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Abstract
Stimulated emission luminescent solar concentrators (SELSCs) have the potential to reduce escape cone losses in luminescent solar concentrators (LSCs). However, a functional SELSC is yet to be demonstrated. Previous numerical studies and detailed balance limits provide guidance, but they also contradict and likely overestimate performance and underestimate requirements. In this work, we introduce a rate-equation model with inversion requirements compatible with detailed balance limits and apply this model to the numerical modelling of window-sized SELSCs. We find that the optimal pump photon energy for both LSCs and SELSCs is 1.35 eV and the potential improvement of SELSCs over LSCs is found to be 19.3%. The efficiencies found are much lower than those specified in previous work due to the increase in Stokes shift required for a highly luminescent material. We also find that SELSCs are more attractive at higher matrix losses, that emission linewidths <0.05 eV are desirable, and that SELSC devices can potentially achieve performance equal to LSCs at low illumination levels and simultaneously exceed it by up to 16.5% at 1-sun illumination.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A luminescent solar concentrator (LSC) consists of a slab of photoluminescent material where the waveguiding matrix is coupled to photovoltaic (PV) cells on its periphery. Solar illumination is absorbed and re-emitted by the photoluminescent material, where total internal reflection guides the emitted light to the cells. Luminescent solar concentrators in relation to conventional concentrators during the early stages of LSC research were deemed to provide the benefit of distributing the heat loss over the concentrator rather than having the thermalization loss occurring all in the solar the cell [1]. Over time, the low cost of the photoluminescent slab became an argument over conventional PV [2], and more recently by its potential in building-integrated photovoltaics (BIPV) [3], including windows [4,5] and rooftops [6].
However, LSC efficiencies have not advanced markedly and remain at a single digit level. The highest reported power conversion efficiencies are under 8% [7], whereas the efficiency record for a conventional c-Si PV cell is 26.7% [8]. LSCs suffer from several sources of loss including the Stokes shift, re-absorption, escape from the slab, and matrix absorption [9]. The use of large photoluminescent Stokes shifts appears to be a promising approach to reduce re-absorption losses [10–12], but escape cone losses remain a challenge. These can be as high as 25% for a matrix with an index of 1.5. While the use of interference filters appears to be an obvious approach, it is impossible to create a reflection band at longer wavelengths without also increasing reflection of incoming sunlight at the shorter wavelengths corresponding to the higher order reflection peaks [13].
An alternative approach to reducing escape cone losses was proposed by Morgan Solar in 2010 in which a seed laser is shone down an LSC, inducing stimulated emission thereby reducing escape cone losses [14,15]. These are called stimulated emission luminescent solar concentrators (SELSC). Numerical studies have shown the potential for Perylene-Red as an SELSC material, but a device length approaching 100 m would be required [16] while experimental measurements of gain have not been promising [17]. A functional SELSC device has yet to be demonstrated.
In response to these challenges, some recent work has focused on defining the material properties and device parameters required for the performance of SELSC to eclipse that of conventional LSCs. Thermodynamic calculations have shown that in order to achieve the required inversion for gain under solar illumination, high broad band absorption and narrow emission linewidths are necessary [18–20]. An ideal 4-level model has also been used to explore the parameter space further, suggesting that SELSCs are likely to outperform conventional LSCs for low quantum efficiency luminophores and when there are high waveguide matrix losses [21]. Using a circular geometry with a central seed laser iris can greatly reduce the requirements for SELSCs to outperform conventional LSCs [22]. The peak efficiency for SELSCs was found to be at emission photon energies of about 1.0 eV (with pump energies of 1.34 eV) in study [20], while 0.83 eV (with pump energies of 0.96 eV) in studies [21,22]. This difference is due to three factors: a) the overly ideal nature of the 4-level model used in studies [21,22] where the populations of the 4th and 2nd energy levels are approximated as zero (this model does not require a minimum Stokes shift and allows for the breaking of thermodynamic limits); b) the requirements of study [20] are just to reach inversion, not optimize stimulated emission; and c) there are some differences in the choice of index of refraction of the medium as well as other parameters. The potential to break thermodynamic limits in previous numerical studies is of particular concern given how much the results differ from detailed balance studies. A new material model is required to address this concern.
