1. Introduction

Upconversion (UC) describes the generation of one higher energy photon out of two or more lower energy photons. This mechanism is exploited in various research areas like bio-imaging [1,2], theranostics [3] and especially for an increased energy conversion efficiency in photovoltaics. In photovoltaics, UC presents a possibility to utilize sub-bandgap photons for additional current generation. This way, the Shockley-Queisser limit of around 30% [4] for a silicon solar cells is outperformed by a 40% theoretical efficiency limit [5]. Trivalent Erbium (Er³⁺) is an efficient upconverter material that provides suitable energy levels for silicon photovoltaics, showing absorption around 1523 nm and UC emission around 980 nm [6–9]. Currently, the record solar cell efficiency increase due to UC is 0.55% relative [10]. Although this is a 40-fold enhancement compared to the state of the art five years ago [11], for successful applications UC performance still needs to be improved.

One possibility to enhance UC performance is embedding the upconverter into a photonic structure. The first effect of a photonic structure is a locally increased irradiance. This increases absorption and, due to the non-linearity of UC, also the upconversion quantum yield (UCQY). The second effect is a modulated local density of photon states (LDOS) by which UC emission can be enhanced, while unwanted spontaneous emission processes can be suppressed. In Fig. 1(b) the energy levels of Er³⁺ and the most important transitions involved in the UC process, as well as the desired photonic effects are shown. These effects can be achieved with both dielectric photonic structures [12–21], as well as metal plasmonic structures in [22–26]. Because plasmonic structures suffer from parasitic absorption [24], this paper focuses on dielectric structures.

 

Fig. 1 A) The investigated Bragg stack consisting of the active layers made from PMMA containing upconverter nanoparticles (NP) with refractive index $n_{low}$, and TiO₂ ($n_{high}$). The free design parameters are the design wavelength ${\lambda }_D$, which defines the layer thickness ($d_{low}, d_{high}$), and the number of bilayers. The first and last layer feature a thickness ($d_{\lambda /8}$) of one eighth of ${\lambda }_D$ to reduce side lopes of the reflection peak and only consists of undoped PMMA ($n_{\lambda /8}$). Super- and substrate are air and glass, respectively. The reference consists of only the active layers of the respective Bragg stack. B) Energy level diagram, including the most important transitions of the UC processes in β-NaYF₄:25% Er³⁺ for the application in silicon photovoltaics. Two low energy photons at a wavelength of 1523 nm are absorbed. Via an energy transfer process one Er³⁺ ion is lifted into a higher excited state. After a relaxation process, UC emission at 984 nm takes place. Additionally, the desired effects of the Bragg stack are sketched: a locally increased irradiance increases absorption, which non-linearly enhances the probability of an energy transfer process. A modified local density of photon states (LDOS) can enhance the probability of UC emission and suppress loss mechanisms.

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Many different dielectric photonic structures have been tested on a variety of upconverter materials and effects on UC luminescence have been reported, for example in [12–21]. However, in most publications of experimental works, the design of the structure is not, or only partly optimized in simulations. Lin et al. take into account the simulated locally increased irradiance induced by the structure to choose a design that is then experimentally realized [16]. Johnson et al. use an indirect calculation of an increased irradiance at the band edge [12]. Others only use the position of the bandgap in 3D opal photonic crystals and the assumption of an increased irradiance at the band edge to explain measured enhancements of UC luminescence [13–15]. Another possibility is to suppress UC by placing the UC emission itself in the photonic bandgap. This effect was also shown in different 3D photonic crystal structures [18–20]. However, for an increased UC emission, the effects of a modified LDOS as well as the internal dynamics of the UC process are usually neglected. To our knowledge, the only theoretical work that includes a simulation of both photonic effects as well as the UC dynamics for dielectric photonic structures was published by Herter et al. [21,27]. The authors applied a theoretical framework, first developed for the description of the effects of plasmonic structures [22,23] to a dielectric waveguide structure containing the upconverter β-NaYF₄:20% Er³⁺. Under excitation at 1523 nm with an irradiance of 200 W/m², an enhancement of the UCQY by a factor of 1.8 was predicted.

Summarizing the current state of the art, in some experimental works very high enhancement factors are measured but their origin is not fully theoretically understood. This makes it difficult to optimize designs for future applications. In other works, however, the theory is well understood but the gained enhancement factors are too small. Furthermore, no enhancement of solar cell performance has been demonstrated by the application of a dielectric photonic structure to enhance UC of sub-bandgap photons. For increasing solar cell performance, a higher UCQY and absorption of the upconverter-photonic structure device has to be achieved. A promising photonic structure for this application is the Bragg stack for it offers - additionally to the described photonic effects - the possibility to increase absorption, and thereby the external efficiency, by adding more layers to the stack.

In this work, we present a comprehensive simulation-based analysis of the effects of a Bragg stack, on the UC luminescence of the upconverter β-NaYF₄:25% Er³⁺. The Bragg stack investigated in this work consists of alternating layers of Poly(methylmethacrylate) (PMMA), containing β-NaYF₄:25% Er³⁺ nanoparticles (active layers), and TiO₂ as sketched in Fig. 1(a). The free design parameters of the stack are the design wavelength, defining the thicknesses of the layers, and the number of bilayers. Using experimentally determined input parameters (Section 2.1), the local irradiance enhancement is simulated (Section 2.2 and 3.1). The LDOS is calculated from simulations of the photonic band structure and local fields, method and results are described in Sections 2.3 and 3.2, respectively. In a rate equation model describing the dynamics of the UC process (Section 2.4), both photonic effects are considered (Section 2.5). Subsequently, UC luminescence and the UCQY are calculated (Section 2.6). Using this integrated simulation model, the Bragg stack design is optimized, varying both design parameters. The photonic effects on UC, especially the effect of the often neglected LDOS, are thereby analyzed in detail, as explained in Section 3.3. For a range of incident irradiances an optimization of the Bragg stack design is done for each irradiance individually, shown in Section 3.4. Optimal designs are defined for two parameters separately, the UCQY and the UC luminescence. In Section 4 the results are discussed and a comparison to literature is drawn.

