## Abstract

Light scattering at randomly textured interfaces is essential to improve the absorption of thin-film silicon solar cells. Aluminium-induced texture (AIT) glass provides suitable scattering for amorphous silicon (a-Si:H) solar cells. The scattering properties of textured surfaces are usually characterised by two properties: the angularly resolved intensity distribution and the haze. However, we find that the commonly used haze equations cannot accurately describe the experimentally observed spectral dependence of the haze of AIT glass. This is particularly the case for surface morphologies with a large rms roughness and small lateral feature sizes. In this paper we present an improved method for haze calculation, based on the power spectral density (PSD) function of the randomly textured surface. To better reproduce the measured haze characteristics, we suggest two improvements: i) inclusion of the average lateral feature size of the textured surface into the haze calculation, and ii) considering the opening angle of the haze measurement. We show that with these two improvements an accurate prediction of the haze of AIT glass is possible. Furthermore, we use the new equation to define optimum morphology parameters for AIT glass to be used for a-Si:H solar cell applications. The autocorrelation length is identified as the critical parameter. For the investigated a-Si:H solar cells, the optimum autocorrelation length is shown to be 320 nm.

© 2013 Optical Society of America

## 1. Introduction

Due to thin absorber layers in thin-film silicon solar cells, it is essential to enhance the optical absorption properties of the absorber layer to improve the device efficiency. Scattering from a randomly textured interface is a proven and efficient method to enhance the optical pathlength inside the solar cell and increase the optical absorption [1–5].

It was first shown in 1983 that scattering at textured interfaces enhances the optical absorption in a-Si:H thin-film solar cells [1, 2]. Today, scattering surfaces are an inherent part of all thin-film silicon solar cells and a large variety of textures have been investigated. In a-Si:H solar cells, different interfaces are to be textured. Examples are the glass/TCO (transparent conductive oxide) interface [6], the TCO/a-Si:H interface [4, 5, 7], and the interface between the solar cell and the back reflector [8]. Additionally, it was shown that super-imposed (or multi-scale) textures using micro-texturing of the glass/TCO interface and nano-texturing of the TCO/a-Si:H interface result in very efficient light scattering in the long-wavelength range [6, 9].

The AIT (aluminium-induced texture) method is a versatile approach for texturing glass surfaces [10]. The lateral feature sizes resulting from this process are in the range of 200 nm to a few microns. Moreover, as previously shown by us [11], the AIT method is capable of producing a wide variety of feature sizes and roughnesses. Improvements in the optical absorption of polycrystalline silicon thin-film solar cells on glass were demonstrated for AIT glass [12]. Using simulation, we have also shown that the structure is suitable for light scattering in a-Si:H solar cells [13]. However, the criteria for surface morphology factors of AIT glass (i.e. root mean square (rms) roughness or autocorrelation length) that are suitable for efficient light scattering in a-Si:H solar cell are yet to be investigated.

In order to understand the optical absorption enhancement in thin-film solar cells due to
textured surfaces, the scattering properties of the textured surface need to be known. Haze and
angular resolved scattering (ARS) measurements are the most commonly used methods to evaluate
light scattering from randomly textured surfaces. Haze is typically defined as the ratio of
scattered/total light intensity for the transmitted or reflected light. ARS specifies the
directional intensity of the scattered light. Using available instruments, these two properties
can be easily measured for different wavelengths at the interface of the textured surface and
air. While these methods provide reasonable approximations, the correct value of these
parameters at the interface adjacent to the absorbing material (i.e. a-Si:H in the present case)
are difficult to measure. Some methods have been suggested to measure ARS at the TCO/Si
interface, however, the experimental procedure for this measurement is complex and furthermore
results in additional errors due to an extra Si/air interface in the system [14]. Different methodologies have been proposed for
calculating the ARS [14, 15] and haze value [7, 8, 16] of the samples at the interface
between arbitrary media. Although the ARS calculated by these methods well predicts the
distribution function of the scattered light, the proposed methods for haze calculation lack
accuracy and sometimes physical meaning. In these methods, the only morphological characteristic
that defines the haze is the rms roughness (${\sigma}_{rms}$) value (the height distribution is commonly assumed to be a
Gaussian distribution). However, the lateral feature size is not considered. The autocorrelation
length *l* of the surface, which is a measure for the average lateral feature
size, is one of the important factors that affect the haze, especially at short wavelengths.
Porteus [17] showed that the incoherent reflection (or
transmission) is significant where *l* becomes comparable to the wavelength
*λ*, especially for small angle scattering. In this regard, Simonsen
*et al.* [18] discussed that in all the
above approximations, haze is calculated as

