Superconductivity: Phenomenology and BCS

In 1911, while probing the properties of metals at liquid-helium temperatures, Kamerlingh Onnes found that the resistance of mercury dropped abruptly to zero below a critical temperature $T_c=4.2\text{K}$. Below $T_c$ the resistivity is not merely small but exactly zero: currents set up in superconducting rings have persisted for years with no measurable decay.

Critical temperatures range from below $0.1\text{K}$ (hafnium, iridium) to $9.25\text{K}$ (niobium) for elements. An applied magnetic field lowers $T_c$; above a critical field $B_c$ superconductivity vanishes at any temperature. The metal compound ${\text{MgB}}_2$ superconducts at $39\text{K}$, and the cuprate ceramics — beginning with the 1986 discovery and reaching ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_7$ at $92\text{K}$ — superconduct above the $77\text{K}$ boiling point of inexpensive liquid nitrogen.

The Meissner effect

Zero resistance is not the defining property. Consider cooling a material through $T_c$ in a small external field. Zero resistance alone (via Faraday's law) would only freeze the field already inside. What is observed is stronger: the field is actively expelled, and the interior field becomes exactly zero.

Establishing the screening current costs an energy density $B^2/2{\mu }_0$; when $B>B_c$ there is not enough energy available and the material reverts to normal. Superconductors come in two kinds:

• Type I (soft), mostly pure metal elements, show the complete Meissner effect up to a single $B_c$, which is too small for useful magnets.
• Type II (hard), usually alloys, have two critical fields. Below $B_{c1}$ they fully expel flux; between $B_{c1}$ and $B_{c2}$ the field penetrates as quantized flux tubes (vortices) of normal material threading the superconductor; above $B_{c2}$ they go normal. Because $B_{c2}$ can reach tens of tesla, type II materials build high-field magnets.

The flux threading a superconducting loop is quantized. Because no emf can exist in the loop, the enclosed flux is frozen, and quantum mechanics sharpens this to

The quantum of flux ${\Phi }_0$ is a fluxoid; each flux tube in a type II superconductor carries exactly one. The factor of $2e$, not $e$, is the first hint that the charge carriers are electron pairs.

BCS theory and Cooper pairs

The mechanism was found in 1957 by Bardeen, Cooper, and Schrieffer. The decisive clue was the isotope effect (1950): the critical temperature depends on the average isotopic mass $M$ as

with $\alpha \approx 0.5$ for many superconductors. A dependence on the mass of the ions means the lattice vibrations — phonons — are essential. If the ions were infinitely heavy (fixed lattice), $T_c$ would be zero.

The pairing works through the lattice. An electron moving through the lattice attracts the nearby positive ions, displacing them slightly and leaving a region of enhanced positive charge behind it — a propagating lattice distortion, i.e. a phonon. A second electron is attracted to that positive region. The net effect is an attraction between the two electrons, mediated by the lattice.

Below $T_c$ this attraction can exceed the Coulomb repulsion, and the electrons form a bound Cooper pair with opposite spins and opposite momenta — total spin zero, total momentum zero.

Because all the pairs occupy one quantum state and act together, a single pair cannot be scattered by a lattice ion without breaking it. Breaking a pair costs the superconducting energy gap $E_g$, predicted by BCS to be

at $T=0$. For cadmium ($T_c=0.517\text{K}$) this gives $E_g=1.56\times 10^{-4}\text{eV}$, within 4% of the measured value — some four orders of magnitude smaller than a semiconductor gap. When a pair carries net momentum, the whole condensate carries current, and since scattering that changes one pair's momentum is forbidden below the gap, the current flows without resistance.

The gap closes as $T\to T_c$, where $E_g\to 0$ and pairs break freely. The critical field follows a similar law, $B_c(T)=B_c(0)[1-(T/T_c{)}^2]$. Cooper pairs are large: estimating the pair size from the uncertainty principle gives $\Delta x\sim 10^3\text{nm}$, about ten thousand atomic diameters, so the pairs overlap heavily and the condensate is a single coherent quantum state on a macroscopic scale.

The Josephson effects

That macroscopic coherence is visible when two superconductors are separated by a thin insulating barrier — a Josephson junction. Cooper pairs can tunnel across it with no resistance. Josephson predicted in 1962 that with no applied voltage a supercurrent flows,

set by the phase difference of the pair wave functions on the two sides (the dc Josephson effect). With a dc voltage $V$ across the junction, the current instead alternates at

(the ac Josephson effect), again showing the pair charge $2e$.

Because frequency is measurable to extreme precision, the ac Josephson effect fixes the ratio $e/h$ and sets voltage standards. The same coherence underlies SQUID magnetometers sensitive to a single flux quantum. Superconductivity closes this module: it is the point where the quantum statistics of Cooper-pair bosons, the tunneling of the Schrödinger equation, and the lattice physics of solids combine into a single macroscopic quantum state.