Abstract

A non-absorbing transparent shell is proposed to be coated on the outer surface of the core photoactive GaInP nanowire array (NWA) of the III-V nanowire (NW)/Si film two-junction solar cell. Interestingly, the diluted (at the filling ratio of 0.25) GaInP NWA with core / transparent shell structure can absorb more light than that in bare denser (at the filling ratio of 0.5) NWA. This allows for less source material consumption during the fabrication of III-V NWA/Si film two-junction cell. Meanwhile, the condition of current matching between the top III-V NWA and Si film sub cell can be easily fulfilled by tailoring the coating thickness of the transparent coating. Beyond the advantages on light absorption, the surface passivation effects introduced by the addition of some transparent dielectric coatings can reduce the surface recombination rate at the top NWA sub cell surface. This facilitates the effective extraction of photo-generated carriers and enhances output stability of the top NWA sub cell. From electrical simulation, a power conversion efficiency of 29.9% can be obtained at the optimized coating geometry.

© 2015 Optical Society of America

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References

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2015 (1)

N. Anttu, “Shockley–Queisser detailed balance efficiency limit for nanowire solar cells,” ACS Photonics 2(3), 446–453 (2015).
[Crossref]

2014 (3)

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

J. Michallon, D. Bucci, A. Morand, M. Zanuccoli, V. Consonni, and A. Kaminski-Cachopo, “Light trapping in ZnO nanowire arrays covered with an absorbing shell for solar cells,” Opt. Express 22(S4Suppl 4), A1174–A1189 (2014).
[Crossref] [PubMed]

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

2013 (6)

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

Y. Hu, M. Li, J. J. He, and R. R. LaPierre, “Current matching and efficiency optimization in a two-junction nanowire-on-silicon solar cell,” Nanotechnology 24(6), 065402 (2013).
[Crossref] [PubMed]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

G. Mariani, A. C. Scofield, C. H. Hung, and D. L. Huffaker, “GaAs nanopillar-array solar cells employing in situ surface passivation,” Nat. Commun. 4, 1497 (2013).
[Crossref] [PubMed]

N. Anttu, “Geometrical optics, electrostatics, and nanophotonic resonances in absorbing nanowire arrays,” Opt. Lett. 38(5), 730–732 (2013).
[Crossref] [PubMed]

N. Anttu and H. Q. Xu, “Efficient light management in vertical nanowire arrays for photovoltaics,” Opt. Express 21(S3Suppl 3), A558–A575 (2013).
[Crossref] [PubMed]

2012 (2)

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

N. Huang, C. Lin, and M. L. Povinelli, “Limiting efficiencies of tandem solar cells consisting of III-V nanowire arrays on silicon,” J. Appl. Phys. 112(6), 064321 (2012).
[Crossref]

2011 (4)

S. Plissard, G. Larrieu, X. Wallart, and P. Caroff, “High yield of self-catalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning,” Nanotechnology 22(27), 275602 (2011).
[Crossref] [PubMed]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

R. R. LaPierre, “Theoretical conversion efficiency of a two-junction III-V nanowire on Si solar cell,” J. Appl. Phys. 110(1), 014310 (2011).
[Crossref]

B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19(S5Suppl 5), A1067–A1081 (2011).
[Crossref] [PubMed]

2010 (3)

J. Kupec, R. L. Stoop, and B. Witzigmann, “Light absorption and emission in nanowire array solar cells,” Opt. Express 18(26), 27589–27605 (2010).
[Crossref] [PubMed]

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,” Nano Lett. 10(3), 1082–1087 (2010).
[Crossref] [PubMed]

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

2009 (4)

2008 (2)

E. C. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells,” J. Am. Chem. Soc. 130(29), 9224–9225 (2008).
[Crossref] [PubMed]

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

2007 (1)

M. Bosi and C. Pelosi, “The potential of III-V semiconductors as terrestrial photovoltaic devices,” Prog. Photovolt. Res. Appl. 15(1), 51–68 (2007).
[Crossref]

2005 (1)

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005).
[Crossref]

1992 (1)

J. S. Herman and F. L. Terry, “Hydrogen sulfide plasma passivation of gallium arsenide,” Appl. Phys. Lett. 60(6), 716–718 (1992).
[Crossref]

Aberg, I.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Åberg, I.

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

Abrand, A.

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

Abstreiter, G.

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

Ahtapodov, L.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Anttu, N.

