APPLIED PHYSICS LETTERS 93, 261109 共2008兲

Geometrical effects in the energy transfer mechanism for silicon nanocrystals and Er3+ K. Choy, F. Lenz, X. X. Liang, F. Marsiglio, and A. Meldruma兲 Department of Physics, University of Alberta, Edmonton, Alberta T6G2G7, Canada

共Received 17 October 2008; accepted 5 December 2008; published online 29 December 2008兲 Silicon nanoclusters 共NCs兲 strongly sensitize the luminescence of Er3+ ions. Attempts to calculate the interaction distance have assumed that the Förster 关Ann. Phys. 437, 55 共1948兲兴 and Dexter 关J. Chem. Phys. 21, 836 共1953兲兴 relationships for point-to-point energy transfer can be applied to experiments based on multilayered thin-film specimens. Here, the effective finite plane-to-plane relationships are derived for both interaction mechanisms. These relationships show that energy transfer can result from the Förster interaction despite the fact that the measured luminescence intensity varies much more weakly with NC-Er3+ separation than predicted by theory for point dipoles. An effective energy transfer distance is found for the NC-Er3+ system. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3058440兴 Silicon nanoclusters 共NCs兲 sensitize the luminescent 4f-shell transitions in the rare earth ions,1–3 with Er3+ being most widely studied because of its fluorescence near 1.53 ␮m.4 By employing nanocrystals as sensitizers, the rare earths can be excited nonresonantly, and the absorption cross sections—normally in the order of 10−20 cm2—become effectively similar to those of the NCs 共⬃10−16 cm2兲.5 This has led to several technical advances, including the initial reports of optical gain at 1.5 ␮m in erbium-doped silicon nanocomposites pumped with blue light emitting diodes.6 These materials now have many potential applications ranging from waveguide amplifiers to light emitters for microphotonics.7 However, despite these promising developments there has been uncertainty over the nature of the energy transfer mechanism. There are three sensitization mechanisms for the nanocrystal donor 共D兲 and Er3+ acceptor 共A兲 species: radiative energy transfer, multipolar Coulomb 共Förster兲 energy transfer, and the overlapping wave function 共Dexter兲 transfer. Of these, the radiative mechanism is discarded because the excitation of Er3+ would be even less efficient than for direct pumping. The efficiencies of the Förster8 and Dexter9 mechanisms are different functions of the separation r between the D and A species. In the Förster case, the transfer rate 共wTr兲 varies according to r−b where b = 6 for the electric dipoledipole case, b = 8 for the electric dipole-quadrupole case, and so on.9 For the Dexter mechanism, wTr = ␣e−br where b is a constant equal to 2 / L, with L being the sum of the mean van der Waals radii for the D and A states 共about 0.4 nm for Si and Er兲. A discussion of the different mechanisms and their mathematical formalisms can be found in Ref. 10, and the original theory is in Ref. 9. The energy transfer between Si NCs and Er3+ was measured by several groups using multilayered films in which the Er3+ ions can be separated from the NCs. These efforts found that the transfer efficiency decreases exponentially with r,11–15 implying an interaction whose characteristic length is approximately 0.5 nm for amorphous NCs 共Ref. 13兲 or about 2.1 nm for crystalline ones.14 Here, the relationships will be derived for both ena兲

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ergy transfer processes when the D and A species are contained in separate thick planes. The transfer rate between two point dipoles is wTr = wPL共R0 / r兲6, where wPL is the donor photoluminescence 共PL兲 rate and R0 is the distance at which wTr = wPL. Treating the emitter as a single point and integrating the transfer rate over a plane of radius y of acceptor molecules 共of number per unit area ␴兲 separated by a perpendicular distance x from the donor yields wTr = wPL␥␴

冕

⬁

␲wPL␴␥R60 2␲ y , 2 3 dy = 共x + y 兲 2x4

共1兲

2

0

where ␥ is the homogeneously distributed orientation factor.16–18 When the transfer rate is averaged over a thin donor plane, one obtains the trivial result identical to Eq. 共1兲. In order to obtain the average transfer rate between a donor and an acceptor plane of thicknesses tD and tA, respectively, separated by a spacer of thickness x, we have wTr共x兲 = =

