Abstract

The emergence of nanotextures in photovoltaics has resulted in challenges associated with optical modelling. Whilst rigorous methods exist to accurately solve these textures, the computational effort required limits the scope of modeling applications. The effective medium approximation (EMA) is a potential alternative to provide rapid modeling results which can be easily integrated with ray tracing of large complex structures. However, the validity of this technique is strongly dependent on the size of features relative to the wavelength of interest, making the application of EMA ambiguous for many situations. This paper aims to address this issue by comparing the simulated results between EMA and finite element methods for three randomly distributed silicon textures with and without a dielectric layer. Criteria for which the EMA approach is valid are proposed and generalized using ratios between root-mean-square roughness, correlation length and incident wavelength, making these limits broadly applicable, beyond that of just the nanotexture under specific solar spectrum regimes. The results in this work apply to random, isotropic textures under normally incident light. Based on the proposed criteria, the validity of different optical simulation techniques for a set of industrial photovoltaic textures is discussed. This analysis reveals a region within which neither geometric optics nor EMA are adequate for calculating the reflectivity of a textured surface, and hence FDTD or other new approaches are required.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2019 (2)

S. A. Pelaez, C. Deline, S. M. Macalpine, B. Marion, J. S. Stein, and R. K. Kostuk, “Comparison of Bifacial Solar Irradiance Model Predictions with Field Validation,” IEEE J. Photovolt. 9(1), 82–88 (2019).
[Crossref]

A. Cordaro, J. Van De Groep, S. Raza, E. F. Pecora, F. Priolo, and M. L. Brongersma, “Antireflection high-index metasurfaces combining Mie and Fabry-Pérot resonances,” ACS Photonics 6(2), 453–459 (2019).
[Crossref]

2018 (6)

E. F. Pecora, A. Cordaro, P. G. Kik, and M. L. Brongersma, “Broadband antireflection coatings employing multiresonant dielectric metasurfaces,” ACS Photonics 5(11), 4456–4462 (2018).
[Crossref]

A. Riverola, A. Mellor, D. Alonso Alvarez, L. Ferre Llin, I. Guarracino, C. N. Markides, D. J. Paul, D. Chemisana, and N. Ekins-Daukes, “Mid-infrared emissivity of crystalline silicon solar cells,” Sol. Energy Mater. Sol. Cells 174, 607–615 (2018).
[Crossref]

J. Lv, T. Zhang, P. Zhang, Y. Zhao, and S. Li, “Review application of nanostructured black silicon,” Nanoscale Res. Lett. 13(1), 2523 (2018).
[Crossref]

S. Ma, S. Liu, Q. Xu, J. Xu, R. Lu, Y. Liu, and Z. Zhong, “A theoretical study on the optical properties of black silicon,” AIP Adv. 8(3), 035010 (2018).
[Crossref]

N. Tucher, H. T. Gebrewold, and B. Bläsi, “Field stitching approach for the wave optical modeling of black silicon structures,” Opt. Express 26(22), A937–945 (2018).
[Crossref]

Ah. A. ELsayed, Y. M. Sabry, F. Marty, T. Bourouina, and D. Khalil, “Optical modeling of black silicon using an effective medium / multi-layer approach,” Opt. Express 26(10), 13443–13460 (2018).
[Crossref]

2017 (2)

M. M. Plakhotnyuk, M. Gaudig, R. S. Davidsen, J. M. Lindhard, J. Hirsch, D. Lausch, M. S. Schmidt, E. Stamate, and O. Hansen, “Low surface damage dry etched black silicon,” J. Appl. Phys. 122(14), 143101 (2017).
[Crossref]

T. Rahman and S. A. Boden, “Optical modeling of black silicon for solar cells using effective index techniques,” IEEE J. Photovolt. 7(6), 1556–1562 (2017).
[Crossref]

2016 (6)

J. Eisenlohr, N. Tucher, H. Hauser, M. Graf, J. Benick, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Efficiency increase of crystalline silicon solar cells with nanoimprinted rear side gratings for enhanced light trapping,” Sol. Energy Mater. Sol. Cells 155, 288–293 (2016).
[Crossref]

O. Zhuromskyy, “Applicability of effective medium approximations to modelling of mesocrystal optical properties,” Crystals 7(1), 1 (2016).
[Crossref]

V. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
[Crossref]

A. J. Bett, J. Eisenlohr, O. Höhn, P. Repo, H. Savin, B. Bläsi, and J. C. Goldschmidt, “Wave optical simulation of the light trapping properties of black silicon surface textures,” Opt. Express 24(6), A434 (2016).
[Crossref]

R. Couderc, M. Amara, and M. Lemiti, “In-depth analysis of heat generation in silicon solar cells,” IEEE J. Photovolt. 6(5), 1123–1131 (2016).
[Crossref]

M. M. Hossain and M. Gu, “Radiative cooling: Principles, progress, and potentials,” Adv. Sci. 3(7), 1500360–10 (2016).
[Crossref]

2015 (6)

A. Ingenito, O. Isabella, and M. Zeman, “Nano-cones on micro-pyramids: modulated surface textures for maximal spectral response and high-efficiency solar cells,” Prog. Photovoltaics Res. Appl. 23(11), 1649–1659 (2015).
[Crossref]

M. Steglich, M. Oehme, T. Käsebier, M. Zilk, K. Kostecki, E. B. Kley, J. Schulze, and A. Tünnermann, “Ge-on-Si photodiode with black silicon boosted responsivity,” Appl. Phys. Lett. 107(5), 051103 (2015).
[Crossref]

C. K. Lo, Y. S. Lim, and F. A. Rahman, “New integrated simulation tool for the optimum design of bifacial solar panel with reflectors on a specific site,” Renewable Energy 81, 293–307 (2015).
[Crossref]

N. Tucher, J. Eisenlohr, P. Kiefel, O. Höhn, H. Hauser, M. Peters, C. Müller, J. C. Goldschmidt, and B. Bläsi, “3D optical simulation formalism OPTOS for textured silicon solar cells,” Opt. Express 23(24), A1720 (2015).
[Crossref]

Y. Li, Y. Chen, Z. Ouyang, and A. Lennon, “Angular matrix framework for light trapping analysis of solar cells,” Opt. Express 23(24), A1707–A1719 (2015).
[Crossref]

M. Otto, M. Algasinger, H. Branz, B. Gesemann, T. Gimpel, K. Füchsel, T. Käsebier, S. Kontermann, S. Koynov, X. Li, V. Naumann, J. Oh, A. N. Sprafke, J. Ziegler, M. Zilk, and R. B. Wehrspohn, “Black silicon photovoltaics,” Adv. Opt. Mater. 3(2), 147–164 (2015).
[Crossref]

2014 (2)

M. Steglich, T. Käsebier, M. Zilk, T. Pertsch, E. B. Kley, and A. Tünnermann, “The structural and optical properties of black silicon by inductively coupled plasma reactive ion etching,” J. Appl. Phys. 116(17), 173503 (2014).
[Crossref]

Y. Battie, A. En Naciri, W. Chamorro, and D. Horwat, “Generalized effective medium theory to extract the optical properties of two-dimensional nonspherical metallic nanoparticle layers,” J. Phys. Chem. C 118(9), 4899–4905 (2014).
[Crossref]

2013 (4)

P. Repo, A. Haarahiltunen, L. Sainiemi, M. Yli-Koski, H. Talvitie, M. C. Schubert, and H. Savin, “Effective passivation of black silicon surfaces by atomic layer deposition,” IEEE J. Photovolt. 3(1), 90–94 (2013).
[Crossref]

M. Steglich, M. Zilk, A. Bingel, C. Patzig, T. Käsebier, F. Schrempel, E. B. Kley, and A. Tünnermann, “A normal-incidence PtSi photoemissive detector with black silicon light-trapping,” J. Appl. Phys. 114(18), 183102 (2013).
[Crossref]

P. Repo, J. Benick, G. von Gastrow, V. Vähänissi, F. D. Heinz, J. Schön, M. C. Schubert, and H. Savin, “Passivation of black silicon boron emitters with atomic layer deposited aluminum oxide,” Phys. Status Solidi RRL 7(11), 950–954 (2013).
[Crossref]

S. Jeong, M. D. McGehee, and Y. Cui, “All-back-contact ultra-thin silicon nanocone solar cells with 13.7% power conversion efficiency,” Nat. Commun. 4(1), 1–7 (2013).
[Crossref]

