Abstract

One of the keys towards high efficiency thermophotovoltaic (TPV) energy conversion systems lies in spectral control. Here, we present detailed performance predictions of realistic TPV systems incorporating experimentally demonstrated advanced spectral control components. Compared to the blackbody emitter, the optimized two-dimensional (2D) tantalum (Ta) photonic crystal (PhC) selective emitter enables up to 100% improvement in system efficiency. When combined with the well characterized cold side tandem filter and the latest InGaAs TPV cells, a TPV energy conversion system with radiant heat-to-electricity efficiency of 25% and power density of 0.68 W cm−2 is achievable today even at a relatively low temperature of 1320 K. The efficiency could be increased to ∼ 40% (the theoretical 0.62 eV single bandgap TPV thermodynamic limit at 1320 K is 55%) as future implementation of more optimized TPV cells approach their theoretical thermodynamic limit.

© 2013 OSA

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2013

V. Rinnerbauer, S. Ndao, Y. X. Yeng, J. J. Senkevich, K. F. Jensen, J. D. Joannopoulos, M. Soljačić, I. Celanovic, R. D. Geil, “Large-area fabrication of high aspect ratio tantalum photonic crystals for high-temperature selective emitters,” J. Vac. Sci. Technol. B 31, 011802 (2013).
[CrossRef]

V. Rinnerbauer, Y. X. Yeng, W. R. Chan, J. J. Senkevich, J. D. Joannopoulos, M. Soljačić, I. Celanovic, “High-temperature stability and selective thermal emission of polycrystalline tantalum photonic crystals,” Opt. Express 21, 11482–11491 (2013).
[CrossRef] [PubMed]

2011

M. Ghebrebrhan, P. Bermel, Y. X. Yeng, J. D. Joannopoulos, M. Soljačić, I. Celanovic, “Tailoring thermal emission via Q-matching of photonic crystal resonances,” Phys. Rev. A 83, 033810 (2011).
[CrossRef]

Y. X. Yeng, M. Ghebrebrhan, P. Bermel, W. R. Chan, J. Joannopoulos, M. Soljačić, I. Čelanović, “Enabling high temperature nanophotonics for energy applications,” Proc. Natl. Acad. Sci. USA 109, 2280 (2011).
[CrossRef]

M. Araghchini, Y. X. Yeng, N. Jovanovic, P. Bermel, L. A. Kolodziejski, M. Soljačić, I. Celanovic, J. D. Joannopoulos, “Fabrication of two-dimensional tungsten photonic crystals for high-temperature applications,” J. Vac. Sci. Technol. B 29, 061402 (2011).
[CrossRef]

2010

W. Chan, R. Huang, C. A. Wang, J. Kassakian, J. D. Joannopoulos, I. Celanovic, “Modeling low-bandgap thermophotovoltaic diodes for high-efficiency portable power generators,” Sol. Ener. Mater. Sol. Cells 94, 509–514 (2010).
[CrossRef]

P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
[CrossRef] [PubMed]

2008

T. A. Walsh, S. Y. Lin, “Power density and efficiency of thermophotovoltaic energy conversion using a photonic-crystal emitter and a 2-D metal-grid filter,” IEEE Trans. Electron. Dev. 55, 1101–1108 (2008).
[CrossRef]

I. Celanovic, N. Jovanovic, J. Kassakian, “Two-dimensional tungsten photonic crystals as selective thermal emitters,” Appl. Phys. Lett. 92, 193101 (2008).
[CrossRef]

R. Biswas, D. Zhou, I. Puscasu, E. Johnson, A. Taylor, W. Zhao, “Sharp thermal emission and absorption from conformally coated metallic photonic crystal with triangular lattice,” Appl. Phys. Lett. 93, 063307 (2008).
[CrossRef]

2007

L. B. Karlina, M. M. Kulagina, N. K. Timoshina, A. S. Vlasov, V. M. Andreev, “In0.53Ga0.47As/InP conventional and inverted thermophotovoltaic cells with back surface reflector,” AIP Conf. Proc. 890, 182–189 (2007).
[CrossRef]

V. M. Andreev, A. S. Vlasov, V. P. Khvostikov, O. A. Khvostikova, P. Y. Gazaryan, S. V. Sorokina, N. A. Sadchikov, “Solar thermophotovoltaic converters based on tungsten emitters,” J. Sol. Ener. Eng. 129, 298–303 (2007).
[CrossRef]

