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Physics 216: Special topics in many-body physics, Spring 2003
Lecture II
In the previous lecture, it was claimed that a retarded phonon-mediated attraction between electrons in a metal could induce superconductivity, a qualitative change in the electrical conduc- tivity and other properties. We now want to understand how that happens, using a variational approach and the second quantization formalism. More precisely, most of the phenomenology of su- perconductors follows from the Ginzburg-Landau equation. This equation follows from minimizing the so-called Landau free energy, which we will now construct from physical assumptions.
The Landau free energy is written in terms of a single-particle wavefunction. One major con- tribution of BCS theory was explaining why a single-particle wavefunction is a natural description for a system of fermions. With Bose-Einstein condensed bosons, description in terms of a single- particle wavefunction is more natural, since then most of the bosons are in the ground state. The magnitude of this wavefunction gives the fraction of electrons at a point that are superconducting, in a two-fluid model where some electrons are superconducting and some are normal:
|ψ(r)|^2 =
ns(r) n
Here n is the total electron density, assumed constant in space.
First suppose that the wavefunction is constant in space, and let f (ψ, T ) be the difference in free energy density between the superconducting and normal states if ψ is uniform. That is, for a system of volume V , the free energy difference is
∆F = FN − FS = V f (ψ, T ). (2)
Now we make another assumption. If the system is just below the superconducting transition Tc, then only a few electrons are superconducting, which means that we can expand f in a power series in ψ, since |ψ|^2 1:
f (ψ, T ) ≈ a(T )|ψ|^2 +
b(T )|ψ|^4. (3)
We can make one more simplification. First, note that the free energy is minimized when
∂f ∂|ψ|^2 = 0 ⇒ b(T )|ψ|^2 + a(T ) = 0 ⇒ |ψ|^2 = −
a(T ) b(T )
At this magnitude, the free energy difference is
f (ψ, T ) = −
a^2 (T ) b(T )
Recall that one of the defining features of a superconductor is the expulsion of magnetic field (the Meissner effect). The magnetic field reenters and drives the system normal once it is energetically favorable to do so (here we are assuming that the superconductor is type I, so the magnetic field penetrates uniformly above the critical magnetic field Hc). The free energy difference per volume is therefore related to Hc.
f (ψ, T ) = −
a^2 (T ) b(T )
Hc^2 8 π
We can obtain another relation involving a and b if we use the fact that the superfluid density goes as the inverse square of the penetration depth: then
λ^2 (0) λ^2 (T )
|ψ(T )|^2 |ψ(0)|^2
= |ψ(T )|^2 = − a(T ) b(T )
Here we assumed that at zero temperature, all the electrons participate in the superconductivity. These two equations involving a and b can be used to reexpress everything in the Ginzburg-Landau equation in terms of the experimental quantities Hc and λ (left as an exercise).
Now we want to allow for spatial variations in ψ. For slow variations, we can keep just the gradient-squared term, which introducing some constants becomes ∫ (^) n∗ 2 m∗
∣∣ ∣∣^ ¯h i ∇ψ(r) +
e∗ c A(r)ψ(r)
∣∣ ∣∣
2 dr. (8)
Note that we have defined quantities n∗, m∗, e∗^ with the units of number density, mass, and charge. The BCS theory will predict e∗^ = 2e and m∗^ = 2m. We have also assumed that an external vector potential A enters in the same way as for a single particle.
Combining the constant and gradient terms, and the magnetic field energy, gives finally the Landau free energy
F (ψ, T ) =
∫ n∗ 2 m∗
∣∣ ∣∣^ ¯h i
∇ψ(r) + e∗ c
A(r)ψ(r)
∣∣ ∣∣
2 dr
∫ [ a(T )|ψ(r)|^2 +
b(T )|ψ(r)|^4
] dr +
∫ H(r)^2 8 π dr. (9)
Then the minimization of this functional over ψ gives
δF δψ(r)
¯h^2 n∗ 2 m∗
[ ∇ + ie∗ ¯hc
A(r)
] 2 ψ(r) + a(T )ψ(r) + b(T )|ψ(r)|^2 ψ(r) = 0. (10)
Before trying to find a microscopic justification of this theory, let us review what it does contain and what it doesn’t. It accounts for the existence of a supercurrent and the Meissner effect, and even for the existence of vortices in type II superconductors. It also shows the importance of a good phenomenology: only seven years passed between the above “derivation” of the Ginzburg- Landau equation and its justification by BCS theory. Of course, the numbers m∗^ and e∗^ remain unexplained, and it gives no prediction of when the approximation by a single wavefunction breaks down: what happened to the original (fermionic) electrons?
Our starting point will be the interacting electron Hamiltonian
H =
∑
kσ
kc† kσckσ +
∑
k,k′,q;σ,σ′
V (k, k′, q)c† k′−q,σ′ ck′,σ′^ c† k+q,σck;σ (11)
Here the form of the interaction term was simplified slightly by assuming translation invariance of the interaction (conservation of total momentum). For an unscreened Coulomb interaction, V (q) ∼ 4 πe^2 |q|−^2. In general, even with screening, the strongest interaction is at momentum transfer q = 0.
As a simplifying approximation, we keep only one part of the full interaction term. First introduce the pair creation and annihilation operators
b† k = c† k↑c†−k↓, bk = c−k↓ck↑. (12)
Let us assume that there is a self-consistent solution of the famous “gap equation”:
∆k = −
∑
k′
Vkk′
∆k′ 2 Ek′
where Ek =
√ (k − μ)^2 + ∆k^2. This Ek will turn out to be the energy required to create a “quasiparticle” at momentum k. We will choose
ukvk = ∆k 2 Ek
Now the variational equation (from varying W with respect to the gk) can be written simply: (note that variations of uk and of vk are not independent: δuk = −uk^2 vkδgk, δvk = uk^3 δgk)
δW = uk^2 δgk
( 4(k − μ)vkuk + 2(
∑
k′
Vkk′^ ∆k′ 2 Ek′
)(uk^2 − vk^2 )
) = 0. (23)
Then substituting in the gap equation gives
uk^2 − vk^2 = k − μ Ek
which combined with the constraint uk^2 + vk^2 = 1 gives
uk^2 =
( 1 + k − μ Ek
) , vk^2 =
( 1 − k − μ Ek
)
. (25)
We can think of this wavefunction as describing a large number N/2 of “Cooper pairs”, all in the zero-momentum and zero-spin state. However, note that there is still a hint of the fermionic nature of the underlying electrons, since we had to use all the electron states up to the Fermi level in order to make these Cooper pairs. The connection to the Ginzburg-Landau equation is, in words, that the LG equation describes the wavefunction of Cooper pairs, which in the absence of an external potential are all in the same center-of-mass state.
For now, we are going to make a simple choice for V that will give a so-called “s-wave” super- conductor: Vk,k′ = −V < 0 if both k and k′^ are within some small distance δk of the Fermi level, and 0 otherwise. We will see later how other potentials can give exotic superconductors of other symmetries.
Note that the fractional number variations in the BCS state are statistically very small (∼ N 1 /^2 , where N is the total number of electrons). We could choose a similar state of well-defined number, but would find that then the phase appearing in the Ginzburg-Landau equation is not well-defined: in this sense number and phase are conjugate variables in a superconductor.
It is amazing that the small attractive residual interaction, which only changes the electron interaction energy by a factor of about 10−^8 , can so dramatically change the physical properties.
