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The procedure used in our previous publication [Opt. Express 20, 271, (2012)] to calculate how coupling to a spherical gold nanoparticle changes the upconversion luminescence of Er3+ ions contained several errors. The errors are corrected here.
©2013 Optical Society of America
In [1], we presented a theoretical description of the influence of spherical gold nanoparticles on upconversion processes occurring in a surrounding upconverter material consisting of embedded Er3+ ions. We considered two effects of the metal nanoparticle on the upconversion processes: first, the local electric field enhancement, quantified by an enhancement factor γ E, and second the change of transitions rates within the upconverter, described by the Einstein coefficients. The Er3+ ions of the upconverter were approximated as dipole emitters and their coupling to an adjacent spherical gold nanoparticle was modeled using Mie theory. The local-field enhancement and the nanoparticle-induced changes to the Einstein coefficients were then used in a rate equation model of the upconverter material β-NaYF4: 20% Er3+.
The treatment in [1] contained the following three errors which will be corrected here.
Criticisms of the treatment presented in [1]
1) In [1], the local-field enhancement induced by the metal nanoparticle was described by a factor so that
where u plasmon(ωif) and u(ωif) are the spectral energy densities of the light field with and without the nanoparticle, respectively. The nanoparticle-induced changes in the Einstein A coefficients for spontaneous de-excitation processes in the Er3+ ions were described by factors and so thatwhere γ rad is the factor for the nanoparticle-induced enhancement of the spontaneous emission rate and γ nonrad describes the relative change in the Aif due to non-radiative losses in the nanoparticle. Moreover, the well-known equationwhich relates the Einstein coefficients for stimulated emission Bif to Aif, was assumed to equally hold in the presence of the nanoparticle,However, it turns out that Eq. (4) leads to unphysical conclusions. When combined with Eq. (2), Eq. (4) would result in the expressionimplying that the probability of stimulated transitions (absorption and stimulated emission) would be enhanced by non-radiative processes in the metal nanoparticle. Therefore, we must reject Eq. (4).An alternative choice for an expression relating B if,plasmon to the Einstein coefficients in the absence of a nanoparticle is
This means that coupling to the nanoparticle will influence the rates for stimulated processes only through the local-field enhancement contained in Eq. (1), and not through a change in the Einstein B coefficients of the Er3+ ions. It is this choice that we adopt in the improved calculations that we present below.2) Another important aspect concerns the orientation of the optical dipoles of the Er3+ ions. The dipole orientation is of great importance for the rates of the spontaneous emission processes in our rate-equation system because the factors γ rad and γ nonrad from Eq. (2) depend on the orientation of the emitting dipole relative to the surface of the gold nanoparticle, i.e. either parallel (PPOL) or perpendicular (SPOL). In the absorption processes, the dipoles excited in the ions by the local optical field are essentially oriented along the field and, in general, have components along both the SPOL and PPOL directions. In [1], we decomposed the dipole moments of the absorption path into SPOL and PPOL components for each ion, and then solved the rate-equation system separately for SPOL and PPOL orientations of the dipoles. This separation implicitly assumed a perfect correlation between the dipole orientations in the absorption and emission paths, thus neglecting the possibility that an ion excited in SPOL orientation might emit in PPOL orientation and vice versa. The results for the upconversion luminescence intensities were finally averaged over the dipole orientations.
However, the assumption made in [1] of a perfect correlation between the dipole orientations in excitation and emission needs to be dropped in view of the substantial polarization losses that are expected to occur during the multi-phonon relaxation and energy transfer processes prior to emission. In fact, the luminescence from laser-excited Er-doped glasses has been found to be almost completely depolarized [2]. Hence in our current understanding, we must allow for different dipole orientations in the absorption and emission paths. In particular, the factors γ rad and γ nonrad from Eq. (2) should be averaged over the PPOL and SPOL orientations before entering them in the rate-equation system for the upconverter.
3) An important upconversion process included in our rate-equation system is energy transfer upconversion (ETU). This process is based on Förster energy transfer between neighboring excited Er3+ ions. There is an ongoing discussion in the literature on whether the rate of Förster energy transfer is influenced by the local density of photon states and could thus be altered by suitable photonic or plasmonic environments [3–5]. In our implementation of ETU in [1], we assumed the Förster energy transfer rate to be proportional to the radiative decay rates γ rad, if of the involved transitions in both the donor and acceptor of the Er3+ ion pair, and thus proportional to the square of the local density of electromagnetic states, in agreement with [6]. This assumption yielded considerable plasmon-induced enhancements of the ETU as compared to the case without a nanoparticle. In contrast, the theoretical works of [7–9] and the recent experimental study of [5] have argued that the Förster transfer rate is independent of the local electromagnetic density of states. We have therefore checked our earlier results by performing electrodynamic computations of the Förster transfer rate based on the method from [8].The computations demonstrated that the absolute changes of the Förster transfer rate of Er3+ ions coupled to the spherical gold nanoparticle studied in [1] are actually too small to have any sizeable effect on the upconversion luminescence intensities. The discrepancy with respect to our earlier treatment of the Förster transfer rate makes it necessary to revise the rate-equation simulations presented in [1]. A publication on our detailed theoretical work on Förster energy transfer in the presence of metal nanoparticles is currently in preparation.
