Review on the magnetic topological insulator MnBi_{2<bold><italic>n</italic></bold>}Te_{3<bold><italic>n</italic></bold>+1}

The emergence of topological insulators (TIs) has brought about a paradigm shift in our understanding of quantum phenomena. Magnetic topological insulators (MTIs) of the MnBi2Te4(Bi2Te3)n family, including MnBi2Te4 (n=0), MnBi4Te7 (n=1), MnBi6Te10 (n=2), and MnBi8Te13 (n=3), etc., have attracted extensive research interest in the field of condensed matter physics due to their combination of intrinsic magnetism and nontrivial band topology. This class of intrinsic magnetic topological insulators provides an ideal platform for studying topological order and its interactions with magnetic order, spin-orbit coupling, and charge degrees of freedom, as well as the modulation of its magnetic structure. The understanding and manipulation of its magnetism is the basis for further exploring its novel states, and is of great significance for the in-depth understanding of quantum physics and the development of quantum technology applications. This paper reviews the experimental and theoretical studies on magnetism in crystals and thin films of MnBi2Te4(Bi2Te3)n (n=0, 1, 2, 3) family of materials in recent years, including the evolution of their magnetic ground states, the modulation of magnetism by substitutional defects, the effective regulation of magnetic structures by external fields, and the exchange-bias phenomenon under various mechanisms. Firstly, crystal growth via flux or chemical vapor transport inherently introduces antisite defects, vacancies, and stacking faults, profoundly impacting magnetism. As n increases, neutron diffraction and STEM-EELS reveal rising defect complexity, driving magnetic ground-state evolution: A-type antiferromagnetic (AFM) order dominates in MnBi2Te4​, while coexisting ferromagnetic (FM)-AFM phases emerge in MnBi4Te7 and MnBi6Te10, transitioning to pure FM in MnBi8Te13 due to weakened interlayer exchange. High-field magnetization uncovers defect-mediated interactions, such as strong AFM coupling between Mn and MnBi​ sites requiring ~50 T for alignment. Sb doping further modulates carrier types and exchange interactions, enabling topological phase transitions (e.g., AFM TI to FM axion insulator). Secondly, in the 2D limit, mechanical exfoliation enables few-layer samples exhibiting odd-even layer-dependent magnetism. Advanced probes—including magneto-optical Kerr effect (MOKE), reflective magnetic circular dichroism (RMCD), and magneto transport measurements—quantify layer-resolved spin configurations, revealing multi-step spin-flop transitions and critical fields sensitive to n and thickness. Odd-layer MnBi2Te4 hosts uncompensated moments, facilitating zero-field QAH (quantized to 0.97h/e2 at 1.4 K), whereas even-layers realize axion insulator states protected by PT symmetry. However, challenges persist: chemical disorder and surface degradation cause discrepancies between theory and experiments. Thirdly, external stimuli provide powerful control knobs: electric gating shifts Fermi levels to stabilize QAH; circularly polarized light selectively flips AFM domains via axion-field interactions; hydrostatic pressure tunes interlayer exchange, inducing AFM to FM transitions in MnBi6​Te10​ at 1.5 GPa; exchange bias (EB) arises in odd-layer devices originated from the interactions between intrinsic and defect-induced magnetic domains without the need of a field-cooling process, enabling programmable HE via field-sweep protocols. Outstanding puzzles include the origin of uncompensated moments in even-layer systems, the precise interplay between magnetic phase transitions and topological states, and defect impacts on few-layer magnetism. Future efforts must prioritize high-purity crystal growth, interfacial engineering in heterostructures, and techniques like NV-center microscopy to correlate defects with quantum transport. Resolving these issues will accelerate applications in spintronics and topological quantum computation.