Abstract

A comprehensive study of the plasmonic thin-film solar cell with the periodic strip structure is presented in this paper. The finite-difference frequency-domain method is employed to discretize the inhomogeneous wave function for modeling the solar cell. In particular, the hybrid absorbing boundary condition and the one-sided difference scheme are adopted. The parameter extraction methods for the zeroth-order reflectance and the absorbed power density are also discussed, which is important for testing and optimizing the solar cell design. For the numerical results, the physics of the absorption peaks of the amorphous silicon thin-film solar cell are explained by electromagnetic theory; these peaks correspond to the waveguide mode, Floquet mode, surface plasmon resonance, and the constructively interference between adjacent metal strips. The work is therefore important for the theoretical study and optimized design of the plasmonic thin-film solar cell.

© 2010 Optical Society of America

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    [CrossRef]

2009

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

2008

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, "Plasmonic nanostructure design for efficient light coupling into solar cells," Nano Lett. 8, 4391-4397 (2008)
[CrossRef]

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, "Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern," Appl. Phys. Lett. 93, 143501 (2008).
[CrossRef]

A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, "The failure of perfectly matched layers and towards their redemption by adiabatic absorbers," Opt. Express 16, 11376-11392 (2008).
[CrossRef] [PubMed]

2007

X.W. Chen, W. C. H. Choy, and S. L. He, "Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes," J. Disp. Technol. 3, 110-117 (2007).
[CrossRef]

W. Zhou, M. Tao, L. Chen, and H. Yang, "Microstructured surface design for omnidirectional antireflection coatings on solar cells," J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

C. Haase, and H. Stiebig, "Thin-film silicon solar cells with efficient periodic light trapping texture," Appl. Phys. Lett. 91, 061116 (2007).
[CrossRef]

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

2006

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
[CrossRef] [PubMed]

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]

2004

K. L. Chopra, P. D. Paulson, and V. Dutta, "Thin-film solar cells: An overview," Prog. Photovoltaics 12, 69-92 (2004).
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004).
[CrossRef]

S. Zhao and G. W. Wei, "High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces," J. Comput. Phys. 200, 60-103 (2004).
[CrossRef]

C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals," Opt. Express 12, 1397-1408 (2004).
[CrossRef] [PubMed]

2002

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Z. M. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002)
[PubMed]

2001

2000

C. M. Rappaport, M. Kilmer, and E. Miller, "Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation," Int. J. Numer. Model.-Electron. Netw. Device Fields 13, 471-482 (2000).
[CrossRef]

M. Qiu, and S. L. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

1999

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: Review," Sens. Actuators B. 54, 3-15 (1999).
[CrossRef]

1997

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).
[CrossRef]

1996

D. F. Kelley and R. J. Luebbers, "Piecewise linear recursive convolution for dispersive media using FDTD," IEEE Trans. Antennas Propag. 44, 792-797 (1996).
[CrossRef]

1995

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 12, 1068-1076 (1995).
[CrossRef]

1994

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic-waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

1993

M. E. Veysoglu, R. T. Shin, and J. A. Kong, "A finite-difference time-domain analysis of wave scattering from periodic surfaces: Oblique-incidence case," J. Electromagn. Waves Appl. 7, 1595-1607 (1993).
[CrossRef]

1990

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

1981

G. Mur, "Absorbing boundary-conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Alamariu, B. A.

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]

Atwater, H. A.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, "Plasmonic nanostructure design for efficient light coupling into solar cells," Nano Lett. 8, 4391-4397 (2008)
[CrossRef]

Aussenegg, F. R.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Avniel, Y.

Barnard, E.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

Berenger, J. P.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic-waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

Bermel, P.

Brongersma, M. L.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004).
[CrossRef]

Brown, T. G.

Burr, G. W.

Catrysse, P. B.

Chang, H. C.

Chen, L.

W. Zhou, M. Tao, L. Chen, and H. Yang, "Microstructured surface design for omnidirectional antireflection coatings on solar cells," J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Chen, X.W.

X.W. Chen, W. C. H. Choy, and S. L. He, "Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes," J. Disp. Technol. 3, 110-117 (2007).
[CrossRef]

Chew, W. C.

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).
[CrossRef]

W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Chopra, K. L.

