Abstract

We compute the short-circuit diffusion current of excitons in an organic solar cell, with special emphasis on fluorescence losses. The exciton diffusion length is not uniform but varies with its position within the device, even with moderate fluorescence quantum efficiency. With large quantum efficiencies, the rate of fluorescence can be strongly reduced with proper choices of the geometrical and dielectric parameters. Hence, through proper micro-cavity design, the diffusion length can be increased and the device performance significantly improved without recourse to triplet excitonic states.

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2012 (4)

L. Penninck, F. Steinbacher, R. Krause, K. Neyts, “Determining emissive dipole orientation in organic light emitting devices by decay time measurement,” Org. Electron. 13, 3079 – 3084 (2012).
[CrossRef]

R. Betancur, A. Martínez-Otero, X. Elias, P. Romero-Gómez, S. Colodrero, H. Miguez, J. Martorell, “Optical interference for the matching of the external and internal quantum efficiencies in organic photovoltaic cells,” Sol. Energy Mater. Sol. Cells 104, 87 – 91 (2012).
[CrossRef]

G. Chen, D. Yokoyama, H. Sasabe, Z. Hong, Y. Yang, J. Kido, “Optical and electrical properties of a squaraine dye in photovoltaic cells,” Appl. Phys. Lett. 101, 083904 (2012).
[CrossRef]

O. D. Miller, E. Yablonovitch, S. R. Kurtz, “Strong internal and external luminescence as solar cells approach the shockley–queisser limit,” IEEE J. Photovolt. 2, 303–311 (2012).
[CrossRef]

2011 (3)

B. Verreet, B. P. Rand, D. Cheyns, A. Hadipour, T. Aernouts, P. Heremans, A. Medina, C. G. Claessens, T. Torres, “A 4% efficient organic solar cell using a fluorinated fused subphthalocyanine dimer as an electron acceptor,” Adv. Energy Mater. 1, 565–568 (2011).
[CrossRef]

D. Rezzonico, B. Perucco, E. Knapp, R. Häusermann, N. A. Reinke, F. Müller, B. Ruhstaller, “Numerical analysis of exciton dynamics in organic light-emitting devices and solar cells,” J. Photon. Energy 1, 011005 (2011).
[CrossRef]

K. J. Bergemann, S. R. Forrest, “Measurement of exciton diffusion lengths in optically thin organic films,” Appl. Phys. Lett. 99, 243303 (2011).
[CrossRef]

2010 (1)

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, A. Tünnermann, “Orientation of emissive dipoles in oleds: Quantitative in situ analysis,” Org. Electron. 11, 1039 – 1046 (2010).
[CrossRef]

2009 (3)

L. T. Vuong, G. Kozyreff, R. Betancur, J. Martorell, “Cavity-controlled radiative recombination of excitons in thin-film solar cells,” Appl. Phys. Lett. 95, 233106 (2009).
[CrossRef]

R. R. Lunt, N. C. Giebink, A. A. Belak, J. B. Benziger, S. R. Forrest, “Exciton diffusion lengths of organic semiconductor thin films measured by spectrally resolved photoluminescence quenching,” J. Appl. Phys. 105, 053711 (2009).
[CrossRef]

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, K. Meerholz, “In situ measurement of the internal luminescence quantum efficiency in organic light-emitting diodes,” Appl. Phys. Lett. 95, 263306 (2009).
[CrossRef]

2007 (3)

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

H. H. P. Gommans, D. Cheyns, T. Aernouts, C. Girotto, J. Poortmans, P. Heremans, “Electro-optical study of subphthalocyanine in a bilayer organic solar cell,” Adv. Funct. Mater. 17, 2653–2658 (2007).
[CrossRef]

K. Celebi, T. D. Heidel, M. A. Baldo, “Simplified calculation of dipole energy transport in a multilayer stack using dyadic green’s functions,” Opt. Express 15, 1762–1772 (2007).
[CrossRef] [PubMed]

2006 (1)

M. Wojdyla, B. Derkowska, W. Bala, A. Bratkowski, A. Korcala, “Investigation of optical transition energy in copper phthalocyanine by transmission, reflection and photoreflectance spectroscopy,” Opt. Mat. 28, 1000 –1005 (2006).
[CrossRef]

2005 (2)

