Effect of temperature on field emission from a micrometer-long single-walled carbon nanotube

Charge distribution along a micrometer-long carbon nanotube subject to an external applied electric field at different temperatures is determined via a quantum mechanical method. Corresponding emission current is evaluated via the Wentzel-Kramere-Brillouin approximation. For a capped micrometer-long (5,5) nanotube, an external field of $10\mathrm{V}∕\mathrm{\mu }\mathrm{m}$ or above is required to produce the measurable current. Because of the overwhelming effects of the accumulation charges of the tip, the temperature alone has small effects on the field emission that can be observed in the experiments, and the strong self-heating effects are thus attributed mainly to the induced structural changes of carbon nanotubes.

Figure 1

The model of field emission from a SWNT. The gray part is the mirror of the nanotube, the potential of the cathode plane is set as zero. Negative induced charge accumulates on the surface of the nanotube. The zero of the coordinates of $z$ axis is set at the cathode plane and the zero of the $x$ axis is the central axis of the nanotube.

Figure 2

The model of the classical field emission. The horizontal axis is the distance from the emitter surface. The region where $x$ is less than 0 means the inner part of the metal emitter and that where $x$ is larger than 0 means the outer part. The vertical axis is the potential barrier. The zero potential is set as the vacuum level. $\Phi$ is the work function of the emitters. $V\left(x\right)$ is the potential barrier near the emitter. $E_F$ is the Fermi level. $E_B$ is the binding energy of the emitter. $E$ is the total energy of the electrons. $V_{max}$ is the maximum point of the barrier potential curve and $x_1$, $x_2$ are the coordinates of the point $V\left(x\right)$ equals $E$.

Figure 3

(Color online) Induced charge density and electrostatic potential along the surface for the (5,5) CNT. (a) $E_{appl}=10\mathrm{V}∕\mathrm{\mu }\mathrm{m}$; (b) $E_{appl}=7.5\mathrm{V}∕\mathrm{\mu }\mathrm{m}$; (c) $E_{appl}=5\mathrm{V}∕\mathrm{\mu }\mathrm{m}$; (d) $E_{appl}=2.5\mathrm{V}∕\mathrm{\mu }\mathrm{m}$. The horizontal axis is the coordinate from the cathode plane. The vertical axis is the potential barrier. Data are extracted along the central axis of the carbon nanotube. The potential curve is $298\mathrm{K}$ (dash-dot line), $750\mathrm{K}$ (dotted line), $1250\mathrm{K}$ (dashed line) and $1500\mathrm{K}$ (solid line). A is the valley point and B is the peak point of the potential barrier. The insets are the induced charge density along the axis. The horizontal axis is the coordinate from the cathode plane. The vertical axis is the induced charge density $(\mathrm{e}∕{\mathrm{\AA }}^3)$. All charge densities in (a) and (b) are in units of $\mathrm{electron}∕{\mathrm{\AA }}^3$.

Figure 4

(Color online) Potential barrier above Fermi level under different applied fields $(\mathrm{V}∕\mathrm{\mu }\mathrm{m})$. The horizontal axis is the different temperature. The vertical axis is the relative potential barrier (eV) of the surface. The value equals the minus of the potential barrier of the peak point (B) and Fermi level $E_{\mathrm{F}}$ $(-4.5\mathrm{eV})$.

Figure 5

(Color online) Potential energy contour plots for the (5,5) CNT. (a) $E_{appl}=10\mathrm{V}∕\mathrm{\mu }\mathrm{m}$, $T=298$ and $1500\mathrm{K}$; (b) $E_{appl}=7.5\mathrm{V}∕\mathrm{\mu }\mathrm{m}$, $T=298$ and $1500\mathrm{K}$. (c) $E_{appl}=5\mathrm{V}∕\mathrm{\mu }\mathrm{m}$, $T=298$ and $1500\mathrm{K}$; (d) $E_{appl}=2.5\mathrm{V}∕\mathrm{\mu }\mathrm{m}$, $T=298$ and $1500\mathrm{K}$. The zero of the horizontal axis is the central axis. The vertical axis is the coordinate from the cathode plane.