solid_state_physics/tutorial/2n_01s.tex

   1 \pdfoutput=0
   2 \documentclass[a4paper,11pt]{article}
   3 \usepackage[activate]{pdfcprot}
   4 \usepackage{verbatim}
   5 \usepackage{a4}
   6 \usepackage{a4wide}
   7 \usepackage[german]{babel}
   8 \usepackage[latin1]{inputenc}
   9 \usepackage[T1]{fontenc}
  10 \usepackage{amsmath}
  11 \usepackage{ae}
  12 \usepackage{aecompl}
  13 \usepackage[dvips]{graphicx}
  14 \graphicspath{{./img/}}
  15 \usepackage{color}
  16 \usepackage{pstricks}
  17 \usepackage{pst-node}
  18 \usepackage{rotating}
  19 \usepackage{epic}
  20 \usepackage{eepic}
  21
  22 \setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
  23 \setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
  24 \setlength{\oddsidemargin}{-10mm}
  25 \setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm}
  26 \setlength{\textheight}{26cm} \setlength{\headsep}{0cm}
  27
  28 \renewcommand{\labelenumi}{(\alph{enumi})}
  29
  30 \begin{document}
  31
  32 % header
  33 \begin{center}
  34  {\LARGE {\bf Materials Physics II}\\}
  35  \vspace{8pt}
  36  Prof. B. Stritzker\\
  37  SS 2011\\
  38  \vspace{8pt}
  39  {\Large\bf Tutorial 1 - proposed solutions}
  40 \end{center}
  41
  42 \section{Indirect band gap of silicon}
  43
  44 \begin{enumerate}
  45  \item \begin{itemize}
  46         \item Photon wavelength:\\
  47               $E_g=\hbar\omega=\hbar\frac{2\pi}{T}=\hbar 2\pi v
  48                   \stackrel{c=v\lambda}{=}\hbar 2\pi\frac{c}{\lambda}$
  49               $\Rightarrow \lambda=\frac{\hbar 2\pi c}{E_g}
  50                                   =\frac{hc}{E_g}=\ldots=1.11 \, \mu m$
  51         \item Photon momentum:\\
  52               $p=\hbar k=\hbar\frac{2\pi}{\lambda}=\frac{h}{\lambda}
  53                 =\ldots=5.97 \cdot 10^{-28} \, kg\frac{m}{s}$
  54        \end{itemize}
  55  \item Phonon momentum necessary for transition:\\
  56        $\Delta p=\hbar \cdot \Delta k=\hbar \cdot 0.85 \, \frac{2\pi}{a}
  57          =\frac{0.85 \, h}{a}=\ldots=1.04 \cdot 10^{-24} \, kg\frac{m}{s}$\\
  58        $\rightarrow$ Phonon momentum 3 orders of magnitude below
  59                      the momentum necessary for transition!
  60  \item \begin{itemize}
  61         \item Photon momentum insufficient.
  62               Momentum contribution of phonon (lattice vibration) required.\\
  63               $\Rightarrow$ Probability of transition very small.
  64         \item Recombination energy of electron-hole pairs most probably
  65               released as vibrational energy of the lattice.\\
  66               $\Rightarrow$ Only direct band gap semiconductors suitable for
  67                             effective photon generation.
  68        \end{itemize}
  69 \end{enumerate}
  70
  71 \section{Charge carrier density of semiconductors}
  72
  73 \begin{itemize}
  74  \item Calculation of $n$:\\
  75 $\forall \epsilon$ of states within conduction band:
  76 $\epsilon-\mu_{\text{F}} >> k_{\text{B}}T$
  77 $\Rightarrow$
  78 $f(\epsilon,T)=
  79  \frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
  80  \exp(-\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
  81 Parabolic approximation:
  82 $D_c(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}(\epsilon-E_c)^{1/2}$
  83 $\Rightarrow$\\
  84 $n=\int_{E_{\text{c}}}^{\infty}D_c(\epsilon)f(\epsilon,T)d\epsilon\approx
  85  \frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}
  86  \exp(\frac{\mu_{\text{F}}}{k_{\text{B}}T})
  87  \int_{E_{\text{c}}}^{\infty}(\epsilon-E_c)^{1/2}
  88  \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
  89 Use: $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$
  90      $\Rightarrow\epsilon=xk_{\text{B}}T+E_{\text{c}}$ and
  91      $d\epsilon=k_{\text{B}}Tdx$\\
  92 $\Rightarrow$
  93 $n=\frac{1}{2\pi^2}(\frac{2m_nk_{\text{B}}T}{\hbar^2})^{3/2}
  94  \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})
  95  \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
  96  \underbrace{2(\frac{m_nk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{c}}}
  97  \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})=
  98  N_{\text{c}}\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})$
  99  \item In the same way, calculate $p$:\\
 100 $\forall \epsilon$ of states within conduction band:
 101 $\mu_{\text{F}}-\epsilon >> k_{\text{B}}T$
 102 $\Rightarrow$
 103 $1-f(\epsilon,T)=
 104  1-\frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
 105  \exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
 106 Parabolic approximation:
 107 $D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$
 108 $\Rightarrow$\\
 109 $p=\int_{-\infty}^{E_{\text{v}}}D_v(\epsilon)(1-f(\epsilon,T))d\epsilon\approx
 110  \frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
 111  \exp(-\frac{\mu_{\text{F}}}{k_{\text{B}}T})
 112  \int_{-\infty}^{E_{\text{v}}}(E_v-\epsilon)^{1/2}
 113  \exp(\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
 114 Use: $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$
 115      $\Rightarrow\epsilon=E_{\text{v}}-xk_{\text{B}}T$ and
 116      $d\epsilon={\color{red}-}k_{\text{B}}Tdx$\\
 117 $\Rightarrow$
 118 $p=\frac{1}{2\pi^2}(\frac{2m_pk_{\text{B}}T}{\hbar^2})^{3/2}
 119  \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})
 120  \underbrace{\int_{{\color{red}0}}^{{\color{red}\infty}}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
 121  \underbrace{2(\frac{m_pk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{v}}}
 122  \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})=
 123  N_{\text{v}}\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})$
 124 \end{itemize}
 125
 126 \end{document}