## Local refractive index probed via the fluorescence decay of semiconductor quantum dots |

Optics Express, Vol. 20, Issue 3, pp. 3200-3208 (2012)

http://dx.doi.org/10.1364/OE.20.003200

Acrobat PDF (806 KB)

### Abstract

We present a novel approach for convenient tuning of the local refractive index around nanostructures. We apply this technique to study the influence of the local refractive index on the radiative decay time of CdSe/ZnS quantum dots with three distinct emission wavelengths. The dependence of the luminescence decay time on the environment is well described by an effective medium approach. A critical distance of about 80 nm is found for the determination of the effective local index of refraction. An estimation for the emitting-state quantum efficiency can be extracted.

© 2012 OSA

## 1. Introduction

*k*

_{r}of a quantum system in an isotropic medium can be written as [1

1. E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. **4**, 87–132 (1932). [CrossRef]

3. D. Toptygin, “Effects of the solvent refractive index and its dispersion on the radiative decay rate and extinction coefficient of a fluorescent solute,” J. Fluoresc. **13**, 201–219 (2003). [CrossRef]

*τ*

_{r}is the radiative decay time,

*ω*

_{12}is the angular emission frequency,

*ρ*(

*ω*

_{12}) is the density (per unit energy) of field oscillators at frequency ω

_{12}, 𝓔

_{loc}is the local zero-point field at the position of the emitting center, and

*μ*

_{12}is the transition dipole moment between its ground and excited states. The refractive index of the surrounding medium affects both

*ρ*(

*ω*

_{12}) and 𝓔

_{loc}. An additional influence of the environment can result from optical coupling of the emitter to a cavity, which changes the density of field oscillators

*ρ*(

*ω*

_{12}). This effect can be used for emission enhancement [4

4. A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, A. J. Meixner, and J. Enderlein, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. **102**, 073002 (2009). [CrossRef] [PubMed]

5. A. Belarouci, F. Menchini, H. Rigneault, B. Jacquier, R. M. Montereali, F. Somma, and P. Moretti, “Spontaneous emission properties of color centers based optical microcavities,” Opt. Commun. **189**, 281–287 (2001). [CrossRef]

6. J. Vučković, D. Fattal, C. Santori, and G. S. Solomon, “Enhanced single-photon emission from a quantum dot in a micropost microcavity,” Appl. Phys. Lett. **82**, 3596–3598 (2003). [CrossRef]

7. A. I. Chizhik, A. M. Chizhik, D. Khoptyar, S. Bär, A. J. Meixner, and J. Enderlein, “Probing the radiative transition of single molecules with a tunable microresonator,” Nano Lett. **11**, 1700–1703 (2011). [CrossRef] [PubMed]

8. G. Lamouche, P. Lavallard, and T. Gacoin, “Optical properties of dye molecules as a function of the surrounding dielectric medium,” Phys. Rev. A **59**, 4668–4674 (1999). [CrossRef]

9. X. Brokmann, L. Coolen, M. Dahan, and J. P. Hermier, “Measurement of the radiative and nonradiative decay rates of single cdse nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. **93**, 107403 (2004). [CrossRef] [PubMed]

10. S. F. Wuister, C. D. Donega, and A. Meijerink, “Local-field effects on the spontaneous emission rate of cdte and cdse quantum dots in dielectric media,” J. Chem. Phys. **121**, 4310–4315 (2004). [CrossRef] [PubMed]

11. R. S. Meltzer, S. P. Feofilov, B. Tissue, and H. B. Yuan, “Dependence of fluorescence lifetimes of y2o3 : Eu3+ nanoparticles on the surrounding medium,” Phys. Rev. B **60**, 14012–14015 (1999). [CrossRef]

13. D. W. Cooke, J. K. Lee, B. L. Bennett, J. R. Groves, L. G. Jacobsohn, E. A. McKigney, R. E. Muenchausen, M. Nastasi, K. E. Sickafus, M. Tang, J. A. Valdez, J. Y. Kim, and K. S. Hong, “Luminescent properties and reduced dimensional behavior of hydrothermally prepared y(2)sio(5): ce nanophosphors,” Appl. Phys. Lett. **88**, 103108 (2006). [CrossRef]

_{2}O

_{3}:Eu

^{3+}) [14

14. V. LeBihan, A. Pillonnet, D. Amans, G. Ledoux, O. Marty, and C. Dujardin, “Critical dimension where the macroscopic definition of refractive index can be applied at a nanometric scale,” Phys. Rev. B **78**, 113405 (2008). [CrossRef]

