Boltzmann transport equation for surface phonon polaritons
The one-dimensional BTE is expressed with the relaxation time approximation22 as
$$\frac{\partial f}{{\partial t}} + v_{g} \cdot \nabla f = \frac{{f_{0} – f}}{\tau },$$
(1)
where f is the distribution function of the SPhPs, vg is the group velocity of the SPhPs, f0 is the equilibrium distribution function (Bose-Einstein distribution function), and τ is the relaxation time. Assuming that the film is sufficiently thin such that the heat transfer in the thickness direction can be neglected, the one-dimensional steady-state BTE along the in-plane direction (x-direction in Fig. 1) can be rewritten using the intensity notation as
$$\cos \theta \frac{{\partial I_{\omega } }}{\partial x} = \frac{{I_{\omega }^{0} – I_{\omega } }}{{\Lambda_{e} }},$$
(2)
where Λe is the effective propagation length of the SPhP obtained using Matthiessen’s rule; that is, Λe−1 = ΛSPhP−1 + L−1, where ΛSPhP is the propagation length of the SPhP and L is the film length1. In addition, θ is the polar angle along the x-axis, and \({I}_{\omega }\) and \({I}_{\omega }^{0}\) are the directional SPhP intensity for a particular frequency per unit length (W s m−1 rad−1) and the equilibrium SPhP intensity, respectively, and are defined as (Supplementary information S1)
$$I_{\omega } \left( {x,\omega } \right) = \left| {v_{g} } \right|\hbar \omega f\left( x \right)\,D_{2D} \left( \omega \right)/2\pi$$
(3)
$$I_{\omega }^{0} \left( {x,\omega } \right) = \frac{1}{2\pi }\int_{0}^{2\pi } {I_{\omega } \left( {x,\omega } \right)d\theta } ,$$
(4)
where ħ is the Planck constant divided by 2π, ω is the SPhP frequency, and D2D(ω) is the SPhP density of states per unit area. Once the SPhP intensity is determined, the temperature distribution and heat flux (qSPhPs) can be obtained by integrating over the SPhP frequency range as
$$\int_{{\omega_{L} }}^{{\omega_{H} }} {I_{\omega }^{0} \left( {x,\omega } \right)d\omega } = \int_{{\omega_{L} }}^{{\omega_{H} }} {\frac{{\left| {v_{g} } \right|}}{2\pi }\frac{{\hbar \omega D_{2D} \left( \omega \right)}}{{\exp \left[ {\hbar \omega /k_{B} T\left( x \right)} \right] – 1}}d\omega }$$
(5)
$$q_{SPhPs} = \frac{1}{d}\int_{{\omega_{L} }}^{{\omega_{H} }} {\int_{0}^{2\pi } {I_{\omega } \cos \theta \,d\theta } \,d\omega } ,$$
(6)
where ωH and ωL are the highest and lowest frequencies of the SPhPs, respectively, and d is the film thickness. In Eq. (6), the heat flux is inversely proportional to the film thickness. The heat flux of a surface polariton includes the assumption that the amount of in-plane energy transferred by the SPhPs is constant regardless of the position according to film thickness direction and is a depth-averaged value. Because SPhPs can propagate in any direction on the surface of a thin film, the numerical method used to solve the BTE should include the discretization of space and polar angle. In the present study, the finite element method was used with the discrete ordinates method (DOM) to discretize the spatial and angular terms of the BTE. The DOM numerically integrates angles by assigning a weight function w to a specific angle23. The Gaussian–Legendre quadrature was adopted to integrate in polar angle. The boundary conditions of the BTE were given by the constant equilibrium intensity of the SPhPs at a fixed temperature at the boundary. Because the SPhPs propagate forward (positive x-direction) when cosθ > 0 and backward (negative x-direction) when cosθ < 0, the boundary conditions for the two cases can be expressed as18
$$I_{\omega } \left( 0 \right) = I_{\omega }^{0} \left( {T_{l} } \right)\,\,\,\,\,{\text{when}}\,\,\cos \theta > \,0$$
(7a)
$$I_{\omega } \left( L \right) = I_{\omega }^{0} \left( {T_{r} } \right)\,\,\,\,\,{\text{when}}\,\,\cos \theta < \,0,$$
(7b)
where Tl and Tr are the temperatures at the boundaries of the analysis domain, as shown in Fig. 1.
Effective in-plane thermal conductivity
The effective in-plane thermal conductivity due to the SPhPs in the layered thin film is given as follows8,9:
$$\kappa_{theory} = \frac{1}{4\pi d}\int_{{\omega_{L} }}^{{\omega_{H} }} {\hbar \omega \beta_{R} \Lambda_{e} \frac{{\partial f_{0} }}{\partial T}d\omega } ,$$
(8)
where βR is the real part of the SPhP in-plane wave vector β. Equation (8) expresses the effective in-plane thermal conductivity at an interface of infinite length. The effective propagation length is used in consideration of the boundary scattering caused by the propagation length of the SPhPs, which is significantly longer than the film length1,9.
The optical phonons are thermally excited in the frequency range of 7.6–258 Trad/s for SiO29. In this frequency range, the SPhPs can be classified into three modes: Zenneck, SPhP, and transverse magnetic (TM)-guided modes24. However, the TM-guided mode was excluded because it does not satisfy the existence condition of SPhPs on a SiO2 film9. Therefore, we considered both the SPhP and Zenneck modes in this study.
Dispersion relation of surface phonon polaritons
The dispersion relation estimates the physical properties of energy carriers such as propagation length and group velocity as a relationship between frequency and wave vector. These properties were required to solve the BTE and obtain theoretical solutions.
The dispersion relation for an amorphous SiO2 thin film, assumed to be a nonmagnetic material surrounded by air as shown in Fig. 2, can be derived from Maxwell’s equations for the three layers of the thin film25:
$$\tanh \left( {\frac{{p_{1} d}}{2}} \right) = – \frac{{p_{0} \varepsilon_{1} }}{{p_{1} \varepsilon_{0} }},$$
(9)
where pj is the transverse wave vector of the SPhPs given by \({p}_{j}^{2}={\beta }^{2}-{\varepsilon }_{j}{k}_{0}^{2}\), εj is the relative permittivity of the material (j = 0, 1, 2), and k0 is the wave vector defined as the frequency divided by the speed of light in a vacuum.
For lossy dielectric materials, the permittivity can have a complex form and can be expressed as the relationship between the refractive index n and the extinction coefficient k of the material; that is, ε = (n + i∙k)2, where i is an imaginary number. The permittivity of amorphous SiO2 at room temperature (300 K) was obtained from the experimental data, using its refractive index and extinction coefficient26.