Effective anisotropy of thin nanomagnets: Beyond the surface-anisotropy approach

We study the effective anisotropy induced in thin nanomagnets by the nonlocal demagnetization field (dipole-dipole interaction). Assuming a magnetization independent of the thickness coordinate, we reduce the energy to an inhomogeneous on-site anisotropy. Vortex solutions exist and are ground states for this model. We illustrate our approach for a disk and a square geometry. In particular, we obtain good agreement between spin-lattice simulations with this effective anisotropy and micromagnetic simulations.

Figure 1

(Color online) Arrangement of coordinates in the local reference frame.

Figure 2

(Color online) Spacial dependence of the effective anisotropy constants $\mathcal{A}$ [see Eq. (B8)] and $\mathcal{B}$ [see Eq. (B10)]. (a) The anisotropy constant $\mathcal{A}\left(\xi \right)$ vs $\xi$. (b) The product $\epsilon \bullet \mathcal{B}\left(\xi \right)$ vs $\xi$.

Figure 3

(Color online) Comparison of the vortex profiles for the micromagnetic simulation and the effective anisotropy approximation for a Py nanodisk ($2R=212\mathrm{nm}$ and $h=16\mathrm{nm}$). The red curve corresponds to the spin-lattice simulations for the effective anisotropy model with $\mathcal{H}={\mathcal{H}}_{\mathrm{ex}}+{\mathcal{H}}_d^{\mathit{loc}}$. The blue curve corresponds to the micromagnetic simulations. The black dashed curve to the Gaussian ansatz $\cos \theta =\exp (-r^2∕r_v^2)$.

Figure 4

(Color online) Numerical results for the vortex state Py prism (sides $212\times 212{\mathrm{nm}}^2$ and thickness $h=16\mathrm{nm}$). (a) and (b) represent the spin-field distribution [(a) spin-lattice simulations for the effective anisotropy model Hamiltonian (9) and (b) micromagnetic OOMMF simulation data], and (c) the configurational anisotropy lines. These lines determine the in-plane anisotropy axis orientation, calculated from Eq. (20); the length of the lines corresponds to the anisotropy amplitude in a particular point.

Figure 5

(Color online) The in-plane spin angle $\varphi$ as a function of the polar angle for the vortex state in a prism of Py of sides $212\times 212{\mathrm{nm}}^2$ and thickness $h=16\mathrm{nm}$. The red dashed curves correspond to the effective anisotropy approximation; the blue solid curves to the micromagnetic simulation data. (a) Distance from the vortex center $r=10\ell$. (b) Distance from the vortex center $r=20\ell$.

Figure 6

(Color online) Arrangement of coordinates in the local reference frame for the prism-shaped particle.