## Photonic light-trapping versus Lambertian limits in thin film silicon solar cells with 1D and 2D periodic patterns |

Optics Express, Vol. 20, Issue S2, pp. A224-A244 (2012)

http://dx.doi.org/10.1364/OE.20.00A224

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### Abstract

We theoretically investigate the light-trapping properties of one- and two-dimensional periodic patterns etched on the front surface of c-Si and a-Si thin film solar cells with a silver back reflector and an anti-reflection coating. For each active material and configuration, absorbance *A* and short-circuit current density *J _{sc}* are calculated by means of rigorous coupled wave analysis (RCWA), for different active materials thicknesses in the range of interest of thin film solar cells and in a wide range of geometrical parameters. The results are then compared with Lambertian limits to light-trapping for the case of zero absorption and for the general case of finite absorption in the active material. With a proper optimization, patterns can give substantial absorption enhancement, especially for 2D patterns and for thinner cells. The effects of the photonic patterns on light harvesting are investigated from the optical spectra of the optimized configurations. We focus on the main physical effects of patterning, namely a reduction of reflection losses (better impedance matching conditions), diffraction of light in air or inside the cell, and coupling of incident radiation into quasi-guided optical modes of the structure, which is characteristic of

*photonic*light-trapping.

© 2011 OSA

## 1. Introduction

2. J. Poortmans and V. Arkhipov (editors), *Thin Film Solar Cells* (Wiley, Chichester, 2006). [CrossRef]

*J*, which is calculated according to assumptions of Sect. 3.1 with the standard AM1.5 incident solar spectrum [3

_{sc}3. AM1.5 solar spectrum irradiance data: http://rredc.nrel.gov/solar/spectra/am1.5.

5. D. T. Pierce and W. E Spicer, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B **5**, 3017–3029 (1972). [CrossRef]

6. M. I. Alonso, M. Garriga, C. A. Durante Rincán, E. Hernández, and M. León, “Optical functions of chalcopyrite CuGa* _{x}*In

_{1−x}Se

_{2}alloys,” Appl. Phys. A

**74**, 659–664 (2002). [CrossRef]

13. J.N. Munday and H.A. Atwater, “Large integrated absorption enhancement in plasmonic solar cells by combining metallic gratings and antireflection coatings,” Nano Lett. **11**, 2195–2201 (2011). [CrossRef]

14. J. Krc̆, M. Zeman, O. Kluth, F. Smole, and M. Topic̆, “Effect of surface roughness of ZnO:Al films on light scattering in hydrogenated amorphous silicon solar cells,” Thin Solid Films **426**, 296–304 (2003). [CrossRef]

18. T. Lanz, B. Ruhstaller, C. Battaglia, and C. Ballif, “Extended light scattering model incorporating coherence for thin-film silicon solar cells,” J. Appl. Phys. **110**, 033111 (2011). [CrossRef]

20. M. Steltenpool, J. Rutten, G. van der Hofstad, H. de Groot, J. de Ruijter, A. J. M. van Erven, and G. Rajeswaran, “*Periodic textured TCO for increased light-trapping in thin-film silicon solar cells*,” in Proceedings of the 26th European Photovoltaic Solar Energy Conference and Exhibition (Hamburg, September 5–9, 2011), paper 3AV.1.55.

21. J. Gjessing, A. S. Sudbø, and E. S. Marstein, “Comparison of periodic light-trapping structures in thin crystalline silicon solar cells,” J. Appl. Phys. **110**, 033104 (2011). [CrossRef]

22. C. Heine and R. H. Morf, “Submicrometer gratings for solar energy applications,” Appl. Opt. **34**, 2476–2482 (1995). [CrossRef] [PubMed]

23. S. Hava and M. Auslender, “Design and analysis of low-reflection grating microstructures for a solar energy absorber,” Solar Energy Mat. Solar Cells **61**, 143–151 (2000). [CrossRef]

24. P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express **15**, 16986–17000 (2007). [CrossRef] [PubMed]

42. A. Mellor, I. Tobs, A. Mart, M. J. Mendes, and A. Luque, “Upper limits to absorption enhancement in thick solar cells using diffraction gratings,” Prog. Photovolt: Res. Appl. **19**, 676–687 (2011). [CrossRef]

43. N. Senoussaoui, M. Krause, J. Müller, E. Bunte, T. Brammer, and H. Stiebig, “Thin-film solar cells with periodic grating coupler,” Thin Solid Films **451–452**, 397–401 (2004). [CrossRef]

53. X. Meng, G. Gomard, O. E. Daif, E. Drouard, R. Orobtchouk, A. Kaminski, A. Fave, M. Lemiti, A. Abramov, P. Roca i Cabarrocas, and C. Seassal, “Absorbing photonic crystals for silicon thin-film solar cells: Design, fabrication and experimental investigation,” Solar Energy Mat. Solar Cells **95**, S32–S38 (2011). [CrossRef]

