Field-effect birefringent spin lens in ultrathin film of magnetically doped topological insulators.

We investigate the low-energy electron dynamics in two-dimensional ultrathin film of magnetically doped topological insulators in the context of gate-tuned coherent spin manipulation. Our first-principles calculations for such film unambiguously identify its spin-resolved topological band structure arising from spin-orbit coupling and time-reversal symmetry breaking. Exploiting this characteristic, we predict a negative birefraction for chiral electron tunneling through a gate-controlled p-n interface in the film, analogous to optical birefringence. By fine-tuning the gate voltage, a series of unusual phenomena, including electron double focusing, spatial modulation of spin polarizations, and quantum-interference-induced beating patterns, could be efficiently implemented, offering a powerful platform to establish spin-resolved electron optics by all-electrical means.