Performance optimization of tandem organic solar cells at varying incident angles based on optical analysis method

Xuenan Zhao; Ruoxi Xia; Honggang Gu; Honggang Gu; Xianhua Ke; Yating Shi; Xiuguo Chen; Hao Jiang; Hin-Lap Yip; Hin-Lap Yip; Shiyuan Liu; Shiyuan Liu

doi:10.1364/OE.382245

Optics Express
Vol. 28,
Issue 2,
pp. 2381-2397
(2020)
•https://doi.org/10.1364/OE.382245

Performance optimization of tandem organic solar cells at varying incident angles based on optical analysis method

Xuenan Zhao, Ruoxi Xia, Honggang Gu, Xianhua Ke, Yating Shi, Xiuguo Chen, Hao Jiang, Hin-Lap Yip, and Shiyuan Liu

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Xuenan Zhao, Ruoxi Xia, Honggang Gu, Xianhua Ke, Yating Shi, Xiuguo Chen, Hao Jiang, Hin-Lap Yip, and Shiyuan Liu, "Performance optimization of tandem organic solar cells at varying incident angles based on optical analysis method," Opt. Express 28, 2381-2397 (2020)

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- Solar Energy
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: November 5, 2019
- Revised Manuscript: January 8, 2020
- Manuscript Accepted: January 8, 2020
- Published: January 16, 2020
Virtual Issues
Optics Express Optical Devices and Materials for Solar Energy and Solid-State Lighting (2019)

Abstract

Tandem organic solar cells (OSCs) show great potential due to advantages such as the utilization of wide-spectrum light and low thermalization loss. The current mismatch between sub-cells is one of the major issues reducing the final output efficiency of a tandem device. In this paper, we focus on the current mismatch of tandem OSCs at oblique incidence and aim to reduce its adverse effect on the performances of realistic devices working at varying incident angle. Firstly, we propose an optical analysis method based on the 4×4 matrix formalism to analyze and optimize the performance of tandem solar cells at arbitrary incident angles. Compared with those optimal designs via matching the currents of sub-cells only at normal incidence, the proposed method chooses the optimal structure of the tandem device by maximizing the generated energy density per day with considering the current match at different incident angles during daytime. With the proposed method, a typical tandem organic solar cell is optimized as an example, and the optimized tandem device has a balanced current match at all incident angles during a whole day. Experimental results demonstrate that the generated energy density per day of the optimized tandem device has increased by 4.9% compared to the conventional device optimized only at normal incidence. The proposed method and results are expected to provide some new insights for the performance analysis and optimization of tandem or multi-junction solar cells, especially those devices exhibiting serious current mismatch between sub-cells at varying incident angles in practical applications.

