Condensed matter physics
Superconductivity and BCS theory
In BCS superconductors, an effective attraction pairs electrons into Cooper pairs; their coherent condensate produces a spectral gap and macroscopic electrodynamics.
In BCS superconductors, an effective attraction pairs electrons into Cooper pairs; their coherent condensate produces a spectral gap and macroscopic electrodynamics.
Definition: Cooper pairing
Electrons near the Fermi surface can pair in a spin-singlet channel with nearly zero total momentum. In conventional metals, electron–phonon coupling can provide an effective attraction.
Physical model
Electrons near the Fermi surface can pair in a spin-singlet channel with nearly zero total momentum. In conventional metals, electron–phonon coupling can provide an effective attraction.
The order parameter Δ measures pairing amplitude; E_k=√(ξ_k²+|Δ|²) has a gap. Weak-coupling BCS predicts 2Δ(0)≈3.52k_BT_c.
Example: Worked example
At T_c=10 K, Δ(0)≈1.76k_BT_c≈1.52 meV for k_B=0.08617 meV/K; this is the weak-coupling BCS limit.
Solution
Δ(0)=1.76×0.08617×10 meV≈1.52 meV.
Quick check
The Bogoliubov transformation mixes electron and hole amplitudes, producing quasiparticles with spectrum Eₖ=√(ξₖ²+|Δ|²). The gap blocks low-energy single-particle excitations, while superconducting current carries the common phase of the order parameter. In weak-coupling BCS theory, 2Δ(0)/(kBTc)≈3.52 is universal for an isotropic s-wave model; gap anisotropy, strong coupling, or other mechanisms can change the ratio. Pairing spontaneously breaks U(1) phase symmetry in the thermodynamic description.
The superconducting gap directly affects heat transport and microwave absorption; as T approaches Tc, the order parameter decreases and the gap closes. The Meissner effect shows that superconductivity is more than zero resistance: under appropriate conditions magnetic flux is expelled from the bulk. BCS theory describes many conventional metals well, but it is not a universal explanation for every high-temperature superconductor or pairing symmetry.
The condensate’s phase coherence quantizes flux in a superconducting ring, Φ₀=h/(2e), reflecting the Cooper-pair charge 2e. In a type-II material, magnetic field penetrates as quantized vortices above the lower critical field; each core suppresses superconducting order. This is a macroscopic test independent of measuring the spectral gap.
Which ratio is predicted by weak-coupling BCS theory?
Why does the BCS state have an excitation gap?
References
- Charles Kittel (2004). Introduction to Solid State Physics