Electronic structures of transition metal dipnictides $XP{n}_2$ ($X=\mathrm{Ta}$, Nb; $Pn=\mathrm{P}$, As, Sb)

The electronic structures and topological properties of transition metal dipnictides $XP{n}_2$ ($X=\mathrm{Ta}$, Nb; $Pn=\mathrm{P}$, As, Sb) have been systematically studied using first-principles calculations. In addition to small bulk Fermi surfaces, the band anticrossing features near the Fermi level can be identified from band structures without spin-orbit coupling, leading to nodal lines in all these compounds. Inclusion of spin-orbit coupling gaps out these nodal lines, leaving only a pair of disentangled electron/hole bands crossing the Fermi level. Therefore, the low-energy physics can be in general captured by the corresponding two-band model with several isolated small Fermi pockets. Detailed analysis of the Fermi surfaces suggests that the arsenides and ${\mathrm{NbSb}}_2$ are nearly compensated semimetals while the phosphorides and ${\mathrm{TaSb}}_2$ are not. Based on the calculated band parities, the electron and hole bands are found to be weakly topological nontrivial, giving rise to surface states. As an example, we presented the surface-direction-dependent band structure of the surfaces states in ${\mathrm{TaSb}}_2$.

Figure 1

(a) Conventional (the larger black box) and primitive (the smaller orange box) cells of $XP{n}_2$. Transparent atoms only appears in the conventional cell. (b) Primitive Brillouin zone (BZ), the reciprocal lattice vectors of conventional BZ ($k_a, k_b$, and $k_c$), the BZ of conventional [100] (green solid lines), [010] surface (red solid lines), [001] surface (blue solid lines), and their respective high symmetry points.

Figure 2

Bulk band structure of $XP{n}_2$ calculated without SOC (reproduced with MLWF-fitted TB Hamiltonians). The circles indicates anticrossing features within $E_F\pm 200$ meV range. The Fermi energy is aligned at 0. The blue circled features are present in all compounds, while the green circled features are not.

Figure 3

Bulk band structure of $XP{n}_2$ calculated with SOC. The Fermi energy is aligned at 0.

Figure 4

Fermi surfaces of $XP{n}_2$ calculated with SOC (using MLWF-fitted TB Hamiltonians). In all compounds, there are 3 isolated pockets, including 2 large ones and 1 small one.

Figure 5

Illustration of possible surface states. The rows from top to bottom are phosphorides, arsenides, and antimonides, respectively. The bottom left corner of each panel is $\mathrm{\Gamma }$, and the surface indices are illustrated at the center of the panel. The blue/red circles represent $+/-$ polarizations, respectively; and open circles indicate overwhelming bulk states due to occupations. Topological surface states can be present between solid circles with different colors.

Figure 6

Surface states of ${\mathrm{TaSb}}_2$. The surface direction is labeled in the conventional cell; thus the [110] surface is indeed the [100] surface of the primitive cell.