Set of supplementary notes.

1.2. DRUDE MODEL 19
We neglect B for the time being, as the effects of the magnetic field accompanying an e-m
wave on the electron system are much weaker than those of the electric field. However, B will
be included in transport phenomena at low frequencies, below (Hall effect). Then, we obtain
the simplified equations
( d
dt + 1 τ
)
v = qE(t)/m
( d
dt + 1 τ
)
j = N V
q 2
E(t) (1.15)
m
To extract the frequency dependent conductivity from (1.15), we again insert a trial solution
of the form j = j ω e −iωt and a time-varying field of the form E = E ω e −iωt . This transforms the
above differential equation into an algebraic equation:
j ω = ne2
m
1
1/τ − iω E ω (1.16)
(remember that q = −e is the electron’s charge, and we write the number density N/V as
n). Using the definition of the conductivity j = σE, we obtain:
σ ω = ne2
m
1
1/τ − iω , (1.17)
In the low frequency limit, this expression for the conductivity tends to the DC conductivity
σ 0 = ne2 τ
m
At frequencies larger than 1/τ the conductivity falls off rapidly:
(1.18)
Re(σ ω ) =
σ(0)
1 + ω 2 τ 2 . (1.19)
The DC (ω = 0) conductivity can also be written in terms of the mobility µ = eτ/m
σ = neµ = ne2 τ
m
(1.20)
Inserting (1.17) into Eqn. 1.11 gives
ɛ(ω) = ɛ ∞ −
ω 2 p
ω 2 + iω/τ
, (1.21)
which is exactly the expression we obtained right at the start (Eq. 1.5) by setting ω T → 0,
if we identify 1/τ with γ. Here, as before, ω p is the Plasma frequency:
ω 2 p = ne2
ɛ 0 m
(1.22)