1. Introduction

The absorber thickness of wafer-based crystalline silicon (c-Si) solar cells will continue to decrease, as predicted by the International Technology Roadmap for Photovoltaic [1]. Consequently, also c-Si solar cells based on thin-film approaches are becoming increasingly important with the potential to compete with wafer-based c-Si technologies. Further, they can combine the advantages of thin-film technology with the high material quality of crystalline silicon.

An important approach to build c-Si thin-film solar cells is the transfer method, where a c-Si epitaxial layer is grown onto a porous layer on a Si wafer such that it can be detached from the wafer and transferred onto another substrate [2–4]. For this approach efficiencies of 19% are reported for a 43 µm thick absorber [2]. Further, wafers thinned to several tenths of micrometers are used to make solar cells. While this approach is not directly applicable to industry, it is very valuable as it allows to explore the efficiency limits of such thin solar cells theoretically and experimentally [5–7]. In 2015, 15.7% on a cell with a 10 µm thick absorber was demonstrated [7].

In this work, we focus on structures that are used in thin-film solar cells with a liquid-phase-crystallized silicon (LPC-Si) absorber [8–11]. Because the c-Si absorber in LPC-Si solar cells is present directly on the glass substrate, no handling issues arise in contrast to wafer-based technologies, where the probability for breakage increases when the wafers get thinner and thinner. The current efficiency record for this technology is 12.1% [12]. For making LPC-Si films, an approximately 10 µm thick nanocrystalline Si film is deposited onto a glass substrate and subsequently crystallized by melting using an electron beam or a laser. The crystallization process leads to grain sizes comparable to multi-crystalline silicon used in industry [10].

c-Si thin-film solar cells with an absorber thickness in the range of 10 µm require advanced light management. Besides light trapping, which is very important because of the weak absorptivity of Si due to its indirect bandgap, as much light as possible has to be coupled into the absorber. This in-coupling is especially important, because Si has a high refractive index and hence a large contrast with the surrounding materials. This leads to a high reflectivity, especially when flat interfaces are used. In-coupling of light into LPC-Si devices can be enhanced using nanotextures at the glass-silicon interface that are manufactured with nanoimprint lithography, which is explained in detail in [13,14].

In recent years, 2-dimensional (2D) textures for light trapping have been heavily investigated: often, they are studied at the rear side of solar cells for light trapping [15,16], on the front side for in-coupling [17,18], or on both sides [19,20]. All these 2D textures have sharp edges making them incompatible with LPC-Si technology, because the textures used for in-coupling have to be implemented prior to the crystallization process. It has been shown that sharply-edged nanotextures on the front side induce a very high defect density in the LPC-Si film, which has a detrimental effect on the open circuit voltage and also on the current density – despite a strongly increased absorption due to the nanotexture [11,21]. Hence, in-coupling must be done with gentle nanostructures that allow the crystallization of LPC-Si with a high electric quality.

To combine good in-coupling of the incident light with high material quality, we look at sinusoidal gratings that are very smooth and gentle. In this work we numerically investigate how reflection from Si interfaces can be minimized by using such sinusoidal gratings. After discussing the mathematical description of such gratings and some general properties of two-dimensional hexagonal gratings, we mention important details of our simulations. Then, we discuss the results for two different layer stacks: a simple layer stack, which is very well suited for studying the physics of in-coupling in detail, and a layer stack with additional antireflective layers, as it is used for real LPC-Si solar cells.

2. Mathematical description of sinusoidal nanotextures

According to group theory five types of 2D lattices exist, of which we investigate sinusoidal nanotextures for the two types with the highest symmetry: square and hexagonal.

2.1. Square lattices

A sinusoidal square lattice can be described by multiplying a cosine defined along the x direction with a cosine along the y direction,

 

Fig. 1 Illustrating different sinusoidal gratings in a 3D (top) and a contour plot representation (bottom): (a) a square lattice constructed according to Eq. (1). Three hexagonal lattices constructed according to Eq. (3) for different structure phases ϕ: (b) a positive cosine lattice (“cos”) with ϕ = 0, (c) a sine lattice (“sin”) with ϕ = π/2, and (d) a negative cosine lattice (“−cos”) with ϕ = π. The red quadrilaterals enclose the unit cell.

