Quantum oscillations of the magnetic torque in the nodal-line Dirac semimetal ZrSiS

We report a study of quantum oscillations (QO) in the magnetic torque of the nodal-line Dirac semimetal $\mathrm{ZrSiS}$ in the magnetic fields up to 35 T and the temperature range from 40 K down to 2 K, enabling high resolution mapping of the Fermi surface (FS) topology in the $k_z=\pi$ (Z-R-A) plane of the first Brillouin zone (FBZ). It is found that the oscillatory part of the measured magnetic torque signal consists of low frequency (LF) contributions (frequencies up to 1000 T) and high frequency (HF) contributions (several clusters of frequencies from 7–22 kT). Increased resolution and angle-resolved measurements allow us to show that the high oscillation frequencies originate from magnetic breakdown (MB) orbits involving clusters of individual $\alpha$ hole and $\beta$ electron pockets from the diamond shaped FS in the Z-R-A plane. Analyzing the HF oscillations we unequivocally show that the QO frequency from the dog-bone shaped Fermi pocket ($\beta$ pocket) amounts $\beta =591\left(15\right)$ T. Our findings suggest that most of the frequencies in the LF part of QO can also be explained by MB orbits when intraband tunneling in the dog-bone-shaped $\beta$ electron pocket is taken into account. Our results give a new understanding of the novel properties of the FS of the nodal-line Dirac semimetal $\mathrm{ZrSiS}$ and sister compounds.

Figure 1

(a) The magnetic torque signals from a $\mathrm{ZrSiS}$ crystal at 2 K, for $\theta =-1^{\circ }, \theta =0^{\circ }, \theta =1^{\circ }$. The torque signal consists of a $B^2$ dependent background with QO contributions superimposed. Dashed line shows $a+b{B}^2$ fit to the measured magnetic torque signal. The inset to Fig. 1 shows an enlarged view of isolated HF QO as a function of $1/B$ for $\theta =-1^{\circ }, 0^{\circ }$ and $1^{\circ }$. (b) Image of a typical piezoresistive chip with $\mathrm{ZrSiS}$ crystal glued on the cantilever. (c) A schematic picture of sample and magnetic field configuration. By rotation of the sample around the crystalline $a$ axis the angle $\theta$ between the direction of magnetic field and the crystalline $c$ axis was varied.

Figure 2

The FFT spectrum of QO in the measured magnetic torque of $\mathrm{ZrSiS}$ crystal for $\theta =0^{\circ }$ at 2 K in the magnetic field range from 0 up to 35 T. The FFT frequency spectrum consists of LF contribution (frequencies up to 1 kT) and several clusters of HF contributions (frequencies from 7–22 kT). The high oscillation frequencies are attributed to different electron MB orbits as shown in Fig. 5. The first harmonic of the MB $A+n\alpha$ (yellow bar) and $B+n\alpha$ (blue bar) orbits are clearly visible and highlighted in brown and light blue, respectively.

Figure 3

The LF FFT spectrum of QO in the magnetic torque signal of $\mathrm{ZrSiS}$ crystal at 2 K, obtained for the different magnetic field ranges and angles (a) $\theta =-1^{\circ }\text{and}$ (b) $\theta =0^{\circ }$. Frequencies $\alpha =241\left(4\right)$ T and $\beta =591\left(15\right)$ T are contributions from $\alpha$ and $\beta$ pockets from the FS in the Z-R-A plane of the FBZ. Prominent high field peaks with frequencies $\gamma =415\left(8\right)$ T, $\eta =286\left(8\right)$ T, and 180(5) T and other smaller peaks could be ascribed to various MB orbits, as explained in Sec. 3b. It is interesting to notice that at high fields, a small change in angle $\theta$ leads to a strong change in FFT spectrum and several of the peaks are $\alpha$ apart, giving a strong indication of the MB effect.

Figure 4

(a) The FFT of HF contribution to QO in the magnetic torque signal of $\mathrm{ZrSiS}$ crystal measured at the different temperatures indicated in the main panel. The HF FFT spectrum consists of two main clusters of equidistantly separated peaks by the $\alpha$ hole pocket ($A$ and $B$ clusters, see Fig. 5) and their harmonics [see the inset to Fig. 4]. The calculated FFT spectra is explained by the effect of MB. According to the picture of MB orbits on the FS, $A$ and $B$ clusters of peaks and their harmonics should be separated by $4\beta$ and $8\beta$, respectively. Panel (b) highlights an enlarged view of the HF QO in the magnetic torque signal of $\mathrm{ZrSiS}$ for the temperatures of 2, 4.2, and 10 K as a function of 1/B. (c) FFT of HF contribution to QO in the magnetic torque signal of $\mathrm{ZrSiS}$ for angles $\theta =-1^{\circ }, 0^{\circ }$, and $1^{\circ }$. Figure illustrates strong angle dependence of the MB orbits.

Figure 5

Schematic representation of the FS of $\mathrm{ZrSiS}$ in the Z-R-A plane of the FBZ, with MB orbits highlighted with red lines [41]. FS consist of four $\beta$ electron (light blue dog-bone-shaped pocket) and four $\alpha$ hole pockets (dark blue pocket) in diamond shape configuration separated by a small gap due to the small SOC in $\mathrm{ZrSiS}$. (a) A red line indicates HF MB orbits where an electron encircles the entire FS in the Z-R-A plane. An electron can traverse the $\beta$ pocket over inner ($A$ surface) or outer edge ($B$ surface) and pick up between 0 to $n\alpha$ pockets. This explains the equidistantly separated HF FFT peaks by the $\alpha$ frequency inside $A+n\alpha$ and $B+n\alpha$ clusters. (b) The “figure of eight” MB orbit connecting one electron and one hole pocket resulting in $\beta -\alpha$ frequency of QO. It can be seen as an example of Klein tunneling in momentum space. (c) Example of MB orbit with tunneling through the neck of the dog-bone-shaped $\beta$ pocket. The MB orbit shown explains the observed QO frequency $\gamma =415\left(8\right)$ T, which can represented as a trajectory of the form $\frac{3}{2}\beta -2\alpha$.