Harvey *et al.* [19] also suggested the same in their work. They showed that mid-range spatial frequencies of the power spectral density (PSD) function contribute to small-angle scattering. The PSD function is the frequency spectrum of the surface roughness measured in inverse length. This function gives information about the surface spatial frequencies that produce scattered light. The PSD can be understood as follows: Looking at a rough surface that is flat on average but has many local irregularities, the surface profile as a *continuous function* can be represented by an infinite series of sine waves having different amplitudes, periods (spatial frequencies), and phases. The amplitude of the sine waves is provided by the Fourier transform of the continuous surface in the spatial frequency domain. The PSD function is the square of the Fourier transform of the surface profile. However, in case of the measured profile data, we typically deal with a discrete set of equally spaced (Δ*x*) points rather than a continuous function. The profile length (*L*) is also finite, not infinite. So in this case we use finite-length Fourier series to present the PSD function. As a result, this function is a band limited function and it is defined for spatial frequencies limited to $\left[1/L\text{\hspace{1em}}1/\Delta x\right]$ [20]. It is important to take note of this limitation of the PSD function generated from an Atomic Force Microscope (AFM) image.

In the present paper, we first discuss common methods of calculating haze and their limitations. Then we propose a method based on the developed theories to eliminate the limitations and validate the calculated haze using the measured haze value. We show that our method accurately predicts the transmission haze at the textured glass/air interface for all AIT glass samples investigated in this study. AIT samples are good systems for evaluating the haze equations because glass is non-absorbing within the wavelength range of our interest (400-700 nm) and there is no extra interface before the textured interface (like in the case of a textured TCO film on a planar glass sheet). Subsequently, we use the equation to predict optimum morphology criteria for the maximum scattering of AIT samples for a-Si:H solar cells. Furthermore, with a special look at the scattering region suitable for a-Si:H solar cells, we also investigate how different morphologies of AIT glass sheets promote different light scattering characteristics. For this purpose we introduce four different AIT textures and evaluate their light scattering properties in the red region of the visible spectrum.

## 2. Morphology of the textured samples

The AIT (aluminium-induced texture) method is a versatile approach to texture glass surfaces [10], whereby the resulting lateral feature size is in the order of a few hundred nanometres to a few microns (‘micro-textured surface’). The AIT process was carried out by sputtering a thin layer of aluminium at room temperature onto one side of a glass sheet. An aluminium-glass reaction was then realised by heating the sample to a high temperature of around 600°C for about 120 minutes. The reaction products were then etched off in a solution containing hydrofluoric acid and nitric acid. The textured glass samples were rinsed in water and then dried. Details on the preparation of AIT glass sheets and their properties are discussed elsewhere [10, 21]. The borosilicate glass sheets used in this study were 0.8 mm thick. AFM images of the four AIT glass samples are shown in Fig. 2.The AFM image sizes are10 μm × 10 μm. They were measured with an atomic force microscope (model DAFM-AM from Digital Instruments) in a tapping mode at a scan rate of 0.5 Hz and a field size of 512 × 512 pixels. Samples with different ${\sigma}_{rms}$ and different lateral feature sizes were chosen to evaluate the validity of the explained haze equations for the AIT glass morphology.

