Experimental study of the valence band of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

The valence band of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ is investigated by measuring the Shubnikov–de Haas effect as well as galvanomagnetic and thermoelectric transports. At low hole concentration, the hole Fermi surface is closed and boxlike, but at higher carrier concentrations it develops tubelike extensions that are open, in general agreement with our theoretical calculations. However, the experimentally determined density-of-states effective mass is smaller than density-functional-theory calculations predict; although we cannot give a definitive explanation for this, we suspect that the theory may lack sufficient precision to compute room-temperature transport properties, such as the thermopower, in solids with interlayer van der Waals bonds.

Figure 1

Calculated hole Fermi surface of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ at hole concentrations of $6\times {10}^{18}{\mathrm{cm}}^{-3}$ (top) and $6.7\times {10}^{19}{\mathrm{cm}}^{-3}$ (bottom) using the methods developed in Ref. [11]. (a) has the bag region whereas (b) has both bag and pipe regions. The pipes are connected to the adjacent Brillouin zones.

Figure 2

Experimental temperature dependence of (a) the carrier concentration as measured by the Hall effect, (b) the thermopower, (c) the mobility, and (d) the resistivity of the single-crystal $p$-type ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ samples studied here. The inset in (a) shows the field dependence at 10 K of the Hall resistivity, illustrating a simple linear relation indicative of single-band conduction.

Figure 3

Angular dependence of the Shubnikov–de Haas oscillations trace for all single-crystal samples measured. The angle is the elevation angle between the direction of the applied magnetic field and the basal plane: (1) 0°, (2) 30°, (3) 60°, (4) 90°, (5) 120°, (6) 150°, and (7) 180°. The frames (a)–(d) are for samples S1–S4 (see Table 1).

Figure 4

Polar plot of the cross-sectional area of the Fermi surface as a function of the elevation angle from the basal plane (0°) toward the $c$ axis (90°). The plots are shown for different doping levels as indicated.

Figure 5

Characteristic temperature dependence of the SdH oscillation amplitude. The symbols are the experimental data points, plotted for different values of the magnetic field as indicated. All solid lines are fits to Eq. (2) with a single parameter and an effective mass of $m^{*}=0.25m_e$. The Dingle temperatures vary from sample to sample.

Figure 6

Thermopower at 300 K versus hole concentration as measured by the Hall effect at 100 K for $p$-type ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Symbols and color conventions are given in Table 1.

Figure 7

Measured (data points) and calculated (lines) density-of-states hole effective mass versus doping level. The heavy hole mass $m_H^{*}$ corresponds to the central bag in the Fermi surface shown in Fig. 1, and the light mass $m_L^{*}$ corresponds to the pipes as illustrated in Fig. 1.