Abstract

This work presents a study of the luminescence properties of Er3+ when included into two different TiO2 hosts: a polycrystalline and an amorphous host. The two host environments were produced by depositing two thin films with different Er3+ concentration using radio-frequency magnetron sputtering. Structural analysis revealed the presence of the rutile and anatase phases in the polycrystalline film. Time-resolved and steady-state photoluminescence measurements evidenced the presence of two distinct local Er3+ environments in the polycrystalline host. For the amorphous TiO2 host, only one Er3+ environment was observed, which differed from the two environments in the polycrystalline host. A method for extracting a fast and slow time-resolved emission spectrum from the two observed local environments in the polycrystalline host is also presented.

© 2016 Optical Society of America

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    [Crossref]
  4. A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
    [Crossref]
  5. A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
    [Crossref]
  6. J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
    [Crossref]
  7. T. Trupke, M. A. Green, and P. Würfel, “Improving solar cell efficiencies by up-conversion of sub-band-gap light,” J. Appl. Phys. 92(7), 4117–4122 (2002).
    [Crossref]
  8. M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
    [Crossref] [PubMed]
  9. M. Ishii, S. Komuro, and T. Morikawa, “Study on atomic coordination around Er doped into anatase– and rutile– TiO2: Er–O clustering dependent on the host crystal phase,” J. Appl. Phys. 94(6), 3823–3827 (2003).
    [Crossref]
  10. C. Mignotte, “Structural characterization for Er3+-doped oxide materials potentially useful as optical devices,” Appl. Surf. Sci. 226(4), 355–370 (2004).
    [Crossref]
  11. W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
    [Crossref] [PubMed]
  12. C. Mignotte, “EXAFS studies on erbium-doped TiO2 and ZrO2 sol-gel thin films,” J. Non-Crystal. Solids 291(1–2), 56–77 (2001).
  13. K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
    [Crossref]
  14. S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
    [Crossref]
  15. S.-Y. Chen, C.-C. Ting, and W.-F. Hsieh, “Comparison of visible fluorescence properties between sol–gel derived Er3+–Yb3+ and Er3+–Y3+ co-doped TiO2 films,” Thin Solid Films 434(1–2), 171–177 (2003).
    [Crossref]
  16. D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
    [Crossref]
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    [Crossref]
  18. K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
    [Crossref] [PubMed]
  19. J.-C. G. Bünzli and S. V. Eliseeva, Basics of Lanthanide Photophysics (Springer-Verlag, 2010).
  20. W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
    [Crossref]

2015 (2)

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

2014 (1)

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

2011 (2)

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

2010 (1)

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

2009 (1)

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

2006 (1)

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

2004 (1)

C. Mignotte, “Structural characterization for Er3+-doped oxide materials potentially useful as optical devices,” Appl. Surf. Sci. 226(4), 355–370 (2004).
[Crossref]

2003 (3)

M. Ishii, S. Komuro, and T. Morikawa, “Study on atomic coordination around Er doped into anatase– and rutile– TiO2: Er–O clustering dependent on the host crystal phase,” J. Appl. Phys. 94(6), 3823–3827 (2003).
[Crossref]

S.-Y. Chen, C.-C. Ting, and W.-F. Hsieh, “Comparison of visible fluorescence properties between sol–gel derived Er3+–Yb3+ and Er3+–Y3+ co-doped TiO2 films,” Thin Solid Films 434(1–2), 171–177 (2003).
[Crossref]

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

2002 (2)

T. Trupke, M. A. Green, and P. Würfel, “Improving solar cell efficiencies by up-conversion of sub-band-gap light,” J. Appl. Phys. 92(7), 4117–4122 (2002).
[Crossref]

K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
[Crossref] [PubMed]

2001 (1)

C. Mignotte, “EXAFS studies on erbium-doped TiO2 and ZrO2 sol-gel thin films,” J. Non-Crystal. Solids 291(1–2), 56–77 (2001).

1996 (1)

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

1992 (1)

1991 (1)

W. J. Miniscalco, “Erbium-Doped Glasses for Fiber Amplifiers at 1500-nm,” J. Lightwave Technol. 9(2), 234–250 (1991).
[Crossref]

1989 (1)

W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
[Crossref]

Bahtat, A.

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

Bahtat, M.

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

Balling, P.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

Bartl, M. H.

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
[Crossref] [PubMed]

Bass, M.

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

Bazan, G. C.

