Brown-Zak and Weiss oscillations in a gate-tunable graphene superlattice: A unified picture of miniband conductivity

Abstract

Electrons exposed to a two-dimensional (2D) periodic potential and a uniform, perpendicular magnetic field exhibit a fractal, self-similiar energy spectrum known as the Hofstadter butterfly. Recently, related high-temperature quantum oscillations (Brown-Zak oscillations) were discovered in graphene moiré systems, whose origin lie in the repetitive occurrence of extended minibands/magnetic Bloch states at rational fractions of magnetic flux per unit cell giving rise to an increase in band conductivity. In this work, we report on the experimental observation of band conductivity oscillations in an electrostatically defined and gate-tunable graphene superlattice, which are governed both by the internal structure of the Hofstadter butterfly (Brown-Zak oscillations) and by a commensurability relation between the cyclotron radius of electrons and the superlattice period (Weiss oscillations). We obtain a complete, unified description of band conductivity oscillations in two-dimensional superlattices, yielding a detailed match between theory and experiment.

Patterned bilayer graphene as a tunable, strongly correlated system

“…The superlattice constant, a ≈ 13 nm, is also fixed by the flat band condition, θ ≈ 1.1°and tuning the superlattice strength is only possible by applying hydrostatic pressure 4 . Some prior works have focused on patterning monolayer graphene, either by etching holes directly into the graphene sheet 27 or by patterned electrostatic gating 25,28 . Ref.…”

“…27 demonstrates, theoretically, that patterning introduces an energy gap in the graphene dispersion while Refs. 25,28 measure magnetotransport properties of a real device and show that the result of patterning is essentially a correction to single particle physics. There is not, however, the possibility for generating an isolated flat band or strongly correlated phases in these monolayer graphene systems.…”

Wannier Diagram and Brown-Zak Fermions of Graphene on Hexagonal Boron-Nitride

“…Ultraclean graphene in crystallographic alignment with hexagonal boron nitride (hBN) forms large-scale moiré patterns [1,2]. The area ≈ 165 nm 2 of one supercell is large enough to reach the regime of one magnetic flux quantum Φ 0 = ℎ∕ per moiré unit cell already at = 23.5 T. A plethora of fascinating phenomena emerge such as Hofstadter's butterfly, the fractal energy spectrum of the lattice in the magnetic field [3,4], Weiss oscillations [5,6] and Brown-Zak oscillations [7][8][9], revivals of zero-field conductivity at large magnetic fields periodic in 1∕ . Indeed, Hofstadter's butterfly could recently be experimentally observed [10][11][12][13][14][15][16][17].…”

“…The large supercell of graphene aligned on hexagonal boron nitride with a linear dimension of ≈ 13.6 nm includes tenthousands of atoms, rendering numerical simulations of the conductance challenging. Up to now, calculations therefore have focused on the density of states [25][26][27], a continuum ⋅ Hamiltonian [24,28], scaled graphene [29,30], or on effective 1D models [6]. Here we present, to our knowledge, the first realistic large scale simulation of the conductance for a 250 nm wide graphene on hBN nanoribbon.…”