A planar waveguide with uniformly embedded light sources is a generic concept used in laser phosphor displays [1], solar-pumped lasers [2], and luminescent solar concentrators (LSCs). The latter were first proposed in the early 1970s [3], with the main purpose to concentrate sunlight by means of photoluminescence (PL). The fluorophores embedded in the host matrices absorb a part of sunlight and re-emit light. A significant fraction of PL emission (∼75% for a slab with a refractive index of 1.5) is trapped in the waveguide by total internal reflection until it reaches the edges for conversion to electricity. In recent years, LSCs have gained much attention as semitransparent building-integrated photovoltaic (BIPV) units balancing PV performance and aesthetic quality, commonly known as “solar windows” [4].

The active light-converting materials are typically organic dyes [5], semiconductor quantum dots [4], rare-earth ions [6], or metal nanoclusters [7]. For efficient operation, a high photoluminescence quantum yield (PLQY) and a large Stokes shift are required to reduce reabsorption loss of the emitted light. The typical matrices are glass or polymers such as poly-(methyl-methacrylate) (PMMA) and off-stoichiometry thiol-ene (OSTE) [8]. A triplex laminate structure is also used [8], with a thin polymer interlayer sandwiched in between two glass pieces. The fluorophores should also be well dispersible inside, translating to low scattering and haze for better energy conversion efficiency and glazing function.

Over the years, efforts have been devoted to improving the power conversion efficiency (PCE) of LSCs [9]. In addition, extending the device size to large areas (≥100 cm²) becomes significant for practical applications. For the LSC PV performance characterization, following the conventional method requires a correspondingly large solar simulator. This approach, however, is not feasible when it comes to truly large LSCs (tens of cm) for in-the-lab device measurements. Recently, regional measurements were introduced to analyze the PV performance and the spectral response of large-area LSCs with a shifting mask and a piece of solar cell placed at different edge regions [10,11]. However, scanning numerous excitation points all over the large-area device may not always be practical either.

In this Letter, a new approach for estimating the PV performance of large-area LSCs from point excitations is introduced. These points are named as “optical centers” of an LSC in analogy with the center of mass in mechanics. It is defined as points from which the average optical path to the device edges is equal to that of the whole device under uniform excitation. Here, the response to point excitations for a planar waveguide is analytically derived, and the position of such optical centers is found to be at ∼82% of the center-edge distance for the square geometry. Using selective excitation to this spot, the short-circuit current ${I_{sc}}$ of the whole device can be obtained by linearly scaling with the excitation area. This hypothesis was then successfully verified by a set of measurements on a 20 × 20 cm² LSC with silicon quantum dots (Si QDs) as fluorophores. It is shown that using the average ${I_{sc}}$ measured at the optical center, the PCE of this device can be very well predicted within a measurement error, confirming that this approach is practically useful for the experimental evaluation of the PV performance of large-area LSCs.

In this work, we consider square geometry, which is very common for LSCs. Previously, we found that uniform excitation square and circular geometries were very similar in terms of optical path distribution from an isotropic emitter [12]. Using this established similarity here, we can derive simple expressions for the average optical path for a point excitation.

Consider a square with side length 2a and a circle of the same area with radius R. These two quantities are related by $R = 2a/\sqrt \pi \approx 1.128a$. First, we derive an analytical formula for the average optical path inside a circle for an emitter placed at a distance $\rho$ from its center in 2D [Fig. 1(a), inset]. An optical ray emitted at an angle $\theta$ from the radial direction will experience path r until reaching the perimeter, where it is absorbed (all edges covered with solar cells). These two quantities are related as

The emitter is isotropic, so the angular distribution of its intensity $dI/d\theta$ is a constant. Then one can express emitted intensity distribution over the optical paths as

 

Fig. 1. (a) Average optical path as a function of the point excitation position. Square dots represent numerical simulation values for a square, while the line is for a circle of the same area from Eq. (3). Inset is a schematic illustration. (b) Waveguiding efficiencies for point (green, red, blue, orange solid lines) and uniform (black solid line) excitations of an LSC as a function of the dimensionless variable α_mR. The vertical gray dashed line indicates the LSC device used in this work for the experimental validation.

Download Full Size | PDF

Then the average optical path is a first moment of this distribution. Integration yields

To validate the applicability of this formula for a square geometry, we plotted numerically calculated average optical paths for an isotropic emitter placed at several points inside the same area square (details in the Supplementary material). Those are shown as square dots in the same units in Fig. 1(a), confirming similarity of these two geometries in this sense. So the first conclusion of this analysis is that the position of a point excitation in 2D geometry (x,y) for a square waveguide can be effectively reduced to the function of a single variable $\rho$.

