# Near-field thermal radiation between deep subwavelength membranes driven by corner and edge modes

Jose Ordonez-Miranda, Roman Anufriev, Rafael Liñán-Abanto, Maelie Coral, Masahiro Nomura, Sebastian Volz

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Jose Ordonez-Miranda, Roman Anufriev, Rafael Liñán-Abanto, Maelie Coral, Masahiro Nomura, et al.. Near-field thermal radiation between deep subwavelength membranes driven by corner and edge modes. Journal of Applied Physics, 2026, 139 (8), pp.085108. 10.1063/5.0311645. hal-05527728

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Near-Field Thermal Radiation Between Deep Subwavelength Membranes Driven by
Corner and Edge Modes

Jose Ordonez-Miranda,¹, * Roman Anufriev,²,³ Rafael N. Liñán-Abanto,⁴

Maelie Coral,³ Masahiro Nomura,³,² and Sebastian Volz²,³

¹Sorbonne Université, CNRS, Institut des Nanosciences de Paris, INSP, F-75005 Paris, France

²LIMMS, CNRS-HS IRL 2820, The University of Tokyo, Tokyo 153-8505, Japan

³Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan

⁴Departamento de Física, Facultad de Ciencias, Universidad Nacional

Jorge Basadre Grohmann, Av. 8 Miraflores s/n, Tacna, 23000, Perú

(Dated: January 29, 2026)

We demonstrate that the thermal radiation between deep-subwavelength membranes of silicon carbide (SiC) exhibits a maximum enhancement over that of infinite SiC surfaces separated by the same vacuum gap. Based on fluctuational electrodynamics, we show that this enhancement occurs at a separation distance of 200 nm and increases for thinner and colder membranes. This peak arises from the dominant contribution of electromagnetic modes localized at the corner and vertical edges of sufficiently thin membranes, which enable a strong coupling of surface phonon-polaritons appearing along their top and bottom surfaces. These resonant corner and edge modes effectively extend the emission cross-section area of the membranes over their geometrical one and therefore amplify their thermal radiation. For 10-nm-thick membranes of SiC at 300 K, the thermal conductance reaches $54 \text{ pWK}^{-1}$, which yields a maximum enhancement of 4.5 over the value for infinite SiC surfaces. Our findings thus reveal that the regime of near-field thermal radiation driven by corner and edge modes emerges and is optimized in deep-subwavelength membranes separated by intermediate distances.

# I. INTRODUCTION

Near-field thermal radiation exceeds the blackbody limit between parallel surfaces and subwavelength membranes separated by a gap $d$ much smaller than the Wien's thermal wavelength $\lambda_{th}(\approx 10 \text{ µm at } 300 \text{ K})$. This radiative enhancement, predicted by fluctuational electrodynamics [1-4] and confirmed experimentally [5-7], is driven by the tunnelling and energy concentration of evanescent electromagnetic waves (surface phonon-polaritons (SPhPs) in polar dielectrics and surface plasmon-polaritons (SPPs) in metals) able to dramatically increase the density of states [8-10]. The cross-plane coupling of these energy carriers dominates the near-field radiation ($d \ll \lambda_{th}$) between macroscopic parallel surfaces [11, 12], while their in-plane propagation along the top and bottom surfaces of nanomembranes can enlarge their emission cross-section area more than 100 times and enhance the radiative heat flux over that of blackbody [5, 6, 13, 14]. Further, theoretical and experimental results showed that the SPhPs and SPPs can also enhance the in-plane radiative conduction along a single nanofilm [15-23]. Higher polariton thermal conductivities were observed for thinner, hotter, and longer nanofilms [24-31]. Abundant and recent evidence thus shows that the dimensional confinement in thin structures makes the surface polaritons powerful heat carriers capable of significantly enhancing the thermal radiation and radiative conduction beyond the limits set by the

Planck's law and phonon/electron heat conduction, respectively.

