Monte Carlo Simulation of Quantum-Cutting Nanocrystals as the Luminophore in Luminescent Solar Concentrators

Abstract

Quantum-cutting luminescent solar concentrators (QC-LSCs) have great potential to serve as large-area solar windows. These QC nanocrystals can realize a photoluminescence quantum yield (PLQY) of as high as 200% with virtually zero self-absorption loss. Based on our previous work, we have constructed a Monte Carlo simulation model that is suitable to simulate the performance of the QC-LSCs, which can take into account the band-edge emissions and near-infrared emissions of the QC-materials. Under ideal PLQY conditions, CsPbCl_xBr_3−x:Yb³⁺-based LSCs can reach 12% of the size-independent external quantum efficiency (η_ext). Even if LSCs have a certain scattering factor, the CsPbCl_xBr_3−x:Yb³⁺-based LSCs can still obtain an η_ext exceeding 6% in the window size (>1 m²). The flux gain (FG) of the CsPbCl_xBr_3−x:Yb³⁺-based LSC-PV system can reach 14 in the window size, which is a very encouraging result.

1. Introduction

Building-integrated photovoltaic (BIPV) is proposed to help near-zero-energy buildings to reduce their carbon emissions. BIPV, which aims to integrate photovoltaic elements into the building envelope, can replace the conventional materials of roofs, walls and windows [1,2]. Very recently, luminescent solar concentrators (LSCs) have been widely researched, which are quite suitable for constructing solar windows [3,4,5]. The LSC-PV windows’ surfaces are free of electrodes and can be selected due to their color and transparency, thus better meeting the aesthetic and practical needs of buildings. LSCs are sunlight collectors that absorb incident solar photons, emit photoluminescence photons, and waveguide these photons via total internal reflection to the edges [6]. Therefore, the LSCs do not need additional solar tracking systems, which are expensive. Common luminophores for LSCs often suffer from reabsorption loss and nonradiative recombination. To address this issue, recent work has reported luminophore materials with low reabsorption as well as high photoluminescence quantum efficiency (PLQY). For example, Li et al. reported that they synthesized a 4,4′-(benzo[c][1,2,5] thiadiazole-4,7-diyl) bis (N,N-diphenylaniline) AIE-emitter, which displayed a near-unity emission quantum yield in a matrix with a large Stoke shift of 0.59 eV [7]. Additionally, they engineered the surface texture of the PDMS matrix by using a bioinspired nanolithography method with a natural lotus leaf as the template to build LSCs that inherited the superhydrophobic, self-cleaning properties. Gungor et al. synthesized Zn-doped CuInSe_2−xS_x/CuInS₂ quantum dots with a Stoke shift of 0.15 eV and PLQY of 78% and used this quantum dots to build the LSCs with an external quantum efficiency of 11.7% and dimensions of 9.5 cm × 9.5 cm × 0.77 cm [8]. Park et al. introduced a new, laminated type of LSC structure, where a patterned low-refractive-index medium acts as an optical ‘guard rail’, providing a practically non-decaying path to guide photons. They found that the external quantum efficiencies at 450 nm are 45% for a 100 cm² area [3]. Very recently, Wang et al. fabricated large-area (∼100 cm²) tandem LSCs based on highly stable carbon dots and highly luminescent near-infrared emitting CuInSe_2−xS_x/ZnS QDs [9]. The PL emitters have large Stoke shifts and high PL QYs, of 54% for UV-active CDs and 61% for NIR-emitting CuInSe_2−xS_x/ZnS QDs. Wang et al. synthesized high-quality lead-free Cs₃Cu₂Cl₅ perovskite nano-disks with a high photoluminescence quantum yield of 84.2% and a large Stokes shift of 1.43 eV [10]. LSC luminophores, such as organic dyes [11,12], colloidal quantum dots [8,13,14,15,16,17], and carbon dots [18,19,20,21], have rapidly developed in recent years. However, the PLQY of most of these materials is typically less than 100%, and thus the internal quantum efficiency (η_int) of the LSC is limited to 75%.

