1. Introduction

Wafer-based silicon solar cells provide a high conversion efficiency from incident solar power to electrical power. The main drawback of such solar cells, however, is their high cost resulting from the laborious production of high purity crystalline silicon. Lower amounts of silicon would be advantageous but come at the expense of a lower absorption. An appealing alternative to reduce the costs is the use of micro-crystalline silicon (μc-Si:H), which is less critical to be produced. However, the thickness of μc-Si:H is limited due to bulk recombination. Therefore, regardless of which material system is exactly considered, a problem of primary importance to be solved is the enhancement of light absorption in solar cells with a limited thickness.

To meet these requirements, various photon management concepts have been introduced. A strict definition for this field of research is probably difficult to provide. However, it usually embraces all strategies that target on the enhancement of the efficiency of a solar cell with a given absorbing material of a fixed thickness. One possibility, e.g., is the spectral modification of the incident sun light by either down- or up-conversion [1,2]. Alternatively, a multitude of light guiding, scattering, diffracting, refracting metallic or dielectric structures can be integrated. Their purpose, in general, is to increase the optical path of light in the absorbing solar cell material which enhances the probability for each photon being absorbed.

However, different requirements in different spectral domains exist. Silicon, as considered here, possesses an excellent absorption at short wavelengths. Consequently, the photon management adapted to those wavelengths aims at compensating the impedance mismatch to free space. This enables a sufficiently good in-coupling and suppresses spurious reflections. If this kind of photon management is eventually implemented, the front side of the solar cell, where the light impinges, has to be modified. On the contrary, in the spectral domain near the silicon band gap, where the intrinsic absorption fades away, the problem of photon management is more efficiently solved by the two following strategies. One strategy is to reflect the light from the rear side to at least double the optical path. Another, and usually more promising strategy is to deflect the incident light in the cell to achieve an optical path that is at least twice the optical thickness of the absorber material. If the deflected light even couples to guided modes in the absorbing material, the optical path would be tremendously enhanced. Such a deflecting structure may be integrated into either the front or the rear side. Care, however, must be taken in defining exactly the spectral domain of interest for each approach. A change in the absorbing cell thickness may change the spectral properties which has to be carefully taken into account.

Most of the structures usually discussed in the context of photon management attempt to balance the different requirements by optimizing the geometry. At the front side, where the high index contrast to the silicon layer prevents light from entering the cell due to high Fresnel reflection, antireflection coatings are applied to smoothen the abrupt change of the refractive index. Randomly textured surfaces were shown to equally increase the in-coupling efficiency and in addition to angularly redistribute the light [3, 4].

Simple 1D silica gratings embedded in a silicon layer were investigated by Zanotto et al.[5]. More sophisticated 2D dielectric gratings show even higher enhancements [6, 7]. Enhanced in-coupling and consequently a higher absorptance was achieved by perforating the absorbing layer itself with conical holes that create a gradual change of the refractive index [8].

Other groups introduced metallic particles or gratings in order to excite localized plasmon polaritons. The high field enhancement at the metal-dielectric interface was shown to lead to an absorptance enhancement [9–12].

At the rear side, metal contacts were replaced by 1D, 2D or 3D photonic structures, which not just reflect the light at long wavelengths back into the absorber but which also diffract the light into larger angles; potentially exciting guided modes [13–16]. However, most of these structures are usually considered as isolated elements; and only a few suggestions were made how to combine different schemes of photon management [17]. They are usually associated with tandem or triple junction solar cells. There, the problem is slightly different because in those cells the challenge is usually to in-couple the light in a certain spectral domain into the intended absorbing layer. But the integration and the combination of different schemes of photon management can already be beneficial for single junction solar cells.

Here, we prove such beneficial combination. Specifically, a randomly textured surface was used to accommodate the high index contrast between the surrounding medium, supposed to be zinc oxide (ZnO), and the solar cell. In our work we consider μc-Si:H as the solar cell material. The considered photonic structure at the rear side which acts as a back reflector is an inverted ZnO opal. Inverted opals can be fabricated by depositing polymer spheres on a substrate, infiltrating the voids by e. g. atomic layer deposition and finally dissolving the polymer spheres. As a result, a face centered cubic (fcc) lattice of spherical air voids in a ZnO host builds up. On textured surfaces, however, perfect periodic self assembly of spheres is prevented leading to a distribution of the spheres which follows rather the height profile of the surface texture than the perfect lattice. After a few deposited layers though, the perfect lattice forms even on these surfaces [18].