## 2. Experimental section

### 2.1. Sample preparation

*c*= 10

^{−8}mol/l for each of the three species in the same aqueous solution, was spin-cast onto a suprasil substrate to obtain a film that contained all three sizes of QDs, so that they could be studied under identical conditions. The total luminescence intensity detected from an excitation focus of 500 nm diameter agrees with a monolayer coverage of the substrate with an average of 35–70 nanoparticles of each size in the focus, in accordance with the dilution series carried out to find the optimum concentration of QDs before spin-casting. The QD film was covered with a BK7 lens of focal length

*f*= 12.5 mm, leading to a varying air gap between the substrate and the lens as illustrated in Fig. 1(b). The minimum surface-to-surface distance between lens and substrate,

*d*

_{min}in Fig. 1(b), was calculated from the diameter of the first green interference ring for white-light transmission and the known radius of curvature of the lens, (13.5 ± 0.3) mm;

*d*

_{min}was thus found to be (44 ± 7) nm. The diameter of the largest QD species in the sample is

*d*= 18 nm (including shell and organic ligands) according to the manufacturer. We attribute the difference between this value and our measured

*d*

_{min}to the nonzero probability of occasionally finding an agglomerate with a height of two times the particle diameter.

### 2.2. Confocal microscopy and fluorescence lifetime measurements

## 3. Results and discussion

*d*of the air gap is shown in Fig. 2, which represents the averaged results of twenty different line-scans across the center. As anticipated, one finds the same trend of lengthening luminescence decay time with widening air gap for all three samples, with a saturation of the effect occurring at a lens-substrate distance of about 100 nm, when the lifetime reaches its limiting value.

7. A. I. Chizhik, A. M. Chizhik, D. Khoptyar, S. Bär, A. J. Meixner, and J. Enderlein, “Probing the radiative transition of single molecules with a tunable microresonator,” Nano Lett. **11**, 1700–1703 (2011). [CrossRef] [PubMed]

*Q*is almost 50 times lower than the one of a cavity with metallic mirrors. It is therefore reasonable to neglect the modification of the radiative rate that could arise from the electromagnetic mode structure of the low-quality resonator formed by the two glass surfaces.

15. D. E. Aspnes, “Local-field effects and effective-medium theory - a microscopic perspective,” Am. J. Phys. **50**, 704–709 (1982). [CrossRef]

*R*that is centered on the QD layer. The width of the air gap above the QD layer at a given position can be calculated from the known radius of curvature of the lens, which yields the volume fractions

*f*

_{SiO2}of silica and

*f*

_{BK7}of BK7 in the sphere of interaction. The effective refractive index

*n̄*is then obtained as the positive real solution of the equation where

*n*

_{SiO2}= 1.46,

*n*

_{BK7}=1.52 and

*n*

_{air}= 1 are the refractive indices of silica, BK7 and air, respectively. This solution for

*n̄*was determined numerically in our fitting procedure. The volume fractions

*f*

_{X}of silica and BK7 were calculated by modeling the space occupied by these materials as spherical caps of heights

*h*

_{X}, which have a volume of [16] where

*R*is the radius of the interaction sphere. Division by the total volume of the interaction sphere,

*f*

_{X}=

*V*

_{SC}/

*V*as where the heights

*h*

_{X}of the spherical caps are given by For distances

*d*outside the sphere of interaction, i. e., for

*d*>

*R*+

*d*

_{min}/2, the height

*h*

_{BK7}was set to zero. The effective index of refraction

*n̄*for a given distance

*d*thus depends on the radius

*R*and on

*d*

_{min}, the minimum distance between the lens and the substrate, which was kept fixed at

*d*

_{min}= 44 nm (see experimental section) in all our fits. Given an interaction sphere with a radius

*R*around 100 nm, this means that our experimental geometry explored a range of effective refractive indices

*n̄*between 1.15 and 1.32.

*n*

_{air}for the entire space between lens and substrate as we have done for the calculation of

*n̄*outlined above.

