Phonon-phonon interactions and phonon damping for the curvature modes in carbon nanotubes

There are several aspects driving the rapidly growing interest into carbon nanotubes (CNTs). On the one hand, their unique properties have been explored frequently in recent years, such as resonant Raman or photoluminescence excitation spectra [1–3], electrical [4], thermal [5–8], and water [9,10] transports, etc. On the other hand, there are also various interesting phenomena observed, such as unusually high-thermal conductivity [11,12], nano-electromechanical effects in suspended CNTs [13–15], and quantum size effects of specific heat [16]. In order to understand such properties and effects, it is crucial to analyze phonon modes and their damping.

Various works have theoretically predicted the phonon spectra through the thin-shell hollow cylinder model [17–21] or the force-constant model [22–24]. In addition, just as usual bulk materials, this kind of quasi–one-dimensional materials can also be considered as continuum elastic medium. Therefore, the phonon modes of CNTs can be well described by the continuum elasticity theory [25–29]. Apart from four acoustic phonon branches (including a longitudinal, a twist, and two degenerate flexural modes), we can also predict two degenerate lowest-lying curvature modes with angular-momentum quantum number $|l|=2$ (C modes) via adopting this continuum model with an additional curvature term [18].

This work concentrates on the calculation of C mode damping which is quantified by the relaxation rate $\tau_{C}^{-1}$, i.e., the inverse relaxation time. The intrinsic damping mechanisms include thermoelastic damping [30–32], phonon-phonon interactions [33–35], defect or impurity scattering [36–39], and electron-phonon interactions [40]. In particular, the damping is mainly caused by phonon-phonon interactions in CNTs, and the important calculations have been presented in refs. [41] and [42]. Moreover, an elegant and persuasive idea indicates that the phonon-phonon interactions can be obtained directly from the continuum elasticity theory mentioned above [43]. Applying this theory with curvature effect, we estimate the quality factor $Q_{C}\equiv\omega_{C}/ \tau_{C} ^{-1}$ for the C modes with normal frequency $\omega_{C}$, we show the variation of Q_C with internal size, and manifest the importance of these relaxation processes to heat conduction. These calculations and analyses have not been done yet in previous works. Recently, some experiments detect the mechanical oscillations excited in suspended CNTs [44,45], and our results will be useful for future experiments.

To build a coordinate frame, three unit vectors $\mathbf{e}_{x,y,z}$ are set as the circumferential, axial and normal directions, respectively, in the cylinder surface of CNTs. The position vector $\mathbf{r}=(x, y)$ denotes the coordinate of the two-dimensional cylinder surface with two components along $\mathbf{e}_{x,y}$, respectively. To describe the deformation field in the cylinder surface, we also define the displacement vector $\mathbf{u}(\mathbf{r})=(u_x(x,y),u_y(x,y),u_z(x,y))$ with three components along $\mathbf{e}_{x,y,z}$, respectively.

Considering the continuity and isotropy of the surfaces constructing CNTs, the Hamiltonian for decribing the motion of the deformation field can be written as $H=T+U_e+U_c$. These three terms denote kinetic energy, elastic energy based on elasticity theory [43], and curvature energy for the phonon branches with $|l|>1$ [26], respectively. They are expressed respectively as

$$\begin{eqnarray} T&= \int \text{d}x\text{d}y \sum_{i=x,y,z}{\frac{p_i^2}{2\rho}}, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} U_e&= \int \text{d}x\text{d}y\left[\frac{\lambda}{2}\left(\mathrm{Tr}u\right)^2+\mu \mathrm{Tr} u^2\right]\!, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} U_c&= \frac{a^2\Xi}{2}\int \text{d}x\text{d}y\left[\left(\frac{\partial^2}{\partial x^2}+\frac{1}{R^2}+ \frac{\partial^2}{\partial y^2}\right)u_z\right]^2, \end{eqnarray} \tag{ 3 }$$

