Operating regimes for second generation luminescent solar concentrators

ABSTRACT

In this paper, we introduce the concept of first and second generation luminescent solar concentrators. Traditional, first generation devices are characterised by their randomly oriented molecules that absorb sunlight and emit luminescence isotropically. By applying detailed balance to the absorbed and emitted photon fluxes we derive the Shockley–Queisser limit for these devices. It is found that they have inherently low efficiency due to optical losses (the well known reabsorption problem) and also that device performance is strongly affected by the areal ratio between the top and edge surfaces. This latter property makes it very difficult to achieve significant cost reductions because as the edge area is reduced (to lessen the amount of expensive photovoltaic material required for conversion), the efficiency of the system diminishes. First generation concentrators have now approached the fundamental limits which we predict here, thus to achieve a stand-alone luminescent concentrator that enables significant cost reductions, second generation approaches are now needed. New, second generation devices are characterised by either directional emitters or photonic filters which enhance the waveguiding mechanism, allowing high efficiency and large sizes to be achieved simultaneously. Here we define the fundamental operating regime in which second generation technology must reach to surpass the limit of first generation devices. Copyright © 2011 John Wiley & Sons, Ltd.

INTRODUCTION

Cost and efficiency are the main driving forces behind photovoltaic (PV) research. One purely optical approach to the reduction of costs is the luminescent solar concentrator (LSC). Here, conventional solar panels are replaced with low-cost, planar luminescent waveguides which have a small area of solar cells attached to the edges. The first generation of LSC technology (G1-LSCs) offered realistic savings when they were proposed 31 years ago because of the high cost of silicon PV at that time 1. However, G1-LSCs suffer from optical losses that limit the devices to low efficiency which in today's industry makes them less attractive. The photon transport towards the cells is severely hindered by multiple reabsorption events that allow photons to escape from waveguide modes. For typical dye concentrations and plates sizes a nominal figure of 4 reabsorptions are predicted to occur before collection at the edges, where each introduces a new opportunity for the photon to escape 2. It is well-known that by reducing surface losses for escaping photons the efficiency of an LSC can be theoretically increased 3, 4. However, a clear operating regime for these new ‘second generation’ devices (G2-LSCs) has not yet been unambiguously defined. Moreover, it is likely that new approaches will not instantly reach the high thermodynamic limit identified separately by Rau and Markvart in Refs. 3 and 4. However, to be successful, new LSCs should significantly outperform the thermodynamic limits of the first generation technology. The goal of this article is to clearly define this operating regime with a view to guide and benchmark future devices.

Research interests are now shifting towards methods of enhancing the optical transport, thus enabling power conversion efficiency of LSCs to be significantly increased. This will allow commercialisation of a new and low-cost technology that is competitive with conventional thin-film and crystalline silicon modules.

Second generation devices employ either molecular or optical methods to enhance the photon transport within the waveguide. Molecular approaches rely on controlling the electronic properties of the emitter to prevent photons from being redirected into escaping modes, either through eliminating reabsorption 5, 6, or by controlling the emission direction through alignment 7–10. Optical approaches do not rely on engineering the emitter, but aim to use a filter with an ideal top-hat reflection band, to redirect escaping photons back into the waveguide 11, 12.

In practice LSCs with a photonic band stop filter may only reach fundamentally higher efficiency (and therefore enter the G2 operating regime) when the optical properties of the filters are stringent in both wavelength and angular response. Conversely for the molecular approaches, unless they are also tailored to have zero absorption/emission overlap, the problem of reabsorption remains, which brings with it the loss from non-radiative recombinations. Further, these LSCs may have reduced absorption due to the anisotropic absorptivity of the dipole in the active volume. Both molecular and optical approaches introduce new material challenges, yet have the potential to lead to fundamentally more efficient devices. In this article we wish to establish a general operating regime for second generation approaches, which may not be entered by first generation technology.

