# Phonons and Thermal Transport in Si/SiO\({}_{2}\) Multishell Nanotubes: Atomistic Study
## 3 Results and Discussion
As compared to a NW, the DOS maximum in NTs is shifted toward the lower energy interval, where the drop of the average phonon group velocity is more significant (see Figure 3). The DOS maximum in a MNT with two bilayer shells is more prominent due to a larger number of phonon modes concentrated in SiO\({}_{2}\), which possess a smaller maximal phonon energy than Si. The effect of phonon deceleration in MNTs with different numbers of Si/SiO\({}_{2}\) bilayer shells (\(N\) = 1, 2, 3, 4) is illustrated in Figure 3, where the average phonon group velocity \(\langle v\rangle(\omega)=\sum_{s(\omega)}(v_{z,s}g_{s}(\omega))/\sum_{s(\omega) }g_{s}(\omega)\) is shown as a function of the phonon energy for a Si NW with cross section \(N_{x}\times N_{y}=11\) ML \(\times\) 11 ML and different Si/SiO\({}_{2}\) NTs with the following geometrical parameters: \(N_{cavity,x}\times N_{cavity,y}\) = 45 ML \(\times\) 45 ML, \(N_{Si,x}\) = \(N_{Si,y}\) = 6 ML, \(N_{SiO_{2},x}\) = \(N_{SiO_{2},y}\) = 4 ML (panel (a)) and \(N_{cavity,x}\times N_{cavity,y}\) = 5 ML \(\times\) 5 ML, \(N_{Si,x}\) = \(N_{Si,y}\) = 6 ML, \(N_{SiO_{2},x}\) = \(N_{SiO_{2},y}\) = 4 ML (panel (b)). The phonon group velocity in NTs is smaller than that in a NW over a wide energy range (0 to 30 meV). The reduction of the phonon group velocities in NTs is explained by an acoustic mismatch between Si and SiO\({}_{2}\), a stronger phonon confinement and a spectral redistribution of the phonon DOS in NTs. In the energy range from 30 to 37 meV, the phonon group velocities vary near the same low values for all considered NTs. Increase of the number of shells slightly reinforces the drop of phonon group velocity. A comparison of temperature dependencies of thermal conductivity (TC) in a Si NW and Si/SiO\({}_{2}\) MNTs is provided in Figure 4. The geometrical parameters of a NW and MNTs are the same as in Figure 3. The thermal conductivity curves presented in panels (b) and (e) are calculated considering diffusion transport in SiO\({}_{2}\) (see Equation (7)), while those in panels (a) and (d) consider boundary scattering (see Equation (8)) with the specularity parameter \(p\) = 0.6. The values of the thermal conductivity in the NTs (even for the NTs with a greater number of phonon modes) are lower than the thermal conductivity in the NW in the whole temperature range. This decrease of TC in the MNTs is due to modification of phonon energy spectra in MNTs leading to phonon deceleration, as well as enhancement of the phonon scattering at interfaces and diffusion of vibrational excitations in SiO\({}_{2}\) layers. Figure 3: Average phonon group velocities as a function of phonon energies in Si/SiO\({}_{2}\) MNTs with the cavity cross section 45 ML \(\times\) 45 ML (**a**) and 5 ML \(\times\) 5 ML (**b**) and different numbers of Si/SiO\({}_{2}\) bilayer shells, formed by silicon layer with thickness 6 ML and silica layer with thickness 4 ML. Results for Si NW with cross section 11 ML \(\times\) 11 ML are also shown for comparison. The maxima of the thermal conductivity curves are determined by the interplay between three-phonon Umklapp and boundary or effective diffusion scatterings. At low temperatures, the phonon-boundary scattering or the diffusion of vibrational excitations in SiO\({}_{2}\) layers is the main mechanism of scattering, and the increase of TC with temperature is explained by the population of high-energy phonon modes. A subsequent rise of temperature results in the reinforcement of the Umklapp scattering and a reduction of TC. Additional interfaces between the shells effectively scatter phonons, hence the TC decreases with augmentation of the number of shells (see panels (a) and (d)) reaching values as low as 0.2 W/mK at RT for Si/SiO\({}_{2}\) NT with smaller cavity and \(N=4\). A similar dependence of TC on \(N\) was demonstrated experimentally for Si/SiO\({}_{2}\) rolled-up nanotubes of 1.9 \(\upmu\)m to 3.2 \(\upmu\)m radii and a 24-nm-thick shell [43].