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References

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Year
Author
Publication

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[Crossref]
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[Crossref]
P. Cheng, Y. Liu, S.-Y. Chang, T. Li, P. Sun, R. Wang, H.-W. Cheng, T. Huang, L. Meng, S. Nuryyeva, C. Zhu, K.-H. WEi, B. Sun, X. Zhaon, and Y. Yang, “Efficient Tandem Organic Photovoltaics with Tunable Rear Sub-cells,” Joule 3(2), 432–442 (2019).
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[Crossref]
R. Schueppel, R. Timmreck, N. Allinger, T. Mueller, M. Furno, C. Uhrich, K. Leo, and M. Riede, “Controlled current matching in small molecule organic tandem solar cells using doped spacer layers,” J. Appl. Phys. 107(4), 044503 (2010).
[Crossref]
Z. Wang, C. Zhang, D. Chen, J. Zhang, Q. Feng, S. Xu, X. Zhou, and Y. Hao, “Investigation of Controlled Current Matching in Polymer Tandem Solar Cells Considering Different Layer Sequences and Optical Spacer,” Jpn. J. Appl. Phys. 51(12R), 122301 (2012).
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G. Zhang, R. Xia, Z. Chen, J. Xiao, X. Zhao, S. Liu, H.-L. Yip, and Y. Cao, “Overcoming Space-Charge Effect for Efficient Thick-Film Non-Fullerene Organic Solar Cells,” Adv. Energy Mater. 8(25), 1801609 (2018).
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B. Lipovšek, A. Čampa, F. Guo, C. J. Brabec, K. Forberich, J. Krč, and M. Topič, “Detailed optical modelling and light-management of thin-film organic solar cells with consideration of small-area effects,” Opt. Express 25(4), A176–A190 (2017).
[Crossref]
L. Meng, Y. Zhang, X. Wan, C. Li, X. Zhang, Y. Wang, X. Ke, Z. Xiao, L. Ding, R. Xia, H. L. Yip, Y. Cao, and Y. Chen, “Organic and solution-processed tandem solar cells with 17.3% efficiency,” Science 361(6407), 1094–1098 (2018).
[Crossref]
M. Li, K. Gao, X. Wan, Q. Zhang, B. Kan, R. Xia, F. Liu, X. Yang, H. Feng, W. Ni, Y. Wang, J. Peng, H. Zhang, Z. Liang, H.-L. Yip, X. Peng, Y. Cao, and Y. Chen, “Solution-processed organic tandem solar cells with power conversion efficiencies >12%,” Nat. Photonics 11(2), 85–90 (2017).
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K. Zhang, K. Gao, R. Xia, Z. Wu, C. Sun, J. Cao, L. Qian, W. Li, S. Liu, F. Huang, X. Peng, L. Ding, H. L. Yip, and Y. Cao, “High-Performance Polymer Tandem Solar Cells Employing a New n-Type Conjugated Polymer as an Interconnecting Layer,” Adv. Mater. 28(24), 4817–4823 (2016).
[Crossref]
K. Zhang, B. Fan, R. Xia, X. Liu, Z. Hu, H. Gu, S. Liu, H.-L. Yip, L. Ying, F. Huang, and Y. Cao, “Highly Efficient Tandem Organic Solar Cell Enabled by Environmentally Friendly Solvent Processed Polymeric Interconnecting Layer,” Adv. Energy Mater. 8(15), 1703180 (2018).
[Crossref]
A. Meyer and H. Ade, “The effect of angle of incidence on the optical field distribution within thin film organic solar cells,” J. Appl. Phys. 106(11), 113101 (2009).
[Crossref]
A. Mertens, J. Mescher, D. Bahro, M. Koppitz, and A. Colsmann, “Understanding the angle-independent photon harvesting in organic homo-tandem solar cells,” Opt. Express 24(10), A898–A906 (2016).
[Crossref]
B. V. Andersson, U. Wuerfel, and O. Inganäs, “Full day modelling of V-shaped organic solar cell,” Sol. Energy 85(6), 1257–1263 (2011).
[Crossref]
S. Lee, I. Jeong, H. P. Kim, S. Y. Hwang, T. J. Kim, Y. D. Kim, J. Jang, and J. Kim, “Effect of incidence angle and polarization on the optimized layer structure of organic solar cells,” Sol. Energy Mater. Sol. Cells 118, 9–17 (2013).
[Crossref]
J. Kim, S. Jung, and I. Jeong, “Optical Modeling for Polarization-dependent Optical Power Dissipation of Thin-film Organic Solar Cells at Oblique Incidence,” J. Opt. Soc. Korea 16(1), 6–12 (2012).
[Crossref]
K. Kang, S. Lee, and J. Kim, “Effect of an Incoherent Glass Substrate on the Absorption Efficiency of Organic Solar Cells at Oblique Incidence Analyzed by the Transfer Matrix Method with a Glass Factor,” Jpn. J. Appl. Phys. 52(5R), 052301 (2013).
[Crossref]
J. Bergqvist, H. Arwin, and O. Inganäs, “Uniaxial Anisotropy in PEDOT:PSS Electrodes Enhances the Photocurrent at Oblique Incidence in Organic Solar Cells,” ACS Photonics 5(8), 3023–3030 (2018).
[Crossref]
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[Crossref]
D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4×4-Matrix Formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972).
[Crossref]
S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical Modeling and Analysis of Organic Solar Cells with Coherent Multilayers and Incoherent Glass Substrate Using Generalized Transfer Matrix Method,” Jpn. J. Appl. Phys. 50(12R), 122301 (2011).
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[Crossref]
H. Yao, L. Ye, J. Hou, B. Jang, G. Han, Y. Cui, G. M. Su, C. Wang, B. Gao, R. Yu, H. Zhang, Y. Yi, H. Y. Woo, H. Ade, and J. Hou, “Achieving Highly Efficient Nonfullerene Organic Solar Cells with Improved Intermolecular Interaction and Open-Circuit Voltage,” Adv. Mater. 29(21), 1700254 (2017).
[Crossref]
H. Yao, Y. Cui, R. Yu, B. Gao, H. Zhang, and J. Hou, “Design, Synthesis, and Photovoltaic Characterization of a Small Molecular Acceptor with an Ultra-Narrow Band Gap,” Angew. Chem., Int. Ed. 56(11), 3045–3049 (2017).
[Crossref]
Z. Wu, C. Sun, S. Dong, X. F. Jiang, S. Wu, H. Wu, H. L. Yip, F. Huang, and Y. Cao, “n-Type Water/Alcohol-Soluble Naphthalene Diimide-Based Conjugated Polymers for High-Performance Polymer Solar Cells,” J. Am. Chem. Soc. 138(6), 2004–2013 (2016).
[Crossref]
S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]
H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016).
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[Crossref]