2. Methods

2.1 Generation of simulation input parameters

Experimentally determined refractive indices of the applied materials are required as input in the simulations. Therefore, 270 nm thin layers of Poly(methylmethacrylate) (PMMA), as well as PMMA layers containing 25wt% of upconverter nanoparticles were produced via spin-coating from a solution of nanoparticles and PMMA in toluene. Upconverter nanoparticles of hexagonal sodium yttrium tetraflouride (β-NaYF₄) with a doping concentration of 25% of trivalent erbium (Er³⁺) (β-NaYF₄:25% Er³⁺) were purchased from CAN GmbH Hamburg, Germany. The nanoparticles feature a diameter of 20 ± 7 nm. PMMA with an average molecular weight of 120,000 g/mol was purchased from Sigma-Aldrich. The spin-coating was done with a spin-coater SCS G3P-8 from Specialty Coating Systems (Alura Group BV). The refractive index of the produced layers was determined using a spectroscopic ellipsometer (M-2000F, J.A. Woollam Co., Inc). At 1523 nm the refractive index of low temperature processed, spin-coated TiO₂ was determined to be 1.73 in earlier works within our group [28], the refractive index of plain PMMA and PMMA with 25wt% embedded upconverter nanoparticles was determined to be 1.48 and 1.49, respectively. The absorptance of the upconverter nanoparticle films is very low, even in the active region of the upconverter around 1523 nm. Based on the absorption coefficient of a bulk upconverter material [29], the absorption of one upconverting layer of the regarded Bragg stack is smaller than 0.01%. Therefore, it proved difficult to reliably determine the imaginary refractive index. Hence, in the simulation of the irradiance enhancement in Section 2.2, absorption was neglected by setting the imaginary refractive index to zero.

2.2 Simulation of irradiance enhancement

For simulating the irradiance enhancement within the active layers of the Bragg stack an implementation [30] of the scattering matrix method [31,32] was applied. Within this method, the exact design of a Bragg stack can be simulated, as shown in Fig. 1(a). Experimentally determined refractive indices are used as input parameters. The layers feature a thickness of $d={\lambda }_D/4n$ of a certain design wavelength ${\lambda }_D$ for which the stack shows maximum reflectance. The first and last layer are always undoped PMMA layers with a thickness of ${\lambda }_D/8n$ in order to reduce side lobes of the reflection peak [33]. Super- and substrate are air (n = 1) and glass (n = 1.5), respectively. The two parameters that can be optimized within this design are firstly ${\lambda }_D$ (which is proportional to the optical thickness) and secondly the number of bilayers (BL) of the stack. For each design of ${\lambda }_D$ and BL, we defined as reference one layer of PMMA with embedded upconverter nanoparticles, featuring a thickness equivalent to the sum of the thicknesses of all active layers in the corresponding Bragg stack.

The Bragg stack is a one-dimensional photonic structure, hence a one-dimensional simulation along the x-axis fully describes the effects of the structure. The irradiance at each position in the Bragg stack $I(x)$ was obtained from the time averaged Poynting-vector $\overline{S}\left(r\right)$, with a sampling of 0.5 nm. For plane electromagnetic waves with the amplitudes $E\left(r\right)$ and $H\left(r\right)$ and under the assumption that the relative permeability $\mu (r)=1$, $\overline{S}\left(r\right)$ takes on the form [34]

The relative enhancement factor ${\overline{\gamma }}_I$, averaged over all positions in the Bragg stack, is obtained from the quotient of the integrated irradiances

2.3 Simulation of the local density of photon states

To calculate the photonic band structure MIT Photonic Bands [35,36] was used, which simulates infinite photonic crystals only. In this approach the Maxwell equations, formulated in an Eigenvalue problem, are solved for a discrete set of lattice vectors $k$. The calculation uses dimensionless quantities for the reciprocal lattice vector $k^{\text{'}}=ka/2\pi$, as well as the frequency ${\omega }^{\text{'}}=\omega a/2\pi c$, where $c=c_0/n$ is the speed of light in medium and the size of the unit cell. As refractive indices for the high and low refractive index layer of the Bragg stack, the constants $n_{low}$ = 1.5 and $n_{high}$ = 1.8 were used. A two dimensional calculation was done in MIT Photonic Bands and later expanded to three dimensions, as explained below. As the Bragg stack is a one dimensional photonic structure, it only possesses a Brillouin zone in one direction, defined as the $k_x$ direction. In the second dimension in reciprocal space, $k_y$ was calculated up to the value where $k\text{'}=\sqrt{k_x^2+k_y^2}a/2\text{π}$ = 2.1 was fulfilled. This way, all modes for the first four bands of the band structure up to $\omega \text{'}$ = 1.2 were obtained. The sampling width used in this work is $\Delta k\text{'}=1\cdot {10}^{-3}$ in k-space. The reference for the Bragg structure is a homogeneous medium of the active layers. However, for a homogeneous medium no Wigner-Seitz unit cell exists and therefore MIT Photonic Bands cannot calculate a photonic band structure. The program though has an accuracy for refractive indices of ${10}^{-7}$ [35]. This offers the opportunity to calculate a quasi-homogeneous medium in the same way as the Bragg structure, with the refractive indices $n_{low}=1.5$ and $n_{high}=1.5+1\cdot {10}^{-8}$.