where λ and μ are the Lamé constants (the bulk modulus $K=\mu+\lambda$), and the coefficient Ξ describes the surface curvature deviation of CNTs. Their values are $K /\rho=2.90 \times 10^8\ \text{m}^2/\text{s}^2$, $\mu/\rho=1.51 \times 10^8\ \text{m}^2/\text{s}^2$, and $\Xi/ \rho=6.19 \times 10^6\ \text{m}^2/\text{s}^2$ [26,46], in agreement with the empirical force-constant model [47]. In eq. (1), $\rho=3.80\times10^{-7}\ \text{kg/m}^2$ is the mass density of graphenes, and p_i is the canonical momentum conjugate to the field variable u_i. In eq. (2), according to the elastic model, the strain tensor u consists of linear and nonlinear parts [25], i.e., $u=u^{\mathrm{lin}}+u^{\mathrm{nlin}}$. Its components u_ij are expressed as

$$\begin{eqnarray} u^{\mathrm{lin}}_{ij} &= \frac{1}{2} \left(D_i u_j+D_j u_i\right)\!, \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} u^{\mathrm{nlin}}_{ij}&= \frac{1}{2}\sum_{k=x,y,z}\left(D_i u_k\right)\left(D_j u_k\right), \end{eqnarray} \tag{ 5 }$$

where the covariant derivatives are used for the cylindrical surface structure [25,43]. Furthermore, in eq. (3), R and $a=0.246$ nm denote the tube radius and the ordinary length of a single lattice in CNTs, respectively.

Substituting eqs. (4) and (5) into eq. (2), we obtain the harmonic Hamiltonian as

$$\begin{eqnarray} H_0&= \int \text{d}x\text{d}y \Biggl\{\sum_i \frac{p_i^2}{2\rho} + \frac{\lambda}{2} \left(\mathrm{Tr} u^{\mathrm{lin}} \right)^2+\mu \mathrm{Tr}\left[\left( u^{\mathrm{lin}} \right)^2 \right] \nonumber\\ & \quad+\,\frac{a^2\Xi}{2}\left[\left(\frac{\partial^2}{\partial x^2}+\frac{1}{R^2}+ \frac{\partial^2}{\partial y^2} \right)u_z\right]^2\Biggr\}. \end{eqnarray} \tag{ 6 }$$

In addition, the three-phonon coupling Hamiltonian is given by

$$\begin{eqnarray} H_3&= \int \text{d}x\text{d}y\Bigl[\lambda \mathrm{Tr} u^{\mathrm{lin}} \mathrm{Tr} u^{\mathrm{nlin}} +2\mu \mathrm{Tr}\left(u^{\mathrm{lin}}u^{\mathrm{nlin}}\right) \nonumber\\ & \quad+\,\eta_1 \mathrm{Tr} u^{\mathrm{lin}} \mathrm{Tr}(u^{\mathrm{lin}})^2 +\eta_2 (\mathrm{Tr} u^{\mathrm{lin}})^3\Bigr]\!, \end{eqnarray} \tag{ 7 }$$

where the first two terms are obtained by substituting eqs. (4) and (5) into eq. (2), and are the coupling terms of linear and nonlinear strain tensors. The higher-order nonlinear terms can also be obtained and involve four-phonon coupling. However, in this paper, these high-order interactions are not considered since the leading processes of the C mode are three-phonon interactions. Besides, a complete cubic-order Hamiltonian should include the additional coupling of three linear parts, namely the last two terms with two new coupling coefficients $\eta_{1,2}$ in eq. (7). As shown below, it is the nonlinearity in eq. (2) that dominates in the C mode damping, and $\eta_{1,2}$ are not required to be estimated for the calculations.

