@@ -88,7 +88,7 @@ While most materials have $R_{\rm H}>0$, interestingly some materials are found
### Exercise 1: Extracting quantities from basic Hall measurements
We apply a magnetic field $\bf B$ perpendicular to a planar (two-dimensional) sample that sits in the $xy$ plane. The sample has width $W$ in the $y$-direction, length $L$ in the $x$-direction and we apply a current $I$ along $x$.
1. Suppose we measure a Hall voltage $V_H$. Express the Hall resistance $R_{xy} = V_H/I$ in terms of the Hall resistivity $\rho_{xy}$. Does $R_{xy}$ depend on the geometry of the sample? Also express $R_{xy}$ in terms of the Hall coefficient $R_H$.
1. Suppose we measure a Hall voltage $V_H$. Express the Hall resistance $R_{xy} = V_H/I$ as a function of magnetic field. Does $R_{xy}$ depend on the geometry of the sample? Also express $R_{xy}$ in terms of the Hall coefficient $R_H$.
2. Assuming we know the charge density $n$ in the sample, what quantity can we extract from a measurement of the Hall resistance? Would a large or a small electron density give a Hall voltage that is easier to measure?
3. Express the longitudinal resistance $R=V/I$, where $V$ is the voltage difference over the sample along the $x$ direction, in terms of the longitudinal resistivity $\rho_{xx}$. Suppose we extracted $n$ from a measurement of the Hall resistance, what quantity can we extract from a measurement of the longitudinal resistance? Does the result depend on the geometry of the sample?
