de Haas-van Alphen spectroscopy and fractional quantization of magnetic-breakdown orbits in moir\'e graphene

Quantum oscillations originating from the quantization of the electron cyclotron orbits provide ultrasensitive diagnostics of electron bands and interactions in novel materials. We report on the first direct-space nanoscale imaging of the thermodynamic magnetization oscillations due to the de Haas-van Alphen effect in moir\'e graphene. Scanning by SQUID-on-tip in Bernal bilayer graphene crystal-axis-aligned to hBN reveals abnormally large magnetization oscillations with amplitudes reaching 500 {\mu}_B/electron in weak magnetic fields, unexpectedly low frequencies, and high sensitivity to the superlattice filling fraction. The oscillations allow us to reconstruct the complex band structure in exquisite detail, revealing narrow moir\'e bands with multiple overlapping Fermi surfaces separated by unusually small momentum gaps. We identify distinct sets of oscillations that violate the textbook Onsager Fermi surface sum rule, signaling formation of exotic broad-band particle-hole superposition states induced by coherent magnetic breakdown.Comment: 30 pages, 5 main text figures, 6 supplementary figure

Imaging de Haas-van Alphen quantum oscillations and milli-Tesla pseudomagnetic fields

A unique attribute of atomically thin quantum materials is the in-situ tunability of their electronic band structure by externally controllable parameters like electrostatic doping, electric field, strain, electron interactions, and displacement or twisting of atomic layers. This unparalleled control of the electronic bands has led to the discovery of a plethora of exotic emergent phenomena. But despite its key role, there is currently no versatile method for mapping the local band structure in advanced 2D materials devices in which the active layer is commonly embedded in various insulating layers and metallic gates. Utilizing a scanning superconducting quantum interference device, we image the de Haas-van Alphen quantum oscillations in a model system, the Bernal-stacked trilayer graphene with dual gates, which displays multiple highly-tunable bands. By resolving thermodynamic quantum oscillations spanning over 100 Landau levels in low magnetic fields, we reconstruct the band structure and its controllable evolution with the displacement field with unprecedented precision and spatial resolution of 150 nm. Moreover, by developing Landau level interferometry, we reveal shear-strain-induced pseudomagnetic fields and map their spatial dependence. In contrast to artificially-induced large strain, which leads to pseudomagnetic fields of hundreds of Tesla, we detect naturally occurring pseudomagnetic fields as low as 1 mT corresponding to graphene twisting by just 1 millidegree over one {\mu}m distance, two orders of magnitude lower than the typical angle disorder in high-quality twisted bilayer graphene devices. This ability to resolve the local band structure and strain on the nanoscale opens the door to the characterization and utilization of tunable band engineering in practical van der Waals devices.Comment: Nature (2023