Update exercises and answers to drude model lecture

src/3_drude_model.md

@@ -243,8 +243,9 @@ While most materials have $R_\mathrm{H}<0$, interestingly some materials are fou
 
 ## Conclusions
 
-1. Drude theory is a microscopic justification of the Ohm's law. Resistivity is caused by electrons that scatter with some characteristic time $τ$.
-2. The Lorentz force leads to the Hall voltage that is perpendicular to the direction of electric current pushed through a material.
+1. Drude theory is a microscopic justification of the Ohm's law.
+2. We can calculate the resitivity from the characteristic scattering time $\tau$.
+3. The Lorentz force leads to the Hall voltage that is perpendicular to the direction of electric current pushed through a material and the magnetic field.
 
 ## Exercises
 
@@ -253,38 +254,46 @@ While most materials have $R_\mathrm{H}<0$, interestingly some materials are fou
 1. How does the resistance of a purely 2D material depend on its size?
 2. Check that the units of mobility and the Hall coefficient are correct.  
    (As you should always do!)
-3. Explain why the scattering rates due to different types of scattering events add up.
+3. Explain why the scattering rates due to different types of scattering events add up in the way they do.
 
 ### 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 $x$.
+We apply a magnetic field $\bf B$ in the positive $z$-direction 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 the positive $x$-direction.
 
   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?
+
+??? question "What is the relation between the electric field and the electric potential?"
+    $V_b - V_a = -\oint_{\Gamma} \mathbb{E} \cdot d\mathbb{l}$ if $\Gamma$ is a path from $a$ to $b$.
+  2. Assuming we control the magnetic field $\mathbb{B}$, what quantity can we extract from a measurement of the Hall resistance? Would a large or a small magnetic field 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 $ρ_{xx}$. 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?
 
 ### Exercise 2: Motion of an electron in a magnetic and an electric field.
-We first consider an electron in free space, moving in a plane perpendicular to a magnetic field $\mathbf{B}$ with velocity $\mathbf{v}$.
+We consider an electron in free space experiencing a magnetic field $\mathbf{B}$ pointing in the positive $z$-direction.
+Assume that the electron starts at the origin with a velocity $v_0$ in the positive $x$-direction.
 
   1. Write down the Newton's equation of motion for the electron, compute $\frac{d\mathbf{v}}{{dt}}$.
   2. What is the shape of the motion of the electron? Calculate the characteristic frequency and time-period $T_c$ of this motion for $B=1$ Tesla.
-  3. Now we accelerate the electron by adding an electric field $\mathbf{E}$ that is perpendicular to $\mathbf{B}$. Adjust the differential equation for $\frac{d\mathbf{v}}{{dt}}$ found in (1) to include $\mathbf{E}$. Sketch the motion of the electron.
-  4. We now consider an electron in a metal. Include the Drude scattering time $τ$ into the differential equation for the velocity you formulated in 4.
+  3. Now we accelerate the electron by adding an electric field $\mathbf{E} = E \hat{x}$. Adjust the differential equation for $\frac{d\mathbf{v}}{{dt}}$ found in (1) to include $\mathbf{E}$. Sketch the motion of the electron.
+<!---
+we felt that this subquestion was out of place. The whole exercise is about a single electron, but this final question is suddenly about the average of an ensemble of electrons
+
+4. We now consider an electron in a metal. Include the Drude scattering time $τ$ into the differential equation for the velocity you formulated in 4.
+--->  
 
 
 ### Exercise 3: Temperature dependence of resistance in the Drude model
    We consider copper, which has a density of 8960 kg/m$^3$, an atomic weight of 63.55 g/mol, and a room-temperature resistivity of $ρ=1.68\cdot 10^{-8}$ $\Omega$m. Each copper atom provides one free electron.
 
   1. Calculate the Drude scattering time $τ$ at room temperature.
-  2. Assuming that electrons move with the thermal velocity $\langle v \rangle = \sqrt{\frac{8k_BT}{\pi m}}$, calculate the electron mean free path $\lambda$.
+  2. Assuming that electrons move with the thermal velocity $\langle v \rangle = \sqrt{\frac{8k_BT}{\pi m}}$, calculate the electron mean free path $\lambda$, defined as the average distance an electron travels in between scattering events.
   3. The Drude model assumes that $\lambda$ is independent of temperature. How does the electrical resistivity $ρ$ depend on temperature under this assumption? Sketch $ρ(T)$.
   5. Compare your sketch of $ρ(T)$ with that in the lecture notes. In what respect do they differ? Discuss possible reasons for differences.
 
