Carsten Timm: Theory of superconductivity

Jellium phonons
For our discussions we need a specific model for phonons. We use the simplest one, based on the jellium approximation
for the nuclei (or ion cores). In this approximation we describe the nuclei by a smooth positive charge
density ρ + (r, t). In equilibrium, this charge density is uniform, ρ + (r, t) = ρ 0 +. We consider small deviations
Gauss’ law reads
ρ + (r, t) = ρ 0 + + δρ + (r, t). (8.65)
∇ · E = 4π δρ + (r, t), (8.66)
since ρ 0 + is compensated by the average electronic charge density. The density of force acting on ρ + is f = ρ + E ∼ =
ρ 0 +E to leading order in δρ + . Thus
∇ · f ∼ = 4π ρ 0 + δρ + . (8.67)
The conservation of charge is expressed by the continuity equation
To leading order this reads
∂
∂t ρ + + ∇ · ρ + v
}{{}
= j +
= 0. (8.68)
⇒
∂
∂t δρ + + ρ 0 +∇ · v ∼ = 0 (8.69)
∂2
∂t 2 δρ ∼ + = −ρ 0 +∇ · ∂
∂t v Newton
= −ρ 0 + ∇ ·
f
ρ m
, (8.70)
where ρ m is the mass density of the nuclei. With the nuclear charge Ze and mass M we obtain
which is solved by
with
∂ 2
∂t 2 δρ + ∼ = − Ze
M ∇ · f = −Ze M 4π ρ0 + δρ + , (8.71)
Ω =
δρ + (r, t) = δρ + (r) e −iΩt (8.72)
√
4π Ze
M ρ0 + =
√
4π Z2 e 2
M n0 +, (8.73)
where n 0 + is the concentration of nuclei (or ions). We thus obtain optical phonons with completely flat dispersion,
i.e., we find the same frequency Ω for all vibrations.
One can also obtain the coupling strength g q . It is clear that it will be controlled by the Coulomb interaction
between electrons and fluctuations δρ + in the jellium charge density. We refer to the lecture notes on many-particle
theory and only give the result:
1
V |g q| 2 = Ω 2 V C(q). (8.74)
Consequently, the electron-electron interaction due to phonon exchange becomes
V ph (q, iν n ) = 1 V |g q| 2 D 0 q(iν n ) = Ω 2 V C(q)
2Ω
(iν n ) 2 − Ω 2 = V Ω 2
C(q)
(iν n ) 2 − Ω 2 . (8.75)
It is thus proportional to the bare Coulomb interaction, with an additional frequency-dependent factor.
retarded form reads
V R
ph(q, ν) = V ph (q, iν n → ν + i0 + ) = V C (q)
Ω 2
(ν + i0 + ) 2 − Ω 2 = V C(q)
The
Ω 2
ν 2 − Ω 2 + i0 + sgn ν , (8.76)
where we have used 2ν i0 + = i0 + sgn ν and have neglected the square of infinitesimal quantities.
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