Herein, to more accurately guide the search for appropriate materials and to assess the validity of the SELSC concept, we develop a rate equation model which is consistent with the threshold inversion requirements laid out in prior studies [18–20]. We then apply the said model to large window-sized circular SELSC devices and conventional LSC devices (using a numerical approach similar to that reported previously [22]) in order to compare them. Variations in pump photon energies, Stokes shift, spontaneous emission rate, device thickness, matrix losses, emission linewidth, and solar concentration are all explored. For the edge PV cells, we will primarily consider the optimal bandgap cells, but in a later section silicon PV cells are also considered.
2. Device layout and material model
For this study, we consider devices the size of a typical solar module ($0.9 \times 1.5\,$ m2) or a large commercial window ($1.5 \times 2.5\,$ m2). These serve as guidelines for areas that are appropriate for installation in buildings and general handling. To simplify calculations, we model a circular geometry with equivalent area (radii of 0.66 m and 1.1 m, respectively). A high level layout of the circular SELSC is shown in Fig. 1. The radial calculation should give similar results to the rectangular case because the average optical path length in an LSC is almost equal for two devices with the same surface area [23], but is much simpler to calculate. For LSCs, the circle is continuous, while SELSCs have a central seed laser iris. This initial radial emission might be produced using a laser plus conical mirror or a central LED (as properties such as coherence are not required). Emitted light is collected at the edges.
Fig. 1. Diagram of a circular SELSC. Like a conventional LSC, sunlight illuminates a photoluminescent matrix where the fluorophores absorb and then re-emit light. Some of this light propagates to the PV cells at the edge (via total internal reflection) where it is absorbed. However, in an SELSC there is also a seed laser which induces stimulated emission. This seed laser pulse is amplified as it propagates through the material until it reaches the PV cells at the edges.
To simulate these arrangements, numerical calculations are performed on the grid shown below in Fig. 2, with the laser and solar fluxes determined at each grid point. The power conversion efficiency of an SELSC is given by:
Fig. 2. SELSC grid layout for the numerical calculations in this work. The seed laser starts with an initial flux at $r=r_0$ and (ideally) increases in intensity until it reaches the PV cell at $R$. The fluorophore concentration ($N$) is determined based on the desired thickness to absorb 99.9% of the initial incident photons capable of being absorbed by the material in thickness $T$. For all calculations, the grid was $512(r) \times 256(z)$.
The total captured stimulated emission is simply the laser intensity at the PV cell. As the seed laser propagates, the laser and solar fluxes at each point are calculated using the absorption and gain derived from the material model described in the next section. Numerically, derivatives are taken at the midpoint (Crank-Nicolson scheme [25]) and the populations are calculated for this offset grid with 4 Picard iterations [25] in order to calculate any needed midpoint averages. The grid is meshed linearly in $z$, but meshed geometrically in the $r$ direction. This is because the intensity of the seed laser and stimulated emission spreads out with a derivative of $-\frac {1}{r}$. Optimum values (such as for laser power) for a given SELSC design are found using Brent’s method [26] (via the Scipy python library [27]). All numerical details of the seed laser and solar flux propagation are given in the supplementary.
The spontaneous emission is calculated at the midpoint of each grid line (as opposed to the grid points themselves) as these are the points used for calculating the rate photons which are added/subtracted from the seed laser flux and solar flux. The emitted light is treated as rays emitted in every direction, with those in the escape cone or emitted out of the laser iris being lost, while those that reach the PV cell have length-dependent propagation losses (defined by $\alpha _m$) associated with the matrix itself. These are numerically integrated over all angles. The detailed integral is provided in the supplementary.