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Before we can apply sinusoidal interfaces to our simulations, we have to scale them laterally and vertically such that the period of the texture, i. e. the unit cell side length, is equal to a desired value P and that the maximal peak-to-valley height has the desired value h. For square lattices the period is set via the following substitutions in Eq. (1):

The maximal peak-to-valley height in Eq. (1) is max(f_sq) − min(f_sq) = 2. Hence, we can set the nanotexture to the desired maximal peak-valley height h by multiplying Eq. (1) with h/2.

2.2. Hexagonal lattices

Hexagonal sinusoidals are described by multiplying three cosines with each other,

We also scale the hexagonal nanotextures in order to get the desired period P and peak-to-valley height h. The period can be set with a substitution similar to Eq. (2),

For hexagonal sinusoidals the maximal peak-to-valley height depends on the phase ϕ. The minimum and maximum of Eq. (3) are given by

After some simplifications we obtain for the maximal peak-to-valley height

For the positive and negative cosine structures, we have $h_{\min }^{\max }(0)=h_{\min }^{\max }(\pi )=9/8$, while the value for sine structures is $h_{\min }^{\max }(\pi /2)=3\sqrt{3}/4\approx \mathrm{1.299.}$ We can set hexagonal nanotextures to the desired maximal peak-to-valley height h by multiplying with $h/h_{\min }^{\max }$.

We also apply a vertical shift to the nanotextures such that the minimum is always at zero level. This is done to ensure that the layer thickness above and below the nanotextures is constant for all simulations.

3. Investigated models

We study two different layer stacks: firstly, a simple layer stack illustrated in Fig. 2(a), which allows us to study the effect of the different geometries on the in-coupling in detail. It consists of a glass substrate from which the light is incident. It is covered by a sol-gel layer that carries the sinusoidal nanotexture. Here, we account for the experimental conditions, where textures are transferred onto the sol-gel using nanoimprint lithography [8]. The thickness of the sol-gel layer at the lowest point of the nanotexture is 200 nm. The last layer is the c-Si absorber.

 

Fig. 2 The layer stack and the mesh used in this work shown for a positive cosine (“cos”) texture with P = 500 nm and h = 250 nm, (a) for the simple layer stack and (b) a layer stack with additional antireflective (AR) SiO_x and SiN_x layers.

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Secondly, we perform simulations on a layer stack with additional anti-reflective (AR) layers as it is used in experimental LPC-Si solar cell structures [10]. As illustrated in Fig. 2(b), onto the front glass a 250 nm thick silicon oxide (SiO_x) layer is deposited, which is covered by a sol-gel layer as described above. Onto the sol-gel, a 70 nm thick silicon nitride (SiN_x) layer, another 20 nm thick SiO_x layer and the Si absorber are deposited.

To simulate coupling of light into the material stack we solve Maxwell’s equations on the 3D unit cell of the grating with appropriate source, material and boundary condition settings, using a time-harmonic finite-element solver (JCMsuite) [22]. As illustrated in Fig. 2, the layer stacks used in the simulations are built from a tetrahedral mesh.

A convergence study was performed to determine the optimal parameter settings for reaching a sufficient accuracy: the mesh side length constraint of the nanotextured sinusoidal interface is set to 50 nm and the maximal edge length of the tetrahedra is 100 nm. Polynomial degrees between 2 and 4 are employed.

Because we want to study exclusively how the nanotextures affect in-coupling of light, scattering effects arising from the back side of a Si absorber with finite thickness must be omitted. Therefore, we treat the front glass and the Si absorber as infinite half spaces, which we realize by using perfectly matched layers (PML) on top and bottom as transparent boundary conditions. The light is incident from the glass half space. All the light that is absorbed in the Si layer as well as light that propagates into the PML covering the Si layer is regarded as coupled into the absorber. We keep the thicknesses of the glass layer and Si layer (above the nanotexture) at 100 nm and 200 nm, respectively. Changing volumes of the thin Si layer due to changing nanotextures do not affect the results, because we treat the Si absorber as an infinite half space. On the sides of the unit cell, we employ periodic boundary conditions.

The complex refractive index data for sol-gel, SiO_x and SiN_x were obtained with ellipsome-try. SiO_x was determined to be completely transparent between 300 nm and 1200 nm, and both SiN_x and sol-gel do not absorb at wavelengths longer than 340 nm. Refractive index data for crystalline Si and glass were provided by Helmholtz-Zentrum Berlin. In the simulations, glass is considered to be non-absorptive, hence the imaginary part of its refractive index is omitted.

