phong-scatterer, meaning that the original matrices of the pyramids are multiplied with the corresponding scatter matrix from the left in order to account for the scattering after the interaction. Figure 6 visualizes the scattering effect for an exemplary 45° symmetry element of a redistribution matrix. In Fig. 6(a) very weak scattering with a scattering angle of 0.3° is used. The single diffraction orders can be clearly distinguished. For a stronger scattering of 5.7° in Fig. 6(b), the diffraction orders smear out and reach a broad scattering distribution for an angle of 11.4° in Fig. 6(c).

Fig. 6 Symmetry element of a transmission redistribution matrix (normal light incidence from air, transmission through the encapsulation into the silicon, pyramids of 1 µm period, wavelength 1100 nm). a) The diffraction orders of the pyramids with 1µm period can be clearly distinguished. Additional scattering with an angle of 5.7° in b) and 11.4° in c) leads to a broader redistribution of the diffracted intensity.

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5. Impact of front side pyramid sizes in a PV module stack

The absorption in a PV module stack is modeled with the help of the OPTOS formalism. The rear side reflectance ( $R_{rear}$ ) as well as the scattering angle are varied, in order to understand the influence of these two important parameters on the different pyramid sizes. A typical value for a scattering angle is – as stated above – 11.4°. Typical values for the rear side reflectance are $R_{rear} ~ 0.6$ for an Al-BSF solar cell and $R_{rear} ~ 0.9$ for a PERC solar cell [25]. Note that the rear side reflectance is assumed to be wavelength independent, which is a good approximation within the modeled spectral range. Light trapping shows the strongest influence between 800 and 1200nm, therefore only photocurrents in this range are compared. As the front side reflectance of the system is almost the same for all investigated pyramid sizes, this comparison is meaningful also for the full spectral range. The modeling results depicted in Fig. 7, show that deviations are small for reasonable $R_{rear}$ (0.6-0.9) and scattering angles (~11.4°) for all pyramid sizes. Taking a closer look at the probably most relevant point of $R_{rear} ~ 0.9$ and 11.4° scattering angle, one can see that the differences in current caused by the different pyramid sizes are modeled to be below 0.15 mA/cm². Only for high $R_{rear}$ and absolutely no scattering, a significant difference in the range of around 1 mA/cm² is found for the different pyramid sizes. As soon as some scattering occurs in the system – no matter where – these differences decrease significantly. They become even smaller when assuming imperfect rear side reflectance.

Fig. 7 Modeled photocurrent (in the range of 800-1200 nm) of system (d) depending on the scatter angle and the rear side reflectance ( $R_{rear}$ ) for different pyramid sizes.

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From a physical point of view the wave optical and the ray optical results must converge, when reaching sufficiently large pyramid sizes in the wave optical case. As the wave optical modeling of pyramids with several micrometers in size is almost impossible due to time and capacity reasons – especially when trying to model full OPTOS matrices – this transition can’t be proven here. Obviously, 1µm large pyramids are far away from the ray optical limit.

In Fig. 8, the currents in dependence of the scatter angle are plotted against the pyramid size for $R_{rear}$ of 0.9 to demonstrate possible trends due to the pyramids size more clearly. For strong scattering (angles of 25° and more), there are trends that converge towards the geometric optical case. For weak scattering (angles 11.4° and 0.3°) no clear trend that could be extended to the ray optical case can be seen. However, for the probably most relevant scattering angle of 11.4° the maximal modeled difference in the current is below 0.15 mA/cm².

Fig. 8 Modeled photocurrent (in the range of 800-1200 nm) of system (d) for a rear side reflectance of 90% depending on the pyramid size for different scatter angles. The scatter angle of 11.4° is highlighted, as this is at the moment the experimentally most relevant.

