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

\section{Hall effect and magnetoresistance}
The Hall effect refers to the potential difference (Hall voltage)
on the opposite sides of an electrical conductor
through which an electric current is flowing,
created by a magnetic field applied perpendicular to the current.
Edwin Hall discovered this effect in 1879.

Consider the following scenario:
An electric field $E_x$ is applied to a wire extending in $x$-direction
and a current density $j_x$ is flowing in that wire.
There is a magnetic field $B$ pointing in the positive $z$-direction.
Electrons are deflected in the negative $y$-direction
due to the Lorentz force $F_L=-evB$
until they run against the sides of the wire.
An electric field $E_y$ builds up opposing the Lorentz force
and thus preventing further electron accumulation at the sides.
The two quantities of interest are:
\begin{itemize}
 \item the magnetoresistance
       \[
       \rho(B) = \frac{E_x}{j_x} \textrm{ and}
       \]
 \item the Hall coefficient
       \[
       R_H(B) = \frac{E_y}{j_xB} \textrm{ .}
       \]
\end{itemize}
In this tutorial the treatment of the Hall problem is based on a simple
Drude model analysis.
\\\\
First of all the effect of individual electron collisions can be expressed
by a frictional damping term into the equation of motion for the momentum 
per electron.

\begin{enumerate}
 \item Recall the Drude model. 
       Given the momentum per electron $p(t)$ at time t
       calculate the momentum per electron $p(t+dt)$
       an infinitesimal time $dt$ later.
       {\bf Hint:} What is the probability of an electron taken at random at
       time $t$ to not suffer a collision before time $t+dt$?
       If not experiencing a collision it simply evolves under the influence
       of the force $f(t)$.
       Combine contributions of the order of $(dt)^2$ to the term
       $O(dt)^2$.
 \item Write down the equation of motion for the momentum per electron
       by dividing the above result by $dt$
       and taking the limit $dt\rightarrow 0$.
 \item Sketch a schematic view of Hall's experiment.
 \item Find an expression for the Hall coefficient.
       {\bf Hint:} Insert an appropriate force into the equation of motion
       for the momentum per electron.
       Consider the steady state and acquire the equations
       for the $x$ and $y$ component of the vector equation.
       To find an expression for the Hall coefficient use the second  equation
       and the fact that there must not be transverse current $j_y$
       while determining the Hall field.
\end{enumerate}

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