# Topological phase transition induced by magnetic proximity effect in two dimensions

## TL;DR

This study demonstrates how magnetic proximity effects in a CrI3/SnI3/CrI3 trilayer induce a topological phase transition in a two-dimensional topological insulator, highlighting the interplay between magnetic order and spin-orbit coupling.

## Contribution

The paper reveals how magnetic stacking configurations influence Dirac cone protection and induces a topological phase transition via magnetic proximity effects in a 2D system.

## Key findings

- Dirac cones are protected in ferromagnetic stacking without spin-orbit coupling.
- Antiferromagnetic stacking opens a gap at Dirac points.
- Magnetic proximity effect combined with spin-orbit coupling causes a topological phase transition.

## Abstract

We study the magnetic proximity effect on a two-dimensional topological insulator in a CrI$_3$/SnI$_3$/CrI$_3$ trilayer structure. From first-principles calculations, the BiI$_3$-type SnI$_3$ monolayer without spin-orbit coupling has Dirac cones at the corners of the hexagonal Brillouin zone. With spin-orbit coupling turned on, it becomes a topological insulator, as revealed by a non-vanishing $Z_2$ invariant and an effective model from symmetry considerations. Without spin-orbit coupling, the Dirac points are protected if the CrI$_3$ layers are stacked ferromagnetically, and are gapped if the CrI$_3$ layers are stacked antiferromagnetically, which can be explained by the irreducible representations of the magnetic space groups $C_{3i}^1$ and $C_{3i}^1(C_3^1)$, corresponding to ferromagnetic and antiferromagnetic stacking, respectively. By analyzing the effective model including the perturbations, we find that the competition between the magnetic proximity effect and spin-orbit coupling leads to a topological phase transition between a trivial insulator and a topological insulator.