Characterization of the topological magneto-electric effect in Bi₂Se₃topological insulator

We investigate the topological magneto-electric effect (TME) in Bi₂Se₃ topological insulator (TI) thin films, with a perpendicular exchange field applied to the top and bottom surfaces in the anti-parallel configuration. Here the system is in the axion insulator phase, which allows a direct probe of the quantized topological θ-term of the axion electrodynamics. The θ-term is a TI bulk property which gives the coefficient of proportionality between an applied electric field and the induced magnetization. We first consider an infinite slab and extract θ in a tight-binding framework, under the condition of weak electric field. In the second part of the work, we consider systems which are finite in one or more dimensions. By analyzing the dependence of θ on the system size and the strengths and profiles of the exchange fields on the two surfaces, we determine when the TME is expected to be quantized. Furthermore, we investigate possible deviations from exact quantization in the regime where the exchange field on one of two surfaces is progressively enhanced beyond a critical value. Our study sheds light on the fundamental issue of whether TME in a TI is completely determined by the bulk properties or it can be affected by how time-reversal symmetry is broken at the surfaces.