Basics of Superconductivity: An Introduction to Theoretical Physics

Preface

1 Characteristic properties

1.1 Electrical resistance

1.2 Meissner effect

1.3 Critical magnetic field

1.3.1 Field energy

1.3.2 Phase transition

1.3.3 Type 1 and type 2 superconductors

1.4 Energy gap

1.5 Isotope effect

1.6 Superconducting loops

2 London Theory

2.1 London equations

2.2 Explanation of the Meissner effect

2.3 Quantum mechanical derivation

2.4 Concluding remarks

3 Thermodynamic basics

3.1 Laws of Thermodynamics

3.2 Heat and electromagnetic energy

3.3 Functional and functional derivation

3.4 Free energy

3.5 Gibbs free energy

3.6 Landau theory of phase transitions

4 Ginzburg-Landau theory

4.1 Homogeneous superconductor

4.1.1 Landau potential

4.1.2 Order parameter

4.1.3 Specific heat capacity

4.2 Inhomogeneous superconductors

4.2.1 Minimization of the free energy

4.2.2 Functional derivative

4.2.3 Ginzburg-Landau coherence length

4.2.4 Proximity effect

4.2.5 Loss of energy due to the boundary

4.3 Superconductor in a magnetic field

4.3.1 Gauge invariance

4.3.2 Set of basic equations

4.4 Property of superconductivity

4.4.1 Supercurrent and superfluid velocity

4.4.2 Relation to the London theory

4.4.3 Supercurrent versus order parameter

4.5 The characteristic length scales

4.5.1 Inhomogeneous system, field-free case

4.5.2 Homogeneous system in a magnetic field

5 Theoretical applications

5.1 Critical magnetic field

Approach to the critical field

5.2 The homogeneous state

5.3 Latent heat

5.4 The inhomogeneous state

5.4.1 Two phase transitions

5.4.2 Types of superconductors

Criterion for the type of superconductors

5.4.3 field equations in the London gauge

5.4.4 Boundary between type 1 and type 2

5.4.5 Inhomogeneous solution

5.4.6 The Ginzburg-Landau parameter

5.5 Model of type 2 superconductors

5.5.1 Flux quantization

5.5.2 Model of a single vortex

5.5.3 Lower critical magnetic field

5.5.4 Upper critical magnetic field

5.5.5 Magnetic induction versus magnetic field



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Preface

Aim of the present textbook is to give a thorough introduction to important theories of superconductivity in such a way that the reader will be able to get a deep understanding of the prominent effects in superconductors. Particular importance is attached to explain the basic properties of superconductivity and its most important applications. Thereby, in the present textbook we focus on the phenomenological theories of superconductivity. This means that the theoretical concepts are based on a few assumptions about the superconducting state, which are motivated by experiments, without going into the details of the underlying microscopic system. This is completely sufficient for a good understanding of the phenomena as this book strives for. The treatment of the microscopic theory of superconductivity, in particular the theory of Bardeen, Cooper, and Schrieffer (BCS), is subject of an upcoming second part of this textbook which is planned to be published directly after the present textbook.

After explaining the fundamental assumptions in great detail, all physical properties and effects in superconducting systems are derived from that. Great importance is placed on a detailed and comprehensible derivation of all formulas. A brief introduction to important concepts of classical physics (mechanics, electrodynamics, thermodynamics) and to the basic idea of quantum mechanics is given. All definitions and basic assumptions are explained in great detail, and all theoretical results are derived from these basic equations in such a way as the reader is able to follow the particular steps without additional literature. Final aim is to give the reader the ability to understand the basic experimental properties of superconductors.

An overview of these experimental findings is given in chapter 1 in form of a rather compact illustration. The theoretical explanations of these findings are then presented in a comprehensive way in the subsequent chapters 2-5. The following topics will be dealt with: Chapter 2 develops the basic properties of the London theory of superconductivity. Particular attention is given to the explanation of the famous Meissner effect and the theoretical description of the magnetic field inside a superconductor.

Chapter 3 gives an introduction to the basics of thermodynamics, which is needed for the concepts in this textbook. Starting with the laws of thermodynamics, the general concept of thermodynamic potentials is introduced and free energy and Gibbs free energy are defined. In this context, the concept of functional derivation is also discussed. The chapter ends with a basic introduction to the Landau theory of phase transitions.

Chapter 4 concentrates on the theory of the phase transition from the superconducting state to the normal state within the Ginzburg-Landau theory. Starting from the Landau theory of phase transitions, the concepts of order parameters and the minimization of the Landau potential are described. These considerations lead to a basic set of field equations describing the superconducting state as well as the magnetic field inside the superconductor. The equations are extensively derived and solved for specific cases. In this context, the characteristic length scales determining the superconducting state are also discussed.

Finally, in chapter 5, prominent applications of the Ginzburg-Landau theory are presented. Particular attention is given to the critical magnetic field. In this context, the chapter develops a systematic access to type 1 and type 2 superconductors and provides the derivation of the characteristic parameters to distinguish the two types of superconductors. Moreover, the lower and upper critical values of the magnetic field in type 2 superconductors are explicitly calculated. The chapter provides a comprehensive description of Abrikosov vortices and their relevance for the field penetration.

Throughout this book, we shall use Gaussian units. Relevant equations within this textbook are numbered and referred in the text by citing the corresponding equation number. All figures are numbered as well.

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  equation.pdf and measuring the associated current  equation.pdf . The electrical resistance  equation.pdf is then defined by the ratio 

, since the Ohm’s law  equation.pdf 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  equation.pdf 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  equation.pdf . 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  equation.pdf values up to 

for MgB2. The well-known copper-based high-temperature superconductors are characterized by relatively large critical temperatures up to  equation.pdf 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  equation.pdf . Perfect realization of the Meissner effect means that the magnetic induction  equation.pdf is zero in the whole material volume. From the general relation 

between external field  equation.pdf and magnetization  equation.pdf 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  equation.pdf . Comparing (1.1) with the defining equation 

of the magnetic susceptibility equation.pdf 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  equation.pdf (see chapter 2). Typical values of  equation.pdf are around  equation.pdf . In the surface area, the magnetic induction  equation.pdf is non-zero but decays exponentially and approaches  equation.pdf in the interior of the material (bulk superconductor) where perfect diamagnetism is found as discussed above. The typical behavior of the magnetic induction  equation.pdf 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  equation.pdf 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.  equation.pdf . This applies, of course, to a state of thermodynamic equilibrium. Thus, Maxwell's law of induction,



( equation.pdf : speed of light in vacuum) leads for  equation.pdf 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  equation.pdf , the magnetic induction  equation.pdf inside the material is suppressed. Thus, there is a strong difference between  equation.pdf and  equation.pdf . 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