Materials and their properties

Many different experimental techniques have been applied to study TI materials, including transport measurements, angle-resolved photo-emission spectroscopy (ARPES), scanning tunneling microscopy (STM), and optical conductivity. For three-dimensional TIs, ARPES allows for the most direct measurement of the dispersion of surface states, which can be used to determine the topological nature of the material. For two-dimensional TIs, the edge states have been studied in various transport experiments. As examples of the interesting physical properties of TIs, in Figure 2 we show some representative experimental data on (a) the conductance of the two-dimensional QSH state in HgTe^{Reference König, Wiedmann, Brüne, Roth, Buhmann, Molenkamp, Qi and Zhang12} and (b) three-dimensional Bi₂Te₃ and Bi₂Se₃ surface states observed in ARPES.^{Reference Xia, Qian, Hsieh, Wray, Pal, Lin, Bansil, Grauer, Hor, Cava and Hasan13,Reference Chen, Analytis, Chu, Liu, Mo, Qi, Zhang, Lu, Dai, Fang, Zhang, Fisher, Hussain and Shen14}

Figure 2. (a) Conductance of HgTe/CdTe quantum wells in the quantum spin Hall phase (II, III, IV) and in the trivial insulating phase (I).^{Reference König, Wiedmann, Brüne, Roth, Buhmann, Molenkamp, Qi and Zhang12} (b) Angle-resolved photo-emission spectroscopy (ARPES) experiments showing the surface state dispersion of Bi₂Se₃ (left¹³) and Bi₂Te₃ (right¹⁴). (c) Hall conductance as a function of magnetic field for Cr-doped Bi(Sb)₂Te(Se)₃ thin film, showing the quantized anomalous Hall effect around zero external field.^{Reference Chang, Zhang, Feng, Shen, Zhang, Guo, Li, Ou, Wei, Wang, Ji, Feng, Ji, Chen, Jia, Dai, Fang, Zhang, He, Wang, Lu, Ma and Xue20}

New physical effects, such as the quantum anomalous Hall effect (QAHE),^{Reference Liu, Qi, Dai, Fang and Zhang15,Reference Yu, Zhang, Zhang, Zhang, Dai and Fang16} topological magnetoelectric effect,^{Reference Qi, Hughes and Zhang17,Reference Essin, Moore and Vanderbilt18} and topological surface superconductivity,^{Reference Fu and Kane19} have been proposed for TIs, thus making TIs interesting candidate systems for new spintronics devices and topological quantum computation. In particular, the QAHE (i.e., an integer quantized Hall effect without an external magnetic field) has recently been experimentally realized in magnetically doped TI thin films in the Bi₂Se₃ family.^{Reference Chang, Zhang, Feng, Shen, Zhang, Guo, Li, Ou, Wei, Wang, Ji, Feng, Ji, Chen, Jia, Dai, Fang, Zhang, He, Wang, Lu, Ma and Xue20} This is a direct physical manifestation of the topological nature of the TI surface states. Figure 2c shows the Hall conductance as a function of magnetic field for the Cr-doped Bi(Sb)₂Te(Se)₃ thin film and the QAHE around zero external field.^{Reference Chang, Zhang, Feng, Shen, Zhang, Guo, Li, Ou, Wei, Wang, Ji, Feng, Ji, Chen, Jia, Dai, Fang, Zhang, He, Wang, Lu, Ma and Xue20}

Research on TIs has recently become one of the most active new fields in condensed matter physics. Materials science efforts are essential for the realization of the physical effects and device concepts proposed for TIs, and for the discovery of new TI materials. The articles in this issue of MRS Bulletin aim to provide an overview of the field of TIs for materials scientists and to inspire new theoretical and experimental developments in this exciting field of research.

One fascinating aspect of TIs is that theory has been providing reliable guidance to materials discoveries.^{Reference Yan and Zhang21,Reference Müchler, Zhang, Chadov, Yan, Casper, Kübler, Zhang and Felser22} Many TIs have first been predicted theoretically and have subsequently been realized experimentally. These include the two-dimensional TIs–HgTe quantum wells and InAs/GaSb quantum wells, and the three-dimensional TIs–Bi_xSb_1–x, Bi₂Se₃, Bi₂Te₃, Sb₂Te₃, TlBiSe₂, and TlBiTe₂.^{Reference Bernevig, Hughes and Zhang4,Reference König, Wiedmann, Brüne, Roth, Buhmann, Molenkamp, Qi and Zhang12,Reference Xia, Qian, Hsieh, Wray, Pal, Lin, Bansil, Grauer, Hor, Cava and Hasan13,Reference Fu and Kane19,Reference Liu, Hughes, Qi, Wang and Zhang23–Reference Takafumi, Segawa, Guo, Sugawara, Souma, Takahashi and Ando29} Potential TI candidates can be preselected by chemical intuition and by using some simple criteria such as semiconductors containing heavy elements. Density functional theory calculations can be applied to predict whether a given material can be a TI. Several different criteria have been proposed for determining the topological nature of a band structure, including surface state calculations,^{Reference Qi, Hughes and Zhang17,Reference Dai, Hughes, Qi, Fang and Zhang30} parity inversion counting at high symmetry points in the Brillouin zone,^{Reference Bernevig, Hughes and Zhang4,Reference Fu and Kane19,Reference Müchler, Zhang, Chadov, Yan, Casper, Kübler, Zhang and Felser22} and Wilson loop spectral flow.^{Reference Yu, Qi, Bernevig, Fang and Dai31}

