ISEGORIA / MATH ENCYCLOPEDIA
Superconductivity: zero resistance and flux expulsion
Cool lead, niobium or certain ceramics below a critical temperature and two things happen at once: electrical resistance vanishes, and a magnetic field is pushed out of the bulk. The second is the deeper of the two. A perfect conductor would merely trap whatever field it held; a superconductor expels it, so the superconducting state is a phase of matter with its own free energy, not just a very good metal. London explained the expulsion with one equation that ties the supercurrent to the vector potential, and found that a field dies away within a penetration depth of the surface. Ginzburg and Landau described the transition with a complex order parameter whose phase the material must choose, and Abrikosov showed that in many materials the field enters as a lattice of vortices, each carrying exactly one flux quantum h/2e.
Before you begin: Electromagnetism (Maxwell equations, vector potential), thermodynamics, complex numbers
Predict, manipulate, then test yourself: every experiment ends with practice questions that are generated fresh at three levels. Graphs illustrate the mathematics; they do not replace a proof.
1. The Meissner effect on a sphere
London proposed that the supercurrent is proportional to the vector potential, \(\mu_0\lambda^2\mathbf J=-\mathbf A\). Combined with Ampère’s law this gives \(\nabla^2\mathbf B=\mathbf B/\lambda^2\): inside the metal the field decays over the penetration depth \(\lambda\), typically 50 to 500 nm. For a sphere in a uniform field the equation can be solved exactly. Outside, the field is the applied one plus that of a magnetic dipole pointing against it; inside, it is a modified spherical Bessel function that is largest at the surface. The two join smoothly at the surface. Here the sphere’s size is fixed in units of \(\lambda_0\), the depth at zero temperature, and warming it toward Tc makes \(\lambda\) grow until the expulsion fails.
Worked example. Take a sphere twelve penetration depths across in radius, \(R=12\lambda_0\), at half the critical temperature. Then \(\lambda=\lambda_0/\sqrt{1-0.5^4}=1.033\lambda_0\) and \(R/\lambda=11.62\). The induced moment is \(1-3/11.62\times\coth11.62+3/11.62^2=0.764\) of a perfect diamagnet’s, because the outer shell, about one \(\lambda\) thick, is partly penetrated. The field just outside the equator is \(1.382B_0\), short of the ideal 1.5, and at the centre it is \(2\times10^{-4}B_0\). Warm to \(t=0.9\) and \(\lambda\) grows to \(1.71\lambda_0\): the moment falls to 0.634 and the centre field rises to \(0.012B_0\). A sphere forty depths in radius keeps 0.927 at \(t=0\).
Watch out. This is the London model with the two-fluid law for \(\lambda(T)\); it treats the superconductor as a local, linear medium and is accurate when \(\lambda\) is much larger than the coherence length (type II, London limit). The applied field is assumed far below the critical field, so the sphere stays wholly superconducting; a real sphere enters an intermediate state when its equator reaches the critical field. Above Tc the normal metal is drawn as nonmagnetic. The screening ring marks the direction of the current, which actually spreads through the outer shell as \(\sin\theta\).
Why is the field strongest at the equator?
The flux that would have passed through the sphere has to go around it, and it is squeezed into the gap beside the equator. For a perfect diamagnet the normal component of B vanishes on the surface and the tangential component is \(\tfrac32B_0\sin\theta\), largest at \(\theta=90^\circ\). This is why a type I sphere first lets field in at its equator, at two thirds of the critical field.
2. Ginzburg–Landau: a ring of minima
Ginzburg and Landau described the superconducting state by a complex order parameter \(\psi\), with \(|\psi|^2\) the density of superconducting electrons and its phase the thing that makes currents flow. Near Tc the free energy can be expanded in powers of \(|\psi|^2\): \(a|\psi|^2+\tfrac b2|\psi|^4\), with \(a\) changing sign at Tc. Above Tc the minimum is at \(\psi=0\). Below Tc the minimum is a circle, \(|\psi|^2=-a/b\), and the material has to settle at one point of it: the phase symmetry is broken. The energy gained falls as \((T_c-T)^2\), so its second derivative, the specific heat, jumps at Tc without any latent heat: a second-order transition.
