  Recent experimental advances in scanning tunneling microscopy make the
measurement of the conductance spectra of isolated and magnetically coupled
atoms on nonmagnetic substrates possible. Notably these spectra are
characterized by a competition between the Kondo effect and spin-flip inelastic
electron tunneling. In particular they include Kondo resonances and a
logarithmic enhancement of the conductance at voltages corresponding to
magnetic excitations, two features that cannot be captured by second order
perturbation theory in the electron-spin coupling. We have now derived a third
order analytic expression for the electron-spin self-energy, which can be
readily used in combination with the non-equilibrium Green's function scheme
for electron transport at finite bias. We demonstrate that our method is
capable of quantitative description the competition between Kondo resonances
and spin-flip inelastic electron tunneling at a computational cost
significantly lower than that of other approaches. The examples of Co and Fe on
CuN are discussed in detail.