L. Smith, J. Wasey, I. Samuel, W. Barnes, “Light out-coupling efficiencies of organic light-emitting diode structures and the effect of photoluminescence quantum yield,” Adv. Funct. Mater. 15, 1839–1844 (2005).
[CrossRef]

S. R. Forrest, “The limits to organic photovoltaic cell efficiency,” MRS Bull. 30, 28–32 (2005).
[CrossRef]

2003 (1)

J. A. Barker, C. M. Ramsdale, N. C. Greenham, “Modeling the current-voltage characteristics of bilayer polymer photovoltaic devices,” Phys. Rev. B 67, 075205 (2003).
[CrossRef]

2001 (1)

T. Stübinger, W. Brütting, “Exciton diffusion and optical interference in organic donor–acceptor photovoltaic cells,” J. Appl. Phys. 90, 3632–3641 (2001).
[CrossRef]

2000 (4)

J.-S. Kim, P. K. H. Ho, N. C. Greenham, R. H. Friend, “Electroluminescence emission pattern of organic light-emitting diodes: Implications for device efficiency calculations,” J. Appl. Phys. 88, 1073–1081 (2000).
[CrossRef]

J. A. E. Wasey, W. L. Barnes, “Birefringence and light emission from the polymer led,” Synth. Met. 111, 213–215 (2000).
[CrossRef]

N. Tessler, “Transport and optical modeling of organic light-emitting diodes,” Appl. Phys. Lett. 77, 1897–1899 (2000).
[CrossRef]

J. Wasey, A. Safonov, I. Samuel, W. Barnes, “Effects of dipole orientation and birefringence on the optical emission from thin films,” Opt. Commun. 183, 109 – 121 (2000).
[CrossRef]

1999 (1)

L. A. A. Pettersson, L. S. Roman, O. Inganäs, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
[CrossRef]

1998 (2)

K. A. Neyts, “Simulation of light emission from thin-film microcavities,” J. Opt. Soc. Am. A 15, 962–971 (1998).
[CrossRef]

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B 58, 3730–3740 (1998).
[CrossRef]

1997 (5)

M. G. Harrison, J. Grüner, G. C. W. Spencer, “Analysis of the photocurrent action spectra of MEH-PPV polymer photodiodes,” Phys. Rev. B 55, 7831–7849 (1997).
[CrossRef]

K. A. Michalski, J. R. Mosig, “Multilayered media green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 45, 508–519 (1997).
[CrossRef]

H. Becker, S. E. Burns, R. H. Friend, “Effect of metal films on the photoluminescence and electroluminescence of conjugated polymers,” Phys. Rev. B 56, 1893–1905 (1997).
[CrossRef]

R. M. Amos, W. L. Barnes, “Modification of the spontaneous emission rate of Eu3+ ions close to a thin metal mirror,” Phys. Rev. B 55, 7249–7254 (1997).
[CrossRef]

K. G. Sullivan, D. G. Hall, “Enhancement and inhibition of electromagnetic radiation in plane-layered media. i.plane-wave spectrum approach to modeling classical effects,” J. Opt. Soc. Am. B 14, 1149–1159 (1997).
[CrossRef]

1994 (1)

S. Saito, T. Tsutsui, M. Era, N. Takada, E.-I. Aminaka, T. Wakimoto, “Design of organic electroluminescent materials and devices,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 253, 125–132 (1994).
[CrossRef]

1993 (1)

A. Desormeaux, J. J. Max, R. M. Leblanc, “Photovoltaic and electrical properties of aluminum/langmuir-blodgett films/silver sandwich cells incorporating either chlorophyll a, chlorophyll b, or zinc porphyrin derivative,” J. Phys. Chem. 97, 6670–6678 (1993).
[CrossRef]

1991 (1)

T. Tsutsui, C. Adachi, S. Saito, M. Watanabe, M. Koishi, “Effect of confined radiation field on spontaneous-emission lifetime in vacuum-deposited fluorescent dye films,” Chem. Phys. Lett. 182, 143 – 146 (1991).
[CrossRef]

1981 (1)

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

1980 (1)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
[CrossRef]

1978 (2)

A. K. Ghosh, T. Feng, “Merocyanine organic solar cells,” J. Appl. Phys. 49, 5982–5989 (1978).
[CrossRef]

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1970 (1)

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 12, 693 – 701 (1970).
[CrossRef]

1961 (1)

W. Shockley, H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32, 510–519 (1961).
[CrossRef]

1956 (1)

H. B. DeVore, “Spectral distribution of photoconductivity,” Phys. Rev. 102, 86–91 (1956).
[CrossRef]

1909 (1)

A. Sommerfeld, “Über die ausbreitung der wellen in der drahtlosen telegraphie,” Ann. der Phys. 28, 665–736 (1909).
[CrossRef]

Adachi, C.

