\pdfoutput=0
\documentclass[a4paper,11pt]{article}
\usepackage[activate]{pdfcprot}
\usepackage{verbatim}
\usepackage{a4}
\usepackage{a4wide}
\usepackage[german]{babel}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{ae}
\usepackage{aecompl}
\usepackage[dvips]{graphicx}
\graphicspath{{./img/}}
\usepackage{color}
\usepackage{pstricks}
\usepackage{pst-node}
\usepackage{rotating}

\setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
\setlength{\oddsidemargin}{-10mm}
\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm}
\setlength{\textheight}{26cm} \setlength{\headsep}{0cm}

\renewcommand{\labelenumi}{(\alph{enumi})}

\begin{document}

% header
\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 5}
\end{center}

\section{Charge carrier density of intrinsic semiconductors}

\begin{enumerate}
 \item Recall the free electron in a box.
       Write down an expression for the density of states $D(E)$
       of the free electron gas.
       {\bf Hint:} The density of states is a function of internal energy $E$
                   such that $D(E)dE$ is the number of states
                   (allowed $k$-values) with energies
		   between $E$ and $E+dE$.
		   For large values of $L$ (side length of the box)
		   the states are quasi-continuous and
		   sums in $k$-space can be replaced by integrals.
                   First calculate the amount of states $dZ'$
		   in-between $k$ and $k+dk$.
		   Therefor calculate the volume of the sperical shell
		   and the volume of a single allowed $k$-point.
		   Neglect terms of the order $(dk^2)$.
		   After that express $dk$ and $k$ by $dE$ and $E$
		   and insert these expressions into $dZ'$.
                   By definition $D(E)=dZ/dE$,
		   where $dZ$ is $dZ'$ devided by the box volume
		   (states per crystal volume).
		   Adjust the expression taking into account
		   the spin of an electron.
 \item The conduction and valence band in a semiconductor can be approximated
       by the same parabolic functions of $k$ close to the bandedges.
       The mass of the electron is replaced by an effective mass
       of the electron in the conduction band or the hole in the valence band.
       Show the relation of the effective mass and the curvature of the band.
       {\bf Hint:} The curvature of a function $f(x)$ is given by the second
                   derivative of this function with respect to $x$.
 \item Sketch the density of states, the Fermi function and the resulting
       density of charge carriers (electrons: $m_n$, holes: $m_p$)
       for $m_n=m_p$ and for $m_n\ne m_p$ for non-zero temperatures.
       {\bf Hint:} The density of states is given by
                   $D_c(E)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}
		         (E-E_c)^{1/2}$ for electrons in the conduction band and
                   $D_v(E)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
		         (E_v-E)^{1/2}$ for holes in the valence band.
		   $E_c$ is the lowest energy level of the conduction and
		   $E_v$ the highest energy level of the valence band.
		   Thus the bandgap energy is given by $E_g=E_c-E_v$.
	           The density of charge carriers is the product of $D(E)$ and
		   the Fermi function $f(E)$.
		   The Fermi energy $E_F$ adjusts itself in such a way that
		   the amount of electrons and holes equals.
		   Keep in mind that the distribution valid for the holes is
		   $1-f(E)$.
\end{enumerate}

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

The valence band of silicon is composed by three subbands.
Two of them contact at the $\Gamma$-point ($k=0$),
the one for heavy holes ($m_{ph}$) and the one for light holes ($m_{pl}$).
An additional split-off hole band ($m_{pso}$) is located
shortly below the first two bands (see Figure).

\begin{enumerate}
 \item Assume parabolic bands near $k=0$.
       Write down the total density of states
       near the maximum of the valence band.
       Only consider heavy and light holes.
 \item Write the above result in terms of a density of states expression
       of a parabolic band with a single uniform effective mass $m_p$.
       Determine this 'density of state mass' $m_p$.
       Calculate $m_p$ using the values $m_{ph}=0.49 \, m_e$ and
       $m_{pl}=0.16 \, m_e$ in which $m_e$ is the electron rest mass.
\end{enumerate}

\vspace{0.5cm}

\begin{picture}(0,0)(0,140)
 \includegraphics[width=5.0cm]{silicon_bs.eps}
\end{picture}

\begin{flushright}
\begin{minipage}{5cm}
\end{minipage}
\begin{minipage}{3cm}
 \includegraphics[height=3cm]{weihnachtsbaum.eps}
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
{\Large\bf
 Merry Christmas\\
 \&\\
 Happy New Year!}
\end{center}
\end{minipage}
\end{flushright}

\end{document}
