The p-orbital magnetic topological states on a square lattice

Abstract

Honeycomb or triangular lattices were extensively studied and thought to be proper platforms for realizing the quantum anomalous Hall effect (QAHE), where magnetism is usually caused by d orbitals of transition metals. Here we propose that a square lattice can host three magnetic topological states, including the fully spin-polarized nodal loop semimetal, QAHE and the topologically trivial ferromagnetic semiconductor, in terms of the symmetry and k · p model analyses that are material independent. A phase diagram is presented. We further show that the above three magnetic topological states can indeed be implemented in the two-dimensional (2D) materials ScLiCl₅, LiScZ₅ (Z=Cl, Br) and ScLiBr₅, respectively. The ferromagnetism in these 2D materials is microscopically revealed from p electrons of halogen atoms. This present study opens a door to explore the exotic topological states as well as quantum magnetism from p-orbital electrons by means of the material-independent approach.

Keywords: p-orbital magnetism; square lattice; topological states.

Figure 1.

The square lattice with three magnetic topological states. (a) Schematic illustration of the double degeneracy at high-symmetry points (red dots). (b) Schematic depiction of hourglass dispersion along Γ − X/Y and Γ − M high-symmetry lines. The labels indicate the eigenvalues of . (c) Schematic phase diagram with respect to the parameters (γ + η)/a₃ and β/b₁ in (1), where yellow, red and green regions represent topological semimetal, topologically trivial ferromagnetic semiconductor and QAHE states, respectively.

Figure 2.

Fully spin-polarized nodal loop semimetal ScLiCl₅ monolayer. (a) Top and side views of 2D materials XYZ₅, where X atoms occupy the Wyckoff position 2b(0; 0; 0.5) colored in blue, Y atoms occupy the Wyckoff position 2c(0.5; 0; 0.56383) colored in green and Z atoms occupy the Wyckoff positions 8g(0.20444, 0.11338, 0.59774) (orange) and 2c(0.5, 0, 0.40678) (red). (b) Partial density of states, (c) band structure and (d) the Weyl loop obtained from density functional theory (DFT) calculations in the absence of SOC for the ScLiCl₅ monolayer. (e) Electronic band structure and (f) the Weyl loop obtained from DFT calculations of ScLiCl₅ with SOC.

Figure 3.

Ferromagnetic semiconductor ScLiBr₅ monolayer. The electronic band structure (a) without SOC and (b) with SOC.

Figure 4.

QAHE in the LiScCl₅ monolayer. (a) Band structure in the absence of SOC and (b) partial density of states. (c) The band structure, (d) anomalous Hall conductivity and (e) projected spectrum on the (100) surface (line for 2D) with SOC. (f) The Kerr angle θ_Kerr as a function of photon energy ω.