In our systematic study that considers this device with an increasing complexity, we show that this unavoidable disorder is actually beneficial. The solar cell with a texture and an opaline photonic structure that naturally grows on the texture on the rear side from an amorphous to a periodic structure is the device that sustains the largest absorption enhancements. This work indicates novel directions for research where multiple schemes for the photon management are fused into single junction solar cells. It moreover underpins recent insights that some degree of disorder from the perfect periodic structure is beneficial [19]. Finally, our work is of immediate practical relevance since it constitutes a blue print for an experiment. It considers strictly geometries and materials as available. A sketch of the investigated design is depicted in Fig. 1. For better readability it shows a two dimensional cross section along the direction of illumination. However, all the herein considered geometries are three dimensional structures. The conformally textured μc-Si:H layer is shown in red. On top of this absorbing layer an inverted opaline structure follows consisting of several layers of air spheres without any long range order. On top of these, a perfect inverted opal of several layers is added. Normal incidence is considered only. In these 3D geometries with textured surfaces, any polarization dependent response can be ruled out, because each polarization direction nominally perceives the same structure.

 

Fig. 1 Two dimensional sketch of the considered geometry. Shown in red is the conformally textured μc-Si:H layer. On top of it spheres are dropped onto the surface. The spheres constitute air voids which are cut out of the Zno host (blue region). The long range order gradually converges from an amorphous arrangement to perfect periodicity. Normal incidence illumination is from below.

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2. Methodology

To get access to a feasible structural implementation for the photonic structure on the rear side of the solar cell, an algorithm was implemented based on the sphere-dropping-method. In this algorithm, a sequence of spheres are dropped onto the surface towards arbitrary locations. There trajectories, as enforced by the surface profile and previously dropped spheres, is evaluated until a stable position for the sphere is obtained. The lowest layer of spheres, therefore, tends to be conformally deposited on the textured surface and to agglomerate in local minima of the surface profile. However, after a few layers the enforced imperfections are usually healed and the structure grows further as an opal. This also allows to add a discrete number of stacks to build up a perfect opal on top of the amorphous photonic structure.

The height profile of the considered randomly textured surface was provided by the Forschungszentrum Jülich [20] and is similar to the one described by Berginski [21]. It was determined by an atomic force microscopy scanning of an existing etched ZnO surface. These textures are used in solar cell modules. The root mean square roughness is 130 nm. From this texture a section of about 4 μm by 4 μm was used for the simulations. The optical data for μc-Si:H was also provided by the Forschungszentrum Jülich [20]. The semi-infinite half spaces on top and at the bottom of the structure are composed of ZnO characterized by a dispersionless permittivity of ε = 4. In the spectral domain of interest the changes of the optical properties of ZnO are very weak and the material can be described by a generic permittivity. In order to gain insight into the effect on the absorption of the different photonic elements at both the front and the rear side, the absorption in a solar cell with different geometries, as indicated further below, were studied numerically. To calculate the absorptance spectrum an incoherent Fourier Modal Method (FMM) was used [22–24]. This modification of the original method proposed by Moharam et al. is in particular useful for comparably thick structures which have a homogeneous layer somewhere inside. The idea of the incoherent treatment is to separate the entire geometry into three parts. Two of them, at the front and the back side, are inhomogeneous regions. These are typically the suggested photonic elements responsible for the photon management. The third domain is the homogeneous layer in the middle. Diffraction efficiencies in reflection and transmission are calculated only for the leading and the trailing sections using the FMM. The outputs are large matrices which indicate how the different diffraction orders couple to each other, in reflection as well as in transmission. The obtained diffraction efficiencies are then incoherently coupled by a simple exponential decaying function with the imaginary part of the propagation constant times the thickness of the homogeneous layer in the exponent. The propagation constant describes the dissipation of light in the solar cell and is in general different for all diffraction orders. Consequently, the contribution of the incoherently treated middle section is expressed as a diagonal matrix, which indicates by which amount a certain diffraction order is damped. The purpose of this simulation is to properly reflect the coherence properties of the incident sunlight. Due to the fact that the coherence length of sunlight is in the order of several hundreds of nanometers [25], the incoherent treatment is necessary for layer thicknesses larger than this value, as considered in this work, if spectrally integrated quantities are considered. Furthermore, when diffracting elements at the front side distribute the light over a large range of angles, the ability to interfere is lost for light that experiences multiple reflections between the front and the rear side of the solar cell. Eventually, the total reflectance is the infinite sum of intensity contributions that passed the homogeneous layer an increasing number of times. The same holds for the transmittance. This procedure is completely analogous to the one of an etalon characterized by a reflectance R and transmittance T at the interfaces as well as a phase factor due to the propagation through the etalon, leading to the well-known Fabry-Perot transmission pattern. The only difference is that R and T are matrices and the phase factors are attenuation factor matrices this time. Since the μc-Si:H layer is assumed to be the only absorbing material, the total absorptance is finally given by 1 − R − T.