T. Tsutsui, C. Adachi, S. Saito, M. Watanabe, M. Koishi, “Effect of confined radiation field on spontaneous-emission lifetime in vacuum-deposited fluorescent dye films,” Chem. Phys. Lett. 182, 143 – 146 (1991).
[CrossRef]

Aernouts, T.

B. Verreet, B. P. Rand, D. Cheyns, A. Hadipour, T. Aernouts, P. Heremans, A. Medina, C. G. Claessens, T. Torres, “A 4% efficient organic solar cell using a fluorinated fused subphthalocyanine dimer as an electron acceptor,” Adv. Energy Mater. 1, 565–568 (2011).
[CrossRef]

H. H. P. Gommans, D. Cheyns, T. Aernouts, C. Girotto, J. Poortmans, P. Heremans, “Electro-optical study of subphthalocyanine in a bilayer organic solar cell,” Adv. Funct. Mater. 17, 2653–2658 (2007).
[CrossRef]

Aminaka, E.-I.

S. Saito, T. Tsutsui, M. Era, N. Takada, E.-I. Aminaka, T. Wakimoto, “Design of organic electroluminescent materials and devices,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 253, 125–132 (1994).
[CrossRef]

Amos, R. M.

R. M. Amos, W. L. Barnes, “Modification of the spontaneous emission rate of Eu3+ ions close to a thin metal mirror,” Phys. Rev. B 55, 7249–7254 (1997).
[CrossRef]

Bala, W.

M. Wojdyla, B. Derkowska, W. Bala, A. Bratkowski, A. Korcala, “Investigation of optical transition energy in copper phthalocyanine by transmission, reflection and photoreflectance spectroscopy,” Opt. Mat. 28, 1000 –1005 (2006).
[CrossRef]

Baldo, M. A.

Barker, J. A.

J. A. Barker, C. M. Ramsdale, N. C. Greenham, “Modeling the current-voltage characteristics of bilayer polymer photovoltaic devices,” Phys. Rev. B 67, 075205 (2003).
[CrossRef]

Barnes, W.

L. Smith, J. Wasey, I. Samuel, W. Barnes, “Light out-coupling efficiencies of organic light-emitting diode structures and the effect of photoluminescence quantum yield,” Adv. Funct. Mater. 15, 1839–1844 (2005).
[CrossRef]

J. Wasey, A. Safonov, I. Samuel, W. Barnes, “Effects of dipole orientation and birefringence on the optical emission from thin films,” Opt. Commun. 183, 109 – 121 (2000).
[CrossRef]

Barnes, W. L.

J. A. E. Wasey, W. L. Barnes, “Birefringence and light emission from the polymer led,” Synth. Met. 111, 213–215 (2000).
[CrossRef]

R. M. Amos, W. L. Barnes, “Modification of the spontaneous emission rate of Eu3+ ions close to a thin metal mirror,” Phys. Rev. B 55, 7249–7254 (1997).
[CrossRef]

Becker, H.

H. Becker, S. E. Burns, R. H. Friend, “Effect of metal films on the photoluminescence and electroluminescence of conjugated polymers,” Phys. Rev. B 56, 1893–1905 (1997).
[CrossRef]

Belak, A. A.

R. R. Lunt, N. C. Giebink, A. A. Belak, J. B. Benziger, S. R. Forrest, “Exciton diffusion lengths of organic semiconductor thin films measured by spectrally resolved photoluminescence quenching,” J. Appl. Phys. 105, 053711 (2009).
[CrossRef]

Benziger, J. B.

R. R. Lunt, N. C. Giebink, A. A. Belak, J. B. Benziger, S. R. Forrest, “Exciton diffusion lengths of organic semiconductor thin films measured by spectrally resolved photoluminescence quenching,” J. Appl. Phys. 105, 053711 (2009).
[CrossRef]

Bergemann, K. J.

