Revising the Nottingham Inversion Instability as a discontinuous transition between two distinct steady states of thermo-field emission from micro-protrusions

Abstract Since the publication of our article: Unveiling the Nottingham Inversion Instability during the thermo-field emission from refractory metal micro-protrusions [1], we have performed new simulations that have led us to a renewed interpretation of this phenomenon. In this follow-up article, we show that the initial instability is solely due to the postitive feedback loop between temperature and resistive heating that can diverge above a threshold electric field. The Nottingham inversion is not the trigger but brings the negative feedback loop with temperature that can balance the thermal runaway, eventually yielding the observed jump in current and temperature. From this new perspective, we therefore suggest replacing the original term "Instability" by the Nottingham Inversion "Jump". This jump represents an intermediate possibility between the two usual scenarios envisaged to date – a stable transition with increasing voltage from low to high temperature steady states up to the melting temperature versus the occurrence beforehand of a resistive instability. To formally compare this third path with the other two, a stability analysis of the micro-protrusion self heating during electron emission is proposed. For the system to exhibit a jump, it must reach a point where the retroaction of a temperature rise on the Joule heating exceeds the combined negative feedback on the Nottingham effect and the heat evacuation, before this situation reverses at higher temperature. Formal revisions of our original article are given accordingly.