1. Introduction

Luminescent solar concentrators (LSCs) redirect light to the edges of a planar waveguide via the total-internal-reflection (TIR) of photons emitted by fluorescent materials [1,2]. Photovoltaic (PV) cells coupled to the edges of LSCs are used to convert fluorescence emission to usable electricity. Fluorescence light concentration reduces solar power conversion costs by reducing the required area of expensive PV cells. Additionally, LSCs eliminate the need for expensive solar tracking devices due to efficient absorption of diffuse light [3,4]. The efficiency of an LSC device is dependent upon the light trapping capability of the waveguide and the optical properties of the fluorescent material. However, since most waveguide materials, such as glass, PMMA, and PET, have similar light trapping capability due to their similar refractive indices near 1.5, a more direct focus is placed on finding high-performance fluorescent materials. These materials should have broad absorption spectra, minimal absorption/emission spectrum overlap, high fluorescence quantum yields (${\eta }_{QY}$), and photoluminescence (PL) emission spectra coupled to the band-gaps of common, high-efficiency photovoltaic absorber materials such as silicon (Si) and gallium-arsenide (GaAs).

While organic dyes used to be the most common fluorophores in LSCs, the recent development of new inorganic materials such as semiconducting quantum dots (QDs) sparked research interests due to their large absorption spectra and tunable absorption/emission characteristics [5–10]. Quantum dots benefit from their broad absorption spectrum and relatively narrow emission spectrum, tunable absorption/emission characteristics. Research on QD LSCs in the past has primarily focused on visible light emitting QD materials such as CdSe/ZnS. The efficiency of a visible-emitting QD LSC is limited by the lack of near-infrared (NIR) photon absorption. Recently, some investigation was also done to characterize the viability of using near NIR emitting PbS quantum dots to maximize light absorption and improve coupling efficiency with Si solar cells [9].

To this date, PbSe QDs have not been studied as the active fluorescent material in LSC devices. This is likely due to the fact that typical PbSe QD emission wavelengths (1100 nm-1600 nm) fall beyond the absorption range of Si PV cells. Thus, a PV material with a smaller band-gap than silicon must be used. Germanium (Ge) is the most likely PV absorber candidate due to its band-gap energy of 0.67eV and cutoff wavelength near 1850nm. To the authors’ knowledge, the present record photovoltaic power conversion efficiency (${\eta }_{PCE}$) for a stand-alone germanium PV cell is 8.4% under AM1.5G sunlight [11]. It is worthwhile to investigate the use of PbSe QDs in LSCs due to their broad absorption spectrum and near infrared absorption/emission characteristics. Tandem LSC configurations, which utilize stacks of LSCs with different fluorescent materials to maximize ${\eta }_{PCE}$ [12], may benefit from the addition of a PbSe QD LSC layer to harvest infrared photons.

In this paper, PbSe QD LSCs are modeled using an LSC Monte Carlo ray-tracing simulation. We computed the absorption efficiency, optical efficiency, power conversion efficiency, flux gain, as well as the contribution of each individual loss mechanism in the PbSe nanocrystal LSC system. A Förster resonance energy transfer (FRET) enhanced infrared QD LSC was proposed and simulated for the potential improvement in light concentrating performance. Although this paper focuses on PbSe QD-based LSCs, the LSC Monte Carlo simulation algorithm developed in the present work can be readily extended to study LSCs based one other infrared quantum dots.

2. Monte Carlo simulation

The PbSe QDs used to model the absorption and emission spectra were fabricated with the synthetic approach described by Yu et al. with resulting absorption and emission peaks at 1465nm and 1515nm respectively [13]. The QDs were then dispersed in tetrachloroethylene with PbSe concentrations (c) of 0.35, 1.1, 5.3, and 39.6μM, and the absorption spectra were measured with a Perkin Elmer UV-VIS-NIR spectrometer to obtain percent absorption vs. wavelength curves for a 5mm-sampling thickness. The emission spectra were measured using a Spectral Products CM110 monochromator and a Stanford Research Systems SR810 lock-in amplifier coupled to a chopper operating at 400 Hz. A 650 nm laser was used as the photo-excitation source, and PL output is channeled to a fiber optic cable located directly adjacent to the laser-excited QDs in a 45 × 12 × 4 mm³ glass container.

Figure 1 plots the absorption spectra and normalized emission spectrum of PbSe QDs for c = 0.35, 1.1, 5.3, and 39.6μM. The measured absorption and emission spectra were used to simulate the optical properties of PbSe QDs. The percent absorption vs. wavelength spectra for any PbSe QD solution concentration and LSC thickness can be modeled using the Beer-Lambert law. It is evident from the Fig. 1 that the percentage of photons absorbed by the fluorescent material in an LSC will increase with increasing PbSe QD concentration for a fixed fluorescent material thickness. However, the large overlap between the absorption and emission spectrum leads to significant self-absorption at high solution concentrations. Hence, selecting the optimal concentration and LSC dimensions to balance absorption efficiency and self-absorption loss is essential to produce high efficiency LSCs in our simulation study.

