The observation temperatures of quantum anomalous Hall effect in semimagnetic and magnetic topological insulator (MTI) heterostructures are always much lower than Curie temperature. Here, we theoretically demonstrate that floating of topological surface states (TSSs) into magnetic insulators is not only able to produce ideal semi-MTI heterostructures, but also provides insight into the origin of extremely low observation temperature of the quantum anomalous Hall effect in different MTIs. We show that the emergence of diving and floating of TSSs can be observed in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4/{\mathrm{Bi}}_2{\mathrm{Se}}_3$ family heterostructures. Within the TSSs floating systems, ${\mathrm{MnBi}}_2{\mathrm{Te}}_4/{\mathrm{Sb}}_2{\mathrm{Te}}_3$ and ${\mathrm{NiBi}}_2{\mathrm{Te}}_4/{\mathrm{Sb}}_2{\mathrm{Te}}_3$ exhibit characteristics of ideal semi-MTIs, featuring a tunable Fermi level, and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4/{\mathrm{Bi}}_2{\mathrm{Te}}_3/{\mathrm{NiBi}}_2{\mathrm{Te}}_4$ $({\mathrm{NiBi}}_2{\mathrm{Te}}_4/{\mathrm{Sb}}_2{\mathrm{Te}}_3/{\mathrm{VBi}}_2{\mathrm{Te}}_4)$ is found to be MTI with weak (strong) long-range ferromagnetic coupling and large (small) surface gap. Our findings reveal that the observation temperature of quantum anomalous Hall effect is governed by the smaller of the two surface gaps, rather than being directly linked to the Curie temperature of MTIs.

• Received 11 September 2023
• Accepted 15 March 2024

DOI:https://doi.org/10.1103/PhysRevB.109.155427