Fundamental Elements of Applied Superconductivity in Electrical Engineering

By Yinshun Wang

Superconducting technology is potentially important as one of the future smart grid technologies. It is a combination of superconductor materials, electrical engineering, cryogenic insulation, cryogenics and cryostats. There has been no specific book fully describing this branch of science and technology in electrical engineering. However, this book includes these areas, and is essential for those majoring in applied superconductivity in electrical engineering.

Recently, superconducting technology has made great progress. Many universities and companies are involved in applied superconductivity with the support of government. Over the next five years, departments of electrical engineering in universities and companies will become more involved in this area. This book:

• will enable people to directly carry out research on applied superconductivity in electrical engineering
• is more comprehensive and practical when compared to other advances
• presents a clear introduction to the application of superconductor in electrical engineering and related fundamental technologies
• arms readers with the technological aspects of superconductivity required to produce a machine
• covers power supplying technologies in superconducting electric apparatus
• is well organized and adaptable for students, lecturers, researchers and engineers
• lecture slides suitable for lecturers available on the Wiley Companion Website

Fundamental Elements of Applied Superconductivity in Electrical Engineering is ideal for academic researchers, graduates and undergraduate students in electrical engineering. It is also an excellent reference work for superconducting device researchers and engineers.

• Language: English
• Publisher: Wiley
• Release date: Apr 17, 2013
• ISBN: 9781118451175

Book preview

1

Introduction

In 1911, the physicist H.K. Onnes, of Leiden Laboratory in the Netherlands, was measuring the resistivity of metals at low temperatures. He discovered that the resistance of mercury completely disappeared when the temperature dropped to that of liquid helium (4.2 K). This phenomenon became known as superconductivity. In 1933, German scientists W. Meissner and R. Ochsenfeld found that the magnetic flux completely disappeared from the interior of materials with zero resistance when cooled to 4.2 K in the magnetic field. This zero magnetic field inside a material became known as perfect diamagnetism and is now called the Meissner effect.

In 1962, B.D. Josephson theoretically predicted the superconducting quantum tunneling effect, known as the Josephson effect. This is where a current flows for an indefinitely long time, without any voltage applied, across a device known as a Josephson junction (JJ) consisting of two superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier (known as a superconductor-insulator-superconductor, or S-I-S) junction, and a short section of non-superconducting (S-N-S) metal. Subsequently, P.W. Anderson and J.M. Rowell experimentally confirmed Josephson's prediction.

Since its discovery, the superconductor and its applications have been one of the most active research fields in modern science and technology, due to its unique physical properties of zero resistance, perfect diamagnetism and the quantum tunneling effect. Superconducting technology is mainly applied in electrical engineering and electronics, and these applications and characteristics are listed in Table 1.1.

Table 1.1 Main applications and characteristics of superconducting technology in electrical engineering

Table 1.1 shows that superconducting technology is of great value in the fields of energy resources, transportation, scientific instruments, medical care, national defence and large scientific project. Since its discovery, widespread application of the superconductor has become the pursuit of scientists and engineers. Before the 1960s, practical NbTi and Nb3Sn superconducting wires were not manufactured until nearly half of a century after the discovery of superconductivity. Since then, superconducting technology and application of superconducting magnets are used for laboratory and practical applications. However, the conventional superconductors have not been widely used in power systems, particularly in alternating current (AC) applications, because of their need to operate at 4.2 K.

With the development of NbTi wires, the Magnetic Resonance Imaging (MRI) system has been increasingly used in hospitals for clinical diagnosis since the 1980s. In 1986, a true breakthrough was made in the field of superconductivity by A. Müller and G. Bednorz, researchers at the IBM Research Laboratory in Rüschlikon, Switzerland. They created a brittle copper oxide ceramic compound, the so-called high temperature superconductor (HTS), which presents superconductivity at temperatures above 40 K. Since then, several kinds of HTS have been discovered and the transition temperature from the normal to superconducting state has reached more than 90 K, which is higher than the liquid nitrogen temperature of 77 K.

Therefore, superconducting apparatus working at temperatures of 77 K made the widespread use of superconducting technology possible. With the great progress in development of HTS materials in the late 1990s, practical HTS tapes were manufactured and commercialized. The application of superconducting power technology was developed on a large scale, with the support of governments and multinational companies. Many prototypes of superconducting power apparatus, such as cable, transformer, FCL, motor/generator and SMES, were developed and demonstrated. At present, commercial superconducting apparatus, particularly the HTSs, are continuing to be developed with increasing investment from governments and companies. It is believed that a major breakthrough in superconducting technology will continue well into the future.

Applications of superconductors in electrical engineering primarily involve superconducting power technology and superconducting magnet technology. With their transition from normal state to superconducting state and then the largely increased current carrying capacity at high current density and zero resistance, superconducting power technology has developed further [1–3]. Table 1.2 lists the main advantages of superconducting apparatus and their influence on the electrical power industry. Other applications include the dynamic synchronous condensers (DSC), magneto-hydrodynamic (MHD) generation of power, cryogenic capacitors, gyrotrons, and superconducting induction heaters [4–14].

