Abstract

Depending on the minimum size of their micro/nanostructure, thin films can exhibit very different behaviors and optical properties. From optical waveguides down to artificial anisotropy, through diffractive optics and photonic crystals, the application changes when decreasing the minimum feature size. Rigorous electromagnetic theory can be used to model most of the components, but, when the size is a few nanometers, quantum theory also has to be used. The materials, including quantum structures, are of particular interest for many applications, in particular for solar cells because of their luminescent and electronic properties. We show that the properties of electrons in periodic and nonperiodic multiple quantum well structures can be easily modeled with a formalism similar to that used for multilayer waveguides. The effects of different parameters, in particular the coupling between wells and well thickness dispersion, on possible discrete energy levels or the energy band of electrons and on electron wave functions are given. When such quantum confinement appears, the spectral absorption and extinction coefficient dispersion with wavelength are modified. The dispersion of the real part of the refractive index can be deduced from the Kramers–Kronig relations. Associated with homogenization theory, this approach gives a new model of the refractive index for thin films including quantum dots. The bandgap of ZnO quantum dots in solution obtained from the absorption spectrum is in good agreement with our calculation.

© 2011 Optical Society of America

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References

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  1. D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
    [CrossRef]
  2. D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
    [CrossRef]
  3. P. S. Zory, “The origin of quantum wells and the quantum well laser,” in C.H.Henry, Jr., ed., Quantum Well Lasers(Academic, 1993), pp. 1–13.
  4. H. Schneider and H. Liu, Quantum Well Infrared Photodetectors: Physics and Applications, Springer Series in Optical Sciences (Springer-Verlag, 2006).
  5. M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
    [CrossRef] [PubMed]
  6. S. Nizamoglu, G. Zengin, and H. V. Demir, “Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index,” Appl. Phys. Lett. 92, 031102 (2008).
    [CrossRef]
  7. S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
    [CrossRef] [PubMed]
  8. T. Christian and E. Ghadiri, “QDs application in solar cells,” presented at the EPFL—SPES Conference (28 May 2009).
  9. S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University, 1998).
    [CrossRef]
  10. H. Li, “Refractive index of interdiffused AlGaAs/GaAs quantum well,” J. Appl. Phys. 82, 6251–6258 (1997).
    [CrossRef]
  11. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique (Hermann, 1997), Vol.  1.
  12. F. Flory, “Characterization: guided wave techniques,” in Thin Films for Optical Systems, F.Flory, ed., Vol. 49 of Optical Engineering Series (Marcel Dekker, 1995).
  13. R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. (Wiley, 1985), Appendix H.
  14. P. Carpena, V. Gasparian, and M. Ortuño, “Number of bound states of a Krönig–Penney finite-periodic superlattice,” Eur. Phys. J. B 8, 635–641 (1999).
    [CrossRef]
  15. C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
    [CrossRef] [PubMed]
  16. W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
    [CrossRef]
  17. L. E. Brus, “Electron-electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. 80, 4403–4410 (1984).
    [CrossRef]
  18. K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
    [CrossRef]

2009 (2)

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

2008 (2)

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

S. Nizamoglu, G. Zengin, and H. V. Demir, “Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index,” Appl. Phys. Lett. 92, 031102 (2008).
[CrossRef]

2006 (1)

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

2005 (1)

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

2002 (1)

S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
[CrossRef] [PubMed]

1999 (1)

P. Carpena, V. Gasparian, and M. Ortuño, “Number of bound states of a Krönig–Penney finite-periodic superlattice,” Eur. Phys. J. B 8, 635–641 (1999).
[CrossRef]

1998 (1)

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

1997 (1)

H. Li, “Refractive index of interdiffused AlGaAs/GaAs quantum well,” J. Appl. Phys. 82, 6251–6258 (1997).
[CrossRef]

1984 (1)

L. E. Brus, “Electron-electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. 80, 4403–4410 (1984).
[CrossRef]

Ackermann, J.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Agus, P. A.

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

Alvistos, A. P.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Bawendi, M.

S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
[CrossRef] [PubMed]

Bruchez, M.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Brus, L. E.

L. E. Brus, “Electron-electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. 80, 4403–4410 (1984).
[CrossRef]

Bulovic, V.

S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
[CrossRef] [PubMed]

Carpena, P.

P. Carpena, V. Gasparian, and M. Ortuño, “Number of bound states of a Krönig–Penney finite-periodic superlattice,” Eur. Phys. J. B 8, 635–641 (1999).
[CrossRef]

Cheng, H.-M.

