Topological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite ${\mathrm{Cu}}_3\mathrm{PdN}$

Based on first-principles calculation and effective model analysis, we propose that the cubic antiperovskite material ${\mathrm{Cu}}_3\mathrm{PdN}$ can host a three-dimensional (3D) topological node-line semimetal state when spin-orbit coupling (SOC) is ignored, which is protected by the coexistence of time-reversal and inversion symmetry. There are three node-line circles in total due to the cubic symmetry. Drumheadlike surface flat bands are also derived. When SOC is included, each node line evolves into a pair of stable 3D Dirac points as protected by $C_4$ crystal symmetry. This is remarkably distinguished from the Dirac semimetals known so far, such as ${\mathrm{Na}}_3\mathrm{Bi}$ and ${\mathrm{Cd}}_3{\mathrm{As}}_2$, both having only one pair of Dirac points. Once $C_4$ symmetry is broken, the Dirac points are gapped and the system becomes a strong topological insulator with (1;111) $Z_2$ indices.

Figure 1

(a) Crystal structure of antiperovskite ${\mathrm{Cu}}_3\mathrm{PdN}$ with $\mathit{Pm}\overline{\mathrm{3}}\mathit{m}$ (No. 221) symmetry. A nitrogen atom is at the center of the cube and is surrounded by octahedral Cu atoms. Pd is located at the corner of the cube. (b) Bulk and projected (001) surface Brillouin zone. The three node-line rings (orange) and three pairs of Dirac points (red), without and with SOC included, respectively, are schematically shown.

Figure 2

(a) Electronic band structure without SOC, where the component of Pd $5p$ ($4d$) orbitals is proportional to the width of the red (blue) curves. Band inversion between $p$ and $d$ orbits happens at the $R$ point. (b) Evolution of energy levels of ${\mathrm{\Gamma }}_5^{+}$ and ${\mathrm{\Gamma }}_4^{-}$ at $R$ under hydrostatic pressure. Band inversion happens when $a<1.11a_0$. (c) Electronic band structure with SOC included. A small gap is opened in the $R\text{−}X$ direction while the Dirac point in the $R\text{−}M$ direction is stable and protected by crystal symmetry $C_4$ rotation.

Figure 3

Band structures and DOS for (001) surface (a) without and (b) with SOC. Without SOC, the nearly flat surface bands are clearly shown in the white dashed box around the $\overline{M}$ point.

Figure 4

Fermi surface of (001) surface shown in Fig. 3 with the chemical potential at 0.12 eV [green dashed line in Fig. 3]. The yellow dots are projected Dirac points. The bigger one at $\overline{M}$ means that there are two Dirac points superposed there.