Structural and Magnetic Properties of Small 4d Transition Metal Clusters: Role of Spin–Orbit Coupling

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 first-principles 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.

Multiferroic Rhodium Clusters

“…Rhodium stands out because it is located at the center of the periodic table with an unfilled 4d shell, which produces rich physical and chemical properties. Rhodium clusters have been intensively studied by using theoretical methods [1][2][3][4][5][6] to understand their catalytic properties and the emergence of the electronic and magnetic properties [7][8][9][10]. Small rhodium clusters were found to possess magnetic moments [11,12] on the order of one Bohr magneton (1μ B ) per atom, making this the only 4d element with magnetic order at the nanoscale and with Mn [13] the only known elements that are magnetic at nanoscale but not in the bulk.…”

Monoxides of small terbium clusters: A density functional theory investigation

“…In addition, the bonding preference of different atoms gives another efficient way to construct good candidates for the global minima or to reduce the configuration space. In our previous works, this method has been used to treat transition metal (TM) clusters [45][46][47][48][49][50][51] and lanthanide clusters, 18,20,28 for which the lowest energy structures we obtained were positively confirmed by the subsequent works using the unbiased global optimizations. 13,[52][53][54][55][56][57] Following above approaches, we consider all possible configurations for small sizes (n = 2-7), but construct a large number of initial configurations for large sizes (n = [8][9][10][11][12][13][14] primarily at the additive and substitutive patterns from the low-lying structures of Tb n±1 , e.g., three different binding modes are intensively examined for the adsorption of oxygen atom: atop, bridge, and hollow sites.…”

“…As SOC is included, the SOC couples the magnetization to the lattice and determines the direction of magnetization, and both spin and orbital moments will be anisotropic. Previously, Guirado-López's work on Ni clusters 77 as well as our recent works on Pt clusters, 78 Fe clusters, 49 and 4d clusters 51 have demonstrated that the anisotropy of the local moments depends strongly on the local environment. Since anisotropic moment changes sign as a function of atom sites, the cancellation of contributions from different atomic sites often results in a relatively small anisotropy of the average per atom.…”

“…77 Generally, spin moment anisotropy is very small and can be neglected, but orbital moment anisotropy is often less than 10% of total orbital moment. 49,51,77,78 The SOC coupling of Tb atom is an order stronger than that of TM atom, which would result in somewhat larger magnetic anisotropy in TbO clusters.…”

Orbit and spin resolved magnetic properties of size selected [ConRh]+ and [ConAu]+ nanoalloy clusters

“…4,5 Not only does this enhancement of the magnetic moment occur for the 3d materials that are ferromagnetic in the bulk (Fe, Co, Ni). Also there has been a certain amount of theoretical work for CoRh, some of which included spin-orbit coupling (SOC) 10,11,[13][14][15] but most without. This is true for example in 4d rhodium clusters.…”