2012 (2)

D. Nečas and P. Klapetek, “Gwyddion: An open-source software for SPM data analysis,” Cent. Eur. J. Phys. 10(1), 181–188 (2012).
[Crossref]

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat. Commun. 3(1), 692 (2012).
[Crossref]

2010 (4)

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” J. Opt. Soc. Am. A 27(5), 1083 (2010).
[Crossref]

Y. Wang, N. Lu, H. Xu, G. Shi, M. Xu, X. Lin, H. Li, W. Wang, D. Qi, Y. Lu, and L. Chi, “Biomimetic corrugated silicon nanocone arrays for self-cleaning antireflection coatings,” Nano Res. 3(7), 520–527 (2010).
[Crossref]

S. Chattopadhyay, Y. F. Huang, Y. J. Jen, A. Ganguly, K. H. Chen, and L. C. Chen, “Anti-reflecting and photonic nanostructures,” Mater. Sci. Eng., R 69(1-3), 1–35 (2010).
[Crossref]

M. Kroll, T. Käsebier, M. Otto, R. Salzer, R. B. Wehrspohn, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Optical modeling of needle like silicon surfaces produced by an ICP-RIE process,” Proc. SPIE 7725, 772505 (2010).
[Crossref]

2009 (1)

Y. Cui, J. Zhu, Z. F. Yu, G. F. Burkhard, C. M. Hsu, S. T. Connor, Y. Q. Xu, Q. Wang, M. McGehee, and S. H. Fan, “Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett. 9(1), 279–282 (2009).
[Crossref]

2007 (1)

M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: Comparison with experiment,” Phys. Status Solidi 4(6), 1986–1990 (2007).
[Crossref]

1996 (1)

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surface,” Int. J. Heat Mass Transfer 40(1), 49–59 (1996).
[Crossref]

1995 (1)

1904 (1)

J. C. Maxwell Garnett, “Colours in metal galsses and in metallic films,” Philos. Trans. R. Soc., A 203(359–371), 385–420 (1904).
[Crossref]

Abbott, M.

D. Payne, M. Abbott, A. C. Lopez, Y. Zeng, T. H. Fung, K. Mclntosh, J. Cruz-Campa, R. Davidson, M. Plakhotnyuk, and D. Bagnall, “Rapid optical modelling of plasma textured silicon,” in 33rd European Photovoltaic Solar Energy Conference and Exhibition (EUPVSEC) (2017).

M. Abbott, K. Mcintosh, B. Sudbury, J. Meydbray, T. H. Fung, M. Umair, Y. Zhang, S. Zou, X. Wang, G. Xing, G. Scardera, and D. Payne, “Annual energy yield analysis of solar cell technology,” in 46th IEEE PVSC (2019), pp. 1–5.

Algasinger, M.

M. Otto, M. Algasinger, H. Branz, B. Gesemann, T. Gimpel, K. Füchsel, T. Käsebier, S. Kontermann, S. Koynov, X. Li, V. Naumann, J. Oh, A. N. Sprafke, J. Ziegler, M. Zilk, and R. B. Wehrspohn, “Black silicon photovoltaics,” Adv. Opt. Mater. 3(2), 147–164 (2015).
[Crossref]

Alonso Alvarez, D.

A. Riverola, A. Mellor, D. Alonso Alvarez, L. Ferre Llin, I. Guarracino, C. N. Markides, D. J. Paul, D. Chemisana, and N. Ekins-Daukes, “Mid-infrared emissivity of crystalline silicon solar cells,” Sol. Energy Mater. Sol. Cells 174, 607–615 (2018).
[Crossref]

Amara, M.

R. Couderc, M. Amara, and M. Lemiti, “In-depth analysis of heat generation in silicon solar cells,” IEEE J. Photovolt. 6(5), 1123–1131 (2016).
[Crossref]

Bagnall, D.

D. Payne, M. Abbott, A. C. Lopez, Y. Zeng, T. H. Fung, K. Mclntosh, J. Cruz-Campa, R. Davidson, M. Plakhotnyuk, and D. Bagnall, “Rapid optical modelling of plasma textured silicon,” in 33rd European Photovoltaic Solar Energy Conference and Exhibition (EUPVSEC) (2017).