2.1 Material model and rate equations
In this subsection we describe the material model used, which is key to ensuring that a wide variety of material parameters can be explored while ensuring thermodynamic limits are met. Core to the model is the inclusion of reverse transitions that ensure that the population of the different energy levels obeys Maxwell-Boltzmann statistics at thermal equilibrium and that the rate of spontaneous emission from the pump band is of the correct magnitude. This is a significant departure from earlier numerical work such as [21] and [22]. Figure 3 shows the energy levels of a luminescent emitter with broadband absorption above a specific pump frequency $\nu _p$ and narrow band emission at $\nu _e$, separated by a Stokes shift of $\nu _s$ ($\nu _e=\nu _p-\nu _s$). It has 4 energy levels: $E_0$, $E_1$, $E_2$, and $E_3$. $E_1$ and $E_2$ are centrally situated between $E_0$ and $E_3$ such that $E_3-E_2=E_1-E_0=\frac {h\nu _s}{2}$. The model is partially idealized in the types of transitions considered. The transitions between $E_0$ and $E_3$ are considered to be entirely radiative. The transitions between $E_3$ and $E_2$ are entirely non-radiative, as are those between $E_1$ and $E_0$. For the transitions between $E_2$ and $E_1$, both radiative and non-radiative transitions are considered. $E>E_3$ is a near-continuum of energy states, allowing for broadband absorption between $E_0$ and $E_3$. The concentration of emitters is denoted by $N$ while the concentration at each energy level is $N_i$ such that $N=\sum N_i$. Throughout, we assume that the Boltzmann approximation is valid and that $h\nu _p>h\nu _e>\frac {h\nu _s}{2}>3kT$.
Fig. 3. 4-level system that can be balanced at thermal equilibrium. $E_i$ are the different energy levels, $W_{ij}$ are the absorption or stimulated emission transition rates from $i \rightarrow j$, $r_{ij,nr}$ are the non-radiative transition rates from $i \rightarrow j$, $r_{ij,sp}$ are the spontaneous emission transition rates from $i \rightarrow j$, $\nu _e$ is the frequency of the emitted light, and $\nu _s$ is the stokes shift frequency. Note that $\nu _{p}=\nu _{e}+\nu _{s}$, where $\nu _{p}$ is the minimum absorption/pump frequency.
For the broadband pump transitions, we set the absorption to be constant across all frequencies greater than $v_p$, as is usually done when working at the thermodynamic limits [18–20]. In our notation $R_{ij,process}$ is the rate of transitions per unit volume for population $N$ corresponding to the particular process. For $E_0 \rightarrow E_3$, if we treat the transition cross-section as being a constant $\sigma _p$ for all $\nu >\nu _p$, then:
where $\phi _p$ is the incident photon flux. There are a large number of parameters in some of the following equations, so they are followed by $\equiv \Lambda N_i$, where $\Lambda$ will be used to represent the parameters in more complex equations. Because we’re using Maxwell-Boltzmann statistics, spontaneous emission will be proportional to $N_3$. We use the results from previous work on thermodynamic limits [19,20,28] to write: where $r_{ij,sp}$ is the spontaneous emission rate from $E_i$ to $E_j$ per electron in $E_i$, and $n$ is the matrix index of refraction. The supplementary shows why use of $N_3$ is appropriate here.For the emission band transitions, we use the emission cross-section for the center of a Lorentzian. This results in radiative transition rates of [29]:
3. Efficiency comparison of SELSC and LSC devices
Our primary goal in introducing the rate equation model is to determine what material and device parameters are required for an SELSC to outperform a conventional LSC. We do this by starting with near ideal cases and adding sources of loss (matrix losses) or device complexity/difficulty (increased device thickness, increased emission linewidths; lower solar illumination).
As mentioned previously, we only consider circular LSC and SELSC devices with a with radii of 0.66 m and 1.1 m. Some parameters are kept constant and left unexplored in this study: $\eta _L=0.5$, $r_{nr} \approx 0$, $r_0=1\,\textrm{mm}$, and $\sigma _p=10^{-16}\, \textrm{cm}^{2}$. The pump absorption cross-section lies between rare-earth ions [32] and quantum dots [33], but is closer to quantum dots). Others are varied depending on their effect on device efficiency. Some parameters will increase/decrease device efficiency monotonically (e.g., emission linewidth), while others will have optimums (e.g., seed laser power). In all cases, the initial seed laser power is optimized, as it is obvious that too high a seed laser power would exceed any gains from stimulated emission. We also set the non-radiative rate between $E_2$ and $E_1$ ($r_{nr}$) to 0 as we are most interested in comparing to the ideal LSC case of $\eta _{QE,21} \approx 1$. A previous numerical study found that the additional radiative pathway introduced by stimulated emission would benefit materials with low quantum efficiencies [21].
