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8.512 Theory of Solids II
Spring 2009
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1
Problem
Set 4Set
Problem
8.512 Theory of Solids II
Due March
9, 2009
Due
March
17, 2008
5
1. This problem reviews the Boltzmann equation and compares the result with the Kubo
formula. For a derivation of the Boltzmann equation, read p.319 of Ashcroft and
Mermin.
(a) Consider an electron gas subject to an electron field
→
→
→
i(
E ( r , t)
=E 0 e
→ →
q · r −iωt)
(1)
The Boltzmann equation in the relaxation time approximation is
→
∂f
∂f
→ ∂f
f − f0
+
→ ·
v + → (−e) E = −
∂t
τ
∂ r
∂ k
where f0 is the equilibrium distribution
f0 (ǫ) =
1
eβǫ + 1
(2)
(3)
Write
→ →
→
f ( r , k , t) = f0 (ǫk ) + Φ( k )ei
→
→
→
q · r −iωt
(4)
→
and work to first order in Φ( k ) and E . Show that the conductivity is given by
�
�
�
→
→
→
τ (ê· v )2
∂f0
e
2
q
−
σ( , ω) = 3 d
k
(5)
→ →
4π
∂ǫk
1 − iτ (ω− q · v )
→
where ê is the unit vector in the direction of E 0 .
(b) A simple way to derive the Kubo formula is to compare the energy dissipation
rate σE02 with the rate of photon absorption. At finite temperature, we need
to include both absorption and emission processes. Show that for free electrons
(including spin)
σ ′ (q, ω) =
→→
→
(f (E ) − f0 (Eβ ))
2e
2 1
�
iq·r
p |α
> |2 0 α
|
<
β|e
ê·
δ (�ω − (Eβ − Eα ))
2
m
V α,β
(Eβ − Eα )/�
(6)
Using the Kramers-Kronig relation, show that the complex conductivity is
→
2e
2 1 � | < β|ei q · r ê· p |α
> |2 (−i) (f0 (Eα ) − f0 (Eβ ))
σ( q , ω)
= 2
m V α,β
(Eβ − Eα )�
(Eβ − Eα − �ω − iη)
→ →
→
(7)
2
(c) For |q| ≪ kF , show that Eq.(7) reduces to Eq.(5) under the assumptions that
|α >, |β > are plane waves and η is identified with τ1 .
→
2. Equation (5) in Problem 1 is valid for any relation between q and ê. In an isotropic
→
material the response can be separated into the longitudinal ( q � ê) and transverse
→
parts ( q ⊥ ê). The latter is appropriate for the propagation of electromagnetic waves.
(a) For T ≪ ǫF , show that the transverse conductivity can be written as an integra­
tion over the Fermi surface.
σ0 3
σ⊥ ( q , ω) =
1 − iωτ 4
→
�
1
dx
−1
1 − x
2
1 + sx
(8)
where
s=
iqvF τ
1 − iωτ
(9)
In Eq.(8) σ0 = ne2 τ /m is the DC Boltzmann conductivity and the integration
variable x stands for cos θ in an integration over the Fermi surface.
(b) The integral in Eq.(8) can be done analytically. For our purposes, find the small
|s| and large |s| limits. The small |s| limit is the Drude conductivity while the
large |s| limit is called the “extreme anomalous region.” It describes the situation
when the electron mean free path ℓ is much greater than the wavelength of light.
Note that it is reduced from σ0 by the factor 1/(qℓ). Produce a simple argument to
show that this reduction factor can be understood on the basis of kinetic theory of
classical particles. (Hint: Consider a low frequency transverse electromagnetic
wave. For qℓ ≪ 1, all the electrons can absorb energy from the electric field.
However, for qℓ ≫ 1, only a fraction travelling almost parallel to ê can do so.
The argument was first given by Pippard.)