  Recently discovered Dirac semimetals (DSMs) with two Dirac nodes, such as
Na$_{3}$Bi and Cd$_{2}$As$_{3}$, are regarded to carry the $\mathbb{Z}_{2}$
topological charge in addition to the chiral charge. Here, we study the Floquet
phase transition of $\mathbb{Z}_{2}$ topological DSMs subjected to a beam of
circularly polarized light. Due to the resulting interplay of the chiral and
$\mathbb{Z}_{2}$ charges, the Weyl nodes are not only chirality-dependent but
also spin-dependent, which constrains the behaviors in creation and
annihilation of the Weyl nodes in pair. Interestingly, we find a novel phase:
One spinband is in Weyl semimetal phase while the other spinband is in
insulator phase, and we dub it Weyl half-metal (WHM) phase. We further study
the spin-dependent transport in a Dirac-Weyl semimetal junction and find a spin
filter effect as a fingerprint of existence of the WHM phase. The proposed spin
filter effect, based on the WHM bulk band, is highly tunable in a broad
parameter regime and robust against magnetic disorder, which is expected to
overcome the shortcomings of the previously proposed spin filter based on the
topological edge/surface states. Our results offer a unique opportunity to
explore the potential applications of topological DSMs in spintronics.