N. Anttu, “Shockley–Queisser detailed balance efficiency limit for nanowire solar cells,” ACS Photonics 2(3), 446–453 (2015).
[Crossref]

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

N. Anttu, “Geometrical optics, electrostatics, and nanophotonic resonances in absorbing nanowire arrays,” Opt. Lett. 38(5), 730–732 (2013).
[Crossref] [PubMed]

N. Anttu and H. Q. Xu, “Efficient light management in vertical nanowire arrays for photovoltaics,” Opt. Express 21(S3Suppl 3), A558–A575 (2013).
[Crossref] [PubMed]

Arbiol, J.

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

Asatryan, A. A.

Asoli, D.

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Atwater, H. A.

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005).
[Crossref]

Bichler, M.

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

Borgström, M.

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

Borgström, M. T.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Bosi, M.

M. Bosi and C. Pelosi, “The potential of III-V semiconductors as terrestrial photovoltaic devices,” Prog. Photovolt. Res. Appl. 15(1), 51–68 (2007).
[Crossref]

Botten, L. C.

Bu, S.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Bucci, D.

Caroff, P.

S. Plissard, G. Larrieu, X. Wallart, and P. Caroff, “High yield of self-catalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning,” Nanotechnology 22(27), 275602 (2011).
[Crossref] [PubMed]

Colombo, C.

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

Consonni, V.

Czaban, J. A.

J. A. Czaban, D. A. Thompson, and R. R. LaPierre, “GaAs core--shell nanowires for photovoltaic applications,” Nano Lett. 9(1), 148–154 (2009).
[Crossref] [PubMed]

de Sterke, C. M.

Deppert, K.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Dheeraj, D. L.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Dimroth, F.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Dossou, K. B.

Fält, S.

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Fauske, V. T.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Fimland, B. O.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Finley, J. J.

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

Fontcuberta i Morral, A.

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

Fuss-Kailuweit, P.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Garnett, E.

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,” Nano Lett. 10(3), 1082–1087 (2010).
[Crossref] [PubMed]

Garnett, E. C.

E. C. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells,” J. Am. Chem. Soc. 130(29), 9224–9225 (2008).
[Crossref] [PubMed]

He, J. J.

Y. Hu, M. Li, J. J. He, and R. R. LaPierre, “Current matching and efficiency optimization in a two-junction nanowire-on-silicon solar cell,” Nanotechnology 24(6), 065402 (2013).
[Crossref] [PubMed]

Heidari, B.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Herman, J. S.

J. S. Herman and F. L. Terry, “Hydrogen sulfide plasma passivation of gallium arsenide,” Appl. Phys. Lett. 60(6), 716–718 (1992).
[Crossref]

Hertenberger, S.

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

Heurlin, M.

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Hu, Y.

Y. Hu, M. Li, J. J. He, and R. R. LaPierre, “Current matching and efficiency optimization in a two-junction nanowire-on-silicon solar cell,” Nanotechnology 24(6), 065402 (2013).
[Crossref] [PubMed]

Huang, J. H.

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Huang, N.

N. Huang, C. Lin, and M. L. Povinelli, “Limiting efficiencies of tandem solar cells consisting of III-V nanowire arrays on silicon,” J. Appl. Phys. 112(6), 064321 (2012).
[Crossref]

Huffaker, D. L.

G. Mariani, A. C. Scofield, C. H. Hung, and D. L. Huffaker, “GaAs nanopillar-array solar cells employing in situ surface passivation,” Nat. Commun. 4, 1497 (2013).
[Crossref] [PubMed]

Huffman, M.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Huh, J.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Hung, C. H.

G. Mariani, A. C. Scofield, C. H. Hung, and D. L. Huffaker, “GaAs nanopillar-array solar cells employing in situ surface passivation,” Nat. Commun. 4, 1497 (2013).
[Crossref] [PubMed]

Kaminski-Cachopo, A.

Kayes, B. M.

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005).
[Crossref]

Kempa, T. J.

B. Tian, T. J. Kempa, and C. M. Lieber, “Single nanowire photovoltaics,” Chem. Soc. Rev. 38(1), 16–24 (2009).
[Crossref] [PubMed]

Kim, D. C.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Koblmüller, G.

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

Kupec, J.

LaPierre, R. R.

Y. Hu, M. Li, J. J. He, and R. R. LaPierre, “Current matching and efficiency optimization in a two-junction nanowire-on-silicon solar cell,” Nanotechnology 24(6), 065402 (2013).
[Crossref] [PubMed]

R. R. LaPierre, “Theoretical conversion efficiency of a two-junction III-V nanowire on Si solar cell,” J. Appl. Phys. 110(1), 014310 (2011).
[Crossref]

J. A. Czaban, D. A. Thompson, and R. R. LaPierre, “GaAs core--shell nanowires for photovoltaic applications,” Nano Lett. 9(1), 148–154 (2009).
[Crossref] [PubMed]

Larrieu, G.

