Theoretical study of thermal conductivity in single-walled boron nitride nanotubes

We perform a theoretical investigation on the thermal conductivity of single-walled boron nitride nanotubes (SWBNT) using the kinetic theory. By fitting to the phonon spectrum of the boron nitride sheet, we develop an efficient and stable Tersoff-derived interatomic potential which is suitable for the study of heat transport in $sp2$ structures. We work out the selection rules for the three-phonon process with the help of the helical quantum numbers $(\kappa ,n)$ attributed to the symmetry group (line group) of the SWBNT. Our calculation shows that the thermal conductivity ${\kappa }_{\mathrm{ph}}$ diverges with length as ${\kappa }_{\mathrm{ph}}\propto L^{\beta }$ with exponentially decaying $\beta \left(T\right)\propto e^{-T/T_c}$, which results from the competition between boundary scattering and three-phonon scattering for flexure modes. We find that the two flexure modes of the SWBNT make dominant contribution to the thermal conductivity, because their zero frequency locates at $\kappa =\pm \alpha$, where $\alpha$ is the rotational angle of the screw symmetry in SWBNT.

Figure 1

Phonon spectrum in the BN sheet calculated from the Tersoff potential, $\text{Tersoff}+\text{UFFOOP}$ potential are compared with experimental results from Ref. 33. The UFFOOP potential enhances merely the two out-of-plane (ZA and ZO) vibration modes.

Figure 2

Fit specularity parameter $p$ to the thermal conductivity from the molecular dynamics simulation.

Figure 3

Physical quantities in SWBNT (3,3). (a) Phonon spectrum parameterized by helical quantum numbers $(\kappa ,n)$. (b) Phonon spectrum parameterized by linear quantum numbers $(k,m)$. (c) Phonon velocity. (d) Grüneisen parameter. In (a), (c), and (d), $n=-1$, 0, 1 are displayed by solid (blue) lines, dashed (red) lines, and dotted (green) lines. The four acoustic phonon modes are highlighted by thicker lines.

Figure 4

Thermal conductivity of SWBNT (5,0) with length 1, 5, 10 $\mu$m from bottom to top.

Figure 5

(a) Thermal conductivity of 1 $\mu$m SWBNT (3,3) contributed from different acoustic phonon modes. (b) Lifetimes for acoustic phonon modes.

Figure 6

The vibrational morphology of four flexure modes in SWBNT (5,0) with frequency increased from 1.7 cm${}^{-1}$ to 26 cm${}^{-1}$ in panels (a)–(d). The values displayed are $(\kappa ,n,\sigma ,\omega )$ for each mode.

Figure 7

A natural logarithm of the thermal conductivity and length. Power function fitting for the thermal conductivity of SWBNT (5,0) with $L\in$ [0.3, 10] $\mu$m, and SWBNT (10,0) with $L\in$ [0.3, 5] $\mu$m at 300 K.

Figure 8

Temperature dependence for the power factors of SWBNT (5,0) and (10,0).

Figure 9

Phonon lifetime of TA modes in 2 $\mu$m SWBNT (10,0) at 300 K. The lifetime due to boundary scattering, ${\tau }_{\text{bs}}$, is shown by blue stars. The three-phonon scattering life time, ${\tau }_{\text{ps}}$, is displayed by red open circles. The total lifetime is shown by black diamonds.