1. Introduction

Building-integrated photovoltaics (BIPVs) are a promising solution for the wide adoption of electricity generation through solar energy. With the trend towards large glass facades in public-and commercial buildings a (semi-) transparent BIPV could expect incorporation in the built environment [1]. A luminescent solar concentrator (LSC) is such a transparent potential BIPV. LSCs were intensely studied during the 1970s [2, 3] and are regaining more interest with the advent of new types of quantum dots (QDs), luminescent dyes and rare-earth materials. The window-like qualities of these new materials, such as letting the shape of the visible spectrum remain unaltered, make them suitable as BIPV. A typical LSC is either a polymer plate (mostly PMMA) doped with luminophores, or a glass sheet carrying a luminescent coating [4–6]. Figure 1 illustrates the luminescent solar concentrating process. Luminescent centers contained in these materials absorb part of the incoming diffuse and direct sunlight, followed by reemission at longer wavelengths. The emitted luminescence is waveguided through the LSC plate to its perimeter, where conventional photovoltaic (PV) cells convert this waveguided light into electricity. As a result, the strip-shaped PV cells will receive sunlight from a larger area than they would when directly exposed to the sun, which enhances their electricity generation. If the luminescent coating is semi-transparent, an electricity generating window ideal for BIPV applications can be achieved.

 

Fig. 1 Schematic cross-section of the working of a single-layer LSC. Orange spheres represent the luminescent centers within a transparent medium. PV cells (gray) surround the perimeter of the concentrator plate. ‘desired’ shows the desired concentrating process without any losses. All other arrows show possible loss mechanisms within an LSC and their associated efficiency: non-unity incoupling due to reflection losses (η_inc), non-unity quantum yield due to non-radiative decay (η_qy), non-unity trapping due to escape-cone losses (η_trap) and losses due to self-absorption (η_sa). Losses due to scattering within the LSC are not considered in this work and therefore not depicted.

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In many LSCs the range of absorbed and emitted wavelengths overlap. Reabsorbed light is often not reemitted with unity efficiency, and, even when reemitted, typically has a 25 % chance of being lost through the escape-cone. This parasitic self-absorption therefore yields increasing losses with bigger LSCs. Hence, an accurate and fast treatment of the effect of self-absorption on LSC efficiency is needed to both calculate LSC efficiency and to optimize this efficiency for large LSCs intended for BIPV use.

Light transport within an LSC was described using a series of absorption and reemission events by Batchelder et al. [3,7], who referred to it as multiple-generation self-absorption. The analytical equations presented were solved for the simplified case of a semi-infinite rod. Most modern approaches to quantification of self-absorption are carried out with thermodynamic calculations [8–10], analytical models based on the Beer-Lambert law [11,12], or using Monte Carlo (MC) simulations [13–20], also referred to as raytracing. While MC simulations are relatively easy to implement, they rely on random numbers to calculate physical effects. As a result, a high demand is put on the collection of ever larger statistics as the size of the simulated system increases. In materials featuring a large amount of self-absorption and a high quantum yield η_qy, tracing a single photon in a system will take longer with increasing optical density. In this work we show that these two factors result in a long calculation time when optimizing a realistically-sized LSC. Furthermore, we present a method that fully describes light transport within an LSC. This method can quickly and accurately optimize large-size LSCs through the variation of luminophore concentration.

This paper consists of three parts. In the first part we present a model that calculates all light transport processes within an LSC, based on multiple-generation light transport (MGLT). We will check the correctness and internal consistency of the model by comparing calculations with an MC model using the same input data. Here the calculation speed when optimizing large, high η_qy LSCs using MGLT as opposed to MC will be made evident. Results are presented for well-known luminescent polymers, like Lumogen Red305 as well as state-of-the-art polymer and QD-based LSCs, where all required data was published. In the second part we put the model to use by optimizing the luminophore concentration for the presented materials, both for lab-scale (100 cm²) and BIPV-scale (1 m²) LSCs. In this optimization the potentially beneficial effects of self-absorption when η_qy of an LSC is near unity will be shown. Finally, the potential as BIPV window is evaluated for all presented LSCs, after optimizing their optical efficiency.

