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

Theoretical studies on M$_{13}$ (M = Fe, Co, Ni) and M$_{13}$Pt$_n$ (for $n$ = 3, 4, 5, 20) clusters including the spin-orbit coupling are done using density functional theory. The magnetic anisotropy energy (MAE) along with the spin and orbital moments are calculated for M$_{13}$ icosahedral clusters. The angle-dependent energy differences are modelled using an extended classical Heisenberg model with local anisotropies. From our studies, the MAE for Jahn-Teller distorted Fe$_{13}$, Mackay distorted Fe$_{13}$ and nearly undistorted Co$_{13}$ clusters are found to be 322, 60 and 5 $\mu$eV/atom, respectively, and are large relative to the corresponding bulk values, (which are 1.4 and 1.3 $\mu$eV/atom for bcc Fe and fcc Co, respectively.) However, for Ni$_{13}$ (which practically does not show relaxation tendencies), the calculated value of MAE is found to be 0.64 $\mu$eV/atom, which is approximately four times smaller compared to the bulk fcc Ni (2.7 $\mu$eV/atom). In addition, MAE of the capped cluster (Fe$_{13}$Pt$_4$) is enhanced compared to the uncapped Jahn-Teller distorted Fe$_{13}$ cluster.


I. INTRODUCTION
Density-functional theory (DFT) is an adequate tool to obtain useful information on the physical properties of small clusters. However, with respect to magnetic properties, the variations of magnetic moments with cluster size and morphology are often too obscure for establishing a clear trend (for example, see the behavior of the exchange coupling of free Fe clusters in [1,2]). Still, certain general statements can be made. It may be safely concluded that the local magnetic moments in outer shells of clusters are enhanced, as compared to the interior of clusters or to the corresponding bulk crystal [3,4,5,6]. This is a typical effect of reduced atomic coordination at the surface [7], well known from numerous experiments or calculations of magnetic slabs. Moreover, an analysis of cluster morphologies, by both experiment and theory [8,9,10,11,12,13,14,15], reveals the importance of icosahedral-like structures, prohibited for bulk or surface phases. On the other hand, the diversity of cluster structures in combination with the surface enhancement of magnetic moments make clusters interesting model objects for tuning magnetic properties at the nanometer scale.
When addressing magnetic properties of clusters from the computational point of view at a realistic level, it is important to take into account two specific issues: (i) a possible noncollinear (NC) setting of magnetic moments (i.e., a smooth variation of magnetic density vector from point to point in space), and (ii) spin-orbit interaction (SOI), along with the existence of orbital moments. Both issues have a long record of incorporation into firstprinciples DFT calculations, and are internally related: They mix the spin-up and spin-down states and must be, in principle, treated alongside on equal footing (see, e.g., Refs. [16,17] for a review). Discussing specifically DFT calculations for clusters, one notes certain technical difficulties in combining NC spin density with the SOI, which acted so far as a limiting factor on the number of calculations performed, and the size of clusters treated: Lack of symmetry, big effect of structure relaxation, slow convergence, large size of simulation cell around a free cluster. Moreover there is a conceptual problem of choosing a "correct" non-collinear solution among many apparently close metastable configurations. Different groups report very different results for the same systems (e.g., Refs. [18,19,20]), so that a preference of one or another result is not obvious. Therefore, not so much the size of cluster is a problem in itself, as the organization of a calculation in a way allowing to extract meaningful results, that is, clearly defining structural and magnetic models and carefully 3 analyzing their consequences.
A better understanding of the origins and details of large orbital magnetic moment as well as large MAE in clusters [21,22] is demanding in order to manipulate materials for improving the technology like in magnetic data storage devices. Binary 3d-5d clusters can be a challenging material in this respect. For example, there are observations that 4d and 5d elements like Rh, Pt and Au, which are nonmagnetic in bulk phase, attain significant moments when alloyed with 3d transition metals like Fe, Co or Ni [23,24,25,26].
In the present work, we analyze the effect of capping icosahedral clusters M 13 of M = Fe, Co, and Ni with Pt atoms. Theoretically, there have been several studies on magnetic anisotropy for supported clusters [27,28,29,30,31,32,33], whereas studies on small free clusters are still limited [34,35,36,37,38]. The motivation for the present study is that icosahedral clusters for M 13 are known to be very stable, [6,12,39] and the alloying of 3d transition metals with Pt results in large magnetic anisotropy as well as orbital moment.
Hence, it is expected that the orbital and the spin moments of M 13 clusters be strongly affected by capping with Pt.
