Fractal butterflies of Dirac fermions in monolayer and bilayer graphene

Bloch electrons in a perpendicular magnetic field exhibit unusual dynamics that has been studied for more than half a century. The single-electron energy spectrum of this system, the Hofstadter butterfly has been the subject of theoretical and experimental investigations for the past two decades. Experimental observation of these unusual spectra in semiconductor nanostructures, however, met with only limited success. The fractal nature of the butterfly spectrum was finally observed in 2013, thanks to the unique electronic properties of graphene. Here, the authors present an overview of the theoretical understanding of Hofstadter butterflies in monolayer and bilayer graphene. First, they briefly discuss the energy spectra in conventional semiconductor systems. The electronic properties of monolayer and bilayer graphene are then presented. Theoretical background on the Moire pattern in graphene and its application in the magnetoconductance probe that resulted in graphene butterflies are explained. They have also touched upon the important role of electron-electron interaction in the butterfly pattern in graphene. Experimental efforts to investigate this aspect of fractal butterflies have just begun. They conclude by discussing the future prospects of butterfly search, especially for interacting Dirac fermions in graphene.