Systematic Analyses of the Ideal Selective Spectrum and the Practical Design Strategies for the Solar Thermophotovoltaic System

Abstract

Solar thermophotovoltaic (STPV) systems can break the Shockley–Queisser (SQ) limit through selective absorbers and emitters, whose ideal emissivity is crucial as a design target. In this paper, we systematically analyze the ideal selective spectrum and solve the conflict between energy and efficiency in photothermal conversion efficiency by modifying the corresponding equation. The ideal emissivity of the absorber is one in [$E_{\mathrm{c}}$, ∞] and zero out of this range. For actual design, if spectra deviation cannot be avoided, $E_{\mathrm{c}}$ is preferable to redshifting. The ideal emissivity of the emitter is one in $[E_{\mathrm{g}},E_{\max }]$ and the decrease in $E_{\min }$ should be suppressed relative to $E_{\max }$. Besides, the optimal bandwidth of the emitter is about 0.08 eV for different photovoltaic cells and working conditions, which gives a rough and valuable guide in practical design. The analytical progress and design strategies will give a reference and direction for the future design of STPV.

1. Introduction

To reduce carbon emissions and achieve carbon neutrality, it is urgent to develop renewable energy sources [1]. Solar energy has been widely used in photothermal and photovoltaic conversion due to its universality, non-polluting and abundant reserves [2,3,4,5,6]. Solar photovoltaic (PV) systems are an effective way to convert solar energy into electricity and have made tremendous progress in recent years [7,8,9]. However, the solar spectrum and the response of PV cells do not match. Then its efficiency is subject to the Shockley–Queisser (SQ) limit, primarily due to inherent loss mechanisms: first, photons with energy below the bandgap of the PV cell lack the capacity to generate electron-hole pairs; second, photons with energy significantly higher than the bandgap, while capable of generating electron-hole pairs, dissipate their excess energy as heat [10]. As a result, the theoretical maximum conversion efficiency of an ideal single-junction silicon cell cannot exceed 33% under one sun of normal incident with a power of 100 mW$·{\mathrm{cm}}^{-2}$ [11]. Therefore, multijunction PV cells [12,13,14] or solar thermophotovoltaic (STPV) systems [15,16] are adopted to break this efficiency limit.

An STPV system generally consists of a concentrator, a broadband absorber, a narrowband emitter, a low bandgap PV cell, and a thermal management [17,18,19]. Presently, to improve efficiency, a great deal of research on STPV focuses on designing metasurfaces or metamaterials. Abbas et al. [20] designed an STPV system consisting of Cr nano-cylindrical arrays. The absorber achieves high absorption of solar radiation in the range of 0.83–3.1 eV, while the emitter achieves greater than 95% emissivity at the bandgap of the InGaAsSb cell. Rana et al. proposed selective absorbers and emitters consisting of Ta nanocrosses [21] and Cr nanocrosses [22], which improve the efficiencies of the PV cell to 41.8% and 43.2%, respectively. For the InGaAsSb PV cell, Cui et al. [23] designed a broadband absorber based on Mo nanosheets and a selective narrowband Mo-based grating emitter with a bandwidth of just about 30 nm. Mehrabi et al. [24] designed a metamaterial absorber consisting of alternating TiN and ${\mathrm{TiO}}_2$ nanocubes with an average absorption of 96% in the range of 200–3000 nm. Although these selective STPV devices have broken the SQ limit, their targeted selective spectra of STPV absorber and emitter are not well-consistent because related quantitative analysis is lacking.

Besides these designs of absorbers or emitters, there are some theoretical analyses. Wang et al. [10] quantitatively analyzed the effect of sidewall emission loss, non-ideal view factor, and emitter–absorber area ratios on the performance of the STPV system. Photon recycling technology [25,26,27], such as radiation shields and a cavity reflector, is introduced to reduce the radiative heat loss, and the corresponding balance is analyzed. However, ideal selective absorber and emitter spectra in the works above are given only based on simple quantitative analysis: (1) For the absorber, high absorptivity above the cutoff energy $E_{\mathrm{c}}$ can maximize the absorbed solar radiation, and low emissivity below $E_{\mathrm{c}}$ can minimize the thermal loss. But the cutoff energy $E_{\mathrm{c}}$ is not well consistent, which is simply thought as 1.24 eV [21,22], 0.62 eV [28,29], or 0.31eV [30,31], respectively. (2) Besides, in the optimization of photothermal conversion efficiency, there is a conflict between energy and efficiency. (3) For the emitter, it should have a narrow band high emissivity above the cell bandgap $E_{\mathrm{g}}$ [32,33,34,35] with the bandwidth as narrow as possible [36,37]. It is worth noting that, if the bandwidth of the emitter is as narrow as possible [36,37], the energy effectively utilized is limited to zero. This is a conflict that was ignored but is crucial. (4) Furthermore, in the actual design and preparation of absorbers and emitters, it is difficult to meet such accurate design requirements. Therefore, a study of the dependence of spectrum deviation on system efficiency and giving deviation strategies for practical design is extremely important and urgent.