This approach also aides a second problem, the comparison of different first generation (G1) devices. The large number of design variables such as absorber bandgap, cell bandgap, device size and geometric ratio, can obscure the underlying device performance. Using the approach described here, an expected efficiency can readily be calculated from fundamental design parameters.

Figure 1 shows the geometry considered throughout this article, a cutaway is used for illustrative purposes to reveal the internal photon streams. Photovoltaic cells are bonded to all four edges of a rectangular planar sheet and a reflector fully covers the back surface to return escaping light to the plate. The geometric concentration is given by C_g = A₁/A₂ where A₁ and A₂ are the top and edge areas, respectively. Solar photons enter and exit the LSC through the top surface. Luminescent photons from the LSC enter the solar cells and electroluminescent photons from the PV cells enter the LSC. By considering the thermodynamic balance between the absorbed and emitted photon rates fundamental efficiency limits can be imposed on the design.

Figure 1. A luminescence solar concentrator panel, a cutaway reveals internal photon streams: the luminescence collected and the electroluminescence emitted by the solar cells. The external photon streams are the absorbed incident sunlight and the luminescence lost from the top surface. A reflector placed on the back surface returns escaping light.

CONSTRUCTING THE THERMODYNAMIC RATE MODEL

The Shockley–Queisser limit provides a useful guide for solar cell performance, because the efficiency of a device is calculated simply from a knowledge of the semiconductor bandgap. Here we seek an equivalent limit for LSCs which is dependent on the bandgap of the absorber and also the geometric concentration of the sheet. The starting point is the generalised Planck equation, which gives the emission rate due to the recombination of electrons and holes in a semiconductor absorber 13–15. This differs from the description of purely thermal radiation by the inclusion of a non-zero photon chemical potential µ to describe the emission from the luminescent material and also the attached solar cells when under forward bias. For µ = 0 the equation reduces to Planck's radiation law and describes the thermal emission spectrum of a blackbody such as the Sun or Earth. The photon emission rate in units of photons/m²/s/eV/Sr is given by,

• (1)

where a(E) is the absorptivity of the material, is the Bose–Einstein distribution and is the photon density of states. In these functions, n is the refractive index, and other symbols are physical constants and have their usual meaning. For practical devices, the emitter has to spectrally match the absorption peak of the attached solar cell. However, the following calculations are performed in the limit of unity absorptivity, where α(E > Eg) = 1 and is otherwise zero. In this limit maximum exchange of particles and minimum thermalisation loss occur in the attached solar cells when both materials share the same bandgap. Thus E_g is the bandgap of the luminescent material and also that of the attached solar cells. All interfaces are planar, therefore angular integration over the incident or exiting light forms a cone of solid angle, yielding an angular factor . For Lambertian emission through a planar surface this yields . Performing the same integration over the solid angle subtended by the solar disk yields an angular factor . The total photon rate (photons/s) passing through an area A, inside a medium of refractive index n is,

• (2)

For an LSC, a surface bounds an interface between two different refractive index materials. Because the number of photon modes scales with n² (for a plastic or glass substrate we assume n = 1.5), not all photons which propagate from the substrate towards the outside can be accommodated across the boundary (this is the origin of total internal reflection). Therefore n, in Equation (2), takes the value of the lower index at any planar interface.

Setting A₁ = 1 m² and following the endoreversible scheme of DeVos 14 the thermodynamic fluxes defined in Figure 1 can be parameterised in terms of the geometric concentration ratio to yield,

• (3)

• (4)

• (5)

• (6)

where T_s = 5762 K and T_p = 300 K are the temperatures of the Sun and the Earth, respectively, V is the applied bias and q is the electronic charge.

In the radiative limit the rate at which photons are absorbed by the LSC must balance with the rate at which photons are emitted, therefore the auxiliary equation for particle conservation must be satisfied,

• (7)

The electrical current generated by the solar cell is found from the difference between the absorbed and emitted photon currents,

• (8)

therefore the electrical power is P = VJ14. Equation (2) has no general analytical solution, therefore the results presented in this paper rely on numerical techniques.