Table 1 shows the sample preparation conditions. As expected from the smaller Al thickness, the lateral feature size of samples AIT-1 and AIT-3 is smaller than that of the other two samples. It can also be seen that an increasing etching time increases the lateral feature size slightly and reduces the roughness.

Table 2 shows the surface characteristics
of the samples. Autocorrelation, *l*, length is presented as a measure for the
average lateral feature size of the randomly textured surfaces [20].${\sigma}_{rms}$is the root mean square roughness calculated based on the height
distribution function of the AFM image. ${\sigma}_{\mathrm{int}}$is the intrinsic roughness and will be defined in Section 4-1. As
can be seen in Table 2, the first two samples have the
highest and lowest ${\sigma}_{rms}$ and ${\sigma}_{rms}/l$ ratio, respectively, as expected. Samples AIT-3 and AIT-4 were
chosen because they have the same${\sigma}_{rms}$roughness but different average lateral feature sizes (i.e.
autocorrelation lengths).

## 3. Available formulas to calculate haze

#### 3.1 Haze from ARS

As mentioned in the introduction, angular resolved scattering (ARS) can be calculated based on
the AFM image of the textured surface and the optical constants of the two media separated by
the textured interface. Figures 3(a) and 3(b) show the ARS of the four samples of Fig.
2. In Fig. 3, squares represent ARS measured
using a goniophotometer (model GPII from pab advanced technologies GmbH) and the solid line
shows the simulated scattering behaviour based on the approach by Domine *et
al*. [15, 22] using the AFM images of the textured surfaces of Fig. 2. Samples AIT-1 and AIT-2 are those with the largest and
smallest${\sigma}_{rms}/l$. It is expected that samples with larger
${\sigma}_{rms}/l$show stronger scattering. The ARS based on scalar scattering
theory seems to overestimate the diffused light in larger angels for sample AIT-1. The
calculated ARS of the other samples reproduces the measured data more closely. Samples AIT-3
and AIT-4 have the same ${\sigma}_{rms}$ but different autocorrelation length. We can see that the
different average lateral feature size leads to a totally different scattering behaviour. In
particular, samples with smaller autocorrelation lengths (AIT-1 and AIT-3) show larger
scattered intensities at larger angles.

Domine *et al*. [15] proposed to use the angle resolved scattering *ARS(φ, θ)* of the scattered light (*φ* and *θ* are azimuth and polar angle in spherical coordinates, respectively) to calculate the haze via Eq. (2). ARS is defined as the ratio of the power scattered into a small solid angle (normalised to that solid angle) and the incident power [23]. A detailed explanation of how the ARS is calculated can be found in Ref [15]. For a random homogenous textured surface (in which for a constant *θ* we expect the same amount of scattering for all *φ*), the integration of the ARS over the whole hemisphere reduces to:

*Δθ*is the maximum scattering angle of the direct transmission in the haze measurement. Domine [22] has pointed out the limitation of the ARS calculation method in his thesis as:

- 1- The method does not include multiple scattering and shadowing effects, which typically occur if $\sigma >>\lambda $.
- 2- The distances from the surface equal to or smaller than λ cannot be assessed as the method is not based on a near-field theory

Using the ARS function of the samples for each wavelength, the haze in transmission for the textured surface from medium 1 into the medium 2 can be calculated using the optical constants of the two media. Figure 4 shows the haze value calculated using ARS (Eq. (2)) along with the measured haze in transmission from glass to air for the samples presented in Fig. 2. The haze value was measured with a spectrophotometer (PerkinElmer, model Lambda 950, UV/Vis/NIR) with an integrating sphere (diameter 150 mm), using double-beam method as explained in Ref [24]. As can be seen, the method predicts the haze value correctly for sample AIT-2 (which has a low roughness). It also seems to differentiate the haze value of samples AIT-3 and AIT-4 (which have the same ${\sigma}_{rms}$ but different autocorrelation length). However the method overestimates the haze value for sample AIT-1, which has a large${\sigma}_{rms}/l$ ratio. This overestimation is due to the paraxial approximation in the ARS estimation. This is also seen in the ARS of Fig. 3.

