Thickness dependence of the critical temperature and magnetic coupling in multilayer Cr2Sn2Te6

In this work, using first-principles calculations and Monte Carlo simulations, we investigate the critical (Curie or N\'eel) temperature ${T}_{\mathrm{c}}$ and magnetic coupling in two-dimensional multilayer ${\mathrm{Cr}}₂{\mathrm{Sn}}₂{\mathrm{Te}}₆$ (CST). The results indicate that the value of ${T}_{\mathrm{c}}$ is strongly dependent on thickness. We find that the ${T}_{\mathrm{c}}$ value of CST increases up to 133 K (bulk CST) from 48 K (monolayer CST), which is significantly greater than that of the homogeneous compounds ${\mathrm{Cr}}₂{\mathrm{Si}}₂{\mathrm{Te}}₆$ and ${\mathrm{Cr}}₂{\mathrm{Ge}}₂{\mathrm{Te}}₆$. Then, from the perspective of magnetic exchange coupling, it is found that the intralayer ferromagnetic coupling in CST is mostly associated with the competition between direct and superexchange interactions. In particular, the interlayer magnetic coupling is primarily attributed to the supersuperexchange interactions (Cr-Te-Te-Cr) between antiferromagnetically and ferromagnetically coupled multi-nearest neighbors within the layers. In addition, these findings demonstrate that manipulation of the interlayer distance using $z$-directional strain is an efficient way to enhance the ${T}_{\mathrm{c}}$ value of CST. When the interlayer Cr-Cr distance is decreased to 6.17 \AA{}, the transition temperature of bulk CST increases to 429 K, and an interlayer magnetic coupling transition is observed in which the ferromagnetic layers are coupled antiferromagnetically.