Stacking-dependent magnetoelectronic properties in multilayer graphene

Magneto-electronic properties of multilayer graphenes

“…[12][13][14][15] Electronic states in a uniform perpendicular magnetic field B = B 0 ẑ are evolved to quantized Landau levels (LLs). [65][66][67][68][69][70][71][72][73][74][75][76] It should be noted that such LLs in AA-stacked graphenes are separated from each other by tens of meV, 33,66 while such LLs in AB-and ABC-stacked graphenes are confined in a narrow energy range of 10 meV around E F = 0; 67,69,73,75,76,84 their energy differences can be verified by quantum transport experiments. 4,138,139 On the other hand, the massive fermions in bilayer graphene give rise to quantized LL energies ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn À 1Þ p B 0 .…”

“…[35][36][37][38][39] In ABC systems, there is always one pair of weakly dispersive bands with surface-localized states near the Fermi level, 39,41,[43][44][45] and peculiar sombrero-shaped dispersion bands near the energy of the vertical nearestneighboring interlayer interaction. 33,34,37,39,49,50,53,56,[63][64][65][66][67][68][69][70][71][72][73][74][75][76]81,84,[107][108][109] In the presence of a uniform perpendicular electric field, the Dirac cones in AA-stacked graphenes are preserved, 33,34,65 while only a rigid shift from E F = 0 takes place. The above-mentioned stackingdependent electronic structures have been experimentally verified by infrared optical spectroscopy [101][102][103][104][105] and angle-resolved photoemission spectroscopy.…”

“…67,143,175 The field-independent LL at E F = 0 for n = 0 confirms that the Dirac points are hole-electron states in charge neutral graphene (available to either holes or electrons). 212 Previous studies based on the comprehensive tight-binding model 75,84 showed that the Landau quantization of the sombrero-shaped subbands in ABC-stacked graphenes leads to the peculiar LL evolution under a variation of the magnetic field that reveals inversely fielddependent energies and a complex pattern of intragroup LL anticrossings, especially in the region of weak fields. Based on the zero-field band structures, the LLs can be classified into several groups, with each being responsible for the magnetic quantization of the respective subband.…”

“…[170][171][172][173][174][175] The orbital Landau quantization of massless Dirac fermions in the monolayer [170][171][172][173][174] and massive Dirac fermions in the AB-stacked bilayer 175 are, respectively, verified by the sequence of peaks with square-root and linear dependencies on B 0 . 177,178 Owing to the LL anticrossings and the symmetry broken down in multilayer graphenes, 38,69,75,[77][78][79][80][81][82][83][84]140 it is interesting to measure the behavior of LLs and wave functions. On the other hand, the spectroscopic-imaging STM has been used to obtain the local DOS (LDOS) of LLs, such as in graphene, 176 2D electron gas 177 and topological insulator, 178 revealing the spatial distributions of the wave functions which consist of n-dependent nodal structures.…”

Introduction

“…Under a magnetic field, the intergroup and intragroup excitation channels among wellbehaved LLs comprise the main part of the absorption spectrum [146,[174][175][176], in which the absorption peaks show a twin-peak structure because the LL energy spectrum is asymmetric about = E 0 F [175,176]. Moreover, it is interesting to study the quantization effects of any two constant energy loops in the sombrero-shaped structure, where the quantized energies of the LLs are proportional or inversely proportional to the magnetic field strength [126,146,147]. This leads to many anticrossings of intragroup LLs in the ABC stacking configuration.…”