 

Fig. 1 Absorption spectra (percent absorption vs. wavelength) at PbSe QD solution concentrations of 0.35 μM, 1.1 μM, 5.3 μM and 39.6 μM and emission spectrum (normalized PL intensity vs. wavelength) of PbSe QDs in a 5 mm thick container.

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The LSC Monte Carlo model of PbSe QD LSCs was created using GoldSim Pro Academic Version to evaluate the device performance. To begin the simulation, a photon wavelength is generated by the AM 1.5G solar spectrum obtained from National Renewable Energy Laboratory [14]. The probability of reflection at the front surface of the LSC waveguide is determined by calculating the Fresnel reflection coefficient of the incident photon at the air-waveguide interface. The probability of absorption is then determined by comparing the wavelength of the generated photon to the percent absorption vs. wavelength spectrum for a given solution concentration and LSC thickness. The absorption efficiency of the LSC (${\eta }_{abs}$) is defined as the fraction of incident photons absorbed by the fluorescent material in an LSC, and it is determined in the simulation by the following Eq. (1):

Next, emission of photons by the quantum dots in the LSC is modeled by taking into account the measured emission spectrum, the quantum yield ${\eta }_{QY}$ as well as the isotropic emission pattern. Entrapment of the emitted photons due to total internal reflection (TIR) is determined by comparing the emission angle to the critical angle of reflection. Due to the assumption of isotropic emission in the present study, each time a photon is emitted by a fluorescent particle the probability of entrapment (${\eta }_{trap}$) can be calculated using the following Eq. (2) [2,15]:

For the photons trapped in the LSC, the optical path-length to the edge (l) is determined using trigonometry. Self-absorption is determined by comparing l to a randomly generated self-absorption path-length (l_SA) determined using the Beer-Lambert law and absorption coefficient of the fluorescent material at the emission wavelength (${\alpha }_E$).

In the model, optical efficiency (${\eta }_{opt}$) is used to quantify the fraction of photons reaching the collecting edges. It is defined as the ratio of the photon counts transmitting through the edges of an LSC to that of the incident photons, and is determined in the simulation by the following Eq. (3):

Light harvest at the edge of the LSC is modeled by calculating, the short-circuit current Eq. (4) and open-circuit voltage with the solar cell diode Eq. (5),

The LSC power conversion efficiency (${\eta }_{PCE}$) can be determined using the following Eq. (7),

Finally, the LSC flux gain ($F_{PCE}$) is computed with Eqs. (8) and (9) as the total gain in power produced by PV cells attached to the edges of the LSC relative to the power produced by the same PV cells in direct sunlight [12]:

The cost reduction factor of an LSC is proportional to 1/$F_{PCE}$ [12]. Thus, for an LSC device to be practical, $F_{PCE}$ must be greater than 1, otherwise there is no effective photon flux concentration and the inherent purpose of the LSC is lost. Optical flux gain ($F_{opt}$) may also be defined practically as the product of G and ${\eta }_{opt}$.

Optical and power conversion efficiencies tend to diminish with increasing geometric gain due to self-absorption loss and reduced absorption efficiency. The prevalence of self-absorption increases as $A_{LSC}$ increases due to a corresponding increase in the total optical path length photons must travel to reach an edge. If $A_{LSC}$ is held constant and $A_{PV}$is reduced by reducing the LSC thickness (t), ${\eta }_{abs}$ will diminish due to the shorter solar absorption path-length. To increase the net solar absorption path length and η_abs without changing t or G, a specular or diffuse back surface reflector (BSR) can be used to reflect light that would otherwise transmit through the back surface of an LSC [16]. Meanwhile, the geometric gain must be balanced to yield a high flux gain without excessively sacrificing optical efficiency, and it has been shown that using a geometric gain on the order of 10 is suitable for achieving this balance [17]. In the present simulation study, when a back-surface reflector (BSR) is employed in the LSC design, a specular silver mirror with a reflectivity of 0.97 is used in the modeling, and a small air gap is assumed to exist between the LSC and the BSR so that TIR may still occur at the back surface.

