Wave-mechanical transmission theory for approximately triangular barriers

The theory of the wave-mechanical transmission of particles across barriers has both fundamental and applied relevance. There are many devices (for example, electron microscope sources) where field-induced electron tunnelling – often called "Fowler-Nordheim (FN) tunnelling" –is involved, and many machines (for example, ion beam accelerators) where vacuum breakdown and arcing need to be prevented.

Simple approximate theories of tunnelling have existed since the 1920s. It is difficult to formulate exact theories, due to mathematical problems in the theory of differential equations that have been under study since the early 1800s. The problem is made worse by lack of transparency/completeness in literature discussions of FN tunnelling.

This project aims to clarify and improve existing discussions of transmission theory, in order to facilitate more accurate numerical treatments. Associated tasks have been/are:

1. Reformulation of exact theory of transmission across the exact triangular (ET) barrier [1,2], and statement of the known difficulties in the FN 1928 treatment [3];
2. In the context of the usual approximate theory of transmission across the widely-used Schottky-Nordheim (SN) barrier, re-development of the theory of the Principal SN Barrier Function v [4], development of exact series expansion for v [5], demonstration [6] that a new approximate expression found for v [7] is superior to all (12 or so) approximations of equivalent complexity previously suggested, and archiving statement [8] of the post-1928 attempts to calculate v accurately.
3. Development of alternative "more general" formula for transmission coefficient [9]; exploration of the Kemble approximation as applied to the ET and SN barriers (to be done); exploration of reliability of existing transmission formulae for Schottky emitters, and comparisons with Mayer's approach [10] (to be done).