Theory of Tunneling Magnetoresistance

Rigorous theory of the tunneling magnetoresistance (TMR) based on the real-space Kubo formula and fully realistic tight-binding bands fitted to an ab initio band structure is decribed. It is applied to calculate the TMR of two Co electrodes separated by a vacuum gap. The calculated TMR ratio reaches ≈ 65% in the tunneling regime but can be as high as 280% in the metallic regime when the vacuum gap is of the order of the Co interatomic distance (abrupt domain wall). It is also shown that the spin polarization P of the tunneling current is negative in the metallic regime but becomes positive P ≈ 35% in the tunneling regime. Using the nonequilibrium Keldysh formalism, the Kubo formula is generalized to a Co junction under a finite bias. The TMR of the Co junction calculated from the Keldysh formula decreases very rapidly with applied bias in good qualitative agreement with experiment. It is also demonstrated that the TMR calculated from the Kubo formula remains nonzero when one of the Co electrodes is covered with a copper layer. It is shown that nonzero TMR is due to quantum well states in the Cu layer which do not participate in transport. Since these only occur in the down-spin channel, their loss from transport creates a spin asymmetry of electrons tunneling from a Cu interlayer, i.e. nonzero TMR. Finally, it is shown that diffuse scattering at the ferromagnet/nonmagnet interface may cause quantum well states to evolve into propagating states, in which case the spin asymmetry of the nonmagnetic layer is lost and with it the TMR.