Letter pubs.acs.org/NanoLett
Time-Asymmetric Photovoltaics Martin A. Green* ARC Photovoltaics Centre of Excellence, University of New South Wales, Sydney, Australia 2052 ABSTRACT: Limits upon photovoltaic energy conversion efficiency generally are formulated using the detailed balance approach of Shockley and Queisser. One key underlying assumption is invariance upon time reversal, underpinning detailed balance itself. Recent proposals for compact, layered, time-asymmetrical, magneto-optical devices make their routine implementation likely. It is shown that such time-asymmetry can alter the relationship between solar cell emission and absorption assumed in the Shockley−Queisser approach, allowing generally accepted photovoltaic performance limits to be exceeded. KEYWORDS: Photovoltaic efficiency limits, Shockley−Queisser limits, time-asymmetrical photovoltaics, solar cell conversion limits
T
number of different bandgaps7−9 (20 different bandgaps give 83.8% efficiency, while 40 give 85.8%9). An alternative approach to calculating solar conversion limits is the purely thermodynamic approach as pioneered by Landsberg.14,15 The “Landsberg limit” on solar conversion to electricity deduced by assuming zero entropy production during conversion is 93.3%, notably higher than from the extended SQ approaches. The difference was once attributed to the Landsberg limit not being attainable, even in principle, by any physical converter.7 However, Ries16 has shown for solar thermal conversion that the Landsberg limit could be attained if time-asymmetry was considered, suggesting a macroscopic implementation using mechanical circulators. The present author extended this macroscopic approach to photovoltaics,6 with additional properties subsequently explored.17 Multiple groups18−20 have recently reported compact nonreciprocal devices based on thin layers of bismuth−iron− garnet (Bi3Fe5O12), often with magneto-optical effects strongly plasmonically enhanced. Figure 1 shows one scheme highlighting the potential for “one-way” transmission through a layered magneto-optical system19 (dynamic nonreciprocity such as possible by spatial-temporal modulation of refractive index offers additional possibilities21). Recalling the SQ methodology, one-way transmission could be ideal for photovoltaics since it allows the absorption of incoming photons from one direction while suppressing emission into that direction. However, thermodynamic constraints might be expected if classical paradoxes from one-way transmission are to be avoided, such as that suggested by Wien and resolved by Rayleigh.22 A framework for investigating this issue23 has been established in investigations of the reciprocity of f(θi, ϕi; θj,
he Shockley−Queisser (SQ) limits on photovoltaic energy conversion efficiency1 can be deduced using an elegant but simple argument, in two parts. First, it is noted that ideal solar cells would absorb all light with photon energy above the bandgap of their constituent semiconductor (or equivalent) material, making them ideal blackbodies for such energies. Spectral absorption and emission from blackbodies at thermal equilibrium are well-known,2 allowing the cell photon emission rate at thermal equilibrium to be readily calculated. Assuming that band-to-band optical processes dominate at above-bandgap energies, a net electron−hole recombination event is the source of these emitted photons, allowing the internal net recombination rate at thermal equilibrium to be calculated without needing to address the complications of internal photon reabsorption or total-internal reflection. Second, it is noted that, when voltage appears across the cell terminals, the internal carrier concentration product increases exponentially. For an ideal solar cell collecting all photogenerated carriers, this increase must be uniform through the cell, as implicitly assumed by SQ, but firmly established subsequently.3,4 It follows that the emitted above-bandgap radiation and net internal recombination rates must also increase exponentially. When used as a cell in the radiative limit, increasing numbers of photogenerated carriers fuel this emission with increasing voltage, limiting power output to that at one specific maximum power point voltage. SQ deduced a 30.8% efficiency limit for a single junction cell (6000 K blackbody sun), a value that increases to 40.7% if only direct sunlight is converted, such as when concentrated.5−9 The SQ approach has been extended successively to more general converters including multiple-junction stacks,7−9 cells where high energy photons create multiple electron−hole pairs (or excitons),10 cells with intermediate bands,11 and cells boosted by up- or down-converters.12,13 The highest efficiency deduced with the extended SQ approach is 86.8% for direct sunlight conversion for a stack involving cells of essentially an infinite © 2012 American Chemical Society
Received: September 18, 2012 Revised: October 11, 2012 Published: October 15, 2012 5985
dx.doi.org/10.1021/nl3034784 | Nano Lett. 2012, 12, 5985−5988
Nano Letters
Letter
case is an appropriately weighted product of two Dirac-δ functions.24 The surface texture effectively invalidates the law of reflection when considered macroscopically. Recent work shows that a plasmonic interface incorporating a phase-shifting array can cause deviations from both Snell’s law and the law of reflection,25,26 even for planar, time-symmetric surfaces. Figure 2a shows the normal time-symmetric case where low emissivity into any direction is obtained only when reflectivity
Figure 1. “One way” transmission through a thin-layered structure relying on plasmonically enhanced magneto-optical effects in bismuth−iron−garnet. Red and green rays correspond to light of the same polarization traveling in opposite directions. Reprinted with permission from Khanikaev et al. (2010). Copyright 2010 American Physical Society.
