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

semiconductor science and technology

The influence of proximity induced ferromagnetism, superconductivity and  Fermi-velocity on evolution of Berry phase in Bi$_{2}$Se$_{3}$ topological  insulator