Higher order curvature corrections to the field emission current density

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 $\phi \in [3-5.5]$eV, Fermi energy ${\cal{E}}_F$ in [5-10]eV, local electric fields $E_l$ in[3-9]eV and radius of curvature $R \geq 5$nm). An easy-to-use correction factor $\lambda_P$ is also provided to approximately map the `exact-WKB' current density to the `exact' current density in terms of ${\cal{E}}_F/\phi$. The average error on using $\lambda_P$ is found to be around $3.5\%$. This is a vast improvement over the average error of $15\%$ when $\lambda_P = 1$. 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 radius of curvature.