Revising the Nottingham Inversion Instability as a bifurcation between two branches of steady states solutions of thermo-field emission from micro-protrusions.

Unstable and destructive behaviors in the thermo-field electron emission by micro- and nano-metric structures typically lead to vacuum breakdowns, hindering the experimental exploration of the phenomenon. To address this challenge, numerical models are employed. In our previous publication, a detailed investigation of the emitter self-heating revealed the possibility of a discontinuity in the increase of the emission current and temperature with the applied electric. This phenomenon is caused by the competition between the usual resistive heating and the Nottingham effect (a more complex energy exchange process between emitted and replacement electrons). Recent simulations revealed a clearer interpretation, as described in this follow-up article. The initial instability causing the discontinuity is solely due to the positive feedback loop between temperature and resistive heating, which can diverge above an electric field threshold. The Nottingham inversion is not the trigger, contrary to our initial claim. A negative feedback loop can sometimes balance the thermal runaway if the transition from heating to cooling due to the Nottingham effect starts. The time evolution of these parameters causes a bifurcation in the space of steady-state solutions, generating two distinct branches. This bifurcation represents an intermediate possibility between the two usual scenarios envisaged to date - a stable transition with increasing electric field (applied voltage) from low to high-temperature steady states up to the melting temperature versus the occurrence of a resistive instability beforehand. A stability analysis of the microprotrusion self-heating during electron emission is proposed to formally compare this third path with the other two. Besides, this bifurcation yields a hysteresis loop.