1. Introduction

Thermophotovotaic (TPV) systems [1] are energy conversion units that convert thermal energy into electricity by irradiating PV cells with thermal emission from heated objects. To increase the output power density and conversion efficiency of the TPV systems, it is important to selectively enhance the thermal emission in a wavelength range just below the bandgap wavelength of the PV cell to suppress transmission loss and intraband relaxation loss, and such narrowband thermal emission has been achieved with various optical nanostructures [2–7]. Recently, the use of near-field thermal radiation transfer has also been attracting increasing attention as a promising approach for enhancing the output power density of a TPV system, where an emitter and a PV cell are separated by a gap smaller than the characteristic wavelength of thermal emission (called near-field TPV). In this configuration, not only the modes propagating in free space but also the modes confined inside the medium by total internal reflection (frustrated modes) and the modes localized at the surface (surface modes) can contribute to the energy transfer via evanescent coupling, which enables the enhancement of thermal radiation beyond the blackbody limit [8–15]. In our previous paper, we revealed that the introduction of a photonic crystal (PC) slab into the emitter could further increase the absolute power of the near-field thermal radiation transfer as well as its frequency selectivity [16]. However, as several detailed numerical analyses have shown [12,14,16], the direct positioning of the emitter and the PV cell in close proximity also involves an exponential increase in heat transfer in the far-infrared range, which is caused by the surface modes supported by heavily-doped contact layers or phonon-polaritons in typical PV cells. Therefore, the simultaneous achievement of high power density and high power conversion efficiency in near-field TPV systems has been a challenge. In this paper, we propose the utilization of an intermediate transparent substrate attached to the top of a PV cell to suppress the long-wavelength heat transfer caused by the surface modes of the PV cell while maintaining the near-field enhancement of thermal radiation transfer in the near-infrared range. We confirm that our scheme is applicable for near-field TPV systems using a Si or tungsten (W) emitter. As a specific example, we designed a near-field TPV system composed of a Si PC thermal emitter and an intermediate Si substrate attached to the top of an InGaAs PV cell, and confirmed that high power density (>5 × 10⁴ W/m²) and high power conversion efficiency (>40%) can be simultaneously achieved in our system.

2. Proposed near-field TPV system

2.1 Materials

Figure 1(a) shows the proposed TPV system, which consists of a thermal emitter, a PV cell, and an intermediate transparent substrate. For the material of the thermal emitter, we investigated the use of undoped Si because it exhibits a step-like increase of absorption coefficients at wavelengths shorter than its bandgap wavelength, which is desirable to realize frequency-selective thermal radiation transfer with suppressed emission at longer wavelengths [6,16]. Figures 1(b) and 1(c) show the theoretical absorption coefficient spectra of undoped Si in the near-infrared and the far-infrared range at various temperatures. Here, we took into account both interband transitions and free-carrier absorption due to thermally excited intrinsic carriers. The changes in bandgap energy [17,18], phonon population, and momentum relaxation time [19] that occur with increasing temperature were taken into account in the calculation. As shown in Fig. 1(b), the bandgap wavelength of Si redshifts as the temperature rises. The below-bandgap absorption coefficient due to thermally excited intrinsic carriers increases as the temperature rises [Fig. 1(c)], but it is still one to two orders of magnitude smaller than that of interband absorption. To further suppress the below-bandgap thermal radiation, we decreased the thickness of the emitter to 2 µm. In this paper, we consider both a planar thermal emitter [w = a or h = 0 in Fig. 1(a)] and a one-dimensional (1D) PC thermal emitter. In the latter case, the band-folding effect in the PC selectively enhances the thermal radiation transfer in the above-bandgap energy so that the ratio between the components above and below the bandgap energy can be further enhanced [16]. We also investigated the use of W, which was commonly used in many previous studies [2,9,14,15], as another material of the thermal emitter. The dielectric function of W (room temperature) was taken from Ref. 20 in our calculations. It should be noted that the increase of the mid-infrared and far-infrared emissivity from W at high temperatures [21,22] was not taken into account as is often the case with many other studies.

 

Fig. 1 (a) Near-field TPV system, which consists of a thermal emitter, an intermediate Si substrate, and an InGaAs PV cell. By inserting the intermediate substrate between the emitter and the PV cell, we can suppress the long-wavelength heat transfer caused by the surface modes of the PV cell while maintaining the near-field enhancement in thermal radiation transfer in the near-infrared range. (b)(c) Absorption coefficient spectra of undoped Si in the near-infrared and far-infrared at various temperatures. (d) Real part of the permittivity of undoped Si and the heavily doped contact layer (n-InP).

