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Abstract
The effect of various design and material parameters on the efficiency of stimulated emission-based luminescent solar concentrators (SELSCs) is studied numerically using a 4-level luminescent material containing concentrator. It is shown that the most efficient SELSCs have emission wavelengths of 1.5-1.8 µm, with a strong dependence on the Stokes shift. Depending on the parameters of the system, spontaneous emission is shown to nevertheless account for a significant fraction of potential energy generation. Assuming a propagation loss constant of −0.1m−1, and a refractive index of 1.5, the optimal length of an SELSC is found to be ~1.5m. Given these losses and an efficiency target of 10% greater than traditional LSCs, the required material emission linewidth varies from 10 to 100nm, with maximum thicknesses of 3-30 µm. Further, when reflection and propagation losses are considered, a single laser pass is preferred over multiple passes. It is also shown that SELSCs are significantly less sensitive to luminescent quantum efficiency when compared to conventional LSCs due to the increased radiative emission rate.
© 2017 Optical Society of America
1. Introduction
Since the late 1970s, luminescent solar concentrators (LSCs) have been explored as a potential inexpensive alternative to conventional photovoltaics [1]. LSCs traditionally consist of a sheet of photoluminescent material. The material absorbs solar radiation incident on the large area of the sheet and re-emits it isotropically such that most of the emitted light is trapped by total internal reflection and is emitted at the edges of the sheet. Photovoltaic cells of much smaller area are then placed at the narrow edges of the sheet rather than the conventional approach of solar cells extending over the large surface area of the sheet. The basic concept of a LSC is illustrated in Fig. 1(a). However, present conversion efficiencies of LSCs rarely exceed 7% [2]. Key losses in LSCs include re-absorption, scattering, thermalization, transmission/reflection, escape cone (emission at angles lower than the critical angle), and non-radiative recombination (low radiative quantum efficiency) [2,3]. Some strategies to reduce these losses include the use of large Stokes shifts (to reduce reabsorption) [4–6], resonance shifting (to reduce reabsorption) [7], multiple layers of different luminescent materials (to reduce thermalization losses) [8], and interference filters (to reduce escape cone losses) [9].
Fig. 1 A) A conventional LSC in which the primary photon emission is spontaneous and into a random mode; B) An SELSC in which photons are emitted into a specific mode (with specific directionality) via stimulated emission.
Another potential way of addressing the losses due to escape cone and non-radiative recombination is stimulated emission. This provides directionality for emission thereby addressing the escape cone loss and increasing the radiative recombination rate.
The most famous use of stimulated emission is the laser. Solar-powered lasers have been extensively investigated, with current demonstrations being primarily YAG-based [10–15], although other concepts are also being explored [16,17]. However, the collection efficiencies remain around 30 W/m2 [14] with slope efficiencies not exceeding 14% [18]. Roxlo et al. have performed a thermodynamic analysis of a potential solar powered laser, establishing minimum threshold requirements of important parameters such as absorption, Stokes shift, and emission linewidth [19].
As a middle-ground alternative to both of the above concepts, stimulated emission luminescent solar concentrators (SELSCs) are currently being examined. These are essentially a solar powered optical amplifier, wherein a seed laser is added to the existing LSC device concept. The seed laser field is amplified by (and increases) the stimulated emission from the luminescent material. The amplified laser emission ultimately impinges upon a photovoltaic cell. This concept is illustrated in Fig. 1(b). These were first discussed by Morgan Solar [20,21] followed by the work of Kaysir et al. describing some of the key efficiencies to be considered in any SELSC [22]. Recent work by Kaysir et. al delves into how loss and gain might be modelled in an SELSC [23]. However, these reports primarily consider a small selection of existing materials, rather than taking a more open approach to establishing a set of general properties—be it of existing materials or new materials that have yet to be developed—and their effect on potential SELSC efficiencies.
