Superconductivity: Zero Resistance and the Meissner Effect
Below a critical temperature, certain materials lose all electrical resistance and expel magnetic fields — two effects that seem to violate common sense. The BCS theory explains the first through quantum mechanics; the second remains only partially understood for high-temperature superconductors discovered in 1986.
1. Discovery and the Resistance Paradox
In 1911, Heike Kamerlingh Onnes cooled mercury to 4.2 K and measured its electrical resistance — it dropped instantaneously to exactly zero. Not "very small" — zero. A current induced in a superconducting ring has been measured to persist for years without any measurable decay (upper bound: decay time > 100 000 years).
In a normal metal, electrons scatter off lattice vibrations (phonons) and impurities, which is the origin of resistance. At low temperature this scattering decreases but never reaches zero — there are always impurities. Superconductors do something fundamentally different: they enter a macroscopic quantum state where scattering is quantum-mechanically forbidden.
ρ = m_e / (n·e²·τ) where τ = mean time between collisions
As T → 0: τ increases (less phonon scattering) but ρ → ρ₀ (residual from impurities)
Superconductor below T_c:
ρ = 0 exactly — no scattering mechanism operates
2. Cooper Pairs: Electrons That Attract
The puzzle: electrons are negatively charged and repel each other. How can they pair up? The answer involves a subtle quantum mechanical process mediated by the crystal lattice.
When an electron moves through the lattice, it attracts nearby positive ions slightly toward it, creating a region of transient positive charge density. A second electron, arriving slightly later, is attracted to this positive region. The net effect is a weak net attraction between the two electrons, mediated by a phonon (lattice vibration quantum).
2Δ ≈ 3.52 · k_B · T_c (BCS weak-coupling result)
electron 1: momentum +k, spin ↑
electron 2: momentum −k, spin ↓
Total momentum: 0 — Total spin: 0 (singlet)
Pairs are bosons (integer spin) → all condense into same quantum ground state (BEC-like)
3. BCS Theory and the Energy Gap
Bardeen, Cooper, and Schrieffer (1957) built on Cooper's insight to construct a complete theory. The key result: at T = 0, all electrons near the Fermi surface form Cooper pairs and condense into a single macroscopic quantum state described by one wavefunction Ψ = |Ψ|·e^(iφ).
This condensation opens an energy gap 2Δ around the Fermi energy — no single-electron states exist there. To break a Cooper pair and scatter, you need energy ≥ 2Δ. At T ≪ T_c, thermal fluctuations k_BT ≪ Δ can't provide this energy — so scattering is frozen out.
Δ = 2ħω_D · exp(−1 / (N(0)·V))
ω_D = Debye frequency (phonon cutoff)
N(0) = density of states at Fermi level
V = phonon-mediated electron-electron interaction
Temperature dependence near T_c: Δ(T) ≈ 1.74·Δ(0)·√(1 − T/T_c)
Critical temperature: k_B·T_c ≈ 1.13·ħω_D · exp(−1/(N(0)·V))
4. The Meissner Effect
Place a normal metal in a magnetic field, then cool it below T_c. You might expect the field to remain trapped inside. Instead, the field is expelled from the bulk — the superconductor spontaneously generates surface currents that exactly cancel the external field. This is the Meissner effect (Meissner & Ochsenfeld, 1933), and it proves superconductivity is a distinct thermodynamic state, not just perfect conductivity.
∂J_s/∂t = (n_s·e²/m_e) · E [first London equation: perfect conductor]
∇ × J_s = −(n_s·e²/m_e) · B [second London equation: Meissner effect]
Solution: B(x) = B_ext · exp(−x/λ_L)
London penetration depth: λ_L = √(m_e / (μ₀·n_s·e²))
Typical values: λ_L ≈ 20–500 nm
Inside bulk (x ≫ λ_L): B = 0 — field completely expelled
The penetration depth λ_L is the distance over which the external field decays exponentially into the superconductor. Surface currents flow in this thin layer to maintain B = 0 inside. This is why a magnet floats above a superconductor — the expelled field exerts an upward force.
5. Type I vs Type II Superconductors
Type I — Abrupt transition
Single critical field H_c. Below H_c: B = 0 (Meissner state). Above H_c: normal state, all superconductivity destroyed instantly. Examples: mercury, lead, tin, aluminium. H_c values are too small for practical magnets.
Type II — Vortex state
Two critical fields H_c1 and H_c2. Below H_c1: full Meissner state. Between H_c1 and H_c2: mixed/vortex state — field penetrates as quantised flux tubes (Abrikosov vortices). Above H_c2: normal metal. Niobium, YBCO, MgB₂ — all practical SC magnets are Type II.