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For the TPV cell, we chose p-InP/p-In_0.53Ga_0.47As/n-In_0.53Ga_0.47As/n-InP lattice-matched systems, which are commonly used for low-bandgap PV cells. In this system, p-InP and n-InP layers have larger photonic bandgaps (1.35 eV) than p-InGaAs/n-InGaAs layers (0.73 eV) to form a hetero-junction, preventing electron-hole pairs generated inside InGaAs layers from surface recombination [23]. Note that the bandgap of InGaAs (0.73 eV, λ = 1.7 µm) is almost matched with that of Si at 1400 K as shown in Figs. 1(b) and 1(c). The thickness and doping density of each layer is determined from the typical values of the previous works [24,25]. In our calculations, optical constants of lattice-matched InGaAs and InP at near-infrared range were taken from Refs. 26 and 27, and those at far-infrared range were modeled by Lorentz-Drude dispersions [28].

The most important aspect of the proposed system shown in Fig. 1(a) is the attachment of a transparent substrate at the top of the PV cell. The purpose of the transparent substrate is the suppression of the far-infrared heat transfer mediated by surface modes while maintaining the enhancement in near-infrared thermal radiation. In the conventional TPV system, where the emitter is directly placed close to the PV cell, the energy loss by heat transfer to the surface modes of the PV cells are significant. Because the surface modes are supported by transverse optical (TO) phonons and high-density free carriers in the contact layers of the PV cells, which induce negative permittivity in the far-infrared range [see n-InP (blue line) in Fig. 1(d)], the above loss can be avoided by attaching an appropriate intermediate substrate at the top of the PV cell. In this system, the enhancement of the thermal radiation transfer in the near-infrared beyond the blackbody limit is still achieved owing to the extraction of the frustrated modes inside the emitter into the intermediate substrate via evanescent coupling. The ideal conditions for the intermediate substrate are summarized as follows.

We chose an undoped Si substrate (300 K) in this paper, as it is an intermediate substrate that satisfies the above three conditions. It should be noted that almost all the wide-bandgap semiconductors including GaN, AlN, InP, GaP, SiC do not meet the condition (I) due to the existence of a reststrahlen band caused by transverse-optical phonons [28,29]. As shown by the black line in Fig. 1(d), the undoped Si has a positive permittivity at all wavelengths [19,30], and its refractive index in the near-infrared region (3.4 at 300 K) is approximately as large as that of InGaAs (3.5). Moreover, as shown in Fig. 1(c), the undoped Si at 300 K is almost transparent in the wavelength range longer than 1.1 µm, which covers most of the blackbody spectrum at 1400 K. It should be also noted that an InGaAs/InP PV cell stacked on a Si substrate can be realized by using a wafer-bonding technique [25].

2.2 Simulation method

To calculate the thermal radiation spectra transferred from the PC thermal emitter to the multi-layered PV cell (with the intermediate substrate) shown in Fig. 1(a), we developed a method that combined the fluctuation–dissipation theorem and rigorous coupled wave analysis [16]. The outline of the method is explained as follows, and the details are described in Appendix A.

In the calculations in the next section, we set the observation planes for the spectral energy flux at each interface inside the PV cell, and calculated the spectral flux absorbed in each layer of the PV cell. We also calculated the transmitted flux through the PV cell by monitoring the flux passing through the top surface of the PV cell (the interface between p-InP and vacuum). We fixed the temperatures of the emitter and the PV cell with the intermediate substrate to be 1400 K and 300 K, respectively. The ignorance of the temperature gradient inside the emitter and the intermediate substrate is valid in our system because a thermal conductivity of Si is relatively large (130 W/m/K) and the heat generation inside the PV cell/intermediate substrate is not severe at 1400 K (the details are discussed in Appendix B). The effect of a back reflective mirror on the PV cell [32] was also investigated in Appendix C.

3. Results

3.1 Thermal radiation transfer spectra

Figures 2(a)–2(d) show the far-field (d = 100 µm) and near-field (d = 0.01 µm) thermal radiation transfer spectra from the Si planar emitter (without a PC structure, 1400 K) to each layer of the PV cell. It should be noted that the summation of the absorbed spectra in all the layers and the transmitted spectrum corresponds to the total thermal radiation spectrum from the emitter (not shown to ensure the visibility). When the emitter and the PV cell directly face each other without the intermediate substrate [Figs. 2(a) and 2(b)], the reduction in the gap from 100 µm to 0.01 µm induces energy transfer exceeding the far-field blackbody limit (black line) in the interband absorption of the InGaAs pn junction (red line, λ<λ_g) as well as in the far-infrared absorption of the n-InP window layer (blue line, λ>20 µm). On the other hand, when the intermediate Si substrate with a thickness of 10 µm is placed between the two layers [Figs. 2(c) and 2(d)], the reduction in the gap does not induce excessive heat transfer in the far-infrared range whereas the enhancement in the interband absorption in the InGaAs pn junction is as large as that in the case without the intermediate Si substrate. The former is owing to the aforementioned condition (I) and the latter is owing to the conditions (II) and (III). Although the emission at the wavelength shorter than 1.0 µm is absorbed in the intermediate Si substrate (yellow-green line), its total power is negligible compared to that in the InGaAs pn junction owing to the short-wavelength cutoff of the blackbody spectrum of Planck’s equation at 1400 K.

