We present detailed studies of the high-field magnetoresistance of the layered organic metal $kappa$-(BETS)$_2$-Mn-[N(CN)$_2$]$_3$ under a pressure slightly above the insulator-metal transition. The experimental data are analysed in terms of the Fermi surface properties and compared with the results of first-principles band structure calculations. The calculated size and shape of the inplane Fermi surface are in very good agreement with those derived from Shubnikov-de Haas oscillations as well as the classical angle-dependent magnetoresistance oscillations. A comparison of the experimentally obtained effective cyclotron masses with the calculated band masses reveals electron correlations significantly dependent on the electron momentum. The momentum- or band-dependent mobility is also reflected in the behavior of the classical magnetoresistance anisotropy in a magnetic field parallel to layers. Other characteristics of the conducting system related to interlayer charge transfer and scattering mechanisms are discussed based on the experimental data. Besides the known high-field effects associated with the Fermi surface geometry, new pronounced features have been found in the angle-dependent magnetoresistance, which might be caused by coupling of the metallic charge transport to a magnetic instability in proximity to the metal-insulator phase boundary.
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