Cooper Pair Binding Energy

What you are seeing: two electrons added to a filled Fermi sea bind into a Cooper pair with energy $E_{bind} = 2\hbar\omega_D e^{-2/N(0)V}$ . Exponentially small at weak coupling, the binding is nonperturbative in the attractive interaction strength.

Figure 1. Left: binding energy

E_b = 2\hbar\omega_D e^{-2/N(0)V}

on a log scale as a function of

N(0)V

, with live readout driven by sliders for coupling

V

, density of states

N(0)

, and Debye cutoff

\hbar\omega_D

. Right: pair wavefunction amplitude

|g(\xi)|

in the energy shell around the Fermi surface, peaking near

\xi = 0

with width set by the cutoff

\hbar\omega_D

. Ashcroft-Mermin Ch. 34.

V (coupling)0.30

N(0) (DOS)1.00

ℏω_D (cutoff)1.00

WHAT TO TRY

Turn on any attraction V, however weak: two electrons above a filled Fermi sea always bind, with energy 2 hbar omega_D exp(-2 / N(0)V). The Fermi sea makes the pair bound for arbitrarily small coupling.
Raise the coupling or the density of states N(0): the binding grows exponentially, the same exponential that sets the superconducting gap and transition temperature.
That non-analytic exp(-1/V) form can never be reached by perturbation theory, which is why superconductivity hid for fifty years until BCS.