Next, we postulate that there is a spot on the LSC (at a distance from the center ${\rho _0}$), where the average optical path for an isotropic emitter equals to that of the whole device under uniform illumination. The latter was derived in [12] for a circle as 8R/3π, therefore,

The numerical solution of this equation can be found using Eq. (3). In the units of length, it is ${\rho _0} = 0.731R$. For a square with the side length $2a$, this is equivalent to

We assign these points to the optical centers of a square planar waveguide.

In general, however, the average optical path may not necessarily represent a valid parameter for light propagation assessment. For example, because the distribution of Eq. (2) is bimodal, the presence of attenuation may result in substantially different waveguiding efficiencies than those predicted using ${r_{avg}}$. Another necessary realistic aspect to consider is the 3D geometry of such planar waveguides.

Due to the total internal reflection, all photons emitted outside the escape cone (42° for a glass or plastic slab) will be waveguided to the device edge. The distribution of optical paths for such photons is therefore quite narrow and an average value for the out-of-plane emission can be used [13]. It was previously shown that all 2D optical paths can be simply extended by a factor $k \approx 1.144$ [12]. Changing variables ($l = kr$) from $p(r)$, the normalized probability density function $q(l)$ for the 3D case becomes

Using this formula, we can compare the waveguiding efficiency for the single spot excitation with the total area illumination case in the presence of attenuation. Let the attenuation coefficient be $\alpha$ [cm⁻¹], which in practice can represent a matrix absorption coefficient ($\alpha \approx {\alpha _m}$) when only considering the matrix absorption loss. Then the waveguiding efficiency for a spot excitation at position $\rho$ is

To verify our hypothesis, a 20 × 20 cm² (a = 10 cm, R ≈ 11.3 cm) low-haze LSC device based on luminescent Si QDs was used for the experimental validation. The device fabrication procedure can be found elsewhere [14]. In the triplex laminate structure, an off-stoichiometry thiol-ene (OSTE) polymer interlayer is of 3-mm thickness, and glass sheets are of 2-mm thickness each. Altogether ∼60 mg of near-infrared-emitting (PL peak position ∼870 nm, PLQY ∼50%) Si QDs are uniformly dispersed in the polymer interlayer. The transmittance, absorption, reflectance, and haze spectra of this device are shown in Fig. S1 (Supplement 1). Note that this device is highly transparent (visible light transparency of ∼90%) and thus not optimized for high power generation. Under the standard AM1.5G, ∼8.3% of solar power is absorbed for the ensuing conversion to electricity (Fig. S2, Supplement 1). As reported previously [12], the matrix absorption coefficient ${\alpha _m}$ is 0.04 cm⁻¹ for OSTE and of similar value for the low-iron glass applied here. Therefore, the dimensionless variable for this device is ${\alpha _m}R \approx 0.45$ [marked as a dashed line in Fig. 1(b)]. Details of the attached solar cells and the solar simulator used for the test can be found in Supplement 1l.

For excitation position-dependent measurements, two directions (center-left and center-right) of the LSC were probed under a series of point excitations by shifting the 1 × 1 cm² opening positions on the mask (details in Supplement 1). As shown in Fig. 2(a), a clear rising trend of ${I_{sc}}$ from the center to the edge along both directions was observed. An analytical curve [gray curve in Fig. 2(a)] can be drawn by considering the matrix absorption loss for the average optical path ${r_{avg}}$ under different point excitation positions [Fig. 1(a)] and the linearity of the short-circuit current with irradiance as ${I_{sc}}(\rho ) = {I_0}{e^{ - {\alpha _m}k{r_{avg}}(\rho )}}$. As can be seen, the experimental data match well with the predicted trend with proportionality coefficient ${I_0}$ being the only fitting parameter.

 

Fig. 2. (a) Short-circuit current under point excitations along two directions (center-left in blue and center-right in red) of a 20 × 20 cm² LSC under 0.9 AM1.5G. The gray solid line indicates the predicted trend. Note that due to the fabrication procedure of layered polymerization, only the directions non-crossing interfacial optical defects were considered. (b) External quantum efficiency under the optical center illumination (red) and uniform illumination of the whole device (blue). Inset is a photograph of the 20 × 20 cm² device.

Download Full Size | PDF

To independently verify that the observed trend indeed originates from the luminescence waveguiding and not from random stray light or direct light on solar cells, we performed external quantum efficiency (EQE) measurements under the optical center (8 cm from the geometrical center) and whole device illumination. For the LSC device, EQE(λ) is the ratio of generated electrons by the edge solar cells to incident photons on the device top area. To block the direct illumination of solar cells, four “roofs” (each 5 mm wide) were applied at the top of the device edges in the case of the whole device illumination (in the case of point excitation, those are redundant thanks to the mask). As shown in Fig. 2(b), the EQE curves are consistent with the shape of the absorption spectrum of Si QDs (Fig. S1, Supplement 1). More importantly, both EQE curves are essentially identical, indicating almost the same short-circuit current density (mA/cm²) after being integrated with the solar spectrum. Moreover, the EQEs are close to zero above 650 nm, implying almost no contribution from the direct illumination or stray light, validating previous results.

