Chapter 10 - Superconductivity PDF

**CHAPTER 10** **SUPERCONDUCTIVITY** **10.1 INTRODUCTION** The subject of superconductivity is a fascinating and challenging field of Physics which offers unique and exciting opportunities for research. The Dutch Physicist Heike Kamerlingh Onnes successfully liquefied helium at a temperature of...

**CHAPTER 10** **SUPERCONDUCTIVITY** **10.1 INTRODUCTION** The subject of superconductivity is a fascinating and challenging field of Physics which offers unique and exciting opportunities for research. The Dutch Physicist Heike Kamerlingh Onnes successfully liquefied helium at a temperature of 4K in 1908. This was the last natural gas to be liquefied. Earlier, oxygen, nitrogen and hydrogen had been turned into liquid at 90.2K, 77.4K and 20.4K respectively. Kamerlingh Onnes thereafter continued his research activity on the study of the properties of matter at low temperature. Three years later, in 1911 while measuring the resistivity in pure samples of mercury, he noticed that the resistance disappeared at about 4.2K (Fig 10.1). According to him, 'mercury at 4.2K has entered a new state, which owing to its particular electrical properties can be called the state of superconductivity'. The temperature at which materials undergo transformation to a state of zero resistance or infinite conductivity is called the critical or transition temperature T~c~. This disappearance of electrical resistivity takes place abruptly. In 1913 Kamerlingh was awarded the Noble Price in Physics for the study of matter at low temperatures and the liquefaction of helium. Subsequently many metals have been found to become superconducting. Niobium (Nb) is the element with the highest transition temperature, T~c~ = 9.26K while in 1972 a compound like Nb~3~Ge was found to have a T~c~ of 23.2K. 0.0020 R(ohms) 0.090 4.00 4.20 4.40 T (K) **Fig 10.1:** Resistance versus absolute temperature for Hg The search for materials with higher transition temperatures continued but it was not until 1986 that J. George Bednorz and Karl Alex Muller had a breakthrough with the discovering of a T~c~ of about 35k for La~2-x~B~ax~CuO~4~ for x ≈ 0.15. This marked the beginning of high temperature superconductivity (HTSC). In 1987 Bednorz and Muller were awarded the Nobel Prize in Physics for their discovery of this new class of superconductors. The transition from normal to superconducting state is a phase transition characterized by the total vanishing of electrical resistance at the critical temperature. This transition usually takes place over a fairly narrow temperature range. The interval in temperature from the beginning of the drop (normal state) to zero resistance state (superconductivity state) is called the width of transition and varies with the purity of the material. Initially superconductivity was regarded as a low temperature phenomenon until quite recently when new oxide superconductors with critical temperature of about 125K and above were discovered. These materials with high transition temperatures (T~c~ › 30K) are of great practical importance in the technological application of superconductivity. We have noted that the defining property of a superconductor is the complete disappearance of its dc electrical resistivity at and below a certain temperature called the critical temperature which is characteristics of that material. A second fundamental characteristics associated with a superconductor is the expulsion of magnetic flux from the interior of the superconductor at temperatures below the critical temperature. **Table 1:** Critical temperature of some superconductor Material T~c~ (K) ----------------------- ---------- HgBa~2~Ca~2~Cu~3~O~8~ 134 Tl-Ba-Ca-Ca-Cu-O 125 Bi-Sr-Ca-Cu-O 105 YBa~2~Cu~3~O~7~ 92 Nb~3~Ge 23.2 Nb~3~Sn 18.05 Nb 9.46 Pb 7.18 Hg 4.2 Sn 3.72 Al 1.19 Zn 0.88 Table 2 Some important dates in the history of discoveries concerning superconductivity are listed below 1911 \- Discovery of superconductivity in mercury: H. Kamerlingh Onnes (1913 Nobel Prize) ------- ---- -------------------------------------------------------------------------------------------------------- 1933 \- Perfect diamagnetism: Meissner and Ochsenfeld 1933 \- London Equation: F and H London 1050s \- Ginzburg -- Landau theory: (2003 Nobel prize with Abrikosov) 1950 \- Isotope effect: H. Frӧhlich 1957 \- BCS theory: J Bardeen, L Cooper and J R Schrieffer (1972 Nobel prize) 1957 \- Tunneling: Josephson (1973 Nobel Prize) 1986 \- High temperature superconductivity: J G Bednorz and K A Muller in Ba-La-Cu-O System (1987 Nobel Prize) **1.2 MEISSNER EFFECT** As mentioned in the previous section, another very important characteristic of superconductors is their response to an applied magnetic field. In 