Empirical-potential study of phonon transport in graphitic ribbons

Thermal properties of graphitic ribbon are investigated based on Brenner’s empirical potential. The reliability and usefulness of the empirical potential to address the thermal properties of covalent-bonded carbon nanostructures are verified through acomparison of phonon dispersion relations and the density of states of carbon nanotubes with first-principles calculations. The analysis reveals unique edge-phonon states that are highly localized at edge carbon atoms of both armchair and zigzag ribbons. Applying the phonon dispersion relations to the Landauer formula of phonon transport, the quantization and universal features of the low-temperature thermal conductance of graphitic ribbon are elucidated, and it is found that the width of the quantization plateau in the low-temperature region is inversely proportional to the ribbon width.

Figure 1

(a) Phonon dispersion curves and (b) density of states for a single-walled carbon nanotube with chiral vector (10, 10). $\mid \mathbf{T}\mid$ denotes the magnitude of the translation vector along the tube axis.

Figure 2

Atomic structure of (a) armchair ribbon $(N_a=7)$ and (b) zigzag ribbon $(N_z=4)$. Broken rectangle represents a unit cell, and $\mathbf{a}$ and $\mathbf{z}$ denote the translation vectors along the ribbon axis.

Figure 3

Phonon dispersion relations of armchair ribbon with $N_a=9$, 11, and 15. Shaded areas represent the phonon dispersion relations of a two-dimensional graphitic sheet projected onto the armchair axis. Edge-localized phonon modes are indicated by arrows.

Figure 4

Atomic displacement of edge-localized phonon states at $k\mid \mathbf{a}\mid ∕\pi =1$ in the out-of-plane modes of an armchair ribbon of $N_a=9$ for (a) the transverse acoustic mode, and (b) the lowest-lying optical mode. (c), (d) Atomic displacement of states in the two branches between the upper and lower shaded areas in Fig. 3. Solid circles represent edge carbon atoms.

Figure 5

Phonon dispersion relations of zigzag ribbons with $N_a=8$, 10, and 16. Shaded areas represent the phonon dispersion relations of a two-dimensional graphitic sheet projected onto the zigzag axis. Edge-localized phonon modes are indicated by arrows.

Figure 6

Atomic displacements of edge-localized phonon states at $k\mid z\mid ∕\pi =1$ in the out-of-plane modes of zigzag ribbon with $N_a=6$ for (a) the transverse acoustic mode and (b) the lowest-lying optical mode. Solid circles represent edge carbon atoms.

Figure 7

(a) Low-temperature phonon-derived thermal conductance in graphitic ribbon. Open circles denote the armchair ribbon ($N_a=9$, 11, and 15), and solid circles denote the zigzag ribbon ($N_z=8$, 10, and 16). (b) Thermal conductances of armchair ($N_a=11$ and 15) and zigzag ($N_a=10$ and 16) ribbons as a function of temperature scaled by the energy gap of the lowest optical mode. Broken curve represents a universal curve given by Eq. (5).