# Long-range, short-wavelength, and ultrafast heat conduction driven by three plasmon modes supported by graphene

Jose Ordonez-Miranda, Yuriy Kosevich, Masahiro Nomura, Sebastian Volz

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Jose Ordonez-Miranda, Yuriy Kosevich, Masahiro Nomura, Sebastian Volz. Long-range, short-wavelength, and ultrafast heat conduction driven by three plasmon modes supported by graphene. Physical Review B, 2023, 108 (16), pp.L161404. 10.1103/PhysRevB.108.L161404. hal-04235202

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<figure><img src="image_1.png" /></figure>
Long-range, Short-wavelength and Ultrafast Heat Conduction Driven by Three
Plasmon Modes Supported by Graphene

Jose Ordonez-Miranda,<sup>1,2,*</sup> Yuriy A. Kosevich,<sup>3,2</sup> Masahiro Nomura,<sup>2,1</sup> and Sebastian Volz<sup>1,2</sup>

<sup>1</sup>LIMMS, CNRS-IIS IRL 2820, The University of Tokyo, Tokyo 153-8505, Japan

<sup>2</sup>Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan

<sup>3</sup>N. N. Semenov Federal Research Center for Chemical Physics of Russian Academy of Sciences,
4 Kosygin Street, Moscow 119991, Russian Federation

(Dated: October 9, 2023)

We demonstrate the existence and propagation of three hybrid modes of surface plasmon-polaritons supported by two graphene monolayers coating a solid film. These modes propagate long distances with short wavelengths, which are suitable features to enhance the heat conduction along the film interfaces. For a Si film with a thickness of 15 nm and a length of 5 mm at 300 K, we find a plasmon thermal conductivity of 13.6 Wm<sup>-1</sup>K<sup>-1</sup>, which represents 67% (26%) of its phonon Si (Si+graphene) counterpart. This thermal energy appears due to the coupling of plasmons propagating at speeds comparable to the speed of light in vacuum. The plasmonic heat conduction driven by two-dimensional materials thus appears as a major and unexpected mechanism for controlling temperature in solid-state systems at rates faster than the ones of phonons and electrons.

Heat dissipation in dielectrics is mainly driven by acoustic phonons with long mean free path and high group velocity as compared with those of optical phonons. Even though optical phonons have larger heat capacities than the acoustic ones, their ability to dissipate heat is limited by their localized nature and low speeds. In polar dielectrics, however, the optical phonons support the propagation of surface phonon-polaritons (SPhPs), which are electromagnetic waves confined to the material interface [1–6]. The in-plane propagation and cross-plane evanescent nature of these surface waves enable them to propagate distances (> 1 mm) much greater than the typical mean free path of phonons and electrons [7, 8] and hence their guiding along interfaces can generate an efficient channel of heat conduction [9–11].

Previous theoretical [1, 2] and experimental [9, 10, 12, 13] works reported that the SPhP contribution to the in-plane thermal conductivity of polar nanofilms can be comparable to or higher than its phonon counterpart. This thermal performance enhancement arises from the coupling of a single SPhP mode propagating along the two nanofilm interfaces [1, 2]. The SPhP coupling also appears in micro-sized structures made up of a thick Si layer sandwiched by polar nanolayers [14]. The weak energy absorption in this symmetric structure support the propagation of multiple SPhP modes, which can enhance the in-plane SPhP heat transport to values ten times higher than the corresponding one of a single polar nanofilm [14, 15]. As the Si thickness scales down, the number of these SPhP modes reduces and reaches its minimum of two modes for nanoscales. This is also the case of surface plasmon-polaritons (SPPs) generated by the hybridization of photons and free electrons in metallic nanofilms. The plasmonic heat conduction along these nanofilms is driven by two fundamental plasmon modes [16, 17], whose contribution to significantly boost the in-plane thermal conductivity of a Ti nanofilm was recently

observed experimentally [18]. These works [15, 16, 18] suggest that the plasmonic heat conduction could naturally be enhanced via highly conductive two-dimensional (2D) materials. Even though the electron thermal conductivity of graphene has been widely studied [19–23] along with its ability to support plasmon modes [24–26], their contribution to the heat conduction is not known yet. As the number of these modes can be multiplied by coating a solid with graphene, they have the potential to enhance the thermal performance of solids, without changing their internal material structure.

