Properties of anisotropic magnetic impurities on surfaces

Using numerical renormalization-group techniques, we study static and dynamic properties of a family of single-channel Kondo impurity models with axial magnetic anisotropy $D{S}_z^2$ terms; such models are appropriate to describe magnetic impurity atoms adsorbed on nonmagnetic surfaces, which may exhibit surface Kondo effect. We show that for positive anisotropy $D$ and for any spin $S$, the systems behave at low temperatures as regular Fermi liquids with fully compensated impurity spin. The approach to the stable fixed point depends on the value of the spin $S$ and on the ratio $D/T_K^{\left(0\right)}$, where $T_K^{\left(0\right)}$ is the Kondo temperature in the absence of the anisotropy. For $S=1$, the screening occurs in two stages if $D<T_K^{\left(0\right)}$; the second Kondo temperature is exponentially reduced in this case. More generally, there is an effective spin-1/2 Kondo effect for any integer $S$ if $D<T_K^{\left(0\right)}$ and for any half-integer $S$ if $D>T_K^{\left(0\right)}$. For negative anisotropy $D$, the system is a non-Fermi liquid with residual anisotropic exchange interaction. However, the presence of transverse magnetic anisotropy $E\left(S_x^2-S_y^2\right)$ restores Fermi-liquid behavior in real systems.

Figure 1

(Color online) Energy level diagrams for the anisotropic magnetic impurities with spin 1, 3/2, and 2 as a function of the absolute value of the anisotropy, $\left|D\right|$. Wide (blue) lines represent pairs of states, while narrow (red) lines represent nondegenerate states.

Figure 2

(Color online) Thermodynamic properties of the anisotropic Kondo model. We plot the impurity contribution to the magnetic susceptibility, ${\chi }_{\text{imp}}$, and the impurity contribution to the entropy, $S_{\text{imp}}$. Full (blue) lines correspond to positive anisotropy, $D>0$, dashed (multicolored) lines correspond to negative anisotropy, $D<0$, while black line is the isotropic case, $D=0$. The arrow indicates which direction the anisotropy is increasing. Note that the vertical axis in the upper panels is additionally rescaled as $1/S\left(S+1\right)$.

Figure 3

The Kondo temperature of the $XXZ$ anisotropic $S=1/2$ Kondo model as a function of the $J_{\perp }/J_z$ ratio.

Figure 4

(Color online) The cross-over temperature (identified with the spin-1/2 Kondo temperature $T_K$ for $D<T_K^{\left(0\right)}$) in integer-spin anisotropic Kondo models for the hard-axis case.

Figure 5

(Color online) Thermodynamic properties of the $S=1$ Kondo model with $XXZ$ exchange anisotropy.

Figure 6

(Color online) Impurity contribution to the entropy of the anisotropic Kondo model with $D{S}_z^2+E\left(S_x^2-S_y^2\right)$ terms with constant ratio (a) $E/\left|D\right|=0.2$ and (b) $E/\left|D\right|=0.01$. We plot $D={10}^{-6},{10}^{-5},\dots ,{10}^{-2}$; the arrow indicates which direction the anisotropy is increasing.

Figure 7

(Color online) The cross-over temperature (identified with the spin-1/2 Kondo temperature $T_K$ for $D<T_K^{\left(0\right)}$) in the $S=2$ anisotropic Kondo model with transverse anisotropy.

Figure 8

(Color online) Curie constant as a function of the $\left|D\right|/T_K^{\left(0\right)}$ ratio for $S=1,3/2,2$ Kondo models with easy-axis anisotropy, $D<0$.

Figure 9

(Color online) Energy levels at iteration $N=80$ $\left(T\sim {10}^{-16}W\right)$ as a function of the anisotropy $D$ in the spin-1 Kondo model with easy-axis anisotropy, $D<0$.

Figure 10

(Color online) Dynamic properties of the anisotropic Kondo model for (a) positive $D$ and (b) negative $D$. We plot the spectral function of the $T$ matrix (“conductance”) and the longitudinal and transversal dynamical susceptibility ${\mathcal{S}}_{\text{zz}}\left(\omega \right)$ and ${\mathcal{S}}_{\text{xx}}\left(\omega \right)$. The curves correspond to $\left|D\right|={10}^{-2},\dots ,{10}^{-6}$. Note that for $S=2$ and $D={10}^{-6}$ the susceptibility curves do not diverge; they start decreasing at an energy scale outside the displayed range.

Figure 11

(Color online) Total, elastic, and inelastic-scattering cross sections for the $S=1$ Kondo model with no anisotropy, positive anisotropy, and negative anisotropy.

Figure 12

(Color online) Dynamic properties of the anisotropic Kondo model with the $E\left(S_x^2-S_y^2\right)$ term. $E=0.2\left|D\right|$. The curve styles are the same as in Fig. 2.