Abstract

We design a scheme for obtaining size distribution information for colloidal quantum dots (QDs) embedded in a host material. The calculation uses the QDs’ optical absorption spectrum and requires the absorption spectra for a series of sample QDs, having a uniform size distribution with specific radii. In an analogy to a quantum state expanded by a basis vector set, the absorption spectrum of a QD ensemble with an unspecified size distribution can be expanded by those of the sample spectra, which are treated as basis vectors. The proportion of QDs with a radius around each sample’s radius can be extracted. We also discuss the case without sufficient sample spectra.

© 2013 Optical Society of America

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  1. D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
    [CrossRef]
  2. J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
    [CrossRef]
  3. U. Woggon and S. V. Gaponenko, “Excitons in quantum dots,” Phys. Status Solidi B 189, 285–343 (1995).
    [CrossRef]
  4. W. C. Chan and S. Nile, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
    [CrossRef]
  5. B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
    [CrossRef]
  6. X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
    [CrossRef]
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    [CrossRef]
  8. X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. A.-S. Keita and A. En Naciri, “Size distribution dependence of the dielectric function of Si quantum dots described by a modified Maxwell–Garnett formulation,” Phys. Rev. B 84, 125436 (2011).
    [CrossRef]
  17. I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. I. Moreels, D. Kruschke, P. Glas, and J. W. Tomm, “The dielectric function of PbS quantum dots in a glass matrix,” Opt. Mater. Express 2, 496–500 (2012).
    [CrossRef]

2012

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

I. Moreels, D. Kruschke, P. Glas, and J. W. Tomm, “The dielectric function of PbS quantum dots in a glass matrix,” Opt. Mater. Express 2, 496–500 (2012).
[CrossRef]

2011

A.-S. Keita and A. En Naciri, “Size distribution dependence of the dielectric function of Si quantum dots described by a modified Maxwell–Garnett formulation,” Phys. Rev. B 84, 125436 (2011).
[CrossRef]

2010

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

M. I. Alonso, I. C. Marcus, M. Garriga, and A. R. Goñi, “Evidence of quantum confinement effects on interband optical transitions in Si nanocrystals,” Phys. Rev. B 82, 045302 (2010).
[CrossRef]

2009

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

2008

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

2006

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

R. Espiau de Lamaëstre and H. Bernas, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B 73, 125317 (2006).
[CrossRef]

2005

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

2003

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

2002

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

A. M. Malyarevich, V. G. Savitski, P. V. Prokoshin, N. N. Posnov, K. V. Yumashev, E. Raaben, and A. A. Zhilin, “Glass doped with PbS quantum dots as a saturable absorber for 1-m neodymium lasers,” J. Opt. Soc. Am. B 19, 28–32 (2002).
[CrossRef]

2001

L.-S. Li, J. Hu, W. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001).
[CrossRef]

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

2000

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

1998

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

W. C. Chan and S. Nile, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef]

1995

U. Woggon and S. V. Gaponenko, “Excitons in quantum dots,” Phys. Status Solidi B 189, 285–343 (1995).
[CrossRef]

1989

1982

D. E. Aspnes, “Local-field effects and effective-medium theory: amicroscopic perspective,” Am. J. Phys 50, 704 (1982).
[CrossRef]

1980

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).

Akiyama, T.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Alivisatos, A. P.

L.-S. Li, J. Hu, W. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001).
[CrossRef]

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Allan, G.

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

Alonso, M. I.

M. I. Alonso, I. C. Marcus, M. Garriga, and A. R. Goñi, “Evidence of quantum confinement effects on interband optical transitions in Si nanocrystals,” Phys. Rev. B 82, 045302 (2010).
[CrossRef]

Alves, E.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Amirav, L.

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

Ashcroft, N. W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).

Aspnes, D. E.

D. E. Aspnes, “Local-field effects and effective-medium theory: amicroscopic perspective,” Am. J. Phys 50, 704 (1982).
[CrossRef]

Bakr, O. M.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Bardecker, J. A.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Bernas, H.

