Emergent Antiferromagnetic Behavior in EuCl$_3$ Monolayer: A Comprehensive First-Principles Study Including Hubbard-SOC Interplay

We present a comprehensive study on the electronic and magnetic properties of the EuCl$_3$ monolayer using first-principles calculations. By taking into account the spin-orbit coupling (SOC) and the Hubbard effects, we elucidate the influence of these interactions on the structural properties of EuCl$_3$ monolayer. Comparing the lattice parameters obtained with the PBE+SOC and PBE+SOC+Hubbard effects reveals an increase in the lattice parameters when including the Hubbard effect. In the absence of the Hubbard interaction, the magnetic ground state of the EuCl$_3$ monolayer exhibits a preference for a ferromagnetic (FM) configuration as determined by the PBE+SOC calculations. However, the introduction of the Hubbard parameter leads to a shift in the magnetic ground state preference towards an antiferromagnetic (AFM) N\'{e}el state. Based on the calculated energy values, Monte Carlo simulations are carried out to determine the N\'{e}el temperature $(T_N)$. Our simulations yield a N\'{e}el temperature of \SI{390}{\kelvin} for the EuCl$_3$ monolayer, indicating the temperature at which the transition from a paramagnetic phase to an antiferromagnetic phase occurs. These results highlight the importance of incorporating the SOC and the Hubbard effect in accurately describing the electronic and magnetic properties of the EuCl$_3$ monolayer. Our results contribute to a deeper understanding of the fundamental physics underlying the behavior of this intriguing two-dimensional material and provide insights into its potential applications in spintronics and magnetic devices.