In this work, we use a 4-level luminescent material model within the construct of a SELSC device concept in order to determine the range of material and system parameters required to achieve an improvement in conversion efficiency over LSCs. Unlike previous work, both stimulated and spontaneous emission are considered. The goal is to provide guidance in developing materials and systems that would result in efficient SELSCs. A large emphasis is placed on the seed laser wavelength, device thickness for complete absorption, and the emission bandwidth—all of which are key parameters in determining the efficiency of these devices but have been largely omitted in earlier studies. In section 2 we describe the models and formulas used to calculate the efficiency of an SELSC. In section 3, the effect of various parameters on the calculated efficiency is illustrated.
2. Models and Formulas for Calculating SELSC Efficiency
To investigate the performance of a SELSC given different material and system properties, we model it as a 2D grid, as illustrated in Fig. 2. An initial laser flux (seed) is launched at the top left such that it propagates in the direction of the x-axis. When it arrives at the boundary on the right side, it is reflected (with some loss). For the sake of this model, the reflection results in a reversal of propagation, as well as an increase in depth (y-axis). This (very roughly) approximates a narrow beam reflecting very slightly off-normal. In this way, the entire thickness is “swept”. In a real device, this “complete sweep” would only be accomplished with some overlap of the beam propagation. Depending on the simulated device, there may only be a single pass (no reflections, laser source and PV cell cover the entire thickness) or m passes (m-1 reflections, while laser source and PV cell each cover 1/m of the thickness). An additional simplification is that beam broadening is ignored, which may affect the results for multiple passes; here the beam width is kept constant.
Fig. 2 2D SELSC device schematic/grid diagram and definition of parameters used in the calculations. The luminophores are optically pumped via solar radiation, like a traditional LSC. A seed laser enters the SELSC device via an iris at x = y = 0 (top left) and propagates through the SELSC device. As it propagates, the beam is amplified via stimulated emission from the luminophores. This amplified beam, as well as some spontaneous emission, is absorbed by a PV cell. An SELSC device may comprise a single pass (no reflection of the laser light) or multiple passes with the laser light reflecting back and forth thereby sweeping the complete thickness (in the y-direction) of the device. With multiples passes, the laser iris and PV cell are a fraction of the thickness. This figure illustrates a 3-pass system.
The seed laser field is amplified via stimulated emission of the luminophores. Both the amplified laser signal and spontaneous emission are collected at the photovoltaic (PV) cell. Since we are considering both stimulated and spontaneous emission, the efficiency of this SELSC is given by:
where Cst and Csp are the number of stimulated and spontaneously emitted photons per second impinging upon the PV cell (also referred to as the collected rate); ηLaser is the efficiency of the laser; ΦLaser is the overall photon flux of the laser; ηCell is the efficiency of the photovoltaic cell; h is Planck’s constant; c0 is the speed of light in a vacuum; λ0 is the vacuum wavelength; L is the length of the SELSC (along the x-axis); and PAM1.5g is the total power due to AM1.5g solar spectrum (1000 W/m2). The numerator is the energy generated by the photovoltaic cell due to the total photon flux impinging on the cell less the energy cost of generating the laser photon flux. The denominator is the solar energy flux incident on the device.The applied laser photons is the integral of the flux strength at x = 0:
where ϕλ is the photon flux of the laser (centered at wavelength λ), ϕλ0 is the initial flux of the laser, and YLaser is the width of the laser beam (determined by the number of passes). In these simulations, the initial laser power flux (W·m−2) is used as a parameter, rather than the initial laser photon flux (photons·m−2).To determine the collected spontaneous emission (Csp), for each grid position the percentage of spontaneous emission that will reach the PV cell is calculated. This is integrated over the entire SELSC:
where D is the total thickness (along y-axis) of the SELSC, rsp is the spontaneous emission rate, and ηcapture is the percentage of spontaneously emitted photons captured from that grid point. Details on how both ηcapture and Csp are discretely calculated are given in section 2.5.To determine the collected stimulated emission (Cst), we integrate the photon flux of the gain-increased laser photon flux at the PV cell:
where YCell is the y-coordinate at which the PV cell begins (determined by the number of passes). The laser flux at the PV cell is calculated numerically by iterating through each grid square. The details of this calculation are given in section 2.4. The gain medium itself is abstracted as a 4-level atomic system, which is further described in the next section.2.1. Single grid region: A four-level atomic system
Because the system is pumped with solar illumination, broadband absorption is desirable. However, as shown in [19] (and Eq. (9) below), narrow band emission is also a desirable material quality. Therefore, to separate these properties in the calculations that follow, we use a four-level system [24] in which the transitions are as shown in Fig. 3. In our case, the 0 → 3 transition (rate of R) is due to absorption of solar light, and the 2 → 1 transition (rate of 1/τ) represents both radiative and non-radiative recombination. The non-radiative transitions from 3 to 2 and 1 to 0 are considered “fast”, so that N3 = N1 = 0; Nm is the luminophore concentration in state m.