Considering that membranes with a deep subwavelength thickness $t \ll \lambda_{th}$ support not only the propagation of polariton modes[23], but also the existence of localized corner and edge modes[32, 33], their near-field thermal radiation is enriched by multiple electromagnetic excitations. The contribution of these latter modes to the near-field radiation is negligible for thick membranes ($t > \lambda_{th}$) and predominant for SiC membranes thinner than 50 nm[7]. For 20-nm-thick SiC membranes separated by a vacuum gap of 100 nm, Tang et al.[7] reported that these corner and edge modes lead to a room-temperature radiative heat transfer coefficient of $830 \text{ Wm}^{-2}\text{K}^{-1}$, which is 5.5 times higher than that of two infinite SiC surfaces separated by the same gap. This significant enhancement increases for thinner membranes and appears when both their thickness and separation distance are much smaller than the dominant thermal wavelength ($t < d \ll \lambda_{th}$). In contrast, Luo et al. [6] measured the near-field thermal radiation between coplanar SiN nanodevices and observed no enhancement beyond that between infinite surfaces, as expected for structures without facing corner and edge coupling. Considering that the extreme near-field thermal radiation ($d \to 0$) is dominated by the strong cross-plane coupling of surface polaritons, the corner and edge modes are expected to drive the near-field thermal radiation for intermediate separation distances ($d \sim 100 \text{ nm}$) only. This observation suggests that the maximum contribution of the corner and edges modes compared to that of the evanescent ones emerges for a critical separation distance, which is not known yet. The study of the thermal radiation be-

<footnote>* jose.ordonez@cnrs.fr</footnote>
tween subwavelength membranes separated by different intermediate separation distances is thus essential to understand and optimize the capabilities of the corner and edge modes for enhancing the radiative transfer beyond the one driven by evanescent modes.

In this work, we perform fluctuational electrodymanic simulations to investigate the enhancement of the near-field thermal radiation between two deep subwavelength SiC membranes with both thicknesses and separation distances much smaller than the Wien's thermal wavelength ($t < d \ll \lambda_{th}$). We find that the transition from the near-field regime to the extreme near-field one is characterized by a pronounced enhancement in the radiative heat rate compared to that exchanged between two infinite SiC surfaces. This peak appears at $d = 200$ nm, is higher for thinner membranes, and arises from the dominant contribution of corner and edge modes. Our investigation thus aims to determine the optimal conditions required to achieve the near-field thermal radiation driven by these modes reported in previous works [7, 32, 33].

<figure><img src="image_2.png" /><figcaption>FIG. 1: Scheme of two subwavelength membranes exchanging thermal radiation driven by propagating (black lines), evanescent (reddish wavy lines), and corner (yellow and light blue clouds) electromagnetic modes. The red and blue colors of the geometrically identical membranes indicate high ($T + \Delta T$) and low ($T$) temperatures, respectively. The ray-like depiction of the propagation modes emphasizes their directional, radiative character, and long propagation length; while the wavy representation of the evanescent modes underscores their bound, non-radiative nature, and localization to the vicinity of the surfaces. The membranes' thickness is smaller than the separation distance and much smaller than the Wien's thermal wavelength ($t < d \ll \lambda_{th}$) to ensure the appearance of corner and edge modes.</figcaption></figure>

## II. THEORETICAL METHODOLOGY

Let us consider two plane membranes exchanging heat by thermal radiation due to their temperature difference

$\Delta T$, as shown in Fig. 1. The length $l = 10$ µm and width $a = 20$ µm of the identical membranes were chosen sufficiently large to ensure a sizable heat exchange and small enough to keep the numerical calculations manageable. The thickness $t \le 200$ nm of these subwavelength membranes was selected small enough to ensure the cross-plane coupling and in-plane propagation of SPhPs [5, 16, 23], while their separation distance $d = 10 - 1000$ nm was varied two orders of magnitude to investigate the optimal conditions for the thermal radiation enhancement driven by corner and edge modes. We focus on this “dual nanoscale regime” ($t$ and $d \ll \lambda_{th}$) because of its favorable conditions for the predominant contribution of these latter modes over that of the propagating and evanescent modes [7].

