@@ -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
%matplotlibinline
...
...
@@ -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?
