Thin films of topological insulators in a parallel or tilted magnetic field

In thin films of topological insulators, the surface states on the opposite edges are coupled by a tunneling coupling $t$. We discuss the energy spectrum and the transport properties of the system in a magnetic field. When an electron tunnels between the edges, the Lorentz force due to the in-plane magnetic field $\mathbf{B =(0,B_y,0)}$ changes the in-plane electron momentum by $\Delta p_x\propto B_y$. As a result, the Fermi circles on the opposite edges shift by $\Delta p_x$ in the momentum space. We propose that this effect can be detected by measuring the tunneling conductance $\sigma_{zz}(B_y)$ between the edges of the system. We show that $\sigma_{zz}(B_y)$ has special signatures due to the helical spin configuration of the surface Dirac cones. In case of a tilted field $\mathbf{B=(0,B_y,B_z)}$, the perpendicular component $B_z$ quantizes the in-plane motion to the Landau levels, while the in-plane component $B_y$ spatially shifts the wave functions on the different edges. As the overlap between the wave functions changes, the tunneling amplitude $t$ is renormalized and acquires dependence on both $B_y$ and $B_z$. This effect can be observed as the dependence of the interlayer conductance and in-plane conductivity on the tilt angle $\theta$ of the magnetic field tan$\theta=B_y/B_z$.