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Letter
Vol. 42, No. 16 / August 15 2017 / Optics Letters
Negative illumination thermoradiative solar cell TIANJUN LIAO,1 XIN ZHANG,1 XIAOHANG CHEN,1,* BIHONG LIN,2
AND
JINCAN CHEN1
1
Department of Physics, Xiamen University, Xiamen 361005, China College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China *Corresponding author:
[email protected]
2
Received 13 June 2017; revised 21 July 2017; accepted 24 July 2017; posted 25 July 2017 (Doc. ID 297833); published 14 August 2017
The negative illumination thermoradiative solar cell (NITSC) consisting of a concentrator, an absorber, and a thermoradiative cell (TRC) is established, where the radiation and reflection losses from the absorber to the environment and the radiation loss from the TRC to the environment are taken into consideration. The power output and overall efficiency of the NITSC are analytically derived. The operating temperature of the TRC is determined through the thermal equilibrium equations, and the efficiency of the NITSC is calculated through the optimization of the output voltage of the TRC and the concentrating factor for a given value of the bandgap. Moreover, the maximum efficiencies of the NITSC at different conditions and the optimal values of the bandgap are determined, and consequently, the corresponding optimum operating conditions are obtained. The results obtained here will be helpful for the optimum design and operation of TRCs. © 2017 Optical Society of America OCIS codes: (350.6050) Solar energy; (040.5350) Photovoltaic. https://doi.org/10.1364/OL.42.003236
The thermophotovoltaic cell (TPVC) consisting of an emitter and a photovoltaic (PV) cell is a solid-state device that converts heat into electricity [1,2]. Thermal emission in the emitter mainly is the spontaneous emission of near infrared and infrared frequency photons due to thermal motion of charges. The PV cell absorbs some of these radiated photons and converts them into electricity. In contrast, the thermoradiative cell (TRC) composed of a p-n junction is a kind of easily manufactured device. When the TRC is operated at a high temperature, it will emit photons to the cold environment [3,4]. Electricity can be generated in this process when a load is connected externally. The conceptual designs, performance evaluation, and working regime of the TRC have been reported in previous works [5–8]. Hsu et al. [6] and Wang et al. [7] demonstrated that the performance characteristics of the TRC can be enhanced by near-field radiative heat transfer. Fernández [8] proposed a three-level TRC model. Calculation and detailed balance analysis were given. However, the solar-driven negative illumination TRC has not been reported. In this Letter, the model of the negative illumination thermoradiative solar cell (NITSC) is proposed. Through performance analysis, we find 0146-9592/17/163236-03 Journal © 2017 Optical Society of America
that the NITSC can obtain high efficiency under a low concentrating factor. The presented theoretical work will be beneficial to develop high-efficiency solar cells. The NITSC consists of a solar concentrator, an absorber, a TRC, and an optical filter, as shown in Fig. 1(a), where the TRC is composed of the p- and n-type semiconductors, T C and T E are the temperatures of the TRC and the environment, qin is the input solar radiation flux, qS is the concentrated solar radiation flux, q1 is the radiation loss from the absorber to environment, q2 is the reflected solar radiation of the absorber, qH is the heat flow from the absorber to the TRC, and q 3 is the radiation heat flow between the TRC and environment. By using the same method as that of the thermophotovoltaic (TPV) system [1,10–12], an ideal optical filter is placed at the bottom of the TRC and can reflect the sub-bandgap photons to the TRC, so that the contribution of sub-bandgap photons to the radiation heat flow q3 may not be taken into account. Figure 1(b) shows the band diagram of the TRC, where e− represents the electron, E c and E v are the bottom level of the conduction and the top level of the valence bands, E fe and E fh denote the quasi-Fermi levels of the electron and hole, and the quasi-Fermi level difference E fe − E fh determines the voltage output V of the TRC [6]. According to Fig. 1, the working principle of the NITSC can be described as follows. The TRC absorbs the heat qH and is heated up to a higher-than-ambient temperature. When the radiative recombination is faster than the radiative generation rate, the photons emitted from the TRC are more than those received from environment, resulting in a negative output voltage, i.e., V < 0 [3,6]. When a load is connected externally, electrons flow into the n-type region, holes flow into the p-type region, and, consequently, the TRC generates an electric current I , which is expressed as [3,4,8,13] Z ∞ 2πAC E 2 dE I 3 2 hc E g expE − eV ∕kT C − 1 Z ∞ E 2 dE ; (1) − E g expE∕kT E − 1 where e is the elementary positive charge, E is the photon energy, E g E c − E v is the bandgap of the semiconductor, and AC is the electrode area of the TRC. By using Eq. (1), the power output of the TRC is given by [3,4,8,13]
Vol. 42, No. 16 / August 15 2017 / Optics Letters
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constant, η0 is the optical efficiency of the concentrator, c is the speed of light, and h is the Planck constant. Note that ελ is equal to αλ based on Kirchhoff’s law [13], and Rλ is equal to 1 − αλ, where αλ is the spectral absorptance of the absorber. The curve of αλ varying with the wavelength [9] is shown in Fig. 1(c). According to Fig. 1(c), the range of the AM 1.5G solar spectrum is 280–4000 nm [14]. The effect of the radiative cooling to the atmosphere resulting from the wavelength larger than 4000 nm on the performance of the device is small and may not be considered. q1 in Eq. (7) depends on the equilibrium temperature T C , which varies with the concentrating factor C. The radiation heat flow q 3 is expressed as [3,4,8] Z ∞ 2π AC E 3 dE q3 3 2 h c E g expE − eV ∕kT C − 1 Z ∞ AC E 3 dE : (9) − E g expE∕kT E − 1 Fig. 1. (a) The schematic diagram of an NITSC. (b) The band diagram of a TRC [3,6]. (c) The curve of the absorber emissivity varying with the wavelength [9].
