Abstract

A III-V multi-junction tandem solar cell is the most efficient photovoltaic structure that offers an extremely high power conversion efficiency. Current mismatching between each subcell of the device, however, is a significant challenge that causes the experimental value of the power conversion efficiency to deviate from the theoretical value. In this work, we explore a promising strategy using CdSe quantum dots (QDs) to enhance the photocurrent of the limited subcell to match with those of the other subcells and to enhance the power conversion efficiency of InGaP/GaAs/Ge tandem solar cells. The underlying mechanism of the enhancement can be attributed to the QD’s unique capacity for photon conversion that tailors the incident spectrum of solar light; the enhanced efficiency of the device is therefore strongly dependent on the QD’s dimensions. As a result, by appropriately selecting and spreading 7 mg/mL of CdSe QDs with diameters of 4.2 nm upon the InGaP/GaAs/Ge solar cell, the power conversion efficiency shows an enhancement of 10.39% compared to the cell’s counterpart without integrating CdSe QDs.

© 2013 OSA

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2013

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

Y.-J. Lee, Y.-C. Yao, C.-H. Yang, “Direct electrical contact of slanted ITO film on axial p-n junction silicon nanowire solar cells,” Opt. Express 21(S1Suppl 1), A7–A14 (2013).
[CrossRef] [PubMed]

2012

2011

F. Hetsch, X. Xu, H. Wang, S. V. Kershaw, A. L. Rohach, “Semiconductor nanocrystal quantum dots as solar cell components and photosensitizers: material, charge transfer, and separation aspects of some device toplogies,” J. Phys. Chem. Lett. 2(15), 1879–1887 (2011).
[CrossRef]

Y.-J. Lee, M.-H. Lee, C.-M. Cheng, C.-H. Yang, “Enhanced conversion efficiency of InGaN multiple quantum well solar cells grown on patterned sapphire substrates,” Appl. Phys. Lett. 98(26), 263504 (2011).
[CrossRef]

2010

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Y.-J. Lee, C.-J. Lee, C.-M. Cheng, “Enhancing the conversion efficiency of red emission by spin-coating CdSe quantum dots on the green nanorod light-emitting diode,” Opt. Express 18(S4), A554–A561 (2010).
[CrossRef] [PubMed]

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

2009

D. Souri, K. Shomalian, “Band gap determination by absorption spectrum fitting method (ASF) and structural properties of different compositions of (60−x) V2O5–40TeO2–xSb2O3 glasses,” J. Non-Cryst. Solids 355(31–33), 1597–1601 (2009).
[CrossRef]

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

K. Tanabe, “A review of ultrahigh efficiency III-V semiconductor compound solar cells: multijunction tandem, lower dimensional, photonic up/down conversion and plasmonic nanometallic structures,” Energies 2(3), 504–530 (2009).
[CrossRef]

2008

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

2007

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express 15(25), 16986–17000 (2007).
[CrossRef] [PubMed]

2006

M. C. Wei, S. J. Chang, C. Y. Tsia, C. H. Liu, S. C. Chen, “SiNx deposited by in-line PECVD for multi-crystalline silicon solar cells,” Sol Energ Mat Sol C. 80(2), 215–219 (2006).

A. Franceschetti, J. M. An, A. Zunger, “Impact ionization can explain carrier multiplication in PbSe quantum dots,” Nano Lett. 6(10), 2191–2195 (2006).
[CrossRef] [PubMed]

2004

R. D. Schaller, V. I. Klimov, “High efficiency carrier multiplication in PbSe nanocrystals: implications for solar energy conversion,” Phys. Rev. Lett. 92(18), 186601 (2004).
[CrossRef] [PubMed]

2002

A. J. Nozik, “Quantum dot solar cells,” Physica E 14(1–2), 115–120 (2002).
[CrossRef]

2000

F. Dimroth, U. Schubert, A. W. Bett, “25.5% efficient Ga0.35In0.65P/Ga0.83In0.17 as tandem solar cells grown on GaAs substrates,” IEEE Electron Dev. 21(5), 209–211 (2000).
[CrossRef]

1998

M. Wolf, R. Brendel, J. H. Werner, H. J. Queisser, “Solar cell efficiency and carrier multiplication in Si1-xGex alloys,” J. Appl. Phys. 83(8), 4213–4221 (1998).
[CrossRef]

1994

H. Kato, S. Adachi, H. Nakanishi, K. Ohtsuka, “Optical properties of (AlxGa1-x)0.5In0.5P quaternary alloys,” Jpn. J. Appl. Phys. 33(1A), 186–192 (1994).

