Chemical pressure effect on the optical conductivity of the nodal-line semimetals $\mathrm{ZrSi}Y\left(Y=\mathrm{S},\mathrm{Se},\mathrm{Te}\right)$ and $\mathrm{ZrGe}Y\left(Y=\mathrm{S},\mathrm{Te}\right)$

ZrSiS is a nodal-line semimetal, whose electronic band structure contains a diamond-shaped line of Dirac nodes. We carried out a comparative study on the optical conductivity of ZrSiS and the related compounds ZrSiSe, ZrSiTe, ZrGeS, and ZrGeTe by reflectivity measurements over a broad frequency range combined with density functional theory calculations. The optical conductivity exhibits a distinct U shape, ending at a sharp peak at around $10000{\mathrm{cm}}^{-1}$ for all studied compounds except for ZrSiTe. The U shape of the optical conductivity is due to transitions between the linearly dispersing bands crossing each other along the nodal line. The sharp high-energy peak is related to transitions between almost parallel bands, and its energy position depends on the interlayer bonding correlated with the $c/a$ ratio, which can be tuned by either chemical or external pressure. For ZrSiTe, another pair of crossing bands appears in the vicinity of the Fermi level, corrugating the nodal-line electronic structure and leading to the observed difference in optical conductivity. The findings suggest that the Dirac physics in $\mathrm{Zr}XY$ compounds with $X=\mathrm{Si},\mathrm{Ge}$ and $Y=\mathrm{S},\mathrm{Se},\mathrm{Te}$ is closely connected to the interlayer bonding.

Figure 1

(a) Crystal structure of the $\mathrm{Zr}XY$ compounds with square nets parallel to the $ab$ plane. (b) Scheme illustrating the chemical pressure effect in ZrSiS, ZrSiSe, ZrSiTe, ZrGeS, and ZrGeTe.

Figure 2

Experimental optical conductivity spectra for all measured $\mathrm{Zr}XY$ compounds as obtained from ambient-condition reflectivity spectra.

Figure 3

(a) and (c)–(f) Comparison of the experimental and theoretical optical conductivities without intraband contributions for ZrSiS, ZrSiSe, ZrSiTe, ZrGeS, and ZrGeTe, respectively. (b) Contributions to the experimental optical conductivity of ZrSiS as obtained from the fitting are shown, in particular, the sharp peak labeled L4, which is due to transitions between almost parallel bands.

Figure 4

Calculated band structures of (a) ZrSiS and (d) ZrSiTe with SOC, panels (b) and (e) and (c) and (f) show contributions of different band combinations to the optical conductivity $\sigma$ and the joint density of states (JDOS), respectively. The black arrows in panel (a) highlight the transitions between almost parallel red and yellow bands in the vicinity of the $X$ and $R$ points, which give rise to the L4 peak in the optical conductivity.

Figure 5

Wannier band structures of ZrSiS (left) and ZrSiTe (right). The red and green colors in panels (a)–(f) reflect the Si $pxpy$ and Si $pz$ weights, respectively. Panels (a)–(f) show the band structures obtained for hopping constrained to: nearest neighbors on the Si plane (a) and (b), nearest-neighbors Si-Si, Si-Zr, and Zr-S/Te (c) and (d), and no constraints (e) and (f). Panels (g) and (h) show the same Wannier band structure (red) as panels (e) and (f) compared to the full DFT band structures (black).

Figure 6

(a) Energy position of the L4 peak as a function of the $c/a$ ratio, obtained from the experimental and theoretical optical conductivity spectra of ZrSiS, ZrSiSe, ZrGeS, and ZrGeTe, in comparison with the energy difference $\mathrm{\Delta }E$ between the ungapped band crossings above and below $E_F$ at the $X$ point from Topp et al. [7]. Please note the offset of 0.11 eV between the left and the right ordinates. (b) Experimental optical conductivity of ZrSiS for selected external pressures in the high-energy range. The inset: Energy position of the L4 peak in ZrSiS as a function of external pressure. The pressure dependence of the $c/a$ ratio was extracted from Ref. [31].