The field emission current formula for metallic nanotips at temperatures above absolute zero

With proper consideration of curvature effects, for a metallic field emitter tip having apex radius Ra≥5 nm operating in low thermal-field emission regime (corresponds to field electron emission at low–moderate temperatures), an analytical expression to precisely predict the emitted current is obtained. The methodology lies within the framework of the free electron model, where the surface integration of low thermal-field current density has been carried out using the cosine law of electrostatic field variation provided the field emitter tip has a parabolic end-cap. It renders an easy-to-use current formula in the form of an algebraic equation that can predict the current emitted from a metallic nanotip with mere knowledge of its apex radius (Ra), apex electrostatic field (Ea), and temperature (T). The current formula is found to perform well for nanotips having apex radius Ra≥5 nm at apex electrostatic fields Ea∈[3,8]V/nm and temperatures T<2000 K for tungsten field emitter tips. Additionally, studies on the variation of the notional emission area of the field emitter tip in low thermal-field emission regime show that it is, in fact, strongly dependent on the apex electrostatic field, but rather shows a weak temperature dependence, particularly at low apex fields.