Ubiquitous order-disorder transition in the Mn antisite sublattice of the (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ magnetic topological insulators

Magnetic topological insulators (TIs) herald a wealth of applications in spin-based technologies, relying on the novel quantum phenomena provided by their topological properties. Particularly promising is the (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ layered family of established intrinsic magnetic TIs that can flexibly realize various magnetic orders and topological states. High tunability of this material platform is enabled by manganese-pnictogen intermixing, whose amounts and distribution patterns are controlled by synthetic conditions. Positive implication of the strong intermixing in MnSb$_2$Te$_4$ is the interlayer exchange coupling switching from antiferromagnetic to ferromagnetic, and the increasing magnetic critical temperature. On the other side, intermixing also implies atomic disorder which may be detrimental for applications. Here, we employ nuclear magnetic resonance and muon spin spectroscopy, sensitive local probe techniques, to scrutinize the impact of the intermixing on the magnetic properties of (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ and MnSb$_2$Te$_4$. Our measurements not only confirm the opposite alignment between the Mn magnetic moments on native sites and antisites in the ground state of MnSb$_2$Te$_4$, but for the first time directly show the same alignment in (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ with n = 0, 1 and 2. Moreover, for all compounds, we find the static magnetic moment of the Mn antisite sublattice to disappear well below the intrinsic magnetic transition temperature, leaving a homogeneous magnetic structure undisturbed by the intermixing. Our findings provide a microscopic understanding of the crucial role played by Mn-Bi intermixing in (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ and offer pathways to optimizing the magnetic gap in its surface states.