Z
AC V E 2 dE E g expE∕kT E − 1 Z ∞ AC V E 2 dE ; − E g expE − eV ∕kT C − 1
P −V I
2π h3 c 2
∞
and the overall efficiency of the NITSC is expressed as P −V I R ; η qin CAA 0∞ Jλdλ
(2)
(3)
where AA is the front surface area of the absorber, Jλ is the spectral incident solar radiation at AM 1.5 G [14], λ is the radiation wavelength, and C is the concentration factor. Equations (2) and (3) show that the power output and efficiency of the NITSC depend on T C for the given values of other parameters such as the optical efficiency η0, concentrating factor C, areas AA and AC , voltage output V , and bandgap E g . T C should be determined by solving the following equations: qS qH q1 q2
(4)
qH P q3 ;
(5)
By substituting Eqs. (5)–(9) into Eq. (4), an absorberto-thermoradiative cell area ratio AA ∕AC is derived. T C can be solved for given values of AA ∕AC , C, V , η0 , and E g . In the present study, we assume that the area ratio AA ∕AC is equal to 1. When a small area ratio is selected, the system requires a concentrator with a high concentrating factor to obtain maximum efficiency. When the area ratio is too large, the radiation and reflection losses from the absorber to the environment influence the performance of the system. Using Eqs. (1)–(9), we can generate three-dimensional projective graphs of the operating temperature T C , current density i, power output density P , and efficiency η varying with the concentrating factor C and voltage output V , as shown in Figs. 2(a)–2(d), respectively, where i I ∕AC and P P∕AC . It is seen from Figs. 2(a) and 2(b) that T C and i are two monotonic functions of the concentration factor C and voltage output V . T C monotonically increases with the increase of C and jV j, while i monotonically increases with the increase of C and V . Figure 2(c) shows that the power output density P increases with the increase of C, but it is not a monotonic function of V . It is
and where qS , q1 , and q2 are, respectively, calculated by [15,16] Z ∞ qS ηo CAA Jλdλ; (6) 0
Z q1
∞ 0
AA ελ2πhc 2 ∕λ5 A ελ2πhc 2 ∕λ5 − A dλ; expℏc∕λkT C − 1 expℏc∕λkT E − 1 (7)
and
Z q 2 η0 CAA
∞
RλJλdλ;
(8)
0
where ελ and Rλ are the spectral emissivity and the reflection coefficient of the surface of the absorber, k is the Boltzmann
Fig. 2. Three-dimensional projective graphs of (a) the temperature T C , (b) the current density i, (c) the power density P , and (d) the efficiency η as a function of C and V , respectively, where AA ∕AC 1, k 8.62 × 10−5 eVK −1 , e 1.60 × 10−19 C, c 3 × 1010 cm s−1 , h 4.135 × 10−15 eVs, η0 0.95, E g 0.12 eV, and T E 300 K.
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observed from Fig. 2(d) that the efficiency η is not a monotonic function of C and V . There exist the optimal values C opt and V opt of C and V for a given E g at which the efficiency attains its maximum ηm;E g . It is seen from Eqs. (2) and (3) that, for a given C, the optimal value V opt of V at the maximum power output density is equal to that of V at the maximum efficiency, because the denominator in Eq. (3) does not include the voltage output V . The result derived here is different from that obtained in Ref. [3] because the denominator in Eq. (4) of Ref. [3] includes the voltage output V . Using the data in Figs. 2(a) and 2(b) and the optimal values C opt and V opt , we can calculate the optimal values T C;opt and i opt of T C and i at the maximum efficiency ηm;E g for a given E g . Obviously, ηm;E g is closely dependent on the bandgap E g . We now discuss the choice problem of E g . Using the above equations, we can calculate ηm;E g , C opt , V opt , i opt , and T C;opt for different values of E g , as shown in Figs. 3(a)–3(c), respectively. Figure 3(a) shows that when the bandgap E g is chosen to be equal to E g;η ≈ 0.09 eV, ηm;E g attains its maximum ηmax . The bandgap of the present semiconductor materials is larger than 0.09 eV, whereas the new semiconductor materials in the range of E g < 0.17 eV are still under development. At present, InSb may be one of the best semiconductor materials used to make the TRC because of its E g 0.17 eV. Figure 3 shows that V opt decreases with the increase of E g , while C opt , i opt , and T C;opt first decrease and then increase with the increase of E g . This means that the larger the bandgap E g is, the larger the value of jV opt j. Equation (9) indicates that when E g increases, the number of up-bandgap photons decreases. The increase of jV opt j and the decrease of up-bandgap photons simultaneously lead to the decrease of q 3 , and consequently, the TRC can obtain a relatively large efficiency under low concentration factors, which is contrary to the TPVCs that need a high concentration ratio. More important, maintaining the space between the emitter and PV cell at vacuum state is difficult, especially for the nano-gap near-field TPVCs. Thus, the manufacture of NITSCs is easier than that of TPVCs because NITSCs do not include a nano-level vacuum gap. According to Fig. 3 and the value of E g;η , we can calculate the optimal values i η , C η , V η , and T C;η of the optimized parameters C opt , V opt , i opt , and T C;opt at the maximum efficiency ηmax.