The uniform temperature of each membrane generates the internal fluctuation of electromagnetic currents, whose energy emission results in thermal radiation [23, 34]. Considering that the temperature difference between the membranes is relatively small ($\Delta T \ll T$), the net heat rate $q = G\Delta T$ between the membranes is determined by the thermal conductance $G$ given by fluctuational electrodynamics, as follows [13, 35]

$$G = \int_{0}^{\infty} \hbar\omega \frac{\partial f_{\omega}(T)}{\partial T} \Phi(\omega) d\omega. \quad (1)$$

where $\omega$ is the radiation frequency, $f_{\omega}(T) = [\exp(\hbar\omega/k_{B}T) - 1]^{-1}$ is the Bose-Einstein distribution function, $\hbar$ and $k_{B}$ are the respective reduced Planck and Boltzmann constants, and $\Phi$ is the transmission function of the energy emitted from the left membrane (emitter) into the right one (receiver). The net heat rate and thermal conductance in Eq. (1) are independent of the medium surrounding the membranes due to the electromagnetic reciprocity of the transmission function [$\Phi(\text{emitter} \rightarrow \text{receiver}) = \Phi(\text{receiver} \rightarrow \text{emitter})$] and their identical geometry [$\Phi(\text{surrounding} \rightarrow \text{emitter}) = \Phi(\text{surrounding} \rightarrow \text{receiver})$] [36]. The frequency spectrum of $\Phi(\omega)$ is determined by the electromagnetic fields generated by the fluctuation of surface and volumetric currents, which we calculate through the boundary-element method [36]. The predictions of this method have been validated by abundant experimental data obtained for both the near-field and far-field regimes of thermal radiation [5, 6, 13, 19, 37], and we used it here via the free software solver Scuff-EM. Equation (1) is equivalent to the Landauer formula [8] with a transmission function $\Phi = (2\pi)^{-1} \sum_{i} \tau_{i}$ determined by the sum of the transmission probabilities $\tau_{i}$ of each channel "i" (i. e. TM and TE polarizations as well as propagating, evanescent, corner, and edge modes) of thermal radiation. Therefore, even though the individual channel probability takes values between 0 and 1, $\Phi$ can be greater than unity, due to the general multi-channel energy transmission. The calculation of $\Phi$ via the solver Scuff-EM yields the frequency spectrum of the total energy transmission across all modes, rather than the transmission probability of each radiation channel.
For the reference case of two infinite and identical surfaces separated by a gap $d$, the thermal conductance $G = G_\infty$ for an element $a \times t$ of cross-section area is still given by Eq. (1) with the following explicit transmission function $\Phi = \Phi_\infty$ [34, 38]

$$ \Phi_{\infty} = \frac{at}{(2\pi)^2} \left[ \int_{0}^{k_0} (\tau_e^f + \tau_m^f) kdk + \int_{k_0}^{\infty} (\tau_e^n + \tau_m^n) kdk \right], \quad (2) $$

where $k_0 = \omega/c$ and $c$ are the respective wave vector and speed of light in vacuum, $k$ is the surface-parallel component of the wave vector, and $\tau_u^v$ is the spectral transmissivity for the far-field ($v = f$) and near-field ($v = n$) radiation of the transverse electric ($u = e$) and transverse magnetic ($u = m$) modes defined by

$$ \tau_u^f = \frac{\left(1 - |r_u(k, \omega)|^2\right)^2}{\left|1 - r_u^2(k, \omega) \exp\left(2i\gamma_0 d\right)\right|^2}, \quad (3a) $$

$$ \tau_u^n = 4 \frac{\mathrm{Im}[r_u(k, \omega)]^2 \exp(-2\gamma d)}{|1 - r_u^2(k, \omega) \exp(-2\gamma d)|^2}, \quad (3b) $$

