Quantum oscillations of the topological surface states in low carrier concentration crystals of Bi$_{2-x}$Sb$_{x}$Te$_{3-y}$Se$_{y}$

We report a high-field magnetotransport study on selected low-carrier crystals of the topological insulator Bi$_{2-x}$Sb${_x}$Te$_{3-y}$Se$_{y}$. Monochromatic Shubnikov - de Haas (SdH) oscillations are observed at 4.2~K and their two-dimensional nature is confirmed by tilting the magnetic field with respect to the sample surface. With help of Lifshitz-Kosevich theory, important transport parameters of the surface states are obtained, including the carrier density, cyclotron mass and mobility. For $(x,y)=(0.50,1.3)$ the Landau level plot is analyzed in terms of a model based on a topological surface state in the presence of a non-ideal linear dispersion relation and a Zeeman term with $g_s = 70$ or $-54$. Input parameters were taken from the electronic dispersion relation measured directly by angle resolved photoemission spectroscopy on crystals from the same batch. The Hall resistivity of the same crystal (thickness of 40~$\mu$m) is analyzed in a two-band model, from which we conclude that the ratio of the surface conductance to the total conductance amounts to 32~\%.