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Electronic properties and quantum transports in functionalized graphene Sierpinski carpet fractals

Recent progress in controllable functionalization of graphene surfaces enables the experimental realization of complex functionalized graphene nanostructures, such as Sierpinski carpet (SC) fractals. Herein, we model the SC fractals formed by hydrogen and fluorine functionalized patterns on graphene surfaces, namely, H-SC and F-SC, respectively. We then reveal their electronic properties and quantum transport features. From calculated results of the total and local density of state, we find that states in H-SC and F-SC have two characteristics: (i) low-energy states inside about |E/t|<1 (with t as the near-neighbor hopping) are localized inside free graphene regions due to the insulating properties of functionalized graphene regions, and (ii) high-energy states in F-SC have two special energy ranges including -2.3<E/t<-1.9 with localized holes only inside free graphene areas and 3<E/t<3.7 with localized electrons only inside fluorinated graphene areas. The two characteristics are further verified by the real-space distributions of normalized probability density. We analyze the fractal dimension of their quantum conductance spectra and find that conductance fluctuations in these structures follow the Hausdorff dimension. We calculate their optical conductivity and find that several additional conductivity peaks appear in high energy ranges due to the adsorbed H or F atoms.