Magnetism, structure and chemical order in small CoPd clusters: A first-principles study

Journal of Magnetism and Magnetic Materials 349 (2014) 109–115

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Letter to the Editor

Magnetism, structure and chemical order in small CoPd clusters: A first-principles study Junais Habeeb Mokkath n King Abdullah University of Science And Technology, Physical Science and Engineering Division, Thuwal 23955-6900, Saudi Arabia

art ic l e i nf o

a b s t r a c t

Article history: Received 9 November 2012 Received in revised form 4 June 2013 Available online 31 August 2013

The structural, electronic and magnetic properties of small Com Pdn ðN ¼ mþ n ¼ 8; m ¼ 0NÞ nanoalloy clusters are studied in the framework of a generalized-gradient approximation to density-functional theory. The optimized cluster structures have a clear tendency to maximize the number of nearestneighbor CoCo pairs. The magnetic order is found to be ferromagnetic-like (FM) for all the ground-state structures. Antiferromagnetic-like spin arrangements were found in some low-lying isomers. The average magnetic moment per atom μ N increases approximately linearly with Co content. A remarkable enhancement of the local Co moments is observed as a result of Pd doping. This is a consequence of the increase in the number of Co d holes, due to CoPd charge transfer, combined with the reduced local coordination. The influence of spin–orbit interactions on the cluster properties is also discussed. & 2013 Elsevier B.V. All rights reserved.

Keywords: Density functional theory Binary nanoclusters Nanoalloy magnetism

1. Introduction The binary metal clusters at the nano/subnanometer scale have attracted significant interest owing to their promising applications in optics, magnetism, and catalysis [1,2]. This is because their physio-chemical properties can be tuned not only by varying size, thereby increasing the surface to volume ratio, but also by varying composition and the chemical order. Concerning the magnetic properties, alloying a ferromagnetic 3d metal with a nonmagnetic 4d and 5d metal at the nanoscale creates very interesting magnetic scenario due to the combined effect of the reduced coordination numbers and atomic mixing. In fact, such phenomena have already been reported experimentally for a number of 3d/4d nanoclusters such as CoRh [3], and 3d/5d nanoclusters such as CoPt [4]. In those experiments, a huge enhancement of the magnetic moment of the binary clusters with respect to the corresponding pure clusters was observed, despite the fact that 4d and 5d elements are nonmagnetic in the bulk. It is well-known that the structure, composition and chemical ordering can play an important role in the magnetism of binary clusters. Therefore, for a correct understanding of the magnetic behavior of the bimetallic nanoclusters it is relevant to investigate the local geometrical and chemical environments within the system, in relation to the local magnetic moments distribution. The elements chosen for the present study, Co and Pd, have very interesting properties at the nanometer scale. Co atom shows enhanced magnetic moment compared to the bulk [5–7], whereas

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Pd atom shows a finite magnetic moment despite being non-magnetic in the bulk [8–11]. The magnetic properties of gas phase CoN clusters were investigated via Stern–Gerlach molecular beam deflection experiment by Bloomfield et al. for Co20–Co215 clusters [12] and by de Heer et al. for Co30–Co300 clusters [13]. These studies revealed that in the temperature range of 77–300 K, the CoN clusters show high field deflections which are characteristic of supermagnetic behavior. A number of theoretical results have been reported for cobalt clusters. Castro et al. [14] performed all-electron density functional calculations using both local density and generalized gradient approximations. However, the size of the clusters was limited only up to five atoms. Later on, Lopez et al. [16] studied CoN clusters ð4 r N r 60Þ, where minimization was done using an evolutive algorithm based on a many-body Gupta potential [17] and magnetic properties have been studied by a spd Tight-Binding method. The magnetism of small Pd clusters is still a subject for controversy. Photo-emission experiments [8] predicted a Ni-like spin arrangement in PdN clusters having N r 6 and a Pt-like non-magnetic behavior for N Z 15. In contrast, dc susceptibility measurements [18] found a magnetic moment of 0:237 0:19 μB =atom in Pd clusters with diameters in the range of 50–70 Å. A recent experiment by using gas-evaporative method [19] in a high purity Ar gas atmosphere has observed a magnetic moment of 0:75 7 0:31 μB =atom for fine Pd particles. The DFT calculations by Moseler et al. [10] have shown that both neutral and anionic clusters ð2 r N r 7 and N ¼ 13) of Pd are magnetic. In view of these interesting behaviors one expects that CoPd clusters should show very interesting structural, electronic and magnetic behaviors. In the present systematic study we have investigated the morphology, the electronic structure, and magnetism of CoPd nanoclusters, as a function of cluster size, structure and composition.

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The remainder of the paper is organized as follows. In Section 2 the main details of the theoretical background and computational procedure are presented. The results of our calculations are reported in Section 3. Finally, we conclude in Section 4 with a summary of the main trends and an outlook to future extensions.