*k*is given by the sum of the radiative and non-radiative decay rates, i. e.,

*k*=

*k*

_{r}+

*k*

_{nr}, where

*k*

_{r}depends on

*n̄*according to Eq. (6). The non-radiative rate

*k*

_{nr}is assumed to be dominated by internal radiationless relaxation processes of the quantum dots, which arise from electron-phonon coupling. The high-quality core-shell structure of the QDs is specifically designed to isolate the exciton in order to suppress energy transfer to the environment and to maximize luminescence emission. We can therefore expect

*k*

_{nr}to remain unaffected by changes in the local environment, hence we introduce it as an

*n̄*-independent parameter. We applied each one of the models to our data to extract

*R*,

*τ*

_{rv}= 1/

*k*

_{rv}, and

*τ*

_{nr}= 1/

*k*

_{nr}as fit parameters for the three QD sizes. The resulting best-fit curves have been included in Fig. 2 and the corresponding sets of parameters are given in Table 1, together with the resulting quantum efficiency Φ =

*k*

_{r}/(

*k*

_{r}+

*k*

_{nr}) of the emitting state.

*k*

_{r}on

*n̄*and thus cannot describe our data even for the maximum possible quantum efficiency of Φ = 1, which corresponds to

*k*

_{nr}= 0. A larger

*n̄*-independent non-radiative contribution would further diminish the predicted relative change in luminescence lifetime and thus further reduce the agreement of the FM model with our data. If we remove the

*k*

_{nr}≥ 0 restriction from the fit algorithm, we find that quantum efficiencies Φ between 1.2 and 1.4 would be required to bring the curve of the FM model as close to our data as those of the other two models. (Such values of Φ > 1 in consequence of

*k*

_{nr}< 0 are of course physically meaningless; we only mention these values here as an indication of the extent of disagreement between our data and the FM model.)

10. S. F. Wuister, C. D. Donega, and A. Meijerink, “Local-field effects on the spontaneous emission rate of cdte and cdse quantum dots in dielectric media,” J. Chem. Phys. **121**, 4310–4315 (2004). [CrossRef] [PubMed]

20. C. K. Duan, M. F. Reid, and Z. Q. Wang, “Local field effects on the radiative lifetime of emitters in surrounding media: Virtual- or real-cavity model?” Phys. Lett. A **343**, 474–480 (2005). [CrossRef]

9. X. Brokmann, L. Coolen, M. Dahan, and J. P. Hermier, “Measurement of the radiative and nonradiative decay rates of single cdse nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. **93**, 107403 (2004). [CrossRef] [PubMed]

21. J. Yao, D. R. Larson, H. D. Vishwasrao, W. R. Zipfel, and W. W. Webb, “Blinking and nonradiant dark fraction of water-soluble quantum dots in aqueous solution,” Proc. Natl. Acad. Sci. U.S.A. **102**, 14284–14289 (2005). [CrossRef] [PubMed]

9. X. Brokmann, L. Coolen, M. Dahan, and J. P. Hermier, “Measurement of the radiative and nonradiative decay rates of single cdse nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. **93**, 107403 (2004). [CrossRef] [PubMed]

20. C. K. Duan, M. F. Reid, and Z. Q. Wang, “Local field effects on the radiative lifetime of emitters in surrounding media: Virtual- or real-cavity model?” Phys. Lett. A **343**, 474–480 (2005). [CrossRef]

10. S. F. Wuister, C. D. Donega, and A. Meijerink, “Local-field effects on the spontaneous emission rate of cdte and cdse quantum dots in dielectric media,” J. Chem. Phys. **121**, 4310–4315 (2004). [CrossRef] [PubMed]

22. P. R. Berman and P. W. Milonni, “Microscopic theory of modified spontaneous emission in a dielectric,” Phys. Rev. Lett. **92**, 053601 (2004). [CrossRef] [PubMed]

22. P. R. Berman and P. W. Milonni, “Microscopic theory of modified spontaneous emission in a dielectric,” Phys. Rev. Lett. **92**, 053601 (2004). [CrossRef] [PubMed]

*R*around 80 nm for both the EC and the VC model, and we cannot detect a clear tendency for the dependence of

*R*on

*λ*in the small wavelength range covered by our experiments.

## 4. Conclusion

*n*≈ 2 are readily available, which could allow us to probe effective refractive indices of up to 1.6 – and furthermore it can be combined with advanced single-particle/molecule techniques such as full determination of the 3D orientation of the transition dipole and a detailed analysis of the emission pattern. As such, it can be expected to serve as a versatile tool to test and improve various models for the influence of the nanoscale environment on the radiative dynamics of different types of emitters, such as organic chromophores, semiconductor quantum dots, and doped-insulator nanoparticles.

