Dirac line nodes and effect of spin-orbit coupling in the nonsymmorphic critical semimetals $M\mathrm{SiS}(M=\mathrm{Hf},\mathrm{Zr})$

Topological Dirac semimetals (TDSs) represent a new state of quantum matter recently discovered that offers a platform for realizing many exotic physical phenomena. A TDS is characterized by the linear touching of bulk (conduction and valance) bands at discrete points in the momentum space [i.e., three-dimensional (3D) Dirac points], such as in $\mathrm{N}{\mathrm{a}}_3\mathrm{Bi}$ and $\mathrm{C}{\mathrm{d}}_3\mathrm{A}{\mathrm{s}}_2$. More recently, new types of Dirac semimetals with robust Dirac line nodes (with nontrivial topology or near the critical point between topological phase transitions) have been proposed that extend the bulk linear touching from discrete points to one-dimensional (1D) lines. In this paper, using angle-resolved photoemission spectroscopy (ARPES), we explored the electronic structure of the nonsymmorphic crystals $M\text{SiS}$ $(M=\mathrm{Hf}$, Zr). Remarkably, by mapping out the band structure in the full 3D Brillouin zone (BZ), we observed two sets of Dirac line-nodes in parallel with the $k_z$ axis and their dispersions. Interestingly, along directions other than the line nodes in the 3D BZ, the bulk degeneracy is lifted by spin-orbit coupling (SOC) in both compounds with larger magnitude in HfSiS. Our paper not only experimentally confirms a new Dirac line-node semimetal family protected by nonsymmorphic symmetry but also helps understanding and further exploring the exotic properties, as well as practical applications of the $M\text{SiS}$ family of compounds.

Figure 1

Dirac line nodes and characterization of HfSiS single crystals. (a) Schematic illustration of Dirac node and Dirac line nodes in the momentum space. (b) Crystal structure of the HfSiS, showing the stacking of Si and Hf-S planes. Lattice constant and most probable cleavage plane is indicated. (c)(i) Photograph of the HfSiS crystal cleaved along $\langle 001\rangle$ direction, (ii)–(iv) The XRD pattern of the (001), (100), and (010) surface of HfSiS and (v) core level spectrum of HfSiS showing characteristic ${\mathrm{S}}_{2p},\mathrm{S}{\mathrm{i}}_{2p}$, and $\mathrm{H}{\mathrm{f}}_{4f}$ core level peaks. (d) Illustration of the Dirac line-node location (upper panel) and the dispersion of the line nodes (lower panel). (e) Calculation results of the bulk band structure of HfSiS along high-symmetry directions. Green lines indicate the positions where the bands are nondegenerate, while the red lines indicate the Dirac line nodes. (f) Broad range photoemission spectral intensity map of the FS covers nine BZs, showing the correct symmetry and the characteristic FS.

Figure 2

General electronic structure of HfSiS. (a) The 3D intensity plot of the photoemission spectra centered around $\overline{\mathrm{\Gamma }}$. (b) Top row: Photoemission spectral intensity map showing the constant energy contours of bands at $E_B=0$ (i), 0.3 eV (ii), 0.6 eV (iii), and 0.9 eV (iv), respectively. Bulk state, SS, and DP are labeled. Bottom row (v)–(viii): Corresponding calculated constant energy contours at the same binding energies as in experiments above. (c) Stacking plots of constant-energy contours in broader binding energy range show the band structure evolution. The position of DP is labeled. (d) High symmetry cut along the $\overline{\mathrm{\Gamma }}\text{−}\overline{X}\text{−}\overline{\mathrm{\Gamma }}$ (i) and $\overline{M}\text{−}\overline{X}\text{−}\overline{M}$ (iii) directions near $E_F$, together with their corresponding calculated band structure (ii) and (iv). The BS and SS are labeled on the data and represented by red and green colors in the calculation. (e) Broad range high symmetry cut along the $\overline{M}\text{−}\overline{X}\text{−}\overline{M}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ directions (i) and corresponding calculated band structure (ii). The BS and SS are represented by red and green colors in the calculation.

