A closed-form model for thermionic trap-assisted tunneling

Recently, we proposed a trap-assisted tunneling model (2006) that includes tunneling of thermally activated electrons above the metal Fermi level for explaining the temperature-dependent leakage current in some semiconductor devices. In the present paper, we develop a closed-form version of this model, which provides physical insight by revealing the peak, energy location and spread of emitted electron distribution. The model also yields characteristic field parameters to identify the thermally activated regime of current versus field behavior and the location of peak emission. The closed-form solution of a complicated equation has been achieved using a geometrical interpretation of the integration operation, and by bisecting the range of trap energies, adopting separate approximations for the bisected segments, and then mathematically combining the two segments into a single continuous function valid for the entire range of trap energies. The closed-form model calculations match well with numerical integration results.