An experimental study of the in-plane azimuthal behaviour and frequency dependence of the ferromagnetic resonance field and the resonance linewidth as a function of BiFeO$_3$ thickness is carried out in a polycrystalline exchange-biased BiFeO$_3$/Ni$_{81}$Fe$_{19}$ system. The magnetization decrease of the Pt/BiFeO$_3$/Ni$_{81}$Fe$_{19}$/Pt heterostructures with BiFeO$_3$ thickness deduced from static measurements has been confirmed by dynamic investigations. Ferromagnetic resonance measurements have shown lower gyromagnetic ratio in a perpendicular geometry compared with that of a parallel geometry. The monotonous decrease of gyromagnetic ratio in a perpendicular geometry as a function of the BiFeO$_3$ film thickness seems to be related to the spin-orbit interactions due to the neighbouring Pt film at its interface with Ni$_{81}$Fe$_{19}$ film. The in-plane azimuthal shape of the total linewidth of the uniform mode shows isotropic behaviour that increases with BiFeO$_3$ thickness. The study of the frequency dependence of the resonance linewidth in a broad band of 3 to 35 GHz has allowed the determination of intrinsic and extrinsic contributions to the relaxation as function of BiFeO$_3$ thickness in perpendicular geometries. In our system the magnetic relaxation is dominated by the spin-pumping mechanism due to the presence of Pt. The insertion of BiFeO$_3$ between Pt and Ni$_{81}$Fe$_{19}$ attenuates the spin-pumping damping at one interface.
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