#### 3.2 Analytical solutions to the haze problem

### a) First approach for the haze calculation: (1 - coherent transmission)

The equations predicting the haze value were initially developed for haze in reflection and then developed to haze in transmission considering the transition of ${\scriptscriptstyle \frac{2\pi \text{\hspace{0.17em}}\sigma \times \text{\hspace{0.17em}}2n}{\lambda}}\to {\scriptscriptstyle \frac{2\pi \text{\hspace{0.17em}}\sigma \times \text{\hspace{0.17em}}\left({n}_{\text{\hspace{0.17em}}1}-{n}_{\text{\hspace{0.17em}}2}\right)}{\lambda}}$ for normal angle of incidence [8].

Assuming that the vertical height function of the surface is distributed normally, haze in reflection for normal light incidence was proposed by Carniglia [25]. Equation (3) shows this equation for haze in transmission for normal incidence based on the proposed haze in reflection

*H _{T2}* is the haze in transmission,

*n*and

_{1}*n*are the refractive indices of the two media on either side of the scattering interface, and

_{2}*λ*is the light wavelength in vacuum. The equation was derived based on the scalar scattering theory for small scattering angles where the paraxial approximation is valid, and it is based on the assumption that the surface autocorrelation length

*l*is much larger than the root mean square roughness of the surface (${\sigma}_{rms}$), i.e. ${\scriptscriptstyle \frac{{\sigma}_{rms}}{l}}<<1$. Multiple scattering is also not considered in this equation. As a result, there is always a deviation between the measured haze value and the one calculated from Eq. (3). This equation is widely used for scattering in thin-film solar cells [8, 26–28]. However, it does not work for textured TCO even with moderately high ${\sigma}_{rms}/l$ ratio. Although in principle the equation is derived based on the assumption that the paraxial approximation is valid, it has been mentioned [22] that the scalar scattering theory is also valid outside the paraxial approximation.

Apart from the assumption of ${\scriptscriptstyle \frac{{\sigma}_{rms}}{l}}<<1$, the equation has two more important limitations:

- 1- The ${\sigma}_{rms}$ value does not contain information on the lateral feature size of the textured surface
- 2- The opening angle of the haze measurement is not considered

As explained earlier, in haze measurements a fraction of the incoherently transmitted light escapes from the integrating sphere and thus falls into the directly transmitted light beam. In our experiments the haze value was measured with an opening angle of about 8 degrees. Equation (3) does not consider this part of incoherently transmitted light which falls within the opening angle. As a result, the haze equation (Eq. (3)) overestimates the measured haze value.

Figure 5 shows the measured haze values (symbols) of the four investigated AIT glass samples, as well as the calculated haze values (solid lines) based on Eq. (3). As can be seen from Fig. 5(a), Eq. (3) accurately reproduces the haze behaviour of the sample with the lowest roughness and the largest autocorrelation length. However, it can’t reproduce the behaviour of the sample with larger${\sigma}_{rms}/l$. In addition, Eq. (3) predicts same haze for samples with the same${\sigma}_{rms}$. However, as can be seen in Fig. 5(b), samples AIT-3 and AIT-4 have different haze values although they have the same${\sigma}_{rms}$.

Since Eq. (3) only predicts the haze value for small ${\sigma}_{rms}/l$ ratios, it was modified according to experimental measurements with some fit factors and used for scattering in thin-film solar cells [8]. For example, Zeman *et al.* [8] introduced two fit factors *C* and *m*, as shown in Eq. (4). Krč *et al.* [7, 29] used similar fit factors for predicting the scattering behaviour in hydrogenated amorphous silicon (a-Si:H) solar cells. They changed the fit factors to *C = C(λ)* and *m =* 3 [29] to achieve a better fit with experimental results. However, the physical meaning of the fit factors is not explained.

