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% header
\begin{center}
 {\LARGE {\bf Materials Physics II}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 SS 2011\\
 \vspace{8pt}
 {\Large\bf Tutorial 1}
\end{center}

\section{Indirect band gap of silicon}

Some facts about silicon:
\begin{itemize}
 \item Lattice constant: $a=5.43 \cdot 10^{-10} \, m$.
 \item Silicon has an indirect band gap.
       \begin{itemize}
        \item The minimum of the conduction band is located at
             $k=0.85 \frac{2 \pi}{a}$.
       \item The maximum of the valance band is located at $k=0$.
       \item The energy gap is $E_g=1.12 \, eV$.
       \end{itemize}
\end{itemize}
\begin{enumerate}
 \item Calculate the wavelength of the light necessary to lift an electron from
       the valence to the conduction band.
       What is the momentum of such a photon?
 \item Calculate the phonon momentum necessary for the transition.
       Compare the momentum values of phonon and photon.
 \item Draw conclusions concerning optical applications.
\end{enumerate}

\section{Charge carrier density of semiconductors}

Calculate the charge carrier densities $n$ and $p$ for $E_{\text{c}}-\mu_{\text{F}} >> k_{\text{B}}T$ and $\mu_{\text{F}}-E_{\text{v}} >> k_{\text{B}}T$.\\\\
{\bf Hint:}
Consider the influence of these two conditions for the energy of the states, which are situated in the conduction and valence band, and the consequences for the respective occupation described by the Fermi distribution.
The parabolic approximation of the density of states of electrons in the conduction band and holes in the valence band (effective masses $m_n$ and $m_p$) is given by
$D_c(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}(\epsilon-E_c)^{1/2}$ and
$D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$.
Simplify the Fermi function before calculating the integral and use the substitutions $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$ and $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$.
Furthermore use the equality $\int_0^{\infty} x^{1/2} e^{-x} dx = 1/2 \sqrt{\pi}$.

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