2. Multiple-generation light transport

To characterize LSC performance an often used metric is the optical efficiency η [21]. η gives the ratio of incident solar photons to photons collected at the side. In case of a waveguide without self-absorption, scattering or coupling losses at the perimeter

Light incident on an LSC first has to enter the waveguide; here losses are quantified by the angular-dependent reflection, described by the Fresnel equations. For simplicity and comparability with earlier computations and experiments, the remainder of this work considers uniform unpolarized light entering perpendicular to the LSC’s surface. In this case, the incoupling efficiency is given by

To contribute to photon transport to the edges, light entering the LSC has to be absorbed over the thickness t of the LSC. The light harvesting efficiency η_lhe gives the fraction of solar light which will be absorbed by the LSC after entering:

The absorbed fraction N_absorbed from an initial count of solar photons N will therefore be N_absorbed = η_incη_lhe N. N_absorbed can now be converted by the luminescent centers. Following the derivation by Ten Kate et al. [22], conversion happens with the normalized emission spectrum of the LSC Φ_em as a rate of power

For an isotropic point emitter, the radiant power I(s), a distance $s=\sqrt{x^2+y^2}$ away from the emitter, is given by the Beer-Lambert law:

The amount of light absorbed in a circular segment (s, s + ds) equals the derivative of the Beer-Lambert law with respect to s. The corresponding absorption density factor a(s, λ) per unit area, a distance s away from the emission center, is therefore

To simplify the description of light transport through an LSC using this circular emission, two aspects need to be taken into account. First, the path length of light traveling to the perimeter has to be independent of depth. Figure 2 shows that, to reach a distance s horizontally, light has to traverse a distance $\frac{s}{\text{sin}\theta }$ in an LSC plate, independent of emission depth. Second, the path the light takes needs to be independent of the polar emission angle θ. Redefining α(λ) to account for all possible emission angles, limited by the escape-cone, yields the effective absorption coefficient

 

Fig. 2 The two rays drawn are both emitted at an angle θ from different depths within the LSC. Both rays have to travel the same distance s/ sin(θ) to reach a column at a distance s, viewed from the top of the LSC. The radiant power I(s) of the light from both rays is therefore equal at s, independent of the depth the light was emitted from.

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The effective absorption coefficient allows us to work with depth- and emission-angle independent columns rather than having to take all separate luminescent centers (points) into account. We can therefore work in a 2D rather than in a 3D plane.

To obtain the reabsorption within a column γ_i(x, y, λ), light emitted from all other columns within the LSC plane needs to be accounted for. A 2D convolution describes this problem. Convolving the absorption density factor and emission distribution provides

η_trap is defined as the ratio of light that will be subject to total internal reflection within the LSC plane after reemission. For an LSC in air

η_ca provides a measure to account for light that is emitted within the escape-cone but will be reabsorbed before escaping through the LSC’s surface [22].

Reabsorbed light can be emitted again to a next generation i + 1. Similar to Eq. (5), we sum over all absorbed photons and emit with the normalized emission spectrum as a rate of energy. Taking the quantum yield into account, this yields

The process of reemission and reabsorption continues for many generations i until all photons have either escaped through the escape-cone, have been quenched, or have been collected at the LSC sides. In order to retrieve the amount of collected photons C_i(λ) at the sides, we can simply take the difference between the emitted and the absorbed intensity of a generation over the entire LSC surface A_LSC. When we correct for escape-cone losses this yields

The total number of collected photons C_Σ can be acquired through summation over all generations and integration over the wavelengths of interest.