Using the fit factors, it is possible to fit the haze equation with the measured haze values. However, since these two fit factors are not physically interpretable, it is not reliable to use them to predict the haze value at the interface with the absorbing materials (e.g. a-Si:H). In section 4-1, we show that an equivalent of the fit factor $C(\lambda )$ can be analytically calculated.

### b) Considering the opening angle of the haze measurement

Considering the incoherent light scattered outside the opening angle, Simonsen *et al.* [18] provided an analytical solution for the haze equation, which was derived from the phase perturbation theory. For the surface profiles with large autocorrelation lengths, this equation is the same as the Kirchhoff approximation (used in Eq. (3)). The general form of the proposed equation for haze is derived for both transmission and reflection, for arbitrary angle of incidence and media. However, in the following we only consider the specific case of normal incidence. In this case the analytical solution for the transmission haze becomes [18]:

*ε*and

_{1}*ε*are the dielectric constants of the two media and

_{2}*Δθ*is half of the opening angle of the haze measurement. In this equation, for normal incidence, $G(l)$ is:

*l*is the autocorrelation length. The analytical solution for the haze equation in this case requires an analytical function for the 1-D power spectral density (PSD) of the surface. In this haze equation, since the incoherent light that falls within the opening angle of the haze measurement is considered, this equation differentiates the haze value of the samples with the same roughness and different autocorrelation lengths. However we found that it cannot yet correctly reproduce the measured haze of the sample with the large ${\sigma}_{rms}/l$ (i.e. sample AIT-1).

Using Eq. (5), it is possible to suggest an analytical solution for the fit factor of Eq. (4) which is widely used for evaluating the haze value. Comparing Eqs. (4) and (5) we see that, for non-absorbing materials, they are identical for *m* = 2 and

As we see, the analytical function for *C(λ)* is also a function of *n _{2}*, or in the general case of absorbing materials, the dielectric function of the second medium. As a result,

*C(λ)*calculated with Eq. (4) for the interface with air cannot be used for calculating the transmission haze into another medium with a different dielectric function. We show in section 4-1 that the correction factor

*C(λ)*affects the haze particularly strongly at short wavelengths.

### c) The relevant roughness

AIT glass samples scatter a significant fraction of the light into small angles. The PSD function of these types of randomly textured surfaces has a high power in low spatial frequencies. For light scattering of such a textured surface, the relevant roughness for scattering of a specific wavelength is defined by Harvey *et al.* [19]. They argue that ${\sigma}_{rms}$ used in Eq. (3) should be calculated within the relevant band width using the PSD function. In general, ${\sigma}_{rms}$can be calculated using either the root mean square of the height distribution profile (distributed around zero mean value) or the discrete sum of the power spectral density (PSD) function of the same height profile averaged over the surface area [30]. As a result, the 2D-integral of the 2D-PSD function is equal to the variance or${\sigma}^{2}$. Harvey *et al.* explain that for normal incidence that portion of the 2D-PSD function related to the spatial frequencies greater than $1/\lambda $ produces evanescent waves which are irrelevant for scattering of wavelength *λ*. As a result the relevant roughness for scattering of wavelength *λ* can be calculated by:

In order to calculate ${\sigma}_{rel}$, again an analytical function for the 2D-PSD is needed. This will be discussed in section 4-1. Using ${\sigma}_{rel}$ instead of ${\sigma}_{rms}$in Eq. (3) we found that, although the equation reproduces the measured haze at long wavelengths for all the samples, there are considerable differences at short wavelengths. The reason is that the model does not consider the opening angle in the haze measurements. The shorter wavelengths are scattered more strongly by the textured interface. As a result, the sum of the incoherent light which falls within the opening angle of the haze measurement is larger. As explained, this incoherent light contributes to the error in calculating the measured haze.

In order to eliminate the limitations of Eq. (3) and reproduce the measured haze more accurately, we show that the opening angle of the haze measurement should be considered using Eq. (5) and ${\sigma}_{rms}$ should be replaced with ${\sigma}_{rel}$ calculated using Eq. (8). The result is shown in section 4.

