$s{d}^{2}$ Graphene: Kagome Band in a Hexagonal Lattice

$s d^{2}$ Graphene: Kagome Band in a Hexagonal Lattice

Miao Zhou, Zheng Liu, Wenmei Ming, Zhengfei Wang, and Feng Liu

Phys. Rev. Lett. 113, 236802 – Published 2 December 2014

Abstract

Graphene, made of $s p^{2}$ hybridized carbon, is characterized with a Dirac band, representative of its underlying 2D hexagonal lattice. The fundamental understanding of graphene has recently spurred a surge in the search for 2D topological quantum phases in solid-state materials. Here, we propose a new form of 2D material, consisting of $s d^{2}$ hybridized transition metal atoms in hexagonal lattice, called $s d^{2}$ “graphene.” The $s d^{2}$ graphene is characterized by bond-centered electronic hopping, which transforms the apparent atomic hexagonal lattice into the physics of a kagome lattice that may exhibit a wide range of topological quantum phases. Based on first-principles calculations, room-temperature quantum anomalous Hall states with an energy gap of $\sim 0.1 eV$ are demonstrated for one such lattice made of W, which can be epitaxially grown on a semiconductor surface of $1 / 3$ monolayer Cl-covered Si(111), with high thermodynamic and kinetic stability.

Received 28 July 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.236802

Authors & Affiliations

Miao Zhou¹, Zheng Liu¹, Wenmei Ming¹, Zhengfei Wang¹, and Feng Liu^1,2,*

¹Department of Materials Science and Engineering, University of Utah, Utah 84112, USA
²Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

^*Corresponding author. fliu@eng.utah.edu

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Vol. 113, Iss. 23 — 5 December 2014

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Images

Figure 1
Schematic illustration of structural and electronic properties of (a) $s p^{2}$ graphene made of C and (b) $s d^{2}$ graphene made of TM. NN hopping ( $t$ ) of C $p_{z}$ orbital in a hexagonal lattice [(a), left-hand panel] that leads to a Dirac band [(a), right-hand panel] is indicated. Also, NN hopping of a TM bond-centered $s d^{2}$ $σ$ state in a hexagonal lattice [(b), left-hand panel], which can be renormalized into NN hoping in a kagome lattice [red dots and dashed lines in (b), left-hand panel] that leads to a kagome band [(b), right-hand panel] is indicated.
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Figure 2
Band structures of W@Cl-Si(111) surface for (a) spin-up and (b) spin-down bands. The Fermi level is set at zero. Band compositions are encoded by color: red for W and blue for Si. Inset of (a) shows the optimized atomic structure of the surface, with purple, green, and yellow balls representing W, Cl, and Si atoms, respectively. (c) 3D band structures of the spin-down component around the Fermi level. Inset shows the surface Brillouin zone. (d) 2D energy dispersion contour of conduction (left-hand panel) and valence band (right-hand panel) cutting at $E = \pm 0.1 eV$ . (e) Group velocity for electrons and holes measured by the velocity along the $K Γ$ direction ( $V_{k} / V_{K Γ}$ ) versus angle ( $θ_{k}$ ) of the $\vec{K}$ vector. Along $K Γ$ , $θ_{K} = 90 °$ , $V_{k}$ is maximum; along $K M$ , $θ_{k} = 30 °$ , $V_{k}$ is minimum, as indicated by two arrows.
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Figure 3
(a) The calculated Wannier functions showing the characteristics of $s$ and $d$ orbitals arranged in order of their energy. (b) Schematics of $s d^{2}$ hybridization giving rise to the bonding ( $σ$ ) and antibonding ( $σ^{*}$ ) state, with the corresponding $σ$ -state Wannier function indicated. (c) Kagome lattice formed by the bond-centered $σ$ state as basis in a hexagonal lattice. The NN, 2NN, and 3NN hopping ( $t_{1}$ , $t_{2}$ , and $t_{3}$ ) are indicated. (d) The model band structure obtained from Eq. (1), illustrating different phases in the parameter space of hopping, magnetization, and SOC. Chern number ( $C$ ) is indicated in V.
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Figure 4
(a) Band structures of W@Cl-Si(111) with SOC. The color scheme is the same as in Fig. 2. An energy gap of 0.1 eV is opened at the $K$ point, with the Chern number of all the occupied bands equal to $- 1$ . (b) Anomalous Hall conductance (AHC) around the Fermi level ( $E_{F}$ ), in units of $e^{2} / h$ .
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Figure 5
(a) Schematic view of a two-step epitaxial growth of 2D hexagonal lattices of TM on a semiconductor substrate: We first grow $1 / 3$ ML of halogen ( $X = Cl$ , Br, I) on Si(111) surface, followed by deposition of TM atoms. (b),(c) Top and side view of the proposed structure, respectively, with the surface unit cell vector ( $a_{1}$ , $a_{2}$ ) indicated in (b) and the adsorption length $d$ in (c).
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