BCS Theory of Superconductivity: Cooper Pairs Explained

Superconductivity — the complete disappearance of electrical resistance below a critical temperature — was a mystery for nearly 50 years until Bardeen, Cooper and Schrieffer explained it in 1957. Their BCS theory says that electrons, which normally repel each other, can form weakly bound pairs called Cooper pairs with the help of lattice vibrations. These pairs glide through the crystal without scattering, which is why the resistance drops to exactly zero.

This matters far beyond a physics curiosity. MRI scanners, maglev trains and particle accelerators all run on superconducting magnets made possible by BCS theory. The theory also predicted the energy gap and the isotope effect before they were fully mapped, and it remains the standard model of conventional superconductivity.

Key Concepts

• Cooper pair: two electrons with opposite momenta and opposite spins bound together by a tiny energy, typically a few meV. Millions of pairs overlap in space — they are not small molecules.
• Phonon-mediated attraction: one electron distorts the lattice of positive ions; a second electron is attracted to that distortion. The net effect overcomes the Coulomb repulsion for electrons near the Fermi surface.
• Energy gap Δ: the minimum energy needed to break a pair. At absolute zero, 2Δ(0) = 3.52·k_B·T_c, a universal BCS prediction.
• Critical temperature T_c: above it, thermal energy breaks the pairs; below it, they condense into a single quantum state. Examples: aluminium 1.2 K, lead 7.2 K, Nb₃Sn 18 K.
• Isotope effect: T_c ∝ M^(−1/2), where M is the ionic mass — heavier ions vibrate slower, weakening the pairing. This was the smoking gun for the phonon mechanism.

Formulas

• BCS gap at zero temperature: Δ(0) = 1.76·k_B·T_c, so the pair-breaking energy is 2Δ(0) = 3.52·k_B·T_c, with k_B = 8.617×10⁻⁵ eV/K.
• Critical magnetic field (approx.): H_c(T) = H_c(0)·[1 − (T/T_c)²]. A strong enough field breaks the pairs and restores normal resistance.
• Coherence length ξ: the size of a Cooper pair, typically 100–1000 nm in pure metals — vastly larger than the atomic spacing, so pairs heavily overlap.

Worked Example

Lead becomes superconducting at T_c = 7.2 K. Estimate the BCS energy gap.

1. Thermal energy at T_c: k_B·T_c = 8.617×10⁻⁵ eV/K × 7.2 K = 6.20×10⁻⁴ eV.
2. Pair-breaking energy: 2Δ(0) = 3.52 × 6.20×10⁻⁴ eV = 2.18×10⁻³ eV = 2.18 meV.
3. This tiny gap — about a hundred times smaller than room-temperature thermal energy — is why lead must be cooled to 7 K before pairs survive.
4. Tunnelling experiments measure about 2.7 meV for lead; the simple BCS value is close, and strong-coupling corrections explain the rest.

Common Misconceptions

• "Cooper pairs are like H₂ molecules." A pair spans hundreds of nanometres and overlaps millions of other pairs; it is a momentum-space correlation, not a localised chemical bond.
• "Superconductors have infinite conductivity." They have zero resistance to steady current, but the useful description is perfect diamagnetism plus persistent currents, and there are critical current and field limits.
• "BCS explains all superconductors." It explains conventional ones (lead, mercury, Nb alloys). High-T_c cuprates and iron-based superconductors need different, still-debated mechanisms.

Real-World Applications

• MRI magnets: niobium-titanium coils carry huge persistent currents with no power loss, creating the tesla-scale fields of hospital scanners.
• Particle accelerators: the Large Hadron Collider's steering magnets are superconducting, made affordable by BCS-designed Nb-Ti and Nb₃Sn wires.
• SQUIDs: superconducting quantum interference devices detect magnetic fields a billion times weaker than Earth's, used in brain imaging (MEG).

BCS theory explains superconductivity through Cooper pairs bound by lattice vibrations, with a universal gap 2Δ(0) = 3.52·k_B·T_c confirmed by the isotope effect. It turned a curiosity into MRI machines, leaving high-temperature superconductors as the next puzzle.