 

Fig. 2 (a)(b) Thermal radiation spectra from the Si planar emitter to each layer of the PV cell without the intermediate Si substrate (t = 0 µm) at a gap of 100 µm (a) and 0.01 µm (b). In this case, the interband absorption in the InGaAs pn junction (red line, λ<λ_g) as well as the far-infrared absorption in the n-InP window layer (blue line, λ>20 µm) exceed the far-field blackbody limit (black line). (c)(d) Thermal radiation spectra from the planar emitter to each layer of the PV cell with the intermediate Si substrate (t = 10 µm) at a gap of 100 µm (c) and 0.01 µm (d). Here, the selective enhancement of the interband absorption in the InGaAs pn junction is achieved.

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To illustrate the origin of the far-infrared absorption of the n-InP window layer and the mechanism of its suppression with the intermediate substrate, we calculated the exchange function of the thermal radiation transfer between the Si emitter and the PV cells in the far-infrared range (exchange function is defined as radiation flux per unit in-plane wavevector normalized by the theoretical maximum flux, Eq. (28) in Appendix A). The results at the gap d = 10 nm are shown in Fig. 3. In the case without an intermediate substrate [Fig. 3(a)], several branches of surface modes appear in the far-infrared range (λ = 21 µm and 36 µm). Since these wavelengths almost agree with those where the real part of the permittivity of n-InP turns to a negative value [shown in Fig. 1(d)], the far-infrared absorption is caused by the surface modes induced at the interface of vacuum and n-InP. On the other hand, when an intermediate substrate is inserted [Fig. 3(b)], no thermal radiation transfer arises outside the lightline for materials (k_//>nω/c), meaning that surface modes are completely suppressed.

 

Fig. 3 (a)(b) Calculated exchange function of the thermal radiation transfer between the Si emitter and the PV cells at the gap d = 10 nm as a function of an in-plane wavenumber. (a) without an intermediate substrate, (b) with an intermediate substrate (t = 10 µm).

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Figures 4(a) and 4(b) show the total energy flux of the thermal radiation transfer from the Si planar emitter to the PV cell with and without the intermediate substrate by varying the gap (d) between them. The red line in each figure shows the power absorbed by the interband absorption in the InGaAs pn junction, which can be converted to electrical power in the TPV system. The other lines show the losses that cannot be converted to electrical power, including the below-bandgap absorption in InGaAs (pink), absorption in the window layers (n-InP: blue, p-InP: green), absorption in the intermediate Si substrate (yellow-green), and transmission loss (gray). In both figures, the interband absorption power in InGaAs increases by one order of magnitude when the gap from the far-field length scales is reduced to below 0.1 µm, owing to the contribution of the frustrated modes to the thermal radiation transfer. However, in the case of no intermediate substrate [Fig. 4(a)], the absorption in the outermost surface (n-InP) exponentially increases when the gap is smaller than 0.1 µm; this is caused by the surface modes, as we have already mentioned. On the other hand, when the intermediate substrate (t = 10 µm) is introduced [Fig. 4(b)], the absorption in n-InP remains one order of magnitude smaller compared to the interband absorption in InGaAs even at a gap of 0.01 µm. Figure 4(c) shows the ratio of the interband absorption power in the InGaAs pn junction to the total thermal radiation power from the Si planar emitter (interband absorption ratio) for cases with and without the intermediate Si substrate. In this calculation, the far-field thermal radiation from the emitter in the direction opposite to the PV cell is not included in the calculation of the total radiation power (such radiation is small compared to the near-field transfer, and can be recycled). The interband absorption ratio decreases at the gap below 0.05 µm when the intermediate substrate is not introduced, whereas the ratio monotonically increases as the gap decreases when the intermediate substrate is introduced. The black line in Fig. 4(d) shows the interband absorption ratio of the planar emitter as a function of the thickness of the intermediate substrate (t) calculated at d = 0.05 µm. The ratio is maximized at a moderate substrate thickness (t = 1~2 µm). The reason for this is explained as follows. When the substrate is very thin (t < 0.1 µm), the ratio is low because the surface modes of the PV cell can couple to the emitter by tunneling the intermediate substrate, leading to the increase in unwanted heat transfer. On the other hand, when the substrate is very thick, the absorption loss in the intermediate substrate increases. We also calculated the results for the 1D PC emitter (red line). The structure (a = 0.4 µm, w = 0.28 µm, h = 1.8 µm) was designed so that the band-folding effect in the PC slab selectively enhances the thermal radiation transfer in the above-bandgap energy (when the lattice constant is too large, the band folding is induced at lower frequencies, which decreases the frequency selectivity of the thermal radiation transfer [16]). We introduced 1D PC in the lower surface [see Fig. 1(a)] because the flat upper surface mitigates the decrease in the photonic density of states caused by the introduction of the PC. In Fig. 4(d), the maximum interband absorption ratio achieved with the 1D PC emitter (76.6%) is higher than that of the planar emitter (70.2%).