In this work, we obtain the ideal selective spectra of the absorber and emitter systematically and theoretically by simulating the energy balance of the STPV system and then optimizing the energy transferred to the PV cell $P_{\mathrm{emit}}$ and the efficiency of the PV cell ${\eta }_{\mathrm{PVC}}$. To solve the conflict in the previous definition of the photothermal conversion efficiency ${\eta }_{\mathrm{int}\_\mathrm{past}}$, we propose a modified efficiency calculation formula (See Equation (12)). Through the analysis, (1) The ideal absorptivity of the absorber is one in [$E_{\mathrm{c}}$, ∞] and zero out of this range, and the ideal absorptivity of the emitter is one in [$E_{\mathrm{g}}$, $E_{\max }$] and zero out of this range. (2) The optimal emitter bandwidth for different PV cells and working environments is approximately 0.08 eV, contradicting prior design strategies advocating an “extremely narrow band” emissivity [10,19,36]. This value (0.08 eV) can serve as an engineering design guideline value for simplicity. (3) If spectra deviation cannot be avoided: For absorber, $E_{\mathrm{c}}$ preferentially redshifts; For emitter, suppressing the decrease in $E_{\min }$ is more critical than deviations in $E_{\max }$. These findings and the analytical progress provide a foundational reference for subsequent STPV system designs.

2. Overview of the STPV System

Figure 1 shows the schematic diagram of the STPV system, which consists of an optical concentrator, an absorber, an emitter, a low-bandgap PV cell, and thermal management. The concentrator focuses the incident sunlight onto the absorber with a concentration of $N_{\mathrm{s}}$. Then, the maximum incident angle increases with the increase in solar concentration $N_{\mathrm{s}}$, which is labeled as ${\theta }_{\mathrm{c}}$ and shown in Figure 1. Different from conventional PV systems, the core components of the STPV system are the absorber and the emitter, which form an intermediate structure and are absent in conventional PV systems.

The energy flow of the whole system comprises two parts. First, the absorber absorbs the solar radiation concentrated by the concentrator and converts it into thermal energy. Then the absorber and emitter are heated, and the emitter transfers the thermal energy to the PV cell through photon re-emission. The efficiency of an STPV system can be calculated by [21]:

$${\eta }_{\mathrm{STPV}}={\eta }_{\mathrm{int}}·{\eta }_{\mathrm{PVC}}$$

(1)

where ${\eta }_{\mathrm{int}}$ is the efficiency of the intermediate and is defined as the ratio of outgoing power from the emitter with respect to the incoming solar power. ${\eta }_{\mathrm{PVC}}$ is the efficiency of the solar cell and is defined as the ratio of electrical power extracted from the cell to the incident power radiated by the emitter.

3. Systematic Analyses of the Intermediate Structure

3.1. Ideal Spectrum of the Absorber

$P_{\mathrm{in}}$ is the incident solar radiative power from the sun at a temperature $T_{\mathrm{sol}}$ (taken as 6000 K), which can be calculated as

$$P_{\mathrm{in}}={\int }_0^{2\pi }d\phi {\int }_0^{{\theta }_{\mathrm{c}}}sin\left(\theta \right)cos\left(\theta \right)d\theta {\int }_0^{\infty }{I}_{\mathrm{BB}}\left(E,T_{\mathrm{sol}}\right)dE$$

(2)

here, ${\theta }_{\mathrm{c}}$ is the solar incidence angle after the optical concentrator, which can be calculated by ${sin}^{-1}\sqrt{N_{\mathrm{s}}{\Omega }_{\mathrm{s}}/\pi }$, where $N_{\mathrm{s}}$ is the solar concentration and ${\Omega }_{\mathrm{s}}$ is the solar stereo angle at non-concentration (taken as $68.5\mu \mathrm{Sr}$) [38]. $I_{\mathrm{BB}}$ is the spectral radiance of a blackbody and can be calculated as

$$I_{\mathrm{BB}}(E,T)=2E^5/h^4{c}^3/\left(e^{E/k_{\mathrm{B}}T}-1\right)$$

(3)

where h is Planck’s constant, c is the speed of light, $k_{\mathrm{B}}$ is Boltzmann’s constant, and T is the temperature of a blackbody.

