This paper addresses the question of how spin-dependent transport through a local orbital of a nanosystem exhibiting uniaxial magnetic anisotropy (as in, e.g., magnetic molecules, magnetic adatoms, or quantum dots coupled to a localized magnetic moment) and exchange coupling of conduction electrons to the system's magnetic core affect the Kondo effect. Spectral functions and linear conductance in the Kondo regime are calculated with the use of the numerical renormalization group method. It is shown that the Kondo effect is suppressed due to the exchange coupling between electrons tunneling through the conducting orbital and the magnetic core. The corresponding conductance also depends significantly on the uniaxial anisotropy, which in turn results in a nontrivial behavior of the tunnel magnetoresistance. Finally, the possibility of restoring the Kondo effect by application of an external magnetic field is discussed.
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