  In the nitrogen-incorporated ultrananocrystalline diamond ((N)UNCD) films,
representing an $n$-type highly conductive two-phase material comprised of
$sp^3$ diamond grains and $sp^2$-rich graphitic grain boundaries, the current
is carried by a high concentration of mobile electrons within the large-volume
grain boundary networks. Fabricated in a simple thin-film planar form, (N)UNCD
was found to be an efficient field emitter capable of emitting a significant
amount of charge starting at the applied electric field as low as a few
V/$\mu$m which makes it a promising material for designing electron sources.
Despite the semimetallic conduction, field emission (FE) characteristics of
this material demonstrate a strong deviation from the Fowler-Nordheim law in a
high-current-density regime when (N)UNCD field emitters switch from a
diode-like to resistor-like behavior. Such phenomenon resembles the
current-density saturation effect in conventional semiconductors. In the
present paper, we adapt the formalism developed for conventional semiconductors
to study current-density saturation in (N)UNCD field emitters. We provide a
comprehensive theoretical investigation of ($i$) the influence of partial
penetration of the electric field into the material, ($ii$) transport effects
(such as electric-field-dependent mobility), and ($iii$) features of a complex
density-of-states structure (position and shape of $\pi-\pi^*$ bands,
controlling the concentration of charge carriers) on the FE characteristics of
(N)UNCD. We show that the formation of the current-density saturation plateau
can be explained by the limited supply of electrons within the impurity
$\pi-\pi^*$ bands and decreasing electron mobility in high electric field.
Theoretical calculations are consistent with experiment.