The purpose of this Letter is to demonstrate the potential of 2D materials for enhancing heat conduction currents. For a film coated with a 2D material, we show that this current is driven by three ultrafast modes featuring long propagation distances and short wavelengths. These features allow them to significantly enhance the thermal performance of a graphene-coated nanofilm.

<figure><img src="image_2.png" /><figcaption>FIG. 1: Scheme of a film supporting the propagation of three SPP modes. The curvy lines stand for the SPP magnetic field, whose discontinuity at the film interfaces appears due to their coating with a graphene monolayer.</figcaption></figure>

Let us consider a suspended film supporting the prop-
agation of SPPs along its two interfaces coated with
graphene monolayers, as shown in Fig. 1. The film
and its surrounding medium have relative permittivities
ε₁ and ε₂, respectively. Considering that the film sur-
faces x = 0 and x = l are in contact with thermal
baths at temperatures T_h and T_c(< T_h), the free electrons
of graphene oscillate and emit an electromagnetic field. These oscillations induce the excitation of neighboring electrons, which keeps the field propagation along the film interfaces mainly. This thermally excited field represents a SPP propagating in a broad range of frequency [18, 25, 27]. Under this condition, the heat propagates along the x axis mostly via the combined dynamics of the film phonons (or electrons) and SPPs. Given that SPPs are bosons [6], the Boltzmann transport equation establishes that their contribution k to the in-plane thermal conductivity of the film is given by [4]

$$k = \frac{1}{4\pi d} \int \hbar\omega \mathrm{Re}(\beta) \Lambda_e \frac{\partial f}{\partial T} d\omega, \quad (1)$$

where $\hbar$ is the Planck's constant divided by $2\pi$, Re($\beta$) is the real part of the in-plane SPP wave vector $\beta$, $f = [\exp(\hbar\omega/k_B T) - 1]^{-1}$ is the Bose-Einstein distribution, $T = (T_h + T_c)/2$ is the average temperature, $k_B$ is the Boltzmann constant, $\omega$ is the spectral frequency, and $\Lambda_e$ is the effective propagation length defined by [4]

$$\Lambda_e = \Lambda \left[ 1 - \frac{4}{\pi \mu} \left( \frac{2}{3} - E_5(\mu) \right) \right], \quad (2)$$

where $\mu = l/\Lambda$, $\Lambda = [2\mathrm{Im}(\beta)]^{-1}$ is the intrinsic SPP propagation length, $l$ is the film length, and $E_5(x) = \int_0^{\pi/2} \cos^3(\theta)e^{-x/\cos(\theta)}d\theta$. The effective propagation length $\Lambda_e$ represents the distance that SPPs can travel along the film interfaces and therefore its values are comparable to or smaller than $l$ and $\Lambda$. In the diffusive regime ($\mu = l/\Lambda \gg 1$), $\Lambda_e = \Lambda$, and Eq. (2) reduces to the previous one derived by Chen et al. [1]. By contrast, for the ballistic limit ($\mu \ll 1$), the SPP propagation is limited by the film length and $\Lambda_e = 2l/\pi$. The SPP heat transport is hence enhanced along a film with length smaller than the SPP propagation length ($l \ll \Lambda$). For an arbitrary $\Lambda_e = \Lambda_e(\Lambda, l)$, Eqs. (1) and (2) show that $k$ depends on the material properties through the product Re($\beta$)$\Lambda_e(0.5/\mathrm{Im}(\beta), l)$ driven by the real and imaginary parts of the SPP wave vector $\beta(\omega)$. As $k$ increases with this product, the optimal material configuration to maximize the SPP heat transport is given by a large wave vector Re($\beta$) (short wavelength $2\pi/\mathrm{Re}(\beta)$) and a long propagation length (small Im($\beta$)). The predictions of Eq. (1) derived from the Boltzmann transport equation agree with the corresponding ones of the fluctuational electrodynamics theory [28] and describe well experimental data for the thermal conductivity of SiO$_2$ and SiN nanofilms [9, 10]. After solving the Maxwell equations for the transverse magnetic polarization required for the existence of SPPs, the following two dispersion relations for $\beta(\omega)$ are obtained[29]

$$\frac{\varepsilon_2}{p_2} + \frac{\varepsilon_1}{p_1} \coth(p_1 d/2) = -\frac{i\sigma}{\omega\varepsilon_0}, \quad (3a)$$

$$\frac{\varepsilon_2}{p_2} + \frac{\varepsilon_1}{p_1} \tanh(p_1 d/2) = -\frac{i\sigma}{\omega\varepsilon_0}, \quad (3b)$$