A parameter that we optimize for all SELSCs is the spontaneous emission rate between $E_2$ and $E_1$ ($r_{21,sp}$ or $r_{sp}$ for simplicity). From Eq. (15), we can see that there will be a maximum spontaneous emission for which inversion is possible. However, it is also desirable to have a high spontaneous emission rate at the seed laser wavelength as this increases stimulated emission. This is a well-established trade-off [34]. Therefore, there is an optimal spontaneous emission value that results in maximum SELSC efficiency. This can be seen in Fig. 4(a), where the peak lies between $10^{3}$ and $10^{4}\, \textrm{s}^{−1}$ for a pump energy of 1.35 eV, Stokes shift of 0.45 eV, and depth of 1 µm. A higher spontaneous emission rate will result in a higher overall quantum efficiency for LSCs, but we can see from 4(a) that the efficiency does not change significantly for rates significantly higher than the inversion condition at $r_{sp}=10^{5}\,\textrm{s}^{−1}$. Therefore, for all other calculations, henceforth we will use this optimal rate for SELSCs and the inversion threshold $r_{sp}$ for conventional LSCs.
Fig. 4. Efficiency graphs showing optimal points for spontaneous emission rate, pump photon energy, and Stokes shift under this material model with $\Delta \nu = 0.01\,\textrm{eV}$, $T=1.0\,\mathrm{\mu}\textrm{m}$. A) The efficiency of circular SELSCs relative to spontaneous emission rate ($r_{sp}$) for $\nu _s=0.35\,\textrm{eV}$; B) The efficiency of circular LSCs relative to minimum pump energy for various Stokes shifts; C) The efficiency of circular LSCs and SELSCs with a radius of 0.66 m relative to minimum pump energy for their optimal Stokes shifts; D) The efficiency of circular LSCs and SELSCs with a radius of 1.1 m relative to minimum pump energy for their optimal Stokes shifts.
3.1 Pump and Stokes shift frequencies
For our first exploration, we find the optimal pump and Stokes shift frequencies. We ignore matrix losses for now and set the emission linewidth to 0.01 eV. The maximum efficiencies possible for a conventional LSC at various Stokes shifts and pump frequencies are shown in Fig. 4(B). Because we require that even conventional LSCs meet the inversion threshold (in-line with having no re-absorption), the efficiency of LSCs does not decrease monotonically with increased Stokes shift as it did in [21] and [22]. One can see that the optimal Stokes shift in most cases is close to 0.4 eV. This is inline with other work. Ries et. al. calculate a Stokes shift of 0.4 eV to be enough to allow for infinite concentration factors [35]. It also matches well with experiment, where a 0.4 eV Stokes shift eliminates measurable re-absorption [3].
For SELSCs we will also have an optimal Stokes shift for a given pump frequency. For both LSCs and SELSCs, rather than pick a specific Stokes shift, we can find an optimum. The maximum efficiencies given optimum Stokes shifts and rates of spontaneous emission for devices with a 0.66 m radius are shown in Fig. 4(C). For both LSCs and SELSCs, there are two peaks at 1.15 eV and 1.35 eV. The optimum Stokes shifts for these pump frequencies were 0.446 eV and 0.471 eV for SELSCs and 0.391 eV and 0.407 eV for conventional LSCs (see Table 1). A larger Stokes shift is optimal for SELSCs as they need to exceed the threshold condition in order to achieve a significant amount of stimulated emission. For radii of 1.1 m, there are some improvements in efficiency, particularly for SELSCs (see Fig. 4(D) and Table 1). While both have the same optimum pump frequencies, the maximum possible efficiencies are 25.3% and 21.2% for SELSCs and LSCs respectively. Because matrix losses are not accounted for, the calculated efficiencies will continuously increase for larger radii. The optimal/threshold Stokes shifts do not change significantly with the increase in radius.
Table 1. Comparison of optimal pump energies, Stokes shifts, emission photon energies, and calculated efficiencies for LSCs and SELSCs with a radius of 0.66 m or 1.1 m.
