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Efficiency and loss mechanisms of plasmonic Luminescent Solar Concentrators

Clemens Tummeltshammer, Mark S. Brown, Alaric Taylor, Anthony J. Kenyon, and Ioannis Papakonstantinou

Open Access Open Access

Abstract

Using a hybrid nanoscale/macroscale model, we simulate the efficiency of a luminescent solar concentrator (LSC) which employs silver nanoparticles to enhance the dye absorption and scatter the incoming light. We show that the normalized optical efficiency can be increased from 10.4% for a single dye LSC to 32.6% for a plasmonic LSC with silver spheres immersed inside a thin dye layer. Most of the efficiency enhancement is due to scattering of the particles and not due to dye absorption/re-emission.

© 2013 Optical Society of America

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References

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  • Year
  • Author
  • Publication
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  11. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
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    [Crossref]
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2012 (1)

S. Chandra, J. Doran, S. J. McCormack, M. Kennedy, and A. J. Chatten, “Enhanced quantum dot emission for luminescent solar concentrators using plasmonic interaction,” Sol. Energ. Mat. Sol. Cells 98, 385–390 (2012).
[Crossref]

2010 (3)

R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32, 850–856 (2010).
[Crossref]

X. Miao, I. Brener, and T. S. Luk, “Nanocomposite plasmonic fluorescence emitters with core/shell configurations,” J. Opt. Soc. Am. B 27, 1561–1570 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

2009 (1)

M. Fikry, M. M. Omar, and L. Z. Ismail, “Effect of host medium on the fluorescence emission intensity of rhodamine B in liquid and solid phase,” J. Fluoresc. 19, 741–746 (2009).
[Crossref] [PubMed]

2008 (3)

2007 (3)

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[Crossref]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
[Crossref] [PubMed]

2006 (1)

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[Crossref] [PubMed]

2003 (2)

S. Agostinelli, “Geant4—a simulation toolkit,” Nucl. Instrum. Methods Phys. Res., Sect. A: Accelerators, Spectrometers, Detectors and Associated Equipment 506, 250–303 (2003).
[Crossref]

E. A. Coronado and G. C. Schatz, “Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach,” J. Chem. Phys. 119, 3926–3934 (2003).
[Crossref]

2000 (1)

A. V. Deshpande and E. B. Namdas, “Correlation between lasing and photophysical performance of dyes in polymethylmethacrylate,” J. Lumin. 91, 25–31 (2000).
[Crossref]

1996 (1)

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

1989 (1)

F. L. Arbeloa, P. R. Ojeda, and I. L. Arbeloa, “Fluorescence self-quenching of the molecular forms of rhodamine B in aqueous and ethanolic solutions,” J. Lumin. 44, 105–112 (1989).
[Crossref]

1987 (1)

H. R. Wilson, “Fluorescent dyes interacting with small silver particles; a system extending the spectral range of fluorescent solar concentrators,” Sol. Energ. Mat. 16, 223–234 (1987).
[Crossref]

1982 (1)

1981 (1)

1977 (1)

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

1976 (1)

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Agostinelli, S.

S. Agostinelli, “Geant4—a simulation toolkit,” Nucl. Instrum. Methods Phys. Res., Sect. A: Accelerators, Spectrometers, Detectors and Associated Equipment 506, 250–303 (2003).
[Crossref]

Anger, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[Crossref]

Arbeloa, F. L.

F. L. Arbeloa, P. R. Ojeda, and I. L. Arbeloa, “Fluorescence self-quenching of the molecular forms of rhodamine B in aqueous and ethanolic solutions,” J. Lumin. 44, 105–112 (1989).
[Crossref]

Arbeloa, I. L.

F. L. Arbeloa, P. R. Ojeda, and I. L. Arbeloa, “Fluorescence self-quenching of the molecular forms of rhodamine B in aqueous and ethanolic solutions,” J. Lumin. 44, 105–112 (1989).
[Crossref]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Barnham, K. W. J.

Batchelder, J. S.

Bende, E. E.

Bharadwaj, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[Crossref]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
[Crossref] [PubMed]

Bindhu, C. V.

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1983).

Bose, R.

Brener, I.

Büchtemann, A.

Budel, T.

Burgers, A. R.

Carter, S. A.

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

Catchpole, K. R.

K. R. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
[Crossref]

Chandra, S.