LIMITING EFFICIENCY FOR FIRST GENERATION CONCENTRATORS

To begin defining useful efficiency regimes, we first investigate the electronic and optical properties of three example G1-LSCs with C_g = 5, 15 and 30. Figure 2(a), compares the J(V) characteristic of these plates to that of a Shockley–Queisser solar cell with the same bandgap (E_g = 2 eV) and shows two counter-intuitive results: (1) the short-circuit current of the LSC plates is lower than the solar cells and (2) the current decreases as the geometric concentration is increased. The former illustrates the poor photon transport in G1-LSCs due to the reabsorption problem, in that most of the entering photons are not collected by the attached cells. The latter illustrates that the photon transport efficiency is dependent on size (i.e. geometric concentration) of the plate. The reabsorption problem is related to how photons occupy phase space. For example, on average, randomly orientated molecules emit isotropically ensuring that photon phase space is filled ergodically after a single emission event (all memory of the initial photon direction is lost) 16. Thus dye alignment schemes 7–10 are promising approaches to improving photon transport, because phase space can be selectively populated. In G1-LSCs the photon transport degrades with increasing geometric concentration because the cells become smaller targets in photon phase space, decreasing the likelihood that a photon can reach a solar cell before its direction is randomised by a reabsorption event. If multiple reabsorption events are allowed to occur, trapped photons will eventually be redirected into escaping modes and leave the plate.

Figure 2. (a) J(V) curve of a 1 m² G1-LSCs with bandgap 2 eV and different geometric concentration ratios compared to that of a Shockley–Queisser solar cell of the same bandgap and area. (b) Optical efficiency of an G1-LSC with 3 different geometric ratios.

It is convenient to define the waveguiding efficiency, to give a measure of how well an LSC plate transports the absorbed photons to the attached solar cells,

• (9)

Thus is independent of the bandgap of the absorber and in this model is only related to the geometric concentration of the sheet. Table I, summarise the waveguiding efficiency of the example plates. It is clear that the waveguiding mechanism of G1-LSCs is inefficient and strongly size dependent. For example, is 31% for the small C_g = 5 example plate, which means almost 70% of the absorbed photons are lost before they can be collected by the attached solar cells. The performance of G1-LSCs dramatically decreases as the geometric concentration is increased. For example, reduces to 7% when the geometric concentration becomes C_g = 30, meaning that over 90% of the absorbed photons are lost from waveguide modes before collection.A second optical parameter is the optical efficiency, ,

• (10)

• (11)

Table I. Summary of the optical and electronic properties of G1-LSCs with different geometric ratios.

Cg	ηwave (%)	ηopt, 2 eV (%)	η, 2 eV (%)
5	31	6	7
15	13	2	3
30	7	1	2

The is therefore the fraction of all incident photons that are collected by the attached solar cells, and varies with bandgap of the absorber as shown in Figure 2(b). It is closely related to the external quantum efficiency (EQE) of the complete device. The EQE of LSC modules is lower than for solar cells, peaking at approximately 40% for small devices 17. In order of increasing size, the 2 eV example plates absorb and deliver to the attached solar cells only 6%, 2% and 1% of the incident solar photons. The increases for low bandgap absorbers because more of the spectrum can be collected, however, low bandgap absorbers do not fundamentally enhance the optical efficiency. Lower bandgaps E_g ∼ 1 eV are well served by PbS nanocrystalline quantum dots 18, 19 or rare-earth emitters 6; however, quantum dots typically have low quantum yields, ∼50% 20 and rare-earths have poor absorption cross sections 11. If they can be made with near unity quantum yield and at very low cost they are an attractive candidate material for LSCs. In the near term, organic dyes offer the highest quantum yield (approaching 100%) over photon energies from 2 to 3 eV, but they lose radiative efficiency for energies below this band 21. Despite this, they are still considered the main candidate for the absorber material because of their low cost and photostability over long periods 22.