Table 1.2 Main characteristics, advantages and influences of several superconducting power apparatus

Table 1.2 shows that utilization of superconducting power technology cannot only improve power quality, enhance safety, stability and reliability of the grid system, reduce voltage level, and make a super large-scale power grid possible, but also greatly increases apparatus capacity and transmission capacity, and simultaneously reduces loss of power to the grid. Furthermore, the quality of power from renewable energy resources can be improved by using SMES to which a large grid can be effectively connected.

In comparison with a conventional magnet, a superconducting magnet has many unique advantages, such as no energy consumption, small volume, light weight, greater efficiency, greater thermal stability, longer magnetic field life and easier cooling, and also the ability to generate a high magnetic field in a relatively large space. Superconducting magnet technology has been extensively applied in large science projects, scientific instruments, inductive heating, magnetic separation, traffic, biomedicine and the defence industry [15–17].

Because superconductivity appears only at low temperatures, cryogenic technology is an essential prerequisite for any superconducting apparatus. Maintaining the cryogenic temperature consumes more energy and, in particular, maintaining the helium temperature will consume even more energy, which is a major impediment to the commercialization of low temperature superconductor (LTS) technology in AC operation. However, the cooling technology greatly influences customer perception of the superconductor product and its operational costs, especially its reliability. Except for certain high-value-added applications, such as MRI or NMR, the potential user is inconvenienced by the requirement to transfer liquid cryogen periodically. HTSs require low maintenance and low cooling costs for commercial acceptability, which is achieved since HTS application requires liquid nitrogen temperatures. The eventual widespread introduction of HTSs to power applications will depend on reliable closed-cycle refrigeration systems.

References

1. Tsukamoto, O. (2005) Roads for HTS power applications to go into the real world: Cost issues and technical issues. Cryogenics, 45, 3–10.

2. Hull, J.R. (2003) Applications of high-temperature superconductors in power technology. Report on Progress in Physics, 66, 1865–1886.

3. Moyses Luiz, A. (2011) Applications of High-Tc Superconductivity, InTech, Vienna.

4. Reis, C.T., Dada, A., Masuda, T., et al. (2004) Planned grid installation of high temperature cable in Albany, NY, Power Engineering Society General Meeting. IEEE Transaction on Applied Superconductivity, 14 (2), 1436–1440.

5. Elschner, S., Bruer, F., Noe, M., et al. (2006) Manufacture and testing of MCP2212 Bifilar coils for a 10 MVA fault current limiter. IEEE Transaction on Applied Superconductivity, 13 (2), 1980–1983.

6. Xin, Y., Gong, W.Z., Niu, X.Y., et al. (2009) Manufacturing and test of a 35 kV/90 MVA saturated iron-core type superconductive fault current limiter for live-grid operation. IEEE Transaction on Applied Superconductivity, 19 (3), 1934–1937.

7. Schwenterly, S.W., McConnel, B.W., Demko, J.A., et al. (1999) Performance of a 1 MVA HTS demonstration transformer. IEEE Transaction on Applied Superconductivity, 9 (2), 680–684.

8. Meinert, M., Leghissa, M., Schlosser, R., et al. (2003) System test of a 1-MVA-HTS-transformer connected to a converter-fed drive for rail vehicles. IEEE Transaction on Applied Superconductivity, 13 (2), 2348–2351.

9. Barnes, P.N., Sumption, M.D., and Rhoads, G.L. (2005) Review of high power density superconducting generators: Present state and prospects for incorporating YBCO windings. Cryogenics, 45, 670–686.

10. Kummeth, P., Frank, M., Nick, W., et al. (2005) Development of synchronous machine with HTS motor. Physica C, 426–431, 1358–1364.

11. Luongo, C.A., Baldwin, T., Ribeiro, P., and Weber, C.M. (2003) A 100 MJ SMES demonstration at FSU-CAPS. IEEE Transaction on Applied Superconductivity, 13 (2), 1800–1805.

12. Kim, M.J., Kim, K.K., Lee, H.G., et al. (2010) Current-lead design for variable electric current in HTS power applications. IEEE Transaction on Applied Superconductivity, 20 (3), 1725–1728.

13. Geri, A., Salvini, A., and Veca, G.M. (1995) MHD linear generator modeling. IEEE Transaction on Applied Superconductivity, 5 (2), 465–468.

14. Hanai, S., Kyoto, M., and Takahashi, M. (2007) Design and test results of 18.1 T cryo-cooled superconducting magnet with Bi-2223 insert. IEEE Transaction on Applied Superconductivity, 17 (2), 1422–1425.

15. Runde, M. and Magnusson, N. (2003) Design, building and testing of a 10 kW superconducting induction heater. IEEE Transaction on Applied Superconductivity, 13 (2), 1612–1615.

16. Ohkura, K., Okazaki, T., and Sato, K. (2008) Large HTS magnet made by improved DI-BSCCO tapes. IEEE Transaction on Applied Superconductivity, 18 (2), 556–559

17. Gupta, R. and Sampson, W. (2009) Medium and low field HTS magnets for particle accelerators and beam lines. IEEE Transaction on Applied Superconductivity, 19 (3), 1095–1099.

