Fully Spin-Polarized Nodal Loop Semimetals in Alkaline Metal

Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 3101−3108

pubs.acs.org/JPCL

Fully Spin-Polarized Nodal Loop Semimetals in Alkaline Metal Monochalcogenide Monolayers Xiaodong Zhou,†,∥ Run-Wu Zhang,†,∥ Zeying Zhang,† Da-Shuai Ma,† Wanxiang Feng,*,†,‡ Yuriy Mokrousov,*,‡,§ and Yugui Yao*,† †

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Key Lab of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), Beijing Key Lab of Nanophotonics Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of Technology, Beijing 100081, China ‡ Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany § Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany S Supporting Information *

ABSTRACT: Topological semimetals in ferromagnetic materials have attracted an enormous amount of attention due to potential applications in spintronics. Using firstprinciples density functional theory together with an effective lattice model, here we present a new family of topological semimetals with a fully spin-polarized nodal loop in alkaline metal monochalcogenide MX (M = Li, Na, K, Rb, or Cs; X = S, Se, or Te) monolayers. The half-metallic ferromagnetism can be established in MX monolayers, in which one nodal loop formed by two crossing bands with the same spin components is found at the Fermi energy. This nodal loop half-metal survives even when considering the spin−orbit coupling owing to the symmetry protection provided by the 4 z mirror plane. The quantum anomalous Hall state and Weyl-like semimetal in this system can be also achieved by rotating the spin from the out-of-plane to the in-plane direction. The MX monolayers hosting rich topological phases thus offer an excellent platform for realizing advanced spintronic concepts. gave an additional boost to the young field of NLSs. Unlike the previously proposed NLSs in nonmagnetic materials, it was recently proposed that the NLSs can be achieved in 2D ferromagnets due to band crossings occurring between the states of opposite spins.11,12 Nevertheless, for applications that rely on strong spin polarization of generated currents, it is desirable to realize the NLSs in fully spin-polarized 2D systems, such as half-metals. In this work, using first-principles calculations together with an effective lattice model, we present a family of fully spinpolarized NLSs, termed nodal loop half-metals (NLHMs), in 2D FM alkaline metal monochalcogenide MX (M = Li, Na, K, Rb, or Cs; X = S, Se, or Te) monolayers (MLs). When the magnetization points to the out-of-plane direction [i.e., z axis (see Figure 1a)], MX MLs are 2D NLHMs in the sense that the nodal points due to two intersecting spin-down bands form a closed loop in the first Brillouin zone. The NLHM discovered here is a topologically protected state due to the presence of 4 z symmetry, and it is robust against the inclusion of spin−orbit coupling (SOC). The nodal loop will be gapped out if the magnetization is turned away from the z axis, but interestingly, the emergent gapped state hosts the quantum anomalous Hall (QAH) effect characterized by the

pintronics is a multidisciplinary field, which utilizes the electron’s spin degree of freedom as an information carrier for data storage and processing.1,2 Compared with conventional semiconductor devices, spintronic devices have a higher integration density, a faster processing speed, and a lower level of power consumption. However, many issues such as the generation and transport of a pure spin current still present profound challenges in realizing three-dimensional (3D) spintronic devices,3,4 hindering the range of possible competitive applications. In this respect, two-dimensional (2D) materials are currently being promoted as a flagship for realizing advanced spintronic concepts, among which the emerging 2D topological quantum states exhibiting striking advantages for spintronics have attracted a tremendous amount of attention in recent years. 2D nodal loop semimetals (NLSs)5,6 provide an exciting avenue for realizing topological quantum phase transitions between the gapped and gapless states. Physically, the 2D NLSs are topologically protected by the crystal symmetry, e.g., mirror or glide mirror symmetry. These proposed protection mechanisms provide guiding principles for predicting real materials for creating various NLSs, including conventional linear nodal line semimetals and exotic higher-order nodal line semimetals.7 Many exotic properties were reported to be associated with NLSs, including high-temperature surface superconductivity,8 a nondispersive Landau energy level,9 and specific long-range Coulombic interactions.10 Recent advances in the domain of 2D ferromagnetic (FM) materials

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© 2019 American Chemical Society

Received: March 31, 2019 Accepted: May 17, 2019 Published: May 22, 2019 3101

DOI: 10.1021/acs.jpclett.9b00906 J. Phys. Chem. Lett. 2019, 10, 3101−3108

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The Journal of Physical Chemistry Letters

Figure 1. (a) Top and side views of MX MLs. The large green and small yellow spheres represent alkaline metal (M) and chalcogenide (X) atoms, respectively. The blue dashed lines indicate the 2D unit cell. The mirror plane 4 z (colored pink) is preserved if the spin magnetic moment points to the out-of-plane direction (z axis). (b) Phonon spectrum of CsS ML. (c and d) Magnetocrystalline anisotropy energy (MAE) by rotating the spin within z−x and x−y planes, respectively. The red solid line in panel c and the violet dashed line in panel d represent the fitted curves, i.e., MAE(θ) = 1.32 × cos2 θ − 0.04 × cos4 θ and MAE(φ) = 0, respectively.