S. Plissard, G. Larrieu, X. Wallart, and P. Caroff, “High yield of self-catalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning,” Nanotechnology 22(27), 275602 (2011).
[Crossref] [PubMed]

Lee, K. D.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Lewis, N. S.

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005).
[Crossref]

Li, M.

Y. Hu, M. Li, J. J. He, and R. R. LaPierre, “Current matching and efficiency optimization in a two-junction nanowire-on-silicon solar cell,” Nanotechnology 24(6), 065402 (2013).
[Crossref] [PubMed]

Li, X.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Lieber, C. M.

B. Tian, T. J. Kempa, and C. M. Lieber, “Single nanowire photovoltaics,” Chem. Soc. Rev. 38(1), 16–24 (2009).
[Crossref] [PubMed]

Lin, C.

N. Huang, C. Lin, and M. L. Povinelli, “Limiting efficiencies of tandem solar cells consisting of III-V nanowire arrays on silicon,” J. Appl. Phys. 112(6), 064321 (2012).
[Crossref]

C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express 17(22), 19371–19381 (2009).
[Crossref] [PubMed]

Magnusson, M. H.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Mariani, G.

G. Mariani, A. C. Scofield, C. H. Hung, and D. L. Huffaker, “GaAs nanopillar-array solar cells employing in situ surface passivation,” Nat. Commun. 4, 1497 (2013).
[Crossref] [PubMed]

McPhedran, R. C.

Michallon, J.

Morand, A.

Morante, J. R.

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

Munshi, A. M.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Pelosi, C.

M. Bosi and C. Pelosi, “The potential of III-V semiconductors as terrestrial photovoltaic devices,” Prog. Photovolt. Res. Appl. 15(1), 51–68 (2007).
[Crossref]

Plissard, S.

S. Plissard, G. Larrieu, X. Wallart, and P. Caroff, “High yield of self-catalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning,” Nanotechnology 22(27), 275602 (2011).
[Crossref] [PubMed]

Poulton, C. G.

Povinelli, M. L.

N. Huang, C. Lin, and M. L. Povinelli, “Limiting efficiencies of tandem solar cells consisting of III-V nanowire arrays on silicon,” J. Appl. Phys. 112(6), 064321 (2012).
[Crossref]

C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express 17(22), 19371–19381 (2009).
[Crossref] [PubMed]

Reinertsen, J. F.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Rudolph, D.

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

Samuelson, L.

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Scofield, A. C.

G. Mariani, A. C. Scofield, C. H. Hung, and D. L. Huffaker, “GaAs nanopillar-array solar cells employing in situ surface passivation,” Nat. Commun. 4, 1497 (2013).
[Crossref] [PubMed]

Siefer, G.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Stoop, R. L.

Sturmberg, B. C. P.

Terry, F. L.

J. S. Herman and F. L. Terry, “Hydrogen sulfide plasma passivation of gallium arsenide,” Appl. Phys. Lett. 60(6), 716–718 (1992).
[Crossref]

Thompson, D. A.

J. A. Czaban, D. A. Thompson, and R. R. LaPierre, “GaAs core--shell nanowires for photovoltaic applications,” Nano Lett. 9(1), 148–154 (2009).
[Crossref] [PubMed]

Tian, B.

B. Tian, T. J. Kempa, and C. M. Lieber, “Single nanowire photovoltaics,” Chem. Soc. Rev. 38(1), 16–24 (2009).
[Crossref] [PubMed]

van Helvoort, A. T.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Wallart, X.

S. Plissard, G. Larrieu, X. Wallart, and P. Caroff, “High yield of self-catalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning,” Nanotechnology 22(27), 275602 (2011).
[Crossref] [PubMed]

Wallentin, J.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Wang, W.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

Wang, Y.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Weman, H.

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Wen, L.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Wickert, P.

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

Witzigmann, B.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

J. Kupec, R. L. Stoop, and B. Witzigmann, “Light absorption and emission in nanowire array solar cells,” Opt. Express 18(26), 27589–27605 (2010).
[Crossref] [PubMed]

J. Kupec and B. Witzigmann, “Dispersion, wave propagation and efficiency analysis of nanowire solar cells,” Opt. Express 17(12), 10399–10410 (2009).
[Crossref] [PubMed]

Xu, H. Q.

