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% header
\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 4 - proposed solutions}
\end{center}

\section{Hall effect and magnetoresistance}
\begin{enumerate}
 \item \begin{itemize}
        \item probability: $1-\frac{dt}{\tau}$
	\item momentum contribution of non-colliding electrons:
	      $f(t)dt+O(dt)^2$
        \item momentum per electron at time $t+dt$:\\
	      \[
	      p(t+dt)=\left(1-\frac{dt}{\tau}\right)
	              \left[p(t)+f(t)dt+O(dt)^2\right]
	             =p(t)-\frac{dt}{\tau}p(t)+f(t)+O(dt)^2
	      \]
       \end{itemize}
 \item \[
       p(t+dt)-p(t)=-\frac{dt}{\tau}p(t)+f(t)dt+O(dt)^2
       \]
       \[
       \frac{p(t+dt)-p(t)}{dt}=-\frac{p(t)}{\tau}+f(t)+\frac{O(dt)^2}{dt}
       \]
       \[
       \stackrel{dt\rightarrow 0}{\Rightarrow} \quad
       \frac{dp(t)}{dt}=-\frac{p(t)}{\tau}+f(t)
       \]
 \item \includegraphics[width=14cm]{hall.eps}
 \item \[
       \frac{dp}{dt}=-e\left(E+\frac{p}{m}\times B\right)-\frac{p}{\tau}
       \]
       steady state ($\frac{dp}{dt}=0$):
       \begin{itemize}
       \item $x$ component: $0=-eE_x-\frac{eB}{m}p_y-\frac{p_x}{\tau}$
       \item $y$ component: $0=-eE_y-\frac{eB}{m}p_x-\frac{p_y}{\tau}$
       \end{itemize}
       setting $j_y$ to zero in the second equation ($\Rightarrow p_y=0$):
       \[
       E_y=-\left(\frac{B}{m}\right)p_x
          \stackrel{j=-ne\frac{p}{m}}{=}-\left(\frac{B}{ne}\right)j_x
       \]
       \[
       \Rightarrow R_H=-\frac{1}{ne}
       \]
 \item \begin{itemize}
        \item electron density: $n=\frac{V\rho}{A_ru}/V=\frac{\rho}{A_ru}$
        \item $R_H=-\frac{1}{ne}=-\frac{A_ru}{e\rho}$
	\item $j_x=\frac{I}{ld}$
	\item $E_{Hall}=E_y=R_HBj_x=\ldots=5.3 \cdot 10^{-5} \, \frac{V}{m}$
	\item $U_{Hall}=E_y l=\ldots=-7.95 \, \mu V$
       \end{itemize}
\end{enumerate}

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