For the calculation of the LDOS from the photonic band structure and the local electromagnetic fields, the histogramming method was applied [37,38]. Gutmann et al. further developed this method such that a quasi-three-dimensional LDOS ${\rho }_{q3D}\left(r,\omega \right)$ can be obtained from a two dimensional band structure calculation [28] via

Thereby $E_{b,k} \left(r\right)$ are the local electromagnetic fields and b is the band index. The quasi three dimensional LDOS is in our case ${\rho }_{q3D}\left(r,{\omega }^{\prime }\right)$ and was calculated for the normalized frequency in the range of $0.03\le \omega \text{'}\le 1.2$. The calculation was done with 5000 ω-bins because a fine binning is required for the implementation of the relative LDOS into the rate equation model (see Section 2.5). The relative LDOS ${\gamma }_{LDOS}\left(x, \omega \text{'}\right)$ was calculated from the quotient of the LDOS in the Bragg stack (Bragg) and reference (Ref) at each position $x$ and frequency $\omega \text{'}$

As a consequence of the fine binning, the calculated ${\gamma }_{LDOS}(x,\omega \text{'})$ shows a non-negligible noise. The noise was removed by smoothing the data along the frequency axis with an FFT-Filter with 20 points in the moving window [39]. For further simulations ${\gamma }_{LDOS}(x,\omega \text{'})$ only in the active layer is needed for a particular transition $\omega =2\pi c/{\lambda }_{if}$ and unit cell size $a={\lambda }_D/4(n_{low}+n_{high})/\left(n_{low}{n}_{high}\right)$, considering a volume averaged effective refractive index. Considering these in ${\gamma }_{LDOS}(x,\omega \text{'})$, via

a new relative factor ${\gamma }_{LDOS,if}(x_r)$ is gained for a transition with transition wavelength ${\lambda }_{if}$. The calculation of ${\gamma }_{LDOS,if}(x_r)$ is done separately for each regarded Bragg stack design with different design wavelengths ${\lambda }_D$. To visualize the relative change in the LDOS for a particular transition, additionally, the mean value ${\overline{\gamma }}_{LDOS,if}$ within an active layer was calculated.

As MIT Photonic Bands simulates infinite photonic crystals, the effect of ${\gamma }_{LDOS,if}(x_r)$ is treated equally in all BL and thus boundary effects are neglected. For Bragg stack designs with a small number of BL this represents an overestimation of the LDOS effect on the UC process [40].

2.4 Simulation of upconversion – the rate equation model

The dynamics of the UC process can be described by a rate equation model (REM) which has been developed within our group [29] and is currently being published in a refined version [41]. In the REM, the occupation density vector $\overrightarrow{n}$ describes the occupation of all energy levels of the ensemble of Er³⁺ ions. The rate of change $\stackrel{\cdot }{\overrightarrow{n}}$ of the occupation density can be written as

In the used REM, the seven lowest energy levels of Er³⁺ in the host material β-NaYF₄ are taken into account, shown in Fig. 2, and the following transitions between these energy levels, described by the matrices $M_i$: $M_{ABS}$ denotes all absorption processes, ground state absorption (GSA) and excited state absorption (ESA), $M_{STE}$ describes all stimulated emission (STE) processes, $M_{SPE}$ spontaneous emission (SPE) and $M_{MPR}$ multi-phonon relaxation (MPR). The vector ${\overrightarrow{v}}_{ET}\left(\overrightarrow{n}\right)$ contains all considered energy transfer processes for both energy transfer upconversion (ETU) and the reversed process cross-relaxation (CR).

 

Fig. 2 Energy level diagram of the first seven energy levels of β-NaYF₄:25% Er³⁺.

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The parameters of this rate equation model have been determined for a bulk upconverter material [29]. Embedding the upconverter into thin layers of a Bragg stack requires the usage of small upconverter nanoparticles, which do not feature the same UCQY. In the past 5 years the efficiency of upconverter nanoparticles has been increased from 1/1000 [42] to up to 1/6 [43] of the corresponding bulk material at same irradiance. Comparable values of the UCQY might be reached soon due to further optimization of the synthesis process and better surface passivation of upconverter nanoparticles. Therefore we used the simulation model of the bulk material, representing the target situation of this development.

2.5 Consideration of irradiance and LDOS variation in the rate equation model

The probabilities of absorption and STE are proportional to the local irradiance [41]. Hence, probabilities of these processes in the Bragg stack (BS) are calculated by multiplying the matrices in a homogeneous reference medium (Ref) with the factor describing the relative irradiance enhancement ${\gamma }_I(x_r)$ [17,21]

According to Fermi’s golden rule the probability of spontaneous emission processes is influenced by a modified LDOS [44,45]. This effect is considered by multiplying the Einstein A-coefficients, describing spontaneous emission, with the relative change of the LDOS ${\gamma }_{LDOS,if}(x_r)$ [17,21]. This again is done for each respective transition from an initial state to a final state f.

There is an ongoing discussion in literature on whether energy transfer processes are also influenced by a photonic environment [46–48]. In this work, we follow the reasoning that the important energy processes between Er³⁺ ions occur over a much smaller distance of about 10 nm [49] than the typical length scale of about 100 nm of the considered Bragg stack. Thus, any influence will be minor and in this work no influence on the energy transfer processes is taken into account.

2.6 Calculation of the upconversion quantum yield

From the solved rate equations, the luminescence rate for a certain transition $Lu{m}_{if}$ can be obtained by multiplying the according Einstein coefficient of spontaneous emission $A_{if}$ with the population of the initial state $N_i$ [29]

The absorption is calculated from the sum of all GSA and ESA processes [29]

Within the calculated Bragg stack or reference, the rate equation was solved for every considered point $x_j$. Thus, to gain the total luminescence and absorption, a sum over all points $x_j$ needs to be calculated

where m is the number of points. From luminescence and absorption, the internal upconversion quantum yield (UCQY) is obtained, which relates the number of photons emitted with a higher energy than that of the absorbed photons to the number of absorbed photons

This way, all important UC emissions for the UCQY are taken into account [29]. The direct luminescence $Lu{m}_{21}$ is not counted here because the energy of the emitted photons is not higher than that of the absorbed ones, which means it is no UC luminescence but common photoluminescence, a loss mechanism in our case.

