Bulk Fermi surface coexistence with Dirac surface state in ${\text{Bi}}_2{\text{Se}}_3$: A comparison of photoemission and Shubnikov–de Haas measurements

Shubnikov-de Haas (SdH) oscillations and angle-resolved photoemission spectroscopy (ARPES) are used to probe the Fermi surface of single crystals of ${\text{Bi}}_2{\text{Se}}_3$. We find that SdH and ARPES probes quantitatively agree on measurements of the effective mass and bulk band dispersion. In high carrier density samples, the two probes also agree in the exact position of the Fermi level $E_F$, but for lower carrier density samples discrepancies emerge in the position of $E_F$. In particular, SdH reveals a bulk three-dimensional Fermi surface for samples with carrier densities as low as ${10}^{17}{\text{cm}}^{-3}$. We suggest a simple mechanism to explain these differences and discuss consequences for existing and future transport studies of topological insulators.

Figure 1

(Color online) (a) Temperature dependence of two typical samples of ${\text{Bi}}_2{\text{Se}}_3$ with carrier densities differing by 2 orders of magnitude. (b) Shows the carrier density $n_e$, (c) the resistivity ${\rho }_0$ at $T=2\text{K}$, and (d) the mobility for samples of different thicknesses. Each sample was a cleave from a parent sample, so that the surface area of each sample was kept constant.

Figure 2

(Color online) (a) ARPES band dispersion on samples of ${\text{Bi}}_2{\text{Se}}_3$ with carrier density $2.3\times {10}^{19}{\text{cm}}^{-3}$ (batch S4). Green lines mark the Fermi level, the bottom of the conduction band, $E-E_F=-150\text{meV}$ and the Dirac point, $E-E_F=-355\text{meV}$. (b) Due to the quantization of the energy spectrum into Landau levels, oscillations appear in the magnetoresistance known as SdH oscillations. The SdH oscillations here are for a sample taken from the same batch as in (a) at $\theta =0$, corresponding to a oscillatory frequency of $F=155\text{T}$, consistent with $E_F\sim 160\text{meV}$. ARPES and SdH are in good agreement for these high carrier density samples.

Figure 3

(Color online) (a) Magnetotransport for samples from S1. As the angle is swept the frequency of the oscillation varies according to the topology of the Fermi surface. For a two-dimensional pocket the expected dependence is $1/\text{cos}\theta$ (shown in green in the inset). The observed angle dependence is clear evidence for a closed ellipsoidal Fermi surface pocket, similar to that observed by Köhler (Ref. 12). Similar SdH data were gathered on batches S2 and S3 on a number of samples. Samples from batch S3 showing the temperature dependence of the derivative of the SdH signal in (b) and a fit to the data shown in (c) from which the effective mass, Dingle temperature, and oscillatory frequency can be extracted.

Figure 4

(Color online) ARPES data on samples of ${\text{Bi}}_2{\text{Se}}_3$ from batch S1. The horizontal lines show the crossing of the Fermi level $\left(E-E_F=0\right)$ and the Dirac crossing. (a) Band structure measured by ARPES results on samples from batch S1 showed the Fermi level near the SdH level of $\sim 17\text{meV}$ from the bottom of the conduction band. (b) Measurement on another sample from batch S1 showed the Fermi level in gap. Some other samples from S2 and S3 also showed the Fermi level in the bulk gap. The variation might be due to the lower carrier density of these samples, and the surface band structure is more susceptible to small amounts of surface contamination. (c) Band structure of a sample also from batch S1 which was cleaved in atmosphere and exposed for 10 s, showing significant $n$-type doping with large bulk conduction-band pocket.

Figure 5

(Color online) (a) A schematic representation of the band structure seen by ARPES (solid) red horizontal line denoting the Fermi level as seen by SdH. (b) We infer band bending of about 75 meV at the surface from a comparison of ARPES (Fig. 4) and quantum oscillations (Fig. 3).