Most of our DFT calculations have been done with the planewave code Vasp (Vienna Ab-Initio Simulation Package) [40,41,42]. For test purposes and for validating the results of calculations on non-capped icosahedral clusters, we performed calculations also with a local-orbitals code, Siesta (Spanish Initiative for Electronic Simulations with Thousands of Atoms) [43]. As the two methods are very different in what regards the technical implementation of the DFT calculation scheme, and both have been earlier used in calculations of magnetic clusters, their direct comparison might be of its own interest. In addition, few test calculations for the binary clusters were done using the all-electron local-orbital code Fplo (Full-Potential Local-Orbital scheme) [44]. In this respect, it is interesting to note that all-electron calculations confirm the results obtained with Vasp and Siesta.
The paper is organized as follows. Section II outlines the calculational methods and setup. Section III deals with the results for monometallic icosahedral clusters, notably comparison between Vasp and Siesta results. Section IV discusses the results for capped clusters, obtained with the Vasp code. We have made a comparison between Vasp and Fplo for one of the capped clusters. Conclusions are drawn in Section V.

II. COMPUTATIONAL METHODS
Most of the calculations have been performed with the Vasp code [40,41,42] based on DFT and within the generalized gradient approximation (GGA). The parameters by Perdew are used for the exchange and correlation functional [45]. Vasp uses the projector augmented wave (PAW) method [42,46] and a planewave basis set. Periodic boundary conditions were imposed onto large enough cubic cells (with the edge of 15Å for M 13 clusters and 20Å for M 13 Pt n clusters) in order to sufficiently reduce the interaction between replicated cluster images. Only the Γ point was used for the Brillouin-zone sampling for the cluster calculations. A k-mesh of (11×11×11) is used for the bulk calculations to compute the equilibrium lattice constants in case of bcc Fe, fcc Ni and Co. The values for local magnetic moments correspond to the integration of continuous magnetization density over atom-centred spheres with radii of 1.302Å (Fe, Co), 1.286Å (Ni) and 1.455Å (Pt). It is a well known fact that the magnetic anisotropy energy for cubic bulk transition metals, defined as the maximum energy difference, per atom, between different settings of the spin moment relative to the atoms framework, is of the order of 10 −6 eV. Moreover, a special care is demanded for the study of the MAE in clusters because any slight error can accumulate and produce misleading results while dealing with energy differences of such a small scale.
In order to calculate the magnetic anisotropy of M 13 icosahedral clusters, we have used the energy convergence criterion for the self-consistency as 10 −10 eV with a Gaussian half-width parameter of 0.01 eV for the discrete energy levels. A very high plane wave cutoff value of 1000 eV as well as a dense Fourier grid spacing of 0.04Å in each of x, y and z directions is taken.
While relaxing the clusters using conjugate gradient algorithm, the same energy convergence criterion is used but a lower plane wave cut-off of 270 eV. The structural relaxation of clusters are done in the scalar relativistic mode. For the calculation of orbital moment and MAE, the SOI is treated as a perturbation in the scalar relativistic Hamiltonian. Benchmark calculations for some M 13 clusters have also been done with the Siesta code [43] within GGA, where the exchange and correlation functional is parameterized by Perdew, Burke and Ernzerhof [47]. It uses the norm-conserving pseudopotentials of Troullier and Martins [48]. The localized basis functions of "double-ζ with polarization orbitals" quality (and triple-ζ for 3d functions) have been constructed according to the standard scheme of 5 the Siesta method [49], with the "Energy Shift" parameter of 0.01 Ry. The treatment of SOI was included as described by Fernandez-Seivane et al. [50]. In the Siesta calculations, we used the atomic coordinates as relaxed by Vasp. This allows for a direct comparison of non-collinear structures, spin and orbital moments. The standard representation of local magnetic moments in Siesta is in terms of Mulliken populations of localized orbitals, that is a quite different definition from that used in Vasp. For the sake of better comparison, we report in the following the magnetic moments as properties integrated over atom-centred spheres for both Vasp and Siesta. In order to systematically pursue a search towards probable relaxation patterns from the ideal icosahedron (ICO) structure, we "drove" the structure along two different paths, which are known from previous studies [12,54,55,56] and are likely to lead to different metastable arrangements. The structural relaxation of M 13 clusters results in Jahn-Teller (JT) distortion as well as partially Mackay transformed (MT) clusters (Fe, Co, Ni), which is shown in Fig. 1 for the case of Fe 13 . For the Fe 13 cluster, the JT-distorted structure is by 123 meV/cluster, the MT cluster by 35 meV/cluster lower in energy compared to the energy of the ideal ICO, see also Ref. [12]. Only the Fe cluster exhibits the JT-distortion and the Mackay distortion as additional local minimum in the energy curve [54,55], whereas Co and Ni show only a tendency for MT.