2019 (6)

J. Yuan, Y. Zhang, L. Zhou, G. Zhang, H.-L. Yip, T.-K. Lau, X. Lu, C. Zhu, H. Peng, P. A. Johnson, M. Leclerc, Y. Cao, J. Ulanski, Y. Li, and Y. Zou, “Single-Junction Organic Solar Cell with over 15% Efficiency Using Fused-Ring Acceptor with Electron-Deficient Core,” Joule 3(4), 1140–1151 (2019).
[Crossref]

P. Cheng, Y. Liu, S.-Y. Chang, T. Li, P. Sun, R. Wang, H.-W. Cheng, T. Huang, L. Meng, S. Nuryyeva, C. Zhu, K.-H. WEi, B. Sun, X. Zhaon, and Y. Yang, “Efficient Tandem Organic Photovoltaics with Tunable Rear Sub-cells,” Joule 3(2), 432–442 (2019).
[Crossref]

G. Liu, J. Jia, K. Zhang, X. Jia, Q. Yin, W. Zhong, L. Li, F. Huang, and Y. Cao, “15% Efficiency Tandem Organic Solar Cell Based on a Novel Highly Efficient Wide-Bandgap Nonfullerene Acceptor with Low Energy Loss,” Adv. Energy Mater. 9(11), 1803657 (2019).
[Crossref]

R. Xia, H. Gu, S. Liu, K. Zhang, H.-L. Yip, and Y. Cao, “Optical Analysis for Semitransparent Organic Solar Cells,” Sol. RRL 3(1), 1800270 (2019).
[Crossref]

R. Xia, C. J. Brabec, H.-L. Yip, and Y. Cao, “High-Throughput Optical Screening for Efficient Semitransparent Organic Solar Cells,” Joule 3(9), 2241–2254 (2019).
[Crossref]

R. Schmager, M. Langenhorst, J. Lehr, U. Lemmer, B. S. Richards, and U. W. Paetzold, “Methodology of energy yield modelling of perovskite-based multi-junction photovoltaics,” Opt. Express 27(8), A507–A523 (2019).
[Crossref]

2018 (7)

J. Bergqvist, H. Arwin, and O. Inganäs, “Uniaxial Anisotropy in PEDOT:PSS Electrodes Enhances the Photocurrent at Oblique Incidence in Organic Solar Cells,” ACS Photonics 5(8), 3023–3030 (2018).
[Crossref]

L. Meng, Y. Zhang, X. Wan, C. Li, X. Zhang, Y. Wang, X. Ke, Z. Xiao, L. Ding, R. Xia, H. L. Yip, Y. Cao, and Y. Chen, “Organic and solution-processed tandem solar cells with 17.3% efficiency,” Science 361(6407), 1094–1098 (2018).
[Crossref]

K. Zhang, B. Fan, R. Xia, X. Liu, Z. Hu, H. Gu, S. Liu, H.-L. Yip, L. Ying, F. Huang, and Y. Cao, “Highly Efficient Tandem Organic Solar Cell Enabled by Environmentally Friendly Solvent Processed Polymeric Interconnecting Layer,” Adv. Energy Mater. 8(15), 1703180 (2018).
[Crossref]

