added 1_05 tutorial + part of solutions

[lectures/latex.git] / solid_state_physics / tutorial / 1_05s.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 - 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}