Within the semiclassical approach, we calculate contributions of the Berry phase and of the Zeeman coupling of an 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.

• Received 4 October 2011

DOI:https://doi.org/10.1103/PhysRevB.85.033301