Geometry dependence of the electrostatic and thermal response of a carbon nanotube during field emission

In this paper we present an analysis to simulate heating within an isolated carbon nanotube (CNT) attached to an etched tungsten tip during field emission of an electron beam. The length, radius, wall thickness and shape of the tip (closed with a hemispherical shape or open and flat) of the CNT and its separation distance from the flat surface are considered as variables. Using a finite element method, we predict the field enhancement, emission current and temperature of the CNT as a function of these parameters. The electrostatic and transient thermal analyses are integrated with the field-emission models based on the Fowler-Nordheim approximation and heating/cooling due to emitting energetic electrons (the Nottingham effect). These simulations suggest that the main mechanism responsible for heating of the CNT is Joule heating, which is significantly larger than the Nottingham effect. Results also indicate that the electrostatic characteristics of CNTs are very sensitive to the considered parameters whereas the transient thermal response is only a function of the CNT radius and wall thickness. Further, the thermal response of the CNT is independent of its geometry, meaning that, as long as a given set of geometrical conditions are present that result in a given emission current, the maximum temperature a CNT attains will be the same.