## 4. Calculating haze considering ${\sigma}_{rel}$ and opening angle *Δθ*

#### 4.1 Power spectral density function (PSD)

As mentioned in section 3-2, an equation needs to be defined for the PSD of the surface in order to calculate the correction factors of the haze equation. A general form of PSD equation for randomly textured surfaces was proposed by Church and Takacs [31]. In this section we fit this general form with the discrete PSD functions of the surfaces in order to find the autocorrelation length and intrinsic roughness. The discrete PSD function of the surface can be calculated using fast Fourier transform (FFT) of the AFM image. The square symbols in Fig. 6 mark the discrete 2D PSD of the AFM images of Fig. 2. The PSD of a typical etched AZO film is included as a reference (grey symbols). The power at low spatial frequencies represents large lateral feature sizes of AIT glass in comparison with textured AZO.

Furthermore, low spatial frequencies contribute to low-angle scattering while large spatial frequencies contribute to large-angle scattering. For this reason, we expect large scattered intensities for low-angle scattering in the angle resolved scattering (ARS) of the AIT samples (Fig. 3). The significance of the PSD can best be seen by comparing samples AIT3 and AIT4. These samples have the same${\sigma}_{rms}$ and, as mentioned earlier, simpler expressions for the haze predict similar scattering characteristics for these samples. The PSD, however, shows that AIT3 induces higher power at high spatial frequencies. This indicates that sample AIT3 should scatter more light into large angles than AIT4. In addition, we observe that the power of sample AIT-4 is larger at low spatial frequencies, which indicates that the textured surface of sample AIT-4 consists of larger lateral features. This is also confirmed when looking at their AFM images (see Fig. 2). The larger average lateral feature size (autocorrelation length) of sample AIT-4 results in smaller haze values at short wavelengths.

In addition to the qualitative comparison of the PSD data of the AFM images, it is possible to define an equation for the PSD function. Using the PSD equation, the correction factors *C(λ)* for the haze equation can be calculated analytically. Using the PSD equation, it is also possible to calculate the auto-covariance (or autocorrelation) function. The auto-covariance function (*ACV(τ)*) is the inverse Fourier transform of the PSD. As a result, by fitting the PSD equation with the PSD function of the surface, we can also find the autocorrelation length *l*.

A useful general equation for the 2D power spectral density of an isotropic rough surface is given in [31]:

*f*is the spatial frequency. This function is particularly useful since it provides a simple analytical expression of the auto-covariance function (

*ACV(τ)*) of the surface as well as the 1D-PSD function [31]:

*τ*is the lag length (for details on the PSD and the ACV functions see Ref [20].). The intrinsic roughness ${\sigma}_{\mathrm{int}}$ is a surface characteristic and a coefficient in the PSD function of Eq. (10a). This value also equals to the volume-integral under the 2D PSD function in the whole space as shown in Eq. (11). The intrinsic roughness is independent of the AFM measurement dimension or resolution.

Fitting the PSD equation to the PSD data of Fig. 6 gives the intrinsic roughness ${\sigma}_{\mathrm{int}}$ and autocorrelation length *l* (see Table 2). Note that AFM measurements typically give a band limited value of${\sigma}_{rms}$. It is called band limited because the AFM image represents a discrete set of data with finite image dimension and finite resolution. As a result, the PSD function is a discrete function defined for a specific interval of $\left[1/L\text{\hspace{1em}}1/\Delta x\right]$ where *Δx* is the AFM image resolution and *L* is the AFM image length and not a continues function on the domain of$\left[0\text{\hspace{1em}}\infty \right]$. As a result ${\sigma}_{rms}$ is a roughness value which only represents the surface roughness for the specified interval. Here, The PSD equation was fitted with the discrete PSD function of the AFM images. The fitted PSD equation reproduces the same band limited${\sigma}_{rms}$.

