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\begin_body

\begin_layout Section
Optical functions
\begin_inset CommandInset label
LatexCommand label
name "sec:Optical-functions"

\end_inset


\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
optics
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
General
\end_layout

\begin_layout Standard
The immediate function is the optical dielectric function (without momentum
 transfer)
\begin_inset Formula 
\[
\boxed{\varepsilon\left(\omega\right)=\varepsilon^{\mathrm{intra-band}}\left(\omega\right)+\varepsilon^{\mathrm{inter-band}}\left(\omega\right)}
\]

\end_inset

where the intra-band expression must be approximated via a Drude model and
 the interband part is the expensive thing the optics-module is all about.
 
\begin_inset Formula $\varepsilon$
\end_inset

 is connected to the optical conductivity via
\begin_inset Formula 
\[
\sigma\left(\omega\right)=-i\frac{1}{4\pi}\omega\left(\varepsilon\left(\omega\right)-\boldsymbol{1}\right)
\]

\end_inset

which makes 
\begin_inset Formula $\sigma$
\end_inset

 a measure of the imaginary part of 
\begin_inset Formula $\varepsilon$
\end_inset

.
\begin_inset Formula 
\[
\boxed{\mathrm{Re}\sigma\left(\omega\right)=\frac{1}{4\pi}\omega\mathrm{Im}\varepsilon\left(\omega\right)}
\]

\end_inset


\begin_inset Formula 
\[
\mathrm{Im}\sigma\left(\omega\right)=\frac{1}{4\pi}\omega\left(\boldsymbol{1}-\mathrm{Re}\varepsilon\left(\omega\right)\right)
\]

\end_inset


\end_layout

\begin_layout Standard
The Loss function is defined as 
\begin_inset Formula 
\[
\boxed{L_{\alpha\beta}=-\mathrm{Im}\left(\frac{1}{\varepsilon\left(\omega\right)}\right)_{\alpha\beta}}
\]

\end_inset


\end_layout

\begin_layout Subsection
Intra-band (Drude)
\end_layout

\begin_layout Standard
The intraband contribution is approximated by
\begin_inset Formula 
\begin{equation}
\boxed{\underline{\varepsilon}^{\mathrm{intra}}\left(\omega\right)=\boldsymbol{1}-\frac{\underline{\omega}_{P}^{2}}{\omega\left(\omega+i\Gamma\right)}}\label{eq:intra-band_eps}
\end{equation}

\end_inset

which gives
\begin_inset Formula 
\[
\boxed{\mathrm{Re}\underline{\varepsilon}^{\mathrm{intra}}\left(\omega\right)=\boldsymbol{1}-\frac{\underline{\omega}_{P}^{2}}{\omega^{2}+\Gamma^{2}}}
\]

\end_inset

with a root at
\begin_inset Formula 
\[
\omega_{0}=\sqrt{\underline{\omega}_{P}^{2}-\Gamma^{2}}
\]

\end_inset

and
\begin_inset Formula 
\[
\mathrm{Im}\underline{\varepsilon}^{\mathrm{intra}}\left(\omega\right)=\frac{\underline{\omega}_{P}^{2}\Gamma}{\omega\left(\omega^{2}+\Gamma^{2}\right)}
\]

\end_inset

which results in
\begin_inset Formula 
\[
\boxed{\mathrm{Re}\sigma^{\mathrm{intra}}\left(\omega\right)=\frac{1}{4\pi}\frac{\underline{\omega}_{P}^{2}\Gamma}{\omega^{2}+\Gamma^{2}}}
\]

\end_inset


\begin_inset Formula 
\[
\mathrm{Im}\sigma^{\mathrm{intra}}\left(\omega\right)=\frac{\omega}{\Gamma}\mathrm{Re}\sigma^{\mathrm{intra}}\left(\omega\right)
\]

\end_inset

For the loss function we get (in the diagonal principal axis frame of 
\begin_inset Formula $\underline{\omega}^{P}$
\end_inset

)
\begin_inset Formula 
\begin{eqnarray}
L_{\alpha\alpha} & = & \omega\Gamma\frac{\omega_{P\alpha}^{2}}{\left(\omega^{2}-\omega_{P\alpha}^{2}\right)^{2}+\omega^{2}\Gamma^{2}}\label{eq:Loss_intra}
\end{eqnarray}

\end_inset

with the maximum at
\begin_inset Formula 
\begin{eqnarray*}
\omega_{\mathrm{peak}} & = & \frac{1}{\sqrt{6}}\omega_{P\alpha}\sqrt{2-\frac{\Gamma^{2}}{\omega_{P\alpha}^{2}}+\sqrt{16-4\frac{\Gamma^{2}}{\omega_{P\alpha}^{2}}+\frac{\Gamma^{4}}{\omega_{P\alpha}^{4}}}}\\
\omega_{\mathrm{peak}} & \approx & \omega_{P\alpha}\left(1-\frac{1}{8}\frac{\Gamma^{2}}{\omega_{P\alpha}^{2}}\right)
\end{eqnarray*}

\end_inset


\end_layout

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placement H
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	filename pic/intra.png
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	width 80col%

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\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Intra band contribution for an unaturally large 
\begin_inset Formula $\Gamma=1$
\end_inset

eV (to resolve the small shifts of 
\begin_inset Formula $\omega_{P}$
\end_inset

.)
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Intra-band and constant interband 
\begin_inset Formula $\varepsilon_{\infty}$
\end_inset


\end_layout

\begin_layout Standard
Now, we include a constant approximation for the interband contribution
 
