  Spin-split superconductors offer new functionality compared to conventional
superconductors such as diode-effects and efficient thermoelectricity. The
superconducting state can nevertheless only withstand a small amount of
spin-splitting. Here, we self-consistently determine the spin transport
properties and the phase diagram of a spin-split superconductor in the presence
of an injected spin accumulation. Energy and spin relaxation are accounted for
in the relaxation time approximation via a single effective inelastic
scattering parameter. We find that the spin-splitting field in the
superconductor enables a spin diode effect. Moreover, we consider the
superconducting phase diagram of a system in contact with a spin accumulation
and in the presence of spin relaxation, and find that the inclusion of energy
and spin relaxation alters the phase diagram qualitatively. In particular,
these mechanisms turn out to induce a superconducting state in large parts of
the phase diagram where a normal state would otherwise be the ground-state. We
identify an FFLO-like state even in the presence of impurity scattering which
can be controllably tuned on and off via the electrically induced spin
accumulation. We explain the underlying physics from how the superconducting
order parameter depends on the non-equilibrium modes in the system as well as
the behavior of these modes in the presence of energy and spin relaxation when
a spin-splitting field is present.
