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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 Optical Society of America
1. Introduction
Tuning the photoluminescence of quantum emitters via modification of their local environment has attracted considerable interest in recent years. The optical properties of nanostructures can vary dramatically, as compared to macroscopic structures, due to the emergence of quantum confinement effects. Hence problems of interpretation can arise when optical phenomena are studied at the nanometer scale. The refractive index is particularly important in this context because it has a strong influence on the radiative transition probability of an emitter, as quantified by Fermi’s golden rule: The spontaneous emission rate kr of a quantum system in an isotropic medium can be written as [1–3]
where τr is the radiative decay time, ω12 is the angular emission frequency, ρ(ω12) is the density (per unit energy) of field oscillators at frequency ω12, 𝓔loc is the local zero-point field at the position of the emitting center, and μ12 is the transition dipole moment between its ground and excited states. The refractive index of the surrounding medium affects both ρ(ω12) and 𝓔loc. 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(ω12). This effect can be used for emission enhancement [4] or for controlling the radiation pattern [5, 6], as well as to determine the quantum yield of a luminescent system at the level of single nano-objects [7]. The influence of the refractive index becomes more pronounced when fluorescence decay times or luminescence quantum yields of nanoparticles are considered. This environmental effect has been demonstrated for many types of emitters embedded in various media, for example molecules [8], semiconductor quantum dots [9, 10] or doped insulators [11–13]. If such experiments deal with nanostructures then it is commonly assumed that the bulk refractive index remains meaningful for the surrounding material at the micrometer scale, which implies that a cut-off distance for a “local” refractive index around the emitters can be established, which allows to define a sphere of interaction. We have previously explored this issue by means of an active nanolayer covered by passive nanolayers of varying thicknesses, and we thus extracted a radius of sensitivity between 100 nm and 150 nm for an emitter at a wavelength of 610 nm (Gd2O3:Eu3+) [14]. However, the applicability of this concept to different types of emitters as well as the wavelength dependence of this radius remain open questions. In this article, we therefore present the application of this approach to CdSe/ZnS quantum dots of three different sizes, which enables us to study the same emitting species at three distinct emission wavelengths. We furthermore explore the possibility to extract the quantum yield of the emitter by means of controlling the surrounding medium.2. Experimental section
2.1. Sample preparation
We used water-soluble CdSe/ZnS core/shell quantum dots (QDs, Invitrogen) in three different sizes, whose luminescence spectra are shown in Fig. 1(a). A dilute mixture of these QDs, with a concentration of 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, dmin 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; dmin 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 dmin to the nonzero probability of occasionally finding an agglomerate with a height of two times the particle diameter.
Fig. 1 Luminescence decay times of CdSe/ZnS quantum dots (QDs) reflect variations in the local refractive index. (a) Emission spectra of the CdSe/ZnS quantum dots under excitation with 473 nm laser light. The three different sizes of QDs have distinct emission wavelengths. (b) Construction of the sample: A dilute aqueous solution of three different QDs is spin-cast onto a flat substrate and then covered by a lens, whose optical axis defines the zero position. Laser excitation and luminescence collection is performed through the substrate. The radius R of the interaction sphere is a fit parameter for calculating the effective local index of refraction. (c) Two typical luminescence decay curves and mono-exponential fits for QD565. The difference in lifetime arises from variations in the air gap above the QD film; the lower curve corresponds to the center of the substrate-lens sandwich, the upper trace was recorded 70 μm off-center.
2.2. Confocal microscopy and fluorescence lifetime measurements
The substrate-lens construct was mounted on a home-built confocal microscope based on a Zeiss Axiovert 135 TV in combination with a feedback-controlled sample scanning stage (Physik Instrumente, E-710.3CD) with nanometer positioning accuracy. A pulsed laser diode operating at 473 nm and 10 MHz repetition rate (Picoquant GmbH, LDH-P-C-470) was used to excite the QDs, and an oil-immersion objective (Zeiss plan-neofluar, 100×, NA=1.25, focal spot size below 500 nm) served to both focus the excitation beam and to collect the QD luminescence. The position-dependent luminescence intensity and decay time were recorded by raster-scanning the sample though the excitation focus; selective collection of the luminescence of each individual QD species was assured by appropriate bandpass filters in the detection path. Time-correlated single-photon counting was performed by focusing the QD luminescence onto the active area of a single-photon counting avalanche photo diode (APD) (Perkin Elmer, SPCM 200) and analyzing the timing of the output pulses of the APD with fluorescence lifetime imaging electronics (Picoquant, Picoharp 300 and SymPhoTime software package). All measured luminescence decay curves were fitted with a mono-exponential function, A exp(–t/τ) + B, after deconvolution with the instrument response function.
