Vibrational modes and low-temperature thermal properties of graphene and carbon nanotubes: Minimal force-constant model

We present a phenomenological force-constant model developed for the description of lattice dynamics of $s{p}^2$ hybridized carbon networks. Within this model approach, we introduce a set of parameters to calculate the phonon dispersion of graphene by fitting the ab initio dispersion. Vibrational modes of carbon nanotubes are obtained by folding the two-dimensional (2D) dispersion of graphene and applying special corrections for the low-frequency modes. Particular attention is paid to the exact dispersion law of the acoustic modes, which determine the low-frequency thermal properties and reveal quantum size effects in carbon nanotubes. On the basis of the resulting phonon spectra, we calculate the specific heat and the thermal conductance for several achiral nanotubes of different diameters. Through the temperature dependence of the specific heat, we demonstrate that phonon spectra of carbon nanotubes show one-dimensional behavior and that the phonon sub-bands are quantized at low temperatures. Consequently, we prove the quantization of the phonon thermal conductance by means of an analysis based on the Landauer theory of heat transport.

Figure 1

The graphene honeycomb lattice with lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. A carbon nanotube can be constructed by rolling up the graphene sheet along $\mathbf{C}$ so that points $O$ and $A$ coincide (as well as $B$ and $B^{\prime }$ do). $\Theta$ denotes the chiral angle. (Figure is taken from Ref. 60.)

Figure 2

(Color online) An atom $A$ and its first nearest-neighbor atoms $Bp$ $\left(p=1,2,3\right)$. ${\varphi }_{\text{r}}$, ${\varphi }_{\text{ti}}$, and ${\varphi }_{\text{to}}$ represent forces in radial, in-plane, and out-of-plane directions.

Figure 3

(Color online) Dispersion relation of graphene calculated with the 4NNFC model (solid lines) in direct comparison with ab initio calculations (dotted lines) of Bohnen and Heid (Ref. 49). (a) 4NNFC approach with the original parametrization of Saito et al. (Ref. 38); (b) 4NNFC approach with our parametrization. The corresponding sets of parameters are listed in Table .

Figure 4

(Color online) Phonon dispersion for a (10,10) CNT calculated with the fourth-nearest-neighbor model with (a) the parametrization of Saito et al. (Ref. 38) and (b) our parametrization. The corresponding parameters are listed in Table . Both parametrizations have been subsequently corrected in order to obtain $\omega =0$ at $q=0$ for the acoustic TW mode.

Figure 5

(Color online) Low-frequency region of the phonon-dispersion relation of a (10,10) CNT, shown near the $\Gamma$ point, calculated within (a) 4NNFC model with the original parametrization, (b) 4NNFC model with our parametrization, (c) force-constant model by Mahan and Jeon (Ref. 40) based on three free parameters. While in (a) all four acoustic branches have linear dispersion for small wave vectors, in (b) and (c) two increase linearly with $q$ (LA, TW), and two are degenerate (TA) and increase quadratically.

Figure 6

(Color online) Phonon dispersion for a (10,0) CNT: [(a) and (c)] 4NNFC model with the parametrization of Saito et al. (Ref. 38); [(b) and (d)] 4NNFC model with our parametrization. The two parametrizations show, respectively, linear and quadratic dispersions for the TA mode. For both, the ${\varphi }_{\text{to}}$ constants have been corrected in order to obtain zero frequency for the acoustic TW mode at the Brillouin-zone center. (e) Low-frequency region of the spectrum, calculated with the model of Mahan and Jeon (Ref. 40), which gets the correct quadratic dispersion of the TA mode.

Figure 7

(Color online) (a) Frequency of the radial-breathing mode of various armchair $\left(n=3–12\right)$ and zigzag $\left(n=6–20\right)$ tubes as a function of the nanotube diameter, calculated with the 4NNFC model with our parametrization. For comparison, the dashed lines show the results for the original parametrization of Saito et al. (Ref. 38). [(b) and (c)] Frequency of the RBM as a function of the inverse tube diameter for armchair and zigzag tubes. The solid lines are a linear fit to the data excluding the small-diameter tubes (3,3), (4,4), (6,0), and (7,0), which are marked by circles. These show a deviation from the predicted behavior, with a decrease in the RBM frequency. Reference 34 explains it as a consequence of the hybridization changes and the decrease in the $\pi$ interaction induced by the curvature.

Figure 8

(a) Specific heat as a function of temperature for graphene (dotted line), calculated with the 4NNFC model and our parameters, and for a (10,10) CNT calculated with the model of Mahan and Jeon (Ref. 40) (dotted-dashed line), the 4NNFC model with our parameters (solid line), and the 4NNFC model with the original constants of Saito et al. (Ref. 38) (dashed line). The inset shows a wider temperature interval: $c_V$ approaches the value $2078\text{mJ}/\text{g}\text{K}$ for high temperatures. (b) Specific heat on a logarithmic scale for the low-frequency region. The assignment of the line styles is the same as in (a).

Figure 9

Low-temperature specific heat for a (10,10) and a (10,0) CNT calculated with the 4NNFC and our parametrization. The inset shows the value of their slope $\alpha =\text{d}\left(\text{log}{c}_V\right)/\text{d}\left(\text{log}T\right)$.

Figure 10

(Color online) Chirality dependence of the specific heat: (a) Temperature dependence of $c_V$ for a (10,10) CNT (straight lines) and a (10,0) CNT (dashed lines). The index 1 refers to the model of Mahan and Jeon (Ref. 40) and the index 2 to the 4NNFC model with our parameters. (b) The effect of tube diameter on zigzag and armchair CNT specific heats. At a given temperature, the specific heat increases with the increase in tube diameter. The upper limit is given by graphene, with $c_V=794\text{mJ}/\text{g}\text{K}$ at 300 K.

Figure 11

(Color online) Phonon dispersion of a (10,10) CNT and the transmission function $\mathcal{T}={\sum }_s{\mathcal{T}}_s\left(\omega \right)$. The latter is a sum of $s=120$ step functions.

Figure 12

(Color online) (a) Phonon thermal conductance for a (10,10) CNT calculated with the model of Mahan and Jeon (Ref. 40) (dotted-dashed line), the 4NNFC model with our parameters (solid line), and the 4NNFC model with the original constants of Saito et al. (Ref. 38) (dashed line). The two latter show, respectively, quadratic and linear dispersions for the TA mode. (b) Thermal conductance for several carbon nanotubes, calculated with our parametrization.