Spin-polarized tunneling current through a thin film of a topological insulator in a parallel magnetic field

We calculate the tunneling conductance between the surface states on the opposite sides of an ultra-thin film of a topological insulator in a parallel magnetic field $B$. The parallel magnetic field produces a relative shift of the in-plane momenta of the two surfaces states. An overlap between the shifted Fermi circles and spinor wave functions result in unusual non-monotonic dependence of the tunneling conductance $\sigma(B)$ on the magnetic field $B$. The conductance $\sigma(B)$ grows with the magnetic field $B$, which corresponds to a negative magnetoresistance observed in an experiment [2], until it drops down abruptly to zero at the critical magnetic field $B_{\rm cr}$. Because spin orientation of the electronic surface states in topological insulators is locked to momentum, spin polarization of the tunneling current can be controlled by the magnetic field.\\[4pt] [1] Sergey S. Pershoguba and Victor M. Yakovenko, Phys. Rev. B {\bf 86}, 165404 (2012).\\[0pt] [2] H. B. Zhang et al., Adv. Mater. {\bf 24}, 132 (2012).