non-zero Chern number (* = ±1). When the magnetization lies within the crystal plane, a Weyl-like semimetal state, possessing two nodal points within a mirror plane, emerges. Furthermore, the QAH and magneto-optical (MO) effects, providing the electric and optical means, respectively, are proposed to detect such rich topological phases in the FM MX MLs. These findings provide a new 2D material platform hosting the exotic NLHM and QAH states. Recently, the alkaline metal monochalcogenides MX have attracted a considerable amount of attention due to their unique FM half-metallicity.13−25 Bulk MXs exhibit various crystalline phases, such as CsCl, NiAs, wurtzite, zinc-blende, and rocksalt structures. In particular, CsS and CsSe with the CsCl-type structure (Figure S1a) are predicted to be the most energetically stable.16,20 Taking CsS as an example, we checked its dynamical stability by calculating the phonon spectrum, finding that it is free of imaginary frequencies (Figure S1b). Interestingly, the FM half-metallicity is preserved at the (100), (110), and (111) surfaces of CsS.16 Because of a strong interest in 2D ferromagnetism, one may naturally ask whether MX MLs exist and whether they will exhibit observable halfmetallicity. We focus on the (111) surface of bulk MX and take CsS as a prototypical example because it shares the same features with other MX compounds. As seen from Figure S1a, the Cs and S atoms projecting onto the (111) plane are noncoplanar with a

buckling height h of 1.22 Å. Upon extraction of the (111) plane from the bulk structure and relaxation of the atomic positions in a slab model, the Cs and S atoms are inclined to form a coplanar structure, as shown in Figure 1a. One can see that the CsS ML has a hexagonal honeycomb lattice that contains one Cs atom and one S atom. The relaxed lattice constants of MX MLs and the corresponding bulk counterparts are listed in Table S1. The practical feasibility of creating MX MLs has been examined through the following important aspects: (i) formation energy, (ii) dynamical stability, and (iii) thermal stability. First, the formation energy of MX MLs, defined as the energy difference (per atom) between the ML structure and its bulk phase,26−30 is calculated, and the results listed in Table S1 show that the experimental synthesis of MX MLs is possible. For example, the formation energy of a CsS ML is 0.27 eV/ atom, which is slightly larger than that of a ZnO ML (0.19 eV/ atom)27 but much smaller than that of silicene (0.76 eV/ atom),26 whereas both a ZnO ML and silicene have recently been experimentally synthesized.31,32 Additionally, the dynamical stability of CsS ML is affirmed by the phonon spectrum (see Figure 1b). Moreover, the thermal stability of CsS ML is tested by molecular dynamics simulations, and the planar hexagonal honeycomb lattice is maintained up to 100 K (see Figure S1c,d). 3102


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Figure 2. (a) Spin-polarized band structure of the CsS ML. (b) Orbital-projected and spin-polarized band structure of CsS ML. (c) Momentum distribution of the band-crossing nodes near the Fermi energy. (d) Band structures calculated by density functional theory and the effective lattice model. “+” and “−” mean the eigenvalues of 4 z symmetry. The SOC is not included in panels a−d.

Figure 3. (a) Band structures of CsS ML calculated by density functional theory and the effective lattice model with SOC. (b) Calculated Chern number as a function of magnetization direction θ (i.e., the polar angle away from the z-axis). (c) Orbital-projected band structure of the CsS ML with SOC. The magnetization in panels a and c is considered along the out-of-plane direction. After SOC is turned on, the nodal loop at the Fermi energy is preserved, whereas two topologically nontrivial band gaps (I and II) arise at the Γ point, in which the AHC is quantized with the non-zero Chern numbers (* = + 1 and * = − 1). (d) Band structures and anomalous Hall conductivities of the CsS ML when θ = 0°, 45°, and 90°.

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Table 1. Magnetic Point Groups of MX MLs as a Function of the Polar (θ) or Azimuthal (φ) Angle When the Spin Is Rotated within the z−x (0 ≤ θ ≤ π, φ = 0) or x−y (θ = π/2, 0 ≤ φ < 2π) Plane z−x x−y

