1. Introduction

Thermal radiation transfer between objects is greatly enhanced when the separation between them becomes small enough to induce the evanescent coupling of light, which is known as near-field thermal radiation transfer [1–4]. In recent years, the spectral control of near-field thermal radiation transfer has attracted significant interest with respect to both fundamental science and its potential for energy-related applications, such as thermophotovoltaics (TPV) [5–9] and electroluminescent cooling [10,11]. One way to realize frequency-selective near-field thermal radiation is the use of polar materials, which support surface waves [12,13], but the frequency of the radiation is fixed by the choice of materials (typically in the far-infrared range). As another promising approach, the introduction of periodic nanostructures to the emitter or the absorber (or both) has been intensively investigated [14–23]. However, most previous studies employing periodic nanostructures have aimed to achieve broadband enhancement of thermal radiation transfer by increasing the number of resonant modes or using hyperbolic modes. One paper [19] reported the frequency-selective radiative transfer spectra by using surface-plasmon-polariton coupling between two metallic gratings, but the spectra contained significant amount of broadband radiation. To obtain ideal spectral control, i.e., narrow-bandwidth near-field thermal radiation transfer concentrated at a target frequency with suppressed background radiation at the other frequencies, we previously investigated near-field thermal radiation transfer between semiconductors, where a one-dimensional (1D) silicon photonic crystal (PC) slab was employed as emitter and a low-bandgap semiconductor photovoltaic (PV) cell as absorber [24,25]. The results revealed that the introduction of the PC structure with the optimized lattice constant simultaneously enhanced near-infrared emission and suppressed below-bandgap emission. However, bandwidths of the near-field thermal radiation spectra in the near infrared obtained from the 1D-PC emitters remained almost identical to those obtained from the planar Si emitters without PCs. This is because 1D-PCs can modify optical dispersions only in one direction, which in principle leads to insufficient spectral control of thermal radiation spectra.

To further explore the potential of the spectral control of near-field thermal radiation using PCs, in this paper, we numerically investigate spectral control via the photonic band engineering of two-dimensional (2D) PC slabs. Inside the 2D-PC slabs, optical dispersions are modified in all in-plane directions, and the unique optical characteristics of a photonic bandgap (PBG) and flat bands are obtained. We use the PBG and the flat bands in the 2D-PC slabs to narrow down the near-field thermal radiation transfer spectra from the emitter to the absorber. We also investigate the minimum gap between the emitter and the absorber at which above-spectral control via photonic band engineering is effective. By designing a 2D-PC slab with appropriate thickness and adjusting the gap between the emitter and the absorber, we obtain narrow-bandwidth near-field thermal radiation (Δλ = 0.14 µm) that overcomes the far-field blackbody limit in the near-infrared range, with well-suppressed radiation at other frequencies.

2. Simulation models for near-field thermal radiation transfer

Figures 1(a) and 1(b) show the system of the near-field thermal radiation transfer that we investigate. It consists of an intrinsic silicon (Si) thermal emitter employing 2D-PC slabs (1400 K) and an InGaAs absorber with an intrinsic Si intermediate substrate (300 K). We chose intrinsic Si as material for the emitter because it exhibits a step-like increase in absorption coefficients above its bandgap, which is advantageous for enhancing near-infrared thermal radiation transfer with suppressed emission at lower frequencies [24–26]. As temperature increases, the bandgap wavelength of Si increases [27,28], as does the below-bandgap absorption coefficient owing to thermally excited intrinsic carriers [29], as shown in Fig. 1(c). For the absorber, we assume n-InP (t = 0.1 µm, n = 2 × 10¹⁸ cm⁻³) /n-In_0.53Ga_0.47As (t = 0.3 µm, n = 1 × 10¹⁸ cm⁻³) /p-In_0.53Ga_0.47As (t = 2.0 µm, n = 1 × 10¹⁷ cm⁻³) /p-InP (t = 0.1 µm, n = 2 × 10¹⁸ cm⁻³) PV cells, where the bandgap of In_0.53Ga_0.47As (0.73 eV, λ_g = 1.7 µm) nearly matches with that of intrinsic Si at 1400 K. To the top of the absorber is attached an intermediate transparent substrate (intrinsic Si, t = 2 µm). By introducing this substrate, we can suppress far-infrared heat transfer mediated by the surface modes, which are supported by transverse optical phonons and high-density free carriers in the contact layers of the PV cells [25]. In the near-infrared range, the enhancement of thermal radiation transfer from the emitter to the PV cell beyond the blackbody limit is still achieved because the modes propagating inside the emitter by total internal reflection (frustrated modes) are evanescently coupled to the propagating modes inside the intermediate substrate, which has a larger photonic density of states compared with the vacuum. For the calculations below, the optical constants of Si were taken from Refs. 27–29, and those of InGaAs and InP were taken from Refs. 30–32.

