Josephson junction through a disordered topological insulator with helical magnetization

We study supercurrent and proximity vortices in a Josephson junction made of disordered surface states of a three-dimensional topological insulator with a proximity induced in-plane helical magnetization. In a regime where the rotation period of helical magnetization is larger than the junction width, we find supercurrent $0-\pi$ crossovers as a function of junction thickness, magnetization strength, and parameters inherent to the helical modulation and surface states. The supercurrent reversals are associated with proximity induced vortices, nucleated along the junction width, where the number of vortices and their locations can be manipulated by means of the superconducting phase difference and the parameters mentioned above.

Figure 1

Schematic of a superconductor (S)-topological insulator (TI)-superconductor junction with a helical magnetization pattern. The magnetization vector follows a helical pattern given by $\mathbf{h}\left(\mathbf{r}\right)=h_0(cosQy,sinQy,0)$. The junction plane resides in the $xy$ plane so that the S-TI interfaces lie in the $y$ direction at $x=0,d_F$. The junction has a length and width of $d_F$ and $W_F$, respectively. The superconducting electrodes are connected to the diffusive surface states of the TI through tunneling barriers.

Figure 2

Normalized critical supercurrent through diffusive TI surface states with an induced helical magnetization. (a) Shows the critical current as a function of rotation parameter $q$ at three different values of junction length: $d_F=2.0{\xi }_S,2.5{\xi }_S,3.5{\xi }_S$. (b) Shows the critical current as a function of junction length $d_F$ for $q=0.1\pi ,0.4\pi$, and $0.7\pi$. The junction ground state oscillates between 0 and $\pi$ superconducting phase difference by varying $q,d_F,h_0$.

Figure 3

Normalized critical supercurrent through the TI surface channel with a slow rotating magnetization, $q=0.3\pi$, as a function of magnetization intensity $h_0$ for different values for the junction thickness $d_F=2.0{\xi }_S,2.5{\xi }_S,3.0{\xi }_S,3.5{\xi }_S$.

Figure 4

Current density $J\left(y\right)$ and the absolute value of the Cooper pair wave function $U\left(y\right)$, normalized by their maximum values, as a function of $y$ coordinate along the junction width at the middle of junction $x=0$ (see Fig. 1) for three different magnetization rotation parameter $q=0.2\pi ,0.4\pi$, and $0.6\pi$. The superconducting phase difference is set fixed at $\varphi =\pi /2$ and the magnetization intensity is equal to $h_0=6.0\pi \mathrm{\Delta }$.