Stable Dirac semimetal in the allotropes of group-IV elements

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Stable Dirac semimetal in the allotropes of group-IV elements

Wendong Cao, Peizhe Tang, Shou-Cheng Zhang, Wenhui Duan, and Angel Rubio

Phys. Rev. B 93, 241117(R) – Published 27 June 2016

Abstract

Three-dimensional topological Dirac semimetals represent a novel state of quantum matter with exotic electronic properties, in which a pair of Dirac points with a linear dispersion along all momentum directions exists in the bulk. Herein, by using first-principles calculations, we discover the metastable allotropes of Ge and Sn in the staggered layered dumbbell structure, named germancite and stancite, to be Dirac semimetals with a pair of Dirac points on their rotation axis. On the surface parallel to the rotation axis, a pair of topologically nontrivial Fermi arcs are observed and a Lifshitz transition is found by tuning the Fermi level. Furthermore, the quantum thin film of germancite is found to be an intrinsic quantum spin Hall insulator. These discoveries suggest novel physical properties and future applications of the metastable allotrope of Ge and Sn.

Received 10 January 2016

DOI:https://doi.org/10.1103/PhysRevB.93.241117

Physics Subject Headings (PhySH)

Dirac fermions

Topological materials

First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Wendong Cao¹, Peizhe Tang^2,*, Shou-Cheng Zhang², Wenhui Duan^1,3,4,†, and Angel Rubio^5,6,‡

¹State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China
²Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA
³Institute for Advanced Study, Tsinghua University, Beijing 100084, People's Republic of China
⁴Collaborative Innovation Center of Quantum Matter, Beijing 100084, People's Republic of China
⁵Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
⁶Nano-Bio Spectroscopy group, Dpto. Física de Materiales, Universidad del País Vasco, Centro de Física de Materiales CSIC-UPV/EHU-MPC and DIPC, Av. Tolosa 72, E-20018 San Sebastián, Spain

^*peizhet@stanford.edu
^†dwh@phys.tsinghua.edu.cn
^‡angel.rubio@mpsd.mpg.de

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Issue

Vol. 93, Iss. 24 — 15 June 2016

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Images

Figure 1
(a) The unit cell of the SLD structure with three private lattice vectors set as $a_{1, 2, 3}$ . The balls in different colors stand for the same kind of atoms in different layers. (b) The side view and (c) top view of the SLD structure. The layers containing dumbbell (DB) structures are labeled. The letters $(A, B, C)$ denote the positions of DB sites and the sign of bar is applied to distinguish between two trigonal lattices transformed to each other by inversion. As an example, the top view of two adjacent layers (marked by dashed blue lines) is shown. The DB structures are labeled by the grey shadow shown in the top view of a single layer, and the atoms in one DB structure are represented by grey balls. (d) The 3D Brillouin zone (BZ) of germancite and stancite. The four inequivalent TRIM points are $Γ$ (0,0,0), $L$ (0, $π$ ,0), $F (π, π$ ,0), and T $(π, π, π)$ . The hexagon and square, connected to $Γ$ by blue lines, show the 2D BZs projected to (111) and $(2 \bar{1} \bar{1})$ surfaces respectively, and the high-symmetry $k$ points are labeled. (e) The phonon dispersion of germancite and stancite along high-symmetry lines of 3D BZ.
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Figure 2
(a) The band structures of germancite (left) and stancite (right) along high-symmetry lines with the corresponding DOS around the Fermi level (dashed horizontal line). In the $k$ path $T − Γ$ , the size of the red dots represents the contribution from the atomic $s$ and $p_{z}$ orbitals. The cyan dots are the Dirac points at (0,0, $k_{z 0}$ ), where $k_{z 0} \approx 0.08$ and $\approx 0.18 Å^{- 1}$ , respectively. Shaded regions denote the calculated energy spectrum without SOC. (b) Schematic diagrams of the lowest conduction bands and highest valence bands from the $T$ point to the $Γ$ point for germancite and stancite. The black lines present the SOC effect at the $T$ and $Γ$ points. Between them, the red and blue lines denote doubly degenerate bands belonging to different irreducible representations, where the solid/dashed red line is for germancite/stancite. And the crossing points (solid cyan dots) correspond to those gapless Dirac points in (a), respectively. (c) Schematic band dispersion based on the effective $k \cdot p$ model for germancite and stancite. The $k_{⊥}$ direction refers to any axis perpendicular to the $k_{z}$ direction in the momentum space and the color becomes warmer as the energy increases.
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Figure 3
The electronic spectrum on the $(111)$ surface and its corresponding Fermi surface for (a) germanctie and (d) stancite, respectively. Two bulk DPs are projected to the $\tilde{Γ}$ point. The electronic spectrum on the $(2 \bar{1} \bar{1})$ surface and its corresponding Fermi surface for (b) germanctie and (e) stancite, respectively. The cyan dots label the projected DPs and the yellow dot represents the band crossing at the $\bar{Γ}$ point. On the Fermi surface, the Fermi arcs connect two projected DPs (cyan dots). For the stancite $(2 \bar{1} \bar{1})$ surface, the constant-energy contour is at $ε_{f} - 5.2$ meV, slightly away from the Fermi level, to distinguish the Fermi arcs. Stacking plots of constant-energy contours at different energies on the $(2 \bar{1} \bar{1})$ surface of (c) germanctie and (f) stancite, respectively. The Fermi level is set to be zero.
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Figure 4
(a) Band structure of 16-layer germancite (111) film. The topologically nontrivial gap at the $\tilde{Γ}$ can be seen in the inset. (b) Schematic device consisting of three germancite thin films with different thickness. The middle one is a QSH insulator, whereas the other two are topologically trivial. In the lower panel, the purple and green vectors stand for the spin-polarized current at the interfaces.
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