Exact analytical theory for inverse tunneling of free vacuum electrons into a solid

This paper presents an exact analytical theory for the inverse tunneling of free vacuum electrons through a triangular potential barrier into a metal. It is found that the inverse tunneling probability is the same as that of the field emission of electrons from metal into vacuum, for the same incident electron energy, metal properties (work function, Fermi energy), and applied electric field strength. For incident electrons with velocities not normal to the vacuum-metal interface, the three-dimensional (3D) oblique tunneling is equivalent to the one-dimensional (1D) normal tunneling, by considering only the longitudinal energy of the electrons normal to the vacuum-metal interface.