According to Kirchhoff’s law of thermal radiation, the absorptivity of a reciprocal object is equal to its thermal emissivity [39]. The absorber absorbs energy from the sun and the absorbed power $P_{\mathrm{aa}}$ is

$$P_{\mathrm{aa}}={\int }_0^{2\pi }d\phi {\int }_0^{{\theta }_{\mathrm{c}}}sin\left(\theta \right)cos\left(\theta \right)d\theta {\int }_0^{\infty }{I}_{\mathrm{BB}}\left(E,T_{\mathrm{sol}}\right){\epsilon }_{\mathrm{abs}}(E,\theta ,\phi )dE$$

(4)

where ${\epsilon }_{\mathrm{abs}}(E,\theta ,\phi )$ is the angular spectral emissivity of the absorber at energy E. The absorbed energy $P_{\mathrm{aa}}$ is converted into thermal energy. The temperature of the absorber as well as the emitter increases and they radiate thermal energy to the outside simultaneously, containing the effective thermal radiative power of the emitter $P_{\mathrm{emit}}$ and thermal radiation loss from the top absorber $P_{\mathrm{ae}}$, the top of the emitter $P_{\mathrm{top}}$, and the sidewalls $P_{\mathrm{side}}$. $P_{\mathrm{ae}}$ and $P_{\mathrm{emit}}$ can be calculated by

$$P_{\mathrm{ae}}={\int }_0^{2\pi }d\phi {\int }_0^{{\displaystyle \frac{\pi }{2}}}sin\left(\theta \right)cos\left(\theta \right)d\theta {\int }_0^{\infty }{I}_{\mathrm{BB}}\left(E,T_{\mathrm{abs}}\right){\epsilon }_{\mathrm{abs}}(E,\theta ,\phi )dE$$

(5)

$$P_{\mathrm{emit}}={\int }_0^{2\pi }d\phi {\int }_0^{{\displaystyle \frac{\pi }{2}}}sin\left(\theta \right)cos\left(\theta \right)d\theta {\int }_0^{\infty }{I}_{\mathrm{BB}}\left(E,T_{\mathrm{emit}}\right){\epsilon }_{\mathrm{emit}}(E,\theta ,\phi )dE$$

(6)

here, ${\epsilon }_{\mathrm{emit}}(E,\theta ,\phi )$ is the angular spectral emissivity of the emitter. $T_{\mathrm{abs}}$ and $T_{\mathrm{emit}}$ are the temperatures of the absorber and emitter, respectively. $P_{\mathrm{top}}$ is equal to $P_{\mathrm{emit}}$. The temperature of the intermediate structure $T_{\mathrm{emit}}$ is much larger than the ambient temperature $T_{\mathrm{amb}}$ and the working temperature of the PV. Therefore, the heat absorption from the ambient and radiation from the solar cell to the emitter is neglected. Under thermal equilibrium conditions ($T_{\mathrm{abs}}$ = $T_{\mathrm{emit}}$), the input energy is equal to the output energy. Therefore, the energy balance at steady state for the intermediate structure is

$$A_{\mathrm{abs}}{P}_{\mathrm{aa}}=A_{\mathrm{abs}}{P}_{\mathrm{ae}}+A_{\mathrm{emit}}{P}_{\mathrm{emit}}+A_{\mathrm{top},\mathrm{emit}}{P}_{\mathrm{top}}+\left(A_{\mathrm{side},\mathrm{abs}}+A_{\mathrm{side},\mathrm{emit}}\right)P_{\mathrm{side}}$$

(7)

here, $A_{\mathrm{abs}}$ and $A_{\mathrm{emit}}$ are the top surface areas of the absorber and emitter, respectively. $A_{\mathrm{top},\mathrm{emit}}$ is the uncovered area of the top surface of the emitter, i.e., $A_{\mathrm{top},\mathrm{emit}}$ = $A_{\mathrm{emit}}-A_{\mathrm{abs}}$. $A_{\mathrm{side},\mathrm{abs}}$ and $A_{\mathrm{side},\mathrm{emit}}$ is the sidewall surface area for the absorber and emitter, respectively. Since the area of the absorber and emitter sidewalls is much smaller than the horizontal plane, the $A_{\mathrm{side},\mathrm{abs}}$ and $A_{\mathrm{side},\mathrm{emit}}$ can be neglected and Equation (7) can be simplified to:

$$A_{\mathrm{abs}}{P}_{\mathrm{aa}}=A_{\mathrm{abs}}{P}_{\mathrm{ae}}+\left(A_{\mathrm{emit}}+A_{\mathrm{top},\mathrm{emit}}\right)P_{\mathrm{emit}}$$

(8)

It is worth noting that $A_{\mathrm{top},\mathrm{emit}}$. $P_{\mathrm{emit}}$ is not considered in Equation (1) of Eden Rephaeli’s work [16]. But this cannot be ignored because the temperature of the intermediate structure $T_{\mathrm{emit}}$ is very high, and the area difference $A_{\mathrm{top},\mathrm{emit}}=A_{\mathrm{emit}}-A_{\mathrm{abs}}$ is not a small amount. Assuming $\beta =A_{\mathrm{emit}}/A_{\mathrm{abs}}$, Equation (8) can be deformed into

$$P_{\mathrm{emit}}={\displaystyle \frac{1}{2\beta -1}}\left(P_{\mathrm{aa}}-P_{\mathrm{ae}}\right)$$

(9)

which can be deformed into

$$P_{\mathrm{emit}}={\int }_0^{\infty }{P}_{\mathrm{emit}\_\mathrm{E}}dE={\displaystyle \frac{1}{2\beta -1}}{\int }_0^{\infty }\left(P_{\mathrm{aa}\_\mathrm{E}}-P_{\mathrm{ae}\_\mathrm{E}}\right)dE$$