ϕj), the bidirectional reflectance distribution function (BRDF). The BRDF describes how bright a surface appears viewed from one direction when illuminated from another, by relating light reflected from an opaque surface in a direction j (defined by zenith and azimuth angles θj and ϕj), to light incident in direction i (defined by angles θi, ϕi):24 f (θi , ϕi ; θj , ϕj) =
dLj(θj , ϕj) Li(θi , ϕi)cos(θi)dωi
(1) −2
−1
dLj is the incremental radiance (MKS units: Watts m sr ) in the outgoing direction, and dωi is the solid angle element within which the incident radiance Li is confined. As a measurement example, if radiance Li is uniform at all points in all directions in an incident beam within a small solid angle (as can be approximated experimentally), the measured reflectance ρ(ωi, ωj) (the ratio of reflected to incident flux) equals:24 ρ(ωi , ωj) =
∫ω ∫ω f (θi , ϕi ; θj , ϕj)dΩidΩj/∫ω dΩi j
i
i
Figure 2. (a) Time-symmetric case. Low emissivity can only be obtained from surfaces with high reflectivity. (b) Time-asymmetric case. For specular surfaces where the law of reflection is obeyed, low emissivity into one direction is possible with high absorptivity in this direction, if the reflection into this direction for light incident in the complementary direction is high.
(2)
ωi and ωj are solid angles associated with the incident and reflected beams, and dΩk is an elemental projected solid angle given by dΩk = cos(θk)dωk. From a thermodynamic balance at thermal equilibrium, Snyder et al.23 deduce the following relationships for the directional emissivity ε(θj,ϕj) and absorptivity α(θj,ϕj): ε(θj , ϕj) = 1 −
∫2π f (θj , ϕj; θi , ϕi)dΩi
(3)
α(θj , ϕj) = 1 −
∫2π f (θi , ϕi ; θj , ϕj)dΩi
(4)
is high and hence absorptivity is low. Figure 2b suggests how time-asymmetry might be used to control emission in systems obeying the law of reflection (as for the compact schemes previously mentioned18−20). Light incident from one direction is strongly reflected with near-zero emission into the direction of reflection. Light incident from this latter direction is strongly absorbed. The situation in Figure 2b is promising for photovoltaics since light could be 100% absorbed from one direction without SQ emission into this direction. The penalty is that no light is absorbed from the opposite direction with full SQ emission into this direction. This asymmetry allows separation of incoming and emitted light paths, allowing emitted light to be directed to another converter and contribute to further power output, similarly to Ries’s proposed circulator implementation.16 The ideal properties for a specular converter of this type can be specified. For photon energies where appreciable SQ emission occurs, mostly within kTA of the band-edge of
For materials invariant under time reversal, for a given wavelength and polarization state, f(θi,ϕi;θj,ϕj) = f(θj,ϕj;θi,ϕi), giving Kirchhoff’s law. The thermal equilibrium thermodynamic arguments resulting in eq 3 show that emissivity in a given direction cannot be zero unless there is strong reflection, for at least one angle of incidence, into this direction. This dashes any hopes of a perfectly absorbing but minimally emitting cell. However, thermodynamics do not stipulate from which direction this incident ray might come, although sample geometry may restrict options. For flat, specular surfaces, the law of reflection confines incident ray options to θi = θj; ϕi = ϕj ± π. The BRDF in this 5986
dx.doi.org/10.1021/nl3034784 | Nano Lett. 2012, 12, 5985−5988
Nano Letters
Letter
converter cells (k is Boltzmann’s constant; kTA equals 26 meV with ambient temperature TA around 298 K), angularly asymmetrical absorption and emission are required, with absorption strong in directions where incident sunlight is strongest and light emitted in directions where it can be converted. Photovoltaic efficiency implications are most apparent for systems converting only direct sunlight involving very directional light (Figure 3). Each “cell” shown is a double-cell, with
In this limiting case, uppermost cells operate incrementally close to open-circuit, with incrementally small power conversion occurring in each allowing, in the limit, conversion with zero entropy production. Importantly, the gain with only one extra converter (Figure 3) is 60% of that possible with an infinite number of converters, de-emphasizing the significance of attaining the zero entropy ideal. This large gain by adding one additional converter (Figure 3) is particularly significant when noting that the additional converter needs not be time-asymmetric, but merely have acceptance and emission angles matched to incoming light to attain the limiting efficiency. This adds flexibility in applying the benefits of time-asymmetry to systems accepting light from a wider range of directions. One such case is where sunlight is concentrated 500−1000 times, representing present practical limits. In such systems, light strikes the cells within a well-defined range of angles. With time-asymmetrical geometries, light emission into these angles can be suppressed, instead being emitted into directions where it can be utilized (possibly also in combination with diffuse light,27 also normally wasted in such systems). Prospects for global spectrum converters seem less promising. Equations 3 and 4 show that emission into any direction cannot be restricted without sacrificing absorption for this or another direction. Best prospects are for enhanced response over the most critical part of the incoming hemispherical distribution at the sacrifice of response at less critical angles. Emitted light collection also poses additional challenges due to its likely angular diversity.
