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Charge distribution offers a unique fingerprint of important properties of electronic systems, including dielectric response, charge ordering, and charge fractionalization. We develop an architecture for charge sensing in two-dimensional electronic systems in a strong magnetic field. We probe local change of the chemical potential in a proximitized detector layer using scanning tunneling microscopy, allowing us to infer the chemical potential and the charge profile in the sample. Our technique has both high energy (<0.3 meV) and spatial (<10 nm) resolution exceeding that of previous studies by an order of magnitude. We apply our technique to study the chemical potential of quantum Hall liquids in monolayer graphene under high magnetic fields and their responses to charge impurities. The chemical potential measurement provides a local probe of the thermodynamic gap of quantum Hall ferromagnets and fractional quantum Hall states. The screening charge profile reveals spatially oscillatory response of the quantum Hall liquids to charge impurities and is consistent with the composite Fermi liquid picture close to the half-filling. Our technique also paves the way to map moiré potentials, probe Wigner crystals, and investigate fractional charges in quantum Hall and Chern insulators. 

Interacting electrons in flat bands give rise to a variety of quantum phases. One fundamental aspect of such states is the ordering of the various flavours—such as spin or valley—that the electrons can possess and the excitation spectrum of the broken-symmetry states that they form. These properties cannot be probed directly with electrical transport measurements. The zeroth Landau level of monolayer graphene with fourfold spin–valley degeneracy is a model system for such investigations, but the nature of its broken-symmetry states—particularly at partial fillings—is still not understood. Here we demonstrate a non-invasive spectroscopic technique with a scanning tunnelling microscope and use it to perform measurements of the valley polarization of the electronic wavefunctions and their excitation spectrum in the partially filled zeroth Landau level of graphene. We can extract information such as the strength of the Haldane pseudopotentials that characterize the repulsive interactions underlying the fractional quantum states. Our experiments also demonstrate that fractional quantum Hall phases are built upon broken-symmetry states that persist at partial filling. Our experimental approach quantifies the valley phase diagram of the partially filled Landau level as a model flat-band platform, which is applicable to other graphene-based electronic systems. 

Scanning tunneling spectroscopy is used to image valley ordering in graphene in the presence of a magnetic field. 

Abstract Moiré superlattices created by the twisted stacking of two-dimensional crystals can host electronic bands with flat energy dispersion in which enhanced interactions promote correlated electron states. The twisted double bilayer graphene (TDBG), where two Bernal bilayer graphene are stacked with a twist angle, is such a moiré system with tunable flat bands. Here, we use gate-tuned scanning tunneling spectroscopy to directly demonstrate the tunability of the band structure of TDBG with an electric field and to show spectroscopic signatures of electronic correlations and topology for its flat band. Our spectroscopic experiments are in agreement with a continuum model of TDBG band structure and reveal signatures of a correlated insulator gap at partial filling of its isolated flat band. The topological properties of this flat band are probed with the application of a magnetic field, which leads to valley polarization and the splitting of Chern bands with a large effective g-factor.