Channel resolved field emission beyond the work function: a quantum statistical approach

Cold field emission is almost universally analysed through the Fowler-Nordheim law, in which the metal enters only as a scalar work function and a free electron supply density. We show that the emitted current is instead controlled by the channel resolved overlap between the vacuum escape kernel and the occupied boundary spectral density of the metal, and that this overlap reorders the material rankings the standard theory predicts. Formulating the emission problem on the Keldysh contour and integrating out the vacuum exterior of a conducting wedge yields an exact retarded vacuum self energy, whose boundary reduction expresses the stationary current as a trace pairing between an escape rate carrying the geometry and the barrier, and a boundary spectral density carrying the surface electronic structure. We evaluate this pairing from first principles across 38 surface orientations of 14 metals and compounds. Materials of nearly identical work function differ by up to an order of magnitude in emitted current, the leading emitters being those whose spectral weight lies in the transverse momentum channels that the vacuum selects most strongly. Surface orientation emerges as an important independent design variable, with facet to facet variations of up to a factor of about six that are almost entirely spectral in origin. On thin barrier surfaces the vacuum self energy depletes the boundary spectrum from which emission is drawn, and the channel resolved emission current exhibits a turnover as the field increases. Work function screening alone is an insufficient criterion for selecting field emitters.