Josephson effect in normal and ferromagnetic topological-insulator junctions: Planar, step, and edge geometries

We investigate Josephson junctions on the surface of a three-dimensional topological insulator in planar, step, and edge geometries. The elliptical nature of the Dirac cone representing the side surface states of the topological insulator results in a scaling factor in the Josephson current in a step junction as compared to the planar junction. In edge junctions, the contribution of the Andreev bound states to the Josephson current vanishes due to spin-momentum locking of the surface states. Furthermore, we consider a junction with a ferromagnetic insulator between the superconducting regions. In these ferromagnetic junctions, we find an anomalous finite Josephson current at zero phase difference if the magnetization is pointing along the junction (and perpendicular to the Josephson current). An out-of-plane magnetization with respect to the central region of the junction opens up an exchange gap and leads to a nonmonotonic behavior of the critical Josephson current for sufficiently large magnetization as the chemical potential increases.

Figure 1

(a) Planar, (b) step, and (c) edge topological-insulator junctions and ferromagnetic (d) planar and (e) step junctions. The $S$ planes are the superconductors inducing an effective $p$-wave superconductivity in the surface of the topological insulator (regions I and III). The weak link (i.e., the central region) of the junction in (a), (b), and (c) is the pristine normal-conducting (N) surface of the topological insulator (region II). In (d) and (e), region II is covered by a ferromagnetic insulator with magnetization $\mathbf{M}$ that induces an exchange coupling in the surface states.

Figure 2

Critical Josephson current $J_c$ (in units of $\widetilde{J_c}$) of a step junction as a function of the chemical potential measured from the Dirac point energy $\Lambda$ [see Eq. (29)]. The dashed red line shows the contribution of the propagating waves only, and the dotted black line is the asymptote for $\left|\Lambda \right|\gg 1$.

Figure 3

Critical Josephson current $J_c$ (in units of $\widetilde{J_c}$) of a ferromagnetic step junction as a function of $\Lambda$ [see Eq. (29)] for different values of $q_x=m_{x}L/\left(\hslash v_F^{yz}\eta \right)$. Inset: $J_c$ as a function of $q_x$ at the Dirac point ($\Lambda =0$).

Figure 4

Superconducting phase difference $\varphi \left(J_c\right)$ (which maximizes the current $J$) as a function of $\Lambda$ [see Eq. (29)] for different values of $q_x=m_{x}L/\left(\hslash v_F^{yz}\eta \right)$. The crosses indicate the values of $\Lambda$ at which the scaled number of modes $N^{\prime }=NL/\left(W\eta \right)=\sqrt{{\Lambda }^2-q_x^2)}/\pi$ takes an integer value.

Figure 5

Josephson current $J$ (in units of $\widetilde{J_c}$) for the ferromagnetic step junction as a function of $\varphi$ for different values of the magnetization parametrized by $q_{x,y,z}=m_{x,y,z}L/\left(\hslash v_F^{yz}\eta \right)$. The black line in (a), (b), and (c) corresponds to $q_x=q_y=q_z=0$. (d) shows the difference between the Josephson current of junctions with magnetization in the $x$ and $z$ directions [$\Delta J=J(q_z,\varphi )-J(q_x,\varphi )$].