Topological crystalline insulator state with type-II Dirac fermions in transition metal dipnictides

The interplay between topology and crystalline symmetries in materials can lead to a variety of topological crystalline insulator (TCI) states. Despite significant effort towards their experimental realization, so far only ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$ has been confirmed as a mirror-symmetry-protected TCI. Here, based on first-principles calculations combined with a symmetry analysis, we identify a rotational-symmetry-protected TCI state in the transition metal dipnictide ${RX}_2$ family, where $R=\mathrm{Ta}$ or Nb and $X=\mathrm{P}$, As, or Sb. Taking ${\mathrm{TaAs}}_2$ as an exemplar system, we show that its low-energy band structure consists of two types of bulk nodal lines in the absence of spin-orbit coupling (SOC) effects. Turning on the SOC opens a continuous band gap in the energy spectrum and drives the system into a $C_{2}T$-symmetry-protected TCI state. On the (010) surface, we show the presence of rotational-symmetry-protected nontrivial Dirac cone states within a local bulk energy gap of $\sim 300\mathrm{meV}$. Interestingly, the Dirac cones have tilted energy dispersion, realizing a type-II Dirac fermion state in a topological crystalline insulator. Our results thus indicate that the ${\mathrm{TaAs}}_2$ materials family provides an ideal setting for exploring the unique physics associated with type-II Dirac fermions in rotational-symmetry-protected TCIs.

Figure 1

Schematic illustration of the Dirac-cone surface states of (a) a mirror-symmetry-protected topological crystalline insulator (TCI) and (b) a $C_{2\left[010\right]}$ rotational-symmetry-protected TCI with $C_{2h}$ point-group symmetry. The (010) mirror plane ($M_y$, shaded gray plane) and high-symmetry crystal axes are shown. The Dirac cones in a mirror-symmetry-protected TCI are pinned to the mirror-invariant line over the surface whereas they lie at generic $k$ points on the surface normal to the rotational axis in a rotational-symmetry-protected TCI. (c) The conventional unit cell of ${RX}_2$ compounds. The $M_y$ (010) mirror-invariant plane and $C_{2y}$ ($C_{2\left[010\right]}$) rotational axis are shown. (d) The primitive Brillouin zone and its projection on the (010) surface.

Figure 2

Bulk band structure of ${\mathrm{TaAs}}_2$ (a) without and (b) with spin-orbit coupling (SOC). The orbital compositions of bands are shown using various colors. The band crossings in (a) are seen resolved along the $\mathrm{\Gamma }\text{−}Y$ and $M\text{−}A$ directions; these are gapped in the presence of the SOC. (c) Fermi surface of ${\mathrm{TaAs}}_2$ with electron (cyan) and hole (purple) pockets. (d) Parity eigenvalues of the valence bands at eight time-reversal-invariant momentum points in the BZ.

Figure 3

(a)–(f) Bulk band structure of ${\mathrm{TaAs}}_2$ around the selected time-reversal-invariant momentum points $Y$, $A$, and $M$ where band inversion takes place. The top row shows band structure without SOC whereas the middle row shows band structure including SOC. (g) Two different views of the nodal-line structures in the BZ. The top figure highlights nodal lines extending across the BZs and the bottom figure highlights nodal rings formed near the $M$ point. (h) Schematic illustration of the formation of nontrivial insulating states with SOC.

Figure 4

(a) Surface band structure of ${\mathrm{TaAs}}_2$ along the high-symmetry lines in the (010) surface BZ. Constant energy contours at (b) $E=E_f$ and (c) $E=E_D$, where $E_D=110\mathrm{meV}$ denotes the energy of the Dirac point (DP). Dirac points are marked with white arrows. (d) Band structure along the $k$ path marked with a dashed white line in (c). (e) Constant energy contours near the DP. The electron and hole pockets are noted in the plots. (f) Schematic illustration of the rotational-symmetry-protected Dirac cone states on the (010) surface of ${\mathrm{TaAs}}_2$.

Figure 5

Calculated bulk band structures of the transition metal dipnictide family of materials without the SOC. Blue text on each figure identifies the material.

Figure 6

Same as Fig. 5 except that here the SOC is included. The gapped nodal lines can be seen along the $\mathrm{\Gamma }\text{−}Y$ and $M\text{−}A$ directions.