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[lectures/latex.git] / solid_state_physics / tutorial / 2n_01s.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}
+\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}