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 ultrathin film of a topological insulator in a parallel magnetic field. 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 the spinor wave functions results in an unusual dependence of the tunneling conductance on the magnetic field. Because the spin orientation of the electronic surface states in topological insulators is locked to momentum, the spin polarization of the tunneling current can be controlled by the magnetic field.

Figure 1

Thin film of a topological insulator of thickness $d$ in a parallel magnetic field $B_y$. The surface states ${\psi }_1$ and ${\psi }_2$ shown in red and blue overlap and couple when the thickness $d$ is comparable with the decay length $\xi$ of the surface state.

Figure 2

Electronic spectrum (7) of the thin film consists of the two Dirac cones spin-polarized in opposite directions, as shown with circulating arrows. The Dirac cones are shifted by $q_x$ due to the parallel magnetic field $B_y$. Tunneling between the Dirac cones is dominated by the electrons with momenta ${\mathit{p}}_1$ and ${\mathit{p}}_2$, where the shifted Fermi circles $|\mathit{p}\pm \mathit{q}/2|=p_F$ intersect.

Figure 3

(a) The tunneling conductances for spin-unpolarized $G_0^{\left(1\right)}$, Eq. (12), and spin-polarized $G_0^{\left(2\right)}$, Eq. (13). Fermi circles are shown by dashed and solid black lines, respectively. The latter curve $G_0^{\left(2\right)}$ corresponds to the negative magnetoresistance. The red curve is the spin polarization (15) of the tunneling current, and the right vertical axis shows its value, where 1 corresponds to $100\%$ spin polarization. (b) Schematic views of the shifted Fermi circles $|\mathit{p}\pm \mathit{q}/2|=p_F$ for the corresponding values of $q$. The tunneling is allowed for the momenta ${\mathit{p}}_1$ and ${\mathit{p}}_2$ where the Fermi circles intersect, also shown in Fig. 2. The spin polarizations for these momenta are shown by the blue and red arrows corresponding to the ${\psi }_1$ and ${\psi }_2$ surfaces. The vector sum of the arrows of the same color defines the net spin polarization of the tunneling current, which grows with the increase of $q$ as illustrated in the bottom of panel (b).