2. Computational aspects The calculations reported in this work are performed by Vienna ab initio simulation package [21]. The exchange and correlation energy is described by using both the spin-polarized local density approximation (LDA) and Perdew and Wang's generalized-gradient approximation (GGA) [22]. The VASP solves the spin-polarized Kohn– Sham equations in an augmented plane-wave basis set, taking into account the core electrons within the projector augmented wave (PAW) method [23]. This is an efficient frozen-core all-electron approach which allows to incorporate the proper nodes of the Kohn–Sham orbitals in the core region and the resulting effects on the electronic structure, total energy and interatomic forces. The 4s and 3d orbitals of Co, and the 5s and 4d orbitals of Pd are treated as valence states. The wave functions are expanded in a plane wave basis set with the kinetic energy cut-off Emax ¼ 300 eV. In order to improve the convergence of the solution of the self-consistent KS equations the discrete energy levels are broadened by using a Gaussian smearing s ¼ 0:02 eV. The validity of the present choice of computational parameters has been verified [24]. The PAW sphere radii for Co and Pd are 1.302 Å and 1.434 Å, respectively. A simple cubic supercell is considered with the usual periodic boundary conditions. The linear size of the cell is a¼ 10–22 Å, so that any pair of images of the clusters are well separated and the interaction between them is negligible. Since we are interested in finite systems, the reciprocal space summations are restricted to the Γ point. The atomic positions are fully relaxed by means of conjugate gradient or quasi-Newtonian methods, without imposing any symmetry constraints, until all the force components are smaller than the threshold 5 meV/Å. The convergence criteria are set to 105 eV=Å for the energy gradient, and 5  104 Å for the atomic displacements [25]. The same procedure applies to all considered clusters regardless of composition, chemical order, or total magnetic moment. The different cluster topologies are sampled by generating all possible graphs for N r 6 atoms as described previously [26,27]. For each graph or adjacency matrix it is important to verify that it can be represented by a true structure in D r 3 dimensions. A graph is acceptable as a cluster structure, only if a set of atomic ! coordinates R i with i ¼ 1; …; N exists, such that the interatomic distances Rij satisfy the conditions Rij ¼ R0 if the sites i and j are connected in the graph (i.e., if the adjacency matrix element Aij ¼ 1) and Rij 4 R0 otherwise (i.e., if Aij ¼ 0). Here R0 refers to the nearest neighbor (NN) distance, which at this stage can be regarded as the unit of length, assuming for simplicity that it is the same for all clusters. Notice that for N r4 all graphs are possible cluster structures. For example, for N ¼4, the different structures are the tetrahedron, rhombus, square, star, triangular racket and linear chain. However, for N Z5 there are graphs, i.e., topologies, which cannot be realized in practice. For instance, it is not possible to have five atoms being NNs from each other in a three dimensional space. Consequently, for N Z 5 there are less real structures than mathematical graphs. The total number of graphs (structures) is 21 (20), 112 (104), and 853 (647) for N ¼ 5; 6, and 7, respectively [26,27]. For clusters having N r 6 atoms all these topologies have indeed been taken as starting points of our structural relaxations. Out of this large number of different initial configurations the unconstrained relaxations using VASP lead to only a few geometries, which can be

regarded as stable or metastable isomers. For larger clusters (N¼7 and 8) we do not aim at performing a full global optimization. Our purpose here is to explore the interplay between magnetism and chemical order as a function of composition for a few topologies that are representative of open and close-packed structures. Taking into account our results for smaller sizes, and the available information on the structure of pure CoN and PdN clusters, we have restricted the set of starting topologies for the unconstrained relaxation of CoPd heptamers and octamers to the following: bicapped trigonal bipyramid, capped octahedra, and pentagonal bipyramid for N¼ 7, and tricapped trigonal bipyramid, bicapped octahedra, capped pentagonal bipyramid and cube for N¼ 8. Although, the choice of topologies for N¼7 and 8 is quite restricted, it includes compact as well as more open structures. Therefore, it is expected to shed light on the dependence of the magnetic properties on the chemical order and composition. In the present study we performed fixed spin-moment (FSM) calculations in the whole physically relevant range. Starting from the non-magnetic state ðSmin ¼ 0Þ we increase Sz until the local spin z moments are fully saturated, i.e., until the Co moments in the PAW sphere reach μCo C 3μB and the Pd moments μPd C 1:5μB (typically, Smax ≳3N=2). The above described global geometry optimizations z are performed independently for all values of Sz. The ground-state of ComPdn corresponds to the most stable structural and magnetic configuration yielding the minimum energy as a function of Sz and of the atomic positions [28].

