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Detailed theoretical derivation is given for the tunnel magnetoresistance viaphonon-assisted hopping through an impurity chain under small magnetic fieldand a large bias window. This derivation provides a rigorous basis for thephysical picture of Pauli blockade switch proposed in our previous paper(arXiv:1404.0633). This picture captures the competition of external magneticfield and internal spin interactions in the tunnel barrier, and reliescritically on the strong on-site Coulomb correlation at the impurities. Themaster equations are obtained by deriving the equations of motion for the Greenfunctions at the impurity sites in a slave-boson representation, and utilizingthe so-called Langreth theorem to finally express the spin-dependent densitymatrix in terms of the equilibrium distributions of the contact electrons andof the phonon reservoir.

Theoretical Details of Tunnel Magnetoresistance via inelastic hopping at regime $g\mu B\ll k_B T \ll eV$