1 Effective and surface anisotropies

In experimental situation and basing on macroscopic measurements, different contributions to anisotropy are difficult to distinguish and the concept of the "effective anisotropy" is used. The meaning of the concept is not clear and it is probably highly dependent on the employed experimental method, such as the magnetization work, the ac- susceptibility measurements, the ferromagnetic resonance, the blocking temperature, etc. It has been suggested [43,36,32] that the effective magnetic anisotropy energy $\mathcal{K}^{eff}$ could be phenomenologically separated in a volume and a surface contributions and approximately obeys the relation:

$$\mathcal{K}^{eff}=K_{V} + K_{S}\cdot S/V$$ (15)

where $K_{V}$ is the volume anisotropy, $K_{S}$ is the "effective" surface anisotropy, S is the surface and V is the volume of the system. This means that the magnetic surface anisotropy in thin films can be determined by measuring the total magnetic anisotropy as a function of the film thickness $t$, showing a $1/t$ dependence [54,55,46,56]:

$$\mathcal{K}^{eff}=K_{V}+ 2K_{S}/t$$ (16)

The factor 2 is due to the existence of two surfaces.

In the case of spherical nanoparticles the same relation has been suggested [36], leading to the formula .

$$\mathcal{K}^{eff}=K_{V}+ 6K_{S}/D$$ (17)

where D is the diameter of the nanoparticle.

Figure: Size dependence of the effective anisotropy constant in Co nanoparticles, the line represents the fit to the expression (1.17). (from Ref.[8])

We would like to emphasize that this formula has been introduced in an "ad hoc" manner, and it is far from evident that the surface should contribute into an effective uniaxial anisotropy in a simple additive manner. Actually one cannot expect $K_{S}$ to coincide with the atomistic single-site surface anisotropy, especially when strong deviations from non-collinearities leading to "hedgehog-like" structures appear. The effective anisotropy $K^{eff}$ appears in the literature in relation to the measurements of energy barriers of nanoparticles, extracted from the magnetic viscosity or dynamic susceptibility measurements. Generally speaking, the surface anisotropy should affect both the minima and the saddle points of the energy landscape in this case. It is clear that while the measurement of viscosity are related to the saddle point, the magnetic resonance measurements depend on the stiffness of the energy minima modified by the surface effects. Thus the meaning of the $K^{eff}$ is different for different measurement techniques.

Despite its rather "ad hoc" character, this formula has become the basis of many experimental studies with the aim to extract the surface anisotropy (see, e.g., Refs. [8,57,32]) because of its mere simplicity.

In Fig. 1.6 we present some experimental results (from Ref. [8]) where the effective anisotropy is plotted as a function of the inverse of the diameter of Co nanoparticle (which is supposed to have a spherical shape) and is fitted to the expression of $K^{eff}$ similar to Eq.(1.17).