With C_Σ the optical efficiency η from the MGLT model is calculated using

To summarize, using only the experimentally measured absorption and emission spectra, index of refraction and quantum yield of an LSC material, the complete light transport through an LSC can be calculated by using multiple generations of reemission and reabsorption of light. It is important to note that the required absorption coefficient α(λ) is measured under the lowest possible luminophore concentration. α(λ) measured under high concentrations might cause self-absorption within the material to yield a different spectrum than the real absorption spectrum of the luminophore. Losses due to scattering are not included in the model as presented, but can to a first order be estimated using an additional absorption term, in a similar fashion as presented in the work by Earp et al. [11]. This additional absorption term will not contribute to reemission in later generations. Since no assumption on the geometry of the LSC has been made, the MGLT model is usable for any shape and size of planar LSC window.

3. Monte Carlo simulation

In order to verify the outcomes of the presented MGLT model, MC simulations were carried out. The MC simulations use the same absorption and emission spectra, wavelength-independent refractive indices, and concentration-independent quantum yields as input. A flowchart of the simulation is shown in Fig. 3. To initialize the simulation, a photon P is generated incident on the surface of the LSC. P has a wavelength λ, pseudo-randomly chosen using the AM1.5 direct spectrum as distribution. The photon’s initial position is determined by

 

Fig. 3 Schematic representation of the MC simulation.

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The path length d the photon will travel is found by rewriting the probability of absorption

The photon’s new position x′ after traveling d, with azimuthal angle ϕ, can be found using

If the photon is emitted outside of the escape-cone (θ_c < θ < π − θ_c) and $\left(\theta -\frac{\pi }{2}\right)\times \text{Ev}(R)>0$, then

Else, if emission is outside of the escape-cone, but $\left(\theta -\frac{\pi }{2}\right)\times \text{Ev}(R)<0$, then

And otherwise, if emission happens within the escape-cone,

When the photon exits the LSC through the sides, it is assumed to be collected by the PV cells. Otherwise, the photon has escaped through the escape-cone or has never been absorbed at all. Each simulation runs with N_input = 10⁷ photons to obtain a proper statistic. The optical efficiency is extracted as

4. Model verification

In order to test the MGLT model, a selection of recent LSC materials was made for which all required data were available as published information. Although research on state-of-the-art materials has already started to shift its attention to Stokes’ shift engineering the absorption and emission spectra, dyes featuring significant spectral overlap have also been selected. These materials with spectral overlap can aid in verifying the model’s capabilities for quantifying self-absorption. See Data File 1 for the required spectroscopic properties and optical efficiencies of all materials, calculated with both MC and MGLT. All calculations are done on a desktop computer with an Intel i5-6500 CPU clocked at 3.2 GHz.

When excluding materials with less than 1 % optical efficiency, excellent agreement between the two models is observed. Based on the simulation results from published data ( Data File 1), the relative difference $\left(\frac{{\eta }^{\text{MC}}-{\eta }^{\text{MGLT}}}{{\eta }^{\text{MC}}}\right)$ between MGLT and MC for 100 cm² plates is 2.1 ± 6.3 %, and for 1 m² plates is 4.4 ± 7.7 %. Materials with η < 1 % have been excluded, as only a very minor difference in absolute calculated efficiency already causes a great increase in the difference in relative efficiency.

To illustrate the in-depth agreement with MC and functionality of the MGLT model, a 10 × 10 × 0.5 cm³ Lumogen Orange (LumoO) [23] LSC plate can be taken as example, since it features a strong self-absorption [see Fig. 4(a)] and a high quantum yield (η_qy = 0.95).

