Reduction of phonon lifetimes and thermal conductivity of a carbon nanotube on amorphous silica

We use molecular dynamics (MD) simulations to examine phonon lifetimes in (10,10) carbon nanotubes (CNTs), both when isolated and when supported on amorphous SiO${}_2$ substrates. We determine the intratube phonon-phonon, interfacial, and CNT-substrate phonon scattering rates from the computed inverse lifetimes. Suspended CNTs have in-plane optical phonon lifetimes between 0.7–2 ps, consistent with recent experiments, but contact with the substrate leads to a lifetime reduction to the 0.6–1.3 ps range, which is of consequence for high-field electrical transport. A significant lifetime reduction was also observed for low-frequency acoustic phonons with strong polarization dependence. As a result, the thermal conductivity of the supported CNT is computed to be ∼33% lower than that of the suspended CNT. The calculation of the scattering rates also enables us to estimate the thermal boundary conductance (TBC) between supported CNT and substrate, in good agreement with the Green-Kubo relation and with experimental results. The results highlight that solid substrates strongly affect and could be even used to tune the thermal properties of CNTs.

Figure 1

Rendering of the simulation setup with a (10,10) CNT supported on an a-SiO${}_2$ block mimicking the substrate. The a-SiO${}_2$ block is 24 Å thick, 51.5 Å wide, and 122.8 Å long. The CNT is 122.8 Å (50 unit cells) long and has a diameter of about 13.7 Å. (Top) Cross-sectional view of the setup. (Bottom) Side view of the setup. The surface of the a-SiO${}_2$ substrate is relatively flat with some surface roughness. Periodic boundary conditions are imposed in the plane parallel to the substrate surface.

Figure 2

We plot Φ(k, $k_{\theta }$, ν) over all k and $k_{\theta }$ values for T = 300 K in (a). Given that there are 6000 peaks, we cannot distinguish between the individual peaks. However, by restricting ourselves to a particular k, we reduce the number of peaks to 120, as shown in (b) where k = 0. Even so, many of the peaks are still closely clustered together, especially in the 47 to 55 THz region (enclosed by red dashed lines). In (c) we restrict $k_{\theta }$ = 0, leading to only 12 distinct peaks and resolving the problem of the closely clustered peaks between 47 and 55 THz. (d) The three peaks are sufficiently separated, enabling us to distinguish the G${}^{-}$ and G${}^{+}$ modes.

Figure 3

Frequency dependence of phonon relaxation time of an isolated and supported (10,10) CNT. The data are assigned to (a) LA, (b) TA, (c) TW, and (d) optical phonons. The polarizations of optical phonons are not indicated for the sake of visual simplicity. The dashed lines indicate $\tau \propto {\nu }^{-2}$ of Klemens.[47] The short-dash line in (c) shows a power law ($\tau \propto {\nu }^{-1.1}$) fitted to the data of TW phonons in the linear dispersion regime ($\nu <15$ THz). The TA modes show the strongest lifetime reduction between suspended and supported CNTs. In (a) to (d), blue circles and red diamonds are used for the isolated CNT and for the SiO${}_2$-supported case respectively.

Figure 4

Frequency dependence of substrate-scattering rate with (${\gamma }_{\mathrm{III}}$ − ${\gamma }_{\mathrm{I}}$) and without (${\gamma }_{\mathit{II}}$ − ${\gamma }_{\mathrm{I}}$) MD of the a-SiO${}_2$ substrate. The noise in the data and the consequent unphysical negative γ are mainly due to the peak assignment error, which is larger for less dispersive phonons.

Figure 5

(a) The substrate-scattering rate in the (ν, k)-space. The contour shows the total substrate scattering rate (γ${}_{\mathrm{SiO}2}$) in THz. (b) The magnitude of radial displacements of CNT eigenvectors. Each eigenvector (k, $k_{\theta }$, β) is represented by a point in (ν, k) space and is colored between silver (minimum radial displacement) and red (maximum radial displacement).

Figure 6

We plot the phonon lifetimes for the isolated CNT (case I, blue dots) and the CNT on SiO${}_2$ (case III, hollow black circles) in the 47 to 55 THz region at 300 K. The window between 49.5 and 54.2 THz (enclosed by red dashed lines) contains the G-mode (Raman active and inactive) and other high frequency in-plane optical phonons. The lifetimes of these points are between 0.7–2 ps for the suspended CNT. Contact with the substrate leads to a significant reduction in their lifetimes to the range τ${}_{\mathrm{III}}$ = 0.6–1.3 ps. The difference in phonon lifetimes (plotted with magenta crosses) shows that contact with the substrate can lead to a lifetime reduction of up to 1.2 ps for phonons near the G-mode frequency.

Figure 7

We plot TBC g′(ν) and the phonon DOS of the CNT. The negative g′(ν) values are unphysical and indicate that the CNT-substrate phonon-phonon scattering rates cannot be accurately determined from ${\gamma }_{\mathrm{III}}$ − ${\gamma }_{\mathit{II}}$. The large positive g′(ν) values between 0 and 9 THz indicate that interfacial thermal transport is dominated by phonons in this frequency range.[29]