Exploration and prediction of topological electronic materials based on first-principles calculations
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Introduction In recent years, we have witnessed rapid progress in topological electronic materials, particularly topological insulators (TIs)—a new state of quantum matter with great potential for applications in future and emerging electronic devices and technologies. Topology, a word mostly used in mathematics, is now frequently used to describe and classify the electronic structure of materials. “Topological electronic materials” refers to materials whose electronic structures have non-trivial topology in momentum space; examples include TIs,1–3 topological semimetals,4–11 and topological superconductors.2,12–15 One of the most important characteristics of topology is its robustness against local deformations, or, in the language of physics, its insensitivity to environmental perturbations, thus making topological electronic materials promising for future applications. The idea of topological electronic states can be traced back to the discovery and systematic studies of the quantum Hall effect (QHE) in the 1980s,16,17 which suggested that symmetry alone cannot describe all possible phases of condensed matters. New quantum numbers, namely topological numbers, are required to go beyond Landau symmetry-breaking theory, which points out that the state of matter can be described by a local order parameter corresponding to the symmetry broken during the phase transition. Different quantum Hall states all
have the same symmetry, but they are different by a topological invariant, called the Chern number18 or TKNN (Thouless– Kohmoto–Nightingale–den Nijs) number,19 which is usually labeled by an integer Z. It physically corresponds to the number of edge states or the winding number of the electron wave function in the two-dimensional (2D) Brillouin zone (BZ).20 For a long time, however, this conceptual breakthrough did not have much relevance to materials science, because the QHE is due to Landau level formation in 2D electron gases under strong external magnetic field. Under such extreme conditions, details of the material electronic band structure become irrelevant to the physics of QHE. In this sense, the lattice model for QHE proposed by Haldane in 1988 is very stimulating.21 His results suggested that certain materials can have topologically non-trivial electronic band structure characterized by a non-zero Chern number (and called Chern insulator). They can support a similar QHE effect even without an external magnetic field (and of course no Landau levels), which is referred to as the quantum anomalous Hall effect (QAHE). This idea of accessing the topology of band structures in real materials all of a sudden seemed feasible for a wide variety of materials compositions. Haldane’s idea was not realized until recently. The QAHE was theoretically predicted in magnetically doped TIs, such
Hongming Weng, Institute of Physics, Chinese Academy of Sciences, China; [email protected] Xi Dai, Institute of Physics, Chinese Academy of Sciences, China; [email protected] Zhong Fang, Institute of Physics, Chinese Academy of Scien
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