The reference calculation in the REM is done for a homogeneous upconverter material, without the influence of a photonic structure and with a homogeneous irradiance distribution, such that each point in the reference features the same UC properties. To determine the effect of the photonic structure, relative factors of the Bragg stack and reference are calculated. These factors, the relative luminescence ${\Gamma }_{Lum,if}$ for each transition, the relative absorption ${\Gamma }_{Abs}$ and the relative UCQY ${\Gamma }_{UCQY}$ were obtained by

3. Simulation results

3.1 Irradiance enhancement

In scattering matrix simulations the irradiance enhancement within the active layers of the Bragg stack (see Section 2.2) was optimized in dependence on the two design parameters: the design wavelength ${\text{λ}}_{\text{D}}$ and the number of bilayers (BL). The simulation was done for the BL 4, 8, 12 and 16, as well as 20 to 100 in steps of 5. The design wavelength was varied between 500 nm and 2000 nm in steps of 5 nm. For ${\text{λ}}_{\text{D}}$ between 1590 nm and 1640 nm a finer step width of 0.5 nm was used to fully resolve the enhancement peaks. For the BL 75 to 100 a yet finer step width of 0.1 nm was required in the range between 1598.5 nm and 1600 nm.

In Fig. 3(a) the average irradiance enhancement ${\overline{\gamma }}_I$ within the active layers of the Bragg stack is shown as a function of ${\text{λ}}_{\text{D}}$ for an exemplary design of 20 BL. The origin of the observed dependence can be understood by having a closer look at the irradiance distribution at the maximum and in the region of the minimum of ${\overline{\gamma }}_I$. The design wavelength at which ${\overline{\gamma }}_I$ reaches its maximum will in the following be referred to as ${\lambda }_{D,max}$. In Fig. 3(b) the irradiance distribution $I\left(x\right)$ within the Bragg stack is plotted for ${\lambda }_{D,max}$. Additionally on the right y-axis, the refractive index is shown. The active layers, over which the irradiance is integrated to determine ${\overline{\gamma }}_I$, are highlighted in grey. Clearly, all maxima of $I\left(x\right)$ are located in the active layers, which explains the high value of ${\overline{\gamma }}_I$ for this design. In Fig. 3(c) the corresponding reflectance is shown for 20 BL and ${\lambda }_{D,max}$ of 1620.5 nm. The reflection peak represents the bandgap of the photonic structure. The reduction of the side lobes on the shorter-wavelength side of the reflection peak appears due to the $\lambda /8$-layers as first and last layer of the Bragg stack. For this design, the excitation wavelength ${\lambda }_{exc}$ of 1523 nm lies right at the short-wavelength edge of the bandgap. Getting back to ${\overline{\gamma }}_I$, the minimum region around 1500 nm design wavelength can now be understood too: for such a design wavelength, the excitation wavelength lies within the reflection peak and a large part of $I\left(x\right)$ is reflected. The incoming irradiance exponentially decays within the Bragg stack and hence ${\overline{\gamma }}_I$ is strongly reduced.

 

Fig. 3 A) Mean irradiance enhancement ${\overline{\gamma }}_I$ in the active layers of a 20 Bilayer Bragg stack, reaching a sharp maximum of ${\overline{\gamma }}_{I,\max }$ at ${\lambda }_{D,max}$. B) Local irradiance $I\left(x\right)$ at each position $x$ in the Bragg stack, plotted for the design wavelength ${\lambda }_{D,max}$. On the right y-axis the refractive index is shown. Active layers, containing the upconverter nanoparticles, are highlighted in grey. All maxima of $I\left(x\right)$ are positioned in the active layers, which leads to the high enhancement of ${\overline{\gamma }}_{I,\max }$. C) Reflectance of the same optimized Bragg stack with a design wavelength ${\lambda }_{D,max}$. For this design, the excitation wavelength ${\lambda }_{exc}$ lies right at the band edge of the photonic structure.

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With the second design parameter, the BL of the Bragg stack, the shape of the peak of ${\overline{\gamma }}_I$ changes, as shown in Fig. 4(a). From the three plotted examples of ${\overline{\gamma }}_I$ for 12, 20 and 50 BL it can be seen how an increasing BL number leads to a sharper and higher peak of ${\overline{\gamma }}_I$. A summary of the peak positions ${\lambda }_{D,max}$ of the maximum as well as the reached height of the peak ${\overline{\gamma }}_{I,\max }$ for all calculated designs is given in Fig. 4(b). A fit on the points of all maxima shows that ${\overline{\gamma }}_{I,\max }$ increases quadratically with an increasing BL number. Thus, very high irradiance enhancement factors can be reached.

 

Fig. 4 A) Average irradiance enhancement ${\overline{\gamma }}_I$ in dependence on the design wavelength ${\text{λ}}_{\text{D}}$. For an increasing BL number, here shown for 12, 20 and 50 BL, the peak value ${\overline{\gamma }}_{I,\max }$ increases and the peak position ${\lambda }_{D,max}$ moves to shorter design wavelengths. B) ${\overline{\gamma }}_{I,\max }$ is plotted for each simulated BL design together with the corresponding design wavelength of the peak position ${\lambda }_{D,max}$. A fit on all determined ${\overline{\gamma }}_{I,\max }$ shows that the maximum possible irradiance enhancement increases quadratically with an increasing number of BL.

