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T. van der Sar authoredT. van der Sar authored
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 \mum @ 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.