From a428db34228d4c41c0ad55d556f88d96a29d43f3 Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Mon, 17 Feb 2020 22:31:06 +0100
Subject: [PATCH] drude intro
---
src/3_drude_model.md | 30 ++++++++++++++++++++++--------
1 file changed, 22 insertions(+), 8 deletions(-)
diff --git a/src/3_drude_model.md b/src/3_drude_model.md
index c6f6a074..a6623e98 100644
--- a/src/3_drude_model.md
+++ b/src/3_drude_model.md
@@ -22,16 +22,26 @@ _(based on chapter 3 of the book)_
After this lecture you will be able to:
- - discuss the basics of Drude theory, which describes electron motion in metals.
- - use Drude theory to analyze transport of electrons through conductors in electric and magnetic fields.
- - describe central terms such as the mobility and the Hall resistance.
+ - Discuss the basics of Drude theory, which describes electron motion in metals.
+ - Use Drude theory to analyze transport of electrons through conductors in electric and magnetic fields.
+ - Describe concepts such as electron mobility and the Hall resistance.
-### Drude theory
-Ohm's law states that $V=IR=I\rho\frac{l}{A}$. In this lecture we will investigate where this law comes from. We will use the theory developed by Paul Drude in 1900, which is based on three assumptions:
+## Drude theory
-- Electrons have an average scattering time $\tau$.
-- At each scattering event an electron returns to momentum ${\bf p}=0$.
-- In-between scattering events electrons respond to the Lorentz force ${\bf F}_{\rm L}=-e\left({\bf E}+{\bf v}\times{\bf B}\right)$.
+The Ohm's law, familiar to most from the high school, states that voltage is proportional to current $V=IR$.
+Since we are dealing with *material properties*, let us rewrite this into a relation that does not depend on the material geometry:
+$$V = I ρ \frac{l}{A} ⇒ E = ρ j,$$
+where $E≡V/l$ is the electric field, $ρ$ the material resistivity, and $j≡I/A$ the current through a unit cross-section.
+Our goal is to understand how this law may arise microscopically, starting from reasonable (but definitely incomplete) assumptions.
+
+- Electrons fly freely, and scatter at random uncorrelated times, with an average scattering time $τ$.
+- After each scattering event, the electron's momentum randomizes with a zero average $⟨\mathbf{p}⟩=0$.
+- The Lorentz force ${\bf F}_{\rm L}=-e\left({\bf E}+{\bf v}\times{\bf B}\right)$ acts on the electrons.
+
+The first assumption here is the least obvious: why the time between scattering events not depend on e.g. electron velocity?
+Also observe that for now we forget that electrons also have fermionic statistics—this will come up in the next lecture, and turns out also helps to justify the first assumption.
+
+Even under these minimal assumptions, our problem seems hard. This is how electron motion looks like under these assumptions:
```python
%matplotlib inline
@@ -102,6 +112,10 @@ plt.axis('off');
HTML(anim.to_html5_video())
```
+Stop here for a second, and ask yourself how you would deal with this problem?
+
+---
+
We start by considering only an electric field (_i.e._ ${\bf B}=0$). What velocity do electrons acquire in-between collisions?
$$
--
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