Unconventional superconductivity in nearly-magnetic metals

Superconductivity

The discovery of superconductivity was one of the most important discoveries of the 20th century with promises of many applications in every day life. A superconductor is a material in which electricity flows with no resistance. It was discovered by H.K. Onnes in 1911 after experimenting with the resistance of mercury at low temperatures. He discovered that the resistance sharply dropped to zero below the critical temperature (Tc) of ~4.2 K. Since then researches have been pushing the boundaries of the critical temperature in an attempt to make superconductors viable for practical uses by understanding and engineering materials that superconduct.

The core behind any superconductor is the so-called Cooper pair. These pairs consist of two electrons with opposite momentum and spin. A Cooper pair carries electricity like a normal charge carrier, but suffers from no scattering and zero resistance. Since the spins add, they are considered bosons and obey such statistics. These electrons have a lower energy than a normal electron, measured by the energy gap (delta E). An energy of 2 delta E is required to break up a Cooper pair.

Conventional Superconductivity

In most conventional superconductors, Cooper pairs form as a result of lattice interactions. These quantized phonon waves change the shape of the lattice, distorting it around an electron and creating a net positive charge. This attracts a second electron nearby, forming a Cooper pair. If the energy of this bound state is greater than the energy of the oscillations of atoms from heat, then the electron pair will remain bound. These new particles are bosons and can be described as a large Bose-Einstein condensate.

The interaction that provides the attraction between the paired electrons can be considered as a virtual phonon. This phonon is virtual because the electron cannot change its energy sufficiently (~ hbar omega D) at temperatures below the Debye temperatures to create a real phonon of short wavelength. The timescale of the phonon is so small (< 1/omega D) that the existence is permitted by the energy-time uncertainty relation. Energy is conserved overall in the process. The BCS theory calculated many of the properties of superconductors by replacing the real interaction with this virtual instantaneous interaction spread out to a range ~vF/omega D to allow for the distance moved by an electron during this characteristic time (1/omega D) of ionic motion.

Magnetism in Superconductors

A superconductor's behavior in a magnetic field leads to the classification of Type I superconductors and Type II superconductors. All pure samples of superconducting elements are Type I superconductors, except Nb. This class of superconductors is characterized by the superconductivity being destroyed by applying a modest applied field Bc, called the critical field. Bc as a function of temperature is approximately shown as follows,

Bc(T) = Bc(0)[1 - (T/Tc)^2]

For fields below the critical temperature, flux lines are expelled from the superconducting material. This is known as the Meissner effect, and is caused by electric currents, called "screening" currents, that are generated on the surface of the superconductor in such a way to create a field equal and opposite of the applied field. We can regard the superconductor as a magnetic material and the screening currents can be replaced by an equivalent magnetization, hence:

B = mu not (H + M) = 0

Which leads to,

M = -H

We can then say that a superconductor has a magnetic susceptibility of X = -1. Thus, a type I superconductor is often called a perfect diamagnet.

Type II superconductors become type II when the electron mean free path is reduced significantly by alloying the material. These superconductors also expel flux lines at a critical field, but this critical point Bc1 is lower than the critical field normally. Above Bc1 there is partial flux penetration into the alloy, although it retains superconducting properties. The phase transition to the normal state where flux penetrates the entire object happens at a much higher field Bc2.

The electrodynamics of the screening currents was first worked out by Fritz and Heinz London. They suggested that the currents are described by,

curl j = - nse^2/m * B

This is known as the London equation. They arrived there by taking the curl of this,

j = - nse^2/m * A

By combining Maxwell's equations,

curl B = mu not j

and

div B = 0,

We get the result for the field B inside a superconductor,

lambda^2 del^2 B = B

where lamda^2 = m/mu not ns e^2.

The magnetic field in the superconductor is of the form B = B(x)z when the field is parallel to the boundary in a vacuum. B(x) satifies

lamda^2 d^2B/dx^2 = B.

The solution of this equation is

B(x) = a exp(-x/lamda) + b exp(x/lamda)

The second term can be rejected, as it described a field increasing exponentially at large distances from the boundary and does not physically describe the system. We require a = Be to satisfy B = Be at x = 0. Therefore,

B(x) = Be exp(-x/lamda)

The characteristic length scale lamda is known as the penetration depth.