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WIREs Comput Mol Sci
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# Emerging topological states in quasi‐two‐dimensional materials

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Inspired by the discovery of graphene, various two‐dimensional (2D) materials have been experimentally realized, which exhibit novel physical properties and support promising applications. Exotic topological states in 2D materials (including quantum spin Hall and quantum anomalous Hall insulators), which are characterized by nontrivial metallic edge states within the insulating bulk gap, have attracted considerable attentions in the past decade due to their great importance for fundamental research and practical applications. They also create a surge of research activities and attract extensive efforts to search for new topological materials in realistic 2D/quasi‐2D systems. This review presents a comprehensive survey of recent progress in designing of topological states in quasi‐2D materials, including various quantum well heterostructures and 2D atomic lattice structures. In particular, the possibilities of constructing topological nontrivial states from commonly used materials are discussed and the ways of enlarging energy gaps of topological states and realizing different topological states in a single material are presented. WIREs Comput Mol Sci 2017, 7:e1296. doi: 10.1002/wcms.1296

(a) The crystal structure of graphene. Carbon atoms arranged in a honeycomb lattice. The Dirac cones are located at the K and K′ points. (b) Band dispersion of graphene without spin‐orbit coupling (SOC) and zoom in of the energy bands close to one of the Dirac points. (c) One‐dimensional energy bands with SOC for a strip of graphene (shown in inset). The bands crossing the gap are spin‐filtered edge states. (d) Schematic diagrams showing (upper panel) two terminal and (lower panel) four terminal measurement geometries. In (upper) a charge current I = (2e 2/h)V flows into the right lead. In (lower) a spin current I s = (e/4π)V flows into the right lead. The diagrams to the right indicate the population of the edge states. Figure (a) reprinted with permission from Ref , copyright 2011 by the American Physical Society. Figure (b) reprinted with permission from Ref , copyright 2009 by the American Physical Society. Figure (c and d) reprinted with permission from Ref , copyright 2005 by the American Physical Society.
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Sample structure and properties of Cr0.15(Bi0.1Sb0.9)1.85Te3 film. (a) A schematic drawing depicting the principle of the quantum anomalous Hall (QAH) effect in a TI thin film with ferromagnetism. (b) A schematic drawing of the expected chemical potential dependence of zero field σ xx and σ xy in the QAH effect. (c) An optical image of a Hall bar device made from a Cr0.15(Bi0.1Sb0.9)1.85Te3 film. (d) Magnetic field dependence of σ yx curves of the Cr0.15(Bi0.1Sb0.9)1.85Te3 film measured at different temperatures (from 80 to 1.5 K). The inset shows the temperature dependence of zero field σ yx , which indicates a Curie temperature of 15 K. Reprinted with permission from Ref , copyright 2013 by the American Association for the Advancement of Science.
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The energy level versus the thickness of the quantum well is shown for (a) A 1 = 0 eVÅ and (b) A 1 = 1.1 eVÅ. Other parameters are taken from Ref . The shaded region indicates the regime for QSH states. The blue dashed line in (b) shows how the crossing between |E 1(H 1)〉 and |H 2(E 2)〉 is changed to anticrossing when A 1 is nonzero. In (c), the density of $| S 1 + 〉$ ($| S 2 − 〉$ has the same density) is plotted for A 1 = 1.1 eVÅ. (d) The band gap and the total parity are plotted as the function of the number of the quintuple layers for Bi2Se3 . The calculation is based on TB model constructed by MLWF from first‐principles calculation. Reprinted with permission from Ref , copyright 2010 by the American Physical Society.
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(a) Oblique view of the crystal structure of an La2MnIrO6 (LMIO) monolayer (ML) sandwiched between LaAlO3 layers. (b) Sketch defining the parameters of the TB model that describes the LMIO ML. Only the local e g orbitals on Mn atoms (purple, at origin) are shown; orbitals on Ir (brown, at the center) are suppressed. The octahedral rotations, denoted by angle θ in (b), are exaggerated for clarity of illustration. (c) Electronic structures of the LMIO/LAO superlattices in a hypothetical structure with 15 rotations about the z axis in the LMIO layers only. Red (blue) color coding highlights the character of the $d z 2$ ($d x 2 − y 2$) orbitals of the Mn atoms, and black indicates the Ir‐5d states. (d) Calculated anomalous Hall conductivity for case (c). Reprinted with permission from Ref , copyright 2014 by the American Physical Society.