 

Fig. 1 (a)(b) Schematic of near-field thermal radiation transfer system, which consists of a Si thermal emitter employing a triangular-lattice PC slab and an InGaAs PV cell with an intermediate Si substrate. (c) Absorption coefficient spectra of intrinsic Si at various temperatures. (d) Calculated photonic band diagram of a 2-µm-thick hole-type PC slab. (e)(f) Calculated photonic band diagrams of 0.2-µm-thick hole-type and 0.3-µm-thick rod-type PC slabs. Gray regions indicate a photonic band gap (PBG) for one polarization mode, yellow region indicates flat photonic bands with increased photonic density of states, and blue region indicates photonic bands with a low effective refractive index (i.e. small photonic density of state).

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As described in Section 1, this work focuses on the spectral control of near-field thermal radiation via the photonic band engineering of 2D-PC slabs. For this purpose, we employed a triangular-lattice 2D-PC slab with air-holes [Fig. 1(a)] or silicon rods [Fig. 1(b)]. Unlike the 1D gratings employed in our previous work [25], the 2D structures can modify optical dispersions in all in-plane directions and, thus, enable more sophisticated photonic band engineering. To utilize the effect of such a unique feature of 2D-PCs on the spectral control of thermal emission, the thickness of the PC (t) should be smaller than the characteristic wavelength of thermal emission inside the materials. Otherwise, multiple longitudinal resonances occur inside the PC slab, which makes it difficult to realize precise spectral control [an example of the photonic band diagram of a “too thick” PC slab is shown in Fig. 1(d)]. Figures 1(e) and 1(f) show the photonic band of thinner 2D-PC slabs (t = 0.2 µm for a hole-type PC, t = 0.3 µm for a rod-type PC) that we investigate in this study. The band diagram was calculated by rigorous coupled wave analysis [33]. It should be noted that some of the resonant modes above the light line are very leaky and, thus, challenging to identify in our calculation. In each figure, dispersions for transverse-electric-like (TE-like) modes (modes that have an in-plane electric field at the center of the slab) and transverse-magnetic-like (TM-like) modes (modes that have a vertical electric field at the center of the slab) are shown in black and red, respectively. The hole-type PC slab (t = 0.2 µm, a = 0.5 µm, r = 0.35 a) supports a large photonic bandgap (PBG) for the TE-like mode (shown in gray), whereas no PBG is formed for the TM-like modes, as shown in Fig. 1(e). The rod-type PC slab (t = 0.3 µm, a = 0.5 µm, r = 0.25 a) supports a PBG for the TM-like mode (shown in gray) and flat bands expanding into the entire Brillouin zone below the PBG (shown in yellow), as shown in Fig. 1(f). Such flat bands are expected to realize frequency-selective near-field thermal radiation transfer owing to the increased photonic density of states at that frequency.

3. Results

In this section, we describe the calculation of near-field thermal radiation spectra from the 2D-PC thermal emitters shown in Fig. 1, and elucidate the effect of the photonic band diagram described in the Section 2. Calculations for near-field thermal radiation from periodic nanostructures were performed by using the method based on the fluctuation–dissipation theorem and rigorous coupled wave analysis. The details of these methods are available in our previous paper [25].

We first compare the near-field thermal radiation spectra from the hole-type 2D-PCs with those from the planar slabs without PCs (the case of the rod-type PCs is discussed later). Figures 2(a) and 2(b) show the thermal radiation transfer spectra from the 2-µm-thick and 0.2-µm-thick hole-type PC slabs, respectively, to the pn junction of the PV cell when the gap (d) is 100 nm (red lines). The transfer spectrum from the planar emitter without PCs with the same volume of Si is shown in blue in each figure. In case of the 2-µm-thick hole-type PC [Fig. 2(a)], near-field thermal radiation that overcomes the blackbody limit (black line) is obtained above the bandgap energy of the PV cell (hc/λ_g), but its bandwidth is almost identical to that obtained from the planar Si slab. This is because the 2-µm-thick hole-type PC slabs have too many optical resonant modes inside them, as shown in Fig. 1(d). On the contrary, the 0.2-µm-thick hole-type PC slab [Fig. 2(b)] exhibits suppressed thermal radiation at a frequency range of the TE-PBG and enhanced radiation above it, realizing a much narrower thermal radiation transfer spectrum than that of the planar Si slab. Although the integrated radiation power becomes smaller than that of the 2-µm-thick PC slab, peak power density at λ = 1.2 µm still overcomes the far-field blackbody limit owing to near-field enhancement. Figures 2(c) and 2(d) show the near-field thermal radiation transfer spectra of the 2-µm-thick and 0.2-µm-thick hole-type PCs at various gap sizes. The 2-µm-thick PC exhibits frequency-independent enhancement of thermal radiation transfer as the gap decreases, as shown in Fig. 2(c). On the contrary, the 0.2-µm-thick PC shows different behavior [Fig. 2(d)]: When the gap decreases to 200 nm (yellow–green line), thermal radiation at a normalized frequency of f = 0.4 c/a is selectively enhanced. When the gap further decreases, the spectral bandwidth as well as the peak power gradually increases.