The efficiency difference between the SELSC and LSC cases is primarily caused by two things: reduced escape cone losses and increased optimum Stokes shift. For the 0.66 m optimum case, the escape cone losses have been reduced from 24.6% to 7% (A 23.3% improvement in performance). For larger devices, this would be further improved. Because SELSCs require a greater Stokes shift than LSCs, this reduces the maximum possible improvement. The amount of energy reaching the solar cell per photon is reduced by 7.2% for SELSCs versus LSCs with radii of 0.66 m. This is a convincing demonstration of why having a thermodynamically valid model with reverse transitions is so important for these types of exploratory calculations. The total losses due to spontaneous emission from the pump band remain similar at around 2-3% for both LSCs and SELSCs, despite a much larger optimum spontaneous emission lifetime for the SELSC. This is due to the increased Stokes shift and additional radiative pathway from stimulated emission. In this ideal case the laser energy cost is less than 0.5% of the energy captured and does not play a significant role.
It is of note that the shapes of the maximum possible efficiencies are very similar to those given in [20] wherein the detailed balance limits for solar pumped devices were derived, whereas they differ significantly with the results presented in [22]. In fact, the peak efficiency is at almost the exact same point to that of [20] at a pump photon energy of 1.35 eV. However, the overall efficiencies are much lower. For both SELSCs and LSCs, this is largely due to the increase in required Stokes shift when a greater amount of emission (both stimulated and spontaneous) is desired (In [22] there was no minimum Stokes shift and in [20] the potential efficiency was calculated at threshold). While this Stokes shift is smaller for LSCs, they also have much larger escape cone losses. In the following sections we continue our investigation with pump energies of 1.35 eV.
3.2 Thickness and matrix losses
We now consider the effect of device thickness and matrix losses on SELSC and LSC performance. We use the optimal pump frequency calculated in the previous section (1.35 eV) and the same emission linewidth (0.01 eV). The absorption of borosilicate glass at the optimal emission frequency (around 0.95 eV) is around 0.1 m−1, so it is a reasonable value to consider in these calculations. This loss constant is also used by Kaysir et. al. in their studies [36]. Optical fibers (both PMMA and glass) have much lower loss constants [37,38], but these are unlikely to occur in a conventional LSC or circular SELSC with window dimensions. However, from previous calculations [21,22], we know that a benefit of SELSCs is not just reducing escape cone losses, but reducing the average path length. Therefore, higher absorption coefficients are expected to benefit SELSCs over LSCs. We therefore consider matrix losses from 0.0001 to 0.1 m−1. Reduced absorption in the pump band (leading to larger $T$) will also have a negative impact on performance. In particular, this increases the required seed laser power needed to induce a certain fraction of stimulated emission. In these calculations we consider devices with thicknesses of 1.0 to 10.0 µm.
When we calculate the maximum possible efficiencies over a range of matrix losses, we see a much larger loss for LSCs than SELSCs (see Fig. 5). The losses are not significant for either device type when the matrix losses are less than 0.01 m−1 as the device radius is only 0.66 or 1.1 m. However, for losses closer to those expected for glass and polymer sheets/slabs the efficiency drops significantly. For a radius of 0.66 m LSCs suffer a 6.3% relative loss and SELSCs a 3.2% loss. For devices with a radius of 1.1 m, the relative losses are 10.1% and 4.8% for LSCs and SELSCs respectively. As matrix losses increase, SELSCs become more and more of an attractive alternative to LSCs. This is consistent with previous work [21].
Fig. 5. Efficiencies of devices of various thickness considering various matrix losses. A) 0.66 m radius LSC B) 0.66 m radius SELSC C) 0.66 m radius LSC D) 0.66 m radius SELSC.
For larger device thicknesses, the maximum possible concentration is reduced and the required laser power for SELSCs is significantly increased. This should lead to reductions in possible performance for both LSCs and SELSCs, but a more significant drop for SELSCs. This is exactly what we see in Fig. 5, with SELSCs with thicknesses 5 µm and above having almost no improvement over LSCs. This is despite the fact that the linewidth has been set to an extremely low 0.01 eV.
While a loss of −0.1 m−1 was used, it is important to note that this is low relative to current LSC experimental results. While there are some claims to reducing scattering losses down to negligible amounts [3,39], matrix absorption for polymers in particular are often in the 2 to 3 m−1 range [3,11]. SELSCs will have an even larger advantage over LSCs in these cases.
