Superconductivity was discovered by Heike Kamerlingh Onnes in 1911, in mercury cooled with liquid helium [9]. (Onnes had liquefied helium three years earlier, in 1908, which made the measurement possible.) Below a critical temperature $T_c$, certain metals show two characteristic signatures.
Zero DC resistance. The DC resistance drops to an unmeasurably small value at $T_c$. Persistent currents in superconducting rings have been observed to flow without decay for years [2].
The Meissner effect (1933). A superconductor expels magnetic flux from its interior. Below $T_c$ the field inside the bulk is zero, independent of whether the field was applied before or after cooling. The expulsion is produced by screening currents on the surface, which generate a field that cancels the applied field inside and opposes it outside. A superconductor is therefore a perfect diamagnet, not merely a perfect conductor: a perfect conductor would only freeze in pre-existing flux, whereas the Meissner state excludes it [2].
Because the final state is independent of cooling history, superconductivity is an equilibrium thermodynamic phase, and flux exclusion—rather than zero resistance—is the more fundamental property [2, 9]. A classical perfect conductor cannot expel a pre-existing field; flux exclusion requires a macroscopic quantum state.
A few materials used in circuit QED:
| Material | $T_c$ (K) | Gap $\Delta_0$ (meV) | Used for |
|---|---|---|---|
| Aluminum (Al) | 1.2 | 0.18 | Junctions, capacitor pads (most common) |
| Niobium (Nb) | 9.2 | 1.4 | Resonators, ground planes |
| Niobium nitride (NbN) | ~16 | ~2.3 | Recent qubit experiments (e.g. planaron) |
| Tantalum (Ta) | 4.5 | 0.69 | Long-coherence transmons |
The gap values follow the BCS relation $\Delta_0 \approx 1.76\,k_B T_c$, derived below [1].
The microscopic theory is the BCS theory of Bardeen, Cooper, and Schrieffer (1957) [1]. Its content can be stated in a few steps.
Cooper's instability (1956). In a normal metal at $T=0$, electrons fill states up to the Fermi energy $E_F$. Cooper showed that any attractive interaction between two electrons just above the Fermi surface, however weak, produces a bound state; the Fermi sea is unstable to pairing [1, 2].
The attractive interaction. The attraction comes from electron-phonon coupling. A moving electron polarizes the lattice of positive ions, leaving a region of excess positive charge that attracts a second electron. This interaction is retarded relative to the direct Coulomb repulsion, and within a narrow window $\hbar\omega_D$ around $E_F$ (with $\omega_D$ the Debye frequency) the net effective interaction is attractive [2].
This dependence on electron-phonon coupling explains an early puzzle. Lead, tin, and mercury are poor normal conductors yet superconduct, while copper, silver, and gold are excellent conductors yet do not. Strong electron-phonon scattering raises normal-state resistance but also drives pairing at low temperature; weak coupling gives low resistance but no pairing. A good normal conductor is therefore further from superconductivity, not closer to it.
The pair wavefunction. A Cooper pair consists of two electrons in $|\mathbf{k},\uparrow\rangle$ and $|-\mathbf{k},\downarrow\rangle$: opposite momentum, opposite spin, total momentum and spin zero. It is a spin singlet, and with integer total spin it is bosonic in character. It is not a point boson: the two electrons are spread over a coherence length $\xi$ much larger than the spacing between pairs, so neighboring pairs overlap heavily (within a coherence volume there are of order a million other pairs). For this reason superconductivity is not literally a Bose-Einstein condensate [2].
The BCS ground state. BCS constructed a many-body ground state in which each pair state $(\mathbf{k}\uparrow,-\mathbf{k}\downarrow)$ is occupied with amplitude $v_\mathbf{k}$ and empty with amplitude $u_\mathbf{k}$ [1]:
$|\Psi_\text{BCS}\rangle = \prod_\mathbf{k}\left(u_\mathbf{k} + v_\mathbf{k}\,c^\dagger_{\mathbf{k}\uparrow}c^\dagger_{-\mathbf{k}\downarrow}\right)|0\rangle, \qquad |u_\mathbf{k}|^2 + |v_\mathbf{k}|^2 = 1.$