Abstract

The energy conversion efficiency of a conventional pn junction solar cell decreases as the temperature increases, and this may eventually lead to failures in the photovoltaic system, especially if it uses concentrated solar radiation. In this work, we show that spectrally selective reflector (SSR) surfaces can be important for reducing the heat buildup on passively cooled solar cells. We outline a computational scheme for optimizing DC magnetron-sputtered TiO2:Nb-based SSRs tailored for silicon solar cells and find good agreement of the reflectance with an experimental realization of the optimal SSR. A figure of merit for SSRs has also been derived and applied to the experimental data.

© 2011 Optical Society of America

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  1. J. J. Wysocki and P. Rappaport, “Effect of temperature on photovoltaic solar energy conversion,” J. Appl. Phys. 31, 571–578 (1960).
    [CrossRef]
  2. A. Royne, C. J. Dey, and D. R. Mills, “Cooling of photovoltaic cells under concentrated illumination: a critical review,” Solar Energy Mater. Solar Cells 86, 451–483 (2005).
    [CrossRef]
  3. C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
    [CrossRef]
  4. M. Mwamburi, E. Wäckelgård, and A. Roos, “Preparation and characterisation of solar selective SnOx:F coated aluminium reflector surfaces,” Thin Solid Films 374, 1–9 (2000).
    [CrossRef]
  5. M. Mwamburi and E. Wäckelgård, “Doped tin oxide coated aluminium solar selective reflector surfaces,” Solar Energy 68, 371–378 (2000).
    [CrossRef]
  6. M. Mwamburi, E. Wäckelgård, A. Roos, and R. Kivaisi, “Polarization-dependent angular-optical reflectance in solar-selective SnOx:F/Al2O3/Al reflector surfaces,” Appl. Opt. 41, 2428–2434 (2002).
    [CrossRef] [PubMed]
  7. M. Mwamburi, A. Hoel, and E. Wäckelgård, “Surface morphologies of spectrally selective and polarization-dependent angular optical reflectors of SnOx:F-coated anodized aluminum,” Solar Energy Mater. Solar Cells 84, 381–394 (2004).
    [CrossRef]
  8. C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
    [CrossRef]
  9. C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
    [CrossRef]
  10. M. W. Edenburn, “Active and passive cooling for concentrating photovoltaic arrays,” in Proceedings of the 14th IEEE Photovoltaic Specialists Conference (IEEE, 1980), pp. 771–776.
  11. A. Cheknane, B. Benyoucef, and A. Chaker, “Performance of concentrator solar cells with passive cooling,” Semicond. Sci. Technol. 21, 144–147 (2006).
    [CrossRef]
  12. S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
    [CrossRef]
  13. Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
    [CrossRef]
  14. C. M. Maghanga, G. A. Niklasson, and C. G. Granqvist, “Optical properties of sputter deposited transparent and conducting TiO2:Nb films,” Thin Solid Films 518, 1254–1258 (2009).
    [CrossRef]
  15. M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
    [CrossRef]
  16. N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
    [CrossRef]
  17. T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
    [CrossRef]
  18. Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
    [CrossRef]
  19. http://rredc.nrel.gov/solar/spectra/am1.5.
  20. D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
    [CrossRef]
  21. M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
    [CrossRef]
  22. M. Brogren, P. Nostell, and B. Karlsson, “Optical efficiency of a PV-thermal hybrid CPC module for high latitudes,” Solar Energy 69, 173–185 (2001).
    [CrossRef]
  23. V. Ondok and J. Musil, “Effect of hydrogen on reactive sputtering of transparent oxide films,” Plasma Processes Polym. 4, S319–S324 (2007).
    [CrossRef]
  24. J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
    [CrossRef]
  25. Z. Knittel, Optics of Thin Films (Wiley, 1976), pp. 40–46.
  26. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 332–340.
  27. D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 389–406.
  28. T. S. Eriksson, A. Hjortsberg, G. A. Niklasson, and C. G. Granqvist, “Infrared optical properties of evaporated alumina films,” Appl. Opt. 20, 2742–2746 (1981).
    [CrossRef] [PubMed]

2010 (4)