To compare the various scenarios, for each geometry the short circuit current density J_SC was calculated. This is done by weighting the simulated absorptance spectrum by the incident solar spectrum [26] and integrating over the spectral domain of interest:

3. Preliminary considerations on the photonic structures

The geometry, which was used as a reference, consisted of an unstructured slab of μc-Si:H equipped with a silver back contact. In this case, however, the total absorptance is not just 1 − R − T, since the silver contact is absorbing as well. In order to determine how much light is absorbed inside the μc-Si:H layer an integration of the time-averaged Poynting vector over the cross section of the solar cell has to be carried out. This is particularly easy in layer stacks, as it is the case in the referential cell. However, a randomly textured μc-Si:H layer, where the back contact is equally deposited conformally is more difficult to treat but might have its own benefits. Finally, every suggestion for improvement must be compared to the referential device. All considered scenarios are useful only if they outperform this simple flat metallic back contact. The texture on the front side is given and is not considered to be a subject of modifications. While relying on measured surface topologies, we assure that the perceived geometries are immediately amenable for a final device to be fabricated. Therefore, the structure that promises the largest impact and which can be tuned is the photonic crystal. Consequently, in a first preliminary study the inverted opal at the rear side was adapted to the thickness of the μc-Si:H layer.

In order to properly design the inverted opal, unstructured μc-Si:H layers of various thicknesses were simulated [Fig. 2(a)]. The layers are embedded in ZnO. From the obtained absorptance spectra the absorption edge was determined. The absorption edge is identified as the wavelength at which the absorptance dropped to half its maximum value. This absorption edge indicates the spectral domain where the photon management on the rear side shall concentrate on. At smaller wavelengths μc-Si:H is sufficiently absorptive such that no care needs to be taken to increase the optical paths. On the contrary, at longer wavelengths the intrinsic absorption is such marginal that even an increase in the absorptance by a tremendous factor does not affect the short circuit current to a noticeable extent. This absorption edge, therefore, should be the spectral position of maximum reflectance of the inverted opal. As can be seen in Fig. 2(a), the absorption edge shifts to longer wavelengths with increasing thickness because a larger part of the light is being absorbed and the longer wavelengths remain.

 

Fig. 2 (a) Absorptance spectrum of an unstructured layer for various thicknesses d. (b) Reflectance spectrum of a four layer inverted opal (red line) and bandstructure of an inverted opal (blue dotted lines).

