# Chapter 9 – Superconductivity – Applied Physics

## Superconductivity

#### 9.1 Introduction

From free electron theory of metals, we know that by decreasing the temperature of a perfect metal, the thermal vibrations and electron scattering of ions (or atoms) decrease and hence the electrical resistance of the substance decreases. One would expect that the electrical resistance of a substance may become zero as its temperature reaches 0 K. Based on this point, experiments were conducted on many substances by scientists. In 1911, Kammerling Onnes found that the electrical resistance of pure mercury vanishes suddenly at 4.2 K as shown in Fig. 9.1. This temperature is called its superconducting transition temperature and denoted as TC. The superconducting transition temperature of a few important materials is shown below.

Material Transition Temperature (TC) (in K)
Hg
4.15
CuS
1.6
Nb
9.3
Nb3 Au
11.5
Nb3Sn
18.0
Nb3Ge
23.2
YBa2Cu3O7
90
Tl2Ga2Ba2Cu4O10
125

Above the superconducting transition temperature, the material possesses normal resistance and is said to be in the normal state. Below this temperature, the resistance of the material becomes zero and its conductivity reaches infinity. This state of the material is called superconducting state. Now, the superconducting transition temperature can be defined as the temperature at which the material changes from normal state to superconducting state as it is cooled. The total disappearance of electrical resistance of these few substances is called superconductivity and the materials which exhibit this property are called superconductors.

Figure 9.1 Variation of resistance with decrease of temperature for mercury The electrical resistivity of a material is caused by electron scattering. This is due to: (a) temperature, (b) impurities and (c) crystal defects. Figure 9.2 shows the variation of electrical resistivity of an impure defective material and pure perfect material with temperature. Curve ‘A’ shows the presence of electrical resistance even at 0 K; this is due to defects and impurities in the material and curve ‘B’ shows the superconductivity of pure perfect crystalline material.

Figure 9.2 Shows the variation of electrical resistivity with decrease of temperature. Curve A for a defective impure crystal and curve B for a perfect pure crystal #### 9.2 General features of superconductors

Superconductors show many features. We could study these features one by one as shown below:

1. Persistent current: The electrical current in a superconductor, in superconducting state remains for a very long time. This can be proved by placing a superconducting loop of material in magnetic field and lowering its temperature to below its superconducting transition temperature (TC) and the magnetic field is removed. This causes dc current in the superconductor loop and the current remains for a very long period without attenuation. File and Mills determined the time taken by the super current to reduce to 1/e of its initial value is more than 1,00,000 years. This indicates that the dc current in a superconducting material is persistent.

2. Normally, superconductivity has been observed in metals having valence electrons between 2 and 8 and not 1.

3. Some good conductors of electricity at room temperature will not show superconductivity at low temperatures. For example, gold, silver, copper, sodium, crystalline iron, ferromagnetic and anti-ferromagnetic materials.

4. The materials which possess high resistance at room temperatures will show superconductivity at low temperatures. For example, amorphous thin films of beryllium, bismuth and iron. Under pressure, antimony, bismuth and tellurium show superconductivity.

5. By reducing the temperature of a material, it changes from normal to superconducting state. This transition is sharp in case of pure perfect metals and is broad for metals containing impurities as shown in Fig. 9.3.

Figure 9.3 Shows the transition width of impure metals 6. Isotopic effect: Transition temperature (TC) of a superconducting substance varies with isotopic mass (M ). For example, the transition temperature of three isotopes of mercury is shown below.

It has been observed that TCM −β (or) TC M β = constant. For large number of materials, β ≈ +0.5. = constant.

so TC M 1/2 = constant.

However, deviations from this value have been observed for some materials.

Isotopic Mass (M) of Hg Transition Temperature [TC in K ]
199
4.161
200
4.153
204
4.126

7. Effect of magnetic field: By appling magnetic field of sufficient strength, the superconductivity of a material can be destroyed. The minimum magnetic field strength required to destroy superconductivity of a substance, below TC is called critical magnetic field (HC) at that temperature. HC varies with temperature. This variation has been shown for a number of superconducting elements in Fig. 9.4.

Figure 9.4 Shows the variation of critical magnetic field with temperature for a few elements From the graph, we see that the critical magnetic field for different elements will be different at different temperatures. Also it shows for a material, the critical magnetic field increases with decrease of temperature below TC. At TC, no magnetic field is required to change the material from superconducting to normal state. Maximum magnetic field is required to destroy superconductivity at 0 K. The critical magnetic field at 0 K is H0. The critical magnetic field (HC) at any temperature ‘T’ below TC can be represented as: 8. Critical currents: Suppose a material carries electric current in superconducting state, this current produces magnetic field. If this magnetic field exceeds critical magnetic field (HC) at that temperature T (< TC ), then normal resistance will be included in the material and it will be in the normal state. Hence, it is not possible to pass large currents through a superconductor. The maximum current that can be passed through a superconductor in superconducting state is called critical current, represented by IC.

9. AC Resistivity: The current in a superconductor in normal state is carried by normal electrons only. When the material changes from normal state to superconducting state, then few normal electrons are converted into super electrons which carry dc current in superconducting state without any electrical resistance. If a constant dc current is flowing in a superconductor, there is no resistance in the material; hence, no electric field in the material. If we apply dc voltage source to a superconductor [below TC], then current will not increase suddenly but at the rate at which the electrons accelerate in the electric field. This indicates the presence of electric field in the material. If we apply ac voltage source to the superconductor, then the superelectrons accelerate in the forward and backward direction; they lag behind the field because of inertia. Also under ac fields, current is carried not only by superelectrons but also by normal electrons; this adds resistance to superconductor [below TC]. Under high frequency ac voltages, a superconductor behaves as a normal material because under ac voltages, electric field exists in the material that excites superelectrons to go into higher states where they behave as normal electrons.

