Set of supplementary notes.

18 CHAPTER 1. CLASSICAL MODELS FOR ELECTRONS IN SOLIDS
Differential equation for the drift velocity
Rather than simply inserting our earlier expression (1.5) into (1.11), which would indeed give us
a correct expression for the frequency dependent conductivity σ ω within the Drude model, we
can also derive an expression for σ ω by considering an equation of motion for the drift velocity
of the electrons, or for the resulting current density. This answers the conceptual issues raised
earlier and gives a more precise meaning to the relaxation time τ. The drift velocity relates to
the total momentum of the electron system via v =
p , where N is the number of electrons
Nm
in the system and m is the electron mass. The momentum, in turn, changes in the presence of
an applied electric or magnetic field. If there were no collisions, which can remove momentum
from the electron system, then we would have ṗ = Nf(t) = N q (E + v × B).
m
Collisions, or scattering, will introduce a further term, which represents the decay of the
electron momentum in the absence of an external force. Note that electron-electron collisions
do not give rise to a decay of momentum in any obvious way: they would appear to conserve
the momentum of the electron system. It turns out that at a more advanced level of analysis,
they do contribute to the relaxation of momentum, but let us for the moment neglect this
contribution. The electron momentum decays, then, because of collisions of the electrons with
lattice imperfections, such as impurities, dislocations etc., and – in the wider sense – lattice
distortions caused by lattice vibrations.
We could model the influence of electron scattering events by making two very simplifying
assumptions:
• Electron collisions randomise an electron’s momentum, so that – on average – the contribution
of an electron to the total momentum is 0 after a collision.
• The probability for a collision to occur, P , is characterised by a single relaxation time τ:
P (collision in [t, t + dt]) = dt/τ.
From these assumptions, we find that the probability that a particular electron has not
scattered in the time interval [t, t + dt] is 1 − dt/τ. As only the electrons which have not
scattered contribute to the total momentum (the momentum of the others randomises to zero
on average), and these electrons continue to be accelerated by the applied force, we obtain a
total momentum after time t + dt:
p(t + dt) = (1 − dt/τ)(p(t) + Nf(t)dt) + ... (1.12)
This gives rise to a differential equation for the momentum:
( d
dt + 1 τ
)
p = Nf(t) (1.13)
and, substituting f = qE, we obtain for the drift velocity and for the current density:
( d
dt + 1 τ
)
( d
dt + 1 τ
)
j = N V
v = q (E + v × B)
m
q 2
m E(t) + q m j × B (1.14)