0°

15°

30°

45°

60°

75°

90°

105°

120°

135°

150°

165°

180°

6̅m′2′ m′m2′

2′ m′

2′ m′m′2

2′ m′

2′ m′m2′

2′ m′

m′m2′ m′m′2

2′ m′

2′ m′m2′

2′ m′

2′ m′m′2

2′ m′

6̅m′2′ m′m2′

nonmagnetic materials, referring to the previously predicted 2D NLSs,39−50 while it can be realized in some exotic materials that properly have nonsymmorphic lattice symmetries.51 Differing from the nonmagnetic system, the nodal loop can be retained in magnetic materials even though SOC is considered, for example, the recently discovered 3D fully spin-polarized nodal loop half-metal Li3(FeO3)2.52 Here, we are excited to find that the nodal loop in the FM CsS ML (with out-of-plane magnetization) also survives after the SOC is turned on (see panels a and c of Figure 3). The underlying mechanism is that in the case of out-of-plane magnetization 4 z is a good symmetry regardless of SOC, and the unchanged 4 z parities of the two crossing bands prevent the hybridization between px,y and pz orbitals. It shares a similar physical origin with the mixed nodal lines proposed in ref 11. To further demonstrate the crucial role of 4 z symmetry, we artificially violate the planar structure of the CsS ML by adding a buckling height between Cs and S atoms (see Figure S1e), which thus breaks 4 z symmetry such that the nodal loop is gapped out (see Figure S1f). By explicitly calculating the Chern number and the edge state, we confirm that the CsS ML with a finite buckling height is a topologically trivial insulator. To capture the main physics in the MX ML with and without SOC, we developed an effective lattice model under space group P6̅m2 (see the Supporting Information). The basis functions are chosen as ψ1 = |px⟩, ψ2 = |py⟩, and ψ3 = |pz⟩ because the three bands near the Fermi energy are dominated by the p orbitals of the S atom. The model parameters, including the on-site energy, hopping integral, and SOC strength, are listed in Table S2. The band structures calculated by the lattice model Hamiltonian reproduce well the lowenergy electronic states obtained by the first-principles calculations, as shown in Figures 2 and 3. In addition, we have identified all NLHM candidates in MX MLs, including LiS, LiSe, LiTe, NaSe, and NaTe (see Figure S2). The abundant topological phases are usually expected upon tuning of the magnetization direction.53−55 We first discuss the situation that rotates the spin within the z−x plane (i.e., 0 ≤ θ ≤ π, φ = 0). The corresponding magnetic point groups computed by the ISOTROPY code56 are listed in Table 1. One can see that the magnetic point group changes with a period of π and only three nonrepetitive elements need to be analyzed: 6̅m′2′ (θ = nπ), m′m2′ [θ = (n + 1/2)π], and 2′ [θ ≠ nπ and θ ≠ (n + 1/2)π] with n ∈ . As mentioned above, when the magnetization direction is out of plane [6̅m′2′ (θ = nπ)], the nodal loop survives because symmetry 4 z is not broken. It is well-known that the Berry curvature Ω = [Ωx, Ωy, Ωz] = [Ωyz, Ωzx, Ωxy] is a pseudovector. Moreover, Ωx and Ωy are vanishing in 2D systems and only Ωz (=Ωxy) is relevant for MX MLs. Because Ωxy is an even function with respect to mirror operation 4 z , the anomalous Hall conductivity (AHC), σxy, is expected to be non-zero in the case of θ = nπ, as shown in Figure 3d. When the magnetization direction is along the x-axis [θ = (n + 1/2)π], group m′m2′ contains mirror operation 4x , which breaks the nodal loop except for two band crossing

Magnetic properties are further explored in MX MLs. In the past, the long-range ferromagnetic order in low-dimensional systems (e.g., 2D case) has long been considered to be impossible to realize due to thermal fluctuations according to the Mermin−Wagner theorem.33 However, this limitation has been challenged by the recent discoveries of intrinsic 2D ferromagnets, such as Cr2Ge2Te6,34 CrI3,35 and Fe3GeTe2.36 After performing a spin-polarized calculation, we learned that the spin magnetic moment carried by CsS ML is 1 μB per cell, being similar to its bulk phase.16 The ferromagnetism mainly originated from the p orbitals of the S atom, whereas the spin magnetic moment on the Cs atom is negligible. To confirm this FM state is the ground state, we compared the total energies among the FM, antiferromagnetic, and nonmagnetic states in a doubled supercell. The FM state is found to be more stable than the antiferromagnetic and nonmagnetic states by 7.06 and 139.77 meV/cell, respectively. Moreover, the SOCinduced magnetocrystalline anisotropy energy (MAE), defined as MAE(θ,φ) = E(θ,φ) − E(θ=90°,φ=0°) (where θ and φ are polar and azimuthal angles, respectively), is analyzed by rotating the spin within the z−x and x−y planes. Figure 1c shows that the MAE of CsS ML in the z−x plane can be fitted very well by the equation MAE = K1 cos2 θ + K2 cos4 θ, where K1 (1.32 meV) and K2 (−0.04 meV) are the magnetocrystalline anisotropy coefficients.37 The positive value of MAE indicates a preferred magnetization along the x-axis rather than along the z-axis. Figure 1d further illustrates that the MAE is isotropic in the x− y plane, suggesting that CsS ML belongs to the category of XY ferromagnets.29 The maximal value of MAE between the outof-plane and in-plane spin orientations reaches 1.28 meV/cell, which is somewhat smaller than that of famous 2D ferromagnets CrI3 (1.37 meV/cell)38 and Fe3GeTe2 (2.76 meV/cell).30 Interestingly, the magnetization direction in 2D ferromagnets can be effectively tuned by applying an external magnetic field, as experimentally realized in, e.g., Fe3GeTe2.36 Neglecting the effect of SOC, we next discuss the emergent half-metallicity and the fully spin-polarized band crossings in MX MLs. In Figure 2a, the spin-polarized band structure of the CsS ML is plotted, in which the green solid (blue dashed) lines denote spin-down (spin-up) bands. One can see that the CsS ML is a half-metal in the sense that the spin-down channel is metallic while the spin-up channel is semiconducting with a gap of ∼3.79 eV. Remarkably, the spin-down bands exhibit two band-crossing nodes at the Fermi energy along the Γ−K and Γ−M paths, respectively. These bands are primarily derived from px,y and pz orbitals of the S atom, as shown in Figure 2b. In Figure 2c, we display the momentum distribution of the band-crossing nodes, which form a Weyl-like (2-fold degenerate) nodal loop centered at the Γ point in the first Brillouin zone. Moreover, we calculated the 4 z parities for the valence and conductance bands near the Fermi energy, and the opposite parities of two crossing bands (see Figure 2d) indicate that the gapless nodal loop is protected by the mirror reflection symmetry. Considering the SOC effect, the nodal loops cannot be solely protected by the mirror reflection symmetry in 3104


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The Journal of Physical Chemistry Letters points sitting in the 4x plane, e.g., along the paths from the Γ point to the M and −M points (see Figures 3d and 4).