45. L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. **89**, 111111 (2006). [CrossRef]

46. L. Zeng, P. Bermel, Y. Yi, B. A. Alamariu, K. A. Broderick, J. Liu, C. Hong, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Demonstration of enhanced absorption in thin film Si solar cells with textured photonic crystal back reflector,” Appl. Phys. Lett. **93**, 221105 (2008). [CrossRef]

44. H. Stiebig, N. Senoussaoui, C. Zahren, C. Haase, and J. Müller, “Silicon thin-film solar cells with rectangular-shaped grating couplers,” Prog. Photovolt: Res. Appl. **14**, 13–24 (2006). [CrossRef]

51. Q. Hu, J. Wang, Y. Zhao, and D. Li, “A light-trapping structure based on Bi_{2}O_{3} nano-islands with highly crystallized sputtered silicon for thin-film solar cells,” Opt. Express **19**, A20–A27 (2011). [CrossRef] [PubMed]

43. N. Senoussaoui, M. Krause, J. Müller, E. Bunte, T. Brammer, and H. Stiebig, “Thin-film solar cells with periodic grating coupler,” Thin Solid Films **451–452**, 397–401 (2004). [CrossRef]

44. H. Stiebig, N. Senoussaoui, C. Zahren, C. Haase, and J. Müller, “Silicon thin-film solar cells with rectangular-shaped grating couplers,” Prog. Photovolt: Res. Appl. **14**, 13–24 (2006). [CrossRef]

48. J. Zhu, Z. Yu, G. F. Burkhard, C. Hsu, S. T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett. **9**, 279–282 (2009). [CrossRef]

50. M. Tsai, H. Han, Y. Tsai, P. Tseng, P. Yu, H. Kuo, C. Shen, J. Shieh, and S. Lin, “Embedded biomimetic nanostructures for enhanced optical absorption in thin-film solar cells,” Opt. Express **19**, A757–A762 (2011). [CrossRef] [PubMed]

52. A. Naqavi, K. Söderström, F. J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Understanding of photocurrent enhancement in real thin film solar cells: towards optimal one-dimensional gratings,” Opt. Express **19**, 128–140 (2011). [CrossRef] [PubMed]

44. H. Stiebig, N. Senoussaoui, C. Zahren, C. Haase, and J. Müller, “Silicon thin-film solar cells with rectangular-shaped grating couplers,” Prog. Photovolt: Res. Appl. **14**, 13–24 (2006). [CrossRef]

48. J. Zhu, Z. Yu, G. F. Burkhard, C. Hsu, S. T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett. **9**, 279–282 (2009). [CrossRef]

49. O. El Daif, E. Drouard, G. Gomard, A. Kaminski, A. Fave, M. Lemiti, S. Ahn, S. Kim, P. Roca i Cabarrocas, H. Jeon, and C. Seassal, “Absorbing one-dimensional planar photonic crystal for amorphous silicon solar cell,” Opt. Express **18**, A293–A299 (2010). [CrossRef] [PubMed]

*I-V curve*measurements [44

**14**, 13–24 (2006). [CrossRef]

46. L. Zeng, P. Bermel, Y. Yi, B. A. Alamariu, K. A. Broderick, J. Liu, C. Hong, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Demonstration of enhanced absorption in thin film Si solar cells with textured photonic crystal back reflector,” Appl. Phys. Lett. **93**, 221105 (2008). [CrossRef]

47. I. Prieto, B. Galiana, P. A. Postigo, C. Algora, L. J. Martnez, and I. Rey-Stolle, “Enhanced quantum efficiency of Ge solar cells by a two-dimensional photonic crystal nanostructured surface,” Appl. Phys. Lett. **94**, 191102 (2009). [CrossRef]

50. M. Tsai, H. Han, Y. Tsai, P. Tseng, P. Yu, H. Kuo, C. Shen, J. Shieh, and S. Lin, “Embedded biomimetic nanostructures for enhanced optical absorption in thin-film solar cells,” Opt. Express **19**, A757–A762 (2011). [CrossRef] [PubMed]

52. A. Naqavi, K. Söderström, F. J. Haug, V. Paeder, T. Scharf, H. P. Herzig, and C. Ballif, “Understanding of photocurrent enhancement in real thin film solar cells: towards optimal one-dimensional gratings,” Opt. Express **19**, 128–140 (2011). [CrossRef] [PubMed]

24. P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express **15**, 16986–17000 (2007). [CrossRef] [PubMed]

33. S. Zanotto, M. Liscidini, and L. C. Andreani, “Light trapping regimes in thin-film silicon solar cells with a photonic pattern,” Opt. Express **18**, 4260–4274 (2010). [CrossRef] [PubMed]

39. R. Esteban, M. Laroche, and J. J. Greffet, “Dielectric gratings for wide-angle, broadband absorption by thin film photovoltaic cells,” Appl. Phys. Lett. **97**, 221111 (2010). [CrossRef]