From eq. (6), we can obtain several low-lying modes in the long-wavelength regime. The displacement field $\mathbf{u}(\mathbf{r})$ is expressed as Fourier transformation of the momentum representations $\mathbf{u}(k,l)$ with the wave vector k along $\mathbf{e}_y$ and the angle momentum l along $\mathbf{e}_x$, i.e., $\mathbf{u}(\mathbf{r})=1/\sqrt{S}\sum_{kl}\mathbf{u}(k,l)e^{i(lx/R+ky)}$, where $S=2\pi R \mathcal{L}$ is the cylindrical surface area with tube length $\mathcal{L}$. Inserting it into eq. (6), we can obtain the equation of motion as

$$\begin{equation} \ddot{\mathbf{u}}(k,l)+\omega_B^2\mathbf{\Lambda}(k,l)\mathbf{u}(k,l)=0. \end{equation} \tag{ 8 }$$

In this equation, $\omega_B \equiv \sqrt{\frac{K+\mu}{\rho R^2}}$ and the matrix $\mathbf{\Lambda}(k,l)$ is parameterized with elastic constants as

$$\begin{equation} \left(\begin{array}{c@{\!\quad}c@{\!\quad}c} \displaystyle\frac{\mu}{K+\mu} Y^2+l^2 & \displaystyle l\frac{K}{K+\mu}Y & -il \\ \displaystyle l\frac{K}{K+\mu}Y & \displaystyle Y^2+\frac{\mu}{K+\mu}l^2 & -i\nu Y \\ il & i\nu Y &\displaystyle1+\frac{\xi}{2}(l^2-1+Y^2)^2 \end{array}\right)\!, \end{equation} \tag{ 9 }$$

with two dimensionless parameters $\xi\equiv\frac{a^2\Xi}{R^2(K+\mu)}$ and $\nu\equiv \frac{K-\nu}{K+\nu}$, and a dimensionless variable $Y\equiv kR$. For a given l, three eigenvalues of $\mathbf{\Lambda}(k,l)$ correspond to three normal-mode frequencies $\omega_\sigma(k,l)$ with three polarizations σ, respectively. Each of them corresponds to a specific phonon mode, and is characterized by a unit polarization vector $\mathbf{e}_\sigma(k,l)$. Obviously, $\mathbf{u}(k,l)$ can be expanded a sum running over all polarizations,

$$\begin{equation} \mathbf{u}(k,l)=\sum_{\sigma}\mathbf{e}_{\sigma}(k,l)u_{\sigma}(k,l). \end{equation} \tag{ 10 }$$

The displacement field is quantized via expressing the normal-mode components $u_{\sigma} (k,l)$ in terms of the boson creation and annihilation operators, namely, $a^{\dagger} _{\sigma}(k,l)$ and $a_{\sigma}(k,l)$,

$$\begin{equation} u_{\sigma}(k,l)=\sqrt{\frac{\hbar}{2\rho\omega_{\sigma}(k,l)}}\left[a_{\sigma}(k,l)+ a^{\dagger}_{\sigma}(-k,-l) \right]\!, \end{equation} \tag{ 11 }$$

with the communication relation $[a_{\sigma}(k,l),a^{\dagger}_{\sigma'}(k',l')]= \delta(k-k') \delta_{\sigma\sigma'}\delta_{ll'}$. Using this result, the harmonic Hamiltonian (6) is expressed in the quantized form

$$\begin{equation*} H_0=\sum_{\sigma k l} \hbar \omega_{\sigma}(k,l)\left(a^{\dagger}_{\sigma}(k,l) a_{\sigma} (k,l)+\frac{1}{2}\right)\!. \end{equation*}$$

From this model, we obtain the l = 0 modes (including the longitudinal $(\sigma=L)$, twist $(\sigma=T)$ and breathing modes $(\sigma=B)$), the lowest-lying $l=\pm1$ modes $(\sigma=F_{\pm})$, and the lowest-lying $l=\pm2$ modes $(\sigma=C_{\pm})$. The general exact results of eq. (9) are complicated. But as for $|k_i| R\ll1$, the high-order terms of k_iR are omitted, and the results leave simple forms. Particularly, the C modes are degenerate and their dispersion relation is

$$\begin{equation} \omega_C=\sqrt{\frac{18(K+\mu)\xi}{5\rho R^2}+\frac{3(31K+43\mu)\xi}{50\rho}k^2}+ \mathcal{O} \left(k^4,\xi^2\right), \end{equation} \tag{ 12 }$$

