Free electron model of metals

Classical Electron Model

Drude transport theory

The Drude theory assumes that movement of electrons can be described classically. The model resembles a pinball machine where the electrons accelerate, hit a scattering surface (positive ion), and then begin accelerating once more.

During the acceleration phase the velocity gained by the electron is described as:

$\Delta v=at=-{\frac {eE}{m}}t$

This gives the electron a total velocity

$v=v_{0}+\Delta v\!$

Then the electron scatters off a surface. The distance between scattering surfaces is called the mean free path, $l$.

Note that on average $v_{0}$ is equal to zero since the movement is in completely random directions. So now we can calculate the average total velocity to be

$<v>\ =\ <v_{0}+\Delta v>\ =\ <v_{0}>+<\Delta v>\ =\ <\Delta v>\!$

$<\Delta v>\ \sim {\overrightarrow {E}}$

Now using the equation for current

${\overrightarrow {j}}=-ne<v>=-ne<\Delta v>\ =-{\frac {ne^{2}}{m}}<t>$

Here $<t>$ is the average time between collisions, which is called the scatter time, $\tau$.

This gives us the Drude formula ,

$\sigma =-{\frac {ne^{2}}{m}}\tau$, where $\tau ={\frac {l}{v_{0}}}$, and $\sigma$ is conductivity.

And from E&M recall that the resistivity of the material is:

$\rho ={\frac {1}{\sigma }}$

Also another often quoted value is the mobility, the proportionality constant between the velocity of the electrons and the electric field it is placed in, of the material which is:

$\mu ={\frac {e\tau }{m_{e}}}={\frac {\sigma }{ne}}$

Hall effect

When a metal is placed in a magnetic field $\mathbf {B}$ and a current density $\mathbf {j}$ is passed through it, a transverse electric field $\mathbf {E_{H}}$ is set up given by

$\mathbf {E_{H}} =R_{H}\mathbf {B} \ \times \ \mathbf {j}$

This is the Hall Effect and $R_{H}$ is known as the Hall coefficient.

Limitations of the classical electron model

Looking at the Drude model again and taking into account the Equipartition Theory:

${\frac {1}{2}}mv_{0}^{2}=k_{B}T$

we can show that the scattering time should be proportional to $T$

$v_{0}={\sqrt {\frac {2k_{B}T}{m}}}\sim {\sqrt {T}}$

$\Rightarrow \tau \sim {\frac {1}{\sqrt {T}}}$

However, this is wrong since velocity is not proportional to $T$. When $T=0$ there is still quantum uncertainty.

Pauli Principle and Fermi-Dirac statistics

Fermi Dirac Probability Distribution

$f(E)={\frac {1}{e^{\frac {E-E_{F}}{k_{B}T}}+1}}$