A unified thermal-field emission theory for metallic nanotips

The role of curvature effects, in the thermal-field (TF) emission of electrons from nanotips, has been investigated. It is found that for field emitter tips having tip radius of curvature Ra<50 nm, the error on using the planar tunneling potential to evaluate the current density is significant compared to the curvature-corrected potential. Furthermore, the error is found to be strongly temperature dependent at low apex fields, whereas at high fields, it is only moderate. For emitter tips having tip radius Ra≥5 nm, a unified expression for current density applicable for field, thermal-field, and thermionic emission is obtained. In the analysis, a single linearization point corresponding to the peak of the normal energy distribution of electrons is chosen, about which the Gamow exponent is expanded. The temperature-dependent curvature-corrected current density equation is numerically validated against a benchmark that performs the current density integral with the Kemble form of the Wentzel–Kramers–Brillouin transmission coefficient up to the peak of the tunneling potential, while above the barrier, the Morse transmission coefficient is used. The agreement is found to be good. It is inferred that the curvature plays a significant role in field and thermal-field emission, whereas in thermionic emission, it is not as important. In the limit where the apex radius of curvature Ra→∞, it is able to retrieve the Murphy–Good equation as well as the Richardson–Laue–Dushman equation in their respective limits.