Impurity State and Variable Range Hopping Conduction in Graphene

The variable range hopping theory, as formulated for exponentially localized impurity states, does not necessarily apply in the case of graphene with covalently attached impurities. We analyze the localization of impurity states in graphene using the nearest-neighbor, tight-binding model of an adatom-graphene system with Green’s function perturbation methods. The amplitude of the impurity state wave function is determined to decay as a power law with exponents depending on sublattice, direction, and the impurity species. We revisit the variable range hopping theory in view of this result and find that the conductivity depends as a power law of the temperature with an exponent related to the localization of the wave function. We show that this temperature dependence is in agreement with available experimental results.

Figure 1

Amplitude of the resonance state at different resonance energies. (a) ${ϵ}_r=t/300$. The amplitudes on the $A$ sublattices vanish. (b) ${ϵ}_r=t/6$. The insets show the amplitudes represented by the radii of circles on the graphene honeycomb lattice. In both plots a circle symbol represents an amplitude on a $B$ site, which is in the direction indicated by a line with the same color in the insets. A square symbol represents an amplitude on an $A$ site, which has a circle drawn with the same color in the inset. The center dot (blue) is where the adatom is attached. $a$ is the nearest neighbor carbon-carbon distance.

Figure 2

The characteristic decay exponents of the two sublattice sites vs the resonance energy. For the $A$ sites, the exponent is taken from the sites that forms a triangular lattice with the impurity site, similar to the top line in Fig. 1b. For the $B$ sites, the exponent is taken from sites in the armchair direction, similar to the green line in Fig. 1.