Optical parameters are useful to highlight loss processes but it is the power conversion efficiency (η) that is of fundamental interest to the operation of PV devices. This is defined as ratio between the entering optical power (taken over the full spectral range), P_in (Watts) to the exiting electronic power P_out,

• (12)

To test the model, we computed the efficiency of a G1-LSC with a bandgap of 2 eV (this corresponds well to the red emitting dyes commonly used in experiments), and made a comparison with values taken from the literature 23, 24. Figure 3 shows the results, in general we see excellent agreement, particularly for plates sizes C_g < 30. For higher values of C_g the G1-LSCs deviate slightly from the prediction. Experimental measurements show that the efficiency reaches a plateau for very large sizes, but the theory predicts a continual reduction. This is probably for two reasons: (1) during experimental characterisation solar photons can couple directly into the edge cells due to imperfect alignment or through scattering and reflections, and, (2) the model assumes complete equilibration over the whole volume and thus the maximum permitted level of reabsorbtion is simulated.

Figure 3. Comparison of predicted device efficiency for E_g = 2 eV with experimental values.

OPERATING REGIMES FOR SECOND GENERATION CONCENTRATORS

The above considered the electronic and optical properties of G1-LSCs. Next we consider the limits to the power conversion efficiency η and use this to establish the second generation operating regime. Following the approach of Shockley and Queisser, the efficiency for G1-LSCs is dependent on the absorber bandgap but also the geometric concentration of the sheet.

The thermodynamic model outlined above has been used to help define the G2-LSC operating regime shown in Figure 4(a) and (b). This region is formed between an upper and lower efficiency bound. For the upper efficiency bound we have used the Shockley–Queisser limit. Indeed it has been shown by others that LSCs with ideal optical filters approach this limit 3, 4. It is clear that new devices must outperform the limits of the first generation technology, thus the lower bound corresponds to the thermodynamic limit of conventional G1-LSCs (which we derive here). We observe a large region of possible efficiency enhancement, which reveals the true potential of this technology.

Figure 4. (a) Operating regimes of first and second generation LSCs. (b) Efficiency contours showing the first generation boundary as a function of plate bandgap and geometric ratio.

Figure 4(a) shows the bandgap-efficiency plots which reveal the first and second generation operating regimes for 2 example plates. The area highlighted with lines corresponds to the first generation regime and area with filled with colour corresponds to the second generation (G2) regime. The efficiency of the G1 regime is strongly size dependent which illustrates how the poor waveguiding efficiency of G1-LSCs affects the generated power. For example, a conventional 2 eV organic dye based LSC should be limited to for sizes C_g > 5% assuming no waveguiding enhancement; this prediction is in excellent agreement with experimental measurements on plates with similar properties 23. Figure 4(b) shows efficiency contours against bandgap and geometric ratio of G1-LSCs. It can be used to find the upper limit of the G1 regime for a device of different size and bandgap to the ones we cover here. It also serves as a benchmark by which different G1-LSCs can be compared. Moreover, efficiency alone is not a good metric for the performance of these devices because of their size dependence; devices should be compared by how close they approach fundamental limits. For example, a high efficiency G1-LSC can trivially be achieved by choosing a low geometric ratio, but this goes against the nature of the device: the reduction of costs through large size.

The G2 region in Figure 4(a) shows the dramatic potential of G2-LSCs provided can be significantly increased. For example, the efficiency of a C_g = 30 plate has the potential to be increased by an order of magnitude if the waveguiding efficiency can approach unity. Thus a target efficiency for G2-LSCs should be in the 10–25% range, which is comparable with state of the art thin-film and flat plate silicon cells.

COST CONSIDERATION FOR FIRST GENERATION CONCENTRATORS

Example plates of different sizes have been used to illustrate the performance limits of LSC technology. However, in practice size cannot be chosen arbitrarily. Instead, the size must be chosen to make the device economical. In the following, fundamental limits are placed on the areal cost of G1-LSC panels by optimising the geometric concentration to minimise cost-per-Watt. This provides further evidence of the need to switch to second generation approaches. The following cost analysis does not apply to G2-LSCs because of the significant increase in generated power and size independent performance. However, it should be pointed out that any optical structure used to prevent surface losses does not come entirely free of cost, thus optical approaches must also seek out potentially low-cost reflectors. For example, see the work of Debije et al. 25 in which organic reflectors are used to minimise surface losses.