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
[CrossRef]

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

2009 (4)

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, and C. G. Granqvist, “Optical properties of sputter deposited transparent and conducting TiO2:Nb films,” Thin Solid Films 518, 1254–1258 (2009).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
[CrossRef]

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

2008 (1)

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

2007 (1)

V. Ondok and J. Musil, “Effect of hydrogen on reactive sputtering of transparent oxide films,” Plasma Processes Polym. 4, S319–S324 (2007).
[CrossRef]

2006 (2)

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

A. Cheknane, B. Benyoucef, and A. Chaker, “Performance of concentrator solar cells with passive cooling,” Semicond. Sci. Technol. 21, 144–147 (2006).
[CrossRef]

2005 (4)

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

A. Royne, C. J. Dey, and D. R. Mills, “Cooling of photovoltaic cells under concentrated illumination: a critical review,” Solar Energy Mater. Solar Cells 86, 451–483 (2005).
[CrossRef]

2004 (1)

M. Mwamburi, A. Hoel, and E. Wäckelgård, “Surface morphologies of spectrally selective and polarization-dependent angular optical reflectors of SnOx:F-coated anodized aluminum,” Solar Energy Mater. Solar Cells 84, 381–394 (2004).
[CrossRef]

2002 (1)

2001 (1)

M. Brogren, P. Nostell, and B. Karlsson, “Optical efficiency of a PV-thermal hybrid CPC module for high latitudes,” Solar Energy 69, 173–185 (2001).
[CrossRef]

2000 (2)

M. Mwamburi, E. Wäckelgård, and A. Roos, “Preparation and characterisation of solar selective SnOx:F coated aluminium reflector surfaces,” Thin Solid Films 374, 1–9 (2000).
[CrossRef]

M. Mwamburi and E. Wäckelgård, “Doped tin oxide coated aluminium solar selective reflector surfaces,” Solar Energy 68, 371–378 (2000).
[CrossRef]

1981 (1)

1960 (1)

J. J. Wysocki and P. Rappaport, “Effect of temperature on photovoltaic solar energy conversion,” J. Appl. Phys. 31, 571–578 (1960).
[CrossRef]

Arenkiel, S. P.

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 332–340.

Baroch, P.

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 332–340.

Benyoucef, B.

A. Cheknane, B. Benyoucef, and A. Chaker, “Performance of concentrator solar cells with passive cooling,” Semicond. Sci. Technol. 21, 144–147 (2006).
[CrossRef]

Brogren, M.

M. Brogren, P. Nostell, and B. Karlsson, “Optical efficiency of a PV-thermal hybrid CPC module for high latitudes,” Solar Energy 69, 173–185 (2001).
[CrossRef]

Chaker, A.

A. Cheknane, B. Benyoucef, and A. Chaker, “Performance of concentrator solar cells with passive cooling,” Semicond. Sci. Technol. 21, 144–147 (2006).
[CrossRef]

Cheknane, A.

A. Cheknane, B. Benyoucef, and A. Chaker, “Performance of concentrator solar cells with passive cooling,” Semicond. Sci. Technol. 21, 144–147 (2006).
[CrossRef]

Chen, Z.

S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
[CrossRef]

Cristofari, C.

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

Dabney, M. S.

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

Dey, C. J.

A. Royne, C. J. Dey, and D. R. Mills, “Cooling of photovoltaic cells under concentrated illumination: a critical review,” Solar Energy Mater. Solar Cells 86, 451–483 (2005).
[CrossRef]

Edenburn, M. W.

M. W. Edenburn, “Active and passive cooling for concentrating photovoltaic arrays,” in Proceedings of the 14th IEEE Photovoltaic Specialists Conference (IEEE, 1980), pp. 771–776.

Eriksson, T. S.

Furubayashi, Y.

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Ginley, D. S.

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

González-Hernández, J.

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

Gorley, P. N.

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

Granqvist, C. G.

