19 Superconductivity and some introductory concepts

Mahavir Singh

Learning Outcomes

After studying this module, you shall be able to

• Learn about the combined effect of electrical and magnetic properties of superconductors and their importance in the applications of superconductors.
• Learn about the physics of superconductors and the basics and principle of superconducting systems.
• Know the a) types of superconductors and b) how they are different from each other c) Soft and hard nature of superconductors.
• Learn about the thermodynamic variables of prime importance like entropy, specific heat and conductivity etc. which are responsible for the development of the theory of superconductivity.
• Learn  about  that  the  critical  temperatures  of  the  superconductors  vary  with  the isotopic masses by relation.
• Learn about the fact that energy gap is totally different from that of insulators because in the insulators, energy gap is tied to the lattice, while in the case of superconductors, it is tied to the Fermi gas.

    1 Introduction

With the liquification of helium in 1908 by Heike Kamerlingh lead to the starting steps in the discovery of superconductity. In 1911, when he was studying the variation of the resistance with the temperature for Mercury. He was shocked that at 4.15 K, the dc resistance decreased sharply. With this experiment superconductivity was discovered. At very-very low temperatures, most of the metals, many alloys and certain chemicals compounds lose their resistance completely, the phenomenon is regarded as superconductivity and such type of material is termed as Superconductor. At further later stage, Meissner and Ochsenfeld that the superconductors are perfectly diamagnetic also. We can clearly differentiate between a normal conductor and superconductor by viewing the graph in Fig.1.

Fig. 1: Variation of resistance with temperature (in K).

1.1 Macroscopic Electromagnetic Properties

These two macroscopic properties accounts for the basic principles of superconductivity. Let them discuss in some detail:

Zero Resistance State (Electrical Effects):

In the superconducting transitions, the resistance of the material falls to a very small value. It is quite interesting to check that the resistance actually falls to zero value? Onnes experimented with the superconducting circuit to check the low resistance through the decay of current flowing through the circuit. And the current decays according to the following equation:

Where I(t) is the value of current at any time , R is the resistance and the L is used for coil inductance.

Suppose for a lead coil, L=1.4X10-3 H, where the magnetic field value decayed to nearly 2% in the 7 h only and the resistivity was deduced to less than 4×10—25 ohm m. Therefore we can easily understand that why resistance of a superconductor can be taken as zero.

Magnetic Field Effects

Under the application of very weak magnetic field (few hundred Orsteds), the quenching of the superconductivity is observed, which results in the failure in generating the high magnetic field value through Joule dissipation. This result is quite similar to the observed by Silsbee in 1916. According to him, critical value of current (Ic) produces critical magnetic field (Hc) on the surface of superconductor. The temperature dependence of critical field is given by the following relation:

The above equation can understood in the following graph easily.

Fig. 2: Variation of critical field with Temperature.

The above relation shows that the graph will be a parabolic curve. According to the relation:

At 0K,   HC (T)=HC (0)

At T=TC, HC (TC) =0

These conditions define the boundary of the curve, below which superconductivity is present and outside it normal state is present.

However the above curve takes the form depending upon the types of superconductors, So it’s better to have a look on the classification of superconductors.

Magnetization behavior

Depending upon various phenomenological parameters superconductors have been classified mainly into two types. One of them is soft (Type I) and other one is hard superconductors. Let us study them briefly before studying their thermal properties.

The superconductors in which magnetization behaves like the curve as given in the following Fig.

3 are termed as Type-1 superconductors.

Fig.3: a) Magnetization versus applied magnetic field for type -1 superconductors and b) for type-II superconductors.

A abrupt fall in the magnetization can be observed from the with the increasing applied magnetic field. Since the critical magnetic field is too low in this case that this type of superconductors do not have too much useful technical applications. However, hard superconductors have magnetization curve as given in the right hand side of Fig. 3. These superconductors do not jump abruptly to the normal state like type-1 superconductors. They show superconducting properties upto a field strength HC2 and exhibit perfect diamagnetism for the fields less than HC1. So is is clear that a large amount of magnetic field is needed to destroy the superconducting properties of the material. The in between state is known as mixed state or vortex state. These superconductors have wide application range like high field magnets in particle acceleration , experimental magnetic levitation, in fusion reactors, SQUID, and MRI also.

2.   Thermal Properties

The thermal properties of superconductors have been extensively studied and compared with those of the same materials in the normal state. There are a number of thermodynamic effects in the normal and the superconducting state of a superconductor which are of great importance in the development of superconductivity theories. Among them Entropy, Specific heat, thermal conductivity and energy gap are of more importance. This section explains these parameters except energy gap which will be covered in the later section.

2.1 Entropy

It is experimentally proven that entropy for all superconductors’ decreases if we go below critical temperature. As we know that entropy is a measure of disordered state of a system so the above statement clearly suggests that the Entropy for superconducting state is less than that of normal conductor. For the clear demonstration, Entropy for Aluminum in normal and superconducting sate is given in the following figure.

Fig.4: Variation of Entropy with Temperature (in K) for Aluminium (for normal as well as for superconducting state.

2.2 Specific heat

The specific heat in a normal conductor consists of two contributions one is electronic and the other one is for lattice. Let us say them C_e and C_l respectively. In the case of electronic contribution, C_e is linearly proportional to the absolute temperature (T) and in the second case; C_l is proportional to the T³. So the total specific heat takes the form as:

C= C_e+ C_l= AT+BT³———————————————3

where A and B are constants. These relationships can be clearly understood in the form of the following graphs.

Fig.5: Temperature variation of heat capacity in normal and superconductivity sates (left side) and electronic contribution of heat capacity Vs TC/T.