L. M. Fraas, L. Minkin, “TPV history from 1990 to present & future trends,” AIP Conf. Proc. 890, 17–23 (2007).
[CrossRef]

2006

D. L. C. Chan, M. Soljačić, J. D. Joannopoulos, “Thermal emission and design in one-dimensional periodic metallic photonic crystal slabs,” Phys. Rev. E 74, 016609 (2006).
[CrossRef]

M. W. Dashiell, J. F. Beausang, H. Ehsani, G. J. Nichols, D. M. Depoy, L. R. Danielson, P. Talamo, K. D. Rahner, E. J. Brown, S. R. Burger, P. M. Fourspring, W. F. Topper, P. F. Baldasaro, C. A. Wang, R. K. Huang, M. K. Connors, G. W. Turner, Z. A. Shellenbarger, G. Taylor, J. Li, R. Martinelli, D. Donetski, S. Anikeev, G. L. Belenky, S. Luryi, “Quaternary InGaAsSb thermophotovoltaic diodes,” IEEE Trans. Electron. Dev. 53, 2879–2891 (2006).
[CrossRef]

D. L. C. Chan, M. Soljačić, J. D. Joannopoulos, “Thermal emission and design in 2D-periodic metallic photonic crystal slabs,” Opt. Express 14, 8785–8796 (2006).
[CrossRef] [PubMed]

D. L. C. Chan, M. Soljačić, J. D. Joannopoulos, “Direct calculation of thermal emission for three-dimensionally periodic photonic crystal slabs,” Phys. Rev. E 74, 036615 (2006).
[CrossRef]

2005

F. O’Sullivan, I. Celanovic, N. Jovanovic, J. Kassakian, S. Akiyama, K. Wada, “Optical characteristics of one-dimensional Si/SiO2photonic crystals for thermophotovoltaic applications,” J. Appl. Phys. 97, 033529 (2005).
[CrossRef]

2004

R. T. Kristensen, J. F. Beausang, D. M. Depoy, “Frequency selective surfaces as near-infrared electromagnetic filters for thermophotovoltaic spectral control,” J. Appl. Phys. 95, 4845–4851 (2004).
[CrossRef]

H. Sai, H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85, 3399–3401 (2004).
[CrossRef]

R. K. Huang, C. A. Wang, M. K. Connors, G. W. Turner, M. W. Dashiell, “Hybrid back surface reflector GaInAsSb thermophotovoltaic devices,” AIP Conf. Proc. 738, 329–336 (2004).
[CrossRef]

B. Wernsman, R. R. Siergiej, S. D. Link, R. G. Mahorter, M. N. Palmisiano, R. J. Wehrer, R. W. Schultz, G. P. Schmuck, R. L. Messham, S. Murray, C. S. Murray, F. Newman, D. Taylor, D. M. Depoy, T. Rahmlow, “Greater than 20% radiant heat conversion efficiency of a thermophotovoltaic radiator/module system using reflective spectral control,” IEEE Trans. Electron. Dev. 51, 512–515 (2004).
[CrossRef]

A. Narayanaswamy, G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70, 125101 (2004).
[CrossRef]

J. Bravo-Abad, F. J. García-Vidal, L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93, 227401 (2004).
[CrossRef] [PubMed]

2003

N. P. Harder, P. Wurfel, “Theoretical limits of thermophotovoltaic solar energy conversion,” Semicond. Sci. Technol. 18, S151–S157 (2003).
[CrossRef]

S. Y. Lin, J. Moreno, J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83, 380–382 (2003).
[CrossRef]

2002

M. U. Pralle, N. Moelders, M. P. McNeal, I. Puscasu, A. C. Greenwald, J. T. Daly, E. A. Johnson, T. George, D. S. Choi, I. El-Kady, R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[CrossRef]

B. Bitnar, W. Durisch, J.-C. Mayor, H. Sigg, H. Tschudi, “Characterisation of rare earth selective emitters for thermophotovoltaic applications,” Sol. Ener. Mater. Sol. Cells 73, 221–234 (2002).
[CrossRef]

2001

L. Ferguson, F. Dogan, “A highly efficient NiO-Doped MgO matched emitter for thermophotovoltaic energy conversion,” Mat. Sci. Eng. B 83, 35–41 (2001).
[CrossRef]