where $i = \sqrt{-1}$ is the imaginary unit, $\varepsilon_0$ is the vacuum permittivity, $\sigma$ is the 2D electrical conductivity of a graphene monolayer, and $p_n$ are the cross-plane wave vectors given by $p_n^2 = \beta^2 - \varepsilon_n k_0^2$, with $k_0 = \omega/c$ and $c$ being the wave vector and speed of light in vacuum, respectively. The modes in Eqs. (3a) and (3b) describe the SPP propagation with a respective symmetric ($H_y(-z) = H_y(z)$) and anti-symmetric ($H_y(-z) = -H_y(z)$) magnetic field, as detailed in the supplementary material (SM) [29]. For a very thick film ($|p_2| d \gg 2$), both Eqs. (3a) and (3b) become independent of the film thickness and reduce to the well-known dispersion relation $\varepsilon_1/p_1 + \varepsilon_2/p_2 = -i\sigma/(\omega\varepsilon_0)$ of a single interface [2]. For very thin films ($|p_2| d \ll 2$), on the other hand, the solution of Eqs. (3a) and (3b) for $\beta = k_0\sqrt{\varepsilon}$ can conveniently be found in terms of the effective permittivity $\varepsilon$ and the normalized thickness $\lambda = k_0 d/2 \ll 1$, as detailed in the SM [29]. For an approximation up to $\lambda^2$, the solutions $n=1$ and $n=2$ thus obtained for Eqs. (3a) and (3b) are respectively given by

$$\varepsilon = \frac{i\varepsilon_1}{S\lambda} + \frac{i\varepsilon_2}{S}\sqrt{\frac{i\varepsilon_1}{S\lambda}} + \varepsilon_1 - \frac{1}{S^2}\left(\frac{\varepsilon_1^2}{3} + \frac{\varepsilon_2^2}{2}\right), \quad (4a)$$

$$\varepsilon = \varepsilon_2 - \left(\frac{\varepsilon_2}{S}\right)^2 \left[1 + 2i \frac{\varepsilon_1 \lambda}{S} - 3 \left(\frac{\varepsilon_1 \lambda}{S}\right)^2\right], \quad (4b)$$

where $S = \sigma/(c\varepsilon_0) \neq 0$ is the normalized and dimensionless electrical conductivity of graphene. Since $\lambda \ll 1$, the leading term in Eq. (4a) indicates that the mode $n=1$ features a large wave vector Re($\beta$) $\rightarrow d^{-1/2}$ and a small propagation length $\Lambda \rightarrow d^{1/2}$. These short wavelength and short-range SPPs appear due to the increasing energy absorption of graphene, as the film thickness scales down. For this branch $n=1$, the SPP propagation along the in-plane and cross-plane directions are driven by the same wave vectors ($\beta \approx p_1 \approx p_2$) [29]. For mode $n=2$, on the other hand, $\beta \rightarrow k_0\sqrt{\varepsilon_2 - (\varepsilon_2/S)^2}$ becomes independent of the film thickness and permittivity, for thin enough films. Therefore, the coupling of the SPP mode $n=2$ propagating along the two film interfaces saturates, as the film thickness sizes down. Note that both modes $n=1$ and $n=2$ exist due to the presence of graphene ($S \neq 0$) and they are respectively driven by the media permittivities $\varepsilon \rightarrow \varepsilon_1$ and $\varepsilon \rightarrow \varepsilon_2$, for a very high conductivity $S \rightarrow \infty$. Both modes can thus exist even if the film and its surrounding medium are made of the same material ($\varepsilon_1 = \varepsilon_2$). However, as a result of the permittivity difference $\varepsilon_1 - \varepsilon_2 \neq 0$, Eq. (3a) predicts the propagation of a third mode ($n=3$) determined by

$$\varepsilon = \varepsilon_2 + \left(1 - \frac{\varepsilon_2}{\varepsilon_1}\right)^2 (\varepsilon_2 \lambda)^2 \left[1 + 2i \left(1 - \frac{\varepsilon_2}{\varepsilon_1}\right) S \lambda\right]. \quad (5)$$

The leading dependence of this third mode on the graphene conductivity is thus weighted by the normalized thickness $\lambda$ to the third power. This weak dependence
of $\varepsilon$ on S indicates that the mode $n = 3$ is not strongly affected by the graphene energy absorption and therefore it can propagate long distances, in absence of energy absorption by the film and its surrounding medium ($\text{Im}(\varepsilon_1) = \text{Im}(\varepsilon_2) = 0$). As the SPP thermal conductivity increases with the SPP propagation length, this third mode is expected to significantly enhance the heat transport along the film interfaces, as confirmed below. Therefore, while the propagation of the first two modes is driven by either the permittivity of the film or its surrounding medium, mainly, the third mode is driven by the permittivity difference of these media.