Besides absorption spectra, we use the maximum achievable photocurrent density $J_{\mathrm{ph}}^{\max }$ as a figure of merit. In this work, $J_{\mathrm{ph}}^{\max }$ is the short-circuit current density that would be achieved with a solar cell with an infinitely thick c-Si absorber if all the light coupled into the absorber would lead to the generation of electron-hole pairs under illumination with the AM1.5 solar spectrum [23]. It is calculated with

A_inc is determined via calculating the net energy flow into the absorber,

We take reflection losses of the front air-glass interface into account by multiplying all the absorptivities in the different layers with (1 − R_air–glass), where R_air–glass is calculated with the Fresnel equations and has a value of about 4% at normal incidence. In this study, we neglect absorptive losses in the glass layer, as already mentioned above.

4. Results

As mentioned in Section 3, we discuss results for a simple layer stack and one with additional antireflective layers as it is used in experiments. While studying the simple layer stack enables us to carefully investigate how the different sinusoidal nanotextures affect the coupling of the incident light into the Si absorber, the additional AR layers in the other stack affect the coupling as well. Hence, both coupling due to the nanotexture and due to the AR layers occur and might affect each other. Therefore, studying only the effect of the nanotexture on coupling the light into absorber is not possible in the layer stack with AR layers.

In this section, we first discuss results on the simple layer stack for normal and oblique incidence and investigate how the observed results compare to scattering theory for 2D hexagonal gratings. Secondly, we discuss results on the stack with AR layers.

4.1. Simple layer stack (without AR layers)

4.1.1. Normal incidence

Figure 3(a) shows the maximum achievable current density $J_{\mathrm{ph}}^{\max }$ for c-Si absorbers on a glass substrate covered with nanotextured sol-gel with different periods P and a constant aspect ratio of a = h/P = 0.5. Light is incident from the glass side, as illustrated in Fig. 2. The results are shown for a square lattice and three different hexagonal lattices: with a positive cosine (“cos”), a sine (“sin”), and a negative cosine (“−cos”) structure. We see that the differences between the four geometries are significant. The negative cosine texture couples light into the absorber strongest. Periods between P = 500 nm and 600 nm show the strongest coupling of light into the absorber for all four geometries.

 

Fig. 3 Simulation results for the simple layer stack without AR layers [see Fig. 2(a)] for the four different geometries presented in Fig. 1 and a constant aspect ratio of a = h/P = 0.5. (a) The maximum achievable current density $J_{\mathrm{ph}}^{\max }$ for five different periods and (b) the spectra of the in-coupled light for the three hexagonal structures and P = 500 nm are shown. A flat reference and, in (a), the current achieved with full absorption are shown as well. All results are corrected for the reflection loss at the air-glass interface (about 4%).

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The square lattice shows a similar trend as the hexagonal sine lattice, but exhibits lower current densities for all periods except 350 nm. Because of its comparatively weak performance, we will not consider the square lattice for the rest of this study. We attribute the stronger performance of the hexagonal textures with respect to the square textures to the six channels into which light can couple per diffraction order. In contrast, for square lattices only four channels per diffraction order are available.

The significant differences between the performance of the three hexagonal structures are explored in Fig. 3(b), which shows spectra of the fraction of light coupled into Si for the three hexagonal geometries at a period of P = 500 nm. The most prominent features are the sharp peaks at approx. 330 nm, 380 nm and 650 nm and the valleys between approx. 400 nm and 600 nm. The negative cosine lattice shows the highest values across the whole spectrum; especially the valley at short wavelengths is most shallow for this geometry.

Figure 4 shows the local distribution of the imaginary part of the electric field energy density 1/4ℑ(E·D*), which is proportional to the absorbed power density [24]. Especially the negative cosine structure, shown in Fig. 4(c), exhibits a very strong focus in absorption with maximum densities almost twice as high as for the positive cosine structure shown in Fig. 4(a). Also for the sine (Fig. 4(b)) a focus is visible, which however is weaker than for the negative cosine. It therefore seems that the negative cosine texture functions as an array of convex microlenses, which might help us to understand why this texture couples light better into the Si than the other textures. However, the connection between the stronger in-coupling and the microlensing effect has yet to be investigated more in depth.