The contribution of each optical loss mechanism in an LSC can be found using the Monte Carlo simulation by separating lost photons into different categories based on when the loss event occurred. In principle, optical losses in an LSC can be attributed to three primary mechanisms: solar transmission, PL transmission, and non-radiative relaxation (NRR). Solar transmission loss occurs when an incident solar photon is not absorbed as it passes through the LSC. The total fraction of transmitted photons is equal to $1-{\eta }_{abs}$. Non-radiative relaxation loss occurs when photons absorbed by the fluorescent material do not result in radiative emission. The probability of NRR loss after an absorption event is equal to $1-{\eta }_{QY}$. Escape-cone transmission occurs when an emitted photon transmits through the waveguide due to lack of entrapment by TIR. The probability of escape-cone transmission after an emission event is equal to$1-{\eta }_{trap}$. Self-absorption arises when photons previously emitted by the fluorescent material are reabsorbed and subsequently lost to either NRR or escape-cone transmission. The probability that a photon will reach an edge after becoming trapped in the LSC system will be referred to as self-absorption efficiency (${\eta }_{SA}$). Thus, self-absorption loss can be quantified as $1-{\eta }_{SA}$. By accounting for the various loss mechanisms present in an LSC, ${\eta }_{opt}$ can be represented by the following Eq. (10):

3. Results and discussion

Monte Carlo simulations were performed to evaluate the efficiencies and losses of PbSe QD LSCs with various QD concentrations and LSC thicknesses. The front surface area of all the simulated LSCs were chosen to be 300 × 300 mm² as this represents a realistic size for a usable, small LSC. In the following results, 50000 photons were generated in each simulation.

The η_QY of high-quality PbSe QDs is often reported to be higher than 80% when fluorescence characterization is accomplished using comparison techniques with other known infrared-emitters such as IR-125 and IR-26 [13,18]. However, according to fluorescence data collected using integrating sphere [19], ${\eta }_{QY}$ of 40% is an optimistic value for PbSe QDs with band gaps smaller than 1.1 eV. For comparison and completeness, two sets of PbSe QDs will be modeled in this simulation with ${\eta }_{QY}$ of 40% and 80% respectively. Figure 2 shows a plot of absorption efficiency (${\eta }_{abs}$) vs. PbSe QD solution concentration (c) for c ranging from 0.125 to 40μM and t = 2.5, 5, and 10 mm. Exceptional ${\eta }_{abs}$ values were obtained due to the broad absorption spectra of PbSe QDs.

 

Fig. 2 Absorption efficiency (η_abs) vs. PbSe QD concentration (c) for simulated 2.5, 5.0, and 10mm thick LSCs with c ranging from 0.1 to 50 μM.

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As expected, thicker LSCs yield greater ${\eta }_{abs}$. However, the concentration capabilities of an LSC reduce as t increases due to a corresponding reduction of geometric gain G. Although superior absorption efficiencies are obtained at higher PbSe QD concentrations, the incidence of self-absorption increases as concentration increases thereby quenching potential optical efficiency. It is worth noting that the saturation regime in Fig. 2 occurs due to the fact that an increasing absorber concentration can only boost the absorption of photons that have wavelengths within the absorption spectrum of the absorber. Eventually, as solution concentration keeps increasing, the fractional absorption will saturate to a value less than 100%. Figure 3 shows a plot of self-absorption efficiency vs. concentration for 300 × 300 × 2.5 mm³ PbSe QD LSCs with ${\eta }_{QY}$ = 0.40 and 0.80. Self-absorption efficiency results for 5mm and 10mm thick LSCs are not shown because they are extremely similar to the results for 2.5 mm thick LSCs.

 

Fig. 3 Self-absorption efficiency (η_SA) vs. PbSe QD concentration (c) for simulated 300 × 300 × 2.5 mm³ LSCs with c ranging from 0.1 to 50 μM and η_QY = 0.40 and 0.80.

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As expected, self-absorption efficiency is substantially better at lower PbSe QD concentrations and for PbSe QDs with higher ${\eta }_{QY}$. The average optical path length to an edge is dependent on length and width, but not thickness. Therefore, self-absorption efficiency is relatively independent of LSC thickness, but strongly dependent on the length and width of an LSC. Optical efficiency reaches the maximum at the concentration where the product of ${\eta }_{abs}$ and ${\eta }_{SA}$ is maximized.

Table 1 shows results for various simulated PbSe QD LSCs with $A_{LSC}$ = 300 × 300 mm² at optimized QD concentrations for highest efficiency. Germanium thermo-photovoltaic (TPV) cells [11] with 6.3% AM1.5 photovoltaic power conversion efficiency were modeled to determine the electrical output, power conversion efficiency, and flux gain of simulated PbSe QD LSCs.

Table 1. LSC Monte Carlo simulation results for PbSe QD LSCs with A_LSC 300 × 300mm² and optimized QD concentrations for highest efficiency.