\begin_inset Formula $\varepsilon_{\infty}$
\end_inset

.
\begin_inset Formula 
\begin{eqnarray*}
\underline{\varepsilon}\left(\omega\right) & = & \boldsymbol{1}-\frac{\underline{\omega}_{P}^{2}}{\omega\left(\omega+i\Gamma\right)}+\varepsilon_{\infty}\\
 & = & \left(1+\varepsilon_{\infty}\right)\left(\boldsymbol{1}-\frac{\underline{\tilde{\omega}}_{P}^{2}}{\omega\left(\omega+i\Gamma\right)}\right)
\end{eqnarray*}

\end_inset

where we defined the renormalized plasmon frequency as
\begin_inset Formula 
\[
\tilde{\omega}_{P}=\frac{\omega_{P}}{\sqrt{1+\varepsilon_{\infty}}}
\]

\end_inset

This means that in all expression we have to replace 
\begin_inset Formula $\omega_{P\alpha}\to\tilde{\omega}_{P\alpha}$
\end_inset

 and devide (multiply) by 
\begin_inset Formula $\frac{1}{1+\varepsilon_{\infty}}$
\end_inset

.
\end_layout

\begin_layout Standard
Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Loss_intra"

\end_inset

) then gives 
\begin_inset Formula 
\[
L_{\alpha\alpha}=\frac{1}{1+\varepsilon_{\infty}}\omega\Gamma\frac{\tilde{\omega}_{P\alpha}^{2}}{\left(\omega^{2}-\tilde{\omega}_{P\alpha}^{2}\right)^{2}+\omega^{2}\Gamma^{2}}
\]

\end_inset

with the peak at
\begin_inset Formula 
\[
\boxed{\omega_{\mathrm{peak}}^{\mathrm{intra}+\varepsilon_{\infty}}=\tilde{\omega}_{P\alpha}\left(1-\frac{1}{8}\frac{\Gamma^{2}}{\tilde{\omega}_{P\alpha}^{2}}\right)}
\]

\end_inset

which is better suited to determine the plasmon peak than extracting the
 divergent Drude part from 
\begin_inset Formula $\sigma$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
\begin_inset Formula 
\[
L=\frac{1}{\mathrm{Im}\varepsilon\left(\omega\right)+\mathrm{Re}\varepsilon\left(\omega\right)\frac{1}{\mathrm{Im}\varepsilon\left(\omega\right)}\mathrm{Re}\varepsilon\left(\omega\right)}
\]

\end_inset


\begin_inset Formula 
\begin{eqnarray*}
\mathrm{Re}\varepsilon\left(\omega\right) & = & \mathrm{Re}\varepsilon^{\prime}\left(\omega_{0}\right)\Delta\omega
\end{eqnarray*}

\end_inset


\begin_inset Formula 
\begin{eqnarray*}
L & = & \frac{\mathrm{Im}\varepsilon\left(\omega_{0}\right)+\mathrm{Im}\varepsilon^{\prime}\left(\omega_{0}\right)\Delta\omega}{\left(\mathrm{Im}\varepsilon\left(\omega_{0}\right)+\mathrm{Im}\varepsilon^{\prime}\left(\omega_{0}\right)\Delta\omega\right)^{2}+\left(\mathrm{Re}\varepsilon^{\prime}\left(\omega_{0}\right)\Delta\omega\right)^{2}}\\
 & = & \frac{1}{\mathrm{Im}\varepsilon\left(\omega_{0}\right)}\left[1+\frac{\mathrm{Re}\varepsilon^{\prime}\left(\omega_{0}\right)-\mathrm{Im}\varepsilon^{\prime}\left(\omega_{0}\right)}{\mathrm{Im}\varepsilon\left(\omega_{0}\right)}\Delta\omega+\frac{1}{\mathrm{Im}\varepsilon\left(\omega_{0}\right)}\left[-\mathrm{Re}\varepsilon^{\prime}\left(\omega_{0}\right)\frac{\mathrm{Im}\varepsilon^{\prime}\left(\omega_{0}\right)}{\mathrm{Im}\varepsilon\left(\omega_{0}\right)}-\mathrm{Im}\varepsilon\left(\omega_{0}\right)\frac{\left|\varepsilon^{\prime}\left(\omega_{0}\right)\right|^{2}}{\left(\mathrm{Im}\varepsilon\left(\omega_{0}\right)\right)^{2}}+\mathrm{Im}\varepsilon\left(\omega_{0}\right)\left(\frac{\mathrm{Im}\varepsilon^{\prime}\left(\omega_{0}\right)}{\mathrm{Im}\varepsilon\left(\omega_{0}\right)}\right)^{2}\right]\left(\Delta\omega\right)^{2}\right]
\end{eqnarray*}

\end_inset


\begin_inset Formula 
\[
0=\frac{I^{2}I^{\prime}}{\left(I^{\prime3}+I^{\prime}R^{\prime2}\right)}+2\frac{II^{\prime2}+IR^{\prime2}}{I^{\prime3}+I^{\prime}R^{\prime2}}x+x^{2}
\]

\end_inset


\begin_inset Formula 
\[
\omega_{\mathrm{peak}}=\omega_{0}+\frac{I}{I^{\prime}}\left(-1\pm\sqrt{1-\frac{I^{\prime}}{\left|\varepsilon^{\prime}\right|^{2}}}\right)
\]

\end_inset


\end_layout

\end_inset


\end_layout

\end_body
\end_document
