Title: Stable phases of freestanding monolayer TiO$_2$: The emergence of out-of-plane ferroelectricity

Abstract: Despite being successfully synthesized [Zhang $et$ $al.$, Nat. Mater. \textbf{20}, 1073 (2021)], the monolayer structure of stable hexagonal TiO$_2$ is unknown, and it is not even clear whether it can exist in a freestanding form. Through first-principles calculations, we have identified two previously uncharted stable structures, namely, distorted 1$\times$$\sqrt{3}$ 1T-TiO$_2$ and $\sqrt{3}$$\times$$\sqrt{3}$ 1T-TiO$_2$, both of which are energetically more favourable than commonly adopted 1H and 1T phases. Here structural distortions are characterized by the out-of-plane shifts of Ti atoms due to the pseudo-Jahn-Teller interactions, which break one and all two inversion symmetries of 1T configuration. As a consequence, the 1$\times$$\sqrt{3}$ 1T remains centrosymmetric while the $\sqrt{3}$$\times$$\sqrt{3}$ 1T exhibits out-of-plane ferroelectricity. Electronic structure calculations show that both two are wide-bandgap semiconductors with bandgaps larger than their bulk counterparts. Our study not only deepens the understanding of structural instability in wide-gap semiconductors but also adds a new member to the rare family of two-dimensional out-of-plane ferroelectrics.