Role of thermalizing and nonthermalizing walls in phonon heat conduction along thin films

Phonon boundary scattering is typically treated using the Fuchs-Sondheimer theory, which assumes that phonons are thermalized to the local temperature at the boundary. However, whether such a thermalization process actually occurs and its effect on thermal transport remains unclear. Here we examine thermal transport along thin films with both thermalizing and nonthermalizing walls by solving the spectral Boltzmann transport equation for steady state and transient transport. We find that in steady state, the thermal transport is governed by the Fuchs-Sondheimer theory and is insensitive to whether the boundaries are thermalizing or not. In contrast, under transient conditions, the thermal decay rates are significantly different for thermalizing and nonthermalizing walls. We also show that, for transient transport, the thermalizing boundary condition is unphysical due to violation of heat flux conservation at the boundaries. Our results provide insights into the boundary scattering process of thermal phonons over a range of heating length scales that are useful for interpreting thermal measurements on nanostructures.

Figure 1

Spatial distribution of the temperature profile and pictorial representation of the boundary conditions [Eq. (2)] used in this work. The thin film is assumed to be infinite in extent along the in-plane ($x$) direction and has a finite thickness ($d$) in the cross-plane ($z$) direction. For steady state transport calculations, the temperature gradient exists only in the in-plane ($x$) direction. For transient transport, the initial temperature distribution is an in-plane sinusoidal distribution with a period $\lambda$, but can develop a cross-plane temperature gradient at later times. In the case of the nonthermalizing wall, the diffusely reflected component $\left[g^{\mathrm{diff}}\right]$ of the distribution function is isotropic but away from local thermal equilibrium at the boundary while for the thermalizing wall, $\left[g^{\mathrm{diff}}\right]$ is equal to the local equilibrium distribution function $\left[g_0\right]$. For both thermalizing and nonthermalizing boundary conditions, the specular reflection component $\left[g^{\mathrm{spec}}\right]$ has its direction reversed compared to the incoming distribution $\left[g^{\mathrm{in}}\right]$ and is also away from the local thermal equilibrium. In our work, we consider either thermalizing or nonthermalizing boundary conditions for both walls of the thin films at a time.

Figure 2

(a) Comparison of the cross-plane distribution of the in-plane steady state heat flux between analytical and Monte Carlo solutions of the BTE at 300 K and film thickness of 100 nm for different boundary conditions. The geometry of the thin film and the coordinate axes used in this work are shown in the inset. (b) Comparison of the steady state thermal conductivity between analytical and Monte Carlo solutions of the BTE at different temperatures and thin film thicknesses. For both (a) and (b), the Monte Carlo solutions are identical for thermalizing and nonthermalizing boundary scattering and agree well with the analytical solution derived in this work for both fully diffuse and partially specular boundary conditions (RMS 0.1 nm). For the partially specular boundary condition, the specularity parameter ($p_{\omega }$) is calculated from Ziman's specularity model [40] for a surface rms roughness of 0.1 nm. (c) Effective MFPs of phonons computed using Matthiessen's rule (MR) and the Fuchs-Sondheimer (FS) theory for different film thicknesses and fully diffuse boundary scattering. Matthiessen's rule underpredicts the effective phonon MFPs in thin films compared to the Fuchs-Sondheimer theory, which is a rigorous BTE solution.

Figure 3

(a) Comparison between time traces from the Monte Carlo (colored noisy lines) and the degenerate kernels solutions (black lines) of the BTE for a grating period of $20\mu \mathrm{m}$ at 500 K. (b) Comparison of the thermal conductivity predictions from the Monte Carlo (symbols) and the degenerate kernels solutions (black lines) of the BTE for different temperatures and grating periods. For both (a) and (b), the Monte Carlo solutions and the BTE solutions from this work are in very good agreement. (c) Plot showing the difference between the incoming and outgoing heat flux normalized by the incoming heat flux at the thin film boundaries as a function of the temporal frequency normalized by the maximum temporal frequency at which the simulations were performed (${\eta }_{\max }$). Specular and nonthermalizing diffuse boundary conditions conserve heat flux to numerical precision while the thermalizing diffuse boundary condition violates heat flux conservation at the film wall under quasiballistic ($T=100$ K, grating period = $1\mu \mathrm{m}$) and diffusive ($T=500$ K, grating period = $1000\mu \mathrm{m}$) transport regimes.

Figure 4

Comparison of the thermal conductivity (a) and the suppression functions [(b) and (c)] calculated from the models FS I, FS II and by solving the BTE for nonthermalizing diffuse boundary conditions at different temperatures, grating periods ($\lambda$), and film thicknesses. In panels (b) and (c), the symbols correspond to the degenerate kernel solution, the solid lines correspond to the FS I model, and the dashed solid lines correspond to the FS II model. For very thin films and long grating periods, the models FS I and FS II are in good agreement with the BTE predictions. For thicker films and shorter grating periods, FS I underpredicts and FS II overpredicts the thermal conductivity at short grating periods (a) and the contribution of phonons with long MFP [(b) and (c)] compared to the complete BTE solution.