with polarization vector $\textbf{e}_C^{\pm}$ as follows:

$$\begin{eqnarray} &&\frac{i}{\sqrt{5}}\!\left(\!\begin{array}{c} \displaystyle1+\frac{18}{25}\xi+\frac{(7K-9\mu)k^2 R^2}{40(K+\mu)} \\ \displaystyle\mp kR \left(\!\frac{1}{2}\!+\!\frac{9(5K\!+\!2\mu)\xi}{50\mu}\!-\!\frac{3(11K\!+\!3\mu)k^2R^2} {80(K+\mu)}\!\right) \\ \displaystyle\mp i \left( 2-\frac{9}{25}\xi-\frac{(3K-\mu)k^2R^2}{20(K+\mu)} \right) \end{array}\!\right)\!,\quad\ \end{eqnarray} \tag{ 13 }$$

where high-order terms of k and ξ are cut off due to $|k| R\ll1$ and the smallness of ξ (about $3.4\times10^{-3}$ at $R=0.5\ \text{nm}$). Apart from C mode, dispersion relations of other modes can be obtained from this model (see fig. 1), which are similar to the results in ref. [26]. The comparison between (a) and (b) in fig. 1 indicates that the curvature term has an obvious effect on various modes and gives gaps to the l > 1 modes.

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If expressing eq. (7) in terms of eq. (10), we find that

$$\begin{eqnarray} H_3&= \frac{1}{\sqrt{S}}\sum_{\sigma_1 \sigma_2 \sigma_3}\sum_{k_1 k_2}\Phi_{\sigma_1 \sigma_2 \sigma_3}(k_1,k_2,k_3) \nonumber\\ & \quad\times\,u_{\sigma_1}(k_1)u_{\sigma_2}(k_2)u_{\sigma_3}(k_3), \end{eqnarray} \tag{ 14 }$$

where l is ignored for simplicity, and $\Phi_{\sigma_1 \sigma_2 \sigma_3}(k_1,k_2,k_3)$ is the three-phonon coupling coefficient. Besides, the summation over wave vectors $(k_1,k_2,k_3)$ leaves $(k_1,k_2)$ on account of momentum conservation $k_3=-k_1-k_2$. Due to the degeneration, we only discuss the $C_+$ mode damping, and the signs are ignored below. In the long-wavelength regime, both decay $C\rightarrow FF$ and combination $CC\rightarrow T$ allow for the momentum and energy conservations. Since the longitudinal polarization direction (y-component in eq. (13)) of C modes is small, their coupling with longitudinal modes is ignorable. Thus, the process $CC\rightarrow L$ is unimportant. Based on the dispersion relations in $|k_i| R\ll1$ (see fig. 1(a)), the process $CF\rightarrow F$ is not allowed, and the frequencies of the l > 2 modes are so high that the C mode cannot merge the $l\leq 2$ modes into them. Besides, the $C\rightleftharpoons C+L/T/B$ channels are absent because they are kinematically forbidden. In conclusion, the channels $C\rightarrow FF$ and $CC\rightarrow T$ are the dominant damping mechanisms for a long-wavelength C phonon.

To derive the coupling coefficient $\Phi_{CFF}(k_1,k_2,k_3)$ for the decay $C(k_1)\rightarrow F(k_2)F(k_3)$, we obtain the linear part of the strain tensor for C modes via inserting eqs. (10) and (13) into eq. (4),

$$\begin{eqnarray} u^\mathrm{lin}_C(\mathbf{r})=\sum_k\frac{1}{\sqrt{5S}}\left(\begin{array}{c@{\quad}c} -\displaystyle\frac{9\xi}{5R} & \displaystyle\frac{9K\xi}{10\mu}k \\ \displaystyle\frac{9K\xi}{10\mu}k \displaystyle\frac{k^2R}{2} \end{array}\right)u_C(k) e^{iky}, \end{eqnarray} \tag{ 15 }$$

while, by combining eq. (5) with the C mode polarization (13) and the F mode polarization in ref. [43], the nonlinear part mixing C and F modes is

$$\begin{eqnarray} u^\mathrm{nlin}_{CF}(\mathbf{r})&= \sum_{k_1,k_2}\frac{i k_2}{2\sqrt{10}S} \left(\begin{array}{c@{\quad}c} -(k_1+3\nu k_2) & \displaystyle\frac{3}{2R} \\ \displaystyle\frac{3}{2R} k_1 \end{array}\right) \nonumber\\ & \quad\times\,\exp[i(k_1+k_2) y]u_C(k_1) u_F(k_2). \end{eqnarray} \tag{ 16 }$$