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As a next step the inverted opal had to be optimized, i. e. the reflectance should be high at the spectral position, where the absorptance of the unstructured μc-Si:H layer is insufficient. From a simulation using the FMM, we immediately get the total reflectance and transmittance, so no further process is required as it is the case for calculating the absorptance. A reflectance spectrum of an inverted opal at a particular radius shows two regimes of high reflectivity [red line in Fig. 2(b)]. One at higher wavelengths centered at about 1.36 μm and a second one at lower wavelengths centered at about 600 nm. Bandstructure simulations were performed using the freely available tool MIT Photonic bands. Again, since the structures were calculated in three dimensions (it is an opaline photonic crystal) no distinction can be made between the polarization states since it is fully vectorial. These simulations revealed, that there is a directional optical band gap in Γ-L direction of the fcc lattice for larger wavelengths (lower energies). In our considered geometry the Γ-L direction is the surface normal. In this direction, the high reflectivity is due to the prohibited propagation of light and the reflection takes place mainly into the zeroth order of the periodic surface [dotted blue lines in Fig. 2(b)]. However, in the second spectral range of high reflectivity a multitude of modes exist. The dispersion relation of these modes is almost flat; suggesting a low group velocity. These modes are equally associated with a field profile that strongly deviates form that of a plane wave. The resulting large impedance mismatch renders these modes to be excited with difficulty from free-space, such that most light is back reflected and only a minor part is in-coupled into the photonic crystal. This is the desired mode of operation. Since Maxwell’s equations are scale invariant, the spectral positions of these two maxima can be tuned by changing the radius of the spheres and, consequently, an optimized radius can be found for each μc-Si:H thickness.

To highlight the ability and the importance to spectrally tune the reflection band, Fig. 3(a) shows the absorptance spectrum of 1 μm thick μc-Si:H layer (blue line). The red solid line in Fig. 3(a) shows the reflectance spectrum of a 4-layer inverted ZnO opal for a sphere radius of 325 nm, while the red dotted line shows the reflectance spectrum for a sphere radius of 140 nm. These two values for the radius are chosen such that the reflectance maximum lies at the absorption edge of the μc-Si:H layer indicated by the black dashed line in Fig. 3(a). Four layers of spheres were chosen because simulated inverted opals consisting of five or more layers showed only minor changes in the reflection behaviour. Plotting the diffraction efficiencies for reflection at this wavelength shows that indeed the zeroth order mode is most efficiently excited for a sphere radius of 140 nm [Fig. 3(b)]. On the other hand, a multitude of higher order modes is excited for a sphere radius of 325 nm [Figure 3(c)]. This implies the excitation of a larger number of diffraction orders that contribute to the enhancement of the optical path. The short circuit current density is indeed higher for larger radii. Therefore, larger radii are preferable in particular for perfectly periodic opaline elements. For this reason a radius of 325 nm was chosen for all simulations which follow below.

 

Fig. 3 (a) Absorptance spectrum of a 1 μm thick μc-Si:H layer (blue line) and reflectance spectra for a 4-layer inverted ZnO opal for a sphere radius of 325 nm (red dashed line) and a radius of 140 nm (red solid line). Diffraction efficiencies in reflection at a wavelength of 639 nm [marked by the black dashed line in (a)] for a radius of 140 nm (b) and 325 nm (c).

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4. Simulation results of the solar cell

The absorption enhancement using the aforementioned photonic elements was evaluated with an increasing complexity of the device. The case of a 1 μm thick μc-Si:H layer is discussed in more detail since all properties emerge there. Eventually, a final discussion on the impact of the solar cell thickness is provided.

At first Fig. 4(a) shows the absorptance spectra for the selected geometries. The blue line in Fig. 4(a) shows the absorptance spectrum of the unstructured layer. A large Fresnel reflection, in particular at short wavelengths, leads to a drop in the absorptance. Applying a perfect inverted opal at the back side does not improve the in-coupling but an additional feature appears at the reflection maximum of the inverted opal [orange line in Fig. 4(a)]. However, a conformally textured layer reduces Fresnel reflection and leads in addition to an overall red shift of the absorption edge due to the angular redistribution of the incident light [red line in Fig. 4(a)]. Combining a textured front side but a perfect inverted opal at the back side shows again an enhanced in-coupling. However, the distinct feature of the perfect inverted opal is not visible anymore [green line in Fig. 4(a)]. This is again due to scattering into a large range of angles. Using additionally to the conformal texture an approximately four layer thick amorphous inverted opal leads to a further redshift of the absorption edge [gray line in Fig. 4(a)]. Due to the non-periodic arrangement no distinct features are visible anymore and the inverted amorphous opal is just increasing the reflectivity over a broad range spectral window. The highest absorption enhancement is obtained for a geometry consisting of a textured surface together with a combination of an amorphous and a perfect opal [black line in Fig. 4(a)]. In this case the properties of both amorphous and perfect opals are combined. These properties are the high reflectance around the previously identified absorption edge, a good deflection of light into higher diffraction orders, which enhances the optical path, and prevention of a narrow-band operation due to the amorphous character.