3. Results and discussion
Our experimental geometry allowed straightforward selection of emitters that experience different effective refractive indices depending on their distance from the optical axis of the lens. The influence of the local refractive index was thus quantified by measuring the resulting changes of the QD luminescence decay time in a home-built confocal microscope with fluorescence lifetime imaging capabilities. Figure 1(c) shows the raw data of two luminescence decay curves measured for QD565. The curves are well described by a mono-exponential decay function; inhomogeneities and QD interactions obviously play a minor role due to the low substrate coverage and the averaging in the small diffraction-limited focal spot. One finds the expected increase of the luminescence lifetime with growing distance from the optical axis of the lens, which reflects the decreasing effective index of refraction caused by the widening air gap between substrate and lens. The evolution of the decay times for the three QD sizes as a function of the width 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.
Fig. 2 Evolution of the luminescence decay time for the three types of QDs as a function of the corrected width of the substrate-lens air gap (d – dmin). The symbols represent the data (circles: QD565, triangles: QD605, squares: QD655); two different data points at the same distance correspond to equivalent positions on either side of the lens center. The QD605 dataset shows one obvious outlier, which is attributed to an inhomogeneity in the film and has therefore been disregarded in the fitting procedure. The curves indicate the optimum fits for the three different models (EC: empty cavity, VC: virtual cavity, FM: fully microscopic). The fit parameters are summarized in Table 1.
Before we apply the effective medium approximation, we have to address a possible source of complication for the interpretation of the data: The field mode density, and thus the radiative lifetime, can also be affected by coupling to a cavity, as has been mentioned in the introduction. We therefore have to ensure that such cavity effects are negligible compared to the role of the refractive index in our experimental geometry. Our samples are assembled with a substrate and a lens that are both uncoated, and the quality factor of a cavity formed by these low-reflectivity surfaces is small. The consequences of this fact are illustrated in Fig. 3, which compares the white-light transmission spectrum of such an uncoated glass cavity to the dramatically different response of a metallic microresonator that was used by Chizhik et al. [7] for a deliberate modification of single-molecule emission rates via the cavity effect. The transmission spectrum of the glass cavity does exhibit a single maximum resulting from constructive interference of the light inside the optical resonator, but its quality factor 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.
Fig. 3 Calculated white-light transmission spectra for an uncoated glass cavity (solid line) and a silver-coated microresonator (dashed line), both with a length of 180 nm. The corresponding cavity quality factors Q are found to be 50 (metallic cavity) and 1 (glass cavity). The thicknesses of the two silver mirrors were taken to be 50 nm and 100 nm, respectively.
To find the critical distance for the effective index of refraction, we apply the Bruggeman effective medium approximation, as described by Aspnes [15], to a sphere of interaction with radius 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 fSiO2of silica and fBK7 of BK7 in the sphere of interaction. The effective refractive index n̄ is then obtained as the positive real solution of the equation
where nSiO2= 1.46, nBK7 =1.52 and nair = 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 fX of silica and BK7 were calculated by modeling the space occupied by these materials as spherical caps of heights hX, which have a volume of [16] where R is the radius of the interaction sphere. Division by the total volume of the interaction sphere, , yields the volume fraction fX = VSC/V as where the heights hX of the spherical caps are given by For distances d outside the sphere of interaction, i. e., for d > R + dmin/2, the height hBK7 was set to zero. The effective index of refraction n̄ for a given distance d thus depends on the radius R and on dmin, the minimum distance between the lens and the substrate, which was kept fixed at dmin = 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.Our procedure for calculating the effective refractive index does not take into account the presence of the nanocrystals themselves, which contain CdSe/ZnS core/shell structures with a high refractive index, each covered by a polymer hull. We now want to show that the volume fraction of CdSe/ZnS on the substrate is indeed negligible: According to the manufacturer, the 565 nm QDs have a spherical CdSe/ZnS core/shell center with a diameter of 4.6 nm. The two other species in our sample contain core/shell structures in the shape of prolate ellipsoids, whose diameters (principal axes) are 4 nm and 9.4 nm for QD605, and 6 nm and 12 nm for QD655. All three species are covered by a low-index polymer, resulting in roughly spherical particles with overall diameters of 14, 16, and 18 nm, respectively. Given this geometry of the quantum dots, one can calculate the combined volume of CdSe/ZnS that is present on our substrate, which contains 35–70 particles of each species in an area with a diameter of 500 nm (the focal spot). When we compare this volume of CdSe/ZnS to the total volume of a film with a thickness of 18 nm (the diameter of the largest particles), we find that the high-index volume fraction remains below 1 %; it is therefore an acceptable approximation to use nair for the entire space between lens and substrate as we have done for the calculation of n̄ outlined above.
