Carsten Timm: Theory of superconductivity

gives a reasonable approximation for simple metals such as alkali metals. Qualitatively, the conclusions are much
more general.
Lattice imperfections and interactions result in the Bloch waves ψ αk (r) not being exact single-particle eigenstates.
(Electron-electron and electron-phonon interactions invalidate the whole idea of single-particle states.)
However, if these effects are in some sense small, they can be treated pertubatively in terms of scattering of
electrons between single-particle states |αk⟩.
4.2 Semiclassical theory of transport
We now want to derive an expression for the current in the presence of an applied electric field. This is a question
about the response of the system to an external perturbation. There are many ways to approach this type of
question. If the perturbation is small, the response, in our case the current, is expected to be a linear function
of the perturbation. This is the basic assumption of linear-response theory. In the framework of many-particle
theory, linear-response theory results in the Kubo formula (see lecture notes on many-particle theory). We here
take a different route. If the external perturbation changes slowly in time and space on atomic scales, we can use
a semiclassical description. Note that the following can be derived cleanly as a limit of many-particle quantum
theory.
The idea is to consider the phase space distribution function ρ(r, k, t). This is a classical concept. From
quantum mechanics we know that r and p = k are subject to the uncertainty principle ∆r∆p ≥ /2. Thus
distribution functions ρ that are localized in a phase-space volume smaller than on the order of 3 violate quantum
mechanics. On the other hand, if ρ is much broader, quantum effects should be negligible.
The Liouville theorem shows that ρ satisfies the continuity equation
(phase-space volume is conserved under the classical time evolution).
dispersion, we have the canonical (Hamilton) equations
∂ρ
∂t + ṙ · ∂ρ
∂r + ˙k · ∂ρ
∂k ≡ dρ
dt = 0 (4.6)
Assuming for simplicity a free-particle
ṙ = ∂H
∂p = p m = k m , (4.7)
˙k = 1 ṗ = − 1 ∂H
∂r = − 1 ∇V = 1 F (4.8)
with the Hamiltonian H and the force F. Thus we can write
( ∂
∂t + k m · ∂
∂r + F )
· ∂
ρ = 0. (4.9)
∂k
This equation is appropriate for particles in the absence of any scattering. For electrons in a uniform and timeindependant
electric field we have
F = −eE. (4.10)
Note that we always use the convention that e > 0. It is easy to see that
ρ(r, k, t) = f
(k + eE )
t
(4.11)
is a solution of Eq. (4.9) for any differentiable function f. This solution is uniform in real space (∂ρ/∂r ≡ 0)
and shifts to larger and larger momenta k for t → ∞. It thus describes the free acceleration of electrons in an
electric field. There is no finite conductivity since the current never reaches a stationary value. This is obviously
not a correct description of a normal metal.
Scattering will change ρ as a function of time beyond what as already included in Eq. (4.9). We collect all
processes not included in Eq. (4.9) into a scattering term S[ρ]:
( ∂
∂t + k )
m ·
ρ = −S[ρ]. (4.12)
∂
∂r + F ·
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∂
∂k