solid_state_physics/tutorial/1_04s.tex

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  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 4 - proposed solutions}
  38 \end{center}
  39
  40 \section{Hall effect and magnetoresistance}
  41 \begin{enumerate}
  42  \item \begin{itemize}
  43         \item probability: $1-\frac{dt}{\tau}$
  44         \item momentum contribution of non-colliding electrons:
  45               $f(t)dt+O(dt)^2$
  46         \item momentum per electron at time $t+dt$:\\
  47               \[
  48               p(t+dt)=\left(1-\frac{dt}{\tau}\right)
  49                       \left[p(t)+f(t)dt+O(dt)^2\right]
  50                      =p(t)-\frac{dt}{\tau}p(t)+f(t)+O(dt)^2
  51               \]
  52        \end{itemize}
  53  \item \[
  54        p(t+dt)-p(t)=-\frac{dt}{\tau}p(t)+f(t)dt+O(dt)^2
  55        \]
  56        \[
  57        \frac{p(t+dt)-p(t)}{dt}=-\frac{p(t)}{\tau}+f(t)+\frac{O(dt)^2}{dt}
  58        \]
  59        \[
  60        \stackrel{dt\rightarrow 0}{\Rightarrow} \quad
  61        \frac{dp(t)}{dt}=-\frac{p(t)}{\tau}+f(t)
  62        \]
  63  \item \includegraphics[width=14cm]{hall.eps}
  64  \item \[
  65        \frac{dp}{dt}=-e\left(E+\frac{p}{m}\times B\right)-\frac{p}{\tau}
  66        \]
  67        steady state ($\frac{dp}{dt}=0$):
  68        \begin{itemize}
  69        \item $x$ component: $0=-eE_x-\frac{eB}{m}p_y-\frac{p_x}{\tau}$
  70        \item $y$ component: $0=-eE_y-\frac{eB}{m}p_x-\frac{p_y}{\tau}$
  71        \end{itemize}
  72        setting $j_y$ to zero in the second equation ($\Rightarrow p_y=0$):
  73        \[
  74        E_y=-\left(\frac{B}{m}\right)p_x
  75           \stackrel{j=-ne\frac{p}{m}}{=}-\left(\frac{B}{ne}\right)j_x
  76        \]
  77        \[
  78        \Rightarrow R_H=-\frac{1}{ne}
  79        \]
  80  \item \begin{itemize}
  81         \item electron density: $n=\frac{V\rho}{A_ru}/V=\frac{\rho}{A_ru}$
  82         \item $R_H=-\frac{1}{ne}=-\frac{A_ru}{e\rho}$
  83         \item $j_x=\frac{I}{ld}$
  84         \item $E_{Hall}=E_y=R_HBj_x=\ldots=5.3 \cdot 10^{-5} \, \frac{V}{m}$
  85         \item $U_{Hall}=E_y l=\ldots=-7.95 \, \mu V$
  86        \end{itemize}
  87 \end{enumerate}
  88
  89 \end{document}