new excercises

diff --git a/solid_state_physics/tutorial/2n_01.tex b/solid_state_physics/tutorial/2n_01.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 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.
+The parabolic approximation of the density of states of electrons in the conduction and holes in the valence band with the 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
+$D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$.
+If you do not calculate the non-simplified Fermi-integral the substitutions $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$ and $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$ can be used. Furthermore use the equality $\int_0^{\infty} x^{1/2} e^{-x} dx = 1/2 \sqrt{\pi}$.
+
+\end{document}

diff --git a/solid_state_physics/tutorial/2n_01s.tex b/solid_state_physics/tutorial/2n_01s.tex
new file mode 100644 (file)
index 0000000..b16cf30
--- /dev/null
+++ b/solid_state_physics/tutorial/2n_01s.tex
@@ -0,0 +1,126 @@
+\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}
+\usepackage{epic}
+\usepackage{eepic}
+
+\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 II}\\}
+ \vspace{8pt}
+ Prof. B. Stritzker\\
+ SS 2011\\
+ \vspace{8pt}
+ {\Large\bf Tutorial 1 - proposed solutions}
+\end{center}
+
+\section{Indirect band gap of silicon}
+
+\begin{enumerate}
+ \item \begin{itemize}
+        \item Photon wavelength:\\
+             $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
+                 \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
+             $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
+                                 =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
+       \item Photon momentum:\\
+             $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
+               =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
+       \end{itemize}
+ \item Phonon momentum necessary for transition:\\
+       $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
+         =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
+       $\rightarrow$ Phonon momentum 3 orders of magnitude below
+                     the momentum necessary for transition!
+ \item \begin{itemize}
+        \item Photon momentum insufficient.
+              Momentum contribution of phonon (lattice vibration) required.\\
+              $\Rightarrow$ Probability of transition very small.
+       \item Recombination energy of electron-hole pairs most probably
+             released as vibrational energy of the lattice.\\
+             $\Rightarrow$ Only direct band gap semiconductors suitable for
+                           effective photon generation.
+       \end{itemize}
+\end{enumerate}
+
+\section{Charge carrier density of semiconductors}
+
+\begin{itemize}
+ \item Calculation of $n$:\\
+$\forall \epsilon$ of states within conduction band:
+$\epsilon-\mu_{\text{F}} >> k_{\text{B}}T$
+$\Rightarrow$
+$f(\epsilon,T)=
+ \frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
+ \exp(-\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
+Parabolic approximation:
+$D_c(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}(\epsilon-E_c)^{1/2}$
+$\Rightarrow$\\
+$n=\int_{E_{\text{c}}}^{\infty}D_c(\epsilon)f(\epsilon,T)d\epsilon\approx
+ \frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}
+ \exp(\frac{\mu_{\text{F}}}{k_{\text{B}}T})
+ \int_{E_{\text{c}}}^{\infty}(\epsilon-E_c)^{1/2}
+ \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
+Use: $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$
+     $\Rightarrow\epsilon=xk_{\text{B}}T+E_{\text{c}}$ and
+     $d\epsilon=k_{\text{B}}Tdx$\\
+$\Rightarrow$
+$n=\frac{1}{2\pi^2}(\frac{2m_nk_{\text{B}}T}{\hbar^2})^{3/2}
+ \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})
+ \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
+ \underbrace{2(\frac{m_nk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{c}}}
+ \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})=
+ N_{\text{c}}\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})$
+ \item In the same way, calculate $p$:\\
+$\forall \epsilon$ of states within conduction band:
+$\mu_{\text{F}}-\epsilon >> k_{\text{B}}T$
+$\Rightarrow$
+$1-f(\epsilon,T)=
+ 1-\frac{1}{\exp(\frac{\mu_{\text{F}}-\epsilon}{k_{\text{B}}T})+1}\approx
+ \exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
+Parabolic approximation:
+$D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$
+$\Rightarrow$\\
+$p=\int_{-\infty}^{E_{\text{v}}}D_v(\epsilon)(1-f(\epsilon,T))d\epsilon\approx
+ \frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
+ \exp(-\frac{\mu_{\text{F}}}{k_{\text{B}}T})
+ \int_{-\infty}^{E_{\text{v}}}(E_v-\epsilon)^{1/2}
+ \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
+Use: $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$
+     $\Rightarrow\epsilon=E_{\text{v}}-xk_{\text{B}}T$ and
+     $d\epsilon=-k_{\text{B}}Tdx$\\
+$\Rightarrow$
+$p=\frac{1}{2\pi^2}(\frac{2m_pk_{\text{B}}T}{\hbar^2})^{3/2}
+ \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})
+ \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
+ \underbrace{2(\frac{m_pk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{v}}}
+ \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})=
+ N_{\text{v}}\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})$
+\end{itemize}
+
+\end{document}