Quantum oscillations in the noncentrosymmetric superconductor and topological nodal-line semimetal ${\mathrm{PbTaSe}}_2$

We observed quantum oscillations in thermoelectric and magnetic torque signals in noncentrosymmetric superconductor ${\mathrm{PbTaSe}}_2$. One oscillatory frequency stems from the orbits formed by magnetic breakdown, while others are from two-dimensional-like Fermi surfaces near the topological nodal rings. Our comprehensive understanding of the Fermi surface topology of ${\mathrm{PbTaSe}}_2$, including nailing down the Fermi level and detecting the Berry phases near the nodal rings, is crucial for searching plausible topological superconductivity in its bulk and surface states.

Figure 1

(a) Magnetothermopower of ${\mathrm{PbTaSe}}_2$ (sample P1) at selected temperatures. Purple arrows indicate the positions of the oscillatory peaks corresponding to the lowest oscillating frequency ($\sim 6$ T) and $a_0$ to $a_4$ denote the corresponding Landau indexes. The blue arrow points to the upper critical field of superconducting transition. Inset: zero-field Seebeck coefficient of ${\mathrm{PbTaSe}}_2$ vs temperature. (b) Magnetic torque signals for sample P2 at selected temperatures in higher magnetic fields. Curves are offset for clarity.

Figure 2

(a) FFT spectrum of the observed QOs in $S_{xx}$ and $\tau$. (b) A sketch of the first BZ for ${\mathrm{PbTaSe}}_2$. (c) FS of ${\mathrm{PbTaSe}}_2$ around the $\mathrm{\Gamma }$ point. The outer FS is translucent to reveal the inner nesting discuslike one. (d) Overall FS of ${\mathrm{PbTaSe}}_2$ in the first BZ. [(e), (f)] Cuts of FS at $k_z=0$ and $\pi$, respectively. Note that these two cuts are plotted in the primitive cell of the 2D reciprocal space in order to show closed orbits. Colored orbits correspond to the frequencies in the FFT spectrums.

Figure 3

(a) Magnetic torque signals for P2 at different angles. $\theta$ is defined as the angle between the magnetic field direction and the crystallographic $c$ axis within the $ac$ plane. (b) Corresponding FFT spectrums. Dashed lines are a guide to the eye. (c) FFT frequencies for the orbits ${\gamma }_1$, ${\gamma }_2$, and $\delta$ vs angle. Solid lines are calculated frequencies.

Figure 4

(a) Fitting of the effective mass ($m^{*}$) using the derivative of LK formula for the five frequencies detected in the QOs of magnetothermopower of P1. The amplitude of frequency $F_{\alpha }$ is fitted by using the oscillation at 4.25 T while the others are from the amplitude of corresponding FFT peaks. (b) Fitting of $m^{*}$ with standard LK formula for torque signals. (c) Band dispersion around $H$ and $K$ points with SOC. There are two nodal lines (NL1 and NL2) surrounding $H$ in the $k_z=\pi$ plane and one more (NL3) around $K$ in the $k_z=0$ plane. The intersections of the nodal rings along the high-symmetry lines are marked with purple arrows. The position of the $\alpha$ pocket is marked green. (d) The toruslike hole pocket plotted in extended BZ. Blue circle represents the position of the spinless nodal ring (NL0) without SOC, where the $\alpha$ orbit also resides. (e) Landau fan diagrams for $\alpha$, $\beta$, and $\delta$ bands subtracted from $S_{xx}$ and ${\gamma }_1$ from $\tau$. Inset: Closeup of fitting near $n=0$ which shows residual phases.