We analytically determine the optimal microwave field that allows for the magnetization reversal of a nanomagnet modeled as a macrospin. This is done by minimizing the total injected energy. The results are in good agreement with the fields obtained
numerically using the optimal control theory. For typical values of the damping parameter, a weak microwave field is sufficient to induce switching through a resonant process. The optimal field is orthogonal to the magnetization direction at any time and modulated both in amplitude and frequency. The dependence of the pulse shape on the applied field and damping parameter is interpreted. The total injected energy is found to be proportionnal to the energy barrier between the initial state and the saddle point and to the damping parameter. This result may be used as a means for probing the damping parameter in real nanoparticles.
We develop an efficient and general method for optimizing the microwave field that achieves magnetization switching with a smaller static field. This method is based on optimal control and renders an exact solution for the 3D microwave field that tri
ggers the switching of a nanomagnet with a given anisotropy and in an oblique static field. Applying this technique to the particular case of uniaxial anisotropy, we show that the optimal microwave field, that achieves switching with minimal absorbed energy, is modulated both in frequency and in magnitude. Its role is to drive the magnetization from the metastable equilibrium position towards the saddle point and then damping induces the relaxation to the stable equilibrium position. For the pumping to be efficient, the microwave field frequency must match at the early stage of the switching process the proper precession frequency of the magnetization, which depends on the magnitude and direction of the static field. We investigate the effect of the static field (in amplitude and direction) and of damping on the characteristics of the microwave field. We have computed the switching curves in the presence of the optimal microwave field. The results are in qualitative agreement with micro-SQUID experiments on isolated nanoclusters. The strong dependence of the microwave field and that of the switching curve on the damping parameter may be useful in probing damping in various nanoclusters.