40. D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Opt. Express **19**, A95–A107 (2009). [CrossRef]

*J*are taken as figures of merit and are calculated over the characteristic spectral range of AM1.5 solar spectrum with RCWA. For several thicknesses of active materials in the range of interest for thin film PV cells, the configuration is optimized with respect to the etching depth, the period of the photonic patterns, and the ARC parameters. The calculations are performed over a broader range of active material thickness with respect to the other works in the literature, where typically only one thickness value is investigated. By a detailed investigation of the optical spectra we get insight into the three main physical phenomena involved in light trapping and resulting in absorption enhancement: (

_{sc}**i**) suppression of reflection losses due to better impedance matching, (

**ii**) diffraction in air and into the cell material, and (

**iii**) coupling with the quasi-guided optical modes supported by the structure, which is often referred to as

*photonic light trapping*. As a main goal of this work, results for the optimized structures are compared with Lambertian limits for light-trapping in the ray optics regime. [7

7. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. **72**, 899–907 (1982). [CrossRef]

9. M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Progr. Photovolt: Res. Appl. **10**, 235–241 (2002). [CrossRef]

## 2. Analyzed structures

*x,y*) plane, with

*z*the direction normal to the cell. The direction of incident radiation is indicated by the red arrow in Fig. 2(a) and can be identified with the pair of polar angle

*θ*and azimuthal angle

*ϕ*. For the present work, we limit ourselves to nearly normal incidence, namely

*θ*=0.1° and

*ϕ*=0° (the choice

*θ*=0.1° is necessary for the convergence of the calculations and it does not substantially modify the results from the normal incidence case, [54

54. D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B **60**, 2610–2618 (1999). [CrossRef]

*θ*and

*ϕ*. This is particularly interesting in the case of 1D patterns, since a stronger angular dependence would be expected. Instead, a typical cos

*θ*dependence of the current response was found, and the same trend occurs when the polarization direction is parallel (

*ϕ*=0) or orthogonal (

*ϕ*=90 degrees) to the grating [36]. For 2D structures, the angular dependence is expected to be further suppressed, due to higher symmetry and isotropy.

*d*, deposited on a silver (for optical data see Ref. [4]) substrate that serves as back reflector and electric contact. The active materials we consider are c-Si and a-Si, whose optical functions are taken from Ref. [4]. The spectral absorbances of these materials are different, and this affects the typical thickness

*d*required to obtain complete light absorption, making c-Si PV cells much thicker than a-Si ones. Being interested in light-trapping in thin film Si solar cells, we limit the range of thicknesses from 250 nm to 4

*μ*m for c-Si PV cells and from 50 to 500 nm for a-Si PV cells. Especially for the case of a-Si PV cells, the film thickness is limited by diffusion lengths for the photogenerated carriers, which are less than a few hundreds nanometres [2

2. J. Poortmans and V. Arkhipov (editors), *Thin Film Solar Cells* (Wiley, Chichester, 2006). [CrossRef]

*a*, the width of the etched region with

*b*, and the etching depth with

*h*. The grooves are supposed to be filled with a transparent dielectric medium with refractive index

*n*=1.65, and the same dielectric is supposed to be deposited until a covering slab of thickness

_{ARC}*l*is formed. This layer acts both as a passivating structure for the etched layer as well as an anti-reflection coating (ARC). The value for

*n*is in between that of fused silica (SiO

_{ARC}_{2}, see Ref. [4] for optical data) and that of transparent conductive oxides, as, for example, Al-doped zinc oxide (AZO, see Ref. [56

56. M. Caglar, S. Ilican, Y. Caglar, and F. Yakuphanoglou, “The effect of Al doping on the optical constants of ZnO thin films prepared by spray pyrolysis method,” J. Mater. Sci: Mater. Electron. **19**, 704–708 (2008). [CrossRef]

57. Y. Yang, X. W. Sun, B. J. Chen, C. X. Xu, T. P. Chen, C. Q. Sun, B. K. Tay, and Z. Sun, “Refractive indices of textured indium tin oxide and zinc oxide thin films,” Thin Solid Films **510**, 95–101 (2006). [CrossRef]

_{2}, as we used in our previous work [33

33. S. Zanotto, M. Liscidini, and L. C. Andreani, “Light trapping regimes in thin-film silicon solar cells with a photonic pattern,” Opt. Express **18**, 4260–4274 (2010). [CrossRef] [PubMed]

*l*is set to 70 nm for all the investigated structures: this value has been optimized in a previous work on 1D lattices [33

33. S. Zanotto, M. Liscidini, and L. C. Andreani, “Light trapping regimes in thin-film silicon solar cells with a photonic pattern,” Opt. Express **18**, 4260–4274 (2010). [CrossRef] [PubMed]

*r*etched in the Si slab, and the materials filling fractions are We shall display the results as a function of the ratio

*r/a*(instead of the filling parameters

*FF*), as it gives a more intuitive description of the pattern structure. Attention should be paid to the fact that the ratio

*b/a*for 1D patterns can span the range [0,1], while the ratio

*r/a*is within the range [0,0.5], the upper limit corresponding to rods’ contact.

