Structural and Magnetic Properties of Small 4d Transition Metal

Article pubs.acs.org/JPCA

Structural and Magnetic Properties of Small 4d Transition Metal Clusters: Role of Spin−Orbit Coupling H. K. Yuan, H. Chen,* A. L. Kuang, B. Wu, and J. Z. Wang School of Physical Science and Technology, Southwest University, Chongqing 400715, People’s Republic of China ABSTRACT: The spin and orbital magnetic moments, as well as the magnetic anisotropy energy (MAE), of small 4d transition metal (TM) clusters are systematically studied by using the spin−orbit coupling (SOC) implementation of the density-functional theory (DFT). The effects of spin−orbit interactions on geometrical structures and spin moments are too weak to alternate relative stabilities of different low-lying isomers. Remarkable orbital contributions to cluster magnetic moments are identified in Ru, Rh, and Pd clusters, in contrast to immediate quenching of the atomic orbital moment at the dimer size in other elemental clusters. More interestingly, there is always collinearity between total spin and orbital moments (antiferromagnetic or ferromagnetic coupling depends on the constituent atoms whose 4d subshell is less or more than half-filled). The clusters preserve the validity of Hund’s rules for the sign of orbital moment. The calculations on MAEs reveal the complicated changes of the easy axes in different structures. The perturbation theory and the firstprinciples calculations are compared to emphasize how MAEs evolve with cluster size. Finally, large orbital moments combined with strong spin−orbit coupling are proposed to account for large MAEs in Ru, Rh, and Pd clusters.

I. INTRODUCTION The primary interest in transition metal (TM) clusters is magnetic properties, which plays a crucial role in shedding light on the fundamental interactions responsible for magnetic states and potential applications in advanced magnetic information storage devices.1,2 With the help of experimental advances in synthesis and theoretical developments in calculations, typical ferromagnetic metals such as Fe, Co, and Ni,3−8 as well as nonmagnetic metals such as Sc, Y, Cr, Rh, and Pt,9−12 have been proved to exhibit enhanced magnetic moments at the nanoscale. These novel magnetic behaviors can be easily explained in terms of the d band narrowing that results from the reduced coordination numbers and high structural symmetries.13−16 Although great successes have been achieved in understanding the magnetism of TM clusters, a stubborn issue remains unsolved: quantitative differences between theoretical predictions and experimental measurements.15−19 Regarding that most theoretical investigations have been performed only by assuming the spin contributions, one thus deduces that the first possibility to remove these differences is the survival of orbital magnetic moments (OMM). After the pioneering work on Ni clusters17 and the following attempts on other ferromagnetic 3d clusters,17−19 it has become clear that the OMM are prominent in 3d clusters.17−21 Recently, FernandezSeivane et al.,22 Błoński and Hafner,23−25 and Fritsch et al.26 have theoretically investigated 4d (Rh, Pd) or 5d (Pt, Au) clusters. Bartolomé et al.12 and Sessi et al.27 have experimentally observed the large orbital-to-spin ratios up to 32% for Pt13 cluster and 60% for Rhn clusters (n̅ = 20). Despite very fascinating perspectives, a majority of theoretical works focus on just 4d Rh and Pd clusters,22−24,37,38 and little information is known, to the best of our knowledge, about the consequences © 2012 American Chemical Society

of orbital moments in other TM clusters. Thus, it is an ideal thing if similar information could be identified in other 4d clusters. One of the most significant consequences of the unquenched orbital moments is a preference to form magnetic anisotropy energy (MAE) via the spin−orbit interaction.28 The MAE of a single-domain particle is required to prevent loss of information, thus as is well-known, it is a crucial factor in technological applications (e.g., magnetic recording or memory devices). It is therefore not surprising that equal efforts have been devoted to MAE when OMM are addressed.20−25,29−33 Experimental confirmations for the enhanced MAE of free or deposited clusters are available from the X-ray magnetic circular dichroism spectroscopy (XMCD) and the X-ray absorption spectra (XAS). A typical measurement has been done on systems of Co/Pt(111) by Gambardella et al.,29 where cobalt atoms have a high MAE of 9 meV/atom. Rohart et al. have extracted the influences of chemical composition on MAE of embedded CoPt clusters.34 In the latest experiment, perpendicular MAE in Ru/Co/Ru(0001) systems and dispersion MAE in size-selected CoPt clusters have been observed.35,36 In contrast to above limited experimental investigations, there are large numbers of theoretical works concerning MAE of TM clusters. On the basis of the earlier studies on Fen (n ≤ 7) clusters by Pastor et al.,30 one concluded that MAEs in clusters are considerably larger than the corresponding values in crystals or thin films. This conclusion has been subsequently confirmed in other elemental clusters, including not only Fe, Co, and Ni 3d clusters20,21,29−33 but also Ir and Pt 5d clusters.22−25,37,38 Received: July 20, 2012 Revised: October 23, 2012 Published: November 9, 2012 11673

dx.doi.org/10.1021/jp307202t | J. Phys. Chem. A 2012, 116, 11673−11684

The Journal of Physical Chemistry A

Article

Figure 1. Isomeric structures of TMn clusters (n = 2−7). Initial magnetization orientations are orderly aligned parallel to three spatial directions x⃗, y⃗, z⃗.

With these views in mind, we perform a systematic theoretical investigations on geometrical structures and magnetic properties of 4d TMn clusters (n = 2−7), encompassing the complete 4d period. The calculations in the presence as well as in the absence of SOC are performed, with stressing topics on OMM and MAE. Three fundamental points are addressed (i.e., how they evolve with cluster size and atomic number; how they depend on the local structural environment; how MAE correlates to magnetic moments). In what follows, we will first describe the computational methodology in section II, and then present our results and discussions in section III. Finally, a summary is given in section IV.

Particularly, huge MAE up to 50 meV/atom for small clusters Ir2 and Pt2,23 as well as 4.3/1.4 meV per atom for large clusters Pt5/Pt6 have been reported.24,25 In parallel to these works, depositing 3d clusters on various 5d metal surfaces,39−43 or mixing clusters of CoRh,44−46 CoPd,37,46 CoPt,34,47,48 CoAu,49 and CoFe clusters,50 have been long-explored, because the combinations of strong spin−orbit interaction and large orbital moment are the necessary prerequisites to obtain large MAE. Here, along with the findings derived from previous calculations, further comprehensive investigations would reveal some novel phenomena. For instance, spin and orbital magnetic moments are found to couple ferromagnetically in late TM clusters,17−24 but it may not be necessary in early TM clusters. In addition, there is no theoretical investigation concerning a complete period row. To extract a general trend, comparative studies based on the same theoretical footings are highly desirable. From a computational point of view at the realistic level, the treatments of SOC as a perturbation or calculations involving constrained relaxations have been often carried out in previous investigations (usually adopt the tight binding method). In these studies, structural relaxations were generally neglected and collinear magnetic arrangements were allowed.17−21,37,38 It is well-known that a minor structural variation can change electronic configurations significantly, which would lead to far deviations of OMM and MAE from the realistic values.22−25 Ab initio methods can provide a more realistic scenario and give more reliable results.