 

Fig. 4 (a) Comparison between MC and MGLT spectra for the simulated LumoO geometry received at the edges of the LSC. Absorption and emission spectra [23] are shown as well, signifying the large amount of redshift. (b) Optical efficiency per generation for the MGLT model. All materials simulated have absorption coefficients as reported in their respective papers (see Data File 1). Materials with significant spectral overlap have many more generations contributing to the total η due multiple absorption and remission events. (c) Calculation time needed in order to optimize the absorption coefficient for a 10 × 10 × 0.5 cm³ Red305 LSC with different quantum yields. Each datapoint has α ranging from 1 × 10⁻¹ cm⁻¹ to 1 × 10⁸ cm⁻¹. (d) Contribution to the optical efficiency per unit area of the LSC geometry, calculated by MGLT and (e) by MC.

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The self-absorption efficiency η_sa does not provide information about whether the photon was lost due to escaping or quenching. Therefore we have chosen to separate η_sa into an effective fraction of trapped photons η_{trap eff} and an effective luminescent quantum yield η_{qy eff}. Here

Table 1. Calculated efficiencies for a 10 × 10 × 0.5 cm³ LumoO LSC using absorption coefficients as reported by Krumer et al. [23].

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The exact shape of the spectrum of the collected photons is highly susceptible to small changes in the treatment of physical input parameters, based on a model’s assumptions. Therefore, this spectrum can serve as a rigorous test of physical correctness of the MGLT model. Figure 4(a) shows the spectrum transmitted through the perimeter of a LumoO-based LSC for both MC and MGLT. The transmitted spectra for MGLT and MC are almost identical, with MGLT displaying no random noise in the spectrum. The noise seen with MC is a direct consequence of the model’s random nature, which would only decrease with a large increase in simulated photons. It should be noted that the spectrum is also the same before normalization, which is a direct consequence of both models yielding the same optical efficiency. A significant redshift is seen even for a window of 100 cm². The redshift can be explained by the overlap between absorption and emission, resulting in light emitted within the overlapping spectral area almost never reaching the window’s edge.

These simulations can also be used to optimize LSC performance by changing the amount of absorption, which in practice would mean changing the luminophore concentration. Figure 4(b) shows that the majority of light transport takes place in the first generation (0 on the horizontal axis) of light reemission for the MGLT model. In this first generation a higher contribution η_i to the total η than in any of the succeeding generations is provided. Calculating just one generation is therefore sufficient for optimization, given η_qy < 1 (see Appendix A). The performance increase MGLT offers over MC is evident from Fig. 4(c). Optimizing a large LSC with MC requires a significant amount of photons to receive enough collected photons for a satisfactory simulation result. When η_qy and α rise, a single photon can remain trapped in the LSC for many reabsorption and reemission events, resulting in a long MC simulation time. MGLT, on the other hand, is generation-based, with a constant time required for each generation, resulting in a more constant and predictable time required to calculate optimal α. In MGLT, performance mainly scales with the amount of generations used, but also with how broad the overlap between the absorption and emission spectra is, as the most integration needs to be done across this overlap. Calculating the final efficiency of an optimized LSC is done quickest using a hybrid approach. First the LSC is optimized using one generation of MGLT. The optimal α is then used in an MC simulation to calculate the final efficiency as, for calculation of the full efficiency, MC is usually quicker than MGLT with many (> 5) generations.

Figure 4(d) (MGLT) and Fig. 4(e) (MC) show that the simulations can help acquiring a more detailed view on what part of the LSC contributes most to edge transport. In Figs. 4(d) and 4(e) each unit area receives the same amount of photons; the brightness of that area indicates how many of those photons will eventually reach the perimeter. While the bulk of the LSC provides the most photons, also most light is lost here. These losses can be explained by the distance between the place photons are absorbed and that of their collection. The self-absorption of LumoO causes photons absorbed in the center to become reabsorbed on their way to the perimeter. LumoO’s high quantum yield causes most photons to still be reemitted while underway, but each extra absorption event adds to the chance of losing that photon through the escape-cone. Therefore, per unit area, the edge of the LSC contributes slightly more to the amount of collected photons than the bulk. The effect of an LSC’s geometry can be studied in detail using MGLT, since the model is formulated around the LSC geometry [see Eq. (9)]. Even after one generation of simulation (not shown) the effect of an LSC’s geometry on light transport is clear. With MC a very large amount of photons is necessary to eliminate random noise. In Fig. 4(e) 10⁷ photons were binned over a 15 × 15 grid to reduce noise, resulting in a lesser resolution than observed with MGLT. Around 5 × 10¹¹ photons would need to be simulated with MC to obtain a resolution comparable to MGLT. The geometry appears not to have a high impact on LSC efficiency. It could, however, still be possible to build an aesthetically pleasing LSC with sufficiently high efficiency by optimizing the geometry and opting for a fully transparent center.