## 3. Theory and numerical methods

*I-V curve*. From a theoretical point of view, this relation can be derived assuming an ideal and exponential I-V curve for the p-n junction forming the PV cell and adding a current contribution proportional to the flux of incident photons with energy larger than the band-gap

*E*(Ref. [1]). Considering the response of the PV cell unit area rather than that of the whole PV cell, the

_{g}*J-V curve*can be expressed as: where

*J*is the short-circuit current density,

_{sc}*J*

_{0}is the generation current density for the p-n junction,

*e*the electron charge,

*K*the Boltzmann constant, and

_{B}*T*the thermalized electron temperature of the cell. At a given operating voltage

*V*, the product

*V*·

*J*(

*V*) gives the electrical power converted by PV cell per unit area. It can be shown that this depends nearly linearly upon

*J*, and for this reason

_{sc}*J*has been assumed as our main figure of merit. The short-circuit current density

_{sc}*J*is a spectrally-integrated quantity defined as [1]: where

_{sc}*A*(

*E*) is the absorbance of the active material,

*IQE*is the internal quantum efficiency for separation and collection of the photogenerated electron-hole pairs. Since in this work we are concerned with the optical rather than transport properties of the cell,

*IQE*has been set equal to one. In this way one can define the

*spectral contribution dJ*to the short-circuit current density (Eq. (3)). This in turn is proportional to the absorbance

_{sc}/dE*A*(

*E*) of the active material in the PV cell, which is taken as our second figure of merit. The integration range has a lower limit 1.12 eV for c-Si, or 1.25 eV for a-Si (the bandgap of a-Si with the data of Ref. [5

5. D. T. Pierce and W. E Spicer, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B **5**, 3017–3029 (1972). [CrossRef]

3. AM1.5 solar spectrum irradiance data: http://rredc.nrel.gov/solar/spectra/am1.5.

**18**, 4260–4274 (2010). [CrossRef] [PubMed]

^{2}, which is that of the AM1.5 solar spectrum. Both the truncation of the integration range and the choice of a blackbody spectrum affect the final results, giving lower

*J*with respect to the full AM1.5 solar spectrum, which is slightly richer in photons in the visible range. This is the reason why the subsequent results for Lambertian limits are slightly smaller than those given in Fig. 1: this choice allows to considerably reduce computing time when optimizing the structures. Of course, a real solar cell should be simulated with the full AM1.5 spectrum.

_{sc}*A*(

*E*) of the active layer and hence

*J*are calculated adopting a rigorous electromagnetic approach, starting from Maxwell equations for fields

_{sc}*E*(

*x,y,z*) and

*H*(

*x,y,z*), and then calculating the scattering matrix

*S*for the configuration under investigation [54

54. D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B **60**, 2610–2618 (1999). [CrossRef]

55. M. Liscidini, D. Gerace, L. C. Andreani, and J. E. Sipe, “Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media,” Phys. Rev. B **77**, 035324 (2008). [CrossRef]

*z*(see Fig. 3(a)), with the light incident from the left, under the form of a plane wave. Following the notations of Refs. [33

**18**, 4260–4274 (2010). [CrossRef] [PubMed]

54. D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B **60**, 2610–2618 (1999). [CrossRef]

*x,y*) ≡

*ρ*, the electric field

*E*can be expressed in each point as a Fourier series: where

*NPW*is the total number of plane waves considered in the simulation,

*G*are the reciprocal lattice vectors (taking

_{n}*G*

_{0}= 0), and

*k*

_{//}is the component of incident wave vector in the (

*x,y*) plane. The Fourier amplitudes

*Ẽ*(

*G*) are obtained solving Maxwell equations in each layer by matrix diagonalization, and then propagating along the structure with proper boundary conditions, with the basis set of plane waves determined by

_{n}, z*NPW*(

*z*dependence is implicitly taken into account, with both propagating and exponentially decaying waves being considered).

*x,y*) plane, and apply Poynting theorem to the volume element enclosed by dashed line in Fig. 3(a). Absorption in the active material is calculated from difference in Poynting vector fluxes across the facets normal to

*z*direction, while any contributions from the other facets vanish due to translational invariance. Normalizing incident power density to unit, energetic balance implies: where

*A*is the absorbance of the active material,

*R*=

*R*

_{n=0}is the zeroth-order reflectance, and

*T*=

*T*

_{n=0}is the zeroth-order transmittance to the Ag substrate. The first sum represents all the power contributions deriving from diffraction in air, and the second sum refers to diffraction in the Ag substrate. In the case of a planar cell, the last two sums in Eq. (5) are absent, due to translational invariance, and thus

*R*and

*T*are the only optical losses (Eq. (5) simply reduces to

*A*+

*R*+

*T*=1).

