5 Effective energy landscapes of spherical nanoparticles

In this section we compute the $3D$ energy potential as a function of the polar angle $\theta$ and azimuthal angle $\varphi$ of net magnetization of the multi-spin particle. First, we do this for a spherical particle with uniaxial anisotropy in the core and NSA, cut from an sc and fcc lattices, and for different values of the surface anisotropy constant, $k_s$.

We apply the Lagrange multiplier technique to analyze the case of a spherical multi-spin particle of ${\cal N} = 1736$ spins on an sc lattice with uniaxial anisotropy in the core ( $k_c = 0.0025$) and NSA.

Figure 2.3: Energy potentials of a spherical multi-spin particle of ${\cal N} = 1736$ spins on an sc lattice with uniaxial anisotropy in the core ( $k_c = 0.0025$) and NSA with constant (a) $k_s = 0.005$, (b) $k_s = 0.112$, (c) $k_s = 0.2$, (d) $k_s = 0.5$.

In Fig. 2.3 we show the obtained energy landscapes. One can see that as $k_s$ increases from $k_s=2 k_c$ to $k_s=200 k_c$, the global minima move away from those defined by the uniaxial core anisotropy, i.e., at $\theta=0,\pi$ and any $\phi$, and become maxima, while new minima and saddle points develop which are reminiscent of cubic anisotropy.

With the aim to extract the value of the effective uniaxial and cubic anisotropy constants, we cut the $3D$ energy landscape at $\varphi =0$ and obtain the $2D$ energy potential, later we fit it to formula (2.23). In Fig. 2.4 we present the $2D$ energy potential ($\varphi =0$) of a multi-spin particle with uniaxial anisotropy in the core ( $k_c = 0.01$) and NSA with $k_s = 0.1$ (left), $0.5$ (right). The solid lines are numerical fits to formula (2.23). From this graph we see that the energy of the multi-spin particle is well recovered by Eq. (2.23) when $k_s$ is small. Consequently, such multi-spin particle can be treated as an EOSP with an energy that contains uniaxial and cubic anisotropies. However, as it is started to be seen in the right panel, and as was shown in Ref. [87], when the surface anisotropy increases, this mapping of the multi-spin particle onto an effective one-spin particle is less satisfactory.

Figure 2.4: $2D$ energy potentials of a spherical multi-spin particle of ${\cal N} = 1736$ spins on an sc lattice with uniaxial anisotropy in the core ( $k_c = 0.01$) and NSA with constant $k_s = 0.1$ (left), $0.5$ (right). The solid lines are numerical fits to formula (2.23).

Figure 2.5: Effective anisotropy constants for a spherical multi-spin particle of ${\cal N} = 1736$ spins cut from an sc lattice against $k_s$. The panel on the right shows the core-surface mixing (CSM) contribution obtained numerically as $k_\mathrm {ua}^\mathrm {eff}$ upon subtracting the original core contribution ${\cal E}_c$ in Eq. (2.3). The thick solid lines are plots of the analytical expressions (2.15), (2.21).

Repeating this fitting procedure for other values of $k_s$ we obtain the plots of $k_\mathrm {ua}^\mathrm {eff}$ and $k_\mathrm{ca}^\mathrm{eff}$ as a function of $k_{s}$, see Fig. 2.5. Here we first see that these effective constants are quadratic in $k_s$, in accordance with Eqs. (2.15) and (2.21). In addition, the plot on the right shows an agreement between the constant $k_\mathrm {ua}^\mathrm {eff}$ obtained numerically and the analytical expression (2.21), upon subtracting the pure core contribution ${\cal E}_c$, see Eq. (2.6). The agreement is better in the regime of small $k_s$. These results confirm those of Refs. [87,97,73] that the core anisotropy is renormalized by the surface anisotropy, though only slightly in the present case.

Spherical particles cut from the sc lattice exhibit an effective four-fold anisotropy with $k_\mathrm{ca}^\mathrm{eff} < 0$, as we can check from the numerical results in Fig. 2.5 and analytical expression Eq. (2.23). As such, the contribution of the latter to the effective energy is positive.

Figure 2.6: Energy potentials of a spherical multi-spin particle with uniaxial anisotropy in the core ( $k_c = 0.0025$) and NSA with constant (a) $k_s = 0.005$, (b)$k_s = 0.1$, (c) $k_s = 0.175$ and (d) $k_s = 0.375$. The particle contains ${\cal N}=1264$ spins on an fcc lattice.

Next we will analyze the case of a spherical particle with fcc lattice structure. First, in the same way that we have done in the case of an sc particle, we calculate the $3D$ energy landscape, see Fig. 2.6. Comparing the energy potential in Fig. 2.3 for the sc and Fig. 2.6 for the fcc lattice one realizes that, because of the different underlying structure and thereby different spin surface arrangements, the corresponding energy potentials exhibit different topologies. For instance, it can be seen that the point $\theta=\pi/2, \varphi=\pi/4$ is a saddle in MSPs cut from an sc lattice and a maximum in those cut from the fcc lattice.

Figure 2.7: $2D$ energy potentials of a spherical multi-spin particle of ${\cal N}=1264$ spins on an fcc lattice with uniaxial anisotropy in the core ( $k_c = 0.0025$) and NSA with constant $k_s = 0.025$ (left), $0.375$ (right). The solid lines are numerical fits to formula (2.23).

In Figs. 2.7, we plot the $2D$ energy potential for a spherical particle with fcc structure, for two values of $k_{s}$: $k_{s}=10 k_{c}$ in the left graph and $k_{s}=120 k_{c}$ in the right graph. The solid line represents numerical fits to formula (2.23). We can observe an agreement between the numerical results and the fittings to formula (2.23). Now, we extract the values $k_\mathrm {ua}^\mathrm {eff}$ and $k_\mathrm{ca}^\mathrm{eff}$ as a function of $k_{s}$ for a fcc spherical particle. The effective cubic constant $k_\mathrm{ca}^\mathrm{eff}$ appears to be positive in contrast to sc case, see Fig. 2.8, and as for the sc lattice, it is quadratic in $k_s$. As mentioned earlier, the coefficient $\kappa$ in Eq. (2.15) depends on the lattice structure and for fcc it may become negative. To check this, one first has to find an analytical expression for the spin density on the fcc lattice, in the same way that the sc lattice density was obtained in Ref. [74] (see Eq. (6) therein). Likewise, the coefficient $\tilde{\kappa}$ in Eq. (2.21) should change on the fcc lattice, thus changing the uniaxial and cubic contributions as well.

Figure 2.8: Effective anisotropy constants against $k_s$ for a spherical particle of ${\cal N}=1264$ spins cut from an fcc lattice with uniaxial core anisotropy $k_c = 0.0025$. The lines are guides for the eyes.

In fact that not only the value of the effective constant $k_\mathrm{ca}^\mathrm{eff}$ but even its sign depend on the underlying structure, more exactly the surface arrangement. It is an important point for general modelling. Very often and for simplicity the nanoparticles are considered cut from sc lattice, disregarding the fact that realistic nanoparticles never have this structure.