K. L. Chopra, P. D. Paulson, and V. Dutta, "Thin-film solar cells: An overview," Prog. Photovoltaics 12, 69-92 (2004).
[CrossRef]

Choy, W. C. H.

X.W. Chen, W. C. H. Choy, and S. L. He, "Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes," J. Disp. Technol. 3, 110-117 (2007).
[CrossRef]

Ditlbacher, H.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Duan, X.

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]

Dutta, V.

K. L. Chopra, P. D. Paulson, and V. Dutta, "Thin-film solar cells: An overview," Prog. Photovoltaics 12, 69-92 (2004).
[CrossRef]

Ebe, T.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Farjadpour, A.

Felidj, N.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Feng, N.

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]

Ferry, V. E.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, "Plasmonic nanostructure design for efficient light coupling into solar cells," Nano Lett. 8, 4391-4397 (2008)
[CrossRef]

Fujiwara, H.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, "Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern," Appl. Phys. Lett. 93, 143501 (2008).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: Review," Sens. Actuators B. 54, 3-15 (1999).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 12, 1068-1076 (1995).
[CrossRef]

Grann, E. B.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 12, 1068-1076 (1995).
[CrossRef]

Grimes, C. A.

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

Haase, C.

C. Haase, and H. Stiebig, "Thin-film silicon solar cells with efficient periodic light trapping texture," Appl. Phys. Lett. 91, 061116 (2007).
[CrossRef]

He, S. L.

X.W. Chen, W. C. H. Choy, and S. L. He, "Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes," J. Disp. Technol. 3, 110-117 (2007).
[CrossRef]

M. Qiu, and S. L. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: Review," Sens. Actuators B. 54, 3-15 (1999).
[CrossRef]

Hong, C.

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]

Hunsberger, F. P.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

Ibanescu, M.

Inganas, O.

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kanamori, Y.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, "Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern," Appl. Phys. Lett. 93, 143501 (2008).
[CrossRef]

Kaneko, F.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Kato, K.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Kelley, D. F.

D. F. Kelley and R. J. Luebbers, "Piecewise linear recursive convolution for dispersive media using FDTD," IEEE Trans. Antennas Propag. 44, 792-797 (1996).
[CrossRef]

Kilmer, M.

C. M. Rappaport, M. Kilmer, and E. Miller, "Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation," Int. J. Numer. Model.-Electron. Netw. Device Fields 13, 471-482 (2000).
[CrossRef]

Kimerling, L. C.

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]

Kondo, M.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, "Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern," Appl. Phys. Lett. 93, 143501 (2008).
[CrossRef]

Kong, J. A.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, "A finite-difference time-domain analysis of wave scattering from periodic surfaces: Oblique-incidence case," J. Electromagn. Waves Appl. 7, 1595-1607 (1993).
[CrossRef]

Krenn, J. R.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Kunz, K. S.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

Lamprecht, B.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Leitner, A.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Liu, J.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

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]

Luebbers, R.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

Luebbers, R. J.

D. F. Kelley and R. J. Luebbers, "Piecewise linear recursive convolution for dispersive media using FDTD," IEEE Trans. Antennas Propag. 44, 792-797 (1996).
[CrossRef]

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).
[CrossRef]

Miller, E.

C. M. Rappaport, M. Kilmer, and E. Miller, "Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation," Int. J. Numer. Model.-Electron. Netw. Device Fields 13, 471-482 (2000).
[CrossRef]

Moharam, M. G.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 12, 1068-1076 (1995).
[CrossRef]

Mor, G. K.

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

Mur, G.

G. Mur, "Absorbing boundary-conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Ong, K. G.

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

Oskooi, A. F.

Pacifici, D.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, "Plasmonic nanostructure design for efficient light coupling into solar cells," Nano Lett. 8, 4391-4397 (2008)
[CrossRef]

Pala, R. A.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

Paulson, P. D.

K. L. Chopra, P. D. Paulson, and V. Dutta, "Thin-film solar cells: An overview," Prog. Photovoltaics 12, 69-92 (2004).
[CrossRef]

Persson, N. K.

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

Pommet, D. A.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 12, 1068-1076 (1995).
[CrossRef]

Qiu, M.

