CHAPTER I INTRODUCTION TO SUPERCONDUCTIVITY

CHAPTER I INTRODUCTION TO SUPERCONDUCTIVITY 1.1 Introduction Superconductivity is a fascinating and challenging field of Physics. Today, superconductivity is being applied to many diverse areas such as: theoretical and experimental science, military, transportation, power production, electronics, medicine as well as many other areas. Scientists and engineers throughout the world have been striving to understand this remarkable phenomenon for many years. In 1911, Kamerlingh Onnes began to investigate the electrical properties of metals in extremely cold temperatures. It had been known for many years that the resistance of metals fell when cooled below room temperature, but it was not known what limiting value the resistance would approach, if the temperature were reduced very close to 0 K. Some scientists, such as William Kelvin, believed that electrons flowing through a conductor would come to a complete halt as the temperature approached absolute zero. Other scientists, including Onnes, felt that a cold wire's resistance would dissipate. This suggested that there would be a steady decrease in electrical resistance, allowing better conduction of electricity. At some very low temperature point, scientists felt that there would be a leveling off as the resistance reached some ill-defined minimum value allowing the current to flow with little or no resistance. Onnes passed the current through a very pure mercury (Hg) wire and measured its resistance as he steadily lowered the temperature. Much to his surprise there was no leveling off of resistance, let alone the stopping of electrons as suggested by Kelvin. At a temperature of 4.2 K, called the superconducting transition temperature T c, the resistance suddenly vanished [1]. Current was flowing through the mercury wire and nothing was stopping it, the resistance was zero. According to Onnes, "Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called the superconductive state". The experiment left no doubt about the disappearance of the resistance of a mercury wire. Kamerlingh Onnes called this newly discovered state, Superconductivity. 2

1.2 Fundamentals of superconductors The theoretical understanding of superconductivity is extremely complicated and involved. Superconductors have the ability to conduct electricity without the loss of energy. When current flows in an ordinary conductor, for example copper wire, some energy is lost. In a light bulb or electric heater, the electrical resistance creates light and heat. In metals such as copper (Cu) and aluminium (Al), electricity is conducted as outer energy level electrons migrate as individuals from one atom to another. These atoms form a vibrating lattice within the metal conductor; the warmer the metal the more it vibrates. As the electrons begin moving through the maze, they collide with tiny impurities or imperfections in the lattice. When the electrons bump into these obstacles they fly off in all directions and lose energy in the form of heat. Inside a superconductor the behaviour of electrons are vastly different. The impurities and lattice are still there, but the movement of the superconducting electrons through the obstacle course is quite different. As the superconducting electrons travel through the conductor they pass freely through the complex lattice. Because they bump into nothing and create no friction, they can transmit electricity with no appreciable loss in the current and no loss of energy. In this section fundamental terms and phenomena of superconductvity will be discussed. 1.2.1 Meissner effect In 1933, Walther Meissner and R. Ochsenfeld discovered that superconductors are more than a perfect conductor of electricity and they also have an interesting magnetic property of excluding a magnetic field. When a superconductor is cooled below its transition temperature in a magnetic field, it excludes the magnetic flux. This phenomenon, known as Meissner effect, was discovered by Meissner and Ochsenfeld [2]. The Meissner effect will occur only if the magnetic field is relatively small. In a weak applied field, a superconductor "expels" nearly all magnetic flux. It does this by setting up electric currents near its surface. The magnetic field of these surface currents cancels the applied magnetic field within the bulk of the superconductor. As the field expulsion, or cancellation, does not change with time, the currents producing this effect (called persistent currents) do not decay with time. Therefore the conductivity 2

can be thought of as infinite: a superconductor. The sequence of events is shown in Figure 1.1. If the magnetic field becomes too great, it penetrates the interior of the metal and the metal loses its superconductivity. Figure 1.1 Meissner effect in a superconducting sphere cooled in a constant applied magnetic field. Below the transition temperature the magnetic flux are ejected from the sphere. 1.2.2 Characteristics of superconductors Critical magnetic field (B c ) A superconductor has the property of perfect diamagnetism, also called the Meissner effect, means that the magnetic susceptibility has the value c = -1. So the superconducting state cannot exist in the presence of a magnetic field greater than a critical value, even at absolute zero. This critical magnetic field is strongly correlated with the critical temperature for the superconductor. The critical magnetic field at any temperature below the critical temperature is given by the relationship, B c (T) = B c (0) [1 (T T c ) 2 ] (1.1) Above this critical magnetic field the flux penetrate into the material and the material goes to normal state. 3

