Hole Fermi surface in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ probed by quantum oscillations

Transport and torque magnetometry measurements are performed at high magnetic fields and low temperatures in a series of p-type (Ca-doped) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ crystals. The angular dependence of the Shubnikov-de Haas and de Haas-van Alphen quantum oscillations enables us to determine the Fermi surface of the bulk valence band states as a function of the carrier density. At low density, the angular dependence exhibits a downturn in the oscillations frequency between $0^{\circ }$ and ${90}^{\circ }$, reflecting a bag-shaped hole Fermi surface. The detection of a single frequency for all tilt angles rules out the existence of a Fermi surface with different extremal cross sections down to 24 meV. There is therefore no signature of a camelback in the valence band of our bulk samples, in accordance with the direct band gap predicted by $GW$ calculations.

Figure 1

(a) Longitudinal resistance $R_{xx}$ versus total magnetic field in perpendicular configuration for a low-doped p-type ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ sample (B3) at $T=1.2$ K. Upper left inset: oscillatory resistance $\mathrm{\Delta }{R}_{xx}$. Lower right inset: fast Fourier transform of $\mathrm{\Delta }{R}_{xx}(1/B)$. (b) Torque $\tau$ and oscillatory torque $\mathrm{\Delta }\tau$ (upper left inset) versus total magnetic field for $\theta ={16}^{\circ }$ in a highly-doped p-type ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ sample (E1). Lower right inset: Fourier transform of $\tau (1/Bcos\theta )$.

Figure 2

Oscillatory magnetoresistance $\mathrm{\Delta }{R}_{xx}$ versus total magnetic field parallel to the $c$ axis for different temperatures (1.4 – 17.5 K) in sample B1. Upper left inset: temperature dependence of $\mathrm{\Delta }{R}_{xx}$ at a resistance extremum at $B=9.6$ T. Lower left inset: temperature dependence of $\mathrm{\Delta }{R}_{xx}$ at the resistance extremum at $B=10.6$ T for sample B2.

Figure 3

Color plot : Normalized amplitude of the quantum oscillations $1/B$ Fourier transform as a function of the frequency and the tilt angle $\theta$, for different samples. At each angle, the FFT signal is normalized to its maximal value (red color). The blue color corresponds to an intensity $\le 70\%$ of the maximum value.

Figure 4

(left panels) Quantum oscillation frequency as a function of the tilt angle $\theta$ for different carrier concentrations in the valence band. The error bar is defined to be the FFT width at 90% of the maximum signal. From bottom to top, sample B2 (SdH), B3 (SdH), B6 (dHvA), E2(dHvA), and E1(dHvA). Theoretical fits obtained by the analytical model of Eq. (2) (see text) (solid lines). (Right panels) Fermi surfaces computed at $E_F=23.7$, 42, and $58.6$ meV.

Figure 5

(a) and (b) Valence band along $k_{x,y}$ and $k_z$, respectively. Experimental data (dots), parabolic dispersion (solid lines), and camelback model (dotted lines).

Figure 6

Oscillatory magnetoresistance $\mathrm{\Delta }{R}_{xx}$ versus total magnetic field at $T=1.2$ K (solid line). Simulations for the $g^{*}=0$ (dotted line) and $g^{*}=34.8$ cases (dashed line). The arrow emphasizes the $B=22.5$ T minimum. Inset: quantity proportional to the $n{B}_n$ value (blue circle), and simulated Fermi energy as a function of the magnetic field for $g^{*}=34.8$ (dashed line) (see text).