Basics of Superconductivity E-Book

Steffen Sykora

• Herausgeber: tredition
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 187

Veröffentlichungsjahr: 2024

Beschreibung

At very low temperatures, a metal passes into the superconducting state. This state is extremely fascinating, as the electrical resistance completely disappears. This property makes superconductivity very interesting for technology, since it is possible to transport electrical current without loss under certain conditions. This textbook provides an introduction to the basics of superconductivity, with a great focus on the underlying theoretical physics. It is intended to provide a basic understanding of the phenomena that occur in connection with superconductivity and its most important applications. The content ranges from an overview of the most important experimental findings of superconductors to a detailed description of the London theory and the Ginzburg-Landau theory. Prominent effects such as the Meissner effect, critical magnetic fields, the formation of vortices and many other phenomena are explained in great detail and described mathematically.

Leseprobe

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 which conditions regarding the magnetic field the superconducting state is thermodynamically stable.

1.3.1 Field energy

At first, let us start with a few energetic considerations. As discussed in the previous section, when applying an external magnetic field , the magnetic induction inside the material is suppressed. Thus, there is a strong difference between and . This occurs because the superconducting state can extremely easily induce currents on the surface whose magnetic field counteracts the external one. However, the induction of these currents and the associated dislocation of the magnetic field costs energy. If the magnetic field strength is relatively small, this energy loss is small compared to the gain in energy caused by the formation of the superconducting state. This will be shown later. Therefore, if the value of the magnetic field is smaller than some critical value, the gain in energy due to the superconducting state can be larger than the energy loss through the displacement of the field. As a result, the superconducting state is stable while the Meissner effect is present.

The amount of field energy which is 'removed' from the interior of the superconducting material can be calculated approximately as follows. If we assume that the London penetration depth  is small compared to the typical dimension of the superconducting material, , where is the sample volume, this energy is equivalent to the total field energy of the magnetic field inside the volume of the superconductor. In this limit it corresponds exactly to the field energy inside the space volume if the superconductor did not exist. The related amount of field energy can be calculated from electrodynamics, . Roughly speaking, if this energy loss  becomes (for particular high field strength) larger than the energy gain associated with the manifestation of the superconducting state over the normal state, the superconducting state becomes unstable.

1.3.2 Phase transition

The above considerations lead us to the insight that at a given temperature  the superconducting phase is thermodynamically stable if the external magnetic field is lower than a certain critical magnetic field . For temperatures larger than , only the metallic phase is stable. An approximate formula for , which is valid for most of the conventional superconductors, can be derived from experiments. It reads

(1.2)

where  is the critical field at zero temperature. The critical magnetic field curve represents the phase boundary between the superconducting phase and the metallic phase and is qualitatively shown in Figure 1.5. Note that there are unconventional superconductors in which the phase transition line described by (1.2) is further divided into two lines. These so-called type 2 superconductors will be treated later.

Generally, one distinguishes different orders of phase transitions. A simple way to specify the order of the phase transition is to investigate the latent heat which is exchanged during this transition. If a finite amount of latent heat is measured, this indicates a first-order phase transition. In the case of a second-order phase transition, there would be no exchange of latent heat. A typical example of a first-order phase transition is the melting of water ice, where latent heat is supplied during the melting process.

In superconductors actual measurements reveal that for all transitions that take place at finite temperatures , which are less than the  of the field-free case, a latent heat is exchanged. The exchanged heat decreases more and more if the temperature, at which the transition takes place, is increased. Thus, along the red curve in the phase diagram in Figure 1.5 the phase transition is of first order. If the transition finally takes place exactly at , i. e. without the presence of a magnetic field (), no latent heat is exchanged. This corresponds to a second-order phase transition. It is marked in Figure 1.5 with a point on the right edge of the phase boundary line. It can be shown theoretically that the phase transition at  on the left edge of the phase boundary line also corresponds to a second-order transition.

In summary, it can be said that in a finite magnetic field  and at finite temperature  the phase transition is of first order. In a field-free situation or at , it is of second order. We will describe and understand all the above-mentioned experimental results with the help of the theories to be developed.

1.3.3 Type 1 and type 2 superconductors

As already mentioned above, we have to distinguish between two different types of superconductors with regard to the phase transition. This is closely linked to the typical length scales that occur in the variation of the particle density and the magnetic induction. In addition to the London penetration depth  as the first important length scale, there is another one determining the properties of superconductors, the coherence length . This quantity will be defined correctly in chapter 4. Here it should be only mentioned that  describes the characteristic length scale in which the superconducting wave function changes significantly. In addition to the London penetration depth, the length  can be used to further classify superconductors.

