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We study the proximity-induced superconducting state in a general bilayerâconventional s-wave superconductor hybrid structure. For the bilayer we include a general parabolic dispersion, Rashba spin-orbit coupling, and finite-layer tunneling as well as the possibility to apply a bias potential and a magnetic Zeeman field, in order to address experimentally relevant bilayer systems, ranging from topological insulator thin films to generic double quantum well systems. By extracting the proximity-induced anomalous Green's function in the bilayer, we show a very rich structure for the superconducting pairing, including different spin states and odd-frequency pairing. Equal-spin spin-triplet (pxÂ±ipy)-wave pairing is induced in both layers in the presence of a finite spin-orbit coupling and opposite-spin spin-triplet s-wave pairing with odd-frequency dependence appears for an applied magnetic Zeeman field. Finite interlayer pairing is also generally present in the bilayer. The interlayer pairing can be either even or odd in the layer index, with a complete reciprocity between parity in frequency and in layer index. We also find that a bilayer offers the possibility of sign reversal of the superconducting order parameters, both between the two layers and between multiple Fermi surfaces.
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