 

Fig. 2 (a)(b) Thermal radiation transfer spectra from the 2-µm-thick (a) and 0.2-µm-thick (b) hole-type PC slabs (1400 K) to the PV cell (300 K) at a gap of 100 nm (red line). The thermal radiation transfer spectrum from the planar Si slab of the same volume is shown in blue in each figure. The far-field blackbody spectrum at 1400 K is shown in black. The frequency range of the PBG of the PC is superimposed in gray. (c)(d) Thermal radiation transfer spectra from the 2-µm-thick (c) and 0.2-µm-thick (d) hole-type PC slabs at various gap sizes.

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To investigate the origin of the frequency-selective enhancement of near-field thermal radiation transfer from the 0.2-µm-thick hole-type PC, we calculated the exchange function of the thermal radiation transfer, which is defined as the radiation flux per unit in-plane wavenumber normalized by that of the blackbody. The results are shown in Figs. 3(a)–3(d). Note that the exchange function can exceed unity in these figures because all in-plane resonant modes of the 2D-PC are folded into the first Brillouin zone of the PC. In the far-field regime [d = 1000 nm, Fig. 3(a)], the modes below the light line do not contribute to radiation transfer. At d = 200 nm [Fig. 3(b)], the optical band below the light line at f = 0.4c/a (shown with an orange ellipse) makes the greatest contribution to radiation transfer, which results in the frequency-selective thermal radiation spectra shown in Fig. 2(d). The other photonic bands at higher frequencies make a smaller contribution to near-field transfer at this gap size because the electric fields of modes in the higher-frequency bands are more strongly confined inside the slab than those in the lower-frequency bands. At d = 100 nm [Fig. 3(c)], the higher-frequency bands can also contribute to near-field transfer, leading to an increase in the spectral bandwidth. At an extremely small gap [d = 10 nm, Fig. 3(d)], some of the photonic bands at lower frequencies vanish because the evanescent coupling is too strong.

 

Fig. 3 (a)–(d) Calculated exchange function of near-field thermal radiation transfer from the 0.2-µm-thick hole-type PC slab in the first Brillouin zone at various gap sizes. The dotted white lines show the light lines of the PC slab and the horizontal dashed lines show the bandgap of In_0.53Ga_0.47As. The optical band shown with an orange ellipse in Fig. 3(b) makes the greatest contribution to radiation transfer, which results in the frequency-selective thermal radiation spectra shown in Fig. 2(d).

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Figure 4(a) shows near-field thermal radiation transfer spectra from the 0.3-µm-thick rod-type PC emitter in the near-infrared range when the gap (d) is 100 nm (red line). The near-field thermal radiation spectrum from the planar emitter without any PC pattern with the same volume of Si is also shown in the same figure (blue line). As in the case of the hole-type PC, the rod-type PC exhibits a frequency-selective thermal radiation transfer spectrum overcoming the far-field blackbody limit. Thermal radiation is enhanced at the frequency of the flat bands (f ~0.45c/a, shown in yellow) owing to the increase in the photonic density of states while being suppressed at the TM-PBG (shown in gray). Interestingly, thermal radiation at lower frequencies (f < 0.42c/a) is also suppressed compared with that of the planar emitter. This is owing to the small effective refractive index (i.e., small photonic density of states) at lower frequencies, as shown in blue in Fig. 1(f). As a result, the thermal radiation spectrum obtained from the rod-type PC has a narrower bandwidth (Δλ = 0.14 µm) than that of the 0.2-µm-thick hole-type PC shown in Fig. 2(b) (Δλ = 0.24 µm).

