Near-field radiative heat transfer between nonpolar epsilon-near-zero dielectric-filled Si gratings

Epsilon-near-zero (ENZ) materials have attracted significant attention in the far- and near fields of thermal radiation in recent years because of their unique optical characteristics. However, it is not considered an optimal carrier for near-field radiative heat transfer (NFRHT) due to the excessively low frustrated mode. In this paper, we address this drawback with a metamaterial composed of artificial hypothetical nonpolar ENZ dielectric-filled Si gratings. The behavior of NFRHT with ENZ has been investigated based on fluctuational electrodynamic and rigorous coupled-wave analysis. An artificial mode named Meta-NP ENZ mode is presented to reveal the significant enhancement of NFRHT, which can be demonstrated by the electric field intensity enhancement. Furthermore, we find that the increasing imaginary part may lead to an anomalous amplification of heat flux despite causing a recession in the Meta-NP ENZ mode, and this mode remains robust with respect to the plasma frequency shift of grating material. Our findings demonstrate that the ENZ dielectric exhibits outstanding performances similar to those observed in far-field radiation, surpassing the limitations of both Si gratings and nonpolar ENZ dielectric in NFRHT.

Figure 1

(a) Schematic of nonpolar ENZ-filled doped Si gratings. (b) Total heat flux of metamaterial. (c) Far-field radiative heat transfer (FFRHT) of Si substrate covered with ENZ plate. (d) Near-field radiative heat transfer (NFRHT) of Si substrate covered with ENZ plate. (e) Total heat flux of FFRHT and NFRHT.

Figure 2

(a) Spectral heat fluxes of ENZ-Si gratings, Si gratings, Si plate, ENZ gratings, and ENZ plate. The vacuum gap $d$ of all scenarios is 60 nm. The temperatures of emitter and receiver are 400 and 300 K respectively. The permittivity of nonpolar ENZ material is $\varepsilon =0.01+0.0001i$. (b) Total heat fluxes of ENZ-Si gratings, Si gratings, Si plate, ENZ gratings, and ENZ plate.

Figure 3

Contour plots of energy transmission coefficients ξ of different ${\varepsilon }_{\mathrm{real}}$ within the first Brillouin zone at $k_y=0$, and the imaginary part is fixed as 0.0001. (a) ${\varepsilon }_{\mathrm{real}}=0.01$, (b) ${\varepsilon }_{\mathrm{real}}=0.1$, (c) ${\varepsilon }_{\mathrm{real}}=0.4$, (d) ${\varepsilon }_{\mathrm{real}}=0.7$, (e) ${\varepsilon }_{\mathrm{real}}=1$, (f) ${\varepsilon }_{\mathrm{real}}=4$, (g) ${\varepsilon }_{\mathrm{real}}=7$, and (h) ${\varepsilon }_{\mathrm{real}}=10$. The dispersion properties calculated by EMT are plotted as the red solid lines. The white dots surrounded by the dashed white circles in (e)–(h) represent the MPs’ frequencies at the maximum wave vectors where ξ exceeds 0.8.

Figure 4

Contour plot of spectral heat flux as a function of ${\varepsilon }_{\mathrm{real}}$. The ${\varepsilon }_{\mathrm{real}}$ range of Meta-NP ENZ mode is from 0.01 to 0.1. And, the ${\varepsilon }_{\mathrm{real}}$ range of 0.1 to 1 is the transition state where the Meta-NP ENZ mode gradually decouples into MPs and SPPs.

Figure 5

Distributions of electric field of different ${\varepsilon }_{\mathrm{real}}$ at the resonance frequencies. (a) ${\varepsilon }_{\mathrm{real}}=0.01$, ω = $2.1\times {10}^{14}$ rad/s; (b) ${\varepsilon }_{\mathrm{real}}=0.4$, ω = $1.98\times {10}^{14}$ rad/s; (c) ${\varepsilon }_{\mathrm{real}}=1$, ω = $1.51\times {10}^{14}$ rad/s; (d) ${\varepsilon }_{\mathrm{real}}=4$, ω = $1.02\times {10}^{14}$ rad/s; and (e) ${\varepsilon }_{\mathrm{real}}=10$, ω = $0.67\times {10}^{14}$ rad/s.

