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Plasmon enhanced upconversion luminescence near

gold nanoparticles – simulation and analysis of the

interactions: Errata

Stefan Fischer,1,* Florian Hallermann,2 Toni Eichelkraut,3 Gero von Plessen,2

Karl W. Krämer,4 Daniel Biner,4 Heiko Steinkemper,1

Martin Hermle,1 and Jan Christoph Goldschmidt1

1 Fraunhofer Institute for Solar Energy Systems, Heidenhofstr. 2, 79110 Freiburg, Germany

2 Institute of Physics (1A), RWTH Aachen University, 52056 Aachen, Germany

3 Institute of Condensed Matter Theory and Solid State Optics, Abbe Center of Photonics, Friedrich-Schiller-

Universität, 07743 Jena, Germany

4 Department of Chemistry and Biochemistry, University of Bern, Freiestrasse 3, 3012 Bern, Switzerland

*stefan.fischer@ise.fraunhofer.de

Abstract: The procedure used in our previous publication (S. Fischer et al.,

Optics Express 20, 271-282, 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

OCIS codes: (190.7220) Upconversion; (240.6680) Surface plasmons; (260.3800)

Luminescence, (350.6050) Solar Energy

References and links

1. S. Fischer, F. Hallermann, T. Eichelkraut, G. von Plessen, K. W. Krämer, D. Biner, H. Steinkemper, M.

Hermle, and J. C. Goldschmidt, "Plasmon enhanced upconversion luminescence near gold nanoparticles -

simulation and analysis of the interactions," Optics Express 20, 271-282 (2012).

A. Rokhmin, N. Nikonorov, A. Przhevuskii, A. Chukharev, and A. Ul’yashenko, "Study of polarized

luminescence in erbium-doped laser glasses," Optics and Spectroscopy 96, 168-174 (2004).

F. Reil, U. Hohenester, J. R. Krenn, and A. Leitner, "Förster-Type Resonant Energy Transfer Influenced

by Metal Nanoparticles," Nano Letters 8, 4128-4133 (2008).

M. Lessard-Viger, M. Rioux, L. Rainville, and D. Boudreau, "FRET Enhancement in Multilayer

Core−Shell Nanoparticles," Nano Letters 9, 3066-3071 (2009).

C. Blum, N. Zijlstra, A. Lagendijk, M. Wubs, A. P. Mosk, V. Subramaniam, and W. L. Vos,

"Nanophotonic Control of the Förster Resonance Energy Transfer Efficiency," Physical Review Letters

109, 203601 (2012).

T. Nakamura, M. Fujii, S. Miura, M. Inui, and S. Hayashi, "Enhancement and suppression of energy

transfer from Si nanocrystals to Er ions through a control of the photonic mode density," Physical Review

B 74, 045302 (2006).

M. J. A. de Dood, J. Knoester, A. Tip, and A. Polman, "Förster transfer and the local optical density of

states in erbium-doped silica," Physical Review B 71, 115102 (2005).

A. O. Govorov, J. Lee, and N. A. Kotov, "Theory of plasmon-enhanced Förster energy transfer in

optically excited semiconductor and metal nanoparticles," Physical Review B 76, 125308 (2007).

U. Hohenester and A. Trugler, "Interaction of Single Molecules With Metallic Nanoparticles," IEEE

Journal of Selected Topics in Quantum Electronics 14, 1430-1440 (2008).

S. Fischer, H. Steinkemper, P. Löper, M. Hermle, and J. C. Goldschmidt, "Modeling upconversion of

erbium doped microcrystals based on experimentally determined Einstein coefficients," Journal of

Applied Physics 111, 013109 (2012).

2.

3.

4.

5.

6.

7.

8.

9.

10.

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 γE,

and second the change of transitions rates within the upconverter, described by the Einstein

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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 γEso that

)()(

E plasmonifif

uu

ωγω=

(1)

where uplasmon(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

γrad and γnonradso that

if if

AA)(

nonrad

γ

radplasmon,

γ+=

, (2)

where γrad is the factor for the nanoparticle-induced enhancement of the spontaneous emission

rate and γnonrad describes the relative change in the Aif due to non-radiative losses in the

nanoparticle. Moreover, the well-known equation

if

if

if

A

c

B

3

32

ω

π

=

, (3)

which relates the Einstein coefficients for stimulated emission Bif to Aif, was assumed to

equally hold in the presence of the nanoparticle,

plasmon,

3

32

plasmon,if

if

if

A

c

B

ω

π

=

(4)

However, it turns out that Eq. (4) leads to unphysical conclusions. When combined with Eq.

(2), Eq. (4) would result in the expression

if

if

if

A

c

B)(

nonrad

γ

rad

3

32

plasmon,

γ

ω

π

+=

, (5)

implying 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 Bif,plasmon to the Einstein coefficients in the

absence of a nanoparticle is

ifif

BB

=

plasmon,

(6)

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.

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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 γrad and γnonrad 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 γrad and γnonrad 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 γrad,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.

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We begin by reiterating that the stimulated processes are modified by the local electric field

enhancement, which is described by an enhancement factor γE. 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

fi

i

f

if

ifif ifif

A

g

g

c

uBuW

E

3

32

E

GSA/ESA

)()(

γ

ω

π

ωωγ

==

(7)

with the spectral energy density of the excitation u(ωif), 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:

if

if

ifif

A

c

uW

E

3

32

STE

)(

γ

ω

π

ω

=

. (8)

The change factor of the radiative transition rate γrad,if for the transition from state i to f

modifies the probability of spontaneous emission (SPE) per unit time to

ifif, if

AW

rad

SPE

γ=

. (9)

In consequence, the luminescence Li of state i is calculated by multiplication with the

occupation of the corresponding state ni

ifif,ii

AnL

rad

γ=

(10)

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 emission

if if, if

AW

nonrad

γ

Loss

=

. (11)

For 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 4I11/2 to 4I15/2 with a center emission wavelength of 980 nm.

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

4I11/2 4I15/2 and 4I9/2 4I15/2, respectively, were determined by averaging over a distance

range from 20 nm to 25 nm to the surface of the gold nanoparticle.

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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 4I11/2, 4I9/2,

state 4I15/2. The white dashed line represents relative luminescence factors of unity.

4F9/2 and 4S3/2 to the ground

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 4I11/2 and 4I9/2 states to

the ground state 4I15/2, respectively. These are the transitions with the highest intensities. Close

to the surface (<20 nm), the upconversion luminescence from 4I11/2, 4F9/2 and 4S3/2 to 4I15/2 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. Fig. 3 shows two additional scenarios. The

second and most frequently discussed case in the literature is that the Förster transfer rate is

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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.

Acknowledgement

The authors would like to thank Johannes Gutmann1, Carsten Rockstuhl3 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.

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