| Summary: | In layered conductors with a sufficiently weak interlayer coupling in-plane magnetic field cause only small diamagnetic currents and the orbital depairing is strongly suppressed. Therefore, the Zeeman effect predominantly governs the spin-singlet superconductivity making the formation of the spatially modulated Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase possible in such materials. Despite decades of strenuous effort, this state still remains a profound mystery. In the last several years, however, there have been observed several hints indicating the experimental realization of the FFLO state in organic layered superconductors. The emergence of the FFLO phase has been demonstrated mainly based on thermodynamic quantities or microscopically with spin polarization distribution that exhibit anomalies within the superconducting state in the presence of the in-plane magnetic field. However, the direct observation of superconducting order parameter modulation is so far missing. Recently, there have been proposed theoretically several hallmark signatures for FFLO phase, which are a direct consequence of its main feature, the spatial modulation of the order parameter, and hence can provide incontrovertible evidence of FFLO. In this article, a review of these signatures and the underlying theoretical framework is given with the purpose to summarize the results obtained so far, omitting duplications, and to emphasize the ideas and physics behind them.
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