II. METHODOLOGY The DFT calculations are performed within exchange− correlation potential of PW9151 to the generalized gradient approximation (GGA) in the Vienna ab initio simulation package (VASP).52−54 The Kohn−Sham orbitals are expanded in a plane-wave basis set and the interactions of the valence electrons with the ionic cores are described by the projected augmented wave (PAW) potentials.54,55 The spin−orbit interactions together with the magnetic noncollinearities as described by Hobbs et al.56 and Marsman et al.57 have been taken into account in the PAW methods. In our calculations, 10 + n electrons (3p64s24dn4s2, n = 1−2) are treated as the valence electrons for Y and Zr; 7+n electrons (3p64dn5s1; n = 4−6) are 11674



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Table 1. Bond Length d (Å), HOMO−LUMO Gap (eV), Spin Moment μ⃗ cell S , and Orbital Moment μ⃗ L of 4d Dimers for Axial and Perpendicular Magnetizations in SOC Calculationsa NSOC

axial (x)

perpendicular (z)

dimer

d

Eso

μ⃗ cell S

gap

μ⃗ cell S

μ⃗L

gap

μ⃗cell S

μ⃗ L

gap

R

ΔExz

Y2 Zr2 Nb2 Mo2 Tc2 Ru2 Rh2 Pd2 Ag2

2.92 2.29 2.12 1.96 1.98 2.07 2.21 2.48 2.57

0.00 0.11 0.01 0.01 −0.07 −0.15 −0.06 −0.02 −0.00

4 2 2 0 2 4 4 2 0

0.48 0.35 0.70 1.85 0.75 0.31 0.24 0.31 2.02

4.00 2.00 2.00 0.06 2.00 4.00 3.98 1.94 0.04

0.00 −0.29 0.00 0.02 −0.00 0.00 1.81 0.93 0.00

0.45 0.35 0.64 1.77 0.64 0.21 0.25 0.11 2.02

4.00 1.99 1.98 0.06 1.95 3.95 3.93 1.97 0.04

−0.07 −0.04 −0.03 −0.00 0.04 0.23 0.50 0.34 0.00

0.48 0.35 0.70 1.77 0.75 0.30 0.26 0.26 2.02

3.6% 20.0% 1.9% 0.0% 2.5% 6.5% 49.7% 28.7% 0.0%

0.51 −1.49 7.93 −0.05 24.84 35.92 −47.86 49.22 0.00

The scalar-relativistic results are listed in the column NSOC. The spin−orbit coupling energy (eV/atom), Eso = (ENSOC − ESOC b b )/n, is defined as the difference of average binding energies in NSOC and SOC calculations. The ratio |μ⃗ L|/|μ⃗ S| is denoted by R, and ΔExz represents the magnetic anisotropy energy (meV). All values for magnetic moments are in Bohr magnetons (μB). Values deviating less than 0.01 from an integer values are rounded to the next integer. a

principal axis Cn; y⃗ directions are mostly along the subsymmetrical principal axis; x⃗ directions are perpendicular to y ⃗ and z⃗ directions. However, it should be noted that some initial directions are chosen regardless of this restriction. In some structures, three initial magnetization directions are not perpendicular to one another. The detailed directions of the initial magnetizations are illustrated on each structure in Figure 1. The MAE, ΔEδγ = Eδ − Eγ, is calculated in terms of energy differences between two independent magnetization orientations δ and γ. The accuracies of our PAW-PW91 schemes are assessed by the benchmark calculations on 4d TM dimers (Table 1 and Figure 2). In NSOC calculations, our bond

for Nb, Mo, Tc; 1 + n electrons (4dn5s1; n = 7−10) are for Ru, Rh, Pd, Ag. For the cutoff energies and Wigner−Seitz radius, we employ the maximal default values recommended by the VASP-PAW potential, i.e., cutoff energies from 271 to 326 eV and Wigner−Seitz radii from 1.402 to 1.815 Å.52−54 We test the PAW potentials by reproducing the magnetic ground states of isolated 4d atoms. A simple cubic supercell with volume of 15 Å3 is used to ensure sufficient separations between the periodic images. Due to the large sizes of unit cells, only the gamma point is sampled in the Brillouin zone. Symmetry unrestricted geometry optimizations are performed by using the conjugate gradient and quasi-Newtonian methods. Our convergence criteria are 10−7 eV for the self-consistent electronic loop and 0.001 eV/Å for the forces on each atom. To improve the convergence of solution of the self-consistent Kohn−Sham equations, discrete energy levels are broadened with a smearing parameter of σ = 0.02 eV by using the first Methfessel−Paxton (MP) method.58 To avoid falling into a local minimum of the energy surface, two-step geometrical optimizations are adopted. In the first step, different isomeric structures within possible spin multiplicities are identified under the scalar relativistic calculations (labeled by NSOC calculations). The local spin moments are constrained to align in collinearity during the optimizations. We also crosscheck our structures and magnetic results with the DMol code (approximately the same accurate as in the VASP code).59 In the second step, the above resulting structures are treated as the starting structures for three fully self-consistent SOC optimizations, where initial magnetization orientations are orderly aligned parallel to three specific spatial directions (δ = x⃗, y,⃗ z⃗). For each magnetization orientation, two or three initial magnetic moments are examined to perform a thorough search of the magnetic ground states. In fact, we have initially imposed many allowable magnetic moments on each atom in small clusters and interestingly found that different initial magnetic settings are finally optimized to the same magnetic values. These SOC optimizations permit noncollinear magnetic orientations of the local spin and orbital moments, where each initial magnetic vector will relax to a stable direction when cluster energy is stationary. As for the initial magnetization orientations, we usually choose z⃗ directions perpendicular to the basal planes of structures, x⃗ and y⃗ directions in the basal planes. On the basis of this principle, z⃗ directions are usually along the symmetrical

Figure 2. Calculated bond lengths (filled circle) and binding energies (filled pentagram) of 4d dimers. For comparison, previous values (hollow pentagram) are shown.60

lengths of all 4d dimers agree very well with available theoretical values given by Li et al.60 and Sun et al.61 We also reproduce the magnetic moments of all 4d dimers given by Li et al.60 In SOC calculations, our bond lengths, spin moments μ⃗S, orbital moments μ⃗L, and MAEs of Ru2 and Rh2 dimers are in agreement with the corresponding values given by Błoński et al.23 and Strandberg et al.63 Thus, we expect that our calculational methods are appropriate for 4d clusters and the results are reliable.

III. RESULTS AND DISCUSSION A. Geometrical Structure. Figure 1 shows presentive structures of 4d TMn clusters with n = 2−7. These structures have been previously reported as the low-lying structures of 11675



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Table 2. Spin−Orbit Coupling Energy Eso (eV/atom) and the Orbital-to-Spin R as in Table 1 but for TM3 Clusters; Total Spin Moment of the Cluster in the Unit Cell μ⃗ cell S ; Spin Moment μ⃗ S (Absolute Spin Moment |μS|), Orbital Moment μ⃗ L (Absolute Orbital Moment |μL|), and Total Moment μ⃗ J for 4d Trimers from SOC Calculations; and In-Plane and Out-of-Plane Anisotropy Energyies ΔExy and ΔEyza cluster

sym

Eso

μ⃗ cell S

Y3 Zr3 Nb3 Mo3 Tc3 Ru3 Rh3 Pd3 Ag3

D3h C2v C2v Cs C2v C2v D3h D3h/C2v D3h

0.00 0.12 0.01 0.01 −0.07 −0.15 −0.03 −0.04 −0.01

1/1.00 2/2.00 1/1.00 2/2.00 5/4.96 6/5.94 3/2.93 2/1.76 1/1.00

μ⃗S (|μS|) 0.71 1.23 0.79 0.84 3.73 5.31 2.73 1.66 0.26

μ⃗L (|μL|) −0.02 −0.14 0.03 −0.02 −0.16 0.20 0.94 0.94 −0.07

(0.71) (1.23) (0.92) (0.84) (3.75) (5.31) (2.73) (1.66) (0.26)

μ⃗J (easy axis↑↓)

(0.02) (0.19) (0.03) (0.03) (0.29) (0.20) (0.94) (0.98) (0.07)

0.69 1.09 0.82 0.82 3.57 5.51 3.67 2.60 0.19

(y↑↓) (y↑↓) (z↑↑) (y↑↓) (y↑↓) (z↑↑) (y↑↑) (y↑↑) (z↑↓)

R

ΔExy

ΔEyz

2.4% 15.1% 3.0% 4.0% 7.7% 3.7% 34.4% 59.5% 27.8%

0.00 1.24 0.16 0.00 0.13 −5.46 0.03 9.74 0.00

−0.02 −0.53 0.05 −0.28 −4.64 8.80 −22.40 −2.92 18.04

In the second and fourth columns, the symmetries and μ⃗ cell S before/behind the slash refer to NSOC/SOC results. The μ⃗ J = μ⃗ S + μ⃗ L is the vectorial sum of cluster spin moment μ⃗ S and the orbital moment μ⃗ L, and the symbol ↑↓ (↑↑) indicates antiferromagnetic (ferromagnetic) couplings between total spin moment and total orbital moment along the easy axis. The absolute spin |μS| (orbital |μL|) moment is obtained by summing the modulus of the local spin ∑3i=1|μ⃗ is| (orbital ∑3i=1|μ⃗ il|) moments. In-plane anisotropy energy ΔExy = Ex − Ey and out-of-plane anisotropy energy ΔEyz = Ey − Ez are listed (meV), where the indexes of x, y, and z refer to three resulting magnetization directions as are illustrated in Figure 1. All values for magnetic moments are in Bohr magnetons (μB). a