G. Zhang, R. Xia, Z. Chen, J. Xiao, X. Zhao, S. Liu, H.-L. Yip, and Y. Cao, “Overcoming Space-Charge Effect for Efficient Thick-Film Non-Fullerene Organic Solar Cells,” Adv. Energy Mater. 8(25), 1801609 (2018).
[Crossref]

X. Che, Y. Li, Y. Qu, and S. R. Forrest, “High fabrication yield organic tandem photovoltaics combining vacuum- and solution-processed subcells with 15% efficiency,” Nat. Energy 3(5), 422–427 (2018).
[Crossref]

P. Cheng, G. Li, X. Zhan, and Y. Yang, “Next-generation organic photovoltaics based on non-fullerene acceptors,” Nat. Photonics 12(3), 131–142 (2018).
[Crossref]

S. Zhang, Y. Qin, J. Zhu, and J. Hou, “Over 14% Efficiency in Polymer Solar Cells Enabled by a Chlorinated Polymer Donor,” Adv. Mater. 30(20), 1800868 (2018).
[Crossref]

2017 (5)

Z. Xiao, X. Jia, and L. Ding, “Ternary organic solar cells offer 14% power conversion efficiency,” Sci. Bull. 62(23), 1562–1564 (2017).
[Crossref]

M. Li, K. Gao, X. Wan, Q. Zhang, B. Kan, R. Xia, F. Liu, X. Yang, H. Feng, W. Ni, Y. Wang, J. Peng, H. Zhang, Z. Liang, H.-L. Yip, X. Peng, Y. Cao, and Y. Chen, “Solution-processed organic tandem solar cells with power conversion efficiencies >12%,” Nat. Photonics 11(2), 85–90 (2017).
[Crossref]

H. Yao, L. Ye, J. Hou, B. Jang, G. Han, Y. Cui, G. M. Su, C. Wang, B. Gao, R. Yu, H. Zhang, Y. Yi, H. Y. Woo, H. Ade, and J. Hou, “Achieving Highly Efficient Nonfullerene Organic Solar Cells with Improved Intermolecular Interaction and Open-Circuit Voltage,” Adv. Mater. 29(21), 1700254 (2017).
[Crossref]

H. Yao, Y. Cui, R. Yu, B. Gao, H. Zhang, and J. Hou, “Design, Synthesis, and Photovoltaic Characterization of a Small Molecular Acceptor with an Ultra-Narrow Band Gap,” Angew. Chem., Int. Ed. 56(11), 3045–3049 (2017).
[Crossref]

B. Lipovšek, A. Čampa, F. Guo, C. J. Brabec, K. Forberich, J. Krč, and M. Topič, “Detailed optical modelling and light-management of thin-film organic solar cells with consideration of small-area effects,” Opt. Express 25(4), A176–A190 (2017).
[Crossref]

2016 (4)

H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016).
[Crossref]

Z. Wu, C. Sun, S. Dong, X. F. Jiang, S. Wu, H. Wu, H. L. Yip, F. Huang, and Y. Cao, “n-Type Water/Alcohol-Soluble Naphthalene Diimide-Based Conjugated Polymers for High-Performance Polymer Solar Cells,” J. Am. Chem. Soc. 138(6), 2004–2013 (2016).
[Crossref]

K. Zhang, K. Gao, R. Xia, Z. Wu, C. Sun, J. Cao, L. Qian, W. Li, S. Liu, F. Huang, X. Peng, L. Ding, H. L. Yip, and Y. Cao, “High-Performance Polymer Tandem Solar Cells Employing a New n-Type Conjugated Polymer as an Interconnecting Layer,” Adv. Mater. 28(24), 4817–4823 (2016).
[Crossref]

A. Mertens, J. Mescher, D. Bahro, M. Koppitz, and A. Colsmann, “Understanding the angle-independent photon harvesting in organic homo-tandem solar cells,” Opt. Express 24(10), A898–A906 (2016).
[Crossref]

2015 (3)

