Spectral splitting optimization for high-efficiency solar photovoltaic and thermal power generation Free

Solar spectral splitting is a strategy to optimize the extraction of exergy from sunlight through the separation of incident photons by energy levels (or wavelengths). This approach generally implements any combination of thermal, electrical, or chemical processes that can increase the efficiency of a device.¹ 

Concentrated solar photovoltaic (PV) and photothermal (PT) conversion strategies each have their unique benefits and disadvantages, and the limiting efficiencies of both strategies have been discussed extensively.^2–5 In order to take advantage of both technologies simultaneously, hybrid converters have been of great interest (CPV/T). The design of a number of these converters have been reported.^6–9 General methodologies for the optimization of these devices have been presented and used to compare different systems^10,11 but the upper bounds of performance have not been explored in great detail.

Previous limiting efficiencies of CPV/T hybrid systems that split incident light into two bands (above and below bandgap) have been calculated and reported.^12,13 Allowing for the thermalization of high-energy photons, however, by introducing a high-energy cutoff in the spectral splitter can vastly improve device performance by trading waste heat generation in the PV converter for useful power generation in the thermal collector.

Here we develop a methodology to understand the theoretical limits and optimum design of a hybrid CPV/T converter that includes both a high- and low-energy conversion cutoff. By investigating how the limiting efficiency is related to the spectral bandwidth illuminating the PV converter, we explain how the highest performing device indeed minimizes the overall entropy generation by diverting each photon to the converter where it generates the least amount of entropy.

Figure 1 depicts the general approach of the hybrid CPV/T converter, where a thermal engine and a photovoltaic cell sit in parallel under concentrated solar radiation. A key question that needs to be addressed when designing such a hybrid system based on spectral splitting is: To which converter should each incident photon be directed in order to reach the highest conversion efficiency? To understand the fundamentally limiting aspects of the conversion process in this device, an idealized hybrid system was studied. The spectral splitting concept considered in this study distributes incident radiation between the $PT$ and $PV$ converters via three bands, as described below. A blackbody source at T_sun = 5777 K (the thermodynamic temperature of the Sun¹⁴) generates a spectrum of electromagnetic radiation following the Planck distribution, denoted as G_s, which illuminates a lossless spectral splitter at an optical concentration, C. The splitter separates incident photons based on their frequency. Low energy photons (hν < E_L) and high energy photons (hν > E_H) are directed towards the thermal collector, where E_L and E_H represent the low and high energy levels of the PV band, respectively. The absorber surface of the PT is assumed to be ideally angularly and spectrally selective. The net heat gained as a result of the interaction, Q_net, can be described by the difference between what is absorbed from the solar spectrum and what is emitted by the thermal collector. Since there are two different “thermal” spectral bands, the net heat gained can be determined by calculating separate integrals

where E_cut, is the transition between high absorption and low emission on the selective absorber surface, $θs$ is the incident cone angle of the concentrated solar spectrum (for ideal optics, $θs$ and C are coupled,^15,16 and full optical concentration is equivalent to full angular selectivity), $θ$ is the zenith angle, and $QBB$ is the blackbody spectrum characterized by $TH$⁠, which has the same functional form as G_s. This heat is then coupled to a thermal engine with a hot-side temperature equivalent to the absorber temperature and the cold-side temperature set at T_amb = 300 K. In the limiting case, thermodynamic work, W_therm, is extracted based on the Carnot efficiency between these two temperatures.

Meanwhile, photons that have energy within the designated spectral bandwidth (i.e., E_L < hν < E_H) are directed towards a PV converter acting in the detailed balance limit. Since the sub-bandgap light is unconvertible regardless of its wavelength, we fixed E_L to the bandgap, E_g. Based on the constraints of our model (i.e., E_L = E_G), incident radiative energy that illuminates the PV converter is entirely above the bandgap of the cell and therefore only interacts with a blackbody surface with an internal quantum efficiency (IQE) of unity. However, due to the applied voltage, radiative recombination is accounted for through the chemical potential, μ_c. Thus the generated photocurrent is given by

where q is the elementary charge and k_b is the Boltzmann constant. The corresponding electrical power that the cell may extract from these carriers can be determined by

We determine the sum of the output work from the thermal engine and the output work from the photovoltaic cell divided by the input power to the device in order to define the efficiency

The model requires three inputs to solve for the efficiency: (1) the illuminating optical concentration, C, (2) the temperature of the thermal collector (T_H), and (3) the p-n junction's electronic bandgap.

For a representative example, Figure 2(a) shows the efficiency of a device with a bandgap of E_G = 1.1 eV (such as that of a typical Si PV), a thermal collector temperature of T_H = 666 K (the dissociation temperature of one of the most common solar thermal heat transfer fluids, Therminol, VP-1¹⁸), and an optical concentration of C = 46 000× (the thermodynamic limit¹⁴). For this case, the thermodynamic limit for conversion efficiency is η_max = 59.82%. See Figure S1 for the limiting efficiencies given the other optical concentrations and thermal collector temperatures.

The efficiency of the thermal collector remains relatively constant as the PV spectral bandwidth is increased (it decreases slightly as a result of lower illumination intensity). For narrow PV spectral bandwidths, the efficiency of converting photons within the PV is significantly higher than the thermal engine, but hardly any power is generated. As the bandwidth increases, the efficiency monotonically decreases due to increased thermalization losses within the cell. No work may be extracted from the heat generated due to the thermalization of the excess energy of an absorbed photon since T_c = T_amb. Because the total conversion efficiency is a weighted average of these two efficiencies, a maximum exists for the hybrid converter at a particular spectral bandwidth.

Figure 2(b) shows the total conversion efficiency as a function of bandgap values and E_H. Similarly, there is a global maximum in performance for a particular set of operating conditions (i.e., T_H and C). For the case of T_H = 666 K, this occurs at a bandgap of E_G = 1.4 eV and yields a value of 60.8%. Interestingly, this efficiency is only one absolute percentage point away from the maximum efficiency for E_G = 1.1 eV. This suggests that the maximum efficiency (at full optical concentration) is relatively insensitive to the PV bandgap so long as the spectral bandwidth is appropriately chosen.

We also solved the model for the conditions of only a low-energy cutoff (i.e., all high energy photons are incident on the PV converter) in order to elucidate the relative performance enhancement due to a high-energy cutoff. In all cases, the presence of a high-energy cutoff enhances the performance. However, for larger PV bandgaps, this relative enhancement decreases. For bandgaps higher than 1.5 eV, the relative enhancement is less than 5% (Figure S2(a)). Additionally, the performance improvement of a CPV/T system relative to that of a purely thermal converter is plotted for different hot-side temperatures and thermal engine efficiencies (e.g., Carnot, Curzon & Ahlborn) in Figures S2(b) and S2(c).

In what follows, we derive a simple criterion for the optimal bandwidth directed to the PV cell based on an entropy minimization argument. To describe the three spectral bands for the spectral splitter (i.e., PT_high (>E_H), PV, and PT_low (<E_L)), the entropy increase of the system for a given photon mode interacting with either the PV converter or the thermal collector is considered. Generally, the fundamental thermodynamic relation states that for a solid-state system:

where dU is the change in internal energy of the system when a photon at hv is injected, dS is the entropy change, and dN is the number of excited carriers. For a finite internal energy change, δU = hν, the increase in entropy can be written as

Equation (6) is general and can be evaluated for both the PV and the PT converter specifically. The variables δN and T are specified for all cases in Table I, below.

Figure 3 shows the δS_PV and δS_PT with respect to the photon frequency, after including the appropriate values from Table I. δS_PV takes on a piecewise function based on the step change in δN, i.e., because the bandgap energy represents an abrupt change between the ability and inability to extract work from a photon. For the purposes of this study, a constant chemical potential with a value given by the open-circuit limit¹⁷ is assumed

In reality, the operating chemical potential will be lower due to recombination effects, but in the detailed balance limit at full optical concentration, the two values are within 2% of one another (Figure S3(b)). Compared to the maximum power circuit conditions, using this assumption (⁠$μc≈μoc$⁠) will tend to slightly overestimate the optimum bandwidth. This can be corrected by applying a more accurate value for $μc$ which can either be calculated using the detailed balance (Figure S3(a)) or empirically determined for an actual cell.

The functional form of δS_PV in Figure 3 is critical to the design of the spectral bandwidth as it implies that there are exactly two crossovers with the linear function of δS_PT. These crossover points delineate the spectral band where a solar-PV interaction generates less entropy than a solar-thermal interaction. The low-energy crossover occurs when $δN$ assumes a non-zero value (i.e., at the bandgap of the PV). The high-energy crossover can be determined by setting δS_PV equal to δS_PT and solving for the corresponding hν_H = E_H. This occurs when

E_H physically represents the point at which the thermalization loss in the ambient temperature PV converter outweighs the chemical potential on the basis of entropy. At this point, purely thermal conversion becomes preferable. Note that Equation (8) has a dependence on the optical concentration for the PV converter but not for the PT converter (see Section 1 of supplementary material).

By integrating over the entire spectrum of interest, and applying the appropriate value for δN according to Table I, the total entropy generation, S_gen can be evaluated as a function of the high-energy cutoff, E_H

Thus

and it indeed exhibits a minimum value when E_H is equal to $μc1−TcTH−1$⁠, the second crossover point of the functions δS_PV and δS_PT. The efficiency, as calculated from the full model (Equations (1)–(4)), is shown in Figure 4(a) for two arbitrary bandgaps (E_G = 1.1 and 2.0 eV) as a function of E_H. The corresponding S_gen (Equation (10)) is shown in Figure 4(b). The point of minimum entropy generation as a result of the spectral interactions as described by Equation (6) corresponds to the point of maximum conversion efficiency. Thus, the spectral bands that maximize the exergy output can be determined by investigating a particular photon mode and its interaction with a particular converter (PT or PV), and simply choosing one which generates the least entropy.

This analysis can be extended to any number of hybrid converters to determine the proper spectral bands for maximum energy conversion efficiency. For more practical operation, the thermal engine is more likely to operate near the endoreversible thermodynamic limit¹⁹ (i.e., Curzon and Ahlborn engine). Following the method previously described, but including the irreversibility of the thermal engine, the cutoff for more practical operation can be derived, corresponding to the maximum output power

Equations (8) and (12) provide a measure of the PV converter efficiency relative to the thermal converter but show no apparent dependence on the distribution of the incident light spectrum. However, energy conservation needs to be verified before implementation (i.e., T_H could never be greater than T_sun). Additionally, for values of E_H greater than ∼5 eV, the estimate breaks down since <1% of the energy of the solar spectrum is above this value and the high-energy cutoff becomes irrelevant.

The efficiency limits for both the ideal hybrid converter at full concentration (the thermodynamic limit) and the endoreversible hybrid converter operating under an optical concentration of 100× are shown in Figure 5(a). Note that for both cases, the assumption of the ideal angular selectivity has been relaxed. The top curve represents the best possible terrestrial performance for a hybrid CPV/T system while the bottom curve represents the best possible performance given more realistic operating conditions, i.e., all non-essential entropy generation is suppressed.

The agreement between the entropy minimization model and the system-level energy conversion model for predicting the optimum bandwidth is shown in Figure 5(b) through a range of PV bandgaps, hot-side temperatures, optical concentrations, and for both ideal and non-ideal thermal-engine efficiencies. This agreement suggests that without knowledge of the total efficiency via a detailed system model, the spectral bands that correspond to the maximum system performance are known. Further modifications can be made to Equation (12) such as a reduction of μ_c in order to account for non-idealities (e.g., non-radiative recombination). This can be a powerful tool to approximate the optimum design for actual systems. It can aid the choice of photonic designs and material set, revealing important length-scales and spectral regions.

We have presented the thermodynamic limits of a hybrid solar power generation device composed of a photovoltaic converter and a thermal engine in parallel. We provide a simple method for determining the optimum spectral bandwidth of an optical splitting element in this device based on a frequency-dependent entropy minimization scheme. In fact, the spectral bands that maximize performances can be determined by considering the interaction of a particular photon mode with a particular converter, and choosing the option that generates the least entropy. The optimal spectral window to the PV converter is inversely proportional to the thermal engine efficiency and scales linearly with the bandgap of the PV converter. We showed that the inclusion of a high-energy cutoff always enhances the theoretical performance relative to a single, low-energy cutoff, although with diminishing returns for higher PV converter bandgaps. This concept is extended to non-ideal systems in order to increase its utility, and we show that with proper modifications, more realistic systems can be understood using this spectral entropy minimization technique.

See supplementary material for more information about (1) the effect of optical concentration, and thermal engine temperature, (2) a benchmark for the performance of the low- and high-energy cutoff scheme, and (3) the value of the chemical potential relative to the bandgap.

This work was supported as part of the Solid-State Solar Thermal Energy Conversion (S3TEC) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. #DE-FG02-09ER46577. The authors thank Wei-Chun Hsu, Lee Weinstein, Svetlana Boriskina, Vazrik Chiloyan, and Bolin Liao from the NanoEngineering Group and Jeremy Cho from the Device Research Lab for useful discussions on thermodynamics.

All authors contributed extensively to this work. D.M.B. and A.L. envisioned the concept and theoretical studies, and analyzed the data. D.M.B. wrote the code to produce the results. E.N.W. supervised and guided the project.

The authors declare no competing financial interests.