R. Espiau de Lamaëstre and H. Bernas, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B 73, 125317 (2006).
[CrossRef]

Brivanlou, A. H.

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

Brumer, M.

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

Bruno, G.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Cai, Q. Y.

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

Capezzuto, P.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Cerqueira, M. F.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Chan, W. C.

W. C. Chan and S. Nile, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef]

Chen, B.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Chen, L. Y.

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

Chen, X.

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

Chen, Y. M.

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

Chenot, S.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

Debnath, R.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Delerue, C.

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

Deppe, D. G.

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

Ding, I. K.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Dubertret, B.

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

En Naciri, A.

A.-S. Keita and A. En Naciri, “Size distribution dependence of the dielectric function of Si quantum dots described by a modified Maxwell–Garnett formulation,” Phys. Rev. B 84, 125436 (2011).
[CrossRef]

Espiau de Lamaëstre, R.

R. Espiau de Lamaëstre and H. Bernas, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B 73, 125317 (2006).
[CrossRef]

Fischer, A.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Fisson, S.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

Gallas, B.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

Gaponenko, S. V.

U. Woggon and S. V. Gaponenko, “Excitons in quantum dots,” Phys. Status Solidi B 189, 285–343 (1995).
[CrossRef]

Garriga, M.

M. I. Alonso, I. C. Marcus, M. Garriga, and A. R. Goñi, “Evidence of quantum confinement effects on interband optical transitions in Si nanocrystals,” Phys. Rev. B 82, 045302 (2010).
[CrossRef]

Geyter, B. D.

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

Giangregorio, M. M.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Ginger, D. S.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Glas, P.

Goñi, A. R.

M. I. Alonso, I. C. Marcus, M. Garriga, and A. R. Goñi, “Evidence of quantum confinement effects on interband optical transitions in Si nanocrystals,” Phys. Rev. B 82, 045302 (2010).
[CrossRef]

Goorskey, D.

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

Hens, Z.

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

Hu, J.

L.-S. Li, J. Hu, W. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001).
[CrossRef]

Huffaker, D. L.

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

Ishikawa, H.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Jen, A. K.-Y.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Kadanavich, A.

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Kao, C.-C.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

Keita, A.-S.

A.-S. Keita and A. En Naciri, “Size distribution dependence of the dielectric function of Si quantum dots described by a modified Maxwell–Garnett formulation,” Phys. Rev. B 84, 125436 (2011).
[CrossRef]

Kigel, A.

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

Kramer, I. J.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Kruschke, D.

Kuwatsuka, H.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Labelle, A. J.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Lamb, W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).

Lewis, M.

Li, L.-S.

L.-S. Li, J. Hu, W. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001).
[CrossRef]

Libchaber, A.

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

Lifshitz, E.

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

Liu, M. S.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Losurdo, M.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Lu, W. J.

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

Luo, J.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Malyarevich, A. M.

Manna, L.

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Marcus, I. C.

M. I. Alonso, I. C. Marcus, M. Garriga, and A. R. Goñi, “Evidence of quantum confinement effects on interband optical transitions in Si nanocrystals,” Phys. Rev. B 82, 045302 (2010).
[CrossRef]

Moreels, I.

I. Moreels, D. Kruschke, P. Glas, and J. W. Tomm, “The dielectric function of PbS quantum dots in a glass matrix,” Opt. Mater. Express 2, 496–500 (2012).
[CrossRef]

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

Mukai, K.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Munro, A. M.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Nakata, Y.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Nazzal, A.

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

Nile, S.

W. C. Chan and S. Nile, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef]

Niu, Y.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Noireaux, V.

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

Norris, D. J.

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

Pan, J.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Park, G.

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

Peng, X.

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Peng, Z. A.

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

Persans, P. D.

Posnov, N. N.

Prokoshin, P. V.

Raaben, E.

Rivory, J.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

Sargent, E. H.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Sashchiuk, A.

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

Savitski, V. G.

Scher, E.

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Shchekin, O. B.

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

Simoyama, T.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Siozade, L.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

Skourides, P.

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

Stenger, I.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

Stepikhova, M.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

Sugawara, M.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Tomm, J. W.