10. Entropy: Entropy is the measure of disorder in a material. By reducing the temperature of a material, it goes into superconducting state. Also thermal vibrations and entropy of the material get reduced. In all superconducting materials, entropy decreases as they change from normal to superconducting state. The electrons in superconducting state are more ordered than in normal state.

11. Heat Capacity and energy gap: At all temperatures in normal state, the heat capacity of a superconductor is almost the same. A sudden change in heat capacity at the transition temperature was observed. Again in superconducting state, heat capacity changes exponentially with temperature. This suggests the existence of an energy gap at the Fermi gas of the material. This energy gap is represented in Fig. 9.5.

Figure 9.5 Represents the existence of energy gap in superconductors This energy gap separates the superconducting electrons and normal electrons. The superconducting electrons lie below the energy gap and the normal electrons are above it. The width of this energy gap is not constant as we see in case of an insulator. In insulators, the energy gap is attached to the lattice and the width of energy gap will not vary with temperature. In case of superconductors, the width of energy gap at Fermi gas increases by decreasing temperature below TC. This energy gap attains maximum at 0 K and reduces to zero at TC. Giaever confirmed the existence of energy gap in superconductors by electron tunnelling observation across the superconducting junctions.

12. Thermal conductivity: It has been observed that the thermal conductivity of a material in superconducting state is less than in normal state. By applying sufficient magnetic field, a material changes from superconducting state to normal state below TC. In normal state, all free electrons participate in thermal conductivity; hence, the thermal conductivity is large. Whereas in superconducting state, the materials have superelectrons and normal electrons, the superelectrons will not participate in thermal conductivity; so, thermal conductivity is less. A sudden drop in thermal conductivity has been observed when a material changes from normal to superconducting state at a temperature below TC. The thermal conductivities of tin as a function of temperature in normal and superconducting state are shown in Fig. 9.6.

Figure 9.6 Shows the variation of thermal conductivity of tin in normal and superconducting states 13. Mechanical effects: Experimentally, it was found that the superconducting transition temperature and critical magnetic field change slightly by applying mechanical stress on it. Small changes in volume, coefficient of thermal expansion and bulk modulus of elasticity were seen when a material changes from normal state to superconducting state.

14. Acoustic attenuation: When sound wave propagates through a metal, then the ions will be slightly displaced from their original positions. These displacements produce microscopic electric fields. These fields increase the energy of electrons present near the Fermi surface. So, the wave is attenuated. This attenuation has been expressed in terms of attenuation coefficient, α of the acoustic waves. The ratio of α in superconducting state to normal state of a material has been expressed as: At low temperatures, .

15. Flux exclusion or Meissner effect: In 1933, Meissner and Ochsen Feld found the exclusion of magnetic flux lines by a superconductor below TC. They reduced the temperature of a long superconductor in magnetic field. They observed that the superconductor pushes the magnetic lines of force out of the body at some low temperature, TC as shown in Fig. 9.7. When the material is in normal state, the magnetic force of lines pass through it. The magnetic induction (B) inside the material is given as .

Figure 9.7 Figure shows the superconductor in applied magnetic field (H) where μ0 is the magnetic permeability of free space. H is the intensity of applied magnetic field, M is the magnetization of the material and χ is the magnetic susceptibility given as χ = M/H.

When the temperature reaches below TC, the material enters into superconducting state and it expels the magnetic force lines. So, B = 0 inside the material. From the above equation, we write 0 = μ0 (H + M) .

In superconducting state, inside the material magnetization takes place which is equal in magnitude and opposite in direction to the applied field. The superconductor is a perfect diamagnetic material (since χ = − 1.0). The exclusion of magnetic lines from a superconductor when it is cooled in magnetic field to below its transition temperature is called Meissner effect.

#### 9.3 Type-I and Type-II superconductors

Depending on the way of transition from superconducting state to normal state by the application of magnetic field, superconductors are divided into Type-I and Type-II superconductors. In case of Type-I superconductors, as we increase the intensity of applied magnetic field, inside the material opposing magnetization takes place up to some applied magnetic field, up to this applied magnetic field, the material is completely diamagnetic and it is in superconducting state. As the applied field reaches the critical value, HC, all of a sudden the magnetic force lines pass through the material and the material changes sharply from superconducting state to normal state as shown in Fig. 9.8. This effect (Type-I superconductivity) was first noted by Silsbee in 1916. So, this effect is also called Silsbee effect. The critical magnetic field, HC for Type-I superconductors is of the order of 0.1 Tesla or less so, high magnetic fields cannot be produced using Type-I superconductors. These are called soft superconductors. Almost all elements show Type-I superconductivity.

Figure 9.8 Shows the relation between magnetization and applied magnetic field for Type-I superconductors In case of Type-II superconductors, as we increase the intensity of applied magnetic field, in the material opposing magnetization takes place up to some applied magnetic field HC1, called lower critical magnetic filed. Up to this magnetic field, the material completely expels the magnetic force lines. The material is completely diamagnetic and it is in superconducting state. If the applied magnetic field exceeds HC1, slowly the magnetic force lines pass through the material and the transition from superconducting to normal state takes place gradually. The penetration of magnetic force lines through the material increases gradually from HC1 to HC2. At HC2, the magnetic force lines completely pass through the material and the material changes completely from superconducting state to normal state. Above HC2, the material is in normal state. The material is in mixed state from HC1 to HC2. The variation of magnetization with applied magnetic field in Type-II superconductors is shown in Fig. 9.9.

Figure 9.9 Figure shows the variation of magnetization with applied magnetic field for Type-II superconductors Type-II superconductivity was discovered by Schubnikov et al. in 1930s. The critical magnetic field HC2 for Type-II superconductors is of the order of 10 Tesla. HC2 is called upper critical field. Type-II superconductors with a large amount of magnetic hysteresis are called hard superconductors. Type-II superconductors are alloys or transition metals with high values of electrical resistivity.

