Increased upconversion quantum yield in photonic structures due to local field enhancement and modification of the local density of states – a simulation-based analysis

Barbara Herter; Sebastian Wolf; Stefan Fischer; Johannes Gutmann; Benedikt Bläsi; Jan Christoph Goldschmidt

doi:10.1364/OE.21.00A883

Optics Express
Vol. 21,
Issue S5,
pp. A883-A900
(2013)
•https://doi.org/10.1364/OE.21.00A883

Increased upconversion quantum yield in photonic structures due to local field enhancement and modification of the local density of states – a simulation-based analysis

Barbara Herter, Sebastian Wolf, Stefan Fischer, Johannes Gutmann, Benedikt Bläsi, and Jan Christoph Goldschmidt

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Barbara Herter, Sebastian Wolf, Stefan Fischer, Johannes Gutmann, Benedikt Bläsi, and Jan Christoph Goldschmidt, "Increased upconversion quantum yield in photonic structures due to local field enhancement and modification of the local density of states – a simulation-based analysis," Opt. Express 21, A883-A900 (2013)

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- Photonic Crystals and Devices
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About this Article
History
- Original Manuscript: June 18, 2013
- Revised Manuscript: July 21, 2013
- Manuscript Accepted: July 21, 2013
- Published: August 28, 2013

Abstract

In upconversion processes, two or more low-energy photons are converted into one higher-energy photon. Besides other applications, upconversion has the potential to decrease sub-band-gap losses in silicon solar cells. Unfortunately, upconverting materials known today show quantum yields, which are too low for this application. In order to improve the upconversion quantum yield, two parameters can be tuned using photonic structures: first, the irradiance can be increased within the structure. This is beneficial, as upconversion is a non-linear process. Second, the rates of the radiative transitions between ionic states within the upconverter material can be altered due to a varied local density of photonic states. In this paper, we present a theoretical model of the impact of a photonic structure on upconversion and test this model in a simulation based analysis of the upconverter material β -NaYF₄:20% Er³⁺ within a dielectric waveguide structure. The simulation combines a finite-difference time-domain simulation model that describes the variations of the irradiance and the change of the local density of photonic states within a photonic structure, with a rate equation model of the upconversion processes. We find that averaged over the investigated structure the upconversion luminescence is increased by a factor of 3.3, and the upconversion quantum yield can be improved in average by a factor of 1.8 compared to the case without the structure for an initial irradiance of 200 Wm⁻².

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Fig. 1 Simulated grating-waveguide structure with an optimized period of 1.74 μm, a grating height of 1.16 μm, a layer below the grating with a height of 0.39 μm and a top layer thickness of 0.9 μm. The refractive indices n_high and n_low used for the simulation are 2 and 1.5, respectively. The infinitely extended line source is sketched by the red, glowing region above the structure.

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Fig. 2 Simulation setup for the evaluation of the transition enhancement factor. The grating part of the structure (black box) is investigated in the following.

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Fig. 3 Energy level diagram of Er³⁺ in the host crystal β -NaYF₄. The ion is excited at a wavelength of 1523 nm. Higher states are occupied either by subsequent absorption of photons (black broken arrows) or energy transfer processes (red broken arrow). The waved arrows depict multi-phonon relaxation processes [49, 50].

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Fig. 4 Enhancement of the upconversion quantum yield due to the irradiance enhancement within the structure. A first, dominant maximum is obtained for a grating period of 1.74 μm indicating a QY enhancement of a factor of 11. The inset shows the peak shape of this maximum. The orange squares denote integer multiples of 0.87 μm. This period corresponds to a resonance of the grating part of the structure.

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Fig. 5 Enhancement factor γ_E of the local irradiance within the grating structure for a grating period of 1.74 μm. The graph shows the grating part of the structure as indicated by the box in Fig. 2. Within the grating region, the irradiance can be increased by up to a factor of 11.5 in the high-index region (left) and up to a factor of 2.9 in the low-index region (right).

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Fig. 6 Variation of the transition probability γ₃₁ for the transition from the ⁴I_11/2 level to the ground state. The grating part of the structure is shown, with the high-index region on the left and the low-index region on the right. One can see that in the high-refractive index region (left), enhancement factors of the transition rate between 0.9 and 4.1 are found. In the low-index region (right), factors between 1.1 and 2.9 are reached.