Critical current density (J c ) Within two years of the discovery of superconductivity in mercury, Onnes recorded that there was a threshold value of the current density in mercury, above which the zero resistance state disappeared. This critical value was temperature dependent which increases as the temperature was reduced below the critical temperature, according to the expression, J c (T) = J c (0) (T c T) T c (1.2) A common way to estimate J c is to measure a hysteresis loop in high field at a constant temperature and use of the Bean-model formula, J c = 1.59 10 6 µ 0 M d (1.3) Where M = M + M _ is the difference in magnetization between the top and bottom of the hysteresis at a particular magnetic field, µ 0 = 4π 10-7 N/A 2 is the permeability of free space and d is the diameter of the sample grains in meter. London penetration depth (λ L ) For a superconductor in an applied magnetic field, the screening currents which circulate to cancel the magnetic flux inside it must flow within a finite surface layer. Consequently, the flux density does not vanish abruptly to zero at the boundary of the superconductor. It penetrates up to a region in which the screening currents flow and the width of this region is known as the London penetration depth of the superconductor. This is illustrated in Figure 1.2. Figure 1.2 A schematic representation of the penetration depth (λ L ) of a superconductor in an applied magnetic field. 4

One of the theoretical approaches to the description of the superconducting state is the London equation. It relates the curl of the current density J to the magnetic field: J = 1 2 µ 0 h L (1.4) The nature of the decay depends upon the superconducting electron density n: λ 1 = J s 0 mc 2 ne 2 (1.5) Where, ε 0 = 8.854 10-12 F/m is the vacuum permittivity m = mass of an electron c = velocity of light n = electron density e = charge of an electron In the superconducting state, the only field allowed is exponentially damped as we go in from the external surface and it is given by, B(x) = B(0) exp(-x/λ L ) (1.6) Where B(0) is the field at the plane boundary. This implies that the magnetic field penetrates up to a length scale λ L from the surface into the interior of the superconductor, giving the penetration depth, a fundamental length scale, from the Londons equations. The temperature dependence of λ L can be expressed by the empirical equation: λ (T) = λ(0) [1 - (T/T c ) 4 ] -1/2 (1.7) The typical value of the penetration depth for most of the elemental superconductors ranges between 10-2000 nm. 5

Coherence length (ξ 0 ) The coherence length is a measure of distance within which the superconducting electron density cannot change drastically in a spatially-varying magnetic field. That is the superconducting electron density cannot change quickly and there is a minimum length over which a given change can be made, lest it destroy the superconducting state. For example, a transition from the superconducting state to a normal state will have a transition layer of finite thickness which is related to the coherence length. This coherence length is related to the Fermi velocity (v F ) for the material and the energy gap (E g ) associated with the condensation to the superconducting state. Ç O = 2ħr F ne g (1.8) 1.2.3 BCS theory of superconductivity The ability of electrons to pass through superconducting material has puzzled scientists for many years. The warmer a substance is the more it vibrates. Conversely, the colder a substance is the less it vibrates. Early researchers suggested that fewer atomic vibrations would permit electrons to pass more easily. However this predicts a slow decrease of resistivity with temperature. It soon became apparent that these simple ideas could not explain superconductivity. It is much more complicated than that. The understanding of superconductivity was advanced in 1957 by three American Physicists-John Bardeen, Leon Cooper and John Schrieffer, through their Theories of Superconductivity, known as the BCS Theory. The BCS theory explains superconductivity at temperatures close to absolute zero. The basis of this theory is that even a very weak attractive interaction between electrons, mediated by phonons, creates a bound pair of electrons, called the Cooper pair, occupying states with equal and opposite momentum and spin (i.e. k, -k ). The formation of the bound states creates instability in the ground state of the Fermi sea of electrons and a gap ( (Τ)) opens up at the Fermi level. The minimum energy E g required to break a Cooper pair to create two quasi-particle excitations is E g = 2 (T). 6