Fig. 3 Four-level atomic system used in this work. The transition from 0→3 is the pump transition (broadband, rate of R) and the transition from 2→1 is the emission transition (rate of 1/τ). The 3→2 and 1→0 transitions are considered infinitely quick. Nm is the population at energy level m.
In this study, we only consider the steady-state (constant flux of both pump and stimulated light) case. The steady-state solution for a given point in x and y is:
where N is the total luminophore concentration. The pump rate and excited state lifetime can be calculated by: where ϕpump is the pump photon flux, σabs is the absorption cross-section for pump frequencies, τnr is the non-radiative decay lifetime, τsp is the spontaneous emission lifetime, and τst is the stimulated emission lifetime.The absorption cross-section in this study is the same for all wavelengths. The individual emission rates (rxx = rnr, rsp, rst) are:
Two of the individual lifetimes (τnr and τsp) are constant across all x and y. In the case of stimulated emission lifetime, it is proportional to the spontaneous lifetime according to the relation [24]:where n is the refractive index, and Δλ is the emission linewidth. In this equation the flux and wavelength are that of the appropriate stimulating wavelength, while the emission bandwidth is that of the emitting material. As can be seen, stimulated emission increases for higher fluxes, longer stimulating wavelengths, and smaller emission bandwidths.2.2. Incident Solar Photons
The solar photon flux that can be absorbed by the SELSC depends on both the laser/emission wavelength, as well as the Stokes shift. Only those photons with an energy greater than the absorption threshold should be considered. This wavelength is:
For this study, it is assumed that the lasing wavelength is the same as the center emission wavelength.The total number of photons incident on the large front surface of the SELSC that can be absorbed is given by the number of photons in the solar spectrum with a wavelength shorter than the absorption wavelength:
where ϕAM1.5G is the global solar spectrum. Note that by modelling the pump flux using this method, all photons above this energy threshold are absorbed with equal probability (same absorption cross-section). While not accurate for real materials, it is suitable for establishing upper limits.2.3. Solar Cell Efficiency
To estimate the efficiency of the solar cell at the end of the LSC, the method of Henry [25] was used to determine the maximum amount of work a single photon could do in a single photovoltaic cell at 300K, assuming Ephoton = EG. It should be noted that there are three assumptions made by Henry that will increase our error slightly due to the high concentration factor and laser intensity. These are:
The first assumption does not appear to result in significant error (~kT) for our cases. Adjusting the model for assumptions ii and iii is considered out of the scope of this present study.