 

Fig. 4 (a)(b) Interband absorption power in InGaAs (red line) and the other losses, which cannot be converted to electrical power, with and without the intermediate substrate. Total emission flux from the emitter is shown in black. (c) Ratio of the interband absorption power in InGaAs to the total radiation power (interband absorption ratio) with and without the intermediate substrate. (d) Interband absorption ratio at d = 0.05 µm for the planar emitter and the 1D PC emitter (a = 0.4 µm, w = 0.28 µm, h = 1.8 µm) as a function of the thickness of the intermediate substrate (t).

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To investigate the applicability of our scheme to other emitters, we also calculated the near-field thermal radiation transfer spectra with a 2-µm-thick W planar emitter (1400 K). The results at the gap d = 0.01 µm are shown in Figs. 5(a) and 5(b). As in the case of the Si emitter (Fig. 2), the W emitter yields far-infrared thermal radiation transfer caused by surface modes when there is no intermediate substrate [Fig. 5(a)], while the far-infrared radiation transfer is completely suppressed by inserting a 10-µm-thick intermediate Si substrate [Fig. 5(b)]. Figures 5(c) and 5(d) shows the calculated interband absorption power in InGaAs solar cells and its ratio to the total emission flux as a function of the gap length for the 2-µm-thick Si and W planar emitter. In Fig. 5(c), the obtained interband absorption power for the Si emitter is comparable with that of the W emitter. In Fig. 5(d), the W emitter shows a higher interband absorption ratio than the Si emitter; however, as we have pointed out in Section 2.1, the increase of the mid-infrared and far-infrared emissivity from W at high temperatures [21,22] was not taken into account in our calculation, so the experimentally achievable interband absorption ratio for the W emitter will become lower. Therefore, we cannot briefly discuss which materials are more suitable to realize a highly-efficient near-field TPV system. Nevertheless, we can safely conclude that the attachment of the intermediate substrate to the top of a PV cell is effective for both emitters.

 

Fig. 5 (a)(b) Thermal radiation spectra from a 2-µm-thick W planar emitter to each layer of the PV cell (a) without and (b) with an intermediate Si substrate (t = 10 µm) at a gap of 0.01 µm. (c) Interband absorption power in the InGaAs PV cell with an intermediate substrate (t = 10 µm) from the 2-µm-thick Si emitter and W emitter. (d) Ratio of the interband absorption power to the total emission flux from the 2-µm-thick Si emitter and W emitter. It should be noted the increase of the mid-infrared and far-infrared emissivity from W at high temperatures [21,22] was not taken into account in this calculation.

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3.2 Power density and power conversion efficiency of near-field TPV system

Finally, we evaluate the electric power density and the power conversion efficiency of the proposed near-field TPV system. The current density in the PV cell is given by the following equation:

For a specific example, we calculated the output power density and the conversion efficiency of the proposed TPV system with the intermediate Si substrate (t = 2 µm) and the Si PC emitter we designed in Section 3.1 (a = 0.4 µm, w = 0.28 µm, h = 1.8 µm). The results with various gap lengths are shown in Fig. 6. Here, we assume two different emitter temperatures (1400 K and 1273 K) considering the experimentally achievable operating temperature of the Si PC emitter [6]. At 1400 K, high power density (>5 × 10⁴ W/m²) and high conversion efficiency (>40%) are simultaneously realized at the gap below 0.05 µm, and the power density over 1 × 10⁴ W/m² and conversion efficiency of 35% are obtained even at 1273 K.

 

Fig. 6 (a) Electric power density of the proposed near-field TPV system with the Si PhC emitter (a = 0.4 µm, w = 0.28 µm, h = 1.8 µm) and the intermediate Si substrate (t = 2 µm) as a function of the gap. (b) Power conversion efficiency of the near-field TPV system as a function of the gap.

