Universal distribution of magnetic anisotropy of impurities in ordered and disordered nanograins

We examine the distribution of the magnetic anisotropy experienced by a magnetic impurity embedded in a metallic nanograin. As an example of a generic magnetic impurity with a partially filled $d$ shell, we study the case of $d^1$ impurities embedded into ordered and disordered Au nanograins, described in terms of a realistic band structure. Confinement of the electrons induces a magnetic anisotropy that is large, and can be characterized by five real parameters, coupling to the quadrupolar moments of the spin. In ordered (spherical) nanograins, these parameters exhibit symmetrical structures and reflect the symmetry of the underlying lattice, while for disordered grains they are randomly distributed and, for stronger disorder, their distribution is found to be characterized by random matrix theory. As a result, the probability of having small magnetic anisotropies $K_L$ is suppressed below a characteristic scale ${\Delta }_E$, which we predict to scale with the number of atoms $N$ as ${\Delta }_E\sim 1/N^{3/2}$. This gives rise to anomalies in the specific heat and the susceptibility at temperatures $T\sim {\Delta }_E$ and produces distinct structures in the magnetic excitation spectrum of the clusters that should be possible to detect experimentally.

Figure 1

Left: Anisotropy parameters in the $E$ plane for an ordered grain of 225 atoms (87 core atoms). Right: Anisotropies in the $E$ plane in case of 100 disordered Au nanograins. An ordered cluster is created by 225 atoms that fill completely the first six shells of the fcc structure, and with 11 extra atoms, randomly placed onto the next shell. The triangular structure of the anisotropy parameter distribution can still be observed.

Figure 2

Radial distribution of the magnetic anisotropy parameters (dots) in the $E$ plane in the case of $N_S=50$ samples with $N=225+25$ atoms. The continuous line presents the predictions of the Gaussian orthogonal ensemble. Inset: Distribution of $(K_1,K_2)$ in the $E$ plane for these 50 nanograins. At this level of disorder the triangular structure is almost entirely lost.

Figure 3

Orientational and disorder averaged excitation spectrum computed for an ensemble of 1000 randomly chosen and randomly oriented disordered nanograins $\left(N=225+40\right)$. The shape of the signal reflects the distribution of magnetic anisotropies while the small peak corresponds to transitions between the lowest Kramers doublets, split by the magnetic field. Inset: Spectra of selected grains, see the main text and Appendix pp5.

Figure 4

The $spd$ density of states (DOS) components of an ordered nanograin $\left(N=225\right)$ normalized to one atom. The calculated Fermi energy is ${\varepsilon }_F=7.4$ eV.

Figure 5

First- and second-order self-energy diagrams of the impurity spin. The dashed and continuous lines denote the propagators of the spin and the conduction electrons, respectively.

Figure 6

Calculated MA constant $K_L$ values for both ordered and disordered nanoballs. The ordered cluster has $N=225$ atoms, and the solid lines correspond to the values of $K_L$ calculated for the ordered case. The number of core sites is $N_c=87$, i.e., the MA is for atoms located in the center site and on the first six core shells—see the different colors. The number of all sites in disordered nanoballs is $N=265$, i.e., 40 extra atoms are put to the next three outermost shells. The number of sample is $N_S=50$.

Figure 7

The magnetic anisotropy parameters in the $T_2$ space in the case of an ordered nanograin $\left(N=225,N_c=87,N_S=1\right)$. The color coding is the same as in Fig. 6.

Figure 8

Radial distribution of the magnetic anisotropy parameters (dots) in $T_2$ space in the case of $N_S=50$ samples with $N=225+25$ atoms $\left(N_c=87\right)$. The continuous line presents the prediction of the Gaussian unitary ensemble. The obtained MA energy scale: ${\Delta }_T=0.13\text{meV}$.

Figure 9

The level spacing distribution for $N_S=50$ disordered nanograins with $N=225+25$ atoms in the $d$ band. The average level spacing is $\Delta =3.9$ meV. The data are well fitted by GSE.

Figure 10

Specific heat (in units of $k_B)$ of nanograins with magnetic impurities, obtained from the numerical integration in Eq. (F4) as a function of the normalized temperature, $\alpha =k_{B}T/{\Delta }_E$. This curve is universal for all disordered nanograins, where the MA constants follow a GOE-like universal distribution, Eq. (F3). The grain-specific information is hidden in the parameter ${\Delta }_E$.