What we refer to in the following as the Jahn-Teller (JT) type distortion [58] is that which maintains the fivefold symmetry around one of the ICO axes; it allows a compression or tension of the cluster along this axis, possibly accompanied by a mutual opposite rotation of two pentagonal rings pierced by the axis in question.
The occurrence of the JT distortion originates from (accidental) quasi-degeneracy of the highest occupied molecular orbitals so that the system may lower its energy by level splitting arising from the induced distortion and corresponding redistribution of electrons.
The partially Mackay distortion is of completely different origin since it is connected with the tendency of the magnetic Fe clusters to transform to the fcc cuboctahedron (CUBO) and by subsequent Bain like transformation to bcc Fe. Whereas for large Fe clusters, the JT distortion is no longer observed, we find in the simulations that the partially Mackay transformed cluster (which were denoted as shell-wise Mackay transformed cluster in [12,56]) still corresponds to a metastable [59] (local energy minimum) state up to cluster sizes having 15 closed atomic shells (n) and magic atom numbers (N) defined by For the Fe 13 cluster, the JT distortion corresponds to a compression of the cluster along the z -axis as depicted in Fig. 1, whereas we have chosen a different orientation for the MT cluster in the same figure (z -axis through mid-points of two opposite bonds) in order to illustrate its more cubic-like appearance. The relaxed atomic coordinates for the MT and JT 13-atom Fe clusters are listed in Table I The peripheral bond lengths for the MT clusters (see Table II) can be arranged in groups: 6 of the 30 bonds being parallel to the cartesian axes (x : 5-6, 7-8; y: 1-3, 4-2; z : 9-11, 10-12; see Fig. 1 for the labeling of the atoms), the remaining 24 bond lengths are also listed in    atoms 1-4 and also between the atoms 9-12.
When comparing the numerical results for spin and orbital magnetic moments from these two different calculational methods, one must take into acount the difference in their definition. In Vasp, the properties (spin and orbital moments) are extracted as projection onto an atomic sphere (see Section II). The "standard" Vasp value of atomic sphere radius for Co does in fact correspond to slightly overlapping spheres in our Co 13 cluster. The Siesta output results are reported in terms of decomposition over projection onto localized, but spatially extended, numerical orbitals, known as Mulliken population analysis. It is known that the local magnetic moments as well as atomic charges in heterogeneous systems do often come out very different, when estimated according to these two different schemes. In order to illustrate this effect, we give in the last column of Table III [60] it was shown that in Ni clusters of up to 13 atoms, the average orbital moment L per atom is found to be 4 to 8 times larger with respect to L bulk ; for larger cluster sizes, L was shown to approach the bulk value. It turns out from our calculation that for Ni 13 clusters, L agrees quite well with the earlier report [60], with orbital moment having larger magnitude than the bulk value. For Fe 13 and Co 13 , the L is also relatively large with respect to the bulk systems. In addition, the average spin moments S for these clusters are also increased with respect to the bulk systems.
In addition, the magnetic anisotropy energy for M 13 where The spin-spin and dipolar interactions are neglected, while comparing ab-initio total energy differences with the Heisenberg model. The θ-dependent energy difference for the relaxed MT Fe 13 cluster shows qualitatively good resemblance with that of the ideal cluster (see Fig. 3  this is lower with respect to bulk fcc Ni (2.7 µeV/atom). The small anisotropy for the relaxed Ni 13 cluster relative to bulk Ni is due to the high symmetry of the cluster, Ni 13 having a nearly ideal icosahedral shape. The change of sign in the energy vs. θ-curves (shown in Fig.   4) for the relaxed M 13 clusters can be explained through their way of structural relaxation.
Namely, the MT relaxation pattern for the Co 13 cluster is just opposite to that of Fe 13 : The Co "dimers" oriented along the edges of the cube in the right panel of Fig. 1

IV. RESULTS FOR CAPPED CLUSTERS M 13 Pt n
Obviously, binary and core-shell transition metal clusters show a yet larger diversity compared to the elemental systems. For example, the intermixing of 3d elements with 4d or 5d elements results in a large magnetic moment of the binary systems [26]. Both in experiment and in calculations, it has been observed that FePt and Co clusters show enhanced spin moments and orbital moments with respect to corresponding bulk values [64,65]. Hence, it would be interesting to study the change in magnetic properties including MAE of M 13 clusters capping with Pt atoms, which is described in the following.
We considered three high-symmetry positions to cap the M 13 clusters by a single Pt atom and found that a Pt position above the centre of a facet is most favorable in all three cases.