1961

W. Shockley, H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510 (1961).
[CrossRef]

Åberg, I.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Adachi, S.

H. Kato, S. Adachi, H. Nakanishi, K. Ohtsuka, “Optical properties of (AlxGa1-x)0.5In0.5P quaternary alloys,” Jpn. J. Appl. Phys. 33(1A), 186–192 (1994).

An, J. M.

A. Franceschetti, J. M. An, A. Zunger, “Impact ionization can explain carrier multiplication in PbSe quantum dots,” Nano Lett. 6(10), 2191–2195 (2006).
[CrossRef] [PubMed]

Anttu, N.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Asoli, D.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Atwater, H. A.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Bermel, P.

Bett, A. W.

F. Dimroth, U. Schubert, A. W. Bett, “25.5% efficient Ga0.35In0.65P/Ga0.83In0.17 as tandem solar cells grown on GaAs substrates,” IEEE Electron Dev. 21(5), 209–211 (2000).
[CrossRef]

Bhuiyan, A. G.

A. G. Bhuiyan, K. Sugita, A. Hashimoto, A. Yamamoto, “InGaN Solar Cells: Present State of the Art and Important Challenges,” Photovoltaics, IEEE Journal of 2(3), 276–293 (2012).
[CrossRef]

Boettcher, S. W.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Borgström, M. T.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Brendel, R.

M. Wolf, R. Brendel, J. H. Werner, H. J. Queisser, “Solar cell efficiency and carrier multiplication in Si1-xGex alloys,” J. Appl. Phys. 83(8), 4213–4221 (1998).
[CrossRef]

Chang, K. H.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Chang, S. J.

M. C. Wei, S. J. Chang, C. Y. Tsia, C. H. Liu, S. C. Chen, “SiNx deposited by in-line PECVD for multi-crystalline silicon solar cells,” Sol Energ Mat Sol C. 80(2), 215–219 (2006).

Chen, C.-C.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Chen, S. C.

M. C. Wei, S. J. Chang, C. Y. Tsia, C. H. Liu, S. C. Chen, “SiNx deposited by in-line PECVD for multi-crystalline silicon solar cells,” Sol Energ Mat Sol C. 80(2), 215–219 (2006).

Chen, Y.-C.

Chen, Y.-F.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Chen, Y.-T.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Cheng, C.-M.

Cho, J.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Deppert, K.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Dimroth, F.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

F. Dimroth, U. Schubert, A. W. Bett, “25.5% efficient Ga0.35In0.65P/Ga0.83In0.17 as tandem solar cells grown on GaAs substrates,” IEEE Electron Dev. 21(5), 209–211 (2000).
[CrossRef]

Duda, A.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Edmondson, K. M.

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Fetzer, C. M.

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Franceschetti, A.

A. Franceschetti, J. M. An, A. Zunger, “Impact ionization can explain carrier multiplication in PbSe quantum dots,” Nano Lett. 6(10), 2191–2195 (2006).
[CrossRef] [PubMed]

Friedman, D.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Fuss-Kailuweit, P.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Geisz, J.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Hashimoto, A.

A. G. Bhuiyan, K. Sugita, A. Hashimoto, A. Yamamoto, “InGaN Solar Cells: Present State of the Art and Important Challenges,” Photovoltaics, IEEE Journal of 2(3), 276–293 (2012).
[CrossRef]

Hetsch, F.

F. Hetsch, X. Xu, H. Wang, S. V. Kershaw, A. L. Rohach, “Semiconductor nanocrystal quantum dots as solar cell components and photosensitizers: material, charge transfer, and separation aspects of some device toplogies,” J. Phys. Chem. Lett. 2(15), 1879–1887 (2011).
[CrossRef]

Hong, W. D.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

Hsu, H.-C.

Huang, C.-Y.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Huffman, M.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Jiang, Y.-T.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Joannopoulos, J. D.

Jones, K.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Karam, N. H.

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Kato, H.

H. Kato, S. Adachi, H. Nakanishi, K. Ohtsuka, “Optical properties of (AlxGa1-x)0.5In0.5P quaternary alloys,” Jpn. J. Appl. Phys. 33(1A), 186–192 (1994).

Kelzenberg, M. D.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Kershaw, S. V.

F. Hetsch, X. Xu, H. Wang, S. V. Kershaw, A. L. Rohach, “Semiconductor nanocrystal quantum dots as solar cell components and photosensitizers: material, charge transfer, and separation aspects of some device toplogies,” J. Phys. Chem. Lett. 2(15), 1879–1887 (2011).
[CrossRef]

Kiehl, J.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Kim, J. K.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Kimerling, L. C.

