Role of Surface Anisotropy for Magnetic Impurities in Electron Dephasing and Energy Relaxation and their Size Effect

Recently the electron dephasing and energy relaxation due to different magnetic impurities have been extensively investigated experimentally in thin wires, and in this Letter these quantities are theoretically studied. It was shown earlier that a magnetic impurity in a metallic host with strong spin-orbit interaction experiences a surface anisotropy of the form $H=K_d(\mathbf{n}\mathbf{S}{)}^2$ which causes size effects for impurities with integer spin. Here we show that the dephasing and the energy relaxation are influenced by the surface anisotropy in very different ways for integer spin having a singlet ground state. That must result also in strong size effects and may resolve the puzzle between the concentrations estimated from the two kinds of experiments.

Figure 1

The magnetic impurity at distance $d$ from the surface.

Figure 2

The diagrams used for calculating (a) the kernel and (b) the Korringa lifetime of the impurity spin. The solid lines denote the conduction electrons, the dotted lines the impurity spin, and the blob is the Kondo coupling.

Figure 3

The calculated distribution function at $x=0.485$ for different strength of the anisotropy constant $K_d$. The other parameters are $U=0.1\mathrm{m}\mathrm{V}$, $c=8\mathrm{p}\mathrm{p}\mathrm{m}$, ${\rho }_0\tilde{J}=0.11$, and ${\tau }_D=2.8\mathrm{n}\mathrm{s}$.

Figure 4

Fit on the experimental data of Cu wires at $x=0.485$ [18] by the calculated distribution function for different $K_d$ and $c$ pairs. The other parameters are $U=0.1\mathrm{m}\mathrm{V}$, ${\rho }_0\tilde{J}=0.11$ and ${\tau }_D=2.8\mathrm{n}\mathrm{s}$.