Uncovering a widely applicable empirical formula for field emission characteristics of metallic nanotips in nanogaps

Field electron emission is a key mechanism in nanoelectronics with nanogaps, offering advantages such as high electron velocity and fast switching speeds. However, nanoscale field emission, affected by geometric asymmetry including quantum tunneling near to the emitter, and quantum space charge effects, remains largely unexplored in experimental studies. Here, we in situ investigated field emission characteristics of pure tungsten nanotips across vacuum nanogaps. We revealed a widely applicable scaling behavior between field emission characteristics and the ratio of apex radius to gap length ( R/d ), and demonstrated that the effects of quantum tunnelling due to emitter shape are the predominant influence. We further proposed a modified field emission equation, incorporating an empirical formula for the apex shape factor, k MG ( k MG  =  f ( R / d ) = 1.680 × ( R / d  + 0.468) −1.066 ), valid for R / d  = 0.04 to 48. These findings provide fundamental insights into the optimization of nanoelectronic device design and the advancement of future technologies.