The spin and orbital moment of Fen (n = 2–20) clusters

Abstract

Complementary to the recent experimental finding that the orbital magnetic moment is strongly quenched in small Fe clusters [M. Niemeyer, K. Hirsch, V. Zamudio-Bayer, A. Langenberg, M. Vogel, M. Kossick, C. Ebrecht, K. Egashira, A. Terasaki, T. Möller, B. v. Issendorff, and J. T. Lau, Phys. Rev. Lett. 108, 057201 (2012)], we provide the theoretical understanding of the spin and orbital moments as well as the electronic properties of neutral and cation Fen clusters (n = 2-20) by taking into account the effects of strong electronic correlation, spin-orbit coupling, and noncollinearity of inter-atomic magnetization. The generalized gradient approximation (GGA)+U method is used and its effluence on the magnetic moment is emphasized. We find that without inclusion of the Coulomb interaction U, the spin (orbital) moments have an average value between 2.69 and 3.50 μB/atom (0.04 and 0.08 μB/atom). With inclusion of U, the magnetic value is between 2.75 and 3.80 μB/atom (0.10 and 0.30 μB/atom), which provide an excellent agreement with the experimental measurements. Our results confirm that the spin moments are less quenched, while the orbital moments are strongly quenched in small Fe clusters. Both GGA and GGA+U functionals always yield collinear magnetic ground-state solutions for the fully relaxed Fe structures. Geometrical evolution, as a function of cluster size, illustrates that the icosahedral morphology competes with the hexagonal-antiprism morphology for large Fe clusters. In addition, the calculated trends of ionization potentials, electron affinities, fragment energies, and polarizabilities generally agree with respective experimental observations.

Spin and orbital magnetic moments of size-selected iron, cobalt, and nickel clusters

“…Therefore, both possible configurations, i.e., antiferromagnetic 74,76 or ferromagnetic with nearly quenched spin of the central atom 75,90 , are compatible with our experimental results within the error bars of the spin magnetic moment. Indirect evidence for antiferromagnetic spin coupling in Fe + 13 might be obtained from the reduced number n h = 2.92 of unoccupied 3d states that is predicted for Fe [74][75][76]90,116,117,119 . This reduction in bond length corresponds to a spherical volume compression of V /V 0 = 0.91 − 0.94 and might be correlated to the magnetic properties via the bulk phase diagram of iron, where a transition from the ferromagnetic body-centered cubic (bcc) α phase to the nonmagnetic hexagonal close packed (hcp) ǫ phase of iron occurs around 12 − 16 GPa at room temperature [56][57][58][59][60][61] .…”

“…In this process, 3d states are predominantly probed because of the large transition matrix element 70,71 , which leads to x-ray absorption cross sections that are larger by one order of magnitude than transitions into higher ns and nd (n > 3) states. This excitation scheme allows to probe the magnetic moments of iron, cobalt, and nickel clusters, which are carried by the 3d electrons while magnetic contributions from 4s and 4p states are negligible for clusters with more than 10 atoms 72 as well as in the bulk [73][74][75][76] . The 2p core hole that is created in the x-ray absorption process relaxes via Auger decay cascades.…”

“…Table II lists the average experimental 3d spin and orbital magnetic moments per atom and per cluster. Even though XMCD spectroscopy at the L 2,3 edges of 3d transition elements is only sensitive to the 3d magnetic moments, these values will be close to the total magnetic moments because 4s and 4p electron contributions to the spin magnetic moment are expected to be negligible [72][73][74][75][76] .…”

“…In contrast to iron, the situation is different for cobalt and nickel clusters. Here, our data give spin magnetic moments of Co find either an antiferromagnetically coupled spin magnetic moment of the central atom 74,76 or a ferromagnetic alignment of the 3d spins at all atoms but a nearly quenched spin magnetic moment of the central atom 75,90 . In spite of these differences, all these studies predict the same average magnetic moment of µ S = 2.69 µ B per atom [74][75][76]90 for Fe [74][75][76]90 determine identical spin magnetic moments as can be seen from Fig.…”

“…Here, our data give spin magnetic moments of Co find either an antiferromagnetically coupled spin magnetic moment of the central atom 74,76 or a ferromagnetic alignment of the 3d spins at all atoms but a nearly quenched spin magnetic moment of the central atom 75,90 . In spite of these differences, all these studies predict the same average magnetic moment of µ S = 2.69 µ B per atom [74][75][76]90 for Fe [74][75][76]90 determine identical spin magnetic moments as can be seen from Fig. 7, although the geometric and electronic ground state structures of Wu et al 75 and Gutsev et al 90 differ from those found by Alvarado-Leyva et al 74 and Yuan et al 76 These distinctions in the calculated geometric and electronic structures lead to different explanations about the origin of the reduced spin moment of Fe + 13 : While Wu et al conclude a symmetry-driven spin quenching in a highly-symmetric icosahedral structure, Alvarado-Leyva et al report a T h distorted icosahedral structure of Fe + 13 in combination with an electronic shell closure that leads to antiferromagnetic spin coupling 74 .…”