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3.2 Modulated local density of photon states

In Fig. 5 the photonic band structure (A) and relative LDOS (B) are plotted. The band structure is shown within the First Brillouin Zone. The bands show the linear dispersion relation, deviating from linearity only at the edge of the First Brillouin Zone to form the bandgaps. The relative change of the LDOS ${\gamma }_{LDOS}(x,{\omega }^{\text{'}})$ is shown within the Wigner-Seitz unit cell a, with the active low refractive index material ($n_{low}$) on the left, the high refractive index material ($n_{high}$) on the right side, both regions featuring the same optical thickness. In the low refractive index region, especially within the first bandgap, ${\gamma }_{LDOS}(x,{\omega }^{\text{'}})$ is decreased. In this region the upconverter is positioned and therefore a suppression of unwanted spontaneous emission, in particular of the transition ${}_{}{}^4\text{I}{}_{13/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$, is the main effect of the LDOS that can be utilized.

 

Fig. 5 A) Photonic band structure within the First Brillouin Zone. B) Relative modulated local density of photon states ${\gamma }_{LDOS}(x,{\omega }^{\text{'}})$ within the Wigner-Seitz unit cell a of an infinite Bragg structure with $n_{low}=$ 1.5 and $n_{high}=$ 1.8. In the active layer where the upconverter is positioned (the region of $n_{low}$), a decrease of ${\gamma }_{LDOS}(x,{\omega }^{\text{'}})$ is reached, especially within the first bandgap. This decrease is a wanted effect to suppress the unwanted emission ${}_{}{}^4\text{I}{}_{13/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$, which is one of the main loss mechanisms in the considered upconverter system.

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3.3 Photonic effects on upconversion

The upconversion quantum yield (UCQY) is influenced both by the modification of the LDOS and the following change in transition probabilities, as well as the changed local irradiance which scales the stimulated processes. Furthermore, the UCQY depends on the incident irradiance $I_{in}^{}$, by which Bragg stack and reference are illuminated. Because at least two excited upconverter ions are needed for an UC process, the UCQY rises non-linearly with an increasing incident irradiance [29,50]. At very high irradiance values, however, this increase saturates and for even higher irradiance values the UCQY decreases again. Therefore, the same irradiance enhancement can lead to different enhancement factors of the UCQY, depending on the incident irradiance. In consequence, when analyzing the Bragg stacks’ impact on the UCQY, the incident irradiance has to be taken into account additionally.

To clarify the contribution of the different effects, we “switched on” the effects one by one. In literature, the effect of the LDOS is mostly neglected. Therefore we put a particular emphasis on the LDOS effect on different transitions as well as on how the maximum of the UCQY is altered by it. For this purpose, in a first simulation run the relative irradiance enhancement was set to a value of unity, and a two-dimensional scan of design wavelength ${\lambda }_D$ and incident irradiance $I_{in}$ was performed (see Fig. 6). The maximum UCQY found in this analysis shows the maximum possible effect that the LDOS can have on the UC process for the chosen material system. In Fig. 6(a) the results of the two-dimensional scan are shown. The maximum UCQY of 16.3% appears at a design wavelength of 1632 nm and an incident irradiance of 5890 W/m².

 

Fig. 6 Change of UC performance, only regarding the effect of the LDOS by setting ${\gamma }_I\left(x\right)$ to unity. A) UCQY in dependence on design wavelength ${\lambda }_D$ and incident irradiance $I_{in}$, reaching a maximum of 16.3%. B) UCQY in dependence on $I_{in}$ (cut through graph (A) at ${\lambda }_D$ = 1632 nm). The maximum UCQY of the Bragg stack (with ${\gamma }_I=$ 1) is higher than that of the reference and is reached at a lower $I_{in}$. C) Mean relative LDOS for the main UC emission L31 (${}_{}{}^4\text{I}{}_{11/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$) and loss mechanism L21 (emission ${}_{}{}^4\text{I}{}_{13/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$). When the respective transition falls into the bandgap (highlighted region 1.BG), ${\overline{\gamma }}_{LDOS,if}$ is strongly reduced. D) Relative luminescence of L31 and L21 influenced by ${\overline{\gamma }}_{LDOS,if}$. ${\Gamma }_{Lum,31}$ is enhanced in the region where ${\overline{\gamma }}_{LDOS,21}$ is suppressed. E) UCQY in dependence on ${\lambda }_D$ (cut through graph (A) at $I_{in}$ = 5890 W/m²). The UCQY within the Bragg stack follows the course of ${\Gamma }_{Lum,31}$. Due to the changed LDOS, the maximum possible UCQY within the Bragg stack is higher than that of the reference.

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A first cut through the plot at $I_{in}=$ 5890 W/m² (dotted line) provides information about the effect of the design wavelength on the UCQY, shown in Fig. 6(e). To understand the behavior of this curve, one first has to look at the change in the LDOS ${\overline{\gamma }}_{LDOS,if}$ and the resulting change in luminescence ${\Gamma }_{Lum,if}$ for the two most important transitions: the main UC emission L31 stemming from the transition ${}_{}{}^4\text{I}{}_{11/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$ and the loss mechanism L21, the emission stemming from the transition ${}_{}{}^4\text{I}{}_{13/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$ (compare to Fig. 1(b)). One can see in Fig. 6(c) that ${\overline{\gamma }}_{LDOS,21}$ is strongly decreased when ${\lambda }_D$ is close to the emission wavelength of ${\lambda }_{21}=$ 1558 nm, because then the transition lies within the first bandgap (1.BG). Towards higher ${\lambda }_D$, the transition moves out of the bandgap and ${\overline{\gamma }}_{LDOS,21}$ rises. For L31 the curve shows the same behaviour, but is shifted towards smaller ${\lambda }_D$ due to the shorter emission wavelength ${\lambda }_{31}=$ 984 nm. The relative luminescence ${\Gamma }_{Lum,21}$, shown in Fig. 6(d), follows the course of the ${\overline{\gamma }}_{LDOS,21}$ curve. For ${\Gamma }_{Lum,31}$ two effects are visible: for smaller ${\lambda }_D$, the reduction of ${\overline{\gamma }}_{LDOS,31}$ leads to a reduction of ${\Gamma }_{Lum,31}$. For larger ${\lambda }_D$, the reduction of ${\overline{\gamma }}_{LDOS,21}$ then leads to an increase in ${\Gamma }_{Lum,31}$. Because of the suppression of the de-excitation L21, more excited ions are available to take part in an energy transfer process, which leads to enhanced UC emission. Finally, the UCQY is plotted in Fig. 6(e). It is directly proportional to the UC luminescence (see Eq. (15)) and hence follows the behaviour of ${\Gamma }_{Lum,31}$.