However, in the superconducting state lattice contribution will remain same and the only change occurs for the electronic contribution Ce. The electronic contribution of specific heat is non linear in case of superconducting state.

3.   Isotope Effect

It is well known fact proved by previous observations that the critical temperatures of the superconductors varies with the isotopic masses. Let us take the example of Mercury; in Mercury critical temperature varies from 4.185 K to 4.146 K for the isotopic masses 199.5 and 203.4 atomic mass units. If we mix different isotopes of same material, the transition temperature varies accordingly. The experimental results reveal the following relation for these two.

Where M is the isotopic mass.

Since there is a complete dependence of Tc on the isotopic masses, so we can clearly find the direct involvement of lattice vibrations and electron-lattice interactions in superconductivity.

Fig.6: Variation of isotopic mass with Temperature.

4.  Manifestation of Energy Gap

In the previous section, it has been observed that the specific heat abruptly changes at the critical temperature i.e Tc. This suggests the existence of energy gap inside the superconductor. However this type of energy gap is totally different from that of insulators because in the insulators, energy gap is tied to the lattice, while in the case of superconductors, it is tied to the Fermi gas. Here the difference lies! Which simply means energy gap exists between the superconducting electron levels i.e. between the lowest excited level and the ground level. The energy gap is given by;

_Eg=2Δ ——————————————————5

where   is termed as energy gap parameter

Δ= 1.4K_bT_c     for Ga

Eg ≈10^-4 Ev

The energy is basically a function of temperature and is demonstrated in the following Fig.7.

Fig.7: Variation of energy gap with temperature in the left, In the right side: energy gap in the normal and superconducting sate.

Ground state electrons are superconducting electrons while electrons in the excited state are normal electrons. With the rise in the temperature a greater number of electrons are excited in the higher band above the energy gap. Therefore at critical temperature all electrons get excited thereby vanishing the energy gap at TC. In the case of superconductors, the existence of energy gap means that the photons with the energy less than the energy gap can’t be absorbed. Therefore, the measurement of energy gap can be done only by directing microwave radiations at the superconductors. Whenever microwave radiations has more energy then the band gap, strong absorption takes place and more number of superconducting electrons are excited to the states above the energy gap.

    5. SUMMARY

Superconductors materials are those which have zero dc resistance below a certain temperature Tc, called the critical temperature. A second property of a type I superconductor is that it behaves as a perfect diamagnet. Applied magnetic flux is expelled from the interior of a type I superconductor. This phenomenon is known as the

Meissner effect. The superconductivity of a type I superconductor gets destroyed when an applied magnetic field exceeds certain critical magnetic field (Bc).

Type- II superconductor consists of two critical fields. When an applied field is quite less than the critical field, Bc1, the material behaves as superconductor and no flux penetration is possible. When the applied field becomes greater than critical field, Bc2, the superconducting state gets completely destroyed and the flux penetrates takes material. The critical temperatures of the superconductors vary with the isotopic masses by relation.

The abrupt variation of specific heat with the critical temperature suggests the presence of energy gap inside superconductors. Under the application of zero applied filed, persistent currents when set up in a superconducting ring, (also called supercurrents) circulates for several years with no measurable losses.

    Value Addition:

Do You Know?

Year 2011 has been marked as the 100th anniversary of discovery of Superconductivity which was discovered by a Dutch scientist Heike Kamerlingh Onnes at Leiden University. He discovered superconductivity while he was liquefying Helium. But before the liquification of Helium, the lowest temperature which was available for researchers was 14K and its was for solid hydrogen. Originally Onnes named his discovery “supra conductivity” but later it is was called as “superconductivity,” that is the term we use today. He first experiment with the gold and platinum and he moved to the Mercury because it is quite earier to work with the pure metal. At that time scientists have a general thinking that pure metals show zero resistance at liquid-helium temperatures. Till date Five Nobel Prizes in Physics have been awarded for research in superconductivity.

Suggested Reading

Ginzburg-Landau theory

This theory is named after Vitaly Lazarevich Ginzburg and Lev Landau. It is mathematical theory used to describe superconductivity. In the initial stage, it was considered as a phenomenological model which describes type-I superconductors without taking their microscopic properties. At a later stage, a new version of Ginzburg–Landau theory came from the three scientists Bardeen, Cooper, Schrieffer microscopic theory which accounts for microscopic interpretation of all its parameters. Abrikosov and Ginzburg were awarded the 2003 Nobel Prize for their work.

For More Details ( on this topic and other topics discussed in Text Module) See

1.  Introduction to Superconductivity by M.Tinkham, Mc-Graw-Hill Inc.

2.    Superconductivity by C P Poole, H A Farach and R J Creswick, Academic Press Inc.

For General Study on Origins of Superconductivity

1.Introduction to solid state physics by C.Kittel,

2.The solid state by H M Roseberg

3. Superconductivity, superfluids and condensates by J Annet

Glossary:

Critical Field

The filed at which superconductivity is destroyed is known as critical field.

Entropy

It is the measure of disordered state or randomness of a system.

Isotope

Isotopes are variant of chemical element which differs in the number of neutrons, however all isotopes have equal number of protons.

Penetration depth

It is the distance from the surface of the specimen upto where magnetic field reduces 1/e times the field at the surface.

Specific heat

The specific heat is defined as amount of heat per unit mass required to raise the temperature by one degree Celsius.

Superconducting state

When material exhibits infinite conductivity when cooled to a sufficiently low temperature, the phenomenon is known as superconductivity and the corresponding state is known as superconducting state.