2 \documentclass[a4paper,11pt]{article}
3 \usepackage[activate]{pdfcprot}
7 \usepackage[german]{babel}
8 \usepackage[latin1]{inputenc}
9 \usepackage[T1]{fontenc}
13 \usepackage[dvips]{graphicx}
14 \graphicspath{{./img/}}
20 \setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
21 \setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
22 \setlength{\oddsidemargin}{-10mm}
23 \setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm}
24 \setlength{\textheight}{26cm} \setlength{\headsep}{0cm}
26 \renewcommand{\labelenumi}{(\alph{enumi})}
32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 4}
40 \section{Hall effect and magnetoresistance}
41 The Hall effect refers to the potential difference (Hall voltage)
42 on the opposite sides of an electrical conductor
43 through which an electric current is flowing,
44 created by a magnetic field applied perpendicular to the current.
45 Edwin Hall discovered this effect in 1879.
47 Consider the following scenario:
48 An electric field $E_x$ is applied to a wire extending in $x$-direction
49 and a current density $j_x$ is flowing in that wire.
50 There is a magnetic field $B$ pointing in the positive $z$-direction.
51 Electrons are deflected in the negative $y$-direction
52 due to the Lorentz force $F_L=-evB$
53 until they run against the sides of the wire.
54 An electric field $E_y$ builds up opposing the Lorentz force
55 and thus preventing further electron accumulation at the sides.
56 The two quantities of interest are:
58 \item the magnetoresistance
60 \rho(B) = \frac{E_x}{j_x} \textrm{ and}
62 \item the Hall coefficient
64 R_H(B) = \frac{E_y}{j_xB} \textrm{ .}
67 In this tutorial the treatment of the Hall problem is based on a simple
70 First of all the effect of individual electron collisions can be expressed
71 by a frictional damping term into the equation of motion for the momentum
75 \item Recall the Drude model.
76 Given the momentum per electron $p(t)$ at time t
77 calculate the momentum per electron $p(t+dt)$
78 an infinitesimal time $dt$ later.
79 {\bf Hint:} What is the probability of an electron taken at random at
80 time $t$ to not suffer a collision before time $t+dt$?
81 If not experiencing a collision it simply evolves under the influence
83 Combine contributions of the order of $(dt)^2$ to the term
85 \item Write down the equation of motion for the momentum per electron
86 by dividing the above result by $dt$
87 and taking the limit $dt\rightarrow 0$.
88 \item Sketch a schematic view of Hall's experiment.
89 \item Find an expression for the Hall coefficient.
90 {\bf Hint:} Insert an appropriate force into the equation of motion
91 for the momentum per electron.
92 Consider the steady state and acquire the equations
93 for the $x$ and $y$ component of the vector equation.
94 To find an expression for the Hall coefficient use the second equation
95 and the fact that there must not be transverse current $j_y$
96 while determining the Hall field.