Intrinsic ferromagnetic axion states and single pair of Weyl fermions in the stable-state $\mathrm{Mn}{X}_2{B}_2{T}_6$ family of materials

The intrinsic ferromagnetic (FM) axion insulators and Weyl semimetals (WSMs) with only a single pair of Weyl points have drawn intensive attention but so far remain rare and elusive in real materials. Here, we propose a new class of $\mathrm{Mn}{X}_2{B}_2{T}_6$-B ($X$ = Ge, Sn, or Pb; $B$ = Sb or Bi; $T$ = Se or Te) family that is the stable structural form of this system. We find that the $\mathrm{Mn}{X}_2{B}_2{T}_6$-B family has not only the intrinsic FM axion insulators ${\mathrm{MnGe}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B, ${\mathrm{MnSn}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B, and ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B, but also the intrinsic WSM ${\mathrm{MnSn}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_6$-B with only a single pair of Weyl points. Thus, the $\mathrm{Mn}{X}_2{B}_2{T}_6$-B family can provide an ideal platform to explore the exotic topological magnetoelectric effect and the intrinsic properties related to Weyl points.

Figure 1

Optimized crystal structures of the $\mathrm{Mn}{X}_2{B}_2{T}_6$ ($X$ = Ge, Sn, or Pb; $B$ = Sb or Bi; $T$ = Se or Te) and its parent $X_2{B}_2{T}_5$ materials. The $1\times 1\times 3$ supercells of (a) $X_2{B}_2{T}_5$-A and (b) $X_2{B}_2{T}_5$-B in stacking sequences A and B, respectively. Conventional cells of (c) $\mathrm{Mn}{X}_2{B}_2{T}_6$-A and (d) $\mathrm{Mn}{X}_2{B}_2{T}_6$-B in stacking sequences A and B, respectively. (e) Brillouin zone (BZ) of the primitive cell of $\mathrm{Mn}{X}_2{B}_2{T}_6$-B.

Figure 2

Total energy evolution for the ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$ system during the magnetic evolutionary search. The red and blue lines represent the most stable structure ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B and metastable ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-A, respectively.

Figure 3

Spin-polarized band structures and density of states (DOS) of the bulk compounds ${\mathrm{MnSn}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_6$-B and ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B calculated without the SOC in the ferromagnetic ground state.

Figure 4

Band structures with orbital weights for (a) ${\mathrm{MnSn}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_6$-B and (d) ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B bulk materials calculated with the SOC in the out-of-plane FM ground states. Single pair of Weyl points along the $-Z\text{−}\mathrm{\Gamma }\text{−}Z$ path near the Fermi level are shown in inset (a). Two ideal Weyl points (${\mathrm{WP}}_1$ and ${\mathrm{WP}}_2$) are separated 0.06 ${\text{Å}}^{-1}$ along the $k_z$ direction, which is comparable to ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ in the FM state. (b) Motions of the sum of Wannier charge centers (WCCs) on a sphere enclosing each Weyl point in the BZ. (c) The energy gap between the lowest unoccupied state and the highest occupied state at $\mathrm{\Gamma }$ as a function of stress along the $c$ axis.

Figure 5

Surface energy bands of the semi-infinite (a) (111) and (b) ($1\overline{1}0$) surfaces of ${\mathrm{MnSn}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_6$-B. (c) The Fermi arc of the two Weyl points of ${\mathrm{MnSn}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_6$-B on the ($1\overline{1}0$) surface. (d) Surface energy bands of the semi-infinite (111) surface for ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$-B. (e) The enlarged view of the surface states around the $\overline{\mathrm{\Gamma }}$ point in Fig. 5. (f) A triangular Fermi surface on the (111) surface of ${\mathrm{MnPb}}_2{\mathrm{Bi}}_2{\mathrm{Te}}_6$.