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

Bouazaoui, M.

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

Carlos, L. D.

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

Carnall, W. T.

W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
[Crossref]

Cassanho, A.

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

Chen, S.-Y.

S.-Y. Chen, C.-C. Ting, and W.-F. Hsieh, “Comparison of visible fluorescence properties between sol–gel derived Er3+–Yb3+ and Er3+–Y3+ co-doped TiO2 films,” Thin Solid Films 434(1–2), 171–177 (2003).
[Crossref]

Chen, X.

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

Chiodo, L.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

de Wild, J.

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

Ferreira, R. A. S.

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

Frindell, K. L.

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
[Crossref] [PubMed]

Fu, C.

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

Garapon, C.

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

Garcia-Lastra, J. M.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Goodman, G. L.

W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
[Crossref]

Green, M. A.

T. Trupke, M. A. Green, and P. Würfel, “Improving solar cell efficiencies by up-conversion of sub-band-gap light,” J. Appl. Phys. 92(7), 4117–4122 (2002).
[Crossref]

Hsieh, W.-F.

S.-Y. Chen, C.-C. Ting, and W.-F. Hsieh, “Comparison of visible fluorescence properties between sol–gel derived Er3+–Yb3+ and Er3+–Y3+ co-doped TiO2 films,” Thin Solid Films 434(1–2), 171–177 (2003).
[Crossref]

Iacomino, A.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Ishii, M.

M. Ishii, S. Komuro, and T. Morikawa, “Study on atomic coordination around Er doped into anatase– and rutile– TiO2: Er–O clustering dependent on the host crystal phase,” J. Appl. Phys. 94(6), 3823–3827 (2003).
[Crossref]

Jacquier, B.

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

Jenssen, H.

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

Jeppesen, B. R.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

Johannsen, S. R.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Julsgaard, B.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Komuro, S.

M. Ishii, S. Komuro, and T. Morikawa, “Study on atomic coordination around Er doped into anatase– and rutile– TiO2: Er–O clustering dependent on the host crystal phase,” J. Appl. Phys. 94(6), 3823–3827 (2003).
[Crossref]

Labrincha, J. A.

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

Lallier, E.

Lauridsen, L. R.

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Li, R.

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

Liu, C.-H.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Liu, Y.

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

Luo, W.

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

Madsen, S. P.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

Mao, C.-B.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Meijerink, A.

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

Mi, C.-C.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Mignotte, C.

C. Mignotte, “Structural characterization for Er3+-doped oxide materials potentially useful as optical devices,” Appl. Surf. Sci. 226(4), 355–370 (2004).
[Crossref]

C. Mignotte, “EXAFS studies on erbium-doped TiO2 and ZrO2 sol-gel thin films,” J. Non-Crystal. Solids 291(1–2), 56–77 (2001).

Milliez, J.

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

Miniscalco, W. J.

W. J. Miniscalco, “Erbium-Doped Glasses for Fiber Amplifiers at 1500-nm,” J. Lightwave Technol. 9(2), 234–250 (1991).
[Crossref]

Morikawa, T.

M. Ishii, S. Komuro, and T. Morikawa, “Study on atomic coordination around Er doped into anatase– and rutile– TiO2: Er–O clustering dependent on the host crystal phase,” J. Appl. Phys. 94(6), 3823–3827 (2003).
[Crossref]

Mugnier, J.

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

Neuvonen, P. T.

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Nygaard, J. V.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

Nylandsted Larsen, A.

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Ossicini, S.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Petek, H.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Popitsch, A.

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
[Crossref] [PubMed]

Pullar, R. C.

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

Rajnak, K.

W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
[Crossref]

Ram, S. K.

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Rana, R. S.

W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
[Crossref]

Rapaport, A.

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

Rath, J. K.

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

Robinson, M. R.

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

Rubio, A.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Schropp, R. E. I.

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

Seabra, M. P.

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

Stucky, G. D.

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
[Crossref] [PubMed]

Ting, C.-C.

S.-Y. Chen, C.-C. Ting, and W.-F. Hsieh, “Comparison of visible fluorescence properties between sol–gel derived Er3+–Yb3+ and Er3+–Y3+ co-doped TiO2 films,” Thin Solid Films 434(1–2), 171–177 (2003).
[Crossref]

Tobaldi, D. M.

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

Trupke, T.

