Optimal Control of Magnetization Reversal in a Monodomain Particle by Means of Applied Magnetic Field

A complete analytical solution to the optimal reversal of a macrospin with easy-axis anisotropy is presented. An optimal control path minimizing the energy cost of the reversal is identified and used to derive the time-dependent direction and amplitude of the optimal switching field. The minimum energy cost of the reversal scales inversely with the switching time for fast switching, follows exponential asymptotics for slow switching, and reaches the lower limit proportional to the energy barrier between the target states and to the damping parameter at infinitely long switching time. For a given switching time, the energy cost is never smaller than that for a free macrospin. This limitation can be bypassed by adding a hard anisotropy axis that activates the internal torque in the desired switching direction, thereby significantly reducing the energy cost. A comparison between the calculated optimal control path and minimum energy path reveals that optimal control does not translate to the minimization of the energy barrier but signifies effective use of the system’s internal dynamics to aid the desired magnetic transition.

Figure 1

Calculated optimal control paths (OCPs) for the reversal of a macrospin pointing along the unit vector $\overrightarrow{s}$. The initial and the final states are at the north and the south poles of the unit sphere, respectively. The damping factor $\alpha$ is 0.1. The switching time $T$ is $10{\tau }_0$ and $100{\tau }_0$ for the paths shown with thick and thin green lines, respectively. The external magnetic field ${\overrightarrow{b}}_m$ at $t=T/4$, $t=T/2$, and $t=3T/4$ is shown for the shorter path with the brown arrows.

Figure 2

Amplitude of the switching field as a function of time for $T=100{\tau }_0$ and several values of $\alpha$ (solid lines). Dashed lines show $\alpha b_i^{\perp }$, which is proportional to the polar component of the torque generated by the internal field.

Figure 3

Minimum energy cost of magnetization switching as a function of the inverse of the switching time. Dashed (dotted) lines show long (infinite) switching time asymptotics. Thin vertical lines indicate switching time $T_{ϵ}$, for which the minimum energy cost is $ϵ=10\%$ larger than the infinite switching time limit ${\mathrm{\Phi }}_{\infty }$.

Figure 4

Distribution of the torque (blue arrows) generated by the internal field of the macrospin system with both easy and hard anisotropy axes, superimposed on the contour plot showing the energy surface of the system defined by Eq. (13). The green line shows the calculated OCP for $T=0.32{\tau }_0$ and $\alpha =0$. Green arrows indicate the velocity of the system. The size of the blue (green) arrows code the magnitude of the torque (velocity). The black dashed line shows the minimum energy path (MEP). The inset shows the energy of the system as a function of displacement along the OCP (green solid line) and MEP (black dashed line).