Interband absorption edge in the topological insulators ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$

We have investigated the optical properties of thin films of topological insulators ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, and their alloys ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ on ${\mathrm{BaF}}_2$ substrates by a combination of infrared ellipsometry and reflectivity in the energy range from 0.06 to 6.5 eV. For the onset of interband absorption in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, after the correction for the Burstein-Moss effect, we find the value of the direct band gap of $215\pm 10$ meV at 10 K. Our data support the picture that ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ has a direct band gap located at the $\mathrm{\Gamma }$ point in the Brillouin zone and that the valence band reaches up to the Dirac point and has the shape of a downward-oriented paraboloid, i.e., without a camel-back structure. In ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, the onset of strong direct interband absorption at 10 K is at a similar energy of about 200 meV, with a weaker additional feature at about 170 meV. Our data support the recent $GW$ band-structure calculations suggesting that the direct interband transition does not occur at the $\mathrm{\Gamma }$ point but near the $Z\text{−}F$ line of the Brillouin zone. In the ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ alloy, the energy of the onset of direct interband transitions exhibits a maximum near $x=0.3$ (i.e., the composition of ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$), suggesting that the crossover of the direct interband transitions between the two points in the Brillouin zone occurs close to this composition.

Figure 1

Symmetric x-ray diffraction scans of our ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ samples ordered from bottom to top with increasing selenium content recorded with Cu $K{\alpha }_1$ radiation. The spectra are shifted for clarity with respect to each other. The corresponding diffraction peaks shift to the higher $Q_z$ values with increasing selenium content $x$ and decreasing lattice constant $c$. Solid and dashed lines show the positions of the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ diffraction peaks, respectively. The symbols denote calculated structure factors of the ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ lattice assuming a random occupation of Te and Se atoms. Thin black lines represent simulated diffraction profiles corresponding to the calculated structure factors.

Figure 2

Measured ellipsometric angles $\mathrm{\Psi }$ and $\mathrm{\Delta }$ at 300 K (symbols) and model spectra (solid lines) of our ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ samples with $x=0\%$ (a), 27% (b), 74% (c), and 100% (d) as a function of photon energy $E$. The angle of incidence was ${70}^{\circ }$.

Figure 3

Far-infrared reflectivity of our ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ thin films on ${\mathrm{BaF}}_2$ substrate at 300 K (symbols) together with model spectra (solid lines). Shown are results for samples with $x=0\%$ (a), 27% (b), 74% (c), and 100% (d). In addition, the data (magenta circles) and the model spectrum (solid line) of the ${\mathrm{BaF}}_2$ substrate are shown in panel (a).

Figure 4

The real part of the optical conductivity at 300 K of the ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ samples in the whole measured frequency range. The inset displays the region of the onset of interband absorption in terms of ${\varepsilon }_2$.

Figure 5

The real part of the dielectric function, ${\varepsilon }_1$, at 300 K of the ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ samples above the phonon energy range. The inset displays the data on a semilogarithmic scale including the large polarizability due to the $\alpha$-phonon.

Figure 6

(a) Temperature dependence of the imaginary part of the dielectric function, ${\varepsilon }_2$, of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The arrows mark the energy of the lowest critical point at 10 and 80 K, respectively. (b) Second derivative of the real (red squares) and imaginary part (blue triangles) of the dielectric function at 10 K together with the spectrum of the critical point model (solid lines) specified in the text.

Figure 7

(a) Temperature dependence of the imaginary part of the dielectric function, ${\varepsilon }_2$, of the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample. The inset shows the spectra on magnified scales. (b) Second derivative of the real (red squares) and imaginary part (blue triangles) of the dielectric function at 10 K together with critical point model spectra (solid lines). The arrows mark the positions of four critical points.

Figure 8

(a) The imaginary part of the dielectric function of our ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ samples at 10 K. The arrows mark the energy $E_{\mathrm{M}}$ of the maximum of ${\varepsilon }_2$. (b) The second derivative of the real (black thin line) and imaginary part (red thin line) of the dielectric function at 10 K of the $x=74\%$ sample together with the critical point model spectra (thick lines). The arrows mark the position of two critical points. (c) The second derivative of the real (black line) and imaginary part (red line) of the dielectric function at 10 K of the $x=27\%$ sample. The arrow marks the position of the main critical point. The structures near 180 and 330 meV are interference artefacts, as discussed in the text.

Figure 9

(a) The energy $E_{\mathrm{M}}$ of the maximum of ${\varepsilon }_2$ of ${\mathrm{Bi}}_2{\left({\mathrm{Te}}_{1-x}{\mathrm{Se}}_x\right)}_3$ marked by the arrows in Fig. 8. The dashed line is a guide to the eye. (b) Extracted values of the onset of direct interband transitions, $E_g^{\mathrm{direct}}$, at 10 K defined as the center position of the main lowest critical point (squares). The values for $x=76\%$ and $100\%$ were corrected for the Burstein-Moss shift as described in the text. The triangles represent the onset of a weaker direct interband absorption below the main critical point. The circles are previously reported [11] absorption edge energies at 300 K corrected for the Burstein-Moss effect. Parts (c) and (d) schematically represent the position of the direct interband transitions (red arrows) in the band structure for Se content $x<x_c$ (c) and $x>x_c$ (d), respectively, where $x_c\approx 30\%$. The green arrow in (c) depicts the indirect fundamental gap (see Refs. [51, 52]).