3. Results and discussions In the following, we present and discuss results for the binding energy and spin magnetic moments for N ¼ m þ n r 8. The most stable structures of the CoPd clusters are shown in Fig. 1. 3.1. Dimers The result for dimers is summarized in Table 1. Quantitatively, one observes that the CoCo dimer yields the highest cohesive energy, followed by the CoPd dimer and the PdPd dimer. This is qualitatively consistent with the known cohesive energies in the corresponding solids. Moreover, the bond length follows the trend in the atomic radii, namely dPdPd 4dCoPd 4 dCoCo . Let us consider the homogeneous cases first. Our calculations yield EB ¼1.53 eV, dCoCo ¼ 1:96 Å, and μ 2 ¼ 2μB for Co2. These results are similar to those reported in the previous VASP/DFT studies by Datta et al. [29]. However, the collision-induced dissociation experiments [30] reported a somewhat lower Eexpt B (Co2)¼1.32 eV. In the case of Pd2, our results are EB ¼ 0.64 eV, dPdPd ¼ 2:48 Å and μ 2 ¼ 1μB . Notice the bond length contractions bulk

bulk

with respect to the bulk values (dCoCo ¼ 2:51 Å and dPdPd ¼ 2:80 Å). These results are in good agreement with earlier DFT studies [10,31]. In contrast, Barretau et al. [32] found a non-magnetic ground state for Pd2 by using a spd tight-binding model. The binding energy calculated in the GGA compares reasonably well with available experimental results which scatter between 0.73 and 1.13 eV/atom [33]. High-level quantum chemistry calculations [34] also predict a triplet ground-state, having dPdPd ¼ 2:48 Å, and a somewhat smaller EB ¼ 0.43 eV. Finally, the calculations based on hybrid functionals [35] yield EB ¼0.48 eV/atom and dPdPd ¼ 2:53 Å. Concerning the local magnetic moments in the WS cell, one observes contrasting behaviors in the Co and Pd dimers. In Co2, the largest part of the spin magnetization is localized close to the atom: μWS Co ¼ 1:93μB in the WS cell while μ 2 ¼ 2μB . In contrast, in the case of Pd2, a significant part of the spin polarization is found

J.H. Mokkath / Journal of Magnetism and Magnetic Materials 349 (2014) 109–115

111

Fig. 1. Illustration of the calculated lowest energy cluster structures of Com Pd8m clusters having m ¼ 08 atoms. Green (blue) spheres represents Co (Pd) atoms. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

in the spill-off region, beyond WS radius r WS Pd ¼ 1:434 Å, since

μ

WS Pd

¼ 0:6μB while μ 2 ¼ 1μB .

For the CoPd dimer we found that EB ¼1.19 eV, dCoPd ¼ 2:23 Å, and μ 2 ¼ 1:5μB . Notice that the local Co moment μCo ¼ 2:42μB is significantly enhanced by 0:49μB , while μPd is reduced by 0:1μB as compared to their respective pure dimers (see Table 1). A significant charge transfer from Co to Pd can be inferred from this. Indeed, an integration of the electronic density in the Bader cells [36] shows that 0.11 electrons are transferred from the Co to the Pd atom. This

behavior is qualitatively in agreement with the higher Pauling electronegativity of the Pd atom (χ Co ¼ 1:88 and χ Pd ¼ 2:20). 3.2. Trimers Table 2 summarizes all the important results obtained for trimers. The ground-state energy structures are found to be the triangle for all the compositions. For Co3, the ground-state structure is an isosceles triangle with EB ¼ 1.87 eV, dCoCo ¼ 2:16 Å, and μ 2 ¼ 2μB . The highest occupied molecular orbital (HOMO) is degenerate, and partially

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Table 1 Structural, electronic and magnetic properties of CoPd dimers. Results are given for the binding energy EB (in eV/atom), point group symmetry (PGS) of the structure, the magnetic stabilization energy ΔEm ¼EðSz ¼ 0ÞEðSz Þ (in eV/atom), the average interatomic distance dαβ (in Å) between atoms α and β (α; β ¼ Co or Pd), the average spin moment per atom μ N ¼ 2Sz =N (in μB ), and the local spin moment μα at the Co or Pd atoms. Cluster

PGS

EB

ΔEm

dαβ

μN

μCo

Co2 CoPd Pd2

D1h D1h D1h

1.53 1.19 0.64

0.67 0.57 0.31

1.96 2.23 2.48

2.0 1.5 1.0

1.93 2.42

μPd

0.50 0.60

Table 2 Structural, electronic and magnetic properties of CoPd trimers as in Table 1. The average interatomic distance dαβ (in Å) ordered from top to bottom as dCoCo , dCoPd and dPdPd . Cluster

