1 Temperature dependence of the effective anisotropies in nanoparticle with sc lattice structure

The first study that we have to do when we analyze the temperature dependence of the effective anisotropy of the nanoparticle is to evaluate this parameter at $T=0K$, similar to chapter 2. In Fig. 5.1 we present the plot of 2D energy landscapes for a spherical nanoparticle with uniaxial anisotropy cut from sc internal lattice as a function of the constrained angle $\theta$ for two values of the constrained angle $\varphi=0, \pi/4$ and the surface anisotropy constants $K_s= 10 K_c, 50 K_c, 100 K_c$. The onset of the additional cubic anisotropy is clearly seen in a different value of the energy in the maximum at $\varphi =\pi /4$ (the saddle point) and $\varphi =0$ (the absolute maximum). For large surface anisotropy value $K_s=100K_c$ the energy landscape has predominantly cubic form, characterized by a four-fold anisotropy. The effective uniaxial and cubic anisotropy constants can be extracted from these figures, as we have shown in chapter 2. The surface introduces an additional cubic anisotropy with $K_{ca}<0$ for spherical particles cut from sc internal structure and $K_{ca}>0$ for the ones from the fcc lattice. The same is true for the truncated octahedron, although depending on the orientation of the facets and strength of the surface anisotropy (see section 2.8) we may need an additional cubic constant $K_{2,\mathrm{ca}}^\mathrm{eff}$ for the fitting.

Figure 5.1: Internal energy for $T=0K$ for $\varphi =0$ and $\varphi =\pi /4$ as a function of the constrained angle $\theta$ in a spherical nanoparticle with uniaxial anisotropy in the core, sc internal structure ($N = 6272$, $N_{c} = 4968$ and $N_{s}= 1304$) and three values of the Néel surface anisotropy (a) $K_s=10K_c$, (b) $K_{s}=50 K_c$ and (c) $K_{s}=100K_{c}$ .

Figure 5.2: The Y-component of the torque for $\varphi =0$ as a function of the constrained angle $\theta$ for various temperatures in a spherical nanoparticle with uniaxial anisotropy in the core, sc internal structure ($N = 6272$, $N_{c} = 4968$ and $N_{s}= 1304$) and two values of the Néel surface anisotropy $K_s=10K_c$ (upper graph) and $K_s=100K_c$ (lower graph). The line is a fitting curve to Eq.(5.2).

In Fig. 5.2 we present the results for the $Y$-component of the average restoring torque at $T\neq 0K$ for the nanoparticles whose energy landscapes are presented in Figs. 5.1. The shapes of the torque curves are well described by the expression (5.2) for all temperatures with the macroscopic anisotropy constants decreasing with temperature. For relatively small strength of the surface anisotropy $K_s=10K_c$ (see Fig. 5.2(a)), practically only uniaxial anisotropy is present. For large strength of the anisotropy constant $K_s=100K_c$ we observed the competition of two anisotropies: uniaxial and additional cubic due to surface effects (see Fig. 5.2(b)). At high temperatures, however, the cubic anisotropy contribution disappears.

The torque curves presented in Figs. 5.2 allow to investigate the temperature dependence of the effective anisotropies in nanoparticles. Fig. 5.3 presents the corresponding temperature dependence of uniaxial and additional cubic anisotropy for the two values of the surface anisotropy constants. The macroscopic uniaxial anisotropy is independent on the surface anisotropy value, as expected. The macroscopic cubic anisotropy, coming from the surface anisotropy, is practically zero for small strength of the surface anisotropy $K_s=10K_c$. In the case of strong surface anisotropy $K_s=100K_c$ this additional cubic anisotropy is negative and its absolute value is decreasing with temperature. As it happens with the bulk cubic anisotropy, the surface-induced cubic anisotropy is decreasing faster with temperature as the uniaxial core contribution. Consequently, at high temperatures the cubic counterpart disappears leaving the uniaxial core anisotropy as the dominant factor. A transition in the magnetic behavior then can take place. It is similar to the observed in thin films with strong surface effects: at low temperatures the surface effects predominate and the magnetization of the film is perpendicular to the thin film plane while at higher temperatures the surface anisotropy vanishes and the magnetization stays in plane, see section 4.3.3 in previous chapter. In the case of nanoparticles a similar effect occurs: at low temperatures the surface effects dominate determining the overall cubic behavior, at high temperatures the surface contribution vanishes and the nanoparticle behaves as a uniaxial one.

Figure 5.3: Temperature dependence of the macroscopic anisotropies in a spherical nanoparticle with uniaxial anisotropy in the core, sc internal structure ($N = 6272$, $N_{c} = 4968$ and $N_{s}= 1304$) and two values of the Néel surface anisotropy (a) $K_s=10K_c$ and (b) $K_s=100K_c$. The values are normalized by the core anisotropy $K_c$.