Electronic and spin structure of topological surface states in MnBi$_4$Te$_7$ and MnBi$_6$Te$_{10}$ and their modification by an applied electric field

The electronic and spin structure of topological surface states in antiferromagnetic topological insulators MnBi$_4$Te$_7$ and MnBi$_6$Te$_{10}$ consisting of a sequence of magnetic MnBi$_2$Te$_4$ septuple layers separated by nonmagnetic Bi$_2$Te$_3$ quintuple layers has been calculated within the density functional theory. Features characteristic of systems with different terminations of the surface (both septuple and quintuple layers) have been analyzed and theoretical calculations have been compared with the measured dispersions of electronic states. It has been shown that a band gap of about 35–45 meV, as in MnBi$_2$Te$_4$, opens at the Dirac point in the structure of topological surface states in the case of the surface terminated by a magnetic septuple layer. In the case of the surface terminated by a nonmagnetic quintuple layer, the structure of topological surface states is closer to the form characteristic of Bi$_2$Te$_3$ with different energy shifts of the Dirac point and the formation of hybridized band gaps caused by the interaction with the lower-lying septuple layer. The performed calculations demonstrate that the band gap at the Dirac point can be changed by varying the distance between layers on the surface without a noticeable change in the electronic structure. The application of an electric field perpendicular to the surface changes the electronic and spin structure of topological surface states and can modulate the band gap at the Dirac point depending on the magnitude and direction of the applied field, which can be used in applications.