Electronic properties and quantum transport in functionalized graphene Sierpinski-carpet fractals

Abstract

Recent progress in the 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 the calculated results of the total and local densities of state, we find that states in H-SC and F-SC have two characteristics: (i) low-energy states inside about $|E/t|\ensuremath{\le}1$ (with $t$ as the nearest-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 $\ensuremath{-}2.3<E/t<\ensuremath{-}1.9$ 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.