Carsten Timm: Theory of superconductivity

4
Normal metals
To be able to appreciate the remarkable poperties of superconductors, it seems useful to review what we know
about normal conductors.
4.1 Electrons in metals
Let us ignore electron-electron Coulomb interaction and deviations from a perfectly periodic crystal structure
(due to defects or phonons) for now. Then the exact single-particle states are described by Bloch wavefunctions
ψ αk (r) = u αk (r) e ik·r , (4.1)
where u αk (r) is a lattice-periodic function, α is the band index including the spin, and k is the crystal momentum
in the first Brillouin zone. Since electrons are fermions, the average occupation number of the state |αk⟩ with
energy ϵ αk is given by the Fermi-Dirac distribution function
n F (ϵ αk ) =
1
e β(ϵ αk−µ)
+ 1 . (4.2)
If the electron number N, and not the chemical potential µ, is given, µ has to be determined from
N = ∑ αk
1
e β(ϵ αk−µ)
+ 1 , (4.3)
cf. our discussion for ideal bosons. In the thermodynamic limit we again replace
∑
∫
d 3 k
→ V
(2π) 3 . (4.4)
k
Unlike for bosons, this is harmless for fermions, since any state can at most be occupied once so that macroscopic
occupation of the single-particle ground state cannot occur. Thus we find
N
V = ∑ ∫
dk 3 1
(2π) 3 e β(ϵ αk−µ)
+ 1 . (4.5)
α
If we lower the temperature, the Fermi function n F becomes more and more step-like. For T → 0, all states with
energies ϵ αk ≤ E F := µ(T → 0) are occupied (E F is the Fermi energy), while all states with ϵ αk > E F are empty.
This Fermi sea becomes fuzzy for energies ϵ αk ≈ E F at finite temperatures but remains well defined as long as
k B T ≪ E F . This is the case for most materials we will discuss.
The chemical potential, the occupations n F (ϵ αk ), and thus all thermodynamic variables are analytic functions
of T and N/V . Thus there is no phase transition, unlike for bosons. Free fermions represent a special case with
only a single band with dispersion ϵ k = 2 k 2 /2m. If we replace m by a material-dependant effective mass, this
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