Electron transport in disordered graphene

We study the electron transport properties of a monoatomic graphite layer (graphene) with different types of disorder. We show that the transport properties of the system depend strongly on the character of disorder. Away from half filling, the concentration dependence of conductivity is linear in the case of strong scatterers, in line with recent experimental observations, and logarithmic for weak scatterers. At half filling the conductivity is of the order of $e^2∕h$ if the randomness preserves one of the chiral symmetries of the clean Hamiltonian, whereas for generic disorder the conductivity is strongly affected by localization effects.

Figure 1

(a) Honeycomb lattice of the carbon atoms of graphene. Solid and open circles denote the atoms of $A$ and $B$ sublattices, respectively. (b) The first Brillouin zone of graphene. The nodal points of the spectrum are located in the corners of the zone. The two nonequivalent nodal points are denoted as $K$ and $K^{\prime }$.

Figure 2

Graphical representation of the $T$ matrix describing the electron scattering off an impurity.

Figure 3

Diagrams for the Drude conductivity including the vertex correction.

Figure 4

Diagrams yielding logarithmic corrections to the conductivity not included in the SCBA.

Figure 5

One-loop RG diagrams responsible for the renormalization of (a) the energy and (b), (c), (d) the disorder couplings.

Figure 6

“Phase diagram” in the $\epsilon$-$\eta$ plane for a fixed $\alpha$ (upper panel) and in the $\epsilon$-$\alpha$ plane for a fixed $\eta$ (lower panel). The solid line is the “phase boundary,” Eq. (66) or (60), where the crossover between the Born and the unitary regimes takes place. At low energies (dashed part) the density of states saturates, while the Drude conductivity reaches a value $\sim e^2∕h$, implying that generically the localization effects should become strong.

Figure 7

Frequency dependence of conductivity in a system with chiral disorder at half filling. At intermediate frequency $\delta \ll \omega \ll \Gamma$, the conductivity acquires some universal value ${\sigma }_{\omega }$ of the order of $e^2∕h$ which is not known analytically. This value depends on the type of chirality and the symmetry class of the system away from the degeneracy point.

Figure 8

Diagrams for (a) the first-order correction to the density of states and (b) the second-order correction to the partition function.