Charge-order-enhanced capacitance in semiconductor moir\'e superlattices

Van der Waals moir\'e materials have emerged as a highly controllable platform to study the electronic correlation phenomena. In particular, robust correlated insulating states have recently been discovered at both integer and fractional filling factors of the semiconductor moir\'e systems. Here, we reveal the thermodynamic properties of these states by measuring the gate capacitance of MoSe2/WS2 moir\'e superlattices. We observe a series of incompressible states for filling factor 0 - 8 and anomalously large capacitance (nearly 60% above the device's geometrical capacitance) in the intervening compressible regions. The anomalously large capacitance is most pronounced at small filling factor, below the melting temperature of the charge-ordered states, and for small sample-gate separation. It is a manifestation of the device-geometry-dependent Coulomb interaction between electrons and phase mixing of the charge-ordered states. We have further extracted the thermodynamic gap of the correlated insulating states and the entropy of the capacitive device. The results not only establish capacitance as a powerful probe of the correlated states in semiconductor moir\'e systems, but also demonstrate control of the extended Coulomb interaction in these materials via sample-gate coupling.