Mabel Condemarin

When a material undergoes a transition from normal to the superconducting phase, significant changes take place in the electrical and magnetic properties. Even the ther- mal properties and the mechanical properties show abrupt changes. In other words, all the physical and chemical properties dependent on the electrons in the material show modifications. These changes indicate that the conduction electrons undergo certain change in the superconducting phase.

(i) The most important characteristic feature of the superconducting phase is the absence of resistance to the flow of electric current through the material. The transition from the normal phase to the superconducting phase occurs at a well defined critical temperature. For pure materials, the transition occurs over a tem- perature range as low as 0.01 K.

(ii) The materials in their superconducting phase repel magnetic lines of force. When a transition takes place from the normal phase to the superconducting phase in presence of a magnetic field, the magnetic flux lines will be expelled from within the body of the superconductor.

(iii) The presence of magnetic materials like iron, cobalt or nickel, even in minute quantities as impurity, destroys superconductivity in metals.

(iv) The property of superconductivity is not restricted to metals or good conductors. It has been observed that certain semiconducting and insulating mixed oxides as well as some polymers exhibit superconductivity.

lithium, sodium and potassium do not exhibit superconductivity even at temper- atures as low as 0.1 K.

4.2.1 Isotope e_ffect

When an element exhibiting superconductivity exists in various isotopic forms, it is observed that the transition temperature decreases as the isotopic mass increases. The variation of transition temperature with isotopic mass is described by the relation,

M1/2Tc = constant. (4.1)

This dependence of transition temperature on the isotopic mass is known as “Isotope effect”. This indicates an involvement of ion cores in the phenomenon of supercon- ductivity since the natural frequency of vibration of the lattice atoms is proportional to M−1/2_{. However, experimental observations have indicated that the relation is not valid}

for many materials. A more general form of the isotope effect may be written as

Tc_∞M−β (4.2)

where β varies from 0 to 0.5

4.2.2 Meissner e_ffect

An important property of the superconducting phase is the expulsion of magnetic flux lines from within the bulk of the superconductor. This is known as Meissner e_ffect. Consider a material in the normal state(Fig.4.2a).

T > T_c T < T_c B > B_c

(a) (b) (c)

Figure 4.2 Demonstration of Meissner effect. (a) The magnetic flux lines passing through the sample for T > T c, (b)flux lines being expelled for T < T c and (c) the flux lines penetrating when the field is increased above the critical value.

When a magnetic field is applied to the material, the magnetic flux lines pass through the material. Now, if the temperature is reduced below the critical tempera- ture the magnetic flux lines will be expelled from inside the superconductor (Fig 4.2b). Hence, we have,

B = µo(H + M) = 0 (4.3) where M is the magnetization in the material due to an applied magnetic field H. The magnetic susceptibility is given by

χ = M

H = −1 (4.4)

which indicates that the material in its superconducting state is a perfect diamagnetic material.

If a superconductor is subjected to a strong magnetic field, the material loses its superconducting property and becomes normal. As the magnetic field is increased in strength, at a particular value of the magnetic field, the magnetic flux starts penetrating into the material and makes it a normal material (Fig.4.2c). Hence, we can define a critical magnetic field, corresponding to a temperature, upto which the material re- mains in the superconducting phase and above which the material becomes normal. At any temperature T below Tc, the critical magnetic field Hc(T ) upto which the material remains in the superconduting phase, is given by the relation

Hc(T ) = Hc(0) " 1 − T 2 T2 c # (4.5)

where Tc is the critical temperature in the absence of a magnetic field and Hc(0) is a constant representing the critical magnetic field at T = 0K (Fig.4.3). The magnetic field that destroys superconductivity in any material need not be an externally applied field. It may also be an induced magnetic field due to an electric current flowing through the material. The magnetic field H induced in a wire of radius r when a current I flows through it is given by

I = 2πrH (4.6)

If this induced magnetic field becomes equal to the critical magnetic field, the material becomes a normal conductor. Hence, we can define a critical current Iccorresponding to the critical magnetic field Hc that is sufficient to destroy superconductivity in the wire.

T, K

H

Figure 4.3 The variation of critical magnetic field with temperature.

Ic = 2πrHc (4.7)

Equation (4.7) is called Silsbee’s rule. This puts a limit on the electric current that can be passed through a superconductor.

4.3

Classification of superconductors

Superconductors are classified as Type I (or soft) superconductors and Type II (or hard) superconductors. The classification is based on the magnetic behaviour of supercon- ductors. Type I superconductors have a small critical magnetic field and the transition from the superconducting to the normal phase at the critical field is abrupt. Assum- ing that there is a magnetization in the superconductor in a direction opposite to the direction of the applied magnetic field, the variation of magnetization with the applied field will be as shown in Fig.4.4. At the critical magnetic field, there will be an abrupt

H

−M

H

decrease in the magnetization and the material becomes normal. For all values of mag- netic field above the critical field, the material shows finite resistivity and the magnetic flux penetration is complete. In other words, Type I superconductors display Meissner effect completely.

In the case of Type II superconductors, the material is a perfect superconductor upto a magnetic field Hc1 (Fig.4.5). The magnetic flux penetration begins at Hc1 and is complete at a magnetic field Hc2. The material becomes a normal conductor for magnetic field greater than Hc2. The material is said to be in a ‘mixed state’ or ‘vortex state’ between the two critical magnetic fields Hc1and Hc2. Inspite of the fact that the magnetic flux lines penetrate the material in the vortex state,

Hc1 Hc2 H

−M

Figure 4.5 Effect of magnetic field on Type II superconductor.

the electrical resistivity continues to be zero upto Hc2. Since the values of Hc2 are rel- atively larger, Type II superconductors are more useful for possible application based on the resistanceless state of materials. A comparison of Type I and Type II supercon- ductors is given in Table 4.2.

Table 4.2 Comparison of the characteristics of Type I and Type II superconductors.

Type I Type II

1. These are usually elements in their pure form.

These are impure elements, alloys or compounds.

2. The transition to normal state occurs abruptly at a critical magnetic field, Hc

The transition to normal state begins at Hc1and is complete only at Hc2.

3. The value of critical magnetic field Hc is usually very small.

Though the value of Hc1 is small, Hc2 is quite large.

4. They exhibit complete Meissner effect upto Hc

Meissner effect is complete upto Hc1 when the magnetic flux starts pene- trating into the superconductor. The flux penetration is complete only at Hc2 when the material becomes nor- mal conductor.

5. The superconductivity or the zero re- sistance state is observed upto critical magnetic field Hc.

The material remains in the resis- tanceless state even in the intermedi- ate state between Hc1 and Hc2 and be- comes normal only at Hc2.

6. The small value of Hc, whether ap- plied or induced, restricts their use

The large value of Hc2 makes it suit- able for applications.