Complex electron-phonon driven lattice dynamics in ultrasmall-radius (5,0) carbon nanotubes

By using first principles density functional perturbation theory, we computed the phonon dispersions of 4 nm diameter (5,0) carbon nanotubes. We investigated the development of phonon anomalies as the Fermi surface sharpens. The soft modes are related to the nesting features in the electronic band structure. We found that two phonon branches are strongly renormalized by electron-phonon coupling, and the anomaly of one phonon branch is traced to interband coupling while the anomaly of another phonon branch is due to intraband coupling. The complex behavior is explained using a simple model.

Figure 1

Band structure of (5,0) for the atomic coordinates relaxed with a Gaussian width of 0.2 eV and 32 $k$ points in the Brillouin zone. $k$ is given in units of $\pi /a$. The Fermi energy is set to 0.

Figure 2

Phonon dispersions of (5,0) calculated with a Fermi surface smearing of $w=0.2\text{eV}$.

Figure 3

Phonon dispersions of (5,0) calculated with a Fermi surface smearing of $w=0.1\text{eV}$ are shown as solid lines. The dotted lines show the phonons calculated with $w=0.2\text{eV}$ for comparison. Only those modes that are noticeably affected by el-ph coupling are shown.

Figure 4

Phonon dispersions of (5,0) calculated with a Fermi surface smearing of $w=0.025\text{eV}$ are shown as solid lines. The dotted lines show the phonons calculated with $w=0.2\text{eV}$ for comparison. Only those modes that are noticeably affected by el-ph coupling are shown. The arrows mark the nesting $q$ vectors corresponding to $q=k_{+}$—$k_{-}=0.13$ and $q=2k_{-}=0.26$.

Figure 5

Atomic distortions corresponding to the eigenvectors of the modes at the zone center for the branches labeled as $C$ and $D$ in Fig. 4.

Figure 6

(Color online) The energy changes when the lattice is distorted by frozen phonon displacements. Dots and squares correspond, respectively, to the frozen phonon displacements of modes $C$ and $D$ at zone center, and the lines are fourth order polynomial fits. The insets show expanded views near $\Delta E=0$. Results in Figs. 6a, 6b are calculated with Gaussian broadening widths of $w=0.1\text{eV}$ and of $w=0.025\text{eV}$, respectively.

Figure 7

(Color online) The solid lines show the phonon frequencies for a model that describes Einstein-like phonons with their bare frequency renormalized due to the coupling to two electronic bands. Different colors represent different smearing at the Fermi level, as labeled in the legend. The dotted lines are the phonon dispersion curves calculated from first principles. Figure 7a shows that the behavior of branch $C$ is obtained when only the interband coupling is taken into account. Figure 7b shows that when only the intraband coupling of the negative mass band is nonzero, we get the behavior of branch $D$.