Inhomogeneous superconducting state in $\beta ''$-(BEDT-TTF)$_2$SF$_5$CH$_2$CF$_2$SO$_3$

We present $^{13}$C nuclear magnetic resonance (NMR) measurements on the quasi-2D organic superconductor $\beta ''$-(BEDT-TTF)$_2$SF$_5$CH$_2$CF$_2$SO$_3$, at ultra-low temperature ($T\sim$100mK). For a magnetic field applied precisely within the conducting layers, the field evolution of the NMR spectrum reveals a phase transition within the superconducting state ($H_{c2}>$12T) near to the Pauli limit $H_P\sim$9.5T. The transition is identified by the significant line-broadening of the spectrum, associated with the electronic spin polarization distribution due to the emergence of spatially inhomogeneous superconductivity, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The character of this novel SC state is studied via static and dynamic (i.e. spin-lattice relaxation) NMR measurements. Moreover, the stability of the putative FFLO phase upon rotation of the field away from the in-plane condition is investigated.