Electronic structure and topological phases of the magnetic layered materials ${\mathrm{MnBi}}_2{\mathrm{Te}}_4, {\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$

First-principles calculations are performed to study the electronic structures and topological phases of magnetic layered materials ${\mathrm{MnBi}}_2{\mathrm{Te}}_4, {\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ under different film thicknesses, strains, and spin-orbit coupling (SOC) strengths. All these compounds energetically prefer the antiferromagnetic (AFM) state. ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ bulks are AFM topological insulators (TIs) in the AFM state, while they become Weyl semimetals in the ferromagnetic (FM) state. ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$ is trivially insulating in both the AFM and FM states, but it becomes an AFM TI or a Weyl semimetal with increasing SOC strength or applying compressive strains. Under equilibrium lattice constants, the FM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ slabs thicker than two septuple layers (SLs), the AFM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ slabs thicker than three SLs, and the FM ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ slabs thicker than five SLs are all Chern insulators. In addition, Chern insulators can also be obtained by compressing the in-plane lattice constants of the FM ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$ slabs thicker than four SLs and the FM ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ slabs of three or four SLs, but cannot be obtained using the same method in the AFM slabs of these two materials. In-plane tensile strains about 1% to 2% turn the Chern insulators into trivial insulators.

Figure 1

Crystal structures of (a) AFM and (b) FM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, where the arrows indicate the directions of the magnetic moments. (c) Brillouin zone of the crystals.

Figure 2

Band structures of AFM (a) ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, (b) ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and (c) ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$, where the magenta symbols denote the Bi-$p$ or the Sb-$p$ projections, and the symbol size indicates the contribution weight. Evolution of WCCs of AFM (d) ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ and (e) ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$ in the $k_z=0$ plane. Surface states of AFM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ in (f) (001) and (g) (101) surfaces.

Figure 3

(a) Energy gaps and topological phases of AFM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4, {\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ at different strains. (b) Evolution of topological phase and energy gaps of AFM ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$ following the SOC strength ${\lambda }_{\mathrm{soc}}$. Positive and negative energy gaps mean that the bulks are trivial insulators with $Z_2=0$ and AFM TIs with $Z_2=1$, respectively, and the absolute values of the gaps correspond to the gap sizes. (c) Band structure of AFM ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$ at ${\lambda }_{\mathrm{soc}}=1.3$.

Figure 4

Band structures of FM (a) ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, (b) ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and (c) ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ along the $\mathrm{\Gamma }-Z$ direction. (d) WCCs evolution of FM ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$. (e) Band structure of FM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ at ${\lambda }_{\mathrm{soc}}=1.1$. (f) Energy dispersion of FM ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ in the $k_{\mathrm{x}}-k_{\mathrm{z}}$ plane. W and ${\mathrm{W}}^{\prime }$ indicate the two Weyl points. (g) Fermi arc of FM ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ in the (101) surface.

Figure 5

(a)–(f) Calculated energy band structures of the AFM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4, {\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ slabs of 3 and 5 SLs, where the Chern numbers are indicated. Calculated (g) chiral edge states and (h) Hall conductance of the AFM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ slab of 5 SLs.

Figure 6

Calculated energy band structures of the FM (a) ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, (b) ${\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and (c) ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ slabs of 5 SLs, where the Chern numbers are indicated. Calculated (d) chiral edge states and (e) Hall conductance of the FM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ slab of 2 SLs. (f) Band structure of the FM ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ slab of 5 SLs at $\epsilon =-3\%$.

Figure 7

Energy gaps and topological phases of (a)–(c) AFM and (d)–(f) FM ${\mathrm{MnBi}}_2{\mathrm{Te}}_4, {\mathrm{MnBi}}_2{\mathrm{Se}}_4$, and ${\mathrm{MnSb}}_2{\mathrm{Te}}_4$ slabs at different strains and varying film thicknesses. The legend in (a) applies to (a), (b), and (c), while that in (d) applies to (d), (e), and (f). Positive and negative energy gaps mean that the slabs are trivial insulators with $C=0$ and Chern insulators with $C=1$, respectively, and the absolute values of the gaps correspond to the gap sizes.