*A*in the active material contributes to the short-circuit current density, thus the photonic pattern has to increase the absorbance with respect to planar cells without introducing additional losses.

*NPW*→ ∞, this analysis is exact. In practice, one has to truncate the sum in Eq. (4) to a certain number of plane waves. As the computational time for a large number of plane waves is proportional to

*NPW*

^{3}because of matrix diagonalization, it is important to choose the

*NPW*that guarantees the convergence of the results in a reasonable computing time. In this respect, symmetry properties of the pattern play a major role, determining both the appropriate number

*NPW*and coupling between incident field and optical quasi-guided modes of the PV structure.

*NPW*is shown in Fig. 3(b), where a typical absorbance spectrum is reported for a c-Si PV cell (

*d*=500 nm) with 2D square pattern. The same spectrum is calculated with four different values for

*NPW*. All spectra have similar features at low energy, but noticeable differences are evident with increasing energy. This is because low energy photons can couple only with few modes of the structure, namely the lowest energy modes, and a relatively small

*NPW*is sufficient to reach convergence. On the contrary, high energy photons can couple with more modes, and thus a higher

*NPW*is needed to take into account all accessible modes and to reach convergence. We studied convergence for 2D structures with different pattern configurations, and we concluded that differences between

*NPW*equal to 121 and 137 are small and located at energies above 3 eV, where the solar photon flux is weaker, so

*NPW*=121 is sufficient to full convergence within the whole investigated range. For 1D cells the computation is much shorter, and

*NPW*= 25 is enough to reach full convergence.

*NPW*. For a low

*NPW*,

*J*is below the convergence value, then it increases with

_{sc}*NPW*, showing small oscillations around the convergence value. Even if there is convergence in the spectra only at low energies, the variations around the convergence spectrum are limited to high-energy, where solar photon flux is lower, and they tend to balance in the integration, giving a faster global convergence as compared to absorption spectra.

*bulk*PV cells. The upper limits for the enhancement of light path inside the active material were first derived under three assumptions [7

7. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. **72**, 899–907 (1982). [CrossRef]

8. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. **29**, 300–305 (1982). [CrossRef]

- The structure has a front Lambertian scatterer or a back Lambertian reflector on the bottom. In both cases transmitted or reflected light is randomized with isotropic angular distribution at every energy. Furthermore, intrinsic reflection losses of the active material are set to zero, and an ideal back reflector is located at the bottom.
- The intrinsic absorption of the active material is sufficiently low for both the
*single-pass*and the*enhanced*absorbance. This situation is referred to as*weak absorption regime*and is described by the condition: where*n*is the real part of the refractive index,*α*is the absorption coefficient, and*d*the thickness of the active material. - The film thickness
*d*is much larger than wavelength*λ/n*inside the active material, and this makes ray optics arguments suitable.

*α*(

*E*)

*d*

_{eff}, with

*d*

_{eff}being the effective light path inside the active material in a given configuration. For example, if we consider a single-pass through a planar slab, neglecting reflection losses, the product

*αd*

_{eff}has a direct interpretation, as the total absorbance

*A*is: Under Yablonovitch hypothesis, it can be shown using ray optics or statistical mechanics arguments that the maximum light path is enhanced by a factor 4

_{sp}*n*

^{2}/sin

^{2}

*γ*, with

*γ*half of the apex angle of the cone subtended by the dielectric medium surrounding the cell. In the case of a planar cell surrounded by an isotropic medium (sin

*γ*=1), the maximum enhancement is given by 4

*n*

^{2}, which corresponds to nearly 50 for silicon near the band gap and gives an active material absorbance equal to: where reflection losses are neglected. Along the manuscript, this first Lambertian limit (i.e., weak absorption and neglecting reflection losses) will be denoted by LL0.

9. M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Progr. Photovolt: Res. Appl. **10**, 235–241 (2002). [CrossRef]

*γ*=1, it can be shown that the maximum absorption enhancement is lower than in the LL0 limit, as the weak absorption hypothesis is relaxed and the enhancement expected by light trapping is smaller when absorption is larger. We refer to [9

9. M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Progr. Photovolt: Res. Appl. **10**, 235–241 (2002). [CrossRef]

*α*, and it is the most relevant for our work, since active material’s absorption is not negligible in most of the investigated configurations. To compare properly with the LL0 limit, we take also the LL

*α*limit in the case of no reflection losses.

*photonic*light-trapping, in contrast with the case of

*bulk*PV cells. Light trapping in the (nano)photonic regime has been investigated in Refs. [31

31. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express **18**, A367–A380 (2010). [CrossRef]

32. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Nat. Ac. Sci. **107**, 17491–17496 (2010). [CrossRef]

*α*limits as references for the calculated short-circuit current and absorbance spectra, since they can be easily calculated and provide a continuous transition to the case of thick cells, where the ray optics treatment is rigorously valid. In fact, the dashed curves of Fig. 1 are calculated for the realistic case of the LL

*α*limit.