1933, Meissner and Ochsenfeld discovered that if a superconductor is cooled in a magnetic field to below the critical temperature, T ‹ T~c~, that at the transition temperature, the magnetic flux is expelled from the interior of the superconductor. This phenomenon is known as the Meissner effect and is shown in Fig10.2. **Figure 10.2 Meissner effect** If the applied field is gradually increased to a critical or threshold value (H~c~), the superconductivity is destroyed. This transition is reversible. The critical field H~c~ is a function of temperature and for a given substance decreases gradually with increasing temperature, becoming zero at the critical temperature, ie H~c~ (T~c~) = O. Figure 10.3 Critical field versus temperature The relationship between critical magnetic field H~c~ and temperature is given in Tuyn's law as where H~c~(0) is the critical field at T = 0K and H~C~(0) and T~c~ are constants characteristic of the material. The superconducting state is found to be most stable at absolute zero; at T = 0k, H~c~ is maximum and at T = T~c~, H~c~ = 0 Therefore, the reverse transition of moving from a superconducting state to a normal state can occur not by raising temperature but by subjecting the material to a sufficiently high magnetic field. The critical field which destroys superconductivity can be as a result of an electric current flowing through the superconductor specimen. Just as we have critical temperature and critical field, we also have critical current. The current at which superconductivity is destroyed is referred to as critical current I~c~. This current can be as a result of the current flowing through the superconductor itself. The critical current can be estimated for pure metals and simple geometries as the current that will create surface magnetic fields just sufficient to exceed the critical field H~c~. The relationship between critical current and critical field was established by Silsbee in 1916. He said that superconductivity is destroyed by a current when the magnetic field produced by the current reaches H~c~. Consider a superconducting ring of radius r carrying a current I and having its own magnetic field H. As the current is increased to a critical value I~c~, the field also becomes critical, H~c,~ which is sufficient to destroy the superconductivity in the ring. The Silsbee's hypothesis shows that is a limit to the current that can flow in a superconductor. Silsbee's postulate is in good agreement with Type I superconductors but cannot interprets the behavior of Type II superconductors. We have stressed that the most spectacular property of superconductivity is zero resistivity. Consider a superconducting sample, from the above simple definition, if ρ 0, while the current I is held finite, it then implies that the electric field **E** must be zero since Ohm's law **E** =Jρ. Furthermore, from Maxwell equation, [\$\\nabla\\ x\\ E = - \\frac{\\partial B}{\\partial t}\$]{.math.inline} 10.3 If **E** = 0, then [\$\\frac{\\partial B}{\\partial t}\$]{.math.inline} = 0 10.4 Therefore **B** is a constant, implying that the flux passing through the specimen remains constant on cooling through the transition. There is no expulsion of flux! This is not in agreement with Meissner effect. Therefore in addition to the condition of zero resistivity (**E**=O), another condition for an ideal superconductor is that the magnetic flux density **B** must be zero (**B** = 0) within the interior of the superconductor; [**B** = *μ*~0~(**H**+**M**) = 0]{.math.inline} 10.5 where **H** is the external applied field and **M** is the magnetization. Equation 10.5 yields, for a superconductor One sees immediately that a superconductor is a perfect diamagnet with an oppositely directed field by the induced magnetization to the applied magnetic field **H**. This oppositely directed field is exactly the same magnitude with the applied field as long as the applied field does not exceed a certain value **H**~c~, for each superconductor. The ratio of the magnetization to the applied field is the magnetic susceptibility χ. Thus Equations 10.7 defines a perfect diamagnet. Therefore Meissner effect implies perfect diamagnetism. The combination of zero resistance and perfect diamagnetism distinctly defines a superconductor. **10.3 TYPE I AND TYPE II SUPERCONDUCTORS** There are two kinds of superconductors distinguished by their qualitatively different responses to a magnetic field, Type I (or soft or Pippard superconductors) and Type II (or hard or London superconductors). Type I superconductors show perfect bulk diamagnetism for H \< H~c~ and are driven normal for H \> H~c.