T. Trupke, M. A. Green, and P. Würfel, “Improving solar cell efficiencies by up-conversion of sub-band-gap light,” J. Appl. Phys. 92(7), 4117–4122 (2002).
[Crossref]

van Sark, W. G. J. H. M.

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

Wang, M.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Wang, W.-X.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Wu, Y.-F.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Würfel, P.

T. Trupke, M. A. Green, and P. Würfel, “Improving solar cell efficiencies by up-conversion of sub-band-gap light,” J. Appl. Phys. 92(7), 4117–4122 (2002).
[Crossref]

Xu, S.-K.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Xu, Z.-R.

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Zhao, J.

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Zhu, H.

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

ACS Nano (1)

M. Wang, C.-C. Mi, W.-X. Wang, C.-H. Liu, Y.-F. Wu, Z.-R. Xu, C.-B. Mao, and S.-K. Xu, “Immunolabeling and NIR-excited fluorescent imaging of HeLa cells by using NaYF4:Yb,Er upconversion nanoparticles,” ACS Nano 3(6), 1580–1586 (2009).
[Crossref] [PubMed]

Angew. Chem. Int. Ed. Engl. (1)

K. L. Frindell, M. H. Bartl, A. Popitsch, and G. D. Stucky, “Sensitized luminescence of trivalent europium by three-dimensionally arranged anatase nanocrystals in mesostructured titania thin films,” Angew. Chem. Int. Ed. Engl. 41(6), 959–962 (2002).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. R. Johannsen, S. P. Madsen, B. R. Jeppesen, J. V. Nygaard, B. Julsgaard, P. Balling, and A. Nylandsted Larsen, “Up-conversion enhancement in Er3+ doped TiO2 through plasmonic coupling: Experiments and finite-element modeling,” Appl. Phys. Lett. 106(5), 053101 (2015).
[Crossref]

Appl. Surf. Sci. (1)

C. Mignotte, “Structural characterization for Er3+-doped oxide materials potentially useful as optical devices,” Appl. Surf. Sci. 226(4), 355–370 (2004).
[Crossref]

Energy Environ. Sci. (1)

J. de Wild, A. Meijerink, J. K. Rath, W. G. J. H. M. van Sark, and R. E. I. Schropp, “Upconverter solar cells: materials and applications,” Energy Environ. Sci. 4(12), 4835–4848 (2011).
[Crossref]

J. Appl. Phys. (2)

T. Trupke, M. A. Green, and P. Würfel, “Improving solar cell efficiencies by up-conversion of sub-band-gap light,” J. Appl. Phys. 92(7), 4117–4122 (2002).
[Crossref]

M. Ishii, S. Komuro, and T. Morikawa, “Study on atomic coordination around Er doped into anatase– and rutile– TiO2: Er–O clustering dependent on the host crystal phase,” J. Appl. Phys. 94(6), 3823–3827 (2003).
[Crossref]

J. Chem. Phys. (1)

W. T. Carnall, G. L. Goodman, K. Rajnak, and R. S. Rana, “A systematic analysis of the spectra of the lanthanides doped into single crystal LaF3,” J. Chem. Phys. 90(7), 3443–3457 (1989).
[Crossref]

J. Disp. Technol. (1)

A. Rapaport, J. Milliez, M. Bass, A. Cassanho, and H. Jenssen, “Review of the Properties of Up-Conversion Phosphors for New Emissive Displays,” J. Disp. Technol. 2(1), 68–78 (2006).
[Crossref]

J. Lightwave Technol. (1)

W. J. Miniscalco, “Erbium-Doped Glasses for Fiber Amplifiers at 1500-nm,” J. Lightwave Technol. 9(2), 234–250 (1991).
[Crossref]

J. Mater. Chem. C Mater. Opt. Electron. Devices (1)

D. M. Tobaldi, R. A. S. Ferreira, R. C. Pullar, M. P. Seabra, L. D. Carlos, and J. A. Labrincha, “Nano-titania doped with europium and neodymium showing simultaneous photoluminescent and photocatalytic behaviour,” J. Mater. Chem. C Mater. Opt. Electron. Devices 3(19), 4970–4986 (2015).
[Crossref]

J. Non-Cryst. Solids (1)

A. Bahtat, M. Bouazaoui, M. Bahtat, C. Garapon, B. Jacquier, and J. Mugnier, “Up-conversion fluorescence spectroscopy in Er3+: TiO2 planar waveguides prepared by a sol-gel process,” J. Non-Cryst. Solids 202(1–2), 16–22 (1996).
[Crossref]

J. Non-Crystal. Solids (1)

C. Mignotte, “EXAFS studies on erbium-doped TiO2 and ZrO2 sol-gel thin films,” J. Non-Crystal. Solids 291(1–2), 56–77 (2001).