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4. Conclusion

We have proposed a scheme for near-field thermal radiation transfer using an intermediate transparent substrate for highly efficient TPV energy conversion. We have displayed that the addition of the intermediate transparent substrate between the thermal emitter and the PV cell simultaneously realizes the enhancement of the thermal radiation transfer at wavelengths shorter than the bandgap wavelength of the cell and the suppression of the unwanted heat transfer at longer wavelengths mediated by the surface modes in the conventional near-field TPV system. On the practical side, the addition of the intermediate substrate allows us to keep the PV cell separated from the hot emitter by a gap (1–100 µm, depending on the thickness of the intermediate substrate), which will make it easier to perform experimental studies of near-field TPV energy conversion from the viewpoint of electrical contact and cooling of the PV cell. We believe that our findings will theoretically and experimentally contribute to the realization of highly efficient high-power TPV energy conversion systems.

Appendix A Numerical simulation of near-field thermal radiation transfer between a PC emitter and a PV cell

Here, we describe the simulation method of near-field thermal radiation transfer from a single-layered PC emitter (thickness: t₁, the center z-coordinate: z=0) and a PV cell. In the case of the multi-layered PC shown in Fig. 1(a), we repeat the following procedure for each layer of the emitter. The validity of our method was confirmed with several simulation tests in our previous work [16].

The electromagnetic fields of the Bloch mode inside a PC can be written as the superposition of plane waves as follows:

(9)$$\begin{array}{l}{E}_x(r)=\exp \left(i\beta \cdot \rho \right)\times {\displaystyle \sum _n{E}_{x,n}(z)\exp \left(i{G}_n\cdot \rho \right)}\\ E_y(r)=\exp \left(i\beta \cdot \rho \right)\times {\displaystyle \sum _n{E}_{y,n}(z)\exp \left(i{G}_n\cdot \rho \right)},\\ H_x(r)=\exp \left(i\beta \cdot \rho \right)\times {\displaystyle \sum _n{H}_{x,n}(z)\exp \left(i{G}_n\cdot \rho \right)}\\ H_y(r)=\exp \left(i\beta \cdot \rho \right)\times {\displaystyle \sum _n{H}_{y,n}(z)\exp \left(i{G}_n\cdot \rho \right)}\end{array}$$

where $r=(x,y,z)$, $\rho =(x,y)$, z corresponds to the vertical direction, $\beta$is the in-plane wavevector of the Bloch mode in the first Brillion zone, and $G_n$ is the n-th reciprocal vector of the PC. Let ${\rm E}$ and $H$ be column vectors that are composed of the expansion coefficients in Eq. (9):

(10)$$\begin{array}{l}E(z)={\left(E_{x,1},E_{y,1},E_{x,2},E_{y,2},\cdots ,E_{x,N},E_{y,N}\right)}^T,\\ H(z)={\left(H_{x,1},H_{y,1},H_{x,2},H_{y,2},\cdots ,H_{x,N},H_{y,N}\right)}^T\end{array}$$

where $N$ is the number of plane waves used for the Fourier expansion. When there are no current sources inside the PC, substituting Eqs. (9) and (10) into Maxwell’s equations gives the following matrix equations [31]:

(11)$$\frac{\partial }{\partial z}E=iT{}_{1}H,\frac{\partial }{\partial z}H=iT{}_2{\rm E},$$

(12)$$\frac{{\partial }^2}{\partial z^2}E=-T{}_{1}T{}_2{\rm E}.$$

Here, the $2N$ × $2N$ matrices $T{}_1$ and $T{}_2$ are defined as block matrices composed of the following 2 × 2 submatrices;

(13)$$\begin{array}{l}T{}_1{}^{i,j}=\frac{1}{\omega {\epsilon }_0}\left(\begin{array}{cc}{k}_{x,i}{\epsilon }_{ij}^{-1}{k}_{y,j}& -k_{x,i}{\epsilon }_{ij}^{-1}{k}_{x,j}+k_0^2{\delta }_{ij}\\ k_{y,i}{\epsilon }_{ij}^{-1}{k}_{y,j}-k_0^2{\delta }_{ij}& -k_{y,i}{\epsilon }_{ij}^{-1}{k}_{x,j}\end{array}\right),\\ T{}_2{}^{i,j}=\frac{1}{\omega {\mu }_0}\left(\begin{array}{cc}-k_{x,i}{\delta }_{ij}{k}_{y,j}& k_{x,i}{\delta }_{ij}{k}_{x,j}-k_0^2{\epsilon }_{ij}^{}\\ -k_{y,i}{\delta }_{ij}{k}_{y,j}+k_0^2{\epsilon }_{ij}^{}& k_{y,i}{\delta }_{ij}{k}_{x,j}\end{array}\right)\end{array}$$

where $k_{x(y),i}$ is the x(y) component of the wavevector of the i-th plane wave and ${\epsilon }_{ij}^{}$ and ${\epsilon }_{ij}^{-1}$ are the Fourier component of the periodic permittivity distribution of the PC:

