Spin-Torque-Biased Magnetic Strip: Nonequilibrium Phase Diagram and Relation to Long Josephson Junctions

Spin-torque-biased magnetic dynamics in an easy-plane ferromagnet (EPF) is theoretically studied in the presence of a weak in-plane anisotropy. While this anisotropy spoils U(1) symmetry, thereby quenching the conventional spin superfluidity, we show that the system instead realizes a close analog of a long Josephson junction (LJJ) model. The traditional magnetic-field and electric-current controls of the latter map, respectively, onto the symmetric and antisymmetric combinations of the out-of-plane spin torques applied at the ends of the magnetic strip. This suggests an alternative route towards realizations of superfluidlike transport phenomena in insulating magnetic systems. We study a spin-torque-biased phase diagram, providing an analytical solution for static multidomain phases in the EPF. We adapt an existing self-consistency method for the LJJ to develop an approximate solution for the EPF dynamics. The LJJ-EPF mapping has the potential for producing applications with superconductor-based circuit functionality at elevated temperatures. The results apply equally to antiferromagnets with suitable effective free energy in terms of the Néel order instead of in-plane magnetization.

Figure 1

(a) EPF with spin injection from metal contacts on two sides. The spin polarization of the current is along the $z$ axis, while the spin-current flow is oriented along the $x$ axis. The ferromagnet is sufficiently narrow in the transverse dimensions to treat it as quasi-one-dimensional. (b) Diagram of the inline LJJ, with $W$ much smaller than $L$ as well as the depth of the structure in the $z$ direction. We assume two conventional superconductors (SCs) much larger than the London penetration depth in all dimensions. The magnetic-field screening currents as well as the circulating vortices are schematically depicted with black oriented lines. The vortices in the junction map onto the domain walls depicted in (a).

Figure 2

The regions of stability for equilibrium $p$-vortex solutions of the LJJ boundary-value problem, for $L/\lambda =1$. The spin torques ${\tau }_{r,l}$ are in units of $A/\lambda$. Overlapping regions can have either solution. Outside of these regions, i.e., in the high $|{\tau }_r-{\tau }_l|$ limit, there are no stable time-independent solutions. Inset: The dependence of the critical torque ${\tau }_c$, for which ${\tau }_r+{\tau }_l=0$, on the length $L$ in units of $\lambda$.

Figure 3

Plot of the approximate modulation of the superfluid phase solution, ${\epsilon }^{(0)}(x,t)$, resulting from a weak in-plane anisotropy, with $L=1$, ${\beta }_d=0.1$, $\gamma =0.01$, ${\tau }_l=1.5$, and ${\tau }_r=-2$.