Verification of the signal origin being from QD PL is essential for correct evaluation and application of the optical center method. Another aspect to consider is the size of the solar cell modules, as discussed in detail in Supplement 1. Briefly, to correctly resolve optical path variations under the point excitation, small size solar cells units, connected in parallel, are necessary. Otherwise, the ${I_{sc}}$ of a solar cell will be limited by the lowest ${I_{sc}}$ among the in-series connected p-n junctions (longest optical path). An example is shown in Figs. S3 and S4 in Supplement 1, where four-fold longer solar cells units were attached to the device edges. In that case, the trend, which is clearly visible in Fig. 2(a), becomes disrupted when approaching the edges.

After the optical center concept has been experimentally verified, we are in the position to discuss practical implementations. The most straightforward application is the short-circuit current evaluation of the whole device from the optical center excitation. To minimize the impact from the solar simulator non-uniformity, each ${I_{sc\_oc}}$ value under the optical center excitation was measured at each orthogonal direction under the solar simulator by spinning both the device and the mask, yielding an average ${I_{sc\_oc}}$ value of ∼79 µA (±3 µA) under standard AM1.5G. Multiplying by the opening area ratio (400:1), one can predict ${I_{sc}}$ under uniform illumination to be ∼31.6 mA (±1.2 mA), which is very close to the measured value of ∼33 mA (±0.5 mA) by the conventional method, matching within measurement error (I-V curve shown in Fig. S5, left, Supplement 1). We attribute this small error to the measurements instead of the mathematical assumptions, most likely due to the slight non-uniformity of the irradiance from the solar simulator and the LSC device. This confirms the plausibility of single excitation spot measurements for the whole device evaluation in terms of photocurrents.

Next, we can extend the method of predictive ability to the power conversion efficiency (PCE), which is an important figure of merit for photovoltaic devices. To accomplish that, the open circuit voltage and the fill factor (FF) need to be estimated in addition to the short-circuit current. However, under optical center excitations, the ${V_{oc}}$ is limited by the lowest ${V_{oc}}$ among the solar cells connected in parallel (farthest solar cells from the open window), and we instead suggest two methods for the estimation of ${V_{oc}}$ and $FF$.

On average, for each piece of solar cell, ∼1 mA (32 pieces in parallel) of current can be coarsely estimated. This estimation does not introduce a large error since the open circuit voltage grows logarithmically to the incident irradiance (or to the short-circuit current, which is linear to irradiance) [15].

By tuning the irradiance under the solar simulator (either percentage of AM1.5G or monochromatic light), the I-V curve (Fig. S5, right, Supplement 1) of a single piece of solar cell was recorded to simulate device operation conditions delivering ${I_{sc\_s}}$ of ∼1 mA. Then the open circuit voltage was ∼2.01 V and the fill factor was ∼0.63, which are indeed almost the same as measured ones for the operating LSC (Fig. S5, left, Supplement 1). Alternatively, considering the known logarithmic relationship between the ${V_{oc}}$ of the solar cell and the incident irradiance similar values can be obtained, as detailed in the Supporting Information.

At last, the procedure of whole device PCE estimates from point excitation measurements can be summarized. (i) Measure the average ${I_{sc\_oc}}$ with only the window at the optical center open. Scaling with the whole device area gives the overall ${I_{sc}}$ under uniform illumination. (ii) Consider the solar cell connections and estimate the ${I_{sc\_s}}$ value for a single piece of solar cell. Place a single piece of solar cell under the solar simulator, and tune the irradiance until the solar cell delivers the same ${I_{sc\_s}}$. (iii) Extract the open circuit voltage ${V_{oc}}$ and fill factor $FF$ from the I-V curve under this irradiance. (iv) Apply these values to the whole device efficiency calculations: $PCE = {I_{sc}} \cdot {V_{oc}} \cdot FF/{P_{incident}}$.

To conclude, the concept of an optical center on a square LSC was proposed based on analytical derivations, where the average optical path of emitted photons from the optical center and the whole device are stipulated to be the same. It is found that the optical center is located at ∼82% of the half-length distance from the geometrical center of the planar waveguide. In practice, from the short-circuit current measured at the optical center and the ratio of illumination area, one can predict the short-circuit current of the whole device under uniform illumination. This hypothesis is successfully verified by a 20 × 20 cm² Si QDs-based LSC both for ${I_{sc}}$ and PCE with a negligible error. This methodology can help access the PV performance of a large-area LSC device by point excitations with basic laboratory instruments.