Using the models as a design tool can help with understanding how a change in parameters will influence performance [24]. Here the strengths of both models can be used to complement each other. MGLT is formulated using physical laws, without a dependence on the collection of sufficient statistics. Changing one parameter will therefore after a limited amount (usually one) of generations show what influence such a change has on device performance. Once an optimum is found MC can be used to calculate the total device performance.

5. LSC optimization results

Maximizing the optical efficiency of an LSC would result in a maximally efficient BIPV element. Simulations can help designing such an optimal LSC by calculating the absorption coefficient α^★ with maximal optical efficiency η^★ for the desired geometry or size.

Table 2 provides us with the results for such optimizations for both a 10 × 10 × 0.5 cm³ and a 100 × 100 × 0.5 cm³ LSC. The value of α^★ is calculated through MGLT, while the optical efficiency η^★ with α = α^★ comes from MC. What can be noted from Table 2 is that an optimized lab-scale LSC does not immediately yield an optimized 1 m² LSC.

Table 2. Maximal possible efficiencies for two different LSC sizes for selected materials. ${\alpha }_{\mathbf{max}}^{★}$ is the maximal absorption coefficient of the reported spectrum. When ${\alpha }_{\mathbf{max}}^{★}$ is denoted as ↑, the highest possible luminophore concentration should be taken (in these calculations limited at 10⁵ cm⁻¹), for η_lhe has become the dominant factor in determining η for that material.

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Figure 5 shows GQD as a notable example. The optical efficiency (circles) comes from two competing processes. A higher α means, on the one hand, an increased η_lhe (dashed); on the other, a decreased η_sa (diamonds), resulting in a geometry-specific optimum α_max. For GQD, a lab-scale device has a maximum of α^★ (λ = 474 nm) = 7.44 cm⁻¹ with ${\eta }_{10\times 10}^{★}=2.5\hspace{0.17em}\%$. If that maximum were to be used for the 100 × 100 × 0.5 cm³ window, a lower efficiency of η_100×100 = 0.71 % would be obtained than when using α^★ (λ = 474 nm) = 3.26 cm⁻¹ yielding ${\eta }_{100\times 100}^{★}=0.98\hspace{0.17em}\%$. In line with [11,12], we can conclude that dye concentration optimization needs to be done for the actual geometry of the LSC, as opposed to a lab-scale device. The combination of MGLT with MC allows us to do such optimization quickly. While Table 2 also shows drastic increases in ${\alpha }_{\text{max}}^{★}$ for CY, MSQD and TBA, when comparing 10 × 10 cm² to 100 × 100 cm², the efficiencies for these materials are low throughout the entire range of absorption coefficients. Their efficiencies saturate quickly, since their efficiency is dominated by η_lhe. These materials show only a minute increase in η at the reported maximum.

 

Fig. 5 Optimization track for GQD (η_qy = 0.45) using two different LSC sizes. The displayed efficiency is the total optical efficiency, calculated through MC. Dotted lines indicate the maxima for the two simulated geometries.

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In self-absorbing materials, the trapping efficiency as defined by Eq. (10) is mostly an upper bound for the amount of photons that do not escape from the LSC’s top or bottom surface. Figure 6(a) shows the impact of an increasing absorption coefficient α_max on the effective trapping efficiency η_{trap eff} [Eq. (26)] for a 100 cm² Red305 with different quantum yields. From Fig. 6(a) we can see that η_{trap eff} can be separated into two regions: low and high optical density.

