Topological Tuning in Three-Dimensional Dirac Semimetals

We study with first-principles methods the interplay between bulk and surface Dirac fermions in three dimensional Dirac semimetals. By combining density functional theory with the coherent potential approximation, we reveal a topological phase transition in ${\mathrm{Na}}_3{\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ and ${\mathrm{Cd}}_3[{\mathrm{As}}_{1-x}{\mathrm{P}}_x{]}_2$ alloys, where the material goes from a Dirac semimetal to a trivial insulator upon changing Sb or P concentrations. Tuning the composition allows us to engineer the position of the bulk Dirac points in reciprocal space. Interestingly, the phase transition coincides with the reversal of the band ordering between the conduction and valence bands.

Figure 1

(a) Hexagonal unit cell for $A_{3}B$ compounds, with $A=\mathrm{Na}$, K, Rb and $B=\mathrm{Bi}$, Sb. (b) Bulk and surface projected Brillouin zone for the structure with the high symmetry points marked. The three dimensional Dirac crossing occurs along the $\mathrm{\Gamma }-A$ direction.

Figure 2

Bulk band structures obtained by including spin orbit interaction for (a) ${\mathrm{Na}}_3\mathrm{Sb}$, (b) ${\mathrm{Na}}_3\mathrm{Bi}$, (c) ${\mathrm{K}}_3\mathrm{Bi}$, and (d) ${\mathrm{Rb}}_3\mathrm{Bi}$. Note the Dirac crossing in (b)–(d). The insets in (c) and (d) show a zoom around $\mathrm{\Gamma }$ with the crossing along $\mathrm{\Gamma }-A$.

Figure 3

Band structures for ${\mathrm{Na}}_3\mathrm{Bi}$ thin films of thickness (a) 2–4 layers, (b) 5 layers, (c) 20 layers, and (d) 100 layers. Inset in (a)–(b) shows the energy gap at the center of the Brillouin zone for slabs of thickness 1 to 5 layers. In (b)–(d) Dirac crossings are highlighted in red (gray).

Figure 4

Spectral functions for pristine (a) ${\mathrm{Na}}_3\mathrm{Bi}$ and (b) ${\mathrm{Na}}_3\mathrm{Sb}$. (c) Spectral functions for the ${\mathrm{Na}}_3{\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ alloy with increasing Sb concentration ($x=0.25,0.50,0.75$ from top to bottom). The color scale shows the orbital contribution, with red (dark gray, positive values) denoting the $\mathrm{Bi}/\mathrm{Sb}$ $p$ orbitals and blue (light gray, negative values) representing the Na $s$ orbital (in units of states/eV).