Martin Teichmann Atomes de lithium-6 ultra froids dans la ... - TEL

2.4. BCS THEORY
Using the substitution ξ = x0+
√
2 1+x0 √ , we can write these integrals in
2 2 1+x0 the closed form
√
2
G(ξ) =
(2E(ξ) − K(ξ)) (2.41)
π((1 − ξ)ξ) 1/4
1
N(ξ) =
3 √ ((2ξ − 1)E(ξ) + (1 − ξ)K(ξ)) (2.42)
2((1 − ξ)ξ) 3/4
where K and E are the complete elliptic integrals of the first and second
kind, respectively. They are readily available in most computer algebra
systems.
2.4.5 Results
The BCS theory gives a good qualitative description of the whole crossover.
In figure 2.7, we show the behavior of the chemical potential µ
and the gap ∆ at zero temperature in the crossover. While for a deep
BCS state with weak interactions the chemical potential coincides with
the Fermi energy, it decreases while crossing the resonance and even
becomes negative on the BEC side, and eventually becomes asymptotically
equal to the binding energy of the molecules. The Fermi edge
washes out, as can be seen in figure 2.8, and once the chemical potential
crosses zero it disappears completely. This illustrates that the pairs are
not governed by the Fermi-Dirac distribution anymore, but follow the
Bose-Einstein statistic and thus can be considered bosons. The gap,
normally small in superconductors, opens up.
The critical temperature can also be deduced from BCS theory,
usually by calculating the temperature T ∗ at which the gap vanishes, see
references [65, 66]. This is the dashed curve in figure 2.9. As we already
stated, the pair creation and condensation do not coincide for strong
interactions [59]. This becomes clear once we are on the BEC side of
the crossover: the molecules exist even at higher temperatures and only
condense into a BEC below a critical temperature Tc [60]. A similar
effect is seen in high temperature superconductors and is known as the
pseudogap [67, 68].
This critical temperature was first studied by Nozières and Schmitt-
Rink (NSR) [62]. They realized that the BCS approximation to only take
pairs with zero momentum into account will fail at strong interactions,
and considered thermal motion of bound molecules. A recent theoretical
calculation, which agrees well with NSR, is shown in figure 2.9 together
with experimental measurements. Further studies, which took beyond
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