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Effects of photonic structures on upconversion
*B. Herter, S. Wolf, S. Fischer, M. Peters, B. Bläsi, J.C. Goldschmidt
Fraunhofer Institute for Solar Energy Systems, Heidenhofstr. 2, 79110 Freiburg, Germany
ABSTRACT
Photonic crystals modify the local density of photon states. These variations influence the emission properties of a dipole
embedded within the photonic crystal. Furthermore, field enhancement can be observed within photonic crystals. In this
paper, we investigate how these effects influence upconversion processes in β-NaYF4:Er3+. For this purpose we use
finite-difference time-domain (FDTD) simulations of a grating-waveguide-structure in combination with a rate equation
model of the upconversion processes in β-NaYF4:Er3+. The grating parameters are optimized to achieve large field
enhancements within the structure for the combination of s- and p-polarized light. Furthermore, the variation of the
spontaneous emission rates for dipole emitters within the structure is simulated. The varied transition rates, as well as the
field enhancement, serve as input parameters for the rate equation model for upconversion. Using this approach, the
influence of the structure on the upconversion quantum yield is calculated. For a simulated initial irradiance of
1000 W/m², we find enhancement factors of up to four for the field enhancement in the upconverter region and up to a
factor of three for the upconversion quantum yield. In consequence, the incorporation of upconverting material in
photonic structures in very promising to increase upconversion efficiencies.
Keywords: Photonic Crystals, Upconversion, Fermi’s Golden Rule, FDTD
1. INTRODUCTION
Upconversion of low-energy photons presents an interesting possibility to overcome the Shockley-Queisser-efficiency
limit1 of silicon solar cells and make use of the sub-band-gap-photons which are otherwise transmitted through the cell
and cannot be used for energy conversion. Using an upconverter material below a silicon solar cell, the theoretical
efficiency limit can be increased from close to 30 % up to 40.2 .%2. An increased external quantum efficiency of a
bifacial silicon solar cell due to the addition of an upconverter material at the rear side has already been shown
experimentally3, 4. Unfortunately, the efficiency of upconversion processes is still too low for light intensities typically
applying to solar cells, but upconversion is a non-linear process5. Thus, the upconversion quantum yield increases with
the radiation intensity around the ion. This offers a first leverage to increase upconversion efficiencies.
One possibility to achieve large field enhancements is the use of plasmon resonances in metallic nanoparticles. It has
been shown that the resulting field enhancement can be exploited to achieve higher upconversion luminescence6-8. The
disadvantage of this approach is that an alternative non-radiative decay path is introduced due to the absorption of the
metal. A second approach avoiding these losses is to embed the upconverter material into a photonic crystal.
Photonic crystals show very interesting properties like a photonic band gap. These band gaps can for example be used for
efficient waveguiding in defect modes9, and they modify of the photonic density of states. In 1987, Yablonovitch
described the inhibition of spontaneous emission within a photonic band gap10. Near the band gap, the density of states is
increased. According to Fermi’s golden rule11 this increases the transition probability. In consequence, the emission can
be either inhibited within a band gap or increased near the band gap edge Moreover, high field enhancements can be
achieved within a photonic waveguide structure due to slow-light effects12, 13. All these effects can be exploited to
enhance the efficiency of upconversion processes.
*Barbara.Herter@ise.fraunhofer.de; phone +49 761 4588 5762; fax +49 761 4588 9000; ise.fraunhofer.de
It has already been suggested, that the incorporation of erbium ions into Bragg-stacks could increase the upconversion
luminescence due to large field-enhancements14. First experiments on europium ions implemented into a photonic
structure, in opal structures, have already been carried out15. A suppression of the upconverter emission in the photonic
band gap and an increased luminescence near the band edge was reported. First simulation results were presented that
indicate a possible increase in upconversion efficiency in a grating-waveguide structure16.
In this paper, we investigate the effects of a photonic waveguide structure on the upconversion process. In a first step, we
perform FDTD-simulations using Meep17, a freely available simulation environment, to determine the field
enhancement. Furthermore the modification of the transition rates for a dipole within the structure is determined. The
effects of the altered transition rates has already been investigated for Kerr nonlinearities18. Here, we use the same
approach to determine the effects of the varied transition rates on upconversion processes. The results from the FDTD
simulations are subsequently combined with the rate equation model for the upconverter. The model describes the
upconverter trivalent erbium in a hexagonal sodium-yttrium-tetrafluoride (β-NaYF4) host lattice19 that has shown among
the highest upconversion efficiencies in the past20. The changes of the absorption of the upconverter material, the
upconverter luminescence and the quantum yield are calculated.
