Fulde-Ferrell-Larkin-Ovchinnikov state in a superconducting thin film attached to a ferromagnetic cluster

We study theoretically the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states appearing locally in a superconducting thin film with a small circular magnetic cluster. The pair potential, the pairing correlations, the free-energy density, and the quasiparticle density of states are calculated for several cluster sizes and the exchange potentials by solving the Eilenberger equation in two dimensions. The number of nodes in the pair potential increases with increasing the exchange potential and cluster size. The local FFLO states are stabilized by the superconducting condensate away from the magnetic cluster even though the free-energy density beneath the ferromagnet exceeds locally the normal-state value. The analysis of the pairing-correlation functions shows that the spatial variation of the spin-singlet $s$-wave pair potential generates $p$-wave Cooper pairs, and that odd-frequency Cooper pairs govern the inhomogeneous subgap spectra in the local density of states. We also discuss a way of detecting the local FFLO states based on the calculated quasiparticle density of states.