Coupled dynamics of electrons and phonons in metallic nanotubes: Current saturation from hot-phonon generation

We show that the self-consistent dynamics of both phonons and electrons is the necessary ingredient for the reliable description of the hot phonons generation during electron transport in metallic single-wall carbon nanotubes (SWNTs). We solve the coupled Boltzmann transport equations to determine in a consistent way the current vs voltage (I-V) curve and the phonon occupation in metallic SWNTs which are lying on a substrate. We find a good agreement with measured I-V curves and we determine an optical phonon occupation which corresponds to an effective temperature of several thousands K (hot phonons), for the voltages typically used in experiments. We show that the high-bias resistivity strongly depends on the optical phonon thermalization time. This implies that a drastic improvement of metallic nanotubes performances can be achieved by increasing the coupling of the optical phonons with a thermalization source.

Figure 1

(Color online) Scheme of the model used in this paper.

Figure 2

(Color online) Lower panel: current vs voltage characteristic of a 300-nm-long carbon nanotube (CNT). Calculations are done fixing the phonon occupation to zero (cold phonons) or allowing phonons to heat up (hot phonons, ${\tau }_{th}=5.31\mathrm{ps}$). Measurements are from Refs. 6, 7. Upper panel: phonon effective temperature of the Raman $G$ peak as a function of the voltage.

Figure 3

(Color online) Current vs voltage curve calculated for various tube lengths (L); ${\tau }_{th}=5.31\mathrm{ps}$.

Figure 4

(Color online) High-bias differential resistence for various tubes. Different symbols correspond to different ${\tau }_{th}$. Measurements (dots) are from Ref. 7. Inset: the scattering length $l_{sc}$ [defined in Eq. (5)] as a function of ${\tau }_{th}$. The dashed line corresponds to the experimental value $l_{sc}=8.5\mathrm{nm}$.

Figure 5

(Color online) Phonon occupation $n$ as a function of the position along the tube, $x$, and momentum $k$. Upper panel: $\mathbf{\Gamma }$-${\mathrm{E}}_{2g}$ LO phonon. Lower panel: $\mathbf{K}\text{−}{A}_1^{\prime }$. The tube is the same as in Fig. 2, with voltage $1.4\mathrm{V}$.