Superconductivity

Superconductivity is not perfect conductivity. The distinction is the whole subject, and it is the first thing to get straight: a perfect conductor and a superconductor respond differently to a magnetic field, and only one of them expels it. Everything in this page is downstream of that single experimental fact.

Picture current in a copper wire as electrons scattering off vibrations and impurities, losing a little energy each time — that is resistance. Now picture the electrons paired up and moving as a single coherent object, described by one collective wavefunction that extends through the whole piece of metal. To scatter, an electron would have to break its pair, and at low temperature there is not enough thermal energy to do it. Resistance does not become small; it becomes exactly zero, and current set flowing in a superconducting ring has been observed to persist for years without decay.

The Meissner effect is stranger. Bring a magnet near a superconductor and the superconductor pushes back, levitating the magnet. That is not simply “zero resistance” doing its work: a perfect conductor would freeze in whatever field was present when it cooled, because with $\rho = 0$ the flux through a loop cannot change. A superconductor does the opposite — it expels the field regardless of history. Expulsion is a thermodynamic statement about the ground state, and that is why the London equations, which describe it, are not a consequence of $\rho = 0$.

Discovered by Onnes in 1911, in mercury at $T_c = 4.2\,\mathrm{K}$.

1. Zero resistance. $\rho = 0$ for $T < T_c$.
2. Meissner effect. Magnetic flux is expelled from the interior: $\mathbf{B} = 0$ inside for $T < T_c$ and $B < B_c$.
3. Critical field. Superconductivity is destroyed above $B_c(T) = B_c(0)\big[1 - (T/T_c)^2\big]$.
4. Critical current. Destroyed above a critical current density $J_c$.

The critical field is thermodynamic: it is where the free energy gained by condensing into the superconducting state no longer pays for expelling the field. The condensation energy density is

the difference between normal and superconducting free energies per unit volume, which is why $B_c(T)$ has a definite shape and is not a fitting curve.

The London brothers wrote the phenomenology down as two equations for the supercurrent density $\mathbf{J}_s$, with $n_s$ the density of superconducting electrons:

The first is Newton’s second law for carriers that never scatter: $m_e\,\partial\mathbf{v}/\partial t = -e\mathbf{E}$ with $\mathbf{J}_s = -n_s e\mathbf{v}$. The second is the one that carries new physics, and it is assumed, not derived from $\rho = 0$. To see that the two are genuinely different, take the first London equation and Maxwell’s equation for a conductor with $\sigma \to \infty$; what follows is that $\partial\mathbf{B}/\partial t = 0$ — flux is frozen, whatever it was when the material went superconducting. The second London equation says $\mathbf{B} = 0$ outright, which is a different claim and the one experiment supports.

Deriving the penetration depth. Take $\nabla \times$ of the second London equation and use Ampere’s law $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s$ (neglecting the normal current):

Using $\nabla \times (\nabla \times \mathbf{J}_s) = \nabla(\nabla \cdot \mathbf{J}_s) - \nabla^2\mathbf{J}_s$ and $\nabla \cdot \mathbf{J}_s = 0$ for steady state:

and identically for $\mathbf{B}$:

For a half-space $x > 0$ with field $B_0$ outside, the decaying solution is

so the field falls by $1/e$ over the London penetration depth $\lambda_L$, typically $10$—$100\,\mathrm{nm}$. The Meissner effect is not a surface with field exactly zero beyond it; it is field decaying over a finite depth, and $\mathbf{B} = 0$ in the bulk is the large-body limit of that decay.

The London equations describe the electromagnetic response but say nothing about why there is a superconducting order parameter, or what happens near $T_c$. Ginzburg-Landau theory introduces a complex order parameter $\psi(\mathbf{r})$ with $|\psi|^2 = n_s$ and writes the free energy as

Minimising gives two equations. Varying in $\psi^*$:

which is a Schrödinger equation for $\psi$ with charge $2e$ — the first hint of pairing, written down in 1950, seven years before BCS explained it. Varying in $\mathbf{A}$ gives

which reproduces both London equations in the limit $|\psi|$ constant, and additionally shows the flux through a ring is quantised in units of $h/2e$ — the other hint.

Two lengths fall out of the theory, and their ratio controls everything:

$\xi$ is the coherence length, the smallest distance over which $|\psi|$ can vary — the size of a Cooper pair. $\kappa$ is the GL parameter, and the single criterion

is the whole of the type I/II distinction. That a material’s response to a field is decided by one dimensionless ratio is the theory’s central result.

BCS theory (Bardeen, Cooper, Schrieffer, 1957) explains the order parameter. The mechanism is Cooper pairing: two electrons just above the Fermi surface with opposite momenta and spins form a bound state through the lattice-mediated effective attraction. The pair has charge $2e$ and spin $0$, so it is a boson and can condense.

The gap equation. With $E_{\mathbf{k}} = \sqrt{\xi_{\mathbf{k}}^2 + \Delta^2}$ the quasiparticle energy, $\xi_{\mathbf{k}}$ the normal-state energy measured from $E_F$, and $V_{\mathrm{pair}}$ the effective attraction, self-consistency requires

Dividing out $\Delta$ and replacing the sum by an integral over the density of states $N(E_F)$ (constant near the Fermi level, over the Debye window $\hbar\omega_D$) gives

At $T = 0$ the $\tanh$ is $1$ and the integral is elementary:

so

At $\Delta \to 0$, setting $T = T_c$:

Dividing the two eliminates the unknown coupling entirely:

a universal number with no free parameter, and the cleanest test of the theory. Measured values in conventional superconductors cluster around $3.5$.