M. Zenker, A. Heinzel, G. Stollwerck, J. Ferber, J. Luther, “Efficiency and power density potential of combustion-driven thermophotovoltaic systems using GaSb photovoltaic cells,” IEEE Trans. Electron. Dev. 48, 367–376 (2001).
[CrossRef]

O. V. Sulima, A. W. Bett, “Fabrication and simulation of GaSb thermophotovoltaic cells,” Sol. Ener. Mater. Sol. Cells 66, 533–540 (2001).
[CrossRef]

P. F. Baldasaro, J. E. Raynolds, G. W. Charache, D. M. Depoy, C. T. Ballinger, T. Donovan, J. M. Borrego, “Thermodynamic analysis of thermophotovoltaic efficiency and power density tradeoffs,” J. Appl. Phys. 89, 3319–3327 (2001).
[CrossRef]

2000

A. Heinzel, V. Boerner, A. Gombert, B. Bläsi, V. Wittwer, J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).

1999

C. A. Wang, H. K. Choi, S. L. Ransom, G. W. Charache, L. R. Danielson, D. M. Depoy, “High-quantum-efficiency 0.5 eV GaInAsSb/GaSb thermophotovoltaic devices,” Appl. Phys. Lett. 75, 1305–1307 (1999).
[CrossRef]

T. J. Coutts, “A review of progress in thermophotovoltaic generation of electricity,” Renew. Sust. Energ. Rev. 3, 77–184 (1999).
[CrossRef]

1996

G. W. Charache, D. M. DePoy, P. F. Baldasaro, B. C. Campbell, “Thermophotovoltaic devices utilizing a back surface reflector for spectral control,” AIP Conf. Proc. 358, 339–350 (1996).
[CrossRef]

1994

R. A. Lowe, D. L. Chubb, S. C. Farmer, B. S. Good, “Rare-earth garnet selective emitter,” Appl. Phys. Lett. 64, 3551–3553 (1994).
[CrossRef]

1980

C. H. Henry, “Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells,” J. Appl. Phys. 51, 4494–4500 (1980).
[CrossRef]

1963

B. D. Wedlockt, “Thermo-photo-voltaic energy conversion,” Proc. IEEE 51, 694–698 (1963).
[CrossRef]

1961

W. Shockley, H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32, 510–519 (1961).
[CrossRef]

E. R. G. Eckert, E. M. Sparrow, “Radiative heat exchange between surfaces with specular reflection,” Int. J. Heat Mass Trans. 3, 42–54 (1961).
[CrossRef]

1956

H. H. Kolm, “Solar-battery Power Source,” Tech. Rep., MIT Lincoln Laboratory. Quarterly Progress Report, Group 35, pp. 13 (1956).

Akiyama, S.

F. O’Sullivan, I. Celanovic, N. Jovanovic, J. Kassakian, S. Akiyama, K. Wada, “Optical characteristics of one-dimensional Si/SiO2photonic crystals for thermophotovoltaic applications,” J. Appl. Phys. 97, 033529 (2005).
[CrossRef]

Andreev, V. M.

L. B. Karlina, M. M. Kulagina, N. K. Timoshina, A. S. Vlasov, V. M. Andreev, “In0.53Ga0.47As/InP conventional and inverted thermophotovoltaic cells with back surface reflector,” AIP Conf. Proc. 890, 182–189 (2007).
[CrossRef]