Battie, Y.

Y. Battie, A. En Naciri, W. Chamorro, and D. Horwat, “Generalized effective medium theory to extract the optical properties of two-dimensional nonspherical metallic nanoparticle layers,” J. Phys. Chem. C 118(9), 4899–4905 (2014).
[Crossref]

Benick, J.

J. Eisenlohr, N. Tucher, H. Hauser, M. Graf, J. Benick, B. Bläsi, J. C. Goldschmidt, and M. Hermle, “Efficiency increase of crystalline silicon solar cells with nanoimprinted rear side gratings for enhanced light trapping,” Sol. Energy Mater. Sol. Cells 155, 288–293 (2016).
[Crossref]

P. Repo, J. Benick, G. von Gastrow, V. Vähänissi, F. D. Heinz, J. Schön, M. C. Schubert, and H. Savin, “Passivation of black silicon boron emitters with atomic layer deposited aluminum oxide,” Phys. Status Solidi RRL 7(11), 950–954 (2013).
[Crossref]

Bergner, B. C.

Bergstrom, D.

D. Bergstrom, “Rough surface generation and analysis,” http://www.mysimlabs.com/surface_generation.html (2012).

Bessaïs, B.

M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: Comparison with experiment,” Phys. Status Solidi 4(6), 1986–1990 (2007).
[Crossref]

Bett, A. J.

Bhushan, B.

B. Bhushan, “Surface roughness analysis and measurement techniques,” in Modern Tribology Handbook (Taylor and Francis Inc), (2001), p. Chapter 2.2.2.4.

Bingel, A.

M. Steglich, M. Zilk, A. Bingel, C. Patzig, T. Käsebier, F. Schrempel, E. B. Kley, and A. Tünnermann, “A normal-incidence PtSi photoemissive detector with black silicon light-trapping,” J. Appl. Phys. 114(18), 183102 (2013).
[Crossref]

Bläsi, B.

Boden, S. A.

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Y. Cui, J. Zhu, Z. F. Yu, G. F. Burkhard, C. M. Hsu, S. T. Connor, Y. Q. Xu, Q. Wang, M. McGehee, and S. H. Fan, “Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett. 9(1), 279–282 (2009).
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M. Otto, M. Algasinger, H. Branz, B. Gesemann, T. Gimpel, K. Füchsel, T. Käsebier, S. Kontermann, S. Koynov, X. Li, V. Naumann, J. Oh, A. N. Sprafke, J. Ziegler, M. Zilk, and R. B. Wehrspohn, “Black silicon photovoltaics,” Adv. Opt. Mater. 3(2), 147–164 (2015).
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M. M. Hossain and M. Gu, “Radiative cooling: Principles, progress, and potentials,” Adv. Sci. 3(7), 1500360–10 (2016).
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Y. Cui, J. Zhu, Z. F. Yu, G. F. Burkhard, C. M. Hsu, S. T. Connor, Y. Q. Xu, Q. Wang, M. McGehee, and S. H. Fan, “Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett. 9(1), 279–282 (2009).
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Kik, P. G.

E. F. Pecora, A. Cordaro, P. G. Kik, and M. L. Brongersma, “Broadband antireflection coatings employing multiresonant dielectric metasurfaces,” ACS Photonics 5(11), 4456–4462 (2018).
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M. M. Plakhotnyuk, M. Gaudig, R. S. Davidsen, J. M. Lindhard, J. Hirsch, D. Lausch, M. S. Schmidt, E. Stamate, and O. Hansen, “Low surface damage dry etched black silicon,” J. Appl. Phys. 122(14), 143101 (2017).
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Figures (11)

