Advanced Solid State Physics

13.2 Collisionless Metal

In collisionless metals electrons travel ballistically which means that the scattering time and the mean
free path are infinite. The relationship between velocityvand the electric fieldEis:

m

−e

dv(t)

dt

= E(t) (218)

So if an oscillating field influences the material, the electrons start to oscillate but they oscillate out
of phase. The derivative of the velocity is proportional to the electric field (eqn. (218)).
To solve this problem, the solution of a harmonic oscillator is assumed (E(ω)e−iωt,v(ω)e−iωt). From
eqn. (218):

(

iωm

e

)v(ω) = E(ω)

The conductivity:

j = σE⇒σ =

j

E

=

−nev(ω)

E(ω)

=

ine^2

ωm

(219)

Soσis purely imaginary because the motion of the electrons isout of phase, but of course there
must be a real part. With the Kramers-Kronig relation it is possible to calculate the real part:

σ′′(ω) =

1

π

∫∞

−∞

σ′(ω′)

ω′−ω

dω′ (220)

iσ′′(ω) =

−nev(ω)

E(ω)

=

ine^2

ωm

(221)

To calculate the real part we guess the solution (the reason for this is that it is easier)

σ′(ω) =

πne^2

m

δ(ω) (222)

Equation (222) in eqn. (220) gives:

σ′′(ω) =

ne^2

m

∫∞

−∞

δ(ω′)

ω′−ω

dω′ (223)

After integrating this and comparing with eqn. (221) the real part is derived. Now its possible to write
the imaginary and the real part of the conductivity:

σ(ω) =

ne^2

m

(πδ(ω) +

i

ω

) (224)

Furthermore it is possible to calculate the dielectric constant(ω). For the dielectric constant the
susceptibility is needed:

χ(ω) =

P(ω)

 0 E(ω)

=

nex(ω)

 0 E(ω)

=

nev(ω)

 0 E(ω)(−iω)

(225)