added 1_05 tutorial + part of solutions

diff --git a/solid_state_physics/tutorial/1_05.tex b/solid_state_physics/tutorial/1_05.tex
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+\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}

diff --git a/solid_state_physics/tutorial/1_05s.tex b/solid_state_physics/tutorial/1_05s.tex
new file mode 100644 (file)
index 0000000..39d9bdd
--- /dev/null
+++ b/solid_state_physics/tutorial/1_05s.tex
@@ -0,0 +1,96 @@
+\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 - 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}
+
+\end{document}