Sharp Zero-Energy Landau Levels in Multilayer Graphene

Abstract

The existence of the zero-energy Landau levels and their stability against disorder are hallmarks of the quantum Hall effect in graphene [1,2]. Mathematically this phenomenon is related to the index theorem [3], while the stability of zero-energy Landau levels has been numerically investigated for monolayer graphene and its bilayer based on the honeycomb lattice model [4,5], where the chiral symmetry and the absence of the valley mixing are both shown to be essential for the sharp Landau level. In multilayer graphene, the different stacking yields different low-energy dispersions, where the number of zero-energy Landau levels increases with the number of layers. In the present paper, we consider a trilayer graphene to examine the stability of zero-energy Landau levels. Specifically, we investigate whether or not the ABA and ABC stackings exhibit different behavior in the sharpness of zero-energy Landau levels. There, the effect of the chiral-symmetry breaking interlayer couplings on the sharpness is examined. For the ABC stacking, the dispersion at low energies is proportional to k 3 , whereas it is a superposition of those of monolayer and bilayer graphene for the ABA stacking. In both cases, there exist three-fold degenerated zero-energy Landau levels (per spin and valley) in a magnetic field [6]. We examine the sensitivity of the zero-energy Landau levels to the spatial correlation length of the bond disorder, which respects the chiral symmetry. We shall find, in both cases, all the zero-energy Landau levels are anomalously sensitive to the disorder correlation length and become very sharp as soon as the correlation length exceeds only a few lattice constants. We further investigate the influence of the additional interlayer couplings on such a sharp zero-energy Landau level. The result indicates that, while the interlayer couplings that preserve the chiral symmetry retain the sharpness of the zero-energy Landau level, those that break the symmetry degrade the sharpness.