Ferromagnetic insulators can induce a strong exchange field in an adjacent superconductor via the magnetic proximity effect. This manifests as spin-splitting of the BCS density of states of the superconductor, an important ingredient for numerous superconducting spintronics applications and the realization of Majorana fermions. A crucial parameter that determines the magnitude of the induced spin-splitting in FI/S bilayers is the thickness of the S layer $d$: In very thin samples, the superconductivity is suppressed by the strong magnetism. By contrast, in very thick samples, the spin splitting is absent at distances away from the interface. In this work, we calculate the density of states and critical field of FI/S bilayers of arbitrary thickness. From here, we determine the range of parameters of interest for applications, where the exchange field and superconductivity coexist. We compare our theory with the tunneling spectroscopy measurements in several EuS/Al/AlO$_x$/Al samples. If the Al film in contact with the EuS is thinner than a certain critical value, we do not observe superconductivity, whereas, in thicker samples, we find evidence of a first-order phase transition induced by an external field. The complete transition is preceded by a regime in which normal and superconducting regions coexist. We attribute this mixed phase to inhomogeneities of the Al film thickness and the presence of superparamagnetic grains at the EuS/Al interface with different switching fields. Our results demonstrate on the one hand, the important role of the S layer thickness, which is particularly relevant for the fabrication of high-quality samples suitable for applications. On the other hand, the agreement between theory and experiment demonstrates the accuracy of our theory, which, originally developed for homogeneous situations, is generalized to highly inhomogeneous systems.
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