We can begin by considering the situation where absorptivity for both the selective absorber and emitter is unity within a certain frequency range, and δ
otherwise. The ranges for the selective absorbers and emitters are optimized separately, and the lower end of the selective emitter frequency range equals the TPV diode bandgap frequency ωg
. If we consider the case of unconcentrated sunlight, the limit δ
→ 0 implies a decoupling between the selective absorber and emitter, where the selective absorber is kept relatively cool to maximize ηt
, while the selective emitter acts as if it were much hotter with a bandgap frequency ωg
well over the blackbody peak predicted by Wien's law. However, this also leads to declining effective emissivity
, and thus AE
. This expectation is supported by the numerical calculations in Figure (see the Methods sections for details), which demonstrate both that efficiency slowly increases with decreasing δ
, while the area ratio increases rapidly as 1/δ
. Clearly the limit where δ
→ 0 and AE
→ ∞ is unphysical, both because the time to establish equilibrium in an arbitrarily large system is arbitrarily long, and a perfectly sharp emissivity cutoff requires a step function in the imaginary part of the dielectric constant of the underlying material. However, the latter is inconsistent with the Kramers-Kronig relations for material dispersion, which derive from causality [12
For an ideal solar TPV system with unwanted emissivity δ: a system efficiency versus δ and b area ratio for selective emitter to selective absorber versus δ.
Based on a comprehensive review of selective solar absorbers [13
], typical spectrally averaged selective solar absorber emissivities
are about 0.05 at temperatures of approximately 373 K. Assuming δ
= 0.05 as well, this implies a maximum system efficiency of 10.5% (T
= 720 K, ηt
= 0.6937, ηp
= 0.1510, AE/As
= 0.75), as illustrated in Figure . While a physically relevant result, this efficiency is unfortunately less than a quarter of the asymptotic efficiency calculated above as δ
Solar TPV system efficiency: a without angular selectivity, b with optimized angular selectivity of functional form given in Equation 5.
To bridge the gap between performance of solar TPV in the cases where δ
= 0.05 and δ
→ 0, we can employ a combination of wavelength and angle selectivity. It has been shown in a large number of previous publications that absorption can be made to peak at a certain target angle or wavevector, over a certain range of wavelengths. While an exact analytical expression is often lacking, it generally resembles a top hat function in wavelength space, and a local maximum in the angular dimension [14
]. Since local maxima can be approximated as inverted parabolas, the analytical expression we use is as follows [14
is the top hat function, equal to 1 if ω1
<ω < ω2
and 0 otherwise. This definition is illustrated in Figure for frequencies within the window of the top hat.
Schematic diagram of the emissivity as a function of angle for all wavelengths.
The system efficiency of our angle-selective design was determined by inserting Equation 5 into Equation 3, then multiplying with the TPV diode back end efficiency of Equation 4. Optimizing over the following parameters--cutoff frequencies, acceptance angles, TPV bandgap and temperature--yields the results in Figure , where the maximum efficiency is 37.0% (T
= 1, 600 K, ηt
= 0.7872, ηp
= 0.4697, AE
= 0.05). This is 3.5 times higher than our previous result, and fairly close to the asymptotic limit where δ
→ 0 from before, without the physically unreasonable requirement of a perfectly sharp emissivity cutoff (which is inconsistent with causality). This result also exceeds the Shockley-Quiesser limit for photovoltaic energy conversion in unconcentrated sunlight of 31% efficiency [8
]. Furthermore, as illustrated in Figure , photovoltaic diodes made from group IV compounds such as silicon and germanium have bandgaps that would allow for the system to continue to exceed the Shockley-Quiesser limit. Finally, the much lower area ratio AE
= 0.05 implies that the angle-selective solar absorber illustrated in Figure would serve as a sort of thermal concentrator, thus allowing for much less thermophotovoltaic area to be used compared to previous designs in the literature.
Solar TPV system efficiency as a function of operating temperature for germanium and silicon with unconcentrated sunlight. Both can exceed the Shockley-Quiesser limit at certain operating temperatures.
Finally, we consider reasonable metamaterial designs for achieving the desired effective emissivity in Equation 5. Most structures with nanoscale features on the surface in both directions have potential to exhibit strong angular sensitivity. The specific structure we examined is a 2D array of cylindrical holes in single-crystal tungsten, as discussed in [5
]. In Figure , using numerical techniques described in the Methods section, we show that an optimal structure with period 800 nm, hole radius 380 nm, and hole depth 3.04 µ
m exhibits decreasing average emissivity with increasing angle away from normal incidence. In particular, at a 75° angle, the average emissivity for wavelengths from 400 nm to 2 μ
m is 30% lower than at normal incidence. Overall, for an absorber in unconcentrated sunlight held at 400 K, the spectrally averaged absorptivity
, while the spectrally average emissivity
. This results in a projected thermal transfer efficiency ηt
= 0.750. Such a result compares favorably with previously proposed selective absorber designs, such as a germanium with a silver back and an anti-reflection coating, with a projected thermal transfer efficiency of 0.678 under identical conditions [5
]. Additionally, increasing the operating temperature to 1,000 K and employing 100 sun concentration (e.g., with a parabolic trough) yields a projected thermal transfer of 0.741; again, above a semiconductor-based design with an anti-reflection coating, displaying a thermal transfer efficiency of 0.710 under identical conditions [5
]. Clearly, suppressing off-angle emission with relatively simple structures such as 2D arrays of holes in tungsten can give rise to improved spectrally selective performance. Future work should focus on modifying these structures to narrow the acceptance angles. This approach should allow one to achieve record-setting thermal transfer efficiencies for selective solar absorbers.
Figure 6 Emissivity spectra for 2D periodic arrays of cylindrical holes in single crystal tungsten at various angles (a = 800 nm, r = 380 nm, and d = 3.04 μm. Notice that the average emissivity gradually decreases with increasing angle away from normal (more ...)