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Looking at the case with the strongest differences and the largest gap between wave and ray optical simulations (minimum scattering 0.3°), the question arises why these differences occur. As shown in [14], there is no simple parameter that solely can explain the full and exact behavior of a full PV module stack. Still, the mode ratio is one parameter that can help to understand trends in the wave optical regime qualitatively. The mode ratio is defined as [14]:

R_{mode} = M_{S i l i c o n} / M_{A i r},

where M describes the number of diffraction modes caused by the small pyramids in silicon and air respectively. This ratio can be understood similarly to the mode ratio of a Labertian scatterer at the same interface. There, the mode ratio is known to be

R_{mode} = n_{S i}^{2} ~ 12

[29], where

n_{S i}

is the refractive index of silicon. If one assumes an equal filling of all modes, this describes the probability for light to escape at the front side. As in the regime close to the silicon band gap several interactions with the front side take place (a lot of light paths through the 180 µm thick silicon are necessary for these wavelengths to be absorbed), this parameter can be dominant. As one can see in Fig. 9, this indicator is qualitatively good and explains the behavior. It also explains, why the pyramids with sizes between 800 and 1000nm behave better than the Lambertian scatterer (

θ_{s c a t t e r} = 90 °

): The mode ratio is larger in the relevant regime than that of a Lambertian scatterer (

R_{mode} = n_{S i}^{2} ~ 12

). Note that this description based on the mode ratio can only be qualitative and of course also the light path length enhancement for one single pass of the different structures plays quite some role especially when only a few passes are needed to absorb all light (for wavelengths < 1100 nm). Additionally, the assumption that all modes are filled equally is a strong simplification that does surely not hold strictly.

Fig. 9 Modeled absorptance and $R_{mode}$ of the system shown in Fig. 1(d) (Full solar module stack without scattering and a rear side reflection of 1). Additionally a system is shown that has a Lambertian light distribution after transmission into the solar cell.

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After pointing out that in real devices scattering will always occur, leading to strongly decreased differences, new questions arise:

1. Does the scattering angle depend on the pyramid size?
2. Would this influence the choice of the optimal pyramid size?

Scattering typically depends on the sizes of scatter centers and they of course will vary for different pyramid sizes. From this it can be concluded that differences in absorption again can occur for different pyramid sizes if the additional scattering of pyramids is size dependent. Then one would choose the pyramids according to their scattering behavior rather than by their size in order to optimize the optical behavior. However, the interrelations of pyramid size and scattering behavior need further investigation which is beyond the scope of this work.

Note again that in the complete study only perfectly periodically arranged pyramids – so called regular pyramids – are modeled. As soon as one assumes random pyramids with different mean sizes of the pyramids, scattering will increase and most probably reduce the current differences for varying pyramid size even further. As it is to the knowledge of the authors up to now not possible to model small random pyramids in a PV module stack with high enough accuracy to still see such small differences, this statement for the random pyramids case can only be finally proven experimentally.

6. Conclusion & Outlook

In summary it can be stated that for realistic rear side reflectances in the range of 0.6 to 0.9 and a realistic scatter angle of 11.4° deviations in absorption due to different pyramid sizes are strongly limited and most probably not significant in real world devices. Influences that occur due to imperfect – e.g. flattened – pyramids are not investigated here and might have a larger influence. For a rear side reflectance of 0.9 and a scattering angle of 11.4°, which is assumed to be the most relevant case – the maximal modeled current difference is below 0.15 mA/cm².

Most probably the small modeled current differences become even smaller when assuming random pyramids with different mean pyramid sizes. As this can up to now not be treated via modeling, an experimental investigation of differences has to be the next step.

However, from this study it can be concluded that for regular front side pyramids greater than 600 nm the pyramid size only leads to small differences in absorption. Therefore, the pyramid size can be chosen according to other factors, e.g. the amount of scattering, achievable electrical quality or the optimal etching process.

Note that this study is a further example that evaluating the reflectance only at solar cell level is in general not meaningful and might even lead to wrong conclusions.

Funding

German Federal Ministry for Economics and Energy in frame of the GROSCHEN project (Contract no. 0324012B).

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Theoretical study of pyramid sizes and scattering effects in silicon photovoltaic module stacks

Abstract

1. Introduction

2. Definition of the investigated Silicon PV Systems

3. Influence of the pyramid size on the weighted reflectance

4. Extension of the OPTOS formalism to account for additional scattering

5. Impact of front side pyramid sizes in a PV module stack

6. Conclusion & Outlook

Funding

References and links

Cited By

Figures (9)

Equations (4)

Optics Express