Table 3. Same as in Table 2 but for TM4 Clustersa cluster Y4 Zr4 Nb4 Mo4 Tc4 Ru4 Rh4 Pd4 Ag4

sym

Eso

ΔEr

μ⃗ cell S

2a (C2v) 2b (C3v) 2a (D2h) 2b (D2d) 2a (D2h) 2b (Td) 2a (D2h) 2b (D2d) 2a (D4h) 2b (D2) 2a (D2h) 2b (D2d) 2a (D2h) 2b (Td) 2a (D2h) 2b (Td/D2d) 2a (D2h)

0.00 0.00 0.12 0.12 0.01 0.01 0.01 0.01 −0.07 −0.07 −0.16 −0.16 −0.04 −0.03 −0.02 −0.02 −0.00

0.00/0.00 −1.00/−1.00 0.00/0.00 −0.77/−0.77 0.00/0.00 −0.90/−0.90 0.00/0.00 −0.44/−0.44 0.00/0.00 −0.10/−0.10 0.00/0.00 −1.06/−1.04 0.00/0.00 −0.80/−0.78 0.00/0.00 −0.83/−0.82 0.00/0.00

4/3.99 2/2.00 2/2.00 4/4.00 2/2.00 0/0.02 2/2.00 0/0.02 0/0.00 0/0.05 4/4.06 4/3.95 6/5.94 0/0.02 2/0.52 2/1.93 0/0.05

μ⃗S (|μS|) 1.98 0.97 1.22 2.65 1.03 0.01 1.39 0.02 0.00 0.04 3.62 3.53 5.57 0.02 0.51 1.83 0.01

(1.98) (0.97) (1.22) (2.65) (1.74) (0.02) (1.39) (1.05) (0.09) (0.04) (3.62) (3.53) (5.57) (0.04) (0.51) (1.83) (0.01)

μ⃗L (|μL|) −0.04(0.04) −0.05 (0.08) −0.11 (0.11) −0.08 (0.08) 0.01 (0.01) 0.02 (0.04) −0.05 (0.05) −0.00 (0.01) −0.00 (0.00) −0.01 (0.01) 0.55 (0.55) −0.01 (0.44) 1.24 (1.24) 0.01 (0.03) 0.36 (0.36) 0.48 (0.55) 0.00 (0.00)

μ⃗J (easy axis↑↓) 1.94 0.92 1.11 2.57 1.04 0.03 1.34 0.02 0.00 0.03 4.17 3.52 6.81 0.03 0.87 2.31 0.01

(z↑↓) (z↑↓) (y↑↓) (z↑↓) (z↑↑) (z↑↑) (y↑↓) (z↑↓) (y↑↓) (z↑↓) (x↑↑) (z↑↑) (x↑↑) (z↑↑) (y↑↑) (z↑↑) (x↑↑)

R

ΔExy

ΔEyz

2.1% 8.5% 8.9% 2.9% 0.7% 145.8% 3.6% 1.1% 2.3% 11.4% 15.1% 12.4% 22.3% 65.8% 71.3% 30.2% 0.0%

−0.16

0.29 0.07 −0.76 0.16 0.09 0.00 −0.31 2.04 −0.10 0.00 −10.30 12.53 −35.63 −0.51 −1.61 1.10 0.00

0.67 −0.04 0.38 0.00 −4.11 −23.86 12.48 0.00

ΔEr represents the energy of tetrahedral structure with respect to rhombic structure. In some columns, the values before/behind the slash refer to NSOC/SOC results.

a

small TM clusters.13−16 Our optimizations demonstrate that different elemental clusters resemble each other in structural shapes but do not in bond lengths and symmetries. Consequently, we have not plotted each elemental structure but described their bond lengths and symmetries in the following paragraphs. There are exceptions, Mo and Tc clusters, whose structures change significantly from the reference structures shown in Figure 1. Although we cannot guarantee our obtained lowest energy structures being the ground state structures, this uncertainty would not affect the physical pictures we are interested in, because our aim is the effects of SOC on structures and magnetic properties of different isomeric structures. As a starting point of discussion, we first focus on the 4d dimers. The relevant data have been listed in Table 1 and plotted in Figure 2, where binding energies reported by Li et al. have been shown for comparison.60 The binding energy, Eb = nEatom − Ecluster, is the cluster energy relative to the sum energy of free atoms. It can be seen from Figure 2 that the bond

lengths decrease from Y2 to Mo2 and then increase to Ag2, accompanying a similar variations of their binding energies. This evidence is attributed to the following reasons: A pair of TM atoms can interact to form bonding and antibonding sates that arise from the hybridizations of 4d and 5s states between two atoms. In early TM series from Y to Mo, there is gradually increasing 4d electrons participating in bonding, leading to the contractions of bond length, increasing binding energies, and leaving less 4d electrons in representing spin moments. The changes of bond lengths are in line with bonding characters that 4d electrons are localized around the nuclei while 5s electrons are extended far. At a certain point near half-filled 4d shell, antibonding orbitals are occupied and high promotion energies prevent the strong hybridizations, thus weakening the binding energies and causing the extensions of bond length. For trimer series under NSOC calculations (Table 2), Y3, Rh3, Pd3, and Ag3 are predicted to be equilateral triangles (D3h) with bond lengths of 3.157, 2.375, 2.521, 2.707 Å and spin multiplicities of 2, 4, 3, and 2, respectively. Isosceles triangles 11676



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Table 4. Same as in Table 2 but for TM5 Clustersa cluster Y5 Zr5 Nb5 Mo5 Tc5 Ru5 Rh5 Pd5 Ag5 a

sym

ΔEr

μ⃗cell S

μ⃗S (|μS|)

5a (C2v) 5b (C4v/C1) 5a (C2v) 5b (C2v) 5a (Cs) 5a (C2) 5a (C2v) 5b (C4v) 5a (D3h/C2v) 5b (C4v) 5a (D3h) 5b (C4v) 5a (D3h) 5b (C4v) 5a (D3h/C2v)

0.00/0.00 0.65/0.64 0.00/0.00 1.13/1.13 0.00/0.00 0.00/0.00 0.00/0.00 −0.38/−0.33 0.00/0.00 −1.38/−1.36 0.00/0.00 −0.35/−0.32 0.00/0.00 0.03/0.04 0.00/0.00

5/5.00 5/5.00 2/2.00 2/2.00 1/1.00 0/0.05 1/0.03 5/4.96 4/3.89 0/0.04 3/2.94 5/4.89 2/1.94 2/1.94 1/0.95

2.42 (2.42) 2.68 (2.68) 1.21 (1.21) 1.08 (1.08) 0.81 (0.81) 0.04 (0.04) 0.01 (3.82) 3.80 (3.80) 3.48 (3.48) 0.04 (0.07) 2.78 (2.78) 4.57 (4.58) 1.86 (1.86) 2/1.88 (1.88) 0.28 (0.28)

μ⃗L (|μL|) −0.15 −0.07 0.10 −0.04 −0.03 0.00 −0.09 0.13 0.08 0.00 0.41 0.37 0.45 0.34 0.02

(0.15) (0.07) (0.19) (0.04) (0.03) (0.00) (0.09) (0.14) (0.10) (0.01) (0.41) (0.39) (0.45) (0.34) (0.02)

μ⃗ J (easy axis↑↓) 2.27 2.61 1.31 1.04 0.78 0.04 0.08 3.93 3.56 0.04 3.19 4.94 2.31 2.22 0.30

R

ΔExy

ΔEyz

6.1% 2.8% 15.4% 4.0% 3.9% 0.0% 1.9% 3.8% 3.0% 14.7% 14.7% 8.6% 24.1% 17.8% 7.1%

0.02 0.10 0.00 −0.05 0.57 0.00 45.38 −0.04 −0.25 −0.02 0.00 0.18 −0.02 0.17 11.72