Recently, various excellent dopant examples have been demonstrated, including the main group elements [22,23], transition metals [17,24], and rare earth elements [25,26,27]. Rare earth Yb³⁺ ion is not only an important doping candidate for fundamental studies of doping chemistry but also has high application potential. For example, Milstein et al. first described the hot injection synthesis of Yb³⁺:CsPbCl₃ nanocrystals with analytical Yb³⁺ concentrations of as high as 7.5% and replicable PLQYs exceeding 100% [28]. They believed that energy captured by a Yb³⁺-induced defect can subsequently be transferred to two neighboring Yb³⁺ ions in a single concerted step at the picosecond time scale. Ding et al. demonstrated Cr³⁺, Yb³⁺, and Ce³⁺ tri-doped CsPbCl₃ perovskite quantum dots and they integrated them with silicon photodetectors [25]. These quantum dots have a total PLQY of 188% and excellent stability due to the Ce³⁺ doping. The response of silicon photodetectors successfully expanded to the deep UV region, realizing the full spectrum response within 200–1100 nm. Cai et al. reported that they synthesized Mn²⁺/Yb³⁺ co-doped CsPbCl₃ nanocrystals, which showed unique triple-wavelength emissions covering the ultraviolet/blue, visible, and near-infrared regions [24], and achieved a total PLQY of 125.3%. Gao et al. achieved efficient Yb³⁺ infrared emissions from both quantum cutting and up-conversion, demonstrated by adjusting Er³⁺ and Yb³⁺ concentrations [27]. Shen et al. synthesized Yb³⁺-doped CsPbCl_xBr_3−x perovskite nanocrystals to build electroactivated NIR LEDs [29]. Xu et al. directly identified the doped Yb³⁺ in CsPbCl₃ perovskites using state-of-the-art transmission electron microscopy and three-dimensional atom probe tomography at the atomic scale [30]. They proved that Yb³⁺ simultaneously replaces Pb²⁺ and occupies the lattice interstitial sites. This work provides an atomic-level understanding of the doping mechanism in perovskites.

Based on our previous work [31], we first introduce the quantum cutting (QC) materials into the LSCs. Yb³⁺-doped CsPbCl₃ nanocrystal (NC) with an ultrahigh PLQY of 164% and 25 cm² QC-LSC with an η_int of 118.1% are presented in our previous work. In the QC progress, the formation of shallow Yb³⁺-induced defects play a crucial role in facilitating a picosecond, nonradiative, energy-transfer process. This progress can de-excite the photoexcited perovskite nanocrystal and simultaneously excite two Yb³⁺ dopant ions [28,32,33]. The luminescence of Yb³⁺ is ~990 nm, located at the high-EQE region of silicon-based PV cells [34]. Very recently, many new solar cells have emerged, such as organic PV cells, perovskite PV cells, and GaAs PV cells, which may be suitable for LSCs [35,36,37], but we should consider both their cost and efficiency. Some of these PV cells suffer from low power conversion efficiency, such as organic PV cells and perovskite PV cells, while others suffer from a high price, such as GaAs PV cells. Therefore, silicon-based PV cells are the best candidate for commercialized LSCs.

The schematic principle of the QC-LSC-PV system is shown in Figure 1. The system is simply composed of QC luminophores, a waveguide medium, and silicon-based PV cells. The red arrows represent the QC emission progress and the blue arrows represent the band-edge emission progress. When the sunlight is incident on the face of the LSC, some of the photons are absorbed by the luminophores and new photons are emitted according to their emission spectra; the remaining photons that are not absorbed transmit from the bottom surface. As for the QC-LSC, most of the incident photons will undergo the QC process and emit ~990 nm NIR fluorescence and some of them will go through band-edge emission or loss through nonradiative recombination. Some of the emitted photons in the waveguide medium will enter the escape cone or enter the total internal reflection mode. The band-edge emitted photons may be reabsorbed and re-emit new photons depending on the PLQY of the luminophore. In the end, some of the emitted photons will reach the LSC’s edges and be absorbed by the coupled silicon-based solar cells to produce energy.

