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[lectures/latex.git] / solid_state_physics / tutorial / 1_04.tex

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+\pdfoutput=0
+\documentclass[a4paper,11pt]{article}
+\usepackage[activate]{pdfcprot}
+\usepackage{verbatim}
+\usepackage{a4}
+\usepackage{a4wide}
+\usepackage[german]{babel}
+\usepackage[latin1]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{ae}
+\usepackage{aecompl}
+\usepackage[dvips]{graphicx}
+\graphicspath{{./img/}}
+\usepackage{color}
+\usepackage{pstricks}
+\usepackage{pst-node}
+\usepackage{rotating}
+
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+\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
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+\setlength{\textheight}{26cm} \setlength{\headsep}{0cm}
+
+\renewcommand{\labelenumi}{(\alph{enumi})}
+
+\begin{document}
+
+% header
+\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}
+
+\end{document}