2

Superconductivity

D. Dipak

A superconductor has several main macroscopic characteristics, such as zero resistance, the Meissner effect, the Josephson effect, the isotope effect, anomalous specific heat capacity and abnormal infrared electromagnetic absorption. Relating to its main application in an electrical system, this chapter focuses on three basic properties, namely zero resistance, the Meissner effect and the Josephson effect.

2.1 The Basic Properties of Superconductors

2.1.1 Zero-Resistance Characteristic

The zero resistance characteristic of the superconductor refers to the phenomenon that resistance abruptly disappears at a certain temperature. It is able to transport direct current (DC) without resistance in the superconducting state. If a closed loop is formed by a superconductor in which a current is induced, the induced persistent current will show no obvious signs of decay for several years. The upper limit of resistivity measured by the persistent current experiment is less than 10−27 Ω·m, while a good conventional conductor, such as copper, has a resistivity of 10−10 Ω·m at 4.2 K, which is more than 17 orders of magnitude than that of the superconductor. The typically experimental dependence of resistance on temperature in a superconductor is shown in Figure 2.1, in which the resistivity of the superconductor suddenly falls to zero when the temperature reduces to a certain value below the critical temperature Tc.

The zero resistance property of a superconductor is not the minimal resistance in the usual sense but is equal to zero. This is because carriers are not scattered by the crystal lattice, thus there is no energy dissipation in a superconductor carrying a DC current, which suggests that superconductivity is a kind of macroscopic quantum effect.

Figure 2.1 Resistance versus temperature curve of superconductors and normal conductors.

ch02fig001

Since the discovery of superconductivity in 1911, there have been many attempts to establish a theory to explain this phenomenon, and a number of models describing physical characteristics of superconductors have been established. Some simple and easily understandable models belong to phenomenological theories of which the two-fluid model is a relatively intuitive theory. This model can successfully describe motion of carriers and magnetic field distribution within the superconductor. Combined with the constitutive Maxwell's electromagnetism equations, the two-fluid model explains some superconducting phenomena such as zero resistance and the Meissner effect. Based on a series of interaction hypotheses between electrons and lattice in quantum mechanics, in 1957, J. Bardeen, L.N. Cooper and J.R. Schrieffer proposed the concept of Cooper pairs and established the well-known Barden–Cooper–Schrieffer (BCS) theory, that is, the superconducting quantum theory that describes superconductivity from the microscopic point of view and successfully explains most of superconducting phenomena. In order to easily understand superconductivity, this chapter introduces both the phenomenological two-fluid model and the BCS theory.

2.1.1.1 Two-Fluid Model

The two-fluid model is a phenomenological model based on the following three basic assumptions [1]:

(i) Carriers consist of superconducting electrons and normal electrons in the superconducting state, the former transporting current without resistance and the latter transporting current with resistance, respectively. Then the carriers’ density is composed of superconducting electron density and normal electron density.

(ii) Superconducting electrons and normal electrons are defined as follows. In the superconducting state, the carrier density of the superconductor n is combined by normal electrons and superconducting electrons:

(2.1) Numbered Display Equation

where n denotes total carrier density in the superconductor, nS and nN refer to superconducting carrier density and normal carrier density, respectively, while eS and eN represent superconducting electron charge and normal electron charge, respectively.

(iii) In the superconductor, normal current density JN and superconducting current density JS mutually penetrate and independently transmit. Both can be interchangeable according to different temperatures and magnetic fields, and finally constitute total current density J of the superconductor:

(2.2) Numbered Display Equation

(iv) If the velocity of a normal electron is vN, the normal current density JN in the superconductor is:

(2.3) Numbered Display Equation

Since the normal electrons are scattered by lattice vibrations, impurities or defects, the resistance of the conductor is not zero.

If the velocity of the superconducting electron is vS, the resistance of the superconductor is zero, because the electrons are not scattered by lattice vibrations, impurities or defects, then superconducting current density Js in the superconductor is:

(2.4) Numbered Display Equation

Based on classical mechanics, if the superconducting electrons with mass mS and charge eS are not scattered by lattice vibrations, impurities or defects, they will be accelerated in the electric field E and obey Newton's second law:

(2.5) Numbered Display Equation

Combining Equations (2.4) and (2.5), we obtain:

(2.6) Numbered Display Equation

If superconducting current density JS is steady (DC), the left-hand side of Equation (2.6) is zero, thus E = 0. According to Ohm's law, JS = σE, where σ is the conductivity and should be infinite, namely resistivity ρ = 0. However, if superconductor current density JS varies with time, that is, the left-hand side of Equation (2.6) is not zero, thus E ≠ 0, then there is an electric field E in the superconductor and it will drive normal electrons, which will cause Joule loss (losses of superconductors on AC conditions will be introduced in Chapter 5). Therefore, this again illustrates that zero resistance of the superconductor, or E = 0, only occurs at steady operation (DC).