K. J. Bergemann, S. R. Forrest, “Measurement of exciton diffusion lengths in optically thin organic films,” Appl. Phys. Lett. 99, 243303 (2011).
[CrossRef]

Betancur, R.

R. Betancur, A. Martínez-Otero, X. Elias, P. Romero-Gómez, S. Colodrero, H. Miguez, J. Martorell, “Optical interference for the matching of the external and internal quantum efficiencies in organic photovoltaic cells,” Sol. Energy Mater. Sol. Cells 104, 87 – 91 (2012).
[CrossRef]

L. T. Vuong, G. Kozyreff, R. Betancur, J. Martorell, “Cavity-controlled radiative recombination of excitons in thin-film solar cells,” Appl. Phys. Lett. 95, 233106 (2009).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principle of Optics (Pergamon, 1991).

Bratkowski, A.

M. Wojdyla, B. Derkowska, W. Bala, A. Bratkowski, A. Korcala, “Investigation of optical transition energy in copper phthalocyanine by transmission, reflection and photoreflectance spectroscopy,” Opt. Mat. 28, 1000 –1005 (2006).
[CrossRef]

Bräuer, A. H.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, A. Tünnermann, “Orientation of emissive dipoles in oleds: Quantitative in situ analysis,” Org. Electron. 11, 1039 – 1046 (2010).
[CrossRef]

Brütting, W.

T. Stübinger, W. Brütting, “Exciton diffusion and optical interference in organic donor–acceptor photovoltaic cells,” J. Appl. Phys. 90, 3632–3641 (2001).
[CrossRef]

Bulovic, V.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B 58, 3730–3740 (1998).
[CrossRef]

Burns, S. E.

H. Becker, S. E. Burns, R. H. Friend, “Effect of metal films on the photoluminescence and electroluminescence of conjugated polymers,” Phys. Rev. B 56, 1893–1905 (1997).
[CrossRef]

Burrows, P. E.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B 58, 3730–3740 (1998).
[CrossRef]

Celebi, K.

Chance, R. R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Chen, G.

G. Chen, D. Yokoyama, H. Sasabe, Z. Hong, Y. Yang, J. Kido, “Optical and electrical properties of a squaraine dye in photovoltaic cells,” Appl. Phys. Lett. 101, 083904 (2012).
[CrossRef]

Chen, X.-W.

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R. Betancur, A. Martínez-Otero, X. Elias, P. Romero-Gómez, S. Colodrero, H. Miguez, J. Martorell, “Optical interference for the matching of the external and internal quantum efficiencies in organic photovoltaic cells,” Sol. Energy Mater. Sol. Cells 104, 87 – 91 (2012).
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B. Verreet, B. P. Rand, D. Cheyns, A. Hadipour, T. Aernouts, P. Heremans, A. Medina, C. G. Claessens, T. Torres, “A 4% efficient organic solar cell using a fluorinated fused subphthalocyanine dimer as an electron acceptor,” Adv. Energy Mater. 1, 565–568 (2011).
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M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, A. Tünnermann, “Orientation of emissive dipoles in oleds: Quantitative in situ analysis,” Org. Electron. 11, 1039 – 1046 (2010).
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M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, K. Meerholz, “In situ measurement of the internal luminescence quantum efficiency in organic light-emitting diodes,” Appl. Phys. Lett. 95, 263306 (2009).
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M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, A. Tünnermann, “Orientation of emissive dipoles in oleds: Quantitative in situ analysis,” Org. Electron. 11, 1039 – 1046 (2010).
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M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, K. Meerholz, “In situ measurement of the internal luminescence quantum efficiency in organic light-emitting diodes,” Appl. Phys. Lett. 95, 263306 (2009).
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Penninck, L.

