Intrinsic energy spectrum in field emission of carbon nanotubes

The intrinsic electron energy spectra in field emission from the single-wall carbon nanotubes are systematically investigated based on the quantum tunneling theory with the tight-binding approximation. We examine the dependence of the characteristics of the field-emission energy spectrum on the electric field and temperature. For the metallic tubes, as temperature or electric field increases, the main peak of the energy spectrum becomes high and wide, and some sub-peaks occur above or below Fermi energy. For the semiconducting tubes, the energy gap at Fermi energy may occur leading to a double peak spectrum. Electric field competes with temperature to balance the height of the two peaks. Strong field makes the low-energy peak high, while high temperature enhances the high-energy peak. We give a phase diagram on the peak number of the energy spectrum in the space of the electric field and temperature, which summarizes some key characteristics of the energy spectrum.

Figure 1

(a)/(b) The energy spectra of the semiconducting/metallic tubes $(10,0)∕(5,5)$ in the electric field $F=6$, 7, and $8\mathrm{V}∕\mathrm{nm}$ at room temperature. (c)/(d) The energy spectra of the semiconducting/metallic tubes $(10,0)∕(5,5)$ in temperature $T=300$, 1000, and $1500\mathrm{K}$ at the electric field $F=6\mathrm{V}∕\mathrm{nm}$.

Figure 2

(a)/(b) The energy spectra of the semiconducting/metallic tubes $(10,0)∕(5,5)$ in the electric field $F=6$, 7, and $8\mathrm{V}∕\mathrm{nm}$ at temperature $T=1500\mathrm{K}$. (c)/(d) The energy spectra of the semiconducting/metallic tubes $(10,0)∕(5,5)$ in temperature $T=1000$, 1500, and $2000\mathrm{K}$ at the electric field $F=5\mathrm{V}∕\mathrm{nm}$.

Figure 3

(Color online) (a) The full width at half maximum (FWHM) vs temperature at the electric field $F=6\mathrm{V}∕\mathrm{nm}$. (b) The FWHM vs the electric field at room temperature. (c)/(d) The main peak position of the energy spectrum vs temperature/electric field, where $p^{+}$ and $p^{-}$ represent the main peak above and below Fermi energy, respectively.

Figure 4

(a) The net energy exchange vs temperature $⟨E⟩-E_F$. (b) The inverse temperature vs the electric field.

Figure 5

(Color online) The phase diagram of the peak number of the energy spectrum in the space of temperature and the electric field for the semiconducting tubes (10,0) in (a) and metallic tube (5,5) in (b).