Substituting eqs. (15), (16) and the necessary strain tensors involved F mode in ref. [43] (including the linear part of F mode and the nonlinear part with mixing two F modes) into eq. (7), the leading order of the decay coefficient is obtained via comparing with eq. (14),

$$\begin{eqnarray} \Phi_{CFF}(k_1,k_2,k_3)&=& \frac{k_2 k_3}{\sqrt{5}}\Biggl[\frac{9(K+\mu)\xi}{20R} \nonumber\\ &&+\,\frac{K\mu R}{K+\mu}\left(k_1 k_2-3k_2^2+k_1 k_3-3k_3^2\right)\Biggr]\!. \end{eqnarray} \tag{ 17 }$$

In the same way, we obtain the coupling coefficient for the process $C(k_1)C(k_2) \rightarrow T(k_3)$:

$$\begin{equation} \Phi_{CCT}(k_1,k_2,k_3)=-\frac{3i}{5R}\left[\mu (k_1+k_2)k_3+\frac{9(K+\mu)\xi}{5R^2} \right]\!. \end{equation} \tag{ 18 }$$

Apart from these linear and nonlinear coupling terms, the additional third-order terms for process $C\rightarrow FF$, i.e., the η-coefficient terms in eq. (7), are expressed to leading order as

$$\begin{equation*} \frac{k_2^2 k_3^2 R}{\sqrt{5}}\left[\frac{9\left(2\nu^2-2\nu+1\right)\xi}{5}\eta_1 +\frac{\nu^2-2\nu+2}{2}\eta_2 (k_1 R)^2\right]\!, \end{equation*}$$

and for the process $CC\rightarrow T$ as

$$\begin{equation*} \eta_1\frac{9iK}{200 \mu}\xi k_1 k_2 (k_1+k_2) k_3 R. \end{equation*}$$

Compared with eqs. (17), (18), both of them are higher order and negligible due to $|k_i R|\ll1$ and $\xi\ll1$. Therefore, it is unnecessary to estimate $\eta_{1,2}$ for the calculations.

According to Fermi's golden rule [33,48], the single-mode relaxation time for excited phonons with mode K is given by

$$\begin{equation} \begin{array}{rcl} \tau_K^{-1}&=&-\displaystyle\frac{1}{\overline{n}_K} \frac{2\pi}{\hbar}\sum_{f} \left(|\langle f_{+} |H'|i \rangle|^2 - |\langle f_{-}|H'|i \rangle|^2\right)\\ &&\times\,\delta(E_f-E_i), \end{array} \end{equation} \tag{ 19 }$$

where $H'$ denotes a perturbation Hamiltonian, and $|i\rangle$ and $|f_{\pm}\rangle$ are the initial and final states having N_K and $N_K\pm1$ phonons of mode K, respectively. E_i and E_f are the energies of states $|i\rangle$ and $|f\rangle$, respectively. Besides, $\overline{n}_K=N_K-n(\omega_K)$, and $n(\Omega)=[\exp{(\hbar \Omega / k_B T)}-1]^{-1}$ is the thermal equilibrium number of phonons with frequency Ω at temperature T. Using the perturbation Hamiltonian (14) in the quantized field form (11), the relaxation rate of the phonon mode (k, σ) that undergoes three-phonon interactions becomes