PGS

EB

ΔEm

dαβ

μN

μCo

μPd

Co3 Co2Pd

C2v C2v

1.87 1.84

0.57 0.47

2.00 1.33

1.94 1.95

0.06

CoPd2

C2v

1.70

0.19

1.00

2.25

0.33

Pd3

D3h

1.25

0.10

2.16 2.01 2.44 2.31 2.62 2.52

0.67

0.63

occupied. These results are in agreement with the all-electron (AE) DFT calculations by Castro et al. [14]. The linear Co3 is found to be 0.43 eV/atom less stable than the ground-state. We made a comparison between the results obtained using the LSDA and GGA exchange correlation functional on the most stable structure of Co3 cluster. Apparently, GGA provides a better overall description of the cluster properties with respect to the LSDA. For instance, using the LSDA functional, the average bond length calculated is 2.09 Å, while using the GGA, we obtained an increased average bond length of 2.16 Å. The ground-state structure for Pd3 is an equilateral triangle with μ 3 ¼ 0:67μB . This result is similar to the previous DFT study reported by Futschek et al. [9]. Pd3 density of states (DOS) shows that the HOMO is degenerate and partially occupied. Therefore a Jahn–Teller distortion could have been expected. However, our calculation does not display such an instability. In fact, Valerio and Toulhaot [35] have shown that Pd3 is stable against Jahn–Teller distortions at the GGA level, but unstable when a hybrid functional combining DFT and exact exchange is used or if the calculations are performed at the Hartree–Fock plus configuration interaction (HFþ CI) level. For Co2Pd, the ground-state structure is an isosceles triangle with EB ¼1.84 eV, and μ 3 ¼ 1:33μB . The local moments are coupled ferromagnetically. A significant enhancement in the local moments could be expected as in the CoPd dimer. Surprisingly, there is no such an enhancement in the μ Co , which is essentially the same as in Co3. Moreover, the Pd has nearly vanishing moment. The lack of enhancement of μ Co is likely to be related to the rather contracted CoCo bond length dCoCo ¼ 2:01 Å (see Table 2). We found that the linear Pd–Co–Co (Co–Pd–Co) structures are 0.18 (0.48) eV/atom less stable than the ground-state structure. The ground-state structure of CoPd2 is also an isosceles triangle (C2v) with EB ¼1.7 eV and μ 3 ¼ 1μB . Unlike the groundstate Co2Pd, the CoPd2 cluster shows a large enhancement in the μ Co ¼ 2:25μB . This is due to the combined effect of larger CoCo interatomic distance (dCoCo ¼ 2:31 Å) and to the significant charge transfer from Co to Pd. 3.3. Tetramers Table 3 summarizes the relevant geometric and magnetic information obtained for the ground-state tetramers. The ground-state

Table 3 Structural, electronic and magnetic properties of CoPd tetramers as in Table 2. Cluster

PGS

EB

ΔEm

dαβ

μN

μCo

μPd

Co4 Co3Pd

S4 D2h

2.37 2.18

0.72 0.52

2.50 1.75

2.19 2.10

0.32

Co2Pd2

D2h

1.874

0.44

2.43 2.18 2.40 2.39 2.62 2.37 2.71 2.61

1.50

2.30

0.43

0.75

2.18

0.24

CoPd3

S4

1.94

0.41

Pd4

Td

1.67

0.09

0.50

0.47

energy structures are tetrahedra for m¼0, 1 and 4, and rhombi for m¼2 and 3, see Fig. 1. Co4 has a Jahn–Teller distorted ground-state with μ 4 ¼ 2:5μB . In this cluster there are two kinds of bond lengths: two pairs have equal short bond lengths d¼ 2.14 Å, while the third pair has a much larger d¼2.72 Å. The shorter bond lengths should enhance the binding since the localized 3d electrons can also take part in the bonding. The first excited isomer is a rhombi, which is 0.11 eV less stable than the ground-state structure. Our results for Co4 are similar to the previous DFT calculations by Datta et al. [29]. So far, there are, to our knowledge, no experimental result available on the neutral tetramers. However, Yoshida et al. [15] predicted a tetrahedron with a bond length of 2:25 7 0:2 Å as the optimal structure for Co 4 anion. The most stable structure obtained for Pd4 is a tetrahedron with d ¼2.61 Å and μ 4 ¼ 0:50μB . This result is in good agreement with previous DFT studies [9,10,31]. The HOMO (minority spin) is degenerate and partially occupied by only one electron. The corresponding DOS is similar to the one reported by Kumar and Kawazoe [31]. Let us move further to Co3Pd. The optimal structure is a rhombus and the first exited isomer is a tetrahedron lying 41 meV/atom above the ground-state. The rhombus is also found to be the optimal structure for Co2Pd2. This case is an ideal example to study the influence of chemical order on the binding energy and magnetism. Notice that EB has been reduced in the ground-state of Co2Pd2 as compared to the ground-state of Co3Pd and CoPd3. This is probably due to the presence of a Pd dimer, which is weakest among the dimers. The local moments μCo and μPd show somewhat enhanced values 2:30μB and 0:43μB , respectively. This is because Co atoms are farther away and weakly bonded. Consequently, Co d states will be more localized and provide a significant enhancement to the resulting local spin polarization. The first isomer is a tetrahedron and which is only 4 meV/atom less stable. Here the magnetic order is found to be AF-like. The local moments are distributed as μCo1 ¼ 1:73μB , μCo2 ¼ 1:71μB , μPd1 ¼ 0:03μB , μPd2 ¼ 0:02μB . The small but still significant energy difference of 4 meV/atom between FM- and AF-like magnetic order pointing towards a strong competition between the different magnetic isomers to become the groundstate structure. In the rich Pd limit (CoPd3), the optimal structure is a distorted tetrahedron and the first excited isomer is a rhombus. 3.4. Pentamers The results for pentamers are tabulated in Table 4. The optimal structures are trigonal bipyramids (TBP) for all composition expect for Co4Pd, where a square pyramid (SP) is obtained. For Co5, the ground-state structure is a trigonal bipyramid ðD3h Þ having EB ¼2.65 eV, μ 5 ¼ 2:60μB and d¼ 2.41 Å. In this structure there are two types of bond-lengths: all the sides of upper and lower triangular pyramids have the same small length d ¼2.18 Å, while the NN distances d ¼ 2.65 Å along the triangle shared by the two triangular pyramids are much larger. Our results coincide with the

