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We analyze the distributions of electric potential and field, polarization and charge, and the differential capacitance of a silicon metal-oxide-ferroelectric field effect transistor (MOSFET), in which a gate insulator consists of thin layers of dielectric SiO2 and weak ferroelectric HfO2. It appeared possible to achieve a quasi-steady-state negative capacitance of the HfO2 layer, C_(HfO_2 )<0, if the layer thickness is close to the critical thickness of the size-induced ferroelectric-paraelectric phase transition. The quasi-steady-state negative capacitance, being a very slow-varying transient state of the ferroelectric, corresponds to a positive capacitance of the whole system, and so its appearance does not break any thermodynamic principle. Implementation of the quasi-steady-state negative capacitance C_ins of the gate insulator can open the principal possibility to reduce the MOSFET subthreshold swing below the critical value, and to decrease the gate voltage below the fundamental Boltzmann limit. However, we failed to found the parameters for which C_ins is negative in the quasi-steady states; and thus, the negative C_(HfO_2 ) cannot reduce the subthreshold swing below the fundamental limit. Nevertheless, the increase in C_ins, related with C_(HfO_2 )<0, can decrease the swing above the limit, reduce device heating during the operation cycles, and thus contribute to further improvements of the MOSFET performances.