SQUIDs: A Technical Report - Part 1: Superconductivity

Superconductivity

In 1911 H. K. Onnes found that when certain substances are cooled below a critical temperature, T_c, the electrical resistance becomes very small, effectively vanishing. For most metals T_c is of the order of a few Kelvin.

The Meissner Effect

When describing superconductors, it is the magnetic properties that are important. The superconductor is not a perfect conductor, but a perfect diamagnetic material with zero electrical resistance. This is known as the Meissner effect.

The time dependence of the magnetic field in a perfect conductor is described via the classical equation:

where _L is known as the penetration depth. This has the solution:

which physically suggests that apart from a thin surface layer controlled by _L (a decay constant of the order of 10^-6cm), the magnetic field inside the perfect conductor obeys the condition

The above equation implies differences between the perfect conductor and the superconductor. The magnetic state of a perfect conductor is found to be dependent on the order in which the external field is applied (i.e. before or after the material is superconducting).

However, the Meissner-Ochsenfeld effect showed that the superconducting state is found to have no dependence of this kind. In fact the magnetic flux is always expelled, displaying properties of a perfect diamagnetic material.

A refinement came about with the London equation

where J is the current density, n the density of electrons, e/m the electron charge/mass and A the vector potential. One assumption leading to this equation is that only certain electrons (called `superelectrons`) are responsible for the superconducting state. Therefore, the above equation only applies to these `superelectrons`.

The success of this theory is mainly due to the fact that the magnetic flux does not appear anywhere, thus building on the ideas of a perfect conductor to arrive at certain conclusions.

This theory still retains the basic ideas put forward by the Meissner theory though incorporates some other interesting features. The conclusions reached from this theory are:

An external magnetic field will penetrate the superconductor by an order of _L.

The supercurrent (carried by the `superelectrons`) and a normal current exist in equilibrium within the superconductor. The density of the `superelectrons` is zero at T_c, reaching a maximum at 0K.

The supercurrent in a superconductor is confined to the outer region, which is characterised by _L.

Any magnetic flux through a hole in the superconductor is quantised in units of

where q is the carrier charge and is found to be equal to twice the electron charge.

It was found that the critical temperature for any given superconductor varied with the average atomic mass. T_c M^-. This is known as the isotope effect.

The consequences of this empirical relation were far reaching. It gave a strong indication that the electrons were responsible for the superconducting state.

The Classification of Superconductors

Using a thermodynamic argument one can show that an applied external magnetic field will affect the free energy and therefore the stability of the system. Thus, if an external magnetic field applied to a superconductor exceeds a critical field B_c, the flux will penetrate into the material causing it to return to its normal state.

Using the above criterion, the two types of superconductors are;

Type 1. This type has a characteristic as shown in figure 1. The dependence of the critical magnetic field on the temperature can be seen clearly. Typical values for B_c are around 0.04T at 0K.

Type 2. Discovered in 1962 this has two critical magnetic fields associated with its behaviour, enabling it to have a composite N/S structure. (N stands for normal state, S for the superconducting state). See figure 1. Above the first critical value B_c1 the flux penetrates in a series of regular positioned filaments. The core of each filament is in the normal state, thus magnetic flux flows through in quanta given by equation 1. When the second critical field B_c2 is reached, the flux penetrates completely and the material reverts to its normal state.

Figure 1: Superconductor types

Figure 2: Superconductor types

The BCS (Bardeen, Cooper and Schrieffer) Theory

So far we have only concerned ourselves with macroscopic properties to try and describe superconductivity. Since the change in energy per valence electron over the N-S transition is only around 10^-6eV (which is much less than the uncertainty in the calculation of the electron states), the atomic theory is very difficult to understand.

A greater understanding came about in 1956 when L. Cooper explained a process by which two electrons near the Fermi level could couple to form an effective new particle, even under a very weak attractive force. This particle was subsequently called the Cooper pair. It was shown that the most energetically favourable situation for this to occur was when the two electrons had a total spin of zero together with equal and opposite wave functions.

The interaction between the two electrons arises due to lattice dynamics. When any electron travelling through a lattice polarises the lattice a `phonon wave` is set up. Consequently, another electron will experience this disturbance and react to reduce the potential energy, thus causing an effective attraction which overcomes the Coulomb repulsion. Since the dispersion rate of the `phonon wave` is very slow, the attraction can be felt over distances of 10^-6m (200 times the interatomic distance). This process can be represented schematically as in figure 2.

[Figure 2]
Figure 2: The electron with wave vector k₁ polarising the lattice. The phonon created has wave vector q which is absorbed by another electron with wave vector k₂. The time evolution is progressing upwards on the page and the two final electron states are k₁ - q and k₂ + q.

Thus, a superconductor consists of electrons occupying one-electron states and what is known as the condensate. This is composed of Cooper pairs, each having the same energy, momentum and invariant phase relationship. It behaves as a separate entity and is a very coherent state.

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