Inelastic electron scattering in aggregates of transition metal atoms on metal surfaces

Inelastic spin excitations, as observed with a scanning tunneling microscope for Co/Co and Fe/Fe dimers on a Cu2N/Cu(100) surface, have been analyzed theoretically in this paper. In our approach, we use an extended ionic Hamiltonian for the magnetic atom that takes into account first, the role played by the first Hund rule in the atomic states, and second, the cotunneling processes associated with the atomic excitations and the tunneling conductance. This Hamiltonian is solved using the equation of motion method that yields the appropriate Green's functions allowing us to calculate the differential conductance, the inelastic atomic excitations, and possible Kondo resonances. We also analyze an ideal dimer with spin ½ in each atom and discuss the differences and similarities this model has with the Co-Co case.

Figure 1

The complete calculation [Eq. (17)] for the 1/2-1/2 dimer. The thin lines correspond to the conductance neglecting the crossed interaction term (${\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AB}=0$) and the thick lines to the conductance calculated by considering ${\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AB}=0.2{\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AA}$, for several values of the exchange interaction between atoms, $J$. (a) Ferromagnetic interaction. (b) Antiferromagnetic interaction (${\varepsilon }_d=1\mathrm{eV}$ and ${\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AA}=0.05\mathrm{eV}$).

Figure 2

The 1/2-1/2 dimer case taking the crossed interaction term ${\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AB}=0$. Complete version, Eq. (17), (thick lines) compared with the approximated version, Eq. (19), (thin lines); ${\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AA}=0.1\mathrm{eV}$. (a) Ferromagnetic interaction. (b) Antiferromagnetic interaction.

Figure 3

The conductance spectra for a simplified Co dimer in the case of an antiferromagnetic interaction for the $J$ values indicated in the figure and the same energy level and width parameters of Fig. 2 (${\mathrm{\Gamma }}_{\left(\mathrm{surface}\right)}^{AA}=0.1\mathrm{eV}$ and ${\varepsilon }_d=1\mathrm{eV}$).

Figure 4

Conductance through a Co atom as a function of the applied voltage for zero magnetic field. Isolated Co atom: the calculated results (solid line) and the experimental results (crossed circles). The theoretical results considering $J=0.22\mathrm{meV}$ for Co-Co dimer (0,2) of Ref. [13]: black solid line corresponds to the calculation including only the diagonal Green's function components $G_{A_{ij}^1}\left({\widehat{A}}_{ij}^{\left(1\right)}\right)$; gray solid line corresponds to the calculation including diagonals and nondiagonals $G_{A_{ij}^1}\left({\widehat{A}}_{pq}^{\left(1\right)}\right)$ but within a reduced space of (10,4,4,1) states. The crossed circles are the experimental results [13]. The inset is a zoom of the calculated conductance behavior close to zero energy.

Figure 5

The same as in Fig. 4 for a magnetic field applied in the $x$ direction. The symbols are the experimental results of Ref. [13].

Figure 6

Theory (lines) vs experiment (symbols) for the Fe-Fe dimer in the geometry (2,0) of Ref. [14]. In absence of a magnetic field (black curves) and in the presence of a magnetic field in the $z$ direction $B_z=2\mathrm{T}$ (gray curves).