King, R. R.

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Kinsey, G. S.

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Klimov, V. I.

R. D. Schaller, V. I. Klimov, “High efficiency carrier multiplication in PbSe nanocrystals: implications for solar energy conversion,” Phys. Rev. Lett. 92(18), 186601 (2004).
[CrossRef] [PubMed]

Lai, W.-C.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Law, D. C.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Lee, C.-J.

Lee, M. L.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Lee, M.-H.

Y.-J. Lee, M.-H. Lee, C.-M. Cheng, C.-H. Yang, “Enhanced conversion efficiency of InGaN multiple quantum well solar cells grown on patterned sapphire substrates,” Appl. Phys. Lett. 98(26), 263504 (2011).
[CrossRef]

Lee, Y.-J.

Leite, M. S.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

Lewis, N. S.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Liu, C. H.

M. C. Wei, S. J. Chang, C. Y. Tsia, C. H. Liu, S. C. Chen, “SiNx deposited by in-line PECVD for multi-crystalline silicon solar cells,” Sol Energ Mat Sol C. 80(2), 215–219 (2006).

Luo, C.

Magnusson, M. H.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Maiolo, J. R.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Mesropian, S.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

Moriarty, T.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Munday, J. N.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

Nakanishi, H.

H. Kato, S. Adachi, H. Nakanishi, K. Ohtsuka, “Optical properties of (AlxGa1-x)0.5In0.5P quaternary alloys,” Jpn. J. Appl. Phys. 33(1A), 186–192 (1994).

Norman, A.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Nozik, A. J.

A. J. Nozik, “Quantum dot solar cells,” Physica E 14(1–2), 115–120 (2002).
[CrossRef]

Ohtsuka, K.

H. Kato, S. Adachi, H. Nakanishi, K. Ohtsuka, “Optical properties of (AlxGa1-x)0.5In0.5P quaternary alloys,” Jpn. J. Appl. Phys. 33(1A), 186–192 (1994).

Olavarria, W.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Peng, L.-C.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Poxson, D. J.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Putnam, M. C.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Queisser, H. J.

M. Wolf, R. Brendel, J. H. Werner, H. J. Queisser, “Solar cell efficiency and carrier multiplication in Si1-xGex alloys,” J. Appl. Phys. 83(8), 4213–4221 (1998).
[CrossRef]

W. Shockley, H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510 (1961).
[CrossRef]

Rohach, A. L.

F. Hetsch, X. Xu, H. Wang, S. V. Kershaw, A. L. Rohach, “Semiconductor nanocrystal quantum dots as solar cell components and photosensitizers: material, charge transfer, and separation aspects of some device toplogies,” J. Phys. Chem. Lett. 2(15), 1879–1887 (2011).
[CrossRef]

Romero, M.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Samuelson, L.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Schaller, R. D.

R. D. Schaller, V. I. Klimov, “High efficiency carrier multiplication in PbSe nanocrystals: implications for solar energy conversion,” Phys. Rev. Lett. 92(18), 186601 (2004).
[CrossRef] [PubMed]

Schubert, E. F.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Schubert, U.

F. Dimroth, U. Schubert, A. W. Bett, “25.5% efficient Ga0.35In0.65P/Ga0.83In0.17 as tandem solar cells grown on GaAs substrates,” IEEE Electron Dev. 21(5), 209–211 (2000).
[CrossRef]

She, L.-W.

Sherif, R. A.

R. R. King, D. C. Law, K. M. Edmondson, C. M. Fetzer, G. S. Kinsey, H. Yoon, R. A. Sherif, N. H. Karam, “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells,” Appl. Phys. Lett. 90(18), 183516 (2007).
[CrossRef]

Sheu, J. K.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Shockley, W.

W. Shockley, H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510 (1961).
[CrossRef]

Shomalian, K.

D. Souri, K. Shomalian, “Band gap determination by absorption spectrum fitting method (ASF) and structural properties of different compositions of (60−x) V2O5–40TeO2–xSb2O3 glasses,” J. Non-Cryst. Solids 355(31–33), 1597–1601 (2009).
[CrossRef]

Siefer, G.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Sood, A. K.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Souri, D.

D. Souri, K. Shomalian, “Band gap determination by absorption spectrum fitting method (ASF) and structural properties of different compositions of (60−x) V2O5–40TeO2–xSb2O3 glasses,” J. Non-Cryst. Solids 355(31–33), 1597–1601 (2009).
[CrossRef]

Spurgeon, J. M.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Sugita, K.

