Molecular dynamics simulation of thermal boundary conductance between carbon nanotubes and ${\text{SiO}}_2$

We investigate thermal energy coupling between carbon nanotubes (CNTs) and ${\text{SiO}}_2$ with nonequilibrium molecular dynamics simulations. The phonon thermal boundary conductance $\left(g\right)$ per CNT unit length is found to scale proportionally with the strength of the van der Waals interaction $\left(\sim \chi \right)$, with CNT diameter $\left(\sim D\right)$, and as power law of temperature ($\sim T^{1/3}$ between 200 and 600 K). The thermal relaxation time of a single CNT on ${\text{SiO}}_2$ is independent of diameter, $\tau \approx 85\text{ps}$. With the standard set of parameters $g\approx 0.1\text{W}{\text{K}}^{-1}{\text{m}}^{-1}$ for a 1.7 nm diameter CNT at room temperature. Our results are comparable to, and explain the range of experimental values for CNT-${\text{SiO}}_2$ thermal coupling from variations in diameter, temperature, or details of the surface interaction strength.

Figure 1

(Color online) Schematic of heat dissipation from CNTs to ${\text{SiO}}_2$ in (a) vertical CNT array heat sinks (with native ${\text{SiO}}_2$ layer on Si wafer) (Refs. 6, 7) and in (b) interconnect or transistor applications (Refs. 8, 9). The downward red arrows show the direction of heat flow, in particular, via the CNT-${\text{SiO}}_2$ interface. (c) Computed phonon DOS in CNT and ${\text{SiO}}_2$ substrate. The CNT phonon DOS ranges from 0 to 56 THz whereas that of Si and O atoms range from 0 to 40 THz.

Figure 2

(Color online) Generating the simulation structure for a (10,10) nanotube with substrate coupling strength $\chi =1$. (a) We start with a crystalline ${\text{SiO}}_2$ block and anneal it at 6000 K to produce (b) the amorphous ${\text{SiO}}_2$ domain. (c) We delete the top half of the amorphous block to produce the substrate, then place the CNT on it. Inset shows the thermal circuit formed by the CNT heat capacity and the thermal resistance between CNT and substrate $\left(=1/g\right)$. (d) Typical simulated temperature transients. The simulation monitors the temperature difference between the CNT and ${\text{SiO}}_2$ as the temperature drop $\left(\Delta T\right)$ across the interface.

Figure 3

(Color online) (a) Decay of the normalized CNT-substrate temperature difference at 300 K averaged over ten runs, with varying CNT-substrate interaction strength $\left(\chi \right)$. (b) Natural logarithm of the same plot. The decay of the temperature difference can be fitted to a single exponential decay. The decay time is obtained from the fit gradient in the first 20 ps and is used to calculate the TBC as described in the text. (c) Cross-sectional profiles of a (10,10) CNT for different values of CNT-substrate coupling strength $\left(\chi \right)$. As this interaction becomes stronger, the CNT profile is deformed and its cross section becomes more elliptical.

Figure 4

(Color online) (a) Computed TBC for different values of $\chi$ from 200 to 600 K. (b) The TBC normalized with respect to its value at 200 K. Simulations suggest the TBC varies as $\sim T^{1/3}$ for the range of $\chi$ values studied, due to inelastic phonon scattering at the CNT-substrate interface. (c) The TBC normalized with respect to $\chi$. These results suggest there is a nearly linear dependence of the TBC on the strength of the CNT-substrate van der Waals interaction $\left(\chi \right)$. This dependence is not captured by conventional theories such as the acoustic or diffuse mismatch models (Ref. 35).

Figure 5

(Color online) Cross-sectional profiles of (a) (6,6), (b) (8,8), (c) (10,10), and (d) (12,12) CNTs on ${\text{SiO}}_2$, with interaction strength $\chi =1$. The corresponding diameters are 0.81 nm, 1.08 nm, 1.36 nm, and 1.63 nm, respectively. The CNT and the substrate are equilibrated at 500 K and 300 K, respectively, before their temperature difference is allowed to decay. Averaging over ten runs, we extract the decay times and use them to compute the TBC values in Fig. 6.

Figure 6

(Color online) Diameter dependence of the TBC for armchair CNTs at $T=300\text{K}$ and $\chi =1$. Error bars represent range from changing vdW potential cutoff (Ref. 30). (a) The thermal relaxation time is approximately constant over the range of diameters considered. (b) The calculated TBC is proportional to the CNT diameter (dashed line represents linear fit). This implies that the thermal conductance per unit area normalized by the CNT diameter is constant for the range of diameters studied, $\sim 5.8\times {10}^7\text{W}{\text{K}}^{-1}{\text{m}}^{-2}$. (c) Normalized phonon density of states for the (6,6), (8,8), (10,10), and (12,12) CNTs. The distribution of the low- and high-frequency modes is approximately the same for all, suggesting the proportion of low-frequency modes stays approximately constant as the CNT diameter increases.