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We study the magnetic proximity effect on a two-dimensional topologicalinsulator in a CrI$_3$/SnI$_3$/CrI$_3$ trilayer structure. Fromfirst-principles calculations, the BiI$_3$-type SnI$_3$ monolayer withoutspin-orbit coupling has Dirac cones at the corners of the hexagonal Brillouinzone. With spin-orbit coupling turned on, it becomes a topological insulator,as revealed by a non-vanishing $Z_2$ invariant and an effective model fromsymmetry considerations. Without spin-orbit coupling, the Dirac points areprotected if the CrI$_3$ layers are stacked ferromagnetically, and are gappedif the CrI$_3$ layers are stacked antiferromagnetically, which can be explainedby the irreducible representations of the magnetic space groups $C_{3i}^1$ and$C_{3i}^1(C_3^1)$, corresponding to ferromagnetic and antiferromagneticstacking, respectively. By analyzing the effective model including theperturbations, we find that the competition between the magnetic proximityeffect and spin-orbit coupling leads to a topological phase transition betweena trivial insulator and a topological insulator.

Topological phase transition induced by magnetic proximity effect in two dimensions