N. Anttu and H. Q. Xu, “Efficient light management in vertical nanowire arrays for photovoltaics,” Opt. Express 21(S3Suppl 3), A558–A575 (2013).
[Crossref] [PubMed]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Yang, P.

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,” Nano Lett. 10(3), 1082–1087 (2010).
[Crossref] [PubMed]

E. C. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells,” J. Am. Chem. Soc. 130(29), 9224–9225 (2008).
[Crossref] [PubMed]

Zanuccoli, M.

Zeng, X.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Zhao, Y.

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

Zhao, Z.

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

ACS Photonics (1)

N. Anttu, “Shockley–Queisser detailed balance efficiency limit for nanowire solar cells,” ACS Photonics 2(3), 446–453 (2015).
[Crossref]

Appl. Phys. Lett. (3)

J. S. Herman and F. L. Terry, “Hydrogen sulfide plasma passivation of gallium arsenide,” Appl. Phys. Lett. 60(6), 716–718 (1992).
[Crossref]

A. Fontcuberta i Morral, C. Colombo, G. Abstreiter, J. Arbiol, and J. R. Morante, “Nucleation mechanism of gallium-assisted molecular beam epitaxy growth of gallium arsenide nanowires,” Appl. Phys. Lett. 92(6), 063112 (2008).
[Crossref]

S. Bu, X. Li, L. Wen, X. Zeng, Y. Zhao, W. Wang, and Y. Wang, “Optical and electrical simulations of two-junction III-V nanowires on Si solar cell,” Appl. Phys. Lett. 102(3), 031106 (2013).
[Crossref]

Chem. Soc. Rev. (1)

B. Tian, T. J. Kempa, and C. M. Lieber, “Single nanowire photovoltaics,” Chem. Soc. Rev. 38(1), 16–24 (2009).
[Crossref] [PubMed]

J. Am. Chem. Soc. (1)

E. C. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells,” J. Am. Chem. Soc. 130(29), 9224–9225 (2008).
[Crossref] [PubMed]

J. Appl. Phys. (4)

R. R. LaPierre, “Theoretical conversion efficiency of a two-junction III-V nanowire on Si solar cell,” J. Appl. Phys. 110(1), 014310 (2011).
[Crossref]

N. Huang, C. Lin, and M. L. Povinelli, “Limiting efficiencies of tandem solar cells consisting of III-V nanowire arrays on silicon,” J. Appl. Phys. 112(6), 064321 (2012).
[Crossref]

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005).
[Crossref]

S. Hertenberger, D. Rudolph, M. Bichler, J. J. Finley, G. Abstreiter, and G. Koblmüller, “Growth kinetics in position-controlled and catalyst-free InAs nanowire arrays on Si (111) grown by selective area molecular beam epitaxy,” J. Appl. Phys. 108(11), 114316 (2010).
[Crossref]

Nano Lett. (4)

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,” Nano Lett. 10(3), 1082–1087 (2010).
[Crossref] [PubMed]

M. Heurlin, P. Wickert, S. Fält, M. T. Borgström, K. Deppert, L. Samuelson, and M. H. Magnusson, “Axial InP nanowire tandem junction grown on a silicon substrate,” Nano Lett. 11(5), 2028–2031 (2011).
[Crossref] [PubMed]

J. A. Czaban, D. A. Thompson, and R. R. LaPierre, “GaAs core--shell nanowires for photovoltaic applications,” Nano Lett. 9(1), 148–154 (2009).
[Crossref] [PubMed]

A. M. Munshi, D. L. Dheeraj, V. T. Fauske, D. C. Kim, J. Huh, J. F. Reinertsen, L. Ahtapodov, K. D. Lee, B. Heidari, A. T. van Helvoort, B. O. Fimland, and H. Weman, “Position-controlled uniform GaAs nanowires on silicon using nanoimprint lithography,” Nano Lett. 14(2), 960–966 (2014).
[Crossref] [PubMed]

Nano Res. (1)

N. Anttu, A. Abrand, D. Asoli, M. Heurlin, I. Åberg, L. Samuelson, and M. Borgström, “Absorption of light in InP nanowire arrays,” Nano Res. 7(6), 816–823 (2014).
[Crossref]

Nanotechnology (3)

L. Wen, X. Li, Z. Zhao, S. Bu, X. Zeng, J. H. Huang, and Y. Wang, “Theoretical consideration of III-V nanowire/Si triple-junction solar cells,” Nanotechnology 23(50), 505202 (2012).
[Crossref] [PubMed]