1.52 −20.63 1.99 0.05 0.00 0.00 −44.82 −0.51 −20.29 0.03 9.83 3.26 2.09 −0.37 −0.30

R

ΔExy

ΔEyz

(x↑↓) (z↑↓) (x↑↓)

2.1% 2.2% 3.5%

(z↑↓) (z↑↓) (z↑↓) (y↑↑) (z↑↓) (↑↓) (↑↓) (z↑↓) (x↑↑) (x↑↓) (y↑↑) (y↑↓) (y↑↓) (z↑↑) (y↑↑) (z↑↑) (z↑↑) (z↑↑) (y↑↑) (y↑↑)

4.2% 1.9% 4.0% 16.7% 1.0% 4.2% 5.0% 11.6% 5.4% 5.7% 2.3% 5.4% 5.3% 11.3% 10.1% 9.2% 35.5% 30.0% 25.0% 3.6% 0.0%

−0.17 0.00 −0.47 0.00 −0.84 −0.12 −0.01 0.70 −0.06

−0.02 0.13 0.36 0.00 0.87 0.16 0.07 −0.70 0.22

−0.02 −2.41 0.81 3.83 10.25 −0.14 4.20 −2.43 1.82 1.41 5.46 2.36 0.00

−5.55 0.12 −9.46 −2.82 −3.61 1.21 −11.49 4.27 0.00 1.74 −1.73 −5.52 −0.00

(z↑↓) (y↑↓) (z↑↑) (z↑↓) (z↑↓) (z↑↓) (y↑↓) (x↑↑) (x↑↑) (z↑↑) (z↑↑) (z↑↑) (z↑↑) (y↑↑) (y↑↑)

ΔEr represents the energy of structure with respect to the trigonal bipyramid structure.

Table 5. Same as in Table 2 but for TM6 Clustersa cluster Y6

Zr6

Nb6

Mo6 Tc6

Ru6

Rh6

Pd6

Ag6 a

str (sym)

ΔEr

μ⃗ cell S

6a (Oh/D4h) 6b (D3h/C2v) 6c (C2v) 6a (D4h) 6b (D3h) 6c (C2v) 6a (D2h) 6b (D3h) 6c (C2v) 6a (D3) 6c (C2) 6a (Oh/D4h) 6b (D3h) 6c (C2v) 6a (Oh/D3d) 6b (C2v) 6c (C2v) 6a (Oh) 6b (C2v) 6c (C2v) 6a (Oh/D4h) 6b (C2v/D3h) 6c (C2v) 6b (D3h) 6c (C2v)

0.00/0.00 1.91/1.85 0.29/0.25 0.00/0.00 2.47/2.46 0.09/0.09 0.00/0.00 2.12/2.12 −0.11/−0.11 0.00/0.00 0.83/0.00 0.00/0.00 0.87/0.71 1.72/1.82 0.00/0.00 −0.54/−0.52 1.37/1.39 0.00/0.00 0.02/0.00 0.36/0.34 0.00/0.00 0.70/0.75 0.26/0.31 0.00/0.00 −0.65/−0.66

4/4.00 8/8.00 4/4.00 0/0.00 4/4.00 2/2.00 2/2.00 0/0.06 2/2.00 2/1.94 2/1.99 2/1.44 2/1.96 2/1.99 8/7.94 4/3.74 2/1.98 6/5.98 6/6.06 10/9.87 2/1.90 2/1.98 2/1.92 2/1.98 0/0.05

μ⃗S (|μS|) 2.46 4.26 1.98 0.00 3.17 1.10 1.38 0.05 1.27 1.39 1.44 0.96 1.35 1.66 6.56 2.92 1.81 5.41 5.03 8.85 1.88 1.94 1.87 0.64 0.02

(2.46) (4.27) (1.98) (0.00) (3.17) (1.10) (1.71) (0.05) (1.27) (1.39) (1.44) (1.61) (1.35) (2.74) (6.58) (2.99) (3.35) (5.41) (5.03) (8.85) (1.88) (1.94) (1.87) (0.64) (0.02)

μ⃗L (|μL|) −0.04 −0.10 −0.06 0.00 −0.13 −0.02 −0.07 0.01 −0.01 −0.02 −0.11 −0.19 0.07 −0.01 0.05 −0.02 −0.07 0.61 0.46 0.81 0.66 0.49 0.41 0.02 0.00

(0.05) (0.10) (0.07) (0.00) (0.13) (0.02) (0.07) (0.01) (0.01) (0.07) (0.14) (0.19) (0.07) (0.16) (0.15) (0.16) (0.18) (0.61) (0.51) (0.82) (0.66) (0.58) (0.47) (0.02) (0.00)

μ⃗J (easy-axis↑↓) 2.42 4.16 1.92 0.00 3.04 1.08 1.31 0.06 1.26 1.37 1.33 0.77 1.42 1.65 6.61 2.90 1.74 6.02 5.49 9.66 2.54 2.43 2.28 0.66 0.02

ΔEr represents the energy of structure with respect to the octahedral structure.

clusters prefer tetrahedral structures. Nb4 (singlet), Rh4 (singlet), and Pd4 (triplet) clusters are regular Td tetrahedrons, whereas Y4 (triplet), Zr4 (quintet), Mo4 (singlet), Tc4 (singlet), and Ru4 (quintet) clusters are distorted C3v, D2d, D2d, D2, D2d tetrahedrons. Square pyramid structures are unstable for Nb5, Mo5, and Ag5 clusters, and as a result of relaxations, Nb5 and Mo5 are optimized to a trigonal bipyramid and Ag5 to a planar structure. For these clusters, the results corresponding to the square pyramid structures are not listed in Table 4. In going from left to right of the 4d elements, geometric structures evolve from trigonal bipyramid (Y5, Zr5, Nb5, and Mo5 clusters), to square pyramid (Tc5, Ru5, and Rh5 clusters), and finally to trigonal

(C2v) with waists/hemlines of 2.707/2.611, 2.419/2.285, 2.239/ 2.369, and 2.258/2.453 Å and spin multiplicities of 3, 2, 6, 7 are favored for Zr3, Nb3, Tc3, Ru3 trimers, respectively. The Mo3 cluster is a scalene triangle with bond lengths of 2.243/2.244/ 2.249 Å and multiplicity of 3. Our structural information and spin multiplicities of most 4d trimers are well consistent with available theoretical reports at similar calculational levels (e.g., ref 60 and further references cited therein). For TM4 clusters, rhombic and tetrahedral structures are examined (Figure 1: 4a; 4b), and the relevant results are collected in Table 3. The relative energy ΔEr in Table 3 reflects the energy of tetrahedron with respect to rhombus. Except for Tc4 and Ag4 clusters, which respectively favor square and rhombus structures, other 4d 11677



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Table 6. Same as in Table 2 but for TM7 Clusters cluster

sym

μ⃗ cell S

Y7 Zr7 Nb7 Mo7 Tc7 Ru7 Rh7 Pd7

C2v D5h C1 C2v Cs Cs/C2v D5h C2

1/1.00 0/0.00 1/1.00 2/1.97 3/2.99 4/3.90 13/12.86 2/1.89

μ⃗ S (|μS|) 0.50 0.00 0.77 1.10 2.36 3.42 11.20 1.82

(1.11) (0.00) (0.83) (1.11) (3.71) (4.16) (11.20) (1.82)

μ⃗L (|μL|) −0.01 0.00 −0.05 −0.04 0.07 0.11 1.07 0.46

(0.07) (0.00) (0.08) (0.08) (0.07) (0.22) (1.07) (0.49)

μ⃗J (axis↑↓) 0.49 0.00 0.72 1.06 2.43 3.53 12.27 2.28

R

ΔExy

ΔEyz

(x↑↓)

6.3%

−0.00

−0.11

(y↑↓) (y↑↓) (z↑↑) (y↑↑) (x↑↑) (y↑↑)