#### 9.4 Penetration depth

In 1935, F. London and H. London obtained an expression for penetration of applied magnetic field into superconducting material from the surface by adding: (i) Meissner effect i.e., the magnetic induction (B) inside a superconducting material is equal to zero (B = 0) and (ii) zero resistivity i.e., the intensity of electric field (E) in a superconductor in superconducting state is equal to zero (E = 0) to Maxwell's electromagnetic equations. According to them, the applied magnetic field does not drop to zero at the surface of the superconductor [in superconducting state] but decreases exponentially as given by the equation: where H is the intensity of magnetic field at a depth x from the surface, H0 is the intensity of magnetic field at the surface and λ is called London penetration depth. London penetration depth is defined as the distance from the surface of the superconductor to a point inside the material at which the intensity of magnetic field is of the magnetic field at the surface [i.e., H0/e]. The variation of intensity of magnetic field with distance from the surface into the material for tin is shown in Fig. 9.10.

The magnetic field is likely to penetrate to a depth of 10 to 100 nm from the surface of a superconductor. If the superconducting film or filament is thinner than this value, then its properties are significantly different from that of the bulk material. For example, the critical magnetic field increases with a decrease in thinness. The value of λ for some materials is given below.

Figure 9.10 Figure shows the variation of the intensity of magnetic field with distance into the material for tin Material λ(in nm)
Mercury
70
Aluminium
50
Tin
50
39
Indium
64

Figure 9.11 Variation of penetration depth in tin The penetration depth is not constant but varies with temperature as shown in Fig. 9.11 for tin.

From the figure, we know that the penetration depth is independent of temperature, but the penetration depth increases rapidly and approaches infinity as the temperature approaches the transition temperature of the material. The London penetration depth at temperature T (< TC) can be obtained using the equation where λ0 is the London penetration depth at 0 K.

#### 9.5 Flux quantization

We know that electric charge is quantized in terms of integral multiples of charge on an electron [1.6 × 10−19 C]. Similarly, the magnetic flux lines passing through a superconducting loop due to persistent current is quantized in terms of integral multiples of Flux quantization can be proved with the help of a superconducting material in the form of a hollow cylinder or ring.

Figure 9.12 Sample cooled in the presence of magnetic field Magnetic field is applied on the superconducting ring in normal state as shown in Fig. 9.12. As its temperature is reduced to below critical temperature, the material expels the magnetic lines of force and enters into superconducting state. Persistent current is set up in the material this current will remain in the material even if the applied magnetic field is removed. This persistent current sets up magnetic force lines in the ring. This magnetic flux adjusts itself such that the total flux through the cylinder is quantized in integral multiples of . If the persistent current in the superconductor decreases, then the magnetic flux also decreases and adjust to integral multiples of . Here, 2e is the charge on a pair of electrons. Experimentally, it was found that the value of h/2e is equal to 2.07 × 10−15 webers. It confirms the existence of an electron pair in the superconductor in superconducting state. This is very well agreed with the Cooper pair concept. The Cooper pair concept plays a major role in BCS theory.

#### 9.6 Quantum tunnelling

As shown in Fig.9.13(a), if a thick insulating layer is sandwiched between two metals, then electric current will not pass through this insulating layer. If the insulating layer is very thin (10 to 20 Å), then there is a large probability for electrons to pass from one metal to another through this insulating layer by quantum mechanical process called tunnelling. If both the metals are normal conductors, then the V I characteristics is a straight line as shown in Fig. 9.13(b).

In 1961, Giaever took a system in which an insulating layer of 10 nm thick has been sandwiched between a normal metal and a superconductor as shown in Fig. 9.14(a). Gradually, an increasing potential has been applied between the metal and superconductor by connecting them to electrodes. Up to certain voltage, VC, there is no current in the circuit, afterwards the current increases as shown in Fig. 9.14(b).

Figure 9.13 (a) Shows an insulator sandwiched between two metals; (b) V-I characteristics for the system shown in Fig. 9.13(a) Figure 9.14 (a) Figure shows the sandwich of an insulator between a normal metal and a superconductor; (b) V-I characteristics for the system shown in Fig. 9.14(a) The flow of current through this insulating layer has been explained on the basis of quantum mechanical tunnelling of electrons. Quantum theory says that an electron on one side of the insulator [barrier] has a certain probability of tunnelling through it, if there is an allowed available equal energy electron state on the other side of the barrier. The quantum tunnelling for the above system can be explained with the aid of electron states in the energy space. Figure 9.15 shows the energy level diagram for the sandwich consisting of superconductor, insulator and metal at 0 K. The Fermi energy levels of these materials adjust to a same height after sandwiching.

The electron tunnelling in the above sandwich can be explained in the following way: When voltage is not applied across the sandwich, then electrons are filled up to the Fermi energy level EF2 in the normal metal. At this energy, there is a forbidden energy band in superconductor. So, electron states in the superconductor are not available for the electrons present at EF2 in normal metal.

So, electron tunnelling will not take place. Suppose voltage (V) is applied across the sandwich, then the electrons present at the Fermi energy level or near to it in the normal metal gain energy and will go to higher energy level. By the continuous increase of voltage across the sandwich, the electrons present at the Fermi energy level or near to it will go to excited states. The voltage across the barrier is raised such that the electrons in a metal should raise to a height of E11 or higher than that. Then, the electrons in the normal metal can see vacant energy levels in superconductor and tunnel through the insulating layer and reaches the superconducting material. Thus, quantum tunnelling takes place. Obviously, there is no current until the voltage becomes equal to VC so that eVC is equal to energy gap . Thus, the flow of electrons through a thin insulating layer has been explained based on the quantum mechanical tunnelling process.

Figure 9.15 Figure shows the energy level diagram for the sandwich consisting of superconductor―insulator and normal metal #### 9.7 Josephson's effect

In 1962, Josephson passed electrical current consisting of correlated pairs of electrons across an insulating gap (∼ 10 Å) between two superconductors. This effect is known as Josephson effect.