Y. Hu, M. Li, J. J. He, and R. R. LaPierre, “Current matching and efficiency optimization in a two-junction nanowire-on-silicon solar cell,” Nanotechnology 24(6), 065402 (2013).
[Crossref] [PubMed]

S. Plissard, G. Larrieu, X. Wallart, and P. Caroff, “High yield of self-catalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning,” Nanotechnology 22(27), 275602 (2011).
[Crossref] [PubMed]

Nat. Commun. (1)

G. Mariani, A. C. Scofield, C. H. Hung, and D. L. Huffaker, “GaAs nanopillar-array solar cells employing in situ surface passivation,” Nat. Commun. 4, 1497 (2013).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Prog. Photovolt. Res. Appl. (1)

M. Bosi and C. Pelosi, “The potential of III-V semiconductors as terrestrial photovoltaic devices,” Prog. Photovolt. Res. Appl. 15(1), 51–68 (2007).
[Crossref]

Science (1)

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit,” Science 339(6123), 1057–1060 (2013).
[Crossref] [PubMed]

Other (2)

The Ioffe Physico-technical Institute, “n, k database,” http://www.ioffe.ru/SVA/NSM/nk/

The American Society for Testing and Materials, “Reference solar spectral Irradiance: Air Mass 1.5,” http://rredc.nrel.gov/solar/spectra/am1.5/

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Figures (5)

Fig. 1
Fig. 1 (a) The schematic diagram and simulated unit of core-shell GaInP/SiO2 NWA/Si film two-junction solar cell modeled in this study. (b)Absorption (c)Reflection (d)Transmission of GaInP NWA (D/P = 0.2) with various SiO2 coating thicknesses.(e) optical absorption properties of GaInP NWA coated with transparent dielectric materials (MgF2, Si3N4) and transparent conductive oxide (Indium tin oxide, ITO), the coating thicknesses are set at 70nm.
Fig. 2
Fig. 2 (a) The electrical field intensity distribution in uncoated GaInP NWA and GaInP NWA coated with 30nm SiO2 shell at three typical wavelengths. (b) Average electric field <|Exy|2> in the x-y cross section of GaInP NW as a function of axial position (Z, where the bottom of the NW was set as the zero point and the top of the NW as −2μm) of the NW (<|Exy|2> was normalized to the maximum value in the NW with 30nm SiO2 coating shell).
Fig. 3
Fig. 3 (a)-(b) The relative power coupling efficiency of the fundamental mode and key mode for the GaInP NWA without coating and 30nm SiO2 coating. (c)-(d) The electric field intensity distributions of the two modes at four selected wavelengths. (e)-(f) The absorption coefficient of the fundamental mode and key mode of the GaInP NWA with different coating thicknesses. (g) The contribution of the key mode (green solid line), fundamental mode (blue solid line) to the overall absorption, the absorption summed by the absorption of the two Bloch modes(black dotted line) and the FDTD simulation absorption (red dotted line) for the NWA with different SiO2 thicknesses.
Fig. 4
Fig. 4 (a)The ultimate photocurrents of the top GaInP NWA sub cell (yellow curve) and bottom Si sub cell (red curve). (b) The detailed balance efficiencies of GaInP NWA/Si film two-junction cell with different NW geometry.
Fig. 5
Fig. 5 (a) Schematic diagram and optical generation rates in the proposed two-junction solar cell. (b) The variation of Voc and Jsc with SRV. (c) The J-V characteristic of the bottom Si sub cell, top NWA sub cell and series connected two-junction cell at SRV = 2 × 104 cm/s(after passivation).

Tables (1)

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Table 1 Detailed parameters used in the device performance simulation

Equations (8)

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η i = | φ s y s ( x , y ) ϕ i ( x , y ) d x d y | 2 φ s y s ( x , y ) φ s y s ( x , y ) d x d y φ i ( x , y ) φ i ( x , y ) d x d y
A = i = 1 , 2 η i ( 1 e α i L )
J p h = e h c λ A ( λ ) I ( λ ) d λ .
A S i ( λ ) = T ( λ ) { 1 exp [ α S i ( λ ) Z ] }
V i ( J ) = k B T q ln [ ( J p h , i J ) J 0 , i + 1 ]           i = 1 , 2
J 0 , i = 2 π e h 3 c 2 E g , i E 2 exp ( E k B T ) d E
V t o t a l ( J ) = V 1 ( J ) + V 2 ( J )
G opt = r e a l ( S ) 2 ω = ε ' ' | E | 2 2

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