2.4. Integration/Propagation of Stimulated Emission Photons
To calculate the stimulated emission captured at the photovoltaic cell, we iterate along x and y numerically, starting at x = y = 0. Our initial pump rate, laser flux, and lifetime are:
where q is a material parameter defined to simplify the expressions that follow. Note that the pump rate at y = 0 will hold for all x. The initial laser flux values are only set for the given width of the laser, as it may be reflected several times. Next, we fill in all the pump rates for the width of the laser at x = 0 by incrementing along the y-grid. For each step, we get: We then calculate the increase in laser photon flux and lifetime by incrementing along x: where αloss is the propagation loss coefficient. This continues by until all fluxes, pump rates, and lifetimes are filled in for a single width of the laser. At this point it is reflected, and we need to apply a reflection loss and shift our laser in y. The laser flux at each y of the reflected beam path was calculated using:where ηref is the percentage of light reflected. Note that the reflection loss is applied at the end. This is because the other gains and losses are calculated from pre-reflection values. The pump and laser fluxes are then calculated for the reflected beam.Once the beam reaches the PV cell, we can calculate our total photons captured with the discrete formula:
2.5. Calculation of Captured Spontaneous Emission
The amount of spontaneous emission captured is calculated by:
- 1. Projecting all spherically emitted rays onto to x-y axes, accounting for escape cone losses
- 2. From the center of each specific x-y grid square, and for each bundle of rays (angle):
- a. Calculating the propagation length and number of lateral reflections prior to either exiting through the laser entry iris or striking the photovoltaic cell. See Fig. 4.
- b. Calculating the losses associated with that propagation length and number of later reflections. Total internal reflection is considered 100% efficient.
- 3. Multiplying the spontaneous emission rate at each grid square by the normalized sum of the losses for each angle of emission.
- 4. Summing across all grid points.
Fig. 4 Example of a reflected ray that reaches the PV cell compared to an example ray that exists through the laser iris. A ray can either exit through the iris, resulting in no contribution to the energy generation, or can strike the PV cell and contribute. The amount contributed depends on propagation losses, reflection losses, escape cone losses, the discrete number of angles, and the spontaneous emission rate at x,y.
The total number of captured spontaneously emitted photons is then determined by:
where nangles is the number of discrete projected (2D) angles considered at each point, index i denotes a specific angle, δcaptured is whether the ray struck the photovoltaic cell or not (1 or 0), ηesc_i is the fraction of rays at angle i that are not in the escape cone, ηref is the reflection efficiency, nrefli is the number of reflections a ray requires before striking the photovoltaic cell, and li is the propagation distance before reaching the photovoltaic cell.2.6. Beneficial Properties for an SELSC
Before getting into any example calculations, some parameters of an ideal case follow from the above equations and SELSC structure:
How significant a role each of the above plays in achieving a high-efficiency SELSC is less clear, as are their trade-offs with other parameters. Other parameters have inherent trade-offs. These are all explored through numerical calculations in the next section.
3. Effect of Parameters on SELSC Efficiency
In this section, we use the formulations described above to develop an understanding of the properties/parameters required to create SELSCs that outperform traditional LSCs. Because of the large parameter space involved, only a subset of all possible parameters are shown here, with emphasis on properties that will guide to high efficiency SELSCs.
For these simulations, the grid was 1000 × 300 (x by y), the number of angles considered was 4096, and the maximum number of allowed reflections for a single ray before stopping calculations was 98304. When the initial laser flux was 0 m−2, it is assumed that a laser iris is not present. For all studies, ηLaser = 0.5 (reasonable for a laser diode [17,26]), and n = 1.5. For all simulations, the absorption cross-section (σabs) was kept constant at 10−16 cm2 (which is between rare-earth ion [27] and quantum dot [28] cross-sections – but closer to quantum dots). To vary the thickness (D), only the luminophore concentration values (N) were varied, so that σabs·N·D = 10. The spontaneous emission lifetime varies separately, as this is a separate transition. However, as will be seen in section 3.5, they were kept within a few orders of magnitude of core-shell quantum dots [29], as the absorption cross-section is closer to those and spontaneous emission lifetime scales with emission cross-section [24]. The rationale for not covering a wider range will become apparent in section 3.5.
Increasing the laser intensity leads to a corresponding increase in stimulated emission with a net gain, however, only up to the point where any further increase in stimulated emission with laser intensity does not compensate for the applied power. For a given set of device parameters (each point in Fig. 5 through 11), the laser intensity was swept from 104 to 108 W/m2 logarithmically to find the optimal power. These end points were arrived at empirically as the simulations were refined.
Fig. 5 Baseline (traditional LSC, only escape cone losses) for various emission wavelengths and Stokes shifts.