Symmetry 4x forces Ωxy to be zero, and consequently, σxy = 0 (see Figure 3d). In the terminology of ref 11, these points present the mixed Weyl points of type (i). If θ ≠ nπ and θ ≠ (n + 1/2)π, group 2′ does not have any mirror operation, and therefore, all band crossing points are gapped out. Importantly, group 2′ contains a ;*2y operation (; is the time-reversal symmetry, and *2y is the 2-fold rotation around the y-axis), which in principle renders nonvanishing Ωxy. As one can see in Figure 3d, the AHC σxy is indeed non-zero, and strikingly, the quantized value, σxy = e2/h (here, taking θ = 45° as an example), suggests a QAH insulator with Chern number * = +1. The evaluation of the topological phase as a function of polar angle θ is sketched in Figure 3b. Next, we explore the possible topological phases when the spin lies within the x−y plane (θ = π/2, 0 ≤ φ < 2π). From Table 1, one can observe that the magnetic point group has a period of π/3 with respect to φ: m′m2′ ⇒ m′ ⇒ m′m′2 ⇒ m′ ⇒ m′m2′. In particular, group m′m2′ (θ = nπ/3) has a mirror plane that is perpendicular to the spin direction, as shown in Figure 4a. This mirror plane drives the nodal loop into one pair of 2-fold degenerated points sitting in the mirror plane (see panels b and c of Figure 4). If such a mirror symmetry is absent [m′ and m′m′2 (θ ≠ nπ/3)], the nodal loop will be fully gapped out and the insulating phase topologically trivial due to the presence of ;4 z symmetry. The reason is that Ωxy is odd (even) under symmetry ; (4 z ) and thus is odd under symmetry ;4 z . Via integration of Ωxy in the entire Brillouin zone, σxy has to be zero, giving the Chern number * = 0. This is different from the scenario for the QAH insulator with inplane magnetization proposed by Liu et al.,53 because in their systems (e.g., LaCl monolayer) the ;4 z symmetry is lacking.

Figure 4. (a) Magnetic structures upon variation of the spin within the x−y plane. φ is the azimuthal angle starting from the x-axis. The red dashed lines denote the mirror planes. (b) Positions of one pair of 2-fold degenerated points in the Brillouin zone. (c) Relativistic band structures along the Γ−M1, Γ−M2, and Γ−M3 paths.

Figure 5. (a) Relativistic band structures, (b) anomalous Hall conductivities, (c) Kerr rotation angles, and (d) Faraday rotation angles of CsS (doped one hole), CsP, and CsAs MLs. The dashed vertical lines in panels c and d denote the topologically nontrivial band gaps. α in panel d is the fine structure constant. 3105



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The QAH effect plays an important role in the application of spintronics. Herein, the QAH state can also be found in the CsS ML even when the magnetization takes an out-of-plane direction. After SOC is switched on, two topologically nontrivial band gaps (I and II) arise at the Γ point, in which the AHC is quantized with the Chern numbers * = +1 and * = −1 (see panels a and c of Figure 3). Gap I is close to the Fermi energy, and electrostatic doping could be used to engineer the electronic state. Indeed, if one hole is doped into the CsS ML, a QAH state appears with a large nontrivial band gap of 86 meV (see panels a and b of Figure 5). Note that doping one hole into the CsS ML corresponds to a doping concentration of 3.52 × 1014 cm−2, which can be achieved by the current experimental techniques.57,58 Similar to CsS, CsP and CsAs host rich bulk phases and exhibit FM half-metallicity on the low-index surfaces.59−64 The dynamical stability of CsP and CsAs MLs has been determined by calculating their phonon spectra, shown in Figure S3, which indicate the practical feasibility. Because CsP and CsAs are isostructural to CsS but one hole is less in their native states, we propose that CsP and CsAs MLs are intrinsic QAH insulators (see panels a and b of Figure 5). In addition to the anomalous Hall effect, the MO Kerr and Faraday effects, being a kind of noncontact (nondamaging) optical technique, are powerful tools for measuring the magnetism in 2D materials.34,35 With regard to the QAH insulator systems, an additional magnetoelectric term (Θα/ 4π2)E·B (here Θ is the magnetoelectric polarizability and α = e2/ℏc is the fine structure constant) should be added to the usual Lagrangian such that Maxwell’s equations have to be modified.65 In the low-frequency limit (i.e., ω ≪ Eg/ℏ, where Eg is the topologically nontrivial band gap), the MO Kerr and Faraday rotation angles are quantized to θK ≃ −π/2 and θF ≃ *α (* is the Chern number), respectively.66,67 Therefore, in optical terms, the modern MO techniques have been used to characterize the QAH state.68−71 Panels c and d of Figure 5 plot the Kerr and Faraday spectra, respectively, of CsS, CsP, and CsAs MLs, from which one can clearly find the quantized behaviors of Kerr and Faraday rotation angles in the low-frequency limit. In conclusion, using the first-principles calculations and an effective lattice model, we propose a class of nodal loop halfmetal candidates in ferromagnetic alkaline metal monochalcogenide MX (M = Li, Na, K, Rb, or Cs; X = S, Se, or Te) monolayers. The nodal loop is formed by two crossing bands with the same spin components, and in the case of out-of-plane magnetization, the nodal loop is protected by symmetry 4 z that is not influenced by spin−orbit coupling. The topological phase transition from a nodal loop half-metal to a quantum anomalous Hall insulator and to a Weyl-like semimetal is found by rotating the spin from the z-axis to the x-axis. For the in-plane magnetization, the Weyl-like semimetal appears in a period of π/3 starting from the x-axis, and the intermediate phases are topologically trivial insulators. Moreover, we show that the quantum anomalous Hall state can also be realized in the CsS ML by doping one hole or by replacing the S atom with P or As atoms. The exotic quantum MO Kerr and Faraday effects are expected in CsS, CsP, and CsP MLs. Our work reveals a kind of novel 2D ferromagnet hosting rich topological phases, which serve as good candidates for the promising spintronics.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b00906. Details of first-principles calculations, an effective lattice model, and supplementary tables and figures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Xiaodong Zhou: 0000-0002-0378-6729 Yugui Yao: 0000-0003-3544-3787 Author Contributions ∥

