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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 $F_L=-ev\times B$
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 in the equation of motion for the momentum
73 By inserting the forces acting on the elecron into the equation of motion
74 the expression for the Hall coefficient $R_H=-\frac{1}{ne}$ can be found.
77 \item Recall the Drude model.
78 Given the momentum per electron $p(t)$ at time t
79 calculate the momentum per electron $p(t+dt)$
80 an infinitesimal time $dt$ later.
81 {\bf Hint:} What is the probability of an electron taken at random at
82 time $t$ to not suffer a collision before time $t+dt$?
83 If not experiencing a collision it simply evolves under the influence
85 Combine contributions of the order of $(dt)^2$ to the term
87 \item Write down the equation of motion for the momentum per electron
88 by dividing the above result by $dt$
89 and taking the limit $dt\rightarrow 0$.
90 \item Sketch a schematic view of Hall's experiment.
91 \item Find an expression for the Hall coefficient.
92 {\bf Hint:} Insert an appropriate force into the equation of motion
93 for the momentum per electron.
94 Consider the steady state and acquire the equations
95 for the $x$ and $y$ component of the vector equation.
96 To find an expression for the Hall coefficient use the second equation
97 and the fact that there must not be transverse current $j_y$
98 while determining the Hall field.
99 \item Calculate the Hall field for a rectangular wire
100 (width: $l=15\, cm$, thickness: $d=4\, mm$) made out of silver
101 if there is a current $I=200\, A$ flowing
102 and a magnetic field $B=1.5\, Vs/m^2$ applied
103 perpendicular to the current.
104 Silver has the relative atomic mass $A_r=107.87$ and
105 density $\rho=10.5\, g/cm^3$.
106 Assume that there is one valence electron per atom.
107 The atomic mass unit is $u=1.6605 \cdot 10^{-27} \, kg$
108 and $e=1.602 \cdot 10^{-19} \, As$.
109 {\bf Hint:} Start by calculating the Hall coefficient of silver first.