Josephson effect can be explained in the following way. As shown in Fig. 9.16(a), a rectangular superconducting bar is connected in series with a battery (B), plug key (K) and an ammeter (A). A voltmeter (V) is connected across the superconductor. Since the superconductor has zero resistance, so the voltmeter shows zero reading. Whereas the ammeter shows the current through the superconductor. Next, the superconductor is cut into two pieces as shown in Fig. 9.16(b). If the gap between the two pieces is about 1 cm, then current will not flow through the superconductor pieces, so ammeter shows zero reading and the voltmeter shows the open circuit voltage of the battery. If the gap between the superconducting pieces is reduced to 1 nm, then the voltmeter reading drops to zero and the ammeter shows dc current through the superconducting pieces and across the gap between the superconducting pieces [Josephson Junction]. That is without any applied voltage across the gap, dc current passes through the insulating gap. This effect is known as dc Josephson effect.

The gap between the superconducting pieces is slightly increased and the applied voltage is increased, the current passes through the gap also increased so that a small dc voltage exists across the gap. Now, a high-frequency electromagnetic radiation is observed from the gap. That is the gap emits a high-frequency electromagnetic rays. This indicate a high-frequency ac current through the gap. This effect is known as ac Josephson effect.

The VI characteristics of a Josephson junction is shown in Fig. 9.17. With zero applied voltage across the Josephson's junction, a dc current passes across the junction. As shown in figure, the dc current is in between I0 to – I0, where I0 is the maximum dc current under no applied voltage across the junction. This current is due to the quantum mechanical tunnelling of Cooper pairs of electrons across the junction. These electrons tunnel from one superconductor to another across the junction (barrier) and returns to the first conductor through the external circuit. If current exceeds I0, then a potential difference develops across the junction. This indicates resistance in the junction. The change from zero to finite resistance is not related to the elimination of superconductivity.

Figure 9.16 Shows Josephson's effect: (a) Current through superconductor bar and (b) The bar is cut into two pieces with a narrow gap between them Figure 9.17 V-I characteristics of a Josephson's junction The current across the junction can be represented as I = I0 sin , where is the phase difference between the waves associated with Cooper pairs of electrons on two sides of the gap. As shown in Fig. 9.17, whenever current exceeds I0, a potential difference (V) exists across the junction. This shows that the Cooper pairs on both sides of the junction differ by an energy equal to 2eV, where 2e is the charge on a Cooper pair of electrons. Since in superconductors, current is carried by Cooper pairs of electrons, if a Cooper pair passes across the gap, then it emits a photon of energy (hv) equal to 2eV. Then, the frequency of emitted radiation is . This is the oscillating frequency of sinusoidal current across the gap.

The phase difference = 2πt/T = 2πtv Suppose the potential difference across the gap is 1 mV, then the frequency is of the order of 480 GHz, this lies in the microwave region. This enables to construct microwaves resonators.

#### 9.8 BCS theory

The existence of energy gap and long-range electronic order in superconducting state pointed that electrons in superconductor are somehow bound together. The positively charged ions screen the Coulomb repulsive forces between electrons. In 1950, Frohlich and Bardeen concluded that a moving electron inside a crystal distorts the crystal lattice and this distortion is quantized in terms of virtual phonons. That means the reaction between an electron and lattice phonons represent the vibrations of crystal lattice in a solid. The electron-phonon interaction can cause resistance or superconductivity. The interaction of electrons and virtual phonons causes superconductivity. We know generally that superconductors are always poor conductors at room temperature and the best conductor do not become superconductors. For example, gold, silver and copper at low temperatures.

In 1957, Bardeen, Cooper and Schriffer put forward a theory [called BCS theory] [Jhon Bardeen received noble prize twice in physics; in 1947, he invented transistor and later he developed the key concepts of photocopy machine] which explains very well for all the properties shown by superconductors, such as zero resistance, Meissner effect, etc. This theory involves electron interactions through phonons. The basis for BCS theory is: (i) isotopic effect and (ii) specific heat of superconductors. Isotopic effect, TCM 1/2 = constant, infers that the transition to superconducting state must involve the dynamics of motions, lattice vibrations and phonons. Also as Tc → 0, then M approaches infinity. This suggests non-zero transition temperature and finite mass of ions.

#### Description

Suppose an electron approaches a positive ion core in the crystal, then the electron makes an attractive interaction with a positive ion. This attractive interaction sets in motion the positive ion and this ion motion distorts the lattice. This distortion of lattice is quantized in terms of phonons. At that instant, if another electron approaches the distorted lattice, then the interaction between this second electron and distorted lattice takes place; this interaction lowers the energy of second electron. Now, the two electrons interact through the lattice distortion or the phonon field and results in the lowering of energy of the electrons. The lowering of energy indicates that an attractive force exists between the electrons. This attractive interaction is larger if the two electrons have opposite spin and momenta. This interaction is called electron-lattice-electron interaction or electron-electron interaction through phonons as a mediator. Cooper stated that the presence of an attractive interaction even weak in between two electrons in a superconductor makes them to form a bound pair. Cooper showed that the lowering of energy leads to the formation of a bound state. Such bound pairs of electrons formed by the interaction between the electrons with opposite spin and momenta are known as Cooper pairs. This interaction can be represented in terms of the wave vector of electrons as shown in Fig. 9.18. Let an electron having wave vector K1 emits a virtual phonon q and this phonon is absorbed by another electron having wave vector K2, then K1 is scattered as K1−q and K2 as K2+q. Conservation of energy is not satisfied in this reaction. This process is called virtual because virtual phonons are involved in this process.

Figure 9.18 Electron-electron interaction through phonons In this interaction, phonon exchange takes place between electrons. If the phonon energy exceeds electronic energy, then the interaction is attractive and the attractive force between these two electrons exceeds the usual repulsive force. These two electrons which interact attractively in the phonon field are called Cooper pairs. The Cooper pair of electrons are said to be in the bound state or in the condensed state so that their energy is less than in the free state. The difference of energy of these electrons between these two states is equal to the binding energy of Cooper pair. Below critical temperature, the electron-lattice-electron interaction is stronger than electron-electron coulomb interaction, so electrons tend to pair up. The pairing is complete at 0 K and is completely broken at critical temperature. According to quantum theory, a wave function could be associated with a Cooper pair by treating it as a single entity. The Cooper pairs do not get scattered in the material and the conductivity becomes infinite which is named as superconductivity. The best conductors such as gold, silver and copper do not exhibit superconductivity because the electrons in these metals move freely in the lattice that, the electron-lattice interaction is virtually absent and the Cooper pairs will not form. Hence, these metals will not show superconductivity.

