## Abstract

The effect of periodic and disordered photonic structures on the absorption efficiency of amorphous and crystalline Silicon thin-film solar cells is investigated numerically. We show that disordered patterns possessing a short-range correlation in the position of the holes yield comparable, or even superior, absorption enhancements than periodic (photonic crystal) patterns. This work provides clear evidence that non-deterministic photonic structures represent a viable alternative strategy for photon management in thin-film solar cells, thereby opening the route towards more efficient and potentially cheaper photovoltaic technologies.

© 2013 Optical Society of America

## 1. Introduction

Thin-film photovoltaic cells are nowadays recognized as one of the most promising technologies for solar energy harvesting [1], since they benefit from potentially lower production costs and a better adaptability to a wider range of structures and equipments. However, due to the lower amount of material used, a significant amount of the incident radiation is not absorbed by the solar panel, thereby reducing the overall efficiency of the device. This limitation has motivated the scientific community to develop light trapping strategies, taking advantage of coherent effects in nanostructured media to enhance the absorption of light by thin and ultra-thin layers of inorganic (e.g. amorphous Silicon, CIGS, etc) or organic materials [2, 3]. For instance, front and back-side gratings [4–10], as well as randomly-textured surfaces [11–13] have been designed to reduce the reflection of the incident light on the surface of the cell and increase the light spreading in the absorbing material. Increasing attention has also been given to the light trapping ability of photonic crystals, that is, periodic lattices of wavelength-scale holes patterned into the active layer of the cell [14–18]. Light incident on such a structure can couple to the quasi-guided modes formed by the in-plane light diffraction. While most of studies have focused on periodic photonic structures, it was realized recently that structural disorder can offer an alternative strategy to improve the absorption efficiency of the patterned thin films [19–21]. Disorder, either consisting in a perturbed periodic pattern [19] or lacking completely any long-range order [20, 21], typically leads to broader spectral and angular responses, which are clearly beneficial for photovoltaics applications. This phenomenon was explained in terms of the disordered optical modes formed by two-dimensional multiple light scattering, and it was shown that the light coupling efficiency could be fined-tuned by imposing short-range correlations in the structural disorder, that is, a minimum distance between neighboring holes [20]. The application of such structures to standard photovoltaic architectures has however not been studied so far.

In this article, we compare the absorption efficiency of realistic single-junction thin-film solar cells containing periodic and disordered (random and short-range correlated) photonic structures. The absorption spectra of the patterned films are calculated numerically using the 3D Finite Difference Time Domain (FDTD) method [22]. Two active materials, already employed in the industry, namely amorphous and crystalline Silicon, are considered. We find that the disordered structures can enhance absorption up to 110% and 70% in crystalline and amorphous Silicon thin films, respectively, in the wavelength range 610 – 900 nm, exhibiting a light trapping efficiency comparable and sometimes superior compared to their periodic counterpart.

## 2. Analyzed structures and numerical methods

The absorptive response of solar cell prototypes is calculated numerically using a freely available FDTD software package, MEEP [23]. The overall architecture of the system is depicted in Fig. 1(a). We study two different devices respectively with amorphous Silicon (a-Si) and crystalline Silicon (c-Si) active layers. The thickness, *t*, is 300 nm for a-Si and 1 μm for c-Si layers. These values are often considered in literature [18, 24, 25] since these thicknesses are a good trade-off between the ability of the charge carriers to reach the contacts and the quantity of photons absorbed for that specific thickness. A *λ*/4 Anti-Reflection Coating (ARC) is placed above the active layer having a refractive index *n* = 1.52. This kind of architecture is commonly used in commercially available solar cells and in our case has a thickness *a* = 79 nm. A Silver (Ag) layer is used instead as back reflector, and its thickness is *b* = 200 nm, which ensures no light is transmitted.

In our calculations we consider only a reduced range (610 – 900 nm) of the solar radiation spectrum for the following reasons. For wavelengths smaller than 610 the needed spatial resolution for the computational grid requires an amount of memory not available on our computation facility. At wavelength longer than 900 nm instead, both in-plane transport mean free path of light [20] and intrinsic absorption length of the material exceed the computational cell. This could lead to computational artifact in relation to the boundary. However the spectral range we considered is also the most relevant for light trapping techniques.

Three photonic structures, made by patterning holes into the a-Si or c-Si active layer, are considered in this work, namely periodic (hexagonal), random and short-range correlated patterns. To generate the random pattern a Random Sequential Algorithm is used. Air cylinders are introduced one after the other in a box with a non-overlapping constraint and periodic boundary conditions. The Lubachevsky-Stillinger Algorithm [26] instead is chosen to generate short-range correlated-disordered patterns, that is, with a typical distance *D* between neighbouring cylinders. The main advantage of this method is the large control on the degree of correlation of the generated patterns. A system of hard disks in a box with periodic boundary conditions is left to evolve freely. The radius of the disks grows with time until a certain packing fraction, that is the ratio of the surface occupied by the disks over the total surface, is reached. The final structure is the obtained by recording the position of the center of the disks and using these coordinates for placing the holes. Their radius is chosen as to reach the desired hole filling fraction, *f*.

