Microwave-assisted switching of a nanomagnet: analytical determination of the optimal microwave field

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.