Hot phonon effects on the high-field transport in metallic carbon nanotubes

We present a kinetic transport model for hot electrons and hot phonons in metallic single-wall carbon nanotubes. The transport model is based on a coupled system of Boltzmann equations of linearly dispersive electrons and optical phonons. An efficient and accurate deterministic numerical scheme is developed to solve the set of kinetic equations. With this numerical tool we study in detail the high-field transport properties of ohmically contacted molecular nanotube wires lying on a substrate. The simulations demonstrate that the optical phonons are strongly driven out of thermal equilibrium. Nonequilibrium optical phonons are found to influence considerably the electron transport in the high-field regime. We observe that the steady-state current at high bias is sensitive to the anharmonic lifetime of the optical phonons and to the phonon group velocities. Comparisons of experimental current-voltage characteristics with the theoretical results obtained with electron-phonon coupling coefficients as predicted by density functional calculations exhibit very good agreement.

Figure 1

(Color online) Current $J$ versus applied voltage $U$ for a one micron long SWCNT. The symbols 엯 display the theoretically obtained data of Yao (Ref. 10) and the solid line represents our results.

Figure 2

(Color online) Current-voltage (J-U) characteristics of ohmically contacted SWCNT’s of different lengths and diameters $d\sim 2–2.5\mathrm{nm}$. Solid lines represent the measurements presented in Ref. 11 and the markers 엯 display the results of the performed transport simulation.

Figure 3

(Color online) Influence of the nonequilibrium behavior of the optical phonons on the current-voltage characteristics (J-U) of metallic SWCNTs with $L=700\mathrm{nm}$ and $L=150\mathrm{nm}$. The symbols ◇ display results obtained with equilibrium phonon distributions at adapted temperatures and the markers 엯 represent results obtained with nonequilibrium phonons. Solid lines depict the experimental data presented in Ref. 11.

Figure 4

(Color online) Steady-state results for a $150\mathrm{nm}$ long metallic SWCNT at $1\mathrm{V}$ bias: nonequilibrium distribution $g_1$ of K phonons as function of $x$ and $q$.

Figure 5

(Color online) Steady-state results for a $150\mathrm{nm}$ long metallic SWCNT at $1\mathrm{V}$ bias: nonequilibrium distribution $g_2$ of $\Gamma$ phonons as function of $x$ and $q$.

Figure 6

(Color online) Steady-state results for a $150\mathrm{nm}$ long metallic SWCNT at $1\mathrm{V}$ bias: difference $f_2-f_1$ of the distribution functions of left and right propagating electrons interacting with nonequilibrium phonons as function of $x$ and $\epsilon$.

Figure 7

(Color online) Steady-state results for a $150\mathrm{nm}$ long metallic SWCNT at $1\mathrm{V}$ bias: distributions ${⟨g⟩}_1$ and ${⟨g⟩}_2$ of thermalized K phonons (solid line) and $\Gamma$ phonons (dashed line) as functions of $x$; $q$-averaged nonequilibrium distributions of K phonons (symbols 엯) and $\Gamma$ phonons (symbols ◇) as functions of $x$.

Figure 8

(Color online) Steady-state results for a $150\mathrm{nm}$ long metallic SWCNT at $1\mathrm{V}$ bias: difference $f_2-f_1$ of the distribution functions of left- and right-propagating electrons interacting with thermalized phonons as function of $x$ and $\epsilon$.

Figure 9

(Color online) Current-voltage (J-U) characteristics of a $300\mathrm{nm}$ long ohmically contacted SWCNTs with diameter $d=2\mathrm{nm}$ calculated for different phonon relaxation times $\tau$. The solid line represents the experimental data presented in Ref. 11.