Peierls-Type Instability and Tunable Band Gap in Functionalized Graphene

Functionalizing graphene was recently shown to have a dramatic effect on the electronic properties of this material. Here we investigate spatial ordering of adatoms driven by the RKKY-type interactions. In the ordered state, which arises via a Peierls-instability-type mechanism, the adatoms reside mainly on one of the two graphene sublattices. Bragg scattering of electron waves induced by sublattice symmetry breaking results in a band gap opening, whereby Dirac fermions acquire a finite mass. The band gap is found to be immune to the adatoms’ positional disorder, with only an exponentially small number of localized states residing in the gap. The gapped state is stabilized in a wide range of electron doping. Our findings show that controlled adsorption of adatoms or molecules provides a route to engineering a tunable band gap in graphene.

Figure 1

Peierls-type instability resulting from adatom ordering over sublattices $A$ and $B$. A gap in the density of electronic states opens up due to Bragg scattering on the $A/B$ modulation when the occupation probabilities are unequal, $n_A\ne n_B$. The states in the gap move down in energy into a peak positioned at the energy of a single adatom resonance. Inset: the ordered state is stabilized in a wide range of carrier densities, for which the energy gain per adatom is positive. For details of calculation see discussion following Eq. (8). Different curves correspond to the occupancy fraction $n_A/(n_A+n_B)=0,0.1,\dots ,0.5$ with $(n_A+n_B)/2=0.04$, $U=6t_0$, $W=3t_0$.

Figure 2

The density of electronic states as a function of energy for different sublattice occupancy ratios, obtained by numerical diagonalization of the Hamiltonian (2), averaged over 160 realizations of disorder. The band gap, which opens at $n_A\ll n_B$ and $n_B\ll n_A$, is immune to disorder: no electronic states are found inside the gap. Sublattice ordering results in the energy gain at positive dopings (inset). Parameters used: system size $62\times 34$, $U=6t_0$, $(n_A+n_B)/2=0.034$.

Figure 3

Two phases of adatoms on graphene: disordered (a), where adatoms are randomly distributed over two sublattices of the hexagonal lattice, $A$ and $B$, and ordered (b), where adatoms preferentially occupy one of the sublattices. The adatoms residing on sublattice $A$ ($B$) are shown in red [medium gray] (green [light gray]). Schematics of the energy spectrum in the two phases, are also shown. In the ordered phase, the sublattice symmetry breaking leads to a band gap.