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Formation of the honeycomb lattice in a (111) bilayer in the cubic lattice. (a) Perovskite structure ABO3 . (b) A (111) bilayer consisting of the top layer indicated by red circles and the bottom layer indicated by blue circles. The lattice constant is a 0. The bilayer shown as solid lines in (b) forms the honeycomb lattice when projected on the [111] plane with the lattice constant $a ˜ = 2 / 3 a 0$. (c) The real space coordinates are labeled by (x, y, z) in the original cubic lattice, while it is labeled by (X, Y) in the [111] plane. (d) ABO3 monolayer is grown on AO3 ‐terminated ABO3 substrate capped by ABO3 . The direction of crystal growth is indicated by an arrow. Reprinted with permission from Ref , copyright 2011 by the Nature Publishing Group. (e) Density functional theory results of the dispersion relations of the (111) bilayer of LaAuO3 grown between LaAlO3 .
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(a) Atomic structure of (TiO2 )5(VO2 )3 superlattice. (b) Band structure and Berry curvature − Ω(k) along and perpendicular to the diagonal of the Brillouin zone (BZ). (c) Berry curvature − Ω(k) in the entire BZ. (d) Anomalous Hall conductivity σ xy plotted with respect to the position of the Fermi energy E F . The inset shows σ xy with E F varying from − 400 to 400 meV. (e) Energy and momentum dependence of the LDOS at the edge along the [11] direction. Note that the color maps are logarithmic; deep blue patches in (c) are numerical artifacts. Figure (b–e) reprinted with permission from Ref , copyright 2015 by the American Physical Society.
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(a) Schematic diagram showing the structure of an ultrathin Ge layer sandwiched between two thick GaAs layers (upper‐left panel). The upper‐right panel shows an enlarged atomic configuration of the GaAs/Ge/GaAs quantum well. The Ga and As atoms were located at the opposite sides of the interface, which led to charge accumulation. (b) The Brillouin zone (BZ) of the bulk Ge and the folded BZ of the GaAs/Ge/GaAs quantum well along the (111) crystallographic direction. (c) The charge accumulation at the two opposite interfaces obtained from first‐principles calculations. (d) The band structure of the spin Hall bar. The central inset shows a schematic of the edge states. The lower insets show the spatial distributions of the helical edge states. Reprinted with permission from Refs and , copyright 2013 by the American Physical Society.
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(a) Band gap and band offset diagram for asymmetric AlSb/InAs/GaSb quantum wells. The left AlSb barrier layer is connected to a front gate while the right barrier is connected to a back gate. The E 1 subband is localized in the InAs layer and H 1 is localized in the GaSb layer. Outer AlSb barriers provide an overall confining potential for electron and hole states. (b) Schematic band structure diagram. The dashed line shows the crossing of E 1 and H 1 in the inverted regime. Hybridization between E 1 and H 1 opens the gap E g . (c–f) Helical edge transport in meso‐ and macroscale devices. (c) Wide conductance plateaus quantized to 2e 2/h and 4e 2/h, respectively, for two device configurations shown in inset. The inset shows that the plateau persists to 4 K, and conductance increases at higher temperature. (d) Electrical charge transport in large devices is due to edge channels. The inset shows that the resistance scales linearly with the edge length. Figure (a and b) reprinted with permission from Ref , copyright 2008 by the American Physical Society. Figure (c and d) reprinted with permission from Ref , copyright 2015 by the American Physical Society.
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Evolution of band structure and edge states upon increasing spin splitting. For (a) G E < 0 and G H > 0, the spin‐down states |E 1〉 and H 1〉 in the same block of the Hamiltonian first touch each other, and then enter the normal regime. (b) Behavior of the edge states during the level crossing. Figure reprinted with permission from Ref , copyright 2008 by the American Physical Society.
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(a) Bulk energy bands of HgTe and CdTe near the Γ point and schematic illustration of quantum well geometry and lowest subbands for two different thicknesses. (b) Schematic of the spin‐polarized edge channels in a quantum spin Hall insulator. (c) The longitudinal four‐terminal resistance, R 14,23, of various normal (d = 5.5 nm) (I) and inverted (d = 7.3 nm) (II, III, and IV) quantum well structures as a function of the gate voltage measured for B = 0 T at T = 30 mK. Figure (a) reprinted with permission from Ref , copyright 2006 by the American Association for the Advancement of Science. Figure (b and c) reprinted with permission from Ref , copyright 2007 by the American Association for the Advancement of Science.
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(a) Top and side view of the optimized geometries of the TM intercalated graphene/SiC(0001) systems. Yellow, gray, and white balls are Si, C, and H, respectively, while others are TM elements. For clarity, graphene is shown as hexagonal mesh. (b) Schematic view of a two‐step epitaxial growth of 2D hexagonal lattices of TM on a semiconductor substrate: 1/3 ML of halogen (X = Cl, Br, and I) on Si(111) surface are first grown, followed by deposition of TM atoms. (c) Top view and side view of the tripled surface unit cell of MnTe with 2/3 ML Pb. Pb is in green, Mn in gray, Te is dark blue and I in cyan. Figure (a) reprinted with permission from Ref , copyright 2012 by the American Physical Society. Figure (b) reprinted with permission from Ref , copyright 2014 by the American Physical Society. Figure (c) reprinted with permission from Ref , copyright 2013 by the American Physical Society.