In order to describe variation of superconducting electron density with temperature T, an order parameter relating to temperature ω(T) is introduced:

(2.7) Numbered Display Equation

When temperature T is higher than critical temperature Tc, the superconductor is in a normal state, and the superconducting electron density nS(T) is 0, then ω(T) = 0. While temperature T = 0 K, all electrons convert into superconducting electrons, ω(T) = 1, and the superconducting electron density nS(0) is equal to n. If temperature T is in the range of 0 < T < Tc, and the superconductor is in a superconducting state, then the range of the order parameter ω(T) and carriers density is 0 < ω(T) < 1 and 0 < n < nS, respectively, and superconducting electrons appear in the superconductor. Thus, the critical temperature Tc, at which the superconductor transfers from the normal state to the superconducting state, may be considered as the corresponding temperature when Gibbs free energy density takes the minimum value within the range of 0 < ω(T) < 1 with T < Tc.

According to the relationship between Gibbs free energy density and the order parameter ω(T), if T < Tc, dependence of the order parameter of a stable system on temperature is:

(2.8) Numbered Display Equation

Substituting Equation (2.8) into Equation (2.7), we obtain:

(2.9) Numbered Display Equation

where t = T/Tc is the normalized temperature.

When the temperature reduces to absolute zero, namely T = 0 K, all electrons convert into superconducting electrons. Superconducting electron density decreases as temperature increases. When the temperature rises to the critical temperature, T = Tc, and the superconducting electron density is equal to zero, then the superconductor transits to the normal state from the superconducting state, namely, the superconductor quenches.

2.1.1.2 Microscopic Theory – BCS Theory

Although the two-fluid model simply and qualitatively explains the macroscopic superconducting phenomena, as a phenomenological model it cannot fundamentally explain the mechanism of superconductivity. In order to deeply understand superconductivity with the zero-resistance effect, BCS theory that describes superconductivity from a microscopic point of view is introduced according to the Bose–Einstein condensation and interaction theory between electrons and lattice in quantum mechanics. This theory can explain most of superconducting phenomena.

The physical world is composed of two types of elementary particle. One type is the fermion with spins of half an odd integer s = ±1/2, ±3/2, ±5/2, … ; and the other is the boson with spins of integer s = 0, ±1, ±2, ±3 … . Electrons are fermions with spin s = ±1/2, while photons are bosons with spin s = 0. The spin parameter s is an important basic parameter for describing these microscopic particles. Based on the theory of quantum statistical mechanics, a fermion can only occupy one state and obeys the Fermi–Dirac distribution:

(2.10) Numbered Display Equation

where μ denotes chemical potential, E is the electron energy and kB refers to the Boltzmann constant.

In the case of bosons, more than one boson can occupy one and the same energy state, and obey the Bose–Einstein distribution:

(2.11) Numbered Display Equation

If all bosons occupy the same energy state, this phenomenon is known as the Bose–Einstein condensation.

Macro-mechanical laws comply with Newton's laws of mechanics, and macroscopic electromagnetic phenomena obey Maxwell's equations. However, the micro-particle complies with the Schrodinger equation. In classical physics, quantities such as mass, energy, momentum, force, angular momentum and displacement, etc., describe the phenomenon of macroscopic physics. Nevertheless, in microscopic physics, quantities such as mass and spin describe the microscopic particles, while the physical quantities of a single particle such as energy, momentum, force, angular momentum and displacement are meaningless. The laws of microscopic particles are described by wave functions, and the square of absolute value of wave functions represents the probability density of microscopic particles (if the wave function is normalized).

To find the possible energy state of a two-electron system, the wave function of the combined system with an opposed wave vector (± k) is:

(2.12)

Numbered Display Equation

where r1 and r2 denote space coordinate vectors of two electrons, respectively, while ak is the expansion coefficient of the intrinsic plane wave function of free particles. Wave function (2.12) must satisfy the two-electron Schrödinger equation:

(2.13) Numbered Display Equation

where V(r1, r2) represents the effective interaction potential between two electrons, inline = h/(2π), h is Planck's constant and E denotes the Eigenvalue. Substituting Equation (2.12) into Equation (2.13), multiplying both sides by inline and integrating over all space by considering orthonormality of the Eigenstates, gives:

(2.14) Numbered Display Equation

where inline k is the Eigenstate energy of the single particle and Vkk′ denotes the expectation (average) value of the interaction potential between a pair of opposite momentum states.

If k is below the Feimi level, kF, ak disappears due to the Pauli exclusion principle in electron systems. Furthermore, Vkk′ is small when k is more than some cut-off term kC, because the ion-core remains approximately stationary when the electron exits it rapidly. Assuming that the potential Vkk′ is zero below the Fermi energy EF and above EF + EC(kC) and is constant –V between EF and EF + (kC), this indicates that two electrons attract each other and the interaction potential energy is negative (V > 0). These two electrons are called the Cooper pair, in which two electrons or holes have opposing wave vectors and are bound by an attractive interaction to form an integer spin quasi-particle in a superconductor, in spite of their electric charges having the same sign [2, 3]. For simplicity, the average of interaction potential is assumed to be:

(2.15) Numbered Display Equation

Substituting Equation (2.15) into Equation (2.14) and taking the potential energy term out of the summation sign, we find:

