The Berry phase and the phase of the Shubnikov-de Haas oscillations in three-dimensional topological insulators

Within the semiclassical approach, we calculate contributions of the Berry phase and of the Zeeman coupling of the electron moment with the magnetic field to the phase of the Shubnikov - de Haas oscillations for the surface electrons in the Bi$_2$X$_3$ family of three-dimensional topological insulators (X stands for Te or Se). We also discuss a relation of the obtained results with published experimental data on the Shubnikov-de Haas oscillations for this family of topological insulators.