# Phonons and Thermal Transport in Si/SiO\({}_{2}\) Multishell Nanotubes: Atomistic Study
## 3 Results and Discussion  
We calculated the phonon energy spectra in Si/SiO\({}_{2}\) MNTs and Si NWs by solving Equation (1) numerically (eigenvalue problem). Diagonalization of the matrix from Equation (1) was performed using LAPACK library [56] under GCC 8.4 compiler [57]. The calculations were performed for all \(q_{z}\) in the interval (0, \(\frac{\pi}{a}\)). In 1D case, the partial phonon density of states (DOS) per unit length in real space can be found from the relation: \(g_{s}(\omega)d\omega=\frac{1}{2\pi}dq_{z,s}\). Hence, the (total) phonon DOS per unit length in real space is given by:  
\[g(\omega)=\sum_{s(\omega)}g_{s}(\omega)=\sum_{s(\omega)}\frac{1}{2\pi v_{z,s}}, \tag{9}\]  
where summation is performed over all phonon modes \(s(\omega)\) with frequency \(\omega\). The phonon energy spectra and total DOS for Si NW and Si/SiO\({}_{2}\) MNTs are shown in Figure 2. The Si NW has dimensions 11 ML \(\times\) 11 ML (2.97 nm \(\times\) 2.97 nm; 1 monolayer (ML) = 0.27 nm). The geometric parameters of the Si/SiO\({}_{2}\) NTs are as follows: the cavity cross section is 45 ML \(\times\) 45 ML (12.15 nm \(\times\) 12.15 nm), the thickness of the Si layer is 6 ML (1.62 nm) and the thickness of the SiO\({}_{2}\) layer is 4 ML (1.08 nm). Panel (a) represents results for the Si NW, panel (b)-for the NT with a single Si/SiO\({}_{2}\) bilayer shell (\(N=1\)) and panel (c)-for the NT with two Si/SiO\({}_{2}\) bilayer shells (\(N=2\)). Similarly to rectangular nanowires [21; 58], phonon modes in rectangular MNTs can be classified into four types according to the spatial symmetry of the displacement vector components: Dilatational, Flexural\({}_{1}\), Flexural\({}_{2}\) and Shear. For the Si NW, we show phonon branches with numbers \(s\) = 1-4, 11, 17, \(\ldots\), 47, 51, 76, \(\ldots\), 301, 320 for Shear polarization; \(s\) = 1-4, 11, 17, \(\ldots\), 47, 51, 76, \(\ldots\), 326, 342 for Dilatational polarization and \(s\) = 1-4, 11, 17, \(\ldots\), 47, 51, 76, \(\ldots\), 326, 331 for Flexural\({}_{1}\) and Flexural\({}_{2}\) polarizations. For the Si/SiO\({}_{2}\) NT with \(N=1\), we present phonon branches with \(s\) = 1-5, 30, 55, 480, 500, 550, \(\ldots\), 1000, 1075, \(\ldots\), 1600, 1661 for Dilatational polarization; \(s\) = 1-5, 30, 55, 480, 500, 550, \(\ldots\), 1000, 1075, \(\ldots\), 1600, 1640 for Shear polarization and \(s\) = 1-5, 30, 55, 480, 500, 550, \(\ldots\), 1000, 1075, \(\ldots\), 1600, 1650 for Flexural\({}_{1}\) and Flexural\({}_{2}\) polarizations. For the Si/SiO\({}_{2}\) NT with \(N=2\), the following phonon branches are depicted: \(s\) = 1-5, 80, 155, 230, 305, 380, 455, 500, 675, 850, \(\ldots\), 2950, 3000, 3100 \(\ldots\), 3900, 3920 for Dilatational polarization; \(s\) = 1-5, 80, 155, 230, 305, 380, 455, 500, 675, 850, \(\ldots\), 2950, 3000, 3100 \(\ldots\), 3800, 3880 for Shear polarization and \(s\) = 1-5, 80, 155, 230, 305, 380, 455, 500, 675, 850, \(\ldots\), 2950, 3000, 3100, \(\ldots\), 3900 for Flexural\({}  
Figure 2: Phonon dispersions and phonon density of states in the Si NW with the cross section 11 ML \(\times\) 11 ML (panel (a)); in the Si/SiO\({}_{2}\) NTs (nanotubes) with the cavity cross section 45 ML \(\times\) 45 ML and a single Si/SiO\({}_{2}\) bilayer shell (panel (b)) and two Si/SiO\({}_{2}\) bilayer shells (panel (c)). The thicknesses of silicon and silica layers in the Si/SiO\({}_{2}\) bilayer are 6 ML and 4 ML, correspondingly. The number of confined phonon branches in the NTs (\(N_{b}\) = 6600 for a single-shell NT and \(N_{b}\) = 15,600 for a double-shell NT) is substantially larger than that in a NW (\(N_{b}\) = 1323). The slope of the lowest phonon branches in NTs is smaller than that in a NW due to acoustic mismatch between silicon and silicon dioxide. A great number of phonon modes in the NTs with energy \(\hbar\omega>\) 10 meV are nearly dispersionless and possess group velocities close to zero.