2 Temperature dependence of the effective anisotropies in nanoparticles with fcc lattice structure

In the case of nanoparticles with cubic core anisotropy, the surface anisotropy induces an additional contribution which is also cubic in nature being positive for nanoparticles cut from fcc lattice and negative for nanoparticles cut from sc lattice. The contribution of this anisotropy is additive to the bulk one, although one should note [74], see also section 2.3.1, that it is proportional to $K_s^2$, rather than to $K_s$. An example of the average torque curves is plotted in Fig. 5.4. for various temperatures. The shape of the curves remains corresponding to the cubic anisotropy for all temperatures and surface anisotropy values.

Figure 5.4: Angular dependence of the $Y$-component of the total system torque in a truncated octahedral nanoparticle with fcc internal structure ($N = 1289$, $N_{c} = 482$ and $N_{s}= 807$), Néel surface anisotropy $K_s=10K_c$ and various values of the temperature.

In Fig. 5.5 we present the temperature dependence of the effective cubic anisotropy constant in a truncated octahedral nanoparticle cut from fcc crystalline structure for various values of the Néel surface anisotropy constant. The appearance of an additional positive cubic anisotropy contribution coming from the surface is seen starting from the value $K_s \approx 20 K_c$. The low-temperature values of the effective anisotropy coincide with the ones obtained through the Lagrange multiplier technique at $T=0K$. In Fig. 5.6 we present the temperature dependence of total cubic anisotropy in spherical and truncated octahedral nanoparticles both with fcc internal structures and for various values of the Néel surface anisotropy $K_s$. These values are normalized by the value of the effective anisotropy at $T=0K$ ( $K_{ca}^{eff}$) which is different for each case. The universal dependence on temperature corresponding to the overall cubic anisotropy is observed.

Figure 5.5: Temperature dependence of the effective cubic anisotropy, normalized to the core anisotropy value in truncated octahedra nanoparticles with cubic anisotropy in the core, fcc internal structure ($N = 1289$, $N_{c} = 482$ and $N_{s}= 807$) and various values of the Néel surface anisotropy.

Figure 5.6: Temperature dependence of the effective cubic anisotropy, normalized to its value at $T=0K$ in spherical ($N=1505$) and truncated octahedra ($N = 1289$) nanoparticles with cubic anisotropy in the core, fcc internal structure and various values of the Néel surface anisotropy.

Figure 5.7: The effective cubic anisotropy as a function of the Néel surface anisotropy parameter in truncated octahedra nanoparticle with cubic anisotropy in the core, fcc internal structure ($N = 1289$, $N_{c} = 482$ and $N_{s}= 807$) and for various temperatures.

Fig. 5.7 presents the dependence of the effective anisotropy on the value of the surface anisotropy $K_s$ for various temperatures for truncated octahedra cut from the fcc lattice with cubic anisotropy in the core. As it is mentioned above, in this case the surface anisotropy contribution is of the same cubic nature as the core one. The additional surface contribution is expected to be proportional to $K_s^2$, see section 2.3.1 in chapter 2. Consequently, all the data were fitted to this theoretically predicted dependence. The extracted values $K_c^{eff}(T)$ are consistent with the ones calculated independently. The corresponding formula $K^{eff}(T)=K_c^{eff}(T)+A K_{Surf}(T)$ ( $K_{Surf}(T) \sim K_s^2$) may be viewed as the one substituting the original formula (3.5). Unfortunately, the parameter $A$ dependence on the system size is not trivial as is discussed in section 3.5 in chapter 3, since it depends on the surface density of spins. The latter is not smooth as a function of the nanoparticle diameter, due to the fact that small nanoparticles do not have uniform spin density on their surfaces.