J.H. Mokkath / Journal of Magnetism and Magnetic Materials 349 (2014) 109–115

Table 4 Structural, electronic and magnetic properties of CoPd pentamers as in 2. Cluster

PGS

EB

ΔEm

dαβ

μN

μCo

μPd

Co5 Co4Pd

D3h C4v

2.65 2.50

0.61 0.56

2.60 2.00

2.24 2.12

0.37

Co3Pd2

D3h

2.35

0.42

1.40

2.05

0.25

Co2Pd3

D3h

2.19

0.30

1.20

2.16

0.31

CoPd4

D3h

2.05

0.20

0.60

2.16

0.19

Pd5

D3h

1.87

0.05

2.41 2.19 2.48 2.24 2.48 2.73 2.10 2.47 2.70 2.43 2.76 2.65

0.40

0.38

Table 5 Structural, electronic and magnetic properties of CoPd hexamers as in Table 2. Cluster

EB

ΔEm

dαβ

μN

μCo

μPd

Co6 Co5Pd

3.03 2.73

0.77 0.55

2.33 1.83

2.08 1.99

0.31

Co4Pd2

2.63

0.47

1.67

2.12

0.31

Co3Pd3

2.44

0.38

1.50

2.24

0.43

Co2Pd4

2.31

0.27

1.00

2.18

0.32

CoPd5

2.15

0.15

0.50

2.13

0.13

Pd6

1.95

0.10

2.27 2.30 2.46 2.30 2.47 2.24 2.48 2.70 2.22 2.49 2.68 2.43 2.70 2.65

0.32

0.32

previous DFT calculations by Datta et al. [29]. The optimal Pd5 is a TBP with EB ¼1.87 eV, and μ 5 ¼ 0:40μB . The length of the bonds shared by both triangular pyramids along the triangle is 2.66 Å, whereas the rest of bonds are slightly shorter by 0.03 Å. This result is in agreement with previous DFT results [10]. The first isomer is a SP structure, which is 5.9 meV less stable. The optimal Co4Pd is a SP and the average moment μ 5 is reduced by 0:60μB compared to the pure Co5. The Pd atom is located on the most coordinated cap position with μPd ¼ 0:37μB . The first excited isomer is a TBP, which is 27 meV less stable. In the optimal Co3Pd2, Co atoms form an isosceles triangle on the triangular face of TBP. SP is found to be the first excited isomer, which is 31 meV less stable than the ground-state structure. The binding energy per atom shows an approximately linear decrease as a function of the number of Pd atoms, since more the Pd bonds are, weaker the cluster.

3.5. Hexamers The results for hexamers are summarized in Table 5. The optimal structures for Co6, CoPd5 and Pd6 clusters are octahedra and capped-TBP (CTBP) are found to be optimal structure for the rest of the hexamers. The optimal Co6 structure has EB ¼3.03 eV and μ 6 ¼ 2:33μB . This result is in good agreement with those reported previous DFT calculations by Datta et al. [29]. The pentagonal bipyramid (PBP) lies 1.7 eV higher in energy. However, photoelectron spectroscopic studies [15] predicted a PBP structure having bond length of 2:75 70:1 Å to be the most probable structure for the Co 6 anion cluster.