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, and C. G. Granqvist, “Optical properties of sputter deposited transparent and conducting TiO2:Nb films,” Thin Solid Films 518, 1254–1258 (2009).
[CrossRef]

T. S. Eriksson, A. Hjortsberg, G. A. Niklasson, and C. G. Granqvist, “Infrared optical properties of evaporated alumina films,” Appl. Opt. 20, 2742–2746 (1981).
[CrossRef] [PubMed]

Han, J. G.

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

Hasegawa, T.

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Hirose, Y.

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Hitosugi, T.

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Hjortsberg, A.

Hoang, N. L. H.

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Hoel, A.

M. Mwamburi, A. Hoel, and E. Wäckelgård, “Surface morphologies of spectrally selective and polarization-dependent angular optical reflectors of SnOx:F-coated anodized aluminum,” Solar Energy Mater. Solar Cells 84, 381–394 (2004).
[CrossRef]

Horley, P. P.

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

Hu, P.

S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
[CrossRef]

Inaba, K.

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Inokuti, M.

D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 389–406.

Jensen, J.

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

Jiang, S.

S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
[CrossRef]

Kamiyama, T.

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Karlsson, B.

M. Brogren, P. Nostell, and B. Karlsson, “Optical efficiency of a PV-thermal hybrid CPC module for high latitudes,” Solar Energy 69, 173–185 (2001).
[CrossRef]

Kasai, J.

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Kinoda, G.

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Kivaisi, R.

Knittel, Z.

Z. Knittel, Optics of Thin Films (Wiley, 1976), pp. 40–46.

Maghanga, C. M.

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, and C. G. Granqvist, “Optical properties of sputter deposited transparent and conducting TiO2:Nb films,” Thin Solid Films 518, 1254–1258 (2009).
[CrossRef]

Mattei, M.

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

Meneses-Rodríguez, D.

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

Mills, D. R.

A. Royne, C. J. Dey, and D. R. Mills, “Cooling of photovoltaic cells under concentrated illumination: a critical review,” Solar Energy Mater. Solar Cells 86, 451–483 (2005).
[CrossRef]

Min, C.

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

Mo, S.

S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
[CrossRef]

Muselli, M.

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

Musil, J.

V. Ondok and J. Musil, “Effect of hydrogen on reactive sputtering of transparent oxide films,” Plasma Processes Polym. 4, S319–S324 (2007).
[CrossRef]

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

Mwamburi, M.

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
[CrossRef]

M. Mwamburi, A. Hoel, and E. Wäckelgård, “Surface morphologies of spectrally selective and polarization-dependent angular optical reflectors of SnOx:F-coated anodized aluminum,” Solar Energy Mater. Solar Cells 84, 381–394 (2004).
[CrossRef]

M. Mwamburi, E. Wäckelgård, A. Roos, and R. Kivaisi, “Polarization-dependent angular-optical reflectance in solar-selective SnOx:F/Al2O3/Al reflector surfaces,” Appl. Opt. 41, 2428–2434 (2002).
[CrossRef] [PubMed]

M. Mwamburi, E. Wäckelgård, and A. Roos, “Preparation and characterisation of solar selective SnOx:F coated aluminium reflector surfaces,” Thin Solid Films 374, 1–9 (2000).
[CrossRef]

M. Mwamburi and E. Wäckelgård, “Doped tin oxide coated aluminium solar selective reflector surfaces,” Solar Energy 68, 371–378 (2000).
[CrossRef]

Nakao, S.

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Nam, K. H.

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

Niklasson, G. A.

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
[CrossRef]

C. M. Maghanga, G. A. Niklasson, and C. G. Granqvist, “Optical properties of sputter deposited transparent and conducting TiO2:Nb films,” Thin Solid Films 518, 1254–1258 (2009).
[CrossRef]

T. S. Eriksson, A. Hjortsberg, G. A. Niklasson, and C. G. Granqvist, “Infrared optical properties of evaporated alumina films,” Appl. Opt. 20, 2742–2746 (1981).
[CrossRef] [PubMed]

Nostell, P.

M. Brogren, P. Nostell, and B. Karlsson, “Optical efficiency of a PV-thermal hybrid CPC module for high latitudes,” Solar Energy 69, 173–185 (2001).
[CrossRef]

Notton, G.

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

Nuofu, C.

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

Oka, N.