(10)

where $P_{\mathrm{aa}\_\mathrm{E}}=\pi ·{sin}^2{\theta }_c·{\epsilon }_{\mathrm{abs}}\left(E\right)I_{\mathrm{BB}}(E,T_{\mathrm{sol}})$ and $P_{\mathrm{ae}\_\mathrm{E}}=\pi ·{\epsilon }_{\mathrm{abs}}\left(E\right)I_{BB}(E,T_{\mathrm{abs}})$ are the spectral absorbed and emitted power from the top surface of the absorber, respectively. At a fixed $\beta$, $P_{\mathrm{emit}}$ depends on the absorber properties, including the emissivity ${\epsilon }_{\mathrm{abs}}\left(E\right)$ and the equilibrium temperature $T_{\mathrm{abs}}$. Then it can be discussed somewhat independently, with ${\eta }_{\mathrm{int}}$ controlled by the property of the absorber, and ${\eta }_{\mathrm{PVC}}$ controlled by the property of the emitter. Therefore, we discuss ${\eta }_{\mathrm{int}}$ and ${\eta }_{\mathrm{PVC}}$, respectively.

Assuming the absorber is a blackbody, i.e., ${\epsilon }_{\mathrm{abs}}$ = 1. Figure 2a shows the spectral absorbed power $P_{\mathrm{aa}\_\mathrm{E}}$ at different solar concentrations $N_{\mathrm{s}}$ ranging from 1 to 40,000. As $N_{\mathrm{s}}$ increases, $P_{\mathrm{aa}\_\mathrm{E}}$ increases in the whole spectrum range. The energy is mainly concentrated after 0.5 eV and has large values after 1 eV. Figure 2b shows the spectral radiated power of the absorber $P_{\mathrm{ae}\_\mathrm{E}}$ increases as the equilibrium temperature $T_{\mathrm{abs}}$ increases with peak blueshifts. Figure 2c shows $P_{\mathrm{aa}\_\mathrm{E}}$ and $P_{\mathrm{ae}\_\mathrm{E}}$ at $T_{\mathrm{abs}}$ = 1673 K and $N_{\mathrm{s}}$ = 1000. $P_{\mathrm{ae}\_\mathrm{E}}$ is larger than $P_{\mathrm{aa}\_\mathrm{E}}$ when E < 0.7 eV and smaller than $P_{\mathrm{aa}\_\mathrm{E}}$ when E > 0.7 eV. They have an intersection at 0.7 eV. In Equation (10), if $\beta$ is a fixed value, maximizing $P_{\mathrm{emit}}$ is equivalent to maximizing the integral of ($P_{\mathrm{aa}\_\mathrm{E}}$ − $P_{\mathrm{ae}\_\mathrm{E}}$). Then, Figure 2d shows the difference $P_{\mathrm{aa}\_\mathrm{E}}-P_{\mathrm{ae}\_\mathrm{E}}$ in a blue line, which is bigger than 0 when E > 0.7 eV, smaller than 0 when E < 0.7 eV and is 0 at 0.7 eV. The cutoff energy is denoted as $E_{\mathrm{c}}$. Therefore, to maximize the integral of ($P_{\mathrm{aa}\_\mathrm{E}}$ − $P_{\mathrm{ae}\_\mathrm{E}}$), the ideal absorptivity of the absorber is zero before 0.7 eV and one after 0.7 eV, which is shown in the magenta line in Figure 2d. The corresponding wavelength is 1.77 µm.

In the same way, we obtain the optimal cutoff energy at different equilibrium temperatures $T_{\mathrm{abs}}$ and solar concentrations $N_{\mathrm{s}}$, which are shown in Figure 3a. The optimal cutoff energy decreases as the solar concentration $N_{\mathrm{s}}$ increases, and increases as the equilibrium temperature $T_{\mathrm{abs}}$ increases.

It is worth noting that the photothermal conversion efficiency in prior studies, denoted as ${\eta }_{\mathrm{int}\_\mathrm{past}}$, is conventionally defined as follows [16,20,21,22]:

$${\eta }_{\mathrm{int}\_\mathrm{past}}={\displaystyle \frac{A_{\mathrm{emit}}{P}_{\mathrm{emit}}}{A_{\mathrm{abs}}{P}_{\mathrm{aa}}}}=\beta {\displaystyle \frac{P_{\mathrm{emit}}}{P_{\mathrm{aa}}}}$$

(11)

