Spin Dynamics in magnetic dots:

The dynamics of spin precessions in small magnetic permalloy materials is important in designing a proper magnetic memory element.

We have studied this problem by considering a spin Hamiltonian which includes exchange and dipolar interactions exactly, and looked at the evolution of spin precessions by integrating numerically the semiclassical equations of motion. In parallel, we also linearized these equations and extracted the dynamical matrix whose eigenvalues and eigenvectors give the magnon frequencies and modes respectively.

For very small sizes, exchange interaction dominates and the ground state is a single domain ferromagnet, whereas for larger size samples the dipole interaction becomes important and forces the spin field to be divergenceless and tangent to the surface of the sample in order to minimize its magnetostatic energy. The ground state spin configuration in this case is a vortex. We found that the critical exchange coupling for which the transition between these two phases occurs scales as the square root of the system size.

The figure below shows how the lowest magnon frequencies change with the strength of the exchange coupling, J. From the single domain ferromagnet (SDF) side, as J is decreased, the lowest mode is called the C mode as the magnetization has the shape of the letter C. From the vortex side at low J, however, the softest mode involves oscillations of a vortex core raised above the plane. As J is increased, the core is enlarged and the out of plane component of the magnetization is increased until all the spins become aligned perpendicular to the plane.

The mechanism of this transition from both sides was also identified and is summarized in this picture (from ♣).

Click below to see the animation of the few lowest frequency modes for the vortex and single domain ferromagnet configuration.

[♣]  “Spin dynamics characterization in magnetic dots”

M. R. Mozaffari and K. Esfarjani;

Physica B 399, 81 (2007).