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# + [markdown] nbpresent={"id": "1a745ba6-5e68-4ad1-b6a8-38c95d3a080c"} slideshow={"slide_type": "slide"}
# # 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)

# + [markdown] slideshow={"slide_type": "slide"}
# # 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**: $e < 0$!

# + [markdown] slideshow={"slide_type": "fragment"}
# * $\mathbf{B} = 0\quad \Rightarrow\quad \mathbf{p} = e \mathbf{E} t,$ electrons accelerate forever!

# + [markdown] slideshow={"slide_type": "fragment"}
# * Electrons collide with impurities, lattice vibrations (phonons)  
#   $\Rightarrow$ scattering makes $\mathbf{p}$ random.

# + [markdown] nbpresent={"id": "b8928c20-20fa-4776-b2c4-c45342ce0f14"} slideshow={"slide_type": "fragment"}
# * Simplest model for scattering:  
#   $$\frac{d \mathbf{p}}{dt} = - \frac{\mathbf{p}}{\tau}$$
#   $\tau$ average time between scattering events.

# + [markdown] nbpresent={"id": "934aa51c-31a0-4aa6-a9ca-5af7fe96a948"} slideshow={"slide_type": "subslide"}
# # Collision rates
#
# $\tau^{-1}$ is rate of collisions; additive from phonons and crystal disorder:
#
# ![](figures/rates.svg)

# + [markdown] nbpresent={"id": "c53b0a02-ce4f-4588-8840-b4f831e79d54"} slideshow={"slide_type": "subslide"}
# # [Drude Model](https://en.wikipedia.org/wiki/Drude_model):
#
# > Electrons are moved by electric and magnetic fields, 
# > and slowed by friction  
# > – <NAME> 1900, <NAME> 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!

# + [markdown] nbpresent={"id": "8fbc3c99-e758-42b0-94e2-393d586619cf"} slideshow={"slide_type": "subslide"}
# # 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 $\sim 10^8$.

# + [markdown] nbpresent={"id": "c134ea81-3daa-4a91-8b09-a938db828117"} slideshow={"slide_type": "subslide"}
# # 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!

# + [markdown] nbpresent={"id": "2c877278-4eba-4b74-9dde-6e9227a131e8"} slideshow={"slide_type": "slide"}
# # [Hall effect](https://en.wikipedia.org/wiki/Hall_effect)
# Switch to $\mathbf{B} \neq 0$ 
#
# ![](figures/hall_1.svg)

# + [markdown] nbpresent={"id": "aaac3be9-a562-44c1-8118-ba45bae8118d"} slideshow={"slide_type": "subslide"}
# # Determining $E$
#
# Force balance: $ 0 = e\left(\mathbf{E} + \mathbf{v}\times \mathbf{B}\right)-\mathbf{p}/\tau $
#
# ![](figures/hall_2.svg)

# + [markdown] nbpresent={"id": "2c877278-4eba-4b74-9dde-6e9227a131e8"} slideshow={"slide_type": "subslide"}
# # 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)!

# + [markdown] nbpresent={"id": "c617c070-a14c-42ea-841a-38e5d0f0c861"} slideshow={"slide_type": "slide"}
# # Limitations of the Drude-Lorentz model
#
# * Interactions between electrons are very strong:  
#   Addressed by Landau using [Fermi liquid theory](https://en.wikipedia.org/wiki/Fermi_liquid_theory)  
#   Result: can introduce **quasiparticles** that behave like non-interacting electrons!
# * What about Fermi sea?

# + [markdown] nbpresent={"id": "c3962ed4-4f67-418a-b1ea-a48d50903ebf"} slideshow={"slide_type": "slide"}
# # 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:
#
# ![](figures/fermi_drude.svg)

# + [markdown] slideshow={"slide_type": "slide"}
# # 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

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