S. Chandra, J. Doran, S. J. McCormack, M. Kennedy, and A. J. Chatten, “Enhanced quantum dot emission for luminescent solar concentrators using plasmonic interaction,” Sol. Energ. Mat. Sol. Cells 98, 385–390 (2012).
[Crossref]

Chatten, A. J.

Cole, T.

Coronado, E. A.

E. A. Coronado and G. C. Schatz, “Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach,” J. Chem. Phys. 119, 3926–3934 (2003).
[Crossref]

Debije, M. G.

Deshpande, A. V.

A. V. Deshpande and E. B. Namdas, “Correlation between lasing and photophysical performance of dyes in polymethylmethacrylate,” J. Lumin. 91, 25–31 (2000).
[Crossref]

Donegá, C. D. M.

Doran, J.

S. Chandra, J. Doran, S. J. McCormack, M. Kennedy, and A. J. Chatten, “Enhanced quantum dot emission for luminescent solar concentrators using plasmonic interaction,” Sol. Energ. Mat. Sol. Cells 98, 385–390 (2012).
[Crossref]

Farrell, D. J.

Fikry, M.

M. Fikry, M. M. Omar, and L. Z. Ismail, “Effect of host medium on the fluorescence emission intensity of rhodamine B in liquid and solid phase,” J. Fluoresc. 19, 741–746 (2009).
[Crossref] [PubMed]

Goetzberger, A.

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Greube, W.

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Håkanson, U.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[Crossref] [PubMed]

Harilal, S. S.

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Hoeks, T. L.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1983).

Ismail, L. Z.

M. Fikry, M. M. Omar, and L. Z. Ismail, “Effect of host medium on the fluorescence emission intensity of rhodamine B in liquid and solid phase,” J. Fluoresc. 19, 741–746 (2009).
[Crossref] [PubMed]

Issac, R. C.

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

Kennedy, M.

Koole, R.

Kühn, S.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[Crossref] [PubMed]

Lambe, J.

Luk, T. S.

McCormack, S. J.

Meijerink, A.

Meyer, A.

Meyer, T.

Miao, X.

Namdas, E. B.

A. V. Deshpande and E. B. Namdas, “Correlation between lasing and photophysical performance of dyes in polymethylmethacrylate,” J. Lumin. 91, 25–31 (2000).
[Crossref]

Nampoori, V. P. N.

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

Novotny, L.

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
[Crossref] [PubMed]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Ojeda, P. R.

F. L. Arbeloa, P. R. Ojeda, and I. L. Arbeloa, “Fluorescence self-quenching of the molecular forms of rhodamine B in aqueous and ethanolic solutions,” J. Lumin. 44, 105–112 (1989).
[Crossref]

Olson, J. D.

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

Omar, M. M.

M. Fikry, M. M. Omar, and L. Z. Ismail, “Effect of host medium on the fluorescence emission intensity of rhodamine B in liquid and solid phase,” J. Fluoresc. 19, 741–746 (2009).
[Crossref] [PubMed]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

K. R. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
[Crossref]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Quilitz, J.

Reisfeld, R.

R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32, 850–856 (2010).
[Crossref]

Richards, B. S.

Rogobete, L.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[Crossref] [PubMed]

Rowan, B. C.

Sandoghdar, V.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[Crossref] [PubMed]

Schatz, G. C.

E. A. Coronado and G. C. Schatz, “Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach,” J. Chem. Phys. 119, 3926–3934 (2003).
[Crossref]

Sholin, V.

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

Slooff, L. H.

Vallabhan, C. P. G.

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

van Sark, W. G. J. H. M.

Vanmaekelbergh, D.

Varier, G. K.

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

Verbunt, P. P. C.

Weber, W. H.

Wilson, H. R.

H. R. Wilson, “Fluorescent dyes interacting with small silver particles; a system extending the spectral range of fluorescent solar concentrators,” Sol. Energ. Mat. 16, 223–234 (1987).
[Crossref]

Yablonovitch, E.

Zewail, A. H.

