Using static linear response theory to describe field emission field enhancement and a field-induced insulator-conductor transition

Experiments on field electron emission (FE) of single carbon nanotubes (CNTs) indicate that they exhibit a nearly linear Fowler–Nordheim plot, and the field enhancement factor (FEF) near the CNT apex is independent of the applied (macroscopic) field (⁠$FM$⁠) for small radii field emitters. Recent results, based on density functional theory calculations considering CNTs with small radii, retrieved the constancy of the FEF defined in terms of the corresponding induced electron density. As a consequence, it has been reported that the constancy of the FEF with $FM$ could be connected with the linear response of the CNT. In this paper, we reinforce this connection, considering the problem of a floating (6,6) hybrid single-walled nanotube, whose cylindrical body is an insulating one and composed of alternating boron and nitrogen atoms end-capped with carbon atoms. Our results show that the constancy of the FEF is achieved when a linear dependence between the longitudinal component of the induced system dipole moment (⁠$μi,z$⁠) and $FM$ is observed. Two regimes of constant polarizabilities have been found at sufficiently low and high $FM$-values. In the intermediate range $0.3V/nm≲FM≲5$ V/nm, a crossover from insulating-to-conducting behavior, exhibiting a nonlinear dependence of $μi,z$ on $FM$⁠, is found accompanied by an increase of the FEF with $FM$⁠. This result reveals circumstances that could lead to dependence of the FEF on $FM$⁠, being timely for interpretation of FE characteristics in the context of vacuum nanoelectronic devices.