M. Qiu, and S. L. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Rahachou, A.

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

Rappaport, C. M.

C. M. Rappaport, M. Kilmer, and E. Miller, "Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation," Int. J. Numer. Model.-Electron. Netw. Device Fields 13, 471-482 (2000).
[CrossRef]

Rodriguez, A.

Roundy, D.

Sai, H.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, "Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern," Appl. Phys. Lett. 93, 143501 (2008).
[CrossRef]

Salerno, M.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Schider, G.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Schneider, M.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

Selker, M. D.

Shankar, K.

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

Shin, R. T.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, "A finite-difference time-domain analysis of wave scattering from periodic surfaces: Oblique-incidence case," J. Electromagn. Waves Appl. 7, 1595-1607 (1993).
[CrossRef]

Shinbo, K.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Standler, R. B.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

Stiebig, H.

C. Haase, and H. Stiebig, "Thin-film silicon solar cells with efficient periodic light trapping texture," Appl. Phys. Lett. 91, 061116 (2007).
[CrossRef]

Sweatlock, L. A.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, "Plasmonic nanostructure design for efficient light coupling into solar cells," Nano Lett. 8, 4391-4397 (2008)
[CrossRef]

Tao, M.

W. Zhou, M. Tao, L. Chen, and H. Yang, "Microstructured surface design for omnidirectional antireflection coatings on solar cells," J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Tsuruta, H.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Tvingstedt, K.

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

Varghese, O. K.

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

Veysoglu, M. E.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, "A finite-difference time-domain analysis of wave scattering from periodic surfaces: Oblique-incidence case," J. Electromagn. Waves Appl. 7, 1595-1607 (1993).
[CrossRef]

Wakamatsu, T.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Wei, G. W.

S. Zhao and G. W. Wei, "High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces," J. Comput. Phys. 200, 60-103 (2004).
[CrossRef]

White, J.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

Yang, H.

W. Zhou, M. Tao, L. Chen, and H. Yang, "Microstructured surface design for omnidirectional antireflection coatings on solar cells," J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: Review," Sens. Actuators B. 54, 3-15 (1999).
[CrossRef]

Yi, Y.

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]

Yu, C. P.

Zeng, L.

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]

Zhang, L.

Zhao, S.

S. Zhao and G. W. Wei, "High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces," J. Comput. Phys. 200, 60-103 (2004).
[CrossRef]

Zhou, W.

W. Zhou, M. Tao, L. Chen, and H. Yang, "Microstructured surface design for omnidirectional antireflection coatings on solar cells," J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Zhu, Z. M.

Zia, R.

Zozoulenko, I. V.

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

Adv. Mater.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 3504-3509 (2009)
[CrossRef]

Appl. Phys. Lett.

K. Tvingstedt, N. K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, "Surface plasmon increase absorption in polymer photovoltaic cells," Appl. Phys. Lett. 91, 113514 (2007).
[CrossRef]

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]

C. Haase, and H. Stiebig, "Thin-film silicon solar cells with efficient periodic light trapping texture," Appl. Phys. Lett. 91, 061116 (2007).
[CrossRef]

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, "Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern," Appl. Phys. Lett. 93, 143501 (2008).
[CrossRef]

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, "Fluorescence imaging of surface plasmon fields," Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

IEEE Trans. Antennas Propag.

D. F. Kelley and R. J. Luebbers, "Piecewise linear recursive convolution for dispersive media using FDTD," IEEE Trans. Antennas Propag. 44, 792-797 (1996).
[CrossRef]

IEEE Trans. Electromagn. Compat.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990).
[CrossRef]

G. Mur, "Absorbing boundary-conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Int. J. Numer. Model.-Electron. Netw. Device Fields

C. M. Rappaport, M. Kilmer, and E. Miller, "Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation," Int. J. Numer. Model.-Electron. Netw. Device Fields 13, 471-482 (2000).
[CrossRef]

J. Appl. Phys.

W. Zhou, M. Tao, L. Chen, and H. Yang, "Microstructured surface design for omnidirectional antireflection coatings on solar cells," J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

M. Qiu, and S. L. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

J. Comput. Phys.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic-waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

S. Zhao and G. W. Wei, "High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces," J. Comput. Phys. 200, 60-103 (2004).
[CrossRef]

J. Disp. Technol.

X.W. Chen, W. C. H. Choy, and S. L. He, "Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes," J. Disp. Technol. 3, 110-117 (2007).
[CrossRef]

J. Electromagn. Waves Appl.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, "A finite-difference time-domain analysis of wave scattering from periodic surfaces: Oblique-incidence case," J. Electromagn. Waves Appl. 7, 1595-1607 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A-Opt. Image Sci. Vis.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 12, 1068-1076 (1995).
[CrossRef]

Mater. Sci. Eng. C-Biomimetic Supramol. Syst.