1 tA

冋

冕

x+ta

x

冋

1 tD

冕

x⬘+x0⬘+td

x⬘+x0⬘

册

册

␲wPL␥␴R60 dx⬙ dx⬘ 2共x⬙兲4

␲wPL␥␴R60 M共x兲, 12tDtA

共2兲

where M共x兲 = 1 / 共x + x0兲2 − 1 / 共x + x0 + ta兲2 − 1 / 共x + x0 + td兲2 + 1 / 共x + x0 + ta + td兲2. Here, x0 is the minimum D-A separation distance taken as one bond length 共x0 ⬇ 0.3 nm兲 from the D D plane. Next, normalizing the D intensity ID共x兲 = wRD / 共wPL + wTr兲 to its intensity at x = ⬁, D = Inorm

ID共x兲 = ID共⬁兲

1 . ␲␥␴R60 1+ M共x兲 12tAtD

共3兲

Similarly, the acceptor intensity can be normalized to that at x = x 0, A = Inorm

冋

12tAtD +1 ␲␥␴R60M共x0兲

册冒冋

册

12tDtA + 1 . 共4兲 ␲␥␴R60M共x兲

These equations are shown in Fig. 1共a兲 for different values of R0. The two following features are apparent: 共i兲 the interac93, 261109-1

© 2008 American Institute of Physics

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FIG. 2. 共Color online兲 共a兲 EELS image in which the slit was centered on the silicon plasmon line at 16.5 eV, showing a layer of ⬃4 – 5 nm diameter NCs 共bright spots兲. 共b兲 Cross-sectional TEM image showing Z-contrast data 共red, corresponding to erbium兲, superimposed on an EELS Si plasmon image 共green兲. Some Er clustering is evident despite the low concentration. The brighter yellow-green line 共arrowed兲 represents the layer of NCs magnified in 共a兲. This image was taken from the end of the sample where the buffer layer thickness was nearly zero.

FIG. 1. 共Color online兲 共a兲 D 共red兲 and A 共blue兲 normalized fluorescence intensity as a function of plane-to-plane separation distance 共x − x0兲 for the Förster transfer. The D and A planes were 5 and 140 nm thick, respectively. The numbers beside the curves represent the associated R0, in nanometers. The black line shows the acceptor intensity for point-to-point Förster transfer. 共b兲 is the same as in 共a兲 except with the exchange transfer mechanism with w0 = 109 s−1. Numbers beside the curves represent b from 1 to 9 nm−1 in regular intervals. The value of b = 5 nm−1 corresponds to the overlap of 0.4 nm.

sample was annealed at 1000 ° C in flowing N2H2 gas to precipitate passivated Si NCs in the SiO layer.19 The layer thicknesses and compositions were confirmed by Rutherford backscattering spectrometry 共RBS兲 using the QUARK data analysis program, and TEM was performed on a JEOL 2200 FEGSTEM. PL spectroscopy used standard methods described elsewhere.3 The laser wavelength was 476 nm, which is not resonant with any Er3+ transition.20 Figure 3 shows the donor 共Si-NCs兲 and acceptor 共Er3+兲

tion strength drops off less sharply than r−6 and 共ii兲 the donor emission does not go to zero as R0 becomes small compared with tA and tD. This is due to the possibility for donors located far from the interface to radiate instead of transfer to the acceptor plane. The plane-to-plane transfer rate for the Dexter exchange interaction can be solved by integrating w0e−bx over two planes separated by distance x, yielding wTr =

2␲␴w0e−bx 关r共bx兲 + s兴. b 4t at d

共5兲

Here, r = 共1 − e−btd兲共1 − e−bta兲, s = 3共1 − e−btd兲共1 − e−bta兲 −bta −btd −btd −bta − btae 共1 − e 兲 − btde 共1 − e 兲, and w0 is a prefactor. The normalized donor intensity has a form similar to Eq. 共3兲, D = wPL / 共wPL + wTr兲, with wTr given by Eq. 共5兲, and where Inorm A the normalized acceptor intensity is Inorm = 关wTr共x兲共wPL + wTr共x0兲兲兴 / 关wTr共x0兲共wPL + wTr共x兲兲兴. These equations cannot be further simplified because of the parameter w0. Figure 1共b兲 shows that for a given 1 / b the effects occur over shorter distances if the interacting species are in thick planes. In order to compare with the theory, a multilayer thin film was deposited on a quartz substrate. The layers consisted of, in order from the bottom: 180 nm SiO2, 5 nm SiO 共NC layer兲, 0–40 nm SiO2 共“spacer” layer graded in thickness across the wafer兲, 140 nm SiO2 : Er共1.15 ⫻ 1019 Er3+ / cm3兲, and a 130 nm SiO2 cap 共Fig. 2兲. The