L. Penninck, F. Steinbacher, R. Krause, K. Neyts, “Determining emissive dipole orientation in organic light emitting devices by decay time measurement,” Org. Electron. 13, 3079 – 3084 (2012).
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D. Rezzonico, B. Perucco, E. Knapp, R. Häusermann, N. A. Reinke, F. Müller, B. Ruhstaller, “Numerical analysis of exciton dynamics in organic light-emitting devices and solar cells,” J. Photon. Energy 1, 011005 (2011).
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L. A. A. Pettersson, L. S. Roman, O. Inganäs, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
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H. H. P. Gommans, D. Cheyns, T. Aernouts, C. Girotto, J. Poortmans, P. Heremans, “Electro-optical study of subphthalocyanine in a bilayer organic solar cell,” Adv. Funct. Mater. 17, 2653–2658 (2007).
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B. Verreet, B. P. Rand, D. Cheyns, A. Hadipour, T. Aernouts, P. Heremans, A. Medina, C. G. Claessens, T. Torres, “A 4% efficient organic solar cell using a fluorinated fused subphthalocyanine dimer as an electron acceptor,” Adv. Energy Mater. 1, 565–568 (2011).
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D. Rezzonico, B. Perucco, E. Knapp, R. Häusermann, N. A. Reinke, F. Müller, B. Ruhstaller, “Numerical analysis of exciton dynamics in organic light-emitting devices and solar cells,” J. Photon. Energy 1, 011005 (2011).
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D. Rezzonico, B. Perucco, E. Knapp, R. Häusermann, N. A. Reinke, F. Müller, B. Ruhstaller, “Numerical analysis of exciton dynamics in organic light-emitting devices and solar cells,” J. Photon. Energy 1, 011005 (2011).
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R. Betancur, A. Martínez-Otero, X. Elias, P. Romero-Gómez, S. Colodrero, H. Miguez, J. Martorell, “Optical interference for the matching of the external and internal quantum efficiencies in organic photovoltaic cells,” Sol. Energy Mater. Sol. Cells 104, 87 – 91 (2012).
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D. Rezzonico, B. Perucco, E. Knapp, R. Häusermann, N. A. Reinke, F. Müller, B. Ruhstaller, “Numerical analysis of exciton dynamics in organic light-emitting devices and solar cells,” J. Photon. Energy 1, 011005 (2011).
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G. Chen, D. Yokoyama, H. Sasabe, Z. Hong, Y. Yang, J. Kido, “Optical and electrical properties of a squaraine dye in photovoltaic cells,” Appl. Phys. Lett. 101, 083904 (2012).
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W. Shockley, H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32, 510–519 (1961).
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R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
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[CrossRef]

Steinbacher, F.

L. Penninck, F. Steinbacher, R. Krause, K. Neyts, “Determining emissive dipole orientation in organic light emitting devices by decay time measurement,” Org. Electron. 13, 3079 – 3084 (2012).
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Figures (7)

Fig. 1
Fig. 1

Normalized fluorescent decay rate as a function of distance from a thick Ag electrode in a uniform medium of refractive index n = 1.3. Thin lines: ideal, lossless electrode, nAg = 6.37i. Thick lines: real electrode, nAg = 0.04 + 6.37i[34]. Vaccuum emission wavelength: 900 nm. b||: parallel excitons. b perpendicular excitons.

Fig. 2
Fig. 2

Solar cell geometry and coordinate system. HBL: hole-blocking layer. A: Acceptor. D: Donor; EBL: electron-blocking layer.

Fig. 3
Fig. 3

Decay rates (q = 1), gain functions, and external quantum efficiencies for the parameters given in Table 1. Top: ITO transparent electrode. Bottom: Ag transparent electrode. iso: isotropic case where b = 2 3 b | | + 1 3 b .

Fig. 4
Fig. 4

Absorption efficiency ηA computed for the devices described in Table 1 as a function of sun wavelength. A fixed ITO refractive index nITO = 1.76+0.08i and an absorption length of 70 nm in the active region are assumed over the whole spectral range. Ag refractive index is taken from [34].

Fig. 5
Fig. 5

Gain and decay profiles computed with Table 2 and q = 1. HBL: hole-blocking layer, A: acceptor, D: donor, EBL: electron-blocking layer.

Fig. 6
Fig. 6

b(z) in a structure with refractive indexes nAg/n2/n1/n2/nAg/n3 and thicknesses (nm): 140/29/295/6. n2 = 1.3, n3 = 2.8. Cases a to e: n1 = 2.8, 2.2, 1.7, 1.5, 1.4. Wavelength: 900 nm. Dashed line: numerical. Full line: (n2/n1)5.