K. Kato, H. Tsuruta, T. Ebe, K. Shinbo, F. Kaneko, and T. Wakamatsu, "Enhancement of optical absorption and photocurrents in solar cells of merocyanine Langmuir-Blodgett films utilizing surface plasmon excitations," Mater. Sci. Eng. C-Biomimetic Supramol. Syst. 22, 251-256 (2002)
[CrossRef]

Microw. Opt. Technol. Lett.

W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).
[CrossRef]

Nano Lett.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, "Plasmonic nanostructure design for efficient light coupling into solar cells," Nano Lett. 8, 4391-4397 (2008)
[CrossRef]

Opt. Express

Opt. Lett.

Prog. Photovoltaics

K. L. Chopra, P. D. Paulson, and V. Dutta, "Thin-film solar cells: An overview," Prog. Photovoltaics 12, 69-92 (2004).
[CrossRef]

Sens. Actuators B.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: Review," Sens. Actuators B. 54, 3-15 (1999).
[CrossRef]

Sol. Energy Mater. Sol. Cells

K. G. Ong, O. K. Varghese, G. K. Mor, K. Shankar, and C. A. Grimes, "Application of finite-difference time domain to dye-sensitized solar cells: The effect of nanotube-array negative electrode dimensions on light absorption," Sol. Energy Mater. Sol. Cells 91, 250-257 (2007).
[CrossRef]

Other

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

J. Nelson, The Physics of Solar Cells (Imperial College Press, London, 2003)

P. Würfel, Physics of Solar Cells: From Principles to New Concepts (Wiley-VCH, Berlin, 2004).

G. Veronis and S. Fan, "Overview of Simulation Techniques for Plasmonic Devices," in Surface Plasmon Nanophotonics, M. L. Brongersma and P. G. Kik, eds., (Springer, Dordrecht, The Netherlands, 2007)

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, Third Edition (Artech House, Boston, 2005).

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (Wiley-IEEE Press, New York, 2000).

W. C. Chew, Waves and Fields in Inhomogenous Media (Van Nostrand Reinhold, New York, 1990).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, London, 1998).

Cited By

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Figures (9)

Fig. 1:
Fig. 1:

The schematic diagram of a solar cell structure.

Fig. 2:
Fig. 2:

The unit cell of the plasmonic thin-film SC. The four-layered structure includes indium tin oxide (ITO), absorbing materials, Au (or Ag) electrodes, and substrate with thickness of d1, d2, d3, and d4, respectively. The distance between two adjacent strips is ds and the periodicity is P. The incident light propagates into the structure through the ITO. The PML and the Mur absorbing boundary conditions are employed at the top and the bottom of the SC structure. The periodic boundary conditions (PBC) at the left and right sides of the unit cell are imposed.

Fig. 3:
Fig. 3:

The inhomogeneous material treatment. The squares denote the five difference nodes. The center square is enclosed by the four rectangular regions with different dielectric constants. Γ mm are the contiguous edges of the rectangular regions (See Appendix A).

Fig. 4:
Fig. 4:

The zeroth-order reflectance and transmittance by the FDFD method and the rigorous coupled-wave algorithm.

Fig. 5:
Fig. 5:

The -directed attenuation constants in the A-Si and Au layers. When the incident wavelength goes through the zero-crossing point of 560nm, the eigenstates of Maxwell’s equations for the semi-infinite A-Si/Au structure become surface plasmon waves.

Fig. 6:
Fig. 6:

The contour plot of the eigenstate for Ex field at 735nm, at which the dips of the -directed attenuation constants are achieved as shown in Fig. 5.