(2.16) Numbered Display Equation

Taking the sum on both sides of Equation (2.16) over k and considering the normalization condition, we obtain:

(2.17) Numbered Display Equation

According to quantum static mechanics theory, if the number of quantum states is large, summation can be replaced by integration with the weighted term NF, where NF is the state density at the Fermi energy EF level. Then we have:

(2.18) Numbered Display Equation

By integral calculation and series expansion and taking approximation at first level, Equation (2.18) becomes:

(2.19) Numbered Display Equation

or

(2.20) Numbered Display Equation

From Equation (2.19) we conclude that the energy of the two-electron system is smaller than that of two free electrons if there is attractive interaction between them, no matter how weak that attraction is. Universally, the system is always in a state with minimum energy. Thus, despite the charges of the two electrons having the same sign, as long as there is a net attractive interaction, it will always facilitate the formation of electron bound pairs, the Cooper pairs. It should be noted that the occurrence of Cooper pairs is a collective effect rather than a direct combination of two electrons in the superconductor, and the bonding strength depends on the states of all electrons. In classical electromagnetic theory, there is a Coulomb repulsion force between charges with the same sign, which contradicts the concept of Cooper pairs.

According to BCS theory, a pair of electrons is coupled by the interaction between an electron and a phonon, which is equivalent to direct interactions between two electrons, and this coupling makes each electron move toward surroundings of the ion-core with a positive charge due to the Coulomb attractive force. The electron alters the positive charge distribution adjacent to the ion-core when it moves through the lattice, so that a local region with high positive charge distribution forms, which results in attractive interaction with other adjacent electrons. Therefore, there is a possible attractive interaction between those electrons with opposite wave vectors to form an integer spin quasi-particle in spite of their electric charge with the same sign, and Cooper pairs finally form by using the lattice as a media.

There is a remarkable symmetry in Cooper pairs. According to quantum mechanics, the wave function of electrons must be anti-symmetric since they are fermions. However, there are two electrons in each Cooper pair, thus interchange of two electrons does not alter the symmetry, because the sign is changed twice. Because each electron has a spin s = ±1/2, the spin of the Cooper pair is s = −1, 0 or 1, which indicates that the Cooper pair is a boson. Each electron of the Cooper pair must have two actions. First, it must act as a fermion in order to provide exclusion to make pairing possible. At the same time, it must be a member of the Cooper pair by acting as a boson. Moreover, the Cooper pair is a boson with integer spin s = −1, 0 or 1, which means that an arbitrary number of Cooper pairs can be in the same state, especially in the ground state. If this happens to bosons, it is known as Bose–Einstein condensation. If it happens to a Cooper pair, superconductivity will occur.

To explain superconductivity that results from carriers forming Cooper pairs, we briefly introduce the mechanism of resistance generation in a normal conductor. In conventional conductors, the directional movement of a single electron is affected by inelastic scattering of the lattice, so part of the energy in the electron will be delivered to the lattice, which results in increasing vibration amplitude of the lattice, namely, the temperature rises and produces Joule heat. This is also the origin of resistance in a conventional conductor. Nevertheless, in a superconductor, Cooper pairs act as carriers. When they move directionally, if an electron inelastically collides with the lattice, it will lose part of its energy to the lattice; but the other electron with the opposite wave vector in the Cooper pair will simultaneously obtain the same energy from the lattice by its inelastic collision with the lattice. Consequently, the total net energy of the Cooper pair does not change in the whole scattering process, that is, there is no energy loss. Thus there is no resistance in directional movement of the carriers, and superconductivity occurs.

With temperature increase, thermally excited normal electrons appear near the Fermi surface, and two electrons are generated when each Cooper pair is destroyed. Thus, the number of normal electrons increases with the decreasing number of Cooper pairs. When temperature T reaches its critical temperature Tc, Cooper pairs disappear so that a superconductor transits to normal state from superconducting state, that is, it quenches.

Although BCS theory is able to successfully explain most of low temperature superconductivity with temperatures below 25 K, it has difficulty in explaining the microscopic mechanism of high-temperature superconductivity. Until now, the microscopic mechanism of high-temperature superconductivity has not been clear, but it is certain that its origin still comes from the bound states of electrons in pairs, that is, the idea of carrier pairing still works. Furthermore, the relevance amongst electrons in HTS is strong and thereby is beyond the range of the weak interactions required by the BCS theory.

2.1.2 Complete Diamagnetism – Meissner Effect

When the superconductor is subjected to a magnetic field, in a non-superconducting state, the magnetic field can penetrate the superconductor and so the inner magnetic field is not zero in its normal state (Figure 2.2(a)) [4]. However, when the superconductor is in a superconducting state, the magnetic flux within is completely excluded from the superconductor, and the inner magnetic field is zero, that is, the superconductor is completely diamagnetic (Figure 2.2(b)). This phenomenon is called the Meissner effect. The superconductor can be suspended in a magnetic field due to its diamagnetism or the Meissner effect (Figure 2.2(b)), in which a YBCO bulk is in a magnetic field produced by conventional magnetic materials (NdFeB permanent). When the YBCO bulk is in a superconducting state, with liquid nitrogen temperature of 77 K, it is suspended in air because the magnetic flux is completely excluded from the YBCO, which results in a magnetic levitation force. We will explain the Meissner effect of superconductors in this section according to phenomenological theory.