~ That is, for a Type I superconductor, when the applied field H reaches the critical value H~c~, its superconductivity is completely destroyed and the material reverts to a normal system. Thus, the magnetic induction **B** inside the bulk of a Type I superconductor is zero. All known elemental superconductors except Nb and V are type one superconductors. The magnetization curve for Type I, superconductors is shown in Figure 10.4a while that for Type II materials is shown in Figure 10.4b. Sc State ← normal S N Sc states mixed state normal H~c~ H H~c1~ H~c~ H~c2~ H \(a) (b) Fig. 10.4 Magnetization curve for (a) Type I superconductors, (b) Type II superconductors We have noted that for a Type I superconductor, any H \> H~c~ completely destroys the superconductivity and the material turns normal. Type II superconductors present a completely different and more interesting case when subjected to fields beyond some critical value H~c~ characteristic of the material. They have two critical fields usually referred to as the lower critical field H~c1~ and the upper critical field H~c2~. When a magnetic field is applied to a Type II superconductor initially when the field is weak, it is completely expelled. But when the field attains a critical value H~c1~ , (the lower critical field), the flux begins to penetrated the interior of the material. At the same time the magnitude of the induced magnetization starts to decrease gradually until it reaches zero at an applied field value H~c2~, the upper critical field. The magnetization versus applied field for such materials is shown in Figure 10.4b. Between the lower critical field and the upper critical field, there exists an intermediate state called the mixed state or vortex state. In a Type II superconductor, an applied field H \< H~c~ is completely expelled from the bulk of the material, that is Meissner effect (**B**=0) applies. However, as H is increased to the lower critical field H~c1~ the flux starts to penetrate the interior of the material. This continues until H~c2~, the upper critical field, when the material turns normal. Between H~c1~ and H~c2~, **B** ≠ 0 and Meissner effect is incomplete. The system is an inhomogeneous arrangement of superconducting and normal states. The flux penetrates the material at H~c1~ in the form of quantized tubes, each flux tube carrying one flux quantum Փ~0~ = [\$\\frac{h}{2e}\$]{.math.inline}. The number of flux tubes increases continuously with increase in externally applied field and this causes the demagnetization to fall. For H = H~c2~, the flux tubes will start overlapping and beyond H~c2~ the system turns normal. Type II superconductors are usually alloys of transition metals with high resistivity in the normal state. They have the characteristic of retaining their superconductive properties in every high magnetic fields. **10.4 THE ISOTOPE EFFECT** Frӧhlich (1950) carried out a theoretical study of different conducting isotopes of mercury and discovered a relationship between the critical temperature and the isotopic mass. He observed that as the mass number M was varied from 199.5 to 203.4, the transition temperature changed from 4.185 to 4.146K. Mathematically this translates to or T~c~M^1/2^ = Constant 10.8 Equation 10.8 applies to mercury. The same result was observed experimentally by Reynolds *et al* (1950) and Maxwell (1950). For the isotopes of other superconductors the equation can be written as T~c~[*M*^*α*^ ]{.math.inline}= Constant 10.9 where [∝]{.math.inline} is the isotope effect coefficient. The discovery of isotope effect indicates the importance of lattice vibrations and thus electron -- phonon interaction which provides a basis for the microscopic theory of superconductivity. Values of the isotope effect coefficient [∝]{.math.inline} of some selected superconductors are listed in Table 10.3 below; Table 10.3: Experimental values of [∝]{.math.inline} ------------------------------------------------ **Superconductor** \ [**∝**]{.math.display}\ -------------------- --------------------------- Zn 0.45 [+] 0.05 Cd 0.32 [+] 0.07 Sn 0.47 [+] 0.02 Hg 0.50 [+] 0.03 Pb 0.49 [+] 0.02 Ru 0.00 [+] 0.05 Os 0.15 [+] 0.05 Mo 0.33 Nb~3~Sn 0.08 [+] 0.02 Zr 0.00[+] 0.05 ------------------------------------------------ **10.5 HEAT CAPACITY** Another distinctive property of superconductors we will consider is the electronic heat capacity. The heat capacity of a metal consists of contributions from the electrons in the metal and the crystal lattice. The relationship is of the form. C(T) = C~e~ + C~ℓ~ = [*ɤT*]{.math.inline}+ βT^3^ 10.10 The first term on the right hand side (C~e~ = [*ɤ*]{.math.inline}T) is the contribution due to the electrons while the second term (C~ℓ~ =βT^3^) arises from the lattice. At low temperatures the heat capacity is dominated by the electronic parts. But as temperature T decreases, the