Fig. 7:
Fig. 7:

The absorbed power density η by the A-Si for the periodic strip structure and the “artificially” periodic non-strip structure. The lower arrows (from A to B) denote the absorption peaks of the non-strip structure and the upper arrows (from 1 to 6) denote the absorption peaks of the strip structure.

Fig. 8:
Fig. 8:

The generalized reflection coefficients for locating the waveguide modes. The dips of the generalized reflection coefficients correspond to some absorption peaks of Fig. 7.

Fig. 9:
Fig. 9:

The Hzt field distribution for the periodic strip structure.

Equations (61)

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

x ( 1 ε r ( x , y ) H z t x ) + y ( 1 ε r ( x , y ) H z t y ) + k 0 2 H z t = 0
x ( 1 ε r ( x , y ) H z t x ) = 1 Δ x ( H z t ( i + 1 , j ) H z t ( i , j ) ε r ( i + 1 2 , j ) Δ x
H z t ( i , j ) H z t ( i 1 , j ) ε r ( i 1 2 , j ) Δ x ) + O ( Δ x 2 )
1 ε r ( i + 1 2 , j ) 1 2 ( 1 ε r 1 + 1 ε r 4 )
1 ε r ( i 1 2 , j ) 1 2 ( 1 ε r 2 + 1 ε r 3 )
m = 1 5 c m Φ m = 0
c 1 = 1 2 ( 1 ε r 1 + 1 ε r 2 ) · 1 Δ y 2
c 2 = 1 2 ( 1 ε r 2 + 1 ε r 3 ) · 1 Δ x 2
c 4 = 1 2 ( 1 ε r 4 + 1 ε r 1 ) · 1 Δ x 2
c 5 = 1 2 ( 1 ε r 3 + 1 ε r 4 ) · 1 Δ y 2
c 3 = 1 2 ( 1 ε r 1 + 1 ε r 2 + 1 ε r 3 + 1 ε r 4 ) · ( 1 Δ x 2 + 1 Δ y 2 ) + k 0 2
H z t = H z inc + H z s
2 H z s x 2 + 1 s y y ( 1 s y H z s y ) + k 0 2 H z s = 0
s y = { 1 j 0 σ ( y ) ω ε 0 , within PML 1 , other
σ ( j ) = C Δ y ( j 1 / 2 L ) Q , j = 1,2 , , 8
σ ( j + 1 / 2 ) = C Δ y ( j L ) Q , j = 0,1 , , 8
1 s y y ( 1 s y H z s y ) 1 s y ( j ) Δ y [ H z s ( i , j + 1 ) H z s ( i , j ) s y ( j + 1 / 2 ) Δ y
H z s ( i , j ) H z s ( i , j 1 ) s y ( j 1 / 2 ) Δ y ]
[ y j 0 ( k 0 + 1 2 k 0 2 x 2 ) ] H z s y = 0 = 0
f 1 H z s ( i , j ) + f 2 H z s ( i 1 , j ) + f 3 H z s ( i + 1 , j ) + f 4 H z s ( i , j + 1 ) = 0
f 1 = 2 exp ( j 0 k 0 Δ y ) 2 k 0 2 Δ x 2 exp ( j 0 k 0 Δ y ) 2
f 2 = f 3 = 1 exp ( j 0 k 0 Δ y )
f 4 = 2 k 0 2 Δ x 2
H z s ( x + P , y ) = H z s ( x , y ) exp ( j 0 k 0 cos θ · P )
H z s ( x , y ) = H z s ( x + P , y ) exp ( j 0 k 0 cos θ · P )
( 1 ε r 1 y H z s 1 1 ε r 2 y H z s 2 ) y = y h = ( 1 ε r 2 y H z inc 1 ε r 1 y H z inc ) y = y h
