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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 5 - proposed solutions}
\end{center}

\section{Charge carrier density of intrinsic semiconductors}

\begin{enumerate}
 \item \begin{itemize}
        \item Free electron in a box:\\
	      $E(k)=\frac{\hbar^2k^2}{2m}$, $k^2=k_x^2+k_y^2+k_z^2$,
	      $k_i=\frac{\pi}{L}n_i$ with $n_i=1,2,3,\ldots$
	\item Amount of states in-between $k$ and $k+dk$:
	      \begin{itemize}
	       \item Allowed values only in first octant!
	       \item Volume of one $k$-point: $V_k=(\frac{\pi}{L})^3$
	       \item Volume of spherical shell with radius $k$ and $k+dk$:\\
	             $V_{shell}=\frac{4}{3}\pi(k+dk)^3-\frac{4}{3}\pi k^3
		      \stackrel{Taylor}{=}\frac{4}{3}\pi k^3
		      +\frac{3\cdot 4}{3}\pi k^2dk+O(dk^2)-\frac{4}{3}\pi k^3
		      \approx 4\pi k^2dk$
	      \end{itemize}
	      $\Rightarrow dZ'=\frac{\frac{1}{8}4\pi k^2dk}{(\pi/L)^3}$
        \item Express $dk$ and $k$ by $dE$ and $E$ and insert it into $dZ$:
              \begin{itemize}
               \item $\frac{dE}{dk}=\frac{\hbar^2}{m}k \rightarrow
	              dk=\frac{m}{\hbar^2k}dE$
               \item $k=\frac{\sqrt{2m}}{\hbar^2}\sqrt{E}$
	      \end{itemize}
	      $\Rightarrow dZ'=\frac{4\pi k^2m}{(\pi/L)^3\hbar^2k} dE=
	       \frac{4\pi\frac{\sqrt{2m}}{\hbar}\sqrt{E}m}{8(\pi/L)^3\hbar^2}dE
	       =\frac{(2m)^{3/2}L^3}{4\pi^2\hbar^3}\sqrt{E}dE$\\
	      $\Rightarrow dZ=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{E}dE$
	\item Density of states:\\
	      $D(E)=dZ/dE=\frac{(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{E}
	       =\frac{1}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$
	\item Two spins for every $k$-point:\\
	      $\Rightarrow D(E)=
	       \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$
       \end{itemize}
 \item Curvature of the band:\\
       $\frac{d^2E}{dk^2}=\frac{d^2}{dk^2}\frac{\hbar^2k^2}{2m_{eff}}
                         =\frac{\hbar^2}{m_{eff}}$
 \item
\end{enumerate}

\section{'Density of state mass' of electrons and holes in silicon}

\begin{enumerate}
 \item $D_v(E)=\frac{1}{2\pi^2}(\frac{2}{\hbar^2})^{3/2}
               (m_{pl}^{3/2}+m_{ph}^{3/2})(E_v-E)^{1/2}$
 \item
\end{enumerate}

\begin{center}
{\Large\bf
 Merry Christmas\\
 \&\\
 Happy New Year!}
\end{center}

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