FIG. 3. 共Color online兲 共a兲 PL spectra for Er3+ fluorescence as a function of buffer layer thickness from 0 to 20 nm. 共b兲 Integrated Er3+ fluorescence intensity as a function of spacer thickness along with a best fit from Eq. 共4兲 共blue line兲 and for the point-to-point Förster transfer 共black line兲. The inset shows the Si NC PL intensity 共which peaked at ⬃770 nm兲 as a function of buffer thickness, with a fit from Eq. 共3兲.

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emission as a function of the spacer thickness. The structure in the spectrum is due to crystal-field-splitting in silica, in which the F1 and F7 Er3+ subpeaks are characteristically the strongest.21 The intensity follows a trend that can be fit with Eqs. 共3兲 and 共4兲 共R2 ⬇ 0.9兲 with a single fitting parameter R60␴␥. Estimating ␴ = 0.24 nm−2 from the RBS measurements and ␥ = ␲ / 3,16 yielded a mean effective R0 value of 7.0 nm in the fitting of the acceptor PL, or 5.8 nm for the donor PL. The NC PL data trended appropriately but was more scattered than the data for Er3+. Similar or larger R0 values have been reported for Eu3+ and Tb3+ with organic molecular sensitizers.22 By fitting the same data for the 1 / r6 point-to-point energy transfer, R0 becomes 2.8 nm—in close accord with previous estimates; however, with a much steeper dependence on separation distance than observed in the data 关Fig. 3共b兲兴. Effectively, the planar geometry “smoothes” the r−6 dependence and makes it look exponential. In comparison, a characteristic distance of 5.0 nm was found by fitting the point-to-point exchange function to the acceptor PL—also in agreement with previous work11—but with a nonphysically large transfer distance. Even larger exchange distances would be required for plane-to-plane transfer, as shown in Fig. 1共b兲. Finally, the lifetime of the NC emission followed the stretched exponential function exp关−共t / ␶兲␤兴, with ␤ between 0.6 and 0.7 and lifetimes ranged from 40 ␮s for a 1 nm spacer thickness up to 55 ␮s in without the Er-doped layer. This difference is smaller than expected for R0 values estimated here but the analysis does not include the stretched exponential factors or NC-NC interactions.23 Nevertheless, the shorter lifetimes found for a thinner spacer is consistent with the transfer effect. In order to calculate a single transfer time from a random distribution of NCs and Er3+ 共as in most samples兲, one can integrate the transfer rate for acceptors distributed uniformly around a point dipole as wPLR60␳␥兰r⬁ 4␲r2 / r6dr indicating that the ef0 fective transfer time is sensitive to the chosen minimum NC-Er separation. With the distance r0 = 0.3 nm and R0 = 7 nm, we have wTr of ⬃109 s−1. This rate is consistent with the “fast” transfer process reported previously24 but not with the much slower processes also reported.24,25 However, the calculated rate decreases quickly 共r−3 0 兲 for minimum distances greater than 0.3 nm. Although Eqs. 共3兲–共5兲 are analytical solutions, there are experimental sources of error. First, we can only obtain an effective R0 that is averaged over the entire NC size distribution and the resulting phonon-assisted transfer rates. Additional effects that can impact the estimated transfer distances include possible erbium diffusion into the spacer during annealing, NC-NC energy transfer, and the size of the NCs which is significant compared to the transfer distances. Despite these issues, however, Eqs. 共3兲–共5兲 are general for any planar geometry of atoms or molecules interacting via the

