Spontaneous supercurrent and ${\phi }_0$ phase shift parallel to magnetized topological insulator interfaces

Employing a Keldysh-Eilenberger technique, we theoretically study the generation of a spontaneous supercurrent and the appearance of the ${\phi }_0$ phase shift parallel to uniformly in-plane magnetized superconducting interfaces made of the surface states of a three-dimensional topological insulator. We consider two weakly coupled uniformly magnetized superconducting surfaces where a macroscopic phase difference between the $s$-wave superconductors can be controlled externally. We find that, depending on the magnetization strength and orientation on each side, a spontaneous supercurrent due to the ${\phi }_0$ states flows parallel to the interface at the nanojunction location. Our calculations demonstrate that nonsinusoidal phase relations of current components with opposite directions result in maximal spontaneous supercurrent at phase differences close to $\pi$. We also study the Andreev subgap channels at the interface and show that the spin-momentum locking phenomenon in the surface states can be uncovered through density of states studies. We finally discuss realistic experimental implications of our findings.

Figure 1

Schematic of the TI-based Josephson weak-link junction considered. The superconductivity and magnetism are induced in the surface states of the 3D TI by virtue of the proximity effect. The macroscopic phases of the left and right superconductors ${\theta }_{l,r}$ can be controlled externally. The orientation of the uniform in-plane magnetization induced in the surface states ${\mathit{h}}_{l,r}$ also can be calibrated through an external magnetic field. The left and right segments are separated by an insulator at $x=0$, constituting an interface along the $y$ axis. We assume that a phase gradient is applied across the junction normal to the interface and consider two trajectories in the $xy$ plane, parallel with the interface, marked by $A$ and $B$ arrows to analyze the supercurrent flow parallel to the junction interface along the $y$ axis.

Figure 2

Total supercurrent (spontaneous current) at the junction location $x=0$, displayed in Fig. 1, along the $y$ axis as a function of phase gradient perpendicular to the junction $\phi ={\theta }_r-{\theta }_l$. The magnetization in the left segment is assumed zero $h_l=0$ while the magnetization in the right segment is directed along the $x$ axis normal to the interface and its strength varies from $h_r=0$ to $1.8{\mathrm{\Delta }}_0$.

Figure 3

Supercurrent densities along $A$ and $B$ trajectories parallel to the interface shown in Fig. 1 as a function of phase difference perpendicular to the junction interface $\phi$. In panel (a) we set $h_l=0$ and vary $h_r$ while in panel (b) we consider situations where $h_r=\pm h_l$.

Figure 4

The density of states $\mathrm{DOS}(\varepsilon$) as a function of the quasiparticles energy $\varepsilon$ at the interface of weak-link $x=0$. (a) We set $h_r=h_l=0$ and vary the phase difference $\phi =0,0.3\pi ,0.6\pi ,0.8\pi ,\pi$. In panel (b), we examine the effect of magnetization direction on the Andreev subgap states by setting $h_l=0$, $\phi =\pi /2$, and $h_r=0,\pm 0.5{\mathrm{\Delta }}_0$. (c) We consider opposite magnetization directions with identical intensities on the left and right sides of the weak-link $h_l=h_r=0.5{\mathrm{\Delta }}_0$ and vary the phase difference similarly to panel (a).