We report a theoretical study on the spin Josephson effect arising from the exchange coupling of the two ferromagnets (Fs), which are deposited on a two-dimensional (2D) time-reversal-invariant topological insulator. An anomalous spin supercurrent $J_{sz}\sim sin(\alpha +{\alpha }_0)$ is found to flow in between the two Fs and the ground state of the system is not limited to the magnetically collinear configuration $(\alpha =n\pi ,n$ is an integer) but determined by a controllable angle ${\alpha }_0$, where $\alpha$ is the crossed angle between the two F magnetizations. The angle ${\alpha }_0$ is the dynamic phase of the electrons traveling in between the two Fs and can be controlled electrically by a gate voltage. This anomalous spin Josephson effect, similar to the conventional ${\phi }_0$ superconductor junction, originates from the definite electron chirality of the helical edge states in the 2D topological insulator. These results indicate that the magnetic coupling in a topological system is different from the usual one in conventional materials.

• Received 1 August 2016

DOI:https://doi.org/10.1103/PhysRevB.94.155443