  We have investigated the optical properties of thin films of topological
insulators Bi$_{2}$Te$_{3}$, Bi$_{2}$Se$_{3}$ and their alloys
Bi$_2$(Te$_{1-x}$Se$_x$)$_3$ on 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 Bi$_2$Se$_3$, after the correction for
the Burstein-Moss effect, we find the value of direct bandgap of $215\pm10$ meV
at 10 K. Our data supports the picture that Bi$_2$Se$_3$ has a direct band gap
located at the $\Gamma$ 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 Bi$_2$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 $\Gamma$ point but near the Z-F line of the
Brillouin zone. In the Bi$_2$(Te$_{1-x}$Se$_x$)$_3$ alloy, the energy of the
onset of direct interband transitions exhibits a maximum near $x=0.3$ (i.e. the
composition of Bi$_2$Te$_2$Se), suggesting that the crossover of the direct
interband transitions between the two points in the Brillouin zone occurs close
to this composition.

Materials Science