J. Solid State Chem. (1)

K. L. Frindell, M. H. Bartl, M. R. Robinson, G. C. Bazan, A. Popitsch, and G. D. Stucky, “Visible and near-IR luminescence via energy transfer in rare earth doped mesoporous titania thin films with nanocrystalline walls,” J. Solid State Chem. 172(1), 81–88 (2003).
[Crossref]

Phys. Rev. B (1)

L. Chiodo, J. M. Garcia-Lastra, A. Iacomino, S. Ossicini, J. Zhao, H. Petek, and A. Rubio, “Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases,” Phys. Rev. B 82(4), 045207 (2010).
[Crossref]

Small (1)

W. Luo, C. Fu, R. Li, Y. Liu, H. Zhu, and X. Chen, “Er3+ -doped anatase TiO2 nanocrystals: crystal-field levels, excited-state dynamics, upconversion, and defect luminescence,” Small 7(21), 3046–3056 (2011).
[Crossref] [PubMed]

Thin Solid Films (2)

S.-Y. Chen, C.-C. Ting, and W.-F. Hsieh, “Comparison of visible fluorescence properties between sol–gel derived Er3+–Yb3+ and Er3+–Y3+ co-doped TiO2 films,” Thin Solid Films 434(1–2), 171–177 (2003).
[Crossref]

S. R. Johannsen, L. R. Lauridsen, B. Julsgaard, P. T. Neuvonen, S. K. Ram, and A. Nylandsted Larsen, “Optimization of Er3+-doped TiO2-thin films for infrared light up-conversion,” Thin Solid Films 550, 499–503 (2014).
[Crossref]

Other (1)

J.-C. G. Bünzli and S. V. Eliseeva, Basics of Lanthanide Photophysics (Springer-Verlag, 2010).

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Figures (9)

Fig. 1
Fig. 1 (a) XRD pattern at 3° grazing angle of TiO2:Er3+ 1.3 at% (red) and TiO2:Er3+ 6.1 at% (black). (b) Bright-field TEM cross section of TiO2:Er3+ 1.3 at%. The inset is a TEM diffraction pattern of TiO2:Er3+ 1.3 at%. (c) STEM dark-field cross section of TiO2:Er3+ 6.1 at%. The inset is a diffraction pattern of TiO2:Er3+ 6.1 at%.
Fig. 2
Fig. 2 (a) Example of one decay curve for the transition from level 4I13/2 to 4I15/2, obtained at an emission wavelength of 1532 nm and integration time of 5 ms. The areas marked with different colors correspond to the time intervals during which several other decay curves are integrated to form the spectra of panel (b). The inset in panel (a) shows the lower Er3+ energy levels under 800 nm light excitation (red arrow). Possible decay paths are indicated with black arrows. (b) Time-resolved emission spectra (TRES) at different time intervals at 16 K. The time intervals are indicated in the graph, and they correspond to the intervals indicated by colored areas in panel (a).
Fig. 3
Fig. 3 Typical decay curve (black) at 16 K with a double exponential- function fit with rise (red). Insert: Lifetimes as a function of emission wavelength (16 K).
Fig. 4
Fig. 4 Left scale and solid lines: TRES spectrum of the two environments, fast and slow decay, for the polycrystalline host, and the TRES spectrum for the amorphous host. Right scale and triangles: Area of the decay curves for the fast and the slow decay at different emission wavelengths. All data are obtained at 16 K.
Fig. 5
Fig. 5 (a) Emission spectra from TiO2:Er3+ 1.3 at% at 12 K. The excitation wavelengths are indicated in the legend, while the vertical dashed lines indicate the location of the emission peaks. (b) Normalized excitation spectra from TiO2:Er3+ 1.3 at% at 12 K. The studied emission wavelengths are indicated in the legend, and the vertical dashed lines indicate the excitation wavelengths used in panel (a).
Fig. 6
Fig. 6 (a) Emission spectrum at 326 nm excitation (12 K) with Gaussian curves fitted to the spectrum. Shown are the cumulative fits with 8 or 16 Gauss curves respectively, and the 16 individual Gauss curves. (b) Peak area of the Gaussian curves fitted to the emission spectrum at 326 and 340 nm excitation, as a function of the energy of the peak.
Fig. 7
Fig. 7 (a) Excitation spectrum (12 K) monitored at 1539 nm with three Gauss curves fitted to the spectrum. The cumulative fit is also included. (b) Energy level diagram at 12 K for the two types of Er3+ sitting in the TiO2:Er3+ 1.3 at% thin film. The two types are in the diagram denoted by 1529 and 1539 according to the studied emission line used for the excitation spectra. The left side of the diagram is constructed from the Gauss peak positions (thick black lines) and widths (spacing between thin gray lines) obtained from the excitation spectra at 1539 and 1529 nm emission, respectively. Thus the left side shows the excitation to higher energy levels. The A is short for anatase and the R for rutile. The right side show the Er3+ energy levels obtain from ref [11] for Er3+ in an anatase TiO2 host (ground state to 4G11/2). The higher energy levels are found in ref [20], however this is for a LaF3 host because the data are not available for TiO2. Hence only the central position of the energy level are indicated and not the CF splitting.
Fig. 8
Fig. 8 (a) Time-resolved emission spectrum for the TiO2:Er3+ 6.1 at% thin film at 16 K. The different time intervals are indicated in the legend. (b) Area of the decay curve for the amorphous host and polycrystalline (poly) host at 16 K.
Fig. 9
Fig. 9 (a) Normalized emission spectra of TiO2:Er3+ 6.1 at% at 12 K. The excitation wavelengths are indicated in the legend. (b) Normalized excitation spectra of TiO2:Er3+ at 12 K. The emission wavelengths are indicated in the legend.