113

On the other extreme, a perfect octahedron (Oh) with EB ¼ 1.95 eV, μ 6 ¼ 0:33μB and d ¼2.65 Å is found optimal for Pd6, see Fig. 1. The length of the bonds in the middle plane is d ¼2.60 Å, while we find d ¼2.69 Å for the bonds connecting the middle plane and the apex positions. A non-magnetic octahedron with the length of the bond forming the middle plane d ¼2.65 Å and the length of the bonds involving the cap position being d¼ 2.67 Å is nearly degenerate and is found to be only 3 meV less stable. The optimal CTBP structure is 46 meV less stable than the groundstate. These results coincide with those reported previous DFT calculations by Futschek et al. [9] and Kumar and Kawazoe [31]. The optimal Co5Pd is a CTBP structure, Fig. 1. The Co atoms form a TBP structure by pushing Pd atom to occupy the outer position. The average magnetic moment μ 6 is reduced by 0:50μB compared to the pure Co6. The first exited isomer is a capped square pyramid (CSP), with Co atoms forming the square pyramid structure and Pd atom to occupy the cap position. The optimal Co4Pd2 is a CTBP with Co atoms sub-clustered to form a tetrahedron, and Pd atoms are capped without forming a dimer. A CSP structure with a PdPd bond is the first exited isomer which is 67 meV less stable in energy. For the equiatomic composition (i.e., Co3Pd3) a CTBP structure is found to be optimal. Here Co atoms are grouped together to form a triangle. The first excited isomer is a double tetrahedron with Co atoms making a triangle. This is 15 meV/atom less stable. In the Pd rich Co2Pd4 cluster, a CTBP structure with a relatively shorter CoCo bond dCoCo ¼ 2:22 Å is the optimal. The first excited isomer is an octahedron, which is 47 meV less stable. Finally, the optimal CoPd5 is an octahedron and the first exited isomer is a CTBP structure that is 20 meV less stable. It is also very interesting to analyze the site projected local density of states (LDOS), at least for some representative examples. To this aim, we plot in Fig. 2 the spin polarized d-electron LDOS of representative CoPd hexamers having the relaxed structures illustrated in Fig. 1 The DOS for pure Co6 and Pd6 is also shown for the sake of comparison. In all the clusters, the dominant peaks in the relevant energy range (i.e., the occupied valence orbitals and the unoccupied ones near εF ) correspond either to the Co-3d or to the Pd-4d states. It is found that the contributions from s and p states in the DOS is negligible when compared to the d band contribution. The Co6 cluster shows a relatively narrow d-band which dominates the single-particle energy spectrum in the range 4:3 eV r ε  εF r1:15 eV. The majority d-DOS is fully occupied with highest majority state lying about 1.62 eV below εF . In addition there is an appreciable gap (about 0.26 eV) in the corresponding minority spectrum. The spin polarized DOS clearly reflects the FM-like order in the Co6 cluster. For low Pd concentration (e.g., Co4Pd2), the magnetic moments are not saturated. Only spin down (minority) states are found around εF . The Fermi level εF lies on the top of the minority band (see Fig. 2). In the majority band Pd dominates over Co at the higher energies (near to εF ) while Co dominates in the bottom of the band. In the minority band the participation of Pd (Co) is stronger (weaker) below εF and weaker (stronger) above εF , which is consistent with the fact that the Pd local moments are smaller than the Co moments. Finally for the Pd rich limit (e.g., Co2Pd4), the majority states are almost saturated and only minority states are found close to εF . The Co contribution is significant just below and above εF . The Co contribution largely dominates the unoccupied minority spin DOS, in agreement with the larger local Co moments. 3.6. Heptamers and octamers The results for N ¼7 are summarized in Fig. 1 and Table 6. The binding energy per atom increases approximately linearly with increasing Co content. For pure Pd7, PBP structure is the most

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Fig. 2. Electronic density of states (DOS) of CoPd hexamers. Results are given for the total (solid), the Co-projected (dotted), and the Pd-projected (dashed) d-electron DOS. Positive (negative) values correspond to majority (minority) spin. A Lorentzian width λ ¼ 0:02 eV has been used to broaden the discrete energy levels.

Table 6 Structural, electronic and magnetic properties of CoPd heptamers as obtained from a restricted representative sampling of cluster topologies.

Table 7 Structural, electronic and magnetic properties of CoPd octamers as obtained from a restricted representative sampling of cluster topologies.

Cluster

EB

ΔEm

dαβ

μN

μCo

μPd

Cluster

EB

ΔEm

dαβ

μN

μCo

μPd

Co7 Co6Pd

3.04 3.05

0.68 0.60

2.14 2.00

1.96 2.08

0.28

Co8 Co7Pd

3.17 3.11

0.70 0.57

1.88 1.95

0.27

2.88

0.51

1.86

2.15

0.43

Co6Pd2

3.05

0.53

1.75

2.06

0.29

Co4Pd3

2.71

0.45

1.43

2.08

0.37

Co5Pd3

2.93

0.49

1.63

2.13

0.42

Co4Pd4

2.76

0.41

1.50

2.23

0.49

Co3Pd4

2.50

0.30

1.29

2.16

0.43 Co3Pd5

2.62

0.28

1.13

2.19

0.36

Co2Pd6

2.47

0.21

1.00

2.26

0.43

CoPd7

2.27

0.12

0.63

2.31

0.34

Pd8

2.07

0.59

2.31 2.30 2.51 2.31 2.49 2.27 2.48 2.36 2.46 2.29 2.48 2.73 2.34 2.45 2.71 2.42 2.69 2.68

2.00 1.88

Co5Pd2

2.28 2.29 2.48 2.28 2.48 2.26 2.49 2.73 2.30 2.50 2.72 2.22 2.52 2.66 2.42 2.70 2.69

Co2Pd5

2.39

0.26

CoPd6

2.21

0.14

Pd7

1.99

0.04

1.14

0.71 0.29

2.24

2.31

0.50

0.36 0.28

stable of the considered starting geometries. The first isomer is found to be a capped octahedron (CO). Our result shows a good agreement with those of Futschek et al. [9] and Moseler et al. [10]. Heptamers with high Pd concentrations also favor a PBP topology except CoPd6. In the CoPd6 ground-state the Co atom is located on the square ring of the octahedron, while in the Co2Pd5 groundstate the Co atoms are along the axis of the PBP. The results for Co2Pd5 (e.g., structure, local moments and chemical order) match with the previous DFT and tight binding calculations [37]. As the Co content increases, the topology of Com Pdn ground-state geometry changes. In fact, the ground-state structures of Co4Pd3 and Co5Pd2 correspond to a CO. For the pure Co7 cluster, PBP is the most stable of the considered starting geometries. This result is similar with the one reported DFT work by Datta [29]. However, the experimental magnetic moment (2.36 70.25) [5] is higher than the present value. Table 7 summarizes the results for CoPd octamers. The general trends concerning the composition dependence of the binding energy, chemical order, as well as the average and local magnetic

