Theoretical Details of Tunnel Magnetoresistance via inelastic hopping at regime gμ Bll k_B T ll eV

Detailed theoretical derivation is given for the tunnel magnetoresistance via phonon-assisted hopping through an impurity chain under small magnetic field and a large bias window. This derivation provides a rigorous basis for the physical picture of Pauli blockade switch proposed in our previous paper (arXiv:1404.0633). This picture captures the competition of external magnetic field and internal spin interactions in the tunnel barrier, and relies critically on the strong on-site Coulomb correlation at the impurities. The master equations are obtained by deriving the equations of motion for the Green functions at the impurity sites in a slave-boson representation, and utilizing the so-called Langreth theorem to finally express the spin-dependent density matrix in terms of the equilibrium distributions of the contact electrons and of the phonon reservoir.