Section 2 describes the simulation procedures from the field enhancement, over the variations of the transition rates to
the rate equation model and in Section 3 the results of the combined simulation are presented.
2. SIMULATION PROCEDURE
2.1 Field enhancement within a photonic structure
In an appropriately designed photonic structure, it is possible to achieve high field enhancements. An example of such a
structure is shown in Figure 1. For this structure, light at a specific wavelength is refracted into the x-direction and the
dimensions of the grating ensure that only the first order mode is able to propagate. This structure is optimized for a
maximum field enhancement at a wavelength of 1523 nm, which corresponds to the absorption wavelength of erbium.
Figure 1: Sketch of the simulated waveguide structure, the light parts consist of material with refractive index 2
embedded in a material with n=1.5. The bottom part consists of a perfect metallic layer. The light is penetrating
from above and the surrounding material is air (color available online).
The optimization of this structure to obtain high field enhancement has been reported in our previous analysis16.
The structure shown in Figure 1 is simulated in a two-dimensional configuration, thus, it represents a line-grating,
infinitely extended in the z-direction. In the x-direction, periodic boundaries are assumed while in y-direction, a
perfectly-matched layer (PML) is applied.
The whole structure is simulated on top of a perfect metallic mirror. Thus, no absorption is considered within the
structure, the metallic layer acts as a perfect reflector. The first layer above this mirror has a thickness of 600 nm, the
grating itself has a thickness of 1.8 µm covered with a layer of 1.4 µm. The grating is realized by embedding a material
with a refractive index of n = 2.0 into a layer with a refractive index of n = 1.5. The period of the grating is 950 nm.
The energy density within one periodic element of the structure has been simulated using an infinitely extended line
source which is placed several wavelengths away from the structure. The source emits continuously at a wavelength of
1523 nm. Calculating with complex field, the energy density can be determined after the steady state is reached. The
field enhancement can be determined by comparing the values of the energy density to the values obtained when the
same source is placed in vacuum. This analysis was performed at each lattice point for both s- and p-polarized light.
Figure 2 shows the average over both polarization directions.
Figure 2: Field enhancement within one simulation cell of the photonic grating structure. Field enhancements up to a
value of about 10 can be reached in the high-index region, and up to 4 in the low-index region.
For this structure, the maximum intensity enhancement in the high-index region shows a value of 10. In principle it
would be possible to incorporate the upconverter material either in the high-index region, with a refractive index of two
or in the low-index region, having a refractive index of 1.5 which matches the refractive index of the β-NaYF4 host
lattice. This second approach would be desirable for index-matching to avoid scattering.
For further investigation of the influence of this specific photonic structure on upconversion mechanisms only the lowindex part in the middle of the structure is considered. This restriction is due to the fact that the refractive index of
NaYF4 matches 1.5.
2.2 Influence of the photonic structure on the transition probabilities
In addition to the field enhancement, the radiative transition rates between the different energy levels of the trivalent
erbium are altered. This effect is due to changes in the local density of states (LDOS) ρ(ω) of the photons which,
according to Fermi’s Golden Rule 11, 21, is proportional to the transition probability W12 in the weak-coupling limit:
W12 =

2
M 12 ρ (ω )
2π
(1)
with the transition matrix element M12.
Thus, an increased local density of photon states at a specific frequency leads to a higher probability for a radiative
transition.
To calculate the factor of change for the radiative transition, the emission of a dipole emitter is simulated at each lattice
point and compared to the emission in vacuum. Thus, the variation of the local density of states at each lattice point can
be obtained compared to vacuum. This has been done in order to be able to obtain the variations of the 15 most important
transitions considered by the upconversion-model as shown in Figure 4.
Figure 3: The variation of the emission strength of a dipole for the emission wavelength of 980 nm for both polarization
directions is shown over the low-index region of the grating.