Fig. 6 SELSC with minimal losses other than the escape cone for various emission wavelengths and Stokes shifts. A) Maximum possible efficiencies achievable with an SELSC; B) Relative improvement of the SELSC over a traditional LSC with the same parameters; C) Percentage of captured photons which are from the amplified laser, rather than those from spontaneous emission.
Fig. 7 Efficiency of an SELSC at different propagation losses and lengths. A) SELSC efficiency; B) Relative improvement over traditional LSC with the same parameters; C) Percentage of captured photons that are from stimulated emission. The stimulated percentage is the same for 0.0 to 0.5m-1, thus completely overlapping.
Fig. 8 SELSC efficiencies for different emission linewidths and photoluminescent layer thickness. A) SELSC efficiency; B) Relative improvement over traditional LSC with the same parameters; C) Percentage of captured photons that are from stimulated emission.
Fig. 9 SELSC efficiencies for different thickness and number of passes if there was no loss from reflection or propagation. A) SELSC efficiency; B) Relative improvement over a traditional LSC with the same parameters; C) Percentage of captured photons that are due to stimulated emission. The percentage improvement is a comparison between the optimal laser intensity for a SELSC and an identical configuration however without a laser.
Fig. 10 SELSC efficiencies for different thickness and number of passes for a 1.0% reflection loss and a 0.1 m-1 propagation loss constant. A) SELSC efficiency; B) Relative improvement over a traditional LSC with the same parameters. The percentage improvement is a comparison between the optimal laser intensity for an SELSC and an identical configuration however now with a reduced PV cell size due to multiple passes and no laser.
Fig. 11 SELSC efficiency at various spontaneous emission and non-radiative lifetimes. A) SELSC efficiency; B) Relative improvement over traditional LSC with the same parameters.
3.1. Laser Wavelength and Stokes Shift
For the first series of calculations, we looked at cases which involve a minimal number of losses to determine which laser wavelengths would result in the most efficient SELSC. The Stokes shift and initial laser flux were also varied. The laser wavelength range was arrived at empirically; we use the bandgap of crystalline silicon [30] as a starting point and move to longer wavelengths to ensure that the optimal laser wavelength was captured. The Stokes shifts are kept at the same order of magnitude as that by Meinardi et al. given that they have already demonstrated minimal re-absorption losses [4]. The parameters used are given in Table 1.
Table 1. Parameters used to study the effect of emission wavelength and Stokes shift on SELSC efficiency
We first look at the baseline for a traditional LSC. This is equivalent to a single solar cell under high concentration (500,000 × ), with additional losses due to escape cone efficiency and Stokes shift. As can be seen in Fig. 5, efficiencies are maximized for low Stokes shift, which is expected. As the Stokes shift is increased at a given laser (emission) wavelength, the number of photons absorbed is reduced. The shape of the different curves is essentially the same but vertically shifted. Due to the high concentration, the losses in the PV cell are low. The overall efficiencies would be much higher than the 1-Sun solar cell limits, save the escape cone loss of ~25%. The maximum efficiencies are for laser wavelengths of ~1.3µm and ~1.5µm for a small Stokes shift, but increase to 1.8µm for 500.0nm shifts.
If we use the parameters above and the formulas derived by Yablonovitch and Roxlo for solar-pumped lasers [19], we find minimum required Stokes shifts of 40-70 nm to achieve gain. Our simulations are well within this limit. In some of the earlier reports [22,23] it is stated that the required Stokes shift must be greater than the ~0.35eV stated in [19]. However, this specific value was calculated for a different index of refraction and absorption band, and does not apply in general.
To maximize the efficiency, the chosen parameters are very close to ideal. Indeed, when we look at the maximum efficiency of SELSCs shown in Fig. 6, we can see that with increasing wavelength and ultimately at the longer wavelength we get very close to the maximum expected enhancement of 33% relative efficiency; this is principally due to the reduction of escape cone losses. In general, the dependence on emission wavelength and Stokes shift is the same as for the traditional LSC, with a slight redshift in optimal wavelength due to the λ4 dependence of the stimulated emission rate.