Figure 2.2 Meissner effect and levitation of superconductor: (a) normal state; (b) Meissner state; (c) YBCO bulk levitation.

ch02fig002

Since the density and the phase of a Cooper pair varies slowly compared with its size, V.L. Ginzburg and L.D. Landau first proposed that the collective wave function can be defined as inline in superconductors, because all Cooper pairs are in the same local state [5], where |Ψ|² = n is the density of the Cooper pairs, and φ denotes the phase of the wave function. In terms of wave function Ψ, current density associated with Cooper pairs may be found by multiplying velocity v with charge 2e and then taking the mathematical expectation value by integrating over all space:

(2.21) Numbered Display Equation

The momentum of particles in an electromagnetic fields is:

(2.22) Numbered Display Equation

where p, e and m are the momentum of the Cooper pairs, electron charge and mass, respectively, and A denotes the magnetic vector potential.

As mentioned above, momentum and velocity in quantum mechanics should be expressed in the form of operators (quantized):

(2.23) Numbered Display Equation

Then the current density:

(2.24)

Numbered Display Equation

By taking curl calculation on both sides of Equation (2.24), we obtain:

(2.25) Numbered Display Equation

Equation (2.25) is called the London equation. Because ∇ × B = μ0J, μ0 refers to the vacuum permeability, and we obtain:

(2.26) Numbered Display Equation

substituting Equations (2.25) into Equation (2.26) and then rearranging, we obtain:

(2.27) Numbered Display Equation

Defining:

(2.28) Numbered Display Equation

then Equation (2.28) becomes:

(2.29) Numbered Display Equation

A semi-infinite superconductor is used to simply describe the Meissner effect of superconductors (Figure 2.3). By choosing the Cartesian coordinate system, the superconductor infinitely extends along the y- and z- and positive x-z-axes, respectively. When the superconductor is exposed to the external magnetic field, which is uniform and along the z-axis direction, then Equation (2.29) is simplified to a one-dimensional (1D) differential equation with an x component only:

(2.30) Numbered Display Equation

Boundary conditions are:

Figure 2.3 Semi-infinite superconductor in uniform magnetic field.

ch02fig003

(2.31) Numbered Display Equation

The solution of Equation (2.30) is:

(2.32) Numbered Display Equation

where λ is called penetration depth. If the Cooper pair density n is about 10²⁸ m−3, λ is approximately 10−6 cm, and then the current density in the superconductor is:

(2.33) Numbered Display Equation

Figure 2.4 shows dependence of distributions of magnetic field and current density on penetrated depth in the semi-infinite superconductor. It indicates that the magnetic field and current density will exponentially decay with distance x from the surface and exist only in the thin outer layer of the superconductor with penetrated depth x ∼ λ. When x > 5λ, either the magnetic field or current density decays almost to zero. The external magnetic field induces a current on the surface of the superconductor, which flows without resistance, and in turn the magnetic field produced by the induced current exactly offsets the external magnetic field B0. Then the inner magnetic field B is zero, from the macroscopic point of view, because the penetration depth is so small that it can be ignored when compared with conventional finite size. Therefore, the superconductor presents complete diamagnetism, namely the Meissner effect. Although the magnetic field and induced current in the outer thin layer of the superconductor with thickness of penetration depth is not zero, the penetration depth is usually so small that the magnetic field and induced current can be considered as approximately zero, that is, the superconductor fully expels magnetic fields.

Figure 2.4 Dependence of magnetic field and induced current density on penetration depth.

ch02fig004

In classical electromagnetic theory, the level of the discrete magnetic flux lines density is commonly used to describe the magnitude of the magnetic field to explain the concept of magnetic flux density. But there arises the question of whether the flux exists in discrete or continuous forms within the superconductor?

By considering that the magnetic flux Φ goes through a closed superconducting loop and there is no current density inside it, that is, J = 0, according to Equation (2.24), we have:

(2.34) Numbered Display Equation

By integrating along the superconductor loop, we obtain:

Unnumbered Display Equation

so

(2.35) Numbered Display Equation

Since the phase varying along the closed superconducting loop can only be an integer times of 2π, then the following equation must be satisfied:

(2.36) Numbered Display Equation

where m is an integer. Substituting Equation (2.36) into Equation (2.35), we find:

(2.37) Numbered Display Equation

where Φ0 = 2.07 × 10−15 Wb. This means that the magnetic flux does not vary continuously and can only increase or decrease by integer times of Φ0. This is thus called the flux quantization and Φ0 is known as a single quantum.

2.1.3 Josephson Effects

As with semiconductor and thermocouple devices, there is the concept of a junction in a superconductor when it joins with an insulator. Two superconductors are separated by a thin insulating layer whose thickness is so small that Cooper pairs can pass through by the tunneling effect [6, 7]. This geometry is called the Josephson junction (Figure 2.5). S1 and S2 denote two kinds of superconductors, respectively, and I refers to the thin insulating layer. ψ1 and ψ2 are the wave functions of superconductors S1 and S2.