electronic part decreases linearly, C~e~ [∼]{.math.inline}T while the lattice heat capacity C~ℓ~ decreases much faster ([∼]{.math.inline}T^3^). In Figure 10.5 below, we plot the contributions to the total heat capacity by the electrons in the superconducting state and compare with their contributions in the normal state. C~v~ Tc T In the superconducting region experimental results show that at the critical temperature T~c~, the specific heat jumps above the normal state and the transition is not accompanied by any latent heat. As the temperature is decreased below the transition temperature the specific heat falls rapidly and goes below the normal state value as T 0. Since there are no observable changes in the lattice parameters when the superconductivity takes place in a material, it can safely be assumed that the lattice contribution to heat capacity has the same value βT^3^ in the normal and superconducting states. Therefore the difference between the heat capacities of the normal and superconducting states are entirely due to changes to the electronic contributions only. We have earlier noted the linear dependence of the electronic heat capacity on temperature (C~e~[ ∼ ∝]{.math.inline}T). As the temperature is lowered to T~c~, a discontinuity is observed and at temperatures below T~c~, the dependence becomes exponential and assumes the form. [\$\\frac{C\_{e}}{\\gamma T\_{c}\\ }\$]{.math.inline} = ae 10.11 where a and b are constants independent of the temperature. **10.5 THE LONDON THEORY** The electrodynamics of superconductivity, that is, the condition on zero resistance and perfect diamagnetism cannot be adequately explained by the Maxwell's electromagnetic equations. The London brothers (Fritz London and Heinz London) in 1935 proposed two phenomenological equations to describe the electrodynamics of superconductors. They adopted the two fluid model in their theory. They reasoned that there are two types of electrons in a superconductor, namely, the normal electrons and the superelectrons. At absolute zero of temperature, a superconductor contains only superelectrons, but as temperature increases, the ratio of the normal electrons to the superelectrons increases until at the transition temperature when all electrons become normal. Therefore we can say at T= 0K all electrons are superelectrons whereas at T =T~c~ all electrons turn normal. In between these temperatures, normal and superelectrons coexist with the total sum always equal to the conduction electron density in the normal state. If there are n~n~ normal electrons and n~s~ superelectrons, then the total number of conduction electrons n is obtained from and the total current density is given by where v~n~ and v~s~ represent the velocities of the normal and superelectrons. In a superconductor, the superelectrons do not encounter any resistance to their motion. A small constant electric field E is maintained in a superconductor which causes the superelectrons to accelerate steadily. If m and e are the mass and charge of the superelectrons respectively, than the equation of motion for the superconducting electrons is The current density of the superelectrons is Differentiating equation 10.15 with respect to time and applying equation 10.14 we obtain This is the first London equation. Equation 10.16 properly describes the superconducting phenomenon because for **E** = 0, J has a finite value which is constant. This implies that it is possible to have a steady state current in the absence of an electric field. The normal fluid is dissipative, hence This equation implies that in the normal state, if E= 0, then J must be zero also, that is there can be no current in the absence of an electric field. This is quite different from the superconducting phase. Taking the curl of equation 10.16 10.18 Invoking Maxwell's equation 10.19 We have 10.20 Integrating equation 10.20 and taking the constant of integration as zero in order to conform with the experimentally observed Meissner effect, we obtain 10.21 This is the second London equation. **10.6 LONDON PENETRATION DEPTH** The Meissner effect can be deduced from the second London equation. We start with the Maxwell's equation 10.22 Taking the curl of this equation, we get 10.23 Using the vector identity 10.24 We have where [∇]{.math.inline}.B = 0 for a superconductor Using the result for curl J obtained from the London equation 10.21, we get the differential equation. 