Fig. 7
Fig. 7

b(z) and b||(z) computed at λ = 900 nm (thick dashed and thick full lines, respectively), and averaged over the range 860 nm < λ < 940 nm (crosses). Geometrical and dielectric parameters taken in Table 2.

Tables (2)

Tables Icon

Table 1 Two configurations optimized for parallel excitons. A sunlight absorption length of 70 nm is assumed in both photoactive materials.nITO = 1.76 + 0.08i, nAg(900nm) = 0.04 + 6.37i, nAg(750nm) = 0.03 + 5.19i.

Tables Icon

Table 2 Optimized configuration for perpendicular excitons. Absorption length: 70 nm is assumed in both photoactive materials.nAg(900nm) = 0.04 + 6.37i, nAg(750nm) = 0.03 + 5.19i.

Equations (62)

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0 = L 2 d 2 ρ d z 2 ρ + g ( z ) , L 2 = D τ ,
0 = L 2 d 2 ρ d z 2 b ( z ) ρ + g ( z ) , L 2 = D τ ,
0 = D i d 2 ρ d z 2 b ( z ) τ i ρ + α i ϕ i N g ( z ) , i = 1 , 1 ,
ρ = α i ϕ i N τ i ρ ,
0 = L i 2 d 2 ρ d z 2 b ( z ) ρ + g ( z ) , L i 2 = D i τ i ,
d ρ d z | z 1 = 0 , ρ ( z 0 ) = 0 , d ρ d z | z 1 = 0 .
I s c = I 1 + I 1 , I 1 = α 1 ϕ 1 N A τ 1 D 1 d ρ d z | z 0 , I 1 = α 1 ϕ 1 N A τ 1 D 1 d ρ d z | z 0 + ,
I i = ± ( A N ) α i ϕ i L i 2 d ρ d z | z 0 ± .
EQE sc = α 1 ϕ 1 L 1 2 d ρ d z | z 0 + α 1 ϕ 1 L 1 2 d ρ d z | z 0
0 L i 2 d 2 ρ d z 2 b ¯ ρ + g ¯ , b ¯ = 1 d A + D b ( z ) d z , g ¯ = 1 d A + D g ( z ) d z ,
EQE sc α g ¯ d × 2 L d b tanh ( d b 2 L ) η A × η D
ε ¯ i = ( ε i , x 0 0 0 ε i , x 0 0 0 ε i , z ) , z i 1 < z < z i .
k e , z , i = ε i , x k 0 2 ε i , x ε i , z k | | 2 ,
k o , z , i = ε i , x k 0 2 k | | 2 .
b ( z ) = 1 + 3 q 2 ε 1 , x 1 / 2 ε 1 , z 2 Re { 1 k 0 3 0 R ^ 0 p + R ^ 1 p + 2 R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p k | | 3 k e , z , 1 d k | | } ,
R ^ 0 p = R 0 p exp [ 2 i k e , z , 1 ( z z 0 ) ] , R ^ 1 p = R 1 p exp [ 2 i k e , z , 1 ( z 1 z ) ] ,
R N p = R N , N + 1 p ,
R j 1 p = R j 1 , j p + R j p exp ( 2 i k e , z d j ) 1 + R j 1 , j p R j p exp ( 2 i k e , z , j d j ) ,
R i j p = k e , z , i ε j , x k e , z , j ε i , x k e , z , i ε j , x + k e , z , j ε i , x .
b | | ( z ) = 1 + 3 q 4 4 ε 1 , x 1 / 2 3 ε 1 , x + ε 1 , z × Re { 1 k 0 0 ( k e , z , 1 ε 1 , x k 0 2 2 R ^ 0 p R ^ 1 p R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p + 1 k o , z , 1 2 R ^ 0 s R ^ 1 s + R ^ 0 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s ) k | | d k | | } .
R ^ 0 s = R 0 s exp [ 2 i k o , z , 1 ( z z 0 ) ] , R ^ 1 s = R 1 s exp [ 2 i k o , z , 1 ( z 1 z ) ] ,
R N s = R N , N + 1 s , R j 1 s = R j 1 , j s + R j s exp ( 2 i k o , z , j d j ) 1 + R j 1 , j s R j s exp ( 2 i k o , z , j d j ) , R i j s = k o , z , i k o , z , j k o , z , i + k o , z , j .