An important dimensionless parameter in this context is the Ginzburg-Landau parameter, defined by the ratio . Depending on the value of , superconductors behave differently during the phase transition. This property is discussed in detail in chapter 5. In the following, a summary of the main results of this discussion is given.

In materials with , the transition takes place as described in the previous subsection: There is a critical magnetic field  at which the superconductivity breaks down. During the transition, the entire superconducting density disappears at the same time. The phase transition is of first order and accompanied by a latent heat transfer. Materials with  are called type 1 superconductors. In particular, materials with  in which the Meissner effect is realized perfectly, belong to this family.

Materials with  are called type 2 superconductors. They are characterized by the existence of two critical field values, denoted by  and . For external fields , the material behaves like a type 1 superconductor in the Meissner phase. For fields larger than the so-called lower critical field  the material enters an inhomogeneous superconducting phase. Here, the material has the property that a coexistence of superconducting and normal conducting regions in the material are observed. The material forms normal conducting and superconducting domains which coexist. The magnetic field partially penetrates into the material by flooding the normal conducting areas while it is still displaced from the superconducting areas.

In this phase, the magnetic flux passing through the normal conducting domains cannot be arbitrary, i. e. the field penetrates the material in a special way. It forms quantized tubes of magnetic flux which are formed by vortices of current (Abrikosov vortices). In this state, the current circulates around the normal conducting core of the vortex. Such structures were first described theoretically by Abrikosov in 1957 (Nobel Prize 2003).

The size of the Abrikosov vortex is described by the two length scales  and . While the core of the vortex has a characteristic size of the order of the coherence length  the currents flowing around the core decay on a distance of about . The circulating currents induce magnetic fields with the total flux equal to the flux quantum

(1.3)

where  is the Planck's constant,  the velocity of light, and  the elementary charge. Here,



is the total magnetic flux of the circular current of one vortex, i. e. the surface  is chosen such that it encloses completely the current belonging to the particular vortex.

At -values slightly above the lower critical field , the vortices arrange rather disordered. The vortex density increases with increasing magnetic field. An appropriate vortex model and the flux quantization is treated in chapter 5.

In a clean superconductor and for magnetic fields close to the upper critical field, the Abrikosov vortices arrange in form of a triangular lattice as illustrated in Figure 1.6. Each of the vortices carries one flux quantum. Note that, as with other lattices, defects may form as dislocations of the triangular Abrikosov lattice.

If the external magnetic field exceeds the upper critical field , the whole material volume enters the normal state. Typical values for the critical fields  and  are relatively large. For example, in NbSn3 () experiments find  and . Note that the magnetic field of the earth close to the earth's surface is approximately .

1.4 Energy gap

Superconducting materials show an interesting property in the specific heat capacity. This quantity describes the ratio of the heat added to (or removed from) an object to the resulting temperature change divided by its mass. Experiments measuring the specific heat capacity as a function of the temperature show that in superconductors this quantity makes a typical jump at the critical temperature. The qualitative behavior is shown in Figure 1.7. Below we will conclude from thermodynamic arguments that such a jump has to do with the transition into an ordered phase in which latent heat is exchanged. This experimental result thus indicates that the superconducting phase has a higher order than the metallic phase.

The specific heat capacity of a solid is generally composed of different contributions. For example, thermal oscillations of the lattice ions and also the system of the conduction electrons each provide a contribution. Assuming that the superconductivity is caused by the conduction electrons, it makes sense to look at their contribution to the heat capacity.

The typical temperature behavior of the electronic specific heat  in a superconductor is shown in Figure 1.7. As is well-known, in the normal (metallic) phase the specific heat depends linearly on . Instead, in the superconducting phase the material behaves

differently. Experiments observe an exponential -dependence of the form , where  is an energy constant of the order of the thermic energy  (: Boltzmann constant) associated with the critical temperature . The qualitatively different temperature behavior in the superconducting and normal state (green and purple lines in Figure 1.7) results in a characteristic jump of  at .

The energy  is closely related to an energy gap in the spectrum of states of conduction electrons. It is also connected to the binding energy of pairs of conduction electrons (Cooper pairs) that are formed below