Abstract

We develop a ray optics model of a silicon wire array geometry in an attempt to understand the very strong absorption previously observed experimentally in these arrays. Our model successfully reproduces the n2 ergodic limit for wire arrays in free space. Applying this model to a wire array on a Lambertian back reflector, we find an asymptotic increase in light trapping for low filling fractions. In this case, the Lambertian back reflector is acting as a wide acceptance angle concentrator, allowing the array to exceed the ergodic limit in the ray optics regime. While this leads to increased power per volume of silicon, it gives reduced power per unit area of wire array, owing to reduced silicon volume at low filling fractions. Upon comparison with silicon microwire experimental data, our ray optics model gives reasonable agreement with large wire arrays (4 μm radius), but poor agreement with small wire arrays (1 μm radius). This suggests that the very strong absorption observed in small wire arrays, which is not observed in large wire arrays, may be significantly due to wave optical effects.

© 2011 Optical Society of America

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  1. M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
    [CrossRef] [PubMed]
  2. E. Garnett, and P. Yang, "Light trapping in silicon nanowire solar cells," Nano Lett. 10, 1082-1087 (2010).
    [CrossRef] [PubMed]
  3. L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
    [CrossRef]
  4. B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
    [CrossRef] [PubMed]
  5. E. Garnett, and P. Yang, "Silicon nanowire radial p-n junction solar cells," J. Am. Chem. Soc. 130, 9224-9225 (2008).
    [CrossRef] [PubMed]
  6. B. Kayes, H. Atwater, and N. Lewis, "Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells," J. Appl. Phys. 7, 114302 (2005).
    [CrossRef]
  7. M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
    [CrossRef]
  8. L. Hu, and G. Chen, "Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).
    [CrossRef] [PubMed]
  9. C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
    [CrossRef]
  10. K. Peng, and S. Lee, "Silicon nanowires for photovoltaic solar energy conversion," Adv. Mater. 20, 1-18 (2010).
  11. O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).
  12. J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
    [CrossRef]
  13. C. Lin, and M. Povinelli, "Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).
  14. E. Yablonovitch, "Statistical ray optics," J. Opt. Soc. Am. 72, 899-907 (1982).
    [CrossRef]
  15. M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
    [CrossRef]
  16. M. Born, and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 1999).
  17. We find our model very slightly exceeds the ergodic limit across all aspect ratios for the smallest filling fraction. This is observed across aspect ratios, with no trend with increasing aspect ratios. The maximum amount by which the ergodic limit is exceeded is approximately 1% and is likely due to small inaccuracies in the model.
  18. This should not be confused with the areal filling fraction of the wire array. In solar cells, the power can be calculated by multiplying the short circuit current, the open circuit voltage, and the fill factor, where the fill factor accounts for the fact that the current-voltage curve is not square in the power-producing region.
  19. K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
    [CrossRef]
  20. C. Bohren, and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

2010

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

E. Garnett, and P. Yang, "Light trapping in silicon nanowire solar cells," Nano Lett. 10, 1082-1087 (2010).
[CrossRef] [PubMed]

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

K. Peng, and S. Lee, "Silicon nanowires for photovoltaic solar energy conversion," Adv. Mater. 20, 1-18 (2010).

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

2009

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

2008

E. Garnett, and P. Yang, "Silicon nanowire radial p-n junction solar cells," J. Am. Chem. Soc. 130, 9224-9225 (2008).
[CrossRef] [PubMed]

2007

L. Hu, and G. Chen, "Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).
[CrossRef] [PubMed]

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

C. Lin, and M. Povinelli, "Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).

2005

B. Kayes, H. Atwater, and N. Lewis, "Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells," J. Appl. Phys. 7, 114302 (2005).
[CrossRef]

1982

Atwater, H.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

B. Kayes, H. Atwater, and N. Lewis, "Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells," J. Appl. Phys. 7, 114302 (2005).
[CrossRef]

Balch, J.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Barber, G.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Boettcher, S.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

Briggs, R.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

Brunschwig, B.

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

Burkhard, G.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Chen, G.

L. Hu, and G. Chen, "Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).
[CrossRef] [PubMed]

Codella, P.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Connor, S.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Cui, Y.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Davuluru, A.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Dickey, E.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Fallahazad, B.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Fan, S.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Fang, Y.

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Filler, M.

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

Fronheiser, J.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Garnett, E.