Several approaches have been proposed to describe the influence of the refractive index on the radiative decay rate: Macroscopic observable properties such as electric field strength and polarization density of the medium can be related to the local field at the position of the emitter by introducing a cavity around the emitter to establish a (conceptual) boundary between its local environment and the bulk of the medium, which is treated as a classical dielectric. Two limiting cases for setting up this cavity are the virtual cavity (VC) model [17], which assumes that the cavity is completely filled with the host dielectric, and the the empty cavity (EC) model [18], in which the cavity only contains the emitter. An alternative approach is the fully microscopic (FM) model [19] for local-field effects, which treats the atoms of the dielectric host as two-level systems and quantifies their interaction with the emitter at the microscopic level. The expressions for the dependence of the radiative depopulation rate kr on the effective refractive index n̄ resulting from these three models are
where krv = kr(n = 1) is the radiative decay rate in vacuum. The predicted increase of kr with increasing refractive index is most pronounced for the VC model, less strong for the EC model and even weaker for the FM model. All these expressions assume an isotropic medium for which there is no preferred direction for the local zero-point field 𝓔loc; this justifies the form of the golden rule in Eq. (1), where the average 〈|𝓔loc ··μ12|2〉osc over all local field oscillators [3] has been replaced by |𝓔loc|2 |μ12|2. Our experimental geometry is not isotropic due to the presence of the silica-air and the air-BK7 interface in the sphere of interaction. However, we can anticipate 35–70 randomly-oriented nanoparticles of each size in the excitation focus contributing to our overall decay curves, so that a sufficient amount of orientational averaging takes place to justify the application of the above formulas to our data.The overall luminescence decay constant k is given by the sum of the radiative and non-radiative decay rates, i. e., k = kr + knr, where kr depends on n̄ according to Eq. (6). The non-radiative rate knr 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 knr 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, τrv = 1/krv, and τnr = 1/knr 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 = kr/(kr + knr) of the emitting state.
Table 1. The optimum fit parameters for adjusting the three different models – virtual cavity (VC), empty cavity (EC), and fully microscopic (FM) – to the experimental data of Fig. 2. R is the radius of the interaction sphere in nm; τrv and τnr are, respectively, the radiative lifetime in vacuum and the non-radiative decay time, both measured in ns. The emitting-state quantum efficiency Φ was calculated based of the fit results for a refractive index of n = 1.44 to allow a direct comparison with [20].
As can be seen in Fig. 2 and Table 1, both the VC and the EC model can reproduce our data with values for the quantum yield Φ between 0.5 and 0.7. The FM model, on the other hand, predicts a much weaker dependence of kr on n̄ and thus cannot describe our data even for the maximum possible quantum efficiency of Φ = 1, which corresponds to knr = 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 knr ≥ 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 knr < 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.)
Typical quantum efficiencies reported for colloidal semiconductor QDs in the literature are around 50% [10, 20], but it has been suggested that the standard cuvette measurement, which is based on a comparison of fluorescence intensities, may systematically underestimate the quantum yield of QDs due to their particular photophysical properties (long-lived dark states, co-existence of “bright” and “dark” sub-populations) [9,21]. On this basis, we conclude that the quantum efficiencies found by applying the EC or the VC model to our data all fall into a reasonable range. However, our efficiencies are lower than the values of more than 95% reported by Brokmann et al. [9], who used a classical-dipole model to interpret changes in luminescence lifetime due to the presence of a dielectric interface close to the emitter. Both our experiment and that of Brokmann et al. rely on lifetime measurements and should therefore be comparable, because such measurements are insensitive to the effects of blinking which complicate comparison with quantum efficiencies measured in the standard way. The lower quantum efficiencies that we find may be caused by a lower structural quality of our nanocrystals and/or they may reflect fundamental differences between the dipole-interface approach and our effective refractive index method.
The most common application of the EC, VC, and FM models to quantum dots relates to measurements of the fluorescence lifetime in different solvents. In this context, we find agreement with the work of Duan et al. [20], who conclude that the the VC model is compatible with their measurements and quantum efficiencies between 55 % and 90 %. Wuister et al. [10], on the other hand, show a clear preference for the FM model and they conclude that the VC and EC models both overestimate the influence of the solvent refractive index. It is important to note that Berman and Milonni [22] have recently criticized the FM model of Crenshaw and Bowden and have proposed an alternative microscopic theory of spontaneous emission in dielectrics. The main improvement introduced in the Berman-Milonni model is the consideration of the magnetic sublevels of the atoms in the dielectric, which is absolutely necessary to obtain an isotropic index of refraction in an isotropic medium [22]. Unfortunately the new microscopic theory only covers effects that are first-order in the density of the dielectric, at which level the VC and EC expressions are indistinguishable, and it does not directly yield a formula that can be applied to higher-density condensed dielectrics such as silica and BK7. Nevertheless, our results may provide a first experimental confirmation of the problems with the FM model that were pointed out by Berman and Milonni.
Regarding the wavelength dependence of the interaction sphere, our study points to a magnitude for 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
We have presented a straightforward and efficient approach to tune the local dielectric environment of nanoparticles and study the resulting changes in their emission dynamics and thereby their quantum yield. We found that our data is well described by adopting an effective medium approach to determine a local effective index of refraction and then applying either the empty cavity or the virtual cavity model for the relation between refractive index and radiative lifetime. The FM model, however, for this relationship was found incapable to describe our experimental results. Our technique permits covering a wider range of effective refractive indices by replacing the BK7 lens with one made of a higher-index material – commercial lenses with 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
This work was conducted in the framework of the European collaboration Nanolum.
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