Förster or Dexter mechanisms. Essentially, both mechanisms can fit the data for Si NCs and Er3+, but with nonphysically long-range interactions for the Dexter case. For Si NCs and Er3+, an R0 in the range of 6–7 nm is larger than a previously estimated interaction distance for this system,14 but is comparable to those for rare earths sensitized by organic molecules,22 and is within the theoretical range for Si NC-NC transfer as well.26 In conclusion, the long interaction distance for Si NCs and Er3+ is consistent with a strong sensitizing effect that could be important in applications ranging from waveguide amplifiers to fluorescence imaging with silicon nanocrystals. The authors are funded by NSERC and iCORE. Thanks to M. Malac and P. Li for TEM data 关NINT Electron Microscopy Facility 共NRC兲兴 and to Dr. P. Bianucci and an anonymous referee for critical reviews. 1

G. Franzo, V. Vinciguerra, and F. Priolo, Appl. Phys. A: Mater. Sci. Process. 69, 3 共1999兲. 2 G. Franzo, V. Vinciguerra, and F. Priolo, Philos. Mag. B 80, 719 共2000兲. 3 A. Meldrum, A. Hryciw, N. MacDonald, C. Blois, T. Clement, R. DeCorby, J. Wang, and Q. Li, J. Lumin. 121, 199 共2006兲. 4 A. Polman and F. C. J. M. van Veggel, J. Opt. Soc. Am. B 21, 871 共2004兲. 5 C. E. Chryssou, A. J. Kenyon, and C. W. Pitt, Mater. Sci. Eng., B 81, 16 共2001兲. 6 H. Lee, J. H. Shin, and N. Park, Opt. Express 13, 9881 共2005兲. 7 L. Pavesi, Advances in Optical Technologies 2008, 416926. 8 Th. Förster, Ann. Phys. 437, 55 共1948兲. 9 D. L. Dexter, J. Chem. Phys. 21, 836 共1953兲. 10 N. L. Vekshin, Energy Transfer in Macromolecules 共SPIE, Bellingham, 1997兲. 11 F. Gourbilleau, C. Dufour, R. Madelon, and R. Rizk, Opt. Appl. 37, 21 共2007兲. 12 F. Gourbilleau, C. Dufour, R. Madelon, and R. Rizk, J. Lumin. 126, 581 共2007兲. 13 J.-H. Jhe, J. H. Shin, K. J. Kim, and D. W. Moon, Appl. Phys. Lett. 82, 4489 共2003兲. 14 T. Kimura, H. Isshiki, S. Ide, T. Shimizu, T. Ishida, and R. Saito, J. Appl. Phys. 93, 2595 共2003兲. 15 B. Garrido, C. Garcia, P. Pellegrino, D. Navarro-Urrios, N. Daldosso, L. Pavesi, F. Gourbilleau, and R. Rizk, Appl. Phys. Lett. 89, 163103 共2006兲. 16 B. Richter and S. Kirstein, J. Chem. Phys. 111, 5191 共1999兲. 17 H. Kuhn, J. Chem. Phys. 53, 101 共1970兲. 18 P. Fromherz and G. Reinbold, Thin Solid Films 160, 347 共1988兲. 19 M. Glover and A. Meldrum, Opt. Mater. 27, 977 共2005兲. 20 A. Hryciw, C. Blois, A. Meldrum, T. Clement, R. DeCorby, and Q. Li, Opt. Mater. 28, 873 共2006兲. 21 J. A. Buck, Fundamentals of Optical Fibers 共Wiley, New Jersey, 2004兲. 22 See J. R. Lakowicz, Principles of Fluorescence Spectroscopy 3rd ed. 共Springer, New York, 2006兲, and references therein. 23 A. Meldrum, R. Lockwood, V. A. Belyakov, and V. A. Burdov, Physica E 共to be published兲, http://dx.doi.org/10.1016/j.physe.2008.08.044. 24 M. Fujii, K. Imakita, K. Watanabe, and S. Hayashi, J. Appl. Phys. 95, 272 共2004兲. 25 For example, see M. Falconieri, E. Borsella, F. Enrichi, G. Franzò, F. Priolo, F. Iacona, F. Gourbilleau, and R. Rizk, Opt. Mater. 27, 884 共2005兲. 26 G. Allan and C. Delerue, Phys. Rev. B 75, 195311 共2007兲.

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