 

Fig. 4 (a) Absorptance spectra for different geometries. The layer thickness is 1 μm and the sphere radius is 325 nm. (b) Normalized short circuit current densities for the absorptance spectra form (a). Circles are values normalized by the flat μc-Si:H layer and crosses show the values normalized using a silver back contact.

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To condense these results, Fig. 4(b) shows the short circuit current density for the simulated geometries. The circles correspond to the values normalized by the flat μc-Si:H layer only, whereas the crosses are the short circuit current densities normalized by the value obtained applying a flat silver back contact to the flat μc-Si:H layer. In order to outperform the metal back contact, it is mandatory to use a surface texture at the front side together with a whatever shaped inverted opal at the rear side. Considering the mixed configuration an enhancement of about 16% is possible compared to the flat metal back contact.

In addition, the purple markers in Fig. 4(b) show the results for the limiting case of about 20 amorphous layers applied. These two points are very close to the results of the mixed configuration, which means that the latter is already close to the optimum.

The simulations were also extended towards other thicknesses of the μc-Si:H layer. Figure 5 shows the short circuit current densities obtained from simulations like the ones in Fig. 4(a), for various thicknesses normalized to the case of a flat layer terminated by a silver back contact. Plotted are only the case of a perfect inverted opal (orange circles), the case of a front texture together with a perfect inverted opal (green triangles) and the case of the mixed configuration explained above (black squares). It can be seen that for larger thicknesses of the absorber material, a surface texture at the front side and a perfect inverted opal at the rear side cause the normalized short circuit current to just reach unity. It suggests that they perform comparably to a cell with a silver back reflector. Only the more sophisticated structures are able to go beyond unity. Furthermore, thicker layers already show a higher intrinsic absorption without any photonic element. Therefore, it gets increasingly challenging to obtain a substantial enhancement. Nonetheless, for thinner solar cells the impact of a suitably adjusted photon management is quite tremendous. The obtained enhancements are comparable to what other people found. In regard to the fact that thinner cells are desirable to reduce material consumption, the suggested geometries seem to fit well to these requirements.

 

Fig. 5 Normalized short circuit current densities as a function of thickness. Normalization is done by the silver back contact geometry.

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5. Conclusion

Different photonic structures, optimized with respect to different requirements depending on the spectral domain, have been successfully integrated into a microcrystalline silicon single junction solar cell. Their purpose was to enhance the short circuit current density. In particular, inverted ZnO opals were designed to match the absorption behaviour of an unstructured μc-Si:H layer in order to harvest unabsorbed light at the rear side. Considering the two possible reflection regimes of an inverted opal lead to the conclusion that a larger sphere radius gives slightly better results because diffraction into higher orders is more pronounced. This becomes especially obvious for flat layers equipped with a perfect inverted opal. However, to ensure good in-coupling randomly textured surfaces are mandatory to compete with flat metallic back contacts.

The natural self-assembly behaviour of the opaline structures on textured surfaces converging to a perfectly periodic lattice turns out to be beneficial for the photon management. In this case the advantages of disorder are combined with those of discrete features of a periodic arrangement. Table 1 shows a summary of some calculated short circuit current densities for a 1 μm thick μc-Si:H layer with a sphere radius of 325 nm. The best result is obtained for a combination of an amorphous and a perfect inverted opal. In this very case an overall enhancement of more than 16% is possible. However, the enhancement is even larger for thinner cells; being possibly more relevant to lower the total cost of fabricated solar cells. In addition, inverted opals can be fabricated by self assembly techniques. These techniques are, in general, characterized by low cost and high throughput.

Table 1. Short circuit current densities in the diffractive mode for a 1 μm thick μc-Si:H layer

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