 

Fig. 6 (a) Optical efficiency η (dashed) and effective trapping efficiency η_{trap eff} (dotted) in Red305 for increasing quantum yields. The white area indicates the region with low optical density, and the shaded area indicates high optical density. (b) Onset of the increase in η for η_qy = 1 Red305 LSCs of varying sizes and thicknesses. This onset is defined as the point where η starts rising again. Lines are drawn as a guide to the eye.

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In the region with low optical density, the expected behavior for a self-absorbing material occurs. Photons are reabsorbed multiple times, and with each reemission there is a ∼ 25 % chance of entering the escape-cone and escaping through the LSC’s surface. A higher α means many reabsorption events, and therefore a higher possibility that a photon will end up escaping. In this region absorption within the escape-cone is outweighed by absorption during total internal reflection. Figure 5 shows that, when considering very low α, the still present escape-cone absorption gives a photon a chance to become absorbed before escaping. This still present trapping yields positive self-absorption, with η_sa > 1. Transport in the region of low optical density is hence governed by reabsorption and remission events during total internal refraction, resulting in a decreasing η_{trap eff} as α_max rises.

In the region with high α_max, however, we see that η_{trap eff} starts splitting up, depending on η_qy. In this region the redshifting of photons starts greatly influencing light transport. A photon will experience many reabsorption events over a short traveled distance. As η_qy rises, the probability for the photon to redshift outside of the LSC’s absorption spectrum before quenching increases as well. When the photon redshifts outside of the LSCs absorption spectrum it cannot be absorbed again. Transport of the photon is then fully governed by total internal reflection. This leads to the effective trapping efficiency rising to near the theoretical η_trap for a large LSC, because when viewed from the perimeter, the absorption and emission spectrum of the LSC do not overlap anymore. Combined with a high η_qy, this effect is accentuated, as almost every reabsorbed photon will be reemitted. For a high absorption maximum α_max, a local maximum of η_{trap eff} coincides with a high amount of absorbed solar light, yielding a second, higher maximum in optical efficiency for strongly self-absorbing materials, with η_qy ≈ 1, like Red305.

In most small-scale experimental setups the second maximum displayed in Fig. 6(a) will not be visible. With high absorption coefficients (α_max > 60 cm⁻¹) luminescence quenching namely occurs [31]. Figure 6(b) shows that as the size or thickness of the LSC increases, the onset of the second maximum can become visible before luminescence quenching occurs. In large BIPV-scale LSCs of around 4 × 4 m² this effect therefore could become perceivable at realistic concentrations of α_max < 60 cm⁻¹.

6. Expected BIPV efficiencies

With the hybrid approach of MGLT for optimization and MC for a full efficiency calculation, it is possible to look at how the selected materials would perform as (colored) BIPV windows. Using the data from Table 2 would yield a theoretical maximum efficiency, but also visually opaque windows with potential luminescence quenching. Therefore as design critera we use (i) a 100 × 100 × 0.5 cm³ window, (ii) maximally 25 % visible light (390 nm to 700 nm) absorption [32], (iii) α_max ≤ 60 cm⁻¹ to avoid luminescence quenching.

An important figure of merit is the LSC’s power efficiency η_power. The power efficiency is the ratio of power produced by the PV cells enclosing the LSC to the incoming sunlight [18]:

The power efficiency does not yet tell if the LSC yields a higher efficiency than exposing the PV cells directly to sunlight. A parameter that captures this is the concentration factor [18]

To match the output spectrum of an LSC with a solar cell, data on the current record GaAs [33] (η_PV = 27.6 %), CIGS [34] (η_PV = 21.0 %), CdTe [35] (η_PV = 21.0 %), CZSSe [36] (η_PV = 7.6 %), CZTS [36] (η_PV = 9.8 %), Perovskite [36] (η_PV = 22.1 %), Dye [37] (η_PV = 11.9 %), Organic [36] (η_PV = 11.2 %), and Si [38] (η_PV = 25.6 %) cells were used to select the cell which would give the best performance.

