Quantum Hall crystals, liquids, and bubbles in graphene from direct measurement of the chemical potential

We describe an experimental technique to measure the chemical potential in atomically thin layered materials with high sensitivity and in the static limit. We apply this technique to a high quality graphene monolayer to map out the evolution of chemical potential with carrier density up to the N=4 Landau level. In the N=0 level, the measured ground state energy per particle shows exceptional agreement with numerical calculations when the screening of the Coulomb interaction by filled Landau levels is accounted for. In the N=1 Landau level, the comparison reveals the importance of valley anisotropic interaction and the presence of valley-textured electron solids near odd integer fillings. Finally, in the N=2,3 and 4 Landau levels, we find a series of oscillations in the chemical potential that match Hartree-Fock calculations based on single- and mixed phases of multi-electron bubble states. We also report our recent results on measuring chemical potential in the flat band of twisted bilayer graphene.