Dispersion of edge states and quantum confinement of electrons in graphene channels drawn on graphene fluoride

Graphene is an excellent conductor, while graphene fluoride is a wide band-gap semiconductor. We propose the formation of graphene channels embedded in graphene fluoride as a method to induce quantum confinement of charge carriers in graphene. In particular, we study the electronic structure of graphene channels drawn on the fluoride along two high-symmetry directions: the armchair and zigzag orientations. The zigzag channels are found to have dispersive one-dimensional edge bands, contrary to the case of ribbons and channels drawn on graphane, where the edge state is flat close to the Fermi level and has a very large effective mass. The effective mass of this one-dimensional edge state can be controlled by electrostatic interactions at the edge of the channel. This result indicates that the mobility of these channels can be controlled by a localized gate voltage. The armchair channel is found to be metallic or semiconducting depending on the width of the channel, in agreement with ribbons and hydrogen-limited channels.

Figure 1

Zigzag (top) and armchair (bottom) channels of graphene limited by a barrier region of graphene fluoride. The carbon network is represented in gray while the fluorine atoms are represented by light blue balls. For each geometry we indicate the convention to count the number of rows in the channel; this is generically called $N$ in the text. The number of rows in the barrier region is generically called $M$ and counted in the same way. For clarity, two unit cells are presented in each case.

Figure 2

Band structure on the left and partial density of states of the barrier and channel region of a zigzag structure zz(6,12). The tight-binding approximation of the edge state around the Fermi level is shown with a continuous line in red. The tight-binding model demonstrates that the dispersion relation of the edge state is controlled by the site energy of the carbon atoms at the edge.

Figure 3

Band gap of armchair channels as a function of the number of rows in the channel $N$, for a different number of rows in the barrier, $M$. The dashed lines correspond to the tight-binding approximation with hopping integral equal to 2.6 eV and a $9\%$ increase of the hopping at the edge.

Figure 4

Band structure and partial density of states in the barrier and channel region of two representative armchair channels. The top panel corresponds to a semiconducting channel, ac(5,13), and the bottom panel to a semimetallic channel, ac(6,14). From the band structure in the top panel, we see that the semiconducting armchair channel has a direct gap. From the density of states we see that the states close to the gap are localized within the channel because the gap in the channel is centered with respect to the gap in the barrier. For the band structure in the bottom panel we see that the semimetallic structures have a one-dimensional linear band, localized within the channel and producing a constant density of states.