10.26 where is called the London penetration depth. This is the effective depth to which the magnetic field penetrates into the superconductor. Table 10.3 shows calculated London penetration depth of selected superconductors at absolute zero. Table 10.3 London Penetration Depth,, at absolute zero **Metal 10^-6^cm** -------------------- ------ Sn 3.4 Al 1.6 Pb 3.7 Cd 11.0 Nb 3.9 Consider the case of a semi-infinite superconductor with its surface lying on the yz-plane in the z direction. If B~z~(0) is the field at the plane boundary, then the solution of equation 10.26 is B~z~(x) = B~z~(0)exp 10.27 where B~z~(x) is the field inside. This solution indicates that the flux density decays exponentially upon penetrating into a superconductor, falling off to 1/e of its initial value at a distance , the London penetration depth. It therefore shows that the magnetic field vanishes in the bulk of the superconductor which is Meissner effect. H H~0~ H~z~/e λ Penetration depth x Figure 10.5: Exponential decay of the magnetic field inside a superconductor. The penetration depth also varies with temperature according to the expression 10.28 where (0) is the value of the penetration depth at zero temperature. Notice that increases with increasing temperature approaching infinity as T T~c~. The London penetration depth and the number of superconducting electrons n~s~ are inversely related to each other and n~s~ is also temperature dependent, therefore n~s~ can be written as 10.29 and 10.30 where w is called the order parameter. Figure 10.6 shows the temperature dependence of the penetration depth (0) T~c~ T Figure 10.6: Penetration depth versus temperature 10.6 **THE BARDEEN-COOPER-SCHRIEFFER (BCS) THEORY** A remarkably successful theory of superconductivity was formulated in 1957 by Bardeen, Cooper and Shrieffer (BCS). The theory was able to explain the microscopic origins and also give a good account of the basic features of superconductivity consistent with all known experimental facts qualitatively and quantitatively. This was the situation until the 1986 breakthrough of High Temperature Superconductivity (HTSC) with T~c~ \> 30K by Alex Muller and George Bednorz which was at variance with the BCS theory. Therefore the region of validity of the BCS theory may afterall be the low temperature superconductors (LTSC). The theory cannot sufficiently explain the features of HTSC. The basis of formulation of BSC theory was the discovery of the isotope effect = constant, from which it can be inferred that the ionic lattice plays an important role in superconductivity. Let us briefly describe the salient steps in the BCS theory qualitatively. Ordinary electrons carry negative charges and any two electrons in a vacuum will repel each other through their coulomb interaction. Due to the screening effect, this repulsive force decreases substantially in metals. However, in superconductivity, we are not considering two isolated electrons but rather electrons situated inside a crystal. The presence of the crystal lattice therefore modifies the sign of interaction between electrons. In the crystals the electrons experience an attractive force which makes them to form pairs, referred to as Cooper Pairs near the Fermi level. We need to explain how electrons which are known to have same sign attract each other. Inside the crystal the positive ions move slower than the electrons because they are heavier and therefore their response to the passage of an electron which moves with greater speed as a result of its light mass is delayed. The persistence of the response after one electron has passed creates a concentration of positive charge that attracts the second electron. When this attractive Cooper pair interaction is strong enough to dominate the repulsive Coulomb force, a net attractive interaction that binds the electrons in repairs results. Therefore, we see that the second electron interacts with the first electron via the lattice deformation forming a Cooper pair. Single electrons are fermions and obey the Pauli exclusion principle but electron pairs behave like bosons and can condense into the same energy level. The formation of electron pair as a result of electron-lattice-electron interaction gives rise to an energy gap of the order of 0.001eV and this prevents the type of collision interactions that result in resistivity. At sufficiently low temperatures when the thermal energy is less than the band gap, the material shows no resistance hence superconductivity. The attractive electron-electron interaction due to virtual exchange of phonons is seen diagrammatically in Figure 10.7. 