  We present a detailed theoretical study of the optoelectronic properties of a
Na$_3$Bi-based three-dimensional Dirac electronic system (3DDES). The optical
conductivity is evaluated using the energy-balance equation derived from a
Boltzmann equation, where the electron Hamiltonian is taken from a simplified
$\mathbf{k}\cdotp \mathbf{p}$ approach. We find that for short-wavelength
irradiation, the optical absorption in Na$_3$Bi is mainly due to inter-band
electronic transitions. In contrast to the universal optical conductance
observed for graphene, the optical conductivity for Na$_3$Bi based 3DDES
depends on the radiation frequency but not on temperature, carrier density and
electronic relaxation time. In the radiation wavelength regime of about 5 $\mu
m<\lambda<$ 200 $\mu m$, an optical absorption window is found. This is similar
to what is observed in graphene. The position and width of the absorption
window depend on the direction of the light polarization and sensitively on
temperature, carrier density, and electronic relaxation time. Particularly, we
demonstrate that the inter-band optical absorption channel can be switched on
and off by applying the gate voltage. This implies that similar to graphene,
Na$_3$Bi based 3DDES can also be applied in infrared electro-optical
modulators. Our theoretical findings are helpful in gaining an in-depth
understanding of the basic optoelectronic properties of recently discovered
3DDESs.