Figure 3 shows the variations in the radiative transition rate due to the photonic crystal environment for the dominant
emission wavelength of 980 nm in the low-index region of the grating. It shows that the transition rate is increased in the
low index-region close to the boundary to the high-index material for a p-polarized emission. For the s-polarization the
reflection from the perfect metallic layer dominates the emission.
2.3 Combination with the rate equation model
Upconversion is based on different mechanisms. The most prominent upconversion processes are energy transfer
between neighboring ions and excited state absorption22. Figure 4 shows the most relevant energy levels for
upconversion and the processes considered in the model. The most prominent spontaneous emission occurs from the
energy level 4I11/2 to the ground state 4I15/2 at a wavelength of 980 nm. The rate equation model includes ground state
absorption, excited state absorption, energy transfer, spontaneous and stimulated emission and multi-phonon relaxation.
The absorption in the near infrared and the emission of photons with energies above the band gap of silicon is especially
interesting for the application in combination with silicon solar cells.
Figure 4: Energy levels of a trivalent erbium ion in β-NaYF4 with all relevant processes which are considered in the rate
equation model23.
The likelihood of the different transitions is determined by the corresponding Einstein coefficients which have been
determined experimentally23. We now modify these Einstein coefficients by the change factors for the different
transitions γif for the transition from the initial to the final state as described in Section 2.2. With this modified model, the
upconversion dynamics were simulated for an illumination at 1523 nm. The irradiance without the waveguide structure
was assumed to be 1000 W/m². At each lattice point, additionally to the changes in the transition rates, the local field
enhancement γE was taken into account.
For the rate equation model, s- and p-polarized emission of the dipole is regarded separately. The emission of the dipole
is considered to be independent of the polarization of the incident light.
3. RESULTS AND DISCUSSION
Including both effects into the rate equation model of the upconverter, the upconversion quantum yield was calculated
for each lattice point.
As shown in Figure 5, the upconversion quantum yield for the most important emission at 980 nm can theoretically be
increased by a factor of around three for both polarizations. These values are considerably higher than what has been
calculated for the influence of plasmon resonances in gold nanoparticles8, 16. It is also clearly visible, that the
enhancement occurs for nearly the whole simulated region.
Towards the edges, the upconversion efficiency is especially enhanced due to the effects of the increased intensity
towards the edges and due to the increased density of states in the proximity of the surface to the high-index material. In
the lower part of the structure, the quantum yield is additionally enhanced, due to the reflections from the mirror below.
From the results shown in Figure 5, an average enhancement of the upconversion quantum yield can be obtained
assuming that the upconverter fills the whole low-index region of the grating. For s-polarization, the UCQY is improved
by a factor of 2.3 and by a factor of 2.1 for p-polarization. This shows a significant improvement of the upconversion
quantum yield for upconverter material embedded in a photonic structure. More importantly, the UCQY is improved
nearly over the whole range and the exact position of the emitters is not critical.
Figure 5: Relative enhancement of the upconversion quantum yieldγUCQY due to the incorporation of the upconverter
material into a photonic crystal environment, a) for p-polarization and b) for s-polarization
4. CONCLUSION AND OUTLOOK
The results shown here are to our knowledge the first simulations on the effects of photonic structures on upconversion
quantum yield. A significant improvement on the upconversion quantum yield has been shown.
It has been shown that photonic structures influence the efficiency of the upconverter system β-NaYF4:Er3+ by two
mechanisms: the local change in the intensity of the incoming radiation and the change of the density of states at the
transition frequencies. These influences were simulated for a given structure. The combination of these results with the
rate equation model for the upconverter gives the influence of the photonic structure on the quantum yield of the
upconversion process. The first results show that the combination of an upconverting material with photonic structures
has a high potential to improve the upconversion quantum yield.
Furthermore, the effect of an increased intensity can be exploited, which played a minor role for this specific structure
due to the accumulation of intensity in the high-index region.
The combination of an upconverter material with a photonic structure has the advantage that, additionally to the intensity
increase, which influences all transitions, theoretically, each transition can be influenced separately via the local density
of states and the transition rates relative to each other can be varied. Hence, the most important emissions at wavelengths
of 980 nm and 800 nm could be enhanced while others are suppressed to increase the upconverter efficiency. Therefore,
with further optimization, it should be possible to achieve even higher enhancement factors which could improve the
applicability of upconverter materials for solar cells.
ACKNOWLEGEMENT
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 number 03SF0401B),
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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