BCS theory explains the energy gap in superconductors in the following way: The energy gap at the Fermi surface is the energy difference between the free state of the electron and its paired state. The energy gap is a function of temperature. The energy gap is maximum at 0 K because pairing is complete at this temperature. At transition temperature, the energy gap reduces to zero because pairing is dissolved. The existence of energy gap in superconductors can be proved by the absorption of electromagnetic waves in microwave region. At temperature close to 0 K, a superconductor does not absorb energy of incident radiation until the energy of the incident radiation exceeds the width of the gap (2Δ) after absorption of energy, the electrons become free or normal.

#### Coherent length

The paired electrons (Cooper pair) are not scattered because they smoothly ride over the lattice points or to the lattice imperfections. The Cooper pairs are not slowed down. Hence, the substance does not posses any electrical resistivity. Superconductivity is due to the mutual interaction and correlation of electrons over a considerable distance called coherent length (0). It is found to be of the order of 10−6 m. The coherent length is defined as the maximum distance up to which the state of paired electrons are correlated to produce superconductivity. The ratio of London penetration depth (λ) to the coherence length (0) is represented as (K) = λ/∈0, is a number. For Type-I superconductors, and for Type-II superconductors, . The intrinsic coherence (0) is realated to the energy gap as where 2Δ is the energy gap.

#### BCS ground state

Fermi gas in the ground state is bounded by Fermi surface, excited state of an electron can be formed by taking an electron from the Fermi surface to just above it. According to BCS theory in superconducting state, there is an attractive interaction between the electrons [Cooper pairs]. In this case, we cannot form an exited state unless we supply an energy which exceeds the energy of attraction between electrons. These electron states are known as BCS ground states. This implies that the energy of Cooper pairs of electrons or BCS ground state is separated by a finite energy gap Eg (= 2Δ) from the lowest excited energy state (Fermi energy). Further, the Cooper pairs are situated within about KBθD of the Fermi energy where θD is the Debye temperature. The energy gap is situated about the Fermi surface of the Fermi state. The probability of occupation of the ground state in terms of one partical states is shown in Fig. 9.19(a).

BCS ground state of superconductor is shown in Fig. 9.20. Figure 9.20(a) shows the ground state of Fermi gas and Fig. 9.20(b) shows the BCS ground state of electrons with an attractive interaction between them, states near EF are filled in accordance with the probability shown in Fig. 9.19(b). The lowest excited state is separated from the ground state in this case by an energy gap Eg.

The total energy (T.E) of the BCS state is lower than that of the Fermi state. The total energy of the BCS state consists of K.E and attractive P.E, whereas that of Fermi state comprises K.E only. Thus, the attractive P.E reduces the T.E of the BCS state. This is in agreement with experimental observations on the superconducting and normal states.

Figure 9.19 (a) Ground state of Fermi gas; (b) BCS ground state Figure 9.20 (a) Ground state of Fermi gas; (b) BCS ground state of an electron gas #### 9.9 Applications of superconductivity

Superconductors find many applications. Some of them are mentioned below:

#### a. Superconducting magnets

Similar to electromagnets, superconducting magnets can also be formed by using coils of wire made up of superconducting material. To obtain magnetic fields from electromagnets, current should be maintained in the coil, whereas in superconducting coils, current once introduced into the coil will remain for a very long time and during this period magnetic field can be obtained, provided the temperature of the coil is maintained below its transition temperature [usually at liquid helium temperature]. The benefit of using superconducting magnets instead of electromagnets is the cost of power required to maintain superconductors at low temperature will be 1000 times less than the cost of power required in case of electromagnets to produce the same magnetic field. The size of superconducting magnets is less than that of electromagnets. Superconducting magnets are made of Type-II superconducting material because strong magnetic fields is of the order of 20 Tesla can be produced. Of the many superconducting materials, niobium-titanium (Nb Ti), a Type-II superconducting material, is mostly used because it can be easily drawn into thin wires.

The superconducting coils are used in electric machines, transformers and magnetic resonance imaging (MRI) instruments. MRI instruments are used in hospitals to obtain human body cross-sectional images. This process is much safer than using X-rays. Superconductor coils are used in magnetically levitated vehicles and in high-resolution nuclear magnetic resonance (NMR) instruments. Using NMR instruments, molecular structure of chemical compounds can be known. Superconducting coils are used in NMR imaging equipment, this equipment is used in hospitals for scanning the whole human body and diagnoze medical problems.

#### b. Magnetic bearings

Meissner effect is used in these bearings. Mutual repulsion between two superconducting materials due to opposite magnetic fields is used in the construction of magnetic bearings. There is no friction in these bearings.

#### a. Loss-less power transmission

DC current through an ordinary metallic wire causes heating effect called Joule's heating that is proportional to i2 R. This means an amount of electrical energy equal to i2R is wasted for every second. This dc power loss can be eliminated by passing current through a superconductor wire. For ac, the superconductors shows resistance.

#### b. Superconductor fuse and breaker

We know some insulating materials that show superconductivity at low temperatures, thin films of such materials can be used instead of fuse because when more than critical current pass through them, then they change into normal state. In normal state, they are insulators. They would not conduct current, so it will act as a fuse. In breaker, a long thin film of superconductor is used. In normal state, this film possesses high resistance. In this, lead is used.

#### c. Cryotron switch

In this device, the resistance of a superconducting material can be made to zero or normal value by appling magnetic field of strength just below and above its critical magnetic field.

