3_drude_model.md

from matplotlib import pyplot

import numpy as np

from common import draw_classic_axes, configure_plotting

configure_plotting()

(based on chapter 3 of the book)

!!! summary "Learning goals"

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.

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:

• 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)
  .

For now we will consider only an electric field (i.e.

{\bf B}=0

). What velocity do electrons acquire in-between collisions?

{\bf v}=-\int_0^\tau\frac{e{\bf E}}{m_{\rm e}}{\rm d}t=-\frac{e\tau}{m_{\rm e}}{\bf E}=-\mu{\bf E}

Here we have defined the quantity

\mu\equiv e\tau/m_{\rm e}

, which is the mobility. If we have a density

n

of electrons in our solid, the current density

{\bf j}

[A/m

^2

] then becomes:

{\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

\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}

.

Scattering is caused by collisions with:

• Phonons:
  \tau_{\rm ph}(T)
  (
  \tau_{\rm ph}\rightarrow\infty
  as
  T\rightarrow 0
  )
• Impurities/vacancies:
  \tau_0

Scattering rate

\frac{1}{\tau}

:

\frac{1}{\tau}=\frac{1}{\tau_{\rm ph}(T)}+\frac{1}{\tau_0}\ \Rightarrow\ \rho=\frac{1}{\sigma}=\frac{m}{ne^2}\left( \frac{1}{\tau_{\rm ph}(T)}+\frac{1}{\tau_0} \right)\equiv \rho_{\rm ph}(T)+\rho_0

Matthiessen's Rule (1864). Solid (dashed) curve:

\rho(T)

for a pure (impure) crystal.

How fast do electrons travel through a copper wire? Let's take

E

= 1 volt/m,

\tau

~ 25 fs (Cu,

T=

300 K).

\rightarrow v=\mu E=\frac{e\tau}{m_{\rm e}}E=\frac{10^{-19}\times 2.5\times 10^{-14}}{10^{-30}}=2.5\times10^{-3}=2.5

mm/s ! (= 50

\mu

m @ 50 Hz AC)

Hall effect

Consider a conductive wire in a magnetic field

{\bf B} \rightarrow

electrons are deflected in a direction perpendicular to

{\bf B}

and

{\bf j}

.

{\bf E}_{\rm H}

= Hall voltage, caused by the Lorentz force.

In equilibrium, assuming that the average velocity becomes zero after every collision:

\frac{mv_x}{\tau}=-eE

The

y

-component of the Lorentz force

-e{\bf v}_x\times{\bf B}

is being compensated by the Hall voltage

{\bf E}_{\rm H}={\bf v}_x\times{\bf B}=\frac{1}{ne}{\bf j}\times{\bf B}

. The total electric field then becomes

{\bf E}=\left(\frac{1}{ne}{\bf j}\times{\bf B}+\frac{m}{ne^2\tau}{\bf j}\right)

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:

\rho_{xy}=\frac{B}{ne}\equiv -R_{\rm H}B

where

R_{\rm H}=-\frac{1}{ne}

is the Hall resistance. So by measuring the Hall resistance, we can obtain

n

, the density of free electrons in a material.

While most materials have

R_{\rm H}>0

, interestingly some materials are found to have

R_{\rm H}<0

. This would imply that the charge carriers either have a positive charge, or a negative mass. We will see later (chapter 17) how to interpret this.