3.3 Linewidth and depth
Two principle material properties that will determine the success of an SELSC are the emission linewidth and the pump absorption. We’d like to determine what the maximum linewidth can be for a given device thickness that allows for SELSCs to outperform conventional LSCs. Given the previous section, we only consider devices 1.0 to 5.0 µm thick, as it is clear that for devices 10 µm thick there is little possibility of an SELSC outperforming an LSC. The emission linewidth is varied between 0.01 and 0.2 eV, while matrix losses are kept at −0.1 m−1 and the pump photon energy at 1.35 eV.
The results of the calculations can be seen in Fig. 6. For devices with a radius of 0.66 m, the linewidth requirements are stringent, with the results levelling off to LSC values very quickly with increased emission linewidths. It just becomes too difficult to achieve sufficient stimulated emission to overcome the increased energy cost of the seed laser. Even for linewidths as narrow as 0.025 eV (linewidth of a Neodymium-Glass laser [40]), a device over 1 µm thick is unlikely to exceed the performance of a conventional LSC.
Fig. 6. Efficiencies of SELSC devices with various depths and linewidths and radii of A) 0.66 m, B) 1.1 m.
3.4 Varied solar concentration
The previous calculations all assumed 1-sun illumination. However, this would be reduced for the early and latter parts of the day, as well as when the device is not installed at the optimum angle. The latter is particularly likely in BIPV. In this section, we consider whether an SELSC can be an improvement over LSCs in low-light conditions for devices 1 µm thick with a linewidth of 0.05 eV. Specifically, we optimize the LSC and SELSC for low-light (0.1 sun) conditions. As before, matrix losses are kept at −0.1 m−1 and the pump photon energy at 1.35 eV.
The results of the calculations are shown in Fig. 7(A). With both the LSC and SELSC optimized to the same low illumination, the fact that their performance is almost exactly the same at these low intensities means that the optimal SELSC material properties are right at the threshold. The laser can just be kept off until there is a high enough solar flux and then the laser can be turned on to improve performance. At 1-sun illumination, the performance of the SELSC is 16.5% better than that of the LSC. Given that escape cone losses account for a 25% loss in this context, this is a reasonable improvement with limited downside at low illumination intensities.
Fig. 7. A) Efficiencies of LSC and SELSC devices under various illuminations with a radius of 1.1 m; B) Efficiencies of LSC and SELSC devices (illuminating silicon PV cells) under various illumination intensities with a radius of 1.1 m; C) Efficiencies of LSC and SELSC devices (illuminating silicon PV cells) under various illumination intensities with a radius of 2.0 m.
3.5 Silicon PV at edge
While silicon PV may not be optimal for LSCs or SELSCs, it is conventional and stands to benefit from continuous improvements and upgrades toward tandems in the foreseeable future. Because of this, we should consider the requirements necessary for SELSCs to be viable within the framework of silicon based PV technology. From Fig. 4(D), the optimal pump photon energies are 1.65 eV with a Stokes shift of 0.51 eV for SELSCs and 1.6 eV with a Stokes shift of 0.43 eV for LSCs. For simplicity, we use a photon pump energy of 1.625 eV to determine the optimum properties for a material compatible with silicon PV. As in the previous section, we optimize the properties of both the LSC and SELSC for 0.1 sun. The other parameters are kept the same as those used in the calculations in the previous section. However, we consider radii of both 1.1 and 2.0 m.
The results of the calculations for a 1.1 m radius device are shown in Fig. 7(B). We observe the exact same pattern as that of the previous case, but without any upside until an illumination of 0.7 suns and a maximum upside of only 4.2%. To achieve greater efficiency enhancement, we need to again consider larger dimensions. With a 2 m radius, we see performance improvements comparable to the previous calculation at the optimal photon pump energy (see Fig. 7(C)). The maximum upside is 14.1% at 1-sun concentration. That said, a 2 m radius is quite a large area and equivalent to a $3.5 \times 3.5\,\textrm{m}$ square. This is substantially larger than typical building windows, but may be viable for window wall and rooftop applications.