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

Christian, T.

T. Christian and E. Ghadiri, “QDs application in solar cells,” presented at the EPFL—SPES Conference (28 May 2009).

Coe, S.

S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique (Hermann, 1997), Vol.  1.

Demir, H. V.

S. Nizamoglu, G. Zengin, and H. V. Demir, “Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index,” Appl. Phys. Lett. 92, 031102 (2008).
[CrossRef]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique (Hermann, 1997), Vol.  1.

Duché, D.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

Eisberg, R.

R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. (Wiley, 1985), Appendix H.

Escoubas, L.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

Fages, F.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Fan, W. J.

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

Ferry, D.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Flory, F.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

F. Flory, “Characterization: guided wave techniques,” in Thin Films for Optical Systems, F.Flory, ed., Vol. 49 of Optical Engineering Series (Marcel Dekker, 1995).

Gaponenko, S. V.

S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University, 1998).
[CrossRef]

Gasparian, V.

P. Carpena, V. Gasparian, and M. Ortuño, “Number of bound states of a Krönig–Penney finite-periodic superlattice,” Eur. Phys. J. B 8, 635–641 (1999).
[CrossRef]

Ghadiri, E.

T. Christian and E. Ghadiri, “QDs application in solar cells,” presented at the EPFL—SPES Conference (28 May 2009).

Gin, P.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Hsieh, W.-F.

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

Hsu, H.-C.

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

Kanehira, D.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Laloë, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique (Hermann, 1997), Vol.  1.

Li, H.

H. Li, “Refractive index of interdiffused AlGaAs/GaAs quantum well,” J. Appl. Phys. 82, 6251–6258 (1997).
[CrossRef]

Lin, K.-F.

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

Lin, L.-J.

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

Liu, H.

H. Schneider and H. Liu, Quantum Well Infrared Photodetectors: Physics and Applications, Springer Series in Optical Sciences (Springer-Verlag, 2006).

Martini, C.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Mathian, G.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

Monestier, F.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

Moronne, M.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Nizamoglu, S.

S. Nizamoglu, G. Zengin, and H. V. Demir, “Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index,” Appl. Phys. Lett. 92, 031102 (2008).
[CrossRef]

Ortuño, M.

P. Carpena, V. Gasparian, and M. Ortuño, “Number of bound states of a Krönig–Penney finite-periodic superlattice,” Eur. Phys. J. B 8, 635–641 (1999).
[CrossRef]

Poize, G.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Resnick, R.

R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. (Wiley, 1985), Appendix H.

Schneider, H.

H. Schneider and H. Liu, Quantum Well Infrared Photodetectors: Physics and Applications, Springer Series in Optical Sciences (Springer-Verlag, 2006).

Simon, J. J.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

Sun, X. W.

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

Tan, S. T.

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

Torchio, P.

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

Vervisch, W.

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

Weiss, S.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Woo, W. K.

S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
[CrossRef] [PubMed]

Xia, J. B.

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

Yoshimoto, N.

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Yu, S. F.

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

Zengin, G.

S. Nizamoglu, G. Zengin, and H. V. Demir, “Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index,” Appl. Phys. Lett. 92, 031102 (2008).
[CrossRef]

Zory, P. S.

P. S. Zory, “The origin of quantum wells and the quantum well laser,” in C.H.Henry, Jr., ed., Quantum Well Lasers(Academic, 1993), pp. 1–13.

Appl. Phys. Lett. (2)

D. Duché, L. Escoubas, J. J. Simon, P. Torchio, W. Vervisch, and F. Flory, “Slow Bloch modes for enhancing the absorption of light in thin films for photovoltaic cells,” Appl. Phys. Lett. 92, 193310 (2008).
[CrossRef]

S. Nizamoglu, G. Zengin, and H. V. Demir, “Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index,” Appl. Phys. Lett. 92, 031102 (2008).
[CrossRef]

Chem. Phys. Chem. (1)

C. Martini, G. Poize, D. Ferry, D. Kanehira, N. Yoshimoto, J. Ackermann, and F. Fages, “Oligothiophene self-assembly on the surface of ZnO nanorods: toward coaxial p-n hybrid heterojunctions,” Chem. Phys. Chem. 10, 2465–2470(2009).
[CrossRef] [PubMed]

Chem. Phys. Lett. (1)

K.-F. Lin, H.-M. Cheng, H.-C. Hsu, L.-J. Lin, and W.-F. Hsieh, “Band gap variation of size-controlled ZnO QDs synthesized by sol-gel method,” Chem. Phys. Lett. 409, 208–211 (2005).
[CrossRef]