X.Z. and R.-W.Z. contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are thankful for the fruitful discussions with Branton J. Campbell. W.F. and Y.Y. acknowledge support from the National Natural Science Foundation of China (Grants 11874085, 11734003, and 11574029) and the National Key R&D Program of China (Grant 2016YFA0300600). W.F. also acknowledges funding from an Alexander von Humboldt Fellowship. Y.M. acknowledges funding under SPP 2137 “Skyrmionics” (Project MO 1731/7-1), collaborative Research Center SFB 1238, and Project MO 1731/5-1 of Deutsche Forschungsgemeinschaft (DFG). The authors acknowledge computing time on supercomputers JUQUEEN and JURECA at Jülich Supercomputing Centre and JARA-HPC of RWTH Aachen University.

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REFERENCES

(1) Wolf, S. A.; Awschalom, D. D.; Buhrman, R. A.; Daughton, J. M.; von Molnár, S.; Roukes, M. L.; Chtchelkanova, A. Y.; Treger, D. M. Spintronics: A Spin-Based Electronics Vision for the Future. Science 2001, 294, 1488−1495. (2) Ž utić, I.; Fabian, J.; Das Sarma, S. Spintronics: Fundamentals and Applications. Rev. Mod. Phys. 2004, 76, 323−410. (3) Awschalom, D. D.; Flatté, M. E. Challenges for Semiconductor Spintronics. Nat. Phys. 2007, 3, 153−159. (4) Felser, C.; Fecher, G.; Balke, B. Spintronics: A Challenge for Materials Science and Solid-State Chemistry. Angew. Chem., Int. Ed. 2007, 46, 668−699. (5) Burkov, A. A.; Hook, M. D.; Balents, L. Topological Nodal Semimetals. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 235126. (6) Yu, R.; Fang, Z.; Dai, X.; Weng, H. Topological Nodal Line Semimetals Predicted from First-principles Calculations. Front. Phys. 2017, 12, 127202. (7) Yu, Z.-M.; Wu, W.; Sheng, X.-L.; Zhao, Y. X.; Yang, S. A. Quadratic and Cubic Nodal Lines Stabilized by Crystalline Symmetry. Phys. Rev. B: Condens. Matter Mater. Phys. 2019, 99, 121106. (8) Kopnin, N. B.; Heikkilä, T. T.; Volovik, G. E. High-temperature Surface Superconductivity in Topological Flat-band Systems. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 220503. (9) Rhim, J.-W.; Kim, Y. B. Landau Level Quantization and Almost Flat Modes in Three-dimensional Semimetals with Nodal Ring Spectra. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 045126. 3106


