Electronic band structure of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$

We report on Landau-level spectroscopy of an epitaxially grown thin film of the topological insulator ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, complemented by ellipsometry and magnetotransport measurements. The observed response suggests that ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ is a direct-gap semiconductor with the fundamental band gap located at the $\mathrm{\Gamma }$ point or along the trigonal axis, and its width reaches $E_g=(190\pm 10)$ meV at low temperatures. Our data also indicate the presence of other low-energy extrema with a higher multiplicity in both the conduction and valence bands. The conclusions based on our experimental data are confronted with and to a great extent corroborated by the electronic band structure calculated using the $GW$ method.

Figure 1

Schematic view of the first BZ of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ with the mirror planes (gray planes) and trigonal, bisectrix, as well as binary axes indicated.

Figure 2

Magneto-optical response of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ in the midinfrared spectral range at $T=4.2$ K. (a) Relative magnetotransmission spectra $T_B/T_0$, plotted at selected values of the applied magnetic field $B$. (b) A false-color plot of relative magneto-absorbance $A_B=-ln[T_B/T_0]$. (c) Extracted positions of interband inter-LL resonances in the lower (blue) and upper (red) series. The light-red color is used for the satellite line discussed in the main text. The solid and dashed lines are fits using a simple two-band (massive Dirac-type) model; see the main text for a description.

Figure 3

Magneto-optical response of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ in the far-infrared spectral range at $T=4.2$ K. (a) Magnetotransmission spectra $T_B$, normalized by the transmission of a bare ${\mathrm{BaF}}_2$ substrate, are plotted for selected values of $B$. The same data in the form of a false-color plot are presented in (b). In both panels, pronounced coupling of the CR mode with the phonon mode at 7 meV is observed ($E_u$ mode [42]).

Figure 4

Magneto-optical response of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ in the midinfrared spectral range at $T=4.2$ K in the Faraday and Voigt (both $B\perp E$) configurations. Relative magnetotransmission spectra $T_B/T_0$, collected in the Faraday and Voigt configurations, are plotted in a form of (a), (c) false-color plots and (b), (d) stacked plots, respectively.

Figure 5

Magneto-transport experiments. (a) Temperature dependence of background-removed Shubnikov–de Haas oscillations in ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ with $B$ applied along the trigonal axis. Inset: The corresponding Hall resistance at $T=1.36$ K, showing linear-in-$B$ dependence. (b) Frequency spectra of $1/B$-periodic Shubnikov–de Haas oscillations at selected temperatures. (c) False-color plot: frequency spectra of $1/B$-periodic Shubnikov–de Haas oscillations as a function of the angle $\theta$ between the trigonal axis and magnetic field. To visualize the angle dependence, individual experimental traces were normalized by their maximum. Inset: The angle dependence of the oscillations' amplitude.

Figure 6

Real part of optical conductivity of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ deduced using ellipsometry at $T=7,100,200$ and 300 K. The vertical dashed lines mark the position of two inflection points (at $T=7$ K) in the imaginary part of the dielectric function, found as zero points of the second derivative; see the upper part. They are associated with two onsets of interband absorption in the data. The gray curves for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ are replotted from Refs. [25] and [8], respectively.

Figure 7

Band structure of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ calculated using the $GW$ method along different paths of the BZ. (a) Standard high-symmetry path of the bulk BZ. (b) Bulk bands projected on the surface (2D) BZ. Each line corresponds to a path in the bulk BZ that projects onto the path $\overline{\mathrm{M}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{K}}$ in the surface BZ. The color of the bands represents the $k_z$ component of the path. The paths and the corresponding colors are indicated in (c). (d) The calculated joint density of states at low photon energies. (e) Bands along the $\mathrm{Z}-\mathrm{F}$ path of the bulk (blue) and three paths slightly shifted with a fixed $k_z$. This shows that the maximum and minimum are indeed located along the $\mathrm{Z}-\mathrm{F}$ line and therefore have sixfold degeneracy ($N=6$). The paths and colors corresponding to (e) are shown in (f).

Figure 8

The electronic band structure of (${\mathrm{Sb}}_{1-x}{\mathrm{Bi}}_x{)}_2{\mathrm{Te}}_3$ calculated using the $GW$ tight-binding method [23] along selected directions for the bismuth content: $x=0,0.2,0.4,0.6,0.8$, and 1.

Figure 9

DFT calculations in the generalized gradient approximation (GGA) of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ performed with the FLAPW code fleur [26]. The SOC strength is reduced from 100% to 54%. After that, the system undergoes a topological phase transition and becomes trivial.