The Drude theory is basically a kinetic theory
for electrons. It is very crude in the sense
that it assumes electrons to move freely in
metals. But despite its crudeness it works
remarkably well. Even to this day for many
discussions it is often the first model we
look at when we want to get an intuition.
So let us get an idea what the Drude model
is about.
In 1900 german physicist Paul Drude developed
this theory for electrons in metals. He realized
that he could apply Boltzmann’s kinetic
theory of gasses to understand the electrons
motion.
There are three assumptions to consider:
First, we introduce a scattering time tau.
It says that the probability of scattering
within a time interval dt is dt/tau. tau should
be understood as a phenomenological parameter.
For the kinetic theory of molecules we could
estimate the scattering time based on the
velocity, density and scattering cross-section.
But, it is hard to estimate the scattering
time for electrons, as they can scatter with
many things besides other electrons in a solid.
Also, electrons mainly interact via long range
Coulomb interaction, so it is hard to define
a cross-section.
Therefore we just assume such a scattering
time tau exists.
Secondly, once the electron scatters it returns
to zero momentum. That is, as soon as an electron
bumps into anything else it fully stops moving.
In reality it would go off in some direction
with a final momentum depending on its scattering
partner due to momentum conservation. But
on average, when we take the mean over a period
of time it will be zero, because clearly it
would be averaged over all kinds of directions.
And lastly, the third assumption is, that
the electron responds to external forces in-between
scattering events. The force can be due to
applied electric and magnetic fields that
couple to the electrons charge, like in the
Lorentz force
F = -e (E + v x B)
Now, lets use these three assumptions:
There is an electron at time t with momentum
p
A bit later in time it has momentum p( t +
dt ). When the electron did not scatter the
momentum can be expanded. We discover the
definition of the force and get a simple expression.
When the probability of scattering is dt/tau,
then the probability of not scattering is
1 - dt/tau. Now in the event of scattering
the momentum becomes zero.
Then we multiply everything out and keep only
terms of linear order in dt. Rearranging a
bit yields the equation of motion for the
Drude model.
dp/dt = F- p/tau
When the force is zero, the solution becomes
obvious. An exponential ansatz results in
a momentum p(t) that starts out at p0 and
rapidly decays. It is as if the scattering
is dragging the electrons down. Simply bringing
them to a hold within some phenomenological
time scale tau. Thats pretty much it.