Revised implementation of the rate-equation calculations
In the following, we present the revised implementation of the rate-equation calculations described in [1]. This implementation is corrected for the errors discussed in criticisms 1, 2 and 3 above. The results obtained with this improved model are shown in Fig. 1 to Fig. 3.
We begin by reiterating that the stimulated processes are modified by the local electric field enhancement, which is described by an enhancement factor γ E. The probability per unit time for ground state (GSA) or excited state absorption (ESA) between the energy levels i and f is then determined by
with the spectral energy density of the excitation u(ω if), the speed of light in vacuum c, the reduced Planck constant ħ, the degeneracies of the initial gi and final state gf and the Einstein coefficient for spontaneous emission from state f to i Aif. The probability per unit time for stimulated emission (STE) is modified in the same way:The change factor of the radiative transition rate γ rad, if for the transition from state i to f modifies the probability of spontaneous emission (SPE) per unit time toIn consequence, the luminescence Li of state i is calculated by multiplication with the occupation of the corresponding state ni Due to the presence of the metal nanoparticle an additional loss channel appears. This loss channel depopulates excited states of the upconverter ions and can be implemented in analogy to the spontaneous emissionFor multi-phonon relaxation and energy transfer processes, we assume that no changes are induced by the metal nanoparticle. These processes are implemented into the rate-equation system as described in [1, 10] for the case without the metal nanoparticle.Results
The spatially resolved results from the corrected simulation are shown in Fig. 1 for all ion positions in the x-z-plane at y = 0 nm. As a consequence of the corrections discussed above, the calculated upconversion luminescence enhancement due to a single spherical gold nanoparticle with a diameter of 200 nm and for an incident irradiance of 1000 Wm−2 at a monochromatic wavelength of 1523 mn, is much lower than presented in [1]. Nevertheless, locally a large enhancement factor of 4.3 is found for the dominant upconversion transition from 4 I 11/2 to 4 I 15/2 with a center emission wavelength of 980 nm.
Fig. 1 Results of the corrected simulations: effect of a spherical gold nanoparticle with a diameter of 200 nm on the luminescence from certain transitions of the upconverter. Shown are the relative luminescence factors, i.e. the ratios of the luminescence intensities with and without the metal nanoparticle, for the transitions from 4 I 11/2, 4 I 9/2, 4 F 9/2 and 4 S 3/2 to the ground state 4 I 15/2. The white dashed line represents relative luminescence factors of unity.
The distance dependence of the upconversion luminescence enhancement depicted in Fig. 2 was determined by averaging the relative luminescence enhancement over spherical shells around the metal nanoparticle. In conclusion, the upconverter should be placed close to the metal nanoparticle in regions where a strong enhancement of the electric field is found. Enhancement factors for the upconversion luminescence of 1.14 and 1.83 for the transitions 4 I 11/2 → 4 I 15/2 and 4 I 9/2 → 4 I 15/2, respectively, were determined by averaging over a distance range from 20 nm to 25 nm to the surface of the gold nanoparticle.
Fig. 2 When averaged over spherical shells around the metal nanoparticle, the luminescence is only increased by factors up to 1.14 and 1.83 for the transitions from the 4 I 11/2 and 4 I 9/2 states to the ground state 4 I 15/2, respectively. These are the transitions with the highest intensities. Close to the surface (<20 nm), the upconversion luminescence from 4 I 11/2, 4 F 9/2 and 4 S 3/2 to 4 I 15/2 is strongly suppressed.
As discussed in criticism 3, there is an ongoing debate on whether the Förster energy transfer is altered in proximity of a plasmonic or photonic structure or not. The results presented above have been calculated for the case where the Förster transfer rates are not changed by coupling to the metal nanoparticle. Figure 3 shows two additional scenarios. The second and most frequently discussed case in the literature is that the Förster transfer rate is proportional to the radiative decay rate of the donor. If we incorporate this proportionality into our rate equation treatment, the enhancement factor for upconversion luminescence at 980 nm will increase to a maximum of 8.6. If both donor and acceptor radiative decay rates enter into the Förster transfer rate, the maximum luminescence enhancement factor reaches 15.5. These results show the importance of Förster energy transfer for upconversion and highlight the need for a further experimental validation of ETU in the presence of plasmonic nanostructures.
Fig. 3 Maximum value of the relative luminescence enhancement factor in the simulation volume. If the Förster energy transfer rate is modified by the local density of photon states, the relative enhancement of the luminescence will increase drastically. The relative enhancement factor is twice as high if only the radiative decay rate of the donor enters into the Förster energy transfer rate and roughly 4 times higher if the radiative decay rates of both donor and acceptor enter.
Acknowledgment
The authors would like to thank Johannes Gutmann,1 Carsten Rockstuhl,3 and Dmitry Chigrin2 for helpful discussions. The research leading to these results has received funding from the German Federal Ministry of Education and Research in the project “InfraVolt – Infrarot-Optische Nanostrukturen für die Photovoltaik” (BMBF, project numbers 03SF0401B and 03SF0401E), and from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n° [246200]. S. Fischer gratefully acknowledges the scholarship support from the Deutsche Bundesstiftung Umwelt DBU.
References and links
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