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M. W. Dashiell, J. F. Beausang, H. Ehsani, G. J. Nichols, D. M. Depoy, L. R. Danielson, P. Talamo, K. D. Rahner, E. J. Brown, S. R. Burger, P. M. Fourspring, W. F. Topper, P. F. Baldasaro, C. A. Wang, R. K. Huang, M. K. Connors, G. W. Turner, Z. A. Shellenbarger, G. Taylor, J. Li, R. Martinelli, D. Donetski, S. Anikeev, G. L. Belenky, S. Luryi, “Quaternary InGaAsSb thermophotovoltaic diodes,” IEEE Trans. Electron. Dev. 53, 2879–2891 (2006).
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P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
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M. W. Dashiell, J. F. Beausang, H. Ehsani, G. J. Nichols, D. M. Depoy, L. R. Danielson, P. Talamo, K. D. Rahner, E. J. Brown, S. R. Burger, P. M. Fourspring, W. F. Topper, P. F. Baldasaro, C. A. Wang, R. K. Huang, M. K. Connors, G. W. Turner, Z. A. Shellenbarger, G. Taylor, J. Li, R. Martinelli, D. Donetski, S. Anikeev, G. L. Belenky, S. Luryi, “Quaternary InGaAsSb thermophotovoltaic diodes,” IEEE Trans. Electron. Dev. 53, 2879–2891 (2006).
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M. W. Dashiell, J. F. Beausang, H. Ehsani, G. J. Nichols, D. M. Depoy, L. R. Danielson, P. Talamo, K. D. Rahner, E. J. Brown, S. R. Burger, P. M. Fourspring, W. F. Topper, P. F. Baldasaro, C. A. Wang, R. K. Huang, M. K. Connors, G. W. Turner, Z. A. Shellenbarger, G. Taylor, J. Li, R. Martinelli, D. Donetski, S. Anikeev, G. L. Belenky, S. Luryi, “Quaternary InGaAsSb thermophotovoltaic diodes,” IEEE Trans. Electron. Dev. 53, 2879–2891 (2006).
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T. D. Rahmlow, D. M. Depoy, P. M. Fourspring, H. Ehsani, J. E. Lazo-Wasem, E. J. Gratrix, “Development of front surface, spectral control filters with greater temperature stability for thermophotovoltaic energy conversion,” in AIP Conference Proceedings: Thermophotovoltaic Generation of Electricity 7th Conference (AIP, 2007), pp. 59–67.

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T. D. Rahmlow, J. E. Lazo-wasem, E. J. Gratrix, P. M. Fourspring, D. M. Depoy, “New performance levels for TPV front surface filters,” in AIP Conference Proceedings: Thermophotovoltaic Generation of Electricity 6th Conference (AIP, 2004), pp. 180–188.

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M. U. Pralle, N. Moelders, M. P. McNeal, I. Puscasu, A. C. Greenwald, J. T. Daly, E. A. Johnson, T. George, D. S. Choi, I. El-Kady, R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
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Hills, J.

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P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
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M. Araghchini, Y. X. Yeng, N. Jovanovic, P. Bermel, L. A. Kolodziejski, M. Soljačić, I. Celanovic, J. D. Joannopoulos, “Fabrication of two-dimensional tungsten photonic crystals for high-temperature applications,” J. Vac. Sci. Technol. B 29, 061402 (2011).
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W. Chan, R. Huang, C. A. Wang, J. Kassakian, J. D. Joannopoulos, I. Celanovic, “Modeling low-bandgap thermophotovoltaic diodes for high-efficiency portable power generators,” Sol. Ener. Mater. Sol. Cells 94, 509–514 (2010).
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M. Araghchini, Y. X. Yeng, N. Jovanovic, P. Bermel, L. A. Kolodziejski, M. Soljačić, I. Celanovic, J. D. Joannopoulos, “Fabrication of two-dimensional tungsten photonic crystals for high-temperature applications,” J. Vac. Sci. Technol. B 29, 061402 (2011).
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T. D. Rahmlow, D. M. Depoy, P. M. Fourspring, H. Ehsani, J. E. Lazo-Wasem, E. J. Gratrix, “Development of front surface, spectral control filters with greater temperature stability for thermophotovoltaic energy conversion,” in AIP Conference Proceedings: Thermophotovoltaic Generation of Electricity 7th Conference (AIP, 2007), pp. 59–67.

T. D. Rahmlow, J. E. Lazo-wasem, E. J. Gratrix, P. M. Fourspring, D. M. Depoy, “New performance levels for TPV front surface filters,” in AIP Conference Proceedings: Thermophotovoltaic Generation of Electricity 6th Conference (AIP, 2004), pp. 180–188.

Li, J.

