Angular variation of the transmission coefficient of electrons from open carbon nanotubes by a 3-dimensional WKB method

We have calculated by a 3-dimensional (3D) WKB method the transmission coefficient T(θ) of open carbon nanotubes (CNTs) as a function of the angle θ and the geometry of the tube, namely the height h, the radius R and the thickness w of a CNT. For the potential barriers we have used the results of our previous work [G.C. Kokkorakis, J.A. Roumeliotis, J.P. Xanthakis, J. Appl. Phys. 95 (2004) 1468]. For the calculation of T(θ) we have used the method of Das and Mahanty [Phys. Rev. B 36 (1987) 898], which is a 3D generalization of the traditional one-dimensional (1D) WKB method. Due to the open shape of the tube and its nanometric scale we have found considerable deviations of T(θ) from the 1D WKB theory. In particular, the paths of the emitted electrons are highly curved at certain directions θ. In addition T(θ) has contributions predominately from the outer wall of a CNT due to the asymmetry of the barrier potential. These two factors will definitely affect the direction of the emitted current. Such a phenomenon does not occur with closed CNTs. Furthermore, we have established that an increase in the magnitude of the T(θ) (by the use e.g., of a higher tube or higher applied voltage) will result in a T(θ) which will be more asymmetric with respect to angle.