with $\gamma_0 = \sqrt{k_0^2 - k^2}$ and $\gamma = \sqrt{k^2 - k_0^2}$. The Fresnel reflection coefficients $r_e$ and $r_m$ at the surface-vacuum interface for the transverse electric and transverse magnetic polarizations are respectively given by

$$ r_e(k, \omega) = \frac{\gamma_0(k, \omega) - \Gamma(k, \omega)}{\gamma_0(k, \omega) + \Gamma(k, \omega)}, \quad (4a) $$

$$ r_m(k, \omega) = \frac{\varepsilon\gamma_0(k, \omega) - \Gamma(k, \omega)}{\varepsilon\gamma_0(k, \omega) + \Gamma(k, \omega)}, \quad (4b) $$

where $\Gamma = \sqrt{\varepsilon k_0^2 - k^2}$ and $\varepsilon$ is the relative permittivity of the radiating bodies. Taking into account that the membranes and elements of the infinite surfaces have the same cross-section area $a \times t$ and are separated by the same distance $d$, the ratio $G/G_\infty$ is expected to capture the physical effects of the corner and edge modes, as they are not present for the infinite surfaces ($t$ and $a \to \infty$). Considering the finite and small thickness $t$ and width $a$ of the considered subwavelength membranes, the transmission function $\Phi_\infty$ in Eq. (2) determines the thermal conductance $G_\infty$ in nWK$^{-1}$, which facilitates its comparison with $G$ and prior experimental data [5, 6]. The analysis in terms of $G$ and $G_\infty$ is equivalent to that in terms of the heat transfer coefficients $h = G/(a \times t)$ and $h = G/(a \times t)$ in WK$^{-1}$cm$^{-2}$, via the finite and physically meaningful ratio $G/G_\infty = h/h_\infty$, while avoiding the artificial divergence of $h$ as $t \to 0$.

The Scuff-EM calculations were done by discretizing both membranes into surface elements, as shown in Fig. 2(a). The mesh was chosen sufficiently fine to generate 43236 fluctuational surface currents and triangular units with an average side (0.24 µm) much smaller than the relevant wavelengths ($\ge 1$ µm) of thermal radiation. All calculations were done for SiC membranes, which can support the propagation of SPhPs and were

<figure><img src="image_3.png" /><figcaption>FIG. 2: (a) Surface mesh generated for the Scuff-EM calculations performed with the (b) frequency spectrum of the permittivity $\varepsilon = \varepsilon_R + i\varepsilon_I$ of SiC [20, 39]. The yellow zone ($\omega/2\pi = 23.7 - 28.9$ THz) represents the Reststrahlen band ($\varepsilon_R < 0$). (c) Convergence analysis of the transmission function with respect to the number of fluctuational currents (unknowns) and mesh size (average side of the triangular units) used in the Scuff-EM simulations. Calculations were done for 20-nm-thick membranes separated by $d = 200$ nm. The superposition of the blue and red points indicates convergence.</figcaption></figure>
widely used in previous studies [5–7, 25, 28]. The fre-
quency spectrum of the real and imaginary parts of the
SiC permittivity ε = ε<sub>R</sub> + iε<sub>I</sub> required for the determi-
nation of Φ are shown in Fig. 2(b). This optical property
of SiC practically does not change with temperature be-
low 500 K [25, 40] and therefore we considered it to be
independent of temperature and limited our calculations
for T < 500 K. The relatively narrow peak of ε<sub>I</sub> and neg-
ative dip of ε<sub>R</sub> indicate that SiC absorbs energy and fa-
vors the SPhP propagation in short frequency intervals,
respectively [23]. The thermal excitation and propaga-
tion of SPhPs in SiC membranes are thus expected to
enhance the thermal radiation via their energy emission
at the membranes’ boundaries. The superposition of the
blue and red points shown in Fig. 2(c) indicates that the
used mesh is fine enough to ensure the convergence of the
transmission function for frequencies up to 32 THz.

