# Phonons and Thermal Transport in Si/SiO\({}_{2}\) Multishell Nanotubes: Atomistic Study
## 2 Theoretical Model of Thermal Conductivity in Si/SiO\({}_{2}\) Multishell Nanotubes
\begin{array}{c}\kappa_{2}\left(\overset{\rightarrow}{r}_{k},\overset{ \rightarrow^{\prime}}{r}_{k}\right)0\ 0\\ 0\ \kappa_{3}\left(\overset{\rightarrow}{r}_{k},\overset{\rightarrow^{ \prime}}{r}_{k}\right)0\\ 0\ \kappa_{3}\left(\overset{\rightarrow}{r}_{k},\overset{\rightarrow^{ \prime}}{r}_{k}\right)\end{array} \right)\text{ for atoms with coordinates }(\pm a,0,\ 0) \tag{4}\] \[\Phi_{\alpha\beta}^{(2)}\left(\overset{\rightarrow}{h}^{(k)}, \overset{\rightarrow}{r}_{k},\overset{\rightarrow^{\prime}}{r}_{k}\right)= \left( \begin{array}{c}\kappa_{3}\left(\overset{\rightarrow}{r}_{k}, \overset{\rightarrow^{\prime}}{r}_{k}\right)0\ 0\\ 0\ \kappa_{2}\left(\overset{\rightarrow}{r}_{k},\overset{\rightarrow^{ \prime}}{r}_{k}\right)0\\ 0\ 0\ \kappa_{3}\left(\overset{\rightarrow}{r}_{k},\overset{\rightarrow^{ \prime}}{r}_{k}\right)\end{array} \right)\text{ for atoms with coordinates }(0,\ 0,\pm a)\]  
This matrix depends on 3 independent force constants \(\kappa_{1}\), \(\kappa_{2}\) and \(\kappa_{3}\), which can be expressed through the elastic moduli \(c_{11}\), \(c_{12}\) and \(c_{44}\) of a bulk cubic crystal [8]: \(\kappa_{1}=\frac{a(c_{12}+c_{44})}{2}\), \(\kappa_{2}=\frac{a(c_{11}-c_{12}-c_{44})}{4}\), \(\kappa_{3}=\frac{a(c_{44}-c_{12})}{8}\), where \(a\) is the lattice constant. Calculation of the thermal conductivity of Si/SiO\({}_{2}\) MNTs and Si NWs is conducted using the following formula:  
\[\kappa_{ph}=\frac{1}{2\pi k_{B}T^{2}S_{NT}}\sum_{s}\int_{0}^{d_{2,max}}[\hbar \omega_{s}(q_{z})v_{z,s}(q_{z})]^{2}\tau_{tot,s}(q_{z})\frac{\exp\left(\frac{ \hbar\omega_{s}(q_{z})}{k_{B}T}\right)}{\left[\exp\left(\frac{\hbar\omega_{s}( q_{z})}{k_{B}T}\right)-1\right]^{2}}dq_{z} \tag{5}\]  
This expression was derived from the Boltzmann transport equation within the relaxation-time approximation [6; 22; 52; 53] and captured the quasi one-dimensional density of the phonon states. In Equation (5), the summation is performed over all phonon branches \(s=1\),..., \(N_{b}\), \(S_{NT}\) is the NT cross-sectional area, \(\omega_{s}\) is the phonon frequency, \(v_{z,s}\) is the \(Z\)-th component of the phonon group velocity, \(q_{z,max}=\pi/a\) is the maximal amplitude of phonon wavevector, \(\tau_{tot,s}\) is the total phonon relaxation time, \(k_{B}\) is the Boltzmann constant, \(\hbar\) is the [PERSON] constant and \(T\) is the temperature. The total phonon relaxation rate was estimated according to the [PERSON]'s rule:  
\[\tau_{tot,s}^{-1}(q_{z})=\tau_{U,s}^{-1}(q_{z})+\tau_{imp,s}^{-1}(q_{z})+\tau_ {x,s}^{-1}(q_{z}). \tag{6}\]  
Here, \(\tau_{U,s}\) is the relaxation time for the Umklapp scattering: \(\tau_{U,s}^{-1}(q_{z})=B[\omega_{s}(q_{z})]^{2}T\exp(-C/T)\)[22] and \(\tau_{imp,s}\) is the relaxation time for the phonon-impurity scattering: \(\tau_{imp,s}^{-1}(q_{z})=A[\omega_{s}(q_{z})]^{4}\), where \(A=1.32\times 10^{-45}\) s\({}^{3}\) is determined analytically from the isotope concentration of natural silicon [8; 22]. Parameters \(B=1.73\times 10^{-19}\) sK\({}^{-1}\) and \(C=137.3\) K of the Umklapp scattering are fitted to reproduce experimental temperature dependence of bulk Si thermal conductivity [54]. In Si/SiO\({}_{2}\) MNTs, both phonon-boundary scattering at Si/SiO\({}_{2}\) interfaces and diffusion of vibrational excitations in amorphous SiO\({}_{2}\) influence the thermal conductivity. To analyze the role of these mechanisms in thermal transport, we performed calculations of the thermal conductivity using different values of the third term in Equation (6).