In this equation, the denominator $P_{\mathrm{aa}}$ (absorbed power from the sun) becomes zero when the absorptivity ${\epsilon }_{\mathrm{abs}}$ of the absorber is zero. In this condition, ${\eta }_{\mathrm{int}\_\mathrm{past}}$ theoretically equals unity, which seems ideal. However, since $P_{\mathrm{aa}}$ = 0, no incident solar energy is utilized in this scenario. This leads to a fundamental conflict between optimizing efficiency ${\eta }_{\mathrm{int}\_\mathrm{past}}$ and maximizing the usable energy $P_{\mathrm{aa}}$. A similar strange conclusion was derived from Equation (7) in Eden Rephaeli’s work [16] that the intermediate efficiency is 1 when the equilibrium temperature is zero. To address this paradox, the photothermal conversion efficiency is modified as

$${\eta }_{\mathrm{int}}={\displaystyle \frac{A_{\mathrm{emit}}{P}_{\mathrm{emit}}}{A_{\mathrm{abs}}{P}_{\mathrm{in}}}}=\beta {\displaystyle \frac{P_{\mathrm{emit}}}{P_{\mathrm{in}}}}={\displaystyle \frac{\beta }{2\beta -1}}\left({\displaystyle \frac{{\int }_0^{\infty }\left(P_{\mathrm{aa}\_\mathrm{E}}-P_{\mathrm{ae}\_\mathrm{E}}\right)dE}{P_{\mathrm{in}}}}\right)$$

(12)

$P_{\mathrm{in}}$ in the denominator of Equation (12) is the total incident energy, which is constant and has nothing to do with the selection spectrum of the intermediate structure. Therefore, ${\eta }_{\mathrm{int}}$ is linearly dependent on the integral of $(P_{\mathrm{aa}\_\mathrm{E}}-P_{\mathrm{ae}\_\mathrm{E}})$, which is the same as $P_{\mathrm{emit}}$. The larger the integral of $(P_{\mathrm{aa}\_\mathrm{E}}-P_{\mathrm{ae}\_\mathrm{E}})$ is, the larger ${\eta }_{\mathrm{int}}$ and $P_{\mathrm{emit}}$ are. Figure 3b shows the optimal photothermal conversion efficiency ${\eta }_{\mathrm{int}}$ of the intermediate structure calculated by the optimal cutoff wavelength acquired from Figure 3a when $\beta =1$. ${\eta }_{\mathrm{int}}$ increases as the solar concentration increases and the equilibrium temperature decreases.

3.2. Effect of Deviation from the Ideal Spectrum

Figure 4a shows $P_{\mathrm{emit}}$, ${\eta }_{\mathrm{int}}$ and ${\eta }_{\mathrm{int}\_\mathrm{past}}$ at different cutoff energy $E_{\mathrm{c}}$ when $T_{\mathrm{abs}}=1673$ K and $N_{\mathrm{s}}=1000$. $P_{\mathrm{emit}}$ and ${\eta }_{\mathrm{int}}$ both increase first and then decrease as $E_{\mathrm{c}}$ increases with a maximum value at 0.7 eV, while ${\eta }_{\mathrm{int}\_\mathrm{past}}$ decreases slowly. Notably, ${\eta }_{\mathrm{int}\_\mathrm{past}}$ reaches its maximum when the bandwidth of ${\epsilon }_{\mathrm{abs}}$ is zero. However, in this case, both $P_{\mathrm{aa}}$ and $P_{\mathrm{emit}}$ approach zero, implying a complete waste of the incident solar energy. This inconsistency confirms that the definition of ${\eta }_{\mathrm{int}\_\mathrm{past}}$ is erroneous and is suitable after modification as Equation (12).

Besides, in the actual design of absorbers, it is difficult to acquire an ideal selective absorptivity. Therefore, we study the spectrum deviation on system efficiency and try to find useful strategies. Figure 4b shows the relative change in $P_{\mathrm{emit}}$, ${\eta }_{\mathrm{int}}$ and ${\eta }_{\mathrm{int}\_\mathrm{past}}$ as $E_{\mathrm{c}}$ deviates from the optimized value of 0.7 eV. The changing trend of $P_{\mathrm{emit}}$ and modified ${\eta }_{\mathrm{int}}$ is the same. When $\delta E_{\mathrm{c}}=(E_{\mathrm{c}}-0.7)$/0.7 × 100% increases by 25%, $P_{\mathrm{emit}}$ and ${\eta }_{\mathrm{int}}$ decrease by 2% relatively. To absorb more energy, the redshift of the cutoff wavelength $E_{\mathrm{c}}$ is more suitable, although the deviation of the selectivity spectrum of the absorber from the ideal spectrum does not affect the absorber very much.

4. Systematical Analysis of ${\mathbf{\eta }}_{\mathrm{PVC}}$ and the Ideal Spectrum of the Emitter

4.1. Systematical Analysis of the Efficiency of the PV Cell ${\eta }_{\mathrm{PVC}}$

The emitted energy of the emitter is absorbed by the solar cell below. ${\epsilon }_{\mathrm{cell}}(E,\theta ,\phi )$ is the angular spectral emissivity of the solar cell. For each photon above the bandgap, the difference between the photon energy and the output energy at somewhat below the bandgap energy is dissipated as heat. Then, the effective efficiency should be multiplied by ${\epsilon }_{\mathrm{cell}}(E,\theta ,\phi )$$E/E_g$. Therefore, the efficiency of the photoelectric conversion process $U_{\mathrm{eff}}$ can be calculated by the ratio of power efficiently utilized by PV cells $P_{\mathrm{emit},\mathrm{E}\ge {\mathrm{E}}_{\mathrm{g}}}$ and the total power radiated from the emitter $P_{\mathrm{emit}}$ [11,16]:

