Magnetic structure of free iron clusters compared to iron crystal surfaces

Electronic and magnetic properties of free Fe clusters of 9 to 89 atoms are investigated theoretically within an ab initio fully relativistic framework and compared to results of crystal surfaces. It is found that the local spin magnetic moments ${\mu }_{\text{spin}}$ and the orbital magnetic moments ${\mu }_{\mathrm{orb}}$ are enhanced for atoms close to the surface of a Fe cluster. The corresponding Friedel-like oscillations in the depth profiles of ${\mu }_{\text{spin}}$ and ${\mu }_{\mathrm{orb}}$ are more pronounced for clusters than for crystal surfaces. The ${\mu }_{\text{spin}}$ in clusters and at crystal surfaces turned out to depend linearly on the effective coordination number $N_{\mathrm{eff}}$. This empirical ${\mu }_{\text{spin}}\text{−}{N}_{\mathrm{eff}}$ inter-relationship is able to account for some features of the experimentally measured dependence of the magnetic moment of free Fe clusters on the cluster size. The spin-polarized density of states (DOS’s) for atoms in clusters is characterized by sharp atomiclike peaks and substantially differs from the DOS in the bulk. The width of the local valence band gets more narrow if one is moving from the center of the cluster to its surface. The DOS averaged over all atoms in a cluster converges to the bulk behavior more quickly with cluster size than the DOS of the central atoms of these clusters.

Figure 1

Dependence of ${\mu }_{\text{spin}}$ in free iron clusters on the distance of the atoms from the center of the cluster. Each cluster is identified by the number of constituting atoms.

Figure 2

${\mu }_{\mathrm{orb}}$ in free iron clusters as a function of the distance of the atoms from the center (point marks). The magnetization $M$ is oriented parallel to the [001], [110], or [111] crystallographic directions, as indicated in the legend. The solid lines denote ${\mu }_{\mathrm{orb}}$ averaged over all atoms in a coordination shell. Each cluster is identified by the number of constituting atoms.

Figure 3

Comparison of ${\mu }_{\text{spin}}$ profiles as calculated by different methods. Solid lines correspond to this work, coarsely dotted line to Yang et al. (Ref. 5), dashed line to Pastor et al. (Ref. 1), dash-dotted line to Vega et al. (Ref. 2), and densely dotted line to Franco et al. (Ref. 3). Each cluster is identified by the number of its atoms.

Figure 4

Layer dependence of ${\mu }_{\text{spin}}$ below the (001) Fe crystal surface (left panel), (110) crystal surface (middle panel), and (111) crystal surface (right panel). The surface structures are schematically depicted in the insets. Together with our results we display results of Ohnishi et al. (Ref. 11), Fu and Freeman (Ref. 12), Hjortstam et al. (Ref. 13). Niklasson et al. (Ref. 15), Eriksson et al. (Ref. 19), Spišák and Hafner (Ref. 14), Izquierdo et al. (Ref. 16), and Rodríguez-López et al. (Ref. 18), as indicated in the legend. The calculations of Ohnishi et al. (Ref. 11) and of Hjortstam et al. (Ref. 13) yield very similar results and span the same range of layers, so that the corresponding lines in the left panel are hard to distinguish from one another.

Figure 5

Layer-dependence of ${\mu }_{\mathrm{orb}}$ below iron crystal surfaces; left panels stand for the (001) surface, right panels for the (110) surface. The magnetization $\mathit{M}$ is either perpendicular (lower panels) or parallel (upper panels) to the surface, its orientation is schematically depicted in the insets. Moments calculated by Rodríguez-López et al. (Ref. 18) (scaled by a factor of 0.57) and in the case of the (001) surface with perpendicular magnetization also by Eriksson et al. (Ref. 19) and by Hjortstam et al. (Ref. 13) (without OP) are shown for comparison. The results of Eriksson et al. (Ref. 19) nearly concide with our results, hence the corresponding line is hard to distinguish in the lower left panel.

Figure 6

${\mu }_{\text{spin}}$ in an 89-atom cluster explored in the [001] (left panel), [110] (middle panel), and [111] (right panel) crystallographic directions compared with ${\mu }_{\text{spin}}$ at and below corresponding crystal surfaces.

Figure 7

${\mu }_{\mathrm{orb}}$ in an 89-atom cluster explored in the [001] (left panel), [110] (middle panel), and [111] (right panel) crystallographic directions compared with ${\mu }_{\mathrm{orb}}$ at and below corresponding crystal surfaces. The magnetization is either perpendicular to the layers (solid lines) or parallel to them (dashed and dotted lines). For the case of an in-plane magnetization, the direction of the magnetic field $\mathit{M}$ is also indicated by the inset drawings. In each panel, the upper graph corresponds to the cluster while the lower graph to the semi-infinite crystal.

Figure 8

${\mu }_{\text{spin}}$ and ${\mu }_{\mathrm{orb}}$ of atoms in clusters and at crystal surfaces as a function of the effective coordination number $N_{\mathrm{eff}}$. Assignment of marks to different clusters and crystal surfaces is indicated by the legend in the upper panel. The straight line in the upper panel is a fit to the data in the region $N_{\mathrm{eff}}<8.5$, with the 9- and 15-atom clusters omitted.

Figure 9

${\mu }_{\text{spin}}$ and ${\mu }_{\mathrm{orb}}$ of atoms in clusters and at crystal surfaces as a function of the valence electronic charge $N_{\mathrm{val}}$. Assignment of marks to different clusters and crystal surfaces is indicated by the legend in the upper panel. The straight line in the upper panel is a fit to the data in the region $N_{\mathrm{val}}<8.6$.

Figure 10

${\mu }_{\text{spin}}$ of atoms in structurally relaxed fcc-like and bcc-like clusters calculated by Postnikov et al. (Ref. 8) displayed as a function of $N_{\mathrm{eff}}$. Assignment of marks to different clusters is indicated by the legend. The straight line is a fit to the data for fcc-like clusters.

Figure 11

Cluster magnetic moment per atom ${\overline{\mu }}_{\mathrm{clu}}$ as a function of the cluster size, as calculated via a model ${\mu }_{\text{spin}}\left(N_{\mathrm{eff}}\right)$ dependence based on Eq. (2). The experiment of Billas et al. (Ref. 20) is shown for comparison.

Figure 12

Spin-polarized DOS at the central atom of a free Fe cluster (thin solid lines) and for an atom in bulk Fe (thick dashed lines). The left panel displays results for a 9-atom cluster, the middle panel for a 27-atom cluster and the right panel for an 89-atom cluster.

Figure 13

Spin-polarized DOS averaged over all atoms of a free Fe cluster of 9, 27, and 89 atoms (thin solid lines) compared with the DOS of bulk Fe (thick dashed lines).

Figure 14

Spin-polarized DOS at individual atoms of an 89-atom cluster (thin solid lines) compared with DOS averaged over all atoms of that cluster (thick dashed lines). The left panel displays the DOS for an atom of the second coordination shell, the middle panel for an atom of the fifth coordination shell and the right panel for an atom of the seventh (i.e., outermost) coordination shell.