3 Dependence of the energy barrier on the value of the surface anisotropy constant $k_s$

As we have shown in the previous chapter, the numerical evaluation of the energy barrier should be done in a multidimensional space and it is a difficult task. With this aim we use the Lagrange-multiplier method described in section 2.4. We evaluate the energy barriers of multi-spin particles (MSPs) by numerically computing the difference between the energy at the saddle point and at the minimum, and compare them with analytical results obtained from the EOSP approximation.

Note that in the wide range of the parameters several energy barriers (corresponding to different paths of magnetization rotation) coexist in the system in accordance with the complex character of the effective potential with two competitive anisotropies, see Figs. 2.3(c) and 2.9(c). In what follows, in most cases we will only discuss the energy barrier which corresponds to the change in $\theta$ direction. In the case corresponding to elongated nanoparticles cut from fcc lattice and uniaxial anisotropy in the core, see Fig.2.9(d), the only energy barrier corresponds to the rotation around $\varphi$ direction.

Figure 3.2: Energy barrier as a function of $k_s$ for a spherical particle cut from an sc lattice. The particle contains ${\cal N}=20479$ spins and has the uniaxial core anisotropy $k_c = 0.0025$. The solid line is a plot of the analytical expression (3.8).

Figure 3.3: Energy barriers against $k_s$ for truncated octahedral multi-spin particle cut from an fcc lattice ( ${\cal N}=1688$) and spherical particles cut from the fcc ( ${\cal N}=1289$) and hcp ( ${\cal N} = 1261$) lattices. Uniaxial core anisotropy with $k_c = 0.0025$ is assumed.

Fig. 3.2 shows the energy barrier of a spherical particle cut from an sc lattice as a function of $k_s$. The nonlinear behavior of the energy barrier with $k_s$ follows quantitatively that of the EOSP potential (3.7). Indeed, the solid line in this plot is the analytical result (3.8), using analytical expressions of Eqs. (2.14-2.15) together with the pure core anisotropy contribution. The discrepancy at the relatively large $k_s$ is due to the fact that the analytical expressions are valid only if the condition (2.22) is fulfilled; the core-surface mixing (CSM) contribution has not been taken into account.

In Fig. 3.3 we represent the energy barriers as a function of $k_s$ for MSPs with different shapes and internal structures. First of all, one can see a different dependence on $k_s$ as compared to MSPs with the sc lattice. In the present case, i.e., $k_\mathrm{ua}^\mathrm{eff} > 0, k_\mathrm{ca}^\mathrm{eff} > 0$, the energy barriers are given by Eqs. (3.9)(a) and (c). For small values of $k_s$, for which $\vert\zeta\vert < 1$, the energy barrier, in the first approximation neglecting the CSM term, is independent of $k_s$. Accordingly, the nearly constant value of the energy barrier, coinciding with that of the core, is observed for multi-spin particles in a large range of $k_s$. For larger $k_s$, the energy barrier increases, since $k_\mathrm {ua}^\mathrm {eff} > 0$ for multi-spin particles cut from an fcc lattice. At very large values of $k_s$, i.e., $k_s\gtrsim 100 k_c$ the energy barriers depend approximately linearly on $k_s$ and may have values larger than that inferred from the pure core anisotropy.

The energy barriers for ellipsoidal multi-spin particles are shown in Fig. 3.4.

Figure 3.4: Energy barriers versus $k_s$ of ellipsoidal multi-spin particles with different aspect ratio ( $a/c=0.6667, {\cal N}=21121$, and $a/c=0.81, {\cal N}=21171$, with uniaxial core anisotropy $k_c = 0.0025$. The solid lines are linear fits.

Note that in this case the effective uniaxial anisotropy constant $k_\mathrm {ua}^\mathrm {eff}$ is a linear function of $k_s$, according to the analytical results (2.16), (2.17) and the numerical results presented in Fig. 2.10. In this case the value of the energy barrier is not symmetric with respect to the change of the sign of $k_s$. This is due to the fact that for $k_s < 0$ the effective uniaxial constant is a sum of the core anisotropy and the first-order contribution owing to elongation. On the contrary, when $k_s > 0$, the "effective core anisotropy" $k_\mathrm {ua}^\mathrm {eff}$ is smaller than the pure core anisotropy ${\cal E}_c$ in Eq. (2.6). This means that at some $k_s$ the effective uniaxial anisotropy constant ( $k_\mathrm {ua}^\mathrm {eff}$) may change sign. At the same time the effective cubic anisotropy $k_\mathrm {ua}^\mathrm {eff}$ remains positive and is proportional to $k_s^2$. Accordingly, at the vicinity of the point at which $k_\mathrm{ca}^\mathrm{eff}\thickapprox 0$, rapid changes of the character of the energy landscape occur as we have shown in Fig. 2.10 in the previous chapter.

The analysis, based on the effective one spin problem potential shows that when $k_\mathrm {ua}^\mathrm {eff} > 0$ the energy barriers of ellipsoidal multi-spin particles are defined by Eqs. 3.9(a) and (c), and for negative $k_\mathrm{ua}^\mathrm{eff} <0$ these are given by Eq. 3.9(b) and (c).

We would like to note that a regime of linear behavior in $k_s$ exists for both $k_s < 0$ and $k_s > 0$ (see Fig. 3.4). In some region of the effective anisotropy constants, e.g., $k_\mathrm {ua}^\mathrm {eff} > 0$, $\vert\zeta\vert < 1$, the energy barrier $\Delta E_\mathrm{EOSP} = k_\mathrm{ua}^\mathrm{eff}$, i.e., it is independent of the cubic contribution (neglecting again the CSM term). Consequently, it is linear in $k_s$, according to Eq. (2.17). The interval of these parameters is especially large in MSPs with $k_s < 0$ for which $k_\mathrm {ua}^\mathrm {eff}$ does not change sign.