Given the calculations performed above, it would be useful to see if it’s possible to determine an approximation for the threshold solar concentration at which SELSC performance is preferred. Because the seed laser is not 100% efficient, the performance of the SELSC will be optimal at the point where any additional seed laser output is exactly compensated by the additionally collected energy, rather than at signal saturation. If this maximum efficiency is reached at a seed laser power of 0, then LSC operation is preferred. Further, with high quantum efficiency fluorophores, it can be thought of as if stimulated emission is replacing an equal amount of spontaneous emission (with associated escape cone losses). This approximately corresponds to a threshold condition of:
4. Potential material and device development path
In this section, we consider the ramifications of the previous calculations and what is needed for the fabrication of a functional and highly efficient SELSC. It is worth restating that the comparisons made in the previous sections are between SELSCs and theoretically optimum conventional LSCs. A functional SELSC is possible as long as the inversion threshold is met Eq. (14) and gain exceeds propagation losses. By nature, this section is speculative than the earlier calculations, but serves to advance the discussion toward practical realization.
The results of section 3.4 lie closest to what would be required for a SELSC that sufficiently outperforms conventional LSCs, with the primary requirements summarized in Table 2.
Table 2. Requirements for a high performing SELSC that will outperform an optimized conventional LSC. Note that as $\alpha _m$ increases, SELSCs tend to outperform LSCs, but overall performance suffers.
In the case where we do not restrict ourselves to a particular PV at the edge of the SELSC (performance shown in Fig. 7(A)), the optimum pump photon energy is 1.35 eV, the optimal Stokes shift is 0.47 eV, and the emission photon energy is 0.88 eV(1410 nm). If the edge cells are required to be c-Si, then a pump energy of 1.625 eV and Stokes shift of 0.49 eV, leading to an emission of 1.135 eV(1100 nm) is optimal. In both cases, the emission linewidth must be less than 0.05 eV (80 nm) and near complete absorption of the solar spectrum must occur in an approximately 1 µm thick active layer. Matrix losses <−0.1 m−1 ensure a high performing device, but the higher the matrix losses, the better SELSCs will perform relative to conventional LSCs.
These are very stringent requirements, and comparable to the challenges faced in developing a solar-pumped laser as the inversion requirement is the same. Core to the challenge is finding materials with appropriate broadband absorption and narrowband emission. For solar-pumped lasers, focus is primarily on lanthanide-based technologies [41–44], with sensitized lanthanides being a current focus [45–51], particularly Neodymium (Nd). This is because lanthanide ions maintain a very narrow emission linewidth even when embedded in a matrix. There are two lanthanides with emission in the range we’d be interested in: Neodymium and Ytterbium. Neodymium (Nd) is pumped at 1.53 eV and emits at 1.17 eV and 0.92 eV [52], while Ytterbium (Yb) is pumped at 1.32 eV and has the lowest emission energy of 1.18 eV [53]. However, only Nd is a true 4-level system, so gain is much easier to achieve. Nd also has emission lines close to our optimal energies, so this seems like the best emitter to focus on going-forward.
Contrary to emission requirements, the possibilities for a broadband absorber that can sensitize emission are more open. Significant research has gone into the sensitization of lanthanides including the use of dyes, cesium, and nanoparticles [54]. Because we desire a material that is highly absorbing of all light above a certain frequency and also couple to the pump level of Neodymium, semiconductor nanoparticles (or quantum dots) are an attractive candidate. The bandgap of semiconducting nanoparticles can be tuned to optimize the bandgap [55], increasing as the nanoparticles become smaller. Semiconducting nanoparticles that have been used to sensitize Nd include silicon [56,57], TiO2 [58], CdS [59], and CdSe [60]. However, other than silicon, the bandgaps of these materials are too large for this purpose. In addition to further studies on Si nanoparticle sensitization, it is suggested that sensitization with alternative materials that have tunable bandgaps closer to the 1.5 eV range be explored, such as Ge1−xSnx quantum dots [61], Cd(Se,Te) quantum dots [62], or Ag-(In,Ga)-(S,Se) quantum dots [63].