0.25

0.25

moments are similar to smaller clusters. The most stable structure that we obtain for Co8 is a BCO having μ 8 ¼ 2μB , and d ¼2.31 Å. The experimentally measured magnetic moment, 2.51 70.15 μB =atom [5], is higher than the present value. In all the mixed octamers, there is a clear tendency for Co atoms to group together. The most stable structure that we obtain for Pd8 is a BCO having EB ¼2.07 eV, μ 8 ¼ 0:25μB and d ¼2.68 Å. The magnetic properties of heptamers and octamers follow the qualitative behavior established in smaller clusters. In all the cases the average magnetic moment per atom μ N and the magnetic stabilization energy ΔEm increase with Co concentration. The spin–orbit coupling (SOC) is not included in the present calculations. It is assumed to have very little effect on the clusters under consideration. Several authors have discussed the influence of SOC for a variety of 3d, 4d, and 5d transition metal clusters (TMs) [38–40]. These investigations have predicted that for clusters of 3d and 4d TMs the SOC was found much smaller whereas for 5d TMs like Pt, SOC can be strong enough so that it can change

J.H. Mokkath / Journal of Magnetism and Magnetic Materials 349 (2014) 109–115

the energetic ordering of structural isomers. We have performed some representative calculations by taking into account spin–orbit coupling (SOC) in order to explore their effect on the ground-state structure, chemical order and spin moments. For example in Co6, Co3Pd3 and Pd6 we find that the changes in the ground-state energy resulting from SO interaction are typically of the order of 0.2 eV for the whole cluster. This is often comparable to or larger than the energy differences between the low-lying isomers. However, the SO energies are very similar for different structures, so that the ground-state structures remain essentially the same as in the scalar relativistic (SR) calculations. The changes in the bond lengths and in the spin moments resulting from SOC are also very small (e.g., jμ SOC μ SR jC 0.04 μB and jdij dij j C 0.001 Å in Co4Pd4). As a result, the conclusions drawn from our SR calculations on the relative stability and local spin moments seem to be unaffected by the spin–orbit contributions. To conclude this section it is interesting to compare the magnetic properties of this small clusters with the available experiments and theoretical results for macroscopic alloys. The bulk calculations from Bozorth predicted FM-like order at all composition of Co content [41]. He assumed a zero magnetic moment for the Pd atom and 1:7μB for the Co atom. In addition, he found that the magnetic moment is increasing from 0.05 to 1:55μB for 20 to 99.9% of the Co content. Our results also predict FM-like order and a non-vanishing magnetic moment for Pd atom for all composition at least in the case of ground-state structures. This is a consequence of the reduction of local coordination number and the associated effective d-band narrowing, which renders the Stoner criterion far easier to satisfy, and which tends to stabilize the high-spin states. SOC

SR

4. Summary and conclusions The structural, electronic and magnetic properties of small Com Pdn clusters having N ¼ m þ n r 8 atoms have been investigated systematically in the framework of a generalized gradient approximation to density-functional theory. A remarkable enhancement of the local Co moments is observed as a result of Pd doping. This is due to the enhancement in the number of Co d holes, due to CoPd charge transfer, combined with the reduced local coordination number. Acknowledgments Computer resources provided by ITS Kassel is gratefully acknowledged. References [1] See, for instance, Faraday Discussions 138 (2008). [2] R. Ferrando, J. Jellinek, R.L. Johnston, Chemical Reviews 108 (2008) 845–910. [3] D. Zitoun, M. Respaud, M.C. Fromen, M.J. Casanove, P. Lecante, C. Amiens, B. Chaudret, Physical Review Letters 89 (2002) 037203. [4] F. Tournus, A. Tamion, N. Blanc, A. Hannour, L. Bardotti, B. Prevel, P. Ohresser, E. Bonet, T. Epicier, V. Dupuis, Physical Review B 77 (2008) 144411. [5] M.B. Knickelbein, Journal of Chemical Physics 125 (2006) 044308. [6] S. Yin, R. Moro, X. Xu, W.A. de Heer, Physical Review Letters 98 (2007) 113401.

