Magnetic Moments of Chromium-Doped Gold Clusters: The Anderson Impurity Model in Finite Systems

The magnetic moment of a single impurity atom in a finite free electron gas is studied in a combined x-ray magnetic circular dichroism spectroscopy, charge transfer multiplet calculation, and density functional theory study of size-selected free chromium-doped gold clusters. The observed size dependence of the local magnetic moment can be understood as a transition from a local moment to a mixed valence regime. This shows that the Anderson impurity model essentially describes finite systems even though the discrete density of states introduces a significant deviation from a bulk metal, and the free electron gas is only formed by less than 10 electrons. Electronic shell closure in the gold host minimizes the interaction of localized impurity states with the confined free electron gas and preserves the magnetic moment of $5{\mu }_{\mathrm{B}}$ fully in ${\mathrm{CrAu}}_2^{+}$ and almost fully in ${\mathrm{CrAu}}_6^{+}$. Even for open-shell species, large local moments are observed that scale with the energy gap of the gold cluster. This indicates that an energy gap in the free electron gas stabilizes the local magnetic moment of the impurity atom.

Figure 1

Left: Linear x-ray absorption spectra (solid lines) of ${\mathrm{CrAu}}_n^{+}$ clusters, $n=2–7$, normalized to the $L_3$ maximum intensity and overlaid with theoretical spectra (dashed lines) of ${\mathrm{CrAu}}_n^{+}$, $n=2,5–7$, from charge transfer multiplet calculations next to relaxed ground state structures [16] (dark atom: chromium; light atoms: gold). Center: Corresponding XMCD spectra (solid lines) of ${\mathrm{CrAu}}_n^{+}$, normalized to the number of unoccupied $3d$ states and theoretical spectra (dashed lines) from charge transfer multiplet calculations, giving the weight of configurations that contribute to the ground state. Right: Measured orbital magnetization at the experimental conditions of $T=(12\pm 4)\mathrm{K}$ and ${\mu }_{0}H=5\mathrm{T}$.

Figure 2

(a) Calculated local chromium spin magnetic moments ${\mu }_S$ of ${\mathrm{CrAu}}_n^{+}$ from a DFT population analysis (black upwards triangles) and from charge transfer multiplet (CTM) calculations fitted to the experimental data (blue downwards triangles); (b) Cr-Au interaction energy $E_{\text{int}}$; (c) Anderson regime criterion $|E_d|/2\mathrm{\Gamma }$; (d) criterion for full spin polarization $E_d>2\mathrm{\Gamma }-(U_0+4J)$, using $J=0.75\pm 0.25\mathrm{eV}$; (e) energy gap $\mathrm{\Delta }E$ of the ${\mathrm{Au}}_n$ host in the geometry of ${\mathrm{CrAu}}_n^{+}$.