Kondo temperature when the Fermi level is near a step in the conduction density of states

The (111) surface of Cu, Ag, and Au is characterized by a band of surface Shockley states with a constant density of states beginning slightly below the Fermi energy. These states as well as bulk states hybridize with magnetic impurities which can be placed above the surface. We calculate the characteristic low-temperature energy scale, the Kondo temperature $T_K$ of the impurity Anderson model, as the bottom of the conduction band $D_s$ crosses the Fermi energy ${\epsilon }_F$. We find simple power laws $T_K\simeq {|D_s-{\epsilon }_F|}^{\eta }$, where $\eta$ depends on the sign of $D_s-{\epsilon }_F$, the ratio between surface and bulk hybridizations with the impurity ${\mathrm{\Delta }}_s/{\mathrm{\Delta }}_b$, and the ratio between on-site and Coulomb energy $E_d/U$ in the model.

Figure 1

Kondo temperature as a function of the bottom of the conduction band. Squares and circles: NCA. Triangles: SBMFA. Dashed (dot) line: fit to the NCA (SBMFA) results. The inset shows the NCA results in log-log scale. Parameters are $E_d=-4,{\mathrm{\Delta }}_b=1,{\mathrm{\Delta }}_s=0.1$.

Figure 2

Same as Fig. 1 for ${\mathrm{\Delta }}_b=0.5$ and ${\mathrm{\Delta }}_s=0.2$.

Figure 3

Same as Fig. 1 for ${\mathrm{\Delta }}_b=0.5$ and ${\mathrm{\Delta }}_s=0.5$.

Figure 4

Same as Fig. 1 for $E_d+U=4,{\mathrm{\Delta }}_b=0.5$, and ${\mathrm{\Delta }}_s=0.2$.

Figure 5

Spectral density of states within NCA for $E_d=-4,{\mathrm{\Delta }}_b={\mathrm{\Delta }}_s=0.5$ and two values of $D_s$, full line: $D_s=-0.5$, dashed line: $D_s=0.5$. The insets show details of the peaks near $\omega =0$.

Figure 6

Same as Fig. 5 for $E_d+U=4$.

Figure 7

Kondo temperature as a function of the bottom of the conduction band calculated with NRG for several values of $\mathrm{\Lambda }$. The inset shows the results in log-log scale. Parameters as in Fig. 3. $T_K^0$ is the Kondo temperature for $D_s=-D$.