Spatial dependence of the temperature profile along a carbon nanotube during thermal-field emission

Abstract

An efficient algorithm is described to calculate the spatial dependence of the temperature distribution along a carbon nanotube (CNT) during field emission (FE). The algorithm considers the effects of Joule heating in the CNT and radiative losses from the CNT sidewall and tip. The CNT emission current density and the rate of heat exchange per unit area at the CNT tip due to either Henderson-cooling or Nottingham-heating effects are calculated using recent analytical expressions derived by Jensen [J. Appl. Phys. 126, 065302 (2019)]. The latter are valid in the thermionic and field emission regimes and in the transition region between these two extremes. The temperature dependence of the electrical resistivity ρ(T) and the thermal conductivity κ(T) of the CNT is also included in the model. It is shown that replacing ρ(T) and κ(T) by their spatial averages over the length of the CNT can lead to an overestimate of the value of the external electric field threshold at which thermal runaway of the CNT occurs. These results should be considered when calculating the field emission characteristics of CNT arrays such as from a carbon nanotube fiber whose FE properties are primarily determined by the FE properties of the array of CNTs at the tip of the fiber. Using the new algorithm, the simulation times to calculate the CNT FE characteristics and the spatial temperature distribution are found to be nearly two orders of magnitude faster compared to those required when both the current and energy exchange at the CNT tip are calculated numerically.