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The gate defect of the ferroelectric HfO$_2$-based Si field-effect transistor (Si FeFET) plays a dominant role in its reliability issue. The first-principles calculations are an effective method for the atomic-scale understanding of gate defects. However, the first-principles study on the defects of FeFET gate stacks, i.e., metal/orthorhombic-Hf$_{0.5}$Zr$_{0.5}$O$_2$/SiO$_2$/Si structure, has not been reported so far. The key challenge is the construction of metal/orthorhombic-Hf$_{0.5}$Zr$_{0.5}$O$_2$/SiO$_2$/Si gate stack models. Here, we use the Hf$_{0.5}$Zr$_{0.5}$O$_2$(130) high-index crystal face as the orthorhombic ferroelectric layer and construct a robust atomic structure of the orthorhombic-Hf$_{0.5}$Zr$_{0.5}$O$_2$/SiO$_2$/Si gate stack without any gap states. Its high structural stability is ascribed to the insulated interface. The calculated band offsets show that this gate structure is of the type-I band alignment. Furthermore, the formation energies and charge transition levels (CTLs) of defects reveal that the oxygen vacancy defects are more favorable to form compared with other defects such as oxygen interstitial and Hf/Zr vacancy, and their CTLs are mainly localized near the Si conduction band minimum and valence band maximum, in agreement with the reported experimental results. The oxygen vacancy defects are responsible for charge trapping/de-trapping behavior in Si FeFET. This work provides an insight into gate defects and paves the way to carry out the first-principles study of ferroelectric HfO$_2$-based Si FeFET.