Electrically Controlled Anomalous Hall Effect and Orbital Magnetization in Topological Magnet ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$

We propose an intrinsic mechanism to understand the even-odd effect, namely, opposite signs of anomalous Hall resistance and different shapes of hysteresis loops for even and odd septuple layers (SLs), of MBE-grown ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ thin films with electron doping. The nonzero hysteresis loops in the anomalous Hall effect and magnetic circular dichroism for even-SLs ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ films originate from two different antiferromagnetic (AFM) configurations with different zeroth Landau level energies of surface states. The complex form of the anomalous Hall hysteresis loop can be understood from two magnetic transitions, a transition between two AFM states followed by a second transition to the ferromagnetic state. Our model also clarifies the relationship and distinction between axion parameter and magnetoelectric coefficient, and shows an even-odd oscillation behavior of magnetoelectric coefficients in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ films.