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PbSe QDs with ${\eta }_{QY}$ of 0.40 cannot function as an effective fluorescent material for LSCs at the specified dimensions. For all configurations listed with ${\eta }_{QY}$ of 0.40 in Table 1, $F_{PCE}$ is less than 1.5 and ${\eta }_{PCE}$ is less than 0.5%. In the case that $F_{PCE}$ is less than 1, the Ge TPV cells attached to the PbSe QD LSC would actually be producing less power than if they were removed and placed in direct sunlight; i.e. the LSC would be effectively functioning as a luminescent solar diffuser. The combination of extreme NRR loss and self absorption, coupled with a relatively low efficiency PV cell results in extremely low device efficiency.

On the other hand, PbSe QDs with ${\eta }_{QY}$ of 0.80 function relatively well as the fluorescent material for LSCs at the specified dimensions, albeit with low ${\eta }_{PCE}$ below 1.5%. For all the simulated LSCs with optimized concentrations and ${\eta }_{QY}$ of 0.80, $F_{PCE}$ is greater than 1. Therefore, these LSCs are functioning effectively as solar concentrators. Furthermore, simulation results show that the addition of a BSR to a PbSe QD LSC can boost ${\eta }_{PCE}$ above 1.0% while simultaneously improving $F_{PCE}$. The addition of a BSR is particularly effective in PbSe QD LSCs due to the high incidence of self absorption. Since the addition of a BSR effectively doubles the solar absorption path length, self-absorption can be reduced while maintaining absorption characteristics by reducing the fluorescent material concentration.

4. Alternative approach

To improve the device efficiency, the material self-absorption loss must be further reduced. Currie et al. have recently used Förster resonance energy transfer (FRET) between different dye molecules to reduce the self absorption of the emissive dye in organic LSCs [12]. By analogy, we examined, in this study, the effect of FRET on decreasing self-absorption for thin film, spin-cast PbSe QD layers incorporating two different sizes of PbSe QDs. The smaller (donor) PbSe QDs exhibit the exciton emission peak at 1350 nm, which substantially overlaps with the 1st order exciton absorption peak of the larger (acceptor) PbSe QDs at 1362nm, rendering the energy resonance condition for FRET. To improve the FRET rate between quantum dots, 1,2-Ethanedithiol (EDT) treatment was used as a ligand exchange to replace the long-chain oleic acid molecules with small EDT molecules over the surface of the PbSe QDs. At a donor-to-acceptor (D/A) ratio of 5:1, the pronounced quench of the donor QD emission was observed due to non-radiative energy transfer between donor and acceptor QDs, as revealed in Fig. 4.By comparing the PL spectra of FRET enabled QDs and individual QD films, the FRET efficiency was estimated to be ~65%.

 

Fig. 4 Absorption and photoluminescence spectra of FRET enhanced PbSe QDs with a D/A ratio of 5:1. FRET coupling between the donor and acceptor QDs results in a single emission peak.

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To this effect, absorption efficiency can be improved without incurring a greater incidence of self-absorption. LSC simulations were performed for 300 × 300 × 5 mm³ FRET enabled PbSe QD LSCs with a D/A ratio of 5:1. Even though the EDT treatment of FRET enabled PbSe QD films has been found to reduce ${\eta }_{QY}$ by typical factor of 25%, it was assumed that ${\eta }_{QY}$ remains at 0.80 in the present work for the sake of comparison with aforementioned data. In this case, simulation results for FRET enhanced PbSe QD LSCs yielded ${\eta }_{PCE}$ of 1.70% and 1.17% for LSCs with and without a BSR, respectively. These results correspond to performance improvements of 47% and 41% relative to single PbSe QD LSCs with and without a BSR, respectively. It is therefore envisioned that a great efficiency enhancement can be achieved by designing donor-acceptor fluorescent QD system in LSCs and employing FRET to reduce the self-absorption losses if the quantum yield of the acceptor QDs retains high. Studies are currently underway to enhance the fluorescence efficiency of the acceptor QDs in the FRET system by surface-engineering the emissive nanocrystals.

5. Conclusions

Based on Monte Carlo simulation results, PbSe QDs can function reasonably well as the fluorescent material in LSCs, but extensive absorption/emission spectrum overlap results in significant self-absorption loss. Nevertheless, simulation results suggest that LSCs using PbSe QDs with the quantum yield (${\eta }_{QY}$) of 0.80 are capable of yielding the power conversion efficiency (${\eta }_{PCE}$) of nearly 1.5% for G of 7.5. To improve PbSe QD LSC efficiency, it is proposed that self-absorption loss can be reduced by utilizing FRET between two different sizes of PbSe QDs. Simulation results for a FRET enhanced PbSe QD LSCs showed a potential 47% performance improvement in relative to a standard PbSe QD LSCs of the same LSC dimensions. The Monte Carlo simulation developed in this paper allows us to model LSCs in detail and will be used to further investigate and optimize LSC systems.