*photonic*light-trapping, it is difficult to introduce an effective light path. Thus we define an absorption enhancement factor

*F*with respect to the single pass absorption

*A*as where

_{sp}*A*(

*E*) is the structure absorption, which can be calculated analytically or numerically depending on the cell geometry. It is worth noticing that the two limits discussed above can be described using Eq. (9). In particular, in the case of weak absorption the expression gives back the typical 4

*n*

^{2}enhancement factor, which can be considered as a reference value for Lambertian light trapping. As it will be evident from the results in the next section, reaching Lambertian light-trapping limits over a broad spectral range is a very difficult task, which can be partly (but not fully) achieved with the periodic structures considered in this work.

## 4. Results and discussion

*d*in range of interest for thin film solar cells, according to observations of Sect. 2. For brevity, only one thickness is analyzed in detail for each active material (1

*μ*m for c-Si and 300 nm for a-Si), and the results for structures of varying thickness are described at the end of each material’s subsection.

*d*, the analysis is performed in the following way. First, a contour plot of

*J*is calculated with the scattering matrix formalism varying at the same time the etching depth

_{sc}*h*and the ratio

*b/a*(for 1D pattern) or

*r/a*(for square pattern). According to observations of Sec. 3, the number

*NPW*of plane waves is chosen to be 45 for 2D square pattern. Optimal pattern configurations are then identified from maxima in the contour plot, and the corresponding spectra are analyzed in detail. For each optimal configuration, optical spectra for reflectance

*R*, transmittance

*T*, absorbance

*A*, and diffracted power are calculated with a

*NPW*that ensures convergence over the whole spectrum (

*NPW*=25 for 1D patters and 121 for 2D square patterns). Finally, the absorption enhancement

*F*with respect to single pass absorption without reflection losses is compared with the Lambertian limits LL0 and LL

*α*to light-trapping.

*A*and

*J*are considered. For absorbance, the comparison is made with the absorbance of a bare slab of active material, without ARC or back reflector and considering the cases with and without reflection losses. This comparison is useful because it directly recalls intrinsic properties of the active material in terms of bare absorption and reflection. The energy range for calculations can thus be subdivided into two ranges: one at low energy, where patterning enhances absorption due to light-trapping, and the other one at high energy, where total absorption occurs even without light-trapping, and, actually, reflection and diffraction in air have to be minimized.

_{sc}*J*and spectral contributions

_{sc}*dJ*, the comparison is made with a reference cell with same thickness

_{sc}/dE*d*of the investigated configuration, planar geometry, Ag back reflector, and single-layer ARC on top. Optical spectra for reference cells are calculated following the same procedure used for patterned cells. The parameters of the ARC are optimized to give the highest possible

*J*from the reference cell. To make a proper comparison, optimizations (like random texturing) that are common in commercial cells are not considered here.

_{sc}### 4.1. Crystalline silicon (c-Si)

*J*for c-Si PV cells of thickness

_{sc}*d*=1

*μ*m patterned with simple 1D and square 2D lattices are shown in Figs. 4(a) and 4(b). One can immediately note that the 2D square pattern is much more performing that the 1D one over a broad range of pattern parameters. Shallow and deep pattern configurations emerge as the maxima in the contour plot. In particular for simple 1D pattern we find: For 2D square pattern we find:

*J*around the optimal configuration with respect to variations in

_{sc}*h*and

*b/a*, or

*r/a: J*decreases no more than 2% upon relative variations of ±10% around the optimal values.

_{sc}

_{n}_{≠0}

*R*is present, but the total reflectance, which includes all diffraction orders, is always lower than the reflectance

_{n}*R*of the reference cell; a similar trend is observed as for the zero-order reflectance.

*R*contribution, that is nearly complementary to absorbance spectrum. At high energies there is a residual reflection, but it is evident that the intermediate patterns provides a spectral band of low reflectance, which is broader than that of the reference cell with single-layer ARC. Diffraction in air is a substantially small and constant term in 1D cells, but it is larger in 2D cells, due to the larger number of diffraction modes allowed by symmetry. Diffraction in air can be strongly reduced by decreasing the lattice period, as it will be evident for a-Si cells.

*d*=1

*μ*m has the potential to absorb nearly all incident light with energy greater than 2.75 eV, at least when a proper ARC is applied on the front surface. Light trapping has to be tailored to give its maximum contribution below this energy threshold. This is done optimizing the structures, and best results reported in Figs. 5(b) and 5(e) show several peaks deriving from coupling with quasi-guided modes. Since the typical spectral width of each peak is narrow compared to the useful solar spectrum bandwidth, a collection from multiple peaks deriving from different coupling processes is needed to obtain relevant increase in

*J*.