Let χ be the ratio of the cost-per-Watt () of the LSC and PV panels,

• (13)

where P is the maximum power and C is the areal cost of either the PV or LSC system depending on the subscript. The power of the G1-LSC panel is calculated from Figure 4(b), and for the PV panel the Shockley–Queisser limit is used. Equal cost-per-Watt (χ = 1) defines the economic break-even point. When χ < 1, the LSC panel is more economical. It is instructive to break the cost of the LSC panel into known areal costs of the attached cells and unknown areal costs of the plate (the substrate and absorber only) as the latter have implications for what materials and construction techniques can be used. Thus, , where . Therefore the fractional cost of the LSC plate is,

• (14)

To make LSCs economical the size of the plate must be chosen to maximise efficiency and minimise cost. The curves in Figure 5(a) show Equation (14), plotted for constant χ = 0.5, 0.75, 1.0 and varying C_g. Here the optimum geometric concentration is the value corresponding to the peak of the curves; it is the maximum plate cost that can be incurred to achieve the desired cost-per-Watt reduction. To give an indication of what costs are required for the LSC to be economical these values are plotted against fractional cost-per-Watt in Figure 5(b). The results show that to break even with solar panels the areal cost of a first generation LSC plate must be 11% the costs of conventional PV. To be half the cost-per-Watt the plate cost must reduce to <0.1% of the cost of the PV panel. Further cost-per-Watt reductions are impossible because the optimum geometric concentration takes values approaching C_g = 1, such that plates cannot be made cheaply enough to offset the costs of the attached solar cells. This limits the size of an economical G1-LSC to C_g < 30. Furthermore, the vanishing plate costs which enable significant cost-per-Watt reductions are difficult to achieve in today's industry. These stringent cost requirements mean that G1 devices are only economically viable if their optical properties can be improved. To enable an LSC based technology which is a stand alone replacement for conventional solar panels G2 approaches are a necessity.

Figure 5. (a) Optimisation of a G1-LSC plate size to find the highest plate cost that can be endured to achieve the stated cost-per-Watt reduction. (b) The maximum plate cost to achieve desired cost-per-Watt.

However, in the short term G1-LSC panels have interesting applications where they are augmented with existing infrastructure. For example in building integrated photovoltaics such as energy producing windows 26, 27 or facades 28 where these costs are mitigated, and cost effectiveness is not the primary goal. For these applications the LSC is an attractive prospect because it is visually appealing and will generate some power compared to passive systems, which just provide shade.

CONCLUSIONS

In G1-LSCs multiple reabsorption events reduce the efficiency of the waveguiding mechanism to low values , which becomes particularly inefficient for large sizes. However, large size is needed to achieve the primary goal of the technology: the reduction of costs. Thus G1-LSCs cannot be both efficient and large. In order to be economically viable, G2-LSC approaches are needed which must enhance the optical efficiency and enable size insensitive device performance. However, any second generation solution has to scale to production and must not introduce excessive costs. For this reason the molecular approaches seem the most promising route. The main result of this work is the recognition that second generation devices must operate in a high efficiency regime which cannot be achieved by the first generation technology. The cost analysis revealed stringent cost requirements on first generation plates; they must be ∼0.1% the areal cost of conventional PV to half the cost-per-Watt of the generated power. Although G1-LSCs are attractive for building integrated photovoltaics and other niche applications, second generation approaches are now needed to enable stand-alone systems that can compete with conventional flat plate and thin-film photovoltaics.

ACKNOWLEDGEMENTS

DJF would like to acknowledge the input of Ned Ekins-Daukes on the construction of thermodynamic rate models, and thank Markus Führer, Peter Spencer, Rahul Bose, Amanda Chatten and Keith Barnham for their constructive comments during the writing of this article.