g ( z ) = Re ( n i , s ) n ext | a i e i n i , s k 0 z + b i e i n i , s k 0 z | 2 .
k z , 1 = n 1 2 k 0 2 k | | 2 , k z , 2 = n 2 2 k 0 2 k | | 2 .
R ^ 0 p R ^ 1 p R = k z , 1 n 2 2 k z , 2 n 1 2 k z , 1 n 2 2 + k z , 2 n 1 2 ,
b ( z ) 1 + 3 q 2 n 1 3 Re { 1 k 0 3 0 2 R 1 R k | | 3 k z , 1 d k | | } ,
= 1 + 3 q 2 n 1 3 Re { 1 k 0 3 0 k z , 1 n 2 2 k z , 2 n 1 2 k z , 1 k z , 2 n 1 2 k | | 3 d k | | } ,
= 1 + 3 q 2 n 1 3 Re { 1 k 0 3 0 ( n 2 2 / n 1 2 k z , 2 1 k z , 1 ) k | | 3 d k | | }
b 1 q + q ( n 2 n 1 ) 5 .
EQE sc = η A ( λ s ) η D ( λ , q ) ,
Γ r ( 1 + γ ( z ) ) ,
Γ n r + Γ r ( 1 + γ ( z ) ) = ( Γ n r + Γ r ) ( 1 + Γ r Γ n r + Γ r γ ( z ) ) .
b ( z ) = 1 + q γ ( z ) .
k 0 2 ε ¯ E = i ω μ J + i ω × B ,
i ω B = × E ,
Z = Z | | + z ^ Z z .
( 2 z 2 + ε x k 0 2 ) E | | = i ω μ J | | + | | E z z i ω z ^ × | | B z ,
( 2 z 2 + ε x k 0 2 ) ( i ω B | | ) = i ω μ z ( z ^ × J | | ) + i ω | | B z z k 0 2 ε x z ^ × | | E z ,
( ε z 2 z 2 + ε x | | 2 + ε x ε z k 0 2 ) E z = i ω μ ε x z ^ ( J + 1 ε x k 0 2 J ) ,
( 2 z 2 + | | 2 + ε x k 0 2 ) B z = μ z ^ ( × J ) .
g e = e i k 0 ε x x 2 + y 2 + ε z z 2 4 π ε x 1 / 2 ε x ( x 2 + y 2 ) + ε z z 2 , g o = e i ε x 1 / 2 k 0 x 2 + y 2 + z 2 4 π x 2 + y 2 + z 2 .
g e = i 4 π ε z 0 J 0 ( k | | ρ ) k e , z e i k e , z | z | k | | d k | | , g o = i 4 π 0 J 0 ( k | | ρ ) k o , z e i k o , z | z | k | | d k | | ,
E z = i ω μ ε x ( z ^ j 0 + 1 ε x k 0 2 z j 0 ) g e + E , B z = i μ z ^ ( j 0 ) g o + B ,
E z = ω μ ε x j 0 4 π ε z 2 k 0 2 0 k | | 3 k e , z J 0 ( k | | ρ ) ( e i k e , z | z | + C e i k e , z z + D e i k e , z z ) d k | | ,
D = R 1 p e 2 i k e , z 1 ( 1 + C ) R ^ 1 p ( 1 + C ) .
C = R 0 p e 2 i k e , z 0 ( 1 + D ) R ^ 0 p ( 1 + D ) .
C = R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p , D = R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p ,
E z = ω μ ε x j 0 4 π ε z 2 k 0 2 0 k | | 3 k e , z J 0 ( k | | ρ ) ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) d k | | .
( 2 z 2 + ε x k 0 2 ) E | | = ω μ ε x j 0 4 π ε z 2 k 0 2 | | z 0 k | | 3 J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) d k | | .
E | | = ω μ j 0 4 π ε z k 0 2 | | z 0 k | | J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z k e , z ε x 1 / 2 k 0 e i ε x 1 / 2 k 0 | z | ) d k | | ,
2 Re { E j 0 * } h ¯ ω = μ ε x | j 0 | 2 2 π h ¯ ε z 2 k 0 2 Re { 0 k | | 3 k e , z ( 1 + 2 R ^ 0 p R ^ 1 p + R ^ 0 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p ) d k | | . }