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[7] M.B. Knickelbein, Physical Review B 75 (2007) 014401. [8] G. Ganteför, W. Eberhardt, Physical Review Letters 76 (1996) 4975. [9] T. Futschek, M. Marasman, J. Hafner, Journal of Physics: Condensed Matter 17 (2005) 5927–5963. [10] M. Moseler, H. Häkkinen, R.N. Barnett, U. Landman, Physical Review Letters 86 (2001) 2545. [11] J. Dorantes-Dávila, H. Dreysse, G.M. Pastor, Physical Review Letters 91 (2003) 197206. [12] J.P. Bucher, D.C. Douglass, L.A. Bloomfield, Physical Review Letters 66 (1991) 3052; D.C. Douglass, J.P. Bucher, L.A. Bloomfield, Physical Review B 45 (1992) 6341; D.C. Douglass, A.J. Cox, J.P. Bucher, L.A. Bloomfield, Physical Review B 47 (1993) 12874. [13] I.M.L. Billas, J.A. Becker, A. Châtelain, W.A. de Heer, Physical Review Letters 71 (1993) 4067; I.M.L. Billas, A. Châtelain, W.A. de Heer, Science 265 (1994) 1682. [14] M. Castro, C. Jamorski, D.R. Salahub, Chemical Physics Letters 271 (1997) 133. [15] H. Yoshida, A. Terasaki, K. Kobayashi, M. Tsukada, T. Kondow, Journal of Chemical Physics 102 (1995) 5960. [16] J.L. Rodriguez-Lopez, F. Aguilera-Granja, K. Michaelian, A. Vega, Physical Review B 67 (2003) 174413. [17] R.P. Gupta, Physical Review B 23 (1983) 6265. [18] T. Taniyamaand, E. Ohta, T. Sato, Europhysics Letters 38 (1997) 195. [19] T. Shinohara, T. Sato, Physical Review Letters 91 (2003) 197201. [21] G. Kresse, J. Furthmüller, Physical Review B 54 (1996) 11169; G. Kresse, D. Joubert, Physical Review 59 (1999) 1758. [22] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Physical Review B 46 (1992) 6671. [23] P.E. Blöchl, Physical Review B 50 (1994) 17953. [24] A number of tests have been performed in order to assess the numerical accuracy of the calculations. For example, increasing the cutoff and supercell size Emax ¼ 300 eV, a ¼ 12 Å to 500 eV, a ¼ 22 Å in Co4 yields a total energy differences of only 1 meV/atom and 0.2 meV/atom, respectively. This implies an increase in the computation time of a factors of 3–7. In the above calculations we obtain the change in average bond length (bond angle) of only 10  3 Å (10  4 degrees). These difference are not significant for our physical conclusions. In fact typical isomerization energies in these clusters are an order of magnitude larger of the order of 10–50 meV/atom. We also found that the total energy is essentially independent of the choice of the smearing parameter values s provided not too large ðs r 0:05 eVÞ. Values from s ¼ 0:01 to 0.1 eV have been checked. We conclude that set of standard parameters (Emax ¼ 300 eV, supercell size a from 10 to 22 Å and smearing parameter 0.02 eV) offers a good accuracy at a reasonable computational costs. [25] G. Kresse, J. Furthmüller, VASP The Guide 〈http://cms.mpi.univie.ac.at/vasp〉. [26] G.M. Pastor, R. Hirsch, B. Mühlschlegel, Physical Review Letters 72 (1994) 3879; G.M. Pastor, R. Hirsch, B. Mühlschlegel, Physical Review B 53 (1996) 10382. [27] J.H. Mokkath, G.M. Pastor, Physical Review B 85 (2012) 054407; J.H. Mokkath, G.M. Pastor, Journal of Physical Chemistry C 116 (2012) 17228–17238. [28] Results for lower lying isomers and excited magnetic configurations will be reported elsewhere. [29] S. Datta, M. Kabir, S. Ganguly, B. Sanyal, T. Saha-Dasgupta, A. Mookerjee, Physical Review B 76 (2007) 014429. [30] D.A. Hales, C.-X. Su, L. Lian, P. B Armentrout, Journal of Chemical Physics 100 (1994) 1049. [31] V. Kumar, Y. Kawazoe, Physical Review B 66 (2002) 144413. [32] C. Barreteau, R.G. López, D. Spanjaard, M.C. Desjonqéres, A.M. Olés, Physical Review B 61 (2000) 7781. [33] S.S. Lin, B. Strauss, A. Kant, Journal of Chemical Physics 51 (1969) 2282. [34] K. Balasubramanian, Journal of Chemical Physics 91 (1989) 307. [35] G. Valerio, H. Toulhoat, Journal of Physical Chemistry 100 (1996) 10827. [36] R.F.W. Bader, Atoms in Molecules, A Quantum Theory, Oxford University Press, Oxford, 1990. [37] F. Aguilera-Granja, A. Vega, J. Rogan, X. Andrade, G. García, Physical Review B 74 (2006) 224405. [38] M. N Huda, M.N. Niranjan, B.R. Sahu, L. Kleinman, Physical Review A 73 (2006) 053201. [39] L. Fernandez-Seivane, J. Ferrer, Physical Review Letters 99 (2007) 183401. [40] P. Blonski, S. Dennler, J. Hafner, Journal of Chemical Physics 134 (2011) 034107. [41] R.M. Bozorth, P.A. Wolff, D.D. Davis, V.B. Compton, J.H. Wernick, Physical Review 122 (1961) 1157–1160.