Each Abrikosov vortex carries exactly one Φ₀.
Vortices form a triangular lattice (Abrikosov lattice).
Ginzburg-Landau parameter: κ = λ_L / ξ
Type I: κ < 1/√2
Type II: κ > 1/√2 (e.g., YBCO: κ ≈ 100)
| Material | Type | T_c (K) | H_c2 (T) | Use |
|---|---|---|---|---|
| Mercury (Hg) | I | 4.15 | 0.04 | Historic |
| Niobium (Nb) | II | 9.3 | 0.44 | RF cavities |
| Nb₃Sn | II | 18 | 30 | LHC magnets |
| MgB₂ | II | 39 | 15 | MRI (cheaper) |
| YBCO | II | 93 | >100 | Research, maglev |
6. High-Temperature Superconductors
In 1986, Bednorz and Müller discovered that lanthanum barium copper oxide (La-Ba-CuO) becomes superconducting at 35 K — far above anything BCS predicted for phonon-mediated pairing. Within a year, YBCO (YBa₂Cu₃O₇) reached 93 K, above the boiling point of liquid nitrogen (77 K), making practical applications via cheap cooling suddenly feasible. This won Bednorz and Müller the Nobel Prize in 1987, one of the fastest awards in physics history.
Despite 40 years of research, the pairing mechanism in cuprates is still debated. Phonons alone cannot explain T_c = 135 K (mercury barium copper oxide under pressure, the record for cuprates). Proposed mechanisms include spin fluctuations, charge density waves, and RVB (resonating valence bonds). In 2023, room-temperature superconductivity claims (LK-99) were tested globally and found not to be superconducting.
δ = 0 (undoped): Mott insulator, antiferromagnetic order
δ ≈ 0.05–0.10 (underdoped): T_c rises, strange metal, pseudogap
δ ≈ 0.16 (optimal doping): maximum T_c
δ > 0.20 (overdoped): T_c falls, Fermi liquid behaviour restored
YBCO (YBa₂Cu₃O₇): T_c = 93 K at δ_opt; Bi-2212 (Bi₂Sr₂CaCu₂O₈): T_c = 96 K
HgBa₂Ca₂Cu₃O₈ under 30 GPa pressure: T_c = 164 K (record cuprate)
7. Applications: MRI, Maglev, and Qubits
- MRI magnets: Hospital MRI scanners use NbTi or Nb₃Sn coils cooled to 4 K by liquid helium. They produce 1.5–3 T fields that persist indefinitely with zero energy input. A persistent-mode 21.1 T NMR magnet at NHMFL holds 700+ MHz ¹H resonance frequency.
- Particle accelerators: The LHC at CERN uses ~1 600 superconducting dipole Nb₃Sn magnets at 1.9 K, producing 8.33 T to bend 7 TeV proton beams through a 27 km ring. Without SC magnets the ring would need to be ~60 km for the same energy.
- Maglev trains: Japan's SCMaglev (L0 series) uses on-board YBCO coils and track ground coils. World speed record: 603 km/h (2015). The Chuo Shinkansen line under construction will use this technology.
- Quantum computing: Transmon qubits are Josephson junctions — two superconductors separated by a thin insulator. Quantum tunnelling of Cooper pairs through the barrier gives rise to anharmonic energy levels used as qubit states. IBM, Google, and others use aluminium Josephson junctions at 15 mK.
- SQUIDs (Superconducting Quantum Interference Devices): The most sensitive magnetic field detectors ever made. Used in brain imaging (MEG), geophysical surveys, and dark matter searches. Sensitivity: 10⁻¹⁷ T Hz^(-1/2).
8. The Open Mystery
Room-temperature superconductivity would transform energy distribution, computing, and transportation. The search continues across multiple fronts:
- Hydrogen-rich hydrides: LaH₁₀ at 215 GPa shows T_c ≈ 250 K (−23°C) — but only under megabar pressures that make practical use impossible. The record is H₃S at T_c ≈ 203 K (2015). These appear to be conventional BCS superconductors with very high phonon frequencies.
- Nickelate superconductors (2019): Nd₀.₈Sr₀.₂NiO₂ shows T_c ≈ 9–15 K — structurally analogous to cuprates, but with Ni instead of Cu, offering new theoretical clues.
- Graphene moiré systems (2018): Twisted bilayer graphene at the "magic angle" (~1.1°) hosts correlated insulating and superconducting phases — the same mechanisms as cuprates in an atomically clean platform.