Magneto-transport and Berry phase in magnetically doped Bi$_{0.97-x}$Sb$_{0.03}$ single crystals

We report large magnetoresistance (MR) and Shubnikov--de Haas (SdH) oscillations in single crystals of magnetically (Ni and Fe) doped Bi$_{0.97-x}$Sb$_{0.03}$ ($x=$ 0, 0.02) topological insulators. The R$\bar{3}$m symmetry and phase have been confirmed by the Rietveld refinement of x-ray diffraction data. Interestingly, a magnetic field induced phase transition from metallic to semi-conducting/insulating type is found in the parent sample. Moreover, we observe linear behavior of MR upto 15~Tesla in transverse mode and SdH oscillations in longitudinal mode where the field direction is with respect to the current and crystal plane. The magnitude, number as well as the frequency of SdH oscillations have been analyzed using the fast Fourier transform method for the parent and Ni substituted samples. For the parent sample, we found the coherence length L$_\phi=$ 12.7~nm through the fitting of MR data in transverse mode with modified H--L--N equation. The determined Fermi surface area is found to be larger in Ni doped as compared to the parent sample possibly due to change in the Fermi energy. The Kohler's plot indicates a single scattering mechanism below 100~K and the analysis with the help of Landau level (LL) Fan diagram reveals the non-zero Berry phase $\phi_{\rm B}= -$(1$\pm$0.03)$\pi$, which indicate the non-trivial topological states near the Dirac point.