From Nodal Chain Semimetal to Weyl Semimetal in HfC

Based on first-principles calculations and effective model analysis, we propose that the WC-type HfC, in the absence of spin-orbit coupling (SOC), can host a three-dimensional nodal chain semimetal state. Distinguished from the previous material ${\mathrm{IrF}}_4$ [T. Bzdusek et al., Nature 538, 75 (2016)], the nodal chain here is protected by mirror reflection symmetries of a simple space group, while in ${\mathrm{IrF}}_4$ the nonsymmorphic space group with a glide plane is a necessity. Moreover, in the presence of SOC, the nodal chain in WC-type HfC evolves into Weyl points. In the Brillouin zone, a total of 30 pairs of Weyl points in three types are obtained through the first-principles calculations. Besides, the surface states and the pattern of the surface Fermi arcs connecting these Weyl points are studied, which may be measured by future experiments.

Figure 1

(a) The primitive cell of WC-type HfC and its (b) top view and side view. Hf and C atoms occupy the $1d$ $(\frac{1}{3},\frac{2}{3},\frac{1}{2})$ and $1a$ (0,0,0) Wyckoff positions. (c) The bulk Brillouin zone and its projection onto the (010) and (001) direction. The high symmetrical crystal momenta and $k_z=0$, $k_y=0$ mirror planes are indicated.

Figure 2

(a) Band structures of WC-HfC without SOC. Two band-crossing structures, one around the $\mathrm{\Gamma }$ point and the other one around the $M$ point, exist near the Fermi energy. (b) Side view and (c) top view of the Fermi surface of WC-HfC, which consists of electron pockets (blue) and hole pockets (red), and forms a lotus-root-like structure.

Figure 3

(a) The band structure of HfC calculated within the GGA including spin-orbit coupling (SOC). (b) 3D view of the nodal chain (in the absence of SOC) and Weyl points (with SOC) in the BZ. The red and blue points indicate the Weyl point with chirality $\chi =+1$ and $\chi =-1$, respectively. (c) Side view from (100) and (d) top view from (001) directions for the nodal chain and Weyl points. Once the SOC is turned on, the nodal chains are gapped and give rise to 30 pairs of Weyl points off the mirror planes.

Figure 4

(a) Surface states of HfC terminated in the (001) direction. (b)–(d) Fermi surface of HfC (001)-surface states. Chemical potential is set at the (b) $W_1$, (c) $W_2$, and (d) $W_3$ type of Weyl point.