On the Christmas Eve of 2011, Mark Hoefer received an email linking one of his theories to experiments taking place in Gothenburg, Sweden. It was a momentous occasion. During his time with the National Institute of Standards and Technology (NIST) as a postdoctoral fellow, Professor Hoefer and colleagues had proposed a hypothesis surrounding the existence of dissipative droplet solitons. He felt strongly he was correct, but lacked part of the puzzle, the physical observations. On the other side of the world, Ezio Iacocca’s former PhD advisor gave a lecture on his research. Afterward, an audience member brought Professor Hoefer’s hypothesis to the speaker’s attention, a discussion that ultimately led to the momentous Christmas Eve email. Theory and experiment had at last intersected. Now a Swedish International Postdoctoral Fellow in CU Boulder’s Applied Mathematics Department, Dr. Iacocca works closely with Professor Hoefer and, with their associates, continue their exciting findings.
In 2013, they published “Spin Torque-Generated Magnetic Droplet Solitons” in Science, one of the world’s top academic journals. The paper spurred excitement among the academic community, and the project maintained its momentum. A new paper, “Magnetic droplet nucleation boundary in orthogonal spin-torque nano-ocillators” has been published in Nature Communications, another widely recognized and prestigious academic journal.
This publication discusses dissipative droplets, which are magnetic solitons that Professor Hoefer and colleagues theoretically predicted to exist in materials where dispersion, anisotropy, dissipation, and forcing balance each other. This balance is possible to achieve experimentally using ferromagnetic materials with perpendicular magnetic anisotropy (PMA) and a current-induced forcing known as spin transfer torque. In addition to their novel physical features, these solitons may be useful for spintronic applications where the electron’s spin, in addition to its charge, is used for signal and information processing. In this paper, the dissipative droplet’s condition for existence, or nucleation boundary, is derived analytically and measured experimentally. This is achieved by weakly nonlinear stability analysis of the Landau-Lifshitz magnetic equations of motion including dissipation and forcing. From a uniform state, localized forcing excites plane waves when spatially uniform dissipation is balanced or exceeded. However, in materials with PMA, the uniform state exhibits a subcritical Hopf bifurcation. The corresponding plane waves that compose this bifurcation are modulationally unstable, a nonlinear instability that results in the coalescence of waves and, ultimately the formation of a large amplitude dissipative droplet. In contrast, the corresponding equation without PMA exhibits a supercritical Hopf bifurcation and the formation of a stable, weakly nonlinear time-periodic solution. Looking for instability and taking into account asymmetries present in the experiment, an analytical expression for the nucleation boundary as a function of applied fields and current (forcing) is obtained. Experiments show a remarkable agreement and fits to the analytical expression enable the in-situ determination of relevant material parameters such as the spin torque efficiency and asymmetry, with good accuracy. These results reconcile different nucleation trends previously observed experimentally under a single, closed-form expression.
Written by Danielle Hawley