KKRsusc program: Theory

This page should give you a brief introduction to the theoretical background of the KKRsusc program. To find more details on the theory and a detailed description on the method follow these links:

 * S. Lounis et al., Phys. Rev. Lett. 105, 187205 (2010)  * S. Lounis et al., Phys. Rev. B 83, 035109 (2011)  * B. Schweflinghaus et al., Phys. Rev. B 89, 235439 (2014)

Projected Green function

The Kohn-Sham (KS) Green function (GF) is the resolvent of the corresponding Hamiltonian, G_{\text{KS}}(E) = (E - \mathcal{H}_{\text{KS}})\^{-1}. In the Korringa-Kohn-Rostoker Green function (KKR-GF) method, space is partitioned into nonoverlapping regions surrounding the atoms, labeled i. These regions are taken as spherical in the atomic sphere approximation (ASA), and the KS potential is also assumed to be spherical around each atom, V\^{\text{KS}}_i(r), with r = |\vec{r}\,| and \hat{r} = \vec{r}/r. Then the KS GF is expressed in terms of energy-dependent scattering solutions for each atomic potential, R_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r}) and H_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r}), which are products of radial functions and (real) spherical harmonics, for each spin \sigma = \;\uparrow,\,\downarrow and angular momentum L = (\ell,m). R_{i\ell}\^\sigma(r;E) is regular at the center of the ASA sphere, and H_{i\ell}\^\sigma(r;E) diverges there.

The Korringa-Kohn-Rostoker Green function is given by

\begin{aligned}
G_{ij}^\sigma(\vec{r},\vec{r}';E) &= \sum_{LL'}Y_L(\hat{r}) \big(\sqrt{E}\,R_{i\ell}^\sigma(r_{\text{min}};E)\,H_{i\ell}^\sigma(r_{\text{max}};E)\delta_{ij}\delta_{LL'} \\
&\quad + R_{i\ell}^\sigma(r;E)\,G^{\sigma,\text{str}}_{iL,jL'}(E)\,R_{j\ell'}^\sigma(r';E)\big) Y_{L'}(\hat{r}')
\end{aligned}

where r_{\text{min}} = \min(r,r') and r_{\text{max}} = \max(r,r'), and G^{\sigma,\text{str}}_{iL,jL'}(E) is the structural GF, describing backscattering effects.

Near the Fermi energy (E_\text{F}) one may approximate R_{i\ell}\^\sigma(r;E) \approx R_{i\ell}\^\sigma(r;E_\text{F}).

Susceptibility

Kohn-Sham susceptibility

The transverse magnetic Kohn-Sham susceptibility is given by

\begin{aligned}
\chi^{\sigma\bar{\sigma}}_{0,ij}(\vec{r},\vec{r}';\omega) &= -\frac{1}{\pi}\int^{E_{\text{F}}}\text{d}E \\
&\quad \Big(G_{ij}^{\bar\sigma}(\vec{r},\vec{r}';E+\omega+\mathrm{i}0)\,\text{Im}\,G_{ji}^{\sigma}(\vec{r}',\vec{r};E) \\
&\quad\quad + \text{Im}\,G_{ij}^{\bar\sigma}(\vec{r},\vec{r}';E)\,G_{ji}^{\sigma}(\vec{r}',\vec{r};E-\omega-\mathrm{i}0)\Big)
\end{aligned}

Here, \chi\^{\uparrow\downarrow} and \chi\^{\downarrow\uparrow} correspond to \chi\^{+-} and \chi\^{-+}, respectively.

Dyson-like equation

Electron Self-energy

Dyson equation

Let us turn our attention to the Dyson equation for the GF, including the self-energy describing the coupling to the magnetic excitations:

\begin{aligned}
G_{ij}^\sigma(\vec{r},\vec{r}';E) &= G_{0,ij}^\sigma(\vec{r},\vec{r}';E) + \sum_{pq}\int\text{d}\vec{r}_1\int\text{d}\vec{r}_2 \\
&\quad \times G_{0,ip}^\sigma(\vec{r},\vec{r}_1;E)\,\Sigma_{pq}^\sigma(\vec{r}_1,\vec{r}_2;E)\,G_{qj}^\sigma(\vec{r}_2,\vec{r}';E)
\end{aligned}