Appl. Opt. (3)

Appl. Phys. (1)

A. Goetzberger and W. Greube, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Appl. Phys. Lett. (1)

K. R. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
[Crossref]

J. Appl. Phys. (1)

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

J. Chem. Phys. (1)

E. A. Coronado and G. C. Schatz, “Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach,” J. Chem. Phys. 119, 3926–3934 (2003).
[Crossref]

J. Fluoresc. (1)

M. Fikry, M. M. Omar, and L. Z. Ismail, “Effect of host medium on the fluorescence emission intensity of rhodamine B in liquid and solid phase,” J. Fluoresc. 19, 741–746 (2009).
[Crossref] [PubMed]

J. Lumin. (2)

A. V. Deshpande and E. B. Namdas, “Correlation between lasing and photophysical performance of dyes in polymethylmethacrylate,” J. Lumin. 91, 25–31 (2000).
[Crossref]

F. L. Arbeloa, P. R. Ojeda, and I. L. Arbeloa, “Fluorescence self-quenching of the molecular forms of rhodamine B in aqueous and ethanolic solutions,” J. Lumin. 44, 105–112 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys. (1)

C. V. Bindhu, S. S. Harilal, G. K. Varier, R. C. Issac, V. P. N. Nampoori, and C. P. G. Vallabhan, “Measurement of the absolute fluorescence quantum yield of rhodamine B solution using a dual-beam thermal lens technique,” J. Phys. D: Appl. Phys. 291074–1079 (1996).
[Crossref]

Nanotechnology (1)

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[Crossref]

Nat. Mater. (1)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Nucl. Instrum. Methods Phys. Res., Sect. A: Accelerators, Spectrometers, Detectors and Associated Equipment (1)

S. Agostinelli, “Geant4—a simulation toolkit,” Nucl. Instrum. Methods Phys. Res., Sect. A: Accelerators, Spectrometers, Detectors and Associated Equipment 506, 250–303 (2003).
[Crossref]

Opt. Express (2)

Opt. Mater. (1)

R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32, 850–856 (2010).
[Crossref]

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. Lett. (1)

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[Crossref] [PubMed]

Sol. Energ. Mat. (1)

H. R. Wilson, “Fluorescent dyes interacting with small silver particles; a system extending the spectral range of fluorescent solar concentrators,” Sol. Energ. Mat. 16, 223–234 (1987).
[Crossref]

Sol. Energ. Mat. Sol. Cells (1)

S. Chandra, J. Doran, S. J. McCormack, M. Kennedy, and A. J. Chatten, “Enhanced quantum dot emission for luminescent solar concentrators using plasmonic interaction,” Sol. Energ. Mat. Sol. Cells 98, 385–390 (2012).
[Crossref]

Other (5)

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1983).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Oregon Medical Laser Center, “Rhodamine B,” http://omlc.ogi.edu/spectra/PhotochemCAD/html/009.html (2013).

National Renewable Energy Laboratory, “Reference solar spectral irradiance: Air mass 1.5,” http://rredc.nrel.gov/solar/spectra/am1.5/ (2013).

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Figures (11)
Fig. 1
Fig. 1 (a) Sphere, (b) shell and (c) disc configurations. Blue, green and silver represent undoped PMMA, dye doped PMMA and the nanoparticles, respectively. While the thicknesses of the waveguide and the dye layer look similar in the Figs., they are orders of magnitude different in the simulations (200nm dye layer vs. 5mm waveguide). (a) Shows the hybrid model setup. A single nanoparticle is simulated on the nanoscale using FDTD. MATLAB determines the behavior of the entire plasmonic layer. Ray tracing then simulates the whole LSC treating the plasmonic layer as an interface. (b) Shows the enhancement region for one single shell.

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Fig. 2
Fig. 2 Setup for ray tracing and various photon paths. (a) Absorbed by a nanoparticle (b) Scattered by a nanoparticle (c) Dye absorbed and then quenched (d) Dye absorbed and re-emitted at different wavelength (e) No interaction with the nanoparticles or the dye molecules.

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Fig. 3
Fig. 3 Plane wave response to the sphere (green), disc (blue) and shell (red) configurations with a radius of 50nm. (a) Absorption (solid) and scattering (dashed) cross sections. (b) Mean electric field intensity enhancement within the enhancement region. The red solid line represents the volume inside the silver shell and the red dashed line the volume outside of the shell.

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Fig. 4
Fig. 4 (a) Notation used to describe the incoming plane wave and the far field information. The plasmonic layer is along the x − y plane. The incoming plane wave vector k and the plasmonic layer normal are at an angle χ. The far field direction vector n is described by the inclination angle θ and the azimuthal angle ϕ. (b) The differential scattering cross section for the 50nm sphere with χ and λ equal to 0° and 541nm and θ ranging from 0° to 90°.

