School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom.
Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1, 70569, Stuttgart, Germany and L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia.
Phys Rev Lett. 2014 Jan 17;112(2):026802. doi: 10.1103/PhysRevLett.112.026802. Epub 2014 Jan 14.
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 σ is equal to the ballistic value 4e2/πh 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 σ≃4×0.4e2/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.
混合谷的 adatoms 和不混合它们的“标量”杂质。我们采用狄拉克方程的散射理论来表示石墨烯样品的电导作为杂质坐标的函数；然后通过数值方法对杂质位置进行平均。只要磁通量子数远远超过杂质数，对于每个无序实现，电导率 σ 等于弹道值 4e2/πh。对于较弱的磁场，在标量杂质存在下的电导率在半填充时按量子霍尔临界点缩放，σ≃4×0.4e2/h，在非半填充时由于安德森局域化的出现而变为零。对于 adatoms，由于谷间散射导致临界能分裂，在半填充时也会出现局域化行为。我们的结果揭示了由不同对称类的固定点控制的复杂标度流：值得注意的是，所有二维安德森局域化和临界现象的关键表现都在单个设置中通过数值方法观察到。
