Quantum capacitance anomalies of two-dimensional non-equilibrium states under microwave irradiation

We report our direct study of the compressibility on ultrahigh mobility two-dimensional electron system ($\mu =$1x10$^{\mathrm{7}}$ cm$^{\mathrm{2}}$/Vs) in GaAs/AlGaAs quantum wells under microwave (MW) irradiation. The field penetration current results show that the quantum capacitance oscillates with microwave induced resistance oscillations (MIRO), however, the trend is opposite with respect to the compressibility for usual equilibrium states in the theoretical proposal. The anomalous phenomena at integer $j=\omega $/$\omega_{\mathrm{C}}$ regime provide a platform for study on the non-equilibrium system under microwave. Moreover, the quantum capacitance indication for multi-photon process at $j=$1/2 can be detected under intensive microwave below 30 GHz, although it does not appear in resistances.