The heavy-fermion superconductor CeCoIn5 is the first material, where different experimental probes show strong evidence pointing to the realization of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The inhomogeneous superconducting FFLO state with a periodically modulated order parameter was predicted to appear in Pauli-limited, sufficiently clean type-II superconductors already more than 40 years ago. On the other hand, CeCoIn5 is supposed to be close to a magnetic quantum critical point (QCP) showing strong antiferromagnetic (AFM) spin fluctuations (SF) at atmospheric pressure. We studied the evolution of the FFLO phase away from the influence of the strong AFM-SF by heat capacity experiments under pressure (0 GPa <= P <= 1.5 GPa, 0 T <= mu_0 H <= 14 T, and 100 mK <= T <= 4 K). Our results prove the stability of the the FFLO phase under pressure. It even expands, while the Pauli-limiting becomes weaker and the AFM-SF are suppressed. This shows the intriguing influence of the AFM-SF on the FFLO state.
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