Figure 2.5 Geometry of Josephson junction.

ch02fig005

If η is the characteristic rate for tunnelling through the central thin insulating layer, according to the time-dependent Schrödinger equation, the rate of change of wave functions can be found as:

(2.38a) Numbered Display Equation

(2.38b) Numbered Display Equation

Let inline , n1 and n2 correspond to the density of Cooper pairs of superconductors on both sides of the junction, φ1 and φ2 are the corresponding phases of two wave functions, and inline = h/(2π) is Planck's constant. Substituting ψ1 and ψ2 into Equation (2.38a), we obtain:

Unnumbered Display Equation

After rearranging, we have:

(2.39) Numbered Display Equation

Its real part is:

(2.40) Numbered Display Equation

Since current density J flowing through the junction is proportional to the rate of change of Cooper pair density n1, J is proportional to ∂n1/∂t. If superconductors S1 and S2 on both sides of the junction are exactly the same, n1 is approximately equal to n2, so by combining all constants into a coefficient J0, we obtain:

(2.41) Numbered Display Equation

This is completely different from the current density of a normal conductor and junction. A current density with patterns of a sine wave relate to the quantum phase difference going through the junction to which there is no applied voltage. This phenomenon is called the DC Josephson effect.

Now applying a DC voltage V to both sides of the junction and adding the energy term to the Hamiltonian of the Cooper pair with a charge of 2e, we have the time-dependent Schrodinger equations:

(2.42a) Numbered Display Equation

(2.42b) Numbered Display Equation

Substituting the wave functions ψ1 and ψ2 into Equation (2.42a), we obtain:

(2.43) Numbered Display Equation

Its imaginary part is:

(2.44) Numbered Display Equation

Similarly, substituting wave functions ψ1 and ψ2 into Equation (2.42b), we obtain:

(2.45) Numbered Display Equation

Let superconductors on both sides of the junction be the same, then n1 is approximately equal to n2 and by subtracting Equations (2.44) from Equation (2.45), we get:

(2.46) Numbered Display Equation

Integrating Equation (2.46) over time and substituting it into Equation (2.43), after reorganizing and taking the real part, we obtain:

(2.47) Numbered Display Equation

Equation (2.47) shows that there is an AC current whose frequency is proportional to the product of the fundamental constant e/h, even when the DC voltage is applied. This phenomenon is called the AC Josephson effect.

If two Josephson junctions are connected in parallel, a loop is formed, known as the superconducting quantum interference device (SQUID) (Figure 2.6). According to Equation (2.35), the phase difference around the loop must be 2eΦ/ inline . In addition, it is known that the total current is the sum of the two branches; each one is a sine function of the phase difference of the Josephson junctions, and the combined current of the two branches is:

(2.48) Numbered Display Equation

where φ0 denotes the overall phase. Due to the interference between the two currents, the total current is a periodic function of the magnetic flux, which is why a device with this structure is called an interferometer. The minimum flux of the SQUID loop is a single flux quantum, Φ0 = h/2e = 2.07 × 10−15 Wb.

Figure 2.6 Geometry of two identical Josephson junctions connecting in parallel to form SQUID.

ch02fig006

The Josephson effect is an important basis of superconducting electronics applications and has widespread applications in many instruments such as voltage reference, superconducting cavities, superconducting filters and SQUIDs. For example, it is the most sensitive method to detect a magnetic field, by using SQUID to measure current oscillations. SQUID can convert many other small signals in the magnetic field, so it can also be employed to measure these small current signals in clinical diagnosis, such as the magnetocardiogram (MCG) and the magnetoencephalogram (MEG). Superconducting tunnelling junctions can also be used as basic devices for high-speed, low-dissipation logic families.

2.2 Critical Parameters

Generally, there are three basic critical parameters, that is, critical temperature Tc, critical field Hc and critical current density Jc, which are most important parameters in applications of superconductors.

2.2.1 Critical Temperature Tc

The superconductor shows superconductivity when its temperature is below a certain value, that is, the temperature at which the superconductor transfers to a superconducting state from a normal state. This temperature is called the critical temperature and is denoted by Tc. In general, the superconducting transition usually occurs in a temperature range near Tc, which is called the temperature transition width represented by ΔTc. In metal or alloy superconductors with high purity, a single crystal and stress free, ΔTc is smaller than 10−3 K; however, transition width ΔTc of practical HTS materials is usually in the range of 0.5 to 1 K, owing to their intrinsic characteristics, such as internal inhomogeneity, weak link, granularity and defects.

