On modeling the induced charge in density-functional calculations for field emitters

Abstract

The default assumption of many density-functional theory codes that the simulation cell is spatially periodic implies that any unbalanced charge in the cell will cause the solution to diverge, unless the imbalance is removed in some unphysical way. Periodic solution thus makes it difficult to model accurately the charge and field that are induced at the apex of a single carbon nanotube (CNT) when a background electric field is applied. We describe how the charge induced in a single cell containing 1.8 nm of the capped end of a (5,5) CNT can be calculated from a macroscopic model of the CNT with an external field acting on the whole CNT. With this method, a cell containing the CNT tip has been analyzed using the program ONETEP, a linear-scaling code that iterates the density kernel and the localized orbitals self-consistently to minimize the Helmholtz free energy. The results shown include (1) the sheath of mobile charge outside the framework of nuclei; (2) Kohn–Sham (KS) orbitals including the localized end states that are occupied when the field is applied; (3) total effective potential distribution as a function of the applied field; and (4) an induced field-enhancement factor of 50 deduced from the change of potential with the applied field. The computation also shows that (5) the charge density in zero field extends into the potential barrier over a distance of at least 0.12 nm beyond the Fermi equipotential, consistent with KS theory for the boundary between emitter and barrier.