Manipulation of the Majorana Fermion, Andreev Reflection, and Josephson Current on Topological Insulators

We study theoretically the transport properties of a normal metal (N)/ferromagnet insulator (FI)/superconductor (S) junction and a S/FI/S junction formed on the surface of a three-dimensional topological insulator, where the chiral Majorana mode exists at the FI/S interface. We find the chiral Majorana mode generated in N/FI/S and S/FI/S junctions is very sensitively controlled by the direction of the magnetization $\mathit{m}$ in the FI region. In particular, the current-phase relation of the Josephson current in S/FI/S junctions has a phase shift of neither 0 nor $\pi$ that can be tuned continuously by the component of $\mathit{m}$ perpendicular to the interface.

Figure 1

Schematic illustration of the junction. (a) N/FI/S junction and (b) S/FI/S junction formed on the surface of a 3D TI. The current is flowing on the surface of the TI.

Figure 2

Left panel: CMM energy dispersion $E_b$ as a function of the incident angle $\theta$. Right panel: Normalized tunneling conductance $\sigma$ in N/FI/S junctions. $m_{z}d/v_{\mathrm{F}}=1$ and $m_y/m_z=0$. Curve $a$ represents $\mu /m_z=1$, $b$ $\mu /m_z=2$, and $c$ $\mu /m_z=0.5$.

Figure 3

Left panel: CMM energy dispersion $E_b$ for curve $a$ $m_y/m_z=0$, $b$ $m_y/m_z=0.4$, and $c$ $m_y/m_z=-0.4$. Right panel: ${\sigma }_{\mathrm{S}}(\theta )$ in N/FI/S junctions. Curve $A$ represents $eV=0.2\Delta$, $m_y=0.4m_z$, $B$ $eV=-0.2\Delta$, $m_y=0.4m_z$, $C$ $eV=0.2\Delta$, $m_y=-0.4m_z$, and $D$ $eV=-0.2\Delta$, $m_y=-0.4m_z$. In all curves, we choose $m_{z}d/v_{\mathrm{F}}=1$ and $\mu /m_z=1$.

Figure 4

Contour plot of CMM energy level $E_J$ as a function of $\theta$ and $\phi$ for (a) $m_x=0$, (b) $m_x=0.4m_z$, (c) $m_x=-0.4m_z$. The resulting Josephson current in S/FI/S junctions is plotted in (d) with curve $a$ representing $m_x/m_z=1$, $b$ $m_x/m_z=0.4$, and $c$ $m_x/m_z=-0.4$. In all panels, we choose $m_{z}d/v_{\mathrm{F}}=1$, $\mu /m_z=1$, and $m_y/m_z=0$. $T=0.05T_C$ with transition temperature $T_C$.