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(31) Tusche, C.; Meyerheim, H. L.; Kirschner, J. Observation of Depolarized ZnO(0001) Monolayers: Formation of Unreconstructed Planar Sheets. Phys. Rev. Lett. 2007, 99, 026102. (32) Vogt, P.; De Padova, P.; Quaresima, C.; Avila, J.; Frantzeskakis, E.; Asensio, M. C.; Resta, A.; Ealet, B.; Le Lay, G. Silicene: Compelling Experimental Evidence for Graphenelike Two-Dimensional Silicon. Phys. Rev. Lett. 2012, 108, 155501. (33) Mermin, N. D.; Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models. Phys. Rev. Lett. 1966, 17, 1133−1136. (34) Gong, C.; Li, L.; Li, Z.; Ji, H.; Stern, A.; Xia, Y.; Cao, T.; Bao, W.; Wang, C.; Wang, Y.; et al. Discovery of Intrinsic Ferromagnetism in Two-dimensional Van Der Waals Crystals. Nature 2017, 546, 265− 269. (35) Huang, B.; Clark, G.; Navarro-Moratalla, E.; Klein, D. R.; Cheng, R.; Seyler, K. L.; Zhong, D.; Schmidgall, E.; McGuire, M. A.; Cobden, D. H.; et al. Layer-dependent Ferromagnetism in a Van Der Waals Crystal down to the Monolayer Limit. Nature 2017, 546, 270− 273. (36) Deng, Y.; Yu, Y.; Song, Y.; Zhang, J.; Wang, N. Z.; Sun, Z.; Yi, Y.; Wu, Y. Z.; Wu, S.; Zhu, J.; et al. Gate-tunable Room-temperature Ferromagnetism in Two-dimensional Fe3GeTe2. Nature 2018, 563, 94−99. (37) Getzlaff, M. Fundamentals of Magnetism; Springer Science & Business Media, 2007. (38) Zhang, W.-B.; Qu, Q.; Zhu, P.; Lam, C.-H. Robust Intrinsic Ferromagnetism and Half Semiconductivity in Stable Two-dimensional Single-layer Chromium Trihalides. J. Mater. Chem. C 2015, 3, 12457−12468. (39) Jin, Y.-J.; Wang, R.; Zhao, J.-Z.; Du, Y.-P.; Zheng, C.-D.; Gan, L.-Y.; Liu, J.-F.; Xu, H.; Tong, S. Y. The Prediction of a Family Group of Two-dimensional Node-line Semimetals. Nanoscale 2017, 9, 13112−13118. (40) Lu, J.-L.; Luo, W.; Li, X.-Y.; Yang, S.-Q.; Cao, J.-X.; Gong, X.G.; Xiang, H.-J. Two-Dimensional Node-Line Semimetals in a Honeycomb-Kagome Lattice*. Chin. Phys. Lett. 2017, 34, 057302. (41) Niu, C.; Buhl, P. M.; Bihlmayer, G.; Wortmann, D.; Dai, Y.; Blügel, S.; Mokrousov, Y. Two-dimensional Topological Nodal Line Semimetal in Layered X2Y (X = Ca, Sr, and Ba; Y = As, Sb, and Bi). Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 95, 235138. (42) Yang, B.; Zhang, X.; Zhao, M. Dirac Node Lines in Twodimensional Lieb Lattices. Nanoscale 2017, 9, 8740−8746. (43) Zhang, H.; Xie, Y.; Zhang, Z.; Zhong, C.; Li, Y.; Chen, Z.; Chen, Y. Dirac Nodal Lines and Tilted Semi-Dirac Cones Coexisting in a Striped Boron Sheet. J. Phys. Chem. Lett. 2017, 8, 1707−1713. (44) Feng, B.; Fu, B.; Kasamatsu, S.; Ito, S.; Cheng, P.; Liu, C.-C.; Feng, Y.; Wu, S.; Mahatha, S. K.; Sheverdyaeva, P.; et al. Experimental Realization of Two-dimensional Dirac Nodal Line Fermions in Monolayer Cu2Si. Nat. Commun. 2017, 8, 1007. (45) Zhang, R.-W.; Liu, C.-C.; Ma, D.-S.; Yao, Y. From Node-line Semimetals to Large-gap Quantum Spin Hall States in a Family of Pentagonal Group-IVA Chalcogenide. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 97, 125312. (46) Gao, L.; Sun, J.-T.; Lu, J.-C.; Li, H.; Qian, K.; Zhang, S.; Zhang, Y.-Y.; Qian, T.; Ding, H.; Lin, X.; et al. Epitaxial Growth of Honeycomb Monolayer CuSe with Dirac Nodal Line Fermions. Adv. Mater. 2018, 30, 1707055. (47) Chen, H.; Zhang, S.; Jiang, W.; Zhang, C.; Guo, H.; Liu, Z.; Wang, Z.; Liu, F.; Niu, X. Prediction of Two-dimensional Nodal-line Semimetals in a Carbon Nitride Covalent Network. J. Mater. Chem. A 2018, 6, 11252−11259. (48) Zhou, P.; Ma, Z. S.; Sun, L. Z. Coexistence of Open and Closed Type Nodal Line Topological Semimetals in Two Dimensional B2C. J. Mater. Chem. C 2018, 6, 1206−1214. (49) Zhang, X.; Jin, L.; Dai, X.; Liu, G. Topological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound Mg3Bi2. J. Phys. Chem. Lett. 2017, 8, 4814−4819.

