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Anisotropic conductivity in magnetic topological insulators

We study the surface conductivity of a three dimensional topological insulator doped with magnetic impurities. The spin-momentum locking of surface electrons makes their scattering from magnetic impurities anisotropic and the standard relaxation time approximation is not applicable. Using the semiclassical Boltzmann approach together with a generalized relaxation time scheme, we obtain closed forms for the relaxation times and analytic expressions for the surface conductivities of the system as functions of the bulk magnetization and the orientation of the aligned surface magnetic impurities. We show that the surface conductivity is anisotropic, and strongly depends both on the direction of the spins of magnetic impurities and on the magnitude of the bulk magnetization. In particular, we find that the surface conductivity has its minimum value when the spin of surface impurities are aligned perpendicular to the surface of TI, and therefore the backscattering probability is enhanced due to the magnetic torque exerted by impurities on the surface electrons.