Why do pairing and condensation occur simultaneously in superconductors?

Question

Bose condensation in the weak-coupling limit (such as BCS superconductors) and the strong-coupling limit (like BECs of atoms) can be unified into a single framework by the BEC-BCS crossover. One key feature distinguishing the BCS side from BEC is that the pairing of fermions near the Fermi surface into Cooper pairs and the condensation of those Cooper pairs happens simultaneously at the same temperature.

1. Is there a simple argument to understand why this is so?

2. For a usual BCS-type interaction, at what interaction strength does this break down? Is there a known expression for the difference, as a function of interaction strength, between the temperature at which pairs become bound and $T_c$ for them to condense?

Answer 1

1) On the Bose side pairs form when the temperature is lower than the binding energy, $k_BT<B$. Note that this is not a sharp phase transition, and there is no order parameter associated with it. Pairs condense at the Einstein temperature $$ T_c = \frac{2\pi\hbar^2}{mk_B} \left( \frac{n}{\zeta(3/2)}\right)^{2/3} $$ where $n$ is the density of pairs, and $m\simeq 2m_a$ is the mass of a pair. This is a sharp phase transition (Bose condensation). The pair formation and BEC temperatures become equal when $T_c\sim B$.

Note that parametrically, $T_c$ is on the order of the would-be Fermi energy $E_F=k_F^2/(2m_a)$ of the fermionic constituents. Here, $k_F$ is defined via the density of the gas, $n_F=k_F^3/(3\pi^2)$. The pair density is $n=n_F/2$. This means that the crossover estimate $T_c\sim B$ is roughly consistent with the dimensional estimate $T_F\sim B$.

2) Coming from the BCS side the interaction is weak, and in 3d there are no bound states in vacuum. This means that Cooper pairs form at the BCS transition temperature $$ T_c = \frac{8e^\gamma E_F}{(4e)^{1/3}e^2\pi}\exp\left(-\frac{\pi}{2k_F|a|} \right) $$ where $a$ is the scattering length. The crossover is reached when $T_c$ is of order $E_F$, corresponding to $a\to \infty$.

3) We know that the crossover is smooth (no other phase transitions intervene). This has been established experimentally, by numerical simulation, and is consistent with simple many-body theories. At $a\to\infty$ there is some discussion about a possible "pseudo-gap" phase above $T_c$. This is hard to settle, because there is no completely sharp definition of what a pseudo-gap is.