The device consists of a thick straight wire (or core), made up of some superconducting material S1, on the surface of this another long thin wire made up of some other superconducting material S2, has been wounded. This is called coil. The superconducting materials S1 and S2 are selected in such a way that the transition temperature of S1 should be less than that of S2, hence the critical magnetic field of S1 is less than the critical magnetic field of S2, at some temperature ‘T ’ below their transition temperatures. This set-up is immersed in a cold enclosure as shown in Fig. 9.21. The temperature in the enclosure should be less than the transition temperature of S1 and S2, so that the core and coil is in superconducting state.

The current flowing through the coil is adjusted such that the magnetic field produced is very close to the critical magnetic field of core at that temperature. Now, by slightly increasing the current through the coil, the core can be changed from superconducting to normal state; again by slightly reducing the current through the coil, the core can be brought back to superconducting state. Because by increasing the current through the coil, the magnetic field produced will exceed the critical magnetic field of core again by decreasing the current through the coil, the magnetic field produced can be brought below the critical magnetic field of core. Even the core changes to normal state, the coil will be in the superconducting state because of its high transition temperature. Thus, the resistance of core is made ON or OFF by external control, so that this arrangement functions as a switch. Cryotron may be used as an element in flip-flop.

To produce low temperature in the enclosure, the liquid helium [TC = 4.2 K] is used, then the core material could be tantalum [TC = 4.38 K] and coil material will be lead [TC = 7.2 K] or niobium [TC = 9.3 K].

Figure 9.21 Cryotron switch #### 9.9.3 Computer applications

A closed superconducting ring [or a circular ring of superconductor] is used in memory cell. When persistent current in superconducting state passes through it, then it is said to be in ‘1’ state. In normal state, current will not pass through it, then it is said to be in ‘0’ state. Thus, the superconducting memory cell is a binary system.

#### 9.9.4 Josephson junction devices

A very small gap between superconductors forms a junction called Josephson Junction. The devices which use such junctions are called Josephson Junction device. The dc Josephson effect is used in the construction of sensitive magnetometers. These devices can measure magnetic fields accurately up to 10−11 gauss. AC Josephson effect is used to generate and detect electromagnetic radiations from radio frequencies to infrared frequencies.

#### 9.9.5 Maglev vehicles

Maglev vehicles means magnetically levitated vehicles. These vehicles are made to stay afloat above the guide way. So, it is not in contact with guide way. High speeds can be achieved with less energy. It is based on Meissner effect.

The Maglev vehicle is shown in Fig. 9.22. It consists of superconducting magnet at its base. There is a segmented aluminium guide way above which the maglev can be made to afloat by magnetic repulsion. The magnetic repulsion is in between the superconducting magnet at the bottom of Maglev and the magnetic field obtained by passing current through electric coils arranged in the aluminium guide way.

During the motion of the vehicle, only the part of the guided way over which the vehicle is located is actuated instantaneously. For this purpose, the guide way is formed into a large number of segments provided with coils. The currents in the segmented guide way not only levitate the vehicle but also help to move. Usually, the vehicle is levitated above the guide way by 10 to 15 cm. The vehicle is provided with retractable wheels. Once the vehicle is levitated in air, the wheels are pulled into the body, while stopping the wheels are drawn out and the vehicle slowly settles on the guide way after running a certain distance. Maglev train has been constructed in Japan, it runs at a speed of 500 km/h.

Figure 9.22 Maglev vehicle #### a. Superconducting sensitive magnetometer

Superconducting quantum interference devices (SQUIDs) are used in the construction of superconducting sensitive magnetometers. Basically, SQUIDs are superconducting rings that are used in magnetic flux storage devices. The quantization of magnetic flux in SQUID is the basis for construction of sensitive magnetometer. With the aid of good electronic feedback circuit, SQUIDs can measure magnetic field strengths that are less than 1/1000 of a quantum of magnetic flux. SQUIDs are used in medical diagnostics of heart and brain. They can measure the magnetic fields generated by heater brain signals. They are of the order of 10−14 Tesla. The SQUIDs are used to measure the voltages associated with brain, chest and cardiac activity. Earth magnetic field can be measured accurately using SQUIDs and a magnetic map can be constructed. This map helps us to detect mineral and oil deposites inside the earth.

#### b. Superconductors in medicine

(i) Human blood contains iron in certain percentage. Iron supplies oxygen to various parts of the body. If the iron content is less, then oxygen supply will be reduced and if iron content is more then it causes heart attack. The disease caused due to the variation of iron content in blood is called haemochromatosis. It is difficult to diagnoze this disease, often overlooked. Doctors can detect this disease easily and quickly using superconducting susceptometer. In this instrument, superconducting magnet and SQUIDs are used.

(ii) A disease that produces disorder in nervous system of brain is called epilepsy. This disorder causes fits and brain malfunctions. The epiliptic attacked part of the brain is short circuited. If the disease is severe in certain part of the brain, there the nerve path ways get jammed and the person receives meaningless signals from that damaged region. The only permanent cure for epilepsy is to operate and remove the damaged portion of the brain. The short-circuited epileptic centre produces distinctive magnetic signals. Doctors can locate the damaged portion of the brain by placing an array of a dozen SQUID magnetometers around the patient head and magnetic signals received by the magnetometers are fed to a computer. Computer analysis gives a three-dimensional picture of the activity within the brain. Doctors can locate the damaged portion of the brain in the image. This technique is known as magnetoencephalography.

#### Formulae

1. TcM1\2 = constant
2. 3. 4. B = μ0 (H + M) = μ0 H (1 + χ)
5. 6. #### Solved Problems

1. The critical field for niobium is 1 × 10 5 amp/m at 8 K and 2 × 10 5 amp/m at absolute zero. Find the transition temperature of the element.