 

Fig. 4 The imaginary part of the electric field energy density $\frac{1}{4}\Im (\mathbf{E}\cdot \mathbf{D}*)$, which is proportional to the absorbed power density. The simulation results are shown for the simple layer stack for the three hexagonal geometries with P = 500 nm and h = 250 nm at λ = 500 nm along cross sections through the unit cells, as they are depicted in Figs. 1(b)–1(d).

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4.1.2. Analysis using diffraction theory

To analyze the origin of the distinct peaks observed in Fig. 3(b) at wavelengths of approx. 330 nm, 380 nm and 650 nm, we look at the Fourier transform of the field reflected back into the glass. Because of the perfect periodicity of the unit cell, light is scattered into well-defined diffraction channels. Figure 5(a) shows the number of channels diffracted back into the glass as a function of the wavelength for a hexagonal grating with period P = 500 nm: for very short wavelength 19 channels are present. At approx. 330 nm, 380 nm and 650 nm 6 channels vanish, such that only one channel (the zeroth diffraction order) is present above 650 nm. In other words, above these wavelengths the third, second, and first diffraction orders vanish, respectively.

 

Fig. 5 (a) The number of diffracted channels reflected back into the glass for a period of P = 500 nm at 330 nm, 380 nm, 650 nm and 1050 nm. To relate these results to the fraction coupled into Si (FCiSi) shown in Fig. 3(b), the curve for the negative cosine (“−cos”) is shown as well. (b) The reciprocal lattice of a hexagonal lattice with period P. The nearest, second-nearest and third-nearest neighbors of an arbitrary point are depicted, which correspond to the first, second and third diffraction orders, respectively.

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To understand why the diffraction orders vanish at these distinct wavelengths we take a look at the scattering theory for hexagonal two-dimensional gratings. The primitive vectors of the reciprocal lattice for a hexagonal lattice with period P are given by

For diffracted waves the k-vector component k^‖ parallel to the lattice plain is equal to a vector connecting two points of the reciprocal lattice. As illustrated in Fig. 5(b), the first, second and third diffraction orders corresponds to the nearest, second-nearest, and third-nearest neighbors, respectively:

The angle sinθ_i into which the i-th refraction order is scattered is given by

Using $k_i^{\parallel }$ from Eq. (10) in Eq. (12), we obtain λ₁ ≈ 650 nm, λ₂ ≈ 375 nm and λ₃ ≈ 325 nm for a period of = 500 nm, where we approximate the refractive index of glass by n_glass ≈ 1.5. These λ_i values are in agreement with the peaks of the spectra shown in Fig. 3(b).

4.1.3. Oblique incidence

It is also very important to study how structures for light management operate under oblique incidence [25, 26]. We study the behavior of our sinusoidal nanotextures at oblique incidence using the simple layer stack, where we take the angle of incidence of light in the glass layer, θ_glass, as parameter. In contrast to the results for normal incidence discussed above, here we do not take the effects of the air-glass interface into account. According to Snell’s law, the maximum angle θ_glass for light incident via a flat air-glass interface would be 41°. However, with (nano)structured anti-reflective coatings between air and glass, also angles > 41° can be achieved. We discuss results for the negative cosine (“−cos”) with P = 500 nm and h = 250 nm and for a flat layer stack.

When studying oblique incidence, also the azimuth ψ of the incident wave has to be taken into account. We chose to average across all azimuths with steps of 15°. Because of the rotational symmetry of the negative cosine texture it is sufficient to simulate for three azimuths: ψ = 0°, 15° and 30°. To average for all azimuths, a weighted average was taken where ψ = 0° and 30° accounted for 25% each and ψ = 15° accounted for 50%. Finally, the average for both polarizations was taken, just as for normal incidence.

In Fig. 6 the maximum achievable photocurrent densitiy multiplied with 1/cosθ_glass is shown, where we take into account that the AM1.5 spectrum is reduced by this cosine factor for oblique incidence. As a reference, we plot the current density achieved when all the light is absorbed. For small angles < 60° the negative cosine clearly outperforms the flat cases. However, for θ_glass = 75°, the flat structure couples light better into the Si.

 

Fig. 6 The maximum achievable photocurrent densitiy $J_{\mathrm{ph}}^{\max }$ multiplied with 1/cosθ_glass for oblique incidence in the simple layer stack for a negative cosine geometry (“−cos”) with P = 500 nm and h = 250 nm and for a flat layer stack.

