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Fig. 1. Amplitude of the resonance state at different resonance energies. top: εr=t/300. The amplitudes on the A sublattices vanish. bottom: εr=t/6. The amplitudes are represented by the radii of circles on the graphene honeycomb lattice. The center dot (blue) is where the adatom is attached.

Impurities cause electrons to localize when they are introduced into a perfectly periodic system. For semiconductors, impurity (dopant) atoms form states in the band gap that are exponentially localized and they drastically change the conduction mechanism in these materials, which is described by the variable range hopping (VRH) theory. This theory captures the two most important aspects of hopping conduction among these localized states: the overlap of the wavefunctions and the energy difference between the states. It concluded that because of the existence of the impurities, the conductivity depends on temperature with an exponential relationship
lnσ ∝ (T0/T)-1/(d+1).

In the case of graphene, due to the lack of a band gap, the impurity states are resonance states in the band continuum, which changes their localization properties. By studying the tight-binding model of graphene with nearest neighbor interactions, we were able to determine that the decay of the wavefunction of the impurity state primarily follows power laws (see Fig. 1), although the exponent depends on details like direction or sublattice. This means the original VRH theory cannot be applied to graphene directly. Using power-law wavefunctions for the overlap of the states, we obtained that the conductivity should depend on temperature also as a power law. Existing experimental data agree with our version of the VRH equation.

Publications

2012 · All
S. Liang and J. O. Sofo, "The impurity state and variable range hopping conduction in graphene," Phys. Rev. Lett. 109, 256601 (2012)
J. O. Sofo : The impurity state and variable range hopping conduction in graphene