First, two generations of MGLT are used to get an indication of what α_max an LSC would be most efficient with when combined with each of the solar cells. From this range the α_max that yields the highest estimated η_power adhering to the design-constraints is selected. Finally accurate values for η_power and Γ are calculated with MC. The results for the five best performing 100 × 100 × 0.5 cm³ LSCs are shown in Table 3. A theoretical maximum, if there was no concentration quenching, for all LSCs from Table 2 can be found in Data File 2. In the theoretical case, the second maximum of Red305 yields ${\eta }_{\text{power}}^{\text{GaAs}}=8.6\hspace{0.17em}\%$ as maximal power efficiency.

Table 3. Power conversion efficiencies and concentration factors for 100 × 100 × 0.5 cm³ LSC materials with PV cells covering the entire perimeter with α_max = 60 cm⁻¹ (except for Red305, where α_max = 5.5 cm⁻¹ and LumoO with α_max = 7.9 cm⁻¹) and where only 25 % of the visible spectrum is absorbed ( ${\eta }_{\mathbf{LHE}}^{\mathbf{vis}}=\mathbf{25}\hspace{0.17em}\mathbf{\%}$). Here ${\lambda }_{\mathbf{max}}^{\mathbf{out}}$ is the LSC’s output spectrum observed at the perimeter and ${\eta }_{\mathbf{power}}^{\mathbf{GaAs}}$ the highest power efficiency using a GaAs solar cell. ${\eta }_{\mathbf{power}}^{\mathbf{alt}}$ is the best alternative (non-GaAs) cell’s power efficiency, with concentration factor Γ^alt. Perovskite has been abbreviated to Per.

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From Table 3 we see that most presented LSCs would already function as efficient PV solar concentrators, evident from Γ ≫ 1. These could be of use in e.g. noise barriers [39], where no clear visible light throughput at the rear of the LSC is required. Also as BIPV windows, in places where no solar cells would otherwise be placed, the LSCs can still perform their function as a BIPV concentrator. Even when a serious constraint of ${\eta }_{\text{LHE}}^{\text{vis}}=25\hspace{0.17em}\%$ is placed on the LSCs, Γ > 1.

One remark that can be made is that the older LSC materials, Perylene perinone (PP) and Red 305, are still among the most efficient. With the constraint of ${\eta }_{\text{LHE}}^{\text{vis}}=25\hspace{0.17em}\%$ the efficiency of PP does however drop significantly. This is due to the overlap between the visible solar spectrum and the absorption of PP being larger than in the other presented LSCs. ${\eta }_{\text{LHE}}^{\text{vis}}=25\hspace{0.17em}\%$ is therefore a more severe constraint on PP than on any of the other presented materials.

One can also notice the significant redshift observed in the output spectrum of PP when increasing the concentration by more than 40× (noted between parentheses in Table 3). This redshifting can be explained by the large overlap between the absorption and emission spectra of PP.

Optimizing dye concentration with respect to the highest possible η is not always the best approach. For all materials there is a non-constant discrepancy between η and η_power. When regarding practical applications it’s better to optimize α_max with respect to the PV cells available, for maximal power efficiency or concentration factor. We can see that optical efficiency of each material only gives an indication of how well an LSC will perform when connected to a PV element compared to others, i.e. materials with a higher η will also perform better when connected to a solar cell. An example of this is TIIiso. TIIiso has η = 6.7 % for α_max = 60 cm⁻¹, while its ${\eta }_{\text{power}}^{\text{GaAs}}=3.3\hspace{0.17em}\%$. This can be explained by the mismatch of the LSC’s output spectrum and any PV cell’s quantum efficiency curve. Most LSCs will have their peak perimeter-output still within the higher ranges of a PV cell’s EQE curve, but due to redshifting from self-absorption, much of the spectrum will fall in the lower EQE region. Another consequence of this redshifted spectrum observed at the perimeter of an LSC, is that the optimal solar cell choice varies widely. This variation puts an emphasis on the need to either design a solar cell for a specific LSC, or, if the LSC were to be designed for use with a specific solar cell, to optimize the redshift of LSC output spectra by altering the LSC’s geometry or maximal absorption coefficient.