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Fig. 7 Enhancement of the luminescence for an initial irradiance of 200 Wm⁻². The luminescence can be increased by up to a factor of 30.0 in the high-index region (left) and up to a factor of 4.0 in the low-index region (right).

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Fig. 8 Enhancement of the absorption of the incident irradiance at a wavelength of 1523 nm. The absorption is increased by up to a factor of 10.0 in the high-index region (left) at the same spots, where the highest luminescence values are found. In the low-index region (right), the absorption enhancement is smaller, with a maximum enhancement of 2.8.

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Fig. 9 Relative upconversion quantum yield enhancement at each lattice position for the transition from the ⁴I_11/2 level to the ground state ⁴I_15/2. The initial irradiance without the structure was set to be 200 Wm⁻². A maximum relative enhancement of the upconversion quantum yield by a factor of 3.9 can be reached. In the low-index region (right), peak enhancement values of 2.0 can be found. Over the whole structure, the UCQY is increased by a factor of 1.8.

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Tables (1)

Table 1 Overview over maximum and averaged enhancement factors of the determined different quantities within the waveguide structure: The maximum values are given for the low and high refractive index region separately; the average was calculated for the whole structure

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Equations (16)

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\begin{array}{l} {GSA}_{struct} (\vec{r}) = γ_{E} (\vec{r}) \times {GSA}_{0} \\ {ESA}_{struct} (\vec{r}) = γ_{E} (\vec{r}) \times {ESA}_{0} \\ {STE}_{struct} (\vec{r}) = γ_{E} (\vec{r}) \times {STE}_{0}, \end{array}

γ_{E} (\vec{r}) = \frac{Φ_{in, struct} (\vec{r}, ω_{in})}{Φ_{in, 0} (\vec{r}, ω_{in})} .

I (\vec{r}) = n \times E {(\vec{r})}^{2},

γ_{E} (\vec{r}) = \frac{n_{struct} (\vec{r})}{n_{0}} \times \frac{I_{struct} (\vec{r})}{I_{0} (\vec{r})} = \frac{n_{struct} (\vec{r})}{n_{0}} \times {(\frac{E_{struct} (\vec{r})}{E_{0} (\vec{r})})}^{2} = \frac{n_{struct} (\vec{r})}{n_{0}} \times {(\frac{| E_{struct, c} (\vec{r}) |}{| E_{0, c} (\vec{r}) |})}^{2},

P_{if} (\vec{r}) = \frac{2 π}{ℏ} {| M_{if} |}^{2} ρ (\vec{r}, ω_{if}),

γ_{if} (\vec{r}) = \frac{P_{if, struct} (\vec{r})}{P_{if, 0} (\vec{r})} = \frac{ρ_{struct} (ω_{if,} \vec{r})}{ρ_{0} (ω_{if}, \vec{r})},

W (ω_{if}, \vec{r}) = \frac{\partial \int_{A} \vec{S} (ω, \vec{r}) d^{2} \vec{r}}{\partial ω} |_{ω_{if}} Δ ω,

γ_{if} (\vec{r}) = \frac{W_{struct} (ω_{if}, \vec{r})}{W_{0} (ω_{if}, \vec{r})} .

\dot{\vec{n}} = [G S A + E S A + S T E + S P E + M P R] \times \vec{n} + {\vec{v}}_{E T} (\vec{n}) .

A_{if, struct} (\vec{r}) = γ_{if} (\vec{r}) \times A_{if} .

\begin{array}{l} A b s = \vec{n} (1) \times G S A + \vec{n} (2) \times E S A + \vec{n} (4) \times E S A \\ - \vec{n} (2) \times S T E - \vec{n} (4) \times S T E - \vec{n} (6) \times S T E . \end{array}

γ_{Abs} = \frac{{Abs}_{struct}}{{Abs}_{0}} .

Lum = \vec{n} (3) \times A_{31} .

γ_{Lum} = \frac{{Lum}_{struct}}{{Lum}_{0}} .

U C Q Y = \frac{\sum_{\vec{r}} Lum}{\sum_{\vec{r}} Abs} .

γ_{UCQY} = \frac{{UCQY}_{struct}}{{UCQY}_{0}} .

	γ_E	γ_if	γ_Lum	γ_Abs	γ_UCQY
max_low	2.9	2.9	4.0	2.8	2.0
max_high	11.5	4.1	30.0	10.0	3.9
average	2.0	1.9	3.3	1.9	1.8

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