It is well established that symmetry has an important influence on the properties of materials, but the topology of electronic states might be an even more fundamental property. Topological insulators (TIs) are new states of matter based on the topology in the electronic band structure. Relativistic effects are the origin of the topologically non-trivial electronic structure, and the new state of matter has been realized in two-dimensional quantum well structures and three-dimensional bulk crystals of heavy elements and compounds. TI materials have an insulating gap in the bulk, and robust metallic edge/surface states on the boundary, which is robust against disorder and leads to unique spin and charge transport properties. Examples of TIs include HgTe/CdTe quantum wells, Bi-Sb-alloys, Bi2Se3, and half-Heusler compounds.

The class of topological insulator materials is one of the frontier topics of condensed matter physics. The great success of this field is due to the conceptual breakthroughs in theories for topological electronic states and is strongly motivated by the rich variety of material realizations, thus making the theories testable, the experiments operable, and the applications possible. First-principles calculations have demonstrated unprecedented predictive power for material selection and design. In this article, we review recent progress in this field with a focus on the role of first-principles calculations. In particular, we introduce the Wilson loop method for the determination of topological invariants and discuss the band inversion mechanism for the selection of topological materials. Recent progress in quantum anomalous Hall insulators, large-gap quantum spin Hall insulators, and correlated topological insulators is also covered.

Ternary semiconducting or metallic half-Heusler compounds with an atomic composition 1:1:1 are widely studied for their flexible electronic properties and functionalities. Recently, a new material property of half-Heusler compounds was predicted based on electronic structure calculations: the topological insulator. In topological insulators, the metallic surface states are protected from impurity backscattering due to spin-momentum locking. This opens up new perspectives in engineering multifunctional materials. In this article, we introduce half-Heusler materials from the crystallographic and electronic structure point of view. We present an effective model Hamiltonian from which the topological state can be derived, notably from a non-trivial inverted band structure. We discuss general implications of the inverted band structure with a focus on the detection of the topological surface states in experiments by reviewing several exemplary materials. Special attention is given to superconducting half-Heusler materials, which have attracted ample attention as a platform for non-centrosymmetric and topological superconductivity.

A wide class of materials that were discovered to carry a topologically protected phase order has led to a highly active area of research called topological insulators (TIs). This phenomenon has radically changed our thinking because of the robust quantum coherent behavior showing two-dimensional Dirac-type metallic surface states (SSs) and simultaneously insulating bulk states. The Dirac SSs are induced by the strong spin–orbit coupling as well as protected by time reversal symmetry (TRS). Breaking TRS in a TI with ferromagnetic perturbation can lead to many exotic quantum phenomena, such as the quantum anomalous Hall effect, topological magnetoelectric effect, as well as image magnetic monopole. This article presents an overview of the current status of TRS breaking in TIs and outlines the prospects for future studies.

Electrons in topological insulators possess unique electronic band structures and spin properties, promising a novel route to engineer material properties for electronics and energy science. Enhancing the surface state signal in electron transport is critical for both fundamental study of the surface states and future applications. Nanostructures of topological insulators naturally have large surface-to-volume ratios, effectively increasing the surface transport compared to the bulk contribution. Moreover, the unique morphology of topological insulator nanostructures results in various quantum effects of electronic states, which can tailor the surface band via quantum confinement. Here we review recent progress in topological insulator nanostructures. Material design and electron transport of topological insulator nanostructures are introduced, with an emphasis on the unique properties of nanostructures. A few examples of applications and future perspective in using these nanostructures are also discussed.

The ability to image and quantify material behavior in real time at nano to near-macro length scales, preferably in three dimensions, is a crucial feature of modern materials science. Here, we examine such an approach to characterize the mechanical properties of three diverse classes of materials: (1) biological materials, principally bone, using both in situ small-/wide-angle x-ray scattering/diffraction to probe nanoscale deformation behavior and x-ray computed microtomography to study microscale damage mechanisms; (2) biomimetic materials, specifically a nacre-like ceramic, where microtomography is used to identify toughening mechanisms; (3) synthetic materials, specifically ceramic textile composites, using in situ microtomography to quantify the salient mechanical damage at ultrahigh temperatures. The mechanistic insights for the understanding of damage evolution and fracture afforded by these techniques are undeniable; as such, they can help provide a basis for the achievement of enhanced damage tolerance in structural materials.