From Fig. 6(c) there appears to be a threshold photon pump requirement before stimulated emission is beneficial. Once the pump exceeds the threshold solar photon flux, then the optimal cases exhibit stimulated emission fractions of greater than 80%. This discontinuity is due to two factors: the sudden inclusion of the laser iris (allowing more escape) and the energy costs of generating a photon via stimulated emission. Assuming no propagation and reflection losses, the break-even between SELSCs and LSCs occurs when:
where Ctotal is the total captured emission, ηesc is the relative efficiency due to escape cone losses, ηiris is the fraction of spontaneous emission that does not escape through the iris, and ηst is the relative efficiency/cost of generating a photon via stimulated emission (0≤ ηst ≤1). ηst is a function of a variety of factors. However, if laser efficiency and PV cell efficiency are both 1, and there are no other losses, then ηst is 1. For ηst = 1, and ηesc = 0.75 and ηiris = 0.5, stimulated emission must be 60% of that captured to overcome the losses from adding the laser iris. The additional parameters (ηlaser, Δλ, D, L, ηcell, λ) determine the efficiency of generating stimulated emission. Throughout, the presented results show these high stimulated emission thresholds.As stated previously, the parameters in the above calculation are very close to ideal. Some key questions to answer in terms of feasibility are:
- • What’s an acceptable Δλ?
- • What loss per unit length is acceptable? When is it less than the increased efficiency?
- • What’s an acceptable thickness?
- • Do multiple passes result in a net benefit?
- • How do changes in photoluminescence lifetime and quantum efficiency affect SELSC performance?
3.2. Propagation Losses and SELSC Length
One possible way to increase the efficiency is to increase the length of the concentrator. This means the same laser beam is amplified over a longer distance. However, propagation losses could easily overwhelm the gain. The significance of these losses is called out in the work by Kaysir et al. [23]. To determine an optimal length given a certain amount of propagation loss, we calculated the efficiency of SELSCs of various lengths given various propagation losses. In an actual device, these losses can vary significantly. At one extreme, optical fiber losses at ~1.5µm are about −0.000025 m−1 [31], while at the other, Meinardi et al. [4] report scattering losses of −1.25 m−1 in their fabricated LSC. The exact parameters used in these calculations are given in Table 2.
Table 2. Parameters used to study the effect of propagation losses and SELSC length on SELSC efficiency
As can be seen in Fig. 7, even minor propagation losses (αloss = −0.1 m−1) limit the optimal length of an SELSC to ~1.5m. As the losses increase, the optimal length becomes smaller and smaller. When a typical scattering loss (αloss = −1.25 m−1) is considered, the optimal length is closer to 30 cm. The sensitivity of these losses is not unique to SELSCs; LSCs suffer from the same problems. Indeed, the relative improvement in performance for SELSCs over conventional LSCs increases with longer lengths. This is shown in Fig. 7.
It’s clear that low propagation losses are required for all LSC technologies. For SELSCs, these losses play a key role in limiting the types of materials that may be used. If there are no propagation losses and the concentrator can be infinitely long, then almost any luminescent material that can be modelled as a 4-level system can be made more efficient than their traditional LSC counterpart. However, we use a more realistic loss term in the following calculations to better explore how tightly the material parameters might be constrained. We take a middle ground of αloss = −0.1 and a length of 1.5m for all following calculations.
3.3. Emission Bandwidth and SELSC Thickness
For these studies, we consider some typical possible parameters for an SELSC and explore potential maximum values of Δλ and thickness required to achieve a working SELSC that improves on traditional LSCs. We use some of the optimal parameters from the previous studies (λEmission = 1.5µm, λStokes = 200.0 nm, αloss = 0.1m−1, L = 1.5m), while the thickness and emission linewidth are varied. The parameters used are given in Table 3.