In the following, we used this finding as a guideline for initial geometries of M 13 Pt n clusters (n = 3, 4, 5, 20): In all cases, the Pt atoms were initially placed above the facet centres at a distance found in the single-Pt capping case. After relaxation, optimised geometries were obtained as shown in Fig. 5 for Co 13 Pt 3 (left) and Co 13 Pt 5 (right) and in Fig. 6 (right side) for Ni 13 Pt 20 . For n = 20, the initial geometries form a core-shell morphology.
Since atom projected quantities like spin and orbital moments depend on the specific code, we compared for the particular case of Ni 13 Pt 3 related data obtained by Vasp and Fplo. The structure optimisation was carried out by VASP and the same geometry was used to evaluate the magnetic moments by both codes.
In Fplo, the magnetic moments are calculated through Mulliken population analysis. whereas for the MT cluster, θ varies through all atoms as depicted in Fig. 1. Energy difference is The energy difference for the Co 13 cluster is multiplied by a factor 10 and the energy difference for the Ni 13 cluster is multiplied by a factor 100.
on the centre Fe atom for Fe 13 Pt n cluster increases with the number of Pt atoms, whereas for Co 13 Pt n and Ni 13 Pt n , this trend is weak. The orbital moment on Pt atoms is found to be very sensitive with respect to the core atomic species. For example, the |L i | on Pt atoms for Ni 13 Pt n clusters are enhanced relative to that of Fe 13 Pt n and Co 13 Pt n .
In Fig. 9  to the corresponding bulk 3d-spin moments values. Unlike the trend in orbital moment, the spin moment on Pt atoms is not much affected by its core species. In the following, a few quantitative statements are made for each of the capped clusters. The values of average orbital moment and average spin moment for each atomic species along with the total orbital moment L tot and total spin moment S tot for the capped clusters are defined and reported in 19 FIG. 9: The plots in the left, middle and right panels show the variation of |S i | with respect to the distance from centre for Fe 13 Pt n , Co 13 Pt n and Ni 13 Pt n clusters, respectively. The same symbols are used here as in Fig. 8.

C. Ni 13 Pt n clusters
In this case, the L tot and S tot also show a regular increase with the number of Pt atoms.
The L M and L Pt for this cluster do not much depend on the number of Pt atoms.
However, a decrease of L Pt and S Pt from Ni 13 Pt 5 to Ni 13 Pt 20 cluster is also observed. This is probably caused by a structural instability occuring for this composition upon the relaxation. The geometry optimization of this cluster converges to a structure with different symmetry, with a ferromagnetic ordering, where the Ni atoms are placed closer to the surface of the cluster as shown in Fig. 6. The reason for the segregation of Ni atoms towards the surface may be due to its lower surface energy compared to Pt [66]. Another aspect related  17.6 to this may be observed in the right panel of Figs. 8 and 9 showing the variation of onsite orbital and spin moments with respect to the distance. The large variations in orbital and spin moments just occur because of the structural distortion for this cluster composition.
Comparing all three cases of capped clusters, we find that the presence of Pt atoms on M 13 affects the orientation of core orbital moments in such a way that they always prefer to be in non-collinear alignment for the M 13 Pt n clusters, which is not the case in the uncapped M 13 clusters. On the other hand, directions of individual spin moments remain always collinear for the same clusters indicating that they are less affected by the Pt atoms.
Regarding the MAE of capped M 13 Pt n clusters, we observe that the symmetry rules the magnitude of the effect like for M 13 systems. For example, in Fe 13 Pt 4 (see Fig. 10) with second-order anisotropy, we find a high MAE value, even larger by a factor of two (∼ 7 meV/cluster) compared to JT-distorted Fe 13 . MAEs for relaxed clusters are found to be affected by the degree of relaxation due to which Fe 13 clusters have a high anisotropy in comparison with Co 13 and Ni 13 . Iron is a special case for which we obtain two local energy minima [12,56], corresponding to the JT and MT cases. The JT-distorted Fe 13 cluster exhibits an anisotropy approximately 5 times larger than the partially MT cluster due to the low symmetry in the former case. The present calculations of MAE agree well with the qualitative prediction of the Heisenberg model for the θ-dependent energy differences. With respect to the spin and orbital moments, both the capped and the free clusters show an increased value of orbital and spin moments compared to the bulk. For the capped clusters, the spin moments on Pt atoms remain unaffected by the host atoms. Finally, we infer that deposited transition metal clusters, with very large effects of relaxations, may even exhibit larger MAE values. Self-assembly of such clusters, like in case of Fe-Pt [13] and Co [14], may then approach the class of functional magnetic materials of use for magnetic storage devices.