I have been using sfepy to simulate charge currents with Poisson's equation and I would like to add in spin dependence using Valet-Fert theory.

The strong form of the continuity equations are

charge: \Delta(\sigma_{up} \mu{up} + \sigma_{down} \mu{down} ) = 0

spin: \Delta(\mu{up} -  \mu{down} ) = (\mu{up} -  \mu{down}) / \lambda

where \mu_{up} and \mu_{down} are the chemical potentials for spin up and spin down electrons, \sigma_{up} and \sigma_{down} are the spin dependent conductivities and \lambda is a material constant.

The problem I am looking at is a charge current being driven from a magnetic metal to a non-magnetic metal. My issue is with the Neumann BCs, the spin current is defined as Js = \sigma_{up} grad \mu_{up} - \sigma_{down} grad \mu_{down} which has to be continuous across the interface. I do not know how to implement this in sfepy as I do not know what the fluxes at the interface will be before running the simulation in order to add them as material parameters as in the examples.

Should I define the continuity of spin fluxes at the interface as a separate equation? is this something achievable in sfepy?

Many thanks,

Joe