Electronic Particle-Density Asymmetry in Doped Bilayer Graphene with Effective Fractional Dimensionality

This work investigates how effective fractional dimensionality and substitutional doping jointly affect particle-density symmetry in AA-stacked bilayer graphene (BLG). The system consists of one pristine graphene layer and one doped layer, where 50% of the atomic sites are substituted. Doping is described within a tight-binding framework through a dimensionless parameter α (with 0<α<1), which uniformly reduces the intralayer hopping energy in the doped layer, while an effective fractional dimension D>2 phenomenologically accounts for structural inhomogeneities such as ripples or corrugations. Analytical expressions for the energy spectrum and number of states are obtained, and the electronic asymmetry between layers is characterized by a symmetry parameter P. We find that substitutional doping breaks the particle-density symmetry of pristine BLG and strongly enhances the number of states near the Fermi level as α decreases, due to the accumulation of low-energy states in the doped layer. Departures from the ideal dimensionality D=2 further amplify this enhancement. The combined effects of reduced α and increased D lead to a pronounced layer asymmetry, reflected in a nonzero P, while stronger interlayer coupling partially counteracts this imbalance. Results show that doping, effective fractional dimensionality, and interlayer coupling offer tunable control over the low-energy electronic properties of BLG.