Here, using the proposed PSD equation, Eqs. (6) and (8) are solved in order to calculate $G\left(l\right)$ and${\sigma}_{rel}$, respectively.

Using Eqs. (7) and (12), *C(λ)* was calculated for the four different
samples of Fig. 2. The results are shown in Fig. 7(a). According to the Fig. 7(a),
*C(λ)* is smaller for the samples with larger autocorrelation length.
Comparing samples AIT-3 and AIT-4 with the same band limited${\sigma}_{rms}$,*C(λ)* drops faster at short wavelengths
for the sample AIT-4, which has the larger autocorrelation length. Therefore, although these
two samples have the same${\sigma}_{rms}$, AIT-4 has a smaller haze value in the short-wavelength
region.

Figure 7(b) shows ${\sigma}_{rel}$ (Eq. (13)) for the samples. Sample AIT-1 and AIT-2 with the largest and smallest${\sigma}_{rms}$, respectively, show the largest and smallest ${\sigma}_{rel}$ for all wavelengths. ${\sigma}_{rel}$decreases towards longer wavelengths for all samples. This is expected since the upper limit of the integral in Eq. (7) is decreasing and the relevant area of the PSD is smaller for larger *λ*.

#### 4.2 The haze equation

An analytical solution of the haze equation that considers the opening angle of a haze measurement is given by Eqs. (5) and (12). Comparing the calculated haze of Eq. (5) as well as *C(λ)* for the samples in Fig. 7(a), we concluded that the opening angle factor reduces the haze value in the short-wavelength region. However, we observed that only considering the opening angle (or, alternatively, *C(λ)*) the haze value cannot be calculated accurately at long wavelengths. It was also observed that considering the opening angle, Eq. (5) successfully distinguishes between the haze values of samples with the same roughness but different autocorrelation length (AIT-3 and AIT-4), however, the method is unable to correctly predict the haze value of sample AIT-1 with large${\sigma}_{rms}/l$ ratio.

As described in section 3-2, according to Harvey *et al*. [19] the correct calculation of the haze value requires the consideration of the relevant roughness ${\sigma}_{rel}$ for scattering of the wavelengths of interest. It can be seen from Fig. 7(b) that the relevant roughness decreases with increasing wavelength for the investigated samples. Consequently, the calculated haze value is overestimated if we use the intrinsic roughness ${\sigma}_{\mathrm{int}}$ in the haze equation. In order to achieve a good fit of the measured haze value, we propose Eq. (14) which considers both the opening angle of the haze measurement (Eq. (5)) and ${\sigma}_{rel}$ (Eq. (13)).

Figure 8 shows the calculated haze value using Eq. (14). The graphs show that this method reproduces the haze correctly for a wide range of roughnesses and autocorrelation lengths of our AIT samples. This indicates that Eq. (14) is a valid description of haze for textured interfaces with relatively large autocorrelation length as well as large scattering into small angles. It should be noted that we have assumed the PSD equation (Eq. (9)) to be valid for the investigated randomly textured surface. Since Eq. (14) is a general solution for arbitrary materials (absorbing and non-absorbing), haze can be calculated with this method for all interfaces in a thin-film solar cell structure.

