Electron-Phonon Interaction and Transport in Semiconducting Carbon Nanotubes

We calculate the electron-phonon scattering and binding in semiconducting carbon nanotubes, within a tight-binding model. The mobility is derived using a multiband Boltzmann treatment. At high fields, the dominant scattering is interband scattering by $LO$ phonons corresponding to the corners $K$ of the graphene Brillouin zone. The drift velocity saturates at approximately half the graphene Fermi velocity. The calculated mobility as a function of temperature, electric field, and nanotube chirality are well reproduced by a simple interpolation formula. Polaronic binding give a band-gap renormalization of $\sim 70\mathrm{m}\mathrm{e}\mathrm{V}$, an order of magnitude larger than expected. Coherence lengths can be quite long but are strongly energy dependent.

Figure 1

Phonon binding and scattering vs electron energy ${\epsilon }_k$ (relative to conduction band edge ${\epsilon }_c$), for a (25,0) tube ($d=2.0\mathrm{n}\mathrm{m}$). (a) Binding energy, Eq. (3); (b-c) inverse lifetime, Eq. (2); and (d) coherence length. The energy range shown includes only the first band. In (a), (b), and (d), solid red curves are for $T=300\mathrm{K}$, dotted blue curves are for $T=10\mathrm{K}$. Note the large change in scale between low and high energy [$\times 50$ in (b) and (c) for low energy, and $\times 250$ in (d) for high energy]. (c) Decomposition of $e\mathrm{\text{−}}ph$ scattering rate for 300 K: total (red solid curve); $\Gamma$-point contributions (green dotted curves); $K$-point contributions (blue dashed curves). ${\Gamma }_{\mathrm{R}\mathrm{B}\mathrm{M}}$ is radial breathing mode. The inset shows 2D graphene Brillouin zone, with the integration areas shown by the circles of radius equal to 0.1 of the $\Gamma –M$ distance, which contribute virtually 100% of the total scattering; lines are only schematic.

Figure 2

(a) Drift velocity vs electric field at $T=300\mathrm{K}$ (red solid curve) and $T=10\mathrm{K}$ (black dotted curve) in (25,0) tube. (b) Inverse mobility at $T=300\mathrm{K}$ vs $E$ field; straight lines show corresponding fits to Eq. (5), with $v_s=5.0\times {10}^7$. Fitted values of ${\mu }_0$ are ${\mu }_0=65000{\mathrm{c}\mathrm{m}}^2/\mathrm{V}\mathrm{s}$ for (25,0) tube (red triangles); ${\mu }_0=35500{\mathrm{c}\mathrm{m}}^2/\mathrm{V}\mathrm{s}$ for (19,0) tube (green squares); and ${\mu }_0=15000{\mathrm{c}\mathrm{m}}^2/\mathrm{V}\mathrm{s}$ for (13,0) tube (blue circles).

Figure 3

(a) Zero-field mobility vs tube diameter, for tubes of many different chiralities. Temperatures are $T=450\mathrm{K}$ (red), $T=300\mathrm{K}$ (green), and $T=150\mathrm{K}$ (blue). Values are based on linear extrapolation of ${\mu }^{-1}$, Eq. (5), for $E$ field between 0.01 and $0.5\mathrm{V}/\mu \mathrm{m}$. Solid black curves are fitted to Eq. (6). (b) Polaron binding energy (black squares) vs tube diameter. Red circles show portion of binding energy from optical phonons, and blue triangles show the much smaller binding energy (shown $\times 15$) from acoustic phonons.

Figure 4

Steady-state distribution function for an electron in a (25,0) tube at $T=300\mathrm{K}$. $E$ fields are $E=0.001\mathrm{V}/\mu \mathrm{m}$ (black dotted curve), 0.01 (blue, next to black dotted curve), 0.03 (light blue, middle), 0.1 (green, next to red dashed curve), and 0.5 (red dashed curve) $\mathrm{V}/\mu \mathrm{m}$. Distributions are shown with respect to (a) energy, and (b) wave vector along tube axis. The top axis in (b) shows corresponding energy. (Curves include Gaussian broadening of 2 meV.)