Co adatoms on Cu surfaces: Ballistic conductance and Kondo temperature

The Kondo zero-bias anomaly of Co adatoms probed by scanning tunneling microscopy is known to depend on the height of the tip above the surface, and this dependence is different on different low index Cu surfaces. On the (100) surface, the Kondo temperature first decreases then increases as the tip approaches the adatom, while on the (111) surface it is virtually unaffected. These trends are captured by combined density functional theory and numerical renormalization-group calculations. The adatoms are found to be described by an $S=1$ Anderson model on both surfaces, and ab initio calculations help identify the symmetry of the active $d$ orbitals. We correctly reproduce the Fano line shape of the zero-bias anomaly for Co/Cu(100) in the tunneling regime but not in the contact regime, where it is probably dependent on the details of the tip and contact geometry. The line shape for Co/Cu(111) is presumably affected by the presence of surface states, which are not included in our method. We also discuss the role of symmetry, which is preserved in our model scattering geometry but most likely broken in experimental conditions.

Figure 1

Unit cell of the scattering region for Co/Cu(100) for $d_{\text{tip-sur}}=7.60\text{Å}$ (a) in the $yz$ plane and (b) in the $xy$ plane. (c) Brillouin zone for the supercell in the $xy$ plane.

Figure 2

Co/Cu(100): (a) Spin-polarized density of states at Co $3d$ and $4s$ atomic orbitals constructed from scattering states at ${\mathbf{k}}_{xy}=\overline{B}=\frac{\pi }{L}(\frac{1}{2},\frac{1}{2})$ (positive DOS, spin up; negative DOS, spin down); (b) $t^{\uparrow }$ and $t^{\downarrow }$ components of the DFT transmission, transmission eigenvalues $t_i^{\uparrow }$ and $t_i^{\downarrow }$ (${\sum }_i{t}_i^{\uparrow }=t^{\uparrow }, {\sum }_i{t}_i^{\downarrow }=t^{\downarrow }$), and the shot-noise Fano factor. Energies are with respect to the Fermi energy.

Figure 3

Unit cell of the scattering region for Co/Cu(111) for $d_{\text{tip-sur}}=7.81\text{Å}$ (a) in the $yz$ plane and (b) in the $xy$ plane. (c) Brillouin zone for the supercell in the $xy$ plane.

Figure 4

Co/Cu(111): (a) Spin-polarized density of states at Co $3d$ and $4s$ atomic orbitals constructed from scattering states at $\overline{K}=\frac{\pi }{L}(\frac{2}{3},0)$ (positive DOS means spin up; negative DOS means spin down); (b) $t^{\uparrow }$ and $t^{\downarrow }$ components of the DFT transmission, transmission eigenvalues $t_i^{\uparrow }$ and $t_i^{\downarrow }$ (eigenvalues close to 1 correspond to the A1 channel while two lower eigenvalues are from the $E$ channels), and the shot-noise Fano factor. Energies are with respect to the Fermi energy.

Figure 5

(a) DFT conductance $g$ and (b) Kondo temperature $T_K,z^2$ on the Cu(100) surface compared with experimental data from Refs. [10] (Choi et al.) and [9] (Néel et al.). (c) DFT conductance $g$ and (d) Kondo temperature on the Cu(111) surface compared with experimental data from Ref. [11] (Vitali et al.). All data are shown as a function of the distance between the tip and the surface $d_{\text{tip-sur}}$: since only relative distances are known experimentally, we have rigidly shifted experimental data by 5 Å for Ref. [10], 9 Å for Ref. [9], and 5.5 Å for Ref. [11] for best fit with theory. Approximate error bars are shown for the theoretical Kondo temperatures: they take into account all the incertitudes of our method when fixing the parameters of the Anderson Hamiltonian Eqs. (5) and (24) from the DFT calculations; once the parameters are known, the Kondo temperature can be extracted with negligible uncertainty from the NRG run. It must be stressed that most of the incertitudes are systematic, so they affect all the data in the same direction, which turns out, when compared to experiments, to be an overestimation of the Kondo temperatures.