Revised theory of the magnetic surface anisotropy of impurities in metallic mesoscopic samples

In several experiments, the magnitude of the contribution of magnetic impurities to the Kondo resistivity shows size dependence in mesoscopic samples. It was suggested ten years ago that magnetic surface anisotropy can be responsible for the size dependence in cases where there is strong spin-orbit interaction in the metallic host. The anisotropy energy has the form $\Delta E=K_d{\left(\mathbf{n}\mathbf{S}\right)}^2$, where $\mathbf{n}$ is the vector perpendicular to the plane surface, $\mathbf{S}$ is the spin of the magnetic impurity, and $K_d>0$ is inversely proportional to distance $d$ measured from the surface. It has been realized that in the tedious calculation, an unjustified approximation was applied for the hybridizations of the host atom orbitals with the conduction electrons, which depend on the position of the host atoms. That is, the momenta of the electrons were replaced by the Fermi momentum $k_F$. That is reinvestigated by considering the $k$-dependence, which leads to singular energy integrals. Contrary to the previous result, $K_d$ is oscillating like $\sin \left(2k_{F}d\right)$ and the distance dependence is similar to $1∕d^3$ in the asymptotic region. As the anisotropy is oscillating, for integer spin the ground state is either a singlet or a doublet depending on distance $d$, but in the case of the doublet there is no direct electron induced transition between those two states at zero temperature. Furthermore, for half-integer $(S>1∕2)$ spin it is always a doublet with direct transition only in half of the cases.

Figure 1

The magnetic impurity at a distance $d$ from the surface in a metallic host with homogeneously dispersed spin-orbit scatterers labeled by $n$.

Figure 2

The self-energy diagram for the impurity spin. The double line represents the spin, while the single one the conduction electrons. The solid circles stand for the exchange interaction and the crosses ×, labeled by $n$ for the effective spin-orbit interaction on the orbital of the host atom at ${\mathbf{R}}_n$.

Figure 3

The level splitting due to the surface anisotropy for (a) integer (e.g., $S=2$) and (b) half-integer (e.g., $S=5∕2$) spins. It is also indicated whether the electrons can be scattered by the degenerate ground states or not.