Electrostatics of Individual Single-Walled Carbon Nanotubes Investigated by Electrostatic Force Microscopy

We report an experimental study of static charge distribution in individual single-walled carbon nanotubes grown on a $\mathrm{Si}+115\mathrm{nm}$ ${\mathrm{SiO}}_2$ substrate. From these experiments, we conclude that charges are distributed uniformly along the nanotubes. We demonstrate that electrostatic force microscopy can accurately measure the amount of charges per unit length. We found that this amount is diameter dependent and in the range of 1 electron per nanometer for a 2.5 nm nanotube at a potential of $-3.5\mathrm{V}$.

Figure 1

(a) AFM topography image of an individual SWNT. (b) EFM image recorded at $V_{\mathrm{EFM}}=-4\mathrm{V}$ before charging. (c),(d) EFM images recorded at $V_{\mathrm{EFM}}=-4$ and $+4\mathrm{V}$, respectively, after charging at 5 V for 30 s. Gray scales are (a) 7 nm, (b) 2.5 Hz, (c) 15 Hz, and (d) 10 Hz.

Figure 2

Relative frequency shift measured as a function of the charging potential for attractive and repulsive (inset) tip potentials ($V_{\mathrm{EFM}}=\pm 4\mathrm{V}$). Since the data obtained with negative and positive charging potentials are varying within the error bars, only average points are presented here as a function of the charging potential absolute value for enhanced clarity. The dashed lines are fits using expression (6), with $\alpha =-0.32{\mathrm{V}}^{-2}$, and $\beta =-3.65{\mathrm{V}}^{-1}$ and $+3.65{\mathrm{V}}^{-1}$ for repulsive and attractive $V_{\mathrm{EFM}}$, respectively.

Figure 3

EFM frequency shift images of a $40\mu \mathrm{m}$ long nanotube network charged at its extremity on the bottom left corners of the images with $V_{\mathrm{ch}}=-5\mathrm{V}$ for $t_{\mathrm{ch}}=30\mathrm{s}$, recorded (a) at $V_{\mathrm{EFM}}=+2\mathrm{V}$ and (b) at $V_{\mathrm{EFM}}=-2\mathrm{V}$ (20 and 8 Hz gray scales, respectively). Tip characteristics: ${\omega }_0=136.3\mathrm{kH}z$ and $k=3.5\mathrm{N}·{\mathrm{m}}^{-1}$.

Figure 4

(a) Relative frequency shifts as a function of the SWNT diameters within the network presented in Fig. 3. From the experimental data we deduced an apparent charge amount of $\approx 230$ electrons for a 2.5 nm SWNT, i.e., an equivalent charging potential of $-3.4\mathrm{V}$. (b) Extracted linear density of charge ($=Q/L_{\mathrm{int}}$) from Figs. 3a, 3b as a function of the nanotube diameter. Solid line: calculated values from the expression $Q/L=CV=4.7/\ln (460/d)\mathrm{\text{electron}}·{\mathrm{nm}}^{-1}$, with $C=C_{\mathrm{calc}}/L$ and a constant potential $V=-3.4\mathrm{V}$.