2.2.2 Critical Field Hc

Superconductors lose their superconductivity when the magnetic field strength exceeds a certain value in the external magnetic field. The magnetic field strength that causes a superconductor to lose its superconductivity is called the critical field strength and is denoted by Hc. When the temperature is less than critical temperature Tc, Hc is a function of temperature and continuously increases with temperature decrease. Like the critical temperature Tc, there is also a field transition width ΔHc in the vicinity of Hc when the superconductor transfers from normal state to superconducting state. For a practical superconductor, there are usually two critical fields, namely the upper critical field Hc2 and the lower critical field Hc1. When the field H is less than Hc1, the superconductor is in the Meissner state; however, when the field H is larger than Hc2, the superconductor is in the normal state; while the field H is between Hc1 and Hc2, the superconductor is in the mixed state. These properties will be discussed in more detail in the next section.

2.2.3 Critical Current Density Jc

Although a superconductor can transport current without resistance, its ability is limited. It also loses its superconductivity if the transport current increases above a certain value, called the critical current Ic. In practical applications, it is more convenient to use current density than transport current, so that the corresponding current density is defined as critical current density Jc. As the transport current increases, transition of the superconductor does not jump to the normal state from the superconducting state. Usually critical current Ic refers to the maximum direct current that can be regarded as flowing without resistance in the superconductor, and the criteria for this condition are that the electric field strength E is 1 μV/cm or the resistivity ρ is 10−13 Ω·m. The critical current Ic continuously decreases with increase of temperature T and magnetic field B.

The three basic critical parameters Tc, Hc, and Jc of superconductors are not independent of each other, there being a strong correlation between them. Figure 2.7 shows the relationship between the three critical parameters. Any point within the volume enclosed by a curved surface (Jc, Tc, Hc) and three planes (Tc, Jc), (Jc, Hc), (Hc, Tc) is in superconducting state S; any point outside the volume is in normal state N, and any point on the curved surface with (Jc, Tc, Hc) is at critical state C. At present, the discovered critical temperature Tc of Tl-based HTS can reach up to 135 K and its critical magnetic field Bc is more than 25 T, while its theoretical maximum value is more than 100 T. Table 2.1 lists some main critical parameters of several types of superconductors [8], which include crystal structure, critical temperature Tc, upper magnetic field Bc2 (Bc2 = μ0Hc2) at temperature T = 0 K predicted by the Ginzburg–Landau (G-L) theory, penetration depth λ and the coherent length ξ.

Figure 2.7 Critical parameters and their relationships.

ch02fig007

2.3 Classification and Magnetization

2.3.1 Coherence Length

In Section 2.1.2, we introduced the Meissner effect of superconductors and the concept of magnetic field penetration depth λ according to the London equations. In order to clearly explain the purpose of classification of superconductors, another important microscopic parameter, coherence length ξ, is described in this section.

Based on the BCS theory, superconductivity results from formation of Cooper pairs, which act as carriers without resistance. However, the binding energy between two electrons of Cooper pairs is weak, but the correlation distance of two electrons ξ is long. ξ is called the coherent length and can reach up to 10−4 cm, which is more than 10⁴ times that of the lattice size based on the calculation of second-order phase transition theory in a superconductor. Therefore, the superconducting correlation is a long-range interaction and can occur in space spanning many lattices. Furthermore, there are possibly many Cooper pairs in the same space.

Table 2.1 Macro- and microscopic characteristic parameters of several superconductors

Table 2-1

By introducing non-local electrodynamics into superconductivity and developing the London theory, Pippard proposed the concept of the superconducting coherence length ξ. According to the London equation, the penetration depth λ of a superconductor is constant and depends on material properties, as well as on temperature. Based on the Ginzburg–Landau theory and experimental results, corrections on the London penetration depth can be made:

(2.49) Numbered Display Equation

where λ(0) refers to the penetrating depth of superconductors when the temperature is 0 K. The second column from the right in Table 2.1 lists penetration depths of several superconducting materials at 0 K.

A Cooper pair reflects the collective effect of the entire electron system with which lattice ions couple, and the coupling strength is determined by the states of all electrons. Theories and experiments show that the superconducting coherence length relates to temperature, and by considering the influence of temperature, the coherence length of a superconductor is approximately:

(2.50) Numbered Display Equation

where ξ(0) denotes the coherence length of a superconductor with temperature 0 K. The magnitudes of coherence length of several superconductors are also presented in last column of Table 2.1.

2.3.2 Classifications

It was experimentally found that some superconductors in magnetic fields do not allow penetration of magnetic flux with magnetic field increase before they lose superconductivity, even if the magnetic field is more than its critical magnetic field. Conversely, other superconductors permit penetration of the magnetic field into their partial regions, which results in their interiors showing local interlacement with the normal state and the superconducting state simultaneously, even though their resistance remains at zero. Thus, the superconductors are classified into two types [9].

According to the Ginzburg–Landau theory, superconductors can be classified into two categories based on the ratio of penetration depth λ to coherence length ξ. By defining the Ginzburg–Landau parameter κ as:

(2.51) Numbered Display Equation

if κ < 1 / inline , superconductors have positive interface energy and are called Type I superconductors; conversely, if κ > 1 / inline , superconductors have negative interface energy and are defined as Type II superconductors. In order to visually understand the coherence length and penetration depth of superconductors, Figure 2.8 shows the coherence length ξ and penetration depth λ of Type I and Type II superconductors. ns(r) (the square of the absolute value of