Electrostatics of straight and bent single-walled carbon nanotubes

Response of a single-walled carbon nanotube to external electric field, $F$, is calculated analytically within the classical electrostatics. Field-induced charge density distribution is approximately linear along the axis of a metallic nanotube and depends rather weakly, as $\ln (h∕r)$, on the nanotube length, $h$ (here $r$ is the nanotube radius). In a semiconducting nanotube with a gap, $E_g$, charge separation occurs as $F$ exceeds the threshold value $F_{th}=E_g∕eh$. For $F>F_{th}$, positively and negatively charged regions at the ends of nanotube are separated by a neutral strip in the middle. Properties of this neutral strip, length and induced charge distribution near the ends, are studied in detail. We also consider a bent nanotube and demonstrate that the number of neutral strips can be one or two depending on the direction of $F$.

Figure 1

Plot of the function $K(z,z^{\prime })$ [scaled with the factor $2e\ln (h∕4r)∕{\epsilon }^{*}]$ for $h∕r={10}^3$ and different values of $z$: $z=0.1h$ (dotted line), $0.2h$ (solid line), $0.3h$ (dashed line), and $0.4h$ (dash-dotted line).

Figure 2

Charge density distribution induced by external field $F$ along a metallic (dashed line) and semiconducting (solid line) NT.

Figure 3

Schematic illustration of charge separation in semiconducting NT in external field: (a) NT of a wiggly shape; (b) and (c) NT of a semicircle shape with $F$ parallel and perpendicular to the diameter, respectively.