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The radiative transition probability is a fundamental property for optical transitions. Extensive research, theoretical and experimental, has been conducted to establish the relation between the photonic environment and electric dipole (ED) transition probabilities. Recent work shows that the nanocrystal (NC)-cavity model accurately describes the influence of the refractive index n on ED transition rates for emitters in NCs. For magnetic dipole (MD) transitions, theory predicts a simple n3 dependence. However, experimental evidence is sparse and difficult to obtain. Here we report Eu3+-(with distinct ED+MD transitions) and Gd3+-(MD transitions) doped β-NaYF4 NC model systems to probe the influence of n on ED and MD transition probabilities through luminescence lifetime and ED/MD intensity ratio measurements. The results provide strong experimental evidence for an n3 dependence of MD transition probabilities. This insight is important for understanding and controlling the variation of spectral distribution in emission spectra by photonic effects.
The spontaneous emission rate of an emitter is governed by Fermi’s golden rule. Both transition dipole moments and photon density of states resonant with transitions affect the emission rate. The transition dipole moment depends on wave functions of initial and final states. For electric dipole (ED) transitions, the electromagnetic field strength interacting with the transition is influenced by the local environment and a local field correction factor χ is required. The transition probability kED is given by
where kED is the radiative decay rate of ED transition for the emitter in a dielectric medium with the refractive index n and kED0 is the decay rate in vacuum (n = 1). The transition rate k is proportional to the local density of states (LDOS) and can be modified by tuning the dielectric medium surrounding the emitter.1 The distance over which dielectric properties affect transition rates is of the order of the wavelength of the light emitted (hundreds of nanometers).2 Extensive theoretical work has resulted in a variety of models for χ. The most prominent are the virtual3−5 and real cavity6−9 models which predict
For emitters doped in a nanocrystal (NC), the refractive index of the NC (nNC) must be taken into consideration, giving rise to the NC-cavity model:10
In contrast to extensive research on photonic effects on ED transition rates, research on the influence of the local surroundings on magnetic dipole (MD) transitions is very limited. No local field effects are expected since the magnetic susceptibility (in a nonmagnetic medium) is the same as for vacuum, and thus a simple cubic dependence (reflecting the variation of photon density of states with n) is predicted:13
where kMD and kMD0 are the radiative decay rates of the MD transition for the emitter in a dielectric medium with refractive index n and in vacuum, respectively. Experimental results on the variation of MD transition rate with n are sparse as pure MD transitions are not often observed. In 1995, Rikken,14 and later in 2002, Werts15 reported on the influence of n on MD transition probabilities. Rikken studied Eu3+ complexes with distinct MD (5D0–7F1) and ED (5D0–7F2 and 5D0–7F4) transitions. For the complexes dissolved in solvents with different n, the influence of n on the MD transition rates was measured and explained using the n3 dependence. A good agreement was obtained taking into account partial quenching and a kMD0 of 10 s–1 for the 5D0–7F1 transition. This rate is, however, significantly lower than the theoretically calculated kMD0 of 14.4 s–1.16 In the work by Werts, radiative lifetimes of Eu3+5D0 emission in a variety of materials were compared and related to the 5D0–7F1 MD over the total (ED+MD) emission intensity ratio. Within the experimental accuracy (largely determined by significant uncertainties in quantum yield QY), a good agreement was obtained assuming an n3 dependence of MD transition rate. In spite of these two insightful studies, a systematic and accurate investigation using model systems to study the relation between kMD and n is still lacking. As shown previously, doped NCs serve as ideal probes for photonic effects.11,12,17−21 The size of NCs is well below the hundreds of nanometers over which photonic effects influence transitions for emitters inside NCs. The local coordination of emitters is fixed in the nanocrystalline host and is the same as in bulk material. Because the local coordination is unaffected by the solvent surrounding NCs, an ideal model system is realized in which only the variation in solvent refractive index influences radiative decay rates of emitters inside NCs. In the present work, we report two NC model systems (NaYF4:Eu3+ and NaYF4:Gd3+) to accurately determine the dependence of MD transition rates on n. The photonic effects in the present study involve a homogeneous dielectric medium with a limited range of n (1.35 to 1.53). In the past decades, a much wider variation in refractive index has been demonstrated in complex photonic and plasmonic structures, which give rise to much stronger variations in the luminescence properties.22−26
NaYF4:Eu3+ and NaYF4:Gd3+ NCs were synthesized using known colloidal synthesis techniques (see Supporting Information (SI) for details). Transmission electron microscope (TEM) images show monodisperse and nearly spherical ~22 nm NaY0.9Eu0.1F4 core NCs (Figure Figure11a) and slightly prolate ~28 × 25 nm NaY0.9Eu0.1F4@NaYF4 core–shell NCs (Figure Figure11b). Oleate ligands on the surface allow the NCs to colloidally stabilize in apolar solvents with different n to investigate the influence of the photonic environment.11,12 Six commonly used solvents are diethyl ether (n = 1.35), hexane (n = 1.38), octane (n = 1.40), chloroform (n = 1.45), toluene (n = 1.50), and chlorobenzene (n = 1.53). Figure Figure11c presents emission spectra for Eu3+-doped core and core–shell NCs for 5D2 excitation. Fast relaxation to the 5D0 state is followed by radiative decay through both forced ED (5D0–7F2 at ~615 nm and 5D0–7F4 at ~695 nm) and MD (5D0–7F1 at ~590 nm) transitions. Variation of solvents does not affect the energy level structure resulting in identical emission spectra except for slight changes in relative intensities due to photonic effects (vide infra).
