Localized emission from flat diamond cathodes

We analyze steady-state field emission from an n-type semiconductor under the assumption that its surface presents no barrier for electron emission from the conduction band into the vacuum. We construct a classical Lagrangian field theory for the coupled electric and quasi-fermi potentials and use it to show that uniform emission is unstable against laterally nonuniform perturbations. We use a two-parameter model of normal emission current to show that the Lagrangian of the linearized system is minimal when all electrons are emitted from a single site. In addition to this intensely localized emission, we show the normal electric field at the surface is moderately enhanced at the solitary emission site even though the surface itself is planar. We use the result to explain the isolated emission sites observed in nanocrystalline n-type diamond films.