Negative Differential Conductance and Hot Phonons in Suspended Nanotube Molecular Wires

Freely suspended metallic single-walled carbon nanotubes (SWNTs) exhibit reduced current carrying ability compared to those lying on substrates, and striking negative differential conductance at low electric fields. Theoretical analysis reveals significant self-heating effects including electron scattering by hot nonequilibrium optical phonons. Electron transport characteristics under strong self-heating are exploited for the first time to probe the thermal conductivity of individual SWNTs ($\sim 3600\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ at $T=300\mathrm{K}$) up to $\sim 700\mathrm{K}$, and reveal a $1/T$ dependence expected for umklapp phonon scattering at high temperatures.

Figure 1

Freely suspended nanotube devices. (a) SEM image, taken at a 45° angle, of the nonsuspended (on nitride) and suspended (over a $\sim 0.5\mu \mathrm{m}$ deep trench, entirety of trench depth not shown in the image) $L\sim 2\mu \mathrm{m}$ nanotube segments of an individual nanotube, both segments with Pt contacts under the tube [3, 4]. (b) A schematic of the device cross section. (c) SEM image of a nanotube suspended over a trench across a $\sim 10\mu \mathrm{m}$ Pt electrode gap. The bright features between the Pt electrodes were metal lines at the bottom of the wide trench used as local gates. The diameters of the nanotubes were measured by AFM in the range of $d\sim 2–3\mathrm{nm}$. Nanotubes grown under the same condition on transmission electron microscope (TEM) grids were confirmed to be single-walled by TEM.

Figure 2

High-field electron transport in suspended nanotubes. (a) Current-voltage ($I\mathrm{\text{−}}V$) characteristics of the same-length ($L\sim 3\mu \mathrm{m}$) suspended and nonsuspended portions of a SWNT ($d\sim 2.4\mathrm{nm}$) at room temperature measured in vacuum. The symbols represent experimental data; the lines are calculations based on average tube temperature (similar within $\sim 5\%$ to that based on actual tube temperature profile and resistance integrated over the $\sim 3\mu \mathrm{m}$ tube length). (b) Computed average acoustic (lattice) and effective optical phonon temperature vs bias voltage for the suspended tube segment in (a) quantifying the degree of self-heating. The figure inset shows the energy flow in our model from electrons (“heated” by the electric field) to optical phonons and then acoustic phonons. We estimated the heat dissipation by radiation (ignored in our model) to be less than 1% of the power input even up to average SWNT temperatures $T\sim 800\mathrm{K}$. Dissipation at the contacts ($I^2{R}_c$) is estimated to be less than 5% at high bias.

Figure 3

Currents in various-length suspended SWNTs. (a) $I\mathrm{\text{−}}V$ of four suspended tubes with lengths: $L\sim 0.8$, 2.1, 3, and $11\mu \mathrm{m}$. Symbols are experimental data; solid lines are calculations. The $I\mathrm{\text{−}}V$ curves are highly reproducible over many sweeps without irreversible changes to the device, indicating that the electrical characteristics are not due to irreversible contact change. The tube diameters used in the calculation (consistent with AFM) were $d\sim 2$, 2.4, 2.4, and 3.2 nm, respectively. The contact resistance $R_c$ was used as a parameter to fit the $I\mathrm{\text{−}}V$ curves at low bias. $R_c\sim 15\mathrm{k}\Omega$ for all tubes except for $L=3\mu \mathrm{m}$ which had $R_c\sim 30\mathrm{k}\Omega$. This contact resistance is much lower than the resistance along the tube under high bias and most ($>95$ percent) of the power dissipation occurs along the tube length rather than at the contacts. (b) Measured peak current (symbols) for suspended SWNTs of various lengths. The peak current scales approximately as $\sim 1/L$ just like the thermal conductance of the suspended nanotubes, an additional indicator that this is a thermally limited effect. The deviation from the $1/L$ behavior is attributed to variations in diameter between the different tubes.

Figure 4

Electrical characteristics of a $2\mu \mathrm{m}$ suspended tube in vacuum (experimental data with symbols) and calculations (lines) based on three different thermal conductivity models with ${\kappa }_0=3600\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ at $T_0=300\mathrm{K}$. The high bias region of the $I\mathrm{\text{−}}V$ curve provides an indirect measurement of the temperature dependence of the thermal conductivity for $400\mathrm{K}<T<700\mathrm{K}$. Here, as in the rest of our Letter, $T$ refers to the lattice (ac phonon) temperature.