Angular-Dependent Phase Factor of Shubnikov–de Haas Oscillations in the Dirac Semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$

We measure the magnetotransport properties of the three-dimensional Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ single crystal under magnetic fields up to 36 T. Shubnikov–de Haas (SdH) oscillations are clearly resolved and the $n=1$ Landau level is reached. A detailed analysis on the intercept of the Landau index plot reveals a significant dependence of the SdH phase factor on the orientation of the applied magnetic field. When the magnetic field is applied in the [001] direction, i.e., along the fourfold screw axis of the tetragonal crystal structure, a nontrivial $\pi$ Berry phase, as predicted for the Dirac fermions, is observed. However, in a magnetic field tilted away from the [001] direction, the $\pi$ Berry phase is evidently reduced, and a considerable enhancement of the effective mass is also revealed. Our observations demonstrate that the Dirac dispersion in ${\mathrm{Cd}}_3{\mathrm{As}}_2$ is effectively modified in a tilted magnetic field, whereas the preserved $\pi$ Berry phase in a magnetic field along the [001] direction can be related to the realization of the Weyl fermions. The sudden change of the SdH phase also indicates a possible topological phase transition induced by the symmetry-breaking effect.

Figure 1

(a) ${\rho }_{xx}(T)$ of two ${\mathrm{Cd}}_3{\mathrm{As}}_2$ single crystal samples. (Inset) ${\rho }_{xx}$ and ${\rho }_{xy}$ measured in a magnetic field up to 14 T. (b) Angular dependence of the transverse magnetoresistance (MR), [${\rho }_{xx}(B)-{\rho }_{xx}(0)]/{\rho }_{xx}(0)$, at 4.2 K with $B$ rotated in the (010) plane. The tilt angle $\theta$ is defined as the angle between $\overrightarrow{B}$ and the [100] direction, as shown by the sketched measurement configuration. All MR data have been symmetrized between the positive and negative fields to cancel the picked-up Hall signal. (c) SdH oscillatory component of ${\rho }_{xx}(B)$ at different tilt angles. The Landau index number $n$ (integer) is assigned to the maxima of $\mathrm{\Delta }{\rho }_{xx}(B)$. (d) Landau index plots $n$ vs $1/B$ at different $\theta$’s. (Inset) Angular dependence of the oscillation frequency $F$ and the total phase $\gamma -\delta$, extracted from the linear fitting from the first LL to the fifth LL. The open symbol represents the result of a nonlinear fitting (see the Supplemental Material [30]) for $\theta =90°$.

Figure 2

(a) Oscillatory component $\mathrm{\Delta }{\rho }_{xx}(B)$ vs $1/B$, measured in a magnetic field up to 14 T at 2 K. (b) FFT spectra of the SdH oscillations at selected tilt angles. (Inset) Angular dependence of the oscillation frequency. The dashed line is the expectation for an ellipsoidal Fermi surface. Error bars are the HWHM of FFT peaks. (c) Angular dependence of the spin-splitting parameter $S$. The navy squares and olive diamonds are data points obtained from high-field (36 T) and low-field (14 T) measurements, respectively. (Inset) Angular dependence of the SdH oscillation phase, which is obtained by linear fitting from the second to the sixth LL. (d) Temperature dependence of the normalized FFT peak amplitude for $B\parallel$ [100] and $B\parallel$ [001]. Fittings by the LK formula are denoted by solid lines. The fitting parameter $B$ is defined as the average inverse field of the FFT interval, $B_{\text{ave}}$.

Figure 3

(a) MR of a ${\mathrm{Cd}}_3{\mathrm{As}}_2$ single crystal measured at 2 K with different in-plane field orientations. $\overrightarrow{B}$ is rotated in the (100) plane and $\varphi$ is the angle between $\overrightarrow{B}$ and the [001] direction. (Inset) Sketch of the experimental setup. (b) LL index plots of the oscillations $\mathrm{\Delta }{\rho }_{xx}$ with $B$ along the two in-plane axes. (Inset) The frequency (blue diamonds) and the phase factor (red circles) of SdH oscillation as a function of $\varphi$. Both parameters are extracted from the linear extrapolation between the third and the sixth LL.