\usepackage{pstricks}
\usepackage{pst-node}
\usepackage{rotating}
+\usepackage{epic}
+\usepackage{eepic}
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=\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
\end{itemize}
\item Phonon momentum necessary for transition:\\
- $p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
+ $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
=\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
$\rightarrow$ Phonon momentum 3 orders of magnitude below
the momentum necessary for transition!
- \item
+ \item \begin{itemize}
+ \item Photon momentum insufficient.
+ Momentum contribution of phonon (lattice vibration) required.\\
+ $\Rightarrow$ Probability of transition very small.
+ \item Recombination energy of electron-hole pairs most probably
+ released as vibrational energy of the lattice.\\
+ $\Rightarrow$ Only direct band gap semiconductors suitable for
+ effective photon generation.
+ \end{itemize}
\end{enumerate}
-\section{\ldots}
-
-\ldots
+\section{Dielectric function of the free electron gas}
\begin{enumerate}
- \item \ldots
- \item \ldots
+ \item \begin{itemize}
+ \item Equation of motion: $m\frac{d^2x}{dt^2}=-eE$
+ \item Ansatz: $x(t)=x_0 e^{-i\omega t}$
+ \item Solution of the equation of motion: $-m\omega^2x=-eE$
+ \item Dipole moment: $p=-ex=\frac{-e^2E}{m\omega^2}$
+ \item Polarization: $P=np=\frac{-ne^2E}{m\omega^2}$
+ \item Dielectric function:
+ $\epsilon(\omega)=1+\frac{P}{\epsilon_0E}
+ =1+\frac{-ne^2E}{m\omega^2\epsilon_0E}
+ =1-\frac{ne^2}{\epsilon_0m\omega^2}$
+ \end{itemize}
+ \item \begin{itemize}
+ \item Using $\omega_p^2=\frac{ne^2}{\epsilon_0m}$\\
+ $\Rightarrow \epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$
+ \item Sketch of dielectric function:\\
+ (page 2)
+ \item Influence on electromagnetic waves:\\
+ $\frac{\omega}{\omega_p}>1\Leftrightarrow \omega>\omega_p$:
+ $\Rightarrow \epsilon=n^2>0$
+ $\Rightarrow$ transparent region\\
+ $\frac{\omega}{\omega_p}<1\Leftrightarrow \omega<\omega_p$:
+ $\Rightarrow \epsilon=n^2<0$
+ $\Rightarrow$ reflective region\\
+ \end{itemize}
+\input{dielectric_pslatex}
\end{enumerate}
\end{document}