8 Effective energy landscape in cobalt nanoparticle

In the present section we aim at the modeling of nanoparticles experimentally prepared in Ref. [8,100,101,10,9]. In those articles it has been reported that the magnetic anisotropy of the Co cluster prepared by sequential sputtering of Co and $Al_{2}O_{3}$ (or Cu; or Au; or Ag) suffers an increment when the diameter of the cluster is reduced which is associated to the surface anisotropy. The same group suggested that such an increment of the effective anisotropy produces an increase of the effective energy barrier, stabilizing the magnetization against thermal fluctuations.

With the aim to study the effective energy landscape of such nanoparticles, we consider a Co nanoparticle with fcc internal structure, cubic magnetocrystalline anisotropy, varying the strength of the surface anisotropy. To compare with the experimental results in this sub-section the anisotropy parameter was taken from the experimental value and is measured as the anisotropy density per volume, $K_{c}=-2.8\times 10^{6} erg cm^{-3}$. The Co nanoparticles are "numerically prepared" in two forms: truncated octahedral, see Fig. 2.13(a) and elongated truncated octahedral, see Fig. 2.13(b). For truncated octahedra, we use a symmetrical construction which allowed the same surface atom arrangements in all rectangular facets, as it was described above. In the case of perfect truncated octahedral nanoparticle, the nanoparticle's diameter is $D=4.5 nm$ ( $\mathcal{N}=2951$), $\mathcal{N}$ is the total number of spin. For the elongated truncated octahedron the planes, perpendicular to z-axis, used to cut the polygon from the fcc lattice, were additionally separated to obtain the desired elongation, in our case that elongation has been chosen as $e=Dz/Dx=1.228$, where $Dz$ and $Dx$ are dimensions in $z$ and $x$ directions, respectively (corresponding to experimentally prepared nanoparticles). The nanoparticle size in $x$ direction was considered $Dx=3.1 nm$ (the total spin number is equal to $\mathcal{N}=1439$).

Figure 2.13: Modeled Co nanoparticles with fcc lattice and (a) octahedral shape. (b) elongated octahedral shape.

Some examples of the $3D$ effective energy landscapes for several values of the Néel surface anisotropy constants are presented in Fig. 2.14.

Figure 2.14: Effective $3D$ energy landscapes of octahedral Co nanoparticle with D=4.5 nm (a) $k_{s}/k_{c}=-50$, (b) $k_{s}/k_{c}=-60$, (c) $k_{s}/k_{c}=-65$ and (d) $k_{s}/k_{c}=-100$.

We have observed that for relatively small values of the surface anisotropy $k_s \lesssim 50 \vert k_c\vert$, see Figs. 2.14(a) and 2.15(a), the overall anisotropy is in agreement with the previous results, i. e. can be described by formula (2.23). For large value of the surface anisotropy $k_s \gtrsim 60 \vert k_c\vert$, see Figs. 2.14(b-d) and 2.15(b-d), we note the change of the character of the energy landscape. Moreover, we observe a change of the character of the special point at $\theta_0=\pi/2, \varphi=\pi/4$ from the saddle one to the maximum. Interestingly, the special point corresponding to the minimum energy path in Figs. 2.14(d) and 2.15(d), is not a usual one: locally near this point in one of the normal mode directions the energy increases and in another one it decreases (similar to $x^3$ expansion). This point is known as a "monkey saddle" and its existence invalidates the standard Kramers-type approach (in particular, leading to the Arrhenius-Neel law), where the full-harmonic expansion of the energy near the saddle point is necessary [53].

Figure 2.15: $2D$ energy landscapes of an octahedral many-spin Co particle with $D=4.5 nm$ for (a) $k_{s}/k_{c}=-50$, (b) $k_{s}/k_{c}=-60$, (c) $k_{s}/k_{c}=-65$ and (d) $k_{s}/k_{c}=-100$.

Figure 2.16: Magnetic moment configurations(on $Y=0$ plane) in Co fcc truncated octahedral nanoparticle with $D=4.5$ nm. and $k_{s}=350\mid k_{c}\mid$: (a) The minimum energy configuration. (b) The saddle point configuration.

In Figs. 2.15 we present 2D energy landscapes, corresponding to $\varphi_0=0$ (squares) and $\varphi_0=\pi/4$ (circles). Namely, the energy landscape cannot be fitted to the cubic anisotropy with the first cubic anisotropy constant only. We have found that these landscapes could be fitted to the effective macro-spin energy, slightly different from that of Eq. (2.23):

$$\mathcal{E}_{\mathrm{EOSP}} = - k_\mathrm{ua}^\mathrm{eff} m_z^2 ... ...sum_{\alpha=x,y,z}m_{\alpha}^{4}+k_\mathrm{2,ca}^\mathrm{eff}m_x^2 m_y^2 m_z^2,$$ (42)

where $k_\mathrm {ua}^\mathrm {eff}$ is the effective uniaxial anisotropy constant; $k_\mathrm{ca}^\mathrm{eff}$ and $k_\mathrm{2,ca}^\mathrm{eff}$ are the effective first and second cubic anisotropy constants.

For symmetric particles, $k_\mathrm{ua}^\mathrm{eff}\approx0$ (see previous sections) and we have observed that $k_\mathrm{2,ca}^\mathrm{eff}$ is relevant when $k_{s}$ has a large value, when $k_{s}>50\vert k_{c}\vert$, as we can see in Fig. 2.17 and Fig. 2.18.

In Fig. 2.16 we present the minimum and the saddle point spin configurations (on $y=0$ plane) for $k_{s}=350\mid k_{c}\mid$. We would like to mention that when the surface anisotropy is increased ( $k_{s}\sim350\mid k_{c}\mid$) in comparison with the exchange parameter J, then the spin arrangement shows a high non-collinearity. Those spin non-collinearities are similar to the ones reported previously for strong surface anisotropy cases [74,90,102,103], and it is possible that the EOSP approximation is no more valid [73].

Figure 2.17: Effective anisotropy constants for fcc Co nanoparticles with octahedral shape and $D=4.5 nm$ as a function of the Néel surface anisotropy $K_{s}$, normalized to the core anisotropy $K_{c}$.

Figure 2.18: Effective anisotropy constants as a function of the surface anisotropy value $k_{s}/k_{c}$ for fcc Co octahedral elongated nanoparticle with $e=1.228$ and $D=3.1nm$ (smaller dimension). (a) Cubic constants (left) and (b) uniaxial constant .

In Figs. 2.17 and 2.18 we present the values of the uniaxial and cubic macroscopic anisotropy constants extracted by fitting the 2D landscapes to expression (2.25). To make the comparison with experimental values easier we supply the values of the macroscopic anisotropy as energy density. The effective uniaxial anisotropy constant is zero in practice in the case of a perfect truncated octahedral particle. Nevertheless, for elongated particle, there is a non-zero uniaxial contribution coming from the surface effect, see Fig. 2.18(b). However its value is much smaller than the additional cubic anisotropy.

In both cases of octahedral and elongated particles the effective macroscopic first cubic anisotropy constant changes the sign, see Figs. 2.17 and 2.18(a), and therefore the easy axis of the system changes its position, as have been shown clearly in Figs. 2.15 and 2.14.