  Recent experiments revealed that monolayer $\alpha$-RuCl$_3$ can be obtain by
chemical exfoliation method and exfoliation or restacking of nanosheets can
manipulate the magnetic properties of the materials. In this present paper, the
electronic and magnetic properties of $\alpha$-RuCl$_3$ monolayer are
investigated by combining first-principles calculations and Monte Carlo
simulations. From first-principles calculations, we found that the spin
configuration FM corresponds to the ground state for $\alpha$-RuCl$_3$,
however, the other excited zigzag oriented spin configuration has energy of 5
meV/atom higher than the ground state. Energy band gap has been obtained as $3$
meV using PBE functionals. When spin-orbit coupling effect is taken into
account, corresponding energy gap is determined to be as $57$ meV. We also
investigate the effect of Hubbard U energy terms on the electronic band
structure of $\alpha$-RuCl$_3$ monolayer and revealed band gap increases
approximately linear with increasing U value. Moreover, spin-spin coupling
terms ($J_1$, $J_2$, $J_3$) have been obtained using first principles
calculations. By benefiting from these terms, Monte Carlo simulations with
single site update Metropolis algorithm have been implemented to elucidate
magnetic properties of the considered system. Thermal variations of
magnetization, susceptibility and also specific heat curves indicate that
monolayer $\alpha$-RuCl$_3$ exhibits a phase transition between ordered and
disordered phases at the Curie temperature $14.21$ K. We believe that this
study can be utilized to improve two-dimensional magnet materials.