To investigate the relation between kMD and n, we first consider the total 5D0 decay rate. Figure Figure22a shows the decay curves of 5D0 emission for core–shell NCs in different solvents (see also Figure S1). Ideally, the decay curves are single-exponential, but as a result of quenching, e.g., through coupling with high energy ligand vibrations, there is a contribution from nonradiative multiphonon relaxation (MPR), especially for Eu3+ ions close to the surface. The highest energy vibrational modes contributing are C–H stretching vibrations (~3000 cm–1) from oleate ligands on the NC surface and solvent molecules. For core–shell NCs, MPR is reduced. It affects only the initial decay and is similar for the different solvents. Two fitting procedures were used, based on the average lifetime τavg (see SI for details) and single exponential fitting. For core NCs, τavg of the 5D0 state decreases from 6.1 to 4.8 ms upon increasing n from 1.35 to 1.53. A similar trend is observed for the core–shell NCs with longer luminescence lifetimes (7.0 ms vs 6.1 ms in core NCs in diethyl ether, Figure Figure22b) as the inert shell suppresses quenching by MPR.27,28 Also, there is a stronger deviation from single exponential decay for Eu3+ in the core-only NCs. This deviation can be appreciated by comparing the fit residuals that show a systematic deviation for the initial part of the core-only decay curves.
The radiative rates kr for individual 5D0–7FJ transitions can be determined from the total radiative decay rate and relative emission intensities (branching ratio) derived from emission spectra. The nonradiative decay rate knr also contributes to the total decay rate and reduces QY. If the 5D0 radiative lifetime for Eu3+ in bulk NaYF4 is known, the upper limit of the QY of Eu3+-doped NCs can be quantitatively determined using eq 6.
IED/Itot and IMD/Itot denote the fractions of ED and MD emission in the total integrated emission intensity for 5D0–7FJ transitions in bulk material. The NC-cavity local field factor χNC is used and τbulk is the 5D0 lifetime of Eu3+ in bulk NaYF4 (6.2 ms from measurements on bulk NaY0.9Eu0.1F4, in excellent agreement with 6.2 ms reported by Tanner29). With τbulk, the observed branching ratios and n of NaYF4 (1.5),30 we find for kMD0 14.3 s–1 (very close to the theoretical value of 14.4 s–1)16 and for kED0 75.3 s–1. To investigate the measured variation in radiative lifetime with n, Figure Figure22c plots τavg as a function of n. Fitting the data with eq 6 gives a good agreement using QYs of 77% and 87% for core and core–shell NCs, respectively. The slightly longer decay times derived from single exponential tail fits for core–shell NCs give an even higher QY of 93%. Note that actual QYs will be lower as the lifetimes reflect only emissive dopants in the ensemble.12,31 Instead of QY, knr can be quantitatively determined to give a better agreement with experiment (Figure S2).
