  Bi$_{2}$Se$_{3}$ is a well known 3D-topological insulators(TI) with a
non-trivial Berry phase of $ \left(2n+1\right)\pi $ attributed to the topology
of the band structure. The Berry phase shows non-topological deviations from $
\left(2n+1\right)\pi $ in presence of a perturbation that destroys time
reversal symmetry and gives rise to a quantum system with massive Dirac
fermions and finite band gap. Such a band gap opening is achieved on account of
the exchange field of a ferromagnet or the intrinsic energy gap of a
superconductor that influences the topological insulator surface states by
virtue of the proximity effect. In this work Berry phase of such gapped systems
with massive Dirac fermions is considered. Additionally, it is shown that the
Berry phase for such a system also depends on the \textit{Fermi}-velocity of
the surface states which can be tuned as a function of the TI film thickness.