Figure 3. Basic scheme for capitalising on time-asymmetry. Each component of the double-cell shown is operated at a different voltage. If each cell is replaced by an infinite stack of cells, the directionality of the emitted light allows the photovoltaic conversion limit to be boosted beyond its time-symmetric value of 86.8%.
each double-cell component operating at a different voltage. The highest performance and the largest gains from timeasymmetry are obtained when each constituent “cell” consists of a large stack of different bandgap cells. With one time-asymmetric converter, the limiting efficiency is 86.8% for conversion of 6000 K direct sunlight, the same as for the time-symmetric case. To gain from time-asymmetry, the different directions of light emission and absorption must be exploited, such as by directing emitted light to a second cell. With two time-asymmetrical elements (Figure 3), with each double-cell component biased optimally, the limiting efficiency increases from 86.8% to 90.7%, the same value as calculated using two macroscopic circulators.17 This limiting efficiency can be further improved by adding additional layers of time-asymmetric converters (Figure 4). A third converter increases limiting efficiency to 91.8% and a fourth to 92.2%.17 In this way, the Landsberg limit of 93.3% is incrementally approached.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS I thank A. Khanikaev for permission to reproduce Figure 1 in this paper. I also thank the Australian Research Council (ARC) and the Australian Solar Institute for support of related work.
■
REFERENCES
(1) Shockley, W.; Queisser, H. J. J. Appl. Phys. 1961, 32, 510. (2) Planck, M. The Theory of Heat Radiation; Dover: New York, 1959. (3) Donolato, C. Appl. Phys. Lett. 1985, 46, 270. (4) Rau, U. Phys. Rev. B 2007, 76, 085303. (5) Campbell, P.; Green, M. A. IEEE Trans. Electron Devices 1986, 33, 234−239. (6) Green, M. A. Third Generation Photovoltaics: Advanced Solar Energy Conversion; Springer-Verlag: Berlin, 2003. (7) Pauwels, H.; De Vos, A. Solid State Electron. 1981, 24, 835−843. (8) Marti, A.; Araujo, G. L. Sol. Energy Mater. Sol. Cells 1996, 43, 203−222. (9) Brown, A. S.; Green, M. A. Prog. Photovolt: Res. Appl. 2002, 10, 299−307. (10) Werner, J. H.; Brendel, R.; Queisser, H. J. Appl. Phys. Lett. 1995, 67, 1028−1030. (11) Luque, A.; Marti, A. Phys. Rev. Lett. 1997, 78, 369. (12) Trupke, T.; Green, M. A.; Würfel, P. J. Appl. Phys. 2002, 92, 4117−4122. (13) Trupke, T.; Green, M. A.; Würfel, P. J. Appl. Phys. 2002, 92, 1668−1674. (14) Landsberg, P. T.; Tonge, G. J. Appl. Phys. 1980, 51, R1−R20.
Figure 4. Extension of the benefits of time-asymmetry. By adding additional cell layers (and readjusting each cell’s bias), the Landsberg efficiency limit of 93.3% is incrementally approached. 5987
dx.doi.org/10.1021/nl3034784 | Nano Lett. 2012, 12, 5985−5988
Nano Letters
Letter
(15) Luque, A.; Marti, A. Phys. Rev. B 1997, 55, 6994. (16) Ries, H. Appl. Phys. B: Laser Opt. 1983, 32, 153. (17) Brown A. S. PhD thesis. The University of New South Wales, Sydney, Australia, 2003. (18) Yu, Z.; Wang, Z.; Fan, S. Appl. Phys. Lett. 2007, 90, 121133. (19) Khanikaev, A. B.; Mousavi, S. H; Shvets, G.; Kivshar, Y. S. Phys. Rev. Lett. 2010, 105, 126804. (20) Zhu, H.; Jiang, C. Opt. Lett. 2011, 36, 1308−1310. (21) Yu, Z.; Fan, S. Nat. Photonics 2009, 3, 91−94. (22) Rayleigh, L. Nature 1901, 64, 577−578. (23) Snyder, W. C.; Wan, Z.; Li, X. Appl. Opt. 1998, 37, 3464−3470. (24) Nicodemus, F. E.; Richmond, J. C.; Hsia, J. J.; Ginsberg, I. W.; Limperis, T. Geometrical Considerations and Nomenclature for Reflectance; National Bureau of Standards, U.S. Department of Commerce: Washington, DC, 1997. (25) Yu, N.; Genevet, P.; Kats, M. A.; Aieta, F.; Tetienne, J.-P.; Capasso, F; Gaburro, Z. Science 2011, 334, 333−337. (26) Aieta, F.; Genevet, P.; Yu, N.; Kats, M. A.; Gaburro, Z.; Capasso, F. Nano Lett. 2012, 12, 1702−1706. (27) Goetzberger, A. As reported in ref 6, p 15.
5988
dx.doi.org/10.1021/nl3034784 | Nano Lett. 2012, 12, 5985−5988
