Two-dimensional (2D) topological insulators (TIs) are special quantum conductors that possess an insulating bulk but metallic edge with quantized charge and spin conductance protected by electron band topology. The concept of topology and TI has not only renewed our fundamental understanding of electronic properties of solid materials, but also opened an exciting avenue towards potential applications of topological quantum devices with minimized heat dissipation and robustness against disorder. In this video article, I will first introduce and review the concept of TIs within the context of transport properties of solid-state materials. I will then use two examples, the organic 2D TIs and the surface-based 2D topological states, to recap the rapid theoretical and experimental developments made in this emerging field. The existence of quantized edge conductance and topological edge states of 2D TIs has been so far confirmed experimentally in several systems, such as semiconductor quantum wells, 2D transition metal dichalcogenides, metallic overlayer of bismuth on a semiconductor surface. However, discovery of high-temperature 2D TIs and construction of functional TI-based quantum devices remain largely elusive. At the end of this video article, I will offer briefly my personal perspective and possible future directions in low-dimensional topological materials.
In this video review article, I will first introduce the concept of topological insulator (TI) within the context of electron transport in solid materials. In terms of electron transport, one used to classify solid materials into three different classes: metal, semiconductor, and insulator, as illustrated in Fig. 1 of slide 3 in the video article. Interestingly, TI offers a new kind of conductor with an insulating bulk but conducting boundary (edge in 2D), as a material analog of quantum Hall effect. It exhibits a zero longitudinal conductance and a quantized transverse Hall conductance defined by Chern number, as illustrated in Fig. 2 of slide 6 in the video article.
Fig. 1. A slide illustrating the band structure of metal, semiconductor and insulator, and their respective electron density distribution in real space. Note the direct correlation between charge localization in real space and band gap opening in momentum space.
Fig. 2. A slide illustrating the analogy between TI and quantum Hall effect, in terms of conductivity tensor.
I will then review the research progress made in the 2D TIs in recent years, focusing on two systems that I am personally involved with the most. One is 2D organic TIs, and the other is the surface-based topological states, as illustrated in Fig. 3 of the slide 14 and Fig. 4 of the slide 40, respectively, in the video article.
Fig. 3. A slide highlighting three classes of 2D organic topological materials: MOFs, COFs and HOFs. The discussion in this video article will focus on MOF-based systems.
Fig. 4. A slide highlighting the idea of surface-based topological states, such as to create surface-based 2D TI by “Epitaxial Promotion of Topological States”. It means the freestanding 2D layer and substrate surface are originally trivial, but become topological after the 2D layer is grown on the substrate, as discussed in this video article.
This video article concludes with a brief summary and the author’s personal perspective on the rapidly developing fields of topological materials, as exemplified in Fig. 5 of slide 60 in the video article.
Fig. 5. A summary slide showing schematically that there are generally three ingredients in designing topological bands and materials: lattice symmetry, spin-orbit coupling, and orbital composition at the Fermi level. Many works remain to be done towards realizing topological quantum devices in the future.
Feng Liu would like to thank all the collaborators who contributed to the works reviewed in this video article. He acknowledges US Department of Energy (DOE)-Basic Energy Sciences (Grant No. DE-FG02- 04ER46148) as the primary funding support for his research group during the last two decades.
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Comment #1 by reviewer:
On slide 9, there is a statement “Quantum spin Hall effect can be viewed as a two copies of QAHE”. Since the concept of QAHE is not mentioned before this slide, the audience may not be able to understand what it means. QSHE could also be described as two superimposed quantum Hall effects under opposite magnetic fields. It might be easier to describe this with a cartoon.
Author’s response:
Thanks for this comment. The concept of QAHE is introduced earlier in slide 7 discussing Haldane’s model of Chern insulator by inserting a Berry flux into a hexagon of a hexagonal lattice. In slide 8, Kane-Mele model of QSHE is discussed by adding the SOC effect in a hexagonal lattice which effectively inserts one Berry flux in each spin channel in opposite direction. In this sense, one can see that “QSHE can be viewed as two copies of QAHE”. Or at the referee put it that QSHE can also be described by two superimposed QHE under opposite magnetic field.
Comment #2 by reviewer:
When the AFM quantum spin Hall effect in FeSe monolayer was discussed, both AFM and FM edge states were mentioned. I am confused about these concepts. Here more discussions should be given. I also couldn’t follow the discussions which indicate the predicted AFM QSHI is supported by the STS data. Since there are much data presented in that slide, it would be helpful to the audience if Prof. Liu could point out which specific data provides key evidence for the AFM QSHI.
Author’s response:
Thanks for this comment. In slide 37, Figure b indicates the AFM edges are along the diagonal direction, while the FM edges are along the x- and y-direction. In a conventional QSH insulator, the dispersion of topological edge states is usually independent of edge orientation. However, due to different magnetic order, the dispersion of topological edge stages of AFM QSH insulator is different along the AFM versus FM edge, which were shown in slide 38 by theoretical calculations. This interesting difference in topological edge states can be used as a unique signature to experimentally detect an AFM QSH insulator, which was shown in slide 39 where the integrated edge states around the bulk topological gap for AFM versus FM edges are compared between theory and experimental STS measurements, showing very good agreement.
Comment #3 by reviewer:
At one slide, Prof. Liu mentioned “magnetic translation” and “flux translation". What do they refer to? Clear definitions should be given for these concepts.
Author’s response:
Thanks for this comment. As mentioned in the response to comment #2, in Haldane’s model of Chern insulator, a Berry flux is imagined to be inserted into a hexagon of a hexagonal lattice, and by translation symmetry, one can view this flux is periodically translated over the infinite lattice, namely the “flux translation”. Similarly, in the classical QHE experiment, a strong perpendicular magnetic field is applied to a 2D electron gas inducing a magnetic flux. Topology arises when the magnetic length (or the radius of magnetic flux) is comparable with the underlying lattice constant, giving rise to a strong interplay between “magnetic translation” and lattice “translation”, as illustrated in slide 5.
Comment #4 by reviewer:
Near the end of presentation, Prof. Liu pointed out that TI can be present even in amorphous materials or liquid. It would be useful to the audience if some possible examples could be given.
Author’s response:
Thanks for this comment. One specific example of amorphous 2D TI is theoretically predicted in amorphous stanane, a hydrogenated monolayer α-Sn [Wang et. al., Phys. Rev. Lett. 128, 056401 (2022)], awaiting for experimental confirmation.
