Generalizing Fowler–Nordheim Tunneling Theory for an Arbitrary Power Law Barrier

Herein, the canonical Fowler–Nordheim theory is extended by computing the zero‐temperature transmission probability for the more general case of a barrier described by a fractional power law. An exact analytical formula is derived, written in terms of Gauss hypergeometric functions, that fully capture the transmission probability for this generalized problem, including screened interaction with the image potential. First, the quality of approximation against the so far most advanced formulation of Fowler–Nordheim, where the transmission is given in terms of elliptic integrals, is benchmarked. In the following, as the barrier is given by a power law, in detail, the dependence of the transmission probability on the exponent of the power law is analyzed. The formalism is compared with results of numerical calculations and its possible experimental relevance is discussed. Finally, it is discussed how the presented solution can be linked in some specific cases with an exact quantum‐mechanical solution of the quantum well problem. The article provides an analytical formula for the tunneling process through a barrier described by a power law with arbitrary exponent, a generalization of triangular barrier, relevant for nanostructured emitting surfaces. Results are then compared with numerics, revealing when multiple reflections need to be included, and with full quantum‐mechanical solution for the special case of a parabolic potential.