Tu, A.

Vuye, G.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

Wada, O.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

Wickham, J.

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Wirtz, L.

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

Woggon, U.

U. Woggon and S. V. Gaponenko, “Excitons in quantum dots,” Phys. Status Solidi B 189, 285–343 (1995).
[CrossRef]

Wood, D. M.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).

Wu, Y.-J.

Xiao, M.

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

Yang, W.

L.-S. Li, J. Hu, W. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001).
[CrossRef]

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Yumashev, K. V.

Zhang, R. J.

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

Zhao, J.

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Zheng, Y. X.

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

Zhilin, A. A.

Zhitomirsky, D.

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

Zou, Z.

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

Adv. Funct. Mater.

M. Brumer, A. Kigel, L. Amirav, A. Sashchiuk, and E. Lifshitz, “PbSe/PbS and PbSe/PbSexS1−x core/shell nanocrystals,” Adv. Funct. Mater. 15, 1111 (2005).
[CrossRef]

Am. J. Phys

D. E. Aspnes, “Local-field effects and effective-medium theory: amicroscopic perspective,” Am. J. Phys 50, 704 (1982).
[CrossRef]

Appl. Phys. Lett.

D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, “1.3  μm room-temperature GaAs-based quantum-dot laser,” Appl. Phys. Lett. 73, 2564–2566 (1998).
[CrossRef]

R. J. Zhang, Y. M. Chen, W. J. Lu, Q. Y. Cai, Y. X. Zheng, and L. Y. Chen, “Influence of nanocrystal size on dielectric functions of Si nanocrystals embedded in SiO2 matrix,” Appl. Phys. Lett. 95, 161109 (2009).
[CrossRef]

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, M. F. Cerqueira, E. Alves, and M. Stepikhova, “Dielectric function of nanocrystalline silicon with few nanometers (<3  nm) grain size,” Appl. Phys. Lett. 82, 2993 (2003).
[CrossRef]

IEEE Photon. Technol. Lett.

T. Akiyama, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, O. Wada, and H. Ishikawa, “Application of spectral-hole burning in the inhomogeneously broadened gain of self-assembled quantum dots to a multiwavelength-channel nonlinear optical device,” IEEE Photon. Technol. Lett. 12, 1301–1303 (2000).
[CrossRef]

J. Appl. Phys.

I. Stenger, B. Gallas, L. Siozade, C.-C. Kao, S. Chenot, S. Fisson, G. Vuye, and J. Rivory, “Evolution of the optical properties of Si nanoparticles embedded in SiO2 as function of annealing conditions,” J. Appl. Phys. 103, 114303 (2008).
[CrossRef]

J. Opt. Soc. Am. B

Nano Lett.

L.-S. Li, J. Hu, W. Yang, and A. P. Alivisatos, “Band gap variation of size- and shape-controlled colloidal CdSe quantum rods,” Nano Lett. 1, 349–351 (2001).
[CrossRef]

D. Zhitomirsky, I. J. Kramer, A. J. Labelle, A. Fischer, R. Debnath, J. Pan, O. M. Bakr, and E. H. Sargent, “Colloidal quantum dot photovoltaics: the effect of polydispersity,” Nano Lett. 12, 1007–1012 (2012).
[CrossRef]

J. Zhao, J. A. Bardecker, A. M. Munro, M. S. Liu, Y. Niu, I. K. Ding, J. Luo, B. Chen, A. K.-Y. Jen, and D. S. Ginger, “Efficient CdSe/CdS quantum dot light-emitting diodes using a thermally polymerized hole transport layer,” Nano Lett. 6, 463–467 (2006).
[CrossRef]

Nature

X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadanavich, and A. P. Alivisatos, “Shape control of CdSe nanocrystals,” Nature 404, 59–61 (2000).
[CrossRef]

Opt. Mater. Express

Phys. Rev. B

X. Chen, A. Nazzal, D. Goorskey, M. Xiao, Z. A. Peng, and X. Peng, “Polarization spectroscopy of single CdSe quantum rods,” Phys. Rev. B 64, 245304 (2001).
[CrossRef]