(14)$${\epsilon }_{ij}^{}={\displaystyle \underset{unitcell}{\iint }}\epsilon (\rho )\exp \left(i\left(G_i-G_j\right)\cdot \rho \right)d\rho ,{\epsilon }_{ij}^{-1}={\displaystyle \underset{unitcell}{\iint }}\frac{1}{\epsilon (\rho )}\exp \left(i\left(G_i-G_j\right)\cdot \rho \right)d\rho .$$

By defining the eigenvalues and eigenvectors of the matrix $T{}_{1}T{}_2$ as ${\gamma }_i{}^2\left(\mathrm{Im}({\gamma }_i)\ge 0\right)$ and $u_i$$\left(i=1,2,\cdots ,2N\right)$, Eqs. (9) and (10) can be rewritten as

(15)$$\begin{array}{l}{E}_0(z)=E_0{}^{+}(z)+E_0{}^{-}(z)\\ E_0{}^{+}(z)={\displaystyle \sum _i\left[C_i^{+}\exp \left(i{\gamma }_{i}z\right)\right]}{u}_i,E_0{}^{-}(z)={\displaystyle \sum _i\left[C_i^{-}\exp \left(-i{\gamma }_{i}z\right)\right]}{u}_i,\end{array}$$

where $E_0{}^{+}(z)$ and $E_0{}^{-}(z)$ are the electric field components of the eigenmodes propagating in the + z and −z directions, respectively, and $C_i^{+}$ and $C_i^{-}$ are unknown coefficients.

When the PC is heated at a temperature of T, random fluctuating currents $j_m(r)(m=x,y,z)$ are induced inside the object. According to the fluctuation–dissipation theorem, these currents obey the following correlation [10]:

(16)$$〈j_m(r,\omega )j_{m\text{'}}{}^{*}(r\text{'},\omega \text{'})〉=\frac{4\omega {\epsilon }_0\mathrm{Im}(\epsilon )}{\pi }\frac{\hslash \omega }{\exp \left(\hslash \omega /kT\right)-1}{\delta }_{mm\text{'}}\delta (r-r\text{'})\delta (\omega -\omega \text{'}).$$

As in Eq. (9), $j_m(r)$can be expressed with the superposition of plane waves. Using Eq. (16), the amplitudes of each plane wave $j_{m,n}(z)$obey the following correlation:

(17)$$\begin{array}{c}〈j_{m,n}(z)j_{m\text{'},n\text{'}}{}^{*}(z\text{'})〉=\frac{1}{16{\pi }^4}{\displaystyle \underset{unitcell}{\iint }d\rho }{\displaystyle \underset{unitcell}{\iint }d\rho \text{'}}〈j_m(r)j_{m\text{'}}{}^{*}(r\text{'})〉\exp \left[i\left(G_{n\text{'}}\cdot \rho \text{'}-G_n\cdot \rho \right)\right]\\ =\frac{\omega {\epsilon }_0}{4{\pi }^5}\frac{\hslash \omega }{\exp \left(\hslash \omega /kT\right)-1}{\delta }_{mm\text{'}}\delta (\omega -\omega \text{'})\delta (z-z\text{'}).\\ \times {\displaystyle \underset{unitcell}{\iint }d\rho }\mathrm{Im}(\epsilon )\exp \left[i\left(G_{n\text{'}}-G_n\right)\cdot \rho \right]\end{array}$$

When the current sources exist, Eqs. (11) are modified as follows:

(18)$$\frac{\partial }{\partial z}E=iT{}_{1}H+J_1,\frac{\partial }{\partial z}H=iT{}_2{\rm E}+J_2,$$

(19)$$\frac{{\partial }^2}{\partial z^2}E=-T{}_{1}T{}_2{\rm E}+i{T}_1{J}_2+\frac{\partial }{\partial z}{J}_1,$$

where

(20)$$\begin{array}{l}{J}_1=\frac{1}{\omega {\epsilon }_0}{\left(k_{x,1}{\left({\epsilon }_{}^{-1}{j}_z\right)}_1,k_{y,1}{\left({\epsilon }_{}^{-1}{j}_z\right)}_1,k_{x,2}{\left({\epsilon }_{}^{-1}{j}_z\right)}_2,k_{y,2}{\left({\epsilon }_{}^{-1}{j}_z\right)}_2,\cdots \right)}^T\\ J_2={\left(j_{y,1},-j_{x,1},j_{y,2},-j_{x,2},\cdots \right)}^T.\\ {\left({\epsilon }_{}^{-1}{j}_z\right)}_i={\displaystyle \sum _{j=1}^N{\epsilon }_{ij}^{-1}{j}_{z,j}}\end{array}$$

