Nonlinear dynamics in a magnetic Josephson junction

We theoretically consider a Josephson junction formed by a ferromagnetic spacer with a strong spin-orbit interaction or a magnetic spin valve, i.e., a bilayer with one static and one free layer. Electron spin transport facilitates a nonlinear dynamical coupling between the magnetic moment and charge current, which consists of normal and superfluid components. By phenomenologically adding reactive and dissipative interactions (guided by structural and Onsager symmetries), we construct magnetic torques and charge pumping, whose microscopic origins are also discussed. A stability analysis of our coupled nonlinear systems generates a rich phase diagram with fixed points, limit cycles, and quasiperiodic states. Our findings reduce to the known phase diagrams for current-biased nonmagnetic Josephson junctions, on the one hand, and spin-torque driven magnetic films, on the other, in the absence of coupling between the magnetic and superconducting order parameters.

Figure 1

Schematics of our magnetic Josephson junctions. The directions of junction layering, applied current $I$, internal Rashba field $\mathbf{E}$ (a), and direction of the static ferromagnetic layer ${\mathbf{m}}_s$ (b) all lie along the $z$ axis. $\phi$ is the phase difference between the superconducting leads.

Figure 2

Stability diagram as a function of the current and applied magnetic field of the decoupled magnet. $\mu =-1.5$, $\nu ,\lambda ,\kappa =0$, and $K=1$. $p$ and $a$ label the parallel and antiparallel states of the magnet, respectively. Inset: decoupled Josephson junction. The S state (unshaded) and R state (shaded) are separated by the line $I=1$. The solid line is the value of $\phi$ for a 0 junction and dashed for the unstable $\pi$ junction.

Figure 3

Stability diagram as a function of the current $I$ and applied magnetic field $h_a$. $\lambda =-0.1$, $\mu ,\nu ,\kappa =0$, $K=1$, $\alpha =1$, and $\sigma =0.1$. h labels the Hopf bifurcation (solid lines), i labels the infinite-period bifurcation (long-dashed lines), and d labels the saddle-node bifurcation (short-dashed lines).

Figure 4

Separation of the S state (white) and R state (grey) of the superconductor by a nonlinear function defined by $I^{\prime }=1$. The parameters of this system are $\mu =-1$, $\lambda =0.6$, $\nu ,\kappa =0$, $K=\alpha =1$, and $\sigma =2$. The $1,2,3$ labels along the dashed line show the three places where the Josephson junction switches between superconducting and resistive states.

Figure 5

Stereographic projection of the magnetization at different currents. Here, $\mu =0.1$, $\lambda =0.5$, $\nu ,\kappa =0$, $K=1$, $\alpha =1$, and $\sigma =1$.