9.6% 7.2% 1.9% 5.3% 9.6% 27.0%

0.03 0.22

−0.01 −3.26 1.20 −19.40 −32.52 −2.36

−0.05

length of 2.49−2.51 Å, the orbital moment would increase from 0 to 1 μB for the axial magnetization.23 To reproduce their findings, we decrease an order of the force-convergence criterion in our optimizations, similar to their calculation settings, and we find that the bond length of the Pd2 dimer does not elongate (holding bond length of 2.49 Å in both axial and perpendicular magnetizations). In this situation, our orbital moments of 0.35/0.02 μB (perpendicular/ axial magnetization) and MAE of 2.4 meV are the same as Błoński’s reports,23 and also comparable to the values given by Fernández-Seivane22 and Fritsch.26 In view of this clarification, our calculations are in line with their results. We conjecture that different force-convergence criteria for the geometrical relaxations result in different orbital moments of axial magnetization, and thus different MAEs. The above analyses also indicate that orbital moments depend critically on bond lengths, and thus a rigid calculation is indispensable for MAE calculation. For TM3 clusters, three different magnetization directions are considered under SOC calculations, i.e., x⃗ direction (parallel to the hemline of the triangle), y ⃗ direction (along the 2-fold axis of symmetry), and z⃗ direction (perpendicular to the trigonal plane). There is no variation of bond lengths for early TM clusters from Y3 to Nb3, but there are changes of bond lengths in the order of 0.001 Å for late TM clusters from Mo3 to Ag3. The Pd3 cluster is “abnormal”, because relative pronounced changes have been always observed. For example, if magnetization is oriented along x⃗ direction, the hemline toward this magnetization direction is stretched by 0.017 Å but the other two bonds are contracted by 0.002 Å; if magnetization is oriented along the y ⃗ direction, the two waist bonds toward this direction are stretched by 0.012 Å but the hemline is contracted by 0.016 Å. The influences of SOC on all TM4 rhombic structures are rather weak (Tc4 is a square structure), only Mo4, Tc4, and Ag4 clusters undergoing 0.001 Å extensions of bond lengths. For tetrahedral structures of TM4, we initialized z⃗ magnetization along the 2-fold axis of symmetry yet x⃗ and y ⃗ magnetizations along two opposite bonds that are perpendicular to each other (Figure 1: 4b). When SOC is included (along z⃗ direction), only tetrahedral Pd4 suffers large structural distortion, corresponding to the symmetrical transformation from Td to D2d. In this case, four bonds toward the magnetization direction z⃗ are contracted by 0.006 Å, while two bonds perpendicular to the z⃗ direction are elongated by 0.017 Å. Similar to our findings in Pd clusters, Huda et al. have observed resembling elongations of bond lengths in Pt clusters.62 The changes of bond lengths in Pd and Pt clusters, distinctive from that in other TM clusters, may be due to more extensions of 4d orbits as a consequence of the strong relativistic effect. For TMn (n = 5−7) clusters, they always maintain their structural symmetries and present faint

bipyramid again (Pd5 cluster). Our geometrical evolution trend is in line with the compact-open-compact transitions observed in 4d TM13 clusters,61 because the trigonal bipyramid structure consists of the tetrahedral units and can form close-packed structures, whereas square pyramid structure is a cubic motif and can form open structures. The evolutional trend can be rationalized on the d-type bondings in TM clusters. In the case of TM6 clusters, we tried octahedron, trigonal prism, and capped trigonal bipyramid structures. The early TM clusters of Y6 (quintet), Zr6 (singlet), Mo6 (triplet), and Tc6 (triplet) are octahedrons with Oh, D4h, D3, and Oh symmetries, whereas the Nb6 (triplet) cluster is a capped trigonal bipyramid structure with C2v symmetry. For Rh6, a distorted trigonal prism with C2v symmetry and an octahedron with Oh symmetry are nearly energy degenerate (only 0.02 eV difference in energy). Within the accuracies of our calculations, both structures of Rh6 can be regarded as the lowest energy structures. The geometrical evolutions on TM6 clusters are consistent with the above conclusions on TM5 and TM13 clusters,61 where middle elemental clusters of Ru6 and Rh6 present open structural motifs. For the TM7 cluster, only pentagonal bipyramid structure is examined. Zr7 and Rh7 clusters prefer the highest symmetry of D5h, whereas Y7 and Mo7 clusters present the subsymmetry of C2v, and other clusters have low symmetries of C1 (or Cs). B. Spin−Orbit Coupling Effect on the Structure and Stability. The structures of NSOC calculations have been reoptimized with SOC calculations. Initial magnetization directions are along x,⃗ y,⃗ z⃗ orientations illustrated in Figure 1. In consideration of the fact that the experimental measurements on bond length, orbital, and spin moment are the lowest energy states, we only list the easy axis values of these clusters in Tables 1−6. The SOC calculations lead to bond lengths unchanged for Y2, Zr2, Nb2, and Ag2, but bond length changes of −0.002 Å for Mo2 and 0.001 Å for Tc2, Ru2, and Rh2. We obtain the resembling results with that of Błoński and Hafner for Ru2 and Rh2 dimers.23 The Pd2 dimer has a bond length of 2.481 Å in NSOC calculations but exhibits 0.003 or 0.025 Å elongations of bond lengths in SOC calculations. Our bond length of 2.484 Å and orbital moment of 0.34 μB for perpendicular magnetization approach Błoński’s values,22,23 but an elongated bond length of 2.506 Å and an increased orbital moment of 0.93 μB for axial magnetization are in contrast to almost quenching μ⃗L of 0.02 μB given by FernándezSeivane et al.22 and Błoński et al.23 who reported the equilibrium bond lengths of 2.53 and 2.49 Å, respectively. Furthermore, our MAE of 49.22 meV is significantly larger than the values for the Pd2 dimer: 2.0 meV reported by FernándezSeivane et al.;22 2.3 meV reported by Błoński et al.;23 5 meV reported by Fritsch et al.26 However, Błoński et al. have pointed out that if Pd2 dimer is slightly stretched from its equilibrium 11678



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from SOC calculations increases first, reaching the maximum at the Mo atom (6 μB), and then decreases monotonically down to the minimum at the Pd atom (0 μB), most of which approach to the values obtained from Hund’s first rule. Still, there is 1.3 μB magnetic difference for Zr atom and 0.15 μB for Rh atom. To understand what brings these magnetic differences, we calculate the different spin states on the basis of NSOC methods. It is demonstrated that the ground spin states of different 4d atoms are the same as Hund’s predictions, identifying our reliable methods in calculating magnetic moments. However, energy difference between triplet state and quintet state for Zr atom, as well as energy difference between doublet state and quartet state for Rh atom, is rather small (less than 0.2 eV), but energy differences between the ground state and their excited states for other atoms are large (more than 1 eV). In the presence of SOC interactions, sz is no longer a good quantum number, and thus two energetically quasidegenerate spin states can mixing to display a spin magnetic moment distributed between them. In addition to the spin magnetic moments μ⃗ S, Figure 3 also shows atomic orbital magnetic moments μ⃗ L. The DFT calculations and Hund’s second rule present a similar variational trends. Nonetheless, DFT values are too small by about a factor of 0.3. The quantitative differences are probably attributed to different magnetic determinations: (1) Hund’s second rule determines μ⃗ L by orbital polarization, whereas DFT method determines μ⃗ L by spin−orbit interaction in an intraatomic approximation. As a matter of experience, standard (quasi) LDA or GGA approximations that do not include the orbital-dependent exchange effects may lead to the underestimation of μ⃗ L,20,39 and orbital-polarization (OP) corrections are used to improve the values of bulk or surface phases.66 As for TM clusters, Nicolas et al. have shown that orbitaldependent interactions in Fe, Co, and Ni clusters can result in a qualitative similar trend with conventional DFT calculations,20 and Błoński et al. have pointed out that orbital-dependent interactions in TM2 dimers introduce a modest change.23 Here, we obtain the resembling trends as they have indicated. (2) Hund’s rule applies only to free atoms in spherical symmetry, which has exactly defined degeneracies. In our calculations, the unit cell model (free atom in a box-shaped unit cell) breaks the spherical symmetry and, furthermore, GGA tends to break it. Thus, symmetrical deteriorations would lead to an underestimated μ⃗L in our DFT calculations for free atoms. However, the requirement of spherical symmetry is automatically eliminated for clusters, so that this underestimation would not occur in cluster calculations. After exploring atomic magnetic moments, we further discuss cluster magnetic properties. In VASP magnetic calculations, the local spin magnetic moments μ⃗is and the local orbital magnetic moments μ⃗ il are determined by projecting the plane-wave components of the eigenstate onto spherical waves inside atomic spheres.24 Due to interspaces among these atomic spheres, the sum of local spin magnetic moments computed around each atom (μ⃗ S = ∑ni=1μ⃗ is) is smaller than the actual spin magnetic moment of a cluster in a unit cell (μ⃗ cell S ); e.g., the spin moment μ⃗ S slightly deviates from an integer number that a cluster should have in NSOC calculations. Even though investigations on Ru2, Rh2, and Pd2 have been performed by Błoński et al.,23 Fritsch et al.,26 and Strandberg et al.,63 a series of tests on complete 4d dimers are indispensable to clarify the general tendencies. By analyzing relevant data from Table 1, it is important to note four things: (1) Except for Mo2 and Ag2

