From 149f00a0e7b93e9a5d836a910b131ca1445eacbd Mon Sep 17 00:00:00 2001 From: hackbard Date: Tue, 3 May 2011 15:29:08 +0200 Subject: [PATCH] new excercises --- solid_state_physics/tutorial/2n_01.tex | 72 ++++++++++++++ solid_state_physics/tutorial/2n_01s.tex | 126 ++++++++++++++++++++++++ 2 files changed, 198 insertions(+) create mode 100644 solid_state_physics/tutorial/2n_01.tex create mode 100644 solid_state_physics/tutorial/2n_01s.tex diff --git a/solid_state_physics/tutorial/2n_01.tex b/solid_state_physics/tutorial/2n_01.tex new file mode 100644 index 0000000..3e5d711 --- /dev/null +++ b/solid_state_physics/tutorial/2n_01.tex @@ -0,0 +1,72 @@ +\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 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} -- 2.20.1