Quantum-Mechanical Investigation of Field-Emission Mechanism of a Micrometer-Long Single-Walled Carbon Nanotube

A quantum-mechanical simulation is carried out to investigate the charge distribution and electrostatic potential along a $1\mu \mathrm{m}$ long (5,5) single-walled carbon nanotube under realistic field-emission experimental conditions. A single layer of carbon atoms is found sufficient to shield most of the electric field except at the tip where strong field penetration occurs. The penetration leads to a nonlinear decrease of potential barrier for emission, which is equally responsible for the low threshold voltage besides the well-known geometrical field enhancement factor.

Figure 1

(a) Field emission setup for a SWNT. (b) Mirror images of the induced charges.

Figure 2

(a) Induced charge distribution at the top layer of the SWNT’s cap. (b) Induced charge distribution in a middle plane bisecting the tube. (c) Number of induced electrons on each layer along the $z$ axis in the tip region. Excessive charges on the entire tube are depicted in the inset. All charge densities in (a) and (b) are in units of $\mathrm{\text{electron}}/{\AA }^3$.

Figure 3

Potential energy contour plots (a) for the (5,5) SWNT and (b) in the vicinity of the cap under $E_{\mathrm{appl}}=14\mathrm{V}\mu {\mathrm{m}}^{-1}$. To keep the image clear, the core potential is cut at $-10\mathrm{eV}$. The dotted line in (b) is the equipotential line for the Fermi energy ($-4.5\mathrm{eV}$).

Figure 4

(a) Potential energy profiles in the vicinity of the cap under different applied fields; data are extracted along the central axis of the carbon nanotube. The electrostatic potential along the entire tube for $E_{\mathrm{appl}}=10\mathrm{V}\mu {\mathrm{m}}^{-1}$ is presented in the inset. (b) Electrostatic potential along the $A\mathrm{\text{−}}B\mathrm{\text{−}}C\mathrm{\text{−}}D$ path which is embedded in the tube wall as specified by the inset ($E_{\mathrm{appl}}=14\mathrm{V}\mu {\mathrm{m}}^{-1}$).