  A model is constructed to describe the thermal-field emission of electrons
from a three-dimensional ($3$D) topological semimetal hosting Dirac/Weyl
node(s). The traditional thermal-field electron emission model is generalised
to accommodate the $3$D non-parabolic energy band structures in the topological
Dirac/Weyl semimetals, such as cadmium arsenide (\ch{Cd3As2}), sodium
bismuthide (\ch{Na3Bi}), tantalum arsenide (\ch{TaAs}) and tantalum phosphide
(\ch{TaP}). Due to the unique Dirac cone band structure, an unusual dual-peak
feature is observed in the total energy distribution (TED) spectrum. This
non-trivial dual-peak feature, absent from traditional materials, plays a
critical role in manipulating the TED spectrum and the magnitude of the
emission current. At zero temperature limit, a new scaling law for pure field
emission is derived and it is different from the well-known Fowler-Nordheim
(FN) law. This model expands the recent understandings of electron emission
studied for the Dirac $2$D materials into the $3$D regime, and thus offers a
theoretical foundation for the exploration in using topological semimetals as
novel electrodes.
