The quantum anomalous Hall effect (QAHE) is a quantum transport phenomenon observed in certain magnetized topological insulators (TIs), where quantized Hall conductance occurs without an external magnetic field. This effect arises due to the interplay between spin-orbit coupling and intrinsic magnetization, leading to a topologically non-trivial electronic structure. A prominent material system exhibiting QAHE is chromium-doped (Bi,Sb)₂Te₃, where magnetic order breaks time-reversal symmetry while preserving the topological nature of the electronic bands.

In QAHE, the Hall conductance is quantized to e²/h, where e is the electron charge and h is Planck’s constant. This quantization stems from the formation of a single chiral edge state circulating around the sample boundary, immune to backscattering due to its topological protection. Unlike the conventional quantum Hall effect (QHE), which requires strong external magnetic fields to generate Landau levels, QAHE relies on spontaneous magnetization within the material. The key requirement is a ferromagnetic state with perpendicular magnetic anisotropy, ensuring the magnetization axis is aligned out-of-plane to open a gap in the Dirac surface states of the TI.

Cr-doped (Bi,Sb)₂Te₃ serves as an ideal platform for QAHE due to several factors. First, the (Bi,Sb)₂Te₃ host is a well-known three-dimensional TI with robust surface states. Introducing Cr atoms introduces localized magnetic moments that couple ferromagnetically via exchange interactions. When the Fermi level lies within the magnetically induced gap, the system enters the QAHE regime. Experimental observations confirm quantization of the Hall resistance to approximately 25.8 kΩ (h/e²) at temperatures below a few Kelvin, limited by magnetic disorder and thermal fluctuations.

The quantum spin Hall effect (QSHE) shares similarities with QAHE but differs fundamentally in symmetry and edge state structure. QSHE occurs in time-reversal-invariant TIs, such as HgTe/CdTe quantum wells, where spin-orbit coupling generates helical edge states—counter-propagating states with opposite spin polarization. These states are protected by time-reversal symmetry, and backscattering is forbidden unless magnetic perturbations are introduced. In contrast, QAHE breaks time-reversal symmetry, resulting in unidirectional edge transport that is insensitive to spin orientation. While QSHE requires spin-momentum locking, QAHE relies on chirality, making it more robust against certain types of disorder.

The experimental realization of QAHE in Cr-doped (Bi,Sb)₂Te₃ involves precise control of material parameters. The doping concentration must be optimized to ensure ferromagnetic ordering without introducing excessive bulk conduction. Typically, Cr concentrations of a few percent are used to achieve a Curie temperature of 10-30 K, below which the QAHE emerges. Thin-film geometries are preferred to minimize bulk contributions, and electrostatic gating is often employed to fine-tune the Fermi level into the magnetic gap.

Transport measurements in QAHE systems reveal distinct signatures. At zero magnetic field, the longitudinal resistance drops to near zero, while the Hall resistance plateaus at the quantized value. The absence of an external field eliminates Landau level formation, distinguishing QAHE from QHE. However, imperfections such as domain walls or inhomogeneous magnetization can lead to residual dissipative channels, reducing the quantization accuracy. Advances in material growth and interface engineering have progressively improved the robustness of QAHE observations.

Theoretical frameworks for QAHE are rooted in the Chern number formalism, where the quantized Hall conductance is tied to the integral of the Berry curvature over the Brillouin zone. In Cr-doped (Bi,Sb)₂Te₃, the exchange interaction between Cr moments and the TI’s Dirac electrons modifies the band structure, creating a non-zero Chern number. This topological invariant guarantees the existence of chiral edge states, analogous to those in integer QHE systems but without Landau levels.

Compared to QSHE, QAHE offers potential advantages for low-power electronics due to its dissipationless edge transport and lack of reliance on external fields. However, the low operational temperatures remain a significant challenge. Research efforts focus on identifying materials with higher magnetic ordering temperatures or engineering heterostructures that enhance exchange interactions. Proposals include leveraging interfacial effects in TI/ferromagnetic insulator hybrids or exploring two-dimensional magnets coupled to TIs.

In summary, the quantum anomalous Hall effect represents a unique convergence of topology and magnetism, exemplified by Cr-doped (Bi,Sb)₂Te₃. Its distinction from the quantum spin Hall effect lies in the symmetry-breaking mechanism and edge state chirality, offering a different route to topologically protected conduction. While challenges persist in achieving room-temperature QAHE, the phenomenon provides a rich platform for exploring emergent quantum states and potential applications in quantum electronics. Future directions may involve integrating QAHE materials with superconducting or spin-based devices to unlock novel functionalities in quantum computing and spintronics.

The study of QAHE continues to bridge fundamental physics and materials science, highlighting the intricate relationship between electronic structure, magnetism, and topology. As synthesis techniques advance and theoretical models refine, further breakthroughs in understanding and harnessing this effect are anticipated. The contrast with QSHE underscores the diversity of topological phases and their distinct mechanisms of protection, enriching the broader landscape of quantum materials research.