We conducted in-plane rf penetration depth measurements by using the tunnel diode oscillator technique. In the metal-free-all-organic superconductor β"-(ET) ₂SF₅CH₂ CF₂SO₃ the penetration depth measurements were carried out by changing the strength and direction of the magnetic field.

When the magnetic field is applied perpendicular to the conducting planes, the Shubnikov-de Haas quantum oscillations are observed revealing the geometry of the Fermi surface. When the magnetic field is applied parallel to the conducting planes, the low temperature upper critical fields are found to exceed the Pauli paramagnetic limiting field calculated by using a semi-empirical method. Furthermore, we found the possible existence of an additional superconducting state which stablizes above the Pauli limiting field when the superconducting order parameter lowers its free energy by becoming spatially inhomogeneous in a high magnetic field. The additional superconducting state is called the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state and was predicted independently by Fulde and Ferrell [10] and by Larkin and Ovchinnikov [11] in 1964. The signature of the FFLO state observed in this thesis occurs as a kink during the superconducting vortex state (SCVS)-to-FFLO phase transition.

In organic superconductors there have been several reports of the possible existence of the FFLO state [23, 24, 25, 62]. Even though these reports are all based on the fact that the upper critical field increases as the temperature decreases down to very low temperature, the obvious phase transition between the SCVS and the FFLO states has not been observed. To our best knowledge, the FFLO observation in β"-(ET)₂SF₅CH₂ CF₂SO₃ in this thesis is the first clear success which shows both the upper critical field enhancement at low temperature and the phase transition between the SCVS and the FFLO states.