#### 4.3 Defining the optimum average lateral feature size based on haze

Equation (14) gives an analytical expression for
the haze value of AIT textured interfaces. This expression allows determining optimum surface
parameters (i.e. ${\sigma}_{\mathrm{int}}$ and *l*) in order to achieve the maximum
scattering of the relevant wavelengths. As the absorption coefficient of a-Si:H drops very
quickly with increasing wavelength, the relevant wavelengths for a-Si:H solar cells of standard
thickness are around *λ =* 650 nm.
Figure 9 shows the haze value calculated using Eq. (14) for textured surfaces with different
${\sigma}_{\mathrm{int}}$ and *l* at *λ =* 650 nm.
Figure 9(a) shows the haze value at the glass/air
interface and Fig. 9(b) shows the haze value at the
AZO/a-Si:H interface. A conformal deposition of AZO is assumed for simplification. A thorough
study of the haze value after AZO deposition can be considered as a future work. As expected,
increasing roughness increases the haze value for all autocorrelation lengths for both
situations depicted in Fig. 9. In addition, we can see
that there is an optimum autocorrelation length for a constant roughness which provides the
largest haze value. According to Fig. 9, the maximum
haze values for *λ =* 650 nm at the glass/air and AZO/a-Si:H interfaces
occur for autocorrelation lengths of 500 nm and 320 nm, respectively. For the sample with
maximum haze (AIT-1), we have experimentally achieved the highest short-circuit current among
the investigated samples. The experimental results are presented elsewhere.

According to Fig. 9, the optimum autocorrelation length is independent of the intrinsic roughness. Solving $\frac{d\text{\hspace{0.17em}}{H}_{T}}{d\text{\hspace{0.17em}}l}=0$ in order to analytically find the optimum autocorrelation length, we find that this value depends on the wavelength, the dielectric constant of the second medium, but not on the intrinsic roughness. This result implies that roughness and autocorrelation length can be optimised independently. In other words, while the maximum surface roughness is normally experimentally determined via the electrical shunting issues of the fabricated solar cells, the criteria for maximum scattering for that particular roughness are independently defined by finding the optimum average lateral feature size (autocorrelation length).

## 5. Summary and conclusion

In this work we introduced a method to model the scattering properties of AIT glass sheets. AIT glass is a suitable superstrate for a-Si:H solar cells and is used to improve absorption and current generation in these solar cells. In a first step, we discussed different methods of haze calculation that are in use in the solar community. We showed the limitations of these equations by using them to calculate the haze of four different AIT glass samples and compared the obtained results to haze measurements.

A limitation of the most widely used method for haze calculation is that the only surface morphology factor that is considered is the surface roughness. The corresponding simple and widely used haze equation is unable to achieve good agreement with the haze values measured on the investigated AIT samples. The reason for the unsatisfactory agreement is that (i) the average lateral feature size of the textured surface and (ii) the opening angle of the haze measurement are both not considered. A method that only considers a height distribution predicts equal haze values for surfaces with equal roughness. However, surfaces with equal roughness can still have a different lateral feature size distribution, which results in different haze values. The average lateral feature size influences the relevant roughness of the textured surface for each wavelength.

In addition to the average lateral feature size, we also considered the opening angle of the haze measurement for the haze calculation. In order to find the haze equation with the correct roughness and opening angle, we used an equation for the power spectral density function of the investigated surfaces. We used this equation to derive correction functions for the opening angle and the relevant roughness. Using these correction functions, we defined a new haze equation which is capable of calculating the haze of AIT glass superstrates for a-Si:H solar cells. We validated the method using experimental data. The developed equation suggests that the correction factor for the opening angle depends on the optical constants of the material into which the light is scattered.

In addition, the developed method can also be used to define the optimum autocorrelation length for achieving maximum scattering. This autocorrelation length can be found for scattering of any specific wavelength into the desired medium. For the case of standard a-Si:H thin-film solar cells, the optimum autocorrelation length for AIT glass was found to be 320 nm.

In this work only the AIT morphology is considered. Very likely the equation can also be applied to other non-Gaussian surfaces since (i) the AIT morphology does not normally have a Gaussian height distribution function and (ii) the PSD function of the surface is used in the equation rather than the ${\sigma}_{rms}$. In our future work, the provided haze calculation method will be experimentally validated for other surface morphologies.

## Acknowledgments

The Solar Energy Research Institute of Singapore (SERIS) is sponsored by the National University of Singapore (NUS) and Singapore’s National Research Foundation (NRF) through the Singapore Economic Development Board (EDB). This work was partly funded by NRF grant NRF2009EWT-CERP001-037 “Advanced superstrates for micromorph silicon solar cells”.

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