An expression for the potential energy and field emission current density function of spherical conducting carbon nanotubes (CNTs), including the effect of image force, corresponding to the expression for the electron transmission coefficient obtained from the solution of the time-independent Schrödinger equation, has been derived. Numerical calculations of the potential energy, transmission coefficient, and the current density function have been carried out for the typical set of CNT parameters. It is found that the potential energy decreases with radial distance when image force is ignored. When image force is incorporated, it increases with radial distance in the beginning and then decreases. The transmission probability and the current density function increase, both with the normalized radial energy and normalized Fermi energy, but both the transmission probability and the current density function are larger in the presence of the image force for given values of the normalized radial energy and the normalized Fermi energy. In addition, the image force decreases the applied value of the field. The field emission current density function of emitted electrons decreases with the spherical CNT tip radius. Some of our theoretical results are in accordance with existing experimental observations.
Theoretical optimization for field emission current density from carbon nanotubes array