 ### Exercise 4: The Hall conductivity matrix and the Hall coefficient
-We apply a magnetic field $\bf B$ perpendicular to a current carrying 2D sample in the xy plane. In this situation, the electric field $\mathbf{E}$ is related to the current density $\mathbf{J}$ by the resistivity matrix:
+We apply a magnetic field $\bf B$ in the positive $z$-directino to a current carrying 2D sample in the xy plane. In this situation, the electric field $\mathbf{E}$ is related to the current density $\mathbf{J}$ by the resistivity matrix:
 
-$$\mathbf{E} = \begin{pmatrix} ρ_{xx} & ρ_{xy} \\ ρ_{yx} & ρ_{yy} \end{pmatrix} \mathbf{J}$$
+$$\mathbf{E} = \begin{pmatrix} ρ_{xx} & ρ_{xy} \\ ρ_{yx} & ρ_{yy} \end{pmatrix} \mathbf{j}$$
 
   1. Sketch $ρ_{xx}$ and $ρ_{xy}$ as a function of the magnetic field $\bf B$.
-  2. Invert the resistivity matrix to obtain the conductivity matrix $$\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} $$, allowing you to express $\mathbf{J}$ as a function of $\mathbf{E}$.
+  2. Assuming it is not singular, invert the resistivity matrix to obtain the conductivity matrix $$\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} $$, allowing you to express $\mathbf{j}$ as a function of $\mathbf{E}$.
   3. Sketch $\sigma_{xx}$ and $\sigma_{xy}$ as a function of the magnetic field $\bf B$.
   4. Give the definition of the Hall coefficient. What does the sign of the Hall coefficient indicate?

src/3_drude_model_solutions.md

@@ -17,9 +17,9 @@ $R_{xy}$ = $-\frac{B}{ne}$, so it does not depend on the sample geometry.
 
 2.
 
-If hall resistance and charge density are known, magnetic field is calculated from $R_{xy} = -\frac{B}{ne}$.
+If hall resistance and magnetic field are known, the charge density is calculated from $R_{xy} = -\frac{B}{ne}$.
 
-As $V_x = -\frac{I_x}{ne}B$, smaller n ensures large hall voltages easy to measure.
+As $V_x = -\frac{I_x}{ne}B$, a stronger field makes large Hall voltages easier to measure.
 
 3.
 
@@ -51,10 +51,10 @@ $$
 
 2.
 
-Compute $v_x(t)$ and $v_y(t)$ by solving the differential equations in 1. Calculate the positions $x(t)$ and $y(t)$ from $v_x(t)$ and $v_y(t)$ respectively with the initial conditions: $v_x=u_x$ and $v_y=0$. It results in
+Compute $v_x(t)$ and $v_y(t)$ by solving the differential equations in 1 using the Ansatz $v_x(t) = A cos (\omega t)$, $v_y(t) = A sin(\omega t)$. Calculate the positions $x(t)$ and $y(t)$ from $v_x(t)$ and $v_y(t)$ respectively with the initial conditions: $v_x=u_0$ and $v_y=0$. It results in
 
 $$
-(x(t) - x_0)^2 + (y(t) - y_0)^2 = \frac{u_x^2}{\omega_c^2}
+(x(t) - x_0)^2 + (y(t) - y_0)^2 = \frac{u_0^2}{\omega_c^2}
 $$
 
 where $\omega_c = \frac{e\bf B}{m}$. This represents the circular motion.
@@ -85,11 +85,14 @@ $$
 
 This represents the [cycloid](https://en.wikipedia.org/wiki/Cycloid#/media/File:Cycloid_f.gif) where the motion is along along x with velocity $\frac{E}{B}$.
 
+<!---
 4.
+See 3_drude_model.md
 
 $$
 m\left(\frac{d\bf v}{dt} + \frac{\bf v}{\tau}\right) = -e(\bf E + \bf v \times \bf B)
 $$
+-->
 
 ### Exercise 3: Temperature dependence of resistance in the Drude model