The formation of the Cooper pairs mediated by the phonons is illustrated in Figure 1.3. An electron with momentum k travelling through the lattice will polarize it, thereby creating a local positive charge. A second electron with momentum k travelling through this lattice will be attracted to the local positive charge, thereby, getting attracted to the first electron. This leads to the formation of the Cooper pairs. Figure 1.3 Schematic illustration of the formation of a cooper pair between two electrons travelling with momentum k and k, mediated by the lattice. The key consequences of the BCS theory include a connection of the gap parameter Δ to the transition temperature T c, 2 = 3.52 k B T c (1.9) and the Debye temperature (Θ D ), T c = 1.14 Θ D e 1 N(O)V (1.10) where N(0) is the density of states at the Fermi level and V is the attractive electronphonon interaction potential. T c is in part determined by the Debye temperature so that an observable shift in Θ D should accompany an alteration of T c. Such a change in Θ D can be accomplished by replacing one element in the material with a different isotope of the same element. Indeed, measurements demonstrating such a shift in T c, termed the isotope effect, provided convincing support for the BCS model of superconductivity. 7

For weak coupling superconductors, the reduced gap (T)/ (0) is a universal function of the reduced temperature T/T c, near the critical temperature T c, 2 (T) = 1.76 (1 T ) (1.11) (O) T c So that the energy gap approaches zero continuously as T T c as shown in Figure 1.4. Figure 1.4 Variation of the reduced gap (T)/ (0) with the reduced temperature T/T c according to the BCS theory. 1.2.4 Type I and Type II superconductors Figure 1.5a shows the magnetization versus applied magnetic field for a bulk superconductor which exhibits complete Meissner effect (perfect diamagnetism). A superconductor with this behaviour is called Type I superconductor. Above the critical field H c the specimen is a normal conductor and the magnetization is too small. Very pure samples of lead (Pb), mercury (Hg) and tin (Sn) are examples of Type I superconductors. Type II superconductors have superconducting electrical properties up to a field denoted by H c2 (Figure 1.5b). Between the lower critical field H c1 and the upper critical field H c2 the flux density B 0 and the Meissner effect is incomplete in this region. The flux starts to penetrate the specimen at a field H c1 lower than the thermodynamic critical field H c. In the region between H c1 and H c2 the superconductor is threaded by flux lines and is said to be in the vortex state. A schematic of the mixed 8

state is shown in Figure 1.6. The normal regions in the mixed state are in the form of cylinders with their axis along the direction of the magnetic field. Current vortices circulate around these normal cores to generate the flux within. The direction of this current is opposite to the main surface shielding current which makes the flux in the superconducting region zero. With the increase in the magnetic field beyond H c1, the distance between the normal cores decreases. At a field equal to the upper critical field H c2, there is complete overlap of the normal cores and the superconductor goes over completely to the normal state. Figure 1.5 Magnetization versus applied magnetic field for (a) Type I and (b) Type II superconductors. Figure 1.6 Schematic representation of the mixed state of a Type II superconductor. The white cylindrical regions denote the normal cores where the flux penetrates. The normal cores are separated by superconducting regions. 9

