Lévy Flights due to Anisotropic Disorder in Graphene

We study transport properties of graphene with anisotropically distributed on-site impurities (adatoms) that are randomly placed on every third line drawn along carbon bonds. We show that stripe states characterized by strongly suppressed backscattering are formed in this model in the direction of the lines. The system reveals Lévy-flight transport in the stripe direction such that the corresponding conductivity increases as the square root of the system length. Thus, adding this type of disorder to clean graphene near the Dirac point strongly enhances the conductivity, which is in stark contrast with a fully random distribution of on-site impurities, which leads to Anderson localization. The effect is demonstrated both by numerical simulations using the Kwant code and by an analytical theory based on the self-consistent $T$-matrix approximation.

Figure 1

The anisotropic disorder model is obtained by randomly placing adatoms on sites indicated by gray horizontal stripes, with equal probability for sites on sublattices $A$ and $B$ (filled and empty red circles, respectively).

Figure 2

Conductivity of graphene with adatom disorder, as evaluated numerically using the Kwant package [22], plotted vs the system size $L=L_{\parallel }$. The impurity density is $n_{\text{imp}}=0.1/a^2$. Filled symbols correspond to single-color disorder. The conductivity along the stripe direction ${\sigma }_{\parallel }={\sigma }_{xx}$ shows a $\sqrt{L}$ increase, in agreement with the prediction of the Drude theory of Eqs. (3)–(5) (solid lines), both for comparatively weak (top panel) and strong (bottom panel) impurities. Insets display the conductivity in the transverse direction ${\sigma }_{\perp }={\sigma }_{yy}$, which shows a saturation, also in agreement with the Drude theory. The saturation value in the case of strong disorder is, however, much lower than predicted by the SCTMA. For comparison, the corresponding data for impurities distributed randomly over sites of all colors are shown by empty triangles. These data show the conventional behavior—diffusion (top panel) and strong Anderson localization (bottom panel).

Figure 3

Conductivity along the stripe direction ${\sigma }_{\parallel }$, calculated using Kwant [22] for graphene with single-color impurities at $ϵ=0.3t$ (triangles) and $ϵ=0.01t$ (circles) as a function of impurity concentration. The impurity strength is $V_0=10t$; the system size is fixed to $L=999.5a$. The maximal impurity concentration $n_{\text{imp}}^{\max }\simeq 0.26/a^2$ corresponds to impurities occupying all allowed sites. The solid lines show the results according to Eqs. (3)–(5) with the fitting parameter $\kappa$ as given in Table 1 and $\mathrm{\Gamma }$, $\zeta$ as calculated in the Supplemental Material [24].