III. RESULTS AND DISCUSSION

We here comparatively show and analyze the predic-
tions of Eqs. (1) and (2) applied to two membranes and
two infinite surfaces of SiC separated by the same vac-
uum gap *d*. Considering that the corner and edge modes
mainly appear in membranes with a deep-subwavelength
thickness (*t* < *d* ≪ λ<sub>th</sub>)[7], we perfomed calculations for
various nanoscale values of *t* and *d*, while keeping their
in-plane dimensions fixed at the microscale (*a* = 2*l* =
20 µm).

The frequency spectrum of the transmission functions
Φ and Φ∞ for two membranes (points) and two infi-
nite surfaces (solid lines) of SiC are shown in Fig. 3(a),
for five representative separation distances. Note that,
for the considered near-field regime (d ≤ 500 nm), the
main contribution to Φ and Φ∞ arises from the Rest-
strahlen band (yellow zone) supporting the appearance
and coupling of SPhPs along the membranes’ surfaces
and across their gap. Since this band contains the SPhP
resonance (minimum of εR at 23.7 THz), the behavior of
both transmission functions indicates that the near-field
radiation between the membranes and infinite surfaces
is dominated by SPhPs, as expected. The stronger cou-
pling of these evanescent modes for smaller separation
distances results in higher transmission functions Φ and
Φ∞. While the major contribution to Φ∞ arises from
its peak at 28.34 THz, Φ takes high values not only at
this frequency, but also at lower frequencies, especially
for the near-field radiation (d ≤ 100 nm). This widening
of Φ with values higher than those of Φ∞ for frequencies
around 24 THz, indicates that the energy transmission
between the membranes can be better than across the in-
finite surfaces. Considering that the infinite surfaces only
support the cross-plane coupling of SPhPs through their
gap, the difference Φ − Φ∞ is explained by the in-plane
propagation of SPhPs along the membranes and their
cross-plane coupling through each membrane. This lat-
ter propagation and coupling generate additional modes

<figure><img src="image_4.png" /><figcaption>FIG. 3: (a) Spectral transmission function Φ and (b) temperature evolution of the thermal conductance *G* between two SiC membranes of thickness *t* = 20 nm. The solid and dashed lines in (a) and (b) represent *Φ*∞ and *G*∞ obtained for two infinite surfaces, respectively. The black arrow in (b) indicates that the dotted lines stand for the ratio *G*/*G*∞. The Reststrahlen band is highlighted in yellow in (a) and the calculations were done for various separation distances.</figcaption></figure>

[7], whose radiative contribution becomes significant for relatively low frequencies (~24 – 26 THz) and intermediate gaps (*d* = 100 – 250 nm), for which the peak of Φ at 28.34 THz is comparable to that of Φ∞. Even though the evanescent cross-plane SPhP modes in both the membranes and infinite surfaces follow the same dispersion relation determined by the single SiC/vacuum interface [7], their coupling across the gap of the membranes leads to an energy transmission lower than the corresponding one across infinite surfaces, as established by the condition Φ ≤ Φ∞ for the SPhP peak at 28.34 THz shown in Fig. 3(a). The ratio *G*/*G*∞ thus highlights the finite-size effects on the thermal radiation between the membranes. Further, the secondary peak of Φ at 28.9 THz is
related to the SPhP confinement to the surface of the SiC membranes and appears even in the far-field regime [41]. The thermal radiation between the subwavelength membranes is thus expected to be driven not only by the classical cross-plane evanescent modes (peak at 28.34 THz), but also by the ones generated by the in-plane propagation of SPhPs (widening around 24 THz).