_{sc}*α*limits. This is evident from absorption enhancement

*F*reported in Fig. 5(c), which reaches a smoothed maximum value of 15, still far away from 4

*n*

^{2}maximum value for the LL0 limit in the weak absorption regime. For c-Si PV cells with 2D square lattice, instead, the calculated absorbance can overcome both LL0 and LL

*α*limits, at least at the exact energies for coupling with quasi-guided modes, in agreement with predictions from temporal coupled-mode theory [31

31. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express **18**, A367–A380 (2010). [CrossRef]

32. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Nat. Ac. Sci. **107**, 17491–17496 (2010). [CrossRef]

*n*

^{2}for Lambertian limit to light-trapping (Fig. 5(f)).

*dJ*to short-circuit current density are shown in Fig. 6. In terms of short-circuit current density, patterning gives a +30% enhancement for the 1D case, and a +55% enhancement for the 2D square case, compared to a reference cell with the same thickness

_{sc}/dE*d*=1

*μ*m.

29. R. Dewan and D. Knipp, “Light trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. **106**, 074901 (2009). [CrossRef]

40. D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Opt. Express **19**, A95–A107 (2009). [CrossRef]

*μ*m thick microcrystalline Si PV cells with frontal 1D grating and back reflector, with optimal period

*a*=600 nm and optimal grating height

*h*=200 nm, that are in good agreement with our results.

*J*are shown for the optimal configurations for each different thickness

_{sc}*d*. Periodic patterns give nearly the same current density gain at every thickness, while

*J*strongly decreases in thinner reference cells (as indicated by the blue curve of Fig. 7(a)). This makes light trapping much more relevant for thinner structures. For example, only a +23% increase in

_{sc}*J*is calculated for a 4

_{sc}*μ*m thick c-Si PV cell with optimized 2D square pattern, but a +82% is obtained with a 250 nm thick cell with the same, properly optimized pattern.

*J*on lattice period is studied, as shown in Fig. 7(b) for the optimized configurations at different

_{sc}*d*and

*a*in the range from 300 to 1000 nm. Upon variations of

*a*, light trapping can be tuned in order to produce a dense distribution of absorption peaks in the desired spectral region. With a short period, only high energy photons can couple into the structure, and the low energy ones experience the textured layer as uniform and characterized by an effective refractive index intermediate between those of c-Si and ARC. Reflection and diffraction losses can be reduced, but light trapping is limited to high energies, where solar photon flux is weak. Light trapping can be tuned toward low energies by increasing the period

*a*, but also diffraction in air is increased. Optimal periods derive from balance of these two mechanisms: the thicker the cell, the more light-trapping has to be tuned towards lower energies.

*d*=250 nm) to be around 450 nm, then it increases up to 600 nm for intermediate structures (

*d*=1

*μ*m) and up to 700÷800 nm for thicker structures (

*d*=2 and 4

*μ*m). Nevertheless, it is interesting to observed that the optimal period has a much smaller increase than the cell thickness, and that the short-circuit current has a relatively weak dependence on the lattice period close to the optimal value, being always considerably improved with respect to the reference cell.

### 4.2. Amorphous silicon (a-Si)

*J*for a-Si PV cells of thickness

_{sc}*d*=300 nm patterned with simple 1D and square 2D lattices are shown in Fig. 8(a) and 8(b), respectively. As for c-Si structures, various configurations emerge.

*R*

_{n=0}decreases in 1D and, especially, in 2D patterned cells. A spectral band with nearly zero reflection is obtained from 2 to 2.75 eV in 1D cells and up to 3.25 eV in 2D cells. Only two residual contributions are observed: one at high energy (

*R*

_{n=0}≈0.1) and one around

*E*=1.6 eV, which is present in all devices (with just a gradual red shift in the reflection peak from reference to 1D and then 2D cells). Diffraction in air, instead, is substantially suppressed due to the short period

*a*, differently from c-Si cells.

*d*=300 nm, light trapping can give a contribution only for energies below 2 eV, where absorption is sufficiently weak.

*F*reported in Fig. 9(c) and 9(f) are lower than 10, and still far from 4

*n*

^{2}Lambertian limit. This is mainly due to the high intrinsic absorption for the single pass, that gives low

*F*according to Eq. (9): for a-Si, the LL

*α*limit is more relevant in most of the spectral range. 2D square pattern gives higher

*J*, with a relative +20% increase. The simple 1D pattern follows with a relative +11% increase in

_{sc}*J*with respect to reference cell.

