  The theories of field electron emission from perfectly planar and smooth
canonical surfaces are well understood, but they are not suitable for
describing emission from rough, irregular surfaces arising in modern nanoscale
electron sources. Moreover, the existing models rely on Sommerfeld's
free-electron theory for the description of electronic distribution which is
not a valid assumption for modern materials with nonparabolic energy
dispersion. In this paper, we derive analytically a generalized Fowler-Nordheim
(FN) type equation that takes into account the reduced space-dimensionality
seen by the quantum mechanically tunneling electron at a rough, irregular
emission surface. We also consider the effects of non-parabolic energy
dispersion on field-emission from narrow-gap semiconductors and few-layer
graphene using Kane's band model. The traditional FN equation is shown to be a
limiting case of our model in the limit of a perfectly flat surface of a
material with parabolic dispersion. The fractional-dimension parameter used in
this model can be experimentally calculated from appropriate current-voltage
data plot. By applying this model to experimental data, the standard
field-emission parameters can be deduced with better accuracy than by using the
conventional FN equation.
