Spatial magnetization profile in spherical nanomagnets with surface anisotropy: Green's function approach

We consider a single spherical nanomagnet and investigate the spatial magnetization profile ${\bf{m}}\left({\bf{r}}\right)$ in the continuum approach, using the Green's function formalism. The energy of the (many-spin) nanomagnet comprises an isotropic exchange interaction, a uniaxial anisotropy in the core and Néel's surface anisotropy, and an external magnetic field. We derive a semi-analytical expression for the magnetization vector field ${\bf{m}}\left({\bf{r}}\right)$ for an arbitrary position r within and on the boundary of the nanomagnet, as a solution of a homogeneous Helmholtz equation with inhomogeneous Neumann boundary conditions. In the absence of core anisotropy, we use the solution of this boundary problem and infer approximate analytical expressions for the components mα, α = x, y, z, as a function of the radial distance r and the direction solid angle. Then, we study the effects of the nanomagnet's size and surface anisotropy on the spatial behavior of the net magnetic moment. In the presence of a core anisotropy, an approximate analytical solution is only available for a position r located on the surface,i.e. r = Rn, where R is the radius of the nanomagnet and n the verse of the normal to the surface. This solution yields the maximum spin deviation as a result of the competition between the uniaxial core anisotropy and Néel's surface anisotropy. Along with these (semi-)analytical calculations, we have solved a system of coupled Landau–Lifshitz equations written for the atomic spins, and compared the results with the Green's function approach. For a plausible comparison with experiments, e.g. using the technique of small-angle magnetic neutron scattering, we have averaged over the direction solid angle and derived the spatial profile in terms of the distance r. We believe that the predictions of the present study could help to characterize and understand the effects of size and surface anisotropy on the magnetization configurations in nanomagnet assemblies such as arrays of well-spaced platelets.