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

M. F. G. Klein, G. Q. G. de Medeiros, P. Kapetana, U. Lemmer, and A. Colsmann, “Modeling approach to derive the anisotropic complex refractive index of polymer:fullerene blends for organic solar cells utilizing spectroscopic ellipsometry,” J. Photonics Energy 5(1), 057204 (2015).
[Crossref]

Y. Lin, J. Wang, Z. G. Zhang, H. Bai, Y. Li, D. Zhu, and X. Zhan, “An electron acceptor challenging fullerenes for efficient polymer solar cells,” Adv. Mater. 27(7), 1170–1174 (2015).
[Crossref]

2014 (1)

W. Cao and J. Xue, “Recent progress in organic photovoltaics: device architecture and optical design,” Energy Environ. Sci. 7(7), 2123–2144 (2014).
[Crossref]

2013 (4)

M. Bonnet-Eymard, M. Boccard, G. Bugnon, F. Sculati-Meillaud, M. Despeisse, and C. Ballif, “Optimized short-circuit current mismatch in multi-junction solar cells,” Sol. Energy Mater. Sol. Cells 117, 120–125 (2013).
[Crossref]

X. Zhao, Z. Li, T. Zhu, B. Mi, Z. Gao, and W. Huang, “Structure optimization of organic planar heterojunction solar cells,” J. Phys. D: Appl. Phys. 46(19), 195105 (2013).
[Crossref]

S. Lee, I. Jeong, H. P. Kim, S. Y. Hwang, T. J. Kim, Y. D. Kim, J. Jang, and J. Kim, “Effect of incidence angle and polarization on the optimized layer structure of organic solar cells,” Sol. Energy Mater. Sol. Cells 118, 9–17 (2013).
[Crossref]

K. Kang, S. Lee, and J. Kim, “Effect of an Incoherent Glass Substrate on the Absorption Efficiency of Organic Solar Cells at Oblique Incidence Analyzed by the Transfer Matrix Method with a Glass Factor,” Jpn. J. Appl. Phys. 52(5R), 052301 (2013).
[Crossref]

2012 (3)

J. Kim, S. Jung, and I. Jeong, “Optical Modeling for Polarization-dependent Optical Power Dissipation of Thin-film Organic Solar Cells at Oblique Incidence,” J. Opt. Soc. Korea 16(1), 6–12 (2012).
[Crossref]

Z. Wang, C. Zhang, D. Chen, J. Zhang, Q. Feng, S. Xu, X. Zhou, and Y. Hao, “Investigation of Controlled Current Matching in Polymer Tandem Solar Cells Considering Different Layer Sequences and Optical Spacer,” Jpn. J. Appl. Phys. 51(12R), 122301 (2012).
[Crossref]

L. Song and A. Uddin, “Design of high efficiency organic solar cell with light trapping,” Opt. Express 20(S5), A606–A621 (2012).
[Crossref]

2011 (2)

B. V. Andersson, U. Wuerfel, and O. Inganäs, “Full day modelling of V-shaped organic solar cell,” Sol. Energy 85(6), 1257–1263 (2011).
[Crossref]

S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical Modeling and Analysis of Organic Solar Cells with Coherent Multilayers and Incoherent Glass Substrate Using Generalized Transfer Matrix Method,” Jpn. J. Appl. Phys. 50(12R), 122301 (2011).
[Crossref]

2010 (1)

R. Schueppel, R. Timmreck, N. Allinger, T. Mueller, M. Furno, C. Uhrich, K. Leo, and M. Riede, “Controlled current matching in small molecule organic tandem solar cells using doped spacer layers,” J. Appl. Phys. 107(4), 044503 (2010).
[Crossref]

2009 (1)

A. Meyer and H. Ade, “The effect of angle of incidence on the optical field distribution within thin film organic solar cells,” J. Appl. Phys. 106(11), 113101 (2009).
[Crossref]

2005 (1)

L. J. A. Koster, V. D. Mihailetchi, R. Ramaker, and P. W. M. Blom, “Light intensity dependence of open-circuit voltage of polymer:fullerene solar cells,” Appl. Phys. Lett. 86(12), 123509 (2005).
[Crossref]

1998 (1)

A. Donges, “The coherence length of black-body radiation,” Eur. J. Phys. 19(3), 245–249 (1998).
[Crossref]

1996 (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53(8), 4265–4274 (1996).
[Crossref]

1980 (1)

C. H. Energy ProcediaHenry, “Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells,” J. Appl. Phys. 51(8), 4494–4500 (1980).
[Crossref]

1972 (1)

D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4×4-Matrix Formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972).
[Crossref]

Achtner, B.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of optical systems (Wiley Online Library, 2005), Vol. 1.

