Nearly massless Dirac fermions and strong Zeeman splitting in the nodal-line semimetal ZrSiS probed by de Haas–van Alphen quantum oscillations

Topological semimetals represent a new class of quantum materials hosting Dirac/Weyl fermions. The essential properties of topological fermions can be revealed by quantum oscillations. Here we present systematic de Haas–van Alphen (dHvA) oscillation studies on the recently discovered topological Dirac nodal-line semimetal ZrSiS. From the angular dependence of dHvA oscillations, we have revealed the anisotropic Dirac bands in ZrSiS and found surprisingly strong Zeeman splitting at low magnetic fields. The Landé $g$ factor estimated from the separation of Zeeman splitting peaks is as large as 38. From the analyses of dHvA oscillations, we also revealed nearly zero effective mass and exceptionally high quantum mobility for Dirac fermions in ZrSiS. These results shed light on the nature of novel Dirac fermion physics of ZrSiS.

Figure 1

(a) Crystal structure of ZrSiS. (b) Isothermal out-of-plane magnetization $\left(M\right)$ for ZrSiS at various temperatures from $T=1.8\mathrm{K}$ to 20 K. Inset: An image of a ZrSiS single crystal. (c) The oscillatory component of the magnetization $\mathrm{\Delta }M$. Upper inset: Zeeman effect at $T=1.8\mathrm{K}$ and 12 K. Lower inset: Susceptibility oscillation $d\mathrm{\Delta }M/dB$. The high-frequency component is filtered out for clarity. Peak splitting is indicated by arrows and discernible above $B=1.7\mathrm{T}$. (d) The high-frequency oscillation component (blue data points) and the fit of the oscillation pattern to the LK formula (red solid line). To better illustrate the fitting quality, the zoomed-in data and fitting are shown in the inset.

Figure 2

Analyses of the dHvA oscillations for $B\parallel c$. (a) The FFT spectra of the oscillatory component of the dHvA oscillations for $B\parallel c$ at various temperatures. (b) The fits of the FFT amplitudes of $F_{\alpha }$ and $F_{\beta }$ to the temperature damping factor $R_T$ of the LK formula. (c) Dingle plot for the low-frequency oscillation component $(F_{\alpha }=8.4$ T) at $T=1.8\mathrm{K}$, 5 K, and 10 K. (d) LL index fan diagram for the low-frequency oscillation component. $n-1/4$ $(n$, integer LL indices) are assigned to the $\mathrm{\Delta }M$ oscillation minima. The black line represents a linear fit to all LL indices, which yields an intercept of $n_0=0.63$. The red line represents the linear fit to the LL indices obtained below 1 T to minimize the influence of the Zeeman effect, which yields an intercept of 0.34. Inset: The variation of the intercept $n_0$ with the magnetic field range used for the linear fit of the LL fan diagram.

Figure 3

Analyses of the dHvA oscillations for $B\parallel ab$. (a) Isothermal in-plane magnetization measurements for ZrSiS at various temperatures from $T=1.8\mathrm{K}$ to 20 K. (b) The oscillatory component of the magnetization $\mathrm{\Delta }M$. (c) The FFT spectra of the oscillatory component of the in-plane magnetization at various temperatures. (d) The fits of the FFT amplitudes to the temperature damping factor $R_T$ of the LK formula. (e) and (f): The low (e) and high (f) frequency oscillation components of the dHvA oscillations. The red lines show the fits of the oscillation pattern to the generalized two-band and three-band LK formula for the low-frequency (e) and high-frequency (f) oscillations. Inset in (f): The enlarged data and fit within a small field range, which clearly shows the data can be best fitted by the three-band LK formula.

Figure 4

Fermi surface morphology of ZrSiS. (a) dHvA oscillations at $T=1.8\mathrm{K}$ for different magnetic field orientations. Data at different field orientations have been shifted for clarity. (b) Schematic of the measurement setup. The field is rotated from an out-of-plane direction $(B\parallel c$, defined as $\theta =0^{\circ })$ to an in-plane direction $(B\parallel ab$, defined as $\theta ={90}^{\circ })$. (c) The angular dependence of the oscillation frequencies obtained from the FFT spectra. Error bars are defined as the half-width at the half-height of FFT peak. The green lines are guides for the eye. The red lines are fits to $F=F_{3\mathrm{D}}+F_{2\mathrm{D}}/\cos \theta$ for $F_{\alpha }$ and $F_{\beta }$ below 60˚.

Figure 5

Transport properties of ZrSiS. (a) Magnetic field dependence of Hall resistivity ${\rho }_{xy}$. SdH oscillations are seen below 20 K. (b) Two-band model fitting to the transverse $\left({\rho }_{xx}\right)$ and Hall $\left({\rho }_{xy}\right)$ resistivity at $T=2\mathrm{K}$ and 50 K.

Figure 6

Comparison of Zeeman effect in dHvA and SdH oscillations. (a) The lower frequency dHvA oscillation component of the out-of-plane $\left(B\parallel c\right)$ magnetization at $T=2\mathrm{K}$, obtained by filtering the higher frequency component. Zeeman splitting is clearly observed. (b) Derivative of the lower frequency oscillation component, $d\mathrm{\Delta }M/dB$. (c) SdH oscillations of the in-plane magnetoresistance $\left(\mathrm{\Delta }{\rho }_{xx}\right)$ at $T=2\mathrm{K}$. Zeeman splitting occurs at almost the same field for both dHvA $\left(dM/dB\right)$ and SdH $\left({\rho }_{xx}\right)$ oscillations, as denoted by dashed lines.