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Fig. 5
Fig. 5 Average quantum yield qav as a function of wavelength and distance from the 50nm sphere with an intrinsic quantum yield q0 of 80%.

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Fig. 6
Fig. 6 (a) Absorption and emission spectra of Rhodamine B [21]. The x-axis is relative to the peak absorption wavelength. (b) Dye absorption probabilities for the 50nm sphere (green), 50nm disc (blue) and 50nm shell (red). Solid lines represent the dye absorption probability in the enhancement region of the nanoparticles, dashed lines additionally consider the remaining dye layer (thus the entire plasmonic layer) and dotted lines the layer without nanoparticles. The dye molar concentration cD is assumed to be 5 × 10−3 M.

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Fig. 7
Fig. 7 For the 50nm sphere and a wavelength of 597nm: (a) Electric field intensity enhancement EF (log-scale). (b) Probability distribution for each position. The size of the rings represents the probability of a photon being absorbed at the respective position.

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Fig. 8
Fig. 8 (a) Normalised optical efficiencies for the sphere (green), disc (blue) and shell (red) configurations with 50nm radius as a function of nanoparticle coverage (b) Normalised optical efficiencies for the sphere (green) and the “bulk” LSC (cyan) as a function of dye concentration.

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Fig. 9
Fig. 9 Area plot for fate of incident photons for the (a) 50nm disc and (b) 50nm sphere configurations. Photons are either nanoparticle absorbed (light blue), dye absorbed (dark blue), picked up by the solar cell (yellow) or lost (dark red). (c) Trapping efficiency for 50nm sphere (green), 50nm disc (blue) and 50nm shell (red) configurations at normal incidence.

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Fig. 10
Fig. 10 (a) Absorption (solid) and scattering (dashed) cross sections of spheres with radii of 30nm (purple), 50nm (green) and 70nm (black). (b) Optical efficiency as a function of geometrical gain for 70nm sphere (black) and “bulk” LSC (cyan) (c) Optical efficiency for 1 (orange), 2 (purple), 3 (black), 4 (green), 9 (blue) and 19 (cyan) plasmonic layers.

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Fig. 11
Fig. 11 Area plot for fate of incident photons which are either nanoparticle absorbed (light blue), dye absorbed (dark blue), picked up by the solar cell (yellow) or lost (dark red) for (a) 1, (b) 3 and (c) 19 plasmonic layers.

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Equations (8)

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σ abs ( λ , χ ) = P abs ( λ , χ ) I ( λ , χ )                                     d σ sca d Ω ( θ , ϕ , λ , χ ) = P sca ( θ , ϕ , λ , χ ) I ( λ , χ )
EF ( x , λ ) : = | E total | 2 | E 0 | 2 ( x , λ ) = | E 0 + E sca | 2 | E 0 | 2 ( x , λ )                             EF ¯ ( λ ) = 1 V ∫ V EF ( x , λ ) d V
q = γ rad γ rad 0 γ rad γ rad 0 + γ abs γ rad 0 + ( 1 − q 0 ) q 0
Pr ( DYE ( θ , ϕ ) | λ ) = P rad ( θ , ϕ | λ ) P rad 0 ( λ ) P rad ( λ ) P rad 0 ( λ ) + P abs ( λ ) P rad 0 ( λ ) + ( 1 − q 0 ) q 0
q av = ( M rad p x + M rad p y + M rad p z ) 3 ( M rad p x + M rad p y + M rad p z ) 3 + ( M abs p x + M abs p y + M abs p z ) 3 + ( 1 − q 0 ) q 0
Pr ( NP abs | λ , χ ) = C NP σ abs ( λ , χ )                               Pr ( NP sca ( θ , ϕ ) | λ , χ ) = C NP d σ sca d Ω ( θ , ϕ , λ , χ )
d P dye ( λ ) = 1 2 ε 0 μ 0 n ( λ ) σ M ( λ ) c D N A | E ( x , λ ) | 2 d V
η ¯ opt = ∫ λ 1 λ 2 A ( λ ) η opt ( λ ) d λ ∫ λ 1 λ 2 A ( λ ) d λ

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