(10) Huh, Y.; Moon, E.-G.; Kim, Y. B. Long-range Coulomb Interaction in Nodal-ring Semimetals. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 035138. (11) Niu, C.; Hanke, J. P.; Buhl, P. M.; Zhang, H.; Plucinski, L.; Wortmann, D.; Blügel, S.; Bihlmayer, G.; Mokrousov, Y. Mixed Topological Semimetals and Orbital Magnetism in Two-dimensional Spin-orbit Ferromagnets. arXiv 2018, 1805.02549. (12) Feng, B.; Zhang, R.-W.; Feng, Y.; Fu, B.; Wu, S.; Miyamoto, K.; He, S.; Chen, L.; Wu, K.; Shimada, K.; et al. Discovery of Weyl Nodal Lines in a Single-layer Ferromagnet. arXiv 2019, 1901.01429. (13) Gao, G. Y.; Yao, K. L.; Song, M. H.; Liu, Z. L. Half-metallic Ferromagnetism in Rocksalt and Zinc-blende MS (M = Li, Na and K): A First-principles Study. J. Magn. Magn. Mater. 2011, 323, 2652− 2657. (14) Moradi, M.; Rostami, M.; Afshari, M. Half-metallic Ferromagnetism in Wurtzite MS (M = Li, Na, and K). Can. J. Phys. 2012, 90, 531−536. (15) Moradi, M.; Afshari, M.; Hashemifar, S. J. First-principles Prediction of Half-metallicity at the Low Index Surfaces of Rocksalt KS. Surf. Sci. 2013, 616, 71−75. (16) Rostami, M.; Afshari, M.; Moradi, M. Bulk and Surface Halfmetallicity of CsS in CsCl Structure: A Density Functional Theory Study. J. Alloys Compd. 2013, 575, 301−308. (17) Ahmadian, F. Half-metallic Ferromagnetism in the Rocksalt and Zincblende NaX (X = O, S, Se, Te, and Po). J. Supercond. Novel Magn. 2012, 25, 1589−1596. (18) Senol, C.; Kervan, S.; Kervan, N. Half-metallic Ferromagnetism in the RbX (X = Sb, Te) Compounds with the Rock Salt and Zincblende Structures: A First-principles Calculation. Intermetallics 2014, 52, 1−8. (19) Afshari, M.; Moradi, M.; Rostami, M. Structural, Electronic and Magnetic Properties of the (001), (110) and (111) Surfaces of Rocksalt Sodium Sulfide: A First-principles Study. J. Phys. Chem. Solids 2015, 76, 94−99. (20) Karaca, M.; Kervan, S.; Kervan, N. Half-metallic Ferromagnetism in the CsSe Compound by Density Functional Theory. J. Alloys Compd. 2015, 639, 162−167. (21) Li, L.; Lei, G.; Gao, Q.; Deng, J.-B.; Hu, X.-R. First-principles Study on the Bulk and (111) Surface Half-metallicity of KS and RbS in CsCl Structure. Mater. Res. Bull. 2015, 68, 308−313. (22) Lei, G.; Liu, X.-X.; Xie, H.-H.; Li, L.; Gao, Q.; Deng, J.-B. Firstprinciple Study of Half-metallic Ferromagnetism in Rocksalt XO (X = Li, K, Rb, Cs). J. Magn. Magn. Mater. 2016, 397, 176−180. (23) Bialek, B.; Lee, J. I. The Half-metallic Properties of (001) and (110) Surfaces of CsSe from the First-principles. Journal of Magnetics 2016, 21, 1−5. (24) Rostami, M.; Moradi, M. Electronic and Magnetic Properties of the Bulk and Surfaces of Rock Salt KM (M = Se and Te): A Density Functional Theory Study. J. Supercond. Novel Magn. 2016, 29, 215− 226. (25) Tabatabaeifar, A.; Davatolhagh, S.; Moradi, M. Surface Halfmetallicity and Stability of Zinc-blende Sodium Monoselenide. Appl. Surf. Sci. 2017, 394, 282−287. (26) Zhuang, H. L.; Hennig, R. G. Electronic Structures of Singlelayer Boron Pnictides. Appl. Phys. Lett. 2012, 101, 153109. (27) Zhuang, H. L.; Hennig, R. G. Computational Identification of Single-layer CdO for Electronic and Optical Applications. Appl. Phys. Lett. 2013, 103, 212102. (28) Zhuang, H. L.; Hennig, R. G. Computational Discovery, Characterization, and Design of Single-Layer Materials. JOM 2014, 66, 366−374. (29) Zhuang, H. L.; Hennig, R. G. Stability and Magnetism of Strongly Correlated Single-layer VS2. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 054429. (30) Zhuang, H. L.; Kent, P. R. C.; Hennig, R. G. Strong Anisotropy and Magnetostriction in the Two-dimensional Stoner Ferromagnet Fe3GeTe2. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 134407. 3107