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(a) Top view and (b) side view of optimized crystal structure of the functionalized MXene M 2 CO2 and its 2D BZ. (c) The top view and (d) side view of transition metal halide MX monolayer. In a unit cell, MX is related to MX′ by an inversion operation. The inversion center is indicted by red star in (c) and (d). Panel (e) First BZ of MX monolayer and the points of high symmetry. (f–h) The crystal structures of ternary transition‐metal carbide halides TaC X (X = Cl, Br, and I) monolayer: (f) the top view; (g and h) the side view along the a and b axes. The unit cell is indicated by red dashed lines. (i) The BZ of 2D TaC X. The locations of the fundamental gap are marked by red dots and labeled by Λ. Figure (a and b) reprinted with permission from Ref , copyright 2015 by the American Physical Society. Figure (c–e) reprinted with permission from Ref , copyright 2015 American Chemical Society. Figure (f–i) reprinted with permission from Ref .
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(a) Atomistic structure of monolayer transition metal dichalcogenide MX 2 with a 1T′ structure, where the distorted M atoms form one‐dimensional zigzag chains indicated by the dashed blue line. The unit cell is indicated by red rectangles. (b) Crystal structure of monolayer MX 2 allotrope with square‐octagonal lattice. (c) The crystal structure of bulk ZrTe5 (HfTe5 ); the side view, top view, and BZ of single layer structure, respectively. In (c), the inversion center is indicated by the red star symbol, and the waved grid of Te square lattice sheet is shown as the pink dotted lines. Figure (a) reprinted with permission from Ref , copyright 2014 by the American Association for the Advancement of Science. Figure (b) reprinted with permission from Ref , copyright 2015 by the American Physical Society. Figure (c) reprinted with permission from Ref .
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Schematics of 2D organometallic lattice. (a) Top and side view of the 2D organometallic superlattice. Dashed lines show the unit cell and the two metal atoms are labeled with 1 and 2. (b) Atomic structure of the Ni3C12S12 lattice. The solid lines show the unit cell, and the dashed lines outline the kagomé lattice. (c) Schematic atomic structure of Cu‐dicyanoanthracene (DCA) film. The top left inset shows the DCA molecule. The red dashed, blue dashed, and black lines outline the honeycomb lattice by the Cu atoms, the kagomé lattice by the DCA molecules, and the unit cell, respectively. Figure (a) reprinted with permission from Ref . Figure (b) reprinted with permission from Ref , copyright 2013 American Chemical Society. Figure (c) reprinted with permission from Ref , copyright 2016 American Chemical Society.
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(a) The hexagonal unit cell of single crystal bismuth. (b) The top view of the Bi lattice. (c) The first Brillouin zone (BZ) of the hexagonal lattice. (d) and (e) The respective lattice geometries for the full‐hydrogenated and half‐hydrogenated Bi honeycomb monolayers. The red arrows in (e) represent the magnetic moments. The black arrow marks the +z direction. (f–h) Crystal structure of Bi4Br4 in 3D (f), 2D (g), and 1D (h) representations. Figure (a–c) reprinted with permission from Ref , copyright 2011 by the American Physical Society. Figure (d and e) reprinted with permission from Ref , copyright 2015 by the American Physical Society. Figure (f–h) reprinted with permission from Ref , copyright 2014 American Chemical Society.
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(a and b) Crystal structure for (a) stanene and (b) decorated stanene (SnX) from the top (side) view [upper(lower)]. X represents the chemical functional group. (c) Structures of stanene with half passivation by I atoms. For half‐passivated stanene, unpassivated Sn sites exhibit magnetic moments in a triangular lattice (dotted lines), which are indicated by blue arrows. (d and e) Band structure for (a) stanene, (b) fluorinated stanene without (black dash‐dotted lines) and with (red solid lines) spin‐orbital coupling. (f) are band structures and DOS of the half‐passivated stanene with spin‐orbit coupling. (g and h) Band structure of armchair‐edge nanoribbons for (a) stanene and (b) fluorinated stanene. Helical edge states are visualized by red lines crossing linearly at the Γ point for stanene and fluorinated stanene. (i) Calculated local DOS of edge states for a semi‐infinite system of half‐passivated stanene. Figure (a, b, d, e, g, and h) reprinted with permission from Ref , copyright 2013 by the American Physical Society. Figure (c, f, and i) reprinted with permission from Ref , copyright 2014 by the American Physical Society.
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