Multiplicity features of carbon nanotube emitters

An electrothermal model is described for the calculation of the multiple steady states of a carbon nanotube during field emission with and without the Henderson–Nottingham effect (HNE). The model is based on a one-dimensional balance between Joule heating and heat dissipation by conduction from the base, radiation from the surface, and heating/cooling from the apex. The emission current density is calculated from the reformulated general thermal-field equations. Both cases have their own distinct multiplicity features. Without the HNE, the problem admits two stable solutions, a cold and a hot branch, respectively, separated by a singular point. The theoretical predictions are in agreement with the experimental data. When the HNE is included, the problem admits up to three solutions, two stable and one unstable. The stable solutions are separated into a cold and a hot branch, where the latter is formed as soon as the emitter temperature exceeds the inversion point, and a stable thermal equilibrium between heat generation and heat dissipation is established at a higher temperature as the cooling from the apex becomes significant. As a result, the two branches induce a hysteresis behavior that is in qualitative agreement with the emission experiments on nanocarbon materials when projected on a Fowler–Nordheim plot, suggesting the possibility of an alternative explanation to the two-barrier model.