finished solution for tutorial 4 + added hall fig

[lectures/latex.git] / solid_state_physics / tutorial / 1_04s.tex

diff --git a/solid_state_physics/tutorial/1_04s.tex b/solid_state_physics/tutorial/1_04s.tex
index b5d60e3..d99b994 100644 (file)
--- a/solid_state_physics/tutorial/1_04s.tex
+++ b/solid_state_physics/tutorial/1_04s.tex
@@ -60,7 +60,30 @@
        \stackrel{dt\rightarrow 0}{\Rightarrow} \quad
        \frac{dp(t)}{dt}=-\frac{p(t)}{\tau}+f(t)
        \]
- \item
+ \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}
 
 \end{document}