Acoustoelectric Probing of Fractal Energy Spectra in Graphene/hBN Moiré Superlattices

Moiré superlattices with long-range periodicity exhibit Hofstadter energy spectra under accessible magnetic fields, enabling the exploration of emergent quantum phenomena through a hierarchy of fractal states. However, higher-order features, located at elevated energies with narrow bandwidths, typically require high carrier densities and remain difficult to resolve using conventional electrical transport due to limited sensitivity and strong background conductivity. Here, we utilize acoustoelectric (AE) transport to probe high-order fractal states and the Hofstadter spectrum in graphene/hBN moiré superlattices. Surface acoustic waves on a ferroelectric LiNbO$_3$ substrate generate an AE voltage proportional to the derivative of electrical conductivity, significantly enhancing sensitivity to weak spectral features. Combined with substrate-induced high electron doping, this technique resolves fractal Brown-Zak oscillations up to the fifth-order and provides the first AE observation of the Hofstadter butterfly, revealing high-order fractal magnetic Bloch states and symmetry-broken Landau levels over a wide carrier density range. Our results establish AE transport as a powerful derivative-sensitive probe for emergent fractal quantum states in moiré-engineered 2D systems.