(Set-2–May 2008)

Sol: Critical magnetic field at 8 K, HC = 1 × 105 amp/m

T = 8 K

Critical magnetic field at 0 K, H0 = 2 × 105 amp/m

Transition temperature, TC = ? 2. A Josephson junction having a voltage of 8.50 μV across its terminals, then calculate the frequency of the alternating current. [Planck's constant = 6.626 × 10−34 J-sec]

(Set-1–May 2008)

Sol: Voltage across the junction, V = 8.50 μV Frequency of alternating current, v =? 3. A super conducting material has a critical temperature of 3.7 K, and a magnetic field of 0.0306 tesla at 0 K. Find the critical field at 2 K.

(Set-2–May 2007)

Sol: TC = 3.7 K

T = 2 K

H0 = 0.0306 T

HC = ? 4. A long superconducting wire produces a magnetic field of 200 × 103 A/m on its surface due to current through it at temperature T (< TC ). Its critical magnetic field at 0 K is 250 × 103 A/m. The critical temperature of the material of wire is 12 K. Find the value of T.

Sol: 5. The superconducting transition temperature of tin is 3.7 K. Its critical magnetic field at 0 K is 0.03 Tesla. What is the critical magnetic field at 2.5 K?

Sol: TC = 3.7 K, T = 2.5 K

H0 = 0.03 T, HC = ? 3. What is the frequency of the electronmagnetic waves radiated from a Josephson junction, if the voltage drop at the junction is 650 μV?

Sol: v = ?

V = 650 × 10−6 V

e = 1.6 × 10−19 C

h = 6.625 × 10−34 JS 6. A lead super conductor with TC = 7.2 K has a critical magnetic field of 6.5 × 103 Am−1 at absolute zero. What would be the value of critical field at 5 K temperature?

Sol: TC = 7.2 K

H0 = 6.5 × 103 A/m

T = 5 K

HC = ? #### Multiple Choice Questions

1. Below transition temperature, the electrical resistance of a superconductor is. ( )
1. finite
2. large
3. zero
4. none
2. For an impure metal, the transition width is. ( )
1. zero
2. finite
3. large
4. none
3. The time required to decay persistent current to 1/e of its initial value is3 ( )
1. more than 1,00,000 years
2. 1000 years
3. 100 years
4. 10 years
4. The following element will not show superconductivity. ( )
1. gold
2. silver
3. coper
4. all
5. Superconducting bearings operate: ( )
1. with contact
2. without contact
3. with lubricant
4. without lubricant
6. The energy gap in a superconductor is maximum at. ( )
1. critical temperature
2. above critical temperature
3. below critical temperature
4. at 0 K
7. Below transition temperature, the London penetration depth. ( )
1. almost constant
2. increases exponentially
3. decreases exponentially
4. none
8. Cooper pairs are broken at____________temperature. ( )
1. 0 K
2. critical temperature
3. below critical temperature
4. above critical temperature
9. The relation between transition temperature (TC) and isotopic mass (M) is. ( )
1. TCM1/2
2. TCM−1/2
3. TCM−1
4. TCM
10. The critical magnetic field (HC) at temperature T K is:. ( )
1. 2. 3. 4. 11. The thermal conductivity of a metal in normal and in superconducting state is. ( )
1. same
2. different
3. both a and b
4. none
12. Type-I superconductivity is also called as: ( )
1. Silsbee effect
2. Subnikov effect
3. Boltzmann effect
4. Planck's effect
13. Type-I superconductors can produce magnetic fields of the order of. ( )
1. 100 Tesla
2. 10 Tesla
3. 1 Tesla
4. 0.1 Tesla
14. Examples for Type-I superconductors are: ( )
1. all elements
2. alloys
3. ferromagnetic materials
4. ceramics
15. The distance from the surface of a superconductor to a point in the superconductor at which the intensity of magnetic field is (1/e) at the surface is called. ( )
1. Josephson penetration depth
2. London penetration depth
3. Maxwell penetration depth
4. none
16. If H0 is the intensity of magnetic field on the surface of a material, then the intensity of field at a depth ‘x’ from the surface is. ( )
1. 2. 3. 4. 17. At temperature T (< TC), the London penetration depth can be expressed as. ( )
1. 2. 3. 4. 18. A quantum of magnetic flux in a superconductor is equal to. ( )
1. 2. 3. 4. 19. The maintenance cost of superconducting magnets is_____________ times less than the maintenance cost of electromagnet to produce same magnetic field. ( )
1. 10,000
2. 1000
3. 100
4. 10
20. Usually, maglev vehicle is raised above the aluminium path by a height of. ( )
1. 10 to 20 cm
2. 1 to 2 cm
3. 50 to 70 cm
4. 70 to 100 cm
21. SQUIDS are used to measure_______________associated with brain and chest. ( )
1. power
2. energy
3. stress
4. voltages
22. Superconducting susceptometer is used to detect. ( )
1. epilipsy
2. haemochromatosis
3. diabeties
4. none
23. The electron pairs in a superconductor are called. ( )
1. Cooper pairs
2. Bardeen pairs
3. BCS-pairs
4. Josephson pairs
24. A material changes from normal to superconducting state below____________temperature. ( )
1. Curie
2. critical
3. Weiss
4. none
25. 25. The transition temperature of mercury is. ( )
1. 4.2 K
2. 7.5 K
3. 12 K
4. 20 K
26. At transition temperature, the electrical resistance of a material. ( )
1. is large
2. is less
3. vanishes
4. none
27. For a superconductor, the critical magnetic field___________with decrease of temperature. ( )
1. increases
2. decreases
3. will not change
4. none
28. In superconducting state, we____________pass large current. ( )
1. can
2. cannot
3. both a & b
4. none
29. The maximum current that can be passed through a superconductor is called. ( )
1. supercurrent
2. optimum current
3. critical current
4. none
30. Superconductivity is not shown for_________. ( )
1. dc current
2. ac current
3. saw-tooth current
4. none
31. A superconductor is in more ordered than___________. ( )
1. a normal metal
2. a semiconductor
3. a dielectric
4. none
32. Below transition temperature, the heat capacity of a superconductor. ( )
1. changes with temperature
2. changes with magnetic field
3. changes with electric field
4. none
33. The width of superconducting energy gap. ( )
1. increases with temperature
2. decreases with increase of temperature
3. will not change with temperature
4. none
34. A superconductor is a perfect__________material. ( )
1. diamagnetic
2. dielectric
3. insulator
4. semiconductor
35. In Type-I superconductors, the transition from superconducting to normal state by the application of magnetic field is. ( )
1. erratic
2. sharp
3. both a & b
4. none
36. In Type-II superconductors, the transition from superconducting to normal state by the application of magnetic field is. ( )
1. sharp
2. not sharp
3. erratic
4. none
37. Type-II superconductors are: ( )
1. alloys
2. transition metals
3. both a & b
4. none
38. The intensity of an applied magnetic field decreases___________with depth from the surface of a superconductor. ( )
1. exponentially
2. linearly
3. logarithmically
4. inversely
39. If dc voltage exists across Josephson junction, then_______________current passes across the junction. ( )
1. dc
2. ac
3. pulsating
4. none
40. The material used in the construction of superconducting magnets are: ( )
1. niobium
2. titanium
3. both a & b
4. none
41. Joules heating in superconductors is: ( )
1. present
2. absent
3. both a & b
4. none
42. The core and coil of a cryotron switch is prepared with___________superconducting material. ( )
1. same
2. different
3. both a & b
4. none
43. 43.___________Josephson effect is used to generate and detect electromagnetic waves of frequencies ranging from radiowave to infrared wave. ( )
1. ac
2. dc
3. both a & b
4. none
44. Maglev vehicles are constructed based on________effect. ( )
1. gravitational
2. electrical
3. Meissner
4. none
45. With the aid of good electronic feedback circuit, SQUIDS can measure magnetic fields that are less than___________of a quantum of magnetic flux. ( )
1. 1/10
2. 1/100
3. 1/1000
4. none