changes of bond lengths (Tables 4−6; the symmetries before/ behind the slash refer to NSOC/SOC results). In few structures, the SOC effect would break structural symmetries, but the average changes of bond lengths are less than ±0.01 Å. For example, when SOC is imposed on octahedral Y6, Tc6, Ru6, and Pd6 clusters, their Oh symmetries transform to D4h, D4h, D3d, D4h symmetries, respectively (Table 5). Finally, referring to the SOC effect on relative stabilities of different isomeric structures, we compare their energy orders in NSOC calculations and look for possible reorders in SOC calculations. In Tables 3−5, relative energy ΔEr is the energy of the structure with respect to na structure, and the values before/behind the slash refer to NSOC/SOC results. From these tables, it is clear that energy orders obtained in SOC calculations are identical to that in NSOC calculations, indicating that SOC effect has negligible influences on relative stabilities of different structures. In addition, we calculate spin− − ESOC orbit coupling energy, Eso = (ENSOC b b )/n, which is defined as the differences of binding energies without and with SOC effect. As is shown in Tables 1−3, Eso are positive (negative) for early (late) TM clusters, which means that the SOC uniformly decreases (increases) stabilities of early (late) TM clusters. This can be interpreted that the SOC makes larger (smaller) contributions to atomic energies than to cluster energies for early (late) TM clusters, resulting in decreasing (increasing) binding energies in SOC calculations. Interestingly, each elemental cluster possesses stable values of Eso regardless of their sizes and structures. The Eso are around 0.12, 0.01, 0.01, −0.07, −0.15, −0.03, −0.02 eV/atom for Zr, Nb, Mo, Tc, Ru, Rh, Pd clusters, respectively. C. Magnetism. In Figure 3, spin moments μ⃗ S and orbital moments μ⃗ L of free atoms are plotted as a function of elements. Corresponding values from Hund’s rules are also shown for comparison. As we move from Y to Ag, atomic μ⃗S obtained

Figure 3. Magnetic moments of free 4d atoms from the SOC calculations and the Hund’s rule. The orbital magnetic moments from Hund’s second rule are obtained by assuming contributions only from the localized d-electron states, and the corresponding values are scaled by 0.3 for comparison. 11679



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Figure 4. Magnetic moments along the easy axes in SOC calculations.

triangular Pd3 cluster, ΔμS(xy) = 1.35 μB for the rhombic Pd4 cluster, and ΔμS(xz) = −0.25 μB for the trigonal prism Pd6 cluster. Previous works have shown that the spin moment anisotropies are also predominant in Pt clusters25 but not large in Fe, Co, and Ni clusters20,21,30,31 and Rh clusters.37,38 Our SOC calculations permit noncollinearities among the local magnetic moments μ⃗ is and μ⃗ il. Thus, one of the most interesting things is exploring whether there is collinearity between moments or not. For a cluster with noncollinear local spin moments, the angles between any two individual moments could be anything between 0° (parallel) and 180° (antiparallel). The degree of noncollinearity can be estimated by the difference between μ⃗ S and |μS|, where μ⃗S = ∑ni=1μ⃗ is is the vectorial sum of the local spin moments but |μS| = ∑ni=1|μ⃗is| is the absolute sum of the local spin moments. Similar definitions can hold for the orbital moments (difference between μ⃗L and |μL|) and total magnetic moments (difference between the vectorial sum μ⃗J = μ⃗ S + μ⃗ L and the absolute sum |μJ| = |μS| + |μL|). Aguilera-Granja et al. have performed a theoretical assessment of spin noncollinear arrangements in Pdn clusters (n = 3−7), but they finally confirmed that the local spin moments are indeed collinearity in the ground states.64 From Tables 1−6, μ⃗ S are commonly equal to |μS| within negligible differences, along with a resembling phenomenon referring to the orbital moments (μ⃗ L and |μL|), which indicates a collinearity among the local moments in most 4d TM clusters. However, there are slight nonlinearities in Nbn, Mon, and Tcn clusters that adopt distorted structures and have weak magnetic moments. An intriguing finding is that total spin moments μ⃗ S and total orbital moments μ⃗L are rigorous collinearity for each structure, even for structures possessing the noncollinear local magnetic moments. This is a consequence of spin−orbit interaction L⃗ ·S⃗, which favors the alignment of μ⃗ L along the magnetization of μ⃗ S. In addition, when these magnetic clusters are placed in an external magnetic field, their magnetic moments would be

dimers, which are nonmagnetic in NSOC calculations but present spin moments 0.04−0.06 μB in SOC calculations, other dimers suffer a weak reduction of spin moments 0.02−0.07 μB as SOC is applied. (2) For all dimers, spin moment differences between two magnetization orientations are entirely negligible, exhibiting weak spin moment anisotropy ΔμS. (3) The μ⃗ L of Zr, Ru, Rh, and Pd dimers are −0.29, 0.00, 1.81, and 0.93 μB for magnetization parallel to the dimer axis, and −0.04, 0.23, 0.50, and 0.34 μB for magnetization perpendicular to the dimer axis, exhibiting strong orbital moment anisotropy ΔμL. (4) The μ⃗ L couples antiferromagnetically with μ⃗S for dimers from Y2 to Mo2 whose 4d shells are less than half-filled, whereas it couples ferromagnetically for dimers from Tc2 to Pd2 whose 4d shells are more than half-filled, which is in line with Hund’s third rule. The above findings can be readily extended to following TMn (n = 3−7) cluster sizes (Tables 2−6). From these tables, we note that cluster spin magnetic moments μ⃗cell S are marginally affected by SOC interaction, exhibiting small deviations from integer numbers of NSOC calculations (less than 3%). Strong μ⃗ L in Zr, Tc, Ru, Rh, and Pd clusters, together with the reversed signs around the middle elemental series, are generally obtained. In addition, spin magnetic moments are very stable and independent of the magnetization directions (i.e., ΔμS < 10−2 μB), which are in contrast to large ΔμL found in late elemental clusters. For example, there are large ΔμL(yz) of −0.22/0.24/0.84/0.13 μB for Tc3/Ru3/Rh3/Pd3 clusters, and large ΔμL(yz) of 0.71/0.99/0.43 μB for rhombic Ru4/Rh4/Pd4 clusters (detailed information are not given as their abounding data). Because these observations always resemble what we have found in the dimers, briefly, we will not expound on them one after another. However, the above conclusions are not generally viable for all 4d TM clusters. The Pd clusters are often distinctive cases, in which the SOC reduces the spin moments, μ⃗ cell S = 0.24/1.48/0.11 μB for Pd3/Pd4/Pd7 clusters, and induces the spin moment anisotropies, ΔμS(xy) = 0.11 μB for the 11680



