Quantum Hall Criticality and Localization in Graphene with Short-Range Impurities at the Dirac Point

We explore the longitudinal conductivity of graphene at the Dirac point in a strong magnetic field with two types of short-range scatterers: adatoms that mix the valleys and “scalar” impurities that do not mix them. A scattering theory for the Dirac equation is employed to express the conductance of a graphene sample as a function of impurity coordinates; an averaging over impurity positions is then performed numerically. The conductivity $\sigma$ is equal to the ballistic value $4e^2/\pi h$ for each disorder realization, provided the number of flux quanta considerably exceeds the number of impurities. For weaker fields, the conductivity in the presence of scalar impurities scales to the quantum-Hall critical point with $\sigma \simeq 4\times 0.4e^2/h$ at half filling or to zero away from half filling due to the onset of Anderson localization. For adatoms, the localization behavior is also obtained at half filling due to splitting of the critical energy by intervalley scattering. Our results reveal a complex scaling flow governed by fixed points of different symmetry classes: remarkably, all key manifestations of Anderson localization and criticality in two dimensions are observed numerically in a single setup.

Figure 1

Dirac-point conductivity of graphene with (a) scalar impurities and (b) adatoms in zero magnetic field as a function of the system size $L$. Insets show the dependence of ${\sigma }_{xx}$ on the impurity strength ${\ell }_a$ and ${\ell }_s$ for a fixed system size $L=20{\ell }_{\mathrm{imp}}$. In the upper insets all impurities have the same sign while in the lower insets the sign of the impurity potential is random. The sketch in (a) shows a typical path contributing to the conductance correction; this closed path connects the left with the right lead along impurity sites () which are characterized by their individual $T$ matrices.

Figure 2

Zero-energy conductivity ${\sigma }_{xx}$ of graphene with scalar impurities (a),(b) and adatoms (c),(d). Left panels: evolution of ${\sigma }_{xx}$ with length $L$ for different values of the ratio ${\ell }_B/{\ell }_{\mathrm{imp}}$ of the magnetic length to the distance between impurities. The symmetry breaking pattern is $\mathrm{DIII}\to \mathrm{AII}\to \mathrm{A}$ for weaker $B$ and $\mathrm{DIII}\to \mathrm{AIII}\to \mathrm{A}$ for stronger $B$ in panel (a) and $\mathrm{BDI}\to \mathrm{AI}\to \mathrm{A}$ for weaker $B$ and $\mathrm{BDI}\to \mathrm{AIII}\to \mathrm{A}$ for stronger $B$ in panel (c). Scaling flow towards the QH critical point in (a) and localization in (a),(c) is shown in the insets by rescaling of the curves to a length ${\xi }_s$ (respectively, ${\xi }_a$). For the considered range of parameters, the obtained values of ${\xi }_s$ and ${\xi }_a$ are well approximated by phenomenological expressions ${\xi }_s=1.17{\ell }_B(1+0.126{\ell }_{\mathrm{imp}}/{\ell }_B)$ and ${\xi }_a={\ell }_{\mathrm{imp}}+8.85\exp (-1.73{\ell }_B/{\ell }_{\mathrm{imp}})$. For impurities of the same sign, an additional rescaling of ${\sigma }_{xx}$ reflects the two-parameter scaling in magnetic field for $\theta \ne 0$, $\pi$. Right panels: ${\sigma }_{xx}$ as a function of ${\ell }_B/{\ell }_{\mathrm{imp}}$ for fixed large $L/{\ell }_B$. For moderately strong $B$ the system is either at QH criticality or gets localized. For stronger $B$, a quasiballistic transport regime with $\sigma =4e^2/\pi h$ emerges. Vertical bars show mesoscopic fluctuations.