Optical readout of the chemical potential of two-dimensional electrons

The chemical potential (μ) of an electron system is a fundamental property of a solid. A precise measurement of μ plays a crucial role in understanding the electron interaction and quantum states of matter. However, thermodynamics measurements in micro- and nanoscale samples are challenging because of the small sample volume and large background signals. Here we report an optical readout technique for μ of an arbitrary two-dimensional material. A monolayer semiconductor sensor is capacitively coupled to the sample. The sensor optical response determines a bias that fixes its chemical potential to the band edge and directly reads the μ value of the sample. We demonstrate the technique in AB-stacked MoTe₂/WSe₂ moiré bilayers. We obtain the μ value with a d.c. sensitivity of about 20 µeV Hz^–1/2 and the compressibility and interlayer electric polarization using a.c. readout. The results reveal a correlated insulating state at a doping density of one hole per moiré unit cell, which evolves from a Mott insulator to a charge-transfer insulator with an increasing out-of-plane electric field. Furthermore, we image μ and quantify the spatial inhomogeneity of the sample. Our work opens the door for high-spatial-resolution and high-temporal-resolution measurements of the thermodynamic properties of two-dimensional quantum materials.