Electronic structure of Na$_3$Bi near the Dirac point: Theory

Band structure calculations have been performed and compared with recent optical experiments. The ground state of the system is found to be not the highly symmetric P6$_{3}$/mmc structure, but instead the P$\bar{3}$c1 that involves buckling of the Na-Bi hexagonal planes. The band structure shows very little change between various symmetry configurations, yet the low-energy optical transition matrix elements are dramatically enhanced in the P$\bar{3}$c1 symmetry compared with P6$_{3}$/mmc, which results in an electronic response that agrees much more closely with optical data. A peak in the joint density of states driven by the particle-hole asymmetry of the band structure along the $\Gamma-A$ momentum direction results in a large peak in the imaginary part of the dielectric function. Systematic changes are observed in the low energy Dirac cone Fermi velocity and Lifshitz gap energy with lattice spacing and spin-orbit coupling. The large anisotropies of the Dirac cone and small energy gaps are discussed.