In order to investigate the usability of carbon nanotubes as electron-beam sources, we carry out dynamic simulations of the field emission of open-tip armchair single-wall carbon nanotubes in a fully first-principles manner. By including a metallic substrate in the structure, the charge transfer from the substrate to the carbon nanotubes under external electric fields is self-consistently taken into account and the description of the potential becomes more realistic. It is found that $s$-like $\ensuremath{\pi}$ states near the Fermi level make a major contribution to the emission current. The emission patterns of the $s$-like $\ensuremath{\pi}$ states take the shape of a small spot at the center and a ring around it.
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