$$\begin{array}{cc}\hfill U_{\mathrm{eff}}& ={\displaystyle \frac{P_{\mathrm{emit},\mathrm{E}\ge E_{\mathrm{g}}}}{P_{\mathrm{emit}}}}={\displaystyle \frac{{\displaystyle {\int }_{E_{\mathrm{g}}}^{\infty }{P}_{\mathrm{emit}\_\mathrm{E},\mathrm{E}\ge E_{\mathrm{g}}}dE}}{{\displaystyle {\int }_0^{\infty }{P}_{\mathrm{emit}\_\mathrm{E}}dE}}}\hfill \\ & ={\displaystyle \frac{{\displaystyle {\int }_0^{2\pi }d\phi {\int }_0^{\frac{\pi }{2}}sin\theta cos\theta d\theta {\int }_{E_{\mathrm{g}}}^{\infty }{I}_{BB}(E,T_{\mathrm{emit}}){\epsilon }_{\mathrm{emit}}(E,\theta ,\phi ){\epsilon }_{\mathrm{cell}}(E,\theta ,\phi )\frac{E_{\mathrm{g}}}{E}dE}}{{\displaystyle {\int }_0^{2\pi }d\phi {\int }_0^{\frac{\pi }{2}}sin\theta cos\theta d\theta {\int }_0^{\infty }{I}_{BB}(E,T_{\mathrm{emit}}){\epsilon }_{\mathrm{emit}}(E,\theta ,\phi )dE}}}\hfill \end{array}$$

(13)

In this Equation, the bigger ${\epsilon }_{\mathrm{cell}}(E,\theta ,\phi )$, the better $U_{\mathrm{eff}}$, and is assumed as 1 for simplicity. For a more concrete description, we use an InGaAsSb PV cell (with a bandgap energy of 0.54 eV) as an example in this paper. Assuming the emitter is a blackbody, i.e., ${\epsilon }_{\mathrm{emit}}=1$, Figure 5a shows that $P_{\mathrm{emit}\_\mathrm{E},\mathrm{E}\ge {\mathrm{E}}_{\mathrm{g}}}$ and $P_{\mathrm{emit}\_\mathrm{E}}$ at two temperatures of 1273 K and 1873 K, respectively. It shows that no matter what $T_{\mathrm{emit}}$ is, the intersection of $P_{\mathrm{emit}\_\mathrm{E},E\ge E_g}$ and $P_{\mathrm{emit}\_\mathrm{E}}$ is located at 0.54 eV. The emissivity of the emitter ${\epsilon }_{\mathrm{emit}}$ is assumed as one in $[E_{\min },\infty )$ and is zero before $E_{\min }$, and the corresponding $U_{\mathrm{eff}}$ at different temperatures is shown in Figure 5b. $U_{\mathrm{eff}}$ increases quickly and then decreases as $E_{\min }$ increases from 0.3 to 3 eV. The peaks of $U_{\mathrm{eff}}$ are consistently at 0.54 eV and do not vary with the temperature, which proves the optimal lower limit of the selective emissivity of the emitter is $E_{\min }=E_{\mathrm{g}}=0.54$ eV. When $E>E_{\mathrm{g}}$, $U_{\mathrm{eff}}$ mainly depends on $E_{\mathrm{g}}$/E, so higher energy photons result in a decrease in $U_{\mathrm{eff}}$.

To find the optimal upper limit $E_{\max }$, we assume the emissivity of the emitter ${\epsilon }_{\mathrm{emit}}$ is one in [$E_{\mathrm{g}}$, $E_{\max }$] and is zero out of this range. Figure 6a shows the correspondent $U_{\mathrm{eff}}$ when $E_{\max }$ increases from $E_{\mathrm{g}}$ to 3 eV. At different temperatures, $U_{\mathrm{eff}}$ decreases quickly and then is stable at a constant value, which hints that the optimal upper limit of the selective emissivity of the emitter may also be $E_{\mathrm{g}}$ = 0.54 eV. That means the emissivity of the emitter should be as narrow as possible at the bandgap $E_{\mathrm{g}}$. However, this will result in power efficiently utilized by PV cells $P_{\mathrm{emit},\mathrm{E}\ge {\mathrm{E}}_{\mathrm{g}}}$ approaches to zero (as shown in Figure 6b), which means a complete waste of the incident solar energy. To find the equilibrium between $U_{\mathrm{eff}}$ and $P_{\mathrm{emit}\_\mathrm{E},\mathrm{E}\ge {\mathrm{E}}_{\mathrm{g}}}$, we discuss the impact of other factors next, such as the open-circuit voltage loss and the impedance matching loss.