Another cut through Fig. 6(a) at ${\lambda }_D$ = 1632 nm (dashed line) allows for a comparison with the non-photonic reference for which the UCQY only changes with $I_{in}$, shown in Fig. 6(b). The UCQY of the reference reaches its maximum of 14.0% at 11500 W/m². The maximum of the UCQY, altered by the LDOS effect, reaches a value of 16.3%. This means that the LDOS actually changes the maximum possible UCQY that can be reached, compared to a non-photonic reference. Additionally, the maximum is already reached at $I_{in}=$ 5890 W/m². The higher maximum UCQY can be explained, as discussed above, by the suppression of the emission L21.

Now we additionally consider the effect of the local irradiance enhancement ${\gamma }_I(x_r)$. For an exemplary BL number of 40, Fig. 7(a) shows a two-dimensional scan of the UCQY in dependence on ${\lambda }_D$ and $I_{in}$. The UCQY reaches its maximum of 15.8% at ${\lambda }_D$ = 1604.5 nm and $I_{in}$ = 600 W/m². The slightly lower maximum UCQY of 15.8%, in comparison to the case with 16.3% when only the LDOS effect was considered, can be explained by the spatial variation of $I\left(x\right)$ (see Fig. 3(b)). Due to this variation, it is not possible to maintain the optimal irradiance at all locations at the same time. Figures 7(c)-7(f) show the influence of ${\lambda }_D$ on the irradiance enhancement ${\overline{\gamma }}_I$, ${\overline{\gamma }}_{LDOS,if}$, ${\Gamma }_{Lum,if}$ and the UCQY, respectively, for an incident irradiance of 600 W/m². The irradiance enhancement shows a sharp peak at ${\lambda }_D$ slightly larger than the excitation wavelength. ${\overline{\gamma }}_{LDOS,21}$ and ${\overline{\gamma }}_{LDOS,31}$ show the same behavior as discussed for Fig. 6. The emission rates are dominated by the irradiance enhancement, especially ${\Gamma }_{Lum,31}$ is strongly enhanced because of the non-linear dependence of the UC emission on the irradiance. While the peak irradiance enhancement is 8, ${\Gamma }_{Lum,31}$ is enhanced by a factor of 21. This high enhancement, however, is not only due to the irradiance enhancement but also due to the suppression of the loss mechanism L21. Finally, the change of the UCQY shows a superposition of both photonic effects. Figure 7(b), showing the UCQY for ${\lambda }_D$ = 1604.5 nm, reveals the high importance of the irradiance enhancement. Due to the high ${\gamma }_I$, the curve appears to be compressed along the irradiance axis. This means that the irradiance, necessary for reaching the highest UCQY, is reached within the Bragg stack already for the low incident irradiance of $I_{in}=$ 600 W/m².

 

Fig. 7 Optimization of UC performance for an exemplary Bragg stack of 40 bilayers (BL). A) UCQY in dependence on design wavelength ${\lambda }_D$ and incident irradiance $I_{in}$, reaching a maximum of 15.8%. B) UCQY in dependence on $I_{in}$ (cut through graph (A) at ${\lambda }_D$ = 1604.5 nm). Compared to the reference, the higher maximum UCQY of the Bragg stack is already reached at $I_{in}$ = 600 W/m², mainly due to the high irradiance enhancement. D) Mean relative LDOS for the main UC emission L31 (${}_{}{}^4\text{I}{}_{11/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$) and loss mechanism L21 (emission ${}_{}{}^4\text{I}{}_{13/2}$ to ${}_{}{}^4\text{I}{}_{15/2}$). E) Relative luminescence of L31 and L21 influenced by ${\overline{\gamma }}_I$ and ${\overline{\gamma }}_{LDOS,if}$. ${\Gamma }_{Lum,31}$ is mainly governed by ${\overline{\gamma }}_I$, rising non-linearly with an increasing ${\overline{\gamma }}_I$. F) UCQY in dependence on ${\lambda }_D$ (cut through graph (A) at $I_{in}$ = 600 W/m²). The UCQY within the Bragg stack is a superposition of the shapes of ${\overline{\gamma }}_I$ and ${\overline{\gamma }}_{LDOS,if}$, but again reaches a similar maximum as in Fig. 6. In conclusion, the increased maximum of the UCQY is due to the changed LDOS, while the shift of the maximum to shorter $I_{in}$ is mainly caused by ${\overline{\gamma }}_I$.