This hypothesis is confirmed by the enhancement $G/G_\infty > 1$ shown in Fig. 3(b), where the thermal conductance $G$ exceeds its infinite-surface counterpart $G_\infty$ due to the contribution of the in-plane modes. The monotonical increase of $G$ and $G_\infty$ for smaller gaps and higher temperatures results in the ratio $G/G_\infty$ with a highly nonlinear correlation with $T$ and $d$. For $d = 100$ nm, $G/G_\infty > 1$ for the vast majority of the considered temperatures and takes its maximum ($\sim 3$) at about 230 K. This three-fold enhancement is consistent with experimental data reported for SiC membranes at 300 K [7] and arises from the broad spectrum of the transmission function $\Phi$ shown in Fig. 3(a). The low-frequency broadening of $\Phi$ (absent in $\Phi_\infty$) increases the thermal conductance $G$ between the membranes at relatively low temperatures mainly, as established by Eq. (1). For the separation gap $d = 100$ nm, this additional low-frequency contribution maximizes the thermal radiation enhancement $G/G_\infty$ around 230 K, as shown in Fig. 3(b). For a longer separation distance ($d = 1000$ nm), the peak of $G/G_\infty$ reduces drastically and dissappears for a short enough gap ($d = 10$ nm). This fact indicates that the radiative enhancement $G/G_\infty$ has a maximum for intermediate ($d = 100 - 1000$ nm) separation distances.

<figure><img src="image_5.png" /><figcaption>FIG. 4: Thermal conductance between two SiC membranes ($G$) and infinite surfaces ($G_\infty$) as a function of their separation distance. The points joined by the dashed lines represent the ratio $G/G_\infty$. Calculations were done for the thickness $t = 20$ nm and two temperatures.</figcaption></figure>

Figure 4 comparatively shows the computed thermal conductances $G$ and $G_\infty$ as functions of the separation

distance $d$, for temperatures of 300 and 500 K. For each temperature, both $G$ and $G_\infty$ monotonically increase as $d$ decreases, due to the near-field coupling of evanescent modes. The increase rate (slope) of $G_\infty$ is generally higher than that of $G$, but its values are lower than those of $G$, for intermediate separation distances ($d \approx 50 - 1000$ nm). This distinctive near-field behavior of $G$ and $G_\infty$ yields an enhancement $G/G_\infty > 1$ that reaches its room-temperature maximum ($\sim 3.5$) for $d = 200$ nm. Lower enhancements are obtained for higher temperatures as well as longer and shorter separation distances, in agreement with the results displayed in Fig. 3(b). Interestingly, within the Reststrahlen band (23.7 - 28.9 THz) shown in Fig. 3(a), this optimal gap $d = 200$ nm for $T = 300$ K, is comparable to the decay length $\delta \sim 1/\text{Re}(k) = (100 - 250)$ nm of corner and edge modes reported for SiC membranes [7]. The coupling condition $d \sim \delta$ thus maximizes the energy transmission and hence the thermal radiation enhancement $G/G_\infty$ by balancing the strong overlap (favored at smaller $d$) of these modes against their suppression by the short-$d$ cross-plane evanescent polaritons. Higher (lower) temperatures shift $d$ to smaller (larger) values, as hotter (cooler) membranes support the excitation of higher-$\omega$ (lower-$\omega$) modes with shorter (longer) $\delta$ via the Bose-Einstein distribution function in Eq. (1).