An important difference between Type I and Type II superconductors is the mean free path of the conduction electrons in the normal state. If the coherence length is longer than the penetration depth, the superconductor will be Type I, with k < 1; k = λ L /ξ. But when the mean free path is short, the coherence length is short and the penetration depth is great with k > 1, and the superconductor will be Type II. Type I superconductors are conventional superconductors and they are well described by the BCS theory. 1.3 High T c Superconductors Until 1986, Physicists had believed that BCS theory forbade superconductivity at temperatures above about 30 K. In that year, Bednorz and Müller discovered superconductivity in a lanthanum-based cuprate (La 2-x Ba x CuO 4 ) perovskite material, which had a transition temperature of 30 K [3]. It was soon found that replacing the lanthanum with yttrium (i.e., making YBCO) raised the critical temperature to 92 K [4], which was important because liquid nitrogen could then be used as a refrigerant (the boiling point of nitrogen is 77 K at atmospheric pressure). This remarkable discovery has renewed the interest in superconductivity research. Soon after that many related materials which came to be known as cuprates were discovered to show superconductivity at high-t c values. The highest critical T c of 135 K was achieved in 1993 in HgBa 2 Ca 2 Cu 3 O 8 [5] compound (T c = 164 K at high pressure). In the year 2001, three interesting discoveries were made: (i) MgCNi 3, a completely surprising analogy to the oxide perovskites. It is based on the combination of a light electropositive metal (Mg) with another light element (C), with the addition of a transition element (Ni), having T c at 8 K [6]. (ii) MgB 2 was found supercondcuting at 39 K [7], this is the highest T c in a simple binary material. (iii) Another family of oxide supercondcutors AuBa 2 Ca n-1 Cu n O 2n+3 (n = 3, 4) with T c at 99 K was discovered by Kopnin et al [8] in the same year. These high T c superconductors or Type II superconductors are called unconventional superconductors and they do not fit with the conventional BCS theory of superconductivity. 10

The cuprate superconductors are technologically important since the T c varies up to 135 K. They are generally considered to be quasi-two-dimensional materials with their superconducting properties determined by electrons moving within weakly coupled copper-oxide (CuO 2 ) layers. Neighboring layers containing ions such as lanthanum (La), barium (Ba), strontium (Sr) or other atoms act to stabilize the structure and doping electrons or holes onto the copper-oxide layers. The basic structure of the cuprate superconductors is a CuO 2 plane separated by intervening planes composed of metal donor ions and oxygens. The simplest of these is the La 2-x Sr x CuO 4, and related materials. The structure of this material (Figure 1.7) consists of single atomic planes of CuO 2 separated by two atomic planes of La-Sr oxide. One of the properties of the crystal structure of oxide superconductors is an alternating multi-layer of CuO 2 planes with superconductivity taking place between these layers. The more layers of CuO 2 the higher the T c. This structure causes a large anisotropy in normal conducting and superconducting properties, since electrical currents are carried by holes induced in the oxygen sites of the CuO 2 sheets. The electrical conduction is highly anisotropic, with a much higher conductivity parallel to the CuO 2 plane than in the perpendicular direction. Generally, critical temperatures depend on the chemical compositions, cations substitutions and oxygen content. Figure 1.7 Crystal structure of La 2-x Sr x CuO 4. 11

The phase diagram of the high T c cuprate superconductors consists of several distinct regions (Figure 1.8). The stoichiometric parent compounds are antiferromagnetic Mott insulators dominated by strong electronic interactions. In this region of the phase diagram each site of the two-dimensional CuO 2 plane is occupied by single charge carrier and the strong Coulomb repulsion efficiently blocks the movement of charges. Despite the immobility of the charges, the kinetic energy gain due to virtual nearest hopping processes favors an antiferromagnetic ordering of the spins. Upon doping of either electrons or holes the charge carriers gain mobility and the material becomes conducting and at higher doping even superconducting. On the hole doped side of the phase diagram the transition region form the antiferromagnetic Mott insulating to the superconducting state is called the pseudogap regime where strong antiferromagnetic fluctuations are still present although long range magnetic order is no longer maintained. The strongly momentum dependent spin fluctuations are believed to be important for the appearance of d-wave superconductivity in the cuprates and are even prominent candidates for mediating the superconducting pairing interaction. Even though enormous experimental as well as theoretical efforts have been made on the pseudogap regime, it is not yet fully understood. Figure 1.8 Schematic phase diagram of high T c cuprates. 12