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aligned with the external field, either by aligning both the spin and orbital moments or by breaking the effect of SOC and aligning the spins separately (resulting in the noncollinearities between μ⃗ S and μ⃗L). In other words, if the alignment of the orbital moment along the external field is weaker than the alignment of the spin moment, spin and orbital moments would be decoupled and interact separately with the external field analogous to the Paschen-Back effect in atoms.65 In Figure 4, we observe that the signs of μ⃗L alternate from negative in early TM series to positive in late TM series, with small values at the beginning, middle, and end of the 4d period table, which matches Hund’s rule. Nonetheless, the μ⃗ L of Nb and Zr clusters should be negative according to this trend, but positive signs are actually detected in Nb4 and Zr5 clusters. More detailed information are provided by their local magnetic moments, and we find that the μ⃗ is and μ⃗ il always represent opposite signs. Concerning the size dependence on the μ⃗L, we observe the decrease of magnetic moments with the increase of cluster sizes. In Zrn clusters from n = 2 to 7, the average μ⃗ L drops from 0.15 μB to 0.05, 0.03, 0.02, 0.00, and 0.00 μB; in Rhn clusters from n = 2 to 4, average μ⃗ L drops from 0.91 μB to 0.31 and 0.00 μB; and in Pdn clusters from n = 2 to 7, average μ⃗ L drops from 0.17 μB to 0.31, 0.12, 0.09, 0.11, and 0.07 μB. The gradually increasing interactions among atoms lead to the enhanced hybridizations, thus quenching the μ⃗ L . The quenching behaviors, however, can be released by structural symmetries, because high symmetries usually lead to high degenerations near the Fermi level and allow a more effective mixing to enhance the μ⃗ L. For example, abnormally increasing μ⃗ L of 0.08/0.10/0.15 μB per atom for Rh5(D3h)/Rh6(Oh)/ Rh7(D5h) and 0.09/0.12 μB per atom for Pd5(D3h)/Pd6(D4h) nicely reflects the increasing rotational symmetries of their structures. For Rh clusters, Guirado-López et al. have theoretically reported μ⃗L = 0.10−0.18 μB/atom,17,37,38 and Sessi et al. have experimentally predicted that μ⃗ L changes unmonotonously with cluster sizes, exhibiting an maximum value of 0.15 μB/atom at sizes of n̅ = 20.27 It seems that our values (0.08−0.15 μB/atom) of Rhn clusters (n = 5−7) approach their reports. For Pdn clusters from n = 2−6, our μ⃗L are well in agrement with theoretical values 0.18/0.33/0.13/0.11/0.11 μB per atom given by Błoński and Hafner.24 Although the μ⃗L of Pd clusters have not been measured, they are expected to be the close values of isoelectronic Pt clusters. The Pdn (n = 5−7) clusters have values about 0.09−0.12 μB/atom that are comparable to the measurement of 0.10−0.13 μB/atom for the Pt13 cluster.12 The maximum μ⃗ L coincides with the maximum μ⃗S for Rh and Zr clusters, and in contrast, small μ⃗ L and μ⃗ S are observed for Nb and Mo clusters. With the same orbital angular momentum l = 3, Rh clusters always present large μ⃗L relative to Zr clusters due to different spin−orbit coupling strengths. It is an excellent indicator that μ⃗ L depends on the spin−orbit coupling strength and the orbital momentum. To obtain the ability of μ⃗S inducing μ⃗ L, orbital-to-spin ratios R = |μ⃗L|/|μ⃗S| are calculated and shown in Figure 5. From the top panel, the Zr, Rh, and Pd clusters exhibit large R. From the lower panel, the R of Rh and Pd clusters are around 10% and 30% at large sizes. Our values are smaller than related experimental measurements 30−60% for Rh clusters27 and 32% for Pt clusters.12 This may be attributed to following two reasons: (1) The depositing clusters instead of the free clusters have been used in experiments, and cluster− substrate interactions make the structures of depositing clusters different from that of the free clusters. (2) The effective spin

Figure 5. Orbital-to-spin ratio R of different clusters.

moments μ⃗eff S in experiments include a contribution from the intra-atomic magnetic dipole, which is different from the intrinsic moments μ⃗S.67 Still, one may conclude that the orbital contributions to total magnetic moments cannot be ignored in Ru, Rh, and Pd clusters. To test how sensitive the magnetic moments are to interatomic distances, we perform uniform compression and expansion on the octahedral Rh6 cluster. In Figure 6, μ⃗S exhibits

Figure 6. Magnetic moment and orbital-to-spin ratio vs bond length for octahedral Rh6 cluster.

three different values in three ranges and finally reaches a saturated value of 1.3 μB/atom at large bond lengths. A similar phenomenon in the icosahedral Rh13 cluster has been reported by Reddy.68 However, μ⃗ L increases with the gradual extensions of bond lengths, and the increasing behaviors even exist in the situations where μ⃗S is saturated. The extensions of bond lengths favor localizations of the 4d electrons, which tend to Hund’s rule and promote the μ⃗ L. D. Magnetic Anisotropy Energy. We proceed to another magnetic property of the MAE, ΔEγδ = Eγ − Eδ, which is 11681



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derived by taking the energy difference between the γ and δ magnetization orientations. According to the definition, a positive (negative) value means the easy axis along an orientation of δ (γ). Except for Zr2 (−1.49 meV), Mo2 (−0.05 meV), and Rh2 (−47.86 meV) dimers that show axial anisotropy energies, other dimers present perpendicular anisotropy energies. Our MAE of 35.92 (−47.86) meV for Ru2 (Rh2) is in agreement with the value of 36.5 (−47.3) meV reported by Błoński and Hafner,23 but an MAE of 49.22 meV for Pd2 is significantly larger than the value of 2 meV given by Fernández-Seivane22 and 2.3 meV given by Błoński.23,24 Moving within the periodic table, early TM dimers from Y2 to Mo2 (0.05−7.93 meV) usually exhibit MAEs an order smaller than late TM dimers from Tc2 to Pd2 (24.84−49.22 meV). The Ag2 dimer has an isotropic energy due to its zero orbital moment, which verifies the spin−orbit interactions for bringing the magnetic anisotropy. For TM3 trimers (Table 2), Nb3, Ru3, and Ag3 show an outof-plane easy axis (perpendicular to the trigonal plane), and other trimers exhibit an in-plane easy axis (along the 2-fold axis of symmetry). Similar to the observations in TM2 dimers, the magnitudes of the out-of-plane ΔEyz are less than 0.53 meV for the early series from Y3 to Mo3 but on the order of 10 meV for the late series from Tc3 to Ag3. In-plane ΔExy are still faint for early TM clusters but modest for late TM clusters with C2v symmetries. For Pd3 cluster, its out-of-plane ΔEyz = −2.92 meV is comparable to −2.55 meV given by Błoński and Hafner who used the VASP-PAW method,24 but significantly smaller than 25 meV given by Fernández-Seivane who used the local orbital pseudopotential code SIESTA.22 For the Rh3 cluster, on the basis of the tight-binding method including the spin−orbit interactions, Guirado-López et al. have reported in-plane ΔExy of 0.07 meV and an out-of-plane ΔEyz of −10.94 meV,38 which are comparable to our respective values of 0.03 and −22.4 meV. To our best knowledge, there is no ab initio work on other 4d TMn clusters (n ≥ 3), and a few attempts are limited to semiempirical approaches.37,38 For rhombic TM4 clusters (Table 3), only Y4 and Nb4 clusters align their easy axes perpendicular to the rhombus plane. The magnitudes of MAEs are around 0.5 meV for early TM series but increase by almost one order for late TM series. The values move from −10.30 meV for Ru4 to −35.63 meV for Rh4 and to −1.61 meV for Pd4. For tetrahedral TM4 clusters, their easy axes are along the 2-fold axis of symmetry (z⃗, Figure 1: 4b), and their MAEs are often very weak (less than 0.5 meV) except for the Ru4 cluster (12.53 meV). For the Pd4 cluster, Błoński et al. have reported an MAE of 1.1 meV and the easy axis (hard axis) perpendicular (parallel) to one tetrahedral edge,24 in excellent agreement with our results. For TM5 clusters in trigonal bipyramid structures (Table 4), Tc5, Ru5, and Ag5 display their easy axes in the equatorial trigonal plane, whereas other elemental clusters align their easy axes perpendicular to equatorial trigonal plane. Very similar to the findings in TM3 clusters, in-trigonal-plane ΔExy depends on the symmetry of equatorial triangle. Low symmetry C2v corresponds to large values of 45.38/11.72 meV for late TM clusters of Tc5/Ag5, but high symmetry D3h corresponds to small values of −0.25/0.00/−0.02 meV for late TM clusters of Ru5/Rh5/Pd5. As for out-of-trigonal-plane ΔEyz, late TM clusters Tc5/Ru5/Rh5/Pd5 show large values of −44.82/− 20.29/9.83/2.09 meV. Our MAEs of late TM clusters are at least one order larger than the values of corresponding 3d counterparts, where Fe, Co, and Ni clusters have MAEs of 0.5−