E. Garnett, and P. Yang, "Light trapping in silicon nanowire solar cells," Nano Lett. 10, 1082-1087 (2010).
[CrossRef] [PubMed]

E. Garnett, and P. Yang, "Silicon nanowire radial p-n junction solar cells," J. Am. Chem. Soc. 130, 9224-9225 (2008).
[CrossRef] [PubMed]

Guha, S.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Gunawan, O.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Hsu, C.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Hu, L.

L. Hu, and G. Chen, "Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).
[CrossRef] [PubMed]

Huang, J.

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Kayes, B.

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

B. Kayes, H. Atwater, and N. Lewis, "Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells," J. Appl. Phys. 7, 114302 (2005).
[CrossRef]

Kelzenberg, M.

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

Kempa, T.

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Kenrick, C.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Korevaar, B.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

LaBoeuf, S.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Lee, S.

K. Peng, and S. Lee, "Silicon nanowires for photovoltaic solar energy conversion," Adv. Mater. 20, 1-18 (2010).

Lewis, N.

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

B. Kayes, H. Atwater, and N. Lewis, "Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells," J. Appl. Phys. 7, 114302 (2005).
[CrossRef]

Lieber, C.

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Lin, C.

C. Lin, and M. Povinelli, "Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).

Maldonado, S.

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

Mallouk, T.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Mayer, T.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

McGehee, M.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Peng, K.

K. Peng, and S. Lee, "Silicon nanowires for photovoltaic solar energy conversion," Adv. Mater. 20, 1-18 (2010).

Petykiewicz, J.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

Pietrzykowski, M.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Plass, K.

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

Povinelli, M.

C. Lin, and M. Povinelli, "Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).

Putnam, M.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

Rand, J.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Redwing, J.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Ropol, U.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Shen, H.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Shih, M.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Spurgeon, J.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

Sulima, O.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Tian, B.

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Tsakalakos, L.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

Turner-Evans, D.

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

Tutuc, E.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Wang, K.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Wang, Q.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Warren, E.

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

Xu, Y.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Yablonovitch, E.

Yang, P.

E. Garnett, and P. Yang, "Light trapping in silicon nanowire solar cells," Nano Lett. 10, 1082-1087 (2010).
[CrossRef] [PubMed]

E. Garnett, and P. Yang, "Silicon nanowire radial p-n junction solar cells," J. Am. Chem. Soc. 130, 9224-9225 (2008).
[CrossRef] [PubMed]

Yoon, H.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Yu, Z.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Yuwen, Y.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

Zhang, Y.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Zheng, X.

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Zhu, J.

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

Adv. Mater.

K. Peng, and S. Lee, "Silicon nanowires for photovoltaic solar energy conversion," Adv. Mater. 20, 1-18 (2010).

K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, "Flexible polymer-embedded si wire arrays," Adv. Mater. 21, 325-328 (2009).
[CrossRef]

Appl. Phys. Lett.

C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, "Radial junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 97, 143108 (2010).
[CrossRef]

M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, "10 μm minority-carrier diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth," Appl. Phys. Lett. 95, 163116 (2009).
[CrossRef]

Energy Environ. Sci.

M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and H. Atwater, "Si microwire-array solar cells," Energy Environ. Sci. 3, 1037-1041 (2010).
[CrossRef]

J. Am. Chem. Soc.

E. Garnett, and P. Yang, "Silicon nanowire radial p-n junction solar cells," J. Am. Chem. Soc. 130, 9224-9225 (2008).
[CrossRef] [PubMed]

J. Appl. Phys.

B. Kayes, H. Atwater, and N. Lewis, "Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells," J. Appl. Phys. 7, 114302 (2005).
[CrossRef]

J. Nanophoton.

L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar, O. Sulima, J. Rand, A. Davuluru, and U. Ropol, "Strong broadband absorption in silicon nanowire arrays with a large lattice constant for photovoltaic applications," J. Nanophoton. 1, 013552 (2007).
[CrossRef]

J. Opt. Soc. Am.

Nano Lett.

E. Garnett, and P. Yang, "Light trapping in silicon nanowire solar cells," Nano Lett. 10, 1082-1087 (2010).
[CrossRef] [PubMed]

L. Hu, and G. Chen, "Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).
[CrossRef] [PubMed]

J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, "Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays," Nano Lett. 9, 279-282 (2009).
[CrossRef]

C. Lin, and M. Povinelli, "Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications," Nano Lett. 7, 3249-3252 (2007).

Nat. Mater.

M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs, N. Lewis, and H. Atwater, "Enhanced absorption and carrier collection in si wire arrays for photovoltaic applications," Nat. Mater. 9, 239-244 (2010).
[CrossRef] [PubMed]

Nature

B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, "Coaxial silicon nanowires as solar cells and nanoelectronic power sources," Nature 449, 885-889 (2007).
[CrossRef] [PubMed]

Prog. Photovolt. Res. Appl.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, "High performance wire-array silicon solar cells," Prog. Photovolt. Res. Appl. 10, 1002 (2010).