A. G. Bhuiyan, K. Sugita, A. Hashimoto, A. Yamamoto, “InGaN Solar Cells: Present State of the Art and Important Challenges,” Photovoltaics, IEEE Journal of 2(3), 276–293 (2012).
[CrossRef]

Tanabe, K.

K. Tanabe, “A review of ultrahigh efficiency III-V semiconductor compound solar cells: multijunction tandem, lower dimensional, photonic up/down conversion and plasmonic nanometallic structures,” Energies 2(3), 504–530 (2009).
[CrossRef]

Tsai, M.-T.

Tsia, C. Y.

M. C. Wei, S. J. Chang, C. Y. Tsia, C. H. Liu, S. C. Chen, “SiNx deposited by in-line PECVD for multi-crystalline silicon solar cells,” Sol Energ Mat Sol C. 80(2), 215–219 (2006).

Tu, S. J.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Turner-Evans, D. B.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Wallentin, J.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Wang, C.-H.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Wang, D.-Y.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Wang, H.

F. Hetsch, X. Xu, H. Wang, S. V. Kershaw, A. L. Rohach, “Semiconductor nanocrystal quantum dots as solar cell components and photosensitizers: material, charge transfer, and separation aspects of some device toplogies,” J. Phys. Chem. Lett. 2(15), 1879–1887 (2011).
[CrossRef]

Wang, Y.-T.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Ward, J.

J. Geisz, D. Friedman, J. Ward, A. Duda, W. Olavarria, T. Moriarty, J. Kiehl, M. Romero, A. Norman, K. Jones, “40.8% efficient inverted triple-junction solar cell with two independently metamorphic junctions,” Appl. Phys. Lett. 93(12), 123505 (2008).
[CrossRef]

Warren, E. L.

S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J. R. Maiolo, H. A. Atwater, N. S. Lewis, “Energy-conversion properties of vapor-liquid-solid-grown silicon wire-array photocathodes,” Science 327(5962), 185–187 (2010).
[CrossRef] [PubMed]

Wei, M. C.

M. C. Wei, S. J. Chang, C. Y. Tsia, C. H. Liu, S. C. Chen, “SiNx deposited by in-line PECVD for multi-crystalline silicon solar cells,” Sol Energ Mat Sol C. 80(2), 215–219 (2006).

Welser, R. E.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Werner, J. H.

M. Wolf, R. Brendel, J. H. Werner, H. J. Queisser, “Solar cell efficiency and carrier multiplication in Si1-xGex alloys,” J. Appl. Phys. 83(8), 4213–4221 (1998).
[CrossRef]

Witzigmann, B.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Wolf, M.

M. Wolf, R. Brendel, J. H. Werner, H. J. Queisser, “Solar cell efficiency and carrier multiplication in Si1-xGex alloys,” J. Appl. Phys. 83(8), 4213–4221 (1998).
[CrossRef]

Woo, R. L.

M. S. Leite, R. L. Woo, J. N. Munday, W. D. Hong, S. Mesropian, D. C. Law, H. A. Atwater, “Towards an optimized all lattice-matched InAlAs/InGaAsP/InGaAs multijunction solar cell with efficiency >50%,” Appl. Phys. Lett. 102(3), 033901 (2013).
[CrossRef]

Wu, C.-J.

Xu, H. Q.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339(6123), 1057–1060 (2013).
[CrossRef] [PubMed]

Xu, X.

F. Hetsch, X. Xu, H. Wang, S. V. Kershaw, A. L. Rohach, “Semiconductor nanocrystal quantum dots as solar cell components and photosensitizers: material, charge transfer, and separation aspects of some device toplogies,” J. Phys. Chem. Lett. 2(15), 1879–1887 (2011).
[CrossRef]

Yamamoto, A.

A. G. Bhuiyan, K. Sugita, A. Hashimoto, A. Yamamoto, “InGaN Solar Cells: Present State of the Art and Important Challenges,” Photovoltaics, IEEE Journal of 2(3), 276–293 (2012).
[CrossRef]

Yan, X.

X. Yan, D. J. Poxson, J. Cho, R. E. Welser, A. K. Sood, J. K. Kim, E. F. Schubert, “Enhanced omnidirectional photovoltaic performance of solar cells by multiple-discrete-layer tailored- and low- refractive-index anti-reflection coatings,” Adv. Funct. Mater. 23(5), 583–590 (2013).
[CrossRef]

Yang, C. C.