We now determine numerically the propagation and thermal properties of SPPs propagating along a Si film ($\varepsilon_1 = 11.7$) surrounded by air ($\varepsilon_2 = 1$). These two media do not absorb energy in a large frequency range [30] and therefore they favor the propagation of SPPs over long distances. In addition, considering that the Si permittivity only increases slightly with temperature [30, 31], for the considered range of temperatures [(300; 700) K], we assume its behavior is well represented by its given mean value. The graphene monolayers on the top and bottom surfaces of the film are good SPP conductors that absorb energy and hence support the SPP heat transport. For temperatures comparable to or higher than 300 K, the 2D conductivity $\sigma$ of a graphene monolayer can be described by the following Drude-like model

$$ \sigma = \frac{\Omega}{\gamma - i\omega}, \qquad (6) $$

where $\gamma$ is the electron scattering rate and $\Omega = (e/\hbar)^2 E_F/\pi$, with $e$ and $E_F$ being the electron charge and Fermi energy, respectively. $\sigma$ is thus determined by two parameters, whose estimated experimental values are $\gamma/2\pi = 1.59$ THz and $E_F = 1$ eV [24, 26]. As the chemical potential $E_F$ drives the density of free carriers, the proportionality $\sigma \propto E_F$ and the dispersion relations in Eqs. (3a) and (3b) indicate that the plasmon heat conduction increases with $E_F$. Equation (6) represents the intraband contribution to $\sigma$, as the interband one becomes negligible for $\hbar\omega < 2E_F$ ($\omega/2\pi < 483$ THz) [24, 25, 32]. This frequency interval covers the vast majority of relevant frequencies for the propagation and heat conduction of plasmons, for temperatures from 300 to 700 K, as shown below.

The in-plane wave vector $\beta_R$ and propagation length $\Lambda$ of SPPs propagating along a graphene-coated Si film are shown in Figs. 2(a) and 2(b), respectively. Mode $n = 1$ (Eq. (4a)) exhibits the largest $\beta_R$ (shortest wavelength) and smallest $\Lambda$, which takes values below 30 µm. The wave vector of mode $n = 1$ increases with frequency and takes higher values for thinner films, while its propagation length exhibits the opposite behavior. This short wavelength mode ($n = 1$) is therefore expected to contribute to the SPP heat transport due to its large wave vector, which is nearly two orders of magnitude larger

<figure><img src="image_3.png" /><figcaption>FIG. 2: Spectra of the in-plane (a) wave vector and (b) propagation length of the 3 SPP modes propagating along a graphene-coated Si film with $d = 25$ and 75 nm.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3: (a) Spectrum of the cross-plane propagation length of the 3 SPP modes and (b) spatial distribution of the in-plane component of their Poynting vector.</figcaption></figure>

than that of mode $n = 3$. For mode $n = 3$, $\beta_R$ linearly increases with frequency and is very close to the wave vector of light in vacuum ($\beta_R \approx k_0$). This photon-like nature of SPPs is accompanied by very long propagation lengths resulting from the weak energy absorption of graphene (see Eq. (5)). The major contribution of mode $n = 3$ to the SPP heat transport is thus expected to arise from its long propagation length. Mode $n = 2$, on the other hand, has intermediate wave vectors and propagation lengths, whose values are between the corresponding ones of the other two modes. In contrast to mode $n = 1$, these two propagation parameters of mode $n = 2$ are nearly independent of the film thickness, for frequencies lower than 10 THz, as established by its weak dependence on $d$ (see Eq. (4b)). According to Fig. 2(a), the three modes propagate with a group velocity $V_g = \partial\omega/\partial\beta_R$ near to the speed of light in vacuum (see the SM [29]) and hence they can drive a plasmonic heat conduction much faster than those of phonons and electrons.