Other

M. Born, and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 1999).

We find our model very slightly exceeds the ergodic limit across all aspect ratios for the smallest filling fraction. This is observed across aspect ratios, with no trend with increasing aspect ratios. The maximum amount by which the ergodic limit is exceeded is approximately 1% and is likely due to small inaccuracies in the model.

This should not be confused with the areal filling fraction of the wire array. In solar cells, the power can be calculated by multiplying the short circuit current, the open circuit voltage, and the fill factor, where the fill factor accounts for the fact that the current-voltage curve is not square in the power-producing region.

C. Bohren, and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of the wire array for isotropic illumination. The blue wires illustrate how light escaping from the side of a wire impinges on a neighboring wire a given distance away. The orange wires illustrate how the sides of the wires are shadowed by neighboring wires for a given distance and angle of incidence. (b) A top-down view of the wire array illustrates the radial escape approximation. The arrows show the directions of light escape being considered, and the yellow areas give the in-plane angle subtended by the neighboring wires, with the distinct shades indicating neighboring wires at two distinct distances. The wires farther away will have greater loss associated than the closer wires.

Fig. 2
Fig. 2

The variation of the light trapping factor, as a multiple of n2, as a function of areal filling fraction, for various aspect ratios (height/radius). n=3.53. Because we assume a cylindrical wire geometry, the maximum attainable packing fraction is approximately 90%, which corresponds to the sides of the wires touching each other. The minimum filling fraction shown is 0.1%. Both cases approach their respective ergodic limits (denoted by gray dashed lines) for large filling fractions. The no back reflector case is also very close to the ergodic limit for very small filling fractions where the radial escape approximation is accurate. Parts a and b show the same data plotted against a linear and log scale.

Fig. 3
Fig. 3

A schematic of the Lambertian back reflector case. The green wires show the effects of scatterers placed at different heights within the array. Note that for the lower scatterer light from a much smaller range of angles is able to escape. The purple wires illustrate the light which bounces off the reflector at a given point r that escapes between the surrounding wires. Between the red wires the shadowing of the reflector for incident light at a given angle and wires at a given distance is shown.

Fig. 4
Fig. 4

The variation of power with filling fraction, for aspect ratio=50. The dotted lines use the asymptotic fits across all filling fractions, so that the goodness of fit can be evaluated. The solid lines use the model results across all filling fractions. Note that while the asymptotic increase produces increased power per volume of silicon it does not produce increased power per unit area in the array.

Fig. 5
Fig. 5

(a)The outlined surface gives the model output for wires with a 4.9% filling fraction, a 1 μm radius, and a height of 44 μm. The upper surface is the experimental result for such an array. (b) The solid lines show the experimental data for various wavelengths, and the dotted lines show the model output. Note that even for the low absorbing 1000 nm curves, the model significantly underpredicts the absorption. (c) The outlined surface gives the model output for wires with a 7.3% filling fraction, a 4 μm radius, and a height of 160 μm. The other surface is the experimental result for such an array. (d) As before, the solid lines show the experimental data and the dotted lines show the model output. Note the reasonable agreement, especially at 1000nm.

Equations (34)

