  Although multilayer systems possess global inversion symmetry, some of the
layers lack local inversion symmetry because no global inversion centers are
present on such layers. Such locally non-centrosymmetric systems exhibit
spatially modulated Rashba spin-orbit coupling. In this study, the
superconductivity in multilayer models exhibiting inhomogeneous Rashba
spin-orbit coupling is investigated. We study the electronic structure,
superconducting gap, and spin susceptibility in the superconducting state with
mixed parity order parameters. We show the enhancement of the spin
susceptibility by Rashba spin-orbit coupling and interpret it on the basis of
the crossover from a centrosymmetric superconductor to a non-centrosymmetric
superconductor. It is also shown that the spin susceptibility is determined by
the phase difference of the order parameter between layers and is nearly
independent of the parity mixing of order parameters. An intuitive
understanding is given on the basis of the analytic expression of
superconducting order parameters in the band basis. The results indicate that
not only a broken global inversion symmetry but also a broken local inversion
symmetry leads to unique properties of superconductivity. We discuss the
superconductivity in artificial superlattices involving CeCoIn5 and multilayer
high-Tc cuprates.
