Microscopic Eilenberger theory of Fulde-Ferrell-Larkin-Ovchinnikov states in the presence of vortices

We theoretically investigate the Fulde-Ferrell-Larkin-Ovchinnikov state by using the microscopic quasiclassical Eilenberger equation. The Pauli paramagnetic effects and the orbital depairing effects due to vortices are treated on an equal footing for three-dimensional spherical Fermi surface model and s-wave pairing. The field evolution of the Larkin-Ovchinnikov (LO) state is studied in detail, such as the H-T phase diagram, spatial structures of the order parameter, the paramagnetic moment, and the internal field. Field dependences of various thermodynamic quantities-the paramagnetic moment, entropy, and the zero-energy density of states-are calculated. Those quantities are shown to start quickly growing upon entering the LO state. We also evaluate the wavelength of the LO modulation, the flux line lattice form factors for small-angle neutron scattering, and the NMR spectra to facilitate the identification of the LO state. Two cases of strong and intermediate Pauli paramagnetic effect are studied comparatively. The possibility of the LO phase in Sr2RuO4, CeCoIn5, CeCu2Si2, and the organic superconductors is critically examined and crucial experiments to identify it are proposed.