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Over a long period of exploration, the successful observation of quantized version of anomalous Hall effect (AHE) in thin film of magnetically-doped topological insulator completed a quantum Hall trio---quantum Hall effect (QHE), quantum spin Hall effect (QSHE), and quantum anomalous Hall effect (QAHE). On the theoretical front, it was understood that intrinsic AHE is related to Berry curvature and U(1) gauge field in momentum space. This understanding established connection between the QAHE and the topological properties of electronic structures characterized by the Chern number. With the time reversal symmetry broken by magnetization, a QAHE system carries dissipationless charge current at edges, similar to the QHE where an external magnetic field is necessary. The QAHE and corresponding Chern insulators are also closely related to other topological electronic states, such as topological insulators and topological semimetals, which have been extensively studied recently and have been known to exist in various compounds. First-principles electronic structure calculations play important roles not only for the understanding of fundamental physics in this field, but also towards the prediction and realization of realistic compounds. In this article, a theoretical review on the Berry phase mechanism and related topological electronic states in terms of various topological invariants will be given with focus on the QAHE and Chern insulators. We will introduce the Wilson loop method and the band inversion mechanism for the selection and design of topological materials, and discuss the predictive power of first-principles calculations. Finally, remaining issues, challenges and possible applications for future investigations in the field will be addressed.
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The quantum anomalous Hall (QAH) state is a two-dimensional bulk insulator with a non-zero Chern number in absence of external magnetic fields. Protected gapless chiral edge states enable dissipationless current transport in electronic devices. Doping topological insulators with random magnetic impurities could realize the QAH state, but magnetic order is difficult to establish experimentally in the bulk insulating limit. Here we predict that the single quintuple layer of GdBiTe3 film could be a stoichiometric QAH insulator based on ab-initio calculations, which explicitly demonstrate ferromagnetic order and chiral edge states inside the bulk gap. We further investigate the topological quantum phase transition by tuning the lattice constant and interactions. A simple low-energy effective model is presented to capture the salient physical feature of this topological material.
The Hall effect, the anomalous Hall effect and the spin Hall effect are fundamental transport processes in solids arising from the Lorentz force and the spin-orbit coupling respectively. The quant
Instability of quantum anomalous Hall (QAH) effect has been studied as function of electric current and temperature in ferromagnetic topological insulator thin films. We find that a characteristic current for the breakdown of the QAH effect is roughly proportional to the Hall-bar width, indicating that Hall electric field is relevant to the breakdown. We also find that electron transport is dominated by variable range hopping (VRH) at low temperatures. Combining the current and temperature dependences of the conductivity in the VRH regime, the localization length of the QAH state is evaluated to be about 5 $mu$m. The long localization length suggests a marginally insulating nature of the QAH state due to a large number of in-gap states.
The observation of the anomalous quantum Hall effect in exfoliated graphene flakes triggered an explosion of interest in graphene. It was however not observed in high quality epitaxial graphene multilayers grown on silicon carbide substrates. The quantum Hall effect is shown on epitaxial graphene monolayers that were deliberately grown over substrate steps and subjected to harsh processing procedures, demonstrating the robustness of the epitaxial graphene monolayers and the immunity of their transport properties to temperature, contamination and substrate imperfections. The mobility of the monolayer C-face sample is 19,000 cm^2/Vs. This is an important step towards the realization of epitaxial graphene based electronics.
The quantum anomalous Hall (QAH) effect is a quintessential consequence of non-zero Berry curvature in momentum-space. The QAH insulator harbors dissipation-free chiral edge states in the absence of an external magnetic field. On the other hand, the topological Hall (TH) effect, a transport hallmark of the chiral spin textures, is a consequence of real-space Berry curvature. While both the QAH and TH effects have been reported separately, their coexistence, a manifestation of entangled chiral edge states and chiral spin textures, has not been reported. Here, by inserting a TI layer between two magnetic TI layers to form a sandwich heterostructure, we realized a concurrence of the TH effect and the QAH effect through electric field gating. The TH effect is probed by bulk carriers, while the QAH effect is characterized by chiral edge states. The appearance of TH effect in the QAH insulating regime is the consequence of chiral magnetic domain walls that result from the gate-induced Dzyaloshinskii-Moriya interaction and occur during the magnetization reversal process in the magnetic TI sandwich samples. The coexistence of chiral edge states and chiral spin textures potentially provides a unique platform for proof-of-concept dissipationless spin-textured spintronic applications.
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