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The clean isotropic weak-coupling BCS coherence length is conventionally ${\xi }_0=\hslash v_F/(\pi \Delta (0))$, with Fermi velocity $v_F$. It characterizes the spatial extent of pairing correlations.^[ref]

The scale follows from the same spectral calculation. Linearizing the normal energy near the Fermi surface gives ${\xi }_k\simeq \hslash v_F(k-k_F)$. The region $|{\xi }_k|\lesssim \Delta$ therefore has radial width $\delta k\sim \Delta /(\hslash v_F)$. Fourier analysis associates this width with a distance $\ell \sim 1/\delta k\sim \hslash v_F/\Delta$. This argument establishes the scale, not the conventional factor $1/\pi$. A small superconducting energy gap consequently permits pairing correlations over many interparticle spacings.

For a particle-hole-symmetric normal density of states $g_F$ and isotropic superconducting energy gap $\Delta >0$, the electron spectral density is zero for $|E|<\Delta$ and has the displayed form outside the gap. The BCS coherence factors weight the two roots $\xi =\pm \sqrt{E^2-{\Delta }^2}$, where $\xi$ is normal-state energy relative to the Fermi energy. At positive $E$, their electron weights $(1+\xi /E)/2$ sum to one. Multiplying their sum by the Jacobian $|d\xi /dE|=E/\sqrt{E^2-{\Delta }^2}$ proves the formula. This electron spectral density differs from the unweighted count of all positive-energy quasiparticles, which receives both roots. Its gap-edge divergence is integrable and approaches the normal electronic density at large $|E|$.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

A nonzero superconducting energy gap suppresses the population of thermally excited quasiparticles. When $k_{B}T\ll \Delta$, their Boltzmann factor produces an approximately activated electronic heat capacity,where the prefactor $A(T)$ varies algebraically rather than exponentially. For the rough estimate requested, neglect its change between the two quoted temperatures. Taking a ratio eliminates its unknown magnitude:ThereforeThe corresponding excitation energy is $\Delta \simeq 2.19\times 10^{-22}\mathrm{J}\simeq 1.36\mathrm{meV}$. The units $\mathrm{mJ}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ of the two heat-capacity readings cancel in their ratio.

This is a heat-capacity activation estimate, not an exact determination of a gap from two data points. More generally, if $C_e=A{T}^p{e}^{-\Delta /(k_{B}T)}$, the same measurements giveFor example, the low-temperature BCS theory form for a fully gapped isotropic superconductor has $p=-3/2$ and would give about $19.5\mathrm{K}$. The stated approximate data and constant-gap assumption support the simpler $16\mathrm{K}$ estimate; they do not specify a detailed density of states or heat-capacity prefactor. Here $\Delta$ means the minimum quasiparticle excitation energy, rather than automatically twice that value.

Single-electron superconducting tunneling transfers individual Electrons across a barrier into or out of quasiparticle states of a material with superconductivity. In a normal-metal–insulator–superconductor junction, abbreviated NIS, only one electrode has superconductivity. Such a junction can reveal a superconducting energy gap without supporting a DC Josephson effect between two superconducting reservoirs.

For an elementary interpretation, assume an opaque barrier, approximately constant normal-electrode density of states, and an energy-independent tunneling matrix element over the range probed. Write the superconducting density of states divided by its normal-state value as $n_s(E)$, with $E$ measured from its chemical potential. A signed voltage convention chosen for positive normal-electrode injection giveswhere $f$ is the Fermi-Dirac distribution at a common temperature and $R_N$ is the normal-state junction electrical resistance. This current convention is local to the NIS measurement. The expression counts occupied initial and empty final states; it is a quasiparticle transport model, not a current-phase relation of a superconducting weak link.

An original analytic illustration uses the ideal fully gapped BCS theory spectrum at zero temperature, with positive gap edge $\Delta$:For positive $V$, the occupation difference restricts the integral to $0<E<eV$. Since $d\sqrt{E^2-{\Delta }^2}/dE=E/\sqrt{E^2-{\Delta }^2}$, integration and odd extension giveThe threshold is $\Delta /e$ for this NIS model. In the corresponding zero-temperature single-electron process between two fully gapped superconductors, creating excitations on both sides instead requires $|eV|\ge {\Delta }_1+{\Delta }_2$. That quasiparticle threshold does not forbid coherent pair electric current below it. At finite temperature, thermally populated quasiparticles soften the simple threshold argument.

At zero temperature in the NIS approximation, $R_{N}d{I}_{\mathrm{qp}}/dV=n_s(eV)$ away from the singular gap edge. Thus a voltage sweep probes the excitation density of states. The ideal divergence is rounded in real measurements, and at nonzero temperature the derivative samples a thermally broadened range of energies. For $|eV|>\Delta$, the model has $V{I}_{\mathrm{qp}}>0$: electrical work creates excitations and is dissipated on relaxation. This distinguishes its steady transport from the conservative Josephson coupling energy.