The propagation length of the three SPP modes has a similar behavior than the SPP skin depth $\delta_1 = [2\text{Re}(p_1)]^{-1}$ inside the Si film, as shown in Fig. 3(a). The SPP energy absorption by the film hence reduces both the in-plane ($\Lambda$) and cross-plane ($\delta_1$) propagation distances of SPPs. For a given frequency, the intrafilm skin depth satisfies the relation $\delta_1(n = 1) < \delta_1(n = 2) < \delta_1(n = 3)$ and its values are generally longer than the film thickness ($\delta_1 > d$). The SPPs propagating along the bottom and top film surfaces are thus coupled and exist in a
<figure><img src="image_5.png" /><figcaption>FIG. 4: (a) Spectrum of the SPP thermal conductivity of a 1-mm-long Si film and its (b) integrated counterpart as a function of temperature. Calculations were done for the three SPP modes, two film thicknesses, and T = 300 K in (a).</figcaption></figure>

broad band of frequencies. This existence is further confirmed by the spatial distribution of the Poynting vector shown on Fig. 3(b). The energy density of modes $n = 1$ and $2$ is confined along the film interfaces mainly, while the intra-film energy of $n = 3$ is much smaller the extra-film one, due to the high sensitivity of this third mode to the permittivity of the non-absorbing air (see Eq. (4b)).

The contributions of the three SPP modes to the spectrum $k_\omega$ of the SPP thermal conductivity ($k = \int k_\omega d\omega$ in Eq.(1)) of the graphene-coated Si film are shown in Fig. 4(a). The maxima of $k_\omega$ and their frequencies of occurrence increase as the film thickness decreases, such that $k_\omega(n = 1) > k_\omega(n = 3) > k_\omega(n = 2)$. The short-wavelength mode ($n = 1$) thus carries more thermal energy than the long-range one ($n = 3$), as a result of the limited length ($l = 1$ mm) of the considered film. Mode $n = 2$ exhibits the lowest contribution because the peak of $k_\omega$ results from the maximization of both $\beta_R$ and $\Lambda$, which take intermediate values only, as shown in Fig. 2. For frequencies higher than the peak ones, the sharp reduction of $k_\omega$ is driven by the exponential decrease of the Bose-Einstein distribution function (see Eq. (1)), such that its values nearly vanish for $\omega/2\pi > 100$ THz. The three SPP modes thus exhibit broad frequency spectra and their contribution to the SPP thermal conductivity fulfill the relation $k(n = 1) > k(n = 3) > k(n = 2)$, as shown in Fig. 4(b). The near-linear increase of $k$ with temperature deviates from the quadratic one predicted for the ballistic regime ($l \ll \Lambda$) in polar films [4], because the propagation length of modes $n = 1$ and $n = 2$ shown in Fig. 2(b) takes smaller and greater values than the film length $l = 1$ mm. Therefore, in the ballistic-diffusive regime, the SPP thermal conductivity increases with temperature slower than in the pure ballistic one.

The sum of the contributions of the three SPP modes to the SPP thermal conductivity $k$ of the graphene-coated Si film is shown in Fig. 5(a), as a function of its thickness $d$. Note that $k$ monotonically increases as $d$ decreases following nearly the power law $k \propto 1/d$. The

<figure><img src="image_6.png" /><figcaption>FIG. 5: (a) SPP thermal conductivity and (b) thermal conductance per unit width of a Si film, as functions of its thickness. Calculations were done by summing the contributions of the three SPP modes, for film lengths of 1 and 5 mm; and three representative temperatures. The black points in (a) stand for the phonon thermal conductivity of a suspended Si film at $T = 300$ K [33], while the dotted lines in (b) represent the SPhP quantum of thermal conductance</figcaption></figure>