Worked example. In units with \(a_0T_c=b=1\), at \(t=T/T_c=0.6\) the coefficient is \(a=-0.4\), so the minimum sits at \(|\psi|^2=0.4\), \(|\psi|=0.632\), with \(f_s-f_n=-0.4^2/2=-0.08\). Halving the distance to Tc, \(t=0.8\), halves \(|\psi|^2\) to 0.2 and quarters the condensation energy to \(-0.02\). Differentiating \(-a_0^2(T_c-T)^2/(2b)\) twice and multiplying by \(-T\) gives a heat capacity jump of \(a_0^2T_c/b\) at Tc; in aluminium it is about 1.4 times the normal-state \(\gamma T_c\), close to the value 1.43 that the BCS theory predicts. The coherence length grows as \(\xi_0/\sqrt{1-t}\): 1.58\(\xi_0\) at \(t=0.6\).
Watch out. The expansion is valid only near Tc, where \(|\psi|\) is small; the curves are drawn down to \(t=0\) to show their shape, but the true order parameter saturates and the specific heat falls exponentially at low temperature. The spatial gradient term and the magnetic field, which give the coherence length and the penetration depth, are left out of the surface; only uniform states are shown. The heat capacity is scaled to the weak-coupling BCS jump.
If every phase has the same energy, why does the phase matter?
A uniform phase costs nothing, but a phase that varies in space does: the gradient term \(\frac{\hbar^2}{2m^*}|(\nabla-\tfrac{2ie}{\hbar}\mathbf A)\psi|^2\) charges for every twist. A supercurrent is exactly a phase gradient, \(\mathbf J\propto|\psi|^2(\hbar\nabla\theta-2e\mathbf A)\), and because the phase must return to itself around any loop, the flux through a ring of superconductor is quantised in units of \(h/2e\).
3. Vortices and the flux quantum
In a type II superconductor, where the ratio \(\kappa=\lambda/\xi\) of the penetration depth to the coherence length exceeds \(1/\sqrt2\), it pays for the field to enter above a lower critical field Hc1 in thin tubes. Each vortex has a normal core of radius \(\xi\), and around it a supercurrent that confines exactly one flux quantum \(\Phi_0=h/2e\) within a few \(\lambda\). In London theory the field of one vortex is \(\Phi_0K_0(r/\lambda)/(2\pi\lambda^2)\). Vortices repel, and Abrikosov showed they form a lattice; the triangular one has the lowest energy. The mean field is one flux quantum per unit cell, so the spacing follows from the field alone. As the field rises toward Hc2 the cores crowd together until superconductivity is lost.
Worked example. Niobium–titanium, the wire of MRI magnets, has \(\lambda\approx300\) nm and \(\kappa=\lambda/\xi\approx60\), so \(\xi=5\) nm. Its upper critical field is \(\Phi_0/(2\pi\xi^2)=2.068\times10^{-15}/(2\pi\times25\times10^{-18})=13.2\) T, and its lower one about \(\Phi_0\ln60/(4\pi\lambda^2)=7.5\) mT. At a mean field of 0.0395 T the vortex spacing is \(a=\sqrt{2\Phi_0/(\sqrt3\bar B)}=246\) nm, 0.82\(\lambda\). Because the fields of neighbouring vortices overlap, the field between them only dips to 97% of the mean.
Watch out. The London vortex field diverges logarithmically at the centre; the picture cuts it off at the core radius \(\xi\), and the London model is good only while the cores are far apart (\(\bar B\ll H_{c2}\)). The Hc1 formula is the leading term for large \(\kappa\). Real lattices are pinned and distorted by defects, which is what lets a superconducting wire carry current without the vortices drifting and dissipating energy.
Why is the flux quantum h/2e rather than h/e?
Because the order parameter describes pairs of electrons with charge \(2e\). Going once around a vortex, the phase of \(\psi\) must change by \(2\pi\), and the London current vanishes far from the core only if \(2e\oint\mathbf A\cdot d\boldsymbol\ell=2\pi\hbar\), that is \(\Phi=h/2e\). The measured value, by Deaver and Fairbank and by Doll and Näbauer in 1961, was early direct evidence for Cooper pairing.