Electrical properties of electrons

hi again!

This lecture:

• Drude model and why it works
• Electrical conductivity
• Hall effect

Edit these notes online at http://tiny.cc/solidstatephys/drude.ipynb
(or download from http://tiny.cc/solidstate_src/drude.ipynb)

Electrons in fields

• Start with Lorentz force: $$ \frac{d\mathbf{p}}{dt} = e\left(\mathbf{E} + \frac{1}{m}\mathbf{p}\times \mathbf{B}\right) $$ Attention: $e < 0$!

• $\mathbf{B} = 0\quad \Rightarrow\quad \mathbf{p} = e \mathbf{E} t,$ electrons accelerate forever!

• Electrons collide with impurities, lattice vibrations (phonons)
  $\Rightarrow$ scattering makes $\mathbf{p}$ random.

• Simplest model for scattering:
  $$\frac{d \mathbf{p}}{dt} = - \frac{\mathbf{p}}{\tau}$$ $\tau$ average time between scattering events.

Collision rates

$\tau^{-1}$ is rate of collisions; additive from phonons and crystal disorder:

Drude Model:

Electrons are moved by electric and magnetic fields, and slowed by friction
– Paul Drude 1900, Hendrik Lorentz 1905 (quotation approximate)

$$ \frac{d\mathbf{p}}{dt} = e\left(\mathbf{E} + \frac{1}{m}\mathbf{p}\times \mathbf{B}\right)-\frac{\mathbf{p}}{\tau} $$

No interactions, 25 years before the Pauli exclusion principle!

Charge mobility:

In steady state: $$\mathbf{p} = \text{const};\quad 0 = e\mathbf{E} - m\mathbf{v}/\tau\Rightarrow \mathbf{v} = \frac{e \tau}{m}\mathbf{E}$$

Mobility $\mu = e\tau / m$ [cm$^2$/Vs], varies from $\sim 1$ to $\sim 10^8$.

Electric conductivity

Current density $\mathbf{j} = n e \mathbf{v}$ ($n$ is electron density) $$\mathbf{j} = e n \mu \mathbf{E},$$ so electric conductivity $\sigma = ne\mu = e^2 n\tau/m$.

Sanity check: $j = \frac{I}{A};\quad E = \frac{V}{L}; \quad I = \frac{A}{L}\sigma V$ — this is just Ohm's law!

Hall effect

Switch to $\mathbf{B} \neq 0$

Determining $E$

Force balance: $ 0 = e\left(\mathbf{E} + \mathbf{v}\times \mathbf{B}\right)-\mathbf{p}/\tau $

Hall conductance

$\mathbf{E} = \mathbf{p}/e\tau -\mathbf{v}\times \mathbf{B}$

Two components of $\mathbf{E}$:

• $\mathbf{E}_\text{ext} \parallel \mathbf{j}$ same as with $\mathbf{B} = 0$
• Hall field: $\mathbf{E}_\text{H} = \mathbf{j}\times\mathbf{B}/ne$ measures charge carrier concentration.

Hall coefficient $(ne)^{-1}$ is sometimes positive
$\Rightarrow$ particles with positive charge (or negative mass)!

Limitations of the Drude-Lorentz model

• Interactions between electrons are very strong:
  Addressed by Landau using Fermi liquid theory
  Result: can introduce quasiparticles that behave like non-interacting electrons!
• What about Fermi sea?

Fermi vs Drude

Fermi velocity $v_\text{F} = \sqrt{2E_\text{F} / m} \sim 10^6 \text{m/s}$
Drift velocity $v = \mu E \sim \text{mm/s} \ll v_F$ !

Explanation: most Fermi sea is "inert", we only see effects of the Fermi surface:

Conclusions

• Drude-Lorentz model describes electric properties of conductors
• New concepts: scattering rate/time, mobility, Hall coefficient
• Works surprisingly well with interactions and Fermi statistics