In this work, we present a Monte Carlo simulation model for evaluating the performance of QC-LSCs. The model considers both the band-edge intrinsic emissions and near-infrared QC emissions of fluorescent materials. Additionally, the as-synthesized CsPbCl₃:Yb³⁺ and CsPbCl_xBr_3−x:Yb³⁺ nanocrystals have negligible reabsorption and a weak band-edge emission. The spectral data of the two materials were input into our MC simulation model to predict the performance of the QC-LSCs. The simulation results demonstrate the competition between band-edge luminescence and Yb³⁺ ion luminescence. In an ideal situation, CsPbCl_xBr_3−x:Yb³⁺-based LSC can reach external quantum efficiency (η_ext) of 12.3% and there is the potential to apply the CsPbCl_xBr_3−x:Yb³⁺-based LSC to large-area devices. We also discussed the QC-LSC-PV system and found that the CsPbCl_xBr_3−x:Yb³⁺-based LSC-PV system can reach a flux gain (FG) of as high as 14 when the LSC area exceeds 1 m².

2. Monte Carlo Simulation

Monte Carlo (MC) simulation is a very effective way to explore the performance characteristics of LSCs, as the principle of LSCs does not involve phase-dependent wave effects such as interference and diffraction. MC simulation is actually a representation of a statistical law. When enough events are simulated, it will exhibit physical laws close to the actual situation. When all the practical considerations of LSCs are involved, the simulation results will match the actual situation. In this work, we built an MC simulation model which is suitable for quantum-cutting the luminophore of LSC. Figure 2 illustrates the logic block diagram of the MC simulation.

In order to emulate the solar spectrum, each randomly generated photon will carry information about its wavelength and coordinates. When the number of randomly generated photons reaches about one million to one hundred million, the statistical distribution of photons for different wavelengths will be highly similar to the solar spectrum. By considering the solar spectrum as a probability density function and transforming it into a cumulative distribution function, we can obtain information about the wavelength of each randomly generated photon. In this work, all the sunlight energy is considered to be uniformly incident on the surface of the LSC. To obtain coordinates of the random photons, we grid the surface of LSC, and the length of each grid is L/N, where L represents the length of the LSC and N represents the number of randomly generated photons. The coordinates of all photons are randomly distributed to grid points on the surface.

When each photon is randomly generated, it undergoes scattering, absorption, re-emission, re-absorption, and total reflection. The surface scattering can be calculated as follows:

$$R={\left(\frac{n_1-n_0}{n_1+n_0}\right)}^2={\left(\frac{1.5-1}{1.5+1}\right)}^2=0.04,$$

(1)

where n₁ is the refractive index of LSC waveguide and n₀ is the refractive index of air. Absorbance can be determined using the Lambert–Beer Law and the possible distance, ∆S, travelled by each photon in the LSC waveguide medium can also be obtained by the Lambert–Beer Law:

$$∆S=-\frac{l\times {\log }_{10}(1-\delta )}{OD},$$

(2)

where OD is the optical density of each material and l is the length of the cuvette. δ denotes a random number range of [0, 1]. If scattering coefficient, s, is not ignored, Equation (2) can be rewritten as follows:

$$∆S=-\frac{{\log }_{10}(1-\delta )}{OD/l+s/\ln 10},$$

(3)

The scattering coefficient is defined using the Lambert–Beer Law using exponent e, and Equation (2) is defined using the Lambert–Beer Law using exponent 10. Therefore, it is necessary to transform the value of s into a calculation under the condition that the exponent is 10. When a photon has travelled ∆S, some of the photons are considered to be absorbed, while the rest are scattered, i.e., the direction of motion of the photons is updated. If a photon is absorbed, it is re-emitted according to the PLQY.

