Magnetic properties of small Pt-capped Fe, Co, and Ni clusters: A density functional theory study

We have undertaken theoretical studies of spin and orbital magnetic moments as well as magnetic anisotropy energies for $M_{13}$ $\left(M=\text{Fe},\text{Co},\text{Ni}\right)$ and $M_{13}{\text{Pt}}_n$ $\left(n=3,4,5,20\right)$ clusters including the spin-orbit coupling in the framework of density functional theory. For all $M_{13}$ clusters considered we find tendencies for small structural distortions which can be characterized by either Jahn-Teller (JT) or Mackay transformations (MT). The magnetic anisotropy energy (MAE) along with the spin and orbital moments are calculated for $M_{13}$ icosahedral clusters and the angle-dependent energy differences are modeled using a Néel model with local anisotropies. From our studies, the MAE for JT-distorted $M_{13}$ clusters are found to be larger relative to the MT clusters and more than two orders of magnitude larger compared to the corresponding bcc or fcc bulk values. In addition, we demonstrate for one example that Pt capping may further enhance the MAE compared to the uncapped JT- and the Mackay-distorted ${\text{Fe}}_{13}$ cluster.

Figure 1

(Color online) The ideal icosahedral cluster showing the $x\text{−}z$ plane in which the angle $\theta$ is varied in the MAE calculations. The same setting was used for partially Mackay-transformed clusters (see below). E, F, and A abbreviate directions from the center to the middle of an edge, the middle of a facet, and an outer atom, respectively. As $\theta$ is varied from 0 to $\pi /2$, the magnetization direction scans E-F-A-E in succession.

Figure 2

(Color online) The JT-distorted (left) and the MT (right) ${\text{Fe}}_{13}$ cluster. Arrows indicate the direction of relative shift of atoms with respect to the ideal positions. For the JT-distorted and MT ${\text{Fe}}_{13}$ cluster, the displacements marked by the arrows have been scaled up by factors of 20 and 30, respectively. The box is only guide to the eyes. (The actual simulation box size is ${15}^3{\text{Å}}^3$.)

Figure 3

(Color online) Fixed spin moment results for ${\text{Fe}}_{13}$ showing the variation in energy as a function of $s$ from ICO $\left(s=1\right)$ to CUBO $\left(s=2\right)$. The energy difference is given with respect to the lowest energy JT-distorted ${\text{Fe}}_{13}$ cluster.

Figure 4

The minimum energy $r$ and $s$ values obtained for ${\text{M}}_{13}$ clusters. The dotted lines mark the minimum points for both cases. $s=1$ and $r=1$ represent the perfect ICO for each case.

Figure 5

(Color online) Left: the $\theta$-dependent energy (in meV per cluster) for an ideal ${\text{Fe}}_{13}$ icosahedral cluster (state with a spin moment of $46{\mu }_B$ and a center-shell distance of $2.57\text{Å}$). Filled circles: GGA results (VASP calculations), connected by a solid line as guide to the eye. Dashed line: fit to the anisotropy term according to Eq. (1), for $n=6$. For the VASP calculations, the energy difference is defined as $\Delta E=E\left(\theta \right)-E\left(0\right)$. The letters A, E, and F in the plot refer to the positions defined in Fig. 1. Right: the $\left(\theta ,\varphi \right)$ scan for the MAE of an ideal ${\text{Fe}}_{13}$ cluster: The energies plotted are the sixth-order contributions to the total energy using the value of $D_6$ from Table .