J. K. Sheu, C. C. Yang, S. J. Tu, K. H. Chang, M. L. Lee, W.-C. Lai, L.-C. Peng, “Demonstration of GaN-based solar cells with GaN/InGaN superlattice absorption layers,” IEEE Electron Device Lett. 30(3), 225–227 (2009).
[CrossRef]

Yang, C.-H.

Y.-J. Lee, Y.-C. Yao, C.-H. Yang, “Direct electrical contact of slanted ITO film on axial p-n junction silicon nanowire solar cells,” Opt. Express 21(S1Suppl 1), A7–A14 (2013).
[CrossRef] [PubMed]

Y.-J. Lee, M.-H. Lee, C.-M. Cheng, C.-H. Yang, “Enhanced conversion efficiency of InGaN multiple quantum well solar cells grown on patterned sapphire substrates,” Appl. Phys. Lett. 98(26), 263504 (2011).
[CrossRef]

Yang, Y.-J.

C.-Y. Huang, D.-Y. Wang, C.-H. Wang, Y.-T. Chen, Y.-T. Wang, Y.-T. Jiang, Y.-J. Yang, C.-C. Chen, Y.-F. Chen, “Efficient light harvesting by photon downconversion and light trapping in hybrid ZnS nanoparticles/Si nanotips solar cells,” ACS Nano 4(10), 5849–5854 (2010).
[CrossRef] [PubMed]

Yao, Y.-C.

Yoon, H.

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic plot of an InGaP/GaAs/Ge triple-junction solar cell with CdSe QDs spread on the top surface to tailor the solar spectrum and enhance the photocurrent of the GaAs middle subcell. (b) Simplified structure of the tandem solar cell to facilitate optical calculations.

Fig. 2
Fig. 2

(a) Absorption and (b) photoluminescence spectra of CdSe QDs of different sizes in toluene.

Fig. 3
Fig. 3

Calculated light intensity of the solar spectrum I(1,2,3)i(λ) distributed on each subcell for the device (a) without and (b) with CdSe QDs with diameters of D = 2.1 nm. The quantum efficiency QE(1,2,3)(λ) of each subcell is also plotted in the figure.

Fig. 4
Fig. 4

Calculated J-V characteristics of each subcell by Eq. (8) for the device (a) without and (b) with CdSe QDs with diameters of D = 2.1 nm.

Fig. 5
Fig. 5

(a) Calculated short-circuit current density (JSC) of each subcell and (b) the overall power conversion efficiency (PCE) of the device as a function of the CdSe QD’s diameter under 1-sun illumination. The enhancement ratio of the PCE compared to the device without CdSe QDs is also labeled in the figure. The black dash-line indicates the current-matching condition of InGaP and GaAs subcells.

Fig. 6
Fig. 6

(a) Electrical performance of the bare tandem solar cell after dispersing 7 mg/mL of CdSe QDs with diameters of 4.2 nm compared to the one without CdSe QDs under AM1.5G sunlight illumination. The electrical performances of devices with a traditional antireflection coating (ARC, 100 nm SiNx) and CdSe QDs on top of the SiNx ARC are also plotted for comparison. Inset: photograph of the actual devices with dimensions of 1 cm × 1 cm. (b) J-V characteristics with different concentration of CdSe QDs. A summary of JSC as a function of the QD concentration is also plotted and inserted into the figure.

Fig. 7
Fig. 7

Reflectance of the devices spin-casting CdSe QDs with various concentrations of 5 mg/mL, 7 mg/mL, and 9 mg/mL. The reflectance of the bare tandem solar cell is also plotted in the figure.

Equations (8)

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

A=log( I 0i I 0t )
α 0 = (ln10)A L 0
I 0i = I AM1.5G (1 R 0 ) R 0 = ( n air n QD n air + n QD ) 2
I 1i = I 0t (1 R 1 )+ I PL = I 0i 10 A (1 R 1 )+ I PL
I 1t = I 1i exp[( α 1 L 1 )] α 1 = 4π κ 1 λ
I 2i = I 1t (1 R 2 )= I 1i exp[( α 1 L 1 )](1 R 2 ) I 3i = I 2t (1 R 3 )= I 2i exp[( α 2 L 2 )](1 R 3 )
A (1,2,3) (λ)= I (1,2,3)i I (1,2,3)t = I (1,2,3)i [1exp( α (1,2,3) L (1,2,3) )]
J (1,2,3) (V)= q hc 0 λ I (1,2,3)i { 1exp[( α (1,2,3) L (1,2,3) )] }Q E (1,2,3) (λ)dλ q( n (1,2,3) 2 +1) E g(1,2,3) 2 kT 4π 3 c 2 exp( eV E g(1,2,3) kT )

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