Entropy Redistribution Induced by Substitutional Doping in Bilayer Graphene

This study investigates the thermodynamic consequences of substitutional doping in bilayer graphene using a minimal tight-binding model in which doping is encoded as a reduction in the intralayer hopping amplitude α in one sheet. The interlayer coupling Δ fixes the low-energy window, while 0<α<1 introduces spectral asymmetry without generating new energy scales. The results show that decreasing α redistributes the density of states toward the Fermi level, producing an enhancement of the low-energy density of states within the hybridized inner branches. As a consequence, the total electronic entropy vanishes in the limit of zero temperature (T→0) and increases smoothly with temperature for all α, consistent with the third law of thermodynamics. Layer-resolved analysis reveals that the doped sheet acquires a larger electronic entropy than the pristine one for 0<α<1, giving rise to a finite entropic polarization. The maximum polarization follows a linear scaling, demonstrating that the entropy imbalance is continuously controlled by the hopping asymmetry and does not involve critical behavior. These results establish a direct connection between doping-induced spectral redistribution and thermodynamic layer polarization in bilayer graphene.