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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}