Electron-phonon matrix elements in single-wall carbon nanotubes

We have developed the electron-phonon matrix element in single-wall carbon nanotubes by using the extended tight-binding model based on density functional theory. We calculate this matrix element to study the electron-phonon coupling for the radial breathing mode (RBM) and the $G$-band $A$ symmetry modes of single-wall carbon nanotubes. Three well-defined family patterns are found in the RBM, longitudinal optical (LO) mode and transverse optical (TO) mode. We find that among the RBM, LO, and TO modes, the LO mode has the largest electron-phonon interaction. To study the electron-phonon coupling in the transport properties of metallic nanotubes, we calculate the relaxation time and mean free path in armchair tubes. We find that the LO mode, $A_1^{\prime }$ mode, and one of the $E_1^{\prime }$ modes give rise to the dominant contributions to the electron inelastic backscattering by phonons. Especially, the off-site deformation potential gives zero matrix elements for $E_1^{\prime }$ modes while the on-site deformation potential gives rise to nonzero matrix elements for the two $E_1^{\prime }$ modes, indicating that the on-site deformation potential plays an important role in explaining the experimentally observed Raman mode around $2450{\mathrm{cm}}^{-1}$ in carbon.

Figure 1

The nine nonzero off-site deformation potential vectors ${\vec{\alpha }}_p$. The dashed curves represent the atomic potential.

Figure 2

The nine nonzero off-site deformation potential vectors. ${\vec{\beta }}_p$. The dashed curves represent the atomic potential.

Figure 3

The six nonzero on-site deformation potential vectors ${\vec{\beta }}_p$. The dashed curves represent the atomic potential. For ${\lambda }_{ss}$, ${\lambda }_{\sigma \sigma }$, ${\lambda }_{\pi \pi }$ two same orbitals are illustrated by shifting them with respect to each other.

Figure 4

The dependence of the nine off-site deformation potential matrix elements ${\alpha }_p$ on interatomic distance. (a) The matrix elements with the corresponding deformation potential vectors ${\vec{\alpha }}_p$ along the bond of the two atoms. (b) The matrix elements with the corresponding deformation potential vectors ${\vec{\alpha }}_p$ perpendicular to the bond of the two atoms. The nearest-neighbor distance between two carbon atoms is $a_{c-c}=1.42\mathrm{\AA }$.

Figure 5

The six on-site deformation potential matrix elements ${\lambda }_p$ as a function of interatomic distance. (a) The matrix elements with the corresponding deformation potential vectors ${\vec{\lambda }}_p$ along the bond of the two atoms. (b) The matrix elements with the corresponding deformation potential vectors ${\vec{\lambda }}_p$ perpendicular to the bond of the two atoms. The nearest-neighbor distance between two carbon atoms is $a_{c-c}=1.42\mathrm{\AA }$

Figure 6

Family patterns in the el-ph matrix element vs inverse of diameter at $E_{22}^S$ transitions for the RBM, (a) SI and (b) SII tubes. The family numbers for $n-m$ are shown in the figures.

Figure 7

Family patterns in the el-ph matrix element vs chiral angle at $E_{22}^S$ transitions for the RBM, (a) SI and (b) SII tubes. The family numbers for $n-m$ are shown in the figures.

Figure 8

Family patterns in the el-ph matrix element vs inverse of diameter at $E_{11}^M$ transitions for the RBM of metallic tubes, (a) $E_{11L}^M$ and (b) $E_{11H}^M$. The family numbers for $n-m$ are shown in the figures.

Figure 9

Family patterns in the el-ph matrix element vs chiral angle at $E_{11}^M$ transitions for the RBM of metallic tubes, (a) $E_{11L}^M$ and (b) $E_{11H}^M$. The family numbers for $n-m$ are shown in the figures.

Figure 10

The family patterns in the el-ph matrix element at $E_{22}^S$ transitions for the LO mode, (a) the matrix element vs inverse of diameter and (b) the matrix element vs chiral angle. The family numbers for $n-m$ are shown in the figures.

Figure 11

The family patterns in the el-ph matrix element vs inverse of diameter at $E_{22}^S$ transitions for the TO mode, (a) SI and (b) SII tubes. The family numbers for $n-m$ are shown in the figures.

Figure 12

The family patterns in the el-ph matrix element vs chiral angle at $E_{22}^S$ transitions for the TO mode, (a) SI and (b) SII tubes. The family numbers for $n-m$ are shown in the figures.

Figure 13

The electron-phonon scattering processes near the Fermi level (the $K$ and $K^{\prime }$ points) of an armchair tube. The processes 1 and 2 are the intravalley forward and backward scattering, respectively, and the corresponding phonons are around the $\Gamma$ point. The processes 3 and 4 are the intervalley forward and backward scattering, respectively, and the corresponding phonons are around the $K$ point.

Figure 14

The el-ph matrix element for the $E_{22}^S$ transition for the RBM by considering only the $\pi$ orbital for (a) SI and (b) SII tubes. The family numbers for $n-m$ are shown in the figures.