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3.4 Optimization of upconversion performance

In this section, we also vary the BL number and find an optimum design wavelength ${\lambda }_D$ that yields a peak UCQY enhancement at a certain incident irradiance $I_{in}$. In most applications, the incident irradiance will be a given value. To find an optimum design that yields the highest UCQY, both Bragg stack design parameters (${\lambda }_D$ and BL) can be adjusted. Within this work, we performed this optimization for values of $I_{in}$ from 100 W/m² to 1000 W/m² in steps of 100 W/m² and from 1000 W/m² to 5000 W/m² in steps of 1000 W/m². For each $I_{in}$, BL and ${\lambda }_D$ were identified that yield the maximum UCQY $UCQ{Y}_{max}$. The results of the optimization are shown in Fig. 8. For each incident irradiance from 100 W/m² onwards, a Bragg stack design was found that yields close to 16% $UCQ{Y}_{max}$ (Fig. 8(a)). In the blue curves of Fig. 8(b) and Fig. 8(c), the corresponding design parameters BL and ${\lambda }_D$ are plotted, respectively. Designs with a high BL are identified as optimal for low values of $I_{in}$. The explanation is that for a low $I_{in}$, a high irradiance enhancement ${\overline{\gamma }}_I$ is required. As $I_{in}$ increases, lower values of ${\overline{\gamma }}_I$ are favorable, and the optimal BL decreases. The behavior of ${\lambda }_D$ can be understood from Fig. 4. With a higher BL, the optimal position of the photonic bandgap, defined by ${\lambda }_D$, converges to the excitation wavelength ${\lambda }_{exc}$, such that ${\lambda }_{exc}$ is placed right at the edge of the photonic bandgap.

 

Fig. 8 Optimization of the Bragg stack design with respect to maximizing the UCQY or the relative UC luminescence ${\Gamma }_{Lum,31}$ for each incident irradiance separately. A) Maximum UCQY ($UCQ{Y}_{max}$) reached for an optimized Bragg stack design. For each incident irradiance $I_{in}$, a $UCQ{Y}_{max}$ close to 16% can be gained. On the right y-axis ${\Gamma }_{Lum,31}$ is additionally plotted. The blue curve (${\Gamma }_{Lum,31}$) shows the luminescence enhancement when the UCQY is maximized. The green curve (${\Gamma }_{Lum,max,31}$) depicts the maximum possible luminescence enhancement. For both optimizations, the UC luminescence is strongly enhanced, up to a factor of 480 at $I_{in}$ = 100 W/m². In graph B) and C) the number of bilayers (BL) and design wavelength ${\lambda }_D$ of the optimized designs are shown. The blue curve depicts the design when maximizing the UCQY, the green curve when maximizing the UC luminescence.

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A similar optimization can be done with respect to the UC luminescence, which for some applications is of more interest than the UCQY. The green curves in Fig. 8(a)-8(c) represent the results of the optimization for the main UC emission L31. Extremely high relative enhancement factors of ${\Gamma }_{Lum,max,31}$ of up to 480 for $I_{in}$ = 100 W/m² can be reached. ${\Gamma }_{Lum,max,31}$ then rapidly decreases with increasing irradiance but still depicts considerable enhancement factors, reaching a value of ${\Gamma }_{Lum,max,31}$ = 18 at $I_{in}$ = 5000 W/m². The corresponding optimal design is always the one with the highest BL investigated. We limited the BL to 100, because an even higher BL number seems difficult to realize experimentally. For all $I_{in}$, the design wavelength ${\lambda }_D$ = 1600 nm was found to be optimal. In comparison, the blue curve depicts ${\Gamma }_{Lum,31}$ when the design maximizes the UCQY. The results are only slightly lower than for the case when the luminescence is maximized. Hence, also when the design is optimized for maximizing the UCQY, a very high luminescence enhancement is reached.

4. Discussion

Our results identify the Bragg structure as a very promising photonic device to increase UC performance. Depending on the aim of the application the Bragg stack can be optimized in different ways by maximizing the UCQY or the UC luminescence. Both can be done for any given incident irradiance. Another aspect of the Bragg stack is the possibility to include a large amount of upconverter material. If a higher total absorption of the upconverter material is required, more layers can be added to the stack.

The same principle of photonic effects is applicable to any UC material with a similar configuration of energy levels. A high irradiance enhancement is achieved at the band edge, which leads to a linearly increased absorption and subsequently to a non-linearly enhanced UC luminescence. The Stokes-shifted direct emission from the first excited state to the ground state, which is the main loss mechanism of the UC process in the considered system, then lies within the bandgap and is suppressed. This is the most important effect of the local density of photon states that very effectively enhances UC, when the upconverter is embedded into the low refractive index layer of a Bragg stack.

A very important result of this analysis is that the incident irradiance plays a major role in the optimization of the photonic device as well as in the enhancement factors that can be reached with a certain device. For high incident irradiances that come close to the optimum performance of the reference, the enhancement factors are small. The large effect of the photonic structure comes into play at irradiances far below the optimum performance of the reference. For the application of photovoltaics, where an upconverter is placed behind a solar cell, this result is very promising. In the absorption range of Er³⁺, the irradiance of the sunlight is only 30 W/m² [51], which could be increased by external and also spectral concentration [52–55] such that our results from 100 W/m² onwards are applicable and highly significant effects can be expected for the solar cells.

It has to be noted that in our analysis, the total absorption is not taken into account and hence, no absolute luminescence is given. Furthermore, in the rate equation model of the UC dynamics, reabsorption of upconverted photons is neglected. So far in all simulations only normal incidence (0°) of the incoming light is considered. For emission processes, the total angle integrated emission is considered. Despite these simplifications the principle of the photonic effects can be well understood. For applications, the angle of incidence and the resulting absorption, as well as the direction of emission are very important quantities. From earlier works, we know that the photonic bandgap of a Bragg structure moves to smaller wavelengths for larger angles [56]. When considering incident light from all angels, this will result in a trade-off for the Bragg stack optimization between an optimum design for normal incidence and a good functionality for large angles. Another important effect of the Bragg stack is an altered directionality of emission. With established methods to determine the direction of emission from an angle dependent fractional LDOS [28] we will address these questions in our future research.