The analysis shows that the combination of an n3 dependence for kMD and the NC-cavity model for kED can quantitatively explain the observed variation of radiative decay rates. The good agreement between experiment and theory indicates that MD transition rates follow the theoretically predicted n3 dependence. An alternative method to test the validity of the n3 dependence is based on intensity ratios of ED and MD emission lines. Since nonradiative decay quenches both types of emission equally, the variation in ED/MD intensity ratio is a reliable method to test the n dependence of kMD relative to kED without QY as additional fitting parameter. The difference in n dependence for kED and kMD shows that relative intensities of MD and ED transitions will change with n. The stronger n3 dependence predicts an increase in relative intensity for MD emission lines with n. From the emission spectra the relative intensities of MD (5D0–7F1) and ED (5D0–7F2 and 5D0–7F4) transitions were determined. Very weak emission lines corresponding to 5D0–7F0,3,5,6 transitions are neglected in the present analysis. In Figure Figure33a, measured branching ratios for the MD and ED transitions are plotted together with the expected variation assuming an n3 dependence for MD transitions (eq 5) and the NC-cavity model (eq 4) for ED transitions. The agreement is good and values for kMD0 (14.4 s–1) and kED0 (78.2 s–1) are determined, which are consistent with theory and the results from bulk NaYF4. To further verify the n dependence, Figure Figure33b shows the experimentally observed ratio IED/IMD (red dots) and the red line shows the calculated ratio with kMD proportional to n3 with kMD0 = 14.4 s–1 and kED following the NC-cavity model with kED0 (78.2 s–1). Clearly, the n3 dependence for kMD is in excellent agreement with the experimentally observed ratios. The blue and green lines show fits for a fixed n3 dependence for kMD and different n dependencies for kED, viz., the virtual cavity model (eq 2, blue line) and the real cavity model (eq 3, green line). Only the NC-cavity model can explain the experimentally observed variation in IED/IMD assuming an n3 dependence for kMD. The variation in IED/IMD in the narrow refractive index range investigated here is limited (see Figure Figure33). However, intensity ratio changes over an order of magnitude can be realized by making use of high refractive index materials based on metal or semiconductor nanostructures.32−34 This will allow a complete reversal of ED to MD intensities for emitters showing both types of emission.
To provide further evidence for the n3 dependence of kMD, we also investigated the decay times of MD emission from Gd3+. Gd3+ has the 4f7 configuration and the first excited state (6P7/2) for this stable half-filled shell configuration is in the UV. The 6P7/2–8S7/2 emission is around 311 nm and has a large MD transition probability (high value for the MD matrix element (L+2S) of −0.52).35 Using this value, the MD transition rate for the 6P7/2–8S7/2 transition in bulk NaYF4 is kMD = 102 s–1 (τ = 9.8 ms, see SI for details). This value is very close to the observed bulk decay rate (k = 108 s–1 for τ = 9.3 ms). This confirms that the 6P7/2–8S7/2 transition of Gd3+ in NaYF4 has 95% MD character,36,37 which makes the Gd3+6P7/2–8S7/2 emission ideal for investigating the influence of n on MD transition probabilities. The emission spectrum of Gd3+-doped NCs is shown in Figure Figure44a. For the narrow 6P7/2–8S7/2 emission around 311 nm the decay dynamics are shown in Figure Figure44b in solvents with different n (see also Figure S3). Upon increasing n from 1.35 to 1.53 the 6P7/2 lifetime decreases from 9.6 to 6.9 ms (Figure Figure44c). The results are fitted to an n3 dependence (eq 5). An excellent agreement is observed, confirming that the MD transition probability increases with n3 as expected theoretically. Alternatively, this n3 dependence of kMD is confirmed by na fits in Figure S4. This n3 dependence is stronger than the n dependence for ED transitions. As a result, the branching ratio in an emission spectrum consisting of mixed MD and ED transitions will change and the relative intensity of MD transitions will increase with n (see also Figure Figure33a). From the radiative lifetime of Gd3+ emission in bulk NaYF4, we can calculate radiative lifetimes for emission in different solvents (vide supra). The calculated lifetimes are longer than the observed lifetimes, indicating that there is some quenching of the luminescence, probably by trace amounts of UV-absorbing organic chromophores in the solvents.38,39 The QY of Gd3+ emission in core–shell NCs is determined by fitting the experimentally observed lifetimes to
In conclusion, the influence of the photonic environment on MD transition probabilities has been systematically investigated using Eu3+- and Gd3+-doped NaYF4 NCs as model systems. Varying the refractive index of the solvent in which NCs are dispersed reveals a strong increase of the MD transition probability with n. For Eu3+-doped NCs, all experimental results (ED/MD intensity ratios and decay rates) are in excellent agreement with the theoretically predicted n3 dependence for MD transition probabilities if the variation in ED transition probability is assumed to obey the NC-cavity model. For Gd3+, the variation in MD transition probability can be directly assessed as the 6P7/2–8S7/2 transition has 95% MD character. The strong refractive index dependence of the 6P7/2 decay time closely follows the theoretical n3 dependence providing further experimental evidence for the n3 dependence of MD transition probabilities. The present study is the first systematic and accurate investigation providing convincing experimental evidence for the theoretically predicted n3 dependence of MD transition probabilities. Insights in the influence of the local environment on radiative transitions are important for understanding and controlling optical properties through variations in the photonic environment.
We thank F. T. Rabouw and P. T. Prins for the useful discussions. This work was financially supported by the China Scholarship Council (No. 201506380101).
The authors declare no competing financial interest.