1 Characteristic properties

The defining property of a superconductor is the disappearance of electrical resistance at very low temperatures. Associated with the infinitely good conductivity is the ability to completely displace a magnetic field from the interior of the material. The superconductivity is found in all metals and also in other compounds. Therefore, it is a fundamental property of condensed matter that goes far beyond a presence only in some special compounds. The phenomenon was discovered by Onnes in 1911.

In this chapter an overview of the known experimental properties of superconductors is given. Starting with the basic phenomena of the vanishing electrical resistance in section 1.1 and Meissner effect in section 1.2, all important effects are described in the sections 1.3-1.6.

1.1 Electrical resistance

In an ordinary metal at room temperature, the electrical resistance is relatively small, but not zero. Usually, one measures the resistance by applying a bias voltage and measuring the associated current . The electrical resistance  is then defined by the ratio , since the Ohm’s law  applies to metals at room temperature. If the same measurement is carried out at an extremely low temperature below a certain critical temperature, the current becomes ’infinitely large’, so that the value zero is assigned to the resistance. Then it is said from an experimental point of view that the material is superconducting.

The critical temperature  below which the resistance vanishes (usually called transition temperature) depends strongly on material properties and external parameters. One example of a quantity which controls the superconducting transition very well is an externally applied magnetic field.

Figure 1.1 shows the electrical resistance of a superconductor (green line) at low temperatures as typically measured by experiments. The central observation is a sharp transition from a typical behavior of the resistance of normal metals to unmeasurable small values if the temperature is cooled down below . For comparison, the temperature behavior of a usual metal, where the superconducting transition has been suppressed (for example using magnetic fields), is also shown (purple line).

The property of the vanishing resistance was discovered during resistance measurements on mercury. The transition temperature for mercury is relatively small, , but the transition to a superconducting state has been found very soon also for other metallic compounds where the transition temperature might be slightly higher. Figure 1.2 shows measured values of transition temperatures for selected superconducting materials and the corresponding year of discovery. The conventional superconductors (circles) have  values up to  for MgB2. The well-known copper-based high-temperature superconductors are characterized by relatively large critical temperatures up to  at normal pressure.

Using a magnetic field which changes in time, it is possible to induce a steady current in a superconducting loop. This phenomenon is closely related to the vanishing electrical resistance. Experiments could not find any measurable reduction of the steady current over decades, i. e. the half-life is usually measured to be larger than 106 years.

1.2 Meissner effect

If charges in the material can be displaced infinitely easily due to the lack of resistance, it is easy to imagine that they are also extremely sensitive to magnetic fields. The reason for this is that a small change in the magnetic field immediately leads to an induction of an electric field, to which the superconducting charges then react immediately with a large electric current. This current, in turn, generates a magnetic field that counteracts the external field. Experiments show that in most cases even the external field is completely displaced by this effect. This phenomenon, which occurs only in the superconducting state, is called the Meissner effect. It should be noted that metals in their normal state let the magnetic field almost completely into the material.

Thus, as a consequence of the vanishing electrical resistance a superconductor strongly interacts with an external magnetic field. The effect is a displacement of the magnetic field from

the interior of a superconductor during its transition to the superconducting state. A schematic picture of the Meissner effect is given in Figure 1.3. It was discovered in 1933 by Meissner and Ochsenfeld from measurements of the magnetic field distribution outside superconducting tin and lead samples.

The reaction of a material to an external magnetic field by the formation of its own magnetic field in the interior of the material is called magnetization of the material. The superconductor should thus have a very large magnetization directed against the external field. One speaks of ideal diamagnetism. As we will show in the following, the magnetization can be calculated without big effort if the Meissner effect is ideally realized, i. e. the magnetic field is completely displaced from the material. The complete displacement is of course an idealization, but we can very easily calculate further magnetic quantities, such as the magnetic susceptibility, which can be determined experimentally.

We consider a superconducting material that is placed in a homogeneous external magnetic field . Perfect realization of the Meissner effect means that the magnetic induction is zero in the whole material volume. From the general relation between external field and magnetization of the material, we immediately find the relation

 (1.1)

which determines the magnetic susceptibility in the superconducting state. This quantity describes the response of the material to an external magnetic field . Comparing (1.1) with the defining equation of the magnetic susceptibility we find in the superconducting state the value.

It turns out that in real materials the Meissner effect is not realized perfectly, i. e. there is a finite magnetic induction within a narrow region close to the surface of the superconducting material. The magnetic induction in the interior of the superconducting material is screened by surface currents flowing inside the region where the field penetrates. This area has a typical spatial extension of the order of the so-called London penetration depth  (see chapter 2). Typical values of  are around . In the surface area, the magnetic induction is non-zero but decays exponentially and approaches  in the interior of the material (bulk superconductor) where perfect diamagnetism is found as discussed above. The typical behavior of the magnetic induction as a function of the distance to the surface is illustrated schematically in Figure 1.4.

A superconductor is an ideal conductor and therefore any finite electric field causes an infinitely large electric current. Thus, inside a superconducting material energy conservation can only be fulfilled if the interior of the superconductor is free of any electric field, i. e. . This applies, of course, to a state of thermodynamic equilibrium. Thus, Maxwell's law of induction,



(: speed of light in vacuum) leads for  to a time-independent (static) magnetic field inside a superconductor.

1.3 Critical magnetic field

If the superconductor is in an external magnetic field, the superconducting state initially remains stable as long as the field strength is not too large. A further increase in the field strength leads to a phase transition to the normal state. The superconducting state can either disappear completely, or a mixed state of superconducting and normal conducting domains is first formed, which is then replaced by the normal state in a second transition.

In this section, we want to collect important experimental findings on the critical values of the external magnetic field. Here the question of importance is under...