Quantum oscillation studies of the topological semimetal candidate $\mathrm{ZrGe}M(M=\mathrm{S},\mathrm{Se},\mathrm{Te})$

The WHM-type materials ($W=\mathrm{Zr}/\mathrm{Hf}/\mathrm{La}, H=\mathrm{Si}/\mathrm{Ge}/\mathrm{Sn}/\mathrm{Sb}, M=\mathrm{O}/\mathrm{S}/\mathrm{Se}/\mathrm{Te}$) have been predicted to be a large pool of topological materials. These materials allow for fine tuning of spin-orbit coupling, lattice constant, and structural dimensionality for various combinations of $W, H$, and $M$ elements, thus providing an excellent platform to study how these parameters’ tuning affects topological semimetal state. In this paper, we report high field quantum oscillation studies on $\mathrm{ZrGe}M$ ($M=\mathrm{S},\mathrm{Se}$, and Te), from which we have revealed properties consistent with the theoretically predicted topological semimetal states. From the angular dependence of quantum oscillation frequency, we have also studied the Fermi surface topologies of these materials. Moreover, we have compared Dirac electron behavior between the $\mathrm{ZrGe}M$ and $\mathrm{ZrSi}M$ systems, which reveals deep insights to the tuning of Dirac state by spin-orbit coupling and lattice constants in the WHM system.

Figure 1

(a) Crystal structure of $\mathrm{ZrGe}M$ ($M=\mathrm{S},\mathrm{Se},\mathrm{Te}$). (b) Single crystal x-ray diffraction patterns for $\mathrm{ZrGe}M$. (c) Images of $\mathrm{ZrGe}M$ single crystals.

Figure 2

(a) The field dependence of magnetic torque τ for ZrGeS at different temperatures, which displays strong dHvA oscillations. The magnetic field is applied along the nearly out-of-plane direction ($B\parallel c^{\prime }$). The black and purple arrows indicate the splitting of oscillation peaks for the $F_{\alpha }(=12.5\mathrm{T})$ and $F_{\beta 1}(=236\mathrm{T})$ components. Inset: the experiment setup. (b) The oscillatory component of τ for $B\parallel c^{\prime }$. Inset: Zeeman splitting in susceptibility oscillations at $T=1.8,15$, and 30 K. (c) The FFT spectra for the oscillatory component $\mathrm{\Delta }\tau$ for $B\parallel c^{\prime }$. Inset: the temperature dependence of the FFT amplitude of the major fundamental frequencies and the fits to the LK formula (solid lines). (d) The field dependence of magnetic torque τ for ZrGeS at different temperatures. The magnetic field is applied along the nearly in-plane direction ($B\parallel a{b}^{\prime }$). (e) The oscillatory component of τ for $B\parallel a{b}^{\prime }$. Inset: the fit of the oscillation pattern at $T=1.8\mathrm{K}$ to the multiband LK formula. (f) The FFT spectra for the oscillatory component $\mathrm{\Delta }\tau$ for $B\parallel c^{\prime }$. Inset: the temperature dependence of the FFT amplitude of the major fundamental frequencies and the fit to the LK formula (solid lines).

Figure 3

(a) The field dependence of magnetic torque τ for ZrGeSe at different temperatures for $B\parallel c^{\prime }$. Inset: dHvA oscillations in τ under high field, which displays inverse sawtooth pattern and Zeeman splitting. (b) The oscillatory component of τ for $B\parallel c^{\prime }$. Inset: the fit of the oscillation pattern at $T=1.8\mathrm{K}$ to the multiband LK formula. (c) The FFT spectra for the oscillatory component $\mathrm{\Delta }\tau$ for $B\parallel c^{\prime }$. Inset: the temperature dependence of the FFT amplitude of the major fundamental frequencies and the fits to the LK formula (solid lines). (d) The field dependence of magnetic torque τ for ZrGeSe at different temperatures for $B\parallel a{b}^{\prime }$. The black arrows indicate split peaks. (e) The oscillatory component of τ for $B\parallel a{b}^{\prime }$. (f) The FFT spectra for the oscillatory component $\mathrm{\Delta }\tau$ for $B\parallel c^{\prime }$. Inset: the temperature dependence of the FFT amplitude of the major fundamental frequencies and the fits to the LK formula (solid lines).

Figure 4

(a) The field dependence of magnetic torque τ for ZrGeTe at different temperatures for $B\parallel c^{\prime }$. (b) The oscillatory component of τ for $B\parallel c^{\prime }$. Inset: the fit of the oscillation pattern at $T=1.8\mathrm{K}$ to the multiband LK formula. (c) The FFT spectra for the oscillatory component $\mathrm{\Delta }\tau$ for $B\parallel c^{\prime }$. Inset: the temperature dependence of the FFT amplitude of the major fundamental frequencies and the fits to the LK formula (solid lines). (d) The field dependence of magnetic torque τ for ZrGeTe at different temperatures for $B\parallel a{b}^{\prime }$. (e) The oscillatory component of τ for $B\parallel a{b}^{\prime }$. Inset: the fit of the oscillation pattern at $T=1.8\mathrm{K}$ to the multiband LK formula. (f) The FFT spectra for the oscillatory component $\mathrm{\Delta }\tau$ for $B\parallel c^{\prime }$. Inset: the temperature dependence of the FFT amplitude of the major fundamental frequencies and the fits to the LK formula (solid lines).

Figure 5

dHvA oscillations of (a) ZrGeS, (b) ZrGeSe, and (c) ZrGeTe at $T=1.8\mathrm{K}$ under different magnetic field orientations. The data collected under at different θ have been shifted for clarity. The angular dependence of oscillation frequencies for (d) ZrGeS, (e) ZrGeSe, and (f) ZrGeTe. The gray dashed lines indicate the in-plane ($B\parallel c$, 0°) and out-of-plane ($B\parallel ab$, 90°) directions.

Figure 6

In-plane magnetoresistance and SdH oscillations of $\mathrm{ZrGe}M$ ($M=\mathrm{S},\mathrm{Se},\mathrm{Te}$) for $B\parallel c$ and $B\parallel ab$. The data for ZrGeSe and ZrGeTe for $B\parallel c$ have been multiplied by a factor of 2 for clarity. Inset: The FFT spectra for the SdH oscillations; the data have been shifted for clarity.