Figure 6

(a) The configuration of $\mathit{LC}$ circuit in the unit period. (b) Distribution of magnetic field in $xz$ plane, where the white lines indicate the induced current, the arrows indicate the direction, and the lengths indicate the magnitude. (c) Prediction of resonance frequencies by $\mathit{LC}$ circuit compared to RCWA.

Figure 7

Contour plots of energy transmission coefficients ξ as the function of Si grating filling ratio $f$ (a)–(d) and height $h$ (e)–(h) at $k_y=0$. (a) $f_{\mathrm{Si}}=0.2, {\varepsilon }_{\mathrm{real}}=1$; (b) $f_{\mathrm{Si}}=0.8, {\varepsilon }_{\mathrm{real}}=1$; (c) $f_{\mathrm{Si}}=0$.2, ${\varepsilon }_{\mathrm{real}}=0.01$; (d) $f_{\mathrm{Si}}=0.8, {\varepsilon }_{\mathrm{real}}=1$; (e) $h=0.1\mu \mathrm{m}, {\varepsilon }_{\mathrm{real}}=1$; (f) $h=0.5\mu \mathrm{m}, {\varepsilon }_{\mathrm{real}}=1$; (g) $h=0.1\mu \mathrm{m}, {\varepsilon }_{\mathrm{real}}=0.01$; and (h) $h=0.1\mu \mathrm{m}, {\varepsilon }_{\mathrm{real}}=0.01$.

Figure 8

Contour plots of energy transmission coefficients ξ of different ${\varepsilon }_{\mathrm{imag}}$ within the first Brillouin zone at $k_y=0$, and the real part is fixed as 0.01. (a) ${\varepsilon }_{\mathrm{imag}}=0.0001$, (b) ${\varepsilon }_{\mathrm{imag}}=0.001$, (c) ${\varepsilon }_{\mathrm{imag}}=0.01$, (d) ${\varepsilon }_{\mathrm{imag}}=0.1$, (e) ${\varepsilon }_{\mathrm{imag}}=0.4$, (f) ${\varepsilon }_{\mathrm{imag}}=0.7$, (g) ${\varepsilon }_{\mathrm{imag}}=1$, and (h) ${\varepsilon }_{\mathrm{imag}}=2$. The dashed rectangular box plots the range of frequencies and wave vectors where the resonance mode are located, which correspond to the range of spectral heat flux greater than 0.35 nJ/(${\mathrm{m}}^2$ rad).

Figure 9

(a) Contour plot of spectral heat flux as a function of ${\varepsilon }_{\mathrm{imag}}$, and the real part is fixed as 0.01. The ${\varepsilon }_{\mathrm{imag}}$ range of Meta-NP ENZ mode is from 0.0001 to 0.01. (b) Total heat flux as a function of ${\varepsilon }_{\mathrm{imag}}$.

Figure 10

Distributions of losses of different ${\varepsilon }_{\mathrm{imag}}$ at the resonance frequencies. (a) ${\varepsilon }_{\mathrm{imag}}=0.0001$, ω = $2.1\times {10}^{14}$ rad/s; (b) ${\varepsilon }_{\mathrm{imag}}=0.01$, ω = $2.21\times {10}^{14}$ rad/s; (c) ε = 0.4, ω = $2.5\times {10}^{14}$ rad/s; (d) ${\varepsilon }_{\mathrm{imag}}=1$, ω = $2.85\times {10}^{14}$ rad/s; and (e) ${\varepsilon }_{\mathrm{imag}}=10$, ω = $2.98\times {10}^{14}$ rad/s.

Figure 11

(a) Total heat flux as a function of $\mathrm{\Delta }{\omega }_p$. (b) Total heat flux vs vacuum gap $d$ from 0.04 to 1 µm. The temperatures of emitter and receiver are fixed as $T_1=400$ K and $T_2=300$ K in all cases.