2.1 meV.20,21,30,31 For TM5 clusters in square pyramid structures, Y5 and Pd5 align their easy axes along the diagonal of the bottom square, Tc5 along one edge of the bottom square, and Zr5, Ru5, and Rh5 perpendicular to the bottom square. All square pyramid structures exhibit small MAEs. For octahedral TM6 clusters (Table 5), their easy axes along one edge of equatorial square for Y6, or perpendicular to equatorial square plane for Nb6, Tc6, Rh6, and Pd6, are detected. Three initial magnetization orientations are finally relaxed to the same orientations for Mo6 and Tc6. For Pd6, previous calculations have predicted MAE of 1.83 meV (the easy axis parallel to 4-fold axis of symmetry vs hard-axis parallel to one octahedral edge),24 which agrees closely with our ΔExz of 1.82 meV. In combination with our observations that trigonalbipyramid structures exhibit large out-of-trigonal-plane ΔEyz but square-pramid structures exhibit small out-of-square-plane ΔEyz, one may assume that trigonal plane is superior to square plane in exhibiting large out-of-plane MAE. This speculation can be supported in the calculation of trigonal-prism structure, where out-of-trigonal-plane ΔExz of −5.57/1.02/−7.30/3.15/− 3.16 meV are generally larger than out-of-square-plane ΔExy of −0.02/3.83/4.20/1.41/2.36 meV for late TM clusters from Tc6 to Ag6. Among pentagonal-bipyramid TM7 clusters (Table 6), all clusters have their easy axes in the pentagonal plane, except for the Tc7 cluster which orients its easy axis perpendicular to the pentagonal plane. The rotations of in-plane magnetizations to out-of-plane magnetizations need energy about 1.20/19.40/ 32.52/2.36 meV for Tc7/Ru7/Rh7/Pd7, thus resulting in large out-of-plane MAEs. To quantify the role of cluster size on MAE, we show in Figure 7 the size dependences of MAEs for Run, Rhn, and Pdn clusters with n = 2−7 atoms. The values denoted by the black (red) square are the results of na (nb) structures in Figure 1. In each structure, the lowest (highest) energy magnetization of three different magnetizations is regarded as the easy (hard)

Figure 7. MAEs from the DFT and the ξμSΔμL calculations plotted as a function of cluster size for Ru, Rh, and Pd clusters, where the fitting yields the spin−orbit coupling parameters are ξ = 30, 10, and 50 meV for Ru, Rh, and Pd clusters, respectively. The values denoted by the black (red) square are the results of na (nb) structures of Figure 1. The lines are guides to eye. 11682



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IV. CONCLUSION We systematically investigate the effects of SOC on structures and magnetic properties of small 4d clusters by using the firstprinciples calculations without (DFT-PW91) and with SOC (DFT-PW91+SOC). The results show that SOC does not alternate relative stabilities of different isomeric structures of a given cluster. As SOC is included in the calculation, most Pd clusters undergo modest extensions of bond lengths whereas other TM clusters hold their bond lengths unaffected. The SOC usually reduces μ⃗S by a few percent in all 4d clusters but induces a large μ⃗L in Ru, Rh, and Pd clusters. The signs of μ⃗ L are negative (positive) for early (late) TM clusters, which is in line with Hund’s third rule; the magnitudes of μ⃗ L are in proportion to the maximum orbital angular momentum of the constituent atoms, which is compatible with Hund’s second rule. It is revealed that the SOC constant ξ together with the maximum orbital momentum l plays the decisive roles in unquenching μ⃗ L in TM clusters. Except for Pd clusters whose spin and orbital moments are both anisotropic, other 4d clusters often show isotropic spin moments yet anisotropic orbital moments. To check whether MAE correlates to μ⃗ L, MAEs are compared on the basis of both DFT and perturbation calculations. The Ru, Rh, and Pd clusters have large MAEs and are good candidates for future applications. Our μ⃗ L and MAEs are in good agreement with previous calculations and experimental measurements, supporting our reliable results. From a methodological standpoint, however, our calculations could be improved by taking into account the intra-atomic Coulomb interactions (GGA+U) and the orbitalpolarization corrections (GGA+OP).

axis, and the energy difference between the easy and hard axes is the magnetic anisotropy energy ΔE irrespective of in-plane ΔExy or out-of-plane ΔEyz. On the basis of the second-order perturbation theory, together with certain assumptions that majority bands are completely filled and spin moments are isotropic, Bruno has shown that ΔE should be proportional to orbital moment anisotropy ΔμL and spin−orbital coupling constant ξ.28 The late TM clusters may be good candidates to assess this simple theory. Thus, the values from the model ΔE = ξμSΔμL are calculated and plotted for comparison. From Figure 7, it is clear that, if tetrahedral structures instead of rhombic structures have been used, the magnitudes of MAEs decrease monotonously with the increase of cluster size and nearly die down around the size n = 6. However, even for the smallest MAEs at large sizes (1.58/2.77 meV per atom for Ru6 /Ru7; 0.20/4.65 meV per atom for Rh6/Rh7; and 0.30/0.33 meV per atom for Pd6/Pd7), they are larger than the corresponding values of any known bulk materials. In addition, MAEs oscillate between positive and negative by using both self-consistent calculations and model calculations and show similar trends for Rh clusters but some deviations for Ru and Pd clusters. In terms of Bruno’s model, the sign of ΔE is determined by the orbital moment anisotropy ΔμL; i.e., the easy axis should point to the magnetization orientation with the largest μL. In Run (n = 3−5) and Pdn (n = 2−4) clusters, however, opposite signs between ΔE and ΔμL are actually found. We propose that anisotropic spin moment ΔμS may be the main reason that Bruno’s model does not apply to Pd clusters. Though there are some reversal signs for Ru and Pd clusters, the magnitudes of ΔE are generally proportional to the products of ΔμL, μ⃗S, and ξ. To some extent, the application of the perturbation theory is partial valid in TM clusters. Generally speaking, MAEs of late TM clusters such as Tc, Ru, Rh, and Pd clusters are on the order of 10 meV, which are good candidates for future applications. Long-term magnetic data storage requires that spontaneous magnetization reversals should occur significantly less often than once in ten years. This implies that total MAEs of each magnetic particle should exceed 40kBT,69 where kB is the Boltzmann constant and T is the temperature. Recently, it has been experimentally proved that the isolated Fe4 single-molecule magnets with large MAE of 2 meV can be used for storing information at a very low temperature 0.5 K.70 Here, huge MAEs of 36, 48, and 49 meV for Ru2, Rh2, and Pd2 dimers could guarantee 10-year stability of an information bit at the temperature of liquid nitrogen, T = 11, 14, and 14 K. Even for the largest cluster size studied here, Ru7 and Rh7 would provide the required stabilities at 6 and 10 K. Once these clusters are used as the magnetic storage bits, they can be written by simultaneous application of a moderate magnetic field and a strong electric field.71 However, a serious limitation of the direct applications of free clusters is that they will rotate freely in an external magnetic field such that the easy magnetic axis will be aligned with the field. The phenomenon will be prominent if the MAE is smaller than the rotational energy for a given axis, i.e., MAE ≤ 1/2kBT. Otherwise, strong coupling between the magnetic moment and an easy axis magnetization will hinder cluster rotation and will lead to lattice orientations along the preferred axis. On the basis of this constraint, a low limit of the MAE ≥ 0.43 meV per cluster can be estimated at T = 10 K.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: E-mail:[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (Project Nos. 10904125 and 91121013), the Chongqing’s Natural Science Foundation of China (Project Nos. CSTC-2008BB4253 and CSTC-2011BA6004), and the Fundamental Research Funds for the Central Universities (Project No. XDJK2012B008).

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REFERENCES

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Structural and Magnetic Properties of Small 4d Transition Metal

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