Electric charge enhancements in carbon nanotubes: Theory and experiments

We present a detailed study of the static enhancement effects of electric charges in micrometer-long single-walled carbon nanotubes, using theoretically an atomic charge-dipole model and experimentally electrostatic force microscopy. We demonstrate that nanotubes exhibit at their ends surprisingly weak charge enhancements which decrease with the nanotube length and increase with the nanotube radius. A quantitative agreement is obtained between theory and experiments.

Figure 1

(Color online) Charge distribution at the ends of an open-ended and a closed-ended (9, 0) SWCNTs ($L\approx 11.5\text{nm}$, ${\sigma }^{\text{ave}}=-6.6\times {10}^{-4}e/\text{atom}$) in free space [(a) and (b)] and on the ${\text{SiO}}_2$ substrate [(c) and (d)]. The color of the atoms is proportional to the local charge density. The green vectors stand for the induced atomic dipoles. The dark arrows stand for the local electric fields induced by the net charge, their length and color are proportional to the field intensity. (a) The minimum and maximum atomic charge densities in this tube are: ${\sigma }^{\text{min}}=-5.3\times {10}^{-4}e/\text{atom}$ and ${\sigma }^{\text{max}}=-16\times {10}^{-4}e/\text{atom}$, respectively. (b) ${\sigma }^{\text{min}}=-5.2\times {10}^{-4}e/\text{atom}$ and ${\sigma }^{\text{max}}=-34\times {10}^{-4}e/\text{atom}$. (c) ${\sigma }^{\text{min}}=+5.6\times {10}^{-4}e/\text{atom}$ and ${\sigma }^{\text{max}}=-34\times {10}^{-4}e/\text{atom}$. (d) ${\sigma }^{\text{min}}=+5.8\times {10}^{-4}e/\text{atom}$ and ${\sigma }^{\text{max}}=-74\times {10}^{-4}e/\text{atom}$.

Figure 2

Charge profile along three (9, 0) SWCNTs with different tube lengths $L$, in space (hollow symbols) and upon a ${\text{SiO}}_2$ surface (solid symbols), using a separation distance $d=0.34\text{nm}$ (see text). The total net charge density on each tube ${\sigma }^{\text{ave}}$ is fixed to $6.4\times {10}^{-4}e/\text{atom}$ (equivalently, $0.055e/\text{nm}$). Each point is calculated as the average value of the charge carried by the nanotube atoms over $10\%L$.

Figure 3

Ratio of charge enhancement $\phi$ as a function of tube length $L$ (in common logarithmic scale). This ratio is calculated for both an open-ended (circles) and a closed-ended (triangles) (5,5) SWCNTs (radius $R\approx 0.34\text{nm}$) placed upon a ${\text{SiO}}_2$ surface (solid symbols) with $d=0.34\text{nm}$, and is compared with that for the same tubes in space (empty symbols). The symbols present the calculated points, and the lines stand for the extrapolation curves.

Figure 4

(Color online) (a) Schematics of the charge injection and detection with the EFM tip. Charging takes place when the biased tip is put in contact with the CNT. During the data-acquisition cantilever is lifted at distance $z$ above the surface. (b) Topography image of a SWCNT with 0.5 nm radius deposited on 200 nm oxide layer. (c) EFM image acquired before injection. The dark feature corresponds to the uncharged tube. (d) EFM scan after injection experiment. The bright feature corresponds to the charged tube with the uniform linear charge density of $0.055e/\text{nm}$.

Figure 5

(Color online) (a) AFM topography image of single nanotube with $R=0.8\text{nm}$ and $L\approx 2\mu \text{m}$ deposited on silicon dioxide. (b) EFM scan of the same tube made after charge injection. A nonlinear color scale has been used in order to clearly show the weak enhancement at the tube end. The black line is a “guide to the eye” for the physical end of the tube. (c) Experimental charge enhancement along the axis of the nanotube, defined as the ratio of the EFM signal with that measured at the middle of the nanotube.

Figure 6

Ratio of charge enhancement $\phi$ for a number of $\left(n,n\right)$ nanotubes with different radius $R$, on a ${\text{SiO}}_2$ surface. The solid squares stand for $\phi$ derived from the experimental measurements of seven nanotubes (with length $L=1\sim 9\mu \text{m}$) deposited on 200 nm silicon oxide layer. The symbols “$+$” stand for the extrapolation results for micrometer length tubes.