y H z s 1 x = i Δ x 1.5 H z s 1 ( i , j ) 2 H z s 1 ( i , j 1 ) + 0.5 H z s 1 ( i , j 2 ) Δ y
y H z s 2 x = i Δ x 1.5 H z s 2 ( i , j ) + 2 H z s 2 ( i , j + 1 ) 0.5 H z s 2 ( i , j + 2 ) Δ y
E x t ( i , j + 1 / 2 ) 1 2 ( 1 ε r ( i , j + 1 ) + 1 ε r ( i , j ) ) H z t ( i , j + 1 ) H z t ( i , j ) j 0 ω ε 0 · Δ y
E y t ( i + 1 / 2 , j ) 1 2 ( 1 ε r ( i + 1 , j ) + 1 ε r ( i , j ) ) H z t ( i + 1 , j ) H z t ( i , j ) j 0 ω ε 0 · Δ x
η = S a σ a E 2 ds Δ S a = ω ε 0 S a Im ( ε ra ) E 2 ds Δ S a
Ψ p = exp ( j U p x ) exp ( j V p y )
U p = k 0 cos θ + 2 πp P , p = 0 , ± 1 , ± 2 ,
V p = { k 0 2 U p 2 , k 0 2 U p 2 j k 0 2 U p 2 , k 0 2 < U p 2
R p = 1 P 0 P H z s ( x , y r ) exp ( j k 0 cos θ x ) dx 2 A 2
T p = 1 P 0 P H z t ( x , y t ) exp ( j k 0 cos θ x ) dx 2 A 2
H z inc ( x , y ) = exp ( j k 0 ( x cos θ + y sin θ ) )
H z ( x ˜ , y ˜ ) = exp ( j k y Si y ˜ j k x x ˜ ) , k y Si = β y Si + j α y Si , y ˜ < 0
H z ( x ˜ , y ˜ ) = exp ( j k y Au y ˜ j k x x ˜ ) , k y Au = β y Au + j α y Au , y ˜ > 0
R = ε r Au k y Si ε r Si k y Au ε r Au k y Si + ε r Si k y Au
ε r Au k y Si + ε r Si k y Au = 0
( k y Si ) 2 = k 0 2 ε r Si k x 2
( k y Au ) 2 = k 0 2 ε r Au k x 2
k x = k 0 ( ε r Si ε r Au ε r Si + ε r Au ) 1 / 2 , k x = β x + j α x
α y Si < 0 , α y Au < 0
ε r Au ( ω ) 1 ω p 2 ω 2
β y Si < 0 , β y Au < 0
R ˜ i , i + 1 = R i , i + 1 + R ˜ i + 1 , i + 2 e 2 j k i + 1 , y ( d i + 1 d i ) 1 + R i , i + 1 + R ˜ i + 1 , i + 2 e 2 j k i + 1 , y ( d i + 1 d i )
S m H z t n m dl + k m 2 S m H z t ds = 0
k 1 2 Δ x Δ y 4 Φ 3 + Δ y 2 Δ x ( Φ 4 Φ 3 ) + Δ x 2 Δ y ( Φ 1 Φ 3 ) =
Γ 12 Φ n 1 dl Γ 12 Φ n 1 dl
k 2 2 Δ x Δ y 4 Φ 3 + Δ y 2 Δ x ( Φ 2 Φ 3 ) + Δ x 2 Δ y ( Φ 1 Φ 3 ) =
Γ 21 Φ n 2 dl Γ 23 Φ n 2 dl
k 3 2 Δ x Δ y 4 Φ 3 + Δ y 2 Δ x ( Φ 2 Φ 3 ) + Δ x 2 Δ y ( Φ 5 Φ 3 ) =
Γ 32 Φ n 3 dl Γ 34 Φ n 3 dl
k 4 2 Δ x Δ y 4 Φ 3 + Δ y 2 Δ x ( Φ 4 Φ 3 ) + Δ x 2 Δ y ( Φ 5 Φ 3 ) =
Γ 43 Φ n 4 dl Γ 41 Φ n 4 dl
( 1 ε r 1 Φ ( 1 ) n 1 + 1 ε r 2 Φ ( 2 ) n 2 ) Γ 12 , Γ 21 = 0
( 1 ε r 2 Φ ( 2 ) n 2 + 1 ε r 3 Φ ( 3 ) n 3 ) Γ 23 , Γ 32 = 0
( 1 ε r 3 Φ ( 3 ) n 3 + 1 ε r 4 Φ ( 4 ) n 4 ) Γ 34 , Γ 43 = 0
( 1 ε r 4 Φ ( 4 ) n 4 + 1 ε r 1 Φ ( 1 ) n 1 ) Γ 41 , Γ 14 = 0

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