Advanced Solid State Physics

To describe this equation in a simpler way we could define a complex conductivityσor a complex
dielectricityε. Both is possible and both is being done. A complex conductivity is often used for high
frequencies when insulators start to act like metals and a complex dielectric constant is used for very
high frequencies when dielectrics act like metals with poor conductivity.

σ ̃(ω) = σ(ω)−iωε 0 ε(ω) (111)

̃ε(ω) = ε(ω) +i

σ(ω)

ε 0 ω

(112)

We now look at the response of metals, including the inertial term (eqn. (104)) and the ohmic term
(eqn. (105)), the relation between the electric field an the velocity of an electron is

m

−e

dv(t)

︸ ︷︷dt︸

Ballistic

−

v(t)

μ

︸︷︷︸

Diff.

= E(t) (113)

The same equation written as a function ofg(t), the Green’s function

m

−e

dg(t)

dt

−

g(t)

μ

= δ(t)

The solution to this differential equation fort > 0 is

g = μexp(

−et

mμ

), t > 0 (114)

Att = 0the velocity jumps up and then decays, if there would be only the ballistic part the velocity
would also jump up att = 0but then stay at the same level all the time because there is no drag
force or scattering.

Figure 70: responseg(t)toδ(t)driving force

To get the generalized susceptibility the Fourier transform ofg(t)is needed, which is an Lorentzian
for an exponential decay.

G(ω) = μ

1 −iωτ

1 +ω^2 τ^2

, τ =

mμ

e

The generalized susceptibility is the response divided by the driving force

χ(ω) =

v(ω)

E(ω)

= μ

1 −iωτ

1 +ω^2 τ^2

(115)