Theory of local dynamical magnetic susceptibilities from the Korringa-Kohn-Rostoker Green function method

Within the framework of time-dependent density functional theory combined with the Korringa-Kohn-Rostoker Green function formalism, we present a real-space methodology to investigate dynamical magnetic excitations from first principles. We set forth a scheme which enables one to deduce the correct effective Coulomb potential needed to preserve the spin-invariance signature in the dynamical susceptibilities, that is, the Goldstone mode. We use our approach to explore the spin dynamics of $3d$ adatoms and different dimers deposited on a Cu(001) with emphasis on their decay to particle-hole pairs.

Figure 1

(a) Comparison between magnetic moments (in ${\mu }_B$) of adatoms calculated by the full KKR-GF with those calculated from the proposed projection scheme discussed in the text and in Ref. 21. Values of $-U$ (eV/${\mu }_B$) are shown in panel (b) calculated from either from Eq. (47) or from Eq. (46), while in the inset (c) we plot the percentage error defined as the difference between $U_{\mathrm{DFT}}$ and $U_{\mathrm{sumrule}}$ divided by $U_{\mathrm{DFT}}$.

Figure 2

Imaginary part of the transverse dynamical magnetic susceptibility for a Mn adatom/Cu(001) surface. After applying different dc magnetic fields, resonances are obtained and shifted to higher frequencies by increasing the magnitude of the field. The corresponding Zeeman frequencies with $g=2$ for the fields chosen are represented by the black circles. Thus, the $g$ shift is negative for this example.

Figure 3

Different values of $U$ obtained with different schemes for Cr, Mn, Fe, and Co dimers deposited on a Cu(001) surface. See the text for a discussion of the various criteria for choosing $U$.

Figure 4

Local Im${\chi }^{11}$ is shown for the four dimers based on Cr, Mn, Fe, and Co adatoms. To calculate $\chi$ two possible schemes of evaluating are considered: in (a) using $U_{-}$ and in (b) using $U_{+}$. It turns out that $U_{+}$ corresponds to the value obtained from the sum rule [Eq. (47)] derived in the text while $U$ that calculated from a simple iterative scheme out of $U_{\mathrm{DFT}}$ would converge to the wrong $U$ when investigating Cr and Co dimers. The reason is that, for the latter elements, contrary to $U_{+}$, $U_{-}$ is closer to $U_{\mathrm{DFT}}$. The optical modes, estimated for Mn and Fe from a Heisenberg model, are represented as dashed lines.

Figure 5

Local Im$\chi$ for dimers with mixed adatoms are shown in (a) for FeCo dimer and in (b) for MnFe dimer. Equation (47) based on the sum rule derived in the text was used to define $U$. It is interesting to note the presence of resonances at positive frequencies expressing a ferromagnetic ground state for both dimers. Within each dimer, the peaks related to every adatom are not located at the same position since the $g$ shift depends on the nature of the adatom.

Figure 6

The response function Im$({\chi }^{11})$ and Im$({\chi }^{22})$ for two spins of unit length coupled by an exchange interaction of strength $J=1$. Here we mimic Fe and Mn by considering each spin coupled to a reservoir that provides a damping parameter ${\alpha }_{1,2}$ (1 for Mn and 2 for Fe) whose values are given in the inset.