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Ondok, V.

V. Ondok and J. Musil, “Effect of hydrogen on reactive sputtering of transparent oxide films,” Plasma Processes Polym. 4, S319–S324 (2007).
[CrossRef]

Perkins, J. D.

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

Poggi, P.

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

Rappaport, P.

J. J. Wysocki and P. Rappaport, “Effect of temperature on photovoltaic solar energy conversion,” J. Appl. Phys. 31, 571–578 (1960).
[CrossRef]

Roos, A.

M. Mwamburi, E. Wäckelgård, A. Roos, and R. Kivaisi, “Polarization-dependent angular-optical reflectance in solar-selective SnOx:F/Al2O3/Al reflector surfaces,” Appl. Opt. 41, 2428–2434 (2002).
[CrossRef] [PubMed]

M. Mwamburi, E. Wäckelgård, and A. Roos, “Preparation and characterisation of solar selective SnOx:F coated aluminium reflector surfaces,” Thin Solid Films 374, 1–9 (2000).
[CrossRef]

Royne, A.

A. Royne, C. J. Dey, and D. R. Mills, “Cooling of photovoltaic cells under concentrated illumination: a critical review,” Solar Energy Mater. Solar Cells 86, 451–483 (2005).
[CrossRef]

Sanno, Y.

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Sato, Y.

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Shigesato, Y.

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Shiles, E.

D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 389–406.

Shimada, T.

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Smith, D. Y.

D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 389–406.

Tasaki, C.

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Teplin, C. W.

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

van Hest, M. F. A. M.

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

Vlcek, J.

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

Vorobiev, Y. V.

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

Wäckelgård, E.

M. Mwamburi, A. Hoel, and E. Wäckelgård, “Surface morphologies of spectrally selective and polarization-dependent angular optical reflectors of SnOx:F-coated anodized aluminum,” Solar Energy Mater. Solar Cells 84, 381–394 (2004).
[CrossRef]

M. Mwamburi, E. Wäckelgård, A. Roos, and R. Kivaisi, “Polarization-dependent angular-optical reflectance in solar-selective SnOx:F/Al2O3/Al reflector surfaces,” Appl. Opt. 41, 2428–2434 (2002).
[CrossRef] [PubMed]

M. Mwamburi, E. Wäckelgård, and A. Roos, “Preparation and characterisation of solar selective SnOx:F coated aluminium reflector surfaces,” Thin Solid Films 374, 1–9 (2000).
[CrossRef]

M. Mwamburi and E. Wäckelgård, “Doped tin oxide coated aluminium solar selective reflector surfaces,” Solar Energy 68, 371–378 (2000).
[CrossRef]

Wysocki, J. J.

J. J. Wysocki and P. Rappaport, “Effect of temperature on photovoltaic solar energy conversion,” J. Appl. Phys. 31, 571–578 (1960).
[CrossRef]

Xiaoli, Y.

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

Xingwang, Z.

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

Yamada, N.

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

Yamamoto, Y.

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

Yiming, B.

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

Yu, W.

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Furubayashi, T. Hitosugi, Y. Yamamoto, K. Inaba, G. Kinoda, Y. Hirose, T. Shimada, and T. Hasegawa, “A transparent metal: Nb-doped anatase TiO2,” Appl. Phys. Lett. 86, 252101 (2005).
[CrossRef]

J. Appl. Phys. (2)

N. Yamada, T. Hitosugi, J. Kasai, N. L. H. Hoang, S. Nakao, Y. Hirose, T. Shimada, and T. Hasegawa, “Direct growth of transparent conducting Nb-doped anatase TiO2 polycrystalline films on glass,” J. Appl. Phys. 105, 123702 (2009).
[CrossRef]

J. J. Wysocki and P. Rappaport, “Effect of temperature on photovoltaic solar energy conversion,” J. Appl. Phys. 31, 571–578 (1960).
[CrossRef]

J. Semiconductors (1)

C. Min, C. Nuofu, Y. Xiaoli, W. Yu, B. Yiming, and Z. Xingwang, “Thermal analysis and test for single concentrator solar cells,” J. Semiconductors 30, 044011 (2009).
[CrossRef]

