Koopmans' theorem as the mechanism of nearly gapless surface states in self-doped magnetic topological insulators

The magnetization-induced gap at the surface state is widely believed to be the kernel of magnetic topological insulators (MTIs) because of its relevance to various topological phenomena, such as the quantum anomalous Hall effect and the axion insulator phase. However, whether the magnetic gap exists in an intrinsic MTI, such as ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$, still remains elusive, with significant discrepancies between theoretical predictions and various experimental observations. Here, including the previously overlooked self-doping in real MTIs, we find that in general a doped MTI prefers a ground state with a gapless surface state. We use a simple model based on Koopmans' theorem to elucidate the mechanism and further demonstrate it in the self-doped ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}/{({\mathrm{Bi}}_{2}{\mathrm{Te}}_{3})}_{n}$ family through first-principles calculations. Our work sheds light on the design principles of MTIs with magnetic gaps by revealing the critical role of doping effects in understanding the delicate interplay between magnetism and topology.