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

Spin and orbital magnetic moments of cationic iron, cobalt, and nickel clusters have been determined from x-ray magnetic circular dichroism spectroscopy. In the size regime of $n=10–15$ atoms, iron and cobalt clusters show fully spin-polarized unoccupied $3d$ states with maximized spin magnetic moments of $1{\mu }_B$ per hole because of completely filled $3d$ majority-spin bands. The notable exception is ${\mathrm{Fe}}_{13}^{+}$ where an unusually low average spin magnetic moment of $0.73\pm 0.12$ ${\mu }_B$ per unoccupied $3d$ state is detected, an effect which is neither observed for ${\mathrm{Co}}_{13}^{+}$ nor ${\mathrm{Ni}}_{13}^{+}$. This distinct behavior can be linked to the existence and accessibility of antiferromagnetic, paramagnetic, or nonmagnetic phases in the respective bulk phase diagrams of iron, cobalt, and nickel. Compared to the experimental data, available density functional theory calculations generally seem to underestimate the spin magnetic moments significantly. In all clusters investigated, the orbital magnetic moment is quenched to $5\%–25\%$ of the atomic value by the reduced symmetry of the crystal field. The magnetic anisotropy energy in this size range is well below $65\mu \mathrm{eV}$ per atom.

Figure 1

$L_{2,3}$-edge x-ray absorption and XMCD spectra (solid lines) of ${\mathrm{Co}}_{13}^{+}$ and ${\mathrm{Ni}}_{13}^{+}$; integrated $2p\to 3d$ x-ray absorption cross sections (dashed lines); and the two step functions (dotted lines) that approximate direct $2p$ photoionization and $2p\to ns,nd(n>3)$ contributions.

Figure 2

$L_{2,3}$-edge x-ray absorption of ${\mathrm{Fe}}_{10}^{+}$ at different applied magnetic fields. The identical line shape for $0–5$ T shows the absence of natural linear dichroism or spatial alignment. Integrated spectra (inset) are constant within an error of $\pm 6$%.

Figure 3

Experimentally determined variation in the number $n_h$ of unoccupied $3d$ states for ${\mathrm{Fe}}_n^{+}$, ${\mathrm{Co}}_n^{+}$, and ${\mathrm{Ni}}_n^{+}$ clusters. Within the error bars, $n_h$ is constant for clusters of a given element.

Figure 4

X-ray absorption and XMCD spectra taken at ${\mu }_{0}H=5$ T (solid lines) and low-resolution peak intensity scans (triangular markers) of the XMCD and x-ray absorption signal at ${\mu }_{0}H=0.5–5.0$ T, from which magnetization curves are derived.

Figure 5

Upper panel: Total ($m_J$, filled circles), spin ($m_S$, light gray circles), and orbital ($m_L$, open circles) magnetization of ${\mathrm{Co}}_{10}^{+}$ as a function of the applied magnetic field $H$. A Brillouin fit to the total magnetization $m_J$ (solid line) yields the cluster ion temperature $T_{\text{ion}}=14.6\pm 0.3$ K and the total magnetic moment ${\mu }_J=3.15\pm 0.09$ ${\mu }_B$ per atom. The same Brillouin fit is scaled to $m_S$ and $m_L$ as a guide to the eye. Lower panel: The ratio of $m_L/m_S={\mu }_L/{\mu }_S=0.25\pm 0.01$ (filled triangles and dashed line) is constant within the error bars and is used to decompose ${\mu }_J$ into ${\mu }_S$ and ${\mu }_L$. The error bars increase for smaller applied magnetic fields because of decreasing magnetization.

Figure 6

Total (${\mu }_J$, open symbols), spin (${\mu }_S$, filled symbols), and orbital (${\mu }_L$, open symbols with dot) magnetic moments per atom and per unoccupied $3d$ state of iron, cobalt, and nickel clusters. With the exception of ${\mathrm{Fe}}_{13}^{+}$, all iron and cobalt clusters carry $\approx 1$ ${\mu }_B$ per $3d$ hole.

Figure 7

Calculated spin magnetic moments (open symbols) of iron [73, 74, 75, 89, 104, 114, 118], cobalt [104, 140, 141, 142, 143, 144], and nickel [71, 104, 145, 146] clusters compared to spin magnetic moments from this XMCD study and Ref. [53] (filled symbols). The $(+)$ symbol denotes results for cationic clusters. Systematic errors are not shown.

Figure 8

Spin (open symbols) and orbital magnetic moments (filled symbols) scaled to atomic values of the Fe ($3d^6$), Co ($3d^7$), and Ni ($3d^8$) configurations. Dashed lines (Fe, Co: 83%; Ni: 65%) mark the ratio of unoccupied $3d$ states in atoms and clusters.

Figure 9

Upper panel: Comparison of spin, orbital, and total magnetic moments of free cobalt clusters obtained in the present XMCD study (black symbols) to those from Ref. [52] (light gray symbols). Lower panel: Good agreement of the ratio of spin-to-orbital magnetization of both studies.