To solve Eq. (19), we diagonalize the matrix $T{}_{1}T{}_2$ with the matrix $S_a$ whose i-th row vector is $u_i$ . By multiplying $S_a{}^{-1}$ from the left in Eq. (19), we obtain

(21)$$\frac{{\partial }^2}{\partial z^2}{S}_a^{-1}E=-S_a^{-1}T{}_{1}T{}_{2}S{}_a(S_a^{-1}E)+\frac{\partial }{\partial z}{S}_a^{-1}{J}_1+S_a^{-1}iT{}_{1}J{}_2.$$

The equation above can be divided into 2N independent differential equations:

(22)$$\frac{{\partial }^2}{\partial z^2}{f}_i(z)=-{\gamma }_i^2{f}_i(z)+\frac{\partial }{\partial z}{p}_i(z)+i{q}_i(z)\left(i=1,2,\cdots ,2N\right),$$

where $f_i(z),p_i(z),q_i(z)$ are the i-th components of the vectors $S_a^{-1}E,S_a^{-1}{J}_1,S_a^{-1}{T}_1{J}_2$, respectively. $f_i(z)$ corresponds to the amplitude of the i-th eigenvector $u_i$. By solving Eq. (22) using Fourier method, we obtain

(23)$$f_i(z)=\{\begin{array}{c}\frac{\pi }{{\gamma }_i^{}}\left[{\gamma }_i^{}{P}_i({\gamma }_i^{})+Q_i({\gamma }_i^{})\right]\exp (i{\gamma }_i^{}z)(z>0)\\ \frac{\pi }{{\gamma }_i^{}}\left[-{\gamma }_i^{}{P}_i(-{\gamma }_i^{})+Q_i(-{\gamma }_i^{})\right]\exp (-i{\gamma }_i^{}z)(z<0)\end{array},$$

where

(24)$$P_i({\gamma }_i)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{p}_i(z)\exp (i{\gamma }_{i}z)dz},Q_i({\gamma }_i^{})=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{q}_i(z)\exp (i{\gamma }_{i}z)dz}.$$

The integration range of Eq. (24) can be replaced with $-t_1/2\le z\le t_1/2$ since the currents are induced only inside the slab. By comparing these results with Eq. (15), we obtain

(25)$$C_i^{+}=\frac{\pi }{{\gamma }_i}\left[{\gamma }_i{P}_i({\gamma }_i)+Q_i({\gamma }_i)\right],C_i^{-}=\frac{\pi }{{\gamma }_i}\left[-{\gamma }_i{P}_i(-{\gamma }_i)+Q_i(-{\gamma }_i)\right].$$

The plane waves excited by the fluctuation currents at the reference plane $\left(z=0\right)$ inside the emitter then propagate to the PV cell while receiving multiple reflections. The electromagnetic fields at the observation plane $\left(z=z{}_2\right)$ inside the PV cell can be expressed as

(26)$$\begin{array}{l}{E}_2={\displaystyle \sum _i\left[M_E^{+}{C}_i^{+}{u}_i+M_E^{-}{C}_i^{-}{u}_i\right]}\\ H_2={\displaystyle \sum _i\left[M_H^{+}{C}_i^{+}{u}_i+M_H^{-}{C}_i^{-}{u}_i\right]}.\end{array}$$

Here, $M_E^{\pm },M_H^{\pm }$ are the transfer matrices for the plane waves from the reference plane in the emitter to the observation plane in the PV cell, which can be calculated using the RCWA method [31]. Finally, the spectral energy flux propagating through the observation plane inside the PV cell is given by the summation of the Poynting vectors for all in-plane wavevectors:

(27)$$S(\omega ,T)={\displaystyle \underset{1stBrillouin}{\int }d\beta }\frac{1}{2}\mathrm{Re}\left[〈{\displaystyle \sum _{n=1}^N\left(E_{2x,n}{H}_{2y,n}-E_{2y,n}{H}_{2x,n}\right)}〉\right].$$

The calculation of the summation can be performed by substituting Eqs. (20), (24)–(26) into Eq. (27), expanding the summation, and using the correlation of the fluctuation currents [Eq. (17)] for each term. The net thermal radiation transfer between the emitter and the PV cell is calculated as$S(\omega ,T_{emitter})-S(\omega ,T_{cell})$. The exchange function $Z(\omega ,\beta )$ is defined as

(28)$$Z(\omega ,\beta )=\frac{1}{2}\mathrm{Re}\left[〈{\displaystyle \sum _{n=1}^N\left(E_{2x,n}{H}_{2y,n}-E_{2y,n}{H}_{2x,n}\right)}〉\right]/\left(\frac{1}{4{\pi }^3}\frac{\hslash \omega }{\exp \left(\hslash \omega /kT\right)-1}\right),$$

where the denominator of Eq. (28) corresponds to the maximum thermal radiation flux per unit in-plane wavevector from an object at a temperature of T.

