Quantum oscillations in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ in high magnetic fields

${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ has drawn much attention as the leading candidate to be the first topological superconductor and the realization of coveted Majorana particles in a condensed matter system. However, there has been increasing controversy about the nature of its superconducting phase. This study sheds light on ambiguity in the normal-state electronic state by providing a complete look at the quantum oscillations in magnetization in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ at high magnetic fields up to 31 T. Our study focuses on the angular dependence of the quantum oscillation pattern in a low carrier concentration. As the magnetic field tilts from the crystalline $c$ axis to the $ab$ plane, the change of the oscillation period follows the prediction of the ellipsoidal Fermi surface. As the doping level changes, the 3D Fermi surface becomes quasicylindrical at high carrier density. Such a transition is potentially a Lifshitz transition of the electronic state in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 1

Oscillations in the torque data of sample 4 with the polynomial background subtracted. The inset in the upper right-hand corner is a schematic of the experimental setup. The lower inset is a fast Fourier transformation (FFT) of the oscillatory torque after subtracting the polynomial background. The single peak in the FFT spectrum reveals a single Fermi pocket.

Figure 2

(a) Quantum oscillations in torque of sample 4 at different angles after background subtraction. Oscillations are visible in the raw signal up to $90{}^{\circ }$. At high tilt angle, the oscillation signal is multiplied by a factor of 10 for clarity. (b) The FFT spectra of oscillations in panel (a) show a single Fermi pocket with clear angular dependence. The FFT amplitude is normalized by the height of the peak in the range of 200–600 T. For the high tilt angles, the divergence of FFT amplitude in the dc end arises from an incomplete background subtraction.

Figure 3

Angular dependence of the oscillation frequency of the various samples. Dashed lines are ellipsoidal fits for the Fermi surfaces. The inset expands the $y$ axis of the data plot to a higher frequency range to show the extrapolation of the ellipsoidal fits.

Figure 4

(a) Temperature dependence of the oscillation amplitude of sample 4. (b) Temperature dependence of the normalized amplitude with fit to the thermal damping factor. Extracted from this fit is a high-angle effective mass of $0.32m_e$.

Figure 5

Dingle plot of sample 4a. Fitting is of the Dingle damping factor and it yields a Dingle temperature of 57.1 K.

Figure 6

Hall effect of the Cu-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Each sample is cleaved from the same batch as the samples used in the torque measurement. (a) The Hall signal ${\rho }_{xy}$ is plotted against field $H$ up to 5 T at $T=1.5$ K. The Hall signal curves have been antisymmetrized to eliminate the magnetoresistance pickup. The slopes of the ${\rho }_{xy}\text{−}H$ curves are used to determine the Hall carrier density shown in Table 4. (b) Measurements of ${\rho }_{xy}$ in $H$ up to 12 T show quantum oscillations in three samples. In this panel, a polynomial background is subtracted to show the oscillation signal in ${\rho }_{xy}$. The oscillation frequency is found to be the same as the measured frequency in the dHvA effect.

Figure 7

Volume susceptibility measurements of 3 different samples. Sample 4, the sample with lowest carrier concentration, shows a superconducting transition at 3 K and a 16% superconducting volume. Samples 3 and 5 do not show any superconducting property most likely due to sample quality degradation over time.