A large number of disordered structures can be generated with this method imposing hard or periodic boundary conditions, initial temperature, number of the spheres and growth rate of the radii [27]. To avoid the formation of ordered domains, cylinders of two different sizes are used. The ratio between the two radii and the percentage of large spheres are taken to be 90%. All the correlated-disordered structures considered in this work have a packing fraction of 69%. Note that the maximal packing fraction achievable (74%) gives a periodic, hexagonal pattern.

The numerical grid used in the calculation has an extent of 8×8×5 μm^{3} and we impose Periodic Boundary Conditions in the *X* and *Y* directions containing the plane of the structure. The dimension of the cell larger than the transport mean free path of the in-plane multiply scattered wave ensures that the calculations are not affected by the Periodic Boundary Condtions [20]. Perfect Matching Layers (PML) are placed instead in the *Z* direction. A plane wave source, having a normal incidence on the structure, is used and all the calculations are performed with a spatial resolution of 25 nm. The permittivity of a-Si, c-Si and Ag is implemented in the software with a Drude-Lorentz (DL) model

*ε*

_{0},

*σ*, Γ

_{i}*and*

_{i}*w*are obtained by a fitting procedure on the tabulated data [28]. For both active materials a second order DL expansion has been considered, whereas a first order is used for Ag (See Table 1).

_{i}Absorption is then obtained as
$A\left(\lambda \right)=1-\frac{{I}_{r}\left(\lambda \right)}{{I}_{0}\left(\lambda \right)}$ where *I _{r}* is the flux reflected by the structure and

*I*

_{0}is the one for the empty cell.

We define now two figures of merits (FOM) to allow a comparison with other architectures. The first one is useful to emphasize the light trapping ability of the photonic structure we consider. The second one instead is intended to point out the overall efficiency of our device from a technological point of view. The absorption enhancement, defined as the ratio, *E*, between the absorption of patterned films and bare slab, is the most used FOM by the nanophotonic community [29–31]:

*A*(

_{i}*λ*) is the absorption spectrum and

*i*labels the periodic, random or amorphous structure.

*A*(

_{s}*λ*) is the absorption for the bare slab. We also calculate the ratio of the integrated absorption spectra

*A*(

_{i}*λ*) and

*A*(

_{s}*λ*) weigthed by the incident solar photon flux

*P*

_{AM}_{1.5}(

*λ*) [32]:

For this reason we define a FOM, *G*, which compares an increase of the short-circuit current density
${J}_{i}^{\mathit{sc}}$ due to a photonic structure with the maximum increase achievable for
${J}_{s}^{\mathit{sc}}$:

*dJ*∝

^{sc}/dλ*A*(

*λ*)

*P*

_{AM}_{1.5}(

*λ*) [33]. This FOM is useful to characterize different solar cell architectures and light trapping techniques, since it combines the advantages of both absorption enhancement and spectral absorption of the specific material. On one hand,

*G*shows the light trapping ability of the photonic structure through the increase of the short-circuit photocurrent ( ${J}_{i}^{\mathit{sc}}-{J}_{s}^{\mathit{sc}}$). On the other hand, since

*G*has an upper bound (

*max*[

*G*] = 1), it highlights the effectiveness of the device to produce maximum current.

## 3. Results and discussion

In the following we present results from numerical calculations for the two a-Si and c-Si solar cell prototypes. The absorption spectra of the periodic, random, and correlated-disordered structures are presented and compared with that of the bare slab. For comparison all patterns have the same hole depth, *h*, and filling fraction *f*. For each spectrum the absorption enhancement is given, to highlight the light trapping properties of the structures. Finally the integrated FOM values, *IE* and *G*, which also take into account the solar photon flux, are illustrated in tables, to present the overall performance of each photonic structure.

#### 3.1. Amorphous Silicon (a-Si)

We consider at first the a-Si structure. The set of values we choose for thickness, *t*, filling fraction *f*, radius *r* and height, *h*, is:

- Configuration (A1):
*t*= 300 nm,*f*= 0.3,*r*= 200 nm,*h*= 300 nm. - Configuration (A2):
*t*= 100 nm,*f*= 0.3,*r*= 175 nm,*h*= 100 nm.

The same thickness of the a-Si layer in A1 have been considered by other authors [33] and in commercial solar cells [34], whereas the values for *r* and *f* of A2 are the same as in Ref. [20]. The absorption spectra are shown in Fig. 2(a) and 2(c). Interestingly at long wavelengths all three photonic structures lead to considerable absorptions and exceed 80% for most of the wavevelength range. In particular for configuration A2, due to the lower intrinsic absorption at long wavelengths and the presence of Fabry-Perot (FP) fringes, the bare slab response decreases and very relevant enhancements are possible when a light trapping technique is introduced. However both disordered structures show smoother responses than the periodic one and this has to be attributed to additional quasi-guided modes generated by disorder in all the wavelength range [20]. For wavelengths shorter than 750 nm instead correlated-disordered systems and periodic ones have similar performances, while the random structure still considerably improve the absorption.

