solid_state_physics/tutorial/1_06s.tex

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  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}