Carsten Timm: Theory of superconductivity

This is the famous Boltzmann equation. The notation S[ρ] signifies that the scattering term is a functional of ρ.
It is generally not simply a function of the local density ρ(r, k, t) but depends on ρ everywhere and at all times,
to the extent that this is consistent with causality.
While expressions for S[ρ] can be derived for various cases, for our purposes it is sufficient to employ the
simple but common relaxation-time approximation. It is based on the observation that ρ(r, k, t) should relax to
thermal equilibrium if no force is applied. For fermions, the equilibrium is ρ 0 (k) ∝ n F (ϵ k ). This is enforced by
the ansatz
S[ρ] = ρ(r, k, t) − ρ 0(k)
. (4.13)
τ
Here, τ is the relaxation time, which determines how fast ρ relaxes towards ρ 0 .
If there are different scattering mechanisms that act independantly, the scattering integral is just a sum of
contributions of these mechanisms,
S[ρ] = S 1 [ρ] + S 2 [ρ] + . . . (4.14)
Consequently, the relaxation rate 1/τ can be written as (Matthiessen’s rule)
There are three main scattering mechanisms:
1
τ = 1 τ 1
+ 1 τ 2
+ . . . (4.15)
• Scattering of electrons by disorder: This gives an essentially temperature-independant contribution, which
dominates at low temperatures.
• Electron-phonon interaction: This mechanism is strongly temperature-dependent because the available
phase space shrinks at low temperatures. One finds a scattering rate
1
τ e-ph
∝ T 3 . (4.16)
However, this is not the relevant rate for transport calculations. The conductivity is much more strongly
affected by scattering that changes the electron momentum k by a lot than by processes that change it
very little. Backscattering across the Fermi sea is most effective.
k
y
∆k = 2k
F
k x
Since backscattering is additionally suppressed at low T , the relevant transport scattering rate scales as
1
τe-ph
trans ∝ T 5 . (4.17)
• Electron-electron interaction: For a parabolic free-electron band its contribution to the resistivity is actually
zero since Coulomb scattering conserves the total momentum of the two scattering electrons and therefore
does not degrade the current. However, in a real metal, umklapp scattering can take place that conserves the
total momentum only modulo a reciprocal lattice vector. Thus the electron system can transfer momentum
to the crystal as a whole and thereby degrade the current. The temperature dependance is typically
1
τ umklapp
e-e
∝ T 2 . (4.18)
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