Type-I and type-II nodal lines coexistence in the antiferromagnetic monolayer ${\mathrm{CrAs}}_{2}$

Type-I and type-II nodal lines coexistence in the antiferromagnetic monolayer ${CrAs}_{2}$

Busheng Wang, Heng Gao, Qing Lu, WenHui Xie, Yanfeng Ge, Yong-Hong Zhao, Kaicheng Zhang, and Yong Liu

Phys. Rev. B 98, 115164 – Published 28 September 2018

Abstract

Topological nodal line semimetals, hosting one-dimensional Fermi lines with symmetry protection, have become a hot topic in topological quantum matter. Due to the breaking of time reversal symmetry in a magnetic system, nodal lines require protection by additional symmetries. Here, we report the discovery of antiferromagnetic type-I and type-II nodal lines coexisting in the monolayer ${CrAs}_{2}$ based on a systematic first-principles calculation. Remarkably, the type-I nodal line in ${CrAs}_{2}$ is filling enforced by nonsymmorphic analog symmetry and robust against spin-orbital coupling. The type-II nodal lines are protected by the mirror symmetry in the absence of spin-orbital coupling. The antiferromagnetic monolayer ${CrAs}_{2}$ proposed here may provide a platform for the correlation between magnetism and exotic topological phases.

Received 1 May 2018
Revised 11 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115164

Physics Subject Headings (PhySH)

Antiferromagnetism First-principles calculations

Node-line semimetals

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Busheng Wang¹, Heng Gao², Qing Lu³, WenHui Xie^4,*, Yanfeng Ge¹, Yong-Hong Zhao⁵, Kaicheng Zhang⁶, and Yong Liu^1,†

¹State Key Laboratory of Metastable Materials Science and Technology & Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao, 066004, China
²Department of Physics, Shanghai University, 99 Shangda Road, Shanghai, 200444, China
³Institute of Atomic and Molecular Physics, College of Physical Science and Technology, Sichuan University, Chengdu 610065, China
⁴Engineering Research Center for Nanophotonics & Advanced Instrument, Ministry of Education, Department of Physics, East China Normal University, Shanghai, 200062, China
⁵College of Physics and Electronic Engineering, Center for Computational Sciences, Sichuan Normal University, Chengdu, 610068, China
⁶College of Mathematics and Physics, Bohai University, Jinzhou 121013, China

^*whxie@phy.ecnu.edu.cn
^†yongliu@ysu.edu.cn; ycliu@ysu.edu.cn

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Vol. 98, Iss. 11 — 15 September 2018

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Figure 1
(a) Crystal structure of the monolayer ${CrAs}_{2}$ (top view); the unit cell of the AFM configuration is highlighted in orange. There are four Cr atoms (blue spheres) and eight As atoms (green spheres) in each unit cell. The yellow and cyan isosurfaces correspond to the spin-up and spin-down components, respectively, in the spatial spin density distributions. $t = (1 / 2, 0, 0)$ is the half translation along the (100) direction. (b) The cleavage energy and strength calculated using the optB88-vdW functional as a function of the separation d between two fractured parts. $d_{0}$ represents the equilibrium interlayer distance. (c) Phonon dispersion of monolayer ${CrAs}_{2}$ calculated using the PBE functional. (d) Total energy fluctuations with respect to molecular dynamics simulation step observed for the monolayer ${CrAs}_{2}$ at 300 K.
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Figure 2
(a) The electronic band structures of the monolayer ${CrAs}_{2}$ without SOC. Dirac cones are denoted by $L_{1 - 3}$ and $P_{1 - 4}$ , which correspond to type-I and type-II Dirac points, respectively. An zoom-in of the band structure within the green boxes is shown in (b) and (c). The valence band maximum and conduction band minimum are represented as blue and red lines. The Fermi level is set to zero.
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Figure 3
(a) 3D energy band of closed NL (type-I) around $L_{i}$ ( $i = 1, 2, 3)$ . (b) 3D energy band of open NL (type-II) around $P_{i}$ ( $i = 2, 3)$ . (c) Projection of CBM in the BZ. The direction of the reciprocal space is indicated by the arrow magenta, the color represents the energy of the 3D bands. The Type-I and II NLs in hybrid nodal line are denoted by yellow and purple lines, respectively [68]. The Fermi level is set to zero.
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Figure 4
(a) Projected spectrum on the (010) surface of the monolayer ${CrAs}_{2}$ ; the edge states are denoted as e1–e3, and the correspondence between them and the semimetal states are e1 $\to L_{i}$ , e3 $\to P_{i}$ ; e2 is a normal edge state. (b) 2D BZ of the monolayer ${CrAs}_{2}$ and its projected 1D BZ. (c) Enlarged spectrum of edge state e1. The green dots indicate the location of the protected band crossing points on the type-I nodal loop. Color indicates the charge density for each state at the surface layer.
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Figure 5
(a) Top and side view of the lattice structure for the AFM monolayer ${CrAs}_{2}$ with both nonsymmorphic analog $\tilde{T} = \{T | (1 / 2, 0, 0)\}$ and mirror $M_{z}$ symmetry. (d) Side view of the lattice structure after breaking the $M_{z}$ symmetry but maintaining the $\tilde{T}$ symmetry. Red arrows indicate the shift (0.1 $Å$ ) of the Cr atoms along the $z$ axis to break $M_{z}$ . (g) Top view of the lattice structure after breaking the $\tilde{T}$ symmetry but maintaining the $M_{z}$ symmetry. Red arrows indicate the shift (0.1 $Å$ ) of Cr1 and Cr4 atoms along (−1, −1, 0) and (1, 1, 0) direction to break $\tilde{T}$ . The band structured calculated without [(b), (e), (h)] and with [(c), (f), (i)] SOC for the configurations in (a), (d), and (g), respectively. The orbital components are shown in either red or blue. The Fermi level is set to zero.
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Physical Review B

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Type-I and type-II nodal lines coexistence in the antiferromagnetic monolayer ${CrAs}_{2}$

Busheng Wang, Heng Gao, Qing Lu, WenHui Xie, Yanfeng Ge, Yong-Hong Zhao, Kaicheng Zhang, and Yong Liu

Phys. Rev. B 98, 115164 – Published 28 September 2018

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Physical Review B

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Type-I and type-II nodal lines coexistence in the antiferromagnetic monolayer CrAs2

Busheng Wang, Heng Gao, Qing Lu, WenHui Xie, Yanfeng Ge, Yong-Hong Zhao, Kaicheng Zhang, and Yong Liu

Phys. Rev. B 98, 115164 – Published 28 September 2018

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Type-I and type-II nodal lines coexistence in the antiferromagnetic monolayer ${CrAs}_{2}$