$$\begin{eqnarray} \tau_\sigma^{-1}(k)&=& \frac{\hbar}{8\rho^3 R \omega}\int \frac{\text{d}k_1}{2\pi}\frac{| \Phi _{\sigma \sigma_1 \sigma_2}(k,k_1,k_2)|^2}{\omega_1\omega_2} \nonumber\\ &&\times\,\bigl\{[n(\omega_1)+n(\omega_2)+1]\delta(\omega-\omega_1-\omega_2) \nonumber\\ &&+\,2[n(\omega_1)-n(\omega_2)]\delta(\omega+\omega_1-\omega_2)\bigr\}\!, \end{eqnarray} \tag{ 20 }$$

with $k_2=-k-k_1$, $\omega=\omega_{\sigma}(k)$, $\omega_i=\omega_{\sigma_i}(k_i)$ [43]. Its first and second terms indicate the contributions from the decay and merging processes, respectively.

At ultra-low T, only the decay channel $C \rightarrow F F$ is the leading damping for C modes. Eventually, we obtain this decay rate via eq. (20) with the coupling coefficient (17),

$$\begin{eqnarray} \tau^{-1}_{CFF}&= \frac{9\sqrt{3} K^2 \mu^2 a^{1/2} \Xi^{1/4} \hbar} {5^{1/4}10\pi (R v_L)^{9/4} (K+\mu)^2 \rho^{13/4}} \nonumber\\ & \quad\times\,\coth{\left(\frac{3a\hbar}{2R^2 k_B T} \sqrt{\frac{\Xi}{10\rho}}\right)}, \end{eqnarray} \tag{ 21 }$$

with longitudinal sound velocity $v_L=\sqrt{4K\mu/\rho(K+\mu)}$. Keeping the leading order in $|k|R\ll1$, $\tau^{-1}_{CFF}$ is independent of k, and the quality factor is

$$\begin{equation} Q_{CFF}(T)=Q_{CFF}(0) \tanh{\left(\frac{3a\hbar}{2R^2 k_B T} \sqrt{\frac{\Xi}{10\rho}} \right)}, \end{equation} \tag{ 22 }$$

with the zero-temperature factor

$$\begin{equation} Q_{CFF}(0)=(1.15 \times 10^6) \left(R[\mathrm{nm}]\right)^{5/2}. \end{equation} \tag{ 23 }$$

For a typical size $R=0.5\ \text{nm}$, this yields $Q_{CFF}(T=0)\approx2.0 \times 10^5$. Taking $k=\pi/\mathcal{L}$ with $\mathcal{L}=500\ \text{nm}$, we display zero-temperature $\tau_{CFF}^{-1}$ and Q_CFF vs. the tube radius for the decay $C\rightarrow FF$ in fig. 2, as well as $\tau_{LFF}^{-1}$ and Q_LFF for the decay $L\rightarrow FF$. As shown, the decay effect is easier for thinner CNTs, but the C mode is stronger in thicker CNTs due to the monotonic increase of Q_CFF. In the range of radii considered, although the C mode decay is easier than the L mode, it is stronger for the C mode than for the L mode under the comparison of their Q-factors.

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As temperature rises, the merging process $CC\rightarrow T$ becomes another considerable contribution to the C mode damping. Applying eq. (20) with coupling coefficient (18) instead, we then obtain the relaxation rate as

$$\begin{equation} \tau^{-1}_{CCT}=\frac{3(160\mu k^2 R^4+729 a^2 \Xi)\hbar}{400\pi\sqrt{10\mu\Xi}R^5 \rho \sinh{\left(\frac{3a\hbar}{R^2 k_B T } \sqrt{ \frac{2\Xi}{5\rho}}\right)}}. \end{equation} \tag{ 24 }$$

The parameter Ξ in eqs. (21) and (24) manifests that the curvature character of CNTs influences directly the C mode damping. According to dispersion relations without the curvature (see fig. 1(b)), the decay $C\rightarrow FF$ hardly happens since the energy of a C phonon is always lower than that of a F phonon. In contrast, the merging rate of channel $CC\rightarrow T$ becomes extremely high without considering the curvature (see fig. 3). Hence, it is necessary to take into account the curvature term (3) for our calculation.