Ade, H.

A. Meyer and H. Ade, “The effect of angle of incidence on the optical field distribution within thin film organic solar cells,” J. Appl. Phys. 106(11), 113101 (2009).
[Crossref]

Allinger, N.

Andersson, B. V.

B. V. Andersson, U. Wuerfel, and O. Inganäs, “Full day modelling of V-shaped organic solar cell,” Sol. Energy 85(6), 1257–1263 (2011).
[Crossref]

Arwin, H.

Bahro, D.

Bai, H.

Ballif, C.

Bergqvist, J.

Berreman, D. W.

D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4×4-Matrix Formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972).
[Crossref]

Blechinger, F.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of optical systems (Wiley Online Library, 2005), Vol. 1.

Blom, P. W. M.

Boccard, M.

Bonnet-Eymard, M.

Brabec, C. J.

R. Xia, C. J. Brabec, H.-L. Yip, and Y. Cao, “High-Throughput Optical Screening for Efficient Semitransparent Organic Solar Cells,” Joule 3(9), 2241–2254 (2019).
[Crossref]

Bugnon, G.

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ACS Photonics (1)

Adv. Energy Mater. (3)

Adv. Mater. (4)

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Angew. Chem., Int. Ed. (1)

Appl. Phys. Lett. (1)

Energy Environ. Sci. (1)

W. Cao and J. Xue, “Recent progress in organic photovoltaics: device architecture and optical design,” Energy Environ. Sci. 7(7), 2123–2144 (2014).
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Eur. J. Phys. (1)

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J. Am. Chem. Soc. (1)

J. Appl. Phys. (3)

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Joule (3)

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Nat. Energy (1)

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Science (1)

Sol. Energy (1)

B. V. Andersson, U. Wuerfel, and O. Inganäs, “Full day modelling of V-shaped organic solar cell,” Sol. Energy 85(6), 1257–1263 (2011).
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Fig. 1. Schematic diagram of multi-layer structure of the OSC.

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Fig. 2. (a) Simulation conditions without mask. (b) Measurement conditions in lab with mask.

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Fig. 3. (a) The device structure of the used tandem OSC; (b) Refractive index (n) and extinction coefficient (κ) of materials employed in the tandem OSC.

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Fig. 4. (a) Schematic and (b) Platform of the experimental set-up for measuring the performance of solar cells at varying incident angles.

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Fig. 5. Simulated J_sc generated in the tandem OSC as a function of thicknesses of the front and rear sub-cells at normal incidence. (The asterisk symbol represents the optimal structure.)

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Fig. 6. (a) Simulated absorption rate and its integral over wavelength in the conventional device working at normal incidence; (b) Simulated absorption rate of active layers when the conventional device working at oblique incidence; (c) Simulated J_SC in sub-cells and measured J_SC of the conventional device at different incident angles. The inset is the simulated current mismatch between two sub-cells in the conventional device.

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Fig. 7. Corrected experimental performance parameters of the conventional device (d_front = 140 nm, d_rear = 100 nm).

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Fig. 8. Simulated J_sc generated in the tandem OSC as a function of thickness of the front and rear sub-cells at different incident angles. (The asterisk symbol in each subgraph represents the optimal structure at the corresponding incident angle.)

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Fig. 9. Angular response of ρJ_SC existing in optimized device at different incident angle.

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Fig. 10. Simulated generated energy density per day generated in the tandem OSC as a function of thicknesses of the front and rear active layers. (The asterisk symbol represents the optimal structure.)