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

I inc 2 A end T ¯ end + I inc A sides F ¯ = I int 2 A end T ¯ end n 2 + I int A sides L ¯ n 2
I int I inc = n 2 ( 2 A end T ¯ end + A sides F ¯ ) 2 A end T ¯ end + A sides L ¯
T ¯ end = 0 2 π 0 π / 2 T ( ϕ ) cos ( ϕ ) sin ( ϕ ) d ϕ d θ 0 2 π 0 π / 2 cos ( ϕ ) sin ( ϕ ) d ϕ d θ = 0 2 π 0 π / 2 T n cos 2 ( ϕ ) sin ( ϕ ) d ϕ d θ 0 2 π 0 π / 2 cos ( ϕ ) sin ( ϕ ) d ϕ d θ = 2 3 T n
f ( h ) = θ B θ T cos ( θ ) d θ π / 2 π / 2 cos ( θ ) d θ = sin ( θ T ) + sin ( θ B ) 2
g ( d ) = 0 l sin ( θ T ) + sin ( θ B ) d h 2 l = l 2 + d 2 d l
T int ( d ) = 0 l θ B θ T α 1 α 2 T n cos 2 ( ϕ ) d α d θ d l 0 l θ B θ T α 1 α 2 cos ( ϕ ) d α d θ d l
L 2 = T ¯ end g ( 1 T int ) ( 1 g )
L i = T ¯ end ( g ( 1 T int ) ) i 1 ( 1 g )
L ¯ = T ¯ end ( 1 g ) n = 0 ( g ( 1 T int ) ) n = 1 g 1 g ( 1 T int )
u ( d , β ) = l s l = d cot ( β ) l
T 0 ( β ) = π / 2 π / 2 T n cos 2 ( ϕ ) d α π / 2 π / 2 cos ( ϕ ) d α
L 1 ( β ) = u ( β ) ( 1 T 0 ( β ) ) ( 1 g 1 ( β ) )
L i ( β ) = ( 1 g ) u ( β ) ( 1 T 0 ( β ) ) g 1 ( β ) ( 1 T 1 ( β ) ) [ g ( 1 T int ) ] i 2
L t ( β ) = u ( β ) ( 1 T 0 ( β ) ) ( 1 g 1 ( β ) + ( 1 g ) g 1 ( β ) ( 1 T 1 ( β ) ) 1 g ( 1 T int ) )
F ( β ) = u ( β ) L t ( β )
F ¯ = 0 2 π 0 π / 2 F ( β ) sin 2 ( β ) d β d η 0 2 π 0 π / 2 sin 2 ( β ) d β d η
I inc A end T ¯ end + I inc A sides F ¯ + I inc A refl R ¯ = I int A end T ¯ end 2 n 2 + I int A sides L ¯ 2 n 2
I int I inc = 2 n 2 ( A end T ¯ end + A sides F ¯ + A refl R ¯ ) A end T ¯ end + A sides L
1 g = ( 1 g ) / 2 + ( 1 g ) / 2 * L refl
L ( r ) = θ B θ T cos ( θ ) d θ π / 2 π / 2 cos ( θ ) d θ = sin ( tan 1 ( r / l ) ) + sin ( tan 1 ( ( d r ) / l ) ) 2 = r r 2 + l 2 + d r ( d r ) 2 + l 2 2
I ( r ) = 0 l cos ( η 1 ) d h + 0 l cos ( η 2 ) d h = 0 l r r 2 + h 2 d h + 0 l d r ( d r ) 2 + h 2 d h
L refl ( d ) = 0 d I ( r ) [ r r 2 + l 2 + d r ( d r ) 2 + l 2 ] d r 2 0 d I ( r ) d r
T refl ( d ) = 0 d I ( r ) ( θ T π / 2 π / 2 π / 2 T n cos ( θ ) cos 2 ( ϕ ) d α d θ + θ B π / 2 π / 2 π / 2 T n cos ( θ ) cos 2 ( ϕ ) d α d θ ) d r 0 d I ( r ) ( θ T π / 2 π / 2 π / 2 cos ( ϕ ) cos ( θ ) d α d θ + θ B π / 2 π / 2 π / 2 cos ( ϕ ) cos ( θ ) d α d θ ) d r
T int ( d ) = g ( d ) T int ( d ) + ( g ( d ) g ( d ) ) T refl ( d ) g
R = ( 1 g 1 ( d ) ) ( 1 g ( d ) ) / 2
1 g 1 ( d ) = R * L refl + ( 1 g ( d ) ) / 2
T 1 ( d ) = T 1 ( d ) g 1 ( d ) + T refl ( d ) ( g 1 ( d ) g 1 ( d ) ) g 1 ( d )
u ( d , β ) = d l tan ( β ) d
u = 0 π / 2 u ( β ) sin ( β ) cos ( β ) d β 0 π / 2 sin ( β ) cos ( β ) d β
L tot = L inc + ( 1 L inc ) ( 1 T inc ) 1 g 1 g ( 1 T int )
R ¯ = ( 1 L tot ) u
A sides I max sin ( θ ) F ( θ ) + 2 A end I max cos ( θ ) T ( θ ) = 2 α V I int + 2 A end T ¯ end I int n 2 + A sides L ¯ I int n 2
I int = A sides I max sin ( θ ) F ( θ ) + 2 A end I max cos ( θ ) T ( θ ) 2 α V + 2 A end T ¯ end n 2 + A sides L ¯ n 2
A = 2 α V I int A tot I max cos ( θ ) = 2 α V ( A sides F ( θ ) tan ( θ ) + 2 A end T ( θ ) ) A tot ( 2 α V + 2 A end T ¯ end n 2 + A sides L ¯ n 2 )

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