Variations in materials producing similar η_power, but carrying a different concentration factor can be explained by the solar conversion efficiency of the cell. An example of this variation is seen with LumoO and TIIiso. Even with almost the same conversion efficiency for ${\eta }_{\text{LHE}}^{\text{vis}}=25\hspace{0.17em}\%$, the higher η_PV of the Si cell results in a lower concentration factor for LumoO. A similar table could be constructed showing which solar cell would offer the maximum concentration factor. Solar cells with much lower η_PV would dominate such a table, as the difference between η_PV and η_power becomes lower for these cells. Optimization for maximum concentration factor or a compromise situation, yielding quickest return of investment, could be done in a similar fashion.

7. Conclusions

We have shown that, using a multiple-generation light transport model based on Beer-Lambert’s law, light transport without scattering through an LSC of arbitrary geometry can be precisely and fully characterized, with only a relative difference of 4.4 ± 7.7 % between MGLT and MC for 1 m² LSCs. Using the MGLT approach, calculating only the first generation of light transport is sufficient to optimize the luminophore concentration of an LSC. The total efficiency of this optimized LSC can then be calculated using MC. Most insight in most materials can however be gained if both models are used in conjunction with each other. The MGLT model can show in great detail what areas of the LSC will contribute to edge transport, whereas MC can more quickly provide an accurate assessment of what efficiencies one could expect. It has been shown that the optimal luminophore concentration is dependent on the size of an LSC. Therefore, an optimized lab-scale LSC’s concentration is not transferable to LSCs of different sizes. The power conversion efficiency of 1 m² LSCs does not exceed a theoretical 9 % and a practical 6 % for the luminescent materials addressed in this work. These materials can still function very well as BIPV elements, owing to the high concentration factors. As BIPV element, the LSC functions both as a PV solar concentrator and as a semi-transparent power-generating window, which would be complementary to non-transparent solar cells on the building envelope. Finally, we have shown and quantified that for an LSC with a large overlap between its absorption- and emission spectrum, but a high quantum efficiency, self-absorption can be beneficial to its optical efficiency. Photons trapped inside such LSCs will remain trapped until they are redshifted outside of the LSC’s absorption spectrum when the LSC has a sufficiently high absorption maximum. The results obtained in this paper can help in designing efficient, aesthetically pleasing LSCs for use in the built environment.

A. Positive self-absorption in MGLT

Optimizing while taking positive self-absorption into account doesn’t require calculating the total efficiency of the material. The local maxima occurring due to positive self-absorption can also be observed using the MGLT model with 2 generations. Figure 7 shows that, when η_qy = 1 in Red305, the difference in collected fraction Δη_i between two generations decreases as the absorption maximum increases. This indicates that the increasing amount of light absorbed between the first to the second generation does not quench or escape, but rather keeps contributing to collection on the perimeter. For sufficiently high α the second generation starts contributing more to total transport than the first, at the same point that the material’s optical efficiency η starts rising again, giving an indication of a potential efficiency increase due to positive self-absorption.

 

Fig. 7 Optimization track of a 10 × 10 × 0.5 cm³ Red305 LSC with η_qy = 1. A second maximum is seen in the optical efficiency. Here the second MGLT generation contributes more to the edge-transmission than the first generation.

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Funding

Netherlands Organization for Scientific Research (NWO/OCW) (NF16NFS01).

References and links

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