Evolution of the FFLO state of CeCo(In$_{1-x}$Hg$_{x})_{5}$

Specific heat C(T) of the unconventional superconductor CeCoIn$_{5 }$displays an anomaly within the superconducting (SC) state, indicating an additional SC phase in the high field and low temperature corner of the SC state of the H-T phase diagram. This SC state was previously proposed to be an inhomogeneous superconducting FFLO state. We have performed specific heat measurements of very low Hg-doped samples with x=0.001, 0.002 and 0.003, in order to study the evolution of the proposed FFLO state in CeCoIn$_{5}$. In a sample with x=0.001, C(T) still shows the additional transition inside the SC state, but it is broadened. The transition temperature is slightly higher than that of pure compound. C(T) of a higher Hg-doped sample with x=0.003 displays a very shallow and broad hump around the same temperature. Given that only 20{\%} of the nominal Hg concentration goes into the sample, the FFLO state appears to be extremely sensitive to the presence of impurities.