Shubnikov–de Haas oscillations in the three-dimensional Dirac fermion system ${\mathrm{Ca}}_3\mathrm{PbO}$

Cubic antiperovskite ${\mathrm{Ca}}_3\mathrm{PbO}$ is a candidate for three-dimensional (3D) Dirac fermion systems that have recently emerged as a class of topological materials exhibiting linear energy dispersion in the bulk. We report magnetotransport and tunnel diode oscillation (TDO) measurements on Bi-doped ${\mathrm{Ca}}_3\mathrm{PbO}$ in magnetic fields up to 55 T and temperatures between 2 and 78 K. By observing the Shubnikov–de Haas (SdH) oscillations, we resolve the bulk 3D Fermi surface (FS) with distinctive features of Dirac fermions including linear magnetoresistance, light effective mass, and a nontrivial phase shift. TDO measurements under high fields reveal the existence of two primary SdH frequencies, one being twice the other. Together with the low effective mass, the oscillations of these two frequencies account for the emergence of the Landau level splitting that persists up to 43 K. The field-angular-dependence of the oscillation frequencies with three branches confirms that the FS is composed of three pairs of hole pockets with uniaxial anisotropy on the Γ-X path, as predicted for bulk ${\mathrm{Ca}}_3\mathrm{PbO}$ crystal by density functional theory calculation.

Figure 1

SdH oscillations under dc fields and Fermi surface parameters. (a),(b) MR and $R_{xy}$ vs $B$ for Bi-doped ${\mathrm{Ca}}_3\mathrm{PbO}$ up to 9 T at temperatures between 2 and 40 K. The insets show the derivatives $d{R}_{xx}/dB$ and $d{R}_{xy}/dB$ vs $B$ at 2 K, respectively. (c) Oscillatory component of longitudinal resistivity $\mathrm{\Delta }{R}_{xx}$ vs 1/$B$ at various temperatures. The inset shows the FFT spectrum of the SdH oscillations at 2 K. (d) Oscillation amplitudes $\mathrm{\Delta }{R}_{xx}\left(T\right)/\mathrm{\Delta }{R}_{xx}$ (2 K) vs $T$ for the peak at $1/B=0.135{\mathrm{T}}^{-1}$. (e) Dingle plot of the SdH oscillations with the oscillation frequency $F=41.5\mathrm{T}$.

Figure 2

Oscillatory components of the resistance as a function of $1/(Bcos\theta )$ at various angles, measured for another sample with $m^{*}\left(0\right)\sim 0.074m_0$.

Figure 3

(a) Oscillatory component of magnetoconductivity of the Bi-doped ${\mathrm{Ca}}_3\mathrm{PbO}$ single crystal, characterized by transport properties shown in Fig. 1, $\mathrm{\Delta }{\sigma }_{xx}$ vs 1/$B$ fitted to the LK formula. The minima positions are assigned as integer LL indices. (b) LL index plot.

Figure 4

High magnetic field data obtained from TDO measurements. (a) Resonant frequency, $\mathrm{\Delta }{f}_{\mathrm{TDO}}$, vs $B$ up to 55 T at temperatures between 4 and 78 K with the field applied normal to the (001) plane. The inset shows the FFT spectrum of the SdH oscillations at 4 K for fields between 5 and 11 T. (b) FFT spectra of oscillatory component $\mathrm{\Delta }{f}_{\mathrm{TDO}}^{\mathrm{OSC}}$ at 4 K. The peaks are assigned as ${\alpha }_x$ and ${\alpha }_z$ with their second harmonics and their combinations. The inset shows the schematic image of the ellipsoidal FS in the 3D BZ of ${\mathrm{Ca}}_3\mathrm{PbO}$.

Figure 5

Angular dependence of SdH oscillations under high magnetic fields. (a) Oscillatory component $\mathrm{\Delta }{f}_{\mathrm{TDO}}^{\mathrm{OSC}}$ at various angles. (b) Angular dependence of the FFT peak positions. The solid and dashed lines show the simulated results for the 3D ellipsoidal model using harmonics and combinations, respectively. The solid and open dots represent the FFT peaks with high and low amplitudes.