Anomalous magnetic field effects on graphene

Full text: Graphene exhibits anomalous properties in externally applied magnetic field. The orbital susceptibility of graphene has a singularity expressed as a delta function in Fermi energy EF, which diverges at Dirac point (EF =0) and vanishes otherwise. The singular diamagnetism is modified by various external factors such as the disorder potential [1], and the band gap opening [2,3], the finite-size effect [4] and the multilayer stacking [5], and studying those effects give deeper insights into the origin of the diamagnetic singularity of Dirac electron. The delta-function singularity is generally weakened by the electronic coupling between different graphene layers. In usual AB-stacked multilayer graphite, the interlayer coupling changes monlayer's linear band to quadratic, and then the susceptibility peak is broadened into a less singular logarithmic curve [5]. In a turbostratic (randomly-stacked) graphene multilayer, on the other hand, the diamagnetism generally becomes much stronger than in AB-stacked graphite, because the interlayer coupling is significantly reduced in misoriented lattice structure. There the external magnetic field is significantly screened inside the sample in low temperatures, and even a perfect screening is achieved at zero temperature in an ideal sample [4]. When the stacking angle between two graphene layers becomes as small as a few degree, the electronic structure is strongly modified by the long-period lattice structure with a Moire pattern. In increasing magnetic field, the spectrum gradually evolves into a fractal band structure called Hofstadter's butterfly, where the Hall conductivity exhibits a nonmonotonic behavior as a function of Fermi energy [6]. In finite-sized graphene system such as graphene ribbon and graphene nano-islands, the finite-size effect also changes the singular diamagnetism.[4] At T=0, the susceptibility χ(EF) oscillates between diamagnetism and paramagnetism in accordance with the subband structure formed by quantum confinement, while in increasing T, the oscillation rapidly disappears and χ(EF) approaches the bulk limit (thermally broadened delta-function) once the broadening energy kBT exceeds the subband spacing. The electric current supporting the diamagnetism spreads entirely in the sample at T=0 reflecting the absence of the characteristic length scale, while the current distribution becomes more and more localized to the edge as the temperature goes higher. In finite T, there is only length scale parameter, hv/(2 kBT), and it actually characterizes the edge current width from the boundary.(authors) References: [1] M. Koshino and T. Ando, Phys. Rev. B 75, 235333 (2007). [2] M. Koshino and T. Ando, Phys. Rev. B 81, 195431 (2010). [3] M. Koshino, Phys. Rev. B 84, 125427 (2011). [4] Y. Ominato and M. Koshino, Phys. Rev. B 85, 165454 (2012). [5] M. Koshino and T. Ando, Phys. Rev. B 76, 085425 (2007). [6] P. Moon and M. Koshino, Phys. Rev. B 85, 195458 (2012)