J. Vac. Sci. Technol. A (1)

Y. Sato, Y. Sanno, C. Tasaki, N. Oka, T. Kamiyama, and Y. Shigesato, “Electrical and optical properties of Nb-doped TiO2 films deposited by dc magnetron sputtering using slightly reduced Nb-doped TiO2−x ceramic targets,” J. Vac. Sci. Technol. A 28, 851–855 (2010).
[CrossRef]

Phys. Status Solidi A (1)

T. Hitosugi, N. Yamada, S. Nakao, Y. Hirose, and T. Hasegawa, “Properties of TiO2-based transparent conducting oxides,” Phys. Status Solidi A 207, 1529–1537 (2010).
[CrossRef]

Plasma Processes Polym. (1)

V. Ondok and J. Musil, “Effect of hydrogen on reactive sputtering of transparent oxide films,” Plasma Processes Polym. 4, S319–S324 (2007).
[CrossRef]

Proc. SPIE (1)

C. M. Maghanga, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Optical modeling of spectrally selective reflectors based on TiO2:Nb transparent conducting oxide films for silicon solar cell applications,” Proc. SPIE 7407F, 74070F (2009).
[CrossRef]

Renew. Energy (1)

M. Mattei, G. Notton, C. Cristofari, M. Muselli, and P. Poggi, “Calculation of the polycrystalline PV module temperature using a simple method of energy balance,” Renew. Energy 31, 553–567 (2006).
[CrossRef]

Semicond. Sci. Technol. (1)

A. Cheknane, B. Benyoucef, and A. Chaker, “Performance of concentrator solar cells with passive cooling,” Semicond. Sci. Technol. 21, 144–147 (2006).
[CrossRef]

Solar Energy (3)

D. Meneses-Rodríguez, P. P. Horley, J. González-Hernández, Y. V. Vorobiev, and P. N. Gorley, “Photovoltaic solar cells performance at elevated temperatures,” Solar Energy 78, 243–250 (2005).
[CrossRef]

M. Brogren, P. Nostell, and B. Karlsson, “Optical efficiency of a PV-thermal hybrid CPC module for high latitudes,” Solar Energy 69, 173–185 (2001).
[CrossRef]

M. Mwamburi and E. Wäckelgård, “Doped tin oxide coated aluminium solar selective reflector surfaces,” Solar Energy 68, 371–378 (2000).
[CrossRef]

Solar Energy Mater. Solar Cells (4)

A. Royne, C. J. Dey, and D. R. Mills, “Cooling of photovoltaic cells under concentrated illumination: a critical review,” Solar Energy Mater. Solar Cells 86, 451–483 (2005).
[CrossRef]

M. Mwamburi, A. Hoel, and E. Wäckelgård, “Surface morphologies of spectrally selective and polarization-dependent angular optical reflectors of SnOx:F-coated anodized aluminum,” Solar Energy Mater. Solar Cells 84, 381–394 (2004).
[CrossRef]

S. Jiang, P. Hu, S. Mo, and Z. Chen, “Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology,” Solar Energy Mater. Solar Cells 94, 1686–1696 (2010).
[CrossRef]

C. M. Maghanga, J. Jensen, G. A. Niklasson, C. G. Granqvist, and M. Mwamburi, “Transparent and conducting TiO2:Nb films made by sputter deposition: application to spectrally selective solar reflectors,” Solar Energy Mater. Solar Cells 94, 75–79 (2010).
[CrossRef]

Thin Solid Films (4)

C. M. Maghanga, G. A. Niklasson, and C. G. Granqvist, “Optical properties of sputter deposited transparent and conducting TiO2:Nb films,” Thin Solid Films 518, 1254–1258 (2009).
[CrossRef]

M. S. Dabney, M. F. A. M. van Hest, C. W. Teplin, S. P. Arenkiel, J. D. Perkins, and D. S. Ginley, “Pulsed laser deposited Nb doped TiO2 as a transparent conducting oxide,” Thin Solid Films 516, 4133–4138 (2008).
[CrossRef]