The existence of a maximum enhancement indicates that the membranes' in-plane effects, not present for the infinite surfaces, have a predominant contribution to the thermal radiation for intermediate near-field separation distances. This fact is confirmed by Fig. 5 showing the monotonical increase of $G/G_\infty$ for thinner membranes supporting a stronger coupling of SPhPs propagating along their top and bottom surfaces. For $t < 100$ nm, the non-linear behavior of $G > G_\infty$ produces a radiative enhancement that follows the power law $G/G_\infty \propto t^{-0.4}$ (for $t < 50$ nm) and takes values that are consistent with the experimental observations of Tang et al.[7] for room-temperature SiC membranes. This thickness dependence is expected to arise from the emergence of additional modes beyond the evanescent and propagating contributions that dominate the near-field ($d < 100$ nm) and far-field ($d > 1000$ nm) regimes, respectively. Therefore, these extra modes can lead to thermal radiation between thin membranes ($t < 100$ nm) that outperforms the one between two elements of the infinite surfaces with the same cross-section area $a \times t$. Importantly, the results shown in Figs. 4 and 5 for $G$ and $G/G_\infty$ are consistent with those reported by Tang et al.[7] using the discrete system Green's function (DSGF) method. The Scuff-EM and DSGF approaches therefore yield similar predictions, and the remaining differences can be attributed to the distinct dimensions and permittivities of the SiC membranes considered in each calculation.

The spatial distribution of the in-plane component $S_z$ of the Poynting vector around the radiating surfaces of two SiC membranes separated by gaps of 200 nm and 20 nm is respectively shown in Figs. 6(a,b,c) and 6(d,e,f),
<figure><img src="image_6.png" /><figcaption>FIG. 5: Thermal conductances $G$, $G_\infty$, and their ratio $G/G_\infty$ calculated for two SiC membranes as functions of their thickness. The ratio $G/G_\infty = h/h_\infty$ quantifies the finite-size effects independently of the geometrical area normalization involved in the heat transfer coefficients $h = G/(a \times t)$ and $h_\infty = G_\infty/(a \times t)$. Our analysis in terms $G$ and $G_\infty$ is therefore equivalent to that in terms of $h$ and $h_\infty$, while avoiding the artificial divergence of $h$ as $t \to 0$. Calculations were done for the resonant separation distance $d = 200$ nm (see Fig. 4) and two temperatures.</figcaption></figure>

for three representative thicknesses. For the 20-nm-thick membranes separated by $d = 200$ nm (Fig. 6(a)), this energy density is relatively high and concentrated around the corners and vertical edges of the membranes. As the thickness increases (Figs. 6(b) and 6(c)), this cross-plane energy concentration decreases together with the in-plane one distributed along the top and bottom surfaces of the membranes. The observed increase of $S_z$ at the corners and edges is thus directly related to the SPhP energy at the membranes' surfaces, as the SPhP coupling along the $y$ direction becomes stronger for thinner membranes. This localized behavior of the energy density with the thickness of the two membranes is consistent with the one reported at the corners and edges of a single SiC membrane [7] and the appearance of electromagnetic corner and edge modes in a lossy metallic membrane with a finite length $l[32, 33]$. These modes result from the interference between the incoming and reflected SPhPs at the membrane boundaries. The incident SPhPs reflect on the vertical edges due to the permittivity contrast with vacuum. The resulting in-plane interference and cross-plane coupling localize resonant fields at the edges and corners, as predicted by Berini [32, 33] and confirmed by the Poynting vector maps shown in Fig. 6. According to Fig. 3(a), these modes have a predominant contribution to the thermal radiation between SiC membranes within the frequency range 24–26 THz, which spans over the broadened, redshifted peak below that of the

evanescent modes at 28.34 THz. The energy concentration around the corners and edges of the membranes is thus explained by the existence of these localized modes that exhibit fields strongly concentrated around the corners and edges, and decay exponentially along each membrane and across their vacuum gap [32, 33]. This evanescent penetration of the corner and edge fields generates their coupling and energy transmission across the gap, when the gap and thickness of the membranes are much smaller than the thermal wavelength ($t \ll d < \lambda_{th}$). The tunneling of the corner and edge modes is thus analogous to the near-field coupling of SPhPs between infinite surfaces, but with an effective emission cross-section area larger than the geometrical one. This evanescent tunneling and surface enlargement thus enhance the thermal radiation, as shown in Figs. 4 and 5. The radiation is further enhanced by the in-plane emission of energy that transmits through the separation gap at the surface's positions $y = 0$ and $t$ mainly. This in-plane energy confinement through the gap is directly related to the coupling of SPhPs propagating along the top and bottom surfaces of the membranes, as it becomes weaker for thicker membranes and dissappears for $t > 200$ nm. For a shorter separation distance ($d = 20$ nm), on the other hand, Figs. 6(d,e,f) shows that the corner and edge modes are still present, but the energy transmission between the membranes is dominated by the strong coupling of evanescent modes arising from their geometrical cross-section area. Therefore, the corner and edge modes, not present for the infinite surfaces, only dominate the near-field thermal radiation for intermediate separation distances ($d \sim 200$ nm) between SiC membranes with a sufficiently small thickness ($t < 100$ nm).