Γ r = μ ε x | j 0 | 2 2 π h ¯ ε z 2 k 0 2 0 ε z 1 / 2 k 0 k | | 3 k e , z d k | | = μ | j 0 | 2 k 0 3 π h ¯ ε x 1 / 2 .
Γ r ( 1 + 3 ε x 1 / 2 2 ε z 2 k 0 3 Re { 0 k | | 3 k e , z 2 R ^ 0 p R ^ 1 p + R ^ 0 p + R ^ 1 p 1 R ^ 0 p R ^ 1 p d k | | } ) ,
E z = ω μ j 0 4 π ε z k 0 2 2 x z 0 J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) k | | d k | | ,
B z = i μ j 0 4 π y 0 J 0 ( k | | ρ ) k o , z ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z ) k | | d k | | .
δ ( x ) = δ ( z ) 2 π 0 J 0 ( k | | ρ ) k | | d k | | ,
( 2 z 2 + ε x k 0 2 ) E | | = i ω μ j 0 2 π x ^ δ ( z ) 0 J 0 ( k | | ρ ) k | | d k | | ω μ j 0 4 π ε z k 0 2 | | 3 x z 2 0 J 0 ( k | | ρ ) k e , z ( e i k e , z | z | + R ^ 0 p R ^ 1 p R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) k | | d k | | ω μ j 0 4 π z ^ × | | y 0 J 0 ( k | | ρ ) k o , z ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z ) k | | d k | | .
E | | = ω μ j 0 4 π ε x 1 / 2 k 0 x ^ δ ( z ) 0 J 0 ( k | | ρ ) e i ε x 1 / 2 k 0 | z | k | | d k | | ω μ j 0 4 π ε x k 0 2 | | 3 x z 2 0 J 0 ( k | | ρ ) k e , z k | | ( e i k e , z | z | + R ^ 0 p R ^ 1 p + R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z k e , z ε x 1 / 2 k 0 e i ε x 1 / 2 k 0 | z | ) d k | | ω μ j 0 4 π z ^ × | | y 0 J 0 ( k | | ρ ) k o , z k | | ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z k o , z ε x 1 / 2 k 0 e i ε x 1 / 2 k 0 | z | ) d k | | ,
E | | = ω μ j 0 4 π ε x k 0 2 | | 3 x z 2 0 J 0 ( k | | ρ ) k e , z k | | ( e i k e , z | z | + R ^ 0 p R ^ 1 p R ^ 0 p 1 R ^ 0 p R ^ 1 p e i k e , z z + R ^ 0 p R ^ 1 p R ^ 1 p 1 R ^ 0 p R ^ 1 p e i k e , z z ) d k | | ω μ j 0 4 π z ^ × | | y 0 J 0 ( k | | ρ ) k o , z k | | ( e i k 0 , z | z | + R ^ 0 s R ^ 1 s + R ^ 0 s 1 R ^ 0 s R ^ 1 s e i k o , z z + R ^ 0 s R ^ 1 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s e i k o , z z ) d k | | .
2 Re { E x j 0 * } h ¯ ω = μ | j 0 | 2 4 π h ¯ Re { 0 [ k e , z ε x k 0 2 ( 1 + 2 R ^ 0 p R ^ 1 p R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p ) + 1 k o , z ( 1 + 2 R ^ 0 s R ^ 1 s + R ^ 0 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s ) ] k | | d k | | } .
Γ r = μ | j 0 | 2 4 π h ¯ ( 1 ε x k 0 2 0 ε z 1 / 2 k 0 k e , z k | | d k | | + 0 ε x 1 / 2 k 0 1 k o , z k | | d k | | ) = μ | j 0 | 2 k 0 3 π h ¯ ( 3 ε x + ε z 4 ε x 1 / 2 ) .
Γ r ( 1 + 3 4 k 0 4 ε x 1 / 2 3 ε x + ε z Re { 0 ( k e , z ε x k 0 2 2 R ^ 0 p R ^ 1 p R ^ 0 p R ^ 1 p 1 R ^ 0 p R ^ 1 p + 1 k o , z 2 R ^ 0 s R ^ 1 s + R ^ 0 s + R ^ 1 s 1 R ^ 0 s R ^ 1 s ) k | | d k | | } ) ,

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