Thin-slab model for field electron emission

Abstract

The elementary Fowler–Nordheim-type (FN-type) equation is often used to describe cold field electron emission (CFE) from carbon nanotubes (CNTs) including closed single-walled CNTs of small apex radius. FN-type equations were originally derived to describe CFE from the conduction band of a bulk metal with a flat surface and use a “flat thick-slab” potential-energy model. This article identifies many theoretical difficulties with using FN-type equations to describe CFE from CNTs. The most serious arise because a closed CNT apex is one-atom thick and can be sharply curved. This article explores a flat thin-slab model for CFE and derives an emission current equation. The model is parametrized using graphene data. The main conclusions are that (a) the occupied parts of electronic subbands probably do not overlap so CFE would come from a single subband; (b) an electron at the Fermi level has large momentum and kinetic energy parallel to the slab surface; (c) a clear conceptual distinction must be made between local thermodynamic work function (ϕ) and the zero-field height of the tunneling barrier for a Fermi-level electron; (d) for the thin-slab model, this zero-field barrier height is equal (at 0 K) to ϕ+ξf, where ξf is the occupied width of the subband at 0 K; and (e) the emission onset field, for a current density of 1000 A/m2, is estimated as around 9 V/nm. The relevance of these conclusions to emission from closed single-walled CNTs is considered. The predicted onset field for a thin-slab model is significantly higher than that predicted by the elementary FN-type equation (around 3.5 V/nm), so attempts should be made to measure onset field accurately for CFE from closed CNTs. In practice, with closed CNTs, it might be more favorable for emission to occur from quasiatomic or quasimolecular localized states rather than a subband; in this case, the question of how these states are supplied arises. More generally, this work draws attention to the need, with thin emitters, to consider issues relating to how the direction of electron motion prior to emission affects the mechanism of emission and the emission probability.The elementary Fowler–Nordheim-type (FN-type) equation is often used to describe cold field electron emission (CFE) from carbon nanotubes (CNTs) including closed single-walled CNTs of small apex radius. FN-type equations were originally derived to describe CFE from the conduction band of a bulk metal with a flat surface and use a “flat thick-slab” potential-energy model. This article identifies many theoretical difficulties with using FN-type equations to describe CFE from CNTs. The most serious arise because a closed CNT apex is one-atom thick and can be sharply curved. This article explores a flat thin-slab model for CFE and derives an emission current equation. The model is parametrized using graphene data. The main conclusions are that (a) the occupied parts of electronic subbands probably do not overlap so CFE would come from a single subband; (b) an electron at the Fermi level has large momentum and kinetic energy parallel to the slab surface; (c) a clear conceptual distinction must be made between ...