M. I. Alonso, I. C. Marcus, M. Garriga, and A. R. Goñi, “Evidence of quantum confinement effects on interband optical transitions in Si nanocrystals,” Phys. Rev. B 82, 045302 (2010).
[CrossRef]

B. Gallas, I. Stenger, C.-C. Kao, S. Fisson, G. Vuye, and J. Rivory, “Optical properties of Si nanocrystals embedded in SiO2,” Phys. Rev. B 72, 155319 (2005).
[CrossRef]

R. Espiau de Lamaëstre and H. Bernas, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B 73, 125317 (2006).
[CrossRef]

A.-S. Keita and A. En Naciri, “Size distribution dependence of the dielectric function of Si quantum dots described by a modified Maxwell–Garnett formulation,” Phys. Rev. B 84, 125436 (2011).
[CrossRef]

I. Moreels, G. Allan, B. D. Geyter, L. Wirtz, C. Delerue, and Z. Hens, “Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers–Krönig analysis of the absorbance spectrum,” Phys. Rev. B 81, 235319 (2010).
[CrossRef]

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).

Phys. Status Solidi B

U. Woggon and S. V. Gaponenko, “Excitons in quantum dots,” Phys. Status Solidi B 189, 285–343 (1995).
[CrossRef]

Science

W. C. Chan and S. Nile, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef]

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, “In vivo imaging of quantum dots encapsulated in phospholipid micelles,” Science 298, 1759–1762 (2002).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Absorption spectra of eight-sample QD ensemble fabricated of a virtual material; each sample has a different peak. (b) Real part of dielectric function of sample QD ensemble. (c) Imaginary part of dielectric function of sample QD ensemble. (d) Size distribution information extracted from absorption spectrum. Gray bars represent the proportion of QDs in the ensemble with a specific radius. Red diamonds represent the actual size distribution information for the ensemble. Standard deviation σ is set to 1.25.

Fig. 2.
Fig. 2.

(a) Absorption spectra of PbS QDs in glass matrix; spectra A–D represent QD diameters of 3.5 nm, 4.0 nm, 4.4 nm, and 5 nm, respectively. (b) Real part of dielectric function of QDs. (c) Imaginary part of dielectric function of QDs. (d) Size distribution information extracted from virtual combined absorption spectrum. Red diamonds represent actual probability; bars represent extracted one.

Fig. 3.
Fig. 3.

Size distribution information retrieved for incomplete sample set. Gray bars represent actual size distribution of QD ensemble; red, blue, and green diamonds represent cases with 30, 40, and 50 sample spectra, respectively.

Equations (18)

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

μ(λ)=2πεIλ9ns3(εR+2ns2)2+εI2.
Δμ=ΔεRμ(ε)εR|ε0+ΔεIμ(ε)εI|ε0.
εR(λj)=1+2πkjλj2Δλλk(λj2λk2)εI(λk)=1+2πkjj2k(j2k2)εI(λk).
ΔεR=2πA·ΔεI.
ΔεI=(C+2πDA)1Δμμ0,
Cj,j=(1εI,02εI,0(εR,0+2ns2)2+εI,02)Dj,j=2(εR,0+2ns2)(εR,0+2ns2)2+εI,02.
εeff=1+8π3ε2·n·κ14π3ε2·n·κ.
κ=ε1/ε21ε1/ε2+2ε2R3.
η=εeffε2εeff+2ε2=f·ε1ε2ε1+2ε2,
κ¯=0κP(R)dR.
η¯=n4π3ε20κP(R)dR=0η(R)P(R)dR.
η˜=iP(Ri)η(Ri).
|η˜=iPs(Ri)|ηs(Ri).
iP(Ri)|η(Ri)=jPs(Rj)|ηs(Rj).
P(Ri)=j(A1)ijPs(Rj),
Aij=ηs(Ri)|η(Rj).
P(R)=1R2πLog(σ)exp([Log(R/R¯)2Log(σ)]2).
Log(σ)=[jNj(Log(Rj)Log(R¯))2jNj]1/2,

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