On the need for a tunneling pre-factor in Fowler-Nordheim tunneling theory

This paper argues that a tunneling prefactor should appear in expressions for the tunneling probability D relevant to cold field electron emission (CFE) and in Fowler-Nordheim (FN) type equations. Except in the case of the "ideally smooth" parabolic barrier, a prefactor is always present for barriers where D can be found by exact solution of the Schrödinger equation. A review of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approach to solving the Schrödinger equation shows that tunneling barriers should be classified according to whether they are weak or strong and ideally smooth or not: there are four different JWKB-type formulas, depending on the nature of the barrier. CFE tunneling barriers are not ideally smooth but since the 1950s have usually been analyzed using the JWKB formulas for ideally smooth barriers. These analyses, and the standard Murphy-Good FN-type equation, seem mathematically and physically incomplete. The FN-type equations currently used to describe CFE should be revised to explicitly include a tunneling prefactor. Some implications are explored.