Oscillatory tunneling magnetoresistance in magnetic tunnel junctions with inserted nonmagnetic layer

Oscillatory tunneling magnetoresistance (TMR) as a function of spacer thickness is investigated theoretically for a magnetic tunnel junction with a nonmagnetic layer inserted between the tunnel barrier and the ferromagnetic layer. TMRis characterized in an analytical form, that is expressed with the transmission and reflection amplitudes of single interfaces at the Fermi level, and by the extremal wave vectors. Electronic structures with multiple bands are taken into account in the derivation characterizing the TMR, and the proposed analytical expression can be directly applied to real junctions. Based on our model, the features of TMR dependence on spacer thickness are discussed, including selection rules for the oscillation period. Numerical calculations are performed using an envelope-function theory for several cases, and we show that our model is in good agreement with the exact result.