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% header
\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 6 - proposed solutions}
\end{center}

\section{Indirect band gap of silicon}

\begin{enumerate}
 \item \begin{itemize}
        \item Photon wavelength:\\
	      $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
	          \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
	      $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
	                          =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
	\item Photon momentum:\\
	      $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
	        =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
       \end{itemize}
 \item Phonon momentum necessary for transition:\\
       $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
         =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
       $\rightarrow$ Phonon momentum 3 orders of magnitude below
                     the momentum necessary for transition!
 \item \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{Dielectric function of the free electron gas}

\begin{enumerate}
 \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}
