Ferromagnetic resonance and magnetic precessions in φ₀ junction

The Josephson $\varphi_0$ junctions with the current-phase relation $I = I_c \sin (\varphi-\varphi_0)$, where the phase shift $\varphi_0$ is proportional to the magnetic moment perpendicular to the gradient of the asymmetric spin-orbit potential, demonstrate a number of unique features important for superconducting spintronics and modern informational technologies. Here we show that a current sweep along IV-characteristic of the $\varphi_0$ junction may lead to regular magnetization dynamics with a series of specific phase trajectories. The origin of these trajectories is related to a direct coupling between the magnetic moment and the Josephson oscillations in these junctions, and ferromagnetic resonance when Josephson frequency coincides with the ferromagnetic one. We demonstrate that an external electromagnetic field can control the dynamics of magnetic moment within a current interval corresponding to a Shapiro step and produce topological transformation of specific precession trajectories. We demonstrate the appearance of the DC component of superconducting current and clarify its role in the transformation of IV-characteristics in the resonance region. Good agreement between numerical and analytical results has been found in the ferromagnetic resonance region. The presented results might be used for developing novel resonance methods of determination of the spin-orbit coupling parameter in the non-centrosymmetric materials. We discuss experiments which can test our results.