For conversion of the Nd emission to electricity, PV cells with the appropriate bandgap are required. It is desirable to minimize thermalization losses, so bandgaps should be close to the emission energy if possible. For the 1.17 eV emission, appropriately designed c-Si cells are a possibility (although they would need to be quite thick), while for 0.92 eV emission something more novel would be required. InGaAs PV cells would certainly work in this range [64], and there are some possibilities for SiGeSn [65], with the material having an appropriately tunable bandgap and currently being researched in multijunction devices.
For the matrix, an elevated membrane structure is a possibility as sub-micron thick SiO2 [66] and polymer free-standing membranes [67,68] containing nanoparticles have been fabricated. However, we suspect this would be fragile, so a more likely scenario is the matrix being a thin film deposited on an appropriate substrate. While the calculations above were performed assuming a matrix index of 1.5, if the thin film is on a glass substrate higher index material would be required [29]. This requirement would likely be met just by the inclusion of semiconductor sensitizers, as they would raise the effective index of the film.
These requirements are no doubt stringent and present day materials appear limited, however, various alternatives need to be explored. As an example, an optical fiber configuration might be possible, similar to the solar pumped laser in [47]. However, to cover any meaningful area, the fiber would need to be 100s to 1000s of meters in length and it has been previously established that the thresholds required in a linear SELSC configuration are substantially greater than that of a circular SELSC configuration [22]. Also, with advances in metamaterials and photonic crystals, avenues to design materials with desired optical response may open possibilities heretofore not considered.
5. Conclusions
We develop a rate equation model that provides the inversion threshold compatible with detailed balance limits, and then apply these to large area LSCs and SELSCs where the dimensions are comparable to those of windows and solar modules. The optimum pump energy is determined to be 1.35 eV for LSCs while for SELSCs it is 1.15 eV or 1.35 eV, which are close to the detailed balance optimum calculated in [20], but differ from earlier numerical calculations of circular SELSCs such as [22]. The maximum efficiency improvement achieved over conventional LSCs is 20%. This is lower compared to that obtained from the calculations performed with a more ideal 4-level model, while the overall efficiencies are lower than those from the most ideal detailed balance limits. The primary reason for these reductions is that the inversion requirement leads to larger Stokes shifts and spontaneous emission lifetimes for SELSCs than LSCs, and that both have larger Stokes shifts than the detailed balance limits as increased emission is more optimal.
We find that devices thicker than 1 µm are unlikely to significantly improve SELSC performance over conventional LSCs, even with emission linewidths as narrow as 0.05 eV and losses as high as 0.1 m−1. However, when these requirements are met, devices can theoretically be developed that compare with or exceed the performance of conventional LSCs even at low illuminations. For SELSCs with silicon PV cells, a window or solar panel sized device is unlikely to significantly improve over conventional LSCs. However, at dimensions close to 2 m in radius, circular SELSCs with silicon PV become viable.
An important thing to note is that throughout this work we compared optimal LSCs to optimal SLESCs. The luminescent quantum yield has been assumed to be unity and a reduction in this was previously shown to benefit SELSCs over LSCs as a device design [21]. Radii have also been kept constant, rather than optimized. This has been done in such a way so as to underestimate the maximum possible improvement of SELSCs over LSCs and highlight possible direction and challenges.
With such stringent material requirements, the primary contenders for a working SELSC device appear to be sensitized lanthanides. Such materials combine the broad band absorption of one material with the narrow band emission of lanthanide atoms. This is already a direction in which solar-pumped lasers are developing and SELSCs are another technology which stand to benefit. However, it is clear that the material and device requirements for a SELSC that can outperform fully optimized conventional LSCs are quite challenging. The upside is that the types of materials and devices required are already being explored for solar-pumped lasers, medical applications [54], and PV [64,65]. Moreover, the possibilities with designer materials via metamaterial and photonic crystal constructs have yet to be fully explored. We remain hopeful that advances in these areas will lead to SELSCs becoming a viable concept.
Funding
Ontario Research Foundation (ORF-RE09-017); Natural Sciences and Engineering Research Council of Canada (RGPIN-2017-06405).
Acknowledgments
The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Ontario Research Foundation - Research Excellence program, and the University of Toronto.
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research. Code available upon request.
Supplemental document
See Supplement 1 for supporting content.
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