$$\xi <{\eta }_{PL},$$

(4)

where ξ is another random number that refers to quantum yield range of [0, 1]. Considering that the QC luminophore has NIR emissions and band-edge emissions, we redefine the inequality (4). We suppose that the PLQY of band-edge emissions is α and the PLQY of NIR emissions is β. When the ξ is smaller than the α, the emit-photon will change its wavelength information according to the band-edge emission spectrum. When the ξ is larger than the α and smaller than α + β/2, the emitting photon will change its wavelength information according to the NIR emission spectrum and the NIR emissions photon will be duplicated. If ξ is larger than α + β/2, we will consider that nonradiative recombination will occur.

When the photon is re-emitted or scattered, the photon’s position and directional derivatives will be updated. Because the materials are isotropic, the updated photon coordinate is as follows:

$$\begin{array}{l}{x}_{new}=x_{old}+u_x∆S\\ y_{new}=y_{old}+u_y∆S\\ z_{new}=z_{old}+u_z∆S\end{array}$$

(5)

where (x_old, y_old, z_old) are the photon’s old coordinates, (x_new, y_new, z_new) are the photon’s new coordinates, and (u_x, u_y, u_z) are the photon’s direction cosines. u_x, u_y and u_z can expressed as follows:

$$\begin{array}{l}{u}_x=\sin \theta \cos \varphi \\ u_y=\sin \theta \sin \varphi \\ u_z=\cos \theta \end{array}$$

(6)

where θ is the zenith angle, which is obtained by cosθ = 2a − 1, a ϵ [0, 1], and ϕ is the azimuthal angle, which is obtained by ϕ = 2πb, b ϵ [0, 1]. Therefore, the re-emitted and scattered light is isotropically distributed.

When the photon reaches the surfaces of the LSC, whether the photon enters the internal total reflection or exits from the top or bottom surface will be determined. The η_int is defined as the ratio of the number of photons reaching the edges to the number of photons emitted by the luminophores and the η_ext is defined as the ratio of the number of photons reaching the edges to the number of solar photons. In this work, we consider the coupling of the PV cells to the LSC’s edges to be perfect. When photons reach the edges, they are considered to be absorbed by the edge-coupled PV cells, and then the determination of the next photon is carried out. In this work, we used a full polymer LSC with a thickness of d = 0.5 cm as the model for the analysis. The length (L) of the LSCs varies from 5 cm to 150 cm.

3. Experiments

Materials. 1—Octadecene (ODE, technical grade, 90%) and oleic acid (OA, technical grade, 90%) were purchased from Sigma Aldrich, Beijing, China. Oleyl amine (OAm, 80–90%) and ethyl acetate (EtOAc, AR 99%,) were purchased from Macklin, Shanghai, China, lead acetate trihydrate (Pb(OAc)₂·3H₂O, 99.998%), cesium acetate (CsOAc, 99.99%), trimethylsilyl chloride (TMS-Cl, 98%), trimethylsilyl bromo (TMS-Br, 98%), and hexanes (AR, 97%) were purchased from Aladdin, Shanghai, China. Ytterbium acetate hydrate (Yb(OAc)₃·4H₂O, 99.9%) was purchased from Sigma Aldrich. Anhydrous ethanol (ACS; HPLC Certified 99.9%) was purchased from J&K, Beijing, China. All chemicals were used as received without further purification.

CsPbCl₃:Yb³⁺ Nanocrystal (NC) synthesis. CsPbCl₃:Yb³⁺ NCs were synthesized by hot injection following the procedures. Briefly, 5.0 mL ODE, 0.5 mL OAm, 1.0 mL OA, 0.2 mmol Pb(OAc)₂·3H₂O, 0.28 mL of 1 M CsOAc in ethanol, and 0.16 mmol Yb(OAc)₃·4H₂O were added to a 50 mL round-bottom flask. This solution was stirred and degassed at room temperature for 5 min before heating to 120 °C, and the solution was then degassed for 1 h. The reaction vessel was then flushed with N₂ and heated to 240 °C. Upon reaching this temperature, 0.2 mL of TMS-Cl in 0.5 mL ODE was swiftly injected. Immediately after injection, the flask was cooled to room temperature using a water bath. The NCs were separated from the crude solution by centrifuging at 3350 rpm. After being dispersed in hexane, the NCs were washed again with EtOAc.