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Fig. 11. (a) Calculated δJ_SC from noon to sunset and (b) Measured results of the power density during daytime in two kinds of optimized devices

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Tables (1)

Table 1. Optimized active layer thicknesses and J_SC of tandem device

View Table

Equations (27)

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\begin{matrix} {[\begin{array}{cccc} E_{is} & E_{rs} & E_{ip} & E_{rp} \end{array}]}^{T} = T_{c} {[\begin{array}{cccc} E_{ts} & 0 & E_{tp} & 0 \end{array}]}^{T}, \\ T_{c} = {L_{i}}^{- 1} \prod_{j = 2}^{m1} T_{j p} (- d_{j}) L_{t}, \end{matrix}

\begin{matrix} L_{i} {[\begin{array}{cccc} E_{is} & E_{rs} & E_{ip} & E_{rp} \end{array}]}^{T} = Ψ_{2} (0), \\ L_{t} {[\begin{array}{cccc} E_{ts} & 0 & E_{tp} & 0 \end{array}]}^{T} = Ψ_{m} (d_{m}), \\ Ψ_{j} (0) = T_{j p} (- d_{j}) Ψ_{j} (d_{j}) . \end{matrix}

Ψ = {[\begin{array}{cccc} E_{x} & E_{y} & H_{x} & H_{y} \end{array}]}^{T} .

\begin{array}{l} r_{pp} = (\frac{E_{rp}}{E_{ip}})_{E_{is} = 0} = \frac{T_{c 11} T_{c 43} - T_{c 13} T_{c 41}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \\ r_{sp} = (\frac{E_{rs}}{E_{ip}})_{E_{is} = 0} = \frac{T_{c 11} T_{c 23} - T_{c 13} T_{c 21}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \\ r_{ss} = (\frac{E_{rs}}{E_{is}})_{E_{ip} = 0} = \frac{T_{c 21} T_{c 33} - T_{c 23} T_{c 31}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \\ r_{ps} = (\frac{E_{rp}}{E_{ip}})_{E_{ip} = 0} = \frac{T_{c 33} T_{c 41} - T_{c 31} T_{c 43}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \end{array}

\begin{array}{l} t_{sp} = (\frac{E_{ts}}{E_{ip}})_{E_{is} = 0} = \frac{- T_{c 13}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \\ t_{ss} = (\frac{E_{ts}}{E_{is}})_{E_{ip} = 0} = \frac{T_{c 33}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \\ t_{pp} = (\frac{E_{tp}}{E_{ip}})_{E_{is} = 0} = \frac{T_{c 11}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \\ t_{ps} = (\frac{E_{tp}}{E_{is}})_{E_{ip} = 0} = \frac{- T_{c 31}}{T_{c 11} T_{c 33} - T_{c 13} T_{c 31}}, \end{array}

\begin{array}{l} R_{gm - s} = | r_{sp} + r_{ss} |^{2}, \\ R_{gm - p} = | r_{pp} + r_{ps} |^{2}, \end{array}

\begin{array}{l} T_{gm-s} = \frac{N_{m + 1} \cos (θ_{m + 1})}{N_{1} \cos (θ_{1})} | t_{sp} + t_{ss} |^{2}, \\ T_{gm-p} = \frac{N_{m + 1} \cos (θ_{m + 1})}{N_{1} \cos (θ_{1})} | t_{pp} + t_{ps} |^{2} . \end{array}

{[\begin{array}{cccc} U_{is} & U_{rs} & U_{ip} & U_{rp} \end{array}]}^{T} = \bar{T_{c}} {[\begin{array}{cccc} U_{ts} & 0 & U_{tp} & 0 \end{array}]}^{T} .

{[\begin{array}{cccc} U_{0is} & U_{0rs} & U_{0ip} & U_{0rp} \end{array}]}^{T} = \bar{I} \cdot \bar{L} {[\begin{array}{cccc} U_{is} & U_{rs} & U_{ip} & U_{rp} \end{array}]}^{T},

\bar{I} = [\begin{array}{cc} \bar{I_{s}} & 0 \\ 0 & \bar{I_{p}} \end{array}], \bar{I_{s(p)}} = \frac{1}{{| t_{ag-s(p)} |}^{2}} [\begin{array}{cc} 1 & - {| r_{ga-s(p)} |}^{2} \\ {| r_{ag-s(p)} |}^{2} & {| t_{ag-s(p)} t_{ga-s(p)} |}^{2} - {| r_{ag-s(p)} r_{ga-s(p)} |}^{2} \end{array}],

