Topological surface states of $\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$ at finite temperatures and at domain walls

$\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$ has recently been the subject of intensive study, due to the prediction of axion insulator, Weyl semimetal, and quantum anomalous Hall insulator phases, depending on the structure and magnetic ordering. Experimental results have confirmed some aspects of this picture, but several experiments have seen zero-gap surfaces states at low temperature, in conflict with expectations. In this work, we develop a first-principles-based tight-binding model that allows for arbitrary control of the local spin direction and spin-orbit coupling, enabling us to accurately treat large unit-cells. Using this model, we examine the behavior of the topological surface state as a function of temperature, finding a gap closure only above the Néel temperature. In addition, we examine the effect of magnetic domains on the electronic structure, and we find that the domain wall zero-gap states extend over many unit-cells. These domain wall states can appear similar to the high-temperature topological surface state when many domain sizes are averaged, potentially reconciling theoretical results with experiments.

Figure 1

(a)–(b) Side and top views of layered AFM structure of $\mathrm{Mn}{\mathrm{Bi}}_2{\mathrm{Te}}_4$. Mn spin up is red and spin down is blue, Te is yellow, and Bi is green. (c) Brillioun zone. We use hexagonal labels to more easily compare bulk and surface calculations.

Figure 2

(a) Band structure of AFM phase with spins in $z$-direction calculated using DFT. (b) The same but calculated with model. The colors show projections onto Bi Wannier functions. (c) Band gap in eV as a function of SOC fraction. (d) Model band structure at $\text{SOC}=0.59$, the critical value.

Figure 3

(a) DFT and (b) model band structure of three layer slab with surfaces, with out-of-plane AFM spin ordering. The DFT calculation includes surface relaxation. Colors as in Fig. 2.

Figure 4

(a) Band gap (meV) versus temperature. Solid red line: minimum surface gap. Dashed green line: mean bulk gap. Blue points: individual surface calculations. (b)–(d) Unfolded average surface band structure at 1 K, 25 K, and 100 K, respectively.

Figure 5

(a) Surface band gap (meV) of $3\times 3\times 5$ supercell with ordered bulk and partially disordered surface spins. Each point is one spin configuration. (b) Unfolded averaged band structure at 0.5 mixing between ordered and disordered surface spins.

Figure 6

(a) Real-space representation of ${\left|\psi \right|}^2$ localized at domain wall in $24\times 1\times 5$ cell. Larger circles have more weight. Blue circles are Te, green are Bi, and red are Mn. (b) Unfolded band structure in $20\times 1\times 5$ unit-cell with 10 unit-cell domains. (c) Average unfolded band structure.