6 Modeling of energy barriers in cobalt nanoparticle

Experimentally, it has been reported that the surface effects in nanoparticles can lead to a change in the "effective anisotropy" [9,20,14,36,32], and therefore the surface effects can alter the magnetic behavior of the nanoparticles. Luis. et al. [10] have shown that the fcc Co nanoparticles embedded in different non-magnetic matrices experienced an increment of the effective anisotropy, obtaining $ \mathcal{K^{*}}_{S}= 25K_{V}$ (for $ Al_{2}O_{3}$ capping), $ \mathcal{K^{*}}_{S}= 40K_{V}$ (for Cu capping) and $ \mathcal{K^{*}}_{S}= 70K_{V}$ (for Au capping), where $ K_{V}=K_{c}/4$ and $ K_{c}$ correspond to fcc Co bulk anisotropy. Note that in this case and in agreement with the expression (3.6) used in the experimental study, both surface and volume anisotropies are measured in the same units $ erg/cm^3$. To comply with the experimental approach in this subsection we also use this definition. In Fig 3.6 we show the reported experimental value of the dependence of the effective anisotropy with the diameter of the cobalt nanoparticles with different cappings.
Figure 3.9: Size dependence of the effective anisotropy constant, for cobalt fcc nanoparticles with different cappings (from Ref.[10]).
Generally speaking, our results above show that there is no reason to expect this value of the surface anisotropy $ \mathcal{K^{*}}_{S}$ to coincide with the local on-site surface anisotropy value.

In this section we directly model the effective energy landscapes of multi-spin Co nanoparticles, with typical experimental parameters, varying the strength of the local surface anisotropy value. The obtained effective anisotropy, defined as $ K^{eff} = \Delta E/V$ , where $ \Delta E$ is the energy barrier and V is the particle volume, is then compared with those experimentally determined in [10]. Consequently, we get rid of formula (3.6) and obtain the local on-site surface anisotropy from a "direct" comparison of the experimental and numerical effective anisotropy constants.

Figure 3.10: Effective anisotropy values $ K_{eff}=\Delta E/V$, obtained through energy barrier of Co nanoparticle: (a) Particle with $ D=3.1nm$ with (dashed line) and without (solid line) elongation $ e=1.228$, (b) octahedral symmetric particle with $ D=4.5 nm$ without elongation. The straight lines indicate experimental results of Ref.[10]
\includegraphics[totalheight=0.35\textheight]{Keff_R1.eps} \includegraphics[totalheight=0.35\textheight]{Keff_R2.eps}

In Figs. 3.10 we present the effective anisotropy constant $ K^{eff} = \Delta E/V$ evaluated for truncated octahedral nanoparticles with two diameters $ D=3.1$ nm. and $ D=4.5$ nm., and for an elongated nanoparticle with e=1.228 and the smaller dimension $ D=3.1$ nm (experimental parameters). One can see that for $ k_s>100\vert k_{c}\vert$ the energy barriers of multispin particles increase with surface anisotropy increment, confirming multiple experimental results. The horizontal lines in these figures indicate the experimental results for the energy barriers obtained in Refs. [10,8]. From this comparison we have estimated the corresponding local surface anisotropy values. The uncapped Co nanoparticle would have the same value of the effective anisotropy as bulk Co, provided that $ k_s \sim -90kc$ $ (2.5 \times
10^{8} erg cm^{-3})$; this value is only slightly higher than that estimated from the calculations ( $ 2.2 \times 10^{8} erg cm^{-3}$) of Daalderop et al [104]. The results for capped nanoparticles are presented in Tab. 3.2. The estimated surface anisotropy values are 20-40 times higher than those obtained via formula (3.6) and are almost of the order of the exchange parameter $ k_s \thicksim 0.4- 0.7 J$. These values look higher than those normally expected. On the other hand they are in agreement with estimations for the surface anisotropy based on first principles in thin films [105].

Table 3.2: Surface anisotropy constant for Co nanoparticle with different capping, extracted by comparison between simulation and experimental results in Ref.[10].
Capping $ Al_2O_3$ Cu Au
  $ k_\mathrm{s} / \vert k_\mathrm{c}\vert$ $ K_\mathrm{s}/J$ $ k_\mathrm{s} / \vert k_\mathrm{c}\vert$ $ K_\mathrm{s}/J$ $ k_\mathrm{s} / \vert k_\mathrm{c}\vert$ $ K_\mathrm{s}/J$
Octahedral D=3.1nm 228.92 $ 0.447$ 273.01 $ 0.532$ 396.29 $ 0.773$
Octahedral D=4.5nm 237.63 $ 0.467$ 257.60 $ 0.503$ 360.45 $ 0.703$
Elongated particle 187.258 $ 0.365$ 219.37 $ 0.428$ 314.076 $ 0.613$

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