\bar{L} = [\begin{array}{cc} \bar{L_{s}} & 0 \\ 0 & \bar{L_{p}} \end{array}], {\bar{L}}_{s} = \bar{L_{p}} = [\begin{array}{cc} {| e^{- i β_{g} d_{g}} |}^{2} & 0 \\ 0 & {| e^{i β_{g} d_{g}} |}^{2} \end{array}],

\bar{T} = \bar{I} \cdot \bar{L} \cdot \bar{T_{c}} .

\begin{matrix} E_{is (p)} = \sqrt{\frac{T_{s(p)} N_{0} \cos (θ_{0})}{T_{gm - s(p)} N_{1} \cos (θ_{1})}}, \\ E_{rs} = E_{is} r_{ss} + E_{ip} r_{sp}, \\ E_{rp} = E_{is} r_{ps} + E_{ip} r_{pp} . \end{matrix}

\begin{matrix} {[\begin{array}{cccc} E_{is} & E_{rs} & E_{ip} & E_{rp} \end{array}]}^{T} = {L_{i}}^{- 1} \prod_{i = 2}^{j - 1} T_{i p} (- d_{i}) Ψ_{j} (0), \\ Ψ_{j} (0) = T_{j p} (- z) Ψ_{j} (z) (0 \leq z \leq d_{j}), \end{matrix}

\begin{matrix} {S_{z j}}^{s} (z) = - \frac{1}{2} Re {E_{y j} (z) H_{x j} {(z)}^{*}}, \\ {S_{z j}}^{p} (z) = \frac{1}{2} Re {E_{x j} (z) H_{y j} {(z)}^{*}} . \end{matrix}

{Q_{z j}}^{s (p)} (z) = - \frac{d {S_{z j}}^{s (p)}}{d z} .

G = \frac{2 π ε_{0} n κ P_{i n}}{h} | E |^{2},

A_{j} = \frac{1}{{S_{0}}^{s}} \int_{0}^{d_{j}} {Q_{j}}^{s} (z) d z + \frac{1}{{S_{0}}^{p}} \int_{0}^{d_{j}} {Q_{j}}^{p} (z) d z .

\begin{matrix} J_{SC-f} = η_{IQE} \cdot \int \frac{q λ}{hc} P_{in} (λ) A_{f} (λ) d λ = η_{IQE} \cdot q \int \int G_{f} d λ d z, \\ J_{SC-r} = η_{IQE} \cdot \int \frac{q λ}{hc} P_{in} (λ) A_{r} (λ) d λ = η_{IQE} \cdot q \int \int G_{r} d λ d z, \\ J_{SC} = min (J_{SC - f}, J_{SC - r}), \end{matrix}

V_{oc} = (E_{g-f} + E_{g-r} - E_{loss-f} - E_{loss-r}) / q .

P_{out} = F F \cdot J_{SC} \cdot V_{OC},

\cos [θ_{0} (t)] = \cos (θ_{z} - θ_{t}) \cos [(t - 12 h) 15^{\circ} / h] .

E_{day} = \int_{t_{rise}}^{t_{fall}} P (θ (t)) d t .

\cos θ_{0} = \cos δ \cos (ϕ - θ_{t}) \cos θ_{h} + \sin δ \sin (ϕ - θ_{t}) .

S = a \cdot (a - d \tan θ) .

α = S / S_{0} = (a - d \tan θ) / a .

\begin{array}{l} {J_{SC-exp}}^{'} = J_{SC-exp} / α, \\ {V_{OC-exp}}^{'} = V_{OC-exp}, \\ P C {E_{\exp}}^{'} = P C E_{\exp} / α, \\ F {F_{\exp}}^{'} = F F_{\exp} . \end{array}

Incident angle (°)		0	15	30	45	60	75
Optimized thickness (nm)	front	140	138	135	131	127	124
Optimized thickness (nm)	rear	100	101	106	116	121	128
J_SC (mA/cm²)		13.65	13.14	11.65	9.31	6.26	2.69
J_SC of the conventional device (mA/cm²)		13.65	13.10	11.51	9.03	5.92	2.49