Spin-polarized resonant tunneling and quantum-size effect in ferromagnetic tunnel junctions with double barriers subjected to an electric field

Based on the two-band model, we present a transfer-matrix treatment of tunnel magnetoresistance (TMR) for tunneling through double ferromagnetic tunnel junctions with any magnetization angle of the middle ferromagnetic layer subjected to an electric field. Our results show that the TMR changes non-monotonically with the magnetization angle of the middle ferromagnetic layer in double junctions, which is different from that in a single junction. Owing to the comprehensive role of the quantum-size effect and the spin-polarized resonant tunneling, the TMR oscillates with the thickness of the middle ferromagnetic layer and can reach very large values under suitable conditions. Furthermore, the quantum-size effect can also give rise to a positive TMR (inverse spin-valve effect).