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.

电子系统的化学势 ( μ ) 是固体的基本性质。μ的精确测量对于理解电子相互作用和物质的量子态起着至关重要的作用。然而，由于样品体积小和背景信号大，微米和纳米级样品的热力学测量具有挑战性。在这里，我们报告了一种任意二维材料μ的光学读出技术。单层半导体传感器与样品电容耦合。传感器光学响应确定将其化学势固定到带边缘的偏置，并直接读取样品的μ值。我们在 AB 堆叠 MoTe ₂ /WSe ₂莫尔双层中演示了该技术。我们获得了直流灵敏度约为 20 µeV Hz ^–1/2的μ值，并使用交流读数获得了可压缩性和层间电极化。结果揭示了每个莫尔晶胞一个孔的掺杂密度下的相关绝缘态，随着面外电场的增加，它从莫特绝缘体演变为电荷转移绝缘体。此外，我们对μ进行成像并量化样本的空间不均匀性。我们的工作为二维量子材料热力学性质的高空间分辨率和高时间分辨率测量打开了大门。