Letter

The Journal of Physical Chemistry Letters (50) Liu, Z.; Xin, H.; Fu, L.; Liu, Y.; Song, T.; Cui, X.; Zhao, G.; Zhao, J. All-Silicon Topological Semimetals with Closed Nodal Line. J. Phys. Chem. Lett. 2019, 10, 244−250. (51) Wu, W.; Jiao, Y.; Li, S.; Sheng, X.-L.; Yu, Z.-M.; Yang, S. A. Hourglass Weyl Loops in Two Dimensions: Theory and Material Realization in Monolayer GaTeI Family. arXiv 2019, 1902.09283. (52) Chen, C.; Yu, Z.-M.; Li, S.; Chen, Z.; Sheng, X.-L.; Yang, S. A. Weyl-loop Half-metal in Li3(FeO3)2. Phys. Rev. B: Condens. Matter Mater. Phys. 2019, 99, 075131. (53) Liu, Z.; Zhao, G.; Liu, B.; Wang, Z. F.; Yang, J.; Liu, F. Intrinsic Quantum Anomalous Hall Effect with In-Plane Magnetization: Searching Rule and Material Prediction. Phys. Rev. Lett. 2018, 121, 246401. (54) Hanke, J.-P.; Freimuth, F.; Niu, C.; Blügel, S.; Mokrousov, Y. Mixed Weyl Semimetals and Low-dissipation Magnetization Control in Insulators by Spin-orbit Torques. Nat. Commun. 2017, 8, 1479. (55) Zhang, Z.; Gao, Q.; Liu, C.-C.; Zhang, H.; Yao, Y. Magnetization-direction Tunable Nodal-line and Weyl Phases. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 98, 121103. (56) Stokes, H. T.; Hatch, D. M.; Campbell, B. J. ISOTROPY Software Suite. 2018, iso.byu.edu. (57) Efetov, D. K.; Kim, P. Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities. Phys. Rev. Lett. 2010, 105, 256805. (58) Ye, J.; Craciun, M. F.; Koshino, M.; Russo, S.; Inoue, S.; Yuan, H.; Shimotani, H.; Morpurgo, A. F.; Iwasa, Y. Accessing the Transport Properties of Graphene and its Multilayers at High Carrier Density. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 13002−13006. (59) Zhang, Y.; Qi, Y.; Hu, Y. Half-metallic Ferromagnetism in Rocksalt NaN and KN from First Principles. J. Magn. Magn. Mater. 2012, 324, 2523−2527. (60) Lakdja, A.; Rozale, H.; Chahed, A. Half-metallic Ferromagnetism in the Hypothetical RbN and CsN Compounds: First-principles Calculations. Comput. Mater. Sci. 2013, 67, 287−291. (61) Wei, X.-P.; Chu, Y.-D.; Sun, X.-W.; E, Y.; Xing, Y.-Z. FirstPrinciples Study on the (111) Surface of Half-Metallic CsN. J. Supercond. Novel Magn. 2014, 27, 1935−1940. (62) Lakdja, A.; Rozale, H.; Sayede, A.; Chahed, A. Origin of Ferromagnetism in the Half-Heusler XRbCs Compounds (X = N, P and As). J. Magn. Magn. Mater. 2014, 354, 235−238. (63) Wei, X.-P.; Chu, Y.-D.; Lu, H.-W.; Sun, X.-W.; Song, T.; Guo, P. Stability and Half-Metallicity of the (001) and (111) Surfaces of RbN with Cesium Chloride Structure. J. Supercond. Novel Magn. 2015, 28, 1577−1582. (64) Gao, Q.; Li, L.; Xie, H.-H.; Lei, G.; Deng, J.-B.; Hu, X.-R. Firstprinciple Study on Bulk and (111) Surface of MP (M = K and Rb) in Rocksalt Structure. Mater. Chem. Phys. 2015, 163, 460−469. (65) Qi, X.-L.; Hughes, T. L.; Zhang, S.-C. Topological Field Theory of Time-reversal Invariant Insulators. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 195424. (66) Maciejko, J.; Qi, X.-L.; Drew, H. D.; Zhang, S.-C. Topological Quantization in Units of the Fine Structure Constant. Phys. Rev. Lett. 2010, 105, 166803. (67) Tse, W.-K.; MacDonald, A. H. Giant Magneto-Optical Kerr Effect and Universal Faraday Effect in Thin-Film Topological Insulators. Phys. Rev. Lett. 2010, 105, 057401. (68) Wu, L.; Salehi, M.; Koirala, N.; Moon, J.; Oh, S.; Armitage, N. P. Quantized Faraday and Kerr Rotation and Axion Electrodynamics of a 3D Topological Insulator. Science 2016, 354, 1124−1127. (69) Okada, K. N.; Takahashi, Y.; Mogi, M.; Yoshimi, R.; Tsukazaki, A.; Takahashi, K. S.; Ogawa, N.; Kawasaki, M.; Tokura, Y. Terahertz Spectroscopy on Faraday and Kerr Rotations in a Quantum Anomalous Hall State. Nat. Commun. 2016, 7, 12245. (70) Shuvaev, A.; Dziom, V.; Kvon, Z. D.; Mikhailov, N. N.; Pimenov, A. Universal Faraday Rotation in HgTe Wells with Critical Thickness. Phys. Rev. Lett. 2016, 117, 117401. (71) Dziom, V.; Shuvaev, A.; Pimenov, A.; Astakhov, G. V.; Ames, C.; Bendias, K.; Böttcher, J.; Tkachov, G.; Hankiewicz, E. M.; Brüne,

C.; et al. Observation of the Universal Magnetoelectric Effect in a 3D Topological Insulator. Nat. Commun. 2017, 8, 15197. (72) Wang, S.-S.; Yu, Z.-M.; Liu, Y.; Jiao, Y.; Guan, S.; Sheng, X.-L.; Yang, S. A. Two-dimensional Nodal-loop Half Metal in Monolayer MnN. arXiv 2019, 1903.04705. (73) You, J.-Y.; Chen, C.; Zhang, Z.; Sheng, X.-L.; Yang, S. A.; Su, G. Two-dimensional Weyl Half Semimetal and Tunable Quantum Anomalous Hall Effect in Monolayer PtCl3. arXiv 2019, 1903.08373.

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NOTE ADDED IN PROOF After the bulk of our work had been completed, we became aware of two related works that predict a 2D nodal loop halfmetal state in the MnN monolayer72 and a 2D Weyl halfsemimetal state in the PtCl3 monolayer.73

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Fully Spin-Polarized Nodal Loop Semimetals in Alkaline Metal

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