## Acknowledgments

## References and links

1. | E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. |

2. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

3. | D. Toptygin, “Effects of the solvent refractive index and its dispersion on the radiative decay rate and extinction coefficient of a fluorescent solute,” J. Fluoresc. |

4. | A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, A. J. Meixner, and J. Enderlein, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. |

5. | A. Belarouci, F. Menchini, H. Rigneault, B. Jacquier, R. M. Montereali, F. Somma, and P. Moretti, “Spontaneous emission properties of color centers based optical microcavities,” Opt. Commun. |

6. | J. Vučković, D. Fattal, C. Santori, and G. S. Solomon, “Enhanced single-photon emission from a quantum dot in a micropost microcavity,” Appl. Phys. Lett. |

7. | A. I. Chizhik, A. M. Chizhik, D. Khoptyar, S. Bär, A. J. Meixner, and J. Enderlein, “Probing the radiative transition of single molecules with a tunable microresonator,” Nano Lett. |

8. | G. Lamouche, P. Lavallard, and T. Gacoin, “Optical properties of dye molecules as a function of the surrounding dielectric medium,” Phys. Rev. A |

9. | X. Brokmann, L. Coolen, M. Dahan, and J. P. Hermier, “Measurement of the radiative and nonradiative decay rates of single cdse nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. |

10. | S. F. Wuister, C. D. Donega, and A. Meijerink, “Local-field effects on the spontaneous emission rate of cdte and cdse quantum dots in dielectric media,” J. Chem. Phys. |

11. | R. S. Meltzer, S. P. Feofilov, B. Tissue, and H. B. Yuan, “Dependence of fluorescence lifetimes of y2o3 : Eu3+ nanoparticles on the surrounding medium,” Phys. Rev. B |

12. | H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. |

13. | D. W. Cooke, J. K. Lee, B. L. Bennett, J. R. Groves, L. G. Jacobsohn, E. A. McKigney, R. E. Muenchausen, M. Nastasi, K. E. Sickafus, M. Tang, J. A. Valdez, J. Y. Kim, and K. S. Hong, “Luminescent properties and reduced dimensional behavior of hydrothermally prepared y(2)sio(5): ce nanophosphors,” Appl. Phys. Lett. |

14. | V. LeBihan, A. Pillonnet, D. Amans, G. Ledoux, O. Marty, and C. Dujardin, “Critical dimension where the macroscopic definition of refractive index can be applied at a nanometric scale,” Phys. Rev. B |

15. | D. E. Aspnes, “Local-field effects and effective-medium theory - a microscopic perspective,” Am. J. Phys. |

16. | I. N. Bronshtein, K. A. Semendyayev, G. Müsiol, and H. Mühlig, |

17. | J. Knoester and S. Mukamel, “Intermolecular forces, spontaneous emission, and superradiance in a dielectric medium - polariton-mediated interactions,” Phys. Rev. A |

18. | R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A |

19. | M. E. Crenshaw and C. M. Bowden, “Effects of local fields on spontaneous emission in dielectric media,” Phys. Rev. Lett. |

20. | C. K. Duan, M. F. Reid, and Z. Q. Wang, “Local field effects on the radiative lifetime of emitters in surrounding media: Virtual- or real-cavity model?” Phys. Lett. A |

21. | J. Yao, D. R. Larson, H. D. Vishwasrao, W. R. Zipfel, and W. W. Webb, “Blinking and nonradiant dark fraction of water-soluble quantum dots in aqueous solution,” Proc. Natl. Acad. Sci. U.S.A. |

22. | P. R. Berman and P. W. Milonni, “Microscopic theory of modified spontaneous emission in a dielectric,” Phys. Rev. Lett. |

**OCIS Codes**

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: November 28, 2011

Revised Manuscript: January 11, 2012

Manuscript Accepted: January 19, 2012

Published: January 26, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Anne Pillonnet, Pierre Fleury, Alexey I. Chizhik, Anna M. Chizhik, David Amans, Gilles Ledoux, Florian Kulzer, Alfred J. Meixner, and Christophe Dujardin, "Local refractive index probed via the fluorescence decay of semiconductor quantum dots," Opt. Express **20**, 3200-3208 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-3-3200

