Dirac Line Nodes in Inversion-Symmetric Crystals

We propose and characterize a new ${\mathbb{Z}}_2$ class of topological semimetals with a vanishing spin-orbit interaction. The proposed topological semimetals are characterized by the presence of bulk one-dimensional (1D) Dirac line nodes (DLNs) and two-dimensional (2D) nearly flat surface states, protected by inversion and time-reversal symmetries. We develop the ${\mathbb{Z}}_2$ invariants dictating the presence of DLNs based on parity eigenvalues at the parity-invariant points in reciprocal space. Moreover, using first-principles calculations, we predict DLNs to occur in ${\mathrm{Cu}}_3\mathrm{N}$ near the Fermi energy by doping nonmagnetic transition metal atoms, such as Zn and Pd, with the 2D surface states emerging in the projected interior of the DLNs. This Letter includes a brief discussion of the effects of spin-orbit interactions and symmetry breaking as well as comments on experimental implications.

Figure 1

Crystal structure of ${\mathrm{Cu}}_3\mathrm{N}X$. $X$ represents a transition metal atom intercalated at the body center of the cubic unit cell of ${\mathrm{Cu}}_3\mathrm{N}$ in an anti-${\mathrm{ReO}}_3$ structure.

Figure 2

Electronic structures and ${\mathbb{Z}}_2$ indices of (a) ${\mathrm{Cu}}_3\mathrm{NZn}$ and (b) ${\mathrm{Cu}}_3\mathrm{NPd}$. Bands are drawn along the high-symmetry lines of the Brillouin zone (inset). The Dirac points are indicated by red circles. Parity eigenvalues are illustrated at the eight parity-invariant points in the first octant of the BZ.

Figure 3

Dirac line nodes in the Brillouin zone. (a) and (b) ${\mathrm{Cu}}_3\mathrm{NZn}$, and (c) and (d) ${\mathrm{Cu}}_3\mathrm{NPd}$. The DLNs are illustrated by red curves in the 3D BZ [(a) and (c)] and on the 2D boundary plane of the BZ at $k=X^x$ [(b) and (d)].

Figure 4

Two-dimensional surface electronic structure for ${\mathrm{Cu}}_3\mathrm{NZn}$. (a) First octant of three-dimensional Brillouin zone of ${\mathrm{Cu}}_3\mathrm{NZn}$ projected onto the two-dimensional surface BZ of the (100) surface and (b) surface electronic band structure. The Dirac line nodes and the projected interior of DLNs are illustrated with the red and blue schemes, respectively. The slab bands are shown in black lines, and surface states in the enclosed region are shown in blue lines. The shaded region represents bulk bands projected onto the BZ of the (100) surface along $k_x$.