  Motivated by the recent experimental realization of the half-quantized Hall
effect phase in a three-dimensional (3D) semi-magnetic topological insulator
[M. Mogi et al., Nature Physics 18, 390 (2022)], we propose a scheme for
realizing the half-quantized Hall effect and axion insulator in experimentally
mature 3D topological insulator heterostructures. Our approach involves
optically pumping and/or magnetically doping the topological insulator surface,
such as to break time reversal and gap out the Dirac cones. By toggling between
left and right circularly polarized optical pumping, the sign of the
half-integer Hall conductance from each of the surface Dirac cones can be
controlled, such as to yield half-quantized ($0+1/2$), axion ($-1/2+1/2=0$),
and Chern ($1/2+1/2=1$) insulator phases. We substantiate our results based on
detailed band structure and Berry curvature numerics on the Floquet Hamiltonian
in the high-frequency limit. Our paper showcases how topological phases can be
obtained through mature experimental approaches such as magnetic layer doping
and circularly polarized laser pumping and opens up potential device
applications such as a polarization chirality-controlled topological
transistor.