As the temperature increases, carrier recombination accelerates, the bandgap of the PV cell narrows and $V_{\mathrm{op}}$ decreases, resulting in open-circuit voltage loss $V_{\mathrm{eff}}$ [40,41]:

$$\begin{array}{cc}\hfill V_{\mathrm{eff}}& ={\displaystyle \frac{V_{\mathrm{op}}}{V_{\mathrm{g}}}}={\displaystyle \frac{V_{\mathrm{c}}}{V_{\mathrm{g}}}}ln\left({\displaystyle \frac{f{P}_{\mathrm{emit},\mathrm{E}\ge {\mathrm{E}}_{\mathrm{g}}}}{P_{\mathrm{c},\mathrm{E}\ge {\mathrm{E}}_{\mathrm{g}}}}}\right)\hfill \\ \hfill & ={\displaystyle \frac{V_{\mathrm{c}}}{V_{\mathrm{g}}}}ln\left[f{\displaystyle \frac{{\int }_0^{2\pi }d\phi {\int }_0^{\frac{\pi }{2}}sin\left(\theta \right)cos\left(\theta \right)d\theta {\int }_{E_{\mathrm{g}}}^{\infty }{\epsilon }_{\mathrm{emit}}(E,\theta ,\phi )I_{\mathrm{BB}}\left(E,T_{\mathrm{emit}}\right)\frac{E_{\mathrm{g}}}{E}dE}{{\int }_0^{2\pi }d\phi {\int }_0^{\frac{\pi }{2}}sin\left(\theta \right)cos\left(\theta \right)d\theta {\int }_{E_{\mathrm{g}}}^{\infty }{\epsilon }_{\mathrm{cell}}(E,\theta ,\phi )I_{\mathrm{BB}}\left(E,T_{\mathrm{c}}\right)\frac{E_{\mathrm{g}}}{E}dE}}\right]\hfill \end{array}$$

(14)

where $T_{\mathrm{c}}$ is the operating temperature of the PV cell (taken as 300 K) and is much smaller than $T_{\mathrm{emit}}$. f is a non-ideal factor related to the planar geometry of the PV cell and emitter (taken as 0.5). $V_{\mathrm{c}}$ is the thermal voltage of the PV cell, and $V_{\mathrm{c}}=k{T}_{\mathrm{c}}/q$ (q is the electron charge). During the transport process of photogenerated carriers, the open-circuit voltage $V_{\mathrm{op}}$ is the maximum voltage that can be output from the PV cell. When the temperature of the PV cell is 0 K, $V_{\mathrm{op}}=V_{\mathrm{g}}=E_{\mathrm{g}}/q$ [4], where $V_{\mathrm{g}}$ is the bandgap voltage of the PV cell. Figure 7a shows the dependence of $V_{\mathrm{eff}}$ on $E_{\max }$ at different $T_{\mathrm{emit}}$. When $T_{\mathrm{emit}}$ increases, $V_{\mathrm{eff}}$ overall increases. When $T_{\mathrm{emit}}$ is fixed, $V_{\mathrm{eff}}$ monotonically increases and then is stable at a constant value as the upper limit $E_{\max }$ increases from $E_{\mathrm{g}}$ to 3 eV.

When the PV cell is connected to the load, if the output impedance of the battery does not match the input impedance of the load, impedance matching loss will occur, which can be described as the ratio of the maximum output power to the nominal power [11]:

$$Im\left(V_{\mathrm{op}}\right)={\displaystyle \frac{I_{\max }{V}_{max}}{I_{\mathrm{sh}}{V}_{\mathrm{op}}}}={\displaystyle \frac{Z_{\mathrm{m}}^2}{\left(1+Z_{\mathrm{m}}-e^{-Z_{\mathrm{m}}}\right)\left(Z_{\mathrm{m}}+ln\left(1+Z_{\mathrm{m}}\right)\right)}}$$

(15)

where $Z_{\mathrm{m}}$ is related to $Z_{\mathrm{op}}=Z_{\mathrm{m}}+ln(1+Z_{\mathrm{m}})$, and $Z_{\mathrm{op}}=V_{\mathrm{op}}/V_{\mathrm{c}}$. It reflects the effect of charge transfer and power collection within the PV cell. Figure 7b shows $Im\left(V_{\mathrm{op}}\right)$ overall increases when $T_{\mathrm{emit}}$ increases. When $T_{\mathrm{emit}}$ is fixed, $Im\left(V_{\mathrm{op}}\right)$ monotonically increases as the upper energy limit $E_{\max }$ increases, and almost no further increases occurs when the upper energy limit $E_{\max }$ increases to a certain value.

