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# Computational design of two‐dimensional topological materials

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The progress in science and technology is largely boosted by the continuous discovery of new materials. In recent years, the state‐of‐art first‐principles computational approach has emerged as a vital tool to enable materials discovery by designing a priori unknown materials as well as unknown properties of existing materials that are subsequently confirmed by experiments. One notable example is the rapid development of the field of topological materials, where new candidates of topological materials are often predicted and/or designed before experimental synthesis and characterization. Topological phases of condensed mater not only represent a significant advance in the fundamental understanding of material properties but also hold promising applications in quantum computing and spintronics. In this article, we will give an overview of recent progress in computational design of two‐dimensional topological materials and an outlook of possible future research directions. WIREs Comput Mol Sci 2017, 7:e1304. doi: 10.1002/wcms.1304

Schematic band structures of two lattice models for designing two‐dimensional topological insulator materials. (a) The first model with a spin‐orbit coupling (SOC) induced band gap opening at the Dirac point. (b) The second model with an SOC induced band inversion between valence and conduction band with opposite parities. + and − labels even and odd parity. The dashed line denotes the Fermi level.
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(a) The honeycomb lattice of Ir atoms in Na2IrO3 . A large black circle shows an Ir atom surrounded by six O atoms (red small circles). (Reprinted with permission from Ref . Copyright 2009 American Physical Society) (b) The top and side view of MX monolayer, respectively. (Reprinted with permission from Ref . Copyright 2015 American Chemical Society) (c) The top (left) and side (middle) view of the crystal structure of Mo2MC2 (M = Ti, Zr, or Hf) MXene displaying the hexagonal unit cell with lattice vector a 1 and a 2. Trigonal lattice with three d (dz2, dxy, $d x 2 ‐ y 2$) orbitals per lattice site (right). (d) Evolution of band structure of trigonal lattice with d orbitals. Normal band order, left; inverted band order without spin‐orbit coupling (SOC), middle; inverted band order with SOC gap, right. (Reprinted with permission from Ref . Copyright 2016 American Chemical Society)
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Atomistic structures of monolayer transition metal dichalcogenides (TMDCs) for (a) 1H‐MX2 and (b) 1T ′‐MX2 . (c) Band structure of 1T ′‐MoS2 . E g is band gap; 2δ is inverted gap. The inset compares band structures with (red dashed line) and without (black solid line) spin‐orbit coupling (SOC). (d) Calculated edge states at Γ point. (Reprinted with permission from Ref . Copyright 2014 American Association for the Advancement of Science) (e) Atomic configuration of (e) square MX2 (S‐MX2 ) and (f) hexagonal MX2 (H′‐MX2 ). (g) The evolution of orbital‐resolved band structures of 1S‐MoS2 near Γ point when considering SOC. The low energy bands near the Fermi level (E f) are mainly contributed from the $d z 2$ and $d x 2 + y 2$ orbitals of transition metal. (Reprinted with permission from Refs and . Copyright 2015 American Physical Society)
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(a) Top view of FeSe/STO, with a checkerboard antiferromagnetic (AFM) spin configuration on Fe atoms. (b) and (c) Scanning tunneling microscope (STM) images for FM edge and AFM edge of FeSe/STO. The inset shows an atomic‐resolution STM image at the bulk position, showing the topmost Se atom arrangement. (d) Theoretical and angle‐resolved photoemission spectroscopy (ARPES) band structure around the M point. (e) 1D band structure of a FeSe/STO ribbon with FM and AFM edges. (f) Local density of states (LDOS) for edge and bulk states. (g) Scanning tunneling spectroscopy (STS) spectra of edge and bulk states for FM and AFM edges. The light blue band indicates the spin‐orbit coupling (SOC) gap. (Reprinted with permission from Ref . Copyright 2016 Nature Publishing Group)
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(a) Experimental and simulated scanning tunneling microscope images for Au grown on GaAs(111) surface, forming a trigonal lattice. (Reprinted with permission from Ref . Copyright 2006 American Institute of Physics) (b) Band structures of Au/GaAs(111) without spin‐orbit coupling (SOC). The inset is a 3D band plotting of the Dirac band. (c) Three fitted Wannier orbitals and the corresponding schematic un‐hybridized surface orbitals. (d) Band structures of Au/GaAs(111) with SOC. (e) 1D topological edges states and their real space distributions. (Reprinted with permission from Ref . Copyright 2016 Nature Publishing Group)
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(a) and (b) Trigonal lattice with three orbitals (s, px, py) per lattice site and its equivalent three sp2 orbitals. (c)–(e) The first‐type band structures with the increasing spin‐orbit coupling (SOC) strength. (f) and (g) The second‐type band structures without and with SOC. (h) Topological phase diagram in the parameter space, showing normal insulator (NI) and quantum spin Hall (QSH) phases. (Reprinted with permission from Ref . Copyright 2016 Nature Publishing Group)
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(a) Schematic epitaxial growth of heavy metal (HM) atoms on the 1/3 halogen covered Si(111) substrate, forming a hexagonal lattice made of HM atoms. (b) The projected density of states (DOS) and schematic hexagonal lattice made of px and py two orbitals on each site. (c) and (d) Band structures of Bi on Si(111)‐$3 × 3$‐Cl [Bi‐Cl/Si(111)] surface without and with spin‐orbit coupling (SOC). Inset shows the Dirac edge states within the SOC gap of Bi‐Cl/Si(111). (Reprinted with permission from Ref . Copyright 2014 National Academy of Sciences)
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(a)–(c) Structural model, band structure and atomic‐orbital projected density of states (DOS), and the energy diagram (at Γ point), respectively, of a buckled Bi(111) bilayer. The band inversion is highlighted by a dashed rectangle. (d)–(f) Same as (a)–(c) for planar Bi hexagonal lattice with one side saturated by H. (Reprinted with permission from Ref . Copyright 2014 National Academy of Sciences)
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(a) Schematic synthesis process from the triphenyl‐metal molecules to the two‐dimensional (2D) organometallic lattices. (b) Band structure and Dirac edge states of triphenyl‐bismuth lattice with spin‐orbit coupling (SOC). (Reprinted with permission from Ref . Copyright 2013 Nature Publishing Group) (c) Schematic illustration and chemical structure of monolayer nickel bis(dithiolene) complex nanosheet. (Reprinted with permission from Ref 61. Copyright 2013 American Chemical Society) (d) Kagome band structure for the flat and Dirac bands, and the quantized spin Hall conductance within the energy window of two SOC gaps. (Reprinted with permission from Ref . Copyright 2013 American Chemical Society) (e) Scanning tunneling microscope image of the hexagonal network of dicyanoanthracene on Cu(111) and the corresponding atomic model. (Reprinted with permission from Ref 62. Copyright 2013 Wiley) (f) Kagome band structure and Dirac edge states within two SOC gaps. (Reprinted with permission from Ref . Copyright 2016 American Chemical Society)
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