Optical absorption window in Na$_3$Bi based three-dimensional Dirac electronic system

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.