In summary, the efficiency of the PV cell ${\eta }_{\mathrm{PVC}}$ is defined as the electrical power out of the cell into a matched load, divided by the incident energy falling on the cell, which can be expressed as the product of $U_{\mathrm{eff}}$, $V_{\mathrm{eff}}$ and $Im\left(V_{\mathrm{op}}\right)$ [11]:

$${\eta }_{\mathrm{PVC}}=U_{\mathrm{eff}}·V_{\mathrm{eff}}·Im\left(V_{\mathrm{op}}\right)$$

(16)

The dependence of ${\eta }_{\mathrm{PVC}}$ on upper limit energy $E_{\max }$ is shown in Figure 8a. At different temperatures, the efficiency of the PV cell ${\eta }_{\mathrm{PVC}}$ increases first and then decreases with the increase in $E_{\max }$ with a maximum value around 0.62 eV. So the optimal emissivity of the emitter for the InGaAsSb PV cell is temperature independent and is a narrowband of 0.54–0.62 eV. The bandgap of the InAs, GaSb, and InGaSb PV cells is 0.36 eV, 0.72 eV, and 0.6 eV, respectively. To match the requirements of different PV cells, we calculated the optimal upper limit energy of the emitter when matching different PV cells, and the result is shown in Figure 8b. When the STPV system is matched with the InAs, GaSb, InGaSb, and InGaAsSb PV cells, the optimal emissivity of the emitter is one in a narrowband of 0.36–0.45 eV, 0.72–0.79 eV, 0.6–0.68 eV, 0.54–0.62 eV, respectively. It is worth noticing that, for different PV cells and working environments, the optimal bandwidth of emitter absorptivity is around 0.08 eV with an error of ±0.01 eV, which can be used as an engineering design guideline value for simplicity. Here, we show the rigorous analysis process and conclude that the optimal bandwidth of emitter absorptivity is around 0.08 eV and can be used as an engineering design guideline value for simplicity.

4.2. Effect of Deviation from the Ideal Spectrum on Efficiency ${\eta }_{\mathrm{PVC}}$

Since spectrum deviation cannot be avoided, we then study the spectrum deviation of the emitter on system efficiency. Figure 9a shows that ${\eta }_{\mathrm{PVC}}$ at $T_{\mathrm{emit}}$ = 1673 K decreases by 2.8% relatively when the relative error $\delta E_{\min }=(E_{\min }-E_{\mathrm{g}})/E_{\mathrm{g}}\times 100$% increases by 3%, and decreases by 14.3% relatively when $\delta E_{\min }$ increases by 3%. Figure 9b shows that ${\eta }_{\mathrm{PVC}}$ at $T_{\mathrm{emit}}$ = 1673 K decreases by 1.3% relatively when the relative error $\delta E_{\max }=(E_{\max }-E_{\max \_\mathrm{best}})/E_{\max \_\mathrm{best}}\times 100$% increases by 10%, and decreases by 4.8% relatively when $\delta E_{\max }$ decreases by 10%. Therefore, the decrease in $E_{\min }$ has the greatest effect on ${\eta }_{\mathrm{PVC}}$, which indicates that the emissivity before $E_{\mathrm{g}}$ of the emitter should be suppressed as much as possible in the actual design. As the narrowband of the emitter becomes narrower, i.e., $\delta E_{\min }$ increases or $\delta E_{\max }$ decreases, both will degrade the efficiency, contrary to the current general belief that “the emissivity of the ideal emitter must be of an extremely narrow band” [10,19,36], correcting the design idea for future emitter designs.

5. Conclusions

The efficiency of STPV systems can break the SQ limit by reshaping the solar spectrum with the selective emissivity of the intermediate structure. In this paper, we obtain the rigorous selective emissivity of the absorber/emitter through theoretical and systematic analysis of the STPV system. To resolve the inconsistency with the energy $P_{\mathrm{emit}}$, we propose a modified photothermal conversion efficiency in Equation (12). Besides, the radiative loss $A_{\mathrm{top},\mathrm{emit}}$$P_{\mathrm{emit}}$ of the emitter’s top surface in Equation (8), which is ignored in past works, cannot be ignored because the temperature of the intermediate structure $T_{\mathrm{emit}}$ is very high and the area difference $A_{\mathrm{top},\mathrm{emit}}=A_{\mathrm{emit}}-A_{\mathrm{abs}}$ is not small. Through the analysis, we obtained three practical design strategies. First, the ideal emissivity of the absorber and emitter is one in [$E_{\mathrm{c}}$, ∞] and [$E_{\mathrm{g}}$, $E_{\max }$], respectively. The optimal cutoff wavelengths $E_{\mathrm{c}}$ and $E_{\max }$ at different states are given, which can be used directly by designers. Second, if deviation cannot be avoided, it is preferable to shift ${\lambda }_{\mathrm{c}}$ ($E_{\mathrm{c}}$) to a longer wavelength rather than a shorter wavelength when designing an absorber. And the decrease in $E_{\min }$ should be suppressed relative to the deviation from $E_{\max }$ when designing an emitter. Third, the optimal bandwidth of emitter absorptivity is around 0.08 eV for different PV cells and working environments, which can be used as an engineering design guideline value for simplicity. It is worth emphasizing that it is contrary to the current general design strategy, “The emissivity of the ideal emitter must be of an extremely narrow band” [10,19,36]. The analytical progress and design strategies will provide reference and direction for the future design of STPV and real-world applications.