Role of Electronic Correlations on Exchange Interactions and Curie Temperature in Monolayer CrI₃

We investigate the magnetic properties of monolayer CrI₃ by combining first-principles density functional theory calculations with an effective spin model and finite-temperature statistical analysis. The electronic structure is computed within the DFT+U framework, including spin-orbit coupling, allowing us to assess the role of electronic correlations on the structural, electronic, and magnetic properties of the system. Magnetic exchange interactions are extracted using the Green's-function-based Liechtenstein formalism and mapped onto an effective isotropic Heisenberg Hamiltonian. Our results show that the on-site Coulomb interaction U strongly influences the Cr-I bond geometry and the ligand-mediated superexchange pathways, leading to a nonmonotonic dependence of the dominant nearest-neighbor exchange interaction on U. This behavior places monolayer CrI₃ in an intermediate-correlation regime, where electronic localization and hybridization compete to determine the strength of ferromagnetic coupling. Finite-temperature magnetic properties are investigated using the cluster mean-field theory, which partially incorporates short-range magnetic correlations beyond conventional mean-field approaches. The calculated effective Curie temperature within the cluster mean-field (CMF) framework exhibits a pronounced maximum at intermediate values of U, yielding values of the same order of magnitude as experimental measurements for monolayer CrI₃. However, this temperature should be interpreted as an effective energy scale associated with the onset of short-range magnetic correlations within the CMF approximation, rather than a true thermodynamic transition temperature. These results indicate that the dominant energy scale governing magnetic ordering is primarily controlled by isotropic exchange interactions and short-range correlations. Thus, our results show that a minimal Heisenberg model parametrized from first principles, combined with the CMF theory, supplies a well-grounded and computationally efficient framework to describe the magnetic properties of two-dimensional van der Waals magnets.