Appendix B Temperature gradient inside the intermediate substrate

In our calculations, we assume that the rear surface of the emitter opposite to the PV cell is kept at 1400 K with a given heating power, and that the top surface of the solar cell (p-InP layer) opposite to the intermediate substrate is kept at 300 K with a heat sink with a sufficiently large heat transfer coefficient (the realistic heat transfer coefficient is taken into account later). In this situation, the temperature gradient inside the intermediate substrate is negligible even when the thickness is 100 µm owing to a large thermal conductivity of Si (130 W/m/K at room temperature). For example, when we place the Si photonic crystal emitter (a=0.4 µm, w=0.28 µm, h=1.8 µm) at 1400 K and 100-µm-thick Si intermediate substrate at the gap of d=10 nm, the calculated absorption loss inside the Si substrate is 1.93 W/cm², which yields the temperature difference ΔT of only 0.015 K between the top and bottom of the intermediate substrate as shown in the following equation:1.93 W/cm²=(130 W/m/K)/100 µm *ΔT. Therefore, the assumption of the uniform temperature distribution inside the intermediate substrate and the PV cell is valid for our system. We should emphasize that our proposed near-field TPV system with the intermediate substrate is more advantageous for the suppression of the heat generation in the PV cells than conventional near-field TPV systems owing to the complete suppression of the unwanted heat transfer due to the surface modes.

When we consider a heat sink with a finite heat transfer coefficient such as h=1 W/cm²/K [15], we should also take into account the temperature difference between the heat sink and the PV cell. When we consider the gap of d=10 nm (50 nm) between the emitter and the 100-µm-thick intermediate substrate, the total heat generation inside the PV cell (including the free carrier absorption of the PV cell, the absorption of the intermediate Si substrate, and the interband relaxation loss) is 9.1 W/cm² (6.2 W/cm²), which causes a relatively small temperature difference of 9.1 K (6.2 K) between the PV cell and the heat sink. It should be noted that the power density of our emitter (1400 K) at d=10 nm is almost the same as that of a concentration of 200-sun, and that the above thermal management can be achieved for a single PV cell by using passive cooling techniques with a heat sink having a large surface area, as is done in concentrator photovoltaic systems [34]. Therefore, the assumption of the fixed temperature of the emitter and PV cell is valid as long as we consider the maximum emitter temperature of 1400 K.

Appendix C Effect of a back reflector on the PV cell

To investigate the effect of a reflective mirror at the base of the cell, we calculated the thermal radiation transfer from a 2-µm-thick Si planar emitter to an InGaAs PV cell with a Au back reflector [32] with and without a 10-µm-thick Si intermediate substrate [Figs. 7(a) and 7(b)]. The dielectric function of Au is taken from Ref. 35. As in the case of the Si emitter with no Au mirror (Fig. 2), far-infrared thermal radiation transfer is caused by surface modes when there is no intermediate substrate [Fig. 7(a)], while it is suppressed by inserting an intermediate Si substrate [Fig. 7(b)]. The ratio of the interband absorption in InGaAs to the total emission power, and the absolute interband absorption power in the InGaAs layers are shown in Figs. 7(c) and 7(d), respectively. The red and black lines show the case with a Au mirror with and without the 10-µm-thick intermediate substrate, and the blue line shows the case without a Au mirror with the 10-µm-thick intermediate substrate. It is seen from Fig. 7(c) that the addition of the intermediate substrate drastically increases the interband absorption ratio in the near-field (d ≤ 0.1 µm) regardless of the existence of the Au mirror. When the Au back reflector is used, no transmission loss exists but absorption loss by the Au mirror [orange line in Figs. 7(a) and 7(b)] arises, which results in the decrease of interband absorption ratio in the near-field compared to the case without the Au mirror (blue line). It should be noted that the addition of the intermediate substrate or the Au mirror does not change the absolute power of the interband absorption in the InGaAs layers as shown in Fig. 7(d).

 

Fig. 7 (a)(b) Thermal radiation spectra from the Si planar emitter to each layer of the PV cell with an Au reflective mirror (a) without and (b) with the intermediate Si substrate (t = 10 µm) at a gap of 0.01 µm. (c) Interband absorption ratio as a function of the gap length. (d) Interband absorption power in the InGaAs layers as a function of the gap length.

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Funding

Japan Society for the Promotion of Science (JSPS) (17H06125, 17K14665); Keihanshin Consortium for Fostering the Next Generation of Global Leaders in Research (K-CONNEX).

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