Quantum criticality near the upper critical field of Ce$_2$PdIn$_8$

We report low-temperature specific heat measurements in magnetic fields up to 12 T applied parallel and perpendicular to the tetragonal c-axis of the heavy fermion superconductor Ce$_2$PdIn$_8$. In contrast to its quasi-two-dimensional (2D) relative CeCoIn$_5$, the system displays an almost isotropic upper critical field. While there is no indication for a FFLO phase in Ce$_2$PdIn$_8$, the data suggest a smeared weak first-order superconducting transition close to $H_{c2}\approx 2$ T. The normal state electronic specific heat coefficient displays logarithmically divergent behavior, comparable to CeCoIn$_5$ and in agreement with 2D quantum criticality of spin-density-wave type.