Intraband electron-phonon scattering in single-walled carbon nanotubes

The intraband electron-phonon matrix elements of all single-walled carbon nanotubes in the radius range from $3.5\text{to}12\mathrm{\AA }$ were calculated within a nonorthogonal tight-binding model of the electronic band structure. All possible scattering processes for all phonons allowed by the selection rules were considered. The matrix elements for scattering by tangential optical phonons in the linear bands of armchair nanotubes are nonzero for either forward or backward scattering and tend to $12.8\mathrm{eV}∕\mathrm{\AA }$ (intravalley) and $18.1\mathrm{eV}∕\mathrm{\AA }$ (intervalley) in the large-radius limit. For scattering in the lowest-energy parabolic conduction band of any nanotube, the matrix elements have large-radius limits of $9.2\mathrm{eV}∕\mathrm{\AA }$ (intravalley) and $13.4\mathrm{eV}∕\mathrm{\AA }$ (armchair, intervalley). For twist and longitudinal acoustic phonons, they depend on the chiral angle $\theta$ as $\sin 3\theta$ and $\cos 3\theta$ in the large-radius limit. While the matrix elements show radius and chirality dependence, the scattering lengths are almost nanotube independent. For medium-radius metallic tubes, the scattering length in linear bands is $1.0–1.6\mu \mathrm{m}$ (acoustic phonons), $0.16\mu \mathrm{m}$ (longitudinal optical phonons), and $0.06\mu \mathrm{m}$ ($A_1^{\prime }$ $K$-point phonons).

Figure 1

Selected electronic bands of the zigzag tube (18, 0), chiral tube (14, 5), and armchair tube (10, 10) with radii $\approx 7.0$, 6.7, and $6.8\mathrm{\AA }$, respectively. The left panels show bands with minima, which can be at the zone center (zigzag tubes) or at two mirror points inside the zone (armchair and chiral tubes). For a given initial energy of the electron, in zigzag tubes there is one nonequivalent point (I) and two scattering processes—forward and backscattering. For chiral tubes, there are two nonequivalent points (I and II) and two scattering processes (FS and BS). For armchair tubes, there are two nonequivalent points (I and II), two scattering processes (FS and BS), and two possibilities for scattering with respect to the valleys of a given band—intravalley and intervalley scattering. The same sets of processes are present in the case of band crossing at the Fermi energy of metallic tubes (right panels) as well as for bands with minima in semiconducting tubes (not shown). $E_F$ is the Fermi energy.

Figure 2

The calculated intravalley electron-phonon matrix element (in eV/Å) for the armchair tube (10, 10) versus initial electron energy. The arrows point to the common lowest-energy limiting values for the curves for forward and backward scattering. The panels show data for the six possible zone-center phonons for the cases of massive bands and massless bands. The irreguliarities of the curves for the RBM and the LA phonons are due to the crossing of the corresponding phonon branches and mixing of the phonon modes.

Figure 3

The calculated intravalley EPMEs $M^a$ (in eV) and $M^o$ (in eV/Å) for all armchair tubes in the radius range from $3.5\text{to}67\mathrm{\AA }$ versus tube radius. The numbers are the large-radius limits of the EPMEs. The largest EPMEs are due to TW, TO, and LO modes.

Figure 4

The calculated intervalley EPME for the armchair tube (10,10) versus initial electron energy. The panels show data for the six possible $K$-point phonons for the MVB and MLB cases.

Figure 5

The calculated intervalley EPMEs $M^K$ (in eV/Å) for all armchair tubes in the radius range from $3.5\mathrm{\AA }\text{to}67\mathrm{\AA }$ versus tube radius. The numbers are the large-radius limits of the EPMEs. Only the $A_1^{\prime }$ mode gives rise to a significant EPME at large radii.

Figure 6

The calculated intravalley EPME (in eV/Å) for scattering by optical phonons in the MLB of tube (14,5). The points for positions I and II are given with solid and dashed lines, respectively. The panels show the energy region close to the minimal initial electron energy, where there is a noticeable deviation of the curves from straight lines and a convergence of the curves for the FS and BS processes to common limiting points. The width of this energy region is about $0.1\mathrm{eV}$.

Figure 7

The calculated intravalley EPMEs $M^a$ (in eV) and $M^o$ (in eV/Å) for the MLBs of all mod0 zigzag and chiral tubes in the radius range from $3.5\text{to}12\mathrm{\AA }$. The panels present the common lowest-energy limit of the data for FS and BS processes. Such common limits do not exist for optical phonons of armchair tubes. However, the data for armchair tubes are shown on the panels for acoustic modes because only BS processes are possible for these phonons. At large radii, the EPME tends to zero for the RBM and ZO mode and tends to finite values for the TW, LA, TO, and LO modes.

Figure 8

Same as for Fig. 7 but for the MVBs of all mod0 armchair, zigzag, and chiral tubes in the radius range from $3.5\text{to}12\mathrm{\AA }$.

Figure 9

Same as for Fig. 7 but for the MVBs of all mod1 zigzag and chiral tubes in the radius range from $3.5\text{to}12\mathrm{\AA }$.

Figure 10

Same as for Fig. 7 but for the MVBs of all mod2 zigzag and chiral tubes in the radius range from $3.5\text{to}12\mathrm{\AA }$.