Figure 3

Dirac line nodes in HfSiS. (a) Illustration of the location of Dirac line nodes in HfSiS. The 3D BZ also shows the constant energy contours in the $k_y\text{−}{k}_z$ plane (with $k_x=0)$ and $k_x\text{−}{k}_y$ plane (with $k_z=0)$ with energy around the DP. (b) Schematic of the band structure hosting a Dirac line node along the $k_z$ direction. Grey planes represent locations where each cut in (c) is measured, and blue lines represent the Dirac dispersion we would observe in each cut. The Dirac line node is labeled by the red curve. (c) Left panel: schematic of the dispersion of the line node. Right panel: stack plot of the four cuts along $(k_x,k_y)=(0,0)$ to $(k_x,k_y)=(0,\pi /a)$, with $k_z$ values ranging from 15 to $16\pi /c$. Red curve indicate the Dirac line node. (d) (i)–(iv) Detailed analysis of the four cuts shown in (c) with the linear fitting results of the Dirac bands marked by red lines and DP labeled. (v) The Momentum Distribution Curve (MDC) plot for the region around the DP [indicated by the dashed box in (i)]. (vi) The fitting DP positions at various $k_z$ values, with the red curve indicating the calculated dispersion along the $R\text{−}X\text{−}R$ direction. (e) (i)–(iv) Plot and band fitting result of four cuts along $(k_x,k_y)=(0,0)$ to $(k_x,k_y)=(\pi /a,\pi /a)$, with $k_z$ values ranging from 14.6 to $16\pi /c$. Fitted parabolic dispersions and DP positions are labeled. (v) The MDC plot for the region around the DP [indicated by the dashed box in (i)]. (vi) The fitting DP positions at various $k_z$ values, with the red curve indicating the calculated dispersion along the $A\text{−}M\text{−}A$ direction.

Figure 4

Dirac line nodes in ZrSiS. (a) Illustration of the location of Dirac line nodes in ZrSiS. The 3D BZ also shows the constant energy contours in the $k_y\text{−}{k}_z$ plane (with $k_x=0)$ and $k_x\text{−}{k}_y$ plane (with $k_z=0)$ with energy around the DP. (b) Stack plot of the four cuts along $(k_x,k_y)=(0,0)$ to $(k_x,k_y)=(0,\pi /a)$, with $k_z$ values ranging from 15 to $16\pi /c$. Red curve indicates the Dirac line node. (c) (i)–(iv) Detailed analysis of the four cuts shown in (b) with the linear fitting results of the Dirac bands marked by red lines, and DP is labeled. (v) The fitting DP positions at various $k_z$ values, with the red curve indicating the calculated dispersion along the $R\text{−}X\text{−}R$ direction.

Figure 5

Effect of SOC on the electronic structure of HfSiS and ZrSiS. (a) Constant energy contour on $E_F$ of HfSiS (i) and ZrSiS (ii). Green lines indicate the cut directions in (b)–(d). (b) (i)–(ii) The 3D intensity plot of the photoemission spectra centered around $\overline{X}$ and the high symmetry $\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ cut of HfSiS. (iii)–(iv) Same plot for ZrSiS. The SS splitting in both materials are estimated and labeled. (c) Comparison of the high symmetry $\mathrm{\Gamma }\text{−}X$ cut together with the band structure calculation of HfSiS (i)–(ii) and ZrSiS (iii)–(iv). The band splitting in both materials are labeled. (d) Comparison of the high symmetry $\overline{M}\text{−}\overline{\mathrm{\Gamma }}$ cut together with their band structure calculation of HfSiS (i) and (ii) and ZrSiS (iii) and (iv). In the spectrum, red curve labels the band structure at $k_z=\pi$, while the green curve labels the band structure from other $k_z$ values. The band gap between the electron and hole pockets are estimated and labeled. In the calculation results, the band structure of all $k_z$ values are plot together to reflect the uncertainty of $k_z$ values each cut covers. The band splitting between the electron and hole pockets at $k_z=\pi$ are labeled.