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We study the thermodynamic properties of a dirty ferromagneticS$\mid$F$\mid$S Josephson junction with s-wave superconducting leads in thelow-temperature regime. We employ a full numerical solution with a set ofrealistic parameters and boundary conditions, considering both a uniform andnon-uniform exchange field in the form of a Bloch domain wall ferromagneticlayer. The influence of spin-active interfaces is incorporated via amicroscopic approach. We mainly focus on how the electronic specific heat anddensity of states (DOS) of such a system is affected by the \textit{proximityeffect}, which may be tuned via the superconducting phase difference. Our mainresult is that it is possible to \textit{strongly modify the electronicspecific heat} of the system by changing the phase difference between the twosuperconducting leads from 0 up to nearly $\pi$ at low temperatures. Anenhancement of the specific heat will occur for small values $h\simeq\Delta$ ofthe exchange field, while for large values of $h$ the specific heat issuppressed by increasing the phase difference between the superconductingleads. These results are all explained in terms of the proximity-altered DOS inthe ferromagnetic region, and we discuss possible methods for experimentaldetection of the predicted effect.

Phase-controlled proximity-effect in ferromagnetic Josephson junctions: calculation of DOS and electronic specific heat