Fig. 1.
Fig. 1. Line profiles for the artificially created Textures 1, 2 and 3.
Fig. 2.
Fig. 2. Diagram of effective slope index versus wavelength-normalized roughness after Tang et al. [25] with regions of specular reflection (green), geometric optics (blue), and the electrodynamic region. The effective medium region (orange) is proposed in this work. Each dot indicates one of the textures at a 100 nm wavelength interval between 300 and 1200 nm (right to left) for normally incident light.
Fig. 3.
Fig. 3. (a) Schematic diagram on the effective medium approach, where the nano-texture on the surface is modelled as an effective medium which consists of a stack of 20 thin films. (b) The real component of refractive index (λ=600 nm) through the height of the effective medium, calculated by the Bruggeman effective medium model for a silicon surface with Gaussian height distribution in air.
Fig. 4.
Fig. 4. Cumulative height distribution of Texture 1 with and without dielectric (a) and volume fraction of each component in Texture 1 coated with dielectric (b).
Fig. 5.
Fig. 5. Simulated reflectance (a, c and e) and absorption (b, d and f) spectra for bare textured silicon, obtained by FDTD (solid lines) and EMA (dashed). The vertical bar shows ± 2% absolute difference.
Fig. 6.
Fig. 6. (a) Absolute difference in simulated reflectance between EMA and FDTD for Texture 1 to 3. The dashed lines indicate the 2% absolute difference. λcritical represents the wavelength at which the absolute difference reaches 2%. (b) Absolute difference in simulated absorptance within the nanotexture between EMA and FDTD.
Fig. 7.
Fig. 7. FDTD calculation of the electric field intensity distribution through a cross-section (highlighted by red curve) of the three bare silicon textures upon plane-wave illumination at a wavelength of 400 nm (a, c and e) and 1000 nm (b, d and f).
Fig. 8.
Fig. 8. Simulated reflection (a, c and e) and absorption (b, d and f) within the texture for bare silicon coated with dielectric, obtained by FDTD (solid lines) and EMA (dashed).
Fig. 9.
Fig. 9. (a) Absolute difference in simulated reflectance between EMA and FDTD for Texture 1 to 3 coated with dielectric layer. The dashed lines indicate the 2% absolute difference. λcritical represents the wavelength at which the absolute difference reaches 2%. (b) Absolute difference in simulated absorptance within the nanotexture between EMA and FDTD.
Fig. 10.
Fig. 10. Diagram of wavelength-normalized roughness versus wavelength-normalized correlation length, with regions of specular reflection (green), geometric optics (green-blue), and the electrodynamic region (blue) as taken from Tang et al. (same data as in Fig. 2). The criteria proposed in this study are indicated in orange, where the total simulated reflectance from EMA is within 2% absolute difference to FDTD simulation for silicon textures. Each dot indicates one of the textures at a 100 nm wavelength interval between 300 and 1200 nm (right to left). The coloured squares are taken from [42,46], respectively. The diagonal dashed line depicts the σ/τ = 1 line.
Fig. 11.
Fig. 11. Top view, 45° and cross-sectional SEM images of random pyramid (a, b and c), metal-assisted chemical etching (d, e and f), shallow reactive-ion etching (h, i and j) and deep reactive-ion etching (k, l and m).

Tables (1)

Tables Icon

Table 1. Summary of surface characteristics and weighted reflectance between 300 nm – 950 nm for upright random pyramids (RPD), industrial metal-assisted chemical etching (MACE) and reactive-ion etching (RIE) and their regions of validity.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

f S i ( ε S i ε B G ε S i + 2 ε B G ) + ( 1 f S i ) ( 1 ε B G 1 + 2 ε B G ) = 0 ,
0 = A ε B G 2 + B ε B G + C ,
A = 2 ; B = f S i ( 2 ε S i 1 ) + ( 1 f S i ) ( 2 ε a i r ε S i ) ; C = ε S i .
f S i ( ε S i ε B G ε S i + 2 ε B G ) + f d ( ε d ε B G ε d + 2 ε B G ) + ( 1 f s i f d ) ( 1 ε B G 1 + 2 ε B G ) = 0 ,
0 = D ε B G 3 + E ε B G 2 + F ε B G + G ,
D = 4 ,
E = ( 6 f S i 2 ) ε S i + ( 6 f d 2 ) ε d + ( 4 6 f S i 6 f d ) ,
F = ( 3 f S i + 3 f d 1 ) ε S i ε d + ( 2 3 f d ) ε S i + ( 2 3 f S i ) ε d ,
G = ε S i ε d .
P ( z ) = P ( z + 100 ) ,
v S i = 1 P ( z ) ,
v a i r = P ( z ) ,
v d = 1 P ( z ) P ( z ) .

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