CsPbCl_xBr_3−x:Yb³⁺ Nanocrystal synthesis. The CsPbCl₃:Yb³⁺ NCs were dissolved by hexane the solution was placed in a 1 cm path length cuvette under N₂. A certain amount of 0.1 M TMS-Br solution was added to the cuvette. The next day, we could obtain the target product of CsPbCl_xBr_3−x:Yb³⁺ NCs. The solvent was evaporated and the target product was redissolved in hexane for further use.

4. Results and Discussion

We synthesized CsPbCl₃:Yb³⁺ nanocrystal. Figure 3a shows a transmission electron microscopy (TEM) image of CsPbCl₃:Yb³⁺ NCs with average edge lengths of ~15 nm. The absorption spectrum of the nanocrystal is shown in the blue curve of Figure 3b, with the onset of absorption occurring at approximately 410 nm. The emission spectrum is shown in the red curve of Figure 3b, which is dominated by Yb³⁺ ion luminescence in the NIR range centered at ~990 nm. A weak band-edge emission peak at ~400 nm was observed, which may compete with the Yb³⁺ ion luminescence for the charge recombination. The anion-exchange experiment was subsequently carried out, resulting in the production of CsPbCl_xBr_3−x:Yb³⁺. The absorption spectrum of CsPbCl_xBr_3−x:Yb³⁺ is displayed as the green curve in Figure 3b, with the absorption onset occurring at approximately 490 nm. The luminescence of the Yb³⁺ ion remains unchanged. The fundamental concept of QC is illustrated in the inset of Figure 3b. Here, perovskite nanocrystals can absorb high-energy photons and transfer the energy to a pair of Yb³⁺ ions, which are used for NIR luminescence. This QC process overcomes the thermalization loss for coupled Si PV in the LSC-PV system and holds huge potential for breaking the Shockley–Queisser limit.

To predict the performance of CsPbCl₃:Yb³⁺-based LSC and CsPbCl_xBr_3−x:Yb³⁺-based LSC, we simulated four nanocrystal conditions with different PLQYs using MC simulation. Figure 4a,b display the curves of η_ext as a function of η_abs (the LSC absorption efficiency for solar photons). The black dashed line represents the ideal case of a 200% PLQY of NIR emissions, while the gray dashed line shows the results for a 160% PLQY of NIR emissions. The red solid line represents the case of a 10% PLQY for band-edge emissions and a 180% PLQY for QC emissions, and the blue solid line represents the case of a 10% band-edge emissions and a 160% PLQY for QC emissions. If an additional 10% of photons are involved in the band-edge luminescence while there is a 160% PLQY of NIR emissions, leading to 20% band-edge emissions and a 160% PLQY of NIR emissions, then their η_ext will increase. We think most of the photons of band-edge luminescence are reabsorbed and emit light in the NIR or are lost at a sufficient concentration. If all photons are involved in the emissions and 90% of them undergo the QC progress while the remaining 10% undergo band-edge emissions, their η_ext will be very close to the ideal PLQY. However, there will still be a difference due to the competition between band-edge luminescence and Yb³⁺ ion luminescence. When comparing the η_ext of the two materials, CsPbCl_xBr_3−x:Yb³⁺ has an η_ext of 12.3%, which is significantly higher than that of CsPbCl₃:Yb³⁺, at 3.84%. This is due to the stronger absorption efficiency of CsPbCl_xBr_3−x:Yb³⁺ for solar photons. The ray-tracing plots of MC simulations of the two materials are displayed in Figure 4c,d. Both materials have dimensions of 10 cm × 10 cm, and it is visually apparent that CsPbCl_xBr_3−x:Yb³⁺-based LSC has a much better light concentration efficiency due to the higher η_abs.

