solid_state_physics/tutorial/1_03s.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
  20 \setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
  21 \setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
  22 \setlength{\oddsidemargin}{-10mm}
  23 \setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm}
  24 \setlength{\textheight}{26cm} \setlength{\headsep}{0cm}
  25
  26 \renewcommand{\labelenumi}{(\alph{enumi})}
  27
  28 \begin{document}
  29
  30 % header
  31 \begin{center}
  32  {\LARGE {\bf Materials Physics I}\\}
  33  \vspace{8pt}
  34  Prof. B. Stritzker\\
  35  WS 2007/08\\
  36  \vspace{8pt}
  37  {\Large\bf Tutorial 3 - proposed solutions}
  38 \end{center}
  39
  40 \section{Drude theory of metallic conduction}
  41 \begin{enumerate}
  42  \item $U=IR \Rightarrow EL=jA\rho\frac{L}{A}
  43              \Rightarrow E=j\rho$
  44  \item \begin{itemize}
  45         \item distance: $v\,dt$
  46         \item number of electrons crossing $A$: $n(v\,dt)A$
  47        \end{itemize}
  48        $\Rightarrow$ $j=\frac{I}{A}=\frac{dQ/dt}{A}=\frac{-e\,n(v\,dt)A/dt}{A}
  49                        =-nev$
  50  \item \begin{itemize}
  51         \item In the absence of an electric field, electrons are as likely
  52               to be moving in any one direction as in any other.
  53               The velocity averages to zero.
  54               As expected, according to the above equation, there is no
  55               net electric current density.
  56         \item Since electrons emerge in a random direction
  57               there will be no contribution from the thermal velocity
  58               to the average electronic velocity.
  59         \item $v_{average}=at=\frac{F}{m}\tau=-\frac{eE}{m}\tau$
  60        \end{itemize}
  61  \item \begin{itemize}
  62         \item $j=\left(\frac{ne^2\tau}{m}\right)E$\\
  63         \item $j=\sigma E \Rightarrow \sigma=\frac{ne^2\tau}{m}$
  64        \end{itemize}
  65  \item \begin{itemize}
  66         \item Energy transfer: $\frac{m}{2}v_{drift}^2$,
  67                                $\quad v_{drift}$:
  68                                final drift velocity of the accelerated electron
  69         \item $v_{drift}=-\frac{eE}{m}t_0$, $\quad t_0$:
  70               free flight time (no collision) of the electron
  71         \item $v_{average}=\frac{1}{t_0}\int_{0}^{t_0} v(t) dt
  72                           =-\frac{eE}{m}\frac{1}{t_0}[\frac{t^2}{2}]_{0}^{t_0}
  73                           =-\frac{eE}{m}\frac{t_0}{2}=:-\frac{eE}{m}\tau$,
  74               $\qquad t_0=2\tau$, $v_{drift}=2v_{average}$
  75               \includegraphics[width=12cm]{drude_v.eps}
  76         \item Each of the $n$ electrons per unit volume
  77               transfer the kinetic energy $\frac{1}{2}mv^2_{drift}$
  78               once per $t_0$ to the lattice
  79        \end{itemize}
  80        \[
  81        \Rightarrow \frac{P}{V}=\frac{E_{kin}}{Vt_0}
  82                               =\frac{n\frac{1}{2}m\frac{e^2E^2}{m^2}t_0^2}{t_0}
  83                               =n\frac{1}{2}\frac{e^2E^2}{m}2\tau
  84                               =\sigma E^2=jE=j^2\rho=\frac{I^2}{A^2}\frac{A}{L}R
  85                               =\frac{I^2R}{V}
  86        \]
  87        \[
  88        \Rightarrow P=I^2R \textrm{ (Joule heating)}
  89        \]
  90 \end{enumerate}
  91
  92 \end{document}