Figure 6

The plots from top to bottom show the energy differences $\Delta E=E\left(\theta \right)-E\left(0\right)$ in meV/cluster vs $\theta$ of relaxed ${\text{M}}_{13}$ (for two types of relaxations—JT and Mackay distortion) clusters. The magnetization direction $\left(\theta \right)$ varies in the $x\text{−}z$ plane. For the JT case (left panel), this plane passes through atoms 3 and 4 and almost touches atom 12 and the midpoint of the bond 8–10 (see Fig. 2, left panel); for the MT case (right panel), compare Fig. 1. For the JT-distorted $M_{13}$ clusters, the energy difference for the ${\text{Co}}_{13}$ and ${\text{Fe}}_{13}$ clusters is multiplied by factors of 5 and 2, respectively, whereas for the MT clusters, the energy difference for ${\text{Co}}_{13}$ and ${\text{Ni}}_{13}$ is multiplied by factors of 40 and 2, respectively. The slight deviations between the GGA (circles) and fitted data (full lines) appear for ${\text{Co}}_{13}$ (JT) and ${\text{Ni}}_{13}$ (JT) due to the presence of small $D_4$ contributions, were neglected.

Figure 7

(Color online) The lowest energy isomers of ${\text{Co}}_{13}{\text{Pt}}_3$ (left) and of ${\text{Co}}_{13}{\text{Pt}}_5$ (right), found so far. Dark (blue) spheres: Co atoms; light (yellow) spheres: Pt atoms. All Pt atoms of ${\text{Co}}_{13}{\text{Pt}}_5$ lie in the same plane.

Figure 8

(Color online) The initial (left) and final (right) structures of the ${\text{Ni}}_{13}{\text{Pt}}_{20}$ cluster. Ni and Pt atoms are represented by dark (blue) and light (yellow) spheres. The final structure (the lowest energy found so far) demonstrates the importance of atomic relaxations.

Figure 9

(Color online) A comparison of onsite orbital moments $\left|{\mathbf{L}}_i\right|$ (left panel) and spin moments $\left|{\mathbf{S}}_i\right|$ (right panel) for the ${\text{Ni}}_{13}{\text{Pt}}_3$ cluster obtained by VASP and FPLO. In the left panel, the dashed and filled diamonds show the orbital moments of ${\text{Ni}}_{13}$ cluster obtained with FPLO and VASP while the dashed and filled circles are the orbital moments of three capped Pt atoms, respectively. The right panel shows the corresponding results of the individual spin moments. Atomic site 1 is the center atom.

Figure 10

(Color online) The orbital moment $\left|{\mathbf{L}}_i\right|$ of each atomic site of ${\text{Fe}}_{13}{\text{Pt}}_n$ (left), ${\text{Co}}_{13}{\text{Pt}}_n$ (middle), and ${\text{Ni}}_{13}{\text{Pt}}_n$ (right) clusters. The dashed lines show the calculated bulk values of the orbital moments of bcc Fe, fcc Co, and fcc Ni, respectively.

Figure 11

(Color online) The spin moment $\left|{\mathbf{S}}_i\right|$ of each atomic site of ${\text{Fe}}_{13}{\text{Pt}}_n$ (left), ${\text{Co}}_{13}{\text{Pt}}_n$ (middle), and ${\text{Ni}}_{13}{\text{Pt}}_n$ (right) clusters. The dashed lines show the calculated bulk values of the spin moments of bcc Fe, fcc Co, and fcc Ni, respectively. The same symbols as in Fig. 10 are used.

Figure 12

(Color online) Left: the $x\text{−}z$ plane (where $\theta$ is varied) of the ${\text{Fe}}_{13}{\text{Pt}}_4$ cluster, showing the positions of the Pt atoms relative to the ICO in the initial unrelaxed configuration. Light (orange) and dark (blue) spheres are the Fe and Pt atoms, respectively. In the relaxed ${\text{Fe}}_{13}{\text{Pt}}_4$ cluster, two opposite edges (those adjacent to the Pt atoms) of the ${\text{Fe}}_{13}$ cluster moved toward each other. Right: the $\theta$-dependent energy differences for the relaxed ${\text{Fe}}_{13}{\text{Pt}}_4$ cluster. Here, the solid curve is not a fit to the Néel formula because of the heterogeneous form of the cluster but simply a cubic spline fit to the GGA data.