Table 3. Parameters used to study the effect of emission bandwidth and SELSC thickness on SELSC efficiency
As can be seen in Fig. 8, the efficiency of an SELSC depends heavily on the thickness and the emission bandwidth, dropping off significantly as either is increased. This is because the efficiency of the stimulated emission reduces. For longer linewidths, this follows from Eq. (9). For thicker SELSCs, this is due to the pump energy being absorbed in a larger volume and therefore lower possible gain.
From our calculations, it is seen that SELSCs with 100.0 nm emission linewidths are feasible, but require very small thicknesses. If the photoluminescent layer is greater than 45 µm thick, the required laser intensities are too large and it’s better not to have a laser whatsoever. Note that these thickness limits would increase to larger values if the loss per unit length were lower (as per the discussion in section 3.2).
3.4. Number of Passes and Thickness
One way in which this significant thickness restriction might be overcome is via multiple passes, where a laser beam passes back and forth through the SELSC, reflecting off each face until it reaches the cell. Figure 2 provides an illustration of this approach, which was first proposed by Kaysir et al. [22]. To check how much benefit one might achieve, this method is explored for the cases in which no losses from reflection or propagation were considered, and when small losses (αloss = −0.1m−1, ηrefl = 0.99) are included. The number of passes is varied from 1 to 5, and the full parameter list is given in Table 4.
Table 4. Parameters used to study the effect of number of passes and SELSC thickness on SELSC efficiency
As can be seen from Fig. 9 and Fig. 10, the efficiency of the SELSC decreases significantly as the thickness is increased, eventually approaching the conventional LSC efficiency. When propagation and reflection losses are not accounted for, having a greater number of passes improves the efficiency of the SELSC. There is a >8% improvement for a 1µm thick SELSC when the beam traverses the SELSC in five passes versus a single pass. A higher number of passes also results in the SELSC approach being beneficial for thicker devices.
Part of this improvement is just due to the increased concentration factor. This can be seen for a thickness of 25µm, where the optimum for all cases is zero laser intensity, and the case with five passes (smaller PV cell) has the highest performance. Since no losses are accounted for, the five pass case with no seed is a higher-concentration LSC then the single pass case.
However, consideration of propagation and reflection loses reverses this trend, and the advantages of multiple passes disappear. This can be clearly seen in Fig. 10 in which the reflection efficiency is 99% and the propagation loss constant is −0.1 m−1. It is clearly advantageous to have a single pass for thicknesses less than 25 µm even when the losses are minor. For larger thicknesses, the optimal SELSC designs approach conventional LSC (no laser flux) efficiencies with the multiple pass case having a lower efficiency. The reduced efficiency is due to the smaller PV cell area, which results in significantly larger average propagation lengths for spontaneously emitted light. Whether the losses themselves are due to propagation or reflection is not as significant as the overall loss in a single pass, similar to laser pump thresholds.
3.5. Spontaneous Emission and Non-Radiative Lifetimes (Quantum Efficiency)
One element that plays a key role in the efficiency of conventional LSCs is the quantum efficiency of the luminophore. SELSCs will also have their performance affected by this property, but may also have additional sensitivity to the spontaneous emission lifetime, as it affects the rate of stimulated emission as well. For this study, both the spontaneous and non-radiative emission rates of a potential luminophore were varied. The exact parameters are given in Table 5.
Table 5. Parameters used to study the effect of spontaneous emission and non-radiative lifetimes
As can be seen in Fig. 11, efficiency is increased for longer non-radiative lifetimes, as expected. However, the efficiency is very similar between different ratios of spontaneous emission and non-radiative lifetimes. In addition, high efficiencies are possible even when the spontaneous emission and non-radiative lifetimes are equal (QE = 50%), as the stimulated emission significantly increases the overall radiative emission rate. This is very different from conventional LSCs, with SELSCs outperforming them by more than 100% (combined improvements for quantum efficiencies around 50%). Larger than 100% improvements are possible because escape cone losses are reduced as well.