Tables (1)

Tables Icon

Table 1 Lifetimes and rise times of level 4I13/2 for TiO2:Er3+ 1.3 at% (polycrystalline fast and slow decay), and TiO2:Er3+ 6.1 at% (amorphous) at 16 and 300 K. The stated uncertainty was calculated as the standard deviation of the individual lifetimes measured at each emission wavelength as shown in the inset of Fig. 3.

Equations (13)

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

d N 9 / 2 ( t ) d t = Γ 9 / 2 N 9 / 2 ( t )
d N 11 / 2 ( t ) d t = Γ 11 / 2 N 11 / 2 ( t ) + b 11 / 2 9 / 2 Γ 9 / 2 N 9 / 2 ( t )
d N 13 / 2 ( t ) d t = Γ 13 / 2 N 13 / 2 ( t ) + b 13 / 2 9 / 2 Γ 9 / 2 N 9 / 2 ( t ) + b 13 / 2 11 / 2 Γ 11 / 2 N 11 / 2 ( t )
d N 15 / 2 ( t ) d t = b 15 / 2 9 / 2 Γ 9 / 2 N 9 / 2 ( t ) + b 15 / 2 11 / 2 Γ 11 / 2 N 11 / 2 ( t ) + Γ 13 / 2 N 13 / 2 ( t )
N 9 / 2 ( t ) = N 9 / 2 ( 0 ) e Γ 9 / 2 t
N 11 / 2 ( t ) [ N 11 / 2 ( 0 ) + b 11 / 2 9 / 2 N 9 / 2 ( 0 ) ] e Γ 11 / 2 t N ˜ 11 / 2 ( 0 ) e Γ 11 / 2 t
N 13 / 2 ( t ) N ˜ 13 / 2 ( 0 ) e Γ 13 / 2 t 1 a r i s e e Γ r i s e t 1 a r i s e
a r i s e 1 1 + N ˜ 13 / 2 ( 0 ) N ˜ 11 / 2 ( 0 ) Γ 11 / 2 Γ 13 / 2 Γ 11 / 2 b 13 / 2 11 / 2
Γ r i s e = Γ 11 / 2 Γ 13 / 2
N ˜ 13 / 2 ( 0 ) = N 13 / 2 ( 0 ) + b 13 / 2 9 / 2 N 9 / 2 ( 0 )
N 13 / 2 ( t ) = A e Γ 13 / 2 t ( 1 a r i s e e Γ r i s e t )
A = N ˜ 13 2 ( 0 ) 1 a r i s e
N 1 + 2 ( t ) = ( A 1 e Γ 1 t + A 2 e Γ 2 t ) ( 1 a r i s e e Γ r i s e t )

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