Magnetic Bloch states at integer flux quanta induced by super-moiré potential in graphene aligned with twisted boron nitride.

Abstract

Two-dimensional electron systems in both magnetic fields and periodic potentials are described by the Hofstadter butterfly, a fundamental problem of solid-state physics. While moiré systems provide a powerful method to realize this type of spectrum, previous experiments have been limited to fractional flux quanta regime, due to the difficulty of building ~ 50 nm periodic modulations. Here, we demonstrate a super-moiré strategy to overcome this challenge. By aligning monolayer graphene (G) with 1.0° twisted hexagonal boron nitride (t-hBN), a 63.2 nm bichromatic G/t-hBN super-moiré is constructed, made possible by exploiting the electrostatic nature of t-hBN potential. Under magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> , magnetic Bloch states at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi> <mo>/</mo> <msub><mrow><mi>ϕ</mi></mrow> <mrow><mn>0</mn></mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mn>9</mn></math> are achieved and observed as integer Brown-Zak oscillations, expanding the flux quanta from fractions to integers. Theoretical analysis reproduces these experimental findings. This work opens promising avenues to study unexplored Hofstadter butterfly, explore emergent topological order at integer flux quanta and engineer long-wavelength periodic modulations.