Advanced Solid State Physics

13.3 Diffusive Metals

In diffusive metals, there is one component which is out of phase and one component which is in phase
(eqn. 229):

m

−e

dv(t)

︸ ︷︷dt︸

−

v(t)

︸︷︷μ︸

= E(t) (229)

The first term is the initial term(out of phase), the second term is the scattering term(in phase).
Again, the solution of a harmonic oscillator is assumed (E(ω)e−iωt;v(ω)e−iωt):

(

iωm

e

−

1

μ

)v(ω) = E(ω) (230)

As before we calculate the conductivity:

σ(ω) =

j(ω)

E(ω)

=

−nev(ω)

E(ω)

=ne(

−iωm

e

+

1

μ

)−^1

=⇒

σ(ω) = ne(

eμ

e−iμωm

) = neμ(

1

1 −iωτ

) = neμ(

1 +iωτ

1 +ω^2 τ^2

) (231)

For low frequencies(the probability for the electrons to get scattered is high) it is possible to neglect
iωτ
ω^2 τ^2 and as a result the conductivity is:
σ(lowω) = neμ

The solution for high frequenciesis thesameas for collisionless metals:

σ(highω) =

ine^2

ωm

with

μ =

eτ

m

Withσit’s possible to calculate the susceptibility:

χ(ω) =

P(ω)

 0 E(ω)

=

nex(ω)

 0 E(ω)

=

nev(ω)

iω 0 E(ω)

=

σ(ω)

iω 0

=

neμ

iω 0

(

1 +iωτ

1 +ω^2 τ^2

) (232)

And again the dielectric constant:

(ω) = 1 +χ = 1−

neμ

ω 0

(

ωτ−i

1 +ω^2 τ^2

) (233)

=⇒

(ω) = 1−ω^2 p(

ωτ^2 −iτ

ω+ω^3 τ^2

) (234)