$$G_0/a = 12z(3)k_B^3 T^2/ch^2 \text{ of a polar nanofilm [4].}$$

thickness dependence of the plasmon thermal conductivity therefore exhibits an opposite behavior to the one of its phonon counterpart (black line) reported for a bare Si membrane at 300 K [33]. $k = 13.6 \text{ Wm}^{-1}\text{K}^{-1}$ for a Si film at 300 K and a length (thickness) of 5 mm (15 nm). This SPP thermal conductivity represents 67% of its phonon counterpart for a bare Si film (black points in Fig. 5(a)). As higher $k$ values are obtained for longer or hotter films, the sizable plasmon heat conduction driven by graphene is expected to counterbalance the reduction of the phonon counterpart, as the film length or temperature increases. In terms of the thermal conductance ($G = kad/l$), the SPP contribution is $(13.6 \text{ Wm}^{-1}\text{K}^{-1}) \times (15 \text{ nm}) \times a/l = 204 \times (a/l) \text{ nWK}^{-1}$, for a graphene-Si-graphene sandwich at 300 K and a length (thickness) of 5 mm (15 nm). On the other hand, considering that the thermal conductivity of a supported graphene monolayer (thickness $\approx 0.4$ nm) lies between 300 and $600 \text{ Wm}^{-1} \text{K}^{-1}$ [34–36], the estimated upper bound for the phonon+electron thermal conductance of the same structure is $[(600 \text{ Wm}^{-1} \text{K}^{-1}) \times (2 \times 0.4 \text{ nm}) + (20.3 \text{ Wm}^{-1} \text{K}^{-1}) \times (15 \text{ nm})] \times a/l = 784.5 \times (a/l) \text{ nWK}^{-1}$. The SPP contribution to the heat current is therefore $(204/784.5) \times 100\% = 26\%$ of its electron+phonon counterpart. Further, considering that the SPP heat conduction involves the contribution of the long-range mode $n = 3$ (see Fig. 2(b)), which exhibits a ballistic propagation ($l \ll \Lambda$), for any film length of practical interest ($l < 1$ m), its contribution to the SPP thermal conductivity increases linearly with the film length ($k \propto l$), as established by Eqs. (1) and (2). The SPP thermal conductance of this ballistic mode is therefore independent of the film length. Consequently, the SPP contribution to the heat current is expected to become comparable or even higher than that of phonons and electron, as
the sandwich structure length increases. The behavior $k \propto 1/d$ yields a significant SPP thermal conductance nearly independent of the film thickness, as shown in Fig. 5(b). According to Eq. (1), this saturation of $G$ arises from the independence of $\text{Re}(\beta)\Lambda_e$ on $d$. The opposite dependence on $d$ of the SPP wave vector and propagation length (see Fig. 2) therefore cancels out for each mode and film thickness ($d < 100$ nm) giving significant $k$ values. For a given temperature, $G$ increases with the film length and its values are higher than the maximum one $G_0$ (dotted lines) supported by a polar nanofilm [4]. This $G > G_0$ results from the combined contribution of three plasmon modes, while $G_0$ is determined by a single mode supported by polar nanofilms [4]. The graphene coating therefore transforms a Si nanofilm into a plasmonic conductor able to enhance the in-plane heat conduction better than SPhPs in a polar nanofilm, for a given thickness and temperature.

Considering the ultrafast propagation of SPPs, their contribution to the film temperature without the influence of phonons or electrons, could be observed via time-domain thermoreflectance with femtosecond-resolution, as done for other structures [37]. Further, The SPP thermal conductivity could be measured by means of the micro time-domain thermoreflectance or its steady-state version, which have been used to experimentally probe the polariton thermal conductivity of suspended SiN membranes [10] and supported Ti nanofilms [18], respectively. Whether the SPP heat transport is diffusive or ballistic, it enhances the heat dissipation from a heated spot and therefore lowers its temperature with respect to its value in absence of SPPs. This enhancement is thus expected to allow probing the predicted SPP thermal conductivity by fitting the measured temperature to a proper heat transport model.

In summary, the existence and propagation of three plasmon modes along the interface of a solid film coated with graphene monolayers have been demonstrated. These modes propagate long distances with short wavelengths and speeds close to the one of light in vacuum, which are suitable features to enhance the heat conduction at rates orders of magnitude faster than those of phonons and electrons. A plasmon thermal conductivity of $17 \text{ Wm}^{-1}\text{K}^{-1}$ has been found for a Si film with a thickness of 15 nm, a length of 1 mm, and a mean temperature of 500 K. Higher thermal conductivity values are obtained for thinner, longer, or hotter films. The plasmonic heat conduction driven by graphene could be expanded to other 2D materials and represent an efficient mechanism to offset the thermal performance reduction of dielectric films as their thickness decreases or temperature increases, without changing the internal material structure.

ACKNOWLEDGMENTS

This work was supported by the CREST Japan Science and Technology Agency (grant n° JPMJCR19I1), the JSPS Kakenhi (grant n° 21H04635), the core-to-core program (grant n° JPJSCCA20190006), and the Russian Academy of Sciences (Project n° FFZE-2022-0009).

* jose.ordonez@cnrs.fr

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