A core-shell-surface layer model to explain the size dependence of effective magnetic anisotropy in magnetic nanoparticles

The particle size (D) dependence of the effective magnetic anisotropy K
_eff
of magnetic nanoparticles (NPs) usually shows K
_eff
increasing with decreasing D. This dependence is often interpreted using the Eq.: K
_eff
= K
_b
+ (6K
_S
/D) where K
_b
and K
_s
are the anisotropy constants of the spins in the bulk-like core and surface layer, respectively. Here, we show that this model is inadequate to explain the observed size-dependency of K
_eff
for smaller nanoparticles with D <; 5 nm. Instead the results in NPs of maghemite (γ-Fe
₂
O
₃
), NiO and Ni are best described by an extension of the above model leading to the variation given by K
_eff
= K
_b
+ (6K
_S
/D) +K
_Sh
{[1-(2rf/D)]
^-3
-1}, where the last term is due to the spins in a shell of thickness d with anisotropy K
_sh
. The validation of this core-shell-surface layer (CSSL) model for three different magnetic NPs systems viz. ferrimagnetic γ-Fe
₂
O
₃
, ferromagnetic Ni and antiferromagnetic NiO suggests its possible applicability for all magnetic nanoparticles.