_{sc}*d*=300 nm are summarized with spectral contributions

*dJ*shown in Fig. 10. We compared our results for a-Si structures with those in the literature at a few specific thickness, as we made for c-Si cells. For example, Ref. [25

_{sc}/dE25. M. Kroll, S. Fahr, C. Helgert, C. Rockstuhl, F. Lederer, and T. Pertsch, “Employing dielectric diffractive structures in solar cells a numerical study,” Phys. Stat. Sol. (a) **205**, 2777–2795 (2008). [CrossRef]

*a*=360 nm, etching depth

*h*=80 nm and Si fill fraction 1 –

*b/a*=58%. This is in good agreement with the data in the contour plot of Fig. 8(a): we observe the main maximum at a larger etching depth, but also a secondary maximum having features compatible with the aforementioned work. Also Ref. [30

30. Y. Park, E. Drouard, O. El Daif, X. Letartre, P. Viktorovitch, A. Fave, A. Kaminski, M. Lemiti, and C. Seassal, “Absorption enhancement using photonic crystals for silicon thin film solar cells,” Opt. Express **17**, 14312–14321 (2009). [CrossRef] [PubMed]

*a*=450 nm and Si fill fraction 1 –

*b/a*=68%. Also in this case we find good agreement with our results, with optimal period

*a*=500 nm and 1–

*b/a*=65%.

*d*are illustrated in Fig. 11(a). As for c-Si structures, patterned cells have better performance with respect to planar reference ones, with 2D square pattern being better than 1D pattern. In addition, light trapping is more relevant for thinner cells. For example, a +12%

*J*increase is calculated for a 500 nm thick cell with optimized 2D square pattern, but a +49% increase is obtained in a 50 nm thick cell with the same, properly optimized pattern.

_{sc}*J*on thickness

_{sc}*d*and period

*a*, as shown in Fig. 11(b). For thin cells (

*d*=50 and 100 nm), optimal

*a*is of the order of 500 nm and a trend similar to c-Si cells is observed. For thicker cells we do not observe this trend, and, actually, for

*d*=300 nm the shortest

*a*is the best one. Taking into account that a longer

*a*is needed to couple low energy photons inside the structure, we conclude that the absorption enhancement provided by a longer

*a*is overcome by additional losses deriving from diffraction in air. The highest

*J*is thus obtained with a shorter

_{sc}*a*, since diffraction in air is largely suppressed, and reduction of reflection losses becomes the key effect determining the absorption enhancement.

## 5. Conclusions and future developments

*J*and hence higher conversion efficiencies of the PV structures under consideration. For c-Si, relative increase goes from +23% with a 4

_{sc}*μ*m thick cell, up to +82% with a 250 nm thick cell. For a-Si, instead,

*J*relative increase goes from +12% with a 500 nm thick cell, up to +49% with a 50 nm thick cell. These results are rather tolerant with respect to small (up to 10%) deviations from the optimal parameters. Actually, fabrication imperfections might even improve the light trapping performance thanks to the effects of disorder, as discussed below.

_{sc}*α*limit, which is generally valid in the case of arbitrary absorption) we conclude that the

*J*generated by the investigated structures is intermediate between the calculated values for the planar reference cells (with AR coating and back reflector) and the ultimate limits. Two main guidelines can be derived for the future design of more efficiency thin film cells with a photonic pattern:

_{sc}*J*increase.

_{sc}## Acknowledgments

## References and links

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56. | M. Caglar, S. Ilican, Y. Caglar, and F. Yakuphanoglou, “The effect of Al doping on the optical constants of ZnO thin films prepared by spray pyrolysis method,” J. Mater. Sci: Mater. Electron. |

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**OCIS Codes**

(040.5350) Detectors : Photovoltaic

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photovoltaics

**History**

Original Manuscript: October 18, 2011

Revised Manuscript: December 7, 2011

Manuscript Accepted: December 8, 2011

Published: January 30, 2012

**Citation**

Angelo Bozzola, Marco Liscidini, and Lucio Claudio Andreani, "Photonic light-trapping versus Lambertian limits in thin film silicon solar cells with 1D and 2D periodic patterns," Opt. Express **20**, A224-A244 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-S2-A224

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### References

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- J. Poortmans and V. Arkhipov (editors), Thin Film Solar Cells (Wiley, Chichester, 2006). [CrossRef]
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- J. Krc̆, M. Zeman, O. Kluth, F. Smole, and M. Topic̆, “Effect of surface roughness of ZnO:Al films on light scattering in hydrogenated amorphous silicon solar cells,” Thin Solid Films426, 296–304 (2003). [CrossRef]
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- Y. Lee, C. Huang, J. Chang, and M. Wu, “Enhanced light trapping based on guided mode resonance effect for thin-film silicon solar cells with two filling-factor gratings,” Opt. Express16, 7969–7975 (2008). [CrossRef] [PubMed]
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- J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, and D. W. Prather, “Thin film solar cell design based on photonic crystal and diffractive grating structures,” Opt. Express16, 15238–15248 (2008). [CrossRef] [PubMed]
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- Y. Park, E. Drouard, O. El Daif, X. Letartre, P. Viktorovitch, A. Fave, A. Kaminski, M. Lemiti, and C. Seassal, “Absorption enhancement using photonic crystals for silicon thin film solar cells,” Opt. Express17, 14312–14321 (2009). [CrossRef] [PubMed]
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