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We have investigated the optical properties of thin films of topologicalinsulators Bi$_{2}$Te$_{3}$, Bi$_{2}$Se$_{3}$ and their alloysBi$_2$(Te$_{1-x}$Se$_x$)$_3$ on BaF$_{2}$ substrates by a combination ofinfrared 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 forthe Burstein-Moss effect, we find the value of direct bandgap of $215\pm10$ meVat 10 K. Our data supports the picture that Bi$_2$Se$_3$ has a direct band gaplocated at the $\Gamma$ point in the Brillouin zone and that the valence bandreaches up to the Dirac point and has the shape of a downward orientedparaboloid, i.e. without a camel-back structure. In Bi$_2$Te$_3$, the onset ofstrong direct interband absorption at 10 K is at a similar energy of about 200meV, with a weaker additional feature at about 170 meV. Our data support therecent GW band structure calculations suggesting that the direct interbandtransition does not occur at the $\Gamma$ point but near the Z-F line of theBrillouin zone. In the Bi$_2$(Te$_{1-x}$Se$_x$)$_3$ alloy, the energy of theonset of direct interband transitions exhibits a maximum near $x=0.3$ (i.e. thecomposition of Bi$_2$Te$_2$Se), suggesting that the crossover of the directinterband transitions between the two points in the Brillouin zone occurs closeto this composition.

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