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We can understand why the flat structure functions better than the nanotextured one at θ_glass = 75° by looking at the refractive index of sol-gel, which is lower than that of glass: e.g. at 600 nm, n_glass = 1.51 and n_sol-gel = 1.44. While this small difference has only very little effect for normal incidence, at θ_glass = 75° total internal reflection occurs and only evanescent waves can penetrate into the sol-gel layer. While the sol-gel thickness in the flat sample is 200 nm everywhere, for the nanotextured sample the sol-gel thickness is 200 nm only at the minima of the nanotexture. Hence, for the flat case coupling of the evanescent waves into Si is stronger.

4.2. Layer stack with additional AR layers

As already mentioned in Section 3, experimentally a layer stack with additional SiO_x and SiN_x AR layers is employed in order to reduce reflective losses. This stack is illustrated in Fig. 2(b). Here, we analyze, how strong this layer stack affects in-coupling, when sinusoidally nanotex-tured interfaces are used.

Figure 7 shows simulation results for negative cosine (“−cos”) nanostructures with a period of P = 500 nm and different aspect ratios a and hence structure heights h. In Fig. 7(a) the maximum achievable photocurrent density $J_{\mathrm{ph}}^{\max }$ is shown. For all four different structure heights $J_{\mathrm{ph}}^{\max }$ increases when adding the AR layers. This gain is by far largest for the flat stacks with about 5 mA/cm² and gets smaller with increasing h. At h = 250 nm about 1.7 mA/cm² are gained when adding the AR layers. Note that the h = 125 nm structure with the complex layer stack yields almost the same $J_{\mathrm{ph}}^{\max }$ as the simple layer stack with h = 250 nm. Hence, using the complex layer stack can strongly reduce the required height for effective in-coupling. As a consequence, too high aspect ratios, which could deteriorate the electronic silicon material quality, are not necessary.

 

Fig. 7 (a) $J_{\mathrm{ph}}^{\max }$ and (b) fraction of light coupled into Si for the simple layer stack and the complex layer stack with additional AR layers. The results are shown for the negative cosine geometry (P = 500 nm) for different aspect ratios a (or structure heights h).

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In Fig. 7(b) the fraction coupled into Si is shown for the flat and h = 250 nm cases. We observe that the effect of antireflective layers stack becomes smaller if h becomes larger. For h = 250 nm the complex layer stack is very effective to remove the valley between 400 and 600 nm. But also for wavelengths longer than 650 nm the absorbed fraction is higher than for the simple layer stack.

5. Conclusions and outlook

Liquid-phase crystallized silicon (LPC-Si) thin-film solar cells require advanced light-management concepts to maximise the absorption in the absorber layer. In this study we numerically investigated sinusoidal nanotextures that enhance the fraction of light coupled into the Si absorber from the glass side. This investigation was performed for square and hexagonal sinusoidal nanotextures.

Not only the period P and aspect ratio a of hexagonal sinusoidal gratings determines the fraction of light coupled into Si, but also their morphology: for all periods, the negative cosine (“−cos”) nanotexture performs best. For all nanotextures, the optimal period is between 500 nm and 600 nm. The best structure presented in this paper is a “−cos” of P = 500 nm and a = 0.5 with SiN_x and SiO_x antireflective (AR) layers. With this structure the maximal achievable photocurrent density $J_{\mathrm{ph}}^{\max }$ is increased by 3.6 mA/cm² with respect to the flat layer stack with AR layers, which is the current standard for the LPC-Si technology. For the simple layer stack without the AR layers, the gain in $J_{\mathrm{ph}}^{\max }$ is even 7.0 mA/cm². We observed that the hexagonal “−cos” nanotextures, which perform strongest, exhibit a microlensing effect with a strongly increased absorbed power density density in its focal region. Except for the shortest period, square sinusoidal structures are outperformed by all three hexagonal geometries.

Also for oblique incidence, the “−cos” nanotexture with P = 500 nm and a = 0.5 enhances in-coupling of light into the silicon layer with respect to the flat reference. However, the small difference in the refractive index of glass and sol-gel reduces the fraction of light coupled into Si at large angles of incidence due to total internal reflection. While currently the sol-gel is required as a carrier of the nano-imprinted texture it might be an advantage to transfer the nanotexture into the glass carrier or the SiO_x layer.

This work directly impacts manufacturing efficient c-Si thin-film solar cells, as it enables us to define the smooth nanotextures that couple light into silicon best. However, the subtle balance between optical enhancement versus electrical deterioration due to increasing aspect ratio is yet to be investigated experimentally.