Luminescent solar concentrators (LSCs) were first introduced in the early 1970s and were based mainly on organic dyes as fluorophores [1]. By waveguiding their sunlight-generated photoluminescence (PL) inside the LSCs, the device concentrates light harvested by large surface areas to small edges through PL total internal reflection. Recently, LSCs became attractive again due to their semi-transparency, thus of potential interest for a building-integrated photovoltaic (BIPV), commonly described as “solar windows” [2,3]. The number of edges covered by solar cells (sc) versus mirrors varies depending on the device utilization: from a single solar cell in the original concept to half or full perimeter coverage for solar windows.

The main operational principle of an LSC is the total internal reflection, where PL is trapped inside the device until it reaches the solar cell at the edges. The toolbox of light-converting fluorophores by now has expanded from organic dyes [4] to semiconductor quantum dots [2], rare-earth ions [5], and metal nanoclusters [6]. In addition to the performance of the chosen fluorophore, i.e., photoluminescence quantum yield (PLQY) and its reabsorption suppression (Stokes shift), the total efficiency of the LSC device also highly depends on the waveguiding efficiency. In fact, the waveguiding part is the most difficult to evaluate for LSC efficiency prediction, as it is a complex function of different loss mechanisms (absorption, scattering, etc.)

Many efforts have been devoted to estimating the waveguiding efficiency of the LSC devices [7–9]. Ideally, one needs a model that can accurately predict the waveguiding efficiency for different area devices, in particular for large-area LSCs. The latter is not easy to measure experimentally due to the limitation like the need of a correspondingly large solar simulator [10]. As a result, it makes simple mathematical extrapolation from the available small-area LSCs data not very reliable. From the theory side, phenomenological models have mainly been employed previously, such as a polynomial fitting to extensive Monte-Carlo simulations [8,9,11]. Although useful as a first-order approximation, those are void of physical meaning and cannot explicitly reveal transitions between different device operation regimes with increasing geometries.

In this Letter, we introduce an analytical model to describe the waveguiding efficiency of large-area LSCs, where the four edges are covered by a varied number of mirrors and solar cells. These full and simple analytical formulas are not only a straightforward tool for waveguiding efficiency calculation, but also can be used to evaluate the device performance in different material systems for the whole range of αL, where L (cm) is the device length, and α (cm⁻¹) is an attenuation coefficient (e.g., matrix absorption coefficient α_mx). We discuss the physical meaning of the formula, compare our model with the phenomenological expression from [8], and outline reasons for deviations. Importantly, the model predicts a linear dependence in the log-log scale for large αL, which reveals a different waveguiding regime in large-area LSC devices, corresponding to signal contribution from a limited area. Explicit values for critical LSC device sizes before the onset of this less favorable waveguiding regime are presented. The model was experimentally verified by a set of measurements on a 19 × 19 cm² silicon quantum dot-based LSC, where all relevant losses were quantitatively considered, validating the theory without any fitting parameters.

We start by considering a rectangular LSC with two edges covered with solar cells and two others with mirrors (Fig. 1, inset). The key idea of this work is that for an isotropic light emitter placed inside the slab, this geometry is equivalent to two infinitely long solar cells, as far as the optical path is concerned.

 

Fig. 1. Waveguiding efficiency for a rectangular LSC with two mirrors from Eq. (2) (red line) and from the general formula [Eq. (1), black dashes line, $\textrm{w} = 50000\; \textrm{cm}$]. Inset shows device schematics.

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Previously, we have shown that the general expression for the waveguiding efficiency of rectangular LSCs with full perimeter coverage by solar cells is [7]

Then, for the infinitely wide device $({w \gg L,w \to \infty } )$, one can expand the general formula in Eq. (1) by a small parameter L/w (see derivations in Supplement 1), and the waveguiding efficiency becomes a function of a single parameter $\alpha L$,

A simple expression of Eq. (2) allows detailed waveguiding efficiency analysis. From Eq. (2), its trend for large αL can be immediately evaluated as ${f_{2sc\infty }} = 2/\alpha kL\pi $ (due to quickly decaying exponential and exponential integral terms). It is indeed seen as a straight line in the log-log scale in Fig. 1, and the same trend was presented for large device in one of the original LSC reports [12]. It is also obvious from Eq. (2) that for $\alpha L \to 0$, the function ${f_{2sc}}$ becomes unity, as expected for a no-loss scenario.

Now we can compare Eq. (2) with a phenomenological expression ${({1 + \; \beta \alpha L} )^{ - 1}}$ with a parameter $\beta \; \approx \; 1.4$, obtained by polynomial fitting of extensive Monte Carlo simulations [8]. In Fig. 2, this function is compared with Eq. (2) and deviation at large $\alpha L$ is visible. Strictly speaking, the parameter $\beta $ is not a constant, but depends on $\alpha L$. By solving the equation ${f_{2sc}}({\alpha L} )= {({1 + \; \beta \alpha L} )^{ - 1}}$, the actual values of β as a function of $\alpha L$ can be obtained (shown in Fig. 2, inset). It is seen that β saturates for $\alpha L \to \infty $ at ${\beta _\infty } = k\pi /2 \approx \; 1.79$, which is why a value of 1.8 was suggested for this range in [8].

 

Fig. 2. Comparison of Eq. (2) (red line) with the phenomenological equation${\; }{({1 + \beta \alpha L} )^{ - 1}}$, $\beta = 1.4$ (black dotted line). Correct values of $\beta $ are shown in the inset.

