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We present a detailed investigation of the specific heat inCa$_3$(Ru$_{1-x}M_x$)$_2$O$_7$ ($M$ = Ti, Fe, Mn) single crystals. Withdifferent dopants and doping levels, three distinct regions are present,including a quasi-2D metallic state with an antiferromagnetic (AFM) orderformed by ferromagnetic bilayers (AFM-$b$), a Mott insulating state with G-typeAFM order (G-AFM) and a localized state with a mixed AFM-b and G-AFM phase. Ourspecific heat data provide deep insights into the Mott transitions induced byTi and Mn dopings. We observed not only an anomalous large mass enhancement butalso an additional term in the specific heat i.e. $C\propto T^2$ in thelocalized region. The $C\propto T^2$ term is most likely due to thelong-wavelength excitations with both FM and AFM components. A decrease ofDebye temperature is observed in the G-type AFM region, indicating a latticesoftening associated with the Mott transition.

Electron mass enhancement and magnetic phase separation near the Mott transition in double layer ruthenates