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. By solving the Eilenberger equation in two dimensions, we calculate the pair potential, pairing correlations, free-energy density, and quasiparticle density of states for various cluster sizes and exchange potentials. Increasing the exchange potential and cluster size leads to a higher number of nodes in the pair potential. Although the free-energy density beneath the ferromagnet locally exceeds the normal-state value, the FFLO states are stabilized by the superconducting condensate away from the magnetic cluster. Analyzing the pairing-correlation functions, we show that the spatial variation of the spin-singlet s-wave pair potential generates p-wave Cooper pairs. These odd-frequency Cooper pairs play a dominant role in governing the inhomogeneous subgap spectra observed in the local density of states. Furthermore, we propose an experimental method for the detection of local FFLO states by analyzing the quasiparticle density of states.