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The main advantage of Eq. (2) is, however, not a more accurate waveguiding efficiency description for $\alpha L \to \infty $, but the analysis facility. Indeed, for large values of $\alpha L$, it is only the area next to the solar cells that contribute to the waveguiding efficiency, while the part in the middle of the device become inactive (extinct PL). Then the useful area in such a case can be estimated as 2α⁻¹w (two rectangles close to solar cells), while the total area is Lw. In general, the waveguiding efficiency is proportional to the useful fraction of the total area, so it scales here as $2/\alpha L$. Therefore, the analytical expression at large $\alpha L$ range, ${f_{2sc\infty }} = 2/\alpha kL\pi $ reveals an inferior waveguiding regime, where the central part of the device is no longer active (straight line in the log-log scale).

Thus, it is important to identify a transition point where the function ${f_{2sc}}({\alpha L} )$ turns linear in the log-log scale. From Fig. 1, we can set a numerical limit for the LSC with two edges covered by solar cells approximately at $\alpha L = 2$. Here the deviation of the straight line from ${f_{2sc}}$ is only approximately 5%, which can be within experimental measurement error. So, for a typical polymer/glass matrix (PMMA, OSTE, low-iron glass) with an absorption coefficient of 0.04–0.05 cm⁻¹, the maximum mirror length translates to the value ${L_{max}} = 40 - 50$ cm. This result is not straightforward from the phenomenological description method.

Now we turn to a rectangular-shaped LSC with only one edge covered by a fully absorbing photodetector and the rest with mirrors. In fact, it was one of the initially proposed configurations to enhance the collected light concentration about half-a-century ago [12], for which efficiency limits are still being explored numerically today [13]. From the same optical path arguments, it can be shown that for one side solar cell and three side mirrors, the geometry is equivalent to the case of two infinite solar cells separated by mirrors with a length of 2L, ${\; }{f_{1sc}}({\alpha L} )= {f_{2sc}}({2\alpha L} )$:

To complete the description, a rectangular LSC with three side solar cells and one side mirror can also be considered. In this case, the system is similar to a rectangle with fully covered perimeter with a double length. Then the waveguiding efficiency is not a function of a single parameter. It can be expressed from the general formula of Eq. (1) as

 

Fig. 3. Transmittance, absorption, reflection, and haze spectra of the used LSC device. Inset is a photo of the used 19 × 19 cm² LSC device.

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To measure all the configurations mentioned above, the initial device had three mirror-covered edges (attached by a polymer without an air gap) and one solar cell-covered edge. Then the mirrors (metal-coated plastic bars) were replaced by solar cells one by one. “Roofs” (5-mm-wide black paper stripes) covered the top of the device edges for preventing any direct illumination on the solar cell. Ideally, only the portion of sunlight absorbed by Si QDs should contribute to the generated electrons for an intrinsic LSC performance evaluation. To verify this, EQEs were measured. As can be seen in Fig. 4, inset, EQEs are indeed close to zero above 650 nm, implying almost all the measured photocurrent (I-V curves shown in Fig. S2 in Supplement 1) is formed by the photons from the luminescent Si QDs, and not from the direct light (EQE curve follows Si QD absorption curve, cf. Fig. S1 in Supplement 1).

 

Fig. 4. Waveguiding efficiency of the device in the cases of different numbers of edges covered by solar cells: 1, 2, 3, and 4 edges (the remaining edges are covered by mirrors). Lines are from Eq. (5) and points are from the experiment [Eq. (6)]. Inset are external quantum efficiency and short circuit current of the device.

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For more relevant comparison with theory, we note that in practical situations, scattering, re-absorption and matrix absorption losses are all present. In this case, the waveguiding efficiency for N solar cell coverage becomes [7]

Finally, the short-circuit current can be presented as

As seen, the experimental data match well with the predicted values without any fitting parameters. However, there is still a slight systematic error rendering all data points somewhat below theoretical curves. In particular, compared with the 3sc and 4sc cases, the measured waveguiding efficiencies of 2sc and 1sc are further away from the theoretical predictions. We have two hypotheses regarding this: the first one is because the mirrors may not be reflecting completely or vertically to the active layer leading to additional losses; the other possible reason is that, in our device, there are some non-uniformities, such as higher haze areas. This will cause a higher scattering loss when the luminescence light propagates through those regions. Moreover, the waveguiding efficiency in the 2sc case will be affected by these two imperfections together. By contrast, in the 1sc case, the extinction loss is already high (due to the long light path), while in the 3sc case, the mirror-reflected light contribution is minor. Therefore, the experiment error in the 2sc case appears to be the most significant. We also note from Fig. 4 that for the 1sc case, the 19 × 19 cm² device indeed already approaches its critical size (straight line regime in log-log scale).

To conclude, we present an analytical model for the LSCs for different solar cell edge-coverage. Compared with existing models based on extrapolation of small-area LSCs experimental results, numerical simulations, or polynomial fittings, this methodology is not only more accurate, but also contains physical information, making it possible to establish criteria for largest meaningful LSC sizes with attached mirrors. This critical LSC size can be expressed as ${L_{crit}} = N/\alpha $ for N = 1 or 2 solar cells covering the LSC edges for the absorption coefficient $\alpha $. The proposed analytical approach was successfully verified by a 19 × 19 cm² Si QDs-based LSC, with results confirming the theory within the experimental error.