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29
30 \begin{document}
31
32 % header
33 \begin{center}
34  {\LARGE {\bf Materials Physics I}\\}
35  \vspace{8pt}
36  Prof. B. Stritzker\\
37  WS 2007/08\\
38  \vspace{8pt}
39  {\Large\bf Tutorial 6 - proposed solutions}
40 \end{center}
41
42 \section{Indirect band gap of silicon}
43
44 \begin{enumerate}
45  \item \begin{itemize}
46         \item Photon wavelength:\\
47               $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
48                   \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
49               $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
50                                   =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
51         \item Photon momentum:\\
52               $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
53                 =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
54        \end{itemize}
55  \item Phonon momentum necessary for transition:\\
56        $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
57          =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
58        $\rightarrow$ Phonon momentum 3 orders of magnitude below
59                      the momentum necessary for transition!
60  \item \begin{itemize}
61         \item Photon momentum insufficient.
62               Momentum contribution of phonon (lattice vibration) required.\\
63               $\Rightarrow$ Probability of transition very small.
64         \item Recombination energy of electron-hole pairs most probably
65               released as vibrational energy of the lattice.\\
66               $\Rightarrow$ Only direct band gap semiconductors suitable for
67                             effective photon generation.
68        \end{itemize}
69 \end{enumerate}
70
71 \section{Dielectric function of the free electron gas}
72
73 \begin{enumerate}
74  \item \begin{itemize}
75         \item Equation of motion: $m\frac{d^2x}{dt^2}=-eE$
76         \item Ansatz: $x(t)=x_0 e^{-i\omega t}$
77         \item Solution of the equation of motion: $-m\omega^2x=-eE$
78         \item Dipole moment: $p=-ex=\frac{-e^2E}{m\omega^2}$
79         \item Polarization: $P=np=\frac{-ne^2E}{m\omega^2}$
80         \item Dielectric function:
81               $\epsilon(\omega)=1+\frac{P}{\epsilon_0E}
82                                =1+\frac{-ne^2E}{m\omega^2\epsilon_0E}
83                                =1-\frac{ne^2}{\epsilon_0m\omega^2}$
84        \end{itemize}
85  \item \begin{itemize}
86         \item Using $\omega_p^2=\frac{ne^2}{\epsilon_0m}$\\
87               $\Rightarrow \epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$
88         \item Sketch of dielectric function:\\
89               (page 2)
90         \item Influence on electromagnetic waves:\\
91               $\frac{\omega}{\omega_p}>1\Leftrightarrow \omega>\omega_p$:
92               $\Rightarrow \epsilon=n^2>0$
93               $\Rightarrow$ transparent region\\
94               $\frac{\omega}{\omega_p}<1\Leftrightarrow \omega<\omega_p$:
95               $\Rightarrow \epsilon=n^2<0$
96               $\Rightarrow$ reflective region\\
97        \end{itemize}
98 \input{dielectric_pslatex}
99 \end{enumerate}
100
101 \end{document}