M. W. Dashiell, J. F. Beausang, H. Ehsani, G. J. Nichols, D. M. Depoy, L. R. Danielson, P. Talamo, K. D. Rahner, E. J. Brown, S. R. Burger, P. M. Fourspring, W. F. Topper, P. F. Baldasaro, C. A. Wang, R. K. Huang, M. K. Connors, G. W. Turner, Z. A. Shellenbarger, G. Taylor, J. Li, R. Martinelli, D. Donetski, S. Anikeev, G. L. Belenky, S. Luryi, “Quaternary InGaAsSb thermophotovoltaic diodes,” IEEE Trans. Electron. Dev. 53, 2879–2891 (2006).
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M. W. Dashiell, J. F. Beausang, H. Ehsani, G. J. Nichols, D. M. Depoy, L. R. Danielson, P. Talamo, K. D. Rahner, E. J. Brown, S. R. Burger, P. M. Fourspring, W. F. Topper, P. F. Baldasaro, C. A. Wang, R. K. Huang, M. K. Connors, G. W. Turner, Z. A. Shellenbarger, G. Taylor, J. Li, R. Martinelli, D. Donetski, S. Anikeev, G. L. Belenky, S. Luryi, “Quaternary InGaAsSb thermophotovoltaic diodes,” IEEE Trans. Electron. Dev. 53, 2879–2891 (2006).
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B. Wernsman, R. R. Siergiej, S. D. Link, R. G. Mahorter, M. N. Palmisiano, R. J. Wehrer, R. W. Schultz, G. P. Schmuck, R. L. Messham, S. Murray, C. S. Murray, F. Newman, D. Taylor, D. M. Depoy, T. Rahmlow, “Greater than 20% radiant heat conversion efficiency of a thermophotovoltaic radiator/module system using reflective spectral control,” IEEE Trans. Electron. Dev. 51, 512–515 (2004).
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B. Wernsman, R. R. Siergiej, S. D. Link, R. G. Mahorter, M. N. Palmisiano, R. J. Wehrer, R. W. Schultz, G. P. Schmuck, R. L. Messham, S. Murray, C. S. Murray, F. Newman, D. Taylor, D. M. Depoy, T. Rahmlow, “Greater than 20% radiant heat conversion efficiency of a thermophotovoltaic radiator/module system using reflective spectral control,” IEEE Trans. Electron. Dev. 51, 512–515 (2004).
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Figures (7)

Fig. 1
Fig. 1

(a) Conventional thermophotovoltaic (TPV) energy conversion system without spectral control. (b) TPV system with 2D photonic crystal (PhC) selective emitter and cold side filter. Spectral control enables performance enhancement of up to 400% over the conventional TPV system.

Fig. 2
Fig. 2

Simulated high temperature (T = 1478 K) normal spectral emittance of flat Ta and 2D Ta PhCs optimized for GaSb (Design I: r = 0.43 μm, d = 8.00 μm, a = 0.95 μm), InGaAs (Design II: r = 0.51 μm, d = 8.00 μm, a = 1.11 μm), and InGaAsSb (Design III: r = 0.57 μm, d = 8.00 μm, a = 1.23 μm). As can be seen, the cutoff is easily shifted by modifying the dimensions of the PhC.

Fig. 3
Fig. 3

Measured normal incidence transmittance of tandem filters optimized for 0.5 eV and 0.6 eV TPV cells. The tandem filters consist of dielectric stacks of antimony selenide (Sb2Se3) and yttrium fluoride (YF3), terminated with a 1 μm thick heavily doped indium phosphide arsenide (InPAs) layer as the plasma filter. The tandem filters are sourced from Rugate Technologies, Inc.

Fig. 4
Fig. 4

Relevant optical properties for optimized components in an InGaAsSb TPV system. The normal incidence emittance ε and hemispherical emittance εH of the optimized 2D Ta PhC emitter, and 45° reflectance Rθ=45° of the 0.53 eV tandem filter is shown to match the external quantum efficiency (EQE) of InGaAsSb. An ideal cutoff emitter is included in the analysis to elucidate the effect of non-ideal spectral emittance of the optimized 2D Ta PhC.

Fig. 5
Fig. 5

(a) Radiant heat-to-electricity ηTPV for various emitters with or without an optimized tandem filter in combination with InGaAsSb TPV cell at fixed view factor F = 0.99 (10 cm × 10 cm flat plate geometry with separation s = 500 μm). An optimum temperature T exist for each combination. Due to considerable emission below the bandgap of the In-GaAsSb TPV cell for the emitters considered, significant improvement is seen with the use of the tandem filter. (b) Overall spectral efficiency when TPV cavity effects are included, ηCav–Spec. When F = 0.99, use of a selective emitter is not critical if an optimized tandem filter is present. (c) TPV cell efficiency ηCell. For T > 1200 K, degradation of ηCell is seen due to larger series resistance losses from high carrier injection.

