Fractal Spectrum of Magnetic Minibands in Graphene-hBN Heterostructures

Abstract

The fractal spectrum of electrons in a crystal subjected to a strong magnetic field is one of the most spectacular results in the quantum theory of solids. The electronic band structure is fractured into multiple magnetic bands, for each magnetic field providing a rational fraction of magnetic flux quanta, $\phi /\phi _0=p/q$, per unit cell. Since the sparsity of the spectrum increases for larger values of the denominator $q$, the observation of fractal magnetic bands at sustainable magnetic fields requires a system with a long period superlattice, such as the graphene/boron-nitride heterostructure. Here, we find that further generations of weakly gapped Dirac electrons systematically reappear at the edges of the magnetic minibands at rational flux values. Moreover, the fractal spectra for neighbouring values of flux have the form of the Landau levels associated with these gapped Dirac electrons, which determines a specific hierarchy of gaps in the surrounding fractal spectra.