3 Scaling exponent

Finally, we would like to discuss the scaling behavior of the effective anisotropy on the nanoparticle magnetisation $K\propto M^{\gamma}$. The Callen-Callen theory [106] states that in the bulk $\gamma=3$ for uniaxial anisotropy and $\gamma =10$ for the cubic one.

Figure 5.8: Temperature dependence of core and surface magnetization, normalized to $T=0K$ values in a spherical nanoparticle with sc lattice (N=1505).

Similar to what happens in magnetic thin films, in magnetic nanoparticles the surface magnetization has a faster temperature dependence than the core one, see Fig. 5.8, sharing the same Curie temperature. This arises due to a reduction in coordination number at the surface leading to a reduced exchange and a strong surface anisotropy pointing perpendicularly to the surface. At the same time the fully coordinated core effectively polarizes the surface layer, resulting in a shared value for Tc. The surface anisotropy value has very little effect on the temperature dependence of the overall anisotropy, see Fig. 5.6. The total effect is that the scaling exponents are always smaller then the corresponding bulk value and decreases with the surface anisotropy value.

Figure 5.9: Low-temperature scaling of the effective anisotropies with magnetization in a spherical nanoparticles cut from the sc lattice ($N=1505$) , uniaxial anisotropy and Néel surface anisotropy parameter $K_s=100K_c$.

For example, in Fig. 5.9 we present the scaling of the uniaxial and cubic anisotropy constants with the magnetization at low temperatures (up to $\approx 200 K$) in a spherical nanoparticle with uniaxial core anisotropy. Note that no scaling behavior is observed in the whole temperature range. The low temperature scaling exponents are lower than the corresponding bulk values and depend on the surface anisotropy value.

Figure 5.10: Scaling exponents as a function of Néel surface anisotropy constant in spherical ($N=1505$) and truncated octahedra ($N = 1289$) nanoparticles with cubic anisotropy in the core and fcc internal structure.

A similar effect is produced in spherical and truncated octahedral nanoparticles with cubic core anisotropy. In Fig. 5.10 we present the scaling exponent as a function of Néel surface anisotropy constant in nanoparticles with spherical and octahedral shapes, fcc internal structure, cubic anisotropy in the core and approximately the same nanoparticle's diameter $D\approx 3 nm$. The scaling exponents are lower than the corresponding bulk values and weakly depend on the surface anisotropy value, see Fig. 5.10. In fact, the scaling exponent decreases as a function of the surface anisotropy value due to a faster decrease of the magnetization on the surface.