Topological magnetoelectric effect (TME) and quantum anomalous Hall effect (QAHE) of topological-insulator (TI) thin films

We present a theoretical study of the two-dimensional (2D) surface states and orbital magnetoelectric response of TI thin films in which time-reversal symmetry is broken by magnetism at the surface. In the limit of large thickness d and small exchange coupling, the magnetization can be decomposed into independent contributions from top (t) and bottom (b) surfaces, with the single-surface magnetization M_s=t,b(J_s, E_Ds) being a function of its exchange coupling J_s and the Dirac point energy E_Ds relative to chemical potential. In the QAHE phase, appearing for J_t J_b > 0, |∂M_s/∂E_Ds| equals e/2h in the small-J large-d limit when the chemical potential lies in the surface state gap. Since M_s(J_s, E_Ds) = - M_s(-J_s, E_Ds) by time-reversal symmetry, it follows that when J_t J_b < 0 (the axion insulator phase) the 3D magnetization M_3D = (M_t + M_b)/d response to a vertical electric field E is ∂M_3D/∂E = e²/2h, a relationship referred to as the TME. By combining an effective model of the Dirac-cone surface states with tight-binding model calculations, we conclude that the TME is in fact robust against J_s and remains quantized even at exchange strengths for which a half-quantized surface anomalous Hall conductance cannot be properly defined. We comment on how the TME can be realized experimentally.