Higher order curvature corrections to the field emission current density
Debabrata Biswas, Rajasree Ramachandran
Abstract
A simple expression for the Gamow factor is obtained using a second-order curvature-corrected tunneling potential. Our results show that it approximates accurately the “exact-WKB” transmission coefficient obtained by numerically integrating over the tunneling region to obtain the Gamow factor. The average difference in current density using the respective transmission coefficients is about 1.5%, across a range of work functions ϕ∈[3−5.5] eV, Fermi energy EF∈[5−10] eV, local electric fields El∈[3−9] V/nm, and radius of curvature R≥5 nm. An easy-to-use correction factor λP is also provided to approximately map the “exact-WKB” current density to the “exact” current density in terms of EF/ϕ. The average error on using λP is found to be around 3.5%. This is a vast improvement over the average error of 15% when λP=1. Finally, an analytical expression for the curvature-corrected current density is obtained using the Gamow factor. It is found to compare well with the “exact-WKB” current density even at small values of local electric field and the radius of curvature.