  We use first-principles simulations to investigate the topological properties
of Bi$_2$Se$_3$ thin films deposited on amorphous SiO2, Bi$_2$Se$_3$/a-SiO$_2$,
which is a promising substrate for topological insulator (TI) based device
applications. The Bi$_2$Se$_3$ films are bonded to a-SiO$_2$ mediated by van
der Waals interactions. Upon interaction with the substrate, the Bi$_2$Se$_3$
topological surface and interface states remain present, however the degeneracy
between the Dirac-like cones is broken. The energy separation between the two
Dirac-like cones increases with the number of Bi$_2$Se$_3$ quintuple layers
(QLs) deposited on the substrate. Such a degeneracy breaking is caused by (i)
charge transfer from the TI to the substrate and charge redistribution along
the Bi$_2$Se$_3$ QLs, and (ii) by deformation of the QL in contact with the
a-SiO$_2$ substrate. We also investigate the role played by oxygen vacancies
(V$_O$) on the a-SiO$_2$, which increases the energy splitting between the two
Dirac-like cones. Finally, by mapping the electronic structure of
Bi$_2$Se$_3$/a-SiO$_2$, we found that the a-SiO$_2$ surface states, even upon
the presence of V$_O$, play a minor role on gating the electronic transport
properties of Bi$_2$Se$_3$.