Eur. Phys. J. B (1)

P. Carpena, V. Gasparian, and M. Ortuño, “Number of bound states of a Krönig–Penney finite-periodic superlattice,” Eur. Phys. J. B 8, 635–641 (1999).
[CrossRef]

J. Appl. Phys. (2)

W. J. Fan, J. B. Xia, P. A. Agus, S. T. Tan, S. F. Yu, and X. W. Sun, “Band parameters and electronic structures of wurtzite ZnO and ZnO/MgZnO quantum wells,” J. Appl. Phys. 99, 013702 (2006).
[CrossRef]

H. Li, “Refractive index of interdiffused AlGaAs/GaAs quantum well,” J. Appl. Phys. 82, 6251–6258 (1997).
[CrossRef]

J. Chem. Phys. (1)

L. E. Brus, “Electron-electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. 80, 4403–4410 (1984).
[CrossRef]

Nature (1)

S. Coe, W. K. Woo, M. Bawendi, and V. Bulovic, “Electroluminescence from single monolayers of nanocrystals in molecular organic devices,” Nature 420, 800–803 (2002).
[CrossRef] [PubMed]

Science (1)

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alvistos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Solar Energy Mater. Solar Cells (1)

D. Duché, P. Torchio, L. Escoubas, F. Monestier, J. J. Simon, F. Flory, and G. Mathian, “Improving light absorption in organic solar cells by plasmonic contribution,” Solar Energy Mater. Solar Cells 93, 1377–1382 (2009).
[CrossRef]

Other (7)

P. S. Zory, “The origin of quantum wells and the quantum well laser,” in C.H.Henry, Jr., ed., Quantum Well Lasers(Academic, 1993), pp. 1–13.

H. Schneider and H. Liu, Quantum Well Infrared Photodetectors: Physics and Applications, Springer Series in Optical Sciences (Springer-Verlag, 2006).

T. Christian and E. Ghadiri, “QDs application in solar cells,” presented at the EPFL—SPES Conference (28 May 2009).

S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University, 1998).
[CrossRef]

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique (Hermann, 1997), Vol.  1.

F. Flory, “Characterization: guided wave techniques,” in Thin Films for Optical Systems, F.Flory, ed., Vol. 49 of Optical Engineering Series (Marcel Dekker, 1995).

R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. (Wiley, 1985), Appendix H.

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Figures (7)

Fig. 1
Fig. 1

(a) Single-well structure, (b) results from analytical formula [14] and from our program, and (c) wave functions for the three energies levels.

Fig. 2
Fig. 2

Energy levels of electrons for three coupled identical quantum wells of 1 nm width, 1 nm spacing, and 3.2 eV depth.

Fig. 3
Fig. 3

Splitting of the highest energy level of electrons in coupled quantum wells when considering 1, 2, 3, 4, and 5 identical coupled wells. Each well has a depth of 2.9 eV , a width of 1 nm , and the distance between the wells is also 1 nm .

Fig. 4
Fig. 4

Electron density of probability in a structure composed of nine identical coupled wells.

Fig. 5
Fig. 5

Density of probability of electrons in a nonperiodic structure composed of five coupled wells of depth 2.9 eV and widths of 1, 1, 2, 1, and 1 nm , respectively. The distance between each well is 1 nm .

Fig. 6
Fig. 6

Absorption spectrum of ZnO nanocrystals in solution.

Fig. 7
Fig. 7

TEM picture of a ZnO nanocrystal.

Tables (1)

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Table 1 Mean Energy Levels E 1 , E 2 , and E 3 for an Infinite Periodic Structure Calculated with the Kronig–Penney Model and with Our Formalism for Nine Quantum Wells a

Equations (9)

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( h 2 4 π 2 ) 2 m [ 2 + V ( r ) ] ψ ( r ) = E ψ ( r ) .
ψ ( x ) = A e i χ x + A e i χ x for     E > V ,
ψ ( x ) = B e ρ x + B e ρ x for     E < V ,
χ 2 = 2 m 2 ( E V ) ,
ρ 2 = 2 m 2 ( V E ) ,
Q = d ψ d x ψ ,
Q j = k j Q j 1 k j tan ( k j a j ) Q j 1 tan ( k j a j ) + k j ,
Ψ j = C j + [ e i k j x + i k j Q j 1 Q j 1 + i k j e i k j ( 2 x j 1 x ) ] ,
k ( λ ) = λ 4 π d Log ( A ( λ ) ) ,

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