Brown-Zak oscillations and second-order magnetic Bloch states in graphene/hBN superlattices*

We report on a novel type of magneto-quantum oscillation which was recently observed and studied in graphene/hexagonal-boron nitride superlattices [1]. Surprisingly, the oscillations only become clearly noticeable above liquid helium temperatures (100 K), after Shubnikov-de Haas oscillations and the associated spectral gaps become smeared. These so-called BZ oscillations are extremely robust with respect to temperature and were found to exist even at 373 K (100^oC) in a relatively weak magnetic field (B = 4 T). In addition, the oscillations are periodic in 1/B with a fundamental frequency which is independent on carrier density, and that is governed only by the magnetic field value (B₀) when one flux quantum pierces the superlattice unit cell. The effect originates from the magnetic translation group, first proposed by Brown and Zak in the 60’s, which states that translational symmetry is restored in the electron wave function for particular values of B, when the quantum magnetic length is commensurable with the lattice periodicity. The BZ oscillations are also used as a probe to study self-similarity and the zero-effective magnetic field associated with different fractals of the corresponding magnetic Bloch states.
[1] R. Krishna Kumar et al, Science 357, 181-184 (2017)