Universal Scaling of the Magnetocaloric Effect in 2D Ising Monolayers and Bilayer

We report a Monte Carlo study of the magnetocaloric effect (MCE) in two-dimensional ferromagnetic Ising models on square, honeycomb, and triangular lattices with monolayer and bilayer configurations. Using Binder cumulant analysis, we determine the critical temperature ($T_c$) of each structure and find that $T_c$ increases with coordination number, from the honeycomb ($r=3$) to the triangular ($r=6$) lattice. In contrast, the magnetic entropy change ($-\Delta S_M$) decreases with coordination number, reaching its maximum for the honeycomb lattice. After normalization by their peak values and appropriate temperature scaling, both $-\Delta S_M$ and the field exponent $n$ collapse onto universal master curves for different magnetic fields and across all six lattice structures at a fixed low field. This demonstrates universal MCE scaling independent of coordination number and layer count. Critical scaling analysis further supports the observed universality and power-law behavior. Unlike $-\Delta S_M$, the adiabatic temperature change ($\Delta T_{ad}$) increases with coordination number, whereas the magnetic Gr\"{u}neisen parameter ($\Gamma_M$) follows the same trend as $-\Delta S_M$. Although the peak value of $-\Delta S_M$ decreases with coordination number, the relative cooling power and cooling capacity remain nearly unchanged due to compensating broadening of the $-\Delta S_M$ curves. The field dependence of $-\Delta S_M$, relative cooling power, and cooling capacity follows power laws up to $\sim2.6$ T (assuming $J\approx1$ meV). Hysteresis analysis shows that lattices with lower coordination numbers exhibit a faster reduction in loop width with increasing temperature. These results establish the universal scaling behavior of the magnetocaloric effect in two-dimensional monolayer and bilayer magnetic lattices and provide guidelines for designing magnetic refrigerants.