Analytical Approximations for the Special Elliptic Functions of Standard Fowler-Nordheim Theory

摘要

The Standard Fowler-Nordheim (Curve) Equation, as derived from the work of Murphy and Good, may be written in the form J = t{sub}F{sup}(-2)aΦ{sup}(-1) exp{-v{sub}FbΦ{sup}(3/2)/F}, where F is the barrier field, Φ is the local work-function, a and b are the First and Second Fowler-Nordheim Constants, and v{sub}F and t{sub}F are the values, taken for a "forwards-moving state at the Fermi level", of the special elliptic functions v and t. These functions have purely mathematical definitions, and depend only on a single mathematical variable, usually denoted by y. In their application to field electron emission theory, y is termed the Nordheim parameter, and is given by y = (e{sup}3/4πε{sub}0){sup}(1/2)F{sup}(1/2)/Φ,where e is the elementary positive charge and ε{sub}0 is the electric constant.