prior knowledge Drude

src/3_drude_model.md

@@ -10,6 +10,14 @@ configure_plotting()
 
 _(based on chapter 3 of the book)_  
 
+!!! success "Expected prior knowledge"
+
+    Before the start of this lecture, you should be able to:
+
+    - Write down the force acting on an electron in electric and magnetic fields
+    - Solve Newton's equations of motion
+    - Define concepts of voltage, electrical current, and conductivity
+
 !!! summary "Learning goals"
 
     After this lecture you will be able to:
@@ -37,9 +45,9 @@ tau = 1 # relaxation time
 gamma = .3 # dissipation strength
 a = 1 # acceleration
 dt = .1 # infinitesimal
-T = 20 # simulation time 
+T = 20 # simulation time
 
-v = np.zeros((2, int(T // dt), walkers), dtype=float) # 
+v = np.zeros((2, int(T // dt), walkers), dtype=float) #
 
 scattering_events = np.random.binomial(1, dt/tau, size=v.shape[1:])
 angles = np.random.uniform(high=2*np.pi, size=scattering_events.shape) * scattering_events
@@ -70,9 +78,9 @@ trace = r[:, :100]
 nz_scatters = tuple((np.hstack(scatter_pts[0])[~np.isnan(np.hstack(scatter_pts[0]))],
                     np.hstack(scatter_pts[1])[~np.isnan(np.hstack(scatter_pts[1]))]))
 
-plt.axis([min(nz_scatters[0])-1, 
+plt.axis([min(nz_scatters[0])-1,
           max(nz_scatters[0])+1,
-          min(nz_scatters[1])-1, 
+          min(nz_scatters[1])-1,
           max(nz_scatters[1])+1])
 
 lines = []
@@ -106,7 +114,7 @@ $$
 {\bf j}=-en{\bf v}=\frac{n e^2\tau}{m_{\rm e}}{\bf E}=\sigma{\bf E}\ ,\ \ \sigma=\frac{ne^2\tau}{m_{\rm e}}=ne\mu
 $$
 
-where $n$ is the density of electrons, and $\sigma$ is the conductivity, which is the inverse of resistivity $\rho=\frac{1}{\sigma}$. 
+where $n$ is the density of electrons, and $\sigma$ is the conductivity, which is the inverse of resistivity $\rho=\frac{1}{\sigma}$.
 
 If we now take $j=\frac{I}{A}$ and $E=\frac{V}{l}$, we retrieve Ohm's Law: $\frac{I}{A}=\frac{V}{\rho l}$.
 
@@ -141,7 +149,7 @@ In steady state, there is no current flow in the $y$-direction because the $y$-c
 $$
 {\bf E}=\left(\frac{1}{ne}{\bf j}\times{\bf B}+\frac{m}{ne^2\tau}{\bf j}\right).
 $$
-where the second term is associated with the Drude resistivity derived above. 
+where the second term is associated with the Drude resistivity derived above.
 
 We now introduce the _resistivity matrix_ $\tilde{\rho}$ as ${\bf E}=\tilde{\rho}{\bf j}$, where the diagonal elements are simply $\rho_{xx}=\rho_{yy}=\rho_{zz}=\frac{m}{ne^2\tau}$. The off-diagonal element $\rho_{xy}$ gives us:
 
@@ -156,14 +164,14 @@ While most materials have $R_{\rm H}<0$, interestingly some materials are found
 
   1. Drude theory leads to Ohm's law. Resistivity is caused by electrons that scatter with some characteristic time $\tau$.
   2. The Lorentz force leads to a 'Hall voltage' that is perpendicular to the direction of electric current pushed through a material.
- 
+
 
 ## Exercises
 ### 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 $R_{xy} = V_H/I$ 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? 
+  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?
 
 ### Exercise 2: Motion of an electron in a magnetic and an electric field.
@@ -177,7 +185,7 @@ We first consider an electron in free space, moving in a plane perpendicular to
 
 ### 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 $\rho=1.68\cdot 10^{-8}$ $\Omega$m. Each copper atom provides one free electron.
-  
+
   1. Calculate the Drude scattering time $\tau$ 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 $\lambda$.
   3. The Drude model assumes that $\lambda$ is independent of temperature. How does the electrical resistivity $\rho$ depend on temperature under this assumption? Sketch $\rho(T)$.
@@ -187,9 +195,8 @@ We first consider an electron in free space, moving in a plane perpendicular to
 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:
 
 $$\mathbf{E} = \begin{pmatrix} \rho_{xx} & \rho_{xy} \\ \rho_{yx} & \rho_{yy} \end{pmatrix} \mathbf{J}$$
- 
+
   1. Sketch $\rho_{xx}$ and $\rho_{xy}$ 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 $\mathbf{E}$. 
-  3. Sketch $\sigma_{xx}$ and $\sigma_{xy}$ 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 $\mathbf{E}$.
+  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?
-