Size- and temperature-dependent magnetization of iron nanoclusters

The magnetic behavior of bcc iron nanoclusters, with diameters between 2 and 8 nm, is investigated by means of spin dynamics simulations coupled to molecular dynamics, using a distance-dependent exchange interaction. Finite-size effects in the total magnetization as well as the influence of the free surface and the surface/core proportion of the nanoclusters are analyzed in detail for a wide temperature range, going beyond the cluster and bulk Curie temperatures. Comparison is made with experimental data and with theoretical models based on the mean-field Ising model adapted to small clusters, and taking into account the influence of low coordinated spins at free surfaces. Our results for the temperature dependence of the average magnetization per atom $M\left(T\right)$, including the thermalization of the transnational lattice degrees of freedom, are in very good agreement with available experimental measurements on small Fe nanoclusters. In contrast, significant discrepancies with experiment are observed if the translational degrees of freedom are artificially frozen. The finite-size effects on $M\left(T\right)$ are found to be particularly important near the cluster Curie temperature. Simulated magnetization above the Curie temperature scales with cluster size as predicted by models assuming short-range magnetic ordering. Analytical approximations to the magnetization as a function of temperature and size are proposed.

Figure 1

Finite-size scaling for bulk magnetization simulations, using periodic boundary conditions. The data correspond to simulations run at 300 K and error bars show standard deviation resulting from averaging magnetization over the final 0.5 ns.

Figure 2

Time dependence of the components $m_x, m_y, m_z$, of the magnetization along the $x, y$, and $z$ axes and the total magnetization $M$ during a typical simulation. The results correspond to simulations of a 2-nm-wide sphere at 400 K.

Figure 3

Snapshot of one half of a 6-nm diameter Fe NP, obtained with software OVITO [52], inscribed in a cubic region with periodic boundary conditions. The shell and core regions are indicated by the colors, where each small sphere represents an atom.

Figure 4

Snapshots of a typical simulation of a NP with a diameter of 6 nm. In (a), the atomic spins are displayed as arrows, with color indicating magnitude of $m_z$ at 600 K. (b) Same as (a) for thin slab at the center of the nanoparticle. The initial condition of the simulations was $m_z=1$, i.e., all spins pointing along the positive $z$ axis. See also the movies SM1 and SM2 in the Supplemental Material [51].

Figure 5

Snapshots of simulations showing the z-component $m_z$ of the atomic magnetization for nanoparticles having (a) 2 nm and 300 K, (b) 2 nm and 1200 K, (c) 6 nm and 300 K, and (d) 6 nm and 1200 K. The small spheres represent atoms. White indicates that the spins point along the positive $z$ axis, while black indicates that $m_z$ points along the negative $z$ axis. The $z$ axis is vertical with the positive direction pointing upward in the figures.

Figure 6

Total normalized magnetization as a function of temperature for different NPs compared to the bulk values.

Figure 7

Temperature dependence of the magnetization $M\left(T\right)$ as obtained using MD-SD, for (a) Fe bulk, nanoparticle diameters (b) 8 nm, and (c) 2 nm. The results for thermalized and frozen translational degrees of freedom are compared. Bulk results correspond to cubic simulation boxes with $20a_0$ sides.

Figure 8

(a) Experimental results for a Fe nanocluster of about 500–600 atoms as reported by Billas et al. [59] compared to the results of our simulations of a Fe NP composed of 533 atoms using two different exchange functions $J\left(r_{ij}\right)$. The exchange function $J\left(r_{ij}\right)$ fitted from the calculations of Pajda et al. [40] is the one used in this work and it reproduce quite closely the experimental results. The open circles correspond to simulation results using the exchange function proposed by Ma et al. [30]. (b) Magnetization of a Fe Bulk system (cell size $20\times 20\times 20a_0^3$) as obtained using the exchange function proposed by Ma et al. [30] (full circles). Bulk experimental data (triangles) and the results reported by Ma et al. (open circles) are also shown for the sake of comparison.

Figure 9

Size dependence of the total magnetization at different representative temperatures. The normalized magnetization is plotted as a function of the particle inverse diameter ($1/d$). Symbols and full curves correspond to the simulation results, whereas the dashed lines represent the fitted curves. The error bars indicate the standard deviation of the data that are taken into account in the average over the last 0.3 ns. For the fitted curves, the function is $M=a(1-b^T)(1/d)+{\left(\frac{T-T_C}{T}\right)}^{1/3}$ where $a=0.1, b=1.003$, and $T_C=650$ K. The black dash-dotted line corresponds to $M=0.35{(1/d)}^{0.514}$ and is related to the magnetization scaling behavior near the critical temperature in the 3D Heisenberg model as detailed in the text.

Figure 10

Fe NP magnetization as a function of the number of atoms $N$ in the nanoclusters (symbols) at two representative temperatures above the predicted cluster $T_C$, compared with different power law relations, including the one proposed in Ref. [65], for clusters with 15 atoms.

Figure 11

Total normalized magnetization $M$, core magnetization $M_c$ and shell magnetization $M_s$ a functions of temperature for Fe NPs having different diameters. The Fe bulk magnetization is also shown.

Figure 12

Temperature dependence of the total magnetization $M$ of Fe NPs. The results of our MD-SD simulations (symbols) are compared with the core-shell two component model described by Eq. (22) (curves).

Figure 13

Coordination analysis for spherical Fe NPs with diameters (a) $d=2\text{nm}$ and (b) $d=6\text{nm}$ at different temperatures. The data are taken from the last configuration of the system. They indicate lack of melting, even for the smallest considered nanocluster.

Figure 14

Correlation function in the core and shell regions of a 2nm Fe NP at $T=1200K$.

Figure 15

Mean-square displacement of the shell atoms at 600 and 1200 K, as obtained from our simulations for a 4 nm NP. The straight dashed line represents a linear fit of the 1200 K results, giving a diffusion coefficient $D=4.0\times {10}^{-10}{\mathrm{m}}^2$/s. Note that this simulation is 10 times longer (5 ns) than those performed for the magnetization calculations.

Figure 16

Temperature dependence of the core and shell magnetizations as obtained from the semi-infinite Ising model with a free surface for different values of the exchange constant J (Sec. 3b) compared with MD-SD simulations results. The curves are obtained for a Fe NP with a diameter of 6 nm.

Figure 17

Total, core and shell magnetization curves $M, M_c,M_s$, as obtained in the model presented in Sec. 3b (dashed lines) [54] and in the present MD-SD simulations (solid line and symbols) for Fe NPs having different diameters.