Quantum anomalous Hall effect
The quantum anomalous Hall effect (QAHE) is a quantized version of the anomalous Hall effect observed in two-dimensional materials, characterized by a precisely quantized Hall conductance of e^2/h (where e is the elementary charge and h is Planck's constant) and vanishing longitudinal conductance at zero external magnetic field, arising from topologically nontrivial band structures and spontaneous breaking of time-reversal symmetry.[1] This effect manifests as dissipationless chiral edge states that propagate unidirectionally along the sample boundaries, analogous to the conventional quantum Hall effect but without the need for strong magnetic fields or cryogenic cooling to extremely low temperatures for Landau level formation.[1] Theoretically, the QAHE was first conceptualized in F. D. M. Haldane's 1988 model of a honeycomb lattice with complex next-nearest-neighbor hoppings that break time-reversal symmetry, predicting a Chern insulator state with nonzero Hall conductance despite zero net magnetic field.[2] This idea was extended to realistic solid-state systems in the context of topological insulators, where strong spin-orbit coupling inverts band structures, and magnetism induces a topological phase transition to a Chern insulator; a key proposal came from Qi, Hughes, and Zhang in 2008, who described how ferromagnetic ordering in time-reversal-invariant topological insulators could realize the QAHE through an intrinsic topological magnetoelectric response.[1] The effect is fundamentally tied to the Chern number, a topological invariant that counts the integrated Berry curvature over the Brillouin zone, ensuring robust quantization protected against weak disorder.[1] Experimentally, the QAHE was first observed in 2013 by Chang et al. in thin films of chromium-doped bismuth antimony telluride, ({\rm Bi,Sb})_2{\rm Te}_3, a magnetic topological insulator grown by molecular beam epitaxy, where Hall resistance quantized to h/e^2 at approximately 30 mK without applied field.[3] Subsequent realizations expanded to other magnetically doped topological insulators, such as vanadium- or chromium-doped variants achieving higher temperatures up to around 2 K, and intrinsic magnetic topological insulators like few-layer MnBi_2Te_4, which exhibited QAHE up to 1.4 K in zero field, with gating enabling quantization up to 6 K as of 2022, though often requiring modest fields for full observation.[1][4] Materials classes now include magnetically doped three-dimensional topological insulators, van der Waals heterostructures, and moiré superlattices in twisted bilayer graphene or transition metal dichalcogenides, enabling tunable Chern numbers from 1 to 5 in multilayer configurations.[1] The QAHE holds profound significance for both fundamental physics and technology, providing a platform to study topological phases of matter, chiral Majorana modes for quantum computing, and dissipationless charge transport for low-power electronics and metrology standards like precise resistance quantization.[1] Recent advances, including room-temperature proposals in novel two-dimensional magnets like TbCl as of 2025 and higher-Chern-number states, underscore its potential for scalable quantum devices, though challenges like disorder suppression and elevated operating temperatures persist.[1][5]Definition and Principles
Definition
The quantum anomalous Hall effect (QAHE) represents the quantum Hall effect realized in the absence of an external magnetic field, manifesting as a quantized transverse Hall conductivity \sigma_{xy} = n \frac{e^2}{h}, where n is an integer, driven by intrinsic topological properties of the material combined with its magnetization. This quantization arises without the need for Landau levels or external fields, distinguishing it as a topological phenomenon inherent to the electronic band structure.[6] Physically, the QAHE emerges from a band structure characterized by a nonzero Chern number, a topological invariant that quantifies the Berry curvature over the Brillouin zone and enforces the integer quantization of the Hall conductance.[6] This topological order results in the presence of chiral edge states—unidirectional conducting channels at the material's boundaries—that propagate without backscattering, enabling dissipationless charge transport along the edges while the bulk remains insulating.[6] Experimentally, the hallmark signature of the QAHE is the precise quantization of the Hall resistance R_{xy} to \frac{h}{n e^2} at low temperatures and zero magnetic field, coupled with the longitudinal resistivity \rho_{xx} vanishing to near zero, indicating a gapped bulk and robust edge conduction. Such behavior has been observed in systems where internal mechanisms break time-reversal symmetry.[6] The QAHE is typically realized in ferromagnetic topological insulators, where spontaneous magnetization internally disrupts time-reversal symmetry, allowing the topological band inversion to yield the requisite Chern number without external perturbations.[6]Relation to Hall Effects
The classical Hall effect refers to the generation of a transverse voltage across a conductor carrying current in the presence of a perpendicular external magnetic field, arising from the Lorentz force deflecting charge carriers. The Hall conductivity \sigma_{xy} in this case is proportional to the applied magnetic field B and the carrier density and type. The anomalous Hall effect (AHE) extends this phenomenon to ferromagnetic materials, where a transverse Hall voltage emerges due to the material's intrinsic magnetization M, without requiring an external magnetic field. This effect stems from mechanisms such as spin-orbit coupling and disorder scattering, with \sigma_{xy} scaling linearly with M but remaining non-quantized.[7] The integer quantum Hall effect (IQHE) manifests in two-dimensional electron gases subjected to strong perpendicular magnetic fields at low temperatures, yielding a precisely quantized Hall conductivity \sigma_{xy} = n \frac{e^2}{h}, where n is an integer and h is Planck's constant. Quantization arises from the discrete Landau levels formed by the cyclotron motion of electrons in the field. The fractional quantum Hall effect (FQHE), observed under similar conditions but with stronger electron correlations, features fractional quantization \sigma_{xy} = \nu \frac{e^2}{h} with \nu = p/q (p, q integers, q > 1), due to the formation of correlated incompressible states. In distinction from these, the quantum anomalous Hall effect (QAHE) realizes integer quantization of \sigma_{xy} without an external magnetic field, relying instead on the intrinsic Berry curvature of the electronic band structure and spontaneous time-reversal symmetry breaking through intrinsic ferromagnetism. This topological mechanism mimics the chiral edge states of the IQHE but emerges from bulk band topology rather than Landau level filling.| Hall Effect | External Magnetic Field | Quantization | Typical Temperature Range | Representative Materials |
|---|---|---|---|---|
| Classical | Yes | No | >300 K | Metals, semiconductors (e.g., Cu, Si) |
| Anomalous (AHE) | No | No | Up to 300 K | Ferromagnetic metals (e.g., Fe, Co) |
| Integer Quantum (IQHE) | Yes | Yes (integer n) | <4 K | 2D electron gases (e.g., GaAs/AlGaAs) |
| Fractional Quantum (FQHE) | Yes | Yes (fractional \nu) | <1 K | 2D electron gases (e.g., GaAs/AlGaAs) |
| Quantum Anomalous (QAHE) | No | Yes (integer n) | <2 K | Magnetic topological insulators (e.g., Cr-doped (Bi,Sb)_2Te_3, MnBi_2Te_4) |