Factors influencing the energy gap in topological states of antiferromagnetic MnBi$_2$Te$_4$

The experimentally measured angle-resolved photoemission dispersion maps for MnBi$_{2}$Te$_{4}$ samples, which show different energy gaps at the Dirac point (DP), are compared with the results of theoretical calculations to find the conditions for the best agreement between theory and experiment. We have analyzed different factors which influence the Dirac gap width: (i) the surface van der Waals (SvdW) distance between the first and second septuple layers (SLs), (ii) the magnetic moment on Mn atoms, (iii) the spin-orbit coupling (SOC) strength for the surface Te and Bi atoms and related changes in the localization of the topological surface states (TSSs). It was shown that all these factors may change the gap width at the DP in a wide range from 5 to $\sim$90~meV. We show that the Dirac gap variation is mainly determined by the corresponding changes in the TSSs spatial distribution. The best agreement between the presented experimental data (with the Dirac gaps between $\sim$15 and 55~meV) and the calculations takes place for a slightly compressed SvdW interval (of about -3.5~\% compared to the bulk value) with modified SOC for surface atoms (that can occur in the presence of various defects in the near-surface region). We show that upon changing the values of the SvdW interval and surface SOC strength the TSSs spatial distribution shifts between the SLs with opposite magnetizations, which leads to a non-monotonic change in the Dirac gap size.