Fig. 6
Fig. 6

With T fixed at the optimum, the most efficient combination depends on the experimentally achievable F. For F > 0.97 (10 cm × 10 cm flat plate geometry with separation s < 1.7 mm), the use of the optimized tandem filter allows the greybody to slightly outperform the optimized 2D Ta PhC selective emitter. In contrast, it is important to restrict below bandgap emission via selective emitters in TPV systems with smaller view factors.

Fig. 7
Fig. 7

ηTPV for an InGaAsSb TPV system including an ideal cutoff emitter with varying below bandgap hemispherical emittance εlw with or without a 0.53 eV optimized tandem filter at a fixed temperature T of 1250 K. (a) F = 0.99. To outperform the greybody - optimized tandem filter combination, εlw must be smaller than 0.03, to a point where addition of the tandem filter is detrimental given the larger reduction in power density for a small improvement in ηTPV. (b) F = 0.97. To outperform the greybody - optimized tandem filter combination, εlw must be smaller than 0.08. As F is reduced, both aspects of spectral control become important.

Tables (1)

Tables Icon

Table 1 Predicted ηTPV,max for three different TPV cells utilizing experimentally realizable spectral control components at fixed F = 0.99. Optimum temperature indicated in brackets is determined for each TPV system combination using fabricated and characterized TPV cells (GaSb [45], InGaAs [3], and InGaAsSb [45]). Results indicate that current state of the art fabricated TPV cells are ∼ 50% as efficient as their thermodynamically ideal counterparts. It is also interesting to note that spectral control via the optimized 2D Ta PhC and tandem filter enables TPV cells with larger bandgaps (GaSb) to perform as well as TPV cells with smaller bandgaps (InGaAsSb). However, the use of smaller bandgap TPV cells would result in lower optimum temperatures.

Equations (16)

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i BB ( λ , t ) = 2 h c 2 λ 5 [ exp ( h c λ k T ) 1 ]
P em = 0 d λ 0 π 2 d θ 1 0 2 π d ϕ 1 A 1 d A 1 [ i BB ( λ , T ) ε ( λ , θ 1 , ϕ 1 ) cos θ 1 sin θ 1 ]
d Q d A 1 d A 2 = i BB ( λ , T ) ε ( λ , θ 1 , ϕ 1 ) d A 1 d F 2
d F 2 = cos θ 1 cos θ 2 s 2 d A 2
d F l = cos θ 1 cos θ l s l 2 d A l
P re = p = 1 d λ d F 2 p + 1 d A 1 [ i BB ( λ ) R 2 p R 1 p 1 ( 1 R 1 ) ε ( λ , θ 2 p + 1 , ϕ 2 p + 1 ) ]
P re = p = 1 d λ d A 2 p + 1 d A 1 [ i BB ( λ ) R 2 p R 1 p 1 ( 1 R 1 ) ε ( λ , θ 2 p + 1 , ϕ 2 p + 1 ) cos 2 θ 2 p + 1 s 2 p + 1 2 ]
P cell = p = 1 0 λ g d λ d A 2 p d A 1 [ i BB ( R 1 R 2 ) p 1 ( 1 R 2 ) ε ( λ , θ 2 p , ϕ 2 p ) cos 2 θ 2 p s 2 p 2 ]
I sc = 2 q c p = 1 IQE ( λ ) d λ λ 4 [ exp ( h c λ k T ) 1 ] d A 2 p d A 1 [ ( R 1 R 2 ) p 1 ( 1 R 2 ) ε ( λ , θ 2 p , ϕ 2 p ) cos 2 θ 2 p s 2 p 2 ]
I = I sc I o exp [ q ( V + I R s ) m k T c ] V + I R s R sh
η TPV = P elec , max P em P re
η Cav Spec = P cell P em P re
η Cell = P elec , max P cell
ε H ( λ ) = 1 π 0 π 2 d θ 1 0 2 π d ϕ 1 [ ε ( λ , θ 1 , ϕ 1 ) cos θ 1 sin θ 1 ]
FOM = x η TPV + ( 1 + x ) J elec , max PhC J elec , max BB
I o = A 2 q ( n 2 + 1 ) E g 2 k T c 4 π 2 h ¯ 3 c 2 exp ( E g / k T c )

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