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### References

- E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys.4, 87–132 (1932). [CrossRef]
- E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681–681 (1946).
- D. Toptygin, “Effects of the solvent refractive index and its dispersion on the radiative decay rate and extinction coefficient of a fluorescent solute,” J. Fluoresc.13, 201–219 (2003). [CrossRef]
- A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, A. J. Meixner, and J. Enderlein, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett.102, 073002 (2009). [CrossRef] [PubMed]
- A. Belarouci, F. Menchini, H. Rigneault, B. Jacquier, R. M. Montereali, F. Somma, and P. Moretti, “Spontaneous emission properties of color centers based optical microcavities,” Opt. Commun.189, 281–287 (2001). [CrossRef]
- J. Vučković, D. Fattal, C. Santori, and G. S. Solomon, “Enhanced single-photon emission from a quantum dot in a micropost microcavity,” Appl. Phys. Lett.82, 3596–3598 (2003). [CrossRef]
- A. I. Chizhik, A. M. Chizhik, D. Khoptyar, S. Bär, A. J. Meixner, and J. Enderlein, “Probing the radiative transition of single molecules with a tunable microresonator,” Nano Lett.11, 1700–1703 (2011). [CrossRef] [PubMed]
- G. Lamouche, P. Lavallard, and T. Gacoin, “Optical properties of dye molecules as a function of the surrounding dielectric medium,” Phys. Rev. A59, 4668–4674 (1999). [CrossRef]
- X. Brokmann, L. Coolen, M. Dahan, and J. P. Hermier, “Measurement of the radiative and nonradiative decay rates of single cdse nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett.93, 107403 (2004). [CrossRef] [PubMed]
- S. F. Wuister, C. D. Donega, and A. Meijerink, “Local-field effects on the spontaneous emission rate of cdte and cdse quantum dots in dielectric media,” J. Chem. Phys.121, 4310–4315 (2004). [CrossRef] [PubMed]
- R. S. Meltzer, S. P. Feofilov, B. Tissue, and H. B. Yuan, “Dependence of fluorescence lifetimes of y2o3 : Eu3+ nanoparticles on the surrounding medium,” Phys. Rev. B60, 14012–14015 (1999). [CrossRef]
- H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett.89, 257403 (2002). [CrossRef] [PubMed]
- D. W. Cooke, J. K. Lee, B. L. Bennett, J. R. Groves, L. G. Jacobsohn, E. A. McKigney, R. E. Muenchausen, M. Nastasi, K. E. Sickafus, M. Tang, J. A. Valdez, J. Y. Kim, and K. S. Hong, “Luminescent properties and reduced dimensional behavior of hydrothermally prepared y(2)sio(5): ce nanophosphors,” Appl. Phys. Lett.88, 103108 (2006). [CrossRef]
- V. LeBihan, A. Pillonnet, D. Amans, G. Ledoux, O. Marty, and C. Dujardin, “Critical dimension where the macroscopic definition of refractive index can be applied at a nanometric scale,” Phys. Rev. B78, 113405 (2008). [CrossRef]
- D. E. Aspnes, “Local-field effects and effective-medium theory - a microscopic perspective,” Am. J. Phys.50, 704–709 (1982). [CrossRef]
- I. N. Bronshtein, K. A. Semendyayev, G. Müsiol, and H. Mühlig, Handbook of Mathematics, 5th ed. (Springer, 2007).
- J. Knoester and S. Mukamel, “Intermolecular forces, spontaneous emission, and superradiance in a dielectric medium - polariton-mediated interactions,” Phys. Rev. A40, 7065–7080 (1989). [CrossRef] [PubMed]
- R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A43, 467–491 (1991). [CrossRef] [PubMed]
- M. E. Crenshaw and C. M. Bowden, “Effects of local fields on spontaneous emission in dielectric media,” Phys. Rev. Lett.85, 1851–1854 (2000). [CrossRef] [PubMed]
- C. K. Duan, M. F. Reid, and Z. Q. Wang, “Local field effects on the radiative lifetime of emitters in surrounding media: Virtual- or real-cavity model?” Phys. Lett. A343, 474–480 (2005). [CrossRef]
- J. Yao, D. R. Larson, H. D. Vishwasrao, W. R. Zipfel, and W. W. Webb, “Blinking and nonradiant dark fraction of water-soluble quantum dots in aqueous solution,” Proc. Natl. Acad. Sci. U.S.A.102, 14284–14289 (2005). [CrossRef] [PubMed]
- P. R. Berman and P. W. Milonni, “Microscopic theory of modified spontaneous emission in a dielectric,” Phys. Rev. Lett.92, 053601 (2004). [CrossRef] [PubMed]

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