  An extension of the standard spin diffusion theory is presented by
introducing a density-gradient (DG) term that is suitable for describing
interface quantum tunneling phenomena. The magnetoresistance (MR) ratio is
modified by the DG term through an interface electric field. We have also
carried out spin injection and detection measurements using four-terminal Si
devices. The local measurement shows that the MR ratio changes depending on the
current direction. We show that the change of the MR ratio depending on the
current direction comes from the DG term regarding the asymmetry of the two
interface electronic structures.