1.4 Applications of superconductors Soon after Kamerlingh Onnes discovered superconductivity, scientists began dreaming up practical applications for this strange phenomenon. Powerful superconducting magnets could be made much smaller than a resistive magnet, because the windings could carry large currents with no energy loss. Generators wound with superconductors could generate the same amount of electricity with smaller equipment and less energy. Once the electricity was generated it could be distributed through superconducting wires. Energy could be stored in superconducting coils for long periods of time without significant loss. The subsequent discovery of high temperature superconductors brings us a giant step closer to the dream of early scientists. Applications currently being explored are mostly extensions of current technology used with the low temperature superconductors. Current applications of high temperature superconductors include; magnetic shielding devices, medical imaging systems, superconducting quantum interference devices (SQUIDs), infrared sensors, analog signal processing devices and microwave devices. As our understanding of the properties of superconducting material increases, applications such as; power transmission, superconducting magnets in generators, energy storage devices, particle accelerators, levitated vehicle transportation, rotating machinery and magnetic separators will become more practical. The ability of superconductors to conduct electricity with zero resistance can be exploited in the use of electrical transmission lines. Currently, a substantial fraction of electricity is lost as heat through resistance associated with traditional conductors such as copper or aluminum. A large scale shift to superconductivity technology depends on whether wires can be prepared from the brittle ceramics that retain their superconductivity at 77 K while supporting large current densities. The field of electronics holds great promise for practical applications of superconductors. The miniaturization and increased speed of computer chips are limited by the generation of heat and the charging time of capacitors due to the resistance of the interconnecting metal films. The use of new superconductive films 13

may result in more densely packed chips which could transmit information more rapidly by several orders of magnitude. Superconducting electronics have achieved impressive accomplishments in the field of digital electronics. Logic delays of 13 picoseconds and switching times of 9 picoseconds have been experimentally demonstrated. Through the use of basic Josephson junctions scientists are able to make very sensitive microwave detectors, magnetometers, SQUIDs and very stable voltage sources. The use of superconductors for transportation has already been established using liquid helium as a refrigerant. Prototype levitated trains have been constructed in Japan by using superconducting magnets. Superconducting magnets are already crucial components of several technologies. Magnetic resonance imaging (MRI) is playing an ever increasing role in diagnostic medicine. The intense magnetic fields that are needed for these instruments are a perfect application of superconductors. Similarly, particle accelerators used in high-energy physics studies are very dependent on high-field superconducting magnets. The recent controversy surrounding the continued funding for the superconducting super collider (SSC) illustrates the political ramifications of the applications of new technologies. 1.5 Limitations of superconductors Despite many scientists believing that superconductors are the way of the future, there are still a number of limitations to their design. Ø The first of these is the restricted range for operating temperature. Since the world record for the highest critical temperature stands at 135 K, there is still a long way to go before superconductors are available to the average user at room temperature. It is impractical for handheld, consumer devices to have liquid nitrogen running through them. 14

Ø Even if we decide to try and cool some devices continually with liquid nitrogen, it is very impractical to cool thousands of kilometres of underground electrical wiring connected to the power grid. More work must be done before they become a practical room temperature device. Ø Also, like most ceramics, Type II superconductors are extremely brittle and therefore impractical unless methods are developed to reduce the brittle nature of these superconductors. Ø Type I superconductors, whilst not brittle, are not able to be cooled with liquid nitrogen (77 K) and their critical temperatures are nowhere near as feasible as their Type II counterparts. Ø The other noticeable limitation to superconductors is the fact that they are quite sensitive to a changing magnetic field, meaning that AC current will not work effectively with superconductors. As a result, devices such as transformers, which only work with AC current, will be more difficult to implement into a DC oriented world when superconductors become a reality. 1.6 Superconductivity at room temperature Room temperature superconductivity is the holy grail of solid state physics. It is becoming increasingly obvious to scientists all over the world that superconductors are the future in terms of transmission and applications with electricity. To make superconductors a feasible option for the electrical devices, scientists must put their effort into a number of key problem areas. The first of these is getting superconductors to work at room temperature. This may entail creating devices that contain a cooling agent or it could mean that scientists need to find new compounds that work at even higher critical temperatures than those currently available. So while superconductors are a very viable future solution in so many applications, much work must be done before it becomes feasible. This induces me to choose the research problem in superconductivity. In order to achieve room temperature superconductor much efforts have been put by the researcher all around the world. By joining in this race, I made an effort to synthesise new superconducting materials with novel physical and chemical properties. Also, the superconductivity of unconventional superconductors cannot be 15

described by the conventional BCS theory and each requiring a different physical mechanism. The chosen research problem may enlighten the superconducting research by supporting the explanation of the mechanism of already existing superconductors. 16