 

Fig. 4 (a) Thermal radiation transfer spectra from the 0.3-µm-thick rod-type PC slab (1400 K) to the PV cell (300 K) at a gap of 100 nm (red line). Thermal radiation transfer spectrum from the planar Si slab of the same volume is shown in blue. The far-field blackbody spectrum at 1400 K is shown in black. The frequency range of the PBG and flat band of the PC are superimposed in gray and yellow, respectively. (b) Thermal radiation transfer spectra from the rod-type PC slab at various gap sizes. (c)–(f) Calculated exchange function of the near-field thermal radiation transfer from the rod-type PC slab in the first Brillouin zone at various gap sizes. The dotted white lines show the light lines of the PC slab and the horizontal dashed lines show the bandgap of In_0.53Ga_0.47As.

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Figure 4(b) shows the near-field thermal radiation transfer spectra of the rod-type PC at various gap sizes. The reduction of the gap from 1000 nm to 100 nm yields more than a five-fold enhancement in the peak power density. However, further reduction of the gap (10 nm) does not increase peak power, and only increases bandwidth. Figures 4(c)–4(f) show the exchange function of the thermal radiation transfer in the first Brillouin zone of the rod-type PC. At a gap size of 100 nm, the flat band expanding into the entire Brillouin zone at f = 0.45c/a dominantly contributes to radiation transfer, which results in the narrowband thermal radiation spectra shown in Fig. 4(b). A further reduction in gap size to 10 nm blurs the photonic band diagram because the optical coupling between the emitter and the intermediate substrate is too strong.

Thus far, we have focused on the spectral control of thermal radiation transfer in the near-infrared range (above the bandgap energy of the PV cell). In some practical applications such as near-field TPV, the ratio of near-infrared thermal radiation to the total radiation, including below-bandgap radiation loss, should also be considered. Figures 5(a) and 5(b) show the breakdown of thermal radiation power from the 0.2-µm-thick hole-type and 0.3-µm-thick rod-type PC emitters as a function of gap size. The red line in each figure shows the near-infrared radiation power absorbed through interband absorption at the InGaAs pn junction, which can be converted into electrical power in the TPV system. The green line shows thermal radiation power that is absorbed in the PV cell but cannot be converted into electrical power, including the below-bandgap absorption in InGaAs, absorption in the contact layers (n-InP and p-InP), and absorption in the intermediate Si substrate. The blue line shows the unabsorbed power, including the transmission loss of the PV cell and far-field thermal radiation from the emitter in the direction opposite to that of the PV cell. In both cases, we can increase the power of interband absorption by five to 10 times by reducing the gap while the unabsorbed power loss does not increase as it is “far-field” loss. Figure 5(c) shows the ratio of interband absorption power in the InGaAs pn junction to the total thermal radiation power from the emitter (defined as interband absorption ratio) for both emitters. Here, the rod-type PC shows a higher interband absorption ratio than the hole-type PC. This is because the rod-type PC has a small photonic density of states at low frequencies as shown in Fig. 1(f), which leads to the suppression of below-bandgap absorption in the PV cell. Figure 5(d) shows the calculated below-bandgap absorption in the PV cell for the rod-type PC slab and the planar slab of the same volume, which clearly demonstrates the suppression of the below-bandgap heat transfer to the PV cell by the introduction of the rod-type PC slab.

 

Fig. 5 (a)(b) Interband absorption power in the InGaAs pn junction (red line) and other losses that cannot be converted into electrical power, for the hole-type and rod-type PC. The total emission flux from the emitter is shown in black. (c) Ratio of interband absorption power in InGaAs to the total radiation power for both PCs. (d) Below-bandgap absorption in the PV cell for the rod-type PC slab and the planar slab of the same volume.

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4. Conclusion

In this paper, we have numerically investigated the spectral control of near-field thermal radiation between an emitter and a PV cell via the photonic band engineering of 2D-PC slabs. By performing detailed calculations of the near-field thermal radiation spectra in the first Brillouin zone of the 2D-PC with slabs of varying thickness and gaps, we have shown that near-field thermal radiation transfer can be selectively enhanced at the frequency of the flat bands of the 2D-PC slab while being suppressed inside the photonic bandgap and at the frequency of the bands with a small effective refractive index. The narrow-bandwidth near-field thermal radiation spectra we obtained here can be useful in reducing energy loss owing to the intraband relaxation of carriers generated inside a PV cell when the bandgap of the PV cell matches with the emission frequency. We think our findings will introduce a new degree of freedom to the spectral control of near-field thermal radiation transfer for both fundamental physics and various energy-related applications.