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The total relaxation rate $\tau_C^{-1}$ for C modes yields

$$\begin{equation} \tau^{-1}_C(k,T)=\tau^{-1}_{CFF}+\tau^{-1}_{CCT}, \end{equation} \tag{ 25 }$$

while the total quality factor is then Q_C. For ${k=\pi/\mathcal{L}}$ with $\mathcal{L}= 500\ \text{nm}$, the R-dependence of the finite-temperature Q_C is shown in fig. 4. As depicted, this dependence behaviour hardly changes from 0 to 1 K, but the curve becomes flat as T continues rising. Besides, Q_C of thicker tubes is more sensitive to temperature. Moreover, since Q_C at 5 K is even larger than the zero-temperature Q_LFF (see fig. 2), the C mode is stronger than the L mode at $T\leq5\ \text{K}$.

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To analyze the wave vector dependence, we adopt the exact $\tau_C$ from the exact eigen solutions of eq. (9). At ${T=10\ \text{K}}$, the $\tau_C$-dependence on the wave vector is displayed in fig. 5, and agrees well with the analytical formula (25) in the low-k regime. Furthermore, we also consider the η-coefficient terms of eq. (7), and find them ignorable even though the $\eta_{1,2}$ are as large as the elastic parameters in the range of k considered for $R=1\ \text{nm}$. In other words, the relaxation time is basically determined by parameters K, μ and Ξ at $R\leq1\ \text{nm}$ in the low-k regime.

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The thermal conductivity κ at given T is expressed as the sum of various phonon modes [49,50], $\kappa=\frac{1}{V}\sum_{k\sigma} C_{\sigma}(k) v_{\sigma}(k)^2\tau_{\sigma}(k)$, where V denotes the volume of CNT, and $C_{\sigma}$ and $v_{\sigma}$ indicate the specific heat and the magnitude of phonon group velocity for specific polarization σ, respectively. On the basis of the k-dependence of the relaxation time $\tau_{\sigma}(k)$, the T-dependence of thermal conductivity from various polarizations is shown in fig. 6(a) for $R=0.5\ \text{nm}$. Including C modes, the normalized contributions of this gapped-mode relaxation are depicted in fig. 6(b) for various R. At $T\geq8\ \text{K}$, the relative contributions from the C mode damping tend to increase with R reduction. In the case of $R=0.5\ \text{nm}$ and $T\geq11\ \text{K}$, this contribution rate is up to about 0.25 and therefore, C mode contributions are considerable to the total thermal conductivity in this situation.

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While phonon-phonon scattering rates derived from interatomic potentials are more accurate and reveal the differences for various chiralities [51], the continuum elasticity model provides a simplified and analytical framework to describe reasonably the phonon modes with long wavelength in CNTs [26,52]. Moreover, the latter has been used to predict effectively phonon-phonon scattering rates for modes with $|l|=0,1$ [43,53]. In this paper, we first calculate the relaxation rate $\tau_C^{-1}$ of the lowest-lying $|l|=2$ phonons (C modes) based on this model with the curvature-energy term [26].

The simple and approximate form of $\tau_C^{-1}$ is obtained by only keeping low order of kR terms, and used to calculate the upper bounds for the quality factor Q_C in the $|k_i|R\ll1$ regime. Our results suggest that the C mode is stronger than the L mode at $T\leq5\ \text{K}$, and Q_C rises with tube radius increasing at ultra-low temperature. On the other hand, our calculations for thermal conductivity use the exact $\tau_C^{-1}$ because of involving a relatively large wave vector. As a result, the relative thermal conductivity contributions from the C mode damping are up to 25% at $T\geq11\ \text{K}$ and $R=0.5\ \text{nm}$, which reveals the importance of these contributions to heat conduction in CNTs. We believe that our results are important for predicting the quality factor and thermal conductivity and will be helpful for further experimental and theoretical studies.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11535004, 11375086, 11120101005, 11175085, and 11235001), by the 973 National Major State Basic Research and Development of China, Grant No. 2013CB834400 and by the Science and Technology Development Fund of Macau under Grant No. 068/2011/A.