Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that are different from conventional insulators. This distinction is characterized by ${Z}_{2}$ topological invariants, which characterize the ground state. In two dimensions, there is a single ${Z}_{2}$ invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four ${Z}_{2}$ invariants that distinguish the ordinary insulator from ``weak'' and ``strong'' topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the two-dimensional quantum spin-Hall phase and the three-dimensional strong topological insulator, these states are robust and are insensitive to weak disorder and interactions. In this paper, we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the ${Z}_{2}$ invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials that are strong topological insulators, including the semiconducting alloy ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ as well as $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Sn}$ and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments.

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Topological insulators with inversion symmetry

Topologically protected states of matter are receiving widespread attention owing to their unusual electronic properties Using numerical simulations, this study predicts that tin telluride is a physical realization of a new class of materials termed topological crystalline insulators

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Topological crystalline insulators in the SnTe material class

Distortions of the electronic density of states (DOS) are a potent mechanism to increase the thermopower of thermoelectric semiconductors, thereby increasing their power factor. We review band-structure engineering approaches that have been used to achieve this, resonant impurity levels, dilute Kondo effects, and hybridization effects in strongly correlated electron systems. These can increase the thermoelectric power of metals and semiconductors through two mechanisms: (1) the added density of states increases the thermopower in a nearly temperature-independent way; (2) resonant scattering results in a strong electron energy filtering effect that increases the thermopower at cryogenic temperatures where the electron–phonon interactions are weaker. Electronic structure calculation results for Tl:PbTe and Ti:PbTe are contrasted and identify the origin of the thermopower enhancement in Tl:PbTe. This leads to a discussion of the conditions for DOS distortions to produce thermopower enhancements and illustrates the existence of an optimal degree of delocalization of the impurity states. The experimentally observed resonant levels in several III–V, II–VI, IV–VI and V2-VI3 compound semiconductor systems are reviewed.

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Resonant levels in bulk thermoelectric semiconductors

Topological insulators are a class of quantum materials in which time-reversal symmetry, relativistic effects and an inverted band structure result in the occurrence of electronic metallic states on the surfaces of insulating bulk crystals. These helical states exhibit a Dirac-like energy dispersion across the bulk bandgap, and they are topologically protected. Recent theoretical results have suggested the existence of topological crystalline insulators (TCIs), a class of topological insulators in which crystalline symmetry replaces the role of time-reversal symmetry in ensuring topological protection. In this study we show that the narrow-gap semiconductor Pb(1-x)Sn(x)Se is a TCI for x  =  0.23. Temperature-dependent angle-resolved photoelectron spectroscopy demonstrates that the material undergoes a temperature-driven topological phase transition from a trivial insulator to a TCI. These experimental findings add a new class to the family of topological insulators, and we anticipate that they will lead to a considerable body of further research as well as detailed studies of topological phase transitions.

Topological crystalline insulator states in Pb1-xSnxSe

Stable ferroelectricity with high transition temperature in nanostructures is needed for miniaturizing ferroelectric devices. Here, we report the discovery of the stable in-plane spontaneous polarization in atomic-thick tin telluride (SnTe), down to a 1–unit cell (UC) limit. The ferroelectric transition temperature T c of 1-UC SnTe film is greatly enhanced from the bulk value of 98 kelvin and reaches as high as 270 kelvin. Moreover, 2- to 4-UC SnTe films show robust ferroelectricity at room temperature. The interplay between semiconducting properties and ferroelectricity in this two-dimensional material may enable a wide range of applications in nonvolatile high-density memories, nanosensors, and electronics.

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A. J. Strauss

Papers

Band Structure and Laser Action in Pb x Sn 1-x Te

Low Electron Effective Masses and Energy Gap in Cd x Hg 1-x Te