For systems with inversion symmetry, the Bloch states at the inversion symmetric momenta, such as (0,0,0) and (π,π,π), are inversion symmetric and have a definite parity. The parity inversion criteria then counts the parity of occupied states at these points, and looks for “band inversion” defined by the difference between the parity eigenvalues of states at different inversion symmetric momenta. Since this method only applies to inversion symmetric systems, other criteria have been proposed for more generic systems without inversion symmetry. Among them, the Wilson loop spectral flow method is an intuitive way of tracking the motion of bulk electrons. This method finds the center-of-mass position of “hybrid Wannier states,” which are local in real space in one direction (e.g., x direction) and are plane waves in perpendicular directions. The dependence of the center-of-mass position in x to perpendicular momenta describes how bulk electrons move in a perpendicular electric field, and characterize the topological nature of the system. The article by Weng et al. in this issue reviews the prediction and exploration of TIs by the first-principles calculation approach. Different methods of calculating topological invariants are reviewed, with an emphasis on the Wilson loop approach. The authors also provide an overview of recent progress in TI-related materials, including quantum anomalous Hall insulators, large-gap QSH insulators, and correlated TI.

A large number of TIs have been proposed thus far, and many of them have been realized experimentally. Table I lists various TI materials grouped into different families based on their structure, electronic structure, and bonding. The predicted materials cover nearly all areas of condensed matter physics. Among the proposed TIs, there is a large family of half-Heusler compounds, which are multifunctional materials combining properties such as superconductivity and magnetism with the topological band inversion.^{Reference Chadov, Qi, Kübler, Fecher, Felser and Zhang32,Reference Lin, Wray, Xia, Xu, Jia, Cava, Bansil and Hasan33} Yan and de Visser’s article in this issue provides an overview of half-Heusler TIs, including a summary of effective model description and physical properties. An interesting aspect of half-Heusler TIs is that several materials in this family become non-centrosymmetric superconductors at low temperature. The combination of topological band structure and non-centrosymmetric superconductivity may lead to another interesting topological phase—time-reversal invariant topological superconductors with the appearance of Majorana fermions.^{Reference Fu and Kane19} Majorana fermions are fermionic particles that are expected to be their own antiparticles and are potential building blocks for quantum computation. The theoretical discussion about this possibility is also reviewed in the article by Yan and de Visser.

Table I. Proposed topological insulator materials grouped into several different material classes.^{Reference Bernevig, Hughes and Zhang4,Reference König, Wiedmann, Brüne, Roth, Buhmann, Molenkamp, Qi and Zhang12,Reference Xia, Qian, Hsieh, Wray, Pal, Lin, Bansil, Grauer, Hor, Cava and Hasan13,Reference Fu and Kane19,Reference Liu, Hughes, Qi, Wang and Zhang23–Reference Takafumi, Segawa, Guo, Sugawara, Souma, Takahashi and Ando29}

From the list of proposed materials in Table I and beyond, only a limited number have been synthesized, and an even smaller number of them have been proven to be TIs. The need for more high-quality materials in the form of single crystals, nanoparticles, thin films, and in devices is obvious. In the other two articles in this issue, two different growth methods for TIs are reviewed. The article by Chang et al. gives an overview of the current status of thin-film research for TIs with a special focus on time reversal symmetry breaking with ferromagnetic perturbation. The observation of the QAHE in Cr-doped selenides was a breakthrough experiment in 2013.^{Reference Chang, Zhang, Feng, Shen, Zhang, Guo, Li, Ou, Wei, Wang, Ji, Feng, Ji, Chen, Jia, Dai, Fang, Zhang, He, Wang, Lu, Ma and Xue20} The authors compare the physics of different doped films of TIs and heterostructures of TIs with ferromagnetic insulators and outline the prospects for future studies.

The article by Hong et al. gives an overview of TI nanostructures. Compared to bulk materials, nanostructures have a higher surface-to-bulk ratio, which allows topological properties of the surface states to be observed more clearly. This article reviews the growth of physical properties of TI nanostructures, including Bi_2–xSb_xTe_3–ySe_y nanoribbons. An important physical property of the nanoribbon is the Aharonov-Bohm (AB) effect, which demonstrates the existence of quantum coherent surface states. In addition, the AB effect in this system also shows qualitative differences from that in other systems, such as carbon nanotubes and semiconductor nanowires, as a consequence of the unique spin dynamics of the topological surface states. Experimental results on the AB effect in TI nanoribbons and the corresponding theoretical analysis are reviewed in the article by Hong et al.