 1. c 2. b 3. a 4. d 5. b 6. d 7. a 8. b 9. b 10. c 11. b 12. a 13. d 14. a 15. b 16. b 17. a 18. b 19. b 20. a 21. d 22. b 23. a 24. b 25. a 26. c 27. a 28. b 29. c 30. b 31. a 32. a 33. b 34. a 35. b 36. b 37. c 38. a 39. b 40. c 41. b 42. b 43. a 44. c 45. c

#### Review Questions

1. Describe the BCS theory of superconductivity.

(Set-3–June 2005)

2. Write various applications of superconductivity.

(Set-3–June 2005)

3. Explain dc and ac Josephoson's effect.

(Set-3–June 2005)

4. Write notes on the applications of superconducting materials.

(Set-4–May 2008)

5. Describe the differences between Type-I and Type-II superconductors.

(Set-2–May 2008)

6. Explain the critical parameters and their significance in superconductors.

(Set-1–May 2008)

7. Write notes on (i) isotopic effect and (ii) energy gap in superconductors.

(Set-1–May 2008)

8. What is Meissner effect? Explain.

(Set-2, Set-4–May 2008), (Set-2–May 2007)

9. Explain the following (a) critical magnetic field of a superconductor as a function of temperature, (b) Meissner effect and (c) cryotrons.

(Set-4–May 2007)

10. How are superconductors classified? Explain their properties.

(Set-4–May 2008), (Set-1–Sept. 2006), (Set-2–May 2006)

11. What is meant by isotopic effect? Explain with suitable example.

(Set-2–May 2007)

12. Define the terms of superconductivity: (i) critical temperature, (ii) critical magnetic field and critical current.

(Set-3–May 2006)

13. What are Cooper pairs? Explain.

(Set-3–May 2006)

14. Write notes on any four applications of superconductors.

(Set-3–May 2006)

15. Write notes on the applications of superconducting materials.

(Set-4–May 2008), (Set-1–Sept. 2006), (Set-2–May 2006)

16. Explain Meissner effect?

(Set-2–May 2007)

1. What is superconductivity?
2. Explain the two types of superconductors briefly.
17. Discuss the formation of Cooper pairs and energy gap in superconductors on the basis of BCS theory.
18. Explain the phenomenon of superconductivity and Meissner effect.
19. Briefly describe, how Cooper pairs are formed.
20. Explain flux quantization in superconductivity.
21. Write short notes on Type-I and Type-II superconductors.
22. Explain the origin of energy gap of a superconducting material. How this energy gap differs from that of a normal conductor?
23. Explain the properties of a superconductor in detail.
24. Distinguish between Type-I and Type-II superconductors.
25. What is superconductivity? Describe the effect of: (a) magnetic field (b) frequency and (c) isotopes on superconductors. Mention a few industrial applications of superconductors.
26. What is superconductivity? Explain Meissner effect. What are the possible applications of superconductivity?
27. Explain Type-I and Type-II superconductors. What are Josephson's effects?
28. Mention some important characteristics of superconductivity.
29. Explain the BCS theory of superconductivity.
30. Describe Josephson effect and their applications.
31. Perfect diamagnetism is a more fundamental property than perfect conductivity to assert that a material is in superconducting state. Explain this statement.
32. Explain Meissner effect. How is it used to classify the superconductors?
33. Describe dc and ac Josephson effect in superconductors and prove that the current density across a superconducting junction in the former case varies sinusoidally as the phase difference of state function of Cooper pair on either side of it.
34. Describe the phenomena of flux quantization in superconductors and prove that the current oscillates with a frequency equal to times the potential difference across the superconducting junction.
35. Justify that a superconductor can be used as a fuse with the relevant mechanism.
36. Write short notes on Type-II superconductors.
37. Explain the working of a SQUID.
38. Explain BCS theory of superconductors.
39. Describe the Josephson effect underlying a SQUID.
40. Explain Meissner effect. Discuss Type-I and Type-II superconductors. Mention a few applications of superconductors.
41. What are superconductors? Give the qualitative description of the BCS theory.
42. Explain critical temperature, critical field and critical current in a superconductor. Explain BCS theory.