As mentioned in the method section, we used a rate-equation model describing the upconversion process for the bulk upconverter material and not for nanoparticles. The basic theory though of the upconversion process is the same as for the bulk material and the photonic effects influence the upconversion process in the same way. Because of the small difference in the performance of the bulk material and nanoparticles, we expect the relative simulation results to be a good estimate. Furthermore, the most important result of this paper is that upconversion can be enhanced significantly at low incident irradiances with the use of a Bragg stack. Higher recombination losses due to surface quenching in the nanoparticles would result in lower excitation levels, which are comparable to the situation of a lower incident irradiance in the bulk material. Because we found higher relative enhancement factors for lower incident irradiance values, the presented enhancement factors are underestimated due to the use of the bulk material model.

In the following we want to draw a comparison to photonic effects on UC reported literature. With a similar set of simulation methods describing both photonic effects as well as the UC dynamics, Herter et al. investigated a dielectric waveguide structure with the embedded upconverter β-NaYF₄:20% Er³⁺ [17,21,27]. The complete thickness, containing both high- and low refractive index regions, was 2.45 µm. The simulation was done for an excitation wavelength of 1523 nm, and an irradiance of 200 W/m². For this structure, a maximum UC luminescence enhancement of 3.3 and an increase of the UCQY by a factor of 1.8 was reported. In our work, when maximizing the UCQY at 200 W/m² incident irradiance, the optimal design was identified to be 95 BL and gained an UC luminescence enhancement of a factor of 250. The UCQY was enhanced by a factor of 15.8. Each active layer features a thickness of 270 nm, hence, the total thickness of all active layers is 25.7 µm. In comparison, the relative enhancement factors are considerably higher in the Bragg structure. Furthermore, the amount of upconverter material in the complete device is higher in the Bragg stack by a factor of 10, which results in a higher total absorption.

In Johnson et al. [12] the effect of a 30 BL Bragg stack of porous silicon doped with β-NaYF₄:Er³⁺ was investigated. Simulations were done to determine the reflectance of the Bragg stack. The band edge of the realized Bragg stack was chosen to lie at the excitation wavelength of 1550 nm in order to gain an irradiance enhancement. A 6-fold enhancement of the main UC emission was reported. In this analysis, the reference is not well defined and also the irradiance of the excitation is not reported. From the simulations of our work it is evident that with an optimized 30 BL Bragg stack much higher enhancement factors can be reached.

Lin et al. [16] reported a 10⁴-fold enhancement of UC emission by coupling both excitation and UC emission to guided modes of a resonant waveguide grating. UC emission of NaYF₄:Yb³⁺/Tm³⁺ nanocrystals embedded in PMMA was measured with a 976 nm laser at an irradiance of 660.000 W/m². The measured enhancement is remarkable, although it has to be added that it only occurred for angles very close to 31.5° incidence for which the excitation wavelength is in a guided mode. In simulations only the irradiance enhancement was regarded and reported to be in agreement with the experimental results.

The surface effects of an opal 3D photonic crystal were investigated by Niu et al. [15]. NaYF₄:Yb/Er upconverter nanocrystals were placed on top of the opal structure and illuminated with a 980 nm laser at 40.000 W/m². They measured a 30-fold increase of the UC emission, although again with a not clearly defined reference, and explained it by the irradiance enhancement at the opal surface. This enhancement factor is comparable to the UC luminescence enhancement reached in our simulations of a Bragg structure in the irradiance region of around 1000 to 5000 W/m².

In most published works, the effect of the LDOS is neglected and only the irradiance enhancement is taken into account. The results of this work show that the design with the highest irradiance enhancement is, for the system and range of irradiance values considered here, fairly close to the design that also yields the highest increase in the UCQY or UC luminescence. Therefore, the chosen designs in literature, only based on a consideration of the irradiance enhancement can indeed be close to the ones with the highest photonic effect. Nevertheless, to understand and predict the photonic effects completely, especially the change of the UCQY, it is necessary to include the effect of a modified LDOS. Furthermore, it is very important to consider the incident irradiance in the design of the photonic structure for a certain application.

5. Conclusion

We presented a comprehensive theoretical analysis of the photonic effects of a Bragg stack onto the upconversion (UC) processes of β-NaYF₄:25% Er³⁺. The photonic effects of irradiance enhancement and a varied local density of photon states were simulated and analyzed separately. Subsequently, the results of these simulations were coupled into a rate equation model, describing the UC dynamics. The investigated Bragg stack is built up of alternating layers of TiO₂ and PMMA, the latter containing the upconverter in the form of nanoparticles. For the simulations, experimentally determined input parameters were used, while the free parameters for the Bragg stack design were the design wavelength and the number of bilayers.

The incident irradiance plays a major role in defining the optimal design of a photonic structure. We therefore scanned a large irradiance range from 100 W/m² to 5000 W/m² and determined an optimal Bragg stack design, with regard to maximizing the UCQY, for each incident irradiance separately. For each regarded incident irradiance, a Bragg stack design can be found that yields a maximum possible UCQY of 16%. For high incident irradiances, lower numbers of bilayer were optimal, whereas at low incident irradiances a high irradiance enhancement is required and a higher number of bilayers were optimal. Additionally, the UCQY of 16% is higher than the maximum UCQY of 14% that can be reached in the non-photonic reference. This is because of the modulated local density of photon states that suppresses the main loss mechanism of the UC process.

Another possibility is to optimize the Bragg stack design for maximum UC luminescence. Depending on the application, this value can be of more interest than the UCQY. Due to the high irradiance enhancement that can be gained within a Bragg stack, very high UC luminescence enhancements can be achieved, up to a factor of 480 at 100 W/m² incident irradiance. The optimal design in this case was always the design with the most bilayers and therefore highest irradiance enhancement.

The simulation results show clearly, that the Bragg stack is a highly effective photonic structure for enhancing UC. Additionally, Bragg stacks are straight forward to optimize and very flexible. The photonic effects can be tuned by the design wavelength, number of bilayers and by choosing the material of the low- and high refractive index layer. Also, the external UCQY can be enhanced by an increased absorption, which can be reached by adding more layers to the stack.