4. Conclusions
The effect of various design and material parameters on the efficiency of SELSCs was studied via numerical modelling. These parameters included laser emission wavelength, the Stokes shift of the material, the emission linewidth, luminescence quantum efficiency and lifetime, thickness and length of the SELSC, the number of laser passes, propagation loss, and reflection loss.
One of the risks with completely abstracting the material properties is that we might lose sight of how this information can/should be used. Some key insights that can be taken from the above simulations and analysis to guide future work are:
- - Given the requirement of high concentration factors and the need for efficient stimulated emission, NIR emission (especially 1.5-1.8 µm) is desirable.
- - Low waveguide losses allow for efficient SELSCs at longer waveguide lengths. As one gets closer to zero propagation losses and infinite length, a wider variety of luminescent materials are feasible.
- - Assuming a propagation loss constant of −0.1m−1, and a refractive index of 1.5, the optimal length of an SELSC is around 1.5m. In order to achieve an efficiency increase of at least 10% greater than that without a seed laser, the required linewidth for a material that absorbs the entire solar spectrum within a thickness of 30µm is about 10nm. For a 25nm linewidth the maximum thickness is 12µm, and for a 100nm linewidth it is 3µm.
- - Multiple passes are unlikely to significantly improve SELSC efficiency, once propagation losses are considered. It is better to focus on an optimal single-pass waveguide.
- - SELSCs can be made highly efficient, even if the initial luminescent material has a low luminescence quantum efficiency.
Some of these requirements are stringent, despite some of the more optimistic assumptions. Three parameters that have not been well specified and are strongly related to each other are the absorption cross-section, luminophore concentration, and spontaneous emission lifetime. As the absorption cross-section is increased, the concentration required to maintain a constant absorption thickness will also decrease. A corresponding decrease in the spontaneous emission lifetime is also expected. The results of this study should be very similar for a given σabsτsp = constant, and specific luminescent quantum efficiency (as per the results of section 3.5). In retrospect, these might have been a more appropriate parameterization.
Considering the guidance provided above, the authors recommend two avenues toward developing a highly-efficient SELSC:
- 1. Use of broadband absorbing and narrow band NIR emitting materials such as those doped with lanthanides. Some examples are erbium-doped nanoparticles (or nanoparticle systems) [32,33] and organic lanthanide complexes [34]. These are attractive due to the extremely narrow lanthanide emission linewidths (~10nm), coupled with the wider band absorption from the nanoparticles or organic molecule. However, there are likely some key challenges with efficient excited state transfer from the nanoparticles to the lanthanide ions. Another potential challenge is achieving nanoparticle and ion densities large enough (and absorption broadband enough) for effective absorption and stimulated emission without quenching.
- 2. Use of wider band (~100nm) NIR emission materials with extremely low-loss waveguides, and/or <3 µm thickness. While the low-loss approach was not significantly explored beyond mentioning its possibility in section 3.2, this is the approach that Kaysir et al. utilized for Perylene Red [23], albeit with a visible seed laser wavelength. PbSe/PbS core-shell quantum dots [35] are another potential material amenable to this method, as there is evidence that a significant Stokes shift is possible for a thick enough PbS shell. This is similar to what was demonstrated with cadmium-based quantum dots [4], but at NIR wavelengths. The absorption cross-sections and emission lifetimes are also comparable to those used in this study.
It is indeed a challenge to find and develop materials, including those cited here, to meet all the stated requirements, and even more so upon embedding these materials in a suitable matrix. However, these provide clear research goals for the development of efficient SELSCs.
Funding
Natural Sciences and Engineering Research Council Canada; Connaught Challenge Global fund; Canadian Foundation for Innovation; Ontario Research Fund; and Department of Electrical and Computer Engineering at the University of Toronto. While this work was not directly funded by Morgan Solar, the authors have previously worked on projects that were supported by Morgan Solar. Further, there is on-going dialogue between the authors and Morgan Solar pertaining to this field of research.
Acknowledgments
The authors would like to thank Dr. Stefan Myrskog and Mr. John Paul Morgan of Morgan Solar for scientific discussions pertaining to SELSCs.
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