# A Thermal Transport Study of Branched Carbon Nanotubes with Cross and T-Junctions
## 4 Discussion
First, the power spectrum of velocity cross-correlation between a pair of atoms located in two branches was defined as follows:  
\[\mathbf{I}_{b,i}^{mn,n\beta}(\mathbf{\omega})=\int\mathbf{dt}\,\mathbf{e}^{-i\omega t}\Big{<}v _{\mathbf{b},i}^{mn}(\mathbf{t})v_{\mathbf{b},i}^{n\beta}(0)\Big{>}, \tag{4}\]  
where \(v_{\mathbf{b},i}^{mn}\) was the atomic velocity along the \(\mathbf{\alpha}\) direction of the \(b\)th atom at the \(i\)th unit cell located on branch \(\mathbf{m}\). The angle in brackets indicates the time moving average. The power spectrum of the time correlation function can provide information on how one atomic vibration was transmitted to another atom in different frequency ranges. As phonons are a collective motion of the periodic elastic arrangement of atoms inside condensed matter, the power spectrum of velocity cross-correlation between a pair of atoms located in two branches was insufficiently informative. Therefore, a group of atoms within \(\mathbf{n}_{\epsilon}\) unit cells in each branch was selected to calculate the atomic velocity cross-correlation as follows:  
\[\mathbf{C}^{mn,n\beta}(\mathbf{\omega})=\frac{1}{\mathbf{n}_{\epsilon}\mathbf{N}_{b}}\sum_{i }^{\mathbf{n}_{\epsilon}}\sum_{j}^{\mathbf{N}_{b}}\frac{\left|\mathbf{I}_{j,i}^{mn,n\beta} (\mathbf{\omega})\right|^{2}}{\left|\mathbf{I}_{j,i}^{mn,mn}(\mathbf{\omega})\right|\left| \mathbf{I}_{j,i}^{n\beta,n\beta}(\mathbf{\omega})\right|} \tag{5}\]  
where \(\mathbf{N}_{b}\) was the number of atoms in one unit cell. In the conducted analysis, \(\mathbf{n}_{\epsilon}\) was selected as 5. Moreover, the number of atoms in one unit cell, \(\mathbf{N}_{b}\), for a (6, 6) CNT was 24. Our methodology of velocity cross-correlation was similar to the one adopted to calculate coherence length by [PERSON] et al. [36]. Special care was taken to ensure that the distance between each pair of atoms used to calculate the velocity cross-correlation was the same (Figure 10), which was unlike the method used by [PERSON] et al. [36], to ensure that the time for traveling between the calculated atoms would not be different. Different atomic velocity directions were considered to characterize different phonon modes (i.e., the longitudinal and transverse phonon modes). The longitudinal (transverse) phonon mode was identified when the atoms vibrated parallel (perpendicular) to the branch axial direction. The cross-correlation results were averaged over six different runs. Figure 9: Branch length dependence of the heat current ratio in branch 2 of the branched CNT with the T-junction and defect-containing T-junction. The collective power spectra for the longitudinal and transverse vibrations, which were termed the phonon correlation function, were analyzed for cross-junction 1 (Figure 11) and T-junction 1 (Figure 12). As demonstrated in Figures 11a and 12a, the longitudinal vibrations of branch 1 were more easily transmitted to branch 3 than to branch 2 in both the longitudinal and transverse modes, which indicates that the vibrations in branch 2 originated from scattering and their correlation subsequently reduced. At low frequencies (<1 THz), all of the vibrations between branches exhibited higher correlations, which is consistent with the fact that low-frequency phonons have a long mean free path [36; 37]. In addition, the longitudinal vibrations of branch 1 had a higher correlation with the transverse vibration of branch 2 than with the longitudinal vibration of branch 2. This result indicated that the longitudinal phonon may change to a transverse phonon when transported from branch 1 to branch 2. This behavior is similar to the phonon polarization conversion behavior observed by [PERSON] et al. [24; 29; 30]. As depicted in Figures 11b and 12b, the transverse vibrations of branch 1 exhibited higher transmittance strength in branch 3 for a T-junction than for a cross-junction, especially at a frequency of approximately 2 THz.