The exponential in $-1/(N V)$ is the physical content. A weak attraction produces an exponentially small gap, which is why superconductivity was not understood for 46 years: perturbation theory in $V$ produces a series in $NV$, and the answer is not analytic at $NV = 0$. Every term of the perturbation series is finite and they sum to zero.

In the mixed state, magnetic flux penetrates as vortices — tubes of normal material carrying one flux quantum $\Phi_0 = h/2e$ each, with supercurrent circulating around them. Vortices repel, so they form a lattice; pinning them to defects is what lets a type II material carry large currents, which is why superconducting magnets are built from type II materials and why $J_c$ is an engineering quantity rather than a fundamental one.

Flux quantisation in units of $h/2e$ rather than $h/e$ was the experimental confirmation that the carriers have charge $2e$, and it is independent evidence for pairing.

Two superconductors separated by a thin insulating barrier exchange a supercurrent. The DC Josephson effect: with no applied voltage, a current

flows, where $\delta = \theta_1 - \theta_2$ is the phase difference across the junction and $J_c$ depends on the barrier. Current flows with zero voltage — not a small resistance, no resistance.

The AC Josephson effect: with a DC voltage $V$ across the junction, the phase evolves as

so the current oscillates at frequency $\omega = 2eV/\hbar$. Since $e$ and $\hbar$ are the two most precisely known constants in physics, this turns a voltage measurement into a frequency measurement and is the basis of the volt standard.

Both effects follow from one assumption: that $\psi$ is a coherent order parameter with a well-defined phase. Neither has a normal-state analogue, which is why the Josephson junction is the cleanest probe of superconducting phase coherence.

Discovered in 1986 by Bednorz and Müller. Cuprates such as YBCO ($\mathrm{YBa_2Cu_3O_{7-\delta}}$) reach $T_c \approx 135\,\mathrm{K}$, above liquid nitrogen’s $77\,\mathrm{K}$ — the reason they transformed applications.

They are type II with large $\kappa$, layered, and not explained by BCS. Three differences matter:

• $d$-wave pairing. The gap has $d_{x^2-y^2}$ symmetry, $\Delta(\mathbf{k}) = \Delta_0(\cos k_x - \cos k_y)/2$, which changes sign under a $90°$ rotation and vanishes along the diagonals $k_x = \pm k_y$. Conventional superconductors are $s$-wave, with $\Delta$ constant. The sign change is measurable in phase-sensitive experiments and is established.
• Short coherence length. $\xi \sim 1$—$2\,\mathrm{nm}$ against $\sim 100\,\mathrm{nm}$ conventionally, so $\kappa$ is large and the materials are extreme type II. It also makes them sensitive to defects at the scale of a unit cell.
• Strong anisotropy. Properties differ sharply between the $ab$-planes and the $c$-axis, a consequence of the layered perovskite structure.

The pairing mechanism is still open. Phonons cannot be the whole story: the isotope effect, which in conventional superconductors confirms lattice involvement through $T_c \propto M^{-1/2}$, is much reduced or absent in cuprates. Candidate mechanisms involve antiferromagnetic spin fluctuations, and no consensus has formed.

• Calling it perfect conductivity. A perfect conductor freezes in flux; a superconductor expels it. Only the second is observed, and only the London equations give it.
• Treating $\mathbf{B} = 0$ as exact everywhere. The field decays over $\lambda_L$, which is finite; the bulk statement is a limit for samples much larger than $\lambda_L$.
• Reading the gap equation as an equation to solve for $V$. It is a self-consistency condition on $\Delta$: $\Delta$ appears on both sides, and $\Delta = 0$ is always a solution. Superconductivity is the non-trivial solution, and it exists only below $T_c$.
• Expecting perturbation theory to find the gap. The answer is $\exp(-1/NV)$, which is non-analytic at $NV = 0$: every term of the series in $NV$ vanishes. This is why the problem sat open for 46 years.
• Quoting $2\Delta(0)/k_BT_c = 3.53$ as universal. It is universal within weak-coupling BCS. Strong coupling, and all the cuprates, deviate — and the deviation is diagnostic.
• Using $\kappa$ as a small correction. $\kappa$ decides type I versus type II outright at $\kappa = 1/\sqrt{2}$, which decides whether a material is useful for magnets.

• Crystal Structures — the lattice whose vibrations mediate the pairing interaction.
• Lattice Vibrations and Phonons — the Debye frequency $\omega_D$ entering every BCS formula.
• Electronic Band Structure — the Fermi surface and $N(E_F)$ that set the coupling scale.
• Magnetism in Solids — the magnetic response being expelled, and antiferromagnetic fluctuations as the candidate cuprate mechanism.
• Many-Body Physics in Solids — quasiparticles and the broken-symmetry description of the condensate.
• Superconductivity in the BCS Frame — the companion overview page.
• Quantum Mechanics — the coherent order parameter and phase, which is what the Josephson effects measure.
• Statistical Mechanics — the free-energy argument behind $B_c(T)$.

• Advanced Semiconductor Physics
• Advanced Semiconductor Physics (Continued)
• Advanced Topics in Superconductivity
• Crystal Structures