1 2 \(a) (b) **Fig. 10.7** a. The electron-electron interactions b. Electron-electron interaction through lattice phonons The electron 1 of momentum k~1~ emits a phonon of momentum q which is absorbed by electron 2 of momentum k~2~. As a result, the electrons 1 and 2 have momenta k~1~ - q and k~2~ + q respectively. If k~1~ - q = k~1~^ʹ^ and k~2~ +q = k~2~ʹ then k~1~ + k~2~ = k~1~^ʹ^ + k~2~^ʹ^ 10.31 which gives a constant momentum for each pair. The interaction is strongest when the two electrons have opposite momenta and opposite spins. Thus k~1~ = k and k~2~ = -k Phonons are available in the small energy range ħw~D~ = k~B~ϴ~D~ where w~D~ is the Debye frequency and ϴ~D~ is the Debye temperature. The electron-electron interaction is attractive if the energy difference. ε~k1~ - ε~k2~ = ħw~D~ **EXAMPLES** Example 1 If the critical temperature for mercury with isotopic mass 199.5 amu is 4.185k, calculate its critical temperature when the isotopic mass changes to 201.6 amu. Solution: Given, M~1~ = 199.5amu, T~c1~ = 4.185k M~2~ = 201.6amu, T~c2~ = ? For mercury T~c~M^½^ = Constant ie = 4.163K **Example 2:** Calculate the critical field at 3.8K for a superconducting lead material which has a critical temperature of 7.26K at zero magnetic field and a critical field of 8x10^5^ A/m at 0K. Solution; Given, T = 3.8K; T~c~ = 7.26K H~c~(0) = 8x10^5^ A/M; Hc (T) = ? = 5.80 x 10^5^A/m **Example 3** Calculate the value of the London penetration depth at 0K for mercury if it has a penetration depth of 75nm at 3.5K. Find also n~s~ (number of super electrons) (Take critical temperature for mercury = 4.2k). Solution Given: λ= 75nm, T = 3.5K T~c~ = 4.2K, λ~0~ = ? \ []{.math.display}\ ~=~ = 53.97nm ii = 5.035 x 10^27^m^-3^ **Example 4** The critical field for aluminum is 4.28 x 10^4^ A/m. Calculate the critical current that will flow through a superconducting aluminum wire having a radius of 0.001m. Solution: Given: H~c~ = 4.28 x 10^4^ A/m, r = 0.001m I~c~ = ? From: I~c~ = 2[*π*r Hc ]{.math.inline} = [2*πx* 0.001 *x* 4.28 *x* 10]{.math.inline}^4^ = 268.92A **SUMMARY** 1. Superconductivity is a phenomenon of zero electrical resistance and the expulsion of external magnetic fields in some materials when they are cooled below their critical temperature. 2. In the superconducting state, E = 0 and B = 0. 3. A superconductor exhibits a discontinuous increase in its specific heat at T~c~ below which it drops rapidly. 4. Meissner Effect -- if a superconductor is cooled down in the presence of a weak magnetic field, below T~c~ the field is completely expelled from the bulk of the superconductor. 5. Ideal diagmagnetism χ~m~= -1. Weak magnetic fields are completely screened away from the bulk of a superconductor. 6. Flux quantization- The magnetic flux through a superconductivity ring is quantized and constant in time. The superconducting fluxoid is quantized in integer units of Φ~0~ = ħ/2e = 2.00 x 10^-15^ Webers and the total flux trapped in a loop is an integral multiple of a basic quantum of flux Φ = nΦ~0~ = nħ/2e n = 0,1,2....... 7. Type 1- Superconductors -- show perfect bulk diamagnetism for H \< H~c~(T). They are driven normal for H \> H~c~ (T) 8. Type -11 superconductors have two critical fields H~c1~(T) and H~c2~(T). For H \< H~c1~, there is complete flux expulsion and the system is Meissner phase for H~c1~ \< H \< H~c2~ the system is in a mixed state in which quantized vertices of flux o penetrate the system for H\ 1. Describe the Meissner effect in a superconductor. 2. Why do superconductors conduct electricity at zero resistance. Why do they have poor thermal conductivity. 3. What are Cooper pairs. Explain their formation. 4. Derive the London equations. How that the second equation leads to the Meissner effect. 5. Show that the flux in a superconducting ring is quantized in units of ħ/2e 6. Discuss the features of Type I and Type II superconductors. 7. Calculate the critical temperature and the penetration depth for lead at 0K if the penetration depths at 3K and 7K are 39.6nm and 173nm respectively. **REFERENCES** 1. Ajay, K. S., 2006;*Solid State Physics - An Introduction to Solid State Electronic Devices,* Macmillan, New Delhi, India 2. Animalu, A. O. E.,1977; *Intermediate Quantum theory of Crystalline Solids,* Prentice Hall, New Jersey, USA 3. Ketterson J.B. and Song S.N., 1999; *Superconductivity* ,Northwestern University Cambridge University Press 4. Kittel C., 2005; *Introduction to Solid State Physics*, 8^th^ ed., John Wiley & Sons Inc, Berkerly 5. Kresin V. Z. and Wolf S. A., 1990; *Fundamentals of Superconductivity* Plenum Press New York. 6. Okoye C.M.I. 2000; *Lecture note on Fundamentals of Superconductivity and High Temperature Superconductivity,* National Mathematical Centre Abuja 7. Onwuagba B.N., 2000; *Lecture note on Superconductivity,* National Mathematical Centre Abuja 8. Patterson J. and Bailey B, 2010; *Solid State Physics - Introduction to the Theory* 2^nd^ ed, Spinger New York. 9. Pillai S. O., 2010; Solid State Physics (New Revised Sixth Edition), New Age international, New Delhi, India 10. Wahab M.A., 2005; *Solid State Physics-Structure and Properties of Materials,* 2^nd^ Ed., Narosa Publishing House, New Delhi, India