According to Figs. 4–6, the regime of thermal radiation driven by corner and edge modes appears for membranes with a thickness and separation distance much smaller than the Wein’s radiation wavelength ($t < d \ll \lambda_{th}$), and reaches its highest values for $d = 200$ nm. This enhanced thermal radiation modulated by the membranes’ size can be maximized even more with temperatures around $T = 230$ K, as shown in Fig. 3(b). The corner and edge modes thus induce an intermediate (on $d$ and $T$) regime of thermal radiation between the near- and far-field regimes driven by evanescent and propagating modes, respectively. This intermediate regime observed for SiC membranes, also appears for other polar materials, such as SiN and SiO2, which can not only enhance the thermal radiation, but also attenuate it below the level between infinite surfaces [4]. The enhancement or attenuation is driven by the material losses, which reduce the density of available corner and edges modes between the membranes.

## IV. CONCLUSIONS

We have demonstrated that the thermal radiation between SiC membranes is strongly dominated and en-
<figure><img src="image_7.png" /><figcaption>FIG. 6: Density map of the in-plane component $S_z$ of the Poynting vector between two SiC membranes separated by a distance of (a,b,c) 200 nm and (d,e,f) 20 nm. $S_z$ allows visualizing the energy density of the corner and edges modes, as the $z$-direction is parallel to the vertical edges that reflect the SPhPs to form these modes [7]. $S_z$ thus provides the “physical picture” of the modes’ geometry. Calculations were done with the Scuff-EM solver at the middle (half-width $z = 0$) of membranes with three thicknesses and for the peak frequency $\omega/2\pi = 28.34$ THz of the transmission function shown in Fig. 3(a). The in-plane dimensions ($l = 10$ µm and $a = 20$ µm) of the entire membranes (simulation domain size) are much larger than the typical decay length $\delta = (100-250)$ nm of the corner and edge modes [7], which ensures their localization far from the domain boundaries and hence the absence of significant boundary effects. The $S_z$ maps were computed for the energy emitted by the left (hot) membrane only, which explains their asymmetry across the gap between the identical membranes. The dashed white lines represent the positions of the membranes’ surfaces.</figcaption></figure>

hanced by corner and edge modes appearing for thicknesses and separation distances much smaller than the Wien’s thermal wavelength. Using fluctuational electro-dynamics simulations, we have found that the maximum enhancement occurs for a separation distance of 200 nm and monotonically increases as the membranes’ thickness decreases. The energy transmission analysis reveals that these resonant modes localized at the corners and vertical edges of the membranes originate from the strong coupling of surface phonon-polaritons appearing along their top and bottom surfaces. These modes enlarge the emission cross-section area of the membranes and hence enhances their thermal radiation. For 10-nm-thick

membranes of SiC at 300 K, we have obtained a thermal conductance of $54 \text{ pWK}^{-1}$, which yields a maximum enhancement of 4.5 over that of infinite SiC surfaces. These results reveal that extreme subwavelength membranes separated by intermediate distances can substantially enhance the near-field thermal radiation through corner and edge electromagnetic modes.
ACKNOWLEDGMENTS

This work was supported by the CREST JST (Grant
N° JPMJCR19I1) and KAKENHI JSPS (Grant N°
21H04635 and JP20J13729) projects.

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