Fig. 3. Curves of several parameters such as (a) ηm;E g and C opt , (b) V opt and iopt , and (c) T C;opt varying with E g , where C η , V η , i η , and T C;η are the optimal values of the optimized parameters C opt , V opt , i opt , and T C;opt at the maximum efficiency ηmax. The values of other parameters are the same as those used in Fig. 2.
Letter Table 1. Optimal Values of Several Key Parameters at the Maximum Efficiency for Three Different Values of η0 η0
ηmax
jV η j
Cη
E g;η
iη
T C;η
Pη
0.95 0.90 0.85
0.201 0.191 0.180
0.13 0.13 0.13
4.08 4.34 4.60
0.0914 0.0910 0.0910
0.633 0.638 0.639
727 728 728
0.0829 0.0829 0.0829
The parametric characteristics of a NITSC at the maximum efficiency are also dependent on the optical efficiency η0. For different given values of η0 , the optimal values of several key parameters of the NITSC at the maximum efficiency are listed in Table 1. The data in Table 1 show that C η decreases with the increase of η0 , while E g;η , i η , jV η j, and T C;η change negligibly with the increase of η0 . The increase of η0 can effectively enhance the efficiency of the TRC. Thus, the improvement of the optical efficiency of the sunlight concentrator is key to the practical design of NITSCs. It is important that the data in Table 1 may provide some guidance for engineers to design and operate NITSCs. In summary, the parametric characteristics of the NITSC at the maximum efficiency for given values of the optical efficiency η0 have been investigated. The main advantages of the NITSC are that, operated under low concentrated sunlight, it can obtain a relatively large efficiency and that the manufacturing difficulties of the NITSC are less than those of the TPVC. The results obtained here can provide theoretical guidance for engineers to choose some materials with the optimal bandgap E g;η to make the TRC, reasonably design the concentrator and absorber, effectively control the absorber and TRC being operated at the optimal temperature T C;η , adjust the voltage output and current density of the TRC that is located in the optimal values V η and i η , and ensure that the NITSC achieves maximum efficiency. Funding. National Natural Science Foundation of China (NSFC) (11405142); Fundamental Research Fund for the Central Universities (20720170017). REFERENCES 1. I. Celanovic, F. O’Sullivan, M. Ilak, J. Kassakian, and D. Perreault, Opt. Lett. 29, 863 (2004). 2. C. Ungaro, S. K. Gray, and M. C. Gupta, Opt. Lett. 39, 5259 (2014). 3. R. Strandberg, J. Appl. Phys. 117, 055105 (2015). 4. R. Strandberg, J. Appl. Phys. 118, 215102 (2015). 5. P. Santhanam and S. Fan, Phys. Rev. B 93, 161410 (2016). 6. W. C. Hsu, J. K. Tong, B. Liao, Y. Huang, S. V. Boriskina, and G. Chen, Sci. Rep. 6, 34837 (2016). 7. B. Wang, C. Lin, K. H. Teo, and Z. Zhang, J. Quant. Spectrosc. Radiat. Transfer 196, 10 (2017). 8. J. J. Fernández, IEEE Trans. Electron Devices 64, 250 (2017). 9. Y. Nam, Y. X. Yeng, A. Lenert, P. Bermel, I. Celanovic, M. Soljačić, and E. N. Wang, Sol. Energy Mater. Sol. Cells 122, 287 (2014). 10. M. Zenker, A. Heinzel, G. Stollwerck, J. Ferber, and J. Luther, IEEE Trans. Electron Devices 48, 367 (2001). 11. N. P. Harder and P. Wurfel, Semicond. Sci. Technol. 18, S151 (2003). 12. P. Bermel, M. Ghebrebrhan, W. Chan, Y.-X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljacic, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, Opt. Express 18, A314 (2010). 13. R. Strandberg, Appl. Phys. Lett. 106, 033902 (2015). 14. K. Emery, “Reference solar spectral irradiance,” Technical Report, ASTM, 2000, http://rredc.nrel.gov/solar/spectra/am1.5/. 15. T. Liao, X. Chen, Z. Yang, B. Lin, and J. Chen, Energy Convers. Manage. 126, 205 (2016). 16. M. Elzouka and S. Ndao, Sol. Energy 141, 323 (2017).