The *E* FOM shown in Fig. 2(b) and 2(d) highlights these considerations: disordered structures have a broad peak of enhancement reaching 2.3 for configuration A1 and 8 for A2. All three photonic structures have comparable light trapping ability. However higher enhancements are obtained in the A2 configuration due to the coupling to Bloch modes in the film. The overall performance of devices A1 and A2 is summarized in Table 2 where *IE* and the *G* are shown. Periodic and correlated-disordered structures have comparable values for *G*. The fact that *IE* for A1 and A2 differs of around 30% in favour of A2, while these structures possess comparable *G*, highlights the relevance of this FOM for technological applications. In other words, *G* shows that for practical porposes there is no difference between A1 and A2 in both periodic and disordered case.

#### 3.2. Crystalline Silicon (c-Si)

For the c-Si device we consider two different configurations:

- Configuration (C1):
*t*= 1000 nm,*f*= 0.34,*r*= 198 nm,*h*= 190 nm - Configuration (C2):
*t*= 1000 nm,*f*= 0.55,*r*= 252 nm,*h*= 595 nm

These geometries have been chosen for comparison with Ref. [33], which provides optimal structural parameters for periodic systems valid for the wavelength range 610 – 900 nm. The aim of the numerical calculations we present here is to extend those results to random and correlated-disordered systems. The average absorptions of the three structures, shown in Fig. 3(a) and 3(c), are comparable and around 50 – 60% for both configurations. We find a similar spectral response with respect to the previous a-Si device, although in this case, periodic structures yield a larger number of narrower peaks due to the increased number of modes in the film and the lower intrinsic absorption of the material.

Figure 3(b) and 3(d) shows that the absorption can be even eight-fold times higher than the bare slab case for some wavelength and even if *E* strongly oscillates, absorption is enhanced almost at all wavelengths. Interestingly while in C1 the response of the correlated-disordered system is similar to the periodic one, in C2 these two systems are more distant especially at long wavelengths. In the case of higher *f* (C2) a smoother, almost flat, spectral response is observed thanks to quasi-guided modes induced by a stronger scattering and decreasing the Fabry-Perot fringes.

The figures of merit for c-Si are then summarized in Table 3. For both configurations disordered structures have better performances than periodic ones for certain configurations. A comparison between the two kind of architectures is allowed at this point. In contrast with *IE* which would suggest c-Si architecture to be the more performant, *G* shows that the correlated-disordered and periodic photonic structures on a-Si are the most promising system for technological applications. We expect that an a-Si solar cell with *t* ≃ 100 nm would be an excellent device, since *t* is shorter than the diffusion length of the minority carriers in a-Si, thereby providing an excellent carrier extraction. A systematic optimizion of the other geometrical parameters, filling fraction, hole radius, shape and degree of correlation of the holes can be further investigated also based on the specific applications. Depending on the active material, an appropriate tuning of filling fraction and degree of disorder could improve efficiency at long wavelengths where multiple scattering effects are more relevant. Light trapping can benefit from higher filling fractions but at the expenses of the active material that is removed. Smaller degrees of correlation instead could favour enhancements at long wavelengths where light trapping from disorder dominates. At small wavelengths, where the absorption is expected to influence the formation of quasi-guided modes, a geometrical optimization of the single scatterer is probably more relevant. In general, the results we obtained for different degrees of disorder suggest that the optimal values of these parameters in correlated-disordered structures should not differ too much from those of optimized periodic systems.

## 4. Conclusion

This work numerically investigates two (a-Si and c-Si) solar cell prototypes. Three kind of light trapping photonic structures have been considered for each architecture: periodic (hexagonal lattice), random and correlated-disordered. All of them led to strong absorption enhancements in the wavelength range we considered (610 – 900 nm) and show similar overall absorption re-ponses. Enhancement factors up to 70% (a-Si) and 110% (c-Si) have been demonstrated pointing out the strong light trapping ability of these systems. This implies that disordered light trapping structures, fully random or weakly correlated, constitute a valid alternative to periodic ones when used for solar cell applications. We have shown that for wavelengths where the intrinsic absorption of the active material is low, disordered patterns of circular air holes are able to enhance the overall absorption with an efficiency comparable to periodic ones and lead to a smoother spectral response.

In particular for the a-Si case we have demonstrated that a decrease of the active layer is possible without affecting the performance of the device. It turns out that, by making use of disordered structures, the thickness of the active layer can be reduced to one third in this case without significantly affecting the absorption.

We have used a new figure of merit *G* which allowed us to compare different photonic strategies applied on *different* materials to discover the most promising structure for realistic solar cells. We found that 100 nm thick a-Si film with periodic or correlated-disordered patterning is an excellent candidate. The high values obtained for *G* are encouraging since a systematic optimization has not been performed yet on these structures and we expect that further improvements could be obtained.

## Acknowledgments

We thank Gaurasundar Marc Conley for useful suggestions on the FOMs and to the complex photonics systems group at LENS for discussions. This work is financially supported by the European Network of Excellence Nanophotonics for Energy Efficiency, the ERC through the Advanced Grant PhotBots, ENI S.p.A. Novara, CNR-EFOR, and CNR-Fotonica2015.

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