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title: Theory title: Theory
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# KKRsusc program: Theory # KKRsusc program: Theory
This page should give you a brief introduction to the theoretical 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:
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)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.187205) \* [S. Lounis et al., Phys. Rev. Lett. 105, 187205 (2010)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.187205) \* [S. Lounis et al., Phys. Rev. B 83, 035109 (2011)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.035109) \* [B. Schweflinghaus et al., Phys. Rev. B 89, 235439 (2014)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235439)
* [S. Lounis et al., Phys. Rev. B 83, 035109 (2011)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.035109)
* [B. Schweflinghaus et al., Phys. Rev. B 89, 235439 (2014)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235439)
## Projected Green function ## Projected Green function
The Kohn-Sham (KS) Green function (GF) is the resolvent of the The Kohn-Sham (KS) Green function (GF) is the resolvent of the corresponding Hamiltonian, <span dir="">G\_{\\text{KS}}(E) = (E - \\mathcal{H}\_{\\text{KS}})^{-1}</span>. In the Korringa-Kohn-Rostoker Green function (KKR-GF) method, space is partitioned into nonoverlapping regions surrounding the atoms, labeled <span dir="">i</span>. 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, <span dir="">V\\^{\\text{KS}}\_i(r)</span>, with <span dir="">r = |\\vec{r}\\,|</span> and <span dir="">\\hat{r} = \\vec{r}/r</span>. Then the KS GF is expressed in terms of energy-dependent scattering solutions for each atomic potential, <span dir="">R\_{i\\ell}\\^\\sigma(r;E)\\,Y\_{L}(\\hat{r})</span> and <span dir="">H\_{i\\ell}\\^\\sigma(r;E)\\,Y\_{L}(\\hat{r})</span>, which are products of radial functions and (real) spherical harmonics, for each spin <span dir="">\\sigma = \\;\\uparrow,\\,\\downarrow</span> and angular momentum <span dir="">L = (\\ell,m)</span>. <span dir="">R\_{i\\ell}\\^\\sigma(r;E)</span> is regular at the center of the ASA sphere, and <span dir="">H\_{i\\ell}\\^\\sigma(r;E)</span> diverges there.
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 The Korringa-Kohn-Rostoker Green function is given by
...@@ -41,11 +20,24 @@ G_{ij}^\sigma(\vec{r},\vec{r}';E) &= \sum_{LL'}Y_L(\hat{r}) \big(\sqrt{E}\,R_{i\ ...@@ -41,11 +20,24 @@ G_{ij}^\sigma(\vec{r},\vec{r}';E) &= \sum_{LL'}Y_L(\hat{r}) \big(\sqrt{E}\,R_{i\
\end{aligned} \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. where <span dir="">r\_{\\text{min}} = \\min(r,r')</span> and <span dir="">r\_{\\text{max}} = \\max(r,r')</span>, and <span dir="">G^{\\sigma,\\text{str}}\_{iL,jL'}(E)</span> is the structural GF, describing backscattering effects.
Near the Fermi energy (<span dir="">E\_\\text{F}</span>) one may approximate <span dir="">R\_{i\\ell}\\^\\sigma(r;E) \\approx R\_{i\\ell}\\^\\sigma(r;E\_\\text{F})</span>.
We now notice that the quantities present in the Green's function (The regular <span dir="">R(\\mathbf{r};E)</span>, and irregular <span dir="">H(\\mathbf{r};E)</span> component of the radial wave function) depend on both the energy and the position. This would make many calculations unfeasable, thus a separation of the energetic and spatial components is used.
Using a new basis set <span dir="">\\ket{\\phi\_{ilb}}</span>, where <span dir="">i</span> accounts for the lattice site, <span dir="">l</span> for the orbital, and <span dir="">b</span> for the energy. One can disentangle the spatial dependence using
```math
G^{proj}_{iLb,jL^'b^'}(E) = \int_{0}^{r_{MT}}drr^{2}\int_{0}^{r_{MT}}dr^{'}r^{'2} Y_{L}(\hat{r})\phi_{ilb}(r)G_{iL;jL^{'}}(r,r^{'};E)Y_{L^{'}}(\hat{r}^{'})\phi_{jl^{'}b^{'}}(r^{'})
```
The basis function used for the projection is constructed as
```math
Near the Fermi energy ($E_\text{F}$) one may approximate \phi_{ilb} = \frac{R_{il}(R;E_{b}}{\sqrt{\int_{0}^{r_{MT}}dr^{'}r^{'2}(R_{il}(r^{'};E_{b})^{2}}
$R_{i\ell}\^\sigma(r;E) \approx ```
R_{i\ell}\^\sigma(r;E_\text{F})$.
## Susceptibility ## Susceptibility
...@@ -61,10 +53,7 @@ The transverse magnetic Kohn-Sham susceptibility is given by ...@@ -61,10 +53,7 @@ The transverse magnetic Kohn-Sham susceptibility is given by
\end{aligned} \end{aligned}
``` ```
Here, <span dir="">\\chi\_{\\uparrow\\downarrow}</span> and <span dir="">\\chi\_{\\downarrow\\uparrow}</span> correspond to <span dir="">\\chi\_{+-}</span> and <span dir="">\\chi\_{-+}</span>, respectively.
Here, $\chi_{\uparrow\downarrow}$ and
$\chi_{\downarrow\uparrow}$ correspond to $\chi_{+-}$ and
$\chi_{-+}$, respectively.
### Dyson-like equation ### Dyson-like equation
...@@ -72,14 +61,11 @@ $\chi_{-+}$, respectively. ...@@ -72,14 +61,11 @@ $\chi_{-+}$, respectively.
#### Dyson equation #### Dyson equation
Let us turn our attention to the Dyson equation for the GF, including Let us turn our attention to the Dyson equation for the GF, including the self-energy describing the coupling to the magnetic excitations:
the self-energy describing the coupling to the magnetic excitations:
```math ```math
\begin{aligned} \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 \\ 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) &\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} \end{aligned}
``` ```
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