  Topological insulator (TI) thin films differ from the more commonly studied
semi-infinite bulk TIs in that the former possesses both top and bottom
surfaces where the surface states localized at different surfaces can couple to
one another due to the finite thickness of the film. In the presence of an
in-plane mangnetization TI thin films display two distinct phases depending on
which of the inter-surface coupling or the magnetization is stronger. In this
work, we consider a TI thin film system with an in-plane magnetization and
calculate numerically the resulting spin accumulation on both surfaces of the
film due to an in-plane electric field to linear order. We describe a numerical
scheme for performing the Kubo calculation calculation in which we include
impurity scattering and vertex corrections. We find that the sums of the spin
accumulation over the two surfaces in the in-plane direction perpendicular to
the magnetization, and in the out of plane direction, are antisymmetric in
Fermi energy about the charge neutrality point and are non-vanishing only when
the symmetry between the top and bottom TI surfaces is broken. The impurity
scattering, in general, diminishes the magnitude of the spin accumulation and
can also change the sign of the spin accumulation at some Fermi energies where
the accumulation is small.
