Tuning the quantum phase transition of an ultrathin magnetic topological insulator

We explore the effect of thickness, magnetization direction, strain, and gating on the topological quantum phase transition of a thin-film magnetic topological insulator. Reducing the film thickness to the ultrathin regime couples the edge states on the two surfaces, opening a gap known as the hybridization gap, and causing a phase transition from a topological insulator to a normal insulator (NI). An out-of-plane/in-plane magnetization of size proportional to the hybridization gap triggers a phase transition from a normal insulator state to a quantum anomalous Hall (QAH)/semimetal state. A magnetization tilt by angle $\theta$ from the out-of-plane axis influences the topological phase transition in a way that for sufficiently large $\theta$, no phase transition from NI to QAH can be observed regardless of the sample thickness or magnetization, and for $\theta$ close to $\pi /2$ the system transits to a semimetal phase. Furthermore, we demonstrate that compressive/tensile strain can be used to decrease/increase the magnetization threshold for the topological phase transition. Finally, we reveal the effect of a vertical potential acting on the film, be it due to the substrate or applied gating, which breaks inversion symmetry and raises the magnetization threshold for the transition from NI to QAH state.