J. Musil, P. Baroch, J. Vlček, K. H. Nam, and J. G. Han, “Reactive magnetron sputtering of thin films: present status and trends,” Thin Solid Films 475, 208–218 (2005).
[CrossRef]

M. Mwamburi, E. Wäckelgård, and A. Roos, “Preparation and characterisation of solar selective SnOx:F coated aluminium reflector surfaces,” Thin Solid Films 374, 1–9 (2000).
[CrossRef]

Other (5)

Z. Knittel, Optics of Thin Films (Wiley, 1976), pp. 40–46.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 332–340.

D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), pp. 389–406.

http://rredc.nrel.gov/solar/spectra/am1.5.

M. W. Edenburn, “Active and passive cooling for concentrating photovoltaic arrays,” in Proceedings of the 14th IEEE Photovoltaic Specialists Conference (IEEE, 1980), pp. 771–776.

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Figures (7)

Fig. 1
Fig. 1

Ideal spectral reflectance of an SSR appropriate for use with a silicon solar cell. A normalized AM 1.5 solar spectrum is included (shaded).

Fig. 2
Fig. 2

Solar cell temperature versus concentration calculated for constructions with and without SSR for (a)  A r = A c = 4 A 0 , (b)  A r = 4 A 0 and A c = 10 A 0 .

Fig. 3
Fig. 3

Integration of an SSR surface and a PV cell in a compound parabolic concentrator.

Fig. 4
Fig. 4

Optical constants, n and k, of an Nb: TiO 2 film containing 3.7 at . % Nb, from [9].

Fig. 5
Fig. 5

Calculated influence of the thickness of (a) the Al 2 O 3 layer and (b) the TiO 2 :Nb film on the normal spectral reflectance of Ti O 2 : Nb / Al 2 O 3 / Al SSRs. In (a) the thickness of the TiO 2 :Nb film is 90 nm , while in (b) the Al 2 O 3 thickness is 90 nm .

Fig. 6
Fig. 6

Contour plots showing the variation of (a)  R cell and (b)  R therm as a function of the thicknesses of TiO 2 :Nb and Al 2 O 3 in a Ti O 2 : Nb / Al 2 O 3 / Al stack.

Fig. 7
Fig. 7

Calculated normal and measured near-normal spectral reflectance for (a) a TiO 2 :Nb film on Al and for (b), (c), (d) three Ti O 2 : Nb / Al 2 O 3 / Al structures. The TiO 2 films contained 3.7 at . % Nb. Layer thicknesses are given in the insets.

Tables (2)

Tables Icon

Table 1 Parameters Used in Calculations of Solar Cell Temperature

Tables Icon

Table 2 FOM Based on the Experimental Data for SSR Coatings Shown in Figs. 7a, 7b, 7c, 7d

Equations (10)

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

R cell = 300 1100 G ( λ ) R ( λ ) d λ 300 1100 G ( λ ) d λ ,
R therm = 1100 2500 G ( λ ) R ( λ ) d λ 1100 2500 G ( λ ) d λ ,
η = η ref [ 1 β ( τ τ 0 ) ] ,
R α A 0 C q 0 η R α A 0 C q 0 A r ε σ B ( τ 4 τ 0 4 ) A c h ( τ τ 0 ) = 0 ,
R cell α A 0 C q cell ( 1 η ) + R therm α A 0 C q therm ( 1 η ) A r ε σ B ( τ 4 τ 0 4 ) A c h ( τ τ 0 ) = 0 ,
R cell α A 0 C q cell ( 1 η ) + A r ε σ B ( τ 4 τ 0 4 ) A c h ( τ τ 0 ) = 0 .
FOM = R cell q cell + R therm q therm R cell q 0 0.8 + 0.2 R therm R cell .
[ E + ( 0 ) E ( 0 ) ] = { [ m = 1 N I m 1 , m L m ] I N , N + 1 } [ E + ( z + ) 0 ] = { S 11 S 12 S 21 S 22 } [ E + ( z + ) 0 ] .
δ m = ( 2 π / λ ) d N m cos θ m ,
r ( θ , λ ) = E ( 0 ) / E + ( 0 ) = S 21 / S 11 .

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