\textbf{Band-gap opening properties of graphene binding with low-concentration fluorine }

To better understand the effects of low-level fluorine (F) in graphene-based sensors, the structure and impact of low-concentration of fluorine defects on the electrical properties of single- and multi-layer graphene films were investigated by density functional theory with van der Waals dispersion interactions. When F bonds to a carbon atom of graphene, the carbon atom is pulled slightly above the graphene plane creating what is referred to as a C$_{\mathrm{F}}$ defect, and a valence band (B$_{\mathrm{F}})$ near the Fermi level is formed mainly from the $p$ orbitals of the F atoms with some small contribution from the $p$ orbitals of the bonded carbon atoms. Depending on the F binding sites, the B$_{\mathrm{F}}$ can serve as a valence band or a conduction band and only few configurations of the F-binding graphene can open a band gap. Such results indicate that the band gap opening for graphene with low F-adsorption level strongly depends on the F-binding configurations, which is different from the fully or highly partial fluorinated graphene. At low F-adsorption level, the interaction between neighboring pairs of F adatoms is negligible and the most important interaction is between the F and carbon atoms in the C$_{\mathrm{F}}$ defect. Such results are useful for sensor and nano-electronics developments.