Spontaneous supercurrent and $\varphi$0 phase shift parallel to magnetized topological insulator interfaces

Employing a Keldysh-Eilenberger technique, we theoretically study the generation of a sponta- neous supercurrent and the appearance of the $\varphi$0 phase shift parallel to uniformly in-plane mag- netized 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 $\varphi$0-states flows parallel to the interface at the junction location. Our calcula- tions 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 An- dreev 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.