MC simulations were subsequently used to investigate the effect of size on the η_ext of LSCs prepared by CsPbCl_xBr_3−x:Yb³⁺ and CsPbCl₃:Yb³⁺. The results are displayed in Figure 5a. In the ideal case, the η_ext of CsPbCl₃:Yb³⁺-based LSC remains at approximately 4.75% (the purple lines), while that of CsPbCl_xBr_3−x:Yb³⁺-based LSC remains at approximately 12.2% (the dark green lines). To account for the imperfect preparation of LSC, the scattering factor of approximately 0.012 cm⁻¹, reported by Wu et al. [17], was introduced. For CsPbCl₃:Yb³⁺-based LSC, the η_ext decreases from 4% to 2.2% as the size increases from 5 cm to 150 cm (the blue lines). Similarly, for CsPbCl_xBr_3−x:Yb³⁺-based LSC, the η_ext decreases from 12% to 6.3% as the size increases from 5 cm to 150 cm (the bright green lines). It is worth noting that the η_ext of 6.3% is still considered excellent for larger-area LSCs with a size greater than 1 m² [38,39].

The efficiency performance of systems with CsPbCl_xBr_3−x:Yb³⁺-based LSC and CsPbCl₃:Yb³⁺-based LSC coupled with silicon-based PV cells was predicted. The flux gains (FGs) of the two types of LSCs coupled with silicon-based PV systems at different sizes were also computationally predicted. In the LSC-PV system, the FG can represent the enhancement of the silicon-based PV photocurrent, which can be expressed as FG = qGη_ext. Here, q is the spectral reshaping factor, defined as the ratio between the EQE of Si PVs averaged over the LSC PL spectrum and averaged over the whole solar spectrum (q = <EQE>_PL/<EQE>_s). G is the geometric gain of the device, which can be expressed as G = L/4d, where d is the thickness of LSC, which was set as 0.5 cm in this work. Under ideal conditions, when taking the sharp emissions of Yb³⁺-dopants at 990 nm into account (with a shaping factor of about 1.57 for PV cells), the MC simulation results show that CsPbCl₃:Yb³⁺-based LSC can achieve an FG of 5.64 for L = 150 cm (G = 75). Similarly, CsPbCl_xBr_3−x:Yb³⁺-based LSC can achieve an FG of 14.51 for L = 150 cm (G = 75), indicating its potential as a solar window. The results are higher than those for all types of QD-LSC reported by Bradshaw et al. [40].

5. Conclusions

In conclusion, we constructed an MC simulation model which is suitable to simulate the performance of the QC-LSCs. This MC simulation model can take into account the band-edge emissions and near-infrared emissions of the QC-materials. Additionally, we synthesized CsPbCl₃:Yb³⁺ nanocrystals via hot injection and synthesized CsPbCl_xBr_3−x:Yb³⁺ nanocrystal with the absorption onset of 490 nm via anion exchange. We then included the spectral data of the two materials in our MC simulation model to predict the performance of the QC-LSCs. The results demonstrate that additional band-edge emissions can lead to an improvement in the η_ext under the same unideal PLQY of the NIR emissions. However, if all the photons are involved in radiative recombination, the η_ext under the ideal PLQY will be higher due to the competition between band-edge luminescence and Yb³⁺ ion luminescence. The results also show the potential of the CsPbCl_xBr_3−x:Yb³⁺-based LSC, which can obtain an outstanding η_ext exceeding 12%. Due to the imperfections in the technique, the LSC has an indispensable scattering factor. Notably, even with such imperfections accounted for, the CsPbCl_xBr_3−x:Yb³⁺-based LSC can still obtain an η_ext exceeding 6% in the large area (>1 m²). The FG of the CsPbCl_xBr_3−x:Yb³⁺-based LSC-PV system can reach 14 in the large-area LSC, showing the application prospects of QC-LSC.