Variety of scenarios of the magnetic exchange response in topological insulators

Abstract

We present an ab initio relativistic k.p theory of the effect of magnetic exchange field on the band structure in the gap region of bulk crystals and thin films of three-dimensional layered topological insulators. For the field perpendicular to the layers (along $z$), we reveal novel unconventional scenarios of the response of the band-gap edges to the magnetization. The modification of the valence and conduction states is considered in terms of their $\Gamma$-point spin $s^z$ and total angular momentum $J^z$ on the atomic sites where the states are localized. The actual scenario depends on whether $s^z$ and $J^z$ have the same or opposite sign. In particular, the opposite sign for the valence state and the same sign for the conduction state give rise to an unconventional response in Bi$_2$Te$_3$ -- both in the bulk crystal and in ultra-thin films, which fundamentally distinguishes this topological insulator from Bi$_2$Se$_3$, where both states have the same sign. To gain a deeper insight into different scenarios in insulators with both inverted and non-inverted zero-field band structure, a minimal four-band third-order k.p model is constructed from first principles. Within this model, we analyze the field-induced band structure of the insulators and identify Weyl nodes that appear in a magnetic phase and behave differently depending on the scenario. We characterize the topology of the modified band structure by the Chern number $\mathcal{C}$ and find the unconventional response to be accompanied by a large Chern number $\mathcal{C}=\pm3$.