  Multiferroics with intrinsic ferromagnetism and ferroelectricity are highly
desired but rather rare, while most ferroelectric magnets are
antiferromagnetic. A recent theoretical work [Phys. Rev. B {\bf 99}, 195434
(2019)] predicted that oxyhalides VO$X_2$ ($X$: halogen) monolayers are
two-dimensional multiferroics by violating the empirical $d^0$ rule. Most
interestingly, the member VOI$_2$ are predicted to exhibit spontaneous
ferromagnetism and ferroelectricity. In this work, we extend the previous study
on the structure and magnetism of VOI$_2$ monolayer by using density functional
theory and Monte Carlo simulation. The presence of the heavy element iodine
with a strong spin-orbit coupling leads an effective Dzyaloshinskii-Moriya
interaction in the polar structure, which favors a short-period spiral a
magnetic structure.. Another interesting result is that the on-site Coulomb
interaction can strongly suppress the polar distortion thus leading to a
ferromagnetic metallic state. Therefore, the VOI2 monolayer is either a
ferroelectric insulator with spiral magnetism or a ferromagnetic metal, instead
of a ferromagnetic ferroelectric system. Our study highlights the key physical
role of the Dzyaloshinskii-Moriya interaction.

Materials Science;Mesoscale and Nanoscale Physics;Strongly Correlated Electrons