Revealing Hofstadter spectrum for graphene in a periodic potential

Abstract

We calculate the energy bands for graphene monolayers when electrons move through a periodic electrostatic potential in the presence of a uniform perpendicular magnetic field. We clearly demonstrate the quantum fractal nature of the energy bands at reasonably low magnetic fields. We present results for the energy bands as functions of both wave number and magnetic flux through the unit cells of the resulting moiré superlattice. The effects due to pseudospin coupling and Landau orbit mixing by a strong scattering potential have been exhibited. At low magnetic fields when the Landau orbits are much larger than the period of the modulation, the Landau levels are only slightly broadened. This feature is also observed at extremely high magnetic fields. The density of states has been calculated and shows a remarkable self-similarity like the energy bands. We estimate that for modulation period of 10 nm the region where the Hofstadter butterfly is revealed is B 2 T.

Effects of site asymmetry and valley mixing on Hofstadter-type spectra of bilayer graphene in a square-scatter array potential

“…II we present theoretical formalism and acquire a set of characteristic equations, describing electron energy spectrum and corresponding eigenstates for BLG in the presence of both a perpendicular quantizing mag-netic field and a two-dimensional periodic electrostatic modulation potential. These results expand the previously studies for a two-dimensional electron gas 38 and for a monolayer graphene 39,40 . In Sec.…”

“…However, the mirror symmetry with respect to the band center is lost in Fig. 2(b) for monolayer graphene, as discussed in details recently by us 40 .…”

Moiré superlattice effects in graphene/boron‐nitride van der Waals heterostructures

“…Also, transport measurements , and capacitance measurements have revealed the lifting of spin and valley degeneracies of the magnetic minibands as a result of the electron‐electron interaction. There have also been several theoretical works on the magnetic bands of graphene/hBN heterostructure , in addition to a plethora of works on the magnetic bands in the related twisted bilayer system .…”

Ultrafast carrier dynamics in Landau-quantized graphene

“…(2) which in this context is called Peierls substitution [65,77,[86][87][88]. The limit of weak magnetic fields exhibits a negligible Landau level splitting due to the crystal lattice [89], which is interpreted as the cyclotron motion of the electron not being considerably disturbed by the underlying lattice. …”