The above treatments in terms of a black-body Sun allow analytical solutions showing the important dependencies. However, limits applying to the tabulated AM0 and AM1.5 reference spectra at the standard cell test temperature of 25°C (298.15 K) are probably of more practical interest. Figure 2.10 summarises efficiencies versus bandgap computed for the different spectra in the extended SQ
Figure 2.10 Isotropic performance limits for cells under the AM0, AM1.5G and AM1.5D spectral as well as for the AM1.5D spectra, under various concentration levels. The Trivich – Flinn limit for the AM1.5D spectrum is shown for comparison. Source: Green and Ho (2011).
approach. The uppermost curve is the Trivich-Flinn limit for the AM1.5D spectrum, which is approached for small-bandgap cells under high concentration levels (or, more generally, severe angular restriction of the converter response). Note that the limiting efficiency under concentration (or angular restriction) increases more rapidly for small-bandgap cells than for large-bandgap cells (simply because the approximately 60 mV increase in open-circuit voltage for ten times increase in intensity, or decrease in angular acceptance, is proportionately larger for small – bandgap cells).
For the AM0 spectrum for a cell with isotropic response, there is a single peak (30.4% efficiency for Eg = 1.245 eV). This is similar to the value of 31.0% at a bandgap of 1.31 eV calculated for a 6000 K black-body spectrum at a cell temperature of 300 K. For the terrestrial AM1.5 spectra, there are multiple peaks due to the strong water vapour absorption bands apparent in Fig. 2.5. The most important are the absorption bands in the 920-970nm (1.28-1.35eV), 1100-1160nm (1.07-1.13 eV) and 1300-1500 nm (0.83-0.95 eV) wavelength (energy) ranges.
Loss of radiation in the terrestrial spectra due to the first of these bands eliminates the ‘natural’ peak apparent in the AM0 spectrum, creating a local maximum on the high-energy side of the absorption band and a local minimum on the low- energy side.
Note that from Fig. 2.10 the AM0 and AM1.5G spectra give relatively higher efficiency for larger bandgaps than does the AM1.5D spectrum. This is due to blue photons being selectively scattered out of the AM1.5D spectrum during passage
Figure 2.11 Isotropic efficiency limits under the AM1.5G spectrum, and also for devices with 1% and 0.01% radiative efficiency. Also shown are best-certified cell limits efficiencies for various cell technologies. Source: Green and Ho (2011).
through the atmosphere. Combined with the intensity dependence already noted, this shifts the global maximum between three different peaks, each associated with the high-energy side of the three absorption bands already mentioned.
The solid line of Fig. 2.11 shows the limiting efficiency for a conventional cell with isotropic response under the AM1.5G reference spectrum. Compared with an earlier spectrum in use until 2008 (Green et al, 2009), the stronger spectral content over the 650-900 nm range shifts the peak efficiency to higher energy, so that the two peaks around 1.15 eV and 1.35 eV are nearly equal, with peak efficiencies of 33.6% and 33.8% respectively. For the previous reference spectrum (ASTM E892- 87, IEC60904-3:1989), the order of the peaks was reversed and both were about 1% lower in efficiency (Tiedje et al., 1984), consistent with expectations discussed elsewhere (Green et al., 2009).
Also shown in Fig. 2.11 as dashed lines are the limiting efficiencies for 0.01% and 1% radiative efficiencies of the cell, with radiative efficiency defined as the fraction of nett recombination in the device that is radiative. Radiative efficiency is a measure of the state of development of the cell material technology. Limiting efficiencies calculated with reduced radiative efficiency are identical to those calculated under a reduced intensity (further dilution) of sunlight. Hence the previous curves under concentration (Fig. 2.10) can also be used to calculate the effects of non-ideal radiative efficiency in this case as well (for example, the curve for 100 kW m-2 incident intensity represents the limiting performance for a cell under 1000kWm-2 intensity, if the radiative efficiency is 10% rather than 100%).
Given this correspondence, it is not surprising that the efficiency for low – bandgap cells is most sensitive to low values of the radiative efficiency, since a 60 mV/decade voltage reduction with decreasing radiative efficiency has a proportionately more devastating impact. Peak efficiency is pushed to higher-bandgap cells as the radiative efficiency decreases.
Also shown in Fig. 2.11 are best-certified experimental values for a range of different solar cell technologies (Green et al, 2011). ‘Bandgap’ in this case corresponds to the energy on the low-energy side of the cell’s spectral response curve, where external quantum efficiency (EQE) reaches 50% of its peak value. Silicon (c-Si) and c-GaAs lie above the 0.01